Soil Mechanics and Transport in Porous Media: Selected Works of G. de Josselin de Jong (Theory and Applications of Transport in Porous Media)

(J xy'

a yx

shear stresses

effective principal stresses, positive for compression

auxiliary angle defined by equation

0"1,0"3

(40)

Discussion on this Paper closes on 1 April 1989. For further details see p. ii. • Formerly with Delft University of Technology.

'"

angle between x-axis and plane of a'l

533

44

534

d< JOSSEUN d< lONG

ill

a •

!(V,.~ - VK . , ) : material rotation structural rotation, rotation of the clements distinguishes dilatancy from nondilatanc~

time derivative (d/dn v objcdive (;()-rotalional (Jaumann) stress rales

INTRODUCTION

In his article with Randolph (l981~ his Rankine Lecture (1984) and again more recently (1987) Wroth considers the results of simple shear tests on day as obtained by different investigators. As examples, the observation by Borin (1973) is reproduced in Fig. I and the test results of Ladd & Edgers (1972) in Fig. 20. Wroth draws attention to the fact that the observed deformation al the onset of failure is apparently nOI the onc with horizontal planes of maximum stress obliquity, as is oftcn believed (see Instead, failure appears to oa;ur on vertiFig. cal planes of maximum stress obliqui ty and, in addition to sliding on tho§(: vertical planes. a

n

rigid body rotation is execu ted to m~t the boundary conditions. This failure mode is shown in Fig. 2. It can be visualized as a row of books, toppling over sideways when the left-hand support is removed. The possible occurrence of a 'toppling bookrow' mode of failure was predicted (de Jossclin de Jong, 1972) as a consequence of the double sliding, free rotating (DSFR ) model for materials witb internal friction. It was mentioned that the toppling bookrow is one case in a set of possible failure modes in a si mple shear test . The usually aca:ptro mode with horizontal sliding planes is another one of this set. The DSFR model admits both modes and all transition modes between these two extremes. Considering simple shear tests with equal vertical normal effective stress (1' on horizontal planes, the two extreme modes possess different shear stresses at failure. In the toppling bookrow mode the shear stress is equal to u' sin 4> cos 4>1 (1 + sin l 4»; in the horizontal sliding mode it is a' tan 1/>, with 4> the angle of internal friction at failure. Combinations of the two modes give values between these two extremes. For 4> _ 23" the appa rent angle of friction, the tangent of ~ ..

. 23 '

,

"

t.~,.,J

- 0 '\

- 0 ·2

'~=--

- 0 '3

Fil- 1. EfI'eniYe .. ress pili.. .... lbe f.ilure IU le from 1ft 1lIOdnti..... llimple ..... r 1..1 on nor .... U, consoIid.o l" k.oIia (411. from Borin, 1973), lat 10)

", ""--+--++---,, ---~ ,,

",

1'1" 1. T oppli"C Ioook ro.. _hlnism

Selected Works of G. de Josselin de Jong

11 111I11 1

"'. ,n

0'O~

i

~ 45

£L.\STO-PL\STIC VERSION OF DOUBLE SLIDING MODEL

whK:h is the shear stress divided by <1', is only 17·3· for the toppling bookrow mode, while it is 23· for the horizontal !tiding mode. Ignoring the toppling bookrow mechanism leads 10 an underestimation of the angle of shearing resistance. At Ihat time it was not known how to select between the various modes of failure. The DSF R model cannot predicl how a sample will behave in a les l: a behaviour that is presumably unique. This crealed an uncertainty which was allributed 10 an incompleteness in the model Ihus prohibiling its practical usc. II was recognized by Vermeer (1980 and 1981 ) Ihat the incompleteness of the model is due 10 ils being rigid-plastic. By adding some elaslicil y in the pre-failure stage he erealed an elaslo-plaslic version of the DSFR model wh ich produces unique solutions al failure. In this version Ihe sample 'selects' betwcen various modes according 10 its initial Slress Slate. In this respect the DSF R model, which is based on internal friction, differs from perfeo;t plasticity, where the unique response al failure is independent of the initial stress state. II is the purpose of this Paper to demonstrate the use of Ihe elaslo-plastic DSFR model by examinin g Ihe responliC of a sample in the undrained simple shear lesl and establishing the initia l circumslances thai lead either 10 the toppling boobow mechanism or 10 horizonlal sliding planes. The loppling bookrow was obse rved by Drescher (1 976) in phOlo-clastic tes lS on crushed glass. This material can be considered as a model material with properties resembling sa nd. The undrained simple shear teslS mentioned by Wro lh were carried out on clays. In these tests thc average normal effeclive stress reduces during the deformation. This indicales Ihal Ihe mal erial is contractive, i.e. negative dilalan!. The dilalanl version of the elaslo-plaslic DSFR model has reoenlly been implemenled for computer use by Teunissen & Vermcer { 1 988~ Their Formulation is in lerms of matrices and therefore differs in notation from the analysis given here. The relevant expressions arc however. identical. The DSF R model as developed by this Author in 1958, 1959 was specified mathematically in various laler papers (de Josselin de Jong, 1971, 1977a and 1977b) and Ihe principal features of this model will be recalled again here. In the first section of this Paper, the dilatant version of the DSFR model, which was desaibcd by this Author (1977a) using Rowe's (1962) stress dilatancy relalion, is redeveloped using only the laws of friction. This leads 10 th e plastic constitutive eq uations (14). In the second section the elastic part of the constitutive equations is developed. The combined elasto-plaslic constitutive rclations are equation (26~ How to divide the total st rain rates

46

inlO Ihei r plastic and elastic parts is presented vis ually in terms of vectors in the st ress and strain rate spaces. The procedure using a minimum energy principle is set out mathematically in Appendix 2In the third section the response of an elaslOplaslic DSFR sample in an undrained simple shear lest is considered. The mathematical results oblained consisl of e~plicit solutions for the stress paths in the Mohr diagram, lhe resemblance of which to Ihe test results already men tioned is the reason for this publication. SECTION 1 DOUBLE SUDING MECH AN ISM WITH

DILATANCY The double slidi ng model describes the plan e strain deformation of a ma lerial such as a soil, possessing internal friction, at the limit stress slate. It is based on the simplification of subdividing the soil by parallel planes, on which sliding takes place, because on them the frictional shear resistance is e~ hausted . These are the polential sliding planes. The Sliding is called double, because tllere are two conjugate directions for Ihe potential sliding planes, located symmetricall y with respecl 10 the princi pal stress directions. In Fig. 3 the two conjugate directions have both an angle ex with Ihe plane of the major principal compression stress. <1',. In order 10 distinguish belween the IWO conjugate sliding possibilities. Ihey arc called a-sliding and b-sliding respectively as indicated in Fi gs 3(a) and 3(b). The sliding planes divide a block of material into clements, that arc liable to slide with respect to each other as shown in Fig, 3. The plastic deformation of the soil as obse rved from the

");;. ' 1'; , ............

"• '" n" ......

l. Slidift& elelMlltJ . .... poIf'Dlia1 sIkIiDa; pIa_

.,

Soil Mechanics and Transport in Porous Media

536

de: JOSSELIN de lONG

-. - ............ ...

L _ _--") '.!J,",

L____---' "

.~i.:"

'" ou tside is due to this mutual sliding. In the original, rigid pl astic version of the DSF R model the elements were supposed to be rigid. In the dUloplastic version the elements arc clastic and this looks aftcr the clastic pari of the strain rales thai

remain after tile plastic pari is accounted for by sliding of the clements. II is impossible to visua1i~c the two conjugate slidings occurring si multaneously without crealing gaps or overlaps. Mathematically, however. double sliding is introduced in order 10 take account of conjugate slidings that occur successively. as observed in reality. Discrete sliding planes as presented in Fig. 3 form elements of finile sizc. Mathematically it is oomple~

10 describe the diSC(lntinuous deformation of the soil created by their motions. By decreasing the thickness of the elements to infinitesimally thin slices, the sliding becomes shearing, and t he defonnation of a soil region can be described in terms of shear strain rates. The magnitudes of the two conjugate shear strain rates are called and bO in this Paper.' By allowing these variables to become infi nitely large, the di screte sliding of elements of finite size can be fonnulated mathematically. An essential feature of the DSFR model iii that the values of and bO can be different. This is due to the principle that in the limil state of stress sliding displacemen ts have arbitrary magnitooes.

,,0

,,0

I In eadier papen the shear strain rates were denoted by a and b. Throughout this Paper asterisks are used, when dilalancy is involved. The notaTion ,, ' . b' was inlroduood ea.lier (de Jossdin de Jong,. 19771) Ind is uoed here to be IXIn,istenl with Ihat paper.

Selected Works of G. de Josselin de Jong

\.

'" By oonsidering tbe sliding surfaces to be smootb tbe deformation is volume oonserving. Dilatancy is obtained by assuming saw-teetb protuberances on the sliding planes.. having uplift angles (J witb the general direction 11 of tbe poten_ tial sliding planes (see Fig. 4(a)). The angles ex and (J a~e assumed to be oonstants through out II. soil regIOn. The forces between the sliding element s are assumed to be transmitted exclusively by tbe faces along wbicb tbe sliding occurs (see Fig. 4(b)). All forces on tbe teeth are taken to be parallel and inclined at an angle J. to the normal to the saw teeth . Information on ex, 0 is obtained by oonsidering tbe equilibriu m of forces and th e exhaustion of the shear resistance on the sliding planes. This is developed in the .section 'foroe equilibrium· and Appendix I. In Ibe section 'kinematics' tbe geometry of tbe sliding motion is treated. These motions represent the plastic part of the deformations. Expressions are developed for tbe ve locity gradients in termli of the sbear strain rates " . and bO. as required for establishing tbe plastic constitutive equations. The relations obtained in these sections al low us to obtain expressions for the angles ex, 8 in terms of", (Ibe angle of in ternal frH:tion) and v· (tbe angle of dilatancy). In this Paper only the homogeneous situation is considered, pertaining to a soil body in which the stres.ses and strai ns are consta nts over th e entire region. Slich a sitllation occurs in the simple shear test examined in this Paper. When

47

S37

EUS'TO-PU$TIC VERSION OF DOUBLE SUDtNG MODEL

6',

,.,

,,'

'"

the stresses and strains arc not homogeneous the expressions developed here are valid for an infinitesimally small region.

The force K makes an angle ('" - 0 - ;.) with therefore

~:

K. _ K 005(", - 0 - ;') K ( _ K sin(II - O-).)

fORCE EQUILI BRIUM A co-ordinale system (~, {) is taken as shown in Fig. S(b) in the direction of Ihe principal stresses I1j and o') . The two conjugate potential sliding planes are located symmetrically with respect to the 'I-axis: both make an angle (lI with the {-axis. Consider a portion of such a potential plane of length m (Fig. 5(c)). The resultant force K on that plane has venical and horizontal components K ~ . K ( created by the principal stresses on the honzonta] and vertical projections of m. Thcir magnitudes are

K, _ u"mCOSII} K ( _ u', m sin II

"l

The angle If, of maximum stress obliquity in the limit Stress Slate is defined by' sin If, _ (o', - 0",)/(0"',

+ 0", )

(2)

Substituting K • . K~ for 0"'" o', gives . (K.sinl.< - Kccosl.<) sm '" _ (K. sin a + Kc COS a)

(3)

, for the case "'hen cohesion i, present. a lerm ~ cot

are found in principle by .... uming sa.. t... th ofmagnitude ~

48

II.

cohesion

~

on lbe

- <{(I - sin 4> si n vOlll t + sin 4> .in . 0)]'"

,,' sin If, - sin (0

+ ;')jsin (211 -

0 - ;.)

(4)

Because of the mirror symmetry with respect to;> the 'I-axis., the same for()l': K acts at the same angle ). with respect to the normal on the saw teeth surfaces of Ihe conjugate planes (see Fig. 5(a)). Equation (4) gives, for a fixed value of "', a set of possible value combinations for I.< and (0 + ).).

KINEMATICS

Let the regions represented in Figs 6(a) and ((b) consist of many infinitesimally thin elements (like the pages of a book) that are parallel, all at an angle IX. In Fig. 6(b) a constant shear strain rate /I . is assumed to occur, such that all elements slide wilh respect to each other. causing a homogeneous deformation. AU have the same kind of protuberances making an angle 0 with Ihe potential sliding planes. Considering only the plastic part of the defor· mation. the demenlS remain rigid during sliding. So Ihe thin elements conserve their length and all points of tho potentia.] slip _line through Q haY<> the same velocity VQ relative to P. Thus Vo makes an angle (I.< - 0) with the {-axis. because of the teeth uplift angle O. The shear strain rate a· is defined such that the magnitude of VQ is /1 0 ", where n is the perpendicular distance from P to the slip-line.

Soil Mechanics and Transport in Porous Media

SJ8

.;. JOSSELIN de lONG

L

,.,

'" '

fiC. 6. V.locilios of slicn_, plo .....

Consider the line PQ at an arbitrary angle r with the ~-a~is. This tine makes an angle (Ii! + y)

with the slip-line and so its length is n/sin (IE + y). The velocity Vo makes an angle (0; + 1- 0) with PQ. The effoct of Vo on PQ is that the line is elongated with an e~lcnsion rale i;(y) and is rotated with a tilting Tale Ph') (positive counterdock wise) as given by the c~prcssions

+ VQ cos (:>: + l' -O)/ PQ } .. +aOcos(lI+ y- O)sin (I1 +l'1 Vo sin (ex

+y_

O)/ PQ

.. -a o sin (a

+ 1-

0) sin (a

~1;r>-1in"

+ ,1

i.:(yl - + V.cos(lI+ a-},- O)/ PR p
-b·

Sin

(a

-7 -

0)

}

Sin (, - II)

+ (t:- }' -

+ bO) cos ('J -

9) sin

o}

- (0 0 - hO) sin (0 - 0) sin 0

(7)

For y _ !If the extension and Iill rales of lines in the ,,--direction are obtained. These arc iJ(!II) .. +(0 0 - bO) cos (Il - 9) cos

o}(8)

+ bO) sin (tl< - 0) COS "

(5)

in F'8- (,(a) give.

.. - bo oos (a -l' - 0)

- 1;(0) - + (0 '

v..(. - jJ(0) -

V.. . _ + i:(!n) - -(0 0

wh ere a dot represents t he lime derivative d/dT. A similar analysis applied to PR with R on the cnnjug>tle

."

v., .• _ -

I.(y)-

pM _ -

differen tiation with respect to the subscript vari· able following it. For y _ 0 the addition gives

(6)

8)fPR

Sin (y -

(1)

Here b" is Ihe shear strain rate in the conjugate slip direction. This b' may be different from o'

according to the double sliding model. When bolh shear slTain rales 0 ° and b' are aClive, the tOlal elongation ratei:(y) and the total tilt rate j1(:I) are obtained by addition of the above expres. sions. The rales i:(yl and My) are functions of 7. By giving y the val ue uro, the above formulae give the elongalion and lilt rates for lines in the ~. di rection. Lei ~, V. be the veloci ties of material points in the ~, ,,-directions. Then the elongation rale is iJ VdiJ~ and th e til l rale is iJVJo~. These are written as VM and V•.~ where a comma represents

Selected Works of G. de Josselin de Jong

The derivalivC5 V(. ( . , ' eiC. are Ihe so-called veloci ly gradients. In addition to the mutual sliding in two conjugate directions, the elements can also execute a rotation around P. Th is rotation is called the structural rOlation U (positive counter-ctockwisc, see Fig. 7), The indication 'structural' was proposed to tbe Author by D rescher in order to distinguish 0 from the well-known material rotation W. which is delined by w- i(V' .$ - V•. ,) and represents Ihe rotation of the total material as observed from the outside. U creates additional velocity gradients of the

r,=

v..(- U;

Vu - O;

V: .• _ _ U ;

V•.• - O

Combi ni ng these results gives the slTain rale components, which are V(.(

+

Vu Ve.•

v.,. - (0' + bO) sin 9 v.,. - + h sin (m (0

0

O )

} 9)

(9)

- (0 0 - h O) cos (m - 0)

+ V•.(

and the rale of material rOlalion, which is

v..(- V:.• -

- (0

0

-

b cos 9 + 20 .. 2w O )

( 10)

49

ELASTO-PLASTIC VERSION OF DOUBLE SLIDING MODEL

539

Equation (10) shows Ihe differen~ belwC<.:n maleria! rotation", and the structural rotation a. Wl>cn only sliding oo::u rs and the elements themselves do not rotate. so that a ", 0, Ihe total soil body nevertheless appears to rotate, in the case when is not equal 10 b*. For example, when is larger than b*. the soil gives the impression of rotating clockwise, because '" is negative. Henoe the material rotation '" may consist of two parts. One componcnt is the effect of unequal shear strain rates. a' '" b' ; lbe other component a is due to rotation of the structure, i.e. of the clements betwC<.:n the sliding planes.

,,*

,,*

Dilalancy rclolioo

The anglc of dilatancy v" is defined by sin v' - (V:.!

+ V•.• )/W!.! -

v. .• )

(11)

where II(.! and V•.• are the lincar sirain rates of lines in the principal stress directions. Using equations (9) we find that sin v" _ sin Olsin (:z.:. - 8)

( 12)

This relation is independent of the shear strain rates ,,' and b'. Equation (12) gives for a fixed value of y' a set of possible combinations for the values of" and O. Thermodynamic reqllirem£nt

of e~rgy

dissiptllioo In Ihe double sliding model it is assumed Ihat a' can differ from b' . This is due to the principle in plasticity that . at failure. the strain rates are not bounded in magnitud e- they can have any value. So also in the two conjugate directions they can have unequal values. The shear strain rates (1* and b' arc only bounded by the requirement that the laws of friction are not violated. This would oo::ur if the sliding directions were opposed to the shear stresses on the sliding planes. Considering Fig. S it is seen that a' and b' , as defined in Fig. 6, should be positive in order that the directions of the forces on the saw tC<.:th correspond to the sliding directions. So the requirement on the shear strain rates is (13) and this is called the thermodynamic requirement of energy dissipation. Mathematically it is possible to show that (a" + b' ) cannot be negative. The stronger requirements (13) are dictated by the above mentioned friction character of the system.

50

PLASTIC CONSTITUTIVE EQUATIONS The expressions (9) for the strain rates ca n

serve as constitutive equations for the plastic slage of deformation. However. before they can be used, the angles II. 0 hue to be replaced by the material properties 4>. ~" . For this replacemen t, equations (4) and (12) are available. Of these, (4) contains the force inclination angle ;:, which can be eliminated by using the rules of friction to guarantee thaI the shear resistance is exhausted on the surfaces of the teeth. This point is elaborated in Appendix I, leading to equations (76). Using these in equations (9) results in the following plastic constitutive equations

(Y:., + V•..,)" (Y:.! - V•. .,)" -

+(a' +(01'

+ b' ) cos 4> sin · · IF + b' ) cos ~/F

(11( .• + V•.cl' - -(a" -b' )sin

~ COS

I

(14)

.' /F

where F' _ I - sin' 4> sin' y ' and the superscript p denotes plastic. Although not belonging to the plastic constit utive equations proper, equation ( 10) for the rotation can be added to complete the system. giving

(v..! - VI.,) -

-(a ' - b' ) cos

y' /F + W

- 2m (IS)

This is not a constitutive relation since the principle of objectivity prohibits a relation containing rotations to be conslituti~. It is added here b«
Soil Mechanics and Transport in Porous Media

de JOSSEll N
SECTION 2 ELASTIC PART OF THE ELAST()'PL.ASTIC DSFR MODE L

In the clastic-plastic DSFR model the strain rates that are imposed by the boundary conditions are divided into an elas tic part and a plastic part. The plastic part consists of strain rates that obey rel ations (13) and (14) when Ihe material is in the limit stress slate. The remaining part is taken account of by Ihc elastie stress rates. Wilen the stresses are not at the limit Slate the plastic part vanishes, i.e. a- _ 0, b" _ 0, and the enti re strain rates are accounted for by an clastic res ponse to the stress rates. Th e elastic part is dcveloped as follows.

x_axi!l., posit ive coun ter-clockwise. The variablc I canno t be negat ive. These quantities a re shown in Fig. 8, where", is shown with the nega tive value, occurring in th e case of the sim ple shear test considered in th is Paper. The Coulomb-Mohr limi t condition is adopted, with limi t lines at an angle,p to the cr'-axis. Cohesion is assumed to be zero for sim pl icity; the limit lines pass through the origio of the o. r diagram. In terms of I the limit condition is expressed as follows. (II) The material behaves elaslically

whenO ';;I<ssin4>,or

)

t_ssi n ,p and i< 'ssin4>

(17)

then a" ... b" ... O. STRESS ST ATE AND L.IM IT STR ESS STATE Under pl ane strain condi tions in the (x,

u..

Yr

directions. the normal effective stress is a prin cipal stress and inlermedia te belwee n the two principal effective stresses in the (x, yrplane. All derivatives in the z-directi on are zero. Thc norm al effective stresses u. cr'n are taken to be positive for compression~ In order to abbrevialc the notation, the variables $, I, ~ are introduced, such that cr'~~-J-lcos2t/1

a.. - a,~ - - I sin 21/1

) (1 6)


(b) The material behaves elastO-p/aslically

whenl _ ssin4>andi .. jsin ,p } then a"

~ Oand

pta""s

"E:"--f"~-+- "

,y•• ~ .! - icos2"'+lsin 21/1{2"')

)

a., _ a" ~ - i sin 21/1 - t cos 21/1 (2,) a~, _ j + r cos 21/1 -I sin 21/1 (2 ~)

(19)

where a dot represe nts the lime derivative dldT. Wilen rotations or the material are invo lved. the objective co-rota tional (bumann) stress rat es H'.. ... etc. are required. These are related to the time derivatives a~• ... etc. by replacing by (2", - 2w) where w is the material rotation.

f

",

Selected Works of G. de Josselin de Jong

(IS)

~ O.

Siress rilles Considering s. I and", as time dependent variables, the e~ pressions (16) become, when differentiated with respect to time

6~ .

POl. fo<

b"

;'

2'"

'.,

' ...

",

I 51

'"

ELASTO-PLASTIC VERSION OF OOUB LI': SLIDING MODEL

l

They are given by

3;,. .. . -iCOli21/1 + l sin21/1(2~-Uu) ' ., .. I,. ~ - i li n 21/1- 1cos 2+ (~- Uu) f" .. . + f cos 21/1 - 1lin 21/1 (2~ - Uu)

- V••/ .. [(I - w)l'... - ..6'",,1/20 }

t.:.

(~.• +

v..eY -

- +(11° + b· ) cos

~

.i n .o/ F

-1{1 - 2v)/G

+ (V•.• +

v,.J sin 21/1

~

H sin (22)

(23)

Combining equations (20). (2 1). (22) gives the elastic oonstitutive equatioM 2vl/0

cr .. + f (~ -

}

(24)

Uul/G

(26)

cos v· , F

-""

Ekw ic COfUIIt .. flW' cq_iOlU

v..r .. - s( 1 -

~/G

- (a · - bO) si n ~

v..1 - VI., - -(a· - bO) cos .'/F + 20

V,.J cos 21/1

(II( ( - V• • y .. +i/G

.• + V•. ( ..

H sin /(I(~ - 20)/G

21f

(11;..( - ~ •.,) .. (V•.• - V••,) .. 2W

52

+ V••,

+ (V•.• -

For the material rotation the transformation is

(~.• + V,

~.e

11(.( - v.. , - +(a· + b· ) cos /(II F

+ (V•.• +

(V( ( +

The dasto-pl astic concept is that in the li mit stress state. as defined by relalioM ( 18). the total strain rates. as determined from boundary condi. tions. are divided into cl as tic and pl as tic parts sllCh that the plastic consti tutive equations {I 3), (1 4), (IS) are satisfied in the first place. The remainin g part of the strain rates is elastic a nd obeys equations (2$). The total itrain rates arc the .um of elastic and plastic parts as follows

.r v,..r cos 21/1

- (V•.• - V•. J" sin

(25)

In the last upre:ssion wfrom equations (2Q) is replaced by the structural rotation 0 defined by equation (15). The reason is that the rotation which has to be introduced in the co-rotational stress rate formulation must renect the rotation of the mate ria l that deForms elasticall y. In the elasto- plastic version of the DSFR model it is the elements between the sliding plsnes that deform elastically. Thus. only their rotation. i.e. the structural rOlation O. has to enter in the corotational formulation and not the material rotation W. which contains an apparent rota tional part. when ",. is unequal to b· .

v•.

(V(.( + V•.• ), - + (V•.• +

v..,Y -

2v)jG

sin /(I/O

ELASTQ-PLASTIC DSFR MODEL

v..: ...

(Ve.e -

.r" H

(21)

where the supcl'$Cri pt c denotes clastic. The terms t...; V• .: ' V.: are the utension rates (V•. • + is the shea r strain rate y.;. 'fhe shear modulus G is related to the modulus of elasticity E and Poisson's ratio Y in the usual way hy 0 _ Eta, 1 + v). The minus signs arise from the soil ltlC('hanies sign oonvention. from the velocity gradients etc. the lIrain ratel in the principal stress directions ( . " (which are rotated by ~ with respect to "', y) are obtained from the following relations

f...r

v...,)"" - 1( 1 -

( ~.• + V..cY " +ssin /(I{~ - 2D)jG

In plane Itrain the oonstitutive equations are

- v,.: -

+

(VI.( - V..

~,~

" "',..10 .. '&,JO [ - .3'.... + (I - wjol'",,1/20

(Ve.1

(20)

EIIlIlidlY AhhougJ! perfect elasticity is not a good appro~imation for soils, it is adopled here for convenience because the formulation of elastic stress·strain rela tions is well known and undis· puted. It reduces the validi ty of the modd con· sidered here to situations where the clastic strai n ratel are smaJi compared with tbe plastic strain

- (V•.• + V•. J

l

In the limit relatioM (18) apply and the elastic constitutive equations become

Equations (26) apply for a poi nt in (x. y) where t he stress state (in the limit) is kno wn. Th is means t hat ' and 21/1 are given quantities. The question is tben 10 establish how tbe tOial strain rates will change the SirCS'l state in the course of time. i.e. how large j and 2; are. when V(.( . .. etc. are given. The first two relations of (26) give ! uplicitly and W can be eliminated From the lasl two. So there remai n two equat ions for the three unknowns 2~, a· , bO • Additional information is given by the requirements ( I) whiCh st ate that the shear strain rates /I . and b· cannot become

Soil Mechanics and Transport in Porous Media

542

do JOSSELIN lie JONG

vertex V of the cone lies in the horizontal plane for i(a;.. + <1;,.> - O(see Fig. 9~ STRAIN RATE SPACE

Fie. 9. SI.- Spil~ orilb limh ~ negative. So there are two equations and two inequalities for damnining three unknowns. This is an unusual system. Vermeer (1980) proposed 10 solve for the unknowns by requiring tile rale of elastic energy diliSipalion 10 be a minimum. The corresponding procedure is examined mathematically in Appendix 2. Since that an al ysis is rather complex, it is perhaps helpful to present the pr~urc vis ually by means of VC(:IOrs in the stress and strain spaces in the following sections.

STRESS SPACE

The stress space is defined by three perpendicular axes, which have as co-ordinates: !{o:" + 17~) vertical; !(u''1 - 0:.) and " .. _ <1" horizontal (see Fig. 9). A pomt P in this space represents a stress slatc. II is located in a homontal plane at a height s, at the end of a radius vector of length I rotated through an angle 20/1, This corresponds to equations (1 6). In this stress space the variables s, 1,20/1 arc in fact <;:ylindrical co-ordinates. In time, the stress point P moves through the stress space and ill path is called Ihe stress path. The malerial rcac:hes the limit stress state when the stress point reaches the limit cone. This cone CUIS Ihe horizontal (f. 2~) oo-ordinate plane in a circle with radius "100" - J sin I/J. Therefore, the semj-verte~ angle of the limit cone has a magni_ tude of arctan (sin I/J). When cohesion is zero, the

Selected Works of G. de Josselin de Jong

In a similar manner it is possible 10 define a strain rate space with orthogonal co-ordinates: (Y•. • + v,,,l vt:rtical, (Y•.• - v,.,) and (Y•.• + V•••J horizontal. Vectors in this space represent strain rates. Fig. \0 shows su<;:h an cum pIe. The origin of the strain rate space is U5ua11y placed at the stress point P of the stress space, wherever P may be located temporarily. Corresponding stress and strain rate axes are oriented para11el but opposite in dire(:tion because of the sign convention adopted for the stresses. Figure \0 demonstrates the combination of stress and strain rate spaces. Point V is Ihe origin or the stress space and the VC'TIe~ of the limit cone. Point P is the current stress point. Since it lies on the limit conc, this figure represents a limit stress state in which the strain rate consists of a combination of elastic and plastic pariS. This strain rate is plotted as the veclor PQ in the strain rate space with P as origin. It is shown how Ihis PQ is decomposed into its plastic and elastic parts, respectively PR and RQ, It is assumed here that PQ is a veclor of known magnitude and direction. In the horirontal plane through P the axes (Y(,{ - y•.• ). (V:., + Y•. () are shown. According to equalions (22) Inese are rOtaled through 21/1 wilh respect to (V•.• - v,.,l. (V•. , + Vr.• ). They are therefore respeclivt:ly collinear with, and ptTpCn. dicular 10, the I co-ordinate of the stress space. In Fig. \0 the f co-ordinate is divided by the shear modulus G in order thai Ihe stress rates correspond to the elastic strain rales (set' equalions (24)). In the vertical direction the s coordinate hll$ to be adjusted by a faclor (I - lvi/G. By Ihis adaptation of the stress space. Ihe elastic part of the strain rate veo:.:tor coincides with the tangent to the stress path. The adapted limit stress cone is more obtuse; its vene~ angle is equal to arctan (sin I/J/( l - lv)). The stress path through P cannot trespass outside the limit cone. When the material deforms plastically it will remain in its tangent plane, So the elastic pan of the strain rate is a vector paralic! to the plane, Ihat is tangential 10 Ihe adapted cone in P. A vector in Ihis tangenl plane consists of two components: one is in the direclion of the line connecting P to the cone vertex; Ihe other is tangential to the circular eross-section of Ihe cone. The fint may be called the !-component, because only s changes when the stress path has Ihat direction; the second is the .).-(:Omponent, because I{I changes along it.

53

,....

de JOSSELIN de SONG

P!~" .

",.ugh Q p.,.II.1

I. I h. to ng,Bt pl.n. in

~11' 12. Sln i" rale • ...,. for .ilotu l .........

Iid''''' (."",,,, PQ of Im&t~ ~

\0 Ihe left of R•. Decomposing PR YcdoriaUy in directions at angle 4> would gi~ a component R. R in Ihe negative b-direction producing a negative value for b. This is not allowed by the thermodynamic requirement of energy dissipation (13). Instead PR is the vectorial addit ion of the vcdors PRo and R. R, where R. R is a "'-component. Before sllowing lhis it is necessary 10 return to Fig. 10. A case (ii) is shown. wilerI' PQ represents the known lolal strain rate. It is composed of a plastic part PR in the horizontal plane and an elastic pari RQ in the plane through Q paral1c1lo the tan~nl plane through P. The point R lies on Ihe intersection line of these Iwo planes. The requirement of minimal elastic work is satisfied when Q R is the smallest distance between Q and Ihe intersection line, So the conslruction of R is 10 proje<:1 Q perpendicularly on to that line. In Fig. 10 the projection R of Q lies belwccn R. and R,. So a case (ii) is involved and PR can be decomposed inlo two positive shear strain rates .. and b. [n thaI case the clastic stress path of P. being parallel to RQ, consists only of an scomponent Pointing downwards it means that ! decreases in the course of time.

Diwta"t or cOMlr=l i"" behauiour

In Fig. 12 the material i$ contractive, i.e. wit h a negative angle of dilatancy v". According to eq uations (14) the plastic shear strain rates a", b" produ« vectors that al"(: located in the plane R. " PR, " at an angle arctan (sin v") with the horizontal plane. This plane may be called the plan e of the " ", b" shearings. Since v" is negative

54

is .'nin

r

..Ie i........lMd simple shnr IISI)

tile plane tilts upwards, i.e. in the negative (v.;.~ + V•.J-dircction. The shear strain rate .." prod uces II vector along PR." and b" along PR, ", but unfortunalely these: ve<:tors do not have the lengtlls of a" and bO. II is the proje<:lions PRo and PRo of Ihese lines, in the horizontal plane, tllal lIue lengtlls u" and b", respectively. This follows from the last two lines of equations (14) because F defined tllere satisfies F ' ... cos1 4> + sin' 4> cos ' v". Tile apex angle of triangle R. PR, has a magnilude of 2 arctan (tan '" cos va). In Fig. 12 the strain rate, imposed on tile material in the undrained simple shea r test. is sllown. It is, as demonstraled in Ihe seclion below, tile ve<:tor PQ of length P in tile negative (V•. , + V, .•}-dirCClion. In order to de<:omposc PQ inlo plastic and elastic parts, Ihe plane is drawn through Q, parallei lo the tangent plane in P, and the intersection line R, "R.- R" of this plane with the tihed plane for .. ", b" shearings i$ constructed. To determine a O, b" it is then necessary 10 proje<:t R. · and R, " on tbe horiwntal plane. This leads to complicated mathematical expressions, since planes at different angles are involved in this procedure. The formulae arc somewhat simplified in the sections below and Appendix 2 by calling M", N° the co-ordinales of point R in the hori~onta] plane, sucb thai M " ... PT and N" ... TR. In Fig. [2 a situation is shown where the poi nt R" lies outside the allowable range R.-R, " and ils projection R outside the range R. R,. This situation resembles Fig. ] lib) and is accordingly called a case (iii). Minimal e]astic work is produced when tile clastic strain rate vector is the shortest distance from Q to the allowable range R.- R.". III this

Soil Mechanics and Transport in Porous Media

Et.ASTo-Pt.ASTIC VE I'.SION OF DOlI 8I.E SUDING MOOEl

case the veclOr QR, ' is the shortest distanOl.: and so R, ' Q is the elastic part of the strain rate. The plastic part is the ve.:tor PR, ' and thai means that only the shear strain mode a' is active and b" _ O. How this decomposition may be derived mathematically is shown in Appendix 2. The clastic ,-cctor R, "Q can be decomposed into a ~-compon ent R, "R" and an $-component R' Q. Th e !,/I-component is in the nega tive (V( .• + Y.,.(rdirection. which according to the last line of equations (25) produOl.:s a negative (2';' - 2U). The s-component is downward. which means a decrease of s in the course of time. The vector R. · Q is parallel to the tangent of the stress path in P. That path is consequently a downward turning hc1icalline on the cone surface. SECTION J EXAMPLE O F UN DR AINED SIMPLE SHE AR TEST

The und rained simple shea r test is used here as an eumple to show how the elasto·plastic DS FR model predicts stress paths. The deformations in a simple shear test can be considered to be homo· geneously distributed over the sample, when the deformations are not too large. This leads to simple equ ations for the imposed strain ra tes and d05Cd analytic solutions for the stress paths. Figure 13 shows a typical result of a stress path plotted from such a solution. The heavy line is the path of the stress point, representing the elfe.:· li.·e stresses on horizontal planes, as traced in the Mohr diagram. Initially the prin· cipal stresses arc vertical and hori~ontal, equal respectively to q, and q • . These form the initial eonditions. The stress point is in Po. The dottcd initial stress circle is smaller than the limit circle. So the test starts with an clastic stage. In this prefailure stage the Moh r circles remain concentric. but expand, and the stress point moves vertically upwards. The end of the elastic and beginning of the

0;,. 0,.

545

plastic stage is at P, when the corresponding stress circle touches the limit lines. Aner P , the sample behaves plastically: aU stress circles remain tangential to the limi l lines. During the plaSlic stage the total strain rates consist of a plastic part and an elastic part The magnitude of the plastic shear strain rates u· and b' is determined in su~h a way that the remaining elastic strain rates dissipate a minimum amount of energy. Applying the procedure developed in Appendix 2 leads us to idelllify three cases, called respectively case (ii), case (iii) and case (i~). The location of P I on the limit stress circle determines which of those cases is occurring. The relevant regions are indicated in Fig. 13. In all three cases thaI part of the total strain rates which produces volume ch anges, is taken account of by an elastic reduction of the effecti,·e stress level. The model is assumed 10 be contrac· tive (negative dilatant} Since the test is undrained, the volume remains constant and the porewater pressure increases !o prevent the sam ple from contractin g. Aecordingly. the effective st ress level reduces in the course of time, causing the stress path to tum left. Case (ii) is obtained when P , lies on the middle arc of aperture 2;$. where ;$ is an au~iliary angle characteristic for the simple shear test. defined by equation (40). In case (ii) both o' and bO will develop. The non-coaxiality occurring in this case is uniquely determined by the ini tial stresses q" q•. The stress pat hs during the plastic stage arc straight lines (see Fig. 17). Case (iii) is involved when P I lies on the right· hand arc of aperture {!n - ;$~ Then only a ' ·type sliding occurs and b" _ O. The sliding planes are vertical. This leads to a toppling bookrow me.:ha· nism. The solution for s as a function of 21/1 leads to curved stress paths from which one example is shown in Fig. 13. More curves of this family are given in Fig. 18. In case (iii) q, is larger than q.' an ini tial situation which may be called active. The apparent angle of internal friction is smaller than ,p.

,'=:::::::::::::::::'~'~'---------------"'... FII- t3. [urnpa. of5l.-- pI,h iD IIIIII....Ined sirnple .!tear ,..,

Selected Works of G. de Josselin de Jong

55

de IOSSEllN d< lONG

Case (iv) arises when PI lies on the left-hand arc of aperture (ilt -~. Then Q ' _ 0 and only b' -type slidi ng ocx:urs, with horizontal sliding planes. The apparent angle of internal Friction is almost equal to 4>. In this case q. < q. initially. a situation that may be termed passive.

When the sample is completely saturated the pore water prevents volume changes. In the simple shear apparatus a plane strain condition in the .-direction is maintained. Then the sample cross-section in the (x, y).plane conserves its area, while deforming from a rectangle into a paraUelogram {see Fig. 14). The deformation is assumed to be homogeneously distributed over the sample. Then the upper plane remains horizontal and keeps its height, II. Let the velocity of the upper plane be tJ towards the left. The velocity components of points in the sample are then given by '"

- fly;

V, - 0

with

tJ > 0

(28)

In this analysis tJ is taken to be positive and furthermore it is independent of x and y because of the assumed homogenei ty of the deformation. The minus sign allows us to compare the computed stress paths in the Mohr diagram with the test results in Figs I and 20 (taken From Wroth e t aI.) and this is an advantage. A disadYantage is that negati~ values of 21/1 ocx:ur in the analysis below. The terms that represent the velocity gradients v, .• - iJ V)iJx ... etc. are Obtained by partial differentiation of V, and V, with respect to x and y. It follows thai VK•K - V, .K -

v,., - 0;

-tJ

V•• , '"

(29)

The major principal direction of the strain rates accordingly makes an angle 0 with the x-&xi&, where tanU! _ V,. ,+

V... i.e.

56

0 ...

v,.• _

v,..

-ill

~.{+v. .• - O

}

~. ~- V... ... - tJsin2\1i

(3 1)

V~ .• + V•. c - -tJcos21/1

IMPOSED STRAIN RATES IN UNDRAINED SIMPLE SHEAR TEST

VK

The combinations (22) and (23) of the strain rates in the principal stress di re.::lions (~, 'I) are given by

-ct).

' (JO)

v•. ~-v~ .. .. tJ ... 2W Comparing the abo~ results with tbe c0ordinates in Fig. 12, it can be seen that the strain rate vedor imposed on the material in the undrained simple shear test is the vector PQ of length tJ.

'"ifial romliflollS At time T ... To the test starts. At that mom ent tbe principal effective stresses are vertical and horizontal. They are positive and called respectively q, and q. (su Fig. 14~ So the initial stress state is given by

ELASTIC STAGE

In the beginning the material is elutie and the gradients from equations (29) are

~locity

V~.~· '" V,.K" - V,.: - 0;

VK.: -

- tJ

(JJ)

Then equations (2 1) give

3'",, '" 3', , '" 0;

a., ... a,. ... tJo

In order to simplify' the presentat ion, the corotational stress rates
Soil Mechanics and Transport in Porous Media

547

EL\STO-PI.ASTtC VERStON OF OOUIII..E SU OtNG MODEl..

201 exceeds

2w by a factor of the order of (v /s). Disregarding 2W com parod wilh 2~ is reasonable wilen v is large oompa red willi the effective stress level s. The simplified equations are oT,. - <7'lI'- O;

i1 .,_i1,. _pv

Considering the second of equa tions (35), the

values of til mentioned above are accounted for by the expression

2tJi ... -

lJ( + arcsin

[!(q. - q.)/f]

(36)

where I is limited by relations ( 17) 10 the values

These can be integraled diTcctly and with tile

O ,, ' <ssin~.

initi al conditions (eq uation (32)) give

tf.. '" constant .. u'~o) - q. 0;., - constan t ... a;,,(0) .. q. 0", -

}

rT

°

7, -

End of clastic 11IId beginning of plastic $Iage (34)

JT.PVdT -

pG(T - To)

It follows from the constancy of 0;., that the stress path in Fig. 13 is a venical st raIght line. In tile co-rotational solution it is sligh.tly curved to tile right. From equations (34) and ( 16) it follows that

I 005

s - so - t(q.+q.) 21/1 - !(q. - q. )

1

(35)

$, - so ~ t
21/f, - t
Sill

21/1, - -PG(T, - To)/s, SIn~)

Sin

~

1

(31)

Equati on (36) then becomes

I sin 21/1 ... - PG(T - To) Figures 15(a) and 15(b) show tile behaviour of the stresses in tile elastic stage. In Fig. l5(a) an active initial stress st ate is shown, where q, > q•. The principal stresses tum clockwise, i.e:. ~ < o. The angle"", is initially zero and enters the region between 0 and -ill. Figure l 5(b) shows a passive initial stress stale. where q, < q.. The principal SlTesse! turn oounter-clockwise, i.e. ~ > O. T he angle >/I starts from - til and enters the rt:gion between - t,. and - i ll. In the Moh r diagram the pole for planes is indicated, and tl\.( heavy line is the stress path foro'"., 0, • •

,., Fi&-

At time T, the elastic stage ends. The point for • in Fig. 13 has reached P" the stress point for which the accompanyi ng stress circle touches the limit linc. Then from relations (18)1, equals s, sin ~ and the stress stale characteri~ by s" !/I , is from equations (35)

0;. ... 0

21/1, - -tll+arcsin(

q. - q~

(q.+qJSIII ~

) (3S)

SECTION 4 PLASTI C STAGE

After T, Ihe material behaves e1asto-plastically and the equations developed in Appendix 2 are then appl icable. These are expressed in terms of the quantities M- and N · defined by equations (19) and (SO). Using equation (3 1) with I/I - !/I, at

'"'

t5. S t _ i. eluti< SI.p: (.) ..aniDg from .<"Iin milial ........ Ie, f, > f •• (.) ".ni",

from"";n inili.I ......... e, f, <

f.

Selected Works of G. de Josselin de Jong

57

548

... IOSSEUN "" JONG

w,.{" V'l..'1l VI&- I &. Dmil or 1'",. 11 ~

time

7;

cancttr orallXiliarJ alllle.

these quantities art: in the beJinning or

the phlSti(: .tage

M · __ 8sin~,

(1~2') .

(1 - 2v + 51"

~

sm VO)

I

(39)

N° . +(Jcos'4t, At Ihis point it is convenient tn introduce an HllAili.ry angle ~ defined by

tan ~ _ tan ~ cos , ,"(1 - 2v) (I 2v + sin !/l sin . oJ

,"utu. In FiJ. 16, which il1ultC:ll<'S. par' of Fill12, it is t ile angle betwocn PT and the Lines PQ.

and PQ•. Fun ner, it redll(Z$ tn '" in the 1'11'11'1_ dilala nl case when v· _ O. Using this angle f 11K: .ariables M O and N ° rrom eql,lalions (39) are

-p sin 2t/t,

Ian f /lan ~ COl v· }

EJf«litoe Sl ress/evtl Aocord ing to equa tion (77) of A ppend i~ 2, it is found from equations (3]) and (40) that for the undrained simple shear tcst 6

1_ - "G

(40)

The all.mary angle. has 1'11'1 special meaning, but is used here because il simplifies the mathematieal

MO_

the thrcc remaining cases (ii), (iii) and (iv) di stinguished in Appendi~ 2. On ly!. the rate of the effective Itres5level S, is the same (or all thrcc.

(41)

N° . +(Joos21/1,

sin 2", tan ~ tan v' (I

Since sin 2. is negative at time T, > To (equations i has the si gn of p', the angle of dilatancy. When the material il contractive, v' is negative and so the effective stress level reduces in the course of time . The relevant parameters "., b', 20/1 differ for the three different cases. 37)~

CASE (ii)

When II., q. sat isfy the inequalit ies

and the combinations (M " Ian ~ COl . " ± N °) which arc of particular significanoc in AppendiJ: 2

- liin ~ " (q. - qJ/(q.

+ q.).in "',;;sin.

(44)

the value of 2., is (aocord inl to equation (38)) limited by

OR

- ~1I - J:"2",, "- ~1I +~

As sin 21j1, is negative (equations (37)) MO (equations (39)) is always pmitive here. This means (sec Appendix 2) that cue (i) docs nnt occur. Af\;cr T, the material behavcs differently in

58

(43)

2v) tan '"

(4S)

and this implies that - I ~ fin ~L . . -cos ~ and -sin ~ .. cos ~L .. sin~. CODliidenng equations (4 1) it is found thai M ' , N° sa tisfy the inequalities

M ' tan

"'_COS p ' > Psin~}

(<6)

-P sin 4> .. N" " Psin 4>

Soil Mechanics and Transport in Porous Media

549

ELASTO-PL\STIC VERStON OF OOUBLE SLIDING MODEL

So the condition of equation (85) is satisfied and a case (ii) OC\:Ur$.

a/a o, bO, ¥ in ca.se (ii) When case (ii) oc:o::urs the plastic behaviour is dC5Cribcd by equation (89~ Using equations (42) the strain rates are Val~i'J

" - +#=('.,+<11 )<

b' _

F/2 cos ,p Sln,p cos

} VO

-P cos (2!/-, -,p) F/2 cos,p Sin .p cos I"

(47)

)<

Considering relatioos (45) it is verified that both aO. bO are non-negative and that therefore the thermodynamic requirements (1 3) are satisfied. The last line of equations (89) requires 2,), to be equal to 2D and this latter can be determined by the use of the last lines of equations (26~ (31) and (47), giving

Po + cos 2\f1,/'sin,p) (48) This shows that ~ is of the order of t/. Equation (43) shows that .i' is of the order tIC. Since: j is 2D ... 2,), _

negative and s red uces in time. its total change will be smaller than s. So PG(T - T,) < $ and the total change of 2!/- will be of the order s/ C which is assumed to be small. In orde r to simplify the analysis, 2,), will be taken to be zero. This means that 2!/- remains constant and equal to 2+, in the plastic stage. As a consequence the stress paths are straight lines dire<:ted towards the origin in the Mohr diagram. A family of these case (ii) curves is shown in Fig. 17.

Nan-<:oaxlalily il! Caire U') From equation (30) the angle 0 bctwcc:n the x·axis and the major principal dire<:tion of the strain rale tensor equals - ill. The angle belwcc:n

Selected Works of G. de Josselin de Jong

the x-a~is and Ihc minor principal compression stress is 1/1. Th e angle of non-c:oa:c.iality (which was called I in previous work) is therefore gi.'en by 1- 0- '" ... - h - "'. Usi ng equatioo (38) it is deduced that sin 21

=

- cos 2", _ - cos 21/1,

- - (q. - q.)/(q. + qJ sin !/J

(49)

This shows that the non·coa~iality {in case (ii) plaslic behuiour, is uniquely determined by the inilial principal stresses q., q• . Coa~iality (i.e_ ; _ 0) is to be expecled only in the special case when q. '" q., i.e. thai the inilial principal stresses are equa l (indicated in Fig. 17 by an arrow). An e~pression for; in terms of the shear strain rates 0 °, bO is found by the use of equations (78), (8 1) and (41). Since (2,), - 20) is zero in c~ (ii). il is found thaI

As 2; _

~ tlt

- 2+,

it follows that

Ian 21 ... tan ,p(bO- a°)l(aO + bO) This relation reduces for the case of volume conserving non-dilatancy 10 the ....ell_kno .... n tan 21 _ tan .p(b - a)j(a + b)

Apparent u"yle a/lnterllal/ricllan i" case (ii) From Fig. 17 the stress paths for a,.,- <1 • • are straight lines towards the origin. These suggest apparent angles of internal friction,p, defined by tan!/J. -

.,.. ;=""'21/1"." a,., - :S-+c':I'ocos

... - sin'" sin 21/1/ 1 + sin !/J

~os

21/1

In this case 2", is constant and equal to 21/1,. Furthermore, 2!/-, can (depending on q., qJ have any value l)etwcc:n (- til -,p) and (- t .. +!/J)

59

550


(see relations (45)). Therefore tan !fir can have values between

tan <1>. - sin 4> CQ$

~J(1 + sin sin,p)

}

and

(SO)

Ian <1>, - sin cos ,pj(1 - sin



sin .$)

for v' _ 0, ,p _ <1>. and these values for 1/>, cone· spond to the prediction mentioned earlier (de Josselin de Jon8, 1972). By using equation (38) it is possible 10 expres5

the apparent angle of internal friction in tenns of q• • q. only. showing the important influence of Ihe initial tlOriZonlal stress on the lest results. When q• • q. satisfy the inequal ities

I~

value of

2J/f,

+ qJ sin <

I

(5 1)

ds

VO

cos I/> sin.$

s dT

(56)

Since the variables 5 and 2", are separated. each side can be integrated dire<:tly, giving - cos,p log (cos (21/I - ,p)) .. ( I - 2v)

cot cos

VO

.

4> sIn

+ 2t/1

sin ,p

:t log s + const. (57) '+'

log [:s

(~~' - t)] + 2t.t/I -

", IJ tan

.p

is, according \0 equation (38).

limited by

_ (1 _ 2v) ( - ill

+ ~ < 21/1, < 0

and thi s implies Ihal -cos sin ,p < cos 2t/1, < ].

,p < sin 2JI., < 0

(52)

and

Considering equations (41) it is found that MO, N ° then satisry the inequalities MO lan cos v' <

Psin ,p

N*>$sin,p Therefore the condition (86) is satisfied and case (iii) occurs. Vlllues 0/11°, bO, It in cIISe (iiI) W~n case (iii) oox:urs, the plastic behaviour is described by equations (90). With MO, N° given by equations (41) the st rain rates are

aO _ - $ sin 2", tan ,pFlsin 4> cos VO} bO _ 0

(53)

This is single sliding behaviour. , b" in the last of equaUsing these values of aO tions (26~ with 2W _ Pfrom (31~ gives 20 - /1(1 - sin 2", tan ,p/sin 1/» > 0

(54)

The last of equations (90) with MO, N° from equations (41) gives

2.j, - - /1(G/s ) cos (2", - cos.p (SS) In this expression 20 is disregarded compared with the other tcrms, because 2Q is of the order (sl G) smaller. Solution/or tlw Sl rtS3 pIIlru in CaM ( iii) In order to establ ish 2", as a function of time, equation (55) has to be integrated. This is.

60

(1 - 2v) cot

sin 2", d(21/t) cos (2)J- .$) dT

Determining the integration constant by using the bound ary conditions at T, gives the solu tion

CASE (iii), ACTIVE CASE

sin,p < {q. - q.)J(q.

however. not possible dire<:tly, because this expression contains also the variable s. Elimination of p with equation (43) produces the following differential equation

00"·II cos 4>.log [";] q. + q

. cos 4> SIO

(58)

where y" given by equation (38) is a function of q•• q. alone. The stress paths for "'~r 0 . . of Fig. 18 are plotted from this solution, usi ng equation (16) with t .. 5 sin 4>. The material constants are taken to be 1/> - 23· : vo _ -3·; {1-2v) _ (}1. These values give ,p ... 28·05" (equation (40)). Curves have been drawn for (q.!qJ - I·45; 1·7; \·96; 2·28, which correspond to 2t/1, '" -6 1·95" ; - 48·S"; - 33·5'; 0" respectively. Apparent angle o/internal/rielion

As shown by equation (55) 2~ vanishes for 2", _ - !I! +,p. So the curves of Fig. 18 approach this value asymptotically and the curve for 11;, - -!n +,p, which is a straight line, is tlK:ir asymptole. The apparent angle 4>. then has the value tan 4>. '" - sin 4> sin 2",,/( 1 + sin 4> cos 2\11,) ... sin 4> cos ,p/(l

+ sin 4> sin ,p)

(S9)

This is one of the limits for 4>. in case (ii) (see equation SO)). Toppling boolcrow medumiSlll

The plastic behaviour in all case (iii) situations is the toppling book row me<:hanism. This can be seen as follows. According to relation (5 1) the inilial slress sta tes in case (iii) are active. q. > q. and so Fig. 15(a) applies to them. Since bO '"' 0, only /1 0 sliding occurs (see equations (53)). Comparison with Fig 3(a) shows that then only verli_ cal sliding planes develop. This corresponds to Fig. 2.

Soil Mechanics and Transport in Porous Media

EU.STO-PLASTIC VERSION OF DOU BLE SLI DING MODEL

+ ,

: tos. iii ; .-.

~

P...i ...

l_, - - - - - - - - ~. ~

:;-

r--

'"

T ' , ~.

• G'

.:: q~

,, "

, "'

,, , " ~

.'

-o,! •

-(}2 -

Selected Works of G. de Josselin de Jong

61

552

de JOSSELIN de: lONG

Furrhennorc, the structural rotation

a

given

by equation (54) is positive, because sin 20/1 is ini-

1;ally nega tive according to relation (52) and does not change sign afterwards. This means a countcr . dockwisc rotation of the sliding clements as indicated in Fi g. 2. It might be noted that the structural rotation

a

is opposed to the principal stress rotation ~. [n equation (55) eos (20/1 - ~) is positive for 2"', li mited by equation (52) causing 2~ to be negative to start with. The principal stresses start 10 rotate clockwise and con tinue to do so thereafter. This falsifies the postulate that structural rotation and principal stress rotation arc 10 be identified. CASE (iv), PASSIVE CASE

When q,. q. satisfy the inequal ities

- ] < (q. - q.)/(q.

+ II.) sin 4> <

- si n i> (60)

a case (i v) situation occurs. An analysis similar to that for ca.'lC (iiil gives the resul ts (1 .

...

0

- /1 SHi

}

4>FlSHI ~ cos y. (61) 2,), ... -t/(G/sl cos (21/1 + 4>l/SHI ~ cos 4> Th e solution for $ as a function of t/J. giving the b- ...

21/1 tan

stress paths., is cos (21/1, + 4>1] ". [ , .,. 1 cos ( '" + ",)

+ 2(1/1, -!/I) tan

-

~

(62)

Now 2,), vanishes for 2"' ... - ! .. - 4>. so the stress paths approach the line asymptotically al an angle ~r defined by tan ~r ... sin ~ cos

4>Jf. 1 -

sin

4» (63)

Since 4> differs only slightly from ~. the apparent angle of internal friction
Figures 17, 18 and 19 show the families of stress paths in undrained simple shear tests for hypothetical samples that o bey the laws of the

62

co ntractive elasto-plastic DSFR model. The ma terial properties of the samples are the same and they have the same angle of internal frict ion ~. Their behavio ur differs in the tests and this is due only to the differences in their initial stresses. i.e. the ve rtical and ho rizontal normal stresses q, and q•. Co nsider for simplicity the stress paths marked by the arrows in Figs 17. 18 and 19. The marked curve in Fig. 17 starts with q. " q•. In Fig. 18 where q. > q., the ex treme active case is marked. In Fig. 19 Ihe ex treme passive case with q. < q. is mark ed. There is a remarkable resemblance between Ihese three curves and the test results from Ladd & Edgers (1972) reproduced in Fig. 20. In this figure the initial stresses are nOI indicated, but th e overoonsolida tion ratios of I. 2, 4 suggest that the three curves correspond to q. values that are, respecti vely, smaller than, equal to and larger than q •. The theoretical stress palhs start with vertical straight lines and have sharp be nds at the transi tion from elastic to plastic. This is due 10 Ihe oversimplification of perrect elasticity in the prefailure stage. Introducin g dilatancy and a gradual decrease or the shear modulus G may $Often the sharp bends. It was. however, not the objcctive of Ihis sludy to try 10 match leSI results by matching material properl ies. The purpose of this study was only to show Ihal the o rigin al, unaltered DSFR model, if e~tendcd 10 ils elasto-plastic version, predicts the unique failure behaviour observed and Ihal the chosen failure mode depends o n the initial Siress Slate. Beeau.'lC of the preponde rant influence of the ratio q) q., it seems evidenl that in Ihe e~eeu­ lion of simple shear tests th e horizontal stresses should no longer be disregarded .

APPENDIX I : DETERMIN ... TlON OF
The thrt:e parameters~. fl. 1 describe the geometry of the slid ing mechanism and the for= acting upon il. The equations devell>ped ;n sectil>n 1. ..clating th~ parameters tl> the prl>pertics ~, •• 1>1 the particle a~mbly as a whl>l., are equations (4) and ( 12) which repea ted are

h.,.

sin~ _ .i n(9+1I1sin (h _ (J _ l)

sin •• _ sin (Jlsin (2~ - (J)

(64)

(65)

T hese are onty IWI> equation. for solving for the three

unknown parameters a. 9.1 as functil>n. I>f~, ••. wh ~h an: material prl>pcrtiC$ I>bscrvable frl>m the I>u t, ide. The objettive I>f th. analy,is below is to show how ... (J. 1 can be dctennined by use nal infl>rmatil>n that the slKling mechanism is oound tl> I>bey the rules I>f friclil>n. In temt$ of the interparticle friction angle ~. thCS<' rutes are

Soil Mechanics and Transport in Porous Media

EI..AS'TO-f'USTlC VEItSION OF DOUBLE SlIOINO MOOEl

fa) oIidina .... i11 0001 OOXUt .... hen .. is smalln Ih.n _. (b) $Iidinlcan oa:ur .... hm ;, C(jlll.b __ fe) il is impossible for l 10 ucced __ .

llId s'na: " .. 0 for l ... , Ihis gi'o"Cl, ";th

From . 11 sl;.Jin, &e<>tnelries havina value combinlliont of ... 9Ih.1 prod...,. ~o aocordinllO (6S~ only • particular tel of ... /I combin.lions Cln be Clpeded 10 be acli~. ThaI JlVUcuI.r H1 i. the sct WI introduced iIolO U1W1Uon (64) ai_lhc oo.cr.-cd value. rot. YIllue of .I. .... bich U1ual1 _ . However, .me.: l ClIIJK)( cuud _ •• Ibis tel of ... combinations is nOl allowed to I;OIItain values of l elccedina ••. This me:lIn. lit" .. oonYderul as • fllnction of ... 9 mll5t b.ve an utrcmc yal ... .l.,,_ which is a muimum, lhal l .... .. _ •. Elimination of 0 f'om U1ualions (64) and (6S) gi"ts

I

.""h

.... illt ...... (1

+sin.lin yO'+(lin

_+lin ."')COI h

("') '" h>- ,'

.-

.. {sin _

+ sio ."', + (I + lin 4> sill ."') 1>:11 2
.-

sin. _ -cos {h - 8)/1>:11/1

+ sin y' I+sin.sin.o

(7SI

In"erlion from Ih;1 relalion and U1l1ation (6S, Ji "ts Ihc followin,nprcuiOIlJ for 9 alld (2" - f1)

sin /I _ COl. s,n v' j F COl 9 _ COl ,oJF

sin (h - 9) _ COl _IF <XII(lo -

0) _

-an.

""

COl ,o/F

F' _ I _ sin' • sin'."'

(67)

(68)

i J/i2 _ 0 or II .. 0. Th;.

COl 2<1 _ _ sin -

v' ) lin 2a

t.

""

willt ~

+ sin 4> s;n

- ~(I

_

.O.n:

A muimum OOXUrl. when

Ji¥C1l by

AI " is Im.11er Ih.n til. this result IhOWI Ihll 1M MOOnd d~rivlI;YC is nqal've. So 1M solulion .I..".. .. 2a - 28 nprcscnll I maximum and this Illidies lbe rults 01 frif;!ion. IntroduciDI .1..,_ in oq .... cion (64)

This Jives .. as • fu.nction of ... In ts~bli"'in • .1..". by dilfcrmt.i:alion 01 U1 .... lion (66) ,,;11t rc:spccIIO", only II is. variable; the y,lues oI •• nd filed q.... nlilict.. Tlkin, the derivative Ihen givts (d;';,h)!col' A ... 2(sin 4> - lin . ' W",'

~

equalion (68)

(69)

APPEN DI X 2; DECQMPOSlTlON Of TOTAL STRAIN RATES INTO PLASTIC ANO ELASTIC PARTS Equalion. (26) (Jpress lhe componenll of the lo~1 SIRiIl rllts in IcmtI 01. "I. b'. J .nd Q~ The IInkllO"·n J (:In be Iound dircctly ftOll' tile lim lwo by eliminalion 01 (/I' + b O~ This Jiyts

(.J. _

j(l - 2. + lill • sin , O)iG •

-(Vt<

Rcpl8.;ernenl of . in ., sin y' by ..... of U1u. tiont (64) and (6S) giyts

+

v..• )+ (V•., -

v•.• ) sin . '

(77)

The connection of Ihi, ",Ialion .... ilh "ecIO .... in fia. 12 i, th. t Ih~ ri&hl-halld side divided by (I + lin 4> lin ,,O J (I - 2.) U1ual1 RRo, llIe vocnical componcnl of QR' • .... hich ;IXI( is parlllcl lO VP.

SoI
fl« - 28 - .., - 0

(10)

Since II cannOI be zero (In f ip).nd 4) lhe SOlution il

FtOII' lbe firll lWO 01 U1U.IKmI (26) it 1"0110'" lhal

(71)

This rault is U1l1ivaicnl 10 RO.... c·1 "'""' dilal./lt)' rd.· tion (1962~ III order to verify whC1Mr .1..". indeed produoes • mnimum, lhe KCOnd
I,,'

+ bO) cos . IF - ""

+

V•.• ) lin .

(71)

with

MO _ (VI!

(L

+ (VI-! - V,. ,~I - 2.) (79) 2.+sin_sin"')

lAt. q.... nl;ly N° be defined by

H' - -IV,., + V...I

'''''

The third ofUl .... tions (26) then Jives (a' sin _.,.,. vOj F

,,0,

f .. 2 (lin. AI •

is .hvay:tl

sin ,,0) cos'

lar., Ihan

J/""

.. H'

(71)

."'. / is posilive. The

wcolld derivative 01 A il

i'A

+ sin

4>(t/GX2~

- :w)

(81)

UsinlUlWIlionl (78) I nd (81) the war "rain ' i lts ,,0 .nd b O c:an UpressN;n lerms 01 ", 0 .nd H". Thcre

w

(($1I!It

ill

fI{

---/+,,112' II", d2

Selected Works of G. de Josselin de Jong

24' sin _.,.,. .o/F - sin ...J/GX21 - :W) _ Mo lID _ COl

,0+ N '

(82)

63

'"


is 2 an:tan (tan 4> cos y' ) the masnitude of N ' .... ith respect to M' Ian ~ COS y' determines how R i. 10C!ucd "ith respect to that trian&ic- The tbree different pos_ sibilities arc indicaled as cases (ii~ (iii) and (iv) in Fig. 2 1. They are a. fOlio ..... . (ii)

When R lies on th. section R, R. then

- M'

tan 4> cos

y' ..

N' "

M' tan 4> COS

y'

(85)

(iii) When R Ii .. to the left of R, then

N' > (iv)

+M ' tan~cosy'

N' < - M ' tan

~ Fil. 21. HoriJ.oalal ,..... of FiJ. 12, IbowiDg R in lho _ (I), (ii), (iii) aIM (iy) 2b' sin ~ co.

' "IF + .in

q.('IGK2~

_ M" Ian

Eliminalinl (aO - h O) between fourth of equations (26) gives

-

,.pons "r

20)

If> cos." -

eq~alion

N°

(83)

(81) and the

. in If> WG)(2¥ - 20) _ (20 - 2«» sin If> - N° (84) The quantities MO and N°, IJ defined by eq..ations (79) and (80~ are the oo-ordina'.. of R in the hori:.emisl plane of FiJ. 12. II may be verilied Ihal MO _ PT and N ° _ TR. fig. 21 . ho ..... thi. horizontal plane as viewed from above. Th II•• cases Ii). (ii),(iii) and (iv) mentioned in 1hi, Paper,

Cuse (I) Wh..., Ihe compOJIe1lIS of Ihe imposed ,' rain rale are such Ihal (I'l.' - V,.,KI - 20) is smaller than -(V,., + V,.J si n .". the val~ of M O i. oepli,'. acoor (according 10 the first of equalion. (24)~ When the str<:SSn are in the limit state, then a situation oo:urs as indicaled by the second of equation . (17): the . tress poinl P rMnters the elastic region and ddormation i. purely clastic. The corresponding str.in rate vector point. downward. in Fig. 21 inlo the region indicalcd as case: (i). In thai case the Stress history beina purely clastic is dc$cribed by equations (24). Equation, (26) . re not valid and (a ' + b' ) sohcd in the Manner of equation (78) does not apply. Actually (78) with M' negalive i. unaODeptlhlc, because: thai ';olat.. the thcnnoo)'llamic requi rement (13} In the undrained simple: shear test as described by equation. (11) M ' defined by (79) i, always positive due to the third line of (J7). So case (i) docs not occur and th.re remain th .... caxs---(ii). (iii) and (iy)-.... hich arc releyant to this study.

,w"

Ca.<el Oil . (iii) olld (in) M ' and N ' are Ihe co-ordina les of R in the horizon· tal plane in Fig. 12. A. the ape. a ngle of lriangle R, PRo

64

(86)

When R lies to the right of R.then ~

cos y'

(87)

IHtrrmilUllioN of (z¥. - W ) When th. total strain 1111. i, given. M ' and N ' a ", known from equations (79) and (SO). Then only t.... o r.lalion5-( &2) and {83)-ar. availab!. 10 Iklennin. the three unknown. 0' , b' , (2o/r - 2D~ .ince (84) docs not contribut. any useful information. a being also unknown. Vermeer'. proposal for solvinl this system i. to require that Ihe rat. of elaslicenergy dissipation ,t' i. a minimum. U,inS this principle, tOge1!ter with the therm odynamic requirement of eneraY dissip.tion, it i. argued belo .... on logical grounds Ihat one of the thr ... unknown.;' ~ro in each of the th ree cases (ii~ (iii) and (iv) mentioned above. The quantity A" is defined by

A" .. - v•. : ~.. - v•. :~" -

(Y•••

+ v,.•

ra .•

and tbi. can, by using equations (20) and (21) he transformed into

,t' .. [Il - 20),1' + i' + 1'(2o/r - U»')fG Comparinl equation. (24) with the limit values of equalion'12~~ A" becomes for the limit stress Stale

,t' .. (II - 2v + sin'

~)i' +.' sin' 4>(2~ - 2n)'JIG

18S) Since relation (17) establi.hes J independently, • minimum of A' i. obtained by choo.ing the lowest """. sible value nf (2~ _ W)'. Thi. is ta ntamount to requiring Ihat the absolut. value ) 2~ - 20 ) i• • minimum. Together with the thermodynamic requi",· ment (1 3) the conclusions .... hich follnw co.n he drawn. Rt lulll"l1 t X(lTtlSlons

COl
.'l

" , .. (M ' Ian ~ cos v' + N° )F/2 si n ~ cos b' .. (M' tan ~ 001'" - N' )F/2 si n 4> cos v'

(89)

211- - W ... O CQM (Iii). In this case (2o/r - 20) cannot he zero, because (S6) introduced inlO (83) would then produce I ncaative b' , which violat.. requirement. (13~ A minimum for )2o/r - 20 ) is obtained by chooslns b' 10

Soil Mechanics and Transport in Porous Media

El..A$TO-PLASTIC VERSION OF OOUBLE SLIDING MODEL

do: 1000Iin do: Jonlo G .

be Jm). So 1M solution is ,.,·F/c04.;> 0 /I. _0

tJ· _

(2'" - 20) "n4«~G)

... (+M ' tan.a1S.· - N · ) <0

Cax (lv~ In lloimilar mlnne. (87) and (81) requi'" in tbi. cue that 4 ' be zero. So 1M IOlulion i.

_0 } /I. _ M'Ffco'It;;> 0 (2. _ W) $in ¢Is/G) ... ( _ M ' tan '" cos y. - N· );> 0

",

( I ma~

Mathematical dab<>ralion or the double llidin& free /0111;0& model. A,c!tiCC$ of M fC'-Sa 2t, No. 4, S6I-SSlI. de JOlSdin de Jonlo O. (l977b~ Constituli"" rdl tions 10< lho !low oIa .......1.. u....... bly in lhe lim;\ ".Ie of Itml. f>r«. 9r~ 1111. Coo(. Soil M fClt.. Fitt £"(1"(1, T okyo, pp. 87-9S. ()rQcher, A. (1976). An npeOlMnll. inyesl;<ion of 110" rules for ITInu.ar ItIItcrial, usin&optically sen· tili"" 1115. panicb. GftHfCh~lqw 16. No.4. 591 -

.,,-

U9n~ COIISOIl.dllUJ ~nd,aiMJ Ji,«:1 Ji"'p~ SUD' 1t.1I on ""! IUIl/tti clay•. R~arch "'pOrt T 12---82, M alSlthUKIIJ Inll;Wle of Technology. Randolph, M. f . and Wroth C. P. (198n Application of thl: f,ilure SlIle in undrained simple . hear to 1hI: lhaft capacity of driven piles. Gtow:hniqw 31, No.

Ladd, C. C. oft Edit'" L

4·

(91)

REFE RENCES

BtJri n, D. L (197J~ T10t MMl:imit of JDlllttJuJ h>oIl~ ill llot .lJnplt IN"'" 4P_41111 PhD thesis, Uniyersity of Cunbridse. do: Jonelin de Jonlo O. (l9S8~ T1Ic unddinilenm in kinmllliel for friction matcrials. I'roc. Cot{. £tJrf~ Pta"," Probl. BI"IWdr I, 5S-70. do: Jouclin do: Jnnlo G. (1959). SUllies tJJtJ ki....""'flc. in l/w f<'i~ . _ of tJ , ...,.,.u" _mal. Thais, URi\'U"Jity of DdR. do: Jossdin do: Jonlo O . (1971~ T1Ic double slidiD& free rotlti", model for ITInular ........ blies. GftH«It/OIqwo' lI, No 2. US-163. do: Jouclin do: Jonlo O . (1972). DiJcuslion, Session II , It_ Memorial Sympoaium. S,rnJ-Jl,tJin 1It"'..,..,. ofJOiIl (ed. Parry. It. H. G.) pp. 2S&-261. Cam-

bOd.,: Foul ...

Selected Works of G. de Josselin de Jong

1.10- 151. Ro we. P. W. (l962~ The ""'" dilatancy ",Iauon for ' Ialk equilibrium of III auembly of particles in contlC!. I'roc. R.oy. Soc. 11.169, m-S27. Twnisscn, J. A. M. & Venn .... , P. A. ( 1 9I8~ Analysis of doub!<: tbelrin& in frictional ItIIlmlls. Inl. J. Nwnn. d A.wy. Mflll. ~II. 11, )2}-:WO. VeTl1I«f, P. A. (1980). Double Jlidin& witbin I n elUtnplu.tic f........OI"k. l...G. M .-.leUlI"fIt~ 11, 19'}...2O'1. Vmn«r, P. A. ( 1 981~ A fonnulation and an.a.lyois of &ranular no ... "'oc. 1111. C""," M«II. &It<maur Slrvct. Mtfli4. " 3tl-339. WrOlh, C. P. (l9l-4~ T1Ic interpretation of in situ lOiI 1eII..

w.,,«MIqw :M. No.. 4. 44~.

Wroth. C. P. (1987). The behaviour 0( normally consoli· elated day .. obKrved in undrained direct .bear le,U. G#of«Miqw 37. No. 1,31....1.

65

de Jasselin de Jong, G. (1989). Geotechnique 39, No.3, 1989, 565-566

DISCUSSION

Elasto-plastic version of the double sliding model in undrained simple shear tests G. de JOSSELIN de JONG (1988). Geotechnique 38, No.4, 1988,533-555

that case (iii) is shown, since R in Fig. 12 lies in the region indicated as case (iii) in Fig. 21. If this is correct it implies that a stress path of the type shown in Fig. 13 is followed, so that 21/1 1 decreases gradually from 0 to [-(nI2) + 4>]. This gives the impression that the stress point P moves along the cone in Fig. 12 (as viewed from above) in a counterclockwise direction. The following

R. E. Gibson, Golder Associates (UK) Ltd On p. 544 of this Paper the Author states that the vector PQ (in Fig. 12) represents ' ... the strain rate imposed on the material in the undrained simple shear test'. It is, however, not clear to me which of the three cases (ii), (iii) and (iv), is referred to here. Comparing Figs, 12 and 21 it would appear

t(6~x - 6;y)/ Go Fig. 22. Adapted stress space witb limit cone, similar to Fig. 12, but SbOWD from other side

565

66

DISCUSStON question then arises: what stress path will P follow in the remaining cases (ii) and (iv) ? Perhaps the Author will favour us with one of his elegant drawings to elucidate this point. Authors' repl y

The case represented in Fig. 12 on p. 544 is indeed a case iii. In order to demonstrate this explicitly the situation of Fig. 12 is redra"'n here in Fig. 22. In this Fig. the adapted stress space with the limit COile is shown from the other side by rotating the space over 180 degrees to permit a better view of the stress paths. The main rate vecto r, as imposed by the undrained simple shear test, is a vector of length p, parallel to the axis of f1•.IG _ f1 j G and in its d ir«:tion. So it points towards the left, here . It is shown only once as the vector P ,Q for the case that the stress point is located in P, . Let the stress path POP'P l be considered in more detail, fi rst. Point Po represents the initial st ress state, in which according to the st resses. indicated on the co·ordinate axes, a,,(O) a, ,{O) - 0 and a;,(O) < f1~(O). According 10 equations (32) this implies q. > q. and so an active, initial situation denoted as case iii in the paper. is involved. The part PoP, is the stress path during the elastic stage. This part has the diroction of the imposed strain rate voctor of length p. The elastic stage ends when the stress point has reached the limit cone in the point p .. This point corresponds to the point P of Fig. 12 in the paper. The tilted plane R, oP,R." is the plane of the a·, bO shea rings. The imposed strain rale vector P,Q of

Selected Works of G. de Josselin de Jong

p is decomposed into a plastic component P,R,". consisting of only /I" shear, and an elastic component R, oQ, parallel 10 the tangent plane of the cone in P,. The elastic component R. oQ can be deoompooed into a ,J.-c:omponenl R:R o p a rallel to the circular cross seclion of the cone and an scomponent R OQ parallel to the line connecting P to the cone vertex V. The vector R:R* points towards the lefl and Ihal gives the stress poinl P a motion in a counterclockwise direction as viewed from above. This agrees with the corrOCI impression of Professor Gibson. The component R*Q points do"'nward and thaI produces a downward movement, in the form of a helical line on the cone surface, that spirals downwards. AI point P, Ihe langent to this line has the di roction of the vector R, oQ. Two other of those helical lines arc shown on the far side of the cone surface. BOlh belong to case iii initial stress states. Case iv initial stress states produce similar stress paths, that are shown on thi s side of the cone surface. The helical lines on both sides of the cone are similar, but they are each olher mirror image. When the ct:nt~ angle 21{1, of the radius to P , has values between (- tit + i/.) and ( -til - (1,). then case ii initial stress sta tes are in~olved, originally. For such a case ii the cor~­ sponding dotted stress paths, after reaching Ihe cone rim, turn downward and follow a straigh! line path towards the cone vertex. The ~urved helical stress paths of the cases iii or iv initial stress states have the border li nes between case ii and cases iii or iv as asymptotes. length

67

Reprinted (rom Geotet:imique, June 1978

TECHNICAL NOTES

Improvement of the lowerbound solution for the vertical cut off in a cohesive, frictionless soil O. DE JOSSELIN DE l ONG· INTRODUCTION

One of the classical problems in soil mechanics is the determination of the depth h, to which a soil can be excavated by a vertical cut off before collapse OC(:urs. When the soil is rigid plastic with cohesion e and no internal friction, - 0, and its flow properties obey normality, the upper and lowerbound theorems of plasticity are applicable to h. Let a parameter « be defined by

h

~

«elY

where y is the proper weight or the soil, and let «coo, be the value of « correspondi ng to the collapse height hooL , . Then according to the upperbound theorem a value of « larger t han «ooL' is obtained by computing h with a kinematically admissible velocity fie ld, and according to the lowerbound theorem a value 0(<< smaller than "wi, is obtained by usi ng a statically admissible stress field. The solutions known from literature (Heyman, 1973) give ,, _ 2·83 for the 10werbound and « _ 3·83 (or the upperbound. The purpose of this Technical Note is to present a staticaJty admissible stress field corresponding to« - 3'39, which improves the existing lowerbound value. A stress distribution is statically admissible if it obeys the following requirements: (a) The stresses arc everywhere in equilibrium with the soil weight. With lension stresses positive and y in negative y-dire<:tion, the equilibrium equations are (8a.J8x)+(&rsJ8y) - 0 (aa.~/ax)+(aa~~/ay) _ y

(I)

(b) Stress discontinuities are allowed, provided that equilibrium is not violated. With local coordinales n, I normal and tangential to a discon tinuity line separating regions (a) and (b) Ihis is expressed by (2)

(c) The stress state in every point of the interior is within or on the limits imposed by the yield condition (a•• - a.~)~+(a•• +a•• )~ :$ 4c" (d) The stresses comply with loads applied to the boundaries. cut off the stresses on all boundaries are zero.

(3)

In the case of th e vertical

LI MIT STATE OF STRESS EVERYW HERE

The idea was to use a computer for the generation of a statically admissible stress field in ~ 0IL'1oJo ToduIkaJ NOle ...... I Sept.... ber, 1m . For further LIc101b _ '-Ide btl • • "",or. o Department orei.il En.in«,;n., Uni.en.ity orTechnolOlY. Delft, Tho N •• hcrl.ond$.

68

".

TECHNlCAL~

Fla:. 1. DlsroIItlnuity lines, "htl! the tntlre region .d).emt to tbe boundary Is In the lim it slate of ItHIs

order to avoid the usual procedure of predetermining intuitively stress distributions, which restrict the analysis 10 a limited class of solutions. The stress field was generated by using K311er's equ ations, which are based on equilibrium and the limit state of stress everywhere. The equations are integrated numerically along slress characteristics 5, and 52 giving the mean normal stress and the principal stress directions in successive noda l points of the network of characteristics. The shape of the network is obtained by computati on of the locations of the nodal points assuming arcs of circles between them. Being a solution of Kotter's equations the stress distribution satisfies requiremen ts (a) and (c). Where necessary stress discon tinui ty lines were in troduced and the stresses on them were made to satisfy (b). The construction of the stress characteristic network starts from the boundaries, where the stresses are known. On the boundary the solution has to be chosen to be either strong or wea k. Since shear stresses on the boundaries are zero in this case, the principal stresses are pa rallel and perpendicular to the boundaries. The principal stress perpendicular to the boundary is zero, but the principal stress parallel to the boundary can be either a tension or a compression . When the stresses along the boundary are in the limit state, the tension or compression have both the same absolute value, 2e. The choice was made to start along AD with a compression of 2e for the prin cipal stress parallel to AB and along CD with a tension of 2e (Fig. I). So below CD the stress state is given by (4) 0 •• = )'y+2e; O.y = 0YX = 0; On "'" yy Satisfying (I) and (3) with the equality sign. A first trial was made by assuming maximum tension 2e for the principal stress all along BC, also. Constructing the field of characteristics leads in point C to overlaps, which resolve by in troduction of two stress discontinuities CE and CF, making angles .,,/8 wi th CD and CD respectively in the corner point, C. The discontinuity line CE has a bended form and cuts th e vertical in a point B located at a depth 3,24 ely. In this solution the entire region CEB is in th e limit state of stress. In order to satisfy the requirements (a) and (c) in every point of the interior, it is necessary to extend the stress field over the entire soil body. It proved impossible, however, to con-

Selected Works of G. de Josselin de Jong

69

T1'C HN IC"~

'" "

NOT£'s

• I

CO'FF'

CO'FF'I

'-.lr-I I

FF'HH~

\..1I

- - U;u-l r1

i'" I

--- 16 yy -JY)

I I I

I I

I , ,I

H~

I

I I I I

I I , I pp'

: 101 : I

.................. :.; . _. . I

I

, I

(oj

I Jj IKI<.'II"K I I I

I

"J

Fig.! (a). Fk ld or s er, . d\anodnistic:5. "hom lhe rrgioo adjacmll(l BC Is IMlI in lhe limit SlIIlt(lf strtS5; (b)bori_tal and Wfrtiali prladpGll .1rts5ts up I(llnfinily In tbt rtglon I(llbe rigbl (lr FHJK

tinue the solution in the region aroun d B and downwards, because the sheet of stress characteristics folded backwards. Therefore the solUlion mentioned above was not acceptable. STRFSSES IN CEB NOT IN LIMIT STATE

In order to render the situation in B better suited for adjustment to Ihe passive Rankine Stale in the region below AB, it seemed appropriate to consider a non limit stress state in the region CEB expressed by (5) This stress state is statically adm issible because it satisfies (\) everywhe re, and fo r (- 4cl"'l) < y
70

Soil Mechanics and Transport in Porous Media

200

cAlcnding the stress characteristics s. from the line CF lip to the region CEO, wh ere the 51rcu

state given by (5) exists. A next nodal point on CEB is found from a previous one by intersection wi th a subsequent Sa stress characteristic and bending both in such a manner, that the stress state required by Kijtter's equation along the s. line is in equilibrium across the discontinuity line with the known stress state (5) acco rding to re lation (2). The line CEB curves and crosses the vertical through C in a point B located a a depth 3·39 ely. The s2line through B inte rsects the other discontinuity line emanating from C in the point F (see Fig. 2a). For this solution it proved possible to continue below B with a field in the limit stale of stress by introducing two Prandtl wedges with their centres in B. The pattern of stress characteristics obtained by starting from the line AB, the two wedges in B and the line BF, can be extended downwa rds towards infinity. The pauern is bounded above by a SI line shown in Fig. 2(a) as the d otted line, FO. STRESSES BELOW FF'

A conflict arises in elttending the solution in the region above FO, because the pattern of stress characteristics will fold and solving KBtter's equation beyond such a fol d requ ires complicated arra ngements. T he analysis was simplified by assuming a stress field with principal stresses vertical and horizontal, in the region below the horizontal line FF' and to the right of a discontinuity line FHJ K, wit h K at infinity. Since shear stresses are zero on vertical and horizontal planes, integration of the equilibrium equations (I) then give Un W+f(y); U u " U u = 0; Un'" yy+g(lt) (6) with f(y) and g(lt) arbi trary, adjustable functions. In order to have equilibrium on the line FF', between the stress distributions (4) and (6), the value of g(lt) has to be zero along FF'. As a consequence g(lt) is zero in the entire region to the right of FHJ 'K'. From F to H the value of fey) is adjusted in such a manner, that equilibrium exists through the discontinuity line between the stresses in the established field of characteristics BFa and the stresses (6). In every point of BFO there is one possible di rection for such a discontinuity line, and F H follows as a trajectory by plou ing successive tangents. The line FH is unique. At point H the tangent is vertical and the discontinui ty line is also principal stress trajectory. F rom H to J the discontinui ty line is arbitra ri ly chosen to continue as principal stress trajectory. T hen shear stress on HJ is ze ro and equilibrium across HJ requires that f =g on HJ, because only then shear stress to the righ t of HJ is zero, the stress state being isotropic. In order that (6) satisfies requirement (c) it is necessary, that (7) (f(y)-g(x» ;S 2c in the entire region FHJ KK'J' H'F'. In Fig. 2(b) a plot of f(y) full line and g(x) dashed line is given . A vertical line pp' in the region HKK' has a constant value for g(x) and this gives a vertical dashed line in Fig. 2(b). The point J is at such a height, that the line n ° is critical having (f-g) .. 2e. Below 1 the discontinuity line J K is taken at .. /4 with the horizontal giving values of fey) and g(x) as shown in Fig. 2(b). From this figure it follows that (7) is satisfied everywhere. CONCLUSION

Constructing a stress field by characteristics with stresses in the limit state everywhere did not lead to an acceptable solution. It was necessary to assume in the region CEB stresses that were not in the limit state. The stress field obtained then, satisfies the requirements (a) to (d) in the entire soil body and therefore it is statically admissible stress field. T he value 0:<=3'39 is t herefore a lowerbound to O:<",,:n'

Selected Works of G. de Josselin de Jong

71

001

That the solution gives a sare height (ollows from the consideration of the kinematics. A potenti al slipline would be the s~ line BFD. Slip cannot occur along this line, though, be<:ause of the slope discontinuity in F. The solution was verified in 1976 by C. H. Engels with a different computer program, based on the relation for stress discontinuities developed in her graduation study. REFERENCE Heyman, 1. (1913).

Simple Plastic TheQry applied to Soil Me(;hanics. PrtIC, Symp. On Iht role of pIllS/idly Cambridge.

mIDi! multtMJJu, 161- 172.

72

Soil Mechanics and Transport in Porous Media

D~

JO$$I!UN DE JOI'
30. No,

I, 1-16

Application of the calculus of variations to the vertical cut off in cohesive frictionless soil G, DE JOSSELIN DE JONG The collapse heighl of a vertical cut off is computed

hauteur, corrcspondant 1 la ruplure d'un talus vertical, esl dtterrnin« it l'aide du cakul de varialions en prbupposanl I"exl!; tence d'une: liJlle de I lissemeni unique Cn eas de: rupture. La classc de ligneS, COntenanl celie ligne de gl iueme nt, est definie par nquilibre IOUII el local soU! C(l n~ition d'~UlI de eo nlraintes limites. L'exll"l:male de la class posstde Ia forme d'une involute. En vtrifiant la solution, il est fUnlOnU"t que ]'ex ll"l:ma le produit une hauteur de talus plus ~1cWc que la hauteur de rupture, oil une: hauteur qui necorrespond pasdu loull un extremum, Ces resullaU dCcevanu soni engo:nbl
by use of the varia tional c.o.lculus auumin8 tl«: uistence of a real slip line at collapse:, A ciass of lina containinl the real slip line is defined by tOUil and local equilibrium co nditions of the limit stress state. The extremal of lhe dass is found 10 be an involute. Verification of lhe: solution shows lhal the extremal gives ~ithc:r an unsafe eslimate of the collapse lwciaht Or corresponds 10 no ulremum al aU. These disappointing res ults aTe a comequcnce of the inadequate fonnullttion of slope !lability problems, when slip linn are computed by the calculus of variation, .

lNTRODUcnON The calculus of variations provides mathematical procedures to find the shape of an e1ttre mal, the curve that ma1timizes or minimizes the value of an integral along that line. For the readerwho is not familiar with the subject the books by Bolza ( 1973) and Petrov ( 1968) are recommended here, The first because it trea ts all aspects of the parametric solution employed in this article, the second because it gives a comprehensive and convenient descript ion of the subject matter and its practical use. Because it can be used ror establishing the shape of a line wit h pa rticular properties, the ealculus of variations seems of interest for soil mechanics. especially for solving slope stabili ty problems. Since Coulomb, it has become a standard procedu re to solve plane st ra in stability problems by using minimalization procedures in search for the line that represents a railure plane, The variational calculus could be applied to determine the shape of such a line, if the mechanica l requirements of failure along the line can be formulated in the form of integrals. whose e~treme values are related to the stability, This idea bas occurred to several investigators, e.g. Revilla and Castillo (1977). Ramamurthy e/ al. (1977), Baker and Garber (1977, 1979). The approaches proposed by these different writers differ and demonstrate that there is not one unique manner to formulate the minimalization problem. Also th e Author was intrigued a few years ago by the possibility of usi ng tbe variational calculus for establishing sli p lines, but withheld the results from publication when it became dear that the solutions were of a disappointing, unacceptable character. The reaso n for presenting this work now however is that an analysis as pub lished by Bake r and Garber(1978) is incomplete and produces re sults that prove to be not mean ingful when the analysis is properly pursucd. It is the objeetive of thi s Paper to demo nstrate an analysis that is eoncluded by investigating the 1)iJcusslon on Ih...... peT doses I Ju ..... 1980. For furtl>er tkl:alLs I t t l",lde bad: co..... • Department of Ci. il En,ginceri"l. Ulliversity ofTcchnolol)', Delft.

73

2

G.DEJONO

conditions for an c)(tremum and to show the kind of disappointments that are encountered when the calculus of variations is applied to determine slip lines. In order to be specific and to deal with ex plicit results, the relatively simple case is treated here

of a vertical cut off in cohesive, frictionless, non-dilalanl soil. The collapse behaviour of the vertical cut offin such a soil has been studied by reliable procedures based on the rigorous proofs of the theory of plasticity. The exact solution is not yet available, but il is known thaI the collapse he ight h'911 is unique and expressed in terms of the cohesion c and the specific weight Y. is between the following limits 3'64c1l' ~ h'Q" < 3·83c11

The upper limit corresponds to a Fel!cnius solution with a circular slip line, satisfying a kinematically admissible velocity field . The lower limit corresponds to a statically admissi ble stress distribution determined numerically by Pastor (1978). Because of this information on the collapse height, the vertical cut off is an appropriate example for testing the determination of slip lines by variational methods. CONCEPTS BASIC TO THE ANA LYSIS

In its simplest form the calculus of variations is used to de termine the shape of one particular line, wh ich is called extremal, because it can produce an extreme value ofa definite integral. The analysis described in this Paper is an application of this simplest form of variational calculus to the slope stability problem of establishing a safe estimate for the height h of a vertical cut off. The one line ana lysis is based on a few concepts that are decisive for the formulation of the problem and are determinative for the resulting solution. The first concept is the assum ption that at collapse, there exists one particular slip line, which is called here the real slip line. The second concept is a class of potential slip lines, defined in such a manner that it includes the real slip line. The third concept is the presumption that a safe estimateofh can be found by determining the extremal. Real slip line

In the plane strain case of an embankment in z direction, the slip line in a verlical X,)I plane represents a failure plane perpendicular to X,)I. Fai lure planes exist at collapse. They are required 10 separate soil masses that slide with respect to each other and with respect to the stationary soil mass below. When sliding occurs over the failure planes, the stresses along it are in the limit state of stress. Since the soil is taken to be frictionless and non-dilalan t, the lines representing the failu re planes in the two-d imensional x, y-plane are stress characteristics. There are two famili es of conjugate stress characteristics. For slip to develop, it is enough that sliding takcs place along one of the two conjugate lines. This line should intersect the boundaries in at least two points in order to separate a soil mass that is free to move. For sliding it is necessary that the slip line has a continuous slope. The assumption is that at colla pse there is at least one continuous smooth slip line which corresponds to a stress characteristic and which intersects the boundaries in two points. This line is called here the real slip line. The shear stress along the real slip line has the maximum available value which is equal to the cohesion c for a frictionless soil. Class oj potential slip lines

For the analysis a class oflines is required that contains the real slip line. Thisclass is obtained by selecting lines in such a manner that their stress distribution satisfies tbe same equilibrium conditions as the real slip line. At incipienteollapse, the separated soil mass that is about 10 slide

74

Soil Mechanics and Transport in Porous Media

CALCULUS OF VARIATIONS TO VERTICAL CUT OFF IN COHESIVE, FR ICTIO:-lLESS SOIL

3

is still in equilibrium. Its total weight is balanced by the stresses along the real slip line. This is called the requirement of total equilibrium in the following sections. This requirement can be formulated in the form of definite integralS along the line. Furthermore, the real slip line is a stress characteristic. Therefore the stresses along it satisfy Kotler's equation, which represents local equilibrium for the limit stress state, in a direction tangenria! to the stress characteristic. The requirement of local eq uilibrium is satisfied by introducing Kotter's equation in the integral expressions. The requiro;:ments oftotal and local equilibrium produce three definite integralsas shown in the section headed' Determination of the integrals'. The value of one of these integrals corresponds to the he ight of the cut off. The other two are zero. All lines that give these values to the inlegrals are called potential slip lines, because each of them might be the real slip linc. Extremal The calculus of va riations provides methods to find the extremal. That is a particular line of the class of potential slip lines, computed in such a manner, that it can give an extreme value to h, the height of the cut off. In the section headed' Determination of the extremal', the solution for this line is produced by standard variational procedures. The extremal of the potential slip lines is not necessarily the real slip line. It can be any line satisfying the equilibrium I;onditions imposed. The value of h, I;omputed for the extremal is called homr here. This value is an extremum for h, if it is a maximum or a minimum. If it is a minimum, this indicates tbat hom, is smaller than the values of h corresponding to all other lines of the class investigated. If this class is large enough to contain the real slip line (the line that is assumed to correspond to h'~lI)' it may be concluded thaI hom. is smaller than or eq ual to h,.rr. The analysis then procures a safe estimate for the building height of the embankment. The analysis presented here was initiated originally in the supposition that a minimum would result. VerificaliQn

The basic assumptions mentioned in this section may occur to be plausible for soil mechanical inves tigators because they are commonly postulated in slip line analysis. There are two improvements with respect to common procedures: the shape of the line is a result of the analysis and the stress distribution along the slip line satisfies limit stress state conditions by applying Kilner's equation. The assumptions are, however, not self-evident and are even debatable. This is revealed by a verification of the solution using procedures that are standard in the calculus of variations. The character of the extremum is established in the section headed' Investigation of the $Clution' by use of these verifkation methods, Showing that either a weak maximum or no extremum at all is involved. This is disappointing because the conclusion can only be that the analysis was not meaningful. Unfortunately, it is not possible to establish, apriori, whether thc basic assumptions underlying the analysis will lead to a meaningful end result. It is only after the solution has been obtained that its character can be verified. Therefore, first thc ana lysis, based on the three plausible assumptions mentioned in this section. is carried out in the following text. In the final section the result is discussed and the basic assumptions are reconsidered. DETERM INATION OF THE INTEGRALS

Theverticalcutoff is shown in Fig. I . The soil is limited by thefollowingstress free boundaries:

Selected Works of G. de Josselin de Jong

75

4

O. DE

Y, ..

"

-'-'~

,.

" ........... C

"

- 1;~:WdV»!)}AV;;«

,

/

A.

SO~O

•o

•

, A .. ,

Fig. I.

Vtrtlcal cut off wiCh p,mntiaJ slip liM

the vertical plane Be and the horizontal planes AS and CD. The problem is to determine an acce pta ble height II of the cuI off. The soil properties involved are the specific weight y. the co hesion c and a vanis hing angle of internal friction cP _ O. No external loads are acting on the boundary A BCD. Failure is due only to the weight ofthc sliding soil mass. Class of potential slip lines

If the class oflincs considered is to consist of potential sli p lines, the li nes have to intersect the free surface in two points, E and F. The lowe r point E will coincide wi th the lower corner point B, and F is on the upper surface CD. So the lines in the ana lysis are lines SF. The shape of the line BF will be treated in parametric form, such that the horizontal x coordinate and the vertical y coordinate a re both functions of a parameter (( that increases from ((a in B to <XF in F. This can be written as (1)

In the analysis the parameter a will be taken to be the local angle between the line and the x direction (see Fig. I). Differentiati on with respect to a is indicated by a prime such that

x' =

dx/d~,

x" _ d 2 x /d'l\

y' =

dy/d~

y. = d 2y /dct?

(2a) (2b)

A small element of BF with lengt h ds subtcnds in horizontal and vertical directions distances dx, dy (see Fig. 2) given by dx=coSads, dy=sin((ds . (3)

In order to be a potential slip line, it is necessary that the shear stress acting on the line BF has the maximum available valu e, which is eq ual to the cohesio n c. The normal stress on BF will be denoted as p and is taken positive for compression. These quantities will be used to evaluate the integrals that are developed in the following.

76

Soil Mechanics and Transport in Porous Media

CALC ULUS Of VAR IATIONS TO VERTI CAL CUT Off IN COHESIVE, fRICfIONLESS SOil

rom~

Fig. 2. For.,., d R (black arrow) aCllng on t"'mml ds 15 JHHM1II p tis ( .. hlle a rro..,.)

5

of a , h... r compO"',,"1 cds and a !>Oronal com-

Total equilibrium

In the state of incipient failure, the soil and aU its subdivisions satisfy equilibriu m. So the forces acting on the part RCF, separated from the main soil body by the line SF in Fig. I, form an equilibrium system. This means that the force Q created by the weight of the part RCF is in equilibrium with the resistive force R due to the stresses acting along SF. This equilibrium wiU be called total equili brium because it refers to the equilibri um of the total mass BCF, in contrast to local equilibrium which is considered later and refers to the equil ibrium at all points ofthe line BF.

Total equili bri um is satisfied when the forces Q and R annihilate each other. EKpressed in their x , ycompOnenlS: X", YQ ; X R • Y" and their moments M Q, M" around the origin of coordinates, this requires that (4)

Since the soil weight acts only in vertical direction downwards, the horizontal component XQ of Q is zero and the vertical component can be obtained by integrating over slices of width dx as shown in Fig. I, th is gives XQ= O,

yQ = - J:}-(Yr - Y)dX

(5)

The moment of Q around the origin, positive for counter--clockwise rotation, is MQ

= -

f:

Y(yF - y)xdx

(6)

In these cJtpressions x, y are th e coordinates of the line SF an d integration is along SF. Thc components of the resistive force R can also be expressed in the form of integrals along BF. Let dX", dYR be the x, y components of the force d R (black arrow in Fig. 2) produced by the stresses on the stri p of unit width corresponding 10 the line clement ds. This elementary force dR consists of a component c ds tangential to ds and a normal component pds (white arrows in Fig. 2). Decomposed into x, y directions these give according to Fig. 2 and using equation (3)

dX" _ c cos a.ds - p sin r:zds = c dx - pdy

(7a)

dY,, _ csina.ds+pcosr:zds _ cdy+pdx

(7b)

From these the components X". Y" of R are obtained by integration along BF giving

X" -

J: s: dX" -

Selected Works of G. de Josselin de Jong

(cdx-pdy)

(8a)

77

6

G. DEJONG

YA =

J: J: J: dYR =

(cdY+pdx)

(Sb)

The momen t Mil orR around the origin is

MR =

J:

(xdY,II- ydX,II) =

[c(xdy - ydx)+p(xdx+ ydy)]

(9)

The three equations (4), that represent total equilibrium, can now be clIpressed in the form of integra ls. T he fi rst two a rc transformed into the relations ( lOa) a nd (lOb) by use ofthc equations (5) and (8). The last becomes relation (JOe) by using equations (6) and (9). This gives

f:

(cdx-pdy) = 0

(lOa)

(cdy+pdx -Y(YF- y)dx] = 0

(lOb)

[c(xdy-ydx) +p(x dx+ ydy)-rtYF- y)xdx] = 0

(JOe)

f:

f:

These relations represent tota l equilibrium and agree with the equation (5) in the paper by Bake r and Ga rber ( 1979). The analysis continues in the following sections by introducing Kotler's equation along tbe line SF. T his is an essential difference betwecn that paper and the analysis followed hcre. Local equilibrium

In a frictionless mate rial (for which tP .. O)a potential slip line coincides with a stress characteristic. Local equilibrium along stress characteristics is e1:pressed by two Kotter's equations, one for each ofthc two conjugate stress characteristics. The equation corresponding to the line BF is (dp /ds) - 2c(drx/ds)+., si n a = 0 .

( II )

Thisisa differential equation that can be integrated along.)". Adj usti ng the integration constant to fit the si tu at ion in point F gives af"ter integrat ion (12)

in which PF' a F an d YF stand for va lues of t he q uanti ties p, a and y respectively in the point F. Let PF be defined by (13) PF = (PF-2C1f.F)/2c then p can be solved from equation (12) to give P = 2cW r +a) + }"(YF - y)

(14)

It may be rema rked here that it is impossible to solve for P in a similar manner from Kotter's equation, when the soil has internal friction, such that tP is unequal to zero. Therefore, the analysis developed below cannot be applied dircclly for soils with internal friction. Eq uation (14) is used to eli minate p from the integral expressions (10). This gi ves then

f:

78

[cdx - 2c(.8r+ a) dy- Y(Yr- y)dyJ = 0

( ISa)

[cdy+2cWF +a)dx] = 0

(ISb)

f:

Soil Mechanics and Transport in Porous Media

CALCULUS OF VAR IATIONS TO VERTICAL CtTT OFF IN COIIESIVE, FRICTIONLESS SOIL

J:

7

[C(X d), - )'dx)+ 2c(jJ .. +a.) (x dx+ )'d)') + ~(yF - )'»'d)'] = 0

(l Sc)

Since (yf - Y.) is equal to h, the height of the vertical cut off, the parts containing 1 become

f: J:

1(Y.. - )')d)' = hh

1

(l6a)

y(YF - )'»'d)' = iYhl(yF +2)'6)

.

(16b)

By taking the origin of coordinates in point G on one-third of the height of the cut off(Fig. I) such that . (17) )'F=ih, )'6= -ih, x. - O the in tegral (16b) vanishes and the integrals (15) reduce to fo :Z

1", =

f1=

J: J: J:

(dx-2(jJ .. +a.)d)'] = 1'1h

1

1c

(18a) (l8b)

[d),+2(jJF+a.)dx]:z 0

.

[(xdy- ),dx)+2(jJf+ a.) (xdx+yd)'» ) - 0

(18c)

At this stage it is convenient to introduce the parametric manner of describing these integrals. From the relations (2a) it follows that dx, dy can be wri tten as functions of the parameter a. in the following manner (19) dx = x'da., dy=y'da. . Further, relations (3) indicate that tan a _ (dy/dx) and using relations (19) this can be written as tan a.". (y'lx') or _ (20) a. = arc tan (y'lx') So the integral expressions (18) can be written as

f

1

=

J:

J: J: J:

0

=

Go(a) da =

f,

=

G,(«)d<< =

f

G1(a) cia. =

J: J:

[x' - l(jJf +a rc tan),' /x'»"lda =

! '1hl /C

(21 a)

[y' + 2(PF+arc tan )" /x')x']d«

0

(2Ib)

=

[(xy' - x'y)+ 2(.8F+ arc tan),' Ix') (xx' + )'Y')] da. = 0

.

(2 Ic)

The functions Go(et:), G ,(et:), G l(et:) introduced on the left-hand sides of equations (21) are functions of x,)" x', y', defined by the integrals between brackets. Because of equations (I) and (2) they can be considered to be functions ofa only_ These three in tegrals represent the three equilibrium conditions for balancing the total d rivi ng force Q and the total resistive for«: R. In addition, local equilibrium is guara nteed in all points of the potential slip line OF in adirection parallel to the line, the line being a stress cha racteristic.

Selected Works of G. de Josselin de Jong

79

8

O. De JONQ

DETERMI NATION OF THE EXTREMAL

The in tegral exp ressions (21) are suitable for application of the variational calculus in o rder to find an ex treme value of h, the height of the vertical cut off for given values of c and y. The analysis consins of determining the func tions X(II) and y(a) th at represent an extremal in pa rameter form. Such an eXlremal is th e curve th at produces an extreme value for r o. the in tegral

defined by (2Ia), under the addi tional condilions th at the integrals r , and r1 defined by (2Ib) and (lie) vanish. In the calculus of va riations th is is a so-called isoperimetr ic problem that can be soived by th e use of Lagrange mul tipliers. The following analysis is based on th e Weierstrass theory for parameter problems as described by Bolza ( 1904). General expression/or (he extremal Accordin g to th e theory, the curve ),'«(1'), y(1l) is fo und by solving Euler's differential equation, which in Weiers trass'S form is (22) In this eq uation subscripts stand for pa rtial differentiatio n with respe<:1 to the subscript va riable, HI .. iJllliJx, etc. T he quanti ty H is an auxi liary function given by fI _ Go+g,G, +11G1

where Go. G" Gl are the in tegrands of the equa tions (21) and I" 11 are Lagrange multipliers that are constants in th is case. Written out, the function /I has the form

Il .. [(g, + 11X)Y' + (I -glY)x']+ 2(P,,+ are tan y' Ix') [(g,

+glX)x' -( I -

11Y)y'l

(23)

Perfo rming th e partial differen tiat ions on function 1/ required in eq uation (22), th is eq uatio n becomes (24) The solution of thi s differentia l equation satisfying the pa ra metric requi rement y'lx' .. tan a (according to (20» rep resents the e" tremal. It is not so sim ple to solve equation (24) in a straightforward manner_ The solu tion, however, is as follows

X= -(g, I'l)-acosa+(b-aa)sina y _

+ (l 111)-a sin a-(b-aa) cosa

(25a) (25b)

Th is can be ve ri fied by substitu tion, using equation (2) which gives for, ins tance, x' "" dxl drJ. '"' (b -(fa) cos (f,

x'y" -x"y' ... (b _aa)l,

(X' )l

y' .. dYlda. _ (b- a.a) sin (f

+ (y' )1 =

(b _ aa)l

(263)

(26b)

Since equation (24) isa second-o rderditrerentia l eq uation, solution (25) possesses two integration constants. These are the quantitiesaand b that have to be determined from boundary conditions.

S/r(Jpe a/the extremal Equation (25) represents a curve that is called an involu te, with a circle as evolu te. In Fig. 3 the line PoBPaQ FP, ... is such a cun·e. It is the path followed by the end point P ora stri ng of length NoP o .. b, attached in No and wound arou nd the circle with radiu s No M .. o. For an arbitrary value of(f the part NoN. of lhe string with lengt h (fa is con tiguous wi th the ci rcle and the remaini ng part of length (b-aa) is th e straight line N. P•. T he ce ntre of the circle is the point M , with coordinates (27)

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Soil Mechanics and Transport in Porous Media

9

CA LCULUS Of VARIATIONS TO VERTI CAL CUT O FF IN COHESIVE, fRIC110NLESS SOIL

.......... rN,

0,

N . -_ ....

....................., . .......

.......... .j

./

.

>-

\\ . .•.,

• "'''''

i

\

cw~===C"··.'·-.O"'(:6:~~%@

"\!::' \ ../' b

'''':::~:--.' ' .".;. -- . "."....

," .

\,\"

".,./

, kMMv;,

• 0

-----p,

\. \, \

• ,+-~~~==~~~~~B~.,r . ,

Po---- -

--- --- --

... :;,..-

",

.

",-,

Fig. J. Ex trema l BP. QF, sa llsfylng boondary rend itions and consl..,int •. circle N.N.QN, as "oMt

Th~

Ii"" f'oPaQf' , I. an in.-olult wilh

The involute reaches the circle in the point Q, when the value of (I is equal to (lQ = bta

(28)

Beyond that point, th e line continues towa rds FP L •• • etc. as the path of the end point ofa string that unwinds from the circle on the other side. At point Q the curve has a cusp. Determination of the constants of rhe extremal

Of all curves that obey the relations (25), the relevant extremal is obtained by computing the values of th e integration constants a. b and the Lagrange multipliers g, , g2' In addition, the two end points B, F are to be localized on the curve by establishing the value of the parameter (I in th ose points. These are(la and (IF' In total there are six constants D, b,g(, Kl . (la. (lp to be deter· mined and Ihis requires six relations between th ese six variables. The lirst two relations are the constraints which are ex pressed by the requi rement that both r (, r 2 given by the integrals (2 1b a nd c) are zero. Since x , y, x' , y' are all known gonio metric functions of the parameter (I by relations (25) and (26), it is possible to integrate th e integrals (21 b and e) with res~t to« from «8 to (I ••• The resulting formulae are fairly lo ng and therefore,

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10

(I. DE JONG

not written out in full length here. They arc known functions of G, b, II' g:. 01 •• 01". though which atc called r L" and r 2· for brevity. So the first two relations are

r.-

(0,6"1,12,11 1,011')-0

(29&)

r 1•

(a, b, I" 12. 0: •• 0;'1')

0

(29b)

_

The following two relations arc obtained by requiring that the ex tremal passes through the point B. The coordinates of B have the values (17) and combined with equations (25) there

results )1"1 - - (g, /gl)- U ..os Ila +(b-OI.a) sin IXII = 0

y.-

+ (1 {gl)-asin OIa -

(b-erBa)COS (lB =

-ih

(Joa) (30b)

The last two relations arc obtained from the boundary condi tio ns at the point F. This point is

located on the free upper su rface and therefore the magnitude of YF is Imown to be ih (see equation (17». Introduced into equation (2Sb) this gives

Yr. +(l /gJ )-asinlX,-(b -lXpl)coslX,= +ih . The location of the point F on the upper surface is not known, however, so

(31)

x, is an unknown.

The second relation to be obtained from the boundary condition in F is due to the stress state in F, which produces a valuc for the inclination of the potential slip line, by adapting limit stress state to the require me nt, that the horizontal upper s urfa ce is free of stresses, T here are two possible eases represented by the Figs 4(a) and 4(b), which eorrespond , respectively. to maximum compression and maximum tension on vertica l planes. In the com pression case ( Fig. 4(a» the value of p, the normal compressive stress on the potential slip line, is equal to PI' - +c and thedircction of the potential slip line is (32) In the case of maximum tension (Fig. 4(b», assuming a soil which can support a tension o f 2c, the normal compressive stress on the slip line is p, _ - c and the corresponding direction of the potential sli p line is xI' = n(4. It can be s hown that a solution for this second value of aI" satisfying al1 boundary condi tions and the constraints (equation (29», does not exist. For the compression case in F wc have p, _ +c and !x" _ (311 (4), which in troduced in /11' defined by ( 13) gives (33)

r

r:·.

Th is value is needed in the expressions..(29) for I · and Finall y, there are six equations for th e determination of the co nstants a, b,gt, g2' !X., !XI" These are the relations (29a and b), (3Oa and b). (3 1 and 32). Since these relations are implici t goniometric relations, it is not possible to give explicit equations for the constants. A solution satisfying al1 six relations was obtained numerically. The pertinent values of the constants are

a - O·988h. Kt - 0'0424,

IX.'" 0·604

b = 2-t4Oh. Kl - 0'6674/h,

IX" - 2· 356

(34)

Equations (25) for the coordinates, with the va lues of the constants mentioned above, form the solution for the ex tremal satisfying the Weierstrass-Euler condi ti on (22), the constrain ts (21 b) and (2 Ic) and the boundary co ndit ions. Th is solution is unique. The value for h... t , co rresponding to this solu tion of the extremal, is obtained by introducing

82

Soil Mechanics and Transport in Porous Media

CA LCULUS Of VARIATIONS TO VERTICAL CUT

c

"

"

,

"

,

,-,

11

m"F IN COftESIVE, FRICTION LESS SOIL

, 8

"~I

8

Fia:- 4. Bounodary cond itions for stress characte.-Istlc rwessloo; (b) maximum lensJon

.tt~

point F on the

u~r

5tlrf.ce C D : (a) maxImum .,.,....

expressions (25) and (26) into the integral (2 Ia) and integrating. The result is too long to reproduce here. Insertion of the values (34) finally gives homl

= J·783cfy

(35)

Descripliol1 Q/the sQlutiQn/Qr the extremal

The shape of the extremal computed is shown in Fig. 3 as the line BP.QF. This line contains the point Q, which was ment ioned above to be a cusp. That Q lies indeed within the interval BF follows from a verification of the magnitude of «Q' Using the relation (28) and the values (34) il is found that (36) ~ - hlo"" 2·166 . and this is a value between «8 and I1.F given in (34). Further it can be computed that Q lies outside the soil boundaries. Using the value (36) in the expression (25b) for the y coordinate givcs, with the values (34) for the constants, YQ = 0·680Ih. This shows that Q lies a distance 0·01 34h above the free surface which is located aty "" 2h{3. The fact that the extremal BF contains the point Q, where the curve has a cusp and which lies outside the soil mass, indicates that th e solution does not represent the real slip line. For a real slip line it is physically illegitimate to form a I;USp and to exceed the soil body. Therefore the solution (35) is not the collapse height. It is either too high or too low, that depends on Ihe character of the extremum, whether it is a maximum or a minimum. INVESTIGATION OF THE SOLUTION

The solulion of the Weierstrass- Euler equation (22) is a uniqu e curve given in parameter fonn by the expressions (25) with the values (34) for the relevant constants. This curve is ca11ed an extremal because il satisfies condition (22) and that is necessary for thc I;urve to produce an eXlremum for h. However, satisfying condition (22) is not yet sufficient. Before it can be conpluded Ihat the value (35) computed for h.ml wi th the eXlfemal, indeed represents an acceptable extreme value, there arc three additional investigations to be made. These verifications arc required for establishing the character of the solution and completing the analysis. In the first place the Lege ndre condition has to be verified beeause thai condition indicates whether th e solution represents a maximum or a minimum of the height h. In the second place the Jacobi condition has to be verified because that condition indicatcs whether there exists an extremum at all. In the last place the Weierstrass E function has to be investigated because that function shows whether the ext remum (if it exists at aU) is strong or weak.

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O. DE JONG

Legendre CQlldition

The first poin t to in vestigate is t he Legendre condition. This condition is satisfied if the sign of a quantity H" defined below, is the same for every point of the extremal, x(a), y(a), between B and F. For a maximum to be involved it is necessary that H I is negative, for a minim um that HI is positive. The definit io n of the function 11, ca n be written in three different ways which because of homogeneity are eq uivalent, Le. HI

= H~ · , ·I(y' )2

= - H,y/x'y' = H",,/(x' )Z

'. (31)

where H is the function described by (23). Elaboration gives

H, = - 4{(g , +g2X)Y' +(1 -Ir2y)x'11[(x')z+ (y'lW and using relations (24) and (26b) this ca n be wrillen as

H, = -4g l !(x'y"-x"y') = - 4g l !(b-aa)2 According to the solution (34), the value of gz is positive and this indicates that H, is nega tive for every point of the extremal. So the result is H , <0,

for every a .

(38)

Therefo re, the Legendre condition is satisfied and th e extremum involved is presumably a maximum. However, compliance with the Legendre condition is not yet sufficient for the soluti on to be a maximum. It is also necessary thaI the Jacobi condition is satisfied. Jacobi conditiOIl

The second point of investigation is to ve rify whether the Jacobi condition is sa tisfied. According to the theory no extremum, maximum or mi nimum, exists when a function II. defined below, vanishes in a point of the extremal between Band F. The function 1/ is defined by (39a)

with II,

= y'x. ~x'y.

(39b) (39c)

whe re x.' etc. are partial derivatives of the functions (25) with respect to the integration constants a and b. The constants C, and C l are to be chosen in such a manner that for the point B, corn' mon to al l extremals, the values of 1/ and 1/' are as follows: (40)

Elaboration of II, and

I/l

with eq uation (25) gives 11= (~aC,

+ C1 )

(b~aa)

in which the constants C, . Cl ca n be solved by use of relations (40). It is then found that 1/

=

(a~a8)

(b

~

Cla )f(b - ClBa)

(41)

The function /I expressed by equation (4 1) vanishes in the cusp point Q, because there il equals ilO "", bla) (sec equation (28». If, therefore, the extremal BF contains the cusp Q, the Jacob i condition is viola ted and neitber a maximum nor a minimum exists at all.

84

Soil Mechanics and Transport in Porous Media

CALCULUS OF VAR IATIONS TO VERTICAL CU T OFF IN COHESIVE, FfUCTIONlESS SOIL

13

Weierstrass condition The third point of investigation is the Weierstrass £ function, which indicates whether an extremum is weak or strong. For the curve determined here, this investigation could be omitted because tile violation of the Jaeobi condition already showed that no extremum for It is involved at all. However, the analysis of the E-function is mentioned here for completeness. The Weierstrass's E function is defined by E(x, y; x', y'; i',

n = i'[HAx, y: i',y')- II..{x, y; x', y')] +y'r 1I, ,(x, y:

i', ji' ) - It..(x, y; x', y')l

where x', y' are the derivatives of the computed ex tremal and i', y' are the derivatives of another comparison curve. The extremum is strong when the sign of E is independent of the magnitude of (i' - x') and (ji' - y'). It is weak when the sign of E is constant only for small values of (i' - x') and (y' - y'), but changes if the values are large. A nega tive value of E corresponds to a maximum. a positive value to a minimum. Elaboration of the Efunction for H defined in equation (23) results in

+ g,x)x' -( I - K,Y)Y'] [arc tan (jo' Ii') -arc tan (y' lx') + (i')1' _ x'y') (x'" +y" ')] +[(gl +glX) (i' - x')-O- gzy) (y' - y')} [arc tan (j"li')- arc tan (y' lx'») Fo r small values of(i' -x ) and (Y' - y') it is found that E is negative. This indicates that a E .. (gl

mallimum is involved. However, the presence of the term arc tan (Y'/.'f) creates the possibility that the sign of E changes for a comparison eurve with arbitrary .'t', Y' values. Therefore, the computed curve would produce a weak maximum for h, if the Jacobi condition had not been violated. DISCUSS ION OF THE RESULT

The re sult of the variational analysis is an elltremal in the form of an involute. Its mathematical Cllpression is given in parameter form by equations (25). The constants a, b, gl' g2 in these ex pressions have the values (34). The shape of tile line is shown in Fig. 3 as the line BP.QF. The involute is an extremal because it satisfies in every point the Euler condition (equation 22). It is an extremal of the class of potential slip lines that contains the real slip line. This is achieved by defining integrals that impose similar conditions of total and local equilibrium in the limit stress slale on the lines of that class as are satisfied by the real slip line. The extremal, represented by the line BP.QF, is unique. It is the only involute that satisfies the constraints (2 1b and c) and the boundary conditions in Band F. The corresponding height of the vertical cut off has the value h... 1 '" 3'783cl y, which is high, but could be a valid bound for the collapse height, h,.u. All these points are in favour of the analysis al\d suggest that it is an attractive method of slip line determination. There are, however, some disappointments. Verification of the conditions for the extrem um in the previous section ( Investi gation of the solution) shows that the analysis produces a weak maximum in genera!. whereas in the special case that the extremal contains the cusp point Q, there is no extremum at all. These verification results indicate that the analysis is apparently not meaningful. But before th is conclusion can be drawn, it is necessary to consider these two points in more detail. Weak maximum

When a maximum is involved, this means thaI the value of h'm! is higher than every value of h corresponding to any line of the class investigated. When this class contains the real slip line,

Selected Works of G. de Josselin de Jong

85

14

O. DE JONO

then h. m, is larger than the height corresponding 10 the real slip line, which presumably is the collapse height, hcoll - So being a maximum, the computed height he .. , is an unsafe estimate for h,.II' lt is essentia l for this conclusion that the class contains the line that produces heon - If, for instance, a class is considered that gives for all Jines values lowel than hcoli • it would be interesting to es tablish the maximum o f thai class. T hi~ wOIlIe! give the beu. <;afc v:'I l"e for h. In the ca~ considered here, the real slip line, if such a line exists, is still in the class because the stress disTri· butions of the lines in the class satisfy the same equilibrium conditions as the stress distribution along the rea l slip linc. T herefore a maximum corresponds here to an unsafe estimate. The maximum is called weak in variational te rms. This means that it is a maximum with regard to comparison lines that deviate from the extremal only in such a manner that their inclination is almost the same. As a consequence another maximum or even a minimum cou ld exist fo r completely different lines. In textbooks examples are given of unexpected shapes in this respect, consisting of discontinuous solutions. The examination of discontinuous solut ions has not beer ""':"Cted exhaustively for the problem of the slip line analysis. Only the case was consider~ . two involutes with a discontinuous second derivative, forming a smooth S -shaped curve. Such a curve has piecewise di fferent values for the circle centres XM' YM and so according 10 (21) the Lagrange multipliers g" gl are not constant along the line. Since these multipliers have to be constants, such an S shaped curve cannot be an extremal. There are other possibilities for discontinuous solutions. It could be envisaged to study a combination of slip lines consisting piecewise of conjugate stress characteristics. However, such explorations were not pursued. No eXlremwn al all

In the previous section (Investigation of the solution) it is established that an ex tremal consisting of an involute violates the Jacobi condition if the line contains the cusp point Q. It is possible that for embankments or slopes with other boundary conditions the cusp is avoided. For the particular case of the vertical cut off, the cusp is not avoided and is located within the integration interval of the extremal. This indicates that in this particular case no extremum is associated wi th the extremal. Bcing an extremal means that the line satisfies in every point the Euler condition (22). This is necessary for the line to give an extremum, but not yet sufficient. It is also necessary tha i the line satisfies the Jacobi condition and if that second condition is violated, then aecording to the variational calculus no extremum is involved at al l. Since this conclusion seems curious, it is revealing to mention some additional computations that can be reproduced readily to verify the lack of extremum. To that end it is convenient to consider smooth 5 sha ped lines, consisting of two circles, that have the same langent in theiT meeting point. The integrals (2 I a-c) can be evaluated in closed form for such curves. For every height of the meeting point there is one unique combination of cirdes that satisfies the constraints (2 1b and c) and the boundary conditions in B and F. The valufs of" corresponding to these 5 shaped lines are above or below homl fOl lines with a meeting point respectively above or below y:::::-0·33". By this verification it is established tha t the class of potential slip lines is not limited in a meaningful manner. Apparently the conditions of total and local equilibrium of the limit stress state along a line are not restrictive enough. This is further substantiated by an examination of the stress state in the vicinity of the point B. The stress state in the line should be compatible with the stress-free ve rtical boundary Be. Compatibility can be verified by introducing a fan of stress characteristics centred in B. The result is that the S shaped li nes wit h a meeti ng point above y = 0·17h are unacceptable because the stresses near B sur pass the limit stress stale, and

86

Soil Mechanics and Transport in Porous Media

CALCULUS OF VARIA TIONS TO VERTICAL CUT OFF IN COHESIVE, FRICTIONLESS SOIL

15

this limits the height for 5 shaped slip line!; to h~3·69cJy . We will not elaborate this point here further because 5 shaped curves have no special status. They were arbitrarily chosen for convenience to verify the absence of an extremum. Another example of a line Ihat mighl be called an extremal because an Euler eq uation is satisfied, but apparently produces no valid extreme value for h, is encountered in the analysis of Baker and Garber ( 1978). In Ihal analysis the procedure is to dete rmine the critical slip line by eORsidering an extremal, with the location y(x) and the normal stress distribution o-'(x) as the two relevant variables. The stationary value of the integral is determined with respect to these two variables, by two Euler equations. The first involves 0- and 0-', the second involves y and y'. The authors surmise that satisfaction orthe first Euler equation only is enough for a line to form a useful bound for the critical sUp line. Their solution reduces to a circle for the case ora frictionless soil. In fact, this solution is only a regular extremal if also the second Euler equation is satisfied. Introducing the cirde in this relation gives a differential equation for the normal stress 0- which determines the distribution of 0-. Now the curious situation occurs that for the vertical cut off it is impossi ble to find a ci rclethat satisfies Ihe boundary conditions and the constraints, possessing in addition a normal stress distribution that corresponds to the second Euler equation. So there exis ts no circle that can be claimed to be a regular ex tremal. h is possible to indicate circles that satisfy moment equilibrium only and to adjust the normal stress distribution in such a ma nn er that the other two constraints of vertical and horizontal force equilibrium are satisfied. But then the second Euler equation is violated. This class of circles has a minimum hcight corresponding to h m ,. = 3"8c/y. From the theory of plasticity it is known that this height is an upper bound based on a kinematically admissible velocity field. So all ci rcles correspond to heights that are above h
Real slip line Up to this point it has been assumed that there exists a real slip line and that the collapse state is sufficiently characterized by considering the stresses along that line only. For the variational analysis it is a conveRient assumption because th e analysis is then restricted to the determination of the shape of one line and the elementary variational calculus provides all tools for such a determination. The one-line assum ption is, however, questionable because satisfying equilibrium and limit stress state along one line does not guarantee that the stress states in all other points of the soil body are within acceptable limits. An example of considering regions outside the line was mentioned above wi th regard to 5 shaped curves and the regio n in the vicinity of B. But one small region is no! enough. For a line to be considered a potential slip line it is necessary to ve rify the entire soil body. Possibly the disappointments revealed by evaluating the results of the variational ana lysis in this study are due to an inadequate formulation of the proble m. The collapse state may be associated wi th a distribution of stress characteristics and regions of slip lines of a more complex character than assumed so far. Characterizing the stress state and the Sliding mechanism may require the formulation of a more sophisticated system of integrals before the calculus of variations can be used meaningfully.

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16

G. DE JONG

CONCLUSION

The variational calculus seems an intriguing tool for investigating slope stability problems. The formulation of th e problem by defining a class oflines satisfyin g total and local equilibri um and assuming that the ell(rernal or that class is the real slip line is unsati sfactory. T he class is nOI specified in such a manner that a bound is found for the real slip line, and it is even questionable whether or not there exists one single real slip line at collapse. The analysis based on total and local equili brium of a single line leads to a weak maximum (or no extremum at all)and thorefore il produces unsafe predictions for slope stability problems. REFERENCES Baker, R. &. Garber, M. (1977). Variational approach to ~lopc $Iability. Prl)(:. N;nl~ In' , ConI. Soil Mu~ . Fdn Engng, Tokyo 3/2, 9- 12. Baker. R. &; Garber. M. (1978). Tlleo'e!ical analysis of!lIe s!abili!y of slopes. Gbmc/m;que 28. No.4, 395-411. 801o:a. O. (1973). LU I"res On 'ht calculliso/''Ildo'ioM. New York: Chelsea Pobl ishing Company. Paslor. 1. (1978). Analyse limite: D
88

Soil Mechanics and Transport in Porous Media

A variational fa llacy G. DE JOSSELIN DE JONG" directly. but an independen t study b~ Gibson & Morgenstern. which was rather simi lar in character and also resulted in logarithmic spira ls. Their analysis was examined by a mathematician who drew altention toa subt le fa!1ac~ in their reasoning. II was shown that such curves are notlimi tati,·c, but on ly ha ye the propeTl~ that the momcnt is indcpen· dent ofthe nonnalstrcssdistribution. l t was pointed out thai the method break s dow n. but how this is related 10 a defecl in the variational scnse was not explained. Neithertheir ,,·ork norlhe refutation of it was ever published, but rer~rence to it was given by Morge nstern (1971) in his gene ra! report at Ihe Tokyo Conference. The essence: of the defect is that a degenerate functiona l is involYed. Th is can be readily verified since Pctrov's (1968) book appcall:d. In this book, Sections 29 and 30 arc devoted to degenerate cases (which are hardly COII$;dered ;n other textbooks) a nd their properties are extcnsively treated. The degenerat ions are classified according to the theory 01" Kroto .... who dishDgulshes Ii ...e kmds. Pet rov'S description is foUowed here. For Ih e deta ils of the slipline ana lysis it is convenient to refer to Baker & Garbds (! 978) paper. Their treatment is a modification of Kop;\csy's 1961 ana l~is. improved by the introduction of a sarety factor. The essential defect is not removed and the Il:su lt is still a logarithmic spint!o The intermediate runction g, defined byequation (9.1) in the paper by Baker & Ga rber, has two variables: the nonnal stress distribution ""xl and the height of the slip surface y(x). The functional is degenemte, because 9 does not contain II (a prime indicates a derivatil·c with respect to x)a nd is linear in ) 1. In the Krotovscnse Ihisereates adegcneration of the third kind with respect to both ~ and )'. In Petrov's book the simullaneous OCCurren,"", of two degenerations is not tll:aled. but it ca n be expected tha i such a combi nation reinforces the degenerate character, rather tha n ameliorating the situation. Consider thcll:fore on ly 11. Because ortbe ahscnce of lithe funct ional G, defined by equa tion (9.1), haS its extreme value only for particular

It is unfortunate thaI valuable in formation ooncerning fallacies in soil mecllanics can get los t in the course of time. This has happened with Ihe usc of variational calculus in slope stability problems. The variationa l method, for detennining the cri tical slip surface as the surface that minimizes the load at rupture. was presented in soil med1anics by Kopacsy (1957, 1961). He published in the 1957 London eonfell:nce a tllrec-dimensional solulion. and in the 196 ] Paris confell:noe the two-dimcn sional venion of it. The shape of the surface is suitably established in the 1957 paper by voctor ana lysis and described by equation (16) of tha t paper which. wilh defined by equatiOD (20). can be eKpresse wjlh a plane Ihrough the line PQ, which connccls P 10 its projection Q on an axis along II vector ii. fixed in space. This plane is further tilted wi th respect to Ii b}· an angle. which isequalto the angle between the lines PQand PI!.. with I!. located on til(, axis of Ii at a consta nt dist ance of magnitude (X.pl,{p./.
w

" Un;'''I"$;t)· of Technology. Delft.

89

290

Fig. I.

TECHNICAL NOTE

Kopksy'. solution

distribut ions of

<1.

and the possibility exists Ihal

discrete jumps bc::twe<,:n Ihese distributions arc involved . [n o rder 10 establish these distributions it is necessary 10 investigate g•• (the seco nd partial derivative with respecllO ,,). But {} is only linear in tJ.

so g•• vanishes. In that case there is only an

cX!rcmu m of G (maximum or minim um) if IT is bounded, and the extreme va lue is obtained by

giving (J one o f Ihe bounding values. to Ihe mechanical senS!: 11 is not bounded, because Sirt'SS stales arc lim ited in SlresS space by a cone. wh ich widens wilh increasing values of o. There: is no rnuimum value for <1 and there is no upper bound. Therefore. no extremum exists and the method breaks down. There is a lower bound fo r 11, which is zero for cohesionlcss soils or tensile if there is cohesion. This lower bound is irre levan! hen:. because i! produces no valid so lu!io n for the entire slip su rface. A! this poi nl i! can be concluded Iha! Kopacsy's analysis is nOI meaningful. because a func!i on~l is

90

considered which has a degcnera le character and possesses no minimum. Th is is due 10 his par!icu lar formula!ion oflhc slope s!abilily problem in terms oflhe varia!io nalcalculus. 1! is unfortunate !hatthc falsification rem~incd unknown. I! would h~vc prevented the reccn! revival of this variational fallacy. REfERENCES Baker. R. & Garber. M. (1978~ Theon:!icalanaly.isoflhe slabilily of slopes. Giouchniqw l8. No.4. 395,,'111. Kop;\csy. J. \1957~ Three-dilllen";onal st",,,, dimibulion and slip ~urrace5 in earth work, 31 ruptun:. Pnx. 4th Int. COtIf Soil. Mm. .• Lom/Otl la. 17. Kopiicsy. 1. (1961~ D istribution ok, contrai nlcs i la rupture. forme dela surfa~ d. glis.. menl el hauleur tiICori'lUC d "" 1al.... Proc. j,h 1m. Con/. Soil M«II.. Pads 6. 23Morgennem. N. R. 1 1977~ hoc. 9th InI.COtIf So/I. M«h.. 1"C>k)"I> 3. 319. Petrov. Ill. P. (l968~ VarimiOtlol "'~Ihodr in opli",,,,,, <:OtI"o/ Ih,,,,"y. New York: Academic P~.

Soil Mechanics and Transport in Porous Media

R ...... inted from JO\l1I
Consolidation Around Pore Pressure Meters ~ jN .

_

G. DE

I""""'"

DE

101«;

Gt_ _ """i<4, Oodplo_ 25, D
M.",h 26, 1953)

R<SJlOIlO" 01 P"'" ",. .ure ""'er 011 variations in loadin, conditions 01 turnlUndln, "';1 I.....rded by '0 ... COOIoidcwJ:. ri&id type."d • cav,,,,,,,,, 1)'))<. At , . (\ fw unit "q> loading, """oolidation hu Dol y Iot.dif1.!l condition •.

ill."".....'

1. INTRODUCTiON

H E n-..ol tYP"ofpore ~.u,.., ""'t~. in u"" with the Delfl Soil Mechanics Laboratory'" con$lsts of a cylindrical instrument of about 20 em in length and 3,6 em in diameter embedded in the soil masa.. The body of the in.trumenl contain. a diaphragm· type pre5llUre transdueercnnnected by a coe..xial fleJtible cable with the measuring apparalus above $Oil .u rface. The response time of previous nnnelectroni.c Iypt:l amounled to scveral &)'$, beot.usc of the complian~ of the diaphragm a Dd the very grut flow resis\An~ encounte.-.:d in the $Oil by the wale. that b.... tQ acluate the manometer. By C
T

"c.. ;;:i.l3by S. ·Ii ..

N

...... d..

p,",~re-t ... "oduc.r

L.

ODd .I
Boersma. Con ... llioJ Engin....-, DdIt, "The

To . implify the mathematical trealmenl the instrumen l is idealized and instead 01 cyJindrkal symmetry a spberical symmetry isaS$umed. If idulized in this way, the pore pressure meter consists of a spbere with radius " (em). The fGllowipg Iwo typt:l will be CQnside.-.:d. 1.1 The Rigid Type

Here the diaphragm is in contact with a waler ""Iume enclosed in a rigid chanlNr. Holes in the wall fonn the connect",n between this water volume and the water in the po= 'If the soil . We shall represent this Khematically a. a rigid, pervious sphere (Fig . Ia). 1.2 Tbe Cavernous Type Here Ihe diaphragm is in contact witb a wa ter volume tbat fills a cavity in the $Oil. The connection between Ihis waler volume and tbe pore water is direcl . We shall represent Ihis Iochematically .... a spherical cavity (F ig. Ib). Indica ting by av (em') the quanl ity of wa ter na:e!Ilary to account fQ' the defQnnalion 'If the diaphragm under waler pressure, we inttodu~ analog
91

'"

CONSOLIDATION AROUND

P ORE

PRESS URE

METERS

wily that (1.1)

in "'bi<;!> .... (kg/em') indicates t!le pressure in the waler endosed in the sphere. When the surrounding !!Oil m..,.. i. loaded, water pressuru are generated in Ihe p<>Te water and in order that the pore pressure mder may regi$ter this, a volume of water QV must enter the chamber.

For the rigid type Ihe volu me of water enters by percolation through the holes in the wall. This waler can only be supplied by the pore waler squeecd oul of the surrounding !!Oil ma. ... F"r the cavernQUs typl: the volume QV arises principally from the deformation o( the ca.vity, while an additi
......... ... ..... .

2. BOUNDARY CO!W ITIOlfS AT , . "

2.1 W. ter pressure

at

' ''' ' '

The ritid /y;'. 1f we may apply Darcy's law of perc0lation. The "ater volume Q. em' entering through the pervioul wall with a surface area 4.....' is related to the Pl'Q'IUre gradient in the JXlre water iNJ! iH in the 50il a t ,Oo ' .. by the expre5l!ion;

a

(2. LI ) in wbich '}'. (kg/em') denotes the denlity nf water and .. {ern/_} the permesbility nlthe 50il (thepermeabilily of the wall of the JXlre pressure meter is considered large in comparison with .I: nf the ..,il). Since Inr the rigid type toV _Q" we obtain by using (I .I)

-jb1. (&UO!offl- (kh.j(iNJ!(Jrl at

rOo ,.. (2.1.2)

Jphtrial ",vI!y

riqi dperviollS

S,mcn:

Ft<>. I. POfC p ............1.... of ' wo dill...", Iyp<s. 2.2 Soil s keleton

at

rOo T.

In the rj(id Iype the boundary condition at ' _ " is imposed on th e soil skeletnn by the rigidity of the sphere:

... _ 0 at ' _ ' " It is assumed here tbat .... , tbe pmlSUre in the meter, is equal to ... , the JXlre pressuTC in the lOil directly l urrounding the meter, .... (, - •• -O) - ... {. - .o+o). The '_~
tlw.{a..! al} - (1t/'Y.){iJv>!a.) - (all.! al) at

._r..

(2. 1.3)

These expression. relate to the spherically symmetri<::al ~.

In the ease wben the variabl.,. '" and ". are uially I)'tnmetrical and are functions of of (Fig. 2) tbe second member of £<;s. (2.1.2) and (2.1.3) must be integrated over Ihe range O
92

(2.2.1)

In the ''''......",.. Iype the wall of the cavity can d isplace i!$elf freely and the stresses a", determined by the condition that there is equilibrium betWttll the water pressure .... in the cavity and the radial stress in the soil mass a,' (n~ative sign lor compression).

"," --a,-) 0- .,.'

at

''' '0-

(2.2.2)

H ere -~,. indicates the total radial sttellS acting on the soil skeleton aud water combined, 50 we can divide into the radial.t~ on the grains only, and '" lh~ w .. l~r I"=ure in the JXlr.,.t by putting (2. 2.3)

- a:

-a.,

Tb...., stresses act in the $Oil, tha.t is to 1liiY for, > , .. while "," is tbe water pressure in the cavlty ' < '0f W..... II uK"" .uleri>k I" _ . W - ond dutic..,.,_

Olanlt ua<Jcio.ttd wi'" the combined l Y'l<m. -ala' &nd ooil shlet"", OfId "",it it in the ..... of the gniD ... 0000 . \one.

Soil Mechanics and Transport in Porous Media

G. DE lOSSE L 1N

'"

DE lON G

principal.tresses of unequal magnilude: S" S" S•. If we know the inlluence of one principal ItfftS we may obtain by IUpoerpocition tMir combined inHutnee. Tnin. the principal strtSS S in the diRction p _O, . . .e obtain as boundary condition :

al

...... "'.

(l.2)

lIeocause of the simpler lreatment involved ,,·e shall finally derive a IOlution to the cue where a uniform presaure P acts in aU dim::tionl. In luch a cue we have

a."-a,· " at· "- P, •.•• .. ...• ... ...· .0, al

...... ..,.

(3.3)

The condilions (2.1.2), (2.1 .3), (2.2.1), (2.2.2), (2.2.5). (3.1 ), (3.2), a"d (3.3) U1: neeeMU)' and ... /Ii_ cim t Iodetcnnmc thesolution to the problem t.tplicitly in the difle-m.t ases indicaled. 4. SITUIoTIOM AT f_ +0 FOR UNIT STEP W.\I)IMG

It follows from (2.2.2) Ind (2.2.J) that

.... (..-0)_ - ••

+... (.0+0).

(2.2.4)

Nor,....Jly thert! iI no difference betwe<:n .. (•• -0) and ... (.0+0), e..cept when the loading conditions are di .. continUQUI with regard to time, .... for itutane<: at the moment/_O for unit Itep loading. We Iohall treat this !p/!ICial.ituation in Sec." and u.e conditK!n (2.2.2) then. For the (;I.,"" of continuO\l'I~ing conditiono _("-0) equals _(..0+0) and the boundary streM condition

"'"'l. BOUMDARY CON"OITiOMS AT _

(2.2.5) _

We shall UlUme that the dlmetuiono 01 tbe pore prtSIUrt! rneto:r art! 10 sm.all in compa.uon with the distance to the boundary of the lOil rn.ua ,,·hert'in il iI embedded that we may consider the . trcss conditions of Ihe lOil a, homogeneous. To simplify the formulu we Iohall . 110 suppose thai th il homogeno:ou •• trns condition holds uf' to ....... ..,. As th e disturbing effect diminish .. proportional to ,... the errors introduced by th_ ...... mptions an: negligible. In addition we lohaJJ ouppose that in the J\Irroundina: soil ma.u the rate oJ squ..,.in, 001 of port! "1Iter illolow in comparil
Selected Works of G. de Josselin de Jong

Becau,"" of its ducidaloTy cbaraclo:r we IhalI ftrst dtttrmine the I"f$JlOM" of the port! preIIUrc meter to unit Itep loadillfl at thc mom..,t of Ioadins (, .. +0) . Thil iI very simply feasiblc by virtue of the following eonsidtralions. By ~eglecting maSS accdtralion and vilQ.lu. retardalion, we shall ouppose, as il ul ual ill Ihe literature on conlOlidation, that any loading incRases inlllntaneoliloly the stresses in the lOil n-, generalinJ!: eluticaJly the ddonnalions that art! theref~ instantanl!Olll too. Asauming that in comparison with the eompr5libility 01 tl>e pain Ikelcton of the IO~ the port! ...ter is incompr5slble, at the instanl of loading no voIu,,", change of the IO~ .iIl take p\a«. This may be ~ by pullinC (U ) where "o is the Poisoon's ratio of the eombined syst ...., water and lOil Ikeleton. As the ",.. tel" iI gnelually u1)tIIed from an element IIlny poi!)t in the loaded toil, the PoiMon's ratio ". al thil poinl dec_ and tendo to the value of " where .«1) is Ihe value of the PoiMon'. ratio of the grain skeleton alone. A result of tome importance concerning tbe value of E· may he derived in the following "'.y. The wato:r mdoled in tbe pores can prt!,-ent volume ch.o.nga. from ottumng, but not &hurin, suains. Thil implies that sha.r s tm. in thc combined syswn is g.rritd by the grain skeleton alone. It follows dim::tly that and hence that the shear moduLi G' and G a rc equal. Then by a "-elI-kn"",,, relation in dalticily theory,

T·_.,

CO. r-.....j 2(1+"O)-E/2(1+.) .. C II all tima,

(4.2)

and in particular

lE" .. P...j(1+.)

at

'00+0.

(4.3)

93

CONSOLIDATION AR OUN D PORE

The relation.. (4.1) and (4.3) enable us to describe the condition at ' " + 0 as R lunction 01 the dasticity constants 01 the grain shlelon E and •. The relations between the stn:sses in the comhined S)'Slem and grain . keleton only, which have already been stated, may be condensed to

-ao_ -a+..

and

T O_

(4.4)

T.

By addmg the nonnal stresses in three perpendicular directions We obtain

- (",' +a,o+",' ).. - (.. ,+",-t ..,)+h>.

PRESSURE METERS

in which a,' etc., indicate th e twincipal stresses in the combmed s)'Stem. With the aid of th.,.., general consideration. we shan proceed to determine the condition.. at 1- +0 in ou r special case.

Under the inlluence 01 the .tr.,.. .ystem (3.2) and the boundary condition lor the ca,-ity (2.2.5), the stress distribution in the wil can be determmed by the aid of stress lunction...' We ""all not enlarge 00 these compu_ tations because they are easy toamyout and by applymg the lo!lowmg e:cpreWon for the radial displaceme"t as derived lrom a .Iress function, ofo:

A. the volume dilatation " which is known 10 be equal to . - (1- 2.)(a,+",+",J/E, (4.5) is zero at 1_ +0, we obtain, by virtue of the fact that

-!.[~-sin~~l l, a, i), , iJr

·;011,

- (","+" .. +",o)- h>- - (a, o+""o+",,,' ) at 1_ +0.

f (")'+

I+.'! [ "

11,_ --1", - --ros¥ Hit', E*

(4.6)

1+.° ,

f,

Z..,•• in\l- · .......d!f.

i. given by

2(7 5.')

(4.9)

effectively the actual water pn:ssu~ ;n the pores 01 the soil. In the case 01 spberically . ymmetriW stress P at mfinity all expressions have a simpler fonn, and it is easy to verify that the boundary condilions (2.2.2) and (3.3) a~ satisfied, together with those of equilibrium and compatibility, by takmg U, _ -[(l+"o)/2E*]{P- IOt)(r.'/i') -[(1- 2. 0 )/E*]Pr _ - (P-w,),o'/4Gi',

-aV_ I[(I+.')/2E']( -IS+ ..·.) -[( 1-2.°)/3£°)5)4..,,'.

II,

[(6-S.0l+5(.b'- 5)OOS't](,';, )'-[3- 9 COS't]('';' )' ) S . (4.8)

In this expression .... is the pressure in the water contained in the ca.vity and caused by the volume change ~V, which is, in tum, caused by the displ.a.cement of tbe wall 01 the cavity. We may obtain av by integnlotion over the ou rfa.ce of the !!phere

-~V_

the reader may verify that

(4.7)

(4.10)

<1,0 -

-

p+ (P-",.)(. o,I.)',

(4.13)

(4.14)

By the use of (4.1 ) and (4.2) il lollows lhat

",' - - P- !(P-w,)(,oI,)'.

-a v.. 4n.'[-is+WoJ/W

By virtue 01 the identity 01 (4.10) and (4. 13) the resulting waler pre5SUre 01, in the instrument is the same u given by (4.1 2) with P- iS. The water ptes&u~ in Ihe soil mlW il$df is w--Ha,·+2",o)- 1'. The,.., is Iherefo~ a jump in the water pressure at ' _ " ,,-itlt a magnitude 01 t I'lG. It follows thudo,.., that, the water gradient iiwjiJ. is mfinite at Ihat surface an d percolation .tarl5 with infmite velocity giving a vertical slope of the water pressure .egistued by the meter as a function of time. The ""me effect OC(:U ... in the rigid type, but m this case the .tarting point lor the water P""'""U"" at 1_ +0 is zero, because no mcre:t.S
... 4n.'[ -lS+w.J/-IG.

(U I)

However, we ha ve a relatwn (1.1 ) betw..,,, lI V and ..... by which we may diminate av obtaining:

w, - lS/II HbG ).

(" .12)

T his result indicates that the cavernous po~ pressun meter l'egiSle ... a presMl ~ of nearly IS"" long"" b is small in comparison witb I/ G. As in the newest type b is 01 the order 01 I/ IOOG, where G is the s1,ear modulus 01 Ihe . Iiff""t day we have to deal with, the error ;nvolved is less than 1 or 2 per<:mt. We have >bOWD in (4 .6) LhILt the WILler V""".He in the pores of the soil is one-third of the Sum 01 the principal stresses. The~fore, neglecting the influence of the denominator 01 (4.12), it follows that the water prt:ssure, as measured by the cavernous type meter, is

94

(4. 15)

's. Ti"""'h
»any, [nc., t>cw Yorlr., I

~).

p. Jl6.

Soil Mechanics and Transport in Porous Media

C.

DE }OSSELIN

DE }ONC

'26

,~.~. -

,i w (!.

?

[

w

j' ...··~· .. . I

r • ... _

It,.ol

,

~.....; .:.:~:

.... _ .... r

""',,.... f ,G. 3.

It>o l

,..Ole, ~,. . .« lIound I"'''''''''' Sphen ""d lPh
S(~ a nd

5. DETBRMINATION OF THB CONSOUDATION PROCESS AROUND THE PORE PRESSURE METER

To determine the water pressu", registered by the meIer as a funclion of lime we need .. oollJOlidation theory in J dimensions. Such a thoory has been de"eloped by BioI.' For the co.se of saturaled soil the buic ~ualion is derived in the form (5.1)

".compressibility coefficient _ (1 - 2.) (1 + , )/(1- .)E (cm'lkg,) k _ permeability (em/sec), 'r~ _ de""ity of waler (kg/em'), . _ volume dilAtation of grain skeleton, and I_ time. Tbe physical meaning of this result may lie appreciated if we consider the well_known ,..,Iation from elasticity' in wbich ~uilibrium and compatibility conditions a,.., tahn inlo aCCQUnt,

(l/ a)V'_+(ax/ oJx)+ (oJY /ay)+(az/a,) _ o.

(5.2)

The body forces X, Y, Z in our ca.", a,.., furnished by Ihe water gradients X_ -iJoc/iJx elc., SO that

(5.3) Now, by e%lending Darcy's law of percolation to the case of 3 dimensions it is easy \ 0 deduce Ihat the rate of increase ol volume a_/at necessary to 510,.., th e excess of waler at a cerain point is given by (5.4) A combination of (5 .3) and (5.4) givt$ the buic tion (5. 1).

~ua·

In order to obtain an ag!"«Bble t....,.un.."t of the problem we shall consider simple harmonic loading condillons for ....... ..,. P .. Pu;p (iwl) in condo (J.3). • M. Bioi,}. Awl. Ph,... 12, ISS (1141). 19&

's.e nt.""'.. I, p.

Selected Works of G. de Josselin de Jong

All variabll:$ inRuenced by this harmonic eflect as for insanee t, will be wrill.." __ i cxp(i<.I). So W~ can divide all expression~ throughout by exp(iwl) and reain the overdashed characters. We get for (5.1 ) in th ~ ca", of axially symmetrica.l. streu disttib~tion

". _ -;11 ~r~(".'.) ~(''"?) a, ar +_' siruf i¥ i¥ l'-'" whe,..,

(5.5)

'1'-;..,

k ' (",)

which has the solutions • .. ,..-I[ A,] .+1 (; .....)+ ,1.." ,.+I(i..... )}Io.,

(5.6)

where] and N denote, respe<:ti"ely, I.kssel and Neu manns functions" and 'i' ~n
(5.7)

in which S a,.., polynomials in (1/ .....) .• Th;s~uation for .permits us tocompute Ihe solution to the axial symmetrical problem Ihat arises when introducing the tru e fonn o(boundary conditions. For insance location 01 the holes On rigid type, upper part of the cavity fonned by the instrument, etc. Computation would th.." require a treatment with many .lreo;.s functions derived from (5.?) by

.- ((1- 2. )(1+ _)/ E] tc~a/ a, -sim/-r'oJ/i¥JV'1>. • Jahnke and Emd<, TMk, vi N ... YorI< , 11145), pp. t4l, t46 . ' s.., "'<~ 4, p. 101. ' s.e nterul" 4, p. 136.

1i" .,,<Jw.u

(5.8)

(Dow. I'ublict.tiont,

95

CONSOLIDATION AROUND

"" !: "I" 1 1-, I

'"

-

p

~ 'ioJ ,",.•i.y,

PRESS U R E

p

" I

", - - ."

PORE

,/

•

MET E RS

~~,~ ~

" p

~.

:"',

1I

,

I,., ..

p

•

,

• ",

"

1

i

. ,p

F,o. ( . Graphical "P....... ,;oo 0( IOIm"l. (6.6): Ift!lOI'I< a/ po« p....., .. m." .. '"' ';mple h.artn _ IG-", " _ I em, .~..;,, _ 10-> _/<m', . · i,

I" general: in all cases, wh~", the solution of . tress distribution under ,tatk [ooding ;. known as derived from ' IUS/! functions, the consolidation problem is ..,ived by replacing thedilatation -'lress funct;"". which satisfy t"v>¢._ O by COm!$l'onding functions 01 the form (5.7). The function. ~ _ o need no alterations, beca.use ;: _ 0 also satisfies (5.5), neither the rotation_ stress-Functions, We shan limil oursekes, how~\'er, to the S;ause thi.. leads 10 a simplification in the treatment. The lAme solution far w. as" function of time i. then obtained for 3 equal.\~ (P-is) at ...... '" as well as for I principal stress (S) then:. Th is conclusion has already .been proved for the situation at 1_+0 for unit l""ding (""" the idC"l1tity of (4.10) and (4.13» and that it hohls here may he accepted without proof. 6. SPHERICAL S ... MMETR ...

T he solution neffie
._..1..-' e~p( - qr).

(6.1)

We shall nOI use a stress functiCIIl now, bul u, as basic variable as all Slresses and boundary condition. can be wriuen as functions of N. by virtue of the symmetry. for insla nce i - ..-'it/a.-(u,"», which giv~ alter integration: il, _ -A[(q' )-'+(qr)->] exp(-q. )-Br'.

(6.2)

In orner to describe the boundary conditions with il, we muSI firstly disf'O$<' of ,z,. This is s imp le by virtuc of (5 .3) which can be written (1/0)<"'/&">(;. )_&'/&">(..... ), and whkh after integration becomes (I/ a)i - u.-C,-Cr' . We mayomillh e lerm Cr', which conlributes a waler· How radiating from a point !lOllTCe al Ihe origin and i. irrelevant here. Therciore, il 10110""5 thai (l/a)i -tOl -C or -Gl _ C+(I/
We can now uP""'" the boundary couditions in terms 01 ii,. For the rigid type weueffi: (2.1.2), (2.2.1), (3. 1), and (3.3). For thecavemou stype : (2.1.3), (2.2 .5), (3.1), and (3.3). The condition (3. 1) is almldy ""tisf,"" by making A,- ;A" in {S.6}. There then remain three COnditions to be sati!fied for each type. The integration co""talll5 to adjust are J and B in (6.2), and C in {6.3}. It can be "erified that their values all:

If .. - P!/Vro' np{qr.)/N, B - Pllw.'[v.'-I',)(q ••)'+(qr. )+ I ]lN, C- p with N - [J. (qro)'+(qr.)+I], whell: 1' '" 1'' 1' _ 1',

for rigid type _ h/3a, for cavernOU$ type - b/3d+[(I- .)/2(1-2 . )].

With th= values we obtain f'nally for the water pressure,;:.. al • ., (that is the pressure regi.tered by the meltr)

r.

:;:.• - P/I-(b!30)(qr,)'/N ).

(6.6)

The complex "alue of q [see (5.5)] leads toacomple>: "alue of :::...... heing oul of pha.., with P. In F ig. 4 is ,;hown the rdation toetw""u P and ,;:. in the cample. plane aa Q function of the fn::quon
'f'~ ofitr-C----J",. exp(i.:.,t)

It"t- -

2..; __

'"

(6.7)

Putting .r·o in a more manageable (orm

(6.3)

Also ii, and ii,' may be written in terms of Q, in the following way :

i," _ i.-g,_ E/(I + .)[&u./",+ ••/ (1- 2.)]-,;:,. (6.4)

96

(6.S)

- P

I1-3:;•-; --'--I'

'11

f (a-.8ll(q+61 - (q+a) , (M)

Soil Mechanics and Transport in Porous Media

G.

F.... oS. Craphical tq> ......

D E J OSS ELI N 01:: JONG

'"

,.tioo> 01 '""mula (6..9).

the intrgral may be expreued in fUlICtiom in tbe form (,.... Fl(. 5):

R..".,.... 01 """" pr-u'" met ..... unit "

< ";,h .... ,. 'ypieot vatu .. 01 (Jaf4b). lenni <)l

t.abulated

\CIt- P fl - (b/l<>)r.(I-4}1)-l(a exp( ....1j erft( ....t)1 -fJ e..p(fJ',J) erfc(fJ'.J)IJJ, (6.9)

which is VIIlid far all 1>0, where P_value of step loading It ..... .., a - (I+(J-~) 1)l2p..

fJ-(I- (I-4}1)1]l2P..

Selected Works of G. de Josselin de Jong

erfCf- 2"~'f· exp(-),'jA, ' -"/4'Y~

•

(_ (5 .1)).

For the de(,nition of,. _ (6.5) and of bl l<> ,.... (I. I) and (5.1). Puuinl!._}l, and 1_0 the ezp~ (6.9) reduUl 10 (4. 12). Since ~ is Imall in COmparilOn .. ith Ja the expression (6.9) for the rigKI type bttomes appro~ima t ely

....- Pf l - ezp(,.....1j erfe(,.o.J)IJ (6.10)

97

3a/13

Application of Stress Functions to Consolidation Problems Application des Fonctions d'Ai ry aux Problemes de Consolidation by G.

1)£

--.

1bo numbo, of ....... fW>
~~~~~f:"T~~na"t":':~ti':lr!~~~: _"'O. n.: ~ «><>d~""" ...1...,;", '0 ,be t;t>;n <1:.1<10<1. os ",~n os '" 1110 poK " ' ' ' ' . ~ ,hen accou'>1od 'or. n.: """. io -.... fo, ,be cue 01 0 ri~ spbet< <mbedd
'0...,., .,.., re< 1110 oY<"
."... .nd 10.0<1<4 ",-;,b • ",ruc..l ....,~irI/in; .. ..,Iid l<>adtd uniformJ.y

infini .. ",ij

01 •

........

JOSSI'UN DI! JON(), Ir., Sub-dircctor of 'he Soil M echanic$ labora,ory, l}elrt, Ne,bot13nd.

<.a><

surf"",. comid<"';00. ...... porviou>~.

A useful """hoo in .Ia>lici'y for delUmining OIress-d.i>lri· butions is the appli(".Ouion 01 SIr... l\U\<'linns , In or<\<, t o ... 'i«y the C
(n From tbeot lunetitms ,he .......... l lId strain •• ,. oblainehown (by RAVl.l lClI. 1894: I......... I~S) in the case of vibMions type:< of " ..... funetion are n«mle p=co '"'tDcII, by ,hei, UOdienu. aCl '" body fo""," onthe..,il. IIIw been sb<>wn (""J."..., ... ", w J=, 1953) 1m"" Ihis con· ceptK'" """" 'u B"'·.··. (1941) w"i< "'I_,iyn f'" 'vn .... U~ •• iun ;n Ihree dimensio ...

,wo

,ha,

Le -"Ibn< do> (ollOllons de 1<nsioa DCCdSlirc .. oulIi>aDI pou, leo problCmca lie consolidation, .. qme,ric uial< ..., lie La pr<miOte .. ra",..,.un'11a OI)Q'I.".,...,.,. Ia _ lla l'OIaboo .. Ia tn>isitm< ~, i Ia
.ro;'.

«>Crc riJli
d·""

our result" obtain inst.nt.aoeOu, down Imw the ,ime dc:pendtr.ce of ,I>< d'tl f. om the begin"ina_ Ho"".... ', ,he compullliolu be<:ome '00 in'ri<:ltte e'~n ...u.h O'Itr-timplilied IIow pTopefliQ of lhe ",;1Fi .... l1y .... ",ill suppose tho ""in oIi.ltl"" '0 be an ..... ,i<: mcdiW'l"l with shear modulus G ... d Poisson', ratio,.. Thus ,he previou. inITool>l:<> Q _ {I - 2,,)12{1 -,.)G In .ca~'y lhe.e ;.. no .... ilie"'" and Iho n:latioMlUp belwo,," ......... and $tnUu i$ not liDea,.

,,"'rc:

c _ klay. _ consoUdalinn cotffici
Axial S,........!.-y Fo, b ... vity ..., "'ill lte:lt here \he case of ui>.1 .)'mmelry. In term< 0( displaoomenlS~ •• nd ~. (~. _ 0 because of ')'111met.-y) .... can e>puSS Iho volume di!alioo. and \he rolllion

"

.I«It'on;. _ .'olt.. "" dilatation of ""in .ult,on; a nd. _ lime. Fo' , he .... tet pressu'e ... ,,,: then ftruf . •.• (3) 'The"" formulae refer '0 tho ("QmjmSSion and arc bast
tha, the ""'" wal .. is incompressible compared "i,h ,be .0H o}:elt,on. 'The l"'''' water 11>o:,.,orc ",kes pan of (ho normal ,,~, and boca.,... of iu viscosily (he disli· palioo of ,uptrftuo'" wale, is slow a nd the wa ler pressure dcct"eues p:adually. Sin'ilarly tbe .isco<.i'y of 1IIe "",e' "",y inn ... noe the dded water of IP""ltt ,-iscosi(y '''' in clay). W. "ill hov.evu, in Ibis "tidy disrePld lhe time d
''''

98

• .. ; ~(111,)

w" _

(~~Jh)

+ (eu,lez)

...

(~)

_ (1Iu,18z)

w. now introdllOO ,he.,..... funclions Eand D in ,hcf"llow· inK ",,"y: 2Gu, " (, - 2,.K31£I&az) - Z(I -I')(~!JF'~;)

2G~, -

(l -

2,.K~f"/iIz'l + 2(, -

By inst:ni"lllhese e.pressions ""04"" obl.3'n:

f~

2G. _ {I 2Gw" _ 2(1

,,)G ~[~) lhe d;,pla<emcn15 in

, 2,,>t'1"£ •

(5) ~PTC$­

(.

_,,~ ,"'D

which indica'. how thest .... fullOlions ..... separaled aoconJinl 10 compression . and rota'i"" ......

The: "pe""or '1" in ,be

c:ose of ""ial ,ymmc'ry

it
["(' "I ,i'r '1i-! + ~ .1=

Thc: dttermi""ion of the from the "rns-f"IICIi"M is obtained by application of tile ...",,1 expressions fo' the "!USa io "'nIlS of displaoomen .... a. sh'en by lbe lheory of
. . .. "," 2G[!i
1,,»)< + ~w,,~,}. ""'.

Elaboration of t _ re!atio"" tben

,

a

~~( I - 2,,)E- 111 -I,)Q] ~

I

e

", - fi"'II'EJ+ -g;;,,)(1 -

a

T,, " 2; ....1(1 - ,,)D)

~

a:

~,,)E-

+ "r~z.I( 1

'0

.urf=.

~ ,"CS

<1," F: .... '[,,£ + 2(1 - ,,)0) + ~(l - 2,,)£ - 2( 1 -,,)OJ

", " Fr .... '{"E] +

,_try.....

In .lastici'y .be.. are ah';ay •. for ""ial 0 ...... functions and '''"0 bounda'l' cooditio ... , 1 ......... . .. ~'-ea, or I "lUI aDd I di$pl.Kemtnt, 0' 2 displ.a.cemti"" problem there is Doe addiliooa.l boundary con· di.ion "'hid> ... f... .be ~ ....... r; $0 the third ...... func:lion is nda'l' is pe"'ious ;lnd the pa,..·wa«r prQl,ure \bere it >ffO; (8) the boundary is impervious $0 the gradient of "OI.«r pross"", oormal to tbc surf"ce is ",ro.

Rillid Sp/>< •• F.mtot
m -,,)DJ

- 2,,)E - 1(1 -

For .he.,..., of a spherical body embedd.d in a soil m.... we need solu,ions of eq ..... ions 11 and 12 in \be fonn of spherical

"lDl

harmonics. whicll are of lhe .ype : E, _ AR-' ........S •• j(2qRj'l'.

So far compatibility and elastici,y haY< been accounled for. The funher co ... ider.otion or'

+ [(<1, - uJ/rJ + (~"Ial) " ("",/~') + (~"Ii!r) + (T"I,) " Z

(""J?,)

/I.

(0'

where /I. and Z are the body fo= =ned by the ",a"'r gr.odicnlS in radial and ""ial directio ... respecti,-,:Iy. SO that /I. .. iJw/fi> and Z _ awlh

.... combination of •• pras.ions 7 OM 8 liIi"" (I _

(l -

")I ~ "'E- ~ "'DI i:r~: ~~z

_ itw/lr ")I v.' ~ V'E + !,or ~(,.!. ;:, '>",D)I .. 8"'la~

0'

If V''I''D _ O. i, it
,,)(818:)V'E - (8Iet)V'DI

$1, _ BRI - "II'1',

(10)

In eq .... lio ... " 7 and 10 all d;.pl.lcemtpresooo.l in I.rm< of lhe.tn:ss funclions 1:: and D. 10 addition, we _d the wic «["",ions wltich have 10 he sa,isfx,d by E aDd D in order '0 lno ... whid> fontlS
lhe ro'alioo .... 1& a D e>tp""';on fo,,,,",, from S by .Iimination of II. and Z, and by introduelion

( 0)

E, .. {/, .. CRI-·-'J'I'.

w.""opted thefoliowinSn<>Utio"" 11. .. (r' + .')1 _ Spherical coordioal'; S.+I _ palynomialioqH.asdefined byTA'" (19Sof, p. 136); '1'. " Lell'nd .. polynomial of fir$t kind , T ..... (l9~, p. JIl7): .... B. C _ .,bi' rarycoM,"nlS. adju .. able at<;Qfdtdjr; .o boundary condi,ions; q _ V/c)1 whe ... sis .he ,""riabk introd .... d in Heavir o,",r the symbol. l oad> lire supposed .0 be applied step. wi« al I _ O. $O 'he lo&d P whon Iransforn>ed oq""Ia Pis. Consideru.. • rieid .phe.. wi,h R., Ioadtd a l lime , _ 0 wilb a fo to! r in .he Z-di"""ion, .... have to sa,i;fy tbc following boundary condi,ions a, II. _ 110. "The rigidity of .be .phere gi,u ~, _ 0 ..•. (14)

,i""'.ion

Tad,,,,

The fOf« P i. I Uppantd by the .urn of a ll the stresses from

""',.. and ..,il acting loge,ber on .he.urfaoo. This ~\U

,"'2,,11.' $in 'f' d 'I'[.in 'f' .i" +

(Pis) - ~ J.

.lfl1.lcemtain:

")1 (IS)

A ' ""lb =pecliO

deli of Ii

cos '{'(ii, -

"These '''''' coodilio", ref.. to.he lil'"in . kele.oro and .... sulli· cien, in.he.,..., of an elastic mo:>
,pile,.,

~I!.""''') l,v ~ + '''''''' TzI"" _ o Introduction of equa.ion 6 >1>0"" .hat Ibis oondi.ion may be .,ti.fotd if D is a $Olu.ion of 0 ... of \be followina equa.ions:

.wo

(12)

FI"Ofl' tbe .. ~ oq",,'ions II and 12 we obtaio 4 types of ....ess function: E,. E,.D.. D,. "The tWO functioM E, and D, ~.., not diff• ..,nt from ,he funetio., F, ""ti$fyi,.. V' F, .. O.~nd if we take E, _ n,. the ..... i... and .If..... deri,.. d from .hem a.., identical 10 tbose derived from f"" .. is euily verified. So lhe.., are ..... n'jaUy J stress funttiom to sa'i$fy .he boundary oonditions.

Selected Works of G. de Josselin de Jong

(A) .. _ 0

"These

( 8)

~/8R_0

(16)

boundary .oodilions a ... satisfied by ,aktdjr; ,be soIu.;""" n with n .. O.~...u.. E, _ "'W' ~""' . D I .. BR. 1>. _ 0 , _ CR- ' th~

( 17)

CompU,a.ion of the disp1aa:ments and ........ =y be pe,fanned by ,be: in.roduelion of tbe ........ func:tio ... in ,he .."ressions S, 7 aod 10. By introduction of tilde st ...... aod displace""",,,, furlher in ,he condi.ions 14, 15 and 16. il i. easy

99

100

Soil Mechanics and Transport in Porous Media

~~! ~ "

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a";;; -" I .... i!.{'

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!:a~.~P·!lLa -·E' .I ~'!l>d

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•.• • (Z,b)

.. V"",.)((I - ... X2L(O.0) - L(O,I)]

.. f'
f) - (I

"')[U.(~+II-t,()l

'" , .-!)J -L(~+II-!

(,00 .. 1l.1Id Iou • ..-imwn.a\ufo for,. .. 0 which, u, .. "', becomes eqllal 10 .he iInmod"c.c ~ •• Fi&. 2 IohowIIoow the .. ulanenl 01' Ihc C>
c_ _

(I-rIL(.....!!.... + "CIf- 1\" lit 1-,. _ ,,_ 4. '+.4)) If

....!>tn:

If ...

11(' - 9,o)/(1 - ,.Il! and

u,.q) "

.

f...~-_J'O'."
Boat ... fQl' " O.I.(p.q)eq.... Ls 1 fOri '" Olnd ..,oro. I '" .., (if _, < q') ~nd UO.O) .. J independen.ly of lime. bo,h.

In the _ c:ocoolden>d .he loOIu,iono fQl Ihc varialion of wi,1I limo Ihow &0 b ..t.............. "...,boerneDt at .he .......... 01' _ l!>PIi",,,;"" (, .. 0) and .... dte,-dfect. 1/1 ,he c~'mno .,... of eomPR!1$ibilll.y c,. ... 0). lbil after_ effect i, eq ....1 '0 i of Ihc ini,ial ............... n. for .... riP: malbo< ....... ic:al II ....111$ "",f... blo '0 oepua •• both dens alid '0 linl ... h.oth<, ,he consoIlda.ion has atI appreciable _ cut ....... Oft ,he .;,00,.,. 1\(1 ... _ .. ;., 001, ....1ticIt .... iIl_ dqlmdca. of tlIo p:OInCtt)' of .... k*Iod ...a .......

n.

....' ...... 01

lItu"""'",

'mllnle"',

_""i...

""'/ww

7JII' .,W.u /0 "'.~ 'M tiotdat.. G _ I.... IU ItrlpfJ b _ of ,IU IlwJy.

of Ik R. C.

~-

...~ na

and llb Ii... the ........... tkmeM for the -"" of Ihc Ioadod uaU , .. 0""" , ... GO, ~ n:spcn;vdy

2G..," 1'/ .'. and 2G., '" 2(1 - ")1'",,. Hen: apia all ....... ~ ....tkmoa, is obuiDod iD the ooMiott torn:Spood~ 10 Ihc dcf.,.."..,ion 01' the soil under _ . 'Ibo . fter-efl"ea i.apiD ""0 for aD iDc:ompr."..;ble ooil

Selected Works of G. de Josselin de Jong

101

DB JOSS!:W' DB Jom:>, PIlOP. Oil. h G .. 1968.

GlouchiqWf: 18, 195-228.

CONSOUDATION MODELS CONSISTING OF AN ASSE~mLY OF VISCOUS ELEMENTS OR A CAVITY CHAi'< NEL NETWORK PROFESSEUR DR IR

G.

DE JOSSELIN DE JONG-

SYNOPSIS A cavity channel ndwork consisting of ma.ny Un r
I n his book Grl»UimuJranica, Kever!ing Buisman (1940) describes his observation that soils show time settlement effects wrueh differ from the original Terzaghi theory of consolidation and which are especially pronounced in the secondary period. This discovery was mentioned in his paper to the 1936 Conference, but in his book and other publications (1938) he develops in more detail the possible causes and the consequence of this phenomenon. Since, however, all this was written in Dutch and never translated, some of his ideas which formed the basis of the present considerations are reproduced here. Buisman called the secondary settlement the 'secular effect', from the Latin' seculum' which means century. He coined the word' secular' to point out to engineers that settlement could cont inue to develop after excess pore pressure had disappeared, and could continue to do so for a long time, perhaps centuries. Because his work is only thirty years old a centennial verification is not available. However, for practical pu rposes his suggestion to extrapolate the settlement time curve as a straight line, when settlement is plotted on a linear scale and time on a logarithmic scale, gives an estimate which was verified to be reasonable in many instances. He called the corresponding settlement time relation the logarithmic t ime law and gave it t he mathematical form ph(ap+cr.1og/) (1) where, is the settlement, p the effective stress increase by loading, Is the layer t hickness, a. and a. the settlement parameters, and t is the time in days after loading. It was his opinion that the soil skeleton follows such a time settlement law, starting from the moment when an effect ive stress increase is created. Therefore the secular settlement also operates during the primary period of consolidation when excess pore pressures still prevent the effective stress from carrying the total load. Keverling Buisman suggested as a first approximation to use Tenaghi's theory for the determination of excess pore pressure and the effective stresses as a funct ion of time. From these he calculated the settlement as produced by the gradually increasing effective st resses by adopting the validit y of a superposition princi ple. He assumed that the response to uni t step loading always conforms to equation (I) .

,=

• Professor of Soil Mechanics, Civi l Engineering Department, Technolog ical University, Delft. 195

102

196

G. DE JOSSELIl' DE lONG

His analysis to include this secular settlement within the framework of the hydrodynamic theory of consolidation was only a enlde approximation, because at that time the powerful method of integral transforms was not available for the solution of consolidation problems. The importance of his analysis, however, is in the introduction of a time retarded soil skeleton in both primary and secondary periods of consolidation. To give credit to Keverling Buisman, the word' secular ' could be reserved for those effects and mechanisms which delay settlement both in the primary and se<:ondary periods of consolidation, and are responsible for deviations in time-settlement behaviour from Tcrr.aghi's linear theory. I n this Paper' secular' is used in this sense. Another important aspect of consolidation which he examines in his book is the explicit description of t .....o mechanisms that could be responsible for the time-delayed reaction of t he ~oil skeleton. One secular mechanism could be the viscous character of the pore water bound by the particles, the other mentioned is the size difference of the pores which have to transmit the expelled water in order to allow settlement to occur. Both these mechanisms will be shown in this Paper to lead to the deviations from Tel7.aghi's theory whicl1 are frequently observed in tests after the excess pore pressure has reduced to zero. It will be seen t hat the pore syst em with different channel wid ths is to be preferred as a model to the viscous soil skeleton, for the fonner will allow the data to be fitted to the theory over a wider time range than is possible with the latter. Taylor and Merchant (1940) int roduced secular effects in their consolidat ion t heory. As Barden (1965) has pointed out they obtained results which are representative for spring dashpot systems, but they did not specify the mechanism in terms of a rheological model. Tan (1957) substituted a specific spring dashpot system for t he grain contacts in order to introduce a secular effect ac<:ording to Keverling Buisman's viscous secular mechanism. Tan (1957) and Gibson and Lo (1961) give solutions for discrete values of the relaxation properties of the spring dashpot combination. By t aking three kinds of element s with enough difference in the relaxation time, Schiffman, Ladd and Chen (19(4) obtained for the secondary period a time settlement curve, which on a semilogarithmic scale has an undulating shape with three successive waves. Barden (1965) introduced the nOll-linear behaviour of the viscous soil skeleton and using a power law for the relation between shear stress and shear strain rate he obtained a sc<:ondary time settlement curve. By plotting the difference between the final value of settlement and t he settlement at an intennediate time, a power time law is obtained, which differs from the power t ime law used in this Paper. These workers have therefore developed mathematically the first mechanism proposed by Keverling Buisman. The viscous character of the soil skeleton has been incorporated in the consolidation process and e.'(act solutions have been obtained. It may be worth mentioning that the cases considered by Schiffman obey differential equations of so complicated a character that only electronic computers can solve them. In this Paper the notion of a viscous grain skeleton represented by elements containing springs and dashpots is reconsidered. Instead of introducing a limited number of discrete parameters for the elements as was done in the previous work, the treatment will be concerned wi th a distri bution of element parameters which is continuous and stochastic. This means that the properties of the individual elements vary random ly and that by their abundance it is possible to represent the frequency of occurrence 01 t heir parameters by a continuous function. Besides the introduction of a continuous frequency function, t he object of this study is to detennine the fonn of this frequency function for a particular soil by analysing its settlement time response to loading. Actually this is the inverse analysis of the previous investigators. who detennined settlement time response for a given element distribution. The motive for such an inverse procedure is the opportunity it offers us to study soil structure on a microscale.

Selected Works of G. de Josselin de Jong

103

CONSOLIDATION MODELS

197

Detailed information is not to be expected from a settlement observation which is the combined reaction of all elements working simultaneously. Besides this general reason for inaccuracy, the inverse analysis provides limited information, because the settlement time curve is only known by a finite number of observations in a limited time period and the viscous retardat ion medlanism acting at the grain contact point is only sunoised. In order to execut e the inverse analysis, it is necessary to choose a functional relationship between settlement and time which is mathematically convenient and describes t est results closely enough. I n this case the choice was to approximate the settlement time curves by straight lines if plotted in a double logarithmic diagram. Since a straight line at a slope a in such a diagram gives a power law for the settlement time relationship, such that' is proportional to t~, the corresponding law can be called a power time law. In order to include also settlement time curves that are not exactly straight lines in a double logarithmic diagram and settlements that reach an end value within a reasonable time, the more general case is considered that the observed curve can be approximated by a succession of straight line portions. Depending on the mathematical fonos adopted below, these settlement time laws are called the sectional power time law and the product of power time law. Although mathematical convenience was the main reason for adopting power time laws, the choice was inspired by finding that several settlement time observations gave better straight lines when plotted in a double logarithmic diagram than in the single logarithmic diagram used by Keverling Buisman. For practical use the power time law has the additional advantage of giving a safer prediction of settlement than the logarithmic time law because straight line extrapolation on a double logarithmic diagram gives greater values for the settlement than on a single logarithmic diagram. The determination of the frequency function pertaining to the occurrence of spring dashpot elements proved possible by an inverse analysis of power time settlement curves in t he secondary period of consolidation, i.e. after excess pore pressure had vanished. The corresponding theoretical considerations are developed in the following t wo sections. In the subsequent sedion t he consolidation process is considered which occurs when the spring dashpot assembly is considered to be immersed in pore water. The stochastic distribution of elements can be introduced in the consolidat ion theory by usc of the frequency function. The resu lt is a differential equation which resembles the original Terzaghi equation sufficiently to usc the well known solutions as is shown by an example. In the tinal section the cavity channel network suggested by Keveriing Buisman as a second possibility to account for secular effects is considered in detail. This second secular mechanism seems never to have been developed theoretically. While studying settlement curves from peat in 194 1 the Author observed in the secondary period an undulating time settlement behaviour on double logarithmic plots, similar to the curves obtained mathematically by Schiffman, Ladd and Chen (1964). For different load increments three successive waves were found, and for each increment these were repeated at the same times after loading. It was conjectured that interconnected channel systems of different size could be responsible for this behaviour (de J osselin de long, 1942). It is reasonable t o suppose that the stems and veins of foliage have left larger channels embedded in more degraded material containing smaller pores. At the time acrude graphical procedure was used to separate the action of several communicatingchannelsystems. By means of this analysis the observed settlement behaviour could be described by three different interconnecting systems. For clay the undulating character of the settlement curve in the secondary period was not so pronounced. It was, therefore, not possible to separate in clays the action of discrete channel systems. However, it is reasonable to imagine that in natural clays also, channels of different size and conductivity exist. The concept of unequal pore size was used by Olsen (1960) to account for discrepancies observed in the permeability of clays. His work does not cover, however, the consolidation process.

104

Soil Mechanics and Transport in Porous Media

198

G

OF. JOSSELIN DE JONG

In the subsequent analysis the possibility of describing secular effects by a model of cavities with various compressibility interconnected with channels of various conductivity will be examined. Again, the properties of the cavities and the channels will be considered to be stochastic and their occurrence will be described by continuous frequency functions. The curious result obtained wil! be that mathematically the cavity channel elements behave very similarly to the spring dashpot elements. Their time-rlelaying effect on compressibility is in fact identical. The consequence is that, in order to match test results for , large t by secular mechanisms, the same frequency funct ions are requi red for the two different systems. The essential difference between the two systems is the interpretation of excess pore pressure and permeability. In the spring dashpot model the concept of excess pore pressure is the same as in tIle original consolidation theory of Terzaghi. Excess pore pressure has a well defined value at every moment and at every point of the body. This value is computed in the analysis. Discharge of pore water obeys Darcy's law with a well defined permeability cocfficient. In the cavity channel model excess pore pressure is a stochastic variable. It is not possible t o compute the excess pore pressure for the individual cavities since the equations do not describe these various individual pore pressures explicitly. Instead an average excess pore pressure 14 appears in the analysis, which is a weigMed average of the pore pressures in the surrounding cavities. The weighting function turns out to be proportional to the conductivity of the connect ing channels. This quantity 14 will be called the average ambient excess pore pressure. A pore pressure weighted in this manner may be closely related to the pore pressure measured by a piezometric device, although it is probably not the same. The differential equation governing the average ambient excess pore pressure has the same stmcture as the original Terzaghi consolidation equation and resembles the consolidation equation obtained for the spring dashpot assembly . This permits use of all the results obtained for spring dasllpots for the cavity channel network as well. The difference is in the permeability, which in the consolidation equation appears now as a time dependent entity. The effect of a difference in permeability is apparent only at the beginning of the settlement process. The normal theory of consolidation gives for small values of t the familiar root time law as a first approximation. According to the theory with secular effects developed here, the approximation is of the form f l -<)12, where ~ includes time dependence of both compressibility and permeability. This approximate solution was verified with test results obtained from spherical samples loaded by an all round pressure (de Josseli n de Jong and Verruijt, 19(5). It t urned out t hat a positive value for" of o·} gave a close fit of test results with an accuracy of the third order of refinement in the approximation. The coefficients in the teons of the approximate expression were determined by use of Biot's theory of three dimensional consolidation. The fact that " is positive according to observations roles out the model of a spring dashpot assembly submerged in pore water, and therefore favours the cavity channel model. The tests on spherical samples gave for the settlement, in the secondary period, power time laws of the form t" with 0: equal to about 0·1 . The values for 0: and ~ mentioned above as obtained from the spherical compression tests are used as examples. The spherical compression test is specifically adapted to the study of settlement increase during long time periods because the test results are not obscured by side friction as may occur in ocdometer tests. STOCHASTIC ASSEMBLY OF SPRING DASHPOT ELEMENTS

The analysis developed in this P aper departs from previous theories primarily by the introduction of stochastic elements, whose abundance permits us to consider them as being distributed according to a continuous frequency function . In order to explain the theory this

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COXSOLIDATlON MODELS

-_._',. ._. . . rj,p

Fig. 1 (left). Spring d •• hpot element

... ..•. ,

199

I Fig. 2 (right) .

IF

Retardation hmetlon for Une .... prlng daahpot (40)

special aspect is treated first by demonstrating how a continuous distribution of spring dashpot elements can produce a time dependent settlement. Physical muhanism of spring dashpol element

The spring dashpot analogy has been introduced to simulate the response of the solid phase of a soil to an increase of the effective stresses. The dashpot in parallel witll the spring (Fig. I) retards its compression and can therefore be considered to produce a retarding effect similar to that which the physico-chemical bonds between soil particles exercise on the deformation of the grain skeleton. For mathematical convenience the mechanical properties of spring and dashpot are assumed to be linear. This may be a crude approximation to reality, and better mechanisms could be substituted without difficulty when the real phenomena are better understood; a revised formula for the retardation function would then result, but the considerations developed in this Paper would not be altered substantially. Let r be the compressibility of the spring and 1 the fluidity of the dashpot. A reduction in height, of tlle element requires a force in tIle spring and a force (1{I )(dCfdt) in the dashpot. The total force P acting in the element is then

crr

p ""

f+ !~ r 1 dt

,_0

(2)

which can be solved directly. If the initial condition is for 1= 0, then , ., P,[I -exp (- fd)l (3) where p.=l{r. The bracketed expression, the retardation function, will be called F (,.t) , because it is this function which represents the retarding effect of the spring dashpot element (see Fig. 2). Therefore F(p.t ) = l -exp (-,.t) (4) By taking the values of (,.t) to be zero and infinite in this expression it is seen that

°

for ,.t = 0 } (5) for ,.t = a) • Tn general F(p.l) can be any non·decreasing function which exhibits retarding properties. But whatever the mechanism, the possible retardation function will have to obey equations (5) in order to ensure that the settlement is zero at the start and reaches a terminal value at infinity. In order to include also otller possible secular mechanisms, we will retain the generality of F(p.t ) in the analysis as long as the treatment allows. In general, therefore, the settlement of the element will be F(,.t) = F(p.l) - I

,=

PrF(p.l)

(6)

The variable of the function F is tlle product (,J), which is dimensionless, so I" alWlLYS has the dimension of reciprocal t ime. Actually p. is tlle retardat ion parameter whose reciprocal is a measure of the timescale involved in the retardation process. In equation (4) a small value of I" indicatcs;thaqthe clemen(needs a long time belore its compressibility is completely

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mobilized, b&ause F(,.t) -+ 1 only if exp (-fdJ is reduced to a small number, and this is the case if (I't) has be<:ome large. The grain skeleton can be considered to consist of rigid grains connected by many retarding clements at the contact points. nle elements are orientated randomly in all directions, each having ils own I' or 1 value, In order to give expression to this randomness, the clements will be considered as stochastic entities. Since the analysis presented in this Paper deviates from previous theories primarily by tIle introduction of randomness, this aspect will be e xamined fm;t . Stochastic distrilmfioll of elemenls The constituent elements are called stochastic because their occurrence is arbitrary. The probability of finding an clement with special properties at a certain location is proportional to the number of elements possessing that property, expressed as a fraction of the t otal number of clement s present. The frequency of occurrence of clements possessing a certain material property expressed as a fraction of the total is by defmition the frequency function of that property. In this analysis it wiU be assumed t hat the number of elements is so great that the frequency distribution can be considered to be a cont inuous function. This assumption is justified if it is possible to subdivide the soil mass in to volumes in which the stress condit ion is nearly homogeneous but the number of elements large. Such a volume will be called i representative elementary volume and its size will be P. This volume wiil be considered tG contain N3 clements with N a large number. The block ill Fig. 3 is loaded by a vert ical pressure p. This load is transmitted primarily by the elements located at the vertical contact points between the grains. For simplicity the elements are thought of as being arranged in columns, which transmi t the force in a vertical direction. This is an over-simplification because the cross links between the columns are ignored. The effect of the cross links would be to redistribute the forces between the columns, but incorporating them in the analysis leads only to a second order refinement. The forces in the columns are therdor" considered to be equal throughout their ho;:oight.

Til. . . .p ....entaUv. 61.rn.ll\al"y volum. L ' II l ubd.ivided In columne 01 graino. At the COIlt.ac1 pointol b.lw ..,n th. grains s pring dubpol . 1o.=.n'"

Fl&, . 3 .

..... d.lJotributfid I t.ochastically

"

L'

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CONSOLIOATlO~

201

MODELS

On the average there are N2 columns in a cross se<:tion of area on the average an area .1 A _ PIN2 and carry a force P = P(PllV')

L~,

so the columns occupy (7)

(8)

Deviations from the average are also ignored in this study because they also require a second order refinement. StlU~nl

of a swchllslic assembly of relllrded dements Let the lth element of an assembly have a spring compressibility " and a dashpot fluidity I" while its retardation parameter is fL' =1"". The height reduct ion of the ith element is then, using equations (6) and (8) (9) Introducing the volume compressibility p, of the element instead of the spring compressibility r" the settlement of a soil block of height h ean be written as

,.",

, = Ph(N"/P) L p,F(p.,t)

( 10)

The justification of this expression is explained in the marked' paragraphs. The volume compressibility p, is defined as the volume reduction.1 V, of the ith element divided by the vertical pressure Pacting on it. If the element is prohibited from expanding laterally, the entire volume reduction is created by the height reduction {, only, and obtained by multiplying C, by the cross section area occupied by the columns .1A. This gives .1V, "" {,.1A ." {,(PIN") By defmition p, __ .1 V,IP and" = C"P, therefore it follows from equation (8) that .1V, C,PfN'J L4 p, - T - PN"/L" " r, N 4 Changing equation (9) to include t he volume compressibility PI, t he height reduction of the i th element is {, "" P(N"/P)p,F(fLlt)

The height reduction of the total column is the summation of all the height reductions of the elements in the column. For a soil block of height h. the columns contain as an average Nh/L elements. The summation contains therefore Nh fL terms and the height reduction of the block is NMt

, - L {,

' .1

~ P(lP/ P)

I<M t

L

,. ,

p, F (Jilt)

(11)

It is assumed that the number of elements in a column, of any length considered. is so large that although the clements are distributed at random their combined response is a fair representation of the average. This assures that the Nhl L elements contained in h produce a response which is niL times the response of N elemcnts. So the summation in equation (II ) over Nhl L terms is equal to h/ L times the summation over N terms, and this changcs equation (II) into equation (to).

Final s~tllemenl for Il stochastic assembly In t his section an expression for the final settlement of the soil block is developed as equation (15), which contains 4. t he overall compressibility of the soil. This entity Ii is 'Tbe reader can at finot omit the marked paragraphs without impeding comprehension of th
matter .

•

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expressed in relation to the volume compressibility of the individual elements PI by equations (13) and (14).

Because the dashpot continues to retard the process for an infinitely long time, the final settlement is only reached for e_ a:o when eXJl (- PII) is zero and F(~t) = I. Therefore the end value of the settlement is {{co) = Ph(l\~/L3)

!" PI ,.,

(12)

The same fonnula is obtained for any other retardation function provided it obeys equation (5) .

At this stage p, the mean' compressibility, is introduced, defined by P .,. (l IN)

,.!", PI

(13)

The only requirement on N is that it is sufficiently large to ensure that tlle sample contains a representative distribution of P,. Inserted in equation (12) this becomes

{Coo) _ Ph(N3jL")/J The compressibility If of the entire block ill now introduced as the average value of the compressibility p of its component elements multiplied by the number of elements per unit volume, N3fL3 , namely 4. = (NSjU)p (14) which t hen gives for the final settlement (IS) '(00) = pall

FrelJlle1l0' Junctj()tl oj the element dis!,jbutioJI The settlcment of an assembly of elements was obtained in equation (10) as a summation over many elements with different values of PI and 1-'1_ Because of their abundance, the elements can be considered to be distributed according to a continuous frequcncy function. This allows the settlement equation (10) to be converted from a summation into an intcgrnl. How the transition from summation to integral can be developed is explained in the subsequent marked paragraphs, with equation (20) as a result. Fig. 4 helps to visualize the concept of the frequency function in this case. In order to construct a frequency function the large num ber of N'J elements are subdivided into J portions, labelled by j, a number running from 0 to j. Each portion contains the elements possessing a retardation parameter with a value between 1-'1 and 1-'1+<11-'1. The number J of the portions depends on the manner of subdivision. It is irrelevant here how many there are and also how the total interval between 1-'0=0 and 1-', = 00 is subdivided. The size of LlI-'I can, for instance, be different. The number of elements in the jth portion is called LlMJ and so

,

..

,l:

LlMJ = N3

(16)

Tltis summation includes all the values of I-' that are to be found in the N3 elements from to f'J. The teons of the summation in equation (10) will now be rearranged in ascending order of f' . This means that the serial number i is re·allocated in a manner such that 1-'1 < 1-', . ,. Also let M J be the number of clements with f' values smaller than f'f' The interval LlI-'/ is then covered when the serial number i runs from M, to /lff+' =.M,+LlMI according to t he definition of LlMJ above.

1'0

1 The circumflex (~) on a quantity is used to denote some average value of it, defilled 011 each occasioll .

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CONSOLIDAT IO N MODELS

Fig. 4. (a)Blackdot.. .....l em.nt. In Interval iJl',' Nmnboro On lolt aide divided by N' glv. continuous cW"Ve of fnoquency function m (I')' N"m (I')d1' ia nmnb... 01 elementa i.n al' (b) "(1') . cumulative distriliution functlon 10 IDtegral of m(J.) a ccording '" equation (36)

@

In equation (10) the summation contains N tenns, and is after t hat multiplied by N~ . So it contains N3 tenns, from which LIM, t erms have the required values for J1-, in the interval LI"i' Equation (10) can t herefore be rewritten as

, ... (Ph/L3)

Jo [:iM~'p'F(~]

(17)

The interval Lilli will be taken small enough to allow the approximation of considering Flp.,t) a constant in th at interval and equal to F(p,/J. Then 1I, +dM,

!of,+oW,

L

2:

p,F(p.,t) = F(p.,t)

1 ~ Il,

p,

h ll,

As a consequence the number LIM, is only a small fraction of N3 . It is assumed, however, that N3 is so large that although LIM} is only a small fraction of NJ, it is still a large number. Further it is assumed th at there is no correlation between PI and Pi. Then the LIM} elements contain by t heir abundance a fair sample of the p, dist ribution in the entire system. One is then justirled in using the average compressibility fi according to equation (13) with LlMj substituted for N , and writing M, + ~M,

2:

' .11,

p, ~ pLlMj

If there is correlation between PI and 1', this correlation can be t aken care of in the frequency function defined below. It is assumed here that there is no corrciation. These considerations applied to equation (17) yield 1

, = (Ph/V)

or using equation (14)

,-. 2:

"',+AIl,

F(p.1J

L

PI .. (Ph/P)P

.

,-2:

F(p,IJLlMJ (IS)

' _,D

This expression suggests an integral over 1'" because all values of 1'1 arc incorporated, while j ranges from 0 to }. In order to obtain an integral a frequency function m(l') is introduced in SUell a manner that m(p.j)Llp,j=LlM,/Na. Then equations (16) and (18) ~ome

,

2:

1_ 0

110

nI(}<j).dp,, '" I and, - puh

,

I

1_ 0

nI(p.,)F(p.,t)LII',

Soil Mechanics and Transport in Porous Media

G. DE JOSSELlN DE J ONG

If now Lip. is diminished to an infinitesimal quantity, dl', the summation bec::omes an integral which extends over all values of ,. from 0 to 00. T he last t wo expressions then become

So" m(p.)dl' =

I

(19)

md ( = p4h

So"

m(p.)F(p.t)4,.

(20)

The frequency function m(,.), as defined above to be related to t1.MJ can be considered as the fraction of elements whose values of,. cover the interval between,. and ,.+4,., divided by the size of the interval d,.. It indicates how the elements are distributed with respect to I" (s~ Fig.4(a)). Equation (20) describes how the settlement is created by an assembly of elements that are distributed according to m(}t), all having the same retardation function F{pJ). TIME DEPENDENT SETTLE~IENT PRODUCED BY SPRING DASHPQT ASSEMBLY DURING THE SECONDARY PERIOD OF CONSOLIDATION

Equation (20) gives the relation between settlement (I) and m(,.) t he frequency function of tile spring dashpot assembly as an integral expression with m(jL) in the integrand. If one wants to know m(p.) explicitly to study the soil structure, it is necessary to solve equation (20) for m(p.). I n t he subsequent sub-sections two different procedures will be treated which permit determination of m(,..). The first procedure gives a crude approximation but simple expressions. The second is more refined and gives a better approximation, but more complicated expressions. Both procedures require a particular mathematical structure for the settlement-time relationship observed in the secondary period of the oedometer test after c.xcess pore pressures have vanished. Both these will be treated in the next sub-section and in the subsequent sub-sections they will be used for the determination of m(p.). Sdllement tinu law in the jOTttI oj a combination of power tums

The two mathematical procedures required in' the subsequent analysis for the determination of m(p.) demand a curve fitting which in both cases wil! consist of approximating the settlement time curve plotted in a double logarithmic diagram by portions which are straight lines. These straight line portions give power terms of the fonn of equation (22). Let 8" 82 " •• be the times corresponding to t he points where the straight lines intersect and Jet «, ~, y •. . be the slopes of the SC(;tions (see Fig. Sea)) . Then, if :c~log I andy=log ( are the co-ordinates in the double logarithmic t-( diagram, these straight lines are

y ... a:c+log A y ... ~x+ log B y=yx+logC

(21)

The first two tines intersect at t= 0" if Y in the fIrst two expressions is equal for x=log 8,. This gives « log 0, +log A =~ log 0, +log B or B = AO,a- 6. So all successive coefficients, n, C, and so on. can be expressed in terms of A. Since A has an inconvenient dimension, introduce 0o• such that A = h8o-·, with h being the thickness of the sample. That the fonns in (2 1) are power time laws can be seen by reintroducing t and C, instead of x and y. For instance, the first t hen gives C=Ata..,hl"0o -". Written in terms of ( and I

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CONSO LI DATION MODEL S

!

.i

20S

1' :: I.

@

Fie.

II. ( Il) S.tU.ment tim. curv. in doubl. 10g....lthmlc diagram . D ull.d llD. J. ncUonIll pow ... u.ne I llw, aol1d llD. I. product of power" ta ...... tim. la"" (b) Th. cwnulll.l.i .... dietribUUOD lunCtiOD n(p) h •• aectio....u,. th. ....... 1lI0pe in the double logarithmic diagran1 Il. the co ..... .pondlng . .tu .... eut

the equations (21) give a combination of power laws valid in successive time intervals, as follows; (I) .. ht"{Jo-~ O OWl' liml law because it represents a curve consisting of sections, from which each obeys a power law. This representation by power laws will allow a crude approximate determination of mC/t) valid for any retardation function. A more refined theory based on integral transforms is possible only if the retardat ion function is known. The special function pertaining to the linear spring dashpot element will be considered as an example. The analysis then requires for {(I) a function that can be continued analytically in the complex l-plane enclosed by -or < aTS t < +or. A function that satisfies this condition and resembles (22) satisfactorily is the following equation which will be called the product of powl' tm/fS timli4w

_ ('Do)'('+8.)-'"('''')-''' 0;... etc.

crt) - h

~

(23)

This product power law is approximated by (22) if the intervals between B\> B2 , 83 •• and 50 forth on the lOgarithmic diagram are large enough, such that (24)

112

.

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This can be seen by considering first a time t which is small with respect to 9,. Then the tenn (1+ O,lIO, is almost unity. and SO also are the subsequent terms. Thus the settlement curve is practically (25)

If t is large compared with 8" but small compared with 82 , then the second tenn in the product becomes (t/fJ,) --U, whereas the rest remains unity. So the product is approximately (26)

These resul ts show that equation (23) coincides with (22) for the greater part of each of the intervals. Again, the parameters ex. fi. y ... , 9,. 8~, and so on can be obtained directly from the double logarithmic t-{ plot. In the vicinity of 1=0, . 1=82 • and so on functions (22) an d (23) differ. The product power Jaw is a smooth curve, which blends the sharp intersections of straight lines appearing in a double logarithmic diagram, if (22) is plotted (see Fig. 5(a)). The reasons for introducing the sectional power time law or the product power time law are mathematical convenience and c::onvertibility of test results into formulae. The adoption of the eq uations is not dictated by the physic::s of the problem. Tn th e subsequent sub-sections the Laplac::e transform of the settlement will be needed. This will be denoted by a bar over the particular variable and be defined as t(s) ""

('tt)exp (-sl)dl

(27)

rt is not possible to give an explicit expression for the transform of the product of power terms settlement time law. For the sectional power time settlement law, however, the transform can be obtained direc::tly. TIlis gives by integrating sectionally

tis)

= ks-1--Oo--[r (I + .. )-r{( I + .. ;sO)}] +h$ -l-,Oo -"o," - ' [r {( 1 +.8; so,}- r{{l +.8;S02}) +

(28)

where r lx) is the c::omplete gamma func::t ion and r{x;y} is the inc::omplete gamma function, both defined in the appendix. Using t he approximations (1 25) and (126) this can be simplified to t(s) - hr 1 --Oo--r(l + ..) 0 < r l« O,} tis) = hs- 1 - ' Oo-a01'"-' r (I +,B) 0l« r l « O~. (29) ... etc. and if a flllal settlement (tl) _hap is reached for t_ 0() the last t erm of this row will be t(s) _ Mps- 1 $-1 _ 00 (30) The approximations used t o obtain t his result are valid over the entire region of the complex s-plane required in the elaboration of equation (76). ApproxittUlle dekrtninatirm of frequnwy f urv.;/ion and cumulative dislribulion f unction fW generaJ retardation muhanism It is stated above that, if the settlement time relation is represented by a power law, m(p) can be determined for any F (fd) . The result is a power law for m(p) gi ven by equation (42). However, in order for the analysis to be exec::uted the retardation funct ion cannot be completely arbitrary. Physically a retardation function should be a non-de<:reasing 1 function that is zero at t=O, and unity at t=oo as indicated by equation (5) . MOreQver,

I

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CONSOLIDATION MODELS

the analysis also imposes requirements on E, the first derivative of F. The following, not so severe, conditions have to be satisfied: 0< F(A) < a,'>" E ('>") <", A« 1 } (31) J « ,>,. • I> F(A) > l-aaA - ' E('>") < "2,>,.-a where and "a are positive numbers and E{A) ~ dF(A)/dA (32) From (31) it can be deduced that F{A) = 0 E(A) <", (33) F(A) _ 1 E(A) = 0 The retardation functions of linear spring dashpot elements and cavity channel elements obey these requirements. The analysis is based on the fact that the time derivative of the settlement contains the settlement itself, when a power time settlement law is valid. From (22) it is seen, for instance, for the second interval, that dC/dt - hpt._-I(Jo-a(Jlg-I = fi('It) (J,« t« (Ja (34) In order to bring in m{,..), the variable (in equation (34) is replaced by the expression (20). Since in this expression F(P!) is the only {unction containing t, differentiation of ( with ~pect to t requires only the time derivative of F(p.t). Using equation (32) t his derivative

a,

tlF(p.t)fdl - ,..(tlF(,..t){d(p1}] = JiE{p.t)

and so expression (20) gives for the left side of equation (34)

f'

dC/tit - pah

(35)

Jim(Ji)E(,..t)dl-'

On the other hand it is possible to obtain from equation (2Q) by integration by parts an expression for WI) , which resembles equation (35). In order to obtain this expression it is convenient to introduce the cumulative distribution function n(p.) (sa Fig. 4{b», defined. by n(l-') = f'm(p.')dJi'

(36)

.

By this definition n(l-') is the probability that an element possesses a relaxation parameter larger than 1'. From equation (19) it is seen that

:);;:~

!~~ ; : ~}

.

(37)

Differentiating equation (36) with respect to I-' gives dn(Ji) fdl-'

~

-m(p.)

.(38)

So integration by parts of equation (20) using equation (32) gives

' ~ Pdh [ -n(I-')F(,..t{ +1

f"

1I{I')E(,..t)dl-']

The first term between the brackets is zero, because (33) and (37) indicate that the product is zero on both limits. So it follows t hat ({ /I) = pall

f'

n(I')E(!d)dl-'

(39)

A combination of equations (34), (35) and (39) gives

f'

Jim(p.)E(p.t)dl-' =

fi So"" n(p)E(pl)dl-'

and this relation is only valid in the time interval

114

(JI

(40)

« t « 8a.

Soil Mechanics and Transport in Porous Media

G. DE JOSSELiN DE lONG

Although the integrals extend from zero to infmity, only the range I f8~ « ~ « I f8, of the integral contributes to its value. It would take too much space to enter here into all the details of the proof, but with (24). (31) and (43) it is possible to obtain the following conclusions. The part of the integral from I f 0, to infinity gives a small contribution, because E(~) is sma!! there. The part of the integral from zero to I fOa gives a small contribution, because the interval is small. This means that, using the values of fII(,,) and n(,,) valid for I fO~«~« I/O" but extending the integrals over 0<,,< co, an error is introduced which can be disregarded if 8,«t«9a. So equation (40) is satisfied for every E(jd), and F(,.t), if "m(,,) - pn(p) - p

f'

Differentiating with respe<:t to ~ gives ,,[dm(,.)/d,.]+m(,.) _

m(,,')d,,'

~pm(,.)

dm(,.) _ (-l-P)~

m(,.) m(,.) _

"

(4 1) This argument can be repeated for every interval of (21) giving for the frequency function C~~-'- '

C~,,-l- .

1/8,« " < co 1/8,«,.« 1/9,

C3,,-I-.

1/8.«

m(,.) = C,,,-I-I<

~«lf8,

} (42)

... etc.

From these the cumulative distribution function defined by (36) is obtained as n(~) =

(C,/a.),,-"

1/9, « " < co

{C./PJp.-1 (C 3 Jy)p.-·

1 /9~

«"«

}

I f8 1 1/9,« p.« 1/9•... etc.

(43)

The constants C,. C" C3 • •• cannot be detennined unless the function F(,.t) is specified, as shown in the next sub-section. There is, however, the general restriction on these constants that equation (19) has to be satisfied. In Fig. 5(b) n(,.) is ploUed in a double logarithmic diagram such that the ,,-axis runs in the opposite direction to the t-axis. Then the line sections have the same slope as the corresponding intervals of the settlement curve. This remarkable result is not unexpected. The reason is that the retardation mechanisms arc considered to be the same for all clements, so tha t the value of ("u) . for which the retardation function reaches a certain large part of its end value, is the same for all. Let this be for ~ = a. The elements that have completed their task at time I have then retardation parameters which arc larger than a/I. Since settlement is the cumulative reaction of all these elements, it is proportional to the cumulative distribution function of these elements whose ,. is inversely proportional to j. MOTe refined ddl!ffllitlalion of freqllency fllnetion for retardation nuchanism cOlisisting of linear sp,illg dashpols

If the retardation function is the special function (4) valid for linear spring dashpot elements, a more refined analysis is possible, using a Laplace transfonn. The result is equation (46), a complicated fonnula that is approximated by equation (47), an expression resembling (42). This result is obtained as follows. With equation (4) the settlement expression (20) can be written as

I

W) - 1'611

J:

m(p.)[l -exp

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( -~)]dp.

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For t infinite, exp (-fill reduces to zero and {(o:» ~Pdh JO" mij.o.)d"" which by equation (19) coincides with equation (IS). The difference between Ule tenninal value {(o:» of the settlement and an intennediate settlement Wl is therefore

C(o:»-C(I} '"' pdh

f;

m(lL) exp (-,..t)dlL

(44)

This sho\vs that (C(o:» - C(I)] is the Laplace trans form of mv.). Then the t heory of integral transforms gives as a solution for m(.l.) m(lL) cz

_, I . f c+u, C(ro~~ Crt) exp (+,..t) dt

(4S)

1N ) <-10>

The fact that [C(ro)-C(t)] is a Laplace t rall5fonn limits the class of functions t hat can be used as representations of Crt). In particular, W) must be a funct ion of I that is analytic for aU complex values of I whose real part is larger than c, i.e. to t he right of line AB in Fig. 6 . In this case c can be taken as zero. In practice the infonnation about will conI .. Ct) sist of C values for a limited number of I-values. These values of I are aU real and positive. From this infonnation an analytical function has to be constructed which obeys the condition of regularity to the right of A B. i . _..____.... R.(tJ ---'-c'r • • Db The sectional power time law, which by its simplicity was convenient in the previous analysis, is not suitable here because the discontinuities in t=//" //~, and so on introduce singularities to the right of AB. The more complicated power product law equation (23) is preferred here. This function has its IDt.lfl'aUve pa ~ In comple .. singularities in t= - //" - //2 and so on, which are, Fig. 8. actually, branch points. Evaluation of the integral is obtained by shifting t he integral pat h from the vertical line AB towards the line CDEF which runs as a hairpin on both sides of the negative t-axis. The integral paths AC and F B give no contribut ion, because [C{o:»-'f/)] tends t owards zero for large values of t. By this shift the integral is brought into a form which leads to tabulated hypergeometric functioll5, if the power product consists of two terms. If the product (23) contains more terms, two adjacent tenns are separated out in the way (26) was obtained from (23). There, one term. I·, remained by considering //,«/«//, . Here, two terms, (/+81 )$, (t+//,.)-I+', can be dealt with simultaneously by considering 0 < 1«//,. In virtue of (24) all other tenus of equation (23) become constants. The consequence is that the values of ,(I) in t he vicinity of t .. //" are represented more accurately, and therefore also the values of m(lL) in the vicinity of IL - 1///,. The result is

crt)

,

•'-=;-"'~;;-''''~'

" ....

m(.l.) =

h2: r(;(1«);(~~fl) (~r

//'I F ,(1+«; 2+fl; -IL//I)

("j"("j' 0, //2 e _. "1 F,(1+P;2+y;-",//,)

pr(l-p+y) +r(I fl)r(2+y) 8;;

+ r(i(ly);(;!)a) (~r(~r(~r //3 C ··. I F I (1+ y;2+o; + ... etc.

IL//a)

(46)

where I FI is the hypcrgeometric function (see equation (1271).

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This summation of terms reduces to a simple form bm:.ause of (24). By use of the approximation formulae for IF, (equations (128) and (129)) it can be shown that for values of fl between (1/91 ) and co the first term exceeds all other terms and reduces to fl -l-~ multiplied by a constant. In the interval between (I/ (J,) and ( I '(J~) the second term is the largest, giving p.-,-$ and so on. The frequency function (46) can therefore be approximated by

,. ) 1 • (1)_1_. Fo ,. 1 P (8.)'(1)' ->-, pa r(1- (J) So e.

"'\1" - pdr(I-Clj

Ii

1

Y (,.),(,.),( 1)' _._ pd r (I -'ll Fo 0;'" ,

e.

1/83

«

I"

«

I /(J~.··

etc. (47)

A comparison with (42) shows that this is the same function of 1'-. a result that was to be expected, because F('\) .., l -exp (-,\) satisfies the oonditions (3 \ ) required in the analysis there. The difference is that the coefficients C,. C2 . •• are obtained explicitly now.

P()lJ!tI'

ti~

14w with an asymplctic value

I n order to demonstrate the use of t hese results, they can be applied to the example of a settlement curve which follows a power Jaw up to the time 0, and then levels off towards a fmal value. This is a type of settlement observed in tests. In a double logarithmic time settlement diagram the curve can be shown diagrammatically by two straight Jines, the first at a slope, say a, the sewnd horizontal, at a slope zero (su Fig. 7(a)). The product power Jaw in that case would be

I

t(q

~ h(ir('~~r

.,

(48)

.

0<, ••,

._._.-

F ig. 7.

(.) SettluneDt (b) Cwnul.U_ di. t.rihut.ion lunellon n(1') (0) J'Hquency fwlcllonm(l') All .oUd linn ..... from rell.ned theory, da.hed lin.. !rom crude analyaia

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because fJ, y and so on are all zero. The end value for the settlement is obtained for t» 81 giving t(co)-h(O,IOo)", and so according to equation (IS) (8,/80 )'" ,.,. p6 (49) From (46) the frequency function is, since fJ=y'" . . . = 0 and r(2) ... 1

(50)

m(J<) = n8"F,( I +a;2;-,..O,)

This is a tabulated function plotted in Fig. 7(c). The following approximations for m{J<) for large and small values of I" are obtained using equations (128) and (129): m(.u.) = [a/T( l -a))8,-"'I"-' -· m(,..) .., a8,{ I -[(1 +a)/ 1! 21](,..8,) +[(1 +n}(2 +a)/21 3! 1 (p.8,)~ - ... } and for the cumulative distribution function 11(,..) _ [1/r (l -a)](,..O,)-'" 11(1") = =

f."

m(J<')dl'" - I -

r

1/8,«,.. } (5 1)

,..

« 118,

118, « " } . (52)

m(p.')dl'"

{1-a{}A.8,) + [a(1 +a)12 !2!](,..O,)~- ... }

,.. « 118,

The values for large,.. corresponcl well with (42), (43) and (47). The values for small I" are irregular in so far as they do not fit in the pattern of (42), (43) and (47). It could be surmised that the slope being zero, m(,..) would be proportional to 1'-'. Instead m(p.) tends to a8,. The reason for this irregularity is that the analysis based upon equation (40) is not valid for z.ero slope. The analysis of the past sub-section, on the contrary, remains valid whatever the slope is. From this result it can be seen that if an end value of the settlement is reached beyond a time O~ the frequency function and the accumulative distribution function are defined as m(,..) = C~ } 1I(p.) = I -Cn ,.. for,..« I /O~

(53)

C~ a constant, which can be evaluated if the retardation function F(,..t) is known. It is instructive to use the example of this sub-section to show how the constants C" C2 and so on of (42) can be evaluated if F(;d) is known. From the analysis there, together with the infoonation on how to treat an end value with zero slope, with (42) and

with

(53) it is possible to '."rite immediately that tn(l") = C,,..-'-.r(p.) _ (C,I«),..--

1181 «I" < co (54) 0<,..« 1/0, (55) We will now make the crude but simple approximation that (54) and (55) are valid up to,.."" t/fJ,. It is then possible to detennine the settlement, because the retardation function is known and given by equation (4), being F(,..t) - I-exp (-,..t), E(,..t) "" exp (-,..t). Using equation (39) to avoid unsatisfactory integrals gives t(l) -

h (~)~ [1 + c..;.- (1+ ~ + ~,~)

exp (

-f.) + (C,/aw r{(l-« ); (I/O)}]

(56)

Using the approximation (125) for small values of t gives W) ... "(O,/Oo)-I~aC,/r(1-a) t « 8, (57) From equation (48) it follows that this should equal II(t/8 0)· so thatC, should have the value

C, '" [a{r(I -a)JO,·

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This is exactly the same as equation (5\).

The next step is to evaluate C2 • This can be done by use of the requirement (19) on By integrating m(p.) of equation (55) from zero to (lIS,) and of (54) from ( l IB.) to infinity the result must be unity. This gives

m(p.).

(C,JBt) + (C,/a)8'j -

(59)

which together with equation (58) requires that C~

... { 1- [ 1/r(I -a)]}B,

(60)

According to equation (51) this should be aB,. In a typical test result a is 0·1, then r( l -a) _ 1·0682, and so C2 is 0·0638 B, instead of 0·, B,. In order to show the difference between this crude analysis and the more refmed theory, Fig. 7 contains the settlement, the frequency function and the cumulative distribution functions as determined in this subsection. The difference is small enough to permit the use of the simpler analysis.

CQlldusion The above analysis allows us to determine from the secondary period of an oedometer test how the relaxation parameters of an assembly of spring dashpots have to be distributed in order to show a settlement time behaviour similar to that of the soil under the influence of effective stress. Consolidation effects created by excess pore pressures have not been introduced here. The approximate result is that the frequency function (42) and the cumulative function (43) are power laws of 1-', if the settlement obeys a power law of t, (22). These power laws have the same coefficients but a reciprocal character. The more refmed theory gives as a result (46), a complicated formula for the frequency function, which reduces to the same power law by approximation. It may be emphasized here that the information about the frequency function, to be extracted from test results, is limited to values of I-' that are the reciprocal of the observation times. The example of the past sub-section shows that the difference between the crude analysis and the more refined is small. This suggests that t he simpler analysis may be preferred in a st udy of soil stmcture where uncertalnties about the retarding mechanism obviate the necessity for mathematical refinements. CONSOLlDATiON OF SPRING DASHPOT ASSEMBLY

So far the settlement considered was caused by an increase of the effective stress with a

magnitude p for all elements starting at 1=0. If t he spring dashpot elements are immersed in water, the effective stress gradually increases as the excess pore pressure decreases, and during the consolidation process the effective stresses differ at different elevations. I n this section the basic equat ion of consolidation is developed for the one dimensional case, taking account of excess pore pressure and the retarding response to effective stresses caused by the secular mechanism of spring dashpot element s. The effect of secular mechanisms on c(lIlsolidation is discussed by use of an example.

Basic ({juati<m oj ccmsolidation for rdard41 ekments immn-sea in pore U/a/u To develop the basic equation for one dimensional consolidation a layer of thickness H is considered, which is compressed in the z direction. The differential equation will be developed for a slice of thickness LIz which, although of infinitesimal thickness, will contain enough elements to permit the use of the integral expression (20) for evalUating its settlement. Assuming that Darcy's law applies to the excess pore pressure, If (wherek is the permeability

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CONSOLIDATION M ODELS

and rIO is t he specific weight of wate r), the amount of water Ll V expelled from the layer over an area, A , during a time, dt, is equal to d(Ll V) = - (k{y",)(fPu{8z 2 )LlzdtA (61) If the soil is completely saturated this amount of water is equal to the reduction in volume of the grain system. In one dimensional consolidation the elements are prohibited from expanding laterally 50 the volume reduction Ll V divided by the area A is equal to the height reduction d(.dQ of the slice .dz, so equation (61) becomes deLle) = -(kt-r ..)(fPuI8z2)Azdt Introducing the time rate of s::ttlement 8(Ll{){at, which is equal to the height reduction d(AQ divided by the time interval dt necessary to produce that settlement , gives o(LlCl/CJI = -(kfy..)(fPu/CJrjLJz (62) In establishing Terzaghi's equation for consolidation the next step is to relate the rate of settlement CJ(LlCl/CJI to the increase of effective stress by introducing the compressibility as a coefficient which is time independent. I n the analysis here, however, the compressibility is time-
For the transfonnation of 8(LlCl/at use is made of the fact that the settlement is zero at time 1.. 0.

The next step is to produce the relation between the transfonn of the settlement and the transfonn of the effective stress. This can be obtained by starting from equation (2), the differential equation for one element. Instead of a time-independent force P, the load on one clement will now be a time variable effective stress a' which acts over an area AA given by equation (7). The Laplace transfonn of equation (2) then becomes for the ith element, .'(,) p _ l,(s) 11'1"

+s l,(Sj I,

Since S is an inert parameter, this equation, with p._II" be<:omes

'() _ a_'() L2 'I/Ji (64) sN'-s+/A, That this expression is the transfonn of equation (9) can be verified by substituting, in place of a'(I) the time independent pressure, p. The t ransfonn of such a time invariant quantity is Pis, so equation (64) for this case becomes ~,s

'1

',fL,

L2 L2 [' t ,(S) - p N1 S(s+p..)'" P N2" $- S+/Ji The inverse of this is

{,(I) _ P(L2/N2)r,[I_exp (-P-A] and this is the same as <'
•

t(s) ... Ph{N2/P) I

,- , [PtJ<,/s(s+I'.)]

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By introducing the frequency function m(p.), expression (20) was finally obtained. In that analysis neither p nor F(I'l) changed, so all the manipulat ions involved can be applied to the transformed settlement C,(5j from equation (54) as well, without changing v'(s), which stands for PIs, or p.,/{s +/'i) which stands for s times the transfonn of F~t). The transfonned settlement of t he layer of thickness .:lz can then be wri tten

(66) The time dependence of the compressibility caused by the stochastically distributed elemen ts is incorporated in the integral over ",. This will be denoted by g{s) , so g(s) '"' and equati..,n (66)

f'

(p.m(p.) I(s+ p.);dfL

(67)

\.Jeo.;UIIlt:>;

.:It(s) "" u'(s)ag(s)Az (68) This expression describes the transformed relation between settlement and effedive stress and involves the use of a compressibility ag(s) which incorporates the time dependence by the function g(s). Although g(5) is defined here by equation (67) as a function of the frequency function m(p.) , it is not necessary to know m{",) if g(s) is required. The dire<:t way to obtain ages) is by use of settlement observations during a period when the effective stress does not vary any more, i.e. after excess pore pressure has disappeared. Expressed in a sectional power law tlus settlement has a Laplace transform given approximately by equations (29) and (30). Use of equation (68) with u'(s)=P/s and .:lz=h, then gives

g(s) =

;a (~J s-~r(l +«)

J/', ,,,,,,"'

g(5) -

;a (~r (~r r ' r{l +p)

~~~2 « 5 «

1181

I

(69)

. . . ett. HO g(s) .. 1 It is now possible to wntinue the development of the differen ti al equation for the consolidation process, by substituting equation (68) into equation (63), which gives sa'(s)ag(s) _ - (kly .. )(d2a(s)/dz 2 ) The efledive stress can be expressed as the differente of the load p and:excess pore pressure U, whith betomes after Laplace t ransfonm.tion v'(s) _ (p/s)-a(s) (70) So, finally, the transfonned tonsolidation equation can be written (71) (kl y ",)(d"-Uldz 2) = 6g(S)[Sii - PJ and mung (k{y"u) _ t. this becomes [t.{g(s)J(d 2tildz 2 ) .. sa-p (72) Diuussjl)lI of tile cQnSolidalirm equation by use of an example The expression (72) is a differential equation for a(z, 5), whkh can be solved for partkular boundary conditions. While li(z, s) is solved as a fWlttion of z, the other quantities in_ duding s remain wnstant. Therefore the fador [kIY,.ag(s)] in equation (71) is a tonstant

toeffident at this stage, and actually it plays the role of the tonsolidation coefficient c. of the nonnal consolidation theory. TIlerefore (kIY..a) was indicated by t, in equation (72),

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CONSO L IDATION MODE L S

215

and so the consolidation factor there is (i.fg(s)] . This factor excepted, equation (72) is identical to the Laplace transform of the Tenaghi consolidation equation. The complete secular effect is incorporated in the coefficient of consolidation, because this coefficient contains the function g(s), which introduces the retardation mechanism as is seen from its defini_ tion (67). The value of g(s) is obtained direc tly from test results in the form (69). Solution of t he geometric part of the problem requires no specification of g{s) because s is inert at t hat stage. Only at the end, when the solution of Ii(z, s) is inverted to obtain the excess pore pressure u(z, t) as a function of t, is the form g(s) required, This means that aU known transformed solutions of boundary value problems in normal consolidation can be used. Only in the last operation of inversion will the result deviate from the known results. How this works out will be shown by a one dimensional example of a layer of thickness H, resting on an impermeable base at z=O and loaded at t=O with a pressure p. The transformed boundary conditions are ii ... 0,

::= H

du j dz .." 0,

The solution of equation (72) gives -( ) = 1!. u :, s s

[1- sinh cosh (tl:) ] (qz)

with

q ... (sg(s)/l.]'/ll

(74)

The settlement is ~= Sit u'dz which, using equations (70) and (73) becomes Us) - (paM)g(s) tanh (qH) This solution of the transformed setUement is identical to the solution obtained in the nonnal case. The fmal step is to construct from e{s) the settlement {(t) by use of the inversion integral {{tl ... (pa/2ni)

L~::" [g(s)fs4J tanh (rjff) exp (st)as

(76)

Evaluation is effected by detennining the location of t he poles of the integrand. Besides the branch point at so"O these are given by rj." tF..(2tn- l ), m ... I -+co (77) so that the poles aU lie on the imaginary rj axis. Up to this point the analysis is identical to the procedure followed in a normal consolidation problem. and solutions that are worked out along these Jines can be used withou t any revisions. The difference comcs out in the transition from q to s. In the consolidation theory wi thout secular effects g(s) is IUlity and according to equation (74) q is (sQ» ../k)1/~. a function which contains s only as a square root. Therefore t he imaginary q·axis is mapped into the negative s·axis and· aU poles lie on the negative s-axis. In the case of a secular effect, g{s) is some complicated function approximated by the simpler expressions (69). Therefore the relation (74) between rj and s is an intricate function which maps the im· aginary rj-axis on to two undulating lines symmetric to the negative s·axis. These lines, indicated as F ig. 8. IntegraUou path, in oomplu; dotted Jines in Fig. S, carry the poles. , pI"",.,

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Instead of integrating from c-io:> to '+io:> along thtl straight line AB, it is more convenient to shift the integration path to ACDEFB. The parts AC and FB give no contribution, but there remains the hairpin around the negative s-axis and the residues of the poles on the undulating dotted lines. There arc no other singularities in the region swept by the shifting of the integration path because it can be shown that the function g(s), as obtained from the time settlement observations, behaves regularly up to the line CDEF. The solution then becomes t(t) = (p,j/2:ri)

f

[g(s)/s4J tanh (q1l) exp (s/jds

CD£F

8 · -paR ~ ...

'1:

1

I {C(s.) exp (s ·t) g{s-) cxp (s- t) } (2m_ I)2 I +s+[g'(s+)/g(s+)] + I +s -[g'{s-) /g(s-))

with

s+g{s+) = l.[+li7t(2m-l)J2fH2 rg(r) _ 1.[ _F. . (2m_I))2/R2 g'(s) '"'" dg(s) /ds

I n the normal solution g(s) _ I , then from the integral over CDEF in equation (78) only the circle around the branch point in s~O survives to give 2mH, bccau!;C the integrand becomes equal on both sides of the negative s-axis, CD and EF. The summation over the residues of the poles finally gives the well·known solu tion

C(t)

(2m-

8 · I [ - -2-li I " )' ... ]} _pm { 1 -,";i,,~,(2m_l)2exp

(79)

The solution (78) can be evaluated numerically after separating the real and imaginary parts of s· and s-; the result is not shown here. An impression of t he structure of t(t) can be obtained conveniently by considering the solutions for large and small t. These solutions follow directly from approximations of equation (75) for small and large s respectively and taking t he inverse of these approximations. This yields: Approxjmaljrmjor large t is obtained by developing t anh (qll) from equation (75) in a series ex pression for small values of qH, which, because of equation (74) corresponds to small values of s. The transformed settlement then becomes t(s) z (pdM)g(s)[(qII)-i(qH)3+ . . . ] (SO) T he first term between brackets gives pdHs-'g(s). Th is has the fonn of equation (68) and represents t he transformed settlement as created by an effective stress of magni tude p, bccau!;C the transfoml of p is PIs. Its inversion gives the settlement time relation, when excess pore pressure has vanished, as explained in the past sub-section. F rom the second term can be deduced when excess pore pressure has diminished enough to permit disregarding its influence, by considering the value of s that makes the second t erm small with respect to the first. Approximation fqr smaU t is obtained by developing tanh (4Jl) from equation (75) in an expression valid for large values of qH. This expression is [1- 2 exp (-2qH)]. So retaining only the first term, whieh is equal to unity, the transformed settlement in t he first approximation becomes (81)

This can beevaluated by takingg(s) from equation (69). Considering the region 1/0 1 «5 < 0:1 Us) z [t. r( l +0I)JP40o"]'12r{3h)l~

The inverse of t his is (82)

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217

According to this solution the settlement for small t follows a power time law with t to the power (! + tal. In the spherical compression test it was observed that the power of t was less than a half. 50 matching of test results requires a negative value of a, i.e. a positive power of sin g(s). The mathematical implications of this requirement cannot be explained here in detail, but from the theory of 5tieltjes transfonns it can be deduced that no physically acceptable frequency function mijt) exists which introduced in equation (67) produces a positive power of s in g(s). This incompatibility of theory and test results indicates that the present model of spring dashpot elements immersed ill pore water is illca]lable of reproducing reality. COllclusion

The basic equation of consolidation in its L1.place transfonn for a spring dashpot assembly is identical to the transfonn of the original Terzaghi equation, except the coefficient of consolidation, which contains the compressibility ages) as a function of s. Solution of boundary value problems with secular effects therefore follows the same lines as for consolidation without secular effect as tong as only geometry is involved. Only the final inversion of the result is more elaborate because of ages). For large values of t the solution reduces to the case when the load is carried entirely by effective stresses, as treated in the third section. For small values of t the solution reduces to a fonn which contradicts test results, and, therefore, indicates that the present model of spring dashpot clements immersed in pore water is not entirely acceptable. In the follo,\ing soction a model will be treated that consists of cavities interconnected by channels. For this mode! the transformed basic equation 01 consolidation is shown to be slightly different, in so far as the permeability also is a function of s. This second function of s opens the possibility to choose frequency functions for the physical entities involved in a manlier such that for smalll the settlement also will conform to the test results. CONSOLIDATI ON OF STOCHASTIC ASSEMBLY 01' CAVITY CHANNEL ELEMENTS

The second secular mechanism proposed by Keverling Buisman (1938) consists of a network of compressible pores interconnected by channels. Tn tht! following "n"lysis the stochastic character of such a system will again be introduced by considering the cavities to possess different compressibilities and the channels' different conductivities. The difficulty now arises that the pore pressure itself becomes a stochastic variable, because every pore has its own pore pressure. This difficulty is overcome by introducing an average pore pressure, which is not a physical quantity that can be detennincd in a test, but a convenient mathematical device that permits the continued use of the classical theory of consolidation. For further treatment of consolidation problems the time dependence of compressibility and conductivity are involved. PhysicallllechanisnI of cavity chanllel elem ellt

I n a system of cavities interconnected by channels, every cavity is connected by a number of.channels to cavities surrounding it (Fig. 9). Consider a sample cavity with a number k. When finally agrcat number, N, of cavities is considered, a summation is effected where k runs from 1 to N. The compressibility of the kth cavity is called Pk; this means that the cavity expels a volume Pk of fluid when there is a unit pressure decrease of fluid pressure Uk in this cavity. From the kth cavit y v channels emanate and these channels connect the kth cavity to surrounding cavities numbered m. I n a summation m wilt run from I to v, v having an order 01 magnitude of 10 in a t hree dimensional channel network,

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Cavity ehaD.Dei network coruout.i.ng 01 compre._ alble porn iJ:>tercow>&cwd ra.ndomly

Fig. 9.

The conductivity of a channel connecting the kth cavity to an mth cavity will be denoted by This means that a volume of fluid Aho will flow per unit time t hrough this channel if the pressure difference of the fluids in these cavities (u. -u ..l is unity. If the fluid pressure in the kth cavity changes by an amount d,jk ",, (8u./iJljdt during dI.,

"Ie...

the volume of fluid expelled from the cavity during that time is dV/< _ -podu. - -plc(ou.til/j dt (83) The volume of fluid t hat can be discharged through the surrounding channels during it is,

dV..

~

.

.-. • .-. !

["/c .. (,,/c -u .. ))dt

If there is no gas in the fluid, continuity requires that dV /c=dV., p/c(auk!Ot) =

L:

(84)

This gives

Ak. (U .. -U k}

(85)

In this expression tikand u., can be considered to represent the excess pore pressure. Let the initial excess pore pressure be the same for the different pores, and equal to p. Then the Laplace transfonn of equation (85) is p_(suk-P) -

•

.-. ~

(86)

"k .. (U .. -Uk)

Solving for Uk, the transformed excess pore pressure, yields uk[s+( l/pk)

f

.. . I

"k..] =P+(l IPk)

i ("_ . 11..)

.. _ I

I n this equation two summations 01 different character appear. notation the following abbreviations are introduced

. "h.

(87)

In order to simplify the

P.k ..

(I IPk) ~

(88)

4, -

[.~." .'.l / [H

("')

T hese two quan tities have a physical meaning. The character of P.k' as defined by equation (88), is understood by considering t be case when the kth cavity has an excess pore pressure p at time 1_ 0, and all the surrounding cavities are maintained at zero excess pore pressure. Then equation (87) reduces t o 11k --

P!(S+P.k)

with the inverse "k - pexp(-p.J)

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(90)

125

219

CONSOLIDATION MO DELS

From this result it is seen that ,...~ is a relaxation parameter similar to ,... introduced for the spring dashpot elements. It represents the relaxation occurring in the kth cavity, during the decrease of excess pore pressure by discharge of fluid from that cavity towards its surroundings. The character of 12k defmed by eq uation (89) is a pore pressure experienced in the kth cavity as a weighted average of the pore pressures in the surrounding cavities. The weighting {unc· tion is ),h" the conductivity of the connecting channels. The quan tity 12k as defmed by equation (89) will be called the ambienl excess pon presSlire. Substituting equations (88) and (89) in (87) gives (91) This expression indicates the relation between the transformed excess pore pressure in the kth cavity, al<. and the tmnsfonned ambient excess pore pressure, dl<' which is a weighted average of the excess pore pressures in the adjacent cavities. SdJkmeNt of a s~chastic assembly Qf cavi/i/!$ aNd chaN~ls The expression (9 1) permits determination of the settlement of a layer that is composed of an assembly of cavities. It will be assumed that the entire settlement consists of the reduction in volume of the cavities, and that the volume reduction,dV k of the kth cavity is equal to its compressibility Pk multiplied by the decrease in excess pore pressure (P-u k ) after the moment of loading, SO that (92) The number of cavities, present in a volume L3 will be N', with N a large number and L a length as defined in the sub-section on stochastic distribution of elements. These N3 cavities will generat e a volume reduction equal to LlV =

.-, . "'L:

Pk{P-U k)

(93)

If N is large, a number N cavities gives a fair representation of the average, and so a summation over N3 elements is equal to Ni times the summation over N elements. Therefore equation (93) can be written as LlV _ N2

" L:

-,

Pk(P-U k )

(94)

Consider now a layer of small thickness LIz. This layer covers an area Ll A = L'/Llz . if its volume is £3. When this layer is compressed in a manner such that lateral expansion is prohibited, the height reduction LI~ of the layer is

i'

= LlVILl A

=

or, using equation (94) the settlement is LI ~

- Llz(N2/P)

il V(Llz/P)

.-",

Pk{P-U.)

.-" ,

Ph[(P/S) -uJ

L:

Transforming this with the Laplace operation gives [(PIs) -Uk] for fonned settlement LI ~(s) for a layer of thickness LIz is LI~(s} = Ll z(Ni/P)

L:

If now the excess pore pressure in the kth cavity pressure 12k using equation (91) there results

u~

(P-u~) .

So the t rans-

is replaced by its ambient excess pore

L: --11k ] N'"'" [P

,d {(s) .. LI z .

--L.!. L~ . lS+""kS

In this summation (Pis) is a constant, independent of k . because it was stated that the excess pore pressurc at the moment of loading. t = O, is equal for all cavities. So (PIs) can be

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G . DE J OSSELIN DE l ONG

t aken out of the summation, and the transformed settlement becomes

..1~(s) =- .dz N:

[(t. i

P~/L~ ) _(

i Pki"k 11k)]

(95)

L $.1: _ ,$+l"k k _ ,S+P.k The ambient excess pore pressure 12k is different for every .:;avity. because it is a weighted average of the surrounding excess pore pressures, with the conductivity of the connecting ehannels as weighting function. :Make the assumption now that the correlation between 12k and the properties Pi:. 1'-.1: of the particular cavity can be disregarded. It is not obvious that this is a reasonable assumption, since the conductivity of the ad· jacent channels Aka is incorporated in both d k and 1".1: (see equations (88) and (89)). That the correlation will be weak follov.'S from the fact that the excess pore pressure Ii.. in the adjacent cavities is practkally independent of the conductivity of the connecting channel ,\~ .. because t hat channel being only one of the ~ channels emanating from the mth cavity contributes only a portion ( l f~) in the value of u,.. So if.. can be considered nearly random with respect to the kth cavity. Now rearrange the second summation in equation (95) such t hat all cavities with practically the same value for [p''''k/(s+ JL~)] are t aken together, so that they fonn K subgroups, each containing iJN cavities. Let iJN still be a large number such tha t the 6 k of a subgroup represents a fair sample of the total, which is possible because there is no correlation between 4% and Pk or f'k . Then the summation of these iJN values for 6k is iJ N t imes the average value 12, if the average ambient exGess pOI'e pressllfe 12 is defined as

12 _ (l iN)

,

.- . ~

11k

(96)

According to the definition of the volume P in t he section on stochastic distribution of elements the stress condition in the layer with thickness iJz is nearly homogeneous. This means that the excess pore pressure in a continuous reference soil subjected to the same loading as the cavity channel assembly would have been practically a constant throughout this volume. The averaging procedure, applied to obtain 12, smoothes out the difference in excess pore pressures encountered in the individual pores, but does not obliterate pore pressure differences caused by consolidat ion. It is now possible to extract the constant 11 from t he last summation t enn in equation (95), because it can be removed from every subgroup. All subgroups give t he same 12 because of the lack of correlation mentioned. The result is

5:

iJt(S} ... .:IzN: (t_4) [ P.JLk ] L s ~ * lS+"'k

(97)

It may be remarked from this analysis that in general the summation over a product of two stochastic variables is equal to the average of one of these variables multiplied by the summation over the other, if there is no correlation between t hem. This property is used in t he next sub~sections. Similarity btlwun set/lemenl of cavity channel assembly lind ~ring riashPOl system There is a great similarity between expression (97) for the transformed settlement of a cavity channel assembly and equation {65}, the transfonned settlement of a spring dashpot system whose ret ardat ion function obeys equation {4}. The similarit y appears by altering the height h of the block into the thickness.:l: of the layer, and changing PIs into [{Pfs}-d]A similar change was made to obtain equation (66) from equation (65) where, however, instead of [(PJ$) -d] the effective stress ,,'(s) was substituted. Since ,,' = (PIs) - Ii (see equation (70)) the similarity is complete if 12 can be identified with u. For mathematical convenience this identification will be made, but this should not be done though.tlessly, because th.ere is a fundamen t al physical difference between d and u. The quantity if stands for an excess pore

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CONSO LI DA T ION MODE L S

22 \

pressure acting inside a grain skeleton with spring dashpot retardation at the grain contact points. This pore pressure is a deterministic quantity, whose computed value is the physical pore pressure prescnt at the point of consideration. This pore pressure 1i does not interfere with the secular retardation mechanism. On the other hand in a cavity channel system the pressures in the individual pores are stochastic. They may vary substantially between adjacent pores, and therefore it is impossible to detennine the pore pressure at a point. Only a mathematical quantity 4 can enter into the computations as an average ambieut excess pore pressure. This is the average of a complicated quantity, the ambient excess pore pressure, which interferes with the secular mechanism, and is also a stochastic variable. This means that it is not possible to assert the exact magnitude of it at a point, but only the average of the values that might occur there, weighted according to their frequency of occurrence. The definition of the ambient excess pore pressure as given in the sub-section on physical mechanism of cavity-channel element indicates that 4 is an average over many cavities, in the same way as a pore pressure measured by a pierometer is an average. Such a device is usually large with respect to the individual pores, and therefore it cannot determine the pressure in a particular pore but senses some average of the pressures in the surrounding pores. It is plausible to assume that t his average is weighted according to the conductivity of the connecting channels. A similar weighting procedure was obtained for 4k , the ambient excess pore pressure. Since a piezometer is connected to many pores it senses some average; this average is somewhat different from 4 but may be related to it. This complicated notion of an average ambient excess pore pressure is also reflect ed in the effective stress and hence in the transfonned effective stress in the cavity channel syst em. Since this quantity will also involve an averaging it will be denoted by 6', and be defined as 6'(s) ~ (PIs) -4(s)

(98)

A physical meaning is not assigned to 6', bu t it is used as a mathematical entity which t akes the place of ii' in the sub-section on basic equation of consolidat ion for ret arded elements immersed in pore water. Since the similarity has now become complete the transition from equation (65) to equation (68) can be used and applied to equation (97) to obtain At(s) ~ 6'(s)dg(s)Az

with dg(s) _ (N2fP) k

i~

L

Pd'
_ d

(99)

roo &<m(f.)f(s+J.<))dJ.< Jo

(100)

Here /I has the same meaning as given by equation (14), namely an average compressibility of the cavities. The frequency funct ion m(.u) and the retardation parameter,.. also have the same meaning. This justifies the conclusion that it is not possible to decide from test results in the secondary period of consolidation which model gives a better description of a soil behaviour: the spring dashpot model or the cavity channel model. Both give a similar expression for the transformed compressibilit y dg(s) . Basic equaJion of consolidation for a stochastic assembly of cavities and channels

So far the settlement has been expressed in a fonn which accounts for the st ochastic and time dependent compressibility if the effective stress 6' = {PM -II is known. Since the load P is given, it remains t o detennine 4, the average ambient excess pore pressure. This can be done with the nonnal procedures used in the theory of consolidation, because, as is shown, the quantity 4 obeys a partial differential equation which is similar in structure to the basic equation governing Terzaghi's consolidation theory. After the Laplace tmnsfonnation, the basic equation of consolidation cont ains only

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G. D E JOSSELIN DE J O NG

Fig. 10.

Tbe cavity If ie the eentra ol • .urrow>d.lJ!.jf cavities, label_

ledm. Eve .. y cavUym 18 again the centre 01 • _""rounding canUn , labelled n. The num_ b e r .....y vary with the cavi_

ti..... That I
derivatives with respect to the space oo-ordinates, and in the one dimensional case it has the fo~

(101 )

(kfy .. )(tPu/dz"j _ a(su-pJ

A comparable equation is obtained by starting from equation (86) which relat es the t rnnsfonned excess pore pressure, Uk. in the kth cavity to il." the transformed excess pore pressures in the surrounding cavities, labelled m. This equation is

•

.-.!

"k .. (U .. -U k )

=

(102)

p,,(su,,-P)

T he space co-ordina te will come in by considering the location of the mlh cavity, which is one 01 the set of ~ cavities surrounding the kth. Every mlh cavi ty is in tum the centre of its own set of cavities, labelled n, which surround it and are connected to it by channels with conductivity " ..K (see Fig. 10). The mth cavity then has an ambient excess pore pressure 11.. which has adelinition analogous to equation (89) and is related tou .. by an equation analogous to equation (91), being (103) By considering the orient ation in space of the cen tres of these ambient excess pore pressures, 11.. , with respect to the kth cavity, the space derivative of the ambient excess pore pressures can be introduced. T he establishment of the basic equation of oonsolidation req uires two steps. The first is to rearrange equation (102) in a manner such that the right han d side becomes oompa rable to the right hand side of eq uation (101 ). The second step is obtained by rearranging the left hand sides. The first sup starts by rearranging equation (102) in a manner such that a relation between the ambient excess pore pressures 11~ and if.. is obtained, instead of the relation between -a~ and -a .. , the actual pressures in the pores. Introducing equations (9 1) and (103) for u~ and a.. in (102) yields

Pi. "+"k" + ~ ..

~

JL..

'"

"h. ,"+..11.. _ ,+pI'k..L ,,___ •!,"+kllk/-'k ~.. 1' ..

'\1< .. =

~

(sa~ -P)[P~l'k{(S + I'~)]

Rearrangement gives, with equation (88),

~ .1.·..1''' 11 .. _11. L AkA = -p ~ ,\.... .. $+/-,..

..

which can be simplified by adding the factor sd. is then

..

s+l' ..

L (.I.k..{s+ I' ..) to both sides.

~ ['\k.,JL.,.(d.. -ifk)l _ (sI1~-P)L

.. s+ /-'..

"... .. s+fL ..

The result (104)

If an equation of the form (101 ) is to be obtained, the right hand side should oontain the eompressibility, which in the cavity channel case is ag(s) (see equation (100)). In order to produce this here, it is nc£C5Sary to bring equation (\04) into a shape which permits mani-

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CONSOLIDATION MODELS

pulation aswasdonewith equation (95). This requires multiplication of both sides by a factor

(~" · )/(L~) to yi,ld S+f'k

.. s+ f' ..

(105)

Summing over the volume L3 which contains N3 cavities gives, per unit volume, on the right hand side N'

[i (~ ',o".)_p i (~',o)]

L3 k_'

5+1-'~

~_ ,

S+f'k

The similarity of this expression with the greater part of equation (95), (if equation (88) is used), shows that the analysis of the past two sub-sections is applicable to it up to equation (100). So (N~/P) times the summation over k from I to N brings the right hand side of equation (lOS) into the form ag(s) [s6-PJ. That summation applied to the entire equation (105) finally gives

N{'.L

N:.L "".,J.40(lI,._lI k ) L ~ _ , ... ,5+,...

i ' j'.L ~J~ ...,5+1-'

5+f'k

= dg(s)[sd-Pl

(106)

The second sltp is to show that the left hand side of equation (106) represents something !iimilar to Ihe left hand side of equation (101), i.e. the SC«lnd derivative of d multiplied by the permeability. In order to obtain this result usc will be made of the property that the sum of a product of two non-correlated stochastic variables is equal to the product of the average over one of them and the sum of the other. Since it.\: and 11.. are not correlated to >'.., 14k or I'.. it is permissible to extract the average of (d .. -d k ) from the summation over .. and N on the left hand side of equation (106) . In order to bring in th e space cO-Qrdinate we introduce the centre of gravity of the volume oc<:upied by the N3 elements as the location to which the average ambient excess pore pressure is referred. In a soil body, where a one dimensional consolidation process is occurring in the z direction, this location is indicated by its z co-ordinate. The assumption will be made, that the average ambient excess pore pressure 4(z) is a gradually varying function of the co-ordinate z. Then, in view of the one dimensional character of the consolidation process the average ambient excess pore pressure in the neighbourhood of Zo can be expressed by its Taylor expansion in z-direction only, as d(:) _ d(zo) + (:-zo)(dd/dz) + i(z-zo)2(tP6fdz2 + .. .)

(107)

Let 20 be the centre of gravity of the volume occupied by the set of N3 cavities, labelled k. From this set select a subset of cavities which possess adjacent cavities", connected by channels of a length between I and 1+.11, and orientated with respect to the z direction at an a,ngle between" and .,+.1., (see Fig. ll ). The centre of gravity of the mth cavities belonging to this subset lies at a dista.nce 1 cos., above Zo. If tlus subset contains enough cavities, the average ambient excess pore pressure of the corresponding mth cavities is representative for the d at that height. So the average of (d .. -4~) for this subset is with equation (107) subset average (d .. -4~) .. 6(zo+1 cos of) -4(zo)

- (I cos .,)(l/4fdz) +W cos

130

of)~(d2d/d22+

. . _)

(108)

Soil Mechanics and Transport in Porous Media

G. DE lOSSELIN DE lONG

,

Fig. U . FJ-on> tho •• t of C. vitiO. COrultitUtlng a .... p .... HI1t.tive elOUlentary volum., .. eubeot i1I ••lected. lor 'Which tho t:!Utanee betweOJl tho cavitin ,. and. m i. I in a dir
Averaging over all the subsets means averaging over (I 005 ,,) and (I cos y,)2, since (dd/dz) and (d2-dldz 2) are fixed quantities at the height Zo. In this averaging procedure it is assumed that I and" are not correlated and that the soil is isotropic so that the distribution of the channels is homogeneous with respect to direction. By this assum ption the average of (l oos y,) is the average of I multiplied by the average of (cos ,,). This is zero for COS" is an odd function if y, ranges from 0 to.... So the teon with (dd/dz) in equat ion (l OS) vanishes by taking the average of the subsets. The tenn with (d 2d/dz 2 ) does not vanish, because the average of (cos2 y,) over a sphere is (1/411")

I

cos

2 "

2... sin

"d"

=

!

(109)

If 12 denotes the average of /2 the last teon of (l OS) becomes (1/6)12(tP4/dz 2) byaveraging and this is the average of (d .. -d k ) over the vN cavities of the summation in equation (106). Therefore equation (106) can be written as 1 tPdfYJ'II 61~"d"2 p L z k_l

{~[,\","f'.J(S+IL-)] }:"'-} ~ -"-- L [.\b/(Hf'.J]

•

ag(s)[sd-P]

(110)

(S+f'k)

The part of the equation in the braces is a function of s whose dimensions are those of .\, because sand f' both have the dimension (I/time). Since.\ is a conductivity with dimension [length3/force x time] multiplication by (12/P) brings the dimension to [length' / force x time] which is the same as the dimensions of the penneability tenn (kJy ..) in equation (101). Therefore, by analogy with the time dependent compressibility ag(s) introduced in equation (68), we can introduce here a time dependent penneability (k/y .. )f(s), such that

kiN'

I<

{~['\k"f'..!{S+Jt.. )] i. .\... }

- f(s) = - l'a L ~ -"-y.. 6 L k - 1 L ['\k,.f(s+,.,.)] (5 +Jtt)

(1Il)

"

where the time dependence is incorporated in f(s). Substituting this in equation (110) gives finally (kJy,.)f(s)[d2t2/d2f<J = ag($)[$4-pJ (11 2) If again use is made of a coefficient of consolidation t. = kldy.. (1 13) This equation becomes t.[J(s)!g(s)]{tP12ldr) _ s4-p (1 14) Equations (1 12) and (114) have the same structure as the classical consolidation equation (101).

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CONSO LI DATION MODELS

Solution of boundary value problems alld the consequetu:e of the time dependent permeability The settlement equation and the basic equation of consolidation for a cavity channel network (equations (99) and (112)) have the same st ructure as the corresponding equations for a spring dashpot assembly (equations (68) and (7 1)) as well as for the classical theory of consolidation without secular effects. The difference from the equations for spring dashpots is in the penneability, which for the cavity channel network is a function of 5, being (f/y,.)f(s). Solution of boundary value problems follows therefore the same Lines as indicated in t he sub-section on discussion of the consolidation equation. There it was mentioned that the geometric part of the problem remains identical to the solution for nonnal consolidation without secular effects. Only in the last operation of inversion will the result deviate from the known solutions. Because the basic equations are similar, so also most of the results in equations (73) to (82) developed for the solution of the boundary value problem pertaining to a layer of thickness H, will be valid here. Apart from a few differences that are described below, the settlement for sman I will be considered because the functionf(s) in the penneability will enable the fitting of test results at the beginning of consolidation. . The difference in the solution is primarily in 4, which instead of equation (74) will for a cavity channel system be defined as q .. [sg(s)lf(s)t'.)11l1 (lIS) This entails a few changes in equation (7S). in t he summation become

The denominators of the tcnns between braces ~d

I

s-! [g'{S-) j'(S-) ]

+ : g{r)

f(r)

with

s*g(s*)If(s*) = (t'.IH2)[±F1r(2m_I)]2 For small t and large s the solution is the same as the first of (Sl ), which by uS(! of from equation (11 5) gives

Us) ::::: (p4fs1j)g(s) = If againg(s) is taken from (69) with 1/8 1

Pd["~f{5)g(S)J1t2S-3/~

4

(lIG)

« s < co the result is (1I7)

Test results show that the settlement for small t follows a law of the fonn to-·)/2, when f is positive. The transfonned settlement is then a function of s of the form S-(3-.)/~ for large s. For this f{s) must be a function which contains s in the form S·H. Sincef(s) is already defined by the summatioll ( III ), it remains to establish that the summation can provide a function with 5 in the fonn saH. Since the fonn of equation (III) is rather involved, it is neither obvious from its structure that this possibility exists, nor is it easy to prove. Only the outlines are t raced of a proof that was used to verify that f(s) indeed ;;atisfies the requirements. The summation is over many kth cavities, and contains a quotient of two summations over the mt h cavities that surround the kth cavities. If it is assumed that there is no correlat ion between the cavities k and m, the property of a product of non-corre.lated stochastic variables is used to write the right hand side of equation (I l l) as

(liS)

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G. DE JOSSELt!ol DE JO!olG

where use is made of equation (88). From equation (100) we see that the last summation is
•

~.\b

..

I (.\"-'/JL ..)

C

0_1

+s

~

--,-

p

S

(119)

+ p..

The fin;t t enn in (119) can be shown to be small with respect to t he second for those values of s which correspond to the beginning of the consolidation process as follows. In the summation of (.\~,"/p..,) the terms with small p. .. dominate, and this means that the first tenn i~ of thr. order of " timAA the ~mallest v~llle of /'... The smaller values of p. produce t he end value of the settlement and are reciprocal to the time that the final value sets in. Since also S is the reciprocal of the time considered, the fin;t term of (1 19) is small compared with the second, if the end value is reached at a time large compared with the time when the consolidation process is at its starting period. So, finally, the average of the quotient in the brace of equat ion (118) equals approximately (v-I) times the average of sC./ (s+I-'.) over many values of 1-'.. Since the p.. lie in between the actual values 1-',., t heir distribution is a fair representation of the actual distribution of the retardation parameters. Averagi ng over many cavities suggests the introduction of an integral of the form s

L"

[C(I-'l/(s+p.l]ap.

( 12Q)

This integral shonld yield a function of s of the form s2a +<, and this is possible since the factor s preceding it requires that the integral itself leads to a function s-IHaH. Since both a and ~ arc of the order 0·1 in the tests observed, this means that s is raised to a negative power. As is mentioned above, the use of the t heory of Stieltjes transfonns t o ohtain a solution for C(p.) requires a negative power of s. The power (-1 + 2a+f) is therefore in II/,'Teement with the requiremen ts of the theory and t he solution gives for C(I'} a function of the form 1'-1+2«+< .

It is too long to develop here the relation between .\k" and 1-'.. wh ich wil! give this C(I-'l. It turns out to be possible to find a suitable relation, but as it is complicated its presentation is omitted here. The purpose of est ablishingC(JL) was to see whether this theory is n realistic possihility, and it turned out to be so. ConclusiOIl

The basic equation of consolidation in the form of its Laplace translonn for a cavity channel network is identical to the transfonn of the original Terzaghi equation, exccpt that the coefficient of consolidation is itself a function of s, containing as componcnt parts the compres-

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CONSOLIDATION MODELS

227

sibility and the penneability equal respectively to ages) and (Ic/r.,)j(s). Solutions to boundary value problems with secular effects therefore follow the same lines as for consolidation wi thout secular effect as long as only geomet ry is involved. Only t he final inversion of the result is more elaborate because of the tenn (Ic{ay",)[f(s){g(s)). At the geometric stage of the solution, when integration is performed with respect to the co-ordinates, the results are identical to the solutions for normal consolidation. Solutions to Biot 's t hree dimensional equations can also be used without modification. For large va.lu~ of time (t) the wlution reduces to the cal>C when 6' is practically equal to the load p, the situation treated in the t hird section for t he spring dashpot assembly. During this secondary period of consolidation the average ambient excess pore pressure is practically UfO, but nevertheless in a certain number of cavities, connected by badly conducting clmnnels to the rest of the network, the excess pore pressure may still be appreciable. Release of water from these badly connected cavities gives the long term settlement in the secondary period of consolidation. During that period the cavity cllannel model behaves iden tically with the spring dashpot model, and t herefore the frequency distribution of the retardation parameters of the constituent elements is the same for both models. For small values of time (t) the solution reduces to a form which can be made to match test results by a proper choice of t he distribution of conductivities of the channels. This possibility. of arranging the cavity channel model in a manner such tlmt it reproduces observed settlement behaviour of soils. indicates t hat the model contains the required properties. I t is not a verification though that the model is t he correct description of the physical mechanism at work. Other mechanisms may lead to identical effects, in the same way as the cavity channel network produced an identical behaviour to the spring dashpot assembly in the secondary period. The adoption of any mechanism can only be assumed to reflect reality if it has been established that the physics of t he microstruct ure produces t hat particular mechanism, It is sunnised t hat in a consolidating soil the secular mechanism is basically of the same kind as t he spring dashpot and cavity channel systems examined. This may be the case as well in the microstructure consisting of t he assembly of individual particles as on the macroscale, when cracks and channels run through a soil mass. Therefore the concepts developed in this study may serve t he fut ure study of micro- and macrost ructure of soils. ACKNOWLEDGEMENT

The Anthor is grateful for the help given by Professor R. E. Gibson in shaping the t heory and the manuscript, by Ir H. W. Hoogstraten in considering the properties of certain integral t ransforms and by I r A. Verruijt in overcoming difficulties at all stages of t his work. The basic ideas of this study were presented in a lecture in February 1966 at King's College, London. REFERENCES L. (1965). Consolidation 01 clay with non·line.lr vi$l;osity . GJ()I~chniq ... IS, N". 4, 345---362. DE l°S.EL',.. DE lm,o;;, G. (1942), J(aflaa)stel$1;lhypothcse . Report. Lab, Grond. Mech. Delft, K 9153 . liE 10'SEL'N 11& 10><0:: . C. & V ERRUOT, A. (1965). Primary and 5CCondary consolidation 01 a spherical clay &a.mple. Pree. 61h Inl. Con/- S",I Mech., Montrnal, I, 254- 258. G'BSON, R. E. &: Lo, K. Y. (1961). A IIltOty of consolidaHon for &",1$ uJai/)iling &~ccndary cM"preuion . Norwegian Geotechnkal Inst, Puh!.. X o. 4l. KZVKRLlNG Uu'sw ....". A. S. (1936). Re.ultsoflongd"rntion settlement tests. Prc<. 2nd InI.Co"f.S"'1 Meeh., Camllridge, U.S ,A., I. 103-105. KBVZMLlNG BU'S"AN . A. S. {193S}. Enkclc grepen "it de grondmechanica, en meer in het b ij.ondcr uit het zettingsproblwm. D~ J"g.nieu~. No. 32, U 133-155. KE"U~'NG BUISMAN, A. S. (1940). G .ond",ech~"ic~ . Delft: Waltman. OUlIN, H. W. (1960) . Hydraulic flow through &a.tura.ted clays. 9JI> Nat". Clay CrmJ.. Puroue University. S CH'FPWA", R. L., LADD, C. C. &: CHEN. A. "f. F. (19tH). T he secondary consolidation of clay. Proc . Sy...p. Rluol0lY Soil Med .• Grenoble, 273-298. BAR OEN,

134

Soil Mechanics and Transport in Porous Media

228

G. DE J OSSEL1N DE JONG

T.o.N. T. K. j I957). Ouderwekinge" over de rheologi5che eigel13Chappen van kle' . (Investigations on the rheolog.cal properties oj clay.) Dr. thesis Excelsior, ',Crnvcnhage . T AYLOR, D. W. a: MaRC"A!
Some functions used in the text and their approximations fDl" large and amall values of their argument

are given. Cmnpl~"

I'''''''''' jwnc/iI)N: 11"') r(.-) ..

t . A--'

exp(-.l.)d A

»

0

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1'{x,y}-

f."

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(122) (123)

x)

).--'exp(-.l.)d.\

(IU)

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.F.(p;q;~)

F ,y., •.•. " ••

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( 121)

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(125)

'"'' (127)

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I'

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Selected Works of G. de Josselin de Jong

135

[Reprinted from The Journal of The Americau Ceramic Society, Vol. 40, Nu. 2.

February, 19.37.1

Verification of Use of Peak Area for the Quantitative Differential Thermal Analysis by G. de Josselin de Jong Research Department, Soil Mechanics Laboratory, Delft, Netherlands

The amount of reacting material in a sample investigated by differential thermal analysis can be determined from the peak area according to the Boersma equation, which accounts for heat flow through sample and thermocouple wires. This relation has been checked by experiment and it has been found that the use of different thermocouples for the caHbrations may lead to variations of about 30%. The values obtained for heat transfer through a sample and thermocouple have been checked by comparison of computations and observation of base-line offset at the beginning of a run and exponential decay of the amount of heat dissipating out of a sample. It is shown that according to the equation it is not the total amount of reacting material that determines the peak area but merely the density of the material near the thermocouple. The sensitivity of the 'method for quantitative analysis is discussed in relation to the possible variations in the factors involved, namely, density and heat conductivity of the sample and heat transfer through the thermocouple.

I.

Verification of this relation becomes possible if t.he heat dissipation occurring during a test is analyzed and if equations are available which describe this process explicitly. The first theory was advanced by SpeiJ,7 who deduced that Peak area =

if;

Introduction

T

(J

= w~ll/ gA

(1)

=

peak area = !!!. w g'

In the derivation of this equation the writer assumes that g has a constant value throughout the whole process of heat dissipation, an assumption 110t consistent with reality. Moreover, the value of g is not given in its physical components and therefore the equation is still incomplete for verification purposes, although it already shows the influence of the heat conductivity, A, on the calibration valuc. The correct derivation of the calibration factor was given by Kro111g and Snoodijk. 8 They described in detail how the heat dissipates out of a sample, using the theory of heat con~ duction. From this, they determined the differential temperature rise originating in the center of the sample and brought about by the heat produced in various small zones within the sample; thesc individual contributions differ, depending on the place where they originate. By dissipation, the differential temperature win decreasc in the course of time and tend toward zero, giving an area in the temperature-time graph which contributes to the total peak area.

use of differential thermal analysis for quantitative measurements is of great importance and several workers havc studied its possibilities. In this paper the writer considers in particular the method in which the peak area is used to indicate the quantity of reacting material in a sample. In differential thermal analysis the temperature difference between an inert sample and the reacting material is recorded while thc sampIc holder which contains hoth sampIcs is heatcd at a constant rate. The peak area is the area enclosed by the base linc and the Cl1ryC of the differential temperature (0) as recorded versus time (t). ·When the heat of reaction begins after tl and dissipates hdorc t 2, the mathematical representation of the peak area is

f,""

h

By using this equation the unknown quantity of the heat of reaction per gram of sample, W, can be computed from the peak area if the factor .Af/ gX is known. This factor is in fact the calibration value of the test device, but for brevity here the calibration factor if is introduced and equation (1) is rewritten as follows:

HE

Peak area =

f, "e dt

w = heat of reaction per gram of sample (cal./gm.). M = total mass of sample (gm.). g = geometrical factor (cm.) which accounts for temperaturegradient distribution in sample. A = heat conductivity of sample (ca1./deg. cm. sec.).

dt

3

G. M. Schafer and M. B. Russell, "Thermal Method as a

Quantitative Measure of Clay Mineral Content," Soil Sci., 53 [51 31\.3-64 (1942); C,mm. Abs', .• 21 [9[ 199 (1942). 4 L. G. Berg, "Area Measurements in Thermograms for

having the dimension of dcg-. sec. That this peak arca should be related to the amount of heat of reaction released during a test was experimentally establishcd by Kracek et aZ.,! Alexander et aZ.,2 Schafer and Russell,3 Berg,4 Grimshaw and Roherts," and McLaughlin. 6

Quantitative Estimations and the Determination of Heats of Reaction," Compt. rend. acado sci. V.R.S.S., 49, 648-51 (1945) (in English). Ii (a) R. W. Grimshaw, E. Heaton, and A. L. Roberts, "Constitution of Refractory Clays: II, Thermal-Analysis Methods," Trans. Brit. Ceram. Soc., 44 [6J 76-92 (1945); Cera11t. Abstr., 1946, April, p. 66. (b) R. W. Grimshaw and A. L. Roberts, "Quantitative Determination of Some Minerals in Ceramic Materials by Thermal Means," Trans. Brit. Ceram. Soc., 52 11] 50-67 (1903); Ceram. Abstr., 1954, April, p. 78d. 6 R. J. McLaughlin, "Quantitative Mineralogical Analysis of Clay and Silt Fractions by Differential Thermal Analysis"; presented at meeting of the Mineralogical Society, London, March

Recei\'ed l\'larch 4. 1956; revised ('opY rt:ceived May 7, 1956. The author jj assistant director. Re:-;earch Department, Soil l\1echanks Labnratory. 1((1) F. C. Kracek, "Polymorphism of Potassium Kitrate," J. Phys. Chem., 34, 225-47 (i9~0). (b) R. W. Goranson and F. C. Kracek. "Experimental hIvestigation of Phase Relations of KzSi(09 L"nder Pressure," ibtd., 36 r3] 91:3-26 (1\):12); Cemm Abstr., 11 rEi] 325 (lfl:J2). 2 L. T. Alexander. S. R. Hendricks, and R. A. Kelson, "Minerals Present in Soil Colloids: II, Estimation in Some Representative Soils," Soil Sci., 48 [8J 273-79 (1939); Ceram. A bstr. ,

26.19.03.

7 Sidney Speil, "Applications of Thermal Analysis to Clays and Aluminous Minerals," V. S. Bur. Mines Rept. Invest., No.

3764.36 pp. (1944); Cuam. Ab.,"" 23 [111 ZOO (1944). 1\ R. Kronig and F. Snoodijk, "Determination of Heats of Transformation in Ceramic Materials," Appl. Sci. Research. A3 [1] 27 ·30 (1951); Ceram. Abstr .• 1953, January, p. 17a.

19 [7[ 173 (1940).

42

136

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_from 2lA tt> 1116.1 "". S_,ba!. IOu.d 20 ....... bleh ..... p<W:ticaD, tilt _ poU _ .bid! inofi<:,otos tIIa. it io ......... toto! _ , of tho raeliIq; _ to .... , """ u.. _~. 01 tho .... terial ~I ia • di* of wU' bciPl. T'\t;;o . . . Pft(Iioto&tions (2) and (. ) t.eoo .... ,he _pie qh. in tho oy~n.d .....l .... """""Is OUt and ;ntt..d 01 M. U.. lotal m...... ';' .""""n. ured. The"",,", i' tho " "'VI< h.i"" "'11_ • oon";u limit. the ... ",1,.. of ",...ri ... doH no' infllltll<'O It>t , ..ak oreo, _ _ lIy hoin, "", 01 the ;DB......,. """" 01 tho t/><,...,. """pi< (_ oloo B.&rP.d"J. For Lbo ~ry .....n -f'Ie Rich .. tho pNi ..... bo
.... , . _ _ _ _ ... _

, 10 ...

~

' ''' .II,o,

.. *"""000 ...........

" _ _ _ O· .. ,O" C. _

_

,.riaI ""', .......

....

Fttm lhecnph ill

1"",.. G...1Ii<-II_ tbo ............ "' ...

_lion f.".... ... ....... dill...... t doa-ten 01 _pi< jun
tJoenno..

aro it .. o,.f. _ .boII' O.2';vn 11>0 IaquI puk To tilt diu>. • m- the in "",", a....o:. tbe ."... "'-I.

~_

01 juDction 10 t of ...., " th""ll:h . he I..d , ;, t/t<",foro _.nin ~_ It i, ... n in Pi,. ft that lh. dj/f.~ ' J>borillrital 0)......

'''''.-up''

1><.-."

-..,

V.rifk.. 6 ... M A .nd ,l ~y .........1... Orift .<>d h lMna,,"'" D.c .. ,. ,ho nluooo of 1.IIII .l obUinotd ''''''' u.. t;'&libn.. tioo. leSU ..,'" ""....., to vuily , _ vaJ...,.1>y diII"...... ,iaI .h."mal:walysia. TheM lett. romprioed .ho "".."".... tion of ""'·Ii"" 011"... o ..... ho(funing of an upori","n' ond ••. pooonlio! d«Ooy . t .1Io.nd of 0 obemkal ...... .,...

n1 .... de,.rminc;l u perimontally. Wilboul u.. ,........,.. ,bi.otr..,...,.,ld ""w. _n fiw. Ii .... "'rxer. Tbi. tool could ..... rifr I"" <*lib ... tinS proptrti.. 0/ ,lie I .. , for.Vft)' "" and would be _01 •• ~r III lbe tado/ a "",. I .. prK"ti«*.... of .,...... _UOIIabit drif'", of the baoo line.

<010,'" til.."

d<"" (2J

~o-,.

",. Upooot"ll,iaI ~ 01 u.. end of • cbnDi'....". of u.. It"rits "hich ",,,,"",,u,be diJoipootion of lora. ou. of. _pit. This •• "" poua;b " , , _ of 1100 p ..... or ,imo of .be I"'> d"",;.,n of IIt"I;n .IIt . . .pIt . If tilt .11"«1 of ,he IMl"1I>O< ..pootnHol ,. " of Ibis O«<>r
{IJ

a-u...

Orih In Appolldix II (p. 4~) i, is ""ri~ thaI ,he but.'i,.. oIf.. , . ~, ",,"",",--,in, 10 • oIIan(l:O in tho " ..,in, "". of V dt&:Jen p « _ it

.. . v.. [(~'

-j .. l)/ (. +A"~)]

< - ... ___ Tof ........

'"

Hue on .~n'tm ,,,............... ;" Oq"";"" (t ). In Fi,. 7 .... oh
ob,.i".

.,

IV.

To _

't"".

).oad},

..

.v....... .v, _ ,......;Y
f igwt 8 lbo ....... mul .. 0/ ..." w..... """,ndum _dtr of a bo"" !IO"('...........dtnly introd""'" inlO (h~ _plt boIdtr. Tho ....... ,100> ,i ..... in t his to .. ,umod out '0 bo ~ 1010......,.•. A (0) ri_ onluotof . . . Gcm.-'wi l~. " o. ~ .......... 0. 1 "",.,1_ 1>.2 X 10-' tllI. pt1" de•. ~ """'" 6 ....-.. TIoio ...... _ with •• ........... If u.. tbtmoocouplt ofItn bad bt<-Il ..., -.....h.. " - timto Iorg<-r- ""' ..... _ - - . Tho ohonHl "' ..... ,;.", ,;",. ~ lor cloy"';""ralo io

1.100. witb . . . 0.2 W . , ... =.'
=.

140

=. _ .,

Soil Mechanics and Transport in Porous Media

February 1%1

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..... .. hoot

, • !.... .) • ,.,,,01"'" ,.... dO ...." Ii...

,

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g.,

;

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u"...

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by "",taloalloy>i'~' Alo .he abrup' "",, and .. ]"""tioto Ii"",;" ,horn"", OI"'3rt '"""' , han for .10. a ppara· tus i,..II. os;" "" .. oled by Fig. 9. IImo;t QSis fM ".a ~."",a in"''1'''''.'jruo •• bu, they do , .. rify Ihat heat Ioos through the ,he,.,..,.,.,..""" ;, of the <><de. of magnitude d
'''''''ion "'.y

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when ,10."".1

""".''''n

Selected Works of G. de Josselin de Jong

V.

S .n o ltl~11y

of Dj"'' ''''iaf Thwmal Anal y. i. App"r
'''''y

From the I""'i o;n&" it i ' dcd"""d that.be lac'.,... that vary io the diff... nt IOou. ond by thi, van.tioo ;nllue,,,,,, the ""]ibra,i",, of the , ... d.y;"". :ore principll!y tho density, p. of tb. sample. the heat <:OJlducti,;t)". ~. of lbe sample. and th ~ b.at·transf.r ..,.!li";'"I • .\. for the .b.mo""""plo. Th. h.at · in~ ra" doe. not altu the calibrat;"", if the be.t oonductioto of ,h. ,"",pie doe. not hiIerved • Dill......... of abou, :JO% ~ nhoe,-,.w in ~·ig. 11 10< ",,,,,..,1 dilntiotoo of illil< in ... AI.o.; .ath thtrmooouplt 10 . . . milled leaoo",}, to folio\\" a co.librntioto """,e of ito own. AI ... , .. pl.cement of Ih. tbermocouples. ,h • .--.lts "i,h

is."""" ...

141

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Vol. 40, No. Z

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......, .... lor ..·AIA, (,...'lO•. C'lI&l.·~II'O... t"";,,;, •. "" illite ... homODi •• , quarU, ... d f.~ . Yo""" 1.5...",. the .-.11$ from _ dib.,io
ill",

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UoIiliite and _tmorilloaitc rulu,"" ";tb ,,·AI,o, (Fi,.. 12 ..... (3) _ that the diIf. ..- . ""' Itso ~. ~ ~ 01 the tbermoo:>uples ohantes 4. So~ \aU ...,..., ...... ..;u. ,bc...-pIt ~<>. ~ "';U.o... IftICD, JiviD, .. 0laIIdud de,ialiol> 01 2.0 I<> 3.0% roo- 2. taufa r . _ poaI< ...... oI 13110,,",. X ...,. C_ Fie. 14). n.. hoa. tnuISfo. u.. samplrs _ , .......... t" tnt ......... tbdt _ . CWM!ucti"';'y may diII'.... 101' .. ftnO! ... _ Pipn. 3 ~ bow ~ dilren for v.rioI>s materials; ...., u.. ruWu 0I ....'.,.d........ teslJ 0' room \.emptfa'W'e

booIiDit. and ... AJ,O, and lho dilution (1,1) ........... .. coil), ,1>0 __ ~ ,.,. _ d i l \ i d<"oititar ",latiomJlip i>< • ....uI poak..,.,. aDd . . f..-Iooo. This is """"' in FiJ. 12 by ,he "rai,h. fi_ f(lt each ,bc"""""""pl<. Thi. proponklnaH,y f"" boNn;,. mi~ . 'u_ h.. ben '''''00 by ..""..1investigaton. en"""'" and Robtrtl,' de Ilnoyn and ,.,.,. del' ~I_l.' and Wiculs" b a'~ f()Wld .h...... "",po<.ioruIJicy for aolcit•. lUi,., bo .... ' ..". b •• A diff.ri", -.sid< ....bly Iron' U'" A '" .. ·AI,O, • • , ,hown in Pig. 1.-,. TIIc",fot¢, til. dil~l ion. of iltile and .... A1,O, . Cequal d,,"sily olio.' ,'" Ii"".,. te "'.... is _ ,,~ r.,.. OOf>P<' .wro•• (_ " ig. 2). " i&un 3.100 ...",.b tbat the ~ is """,ly prop<>rti<>n.I.o lb. t hold ri_
~

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",. eq.... ,i.ing ofl"oct '" A on , if tho .. ,.,.. putly Tau In Ib" ... $f>OCI. "",,,,,d , hat a bi,ber ,II< ptaIr: ...... al1"""lh tbe dilli<:Wcy 01 obtaini,,, ........u.n. "'o,;'y 1I,,,,,,,"""t lbe sample .....fllStcth·j,y boa.,.. ,h . .. m"r.. .hri nk. ond Iotet """I"", trith "oil .,.. ,benno"""pr..: Ih. Il0. thaI theoe dfeeu betin to ohow ""ly bo)'(lltd dilution no,ioo of 30'7. Witte"''' <>ho<-rv«I. w"",, ......tinr wi,h _ V"""ulII diIY...... _ ''''l tbermaJ aDalysio 1.... IhaC t iro b
.~"';ualed.

denoi.,

j","",utd

vol.,,,.

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M.. _

.,n """

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of OifI..-tiol """"-*I

.... 1_._.1000 ~ II>< Clo, ) 1........... J. A • . C.....

So<.. U jll:l_('II3II~

142

Soil Mechanics and Transport in Porous Media

F~bruary

1957

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tII;t ;nt u.n<e it ddue<:d Mo.t , ...h;"h . t • Ion, M.'in~ ... ,. """Id h.,... dillSipot«l un"', """" r."""hI< OOIIditions. may be ... tud«I. Crimsh . ... and Raben .. pro..,.. \<J ..st ...... p ... "hon tMy ... ditu,«i with On i,..n ma,.riaI , ~.J .• .....~I'O" in • ...,h • ron· ........ tion tbat tl>< hut--<>..< ,,·A1,O,. That i, i< poooib .. 'o . li",i· ""to tho .If...... of sinto,;", ond >hrinkinc of tb~ t.... ,"" ......1 .. h"" tho .... AIA is p ... """ ...... nt i, ... rihod by 'be 1_·

JOin, . VI. Conelu.;,," Tb. most .. now. ~rrors in quantit.,; .... m,,,••,,,,,,,.,.n,, .nth tl>< dilIer. t.o minimize 'hi< di!lurhi"l f,..".,... This con be <1000 by sta"d ... di,ing , be podi", .... thod.., ,hat tbe5O"",

i.....

AP'ENDIX I

01« .,,,,,;,,<11

Tho b.. , """,,,,,,i";,y 01 .11< .... ~ I>'>"""<\e<J .... front ,II< .... ."....,. b.. , """ , bruuP ,o.. .. mpl. ,owan! ,b. .mpooidal ... It 01 ,II< """'O'M" ,h. Ma' "'" ~",d by • • oloc'ric>.Uy .... ,«1 "";1 01 «><>I"",,' .tll..,.,o.ut oIba .... "!be watt 01 ,10< _to .... wao hpt a . a _ . , "m~'W"< (2(I"C) i. a ...... botlo ond ,hot ,..., ...... , .... ,~ 'h< 00; 1 toJist«I'" (-"ton« .. i'h """.h.. , .. ,,~'u .. "",iii· Th • .,.,;) ""' ..... nd on. _ _ ~ '" ,o.. , ....

"*'

'hot

000'.

=.

.w.'~..! ~ ,~:li~ ~....!:'t::' ~~ =-~: '''''' .. tho boo ..... .,. _ ... ,;....,-y """"i,loo, .. "",,,,,,iat

, h
limply _......t
:::: ,t;::;.=.~::'r~ ...T':"ho:~':':t ':!t~~«1~ _'«1 by 'ho.1
"'O"'y

ud tl>< .. me V< oo,.in«l ""d tlltu U... !low noar .h. U'm< dilutod ";tb tho rd''''D« m.n matoriol (.......uy ~ · AJ,o,) ....h . ""g-ee tha t the """tiDg and hta t __ du",ing p..t>p"'. tin.nU b< d<....". durin, tl>< , .., ",pr.n...ofilluinkinc... n... • i"8 . or !iq... fo.ction o(.ht riol. DiMion. h."..." .. ,. mu" "'" P"'''''o' tho ma,. m1 Invntip ,. d from givUll ....,.,gh .b< di!ut;"" t.mniquo 01 Cri.,.h ... and J<.abert.s') . To min;mil< ,ho .If"", of.be ,hc"..".""ple. it .. ~ to wibra .. tho t"" ""vi<.-. . ach ,im< ,II< ,b«. it m>ew«I. This o;alib/"a(ion ""'y I>< ~ wi,h ch.mical. of ""U·kno"n I><.Ia 0( "-mpoOitioo.

1\<.,

'0

t." .......

"""""'p" """dar
....

-.-

Th" ;".-ip., ... ..,.. u .......,.." In «tIIobura,1oo .. ;,b II. W.

va" d« M...... "'''''''I."~ Ex.,... ... "" St .. Ioo. C"",; • ..,.. ); .. h... ta.d .. "!be io indeb'''' ,. H . I..ob<;' """ A.••• d.,. "'_for """''''''in ..... '''"''.

au''''''

A"ENDIX 1I Whoo,o.. "i< • .,bIoet io boa.oda. o .-.t< Vdqrooopu_ ,

"i_,,"

til< , ... ,...,.,~", 01 .... " "'pI< -.riU "'I bohio<1 " " ..... t ... 01 ' .... mplo ... ..." \>oconto Tho boa, .,...--.. tha,,,,,,,,,,,,'1>< ID~oI

,...,,,....,u,,,

,ho.....,.. ....

"",ioaa".

""'Ih' ............. ,h ...... ;i',.nd •• io

01 'hi> hoa. «oooto ,hot ..... h'~ " .."bo, thisQ""n,i,y;" ..... _ . TIl< "'hor " ..,,'

now>

_oI' .....

.-<" -

,..).I", V. I'an ... d .... IJI/1><

. ,........ tu< .

2 ..~·.·ilf/l>< 'hr""'b .... 'h .. """""~ ...... -.ri,. cr--. at 01

oed ..... t .... A. "'"I'h I. 0"<1 .....

M·A·..II.

_""".i., '..

a",'.

n-.t... _~ ..;... ~ h _ A . .. nTli

A"""'-([I", to p" ... ,iot tM<1I'y ,I>< ,om"....,.... dilf..-.", :># ("C.I ......... < .... ,

'ho..,...'''''' """Q{"""'-'" _"""'-' ..1

ft ..... Q(""l . .,..._I.... M

_

21 "

ha, to..,.. ..".,."ieityol;,..... olnprid.

.. -

««ntri
,

'Q

no. d ectri< ......y in.""·""" .... 0 _ 0.31 .... n. "nd ,I>< di· ...... _ . 01 , • • ""prid, -'01 _ I.~cm .. ", _ ' .3:I . • ocr .. _ 1.18. ..... Ii_i •• """troI .... mod. -.ri,. sola'in... bido .... ,o.. ........... """,,,,,,i.ioy., ""'....... ( 1.37 X 10 - ' "",. p« d",. - . cno.1 "", it ,~ . diood .. n.... 01 lon, " - ,h..,.,." """V. "!be ."....,,001 ,om".... ,"'" dill.......,. 01 6 .2 " __

--

"i,,,,,,,,

Selected Works of G. de Josselin de Jong

.. ~ . , _.

, ~ .,,_

.>I

,., ,.,

In' . ... ''" ..... . , .... (4). , ... o!>taift.

.'

[(,, - ., -T lo ';;)/('+A"'"' )]

.>1_ V", . ------.-

,.,

143

l a/13

A Capacitive Cell Apparatus L'Appareil a Cellule Capacitive by G. DE JosstUl' DIO JO)I "" ~enrU .... \a dtfom,'''''''' """'........ Ie d·.iprom.. " .. cYtirocJt;q.a compoirnta " iolanon',' ..."' ....... Iem< du J)'Slm.:. Au coors d'une .... 1e d·....i. oV« 1' . ~;I. on t.udi< "q,.,+men.

E:'t"'~.'i:"",;.JaI!"""~:~ ~~=~';..

dioalolo

\a fooction du"'""po
~* The 'Iudy of cylindrical..,i] samples. under lhe inft ... """ of .. _ as .ff
If,..

Brilish ia] OPflO.r:a.us b<:>ti ... if the ,tnoins may be determinod $<",,,,,.Iy ar>0 borironlllJ otreos app(;ed by a it prewu .. "" the rubber envelop< or .he sample is IlOl. disturbed by lhc dispIaot .... nl de •• rminalioo, Af\cr a descrip60n of tho teo. de_ice and in"''l'~tion of tile .... ults. a diJcussion of typiocal leSt results in <;(>nIJo tonoeplion of sIIca ..... ~ will be ~""n,

o.t .... l...d".. In Fi,_ I ,he apparatus is schematically represoonled. Tho cylindri<;al .. mple i. k>aded """"",lIy by ,...ishu and boti· zontoOy by a ir pressure on the rubber •• lhe ""nical ....,.,.""'01 is .... d on ,he d ial pug<, thno tontri~r.oes bei". of tho """","1;0",,] t)l'O. Thc ........ uremon. of tho horizontal , tnU" i, effe<:t«l by using u.. ..mple . ......""" .. 0 ... of tho t_ .""".-odeo of" rondo ........ •be metal ""usin, btinathe otbc:r If u.. .. mple di[ate5 tho distance bttlO.Un .. mple and oute, wall will decreado .... r boca_ I .... capacity is i.,~ ly proponio ....1 ", tho di>Un ol«lrical capacily tho """",,ntal >lmn may bt deduced, In order to measure "'-" capaci.y a .. r.",_ condoM« is inruIlled in tho bit$< or lbo appal'lt ... ....tticb can bt made ''''''',ically «Iui.... len •• 0 tho oampJe-_U combination. In FiJ. I this ref.ronet rondo .... r i. ahown in I lUbslitu •• circuit comi1ling of a fiud rondo .... ' C,. wbich "'pracntl the maln pan of the capacity from lhe samplo a nd roR""",ina ...... allie partl . • resi$ ..""" If., U I subsli.= for lbe .. mple ... iltan<e ond on odj"""ble "'nile ...... C, wlticlt atWUntl fo,'be remain_ inc ""rio\>1e pan of.be .. mple-wall capaci'y. To oqualize Ihis ole<;trieal . ubo.i' ute 10 .he .. mple·wall capaci.y thtoc t_ circui .. ~ al •• rna •• ly ~ to I zero indicator. For tIIi' pufl>O$t .... is made of on de"ice (,ype 80<..."., C.V.M . Ill , freq ... ncy aI. 1 Me) wbich is ... ",itive 10 ''''-' . mall capacity ,"'''''Iio",,: 0'002 pFci_ a visible dol'lec.ion


on tho Amp, scale. This """';,i'y "".ormi,.,.. tho posaible accuracy for horiwnlal .. rain mca5""' ..... n... With a .. mple: .... igh. (A) _ 10 <:m, diameter _

u, _ 6')!

"'''
an Ind t .... dia·


s:..::g;}/,.,., 01,,_ = p",,,,,.pk,. -

Flub • .,. ."..1Dp.

o-rlptiool of ItIe U I
,,.Iope """

.""",odt. ....

""""ronic

F.., I

So_." "'P'-''''''''' 5<_ ""

or .... "",.,.

r.ppa";t

meter or tho insi
c-

•

1'81<>so I.,.",) - !OpF

An il'lCtU$< or this capacil)' by 0-002 pF cor .... pond, to I "",ironlal . t",in of 'H _ 0 -0) pcr ",n •. beina .he accuracy of .he ....... ur<:men •.

"

144

Selected Works of G. de Josselin de Jong

145

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146

Soil Mechanics and Transport in Porous Media

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H""'~.. r in oierabio ...1"",. ca""inl o""rloadi"ll in o,he< parts of the ooNl"""ioo. In • teo. of &hon dura.ion (in comparison 10 ,he Lir~ of ~ivil e.n&inccri"ll co",'fUClion) .he flow properties of the soil Loading condi.iotlS ha"" 10 be considered because oon.ioual flow may cause the unwan,ed deformations. As a criterion for admissibililY of obtar st ..... It.. 80w rate has to be in.roduced. Ln a ... t .uch as lIIe lriaxial ... t wile", only one .... Iocily is induced .... "blain ... imiih. "r the .i.uaLion of one: contour of D, bul the olope "r the .<1>001 d<>es not .0 .... inl" tho ...1 .... ult. 111;" tklpe ho ........ r dttermincs ""'" rn..:lt tho "bserved 7 has 10 be reduced in ordor '0 obtain ao admissibie value. ....... '.rab'" therefo", is a teS. which by exploralion ofthe $heel .I<>pe lJi_this informa'ion. Such • lest is a Dutclt..."n..... 11K rilPdily of the <: iocn:a... io .teps, bul duriog the lime the ""nical st ..... i... ft constanl th... mpie straill5 and a jradual increase of En <>OCtII"S, thus de<:rea.ing~. This prod ...... Ibo obliq ... Lines in the ...." plane:; &howo io Fi3. 50 "bo. . . Ie$! ....1111 is rep",. _ted. All obliq ... pam I~ f<>rm the till:idily line or ,he appaottUl beina: parallel 10 ~.... of Fig. Sf. n,. corrcspondin,g moVCttlOJ\,. in tho .,.."...D .<1>001 of Fig. 6 follow Ii .... such .. AD, whiclt.um aul I" be nearly perpendicular 10 the con.our Lines (D .. constan.) and so follow tho dirtcLion <>f "'""teot . Iope on.he IIxocI. Requi~ ",.".. for a ao<>d lesl performance an: tltalcach Ioadi", ,tep shouJd WI. long .n<>Uih f<>r observation of tho ornall ""Ioci.ies and thaI tho I<>ading ... ps.hould be ..... UeDOUgh not '0 disturb tho samp'" 11K tcll is cons,ruct<:d so rigidljl thai althaudl a "'""t many .teps of I<>ading and unloading are ....a:utcd and a broad band of the family of .... Ioci .... i. explored, tho l<>tal sbear , train y is. ..ill ~rniled and Ihows DO ,,01 ..... This "" ...10 the intenlion of the o
un"'"

..grnon,.

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"-0&. 6 Ill
",lotion

O ...phiq .... ,mil dimmoioDIlIoo ",Io'ion< .......fJ • dotre.uc <>f 7, The ri$idi'y of the apl"'Ja'us o,..,raled, by Ibo 1IAa'll pointed %i,·zap., the path followed by tho sample beha.;OUr for " ",pealed loadin,g as in tho DuteJ>.ceU 'est. The primary oh<el is ior modiately rejoinod, lhere bein,g no .... ious deviationa. The """""PIion "r this .,.."...D sl>oot ;, of gnoa. help in the interp..,.. ... tioo <>f admissible shear "",istaooo.

...ccuted

Study of Sboat

R_

"Tho .-esisW>oo otr.f "';1 COlUlructio.. 1$ '0 be For jf tho s<>iI ~ a .. di-.ided into compression and shear, i, is weU """"... Iltat lOil can witb$Und ..... ry com--pression """" a . tho expeOl$< of ""rtain def""",,,ion!. bUI thai tho obca: ,tn:sJ. exeeodina: a ""lUin ....1..... cau ... unlimited defotmalioN. "hich brm, .bou' " b<'Ook down of the con·

.iudsod.

""'n,g\h

""""ion.

Selected Works of G. de Josselin de Jong

C_l ....... The copacl.;"" cell apparatus penni,. tho st!>dy "r differenl condilions of "'"'" and .. rain. and by measuring the ""rticaI and It<>rirontal .. rain. ,"""rately comp!'OUive and sbcar ,Ullins may be "btainod. The relaLion be •.....,n $hear st ..... and $hear "",;n i. only uoiqucly do:termiroed if the Bow rate ;" con.idered. The family for dilf.",n. flow .... Iocities in t he ...."pla ... ...."" I<> he challCl0 ",""I a .... Iuc. The ...pl<>ration of the ....Iocilies in diff.",n. Sl
"r cu"'"'

Load"""

TIlt: 4W'Io>£S "'ish 10 ,h,,,,k M, BiNr,malor ,"'"','i"lllhe ..... oja M~I'y _,ItDdI'" lhe _","""",nl oj ""«ttl .,,,,in, M, Wlnkd lor Mrryl"ll oul ,he ,~, .. "nd M, V... de, &/d tutti M, '1

..

Harl lor 1ir<:1, "'s/,,""<~ in lit< detHlopm
of IN "PfHUl"

....

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G."Z&, E. C. W. 1\. and TA:<, T. K. (19~). "Jbe ol><ariolI p"","",ies of"';'" c;kl«~.!. 141 _ (l9Sl). G....rtJ R~p«I. ~. Jr-d '."rItdIUm<>l CO'I/..-mn "" S<>II M..-smol F_';c" E41""d"". Vol. 2, p.ll' I'r«. "I ,10< CMVn-c'K'< "" SN
Ii!< ""_4,,,,..,,,,,,, '"

$<MI,

147

ETUDE PHOTO-ELASTIQUE D'UN EMPILEMENT DE DISQUES par G. DE JOSSEL IN DE JONG Cl A. VE RRUIJT (UN IV ",,"''''''

"~CH~!QU".

D"L~".

PA YS ·DAS)

R E S UME Des essais exec uteS SUI un empllement de dlsque s. consthuant un modele de mllieu pulv erulent. so nt Mcrits. Les disques om lite fabrlques avec un materlau photO- "Iastlquc. ce qui perme! d ' e tudler 1 e s contrainlCS it l' inle,lcu, des disQucs. II est montre comment I'analyse de s cssal~ conduit a la delermlna!lon des fo,ces de contact ent re les dlsQues. auss; bien en grandeu r Qu'en di rection . Les con dilio n!; d'equllib re des diaQues Indl;·lduels sont ve rili ees it I' aid e d' une epure des forces.

I.

I N TR O D UC TI ON

Les proprl etes moIcaniques des milleux puiverulems. Cpe l;elon Is dl,ectlon de la contrainte p. incl pale majeure . 11 CS t rema'Quabl e Qu e. parfols. on vote MSSi un reseau de IIgnes orthogonalc8 nOires. c ' esl-lI · dire un resesu de lignes 11 PC!! prc" dlrlgecs selon la direct ion de la contraln! e l"lndpa) e min eure ( DEJOSSELIN DE JONG. 1960).

_ 13 _

148

methodes decritu ci -dessus, II est Impossible d ~ determiner individuellcment les forces InterLe present article deer!! une m~thode, QUI. en utllls-nt CR 39, un materlau phuto· ~ lastlQue plus sensible Que Ie verre utlllS
le~

~ ranu lal r es.

Apres avoh determine toutes les forces de conlacl. II a etc possible de conSlrulre un dla~ramme de forces QUI. ~tant ferme. prouv e Que toutea In eondltlons d'eQu!llbre sunt sallsfaites. Quelques exemvles d~ resultats uperlmentaul et d ' ~pures delivhs des photographles sont p r~sent~8 ci-dessous.

II.

CONSE QUEN CES PHOro - ELASTIQUES DU CIlARGE.'lENT DES DISQUE!;

Ce paragraph" sera consacre a l'eff"t photo· elasllQue d' un dlsQue d rcula!re charie .. sa pe ripherie par des contralnles exercees par les dis 'lues voisins. Cu rhultars seront utilises pour Indillu er COmment 0 n peut analyser Is dlstrlbutlon des forces dans un assemblale de dlsQUH. Consld(lrons d'ibord un seul dis Que en contact a vec un certain nombre de dlsquu voisins. S I I e8 dimensiOns des plans de contsel sont sufflsamment petites, On peut consldere. In forces a81ssant au bold du disQue comme des forces coneent"lc.';. Pour ... surer I' equlllbre du diSque, II est nrlce$$alre que Is resultanle de toules ces forces 50it nulle et Que ceS forc es ne produlscnt pas de moment resultant. Bien qUe la distribulion du conl raintes II l'lnterleur d' un dlsQue charie pat des forces concentr ~es agissant" as pfrlpherie pul5~e etre calcule e par les methodes de Is theorie d ' ela~tlcitc, it esl Instructl! de commencer par des considerations approxlmatl"es. Pres du polnl d'a ppllcalion d ' une force concent re e, I B distribution des con tralm es s 'approehe ra de celie d ' no plan semi- infi n! charse ~ar un~ fOlce concentree. Ce dernler cas est un prob!~me dUSIQue de la theori e d' l!lastlcl te, resol u rar PLAMA NT ( 1892) pour Ie cas spedal d'une force di rlg"e pc rpendlculalrement au bord, et generalise par BOUSSINESQ ( 1892) pour Ie cas d' une force oblique. De Ie solution de ee Plobleme, 00 pent d,;dulre que les iaochromes (courbes" difference consllnte des deu x cCOII"lntes principales) sont du ce rdes. en ct relu passent par Ie point d'appJlcatloo de la force eoncentree et leurs centres s. trouvenl sur une Ilgne dans Ie prolongement de la force (FROC HT . 19~8). On peut compa rer cette appro.imation avec In resullats oblenus ~ar PORITSKY (19~O) , qui a resolu rlgoureusement Ie probl~me de deux dlsques ,HastiQues en conlact. PuisQue Ie point de wmact, dans ce cas, s'eSI elendu sur une petite surface, les i8oehromes ne 50 01 plus des cercles, mal. la d,h'latlon est assez petite. La ngure I, ci -contre, pr~senle, dans Ie cas partieulier oU la langcnte de I'anille de la force ave c la normale au plan de contact eSI ellale iI 1/ 3, les lloch romes donnets par PORITSKY et les isoch romes empru nt cu l FROCHT. II ressorl de Ja figure I Que les deux sy8!~mes d ' isochrom es son! a peu p", s identiques. eela JU8mle l' appllcaUon su probl~me des deux dlsques en contacl. de certains resu ltat s oblenus pour une force agissant Sur un plan semHnflni. En parllcuiler, Ie fal l mentlon",! cl·dessus Qu e le$ c~nlle~ de s cercles IsoehromatiQues sont sHues dans Ie prolongement de la force, conduit. II determination du diamerre commun des

_14_

Selected Works of G. de Josselin de Jong

149

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I • J$Ochromu pOu, 1. c .. d' une fOle. conce,mle alluont sur un domH,jan. compa,h. an. l oochromu !>OUr Ie ca. de d.... dl aqo u eo <:<>nta cl . O'apr'" FROCHT 0943), l e$p. PORITSKY

(19~).

ccrdcs Isoch'Qmahques gOUI ~n dtduife la dl rectlon de la torce . l,e s Hanes photos.aphiees 80nl eIactemen\ lea !soch,omes e\ 11 n' es! pas difficilc de localise, la dIrection du diamHre commun. La determInatIon de la valeur absol~e de la force uansmise d'"n di sque i. I'autre peut eIre 0 b I enu e .. I'ald e d ' une formule d o ~n" e par f'ROCHT (1941 ). 51 l'up;;,I~n(e est neeulee $u r champ clair. la dlfr.ffenCe des con\,aintes p.inclpales , D, Ie J o n~ d'une l!~nl' ! 8 och rom a tiQu ~ est dnnnee pas .

">_

"'C, oil

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" est la longueur d' onde de la lumie re. t

j' f\ paisseul du modMe,

C

l a c<>nstanle optlQu e dOl

n

J' otdre de l'isochrome (e gal a un nambre enller au centre du (rang es Jumlneuse s ).

mat~,;au,

D'a u!.e ~arl, Jl .e sulle de Ja Hnea.U e des ~'o~.ietes mecanlQuesdu materlau Que 0, - 0, est lionnel ii la valeu r absolue P de Is force, Alors, on peul eerire '

a, _ a,

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cO { ( x, Yl est one fonetlon e "primant Ja (orme gcomet rl<'lue des

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CelIe {onetlon, en gene,al, esl " , ale ment dependanle d'un c erlain nombre de pa rametres lie s au mode d'appUcatlon de la lorce ( par exemp le : l'an~l e d' indlnais.on de la (orcc) , lsoch r ome~ ,

_ 75 _

150

Soil Mechanics and Transport in Porous Media

L.. formu! " (I) et (2) nprtment une rel .Uen llnh.lre entre !. fcrce P et I' ordre de l'llochreme n. Cette ,el &tlon perme! de com~ r e, lea rhult.a de deu> fluls phoor:o-~Il8tlqutl. En efret. On peut d~lcrmlntr I. valeur _blKllut de I. force dans I' un du deul .. ul •. III ' on con".n la force dans I'.ut.e .... I. p.r comp.r&.l lKln d.. ord res dOl Uinta lsochromotlquu en un ce".ln polnl. II ellt cenn" \ PROCHT. 19-48) la foncUon f \1. n est donn~e p&f la

q"".

pou. Ie cu d'une fo.ce concent«;e q!a ..nt IU. un pl ... seml·lnnn!. .

fOf.~le

,

f \1. y)

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Diona celie h aeUon. I CSI enCO, e I '~p al ... eut du m~le. et d Ul Ie di.~llt du ee,ele punnt psr Ie poln l (I. yl el Ie pelnt d' lI.IIplietuion de Is for ee. Ie e""tre de ee cercle ~t&"l 5ltu~ dus I'ue de ia force. Loraqu ' on eompa,e I...CHu ltals de deu l uuls (dhllnf • •elll,eelly,m.nt par lei tndlC" I. 2). U r ~8UlLe dell formulu (I). (2) et (3) que'

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SII'on dllPOte des rClultats d'un enal. dunl On con nan la force P, .t I' ord .. n, d'une I_odIOrne de dlaml! tre d,. on peul r.cUem""1 calc"let I. fo.ce P, d' un leeond cS$.1 avec la formule (4). II .ufm de muurer Ie dllmerre d, d'une llJOChrom~ de I'oldre De ce qu e la fcnclion f (I . 1) eHl donnee p.r I. lormule (3) lIuell e que soit I. direClion de I. force V. II ,,5lu lte qu ' ll n' est IIlerlle pu ""n....I" que lea IOfcel P, et P, lient ",~",e direction.

n,.

,

•

Siturellemen\. la mesute du dl_Ire d' u ne 1_lIteme Cllcull;,e (ou. peu life" d,culal re) n e IMhente "". de dlmcu11~ sfrteuBe . ,..ill •• IrM!lbode permell&fll de dttermlne' l'Ofdre d'"ne teUe lsochrome n'esl Pao! hldenle. Couldi.enl pom eel. un dl"que .tOumls al ' lnfluence de quel ques fo«:u concenlr~es. 'PpUqU~" i. sa perip!lt' r;e. II . ~t~ monu' P" F'ROCHT (948) que. dans Ie c .. d ' un dlsQue ch .. ,~ par deu. force~ d l~m.llra l .m.m ul'I", .. eu. '<>"Ies Ie. coolr.lnln . ' annulent au bOrd du dIHQUC. Alor•. l a dlf"rence de. contralnles prlnclp,ln s ' .nnule a.,,,,,1 dins COl c ..... el. par conselluent . Ie bO,d du dl.que conlltltue l't.ochre,"e d' ordr e z~ro. Des c&ieulll assez III ..ple •• analolun _ u. c.leul" de PROCHT . momrent Que. dans Ie cu d ' un diMlue ch •• " ""r deu forces cOUne.lres. oPPQa\lts el dhi,'ta salon UDe cerde "bltr.lre AS ( flp.c 2).

Fip ... 2 _ DL_e c:IIat,t par den forees collftfol.,..

er 0PP'._.

-

~8-

Selected Works of G. de Josselin de Jong

151

la dlfforo, determine pa r I' ani le .. ec la coroe AB est donnee Pal' la formule :

(J

de Ja liane Be

,

.. Rt

(0)

sin_ + sln(_+28)

R est Ie rayon du disque, l'an,le de la direction OB avec la direction AB,

oil:

Les Miles _ et 0 sont

consld~r's

WS1Uf, s'lls sont orlentes comme IndlQu;\ Sur la fl,ure,

En pasanl .. " 0 dsns II formule (!», on obUenll a relaUon oblenue p.., FROCHT , ", - 0, vslable pcur un couple de forces dlam~trales. Au point D, sllu,; symetrlQuement par r,ppOrt

a

II et B, oil

P

"n "tlll ....nl obllent :

i.

fMmule

~ene .ale (1 ),

".

81n2_ + sin _ .

n etant malntenanl un nombre PC

,

20

illS

0,

on. :

(0)

n"cessalrement entier. on

sin 2 .. I + s in ..

In

Celie formule montre Que, d.ns Ie cas lire sent, I'o.dre IsochromatlQue ne s ' annule pas au bord. Nesnmoins, cels n'emll~che P,as I'an abae, parce Que la valeur de no eSI. en prallque. llmlte e pou. les raisons ulva ntes. L'In,le .. est ecalement I'ancle de la force P avec la n",male au plsn de conlsct. Ii ne peul donc depasse! I'anile de frDltement , qui es! d'envl!on 200 pour Ie m~erlau utlll ..e dans lea essala. Lor a que I'anile .. varie entre _ 200 el + 200 , la quanUte s in 2 .. /1 I + sin .. ) Ville enl/e -0,98 el + 0,48 . D.n s In uuls lea forces P sont au maximum de I' ordre de 300 N et ... Issent su r des dlsques de 1,5 em d ~ r&yon. PoP' Ie m~e rlo.u uUlis~ {CR 39), Is conStanl e C eat d' envlron l )( 10-' cm'/ N . L I lonl\1e u. d' oDde ), Him d' envlron 600)( 10 -, cm, II QuanUte PC /( ~ R), ) est toujours Infe . len re i 0.3. LI formule (7) conduit donc i la conclus ion Que la valeu! absolue de n". c ' est-i-dl re de I' ordre 110chromallque en un pclnt du !>oro du dleque 'quldlstanl des pclnle d'apllllcallon deB forces, est toujon" Inferiep!e a 0.3 . Alnsl. pulSque Ie nomb re entier ie plus proche de la . aleur de n D eat toujoYre z~ro, II eSI i. prholr Que Is recherche de I'ia ochrome d'ord re :rero ne prhe nf,era p... bnuf)Oup de d lfflculte dane Ie cas d' un dlsQPe ch..", p.., deux forces dlrilees selon une cords arbltralre. On nole ... Que ce type de sollicita tion est Ie cn Ie plus ,enerll de sYsteme o!qulUbr ~ comJ)Oa~ de deux forces ee ulem enl . En ext •• pclant ce .esuhal, on peul admetl re qu'U eal probable Que 188 conl ralnles i. I. J)l!rlpMrle d' un dls que chllie pa! plualeu!s forcn concentren (cf. Is fl ,u rs 3, BU verao, qui present. un cert ain nombre de dl aQ un charies de c ette ra.;on ) aeronl ".Iement Irh pelitel , e! que I' tsochrome d 'oro,e ~~ro eers siluee i. proxlmil e du bord du dlsque.

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Soil Mechanics and Transport in Porous Media

J . [)flail de la pho\OIl..,hl . d'ur> elPpll.,..nt ch .. ,~. montrontlu iooch""oe. l l'! n!~r\ .... r du dl${j~e • .

F;,ur~

Qualld 1'llIOChrome d'om re u ro e8e re per
m.

ANALYSE OES FORCES SUR L ES DlSQUES

Comme oous l' avon8 YU. la dIrection d'une torce de conlact est delermlmie par Ie \leu des centres des cercles. et sa valeur absoiue eal d~lermloh Pat Ie dlamelre et I' om re d' une com be lsochromatlque. A lor s, loulea les forces \ransm\aes It l' \nte rleur de I' aaaemblage sont COnnueS en direction IUIui bien qu'en vsleur Bbsolue . II eal alnal poSlilble de v~ r lfler que lea rhuUMa satiatonl Bul cond!t!on a d ' eQulllbre des dl s ques Indlvlduel s . Le pr ocid~ de veTlflcation va ~\re deer!] Itl' alde des flgurea 3. 4 el 5. Lea ftgures 3 el 4 montrenl Ie detail d'un cerlaln empUement char ,~. et I. fl,"", 5 repr~8ente Ie dlBllramme des to rce a 1 co"esl>Ondanl.

c fI,u,," 4 • 1".. Quane dlaque$ au cenl", de la n",re 3 ..ec I.... ra lo.c... de OO
H

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153

c

Pour que l' equlUbre d'un dlsque soli saUsfan, It faut que In forcea aglssant sur c e disque constltuent un pol,ylone ferme. Par exemple, l' equlUbre du dlsQue alto~ a I'extreme lIauche de la figure 4 est satlstall, pulsque Ie chemin AJHGF'A de la figure 5, chemIn compose des forces AJ, JH. HG. OF' et F'A. est

un pol,ylone ferme. Les polnls d'lntersecUon des fOlcu de la flgure 5 (par exemple, les POlnls A et J), correspondent sur la figure 4 ii des rEgions . Cell r~llons onl lite dhl,nees par les m~mes callcl~res (A et J). La frontl~re cOmmone l deux regions eat la Illilne d'actlon de la force, corresPOndanl dans la n,ure 5 au se , ment jol, nant les points A el J. Ces figures donnent Egalement la posslbllite de verifier l'eQulllbre des momenlS des dlsQues. Pour ce falre, les dlrec!!ons des forces de conlact (deduJlu de la fi,ure 3 el deja utlUsees pour construl.e la figure 5) ont ell' uacees sur Is flgu.e 4 i partl. des points de conlact correspondsnts. S'll s'alillt d' un dlsQue charge par troIs forces. II est nccessalr e que lea troi s dlrectlons pa.ssenl par un meme point.

o

r - -__'. E

F

G Dans Ie cas d ' un dlsque charge par plus de trois fo.ces, par exemple Quatre, lJ faut pluSleurS etapes. D' abord, On C(lmpose la rhult""le de deux forces (quelconques PRfml les quar.e) el on const. uil sa ligna d' actlon qui passe par l' lntersection de ces deux forces. En8ulte, on v~,lfl e qu ' elle passe par l'lnlersection des deux forces ,eS\Bntes. De la meme fll<;on, on ve.me I'
IV .

DESCRIPTION

~~~:==~::2::">

K

H 5 • Dlscr",""o dOl tOICOS COllUlXlndanl OU. PiC. 3 et 4, Uluttant 1'~Jl!bre dea dIOQU " .

F; ~ " ,.

DES FSSAlS

L'apparell otllls
panne.u~

de P e,spex,

Entre les pBnneaux Se I.ouve plac~ I' assemblage de dlsques clrculal.es, fab.IQues i partl. du mate .lau photo .t\lutlQu e CR 39. La repartltlon delsmh de cn dlsques etall 1a sulvante: 6 avalent un d la~ tre de 4,OC"' : 1de3,5cm : 16de3,Ocm ; 33de2,5cm: 33de2,Ocm ; 36de 1,8 cm; 28del,5cm at 31 d e 1,0 em, Ces dlSQucs sonl dlsDQses au hasard daus\a cuvette, ll ' exception de ehacun des quat!e coin s de l'usemblage carre, oil I'on a mls, pour des . aisons p,atlqucs, on grand dlsQue. Pour hite . l ' apparl~Jon, au cou.s de la fab.lcalion, de cont •• lntes Inlllales aux bords des dlsQues, II a fallu opolre, avec beaucoop de soln. L' assemblage peut litre chars' horizontalement et veniealement par des plaques rlgldes de m ~ me ep.lsaeu. que les disQues. La plaque Inferl e u!e eSI fixe.. rlilldemenl ; lea trOIs . utrea plaques sont lIb. es de

- 79 -

154

Soil Mechanics and Transport in Porous Media

fir'" 6 - Pbotolrap.ble d' u" empU ...... ,

dI.I&~.

Le det ..U d. 1. III11.e 3 •• Itw ••

R>
.. -
~Oll.

(\U COIlI.o.

se Mplo.cet daM leu . plan. Les plaques laterale. sont cbarch. pat des (ot<:es putemenl ho.lzonlalel. 1,Illea el oppoa~e8. La plaqu e lupl.leurll l ubh une fote e puremenl verUe .. le. La cu velle ell plo.ch dans un apPAleli pbOlo· flasllque de IJ'pe norm ..!. compoB~ successlveme.t d' une aou .ee de luml ~ .e . d' un pola. lBeut, de deux lames qUAlI-d' otHI. eI d' un ...ly. eu •. Le mtHI~le lie lI""v. enlle les deu x lames qUAlt-d' onde. Le mod~le etllnl 1I0P I .and pou. que l'on puis •• en oblenl. une pbDlo,'lIpllle Int
_ 80 _

Selected Works of G. de Josselin de Jong

155

Fil'''' 1 -

Pbc>to,raplll~

d'un e.,pllement char". pO,e6 vertiell e: 1440 N.

pOrc~

horizontal. : 480 N.

La lI~ure 8 cl-contre prhente Ie rhulta/. d ' uR nS11 oii la force vertica le ~Ialt 'gal e It. 2.556 fo j a la force boTi~ontal e . La Bysl~me de Uenn orlllo,onlles aupe rP08~n e81 un rheau IraC
La Il,ure 1, cl·dess"s. montre Ie resultat d' un autre e8lSI Sur Ie m~me ISsembllee, 11 seule diffe rence avec l'UUI de la fl,ure 6 et~t Q.ue Is force vertlClle est lu,mentee de 4,348% el Q.ue la fotce horlzontale elt .... dulle de Is mtme Qua.me. Cecl veul dire Que Ie dhls,eur des COnllalMts. d~Ji. a ase~ J ralKl dan 8 I'e sul de la fllu,e 6. I "ROOle e.e un peu luemen.e. En comparan. In deux photos, on obselve qUe 1'18Ilembillie I Un pe" chance . du tIn de la tupture de quelques plan s de oontlCI. D' lutr es plans de contae. on. ele creh duran.la trans l.lon d'un BysI~me d' eQulllble It. l'.utt e.

- I, -

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Soil Mechanics and Transport in Porous Media

v.

U :S

FORCES

II partir des p h oto~raphles du mod~le eharg~, telJes Que eellu des fi, ures 6 et 7, Ics forces de contact peuve nt et . e dNe. mlnces de 18 maniere decfite plus haul. On voir 'lIl ' au ~ bords de tous les dlsques. !l y a une certalne . e,lon
Lea fl~ure& 10 et 11 mont.ent Ie .eseau des Hi nes de force. L' epalueur des Uenes a el~ cholsle proPQr\lonneile • Ja valeu r abSOlue de cu foreu. De en f1~u ' u, U resulte Que les fo.ces a,lssan! sur un seul dlsQue reallsent l'e
Le fait Que les fo.ces Mte. mlnees .. panlr des phQloiraphlea de I' auemblale oMlssent aux conditions d' eQulllb re des dlsQues lndlvlduels , IndlQue Que la methode utllla~e pour determin er lea fOrCes de contact

f'i,,,,, 8 _ u..r. d~. ro. ou .",rres"".don, . I' .... pll.m..,tdel . Pli. 5. L. lom e A co r· •• ,pond i ceUe de I. PlI, ~

Fi~"

9 _ "por. de. rorc"" cor~.pondant i.l ' ..,plle ",en' dela Pli. 7 .

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157

esl correcle. De ",e"'e. on consl81e Que Ie rBj)lKlr! de la hauleur 10tBle d' une epure it. sa largeu r correslKlnd bien au rllllport de la fo rce ~t8le verUc8le 1 Is force horl~ootale . cependant les di",...,slons KbSolues des epu res dCpassent en gcncr8lles "a leurs des charges exterl emes d' envlron 10 'ii. Cecl dolt eIre Ie resoltat d' une crr.ur systematique, par exemple d' une dlrterence entre lea proprietea mecanlQues des disQues de I 'empilemcnt ct celles du dlsQue de compa raison . Celt" erreur a ~Ie compensee par une modification des eche!les des epu res.

f',,,,,,. d. ~

10 • DIIpUornenl de la !"i l. 6 ..ce les fore .. oontact. \. '''p&ls.rer d •• ll",~. e OI propOrt[<>nnelle la ,'&1 .. , abSOlu e du forces.Lalew .... <.'Ort.s""nO

Fi~" .. Ii Dnpll ... "", de la Fl,. 1 avec I•• fo,e es de oontact. L'cp.I .... rdc.II"'uC., propOrtlonn,ll e II. la yol .. r ab-solu e d .. fo,eea.

• celie de I. Fli. 4.

VI.

D1 SC I ; SS I O~

DF: S

RESU LTATS

Normal emenl. la desc ription du comportement mec.niqu e des male,iau. est basee su' les prlnelpes de la m.\eanlque des milieu. eontlnus. Pour eela. l~ materia" Ut ,emillace. en ima~in81ion. pa r un matcriau fielli. donI loutes les proprl~les SOnl des "arlables contlnuu. La transmission des forces dans un tel mat~­ ,iau contln" se d~efll i I' alde de 13 conceptlon du tensem des conua lnles. et les chancem ents de I. geometr le sont Merits. au cas Ie plus Simple. pa, un autre tenseu! du deuxleme ord re. Ie tenseur des deformations. L.es comlKlunles de e e dernier tenseu r sonl obtenuea par de.ivatlon partlclle des Mpllcements. L.e comPOftement mecanlque du mat~r!au esl deedl en eholslssanl. entte les lenseurs des eon, ralntes et des dHormallons. une relation fonetlonnelle teUe que I'on obtlenne la mellleun! co"eslKlndanee 8,' ec des result atS exP/!, lmentau l . Le comportement m,ksnique d'un empilement de disques possed. un caractere tout·"·fait dl f f~renl de celui d'un m8h;rls" contlnu. La transmission des fo.ces de contact. Bussi bien que les depllcements relatifs des dlsQues. Qui prennent 1a place. respeeti_ement. du tenseu, des contralnt eS Ct du tenseur des dMormations d'un mi11eu ~ontlnu. sont des pMnom~nes ~s"entiellement discontinus . La t ,ansmlsslon des forces esl discontinue patee qu' elle ne s ' effeetue. d' un disque l un au"e, que pa r les pOints de contact. D'autre pa.l. ehaque deforma!!un e5t l"odulte par un Illssement. qui n' a lieu q u · en quelques pOints de camBct. PDU'VU que Is dt!forn,ation ne snit pas ucessh·e. Ccs quelques pointS de contact

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Soil Mechanics and Transport in Porous Media

disparaissent et d'autres ~ont Clees. tandls Que la pluparl des pOints de contact est conservt\e. La leom~t,l e du sysleme de dlsQues chanie, el Is descllplion de ce chan&emenl pOS$~de un caractele dlscontlnu II cause de ia ligldite des dlsQues. II est trh dlmc!]e de remplacer ce compO,temenl tres compllQue d'un empilemenl de dlsQues par des phenomenes continua caraclerises par des lenseurS de contraintes el de dMo,malions. On peUI essayer de clracterlse, la transmission des fo,ces par un tenseur de • conlralntes moyennes •. Les compOsaoles de ce tenseur p~uvent eire oblenues en I,scant une lIgne droite ir. trave,s l' empllemem, el en dlvlnnt II ,esultante des tOlCes Irsnsmlses par Ie materlau ir. l,ave,s ceue lillie, pa r la loniueur de cellec!. En lIene,al, la valeur d'une telle composante depend de la longueur de Is IIlne A t,avers lea disques , et osc!!!e aulour d'une valeu, moyenne. SI ces osc!!!atlons sont trop fortes, II est Inepte d'lnlrodul,e ce tenseu r comme mesn .. de la transmission des fo,ces. C'est seulement lo,sQu'on peut IndlQue, des lonllueurs de Hsnes cOllespOndant II des composantes assez constantes Qu'll est ralsonnable de conslde,er Ie tenseUl de contralntes moyennes COmme mesu,e des tOlces Inle'llranulahes. On appelle • volume elemenlalre representalit. Ie plus peUt volume Qui conUent des IIgnes i. compossnles ralsonnablemenl conslantes. Un procede alllebrlQue pour Ie cal cui des compo sanIes d ' un lenseu, de conl,aintes moyennes a e Ie donne par WEBER (1966). On peul obtenl, une Idee Qualltallve de la variailon de la conlralnle moyenne avec III. lon&ueur de la IIl1ne el~mentaire AI'alde de I'~pu,e des torces. Pour cela , II taul nOler Que I'empllem ent et I'epure sont aeomelliQuement duaux. Un certain n
Selected Works of G. de Josselin de Jong

159

I. lo"&ueu. de I. Hlne consld",,;e, pour un certain nombre de lignes pusanl par ce pOint. Puls~ue l'emp!!emen! ne contlent pas assez de dlsQUea. el pulsQu ' !! esl aaaez dtllicile de m~surer lea d"placementa des dlsQues avec suffisamme nt de predslon, les resullat s Que l'on peul oblenlr des photol raphles de l' empllement ne sont pas d'one tres , rande valeur. C'est pOur cela Que nous ne donnerons ~RS lei de conslMrallons plus drltaill"es. Dea diflleulle s usenHelies se produlsen! lorsque I'on veut dedul re tMorlQuemen!, en pallan! du elementalre de deux dlaquea en contact. Ie compOrtement mecanlQue d'un empllement de dlsquea. exprlme en lermes dea tenseurS de contraintes el de dHormations. Actuellemen!. nOuS ne dlspo&ons pas d'un procede mathematlQue Qui permelle de preyol r Ie mouvement d'un empllement de dlsques &OUS l' lnnuence d' one varlalion des char~es extr!rleures. en part ant du comportement de I'unlle elemenlalre (deux disQues en contact). Il est neanmolna possIble de decrl,e. Qualllativemenl, les eUets Qui se produlsent dans I'empilemenl de Is facon suivant e. sysr~me

Dans un empl1emenl en ' qulllbre. te. forces de contact sonl dlStribuees de telle faeon que I'anile de II force avec la dl reetlon no,male au plan de contact ne depuse l'an,le de lroUeme"1 en auCun des polnls de contact , LorsQue les charges eXle,leures varlent. les aniles d'inclinai&on dn fo,ces varlent. et des qu'en un certain point eel angl e altelnt I' ancle de Irottemenl. ce point Qultte Ie It'elme HastiQue. En ce polnl de conlaCI des deformations Irreverslbles au'oot lieu. c'eal-a-dl re que lea deux disQues ,lIsserDnt Pun par rappo,t i I'autre. OU encore Que Ie contact enlre les deux dlsques dlsparaftla complMement. II est possible que I' empilemenl retrouv. son fQulllbre apres une dHorma!ion irreversible tres peme. mals i1 esl ",alemenl possible Que la deformation Irreversible devlenne si irande. que la ,edistrlbUlion des forcea Qu ' elle entra rn e , amene un aulre polnl de contact it passer au regime IrreverSible, etc.. L'empllement ne presenlera des dOfo,matioos ImpOrtantn Que sl une telle deformation Irreversible a He" en au moins un pOint de coni act. II est a noter Que des nOuveaux pOints de contact peuvenl ~t,e c reh pendant Que l'empllement tend vera sa nouvelle position d' eQulllbre. II eSI POSSible aussl QUe I'empilemenl ne parvlenn. pas a une nouvelle position d'eQuilibre. mals que Ie dr!e,oehement des pOints de contact conslitue une .hcUon en charne. Dans ce cas. on dlt que I'empll ement • depasse j'
VU.

_

C O N CL.U S IO N

II a ete montre Que. par I'emplol des methodes de la phOI<>-elasticHe. it etait pOSS ible de dete rmin e" avec suffiumment de preclalon. les forces de contact dana un emplleme"t de dlsques. Les photographies permette"1 de reconslrulre tes Mplacements relalils des disQues. Qui constitue nl lu deformations leo meIri~ues de l'empilemenl. La mesure de ees dHormations a deja ete faUe par beaocoup de ehercheurs, ii I"aide de l'etude d' un empllement de rouleaux en aluminium. Cependant. des experiences aur un empllement de rouleau~ ne donnent aucune Informllion Su r les fo.ces a l' jnte rleur de ]' empllement. Nous avOnS j' impre&slon Que des recherches. teiles Que celles decriles leI. sont necessaires pour comprendre Ie comportement mecanlQue d ' un emplleme nt de dlsQues.

_ 85-

160

Soil Mechanics and Transport in Porous Media

IJIBLiOGRAl' lIl E

J_ BlAREZ _

Contrlbullon ll'.tude des proprletb mecanlQues des sols et des mate,laux Pulverulents. Thhe de Ooclo,al. Grenoble, 240 pp .. Louis-Jean. Gap. ]962. J. BlAREZ et K. W]ENDIECK. - RemarQue sur I',!,!asttcite etl ' lIllsotrople des materlaux pulverutents. C.R. Acad. 5<:. Paris. 254. P. 2712_2714 . 1962. J. BJAREZ et K. WIENDIECK. - La comparaison Quailtative entre I'anisotropie mecanlQue el l'aniso_ trople de structure des mU ieux PJlverulents. C.R. Acad. 5<:. Paris. 256. P. 1217_1220. 1963. J. BOUSSINESQ. - Des perturbattons lo<:.ales Que prodult au-dessous d'elle une fort e charge. etc. C.R. A<:.ad. 8<:. PariS, 114. p. 1510 -]516, ]892. P. DANTU. - Contribution .. I'etude mecanlque et geometriQue des m1l!eux pu]verulMts. Comptes Rendus du IVcme congres de MecBnlque des Sols et des Fondatlons. P. lH-148. Londres . 195"1. G. DE JOSSELIN DE JONG. - Foto-elastlsch ondenoek van korr elst apelln ~e n. L.C.M. Medcdcllngen, 4, p. 119-13 4, 1960. C. DE JOSSELIN DE JONC. - Statics and kinematics In the fallable zone of a i,anula, mate,lal. OoCIO,'S thesis. Delft, 119 PP., Waltmar Delft, 1959 . M. FLAMANT. - Sur la r"partillon des presslons dans un sollde rectan gulalre charie transversalem ent, C.R. Acad. Sc. PariS, 114. P. 1465-1468, 1892. M.M. F'ROCHT. - Pnotoelastlclty, vol. I, 411 PP., NeW -York, wlI ey , 1941. M.M. F'ROCHT. - Photoe]asUclty, vol. 11. ~O~ PP .. NeW-YOrk, Wlley, 1948. T. WAKABAY ASH!. - Photoelasti<:. method for determination of Siress In powdered mass, Proc. 7 Ih Japan National Con,ress for Applied Mechanics , P. 153-158, Tokyo. 195"1. J. WEBER. Rech er<:.h es concernant les contralnle~ In1er,ranulal . es dans I~s milieux pulverulents. AppllcaUon a la rheologle de en mllleu •. Cahl e rs du Groupe F'ran~als de RMolo gie , n' 3. t. I, P. 161-170, 19-66.

Les auteurs adressent leu.s remerclements ii. Monsieur J .C. Baas, assistani au Laboratolre STEV1N de l'Universite. Qui a effectu,; el
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PART II FLOW AND TRANSPORT IN POROUS MEDIA

"

'.

"\" .,

.

',

,

,

~

~

)

y,

IIJ

I """"

J ff'tO~

162

I"

<1_~

...10. , r:

Soil Mechanics and Transport in Porous Media

4 Flow and Transport in Porous Media

4.1

Introduction to Flow and Transport In Porous Media

When st udying a certain problem, Gerard de Jossclin de JOIIg's way of approach was to create a clear visual concept of the mechanism of the underlying physical process. Each time this WI\8 the most im))Qrtant step in his resean::h. He could make 110 progrCSll without it. In t he early day~ of his career only quite pri",itive two dimensional visualil,ation technique!, basically Hcle - - Shaw models, were available to sec what act ually goes all inside a porous medium. Nowad ays, advanccU three dimengional computer aided techniques have been developed to study flow processes in porous media. An example is given in Figu re 1. It shows snap shots of a fl uorescent dye which was injected into a homogeneous porous medium, occupied by a single fluid moving at constant average velocity. The tracer, initially prescnt in a small ball, dearly develops into an ellipsokl which is elongated in the direct ion of flow and symmetric in the transverse directions. Also note that the ellipsoids are perturbed hy the micro strucUire of the poroll>! material. Jos must have had this p icture in mi nd when he developed the concepts of tracer d i>lpersion in t he mid fifties. In his Dijon (1957) and AGU (1 958) papers he W3ll t he first to prCS€nt a quantitative analysis of the obsc:rvcd longitudinal aud t rausverse dispersion. As a first dl3.racteristic step he visualized a porous medi um as a network of randomly oriented tubes (cana~) of fixed length. Of course he supported this concept by adding a number of artistic impressions of pore systems in the paper. Further he intrOOm:ed a probability distri bution - a choice of path - in the analysis. This second critical step asscrtS t hat a particle arriving at a junction betv.'OOn tubes has a probability to move in a certain direction. He snpposed that this probability is equal to the ratio of water Howing in that direction. The combination of these two steps leads to a scattering mechanism which HlliCmbles a Brownian mot ion with a >luperimposed com'ection in mean flow direction. Following the classical work of Chandrasckhar he then developed a probability density function describing the probability t hat a particle, after N steps, arrives in a given infinitesimal volume during a given infinitesimal time interval. Once this probability densi ty funct ion is know n, he computed t he longitudinal (flow direction) and transverse standard deviations and ~be[Cby quantifying the longitudinal and trans verse dispcrsivitics in terms of the system parameters.

163

4.1 Introduction to Flow and Transport in Porous Media

,. '00

•

•

'00

Figure4.1; Di.!per&ion in a poro1l.! medium. The tmcer UJlI!rime .. t was carned out by Michael Rohr of the Institute for Environmental Physics of the Uni1HlTsity of Heidelberg (2001). We acknowledge Profe88or K. Roth for aI/ouling IU to use thi$ material.

From his approach it is d ear that he considers tracer transport in porous media as a discrete process. Unlike many of his colleagues ill t he field , Jos was never very much in favor of a partial differential equation of convection· diffusion type for the tracers. He describe; transport in terms of integrals based on probability density fUllctious describing t he local phenomena. Much later, in his Socorro (1970) report, he considered uacer dispersion again. In this unpublished work he introduced the concept of Elementary Conveyer Unit (ECU) as opposed to the well-known and much used Representative Elementary Volume (REV). An BCU cal"ries the direction of flow and is char· acterized by repetition and by independence. The latter means that a tracer or fluid partidc has a <.:hok<.l of path at the exit of the elementary unit, which is independent of the situation at the eutrance of the uni t. As a result, again a scattering mechanism results. Using ECU's as elementary units he collllidered in the Socorro report tracer dispersion in fissured rock and he was able to give a complete description of t he Jispersion mcchanism. Salt-water intrusion arises naturally in many coastal regions, and in particular ill a sull sea-level country as The Netherlands. As a consequence a t"l:!rtain tradition was established in t he Dutch hydrological community, in which various kinds of fresh·salt groundwater models were developed and studied. One of the issue!! in t h(~ fifties was the existence or non-existence of a potent ial. It was known that single phase flow of constant density can be considered as potential flow , in the sellse that the specific discharge is proportional to the gradient of a potential (Darcy Law) . Also interface models, with an allrn]>t transition between the dell.~itit'lol or fresh and salt groundwater, can be handled by a (pseudo) potential. But the general case or a smoothly varying dens ity could not be put

164

Soil Mechanics and Transport in Porous Media

4.1 InuodUCIion 10 Row and Tran$pon in Porous Media

"

form , in spite of several severe a~tempts. In his AGU (1960) paper on '~illglllaritiC8 ill multiple fluid flow ', Jos was one of the first to give a proper mathematical formulation of variaole density fluid flow in porous media. In this paper he derived his famous stream function equation in

po~ential

6.tI> +~ify=o I'

ax

for two dimcn~ional incomprC!lSible 1101'0'. He gave an equivalent formulation in terms of pressure as well. In particu\a.r he pointed out that a sharp interface approach leads to a. vortex distribntion along the interface. This ex plains quite elegantly the interface shear flow and the resulting motion of the fluid bodies. Further he introduced the com plex potential as a tool to obtain the specific discharge in a more direct way. In this way he was able to formulate and solve initial bonndary value problems for variable density and for variable viscosity fluid flow. In particular he treated the case of an initially vertical fresh-salt interface in a horizontal porous layer (aquifer); he computed the oorresl)Qnding rl i!
Selected Works of G. de Josselin de Jong

165

4.1 Introduction to flow and Transporl in Porous Media

"

had the rare talent to seleet a characteristic feature out of the complex reality, to create an innovative idea which was then confirmed by experiment. That is true science and that is what we have to teach our studentll.

166

Soil Mechanics and Transport in Porous Media

\'O>.U,U 65. No. II

Nov .... n 1960

Singula rity Distributions for the Analysis of Multiple-Fluid Flow t hrough P or ous Media G.

D~ JOSS£LIN DF. JONG'

/n&titl
U"i1Hl.';tll of California, Bc.kewll, California Alntrae!' In Part I the simultaneous flow of fluids of different propertie~;a treaUld by substituling t.heae lIuid'l by one hypothetical fluid and applying singularitie. lit th,*, points where the propertieo of the IIotual fluids cbange. Their mBgllitude is ohosen 10 that the specific dillcharges in the hypothetiCIII fluid are everywhen! ident.ieBl to the specific di!IC:harges in the actual fluids. Tho How in the hypothetical fluid C"-n be determined by potential theory from the trAnsformed boundary OOI1dition. and the influence of the .ingularitie • . For the determination of U,e discharge a strellm function is ulled which contains singubrities in the form of vort.iee •. For the dctenuination of the fluid prt'llSU!"CS a multiple-fluid po_ tential is defined which contains siniUJarities in the form of lOurce and sink distributions. The IItream and the potential functio"" eseh oombine with su~iliary, mtlny-v alued functions t.o form oomplex potentials. Th""" permit solutions iu the form of one integral in comple~ vari_ ables, valid for any point in the ectire field, il"rl'!;pedive of the fluid pre!lent. TI'e .tOlution for the transition .one between fluids u well lUI the abrupt interface i.B elaborated. In Part 2 the two-dimenBional example of an infinite, confined aquifer with an initial vert.iCIII intcrfaoo between t wo fluids of different specific weight i.B elaborated, giving lUI a result the movemen~ of the fluid. in the entire field a~ the first. momcnt and a fim a pproximation for the rotation or the interface around the oenter a8 II function of time. Theile resultll are verified by a parallel pls te model and an electric rc.sist.ance model. In the lalter model the vortices are replaeed hy IIOnrCCS for the tracing of IItreamline8 and by IOUrct!_ Bink combinations lorming doubleta ror the potentisl lines.

NOTATION

• ,b d

J I.

, - v'=t

• m

"p q

q" q., q., q.

q" q.

4=q.-iq.

,•

Time dependent coefficient describing inclination of interfaee in example Breadth of sb·cam channel ILl Hair height of aquifer ILl Slot width of parallel plate model [L] Thicknll!lll of two-dim(,llSional aquifer ILl ThickDCSII of two-dimensiolUll electric resistance model ILl lmagins.ry unit [Amp L-'] Electric current density vector Specific penncability of aquifer [L'] Integer number Coordinate perpendicular to the interfate ILl /FL-o] P ressure in the fluids ILT-' ] Spedfie di!IChatge Specific discharge compononre in 3, n, 1', 11 dircctiOIlll ILT- '] ContinuoUfl, d iseon tinuous <'omponen18 01 q 011 interfate IDT- 'J Complex specific discharge ILT-'J Coordinate along the interface ILl Time ITI ILT-'] Mean velocity of the fluid Horizontal coordinate ILl

• Since August. 1960 at Technische HQgeIIChool, Delft, The Netberlands.

3739

167

3740

G . DB JOSSELIN DB J ONG

,

[I,,]

: - z+ iy

ILl

A,dA

[V]

A,B,C,D,E,P

«

,-

!yoU]

!Yo/I] [Amp]

I 1m

J,K M N I'

Vertical ooordiMoo Complex ooordill.'l.te A~

Pointa on boundary Electric poteut.i&l Electric potential in current aupply system Electric current Imaginary part of complex expl'C86ion Int.el'!lection point!! of interface and boundary Center point of interface in example Image of M in plane Arbitrary running point Arbitrary running point in r plane Discharge in aquifer Contour or path of line integral Unspedfitd functions of z and y Inclination of interface to horizontal Auxiliary variable Specific weight 01 fluid Doublet distance PorOllity of aquifer Complex coordinate of transformation Strength of lIOuree per unit area Strength of doublet per unit area Dynamic viaeosity of fluid Specific electric re;;istance Exterior angle BP,E Strength of vorticity in z, " plane per unit area. Multiple fluid potential Doublet discharge Auxiliary function Specific discharge stream function Complex specific dil!cha rge potential

r

e, 10T-II

Q

s

ILl

V, IV

" P

[PL-o]

7

'"

• r .. •

ILl ~+ ill

iT-I] iLT-I] [PTL-'] iOhmLi

>"

",

•"e

A"

•

~

U=

ot>+ i'l'

iT-Ii [VT-li [OT-I] i/,oT-II [LIT - 'i

SlIlw:ripts

0 I II 1,2

"SliperlCripts

Point, where a singularity ia present, or point of interface containing a si ngularity Pa.rt of complex potential accounting for !!ingularities Part of complex potential aeeounting for boundary oonditions Pertaining to Duid 1,2 (1 is light, 2 is heavy) Potential functioWl convenient for the stndy of viscosity influencell with electric resistancc analogy In vention for electric n:Ililltanoo model ('racing of stream linea) Doubleta in electric resistance model (tracing of potential linea)

"

PAIIT 1. DERIVATION

Introduct ion. In the study of ground-water flow a theory of the interaetion between differUuids needs to be developed ao tlm~ any arbitrary boundary value problem may be solved rigo rously. Many hydrologil!ta have deen~

168

voted their sttention to this subject, because in sevcral fields of hydrology important p roblems are created by the presence of two Huids. The IIOlu tioll!l obtained, however, are mainly limited to C8.SC!l which yield certaill geometrical approximations. It is the purpoae of this

Soil Mechanics and Transport in Porous Media

MULTlPLE-FI.UID FLOW THROUGH POROUS MEDIA

paper to describe !l. method which penuits the determination of the behavior of two or more fluids, for any form of distribution over !l. fIeld with arbitrary boundary rondition!. Badon Ghijben [1888] and afwrwards Herz· ber{l [1901] stated the hydrostatic equilibrium for a fresh water lens floating on top of salt water in a porous aquifer. Since then several authors have considered the more relevant casc that either one of the two fluids or both are in movemcnt. The first to have gi\·cn a correct account of how the movement of the fluids in· fluencell the behavior of the interface SCCIllll to have been Lorentz [l!H3} . His discussion of the upeoning of salt water unde r a well was lawr inn~stigated ellpcrimentally by Muskat [1937]. Hubbert [1940] established a general desc rip-tion of tho flow of two fluids at either side of a steady interrace between fluids. By applieation of these considerations to a stati onary oil de-poeit above an underlying body of Bowing wawr, the tilt of the interface was derived for a one--dimensional flow system [Hubbert, 1953 ]. Edelman [194lJ] determimxl the shape of the interface between moving fresh water and stationn ry salt water in a dune area adjacent to the sea by use of a graphical flow net analysis. A mathematical solution was obtained for the case in which vertienl flow componenls could be ignored, adopting thQ Dupuit--Forchheimer :la' sumptio". Todd and 1l11i$man [195!l) applied Edelman's method for detennining the influence of recharge and pumping operations to limit o.·erdJ"8.ft in tho drinking water supply aJ"C3 of Amsterdam. If only onc Buid is in motion, the hodograph method can be applied becausc the interlace is circular. Applications of this teebniquc for pa.rticula.r solutiollll have been carried out by llarder, Simp&an, Lau, llote3, alld McGauhey [1953]. Kidder [19Sfu, b], Glover ( 1959], and Henry [195!l]. When both fluids are moving,

the interface, when mapped into the hodograph plane, takes the form of an unpredictable curve, so that the mapping proceduro Oll.lInot be ap-plied. An approximate method based en the Du puit..Forchheimer assumptions was developed fo r one case by Diet~ [1953]. Model studies provide thc only me&llll of studying the location and movement of inter. faces fOf general problems invo!>"in!! movement of both fluids. A ]lllr:"lllel plMe model has been

Selected Works of G. de Josselin de Jong

3741

employed for this purpose by Sallting [ 1951], and sand models have boon used by J/arder, Simp8on, Lau, J/ote3, alld McGauhev, [1953]

and K eukgan [1954]. None of the publieatiollS indicated above C(ln· t:lins a mathematie,"l method by which the cor· rect computation of the movement of both fluida lUI derived from given boundary conditiona e:m be made. The first work which BUg· gested a solution to this problem was an un· published study by Edelman in 1957, 'O rondo wateratroming van een niet homogene vloeistof.' In this study he introduced a concept which proved inspiring for the de\'elopmcnt of the present theory: the replacenlent of the two fluids with their own characte ristics by onc hypothetical fluid wit h the same properties o\'er the entire fidd. In this hypothetical filii!! a. row ef sources coinciding with the posi tion of the interface takes care of the change in properties of the two fluids. EJclman determi ned the source disLribution in such n. lI"ay that the .'elocity field created by these sources is equal \.0 the real velocity dist ribution in one of the two fluids. The velocities in the other fluid can be obtained from the velocities created by the sources by Ihe addition of a fictitious vel ocity. This can· cept was developed only fo r a horilOntal inter· £:\(:c, Il.lld the absencc of sources along a verti· cal intcrf~ce was inferred withollt further proof. The present work [de Jon e/in de Jong, 1959] was based on Ihis coneept of replacing the two different flllids by one hypothetical fluid and of introducing the different fluid properties by singu larities along the interface. In determining the character of the singularities, however, the aim Wall a singularity distribution that would directly create the actu:u velocity distribution in both fluids. By this approach the treatment of the two fluids becomcs equivalent, and !Jc. cause of this equi,·alency the extension to more fluids can be made without further complica. tions. T hc present theory shows tlut it is pas-sible 10 meet these requirements with two kinds of singlila rities, vortices or sou rces and sink6. The choice of the kind to lIJle depends on the objective of the stndy. If the objective is the detennination of the discharges, it is C(lnvenient to introduee vortices. 1f the pressure in the fluids i~ to be dctennined, it is more convenient 10 usc the sources n.nd sink6. Furthennore, in the pres· ent study any inclination of the inlerfnte and

169

"42

C.

lhe cue of a gradual transition wne

D!:

JOSSElJN DIll JONG

~re

eon-

aidercd, 118 well ft!I the introduction of boundary conditionll. The importAnt advantage of the ~ingularities

is the poaibility of rolving any boundnry value problem involving two or more fluids by application of the established methods of potential theo ry for the 1!01ution of boundary value problems of one fluid. Although the singularity method i! completely gene ra! wi th regard to differences of fluid prope rties, the present wo rk derus mostly with the inftuence of differences in density only. The effect of differences in viscosity ctln 11180 be reprcsentOO in terms of aingulll.ritiCII, /Ill indicated, but the procedure is not amenable to mathematical treatment. How an electric lUlalogy can be used to account fo r viscosity dilTe rences in connection with singula rities for the density diD'erenee3 will be shown at tbe

end of Part 2. Ph I/fico! (l.$wmptWII3 oM bo.Iic eqt
170

,

..

~

f p."'''

p

-• ,

1 C-____________

~ .

Fig. I. Coordinate .y&tcm. g!lp6 enn cxis~ ill either !Iuid or lit the inte rfllce. The IlCco nd is thll~ the pressure distribution in the interior of the system allaH be continuous. T his oondition, in which the occurrenoo of ste ll!! in pressure is avoided, is dictated by equilibrium oonditions. It is convenicnt to oonsider only thoee variabl~ which poIIIJCS8 magnitudes to be determined indcpendenUy of the fluid propertiee. T he variable!! are preesure in the fluid p aDd specific d ischarge q (q is e times the mean velocity ii of the liquid , where e is poroeity). To relate p and q, the forees which drive the fluid through the ground " 'ilI be equated to the reaiatance offered by the pore system. Conaider in a vertical eeetion of a twCHiimem.ional aquifer of thickntM I, a small cube (d. dn f) , where • is arc length along a n arbitrary but fu:ed curve, n is the perpendicular to that curve, and a is the direction to the horisontal :r c~ ord inate (Fig. 1). The force actiug on the fluid in the, WIlletion consiat.ll of IL pressure component

-(apia,) ih do I :Lud a gravity com j)Qllcnt

--y(all/IJI) ih dn I By Darcy', law this force ill counteracted by a militance offered by the pore 8y8tem, which is proportional to q" the • component of q. T hiq requires a driving force of magnitude

+iJ./ k)q. ih do I The equation for these three fOn:e!!, afte r division by the clcmentary volume, is

f,p/ k)q . - -(ap/ a.) - "((aIlN.)

(1)

Similarly, in the n direction

f,p/k)q. - - (ap/iJn) - nall/ iJn)

(2)

Soil Mechanics and Transport in Porous Media

MULT I PLE-FLUID FLOW TH ROUG O POROUS M EDlA

'The hio requirement. mentioned above can be written i.o tennl of the variables q and p. The condition of continuity requirtll that di v q _ (aq./ a!)

+ (oq./an)

f. y aw ••

tU

(3)

The condition of equilibrium requires that p be a lingle-valued funetion eontaining no jumJl8. Thill may be wri tten in two ways:

(iJ'p/i)! im) -

(o'p/an 0') or

th = 0

(4)

The contou r integ rnl applies to any clOllCd contour S. &lmitm willi IFOrtict.. If the objective is the determination of the discharge q, it is con,'cnicnt to introduce a , tream function i', whose derivat iVCIJ are tbe component. of q according to

(n /an) -

-q.

«Nt/a.) ...

+q.

+ (a"ir/a. on)

".- 0

(6)

which im plies that it is a single-valued function without jumps. The condition of equilibrium may be used in the contour integral form of (4). Wri ting (1) u a Bimila r contour integral l'C!!ults in

-f .

(p./l)q . ds -

(a pta.) ds + cj!"r(al/ta.)

ds

SubBtituting for q. with (5) and using (4) reduces this to

f.(p/ l )(n

V

v'w d~ dll

area A encloeed by the contour S. If the quantit ies V nnd IV, whieh represent Any two funetions of ~ and y, a re nl)plied to the it term of equation 7 with V _ ,. nnd IV _ it, ,{

,. (Nt

r. k an

tU

if ("az '"az + ;"oy ..oy) 'd, - i If.. ,. v'.., _1.I:

(5)

it iI merely a function ... hich generates the llpeeifie discharge without the material propertiC!l .., or ,. beint involved. Therefore, ita inftuence on the Duid motion is equivalent to whatever Iluid is preeent and may therefore be oollllidemi to be related to the hypothetical fluid. Introduction of (5) into the continuity equation (3) dlo .... tbat

-(a"ir/an a.)

- fL

The lIurfaoo inU!gral extends over the whole

= 0

cj,(OP/ 03)

3743

..

dz dy

To apply Green', theorem to the .., term of (7) it iI neceesa.ry to convert all/a, into an e,,~on involving the pIlrtial derivation of II. ~ , and n are pcrpendieuJar in llUeh a .....y tbat II makes the lIIlIIle angle with IJ U ~ makes with " a,l/o, is equal to -az/ilft. Therefore the "r U!nn of (7) beoomee, lrith V - .., and

w-

~,

- f .., -d, 1iI

."

+ IL "r "1 ' :1: d~ dy

(7)

The conioor inl.egn!.ls in this equation may be re piaoed by IUrface inWgrals by use of Green" theorem, whieh in two dimensions has the form

Selected Works of G. de Josselin de Jong

(9)

This red ueet to !f.. (a..,/a:1:) ~ dy becaUllO (oz/a~) - 1 and (<Jz/a,l) _ V·~ - O. Combi ning (7). (8). find (9) retultll in, for a ny llmall area A, i.e., for every point of the inU!rior,

V 'it __ !!h ,.a~

/iJn) tU - cj! ..,(ay/o.) ds

(8)

_! ,.

(a,. 8it + OjJ.oyayiNf)

(10)

a~ih

Thill i, known in poU!ntial thcory to represent a singularity of \'ortidty ... ( 11)

171

3744

G.

DB

JOSSELIN

Fig. 2. Vortices on interface bel,,'een two fluid.

O~

JONG

Now (k-y/jI.) is the epecific disclmr(l;e for II. head gradient of 1. 10 geneml, Buch high dillcharge vclocities al'1l not pl1)llent; in problems of I!e& water intrusion, gradients in head will be of the order of h. - "'{')/"'{" which are small with respect to 1. Therefore, an analysis in which t.he viseoaity differences are ignored is justified in such problema. Because, howevcr, the magnitude of the terms also depends on the orientation of t.he interface with l'CIlpect to the horizontal and the vector q, it should be verified that their relative importance is not changed because of the sine of the angles involved. In thoae CR!IeII in which the vll.fialion 01 jI. may be disregarded, equation 11 redullCll to

(oy,<,),.), clockwiec rotating in pointa where ')'

v'w

inncllSCl! in z direction.

The first term on the right side reprcsents the tendency of an interface betw{'(!n two fluids of different spceific weight to rotate toward the horizontal position in such a way that the heavier fluid underlies the lighter fluid. This is clarified in Figure 2, where the heavier fluid y., represented by the darker arca, is to thc right of the lighter fluid y,. Here (ily/il:!') is positive. At aU ]Joints of the transition zone between the two fluids, vorticity of ncglllive sign exists, which, according to potential theo ry, impo!!C!l clockwise rotation. The second term on the right sidc of (11) describes the vorticity caused by differences in vise(ll;ity of the tll'O fluids. For simplicity in the !lUbsequcnt trc.'ltment it is convcnient to consider onl}' those e:L'les in which the gravity tenn dominates the viscosity tenn. A compllrison of their ~I)(!eti\'e cont ribution to the vort.icity may be m~de by writing (II) as '" = _k')'oln'Y _ (ol nll /J.

oz

oX

q, _ olnll q,) oy

Iu the study of p roblems of sea water inlnlsi on the propcrtiCl; of fresh water (fluid 1) and salt water (fluid 2) ~re gil'en approximately by the ratios

"'{.h,

= 1.025

jI.,/p., = 1.070

'The componenta of the gradienta (iJ In "'{/iJx). (iJ In jI./iJx), (iJ In jI./ iJy) are therefore of the 8IIme order of magnitude, and the relative importance of the terms del)(!nds on the mllgnitude of q with respcet to the value of (k'Y/p.).

172

= -(k/ll)(fry/iJz)

( 12)

Since the distribution of "'{ is known, the righthand side of (12) is a known function of position and this relation for ..y ill of the type called Poisson's equation, whose solution in terms of vortices is known from potential theory. The IIOlu tion can be written lie

..y = -

fL

(k/2.-p.)(fry/iJx) In T dA

(13)

whel'1l T is the distanee betl'l'een the point of conaiderntion and the point where the vortex of strength -(k/ jI.)(o"'{/ iJx) is pm;cnt. Integration 11M to be effected over all vortiCCll, i.e., the entire area A, where (iJ'Y/ox) "# O. -V is regular in the entire field Bnd also at a point containing a vortex. This is true because, although In T becomes iufinite for T approaching Ulro, the product In r· dA redullCll to zero, since dA is 01 the order of r'. The magnitude of the vortex at a point of the field depends only on the Chllnge in properties of the fluids present at that point. The number 01 fluids involved in the problem is of no importance, and the method is therefore suitable for computiog the flow of several fluids and the wnes of gradual tranaition between them.

Treatment of vorticea b y complex votential. To IIdapt this flow problem to a treatment by complex variables tile stream function i> may be combincd with an auxiliary function 4>, so th~t

_oi> _ _ q. ( 14)

"

Soil Mechanics and Transport in Porous Media

MULTIPLE-FLUID FLOW TH ROUG H POROUS MEDIA The condition of continuity (3) then gives

'V''''

co

(lfi)

0

and from the condition of equilibrium, which fin:!.lly resulted in (12), it follows that

a' ",/ a3 an - ,i/ iJn iJ • .,.

'11'1<

= -(k/ ".)(iJ-r/ iJx)

(16)

Equa.tion 16 aholvs that is not si ngle-valued and therefore has no pa rticular physical meaning. However, the complex combination, to be called 'complex specific discharge potcntinl,'

n - +i>l< ~3Use

g '"'" (- ik/2r".)

.fL

lim (iJ-r/ iJn) dn = -(y. - 'Y ,)

(iJ-r/ax) •• In (: - :,J dA

(18)

Here z is the point of consideration and z. the point containing the vortex. The integration extends over all poin\.!! Zo, where iJ-y/ax ~ O. The expression (18) gives the correet value lor 1', mnce In (z - t.) can be separated into i9], where r real and imaginary parts 1\8 lin r is the distance between z and Zt and 9 the a ngle with the horizontal. T herefore, In r appears in the imaginary part 01 g in the same way as in (13). The real part of g consists of the angles e and is therefore mnny-valued. This manyvaluednC6S, however, only reflectaon the auxiliary function <1>. From g the movement of the fl uid is obt..lined aa the complex specific discharge
+

(20)

,.~

( 17)

the solution (13) call be wri tten as a function of the complex v3riables z = x + ill, as follows:

is convcnient

3745

the analysi.9. The expression (18) lor ft and its derivative are therefore usable instruments in the analysis and give correct a nswers for every point in the entire field, even though the point contaiM vortieity. Vortice, on an abrupt interlace. An abrupt interface may be considered to be the degeneration 01 a t ransition wne betwccn t wo fluid'! of s pecific weight y, and "1'*, respectively. The gradient of y in the zone tends to infinity 118 the thickness of the zone approllches zero in such a way that in the limit

Let, be the coordinate along the interface, and n the normal; then the elementary length of the interface corresponds to an are3 dA

d8

=

dsdn (shaded part of Fig. 3). T he contribution to U,e complex potentilll created by this p.'Lrt of the interface is dft = (-ik/ 2r".)(iJ-r/ az)" In (, - :.) d8 dn where Zo is the complex coordinate 01 the center of the area. If a ill the angle of d8 with the horizontal x llxis, (iJy/ 87;) is equal to - sin a (8y/an) . By usc 01 (20) the complex potcntinl ~omC:ll

dj]

=

( - ik/

27".)(y. - 'Y ,) . [(ain a) •• In (: - ,,J d8]

The tenn between brackets varies along the interface, so that the complex potential origiMted by the whole interface is gi\'en by Sl ... (-ik/ 2r".)('Y. - 'Y,)

.L

(s in a) " In (z - ",) d8

(21)

i, dO iJz

'" ,.

dO

'"

, (19)

The kind of fluid pre.. is many-valued, ita derivat ives arc single-valued and constitute no difficulticls for the determination of i/o Since the discharge is the ohjective of the study, t he aw kwardness of the auxiliary fuaction is of no importance in

Selected Works of G. de Josselin de Jong

.......... . .. !

dy" ..........•...••....

1, 1"ig. 3. In terface 01 infini!.csimallhieklle89 dn.

173

G.

374.6

DE

JOSSELtN

The integral is a line integral running along a line S, which coincides with the interface; the distanee d. represents an incremental length along this line and Zo is any point on the interface. The interface can be represented fill a vortex sheet composed of vortex linea. The interseetion of this interface with the:l', iy plane, which is perpendicular to it, is the line S. As (sin a),. da is equal to dllo, the vertical component of d$ (see F ig. 3), the strength of the vortex contained in d.! is equal to '" d.! = (k!J¥r. - 'Y,) d~o (22) and the complex speeific discharge potenti"l may be simplified to

n=

(-ik/2:r~)("Y1 -

"YI)

.L

In (z - z,,) dyo

(23)

Treatment 01 a boundary vulu~ probfem. T he foregoing introductioll of the stream function '" permits the solution of boundary value prohIeill.'!. Besides the knowledge of conditiOIlli along ule boundary, the position of the interface between the Huids also hilS to be known in order to specify v ove r the entire field. Then the motion of the fluids can be derived for a time interval short enough to negleet the displacement of the interface. From the motion of the fluids t.he movement of the interface, and subsequent positiOIlS, each having a difl"erent How pattern, can be determinod. For the compul.:lt.ion of v at a given moment it is con,·cment to treat tbe influence of the conditions along the boundaries and the velocity created by the interface separately. The superposition principle allows the addition of the particular solutiollli afterwards. The boundary conditiOIlli arc generally given in terms of pressures or discharges. Let 3 be a coordinate running along the boundary with l'! perpendicular to it. If the boundary condition con!ists of discharge, the component Q. is known, and, from equation 5, (a.p/a3) is detemlined. Integration along' gives the value'" for every point along such a boundary. If the boundary condition consi8ts of pre;;sure, it is possible to determine q. along th at boundary ~·ith (1), since p, y, ya re known functions of the coordinate ,. The knowledge of Q., with (5), gives (ai-lUn).

174

DE

JONG

Once the boundary conditions are tran.siated in tenus of the stream function ... , the solution of the problem can proceed lIS follows. The actual solution v is broken down into two parts, to be called land II, respectively.

I take;;

c~re

+

'1'" of the vortices

>I' "" '1', I'~rt

-(klp.)(irr/ih)

'V"'I', -

(24)

(25)

with the solution (13). F rom th is solution the values of '1', :lnd (iNrl/iJn) elln be com puted along the bou ndaries. In geneml, thClie values ",·ill not coincide with the boundary vslues 'I' or (a'll/iJn) required in the problem. The differenCe!! ('I' - '1',) and !(iNr/an) - (iNr,/an)] then constitute new boundary conditiollll which can be IItltislied by part 1I of the !IOlution. The quantity 'I'll must satisfy the deficiencies of part I at the boundarie!! and (26) over the entire field. The detennination of ifl/ constitutes So boundary value problem, with a unique solution to be obtainod by known methods of potential theory. The addition of (25) and (26) ahows that the superpoeition ('II, 'lI'f) eatisliea the requirement (12). It will prove convement to U!lC image vortices of opposite sign in order to create vaiues of if, = colllitant along impermesble boundaries so that, for part II of the sol ution also, these boundaries remain impermesble. Th e thecr jWw at a... abrupt intliTlace. A Rheet. with ~ine")llrities is known to eont!\in rli~_ eDlltinuitiCll. llllltead of showing this by evaluating the principal Cauchy value of the integral expression in (23) as the interface is approached from either side, a sho rter presentation may be given, using equation 12 as a st~rting IlOint. Gauss's gradien t theorem applied to it for a smn.ll reetangle d~dn straddling the interface gh·es, for .:l ,·anishiogly smail width of dn,

+

JJ "9'\rt

d3 dn .,.

f

(iN< jiln,) ds,

+ f (iN' janJ ds.

(27)

where 11.. and n. are the normals to the faces d~, and d,. pointing toward the exterior of the rectangle. These coordinates are related to n

Soil Mechanics and Transport in Porous Media

MULTI PLE-FLUID FLOW THROUCH POROUS MEDiA

with a direction (Fig. 3) according to

+n

~

+11,

+n

=

9,

-n.

n

'"'

\.

'l.;,\

....

From the surface integral of (12) aud the results of the abrupt interface analysis it follows that

.. '

f [(: ),- (: )J" ~ -; ff ~"dO f "" - ; ("Y. - "y ,)

3747

'"

lI in a d.s

... \. \.

A is defined by ( 14). Since, however, in this case, '" is manyvalued, it is an unsuitable function for the detennination of the pTCSllure. If the pressul'C!l are the objeetiyc of the study, it is therefore

Selected Works of G. de Josselin de Jong

Fig. 4.

D isch~rge

discontinui ty at an interface.

con~·enient

to introduce a. quantity which has the eha.racte r of but docs not have its manyvaluedncss. Let this quantity be called 9, the multiple fluid potential, and be defined as (p./k)9 ... p

+ "yy

(30)

where the values of p, 'I, Y aTe those of the point under consideration. Since all these quantities are single-valued also, e is lI. single-valued function of position in the entire field . Taking the derivative in .'lJI arbitrsry direction I, it follow! for p. == constant that

(p./k)(iJ8/ 01) = (iJp/iJs)

+ "'1(0'1/08) + y(o"Y/O&)

Application of (I) reduces this to

(p./k)q, = -(p./k)(iJ9jiJ8)

+ y(iTy/iJs)

which may be written

+

(p./k) q "" -(p./k) grad e 'I grad"y (31) since I is any direction. A comparison with (14) shows that the relation between e and is

e-

(k/,.)y grad"y (32) where the last term takes eare of the many_ valuedncss, so that 0 can be single-valued in the entire field, including the region of vorticity. At the points, where grad 'I is zero, i.e., the points of no vorticity, the quantities and 0 are the same. Application of the condition of continuity (3) to (31) gives grad = grad

175

G.

3748

DE

JQSSELI N

D>:

J ONG

follom. Consider a line n perpendicular to the interface pointing from the Iluid to the -y, Iluid (Fig. 3). At the point of intel"!lection with the interfaOO, lI is a const.ant 11', and the source dilltribution is 11' div grad .,. = 11. \7ly. Since.,. varies in the n diredion only, \7..,. ill equal to iJ'"(/iJn·. Let the inwrface consist of two infiniteeimal bands of width dn at a mutual distance 6". Let (i)-t/on) in the region between the hands be consta.nt and equal to -(.,.. - .,.,)/lI", and let (o....,/iJn·) do in the two bands be equal to ±(-y. - y,)lI". Then the value of y will indeed dccrcaoo from "yo to "y, in tho region o f tho interface . The 'quantity (34) then becomes

"'t

.......................................... .......................................... Fig. S. Doublets 00 in terfllce between two fluids (""<"l'.l. Doublet strength increasea with height of JlO'!ition.

divq = -\7'9

+ (k/,...) div [y grad .,.]

= 0

(33)

O'S/O, iJn - o' S/iJn 0& which confirms the single-valuedncss of 9. The solution for a can be written in anaJogy to (13) :

fL

div [ygrad.,.jln r · dA (34)

Agrun ' 9 can be combined with an auxiliary function having the character of a st ream function in order to fonn II. complex potential. In this case the stream {unction is manr-valued and unsuitable for tho determination of the discharges, but it redu~ to ~ for point\! of no vorticity. T he sourees and sinks can be expressed in tenns of the complex potential br replacing In r in (34) by In(~-:.). D&u/lk18 on G!'I abrupt interfaU and ill?lllldaries. That the expression div [II grad .,.J reduces to a doublet layer if the tmnsition sone reduces to an abrupt interface (Fig. 6), is recognized 1\8

176

If

lIo(O'.,./iJn')(ln r) dn d8

- [kh. -

This is a P... isson·~ equation showing singul:lrities in 9 of magnitude (k/,...) div [y grad y). Since a has the characrer of a. porential in the hypothetical fluid, the singularities mar be termed sources for negative and sinks for positive values of their magnitude. Application of the coodition of equilibrium, br using the first expression of (4) and the condition that the function (yy) is singlevalued, gives

€I .. (k/ 2T,,)

9 - (k/'hjl)

.,.J/27:jl

. f&110[+ In r,

liN]

- In r.] di

(35)

where r, and r. are the dist&llCe8 from the point of consideration to the two sides of the inter· face. This expression describes a source nnd a sink at either side of the mutual distance 8", their strength being equal to d K" = [k (y. y.)II.a'1 yM. If a" ill allowed to sh rink to 2ero, the [In r] term becomes [1l"/rJ cos ~, ~ being the angle between n and the radius r to the point under considerntion. Thill is known to be the potential of a doublet oriented pe rpendicularly to the inwrface. T he expression (35) therefore represents a doublet layer, containing a douhlet of strength. tD." = ~"·(h" = [kh. - ')'1)/,...Jy. di

(36)

for each line segment da of the interface. For the determination of boundary value problems the solution can again be broken down into two parts, €I, and a", where €I, accou nts for the sources and sinks and a" ~atisfies tho rleficienccs at the hound.,.,ies..For boundarics with known pressure, the value of a can be computed with (30). For boundaries with discharge information, the value of q. is known, and from (31) the value of (OO/an) can be determined, since (ay/an) can be obtained from the y distribution, which is assumed to be ayailable as initial information. Since either a or (ae/an) is known a100g the boundaries, 8,

Soil Mechanics and Transport in Porous Media

MULTI PLE-JiLUlD FLOW THROUCH l'OItOUS MEDI A and ell can be evaluated with the usual methods of potential thcory.

3749

consillts only of the first part, and the suffix I may be omitted in the subsequent treatment.

Movement 01 the entire field at t PAlIT 2. MATHEMATICAL ExAMPLE AND ExPI:lRIMI:lNTAL VERIFiCATiON

Introduction. The application of the foregoing bru;ic results Jrul.y be demonstrated by an example of a two-fluid system enelosed hy impenneable boundarics on all sidcs. By this disposition the movement of the fluids is due cxclusil·eJy to the weight differences of the fluids and can be computed by usc of the singu larity method. This eMmple is treated mathematically, with the resul t that the movement of every point in the enti re field is given. Since the distribution of the mo\·ements is the objective here, the treat-ment with vortices will be used. From the mo'·ementa obtained lUI the solution the initial displacement of the interface can be derived. The !lI1hl!equent rotation of the interface in eoul"8C of time cannot be computed rigorously, hut an approximate evaluntion of the rotation of the center part. is developed. As the singularit ies pennit the application of electric resistance analog modei!!, the potential and streamlines for the same example were determined hy the method of electric analogy. I n addition to this verifiCiltion, p.1tallel plate model tests were run in order to obtain photographs of the streamlines traced by particles in the fluids. Both verifications are treated. Given conditioM Jor the mathematical example. The example consists of a two,.fluid system conwined in a confined infinite aquifer of thickness 2c ( Fig. 6). The interface hru;, at time == 0, a vertical position 8ME, and it rotMes in lL clockwise direction from there under the influence of the differences in speeifie weight of the Auids at either si de (fluid I is considered 10 be lighter thn Auid 2). From the two parts of the solution (equation 24), the first part, corte!lponding to the vomCCII along the interface, will include image!! wh ich provide a flow parallel to the boundaries. The second part of the solution th en has to mtillfy V'* 11 - 0 in the interior and i'" => colllltant along aU the boundaries. From potential theory it ill knol"n that i'" must be consum t if it is to aati.afy these oonditions. If the arbitrary value zero ill 8.!lI!igned to the constant, the solution

Selected Works of G. de Josselin de Jong

== o. In_

stead of computing fl from the vortices in the z == x + ill pilUle of lo"igu re 6, it is more convenient fi rst to introduce a transformation in order to obtain a simpler arnngement of the images than to work with the infinite series necessitated in the z plane. By the transfonna_ tion fu nction, ~ "" (2c!-r) In t (3 7) the infinite strip of Figure 6, is mapped into the half plllne co r responding to pos itive real values of , == + iT} ( Fig. 7). The boundaries are mapped into one vertica! line, the 'I axis Ali'. The interface is transformed into a half ci rcle 8ME with radius I. Fluid I occupie!; the region of the half plane inside the circle and flu id 2 the rest of the half plane outside the circle. The image!; arc located on the [eft half circle BNE and have the opposite sign, in order to make Ali' an impcnneable boundary. T he subscript foero will indiClLte thst the interface is concerned. A [inc S<.lgment dz. of the inter face contains, according to (22), a vortex of strength

e

(k/,.)h. - "y,) dyo

(38)

Therefore the vo rtex strcngth associated lI"i th the line segment dll. hIlS to be applied in the tmn~fonned field to the line segment dt. whi ch is the image of dy,. Since x, == 0 is the equation fo r the interface, a line segment dz. is equ al to dx, + idy. =: idy,. Application of the I mnsfonnation (37) gives tk;, - i dy. = (:?eMt. -I dt.

(39)

which rcp~nts dy. lUI a function of " . The st rength of the vo rtex associated with the line segment of the tmnsfo rmed interface is the refore, according to (33) and (39), [2ek (y. - y,)/i7l"1', ,]dt•. The complex specific discharge potential in the t plane is the in tegral of the influences of these ekmentary vortices, each of which is a contribution to the potential of magnitude dn == (if2". ) In (' - , . ) times the vortex strength in ". T his gives

d"

n(t)1.... . . ..... 1

__ ..£ 2ck(1 . -

2....

-f"... In (t -

- 'Y ,) i ....J,I

t,,)!lli

f.

(40)

177

3750

G.

Fig. 6. Infinite confined aquifer in

~

D~

JOSSELI N

plane.

The integration extends over the values of Co along the half circle EMB, counterclockwise. The image vorticca have oppo.!lite sign Bnd the path of integration is the half circle BNE, counterclockwise. H the direetion of integration ill inverted, a second aign invCl'llion is obtained, giving a value for O(C) of the image part, which has the Bame sign liS (4.0). The result after addition ill

(!W,."l

+ nml .....

flCt) ••• tI ... .,.21'

H.. + J.

lIN.

I.

I,



t.)

tJ

7.'


(41)

This integral h8.'l no solution in closed form, and, sinoo the aim of the computation is the movement of the fluid, the specific discharge might aa well be introduced at this moment because it facilit8!.el! the evaluation of the integral. The specific discharge, according to (19), is obtained by a differentiation, and, since it is pennisaible here to carry this out under the integra! sign, (41) yields

178

JONG

Fig. 7. Transfonned infinite confined aquifer in r plane.

+·Cr. - Y.) 2...p

+J.

[J.

u ..

(I

t

'r.10)10

r'r.]

.". (I 10)10 Now integ ration with respect to !IOiution

+ kh.2- I,) IIIn 10 " + [In 10 - In (I -

(2) 4 giVe!! the

t.

In (I - IJ]U1r

10>]".".1 (43) The rea! part, the 2: C(lmponent of the spe<:ilic discharge, is obtained from In !{.! and In !{ t.!, corresponding to the limits Band E. As the interface circle has radius 1, In I{.! = 0 and

d:("(. - "Y,) -

III:

q. '"

k(-y, - ,"),,) .In "'JJ.

11!b1 EP

(44)

r

remains. Here P r i8 the point in the I plane corresponding to the point P of C(lnsideration in the t plane. The loci of pointB Pr for which (BPr/EP r) ill a conatant are circle!! in which the pointB E and B a re invene pointB. Theile circle!! forequal valUe!I of q. are mown in Figure 8. The imaginary part, the " C(lmponent of the specific discharge, is obtained from angle (t.) and angle ({ - {.J. T he lim of these two is the angle swept by the vecto r Co. with starting point at the origin, as its end point traces the half ci rcle EMB and ENB. The angles a re +,.. and

Soil Mechanics and Transport in Porous Media

MULTIPLE·FLUID FLOW THROUGH POROUS l\IEDIA

-", respectively, which cancel each other. There remains, therefore, the angle (t - r.), which iB the angle Bwept by the vector with NrtiDg point on the half circle and end point at the point PI. The toW value of this angle depends on the location of P,. If PI, is located in fluid 1, the path!! of integration BME and BNE p8.IlII on either side, giving (-,h) and (2,.. - ,p,) for the angle swept, respectively, where '1>, iB the exwrior angle EPr,B (Fig. 7). Therefore, the vertical component in fluid I will

...

q"

_ kh.~ "y,) [_,p,

+ (2" _

of,)]

If PI. ill located in fluid 2, the paths of integration both paaa on the left aide, and the angle Bwept ill for both (2Jr - '1>.). The vertical component in fluid 2 will therefore be g•• -

-

k(y, - 1') (2'1' • •

t,)

(46)

T

On the interlace the a.ngles "", and of. both have the value 3".j2, so the difference of the parallel diseharge at either Bide of the ioterfaee iB

(g •• - g• .>u •••• ,...) - (l:/~)("Y. -

"t'J

(47)

which is the value eJlpected for the ahear flow a vertical interface (II = %1l') according to

AI;

(28). Tbe e:rpresslona (45) and (46) show that on

the lineI! of equal value for q. the angles "", or "'. are eorurtants; these lines, therefore are ares of circles tbrougb tbe points Band E (Fig. 8). Tra1l!formed to the z plane these Jines for equal q. and q. fonn the pattern of Figure 9. For each point of the aquifer the specific discharge vector can be determined from the components q. and q., as is shown in Figure 10, where arrows represent the flow in magnitude and direction. The advance of the interface iU./dt at the 0 is equal to the eomponcnt of the moment t mean velocity perpendicular to the interface; i.e" in !.his case, the z component of the specifie discharge divided by the porosity. F rom (t4) it foUows, therefore, that

=

Selected Works of G. de Josselin de Jong

k('Y. - "y,)

In

er>,

3751

IEPr'l BP • e

.. k(-y. - "y,) In or"

Itot.-1 + ~I

By use of r. exp (n./'k) '" exp (iry./2c) this reduooe to

d~o

'"

k(-y.~ "y,)

In {cot

[r(c ~

Vo) ]} (48)

Figu re 11 shows thia ,'e1odty as a function of Y., (l.n S-shaped cur.'e which is representative of th e shape of the interf!lee a small time after t = O. Di~placement of Ihe interface nt t > O. As the interface turns, the shape continues to be in the fonn of an S which is tangent to the imperme(l.ble boundaries. For all subsequent positions a different flow pattern applies, because the vortices, which generate the flolV pattern, shift position with the interface. The detennination of the advancement of the interface necessitates the computation of the specific discharge component perpendicular to the interface. Also this eomponent changes with the shifting interface. Since an exact mathematical !lOlution of this diBplaeement was not obtained, a first approximation is given here, treating the rotation in the center point of a st raight-lined interface. Let tbe inclination of the interface with the hori~ontal be .. a rc cot a, !lO that the interface equation is

=

Zo .. VO cot II - avo

(49)

Then a line segment dz. of the inte rface is equal to

cUo ... dz.

+ i dyo

"" (a

+ i) dyo

(60)

A comparison with (39) ~how8 the only dif· ference to be the replacement of i by a factor (a +i). T he vortex stre ngth associated with a line segment dC. becomes, by aimilar reason ing,

+(2ck(-y. - "Y,)/(a

+ i}.."ro] dr.

(5 1)

The image interface is given by the equation (m = integer) Zo .,. 1I[(4 m

+ 2)c -

yJ

179

G.

3752

Fig.S.

Line~

JOSSELIN

DE

N:

JONG

of coll8tant q. and q. in r pinneo Fig. 9. Lin"" of con~tant q. Rnd q. in z pl=e. Vlllues indicate 12 Q. P/k(~,) and 12 Q. p/k(-Yrr,) , respectively.

. _...."_..._-

. ... ... / 1 ,,, ........ ..... ..

,, - o - - ' - ' - ~ - ~ - ~ - - . - --::r . --

"' - $ .'

~/

, FlUIO!

,

0

, , ,t /

\ t \

, ,\

, "

0 _

0_

, ,_ + _ .

'10. "" "

'

..

•• •

I t \ I I I

I , ,

, ,

........ ".;:: _ _ ' _' _ _ " _ 0 _

,, ,

I " ~ " ",

M"' _

., "

_ _ ~ ..... _ ... _ ... _

4

=

l'ig. 10. Di.,ehtlrge vectors at lime I ><

-, -, -,

'i' -i'

_. -.

b?.. .... . •• "

V -" -j<

(I. -¥.

-,

a.nd

1\

line segment dz. of the image interface is

dzn ... dZ n + i dyo ". -VI - 11 dyo Since the image vortices h\'e opposite signs, their strength associa.ted witll a line segment of tile transformed image interface is

d"

180

Elaboration of the complex speeific dil;eh:u-ge potential, !Ill demonstrated previously, gives the complex specific diseharge in a fonn similar 10 (43);

..

,

ij '" k(y,

,)",)

•+,

r. -

In

(r - rJ].,.,,,

- - '-- - . [In

r. -

In

(r - rJJ.".}

'{- ' -. [In

Fig. I L Distribution of dischnrgc perpendicular to interface.

O.

.- ,

(53)

where the minus sign for the second term between Urnce8 is due to the path iuversion from BN E to ENB. 'fhe derivation (dt/ dz) represe nl.ll the complex deformation of the fluid and the imaginary part of [c'(dq/ d.!) ~"l, the clockwise rotu.tion of 9. line of Huid particle!! oriented at li n lingle

Soil Mechanics and Transport in Porous Media

MULT IPt,E-FJ.UID FI,OW T fi ROUCn POROUS MEnt A with the horizonlAl. If computed for thc point z _ 0, it rcprc:senta the rotntion of the interface at the center point M of the confined byer. This rolAtion ill equnl to the decrease of the interface inclina tion (Z with time, or tf

-da/dt =

t-'

1m [(dof/dz)e"O]._o

By use of (53) the result for z

(54)

= 0 ({ = I) is

l i -l }' .-{ a+i i--+ l -r. a-il-r.ll For the incl ined in terface the point B corresponds to z. = (11 + i)c and the point E to z. = -(a + i)c. Therefore, the integration limits for { . are, respectively,

t~/kh.

"d

!!i

k(y.

dz -

2pc(a'

'Y,)

+ 1)

[Sinh (ra/20:) - i] eOllh (ra/2c)

By introduction of {J = nrc cot [sinh (".aj2c)] and 11 = cot« the expression is simplified to dof/rk "" [kh, - 'Y ,)/2"'::]·[8in' atI-;-] T his value for dlJ/dz used in equa tion (54) giVe!!

oo/dl "" -[kh. - 'Y,)/2ft
tf

s in (2<>- -

fJ)

(55)

T he right side of (55) being a function of a only, the relation between a and t mny be writUln M

t[tt
- / -

[ 1/ 2 Bin' (Z ·sin (2a - /3)] 00 (56)

,1/'"

Ilecanse of the intricate function relating {J to

a, a solution of the integral is not available ill dosed fonn. T hc curve of Figure 12 WM obtained by numerical step-by-step evaluation of (56), and it represents as the finnl result of \hia analysis the decrease of the interface inclination at the center point M with time in II first approximation. Verification tL"ith parallel plate model. To "eri fy the mathematiclIl solution II parallcl plate

Selected Works of G. de Josselin de Jong

3753

model test was carried ont, with the results as represented in the photographs (Figs. 13 and 14). These pidurcs show the center part of a 15 x 60 em Lucite model, with a slot width of d := I mill. Since in the beginning the movcment is primarily concentrllted in the center part, tile de,·iation between this mode! of lim ited length and the infinite oonfincd aquifer trel1.led theoretically can be disregarded. T hc fluids lit either sidc of the origi nally ,·crliell! interface were, to the left, a glyce rine with y, := 1.23 g/em' and, to the right, 60 to 40 pc r ccnt mixture of glycerine and phosphoric acid with y. = 1.40 glcm'. T he viscosity for both fluids at 24~C was p. = 0 .005 g FI!C/ em'. Since f, the porosity, is I fo r II parallel plllle model, the value of the half-Ilquifcr height c i3 7.5 em, and the specific penncability ill k = <1'/12 := 8.3 X 10'" em', the facto r associated with t in (.56) hill! the vlllue

- 'Y ,)

= 265sce

Thc white particles showing in the photographs are flakes of gold leaf, which by their JlIrgC surface-thickness ratio had a sm:ll! settling velocity with rcspeet to the motion of the fluid itself. During the OO-IlCC exposure time the flakes traced lines which are representative of the fluid motion . :Figure 13 is therefore the experimental representation of the specific discharge vectors sh()wn in Figure 10. The resemblance of both pattenlS is apparent. Figure 13 eo,"C r~ the period 0 < t < 90 $CC, }o'igure 14 the period 120 < t < 210 sell. According to the gmph ( Fig. 12), the inclination of the in tcrfllcc is expected to be approximately 9O~ > a > 80~ for Figure 13 and 77· > tf > 68 0 for Figure 14. In the p ictures the area swcpt by the interface can be observed as a lighUl r band, covered by particle tracings of cTOSl!ing directions. These cross paths nrc created by the shcar 1101" at the intcrfaec and are :1.lso present in the rulalytically ob~ined 1I0w pattern of Figure 10. In both figures thc S-shllped interfllce predicted by Figure II i! dist.inguishIIble, and the inelin.'ltion at the center follows the computed ,·alues approximately. Electric analollY. I n the electric model!, which a rc rmtlcu!.'lrly .1pproprinte for the plotting of potential lines and strcfUIllines [Malavard, 1!)5(;], it is impossible to simulate diffcrences of specific weight directly. Howover,

181

G.

3754

1111

JOSSE LIN

01

JONG

"

•

+lJ

~'

~

EE

t .. \ . ·······H ,

"

:, ,. ,

..' ~ .' ....

,

....

-;

...

\~.:.-

'\

'I-,~-.-.

!

\

Z· prAM

'"

1\

/",

( _r'4KC

"

Filii:. 12. Slope of rectilinear illterfac:e

-I

&I ..

(. (t. - r,l / '!"c J

I

!unction of time.

Fig. 13, Photopaph of parallel pb,te model teat, with gnld leal' Hakell tracing ~Iinea : left aide li&bt auid, ri,bt Bide heavy fluid; exposure tim e (1<1<90 ...

Fig. 14. Photogr&ph of pan!1iIl\ plate model test of Figure 13 at a later stage; e%pOllln!! time

182

120
Soil Mechanics and Transport in Porous Media

MULTIP LE-FLUID FLOW T HROUG H POROUS MEDIA by the replacement of lhe fluid!! with different properties by the bypothetical fluid lind tbe introduction of property changes by singularities, a concept is obtained wbieb bas eleetric analogies. I n tbe eleetric resilltanee model, where lines of equal eleetric potential can be traced, sources and sinh can be simulated by electrodes which discharge or withdraw electric cu rrent from the model. In this model the electric resistance body wit.h homogeneous ~peci!ic resistance can represent the hypothetical fluid. H the multiple fluid potential e is to be determined, the strength of the source or sink in the area dA must be p roportional to (k/I') div [y grad rJdA, according to (33). Lines of equal electric powntial are then lines of equal 0, and they are called potential lilies". Fo r a gmdual t rlUlsition lone the source-sink diBtribution is continuous. Exact simulation by a continuous elcetrode arrangement would give the correct potential lines in the entire field, including the transition zone. In practice, a discrete number of electrodes will be used, giving fin eventual distortion of the potent ial lines. At an abrupt interface the 80urce-sink distribution reduces to a line of doublets whose strength pe r line elemen t ds, according to (36), is equal to

If the st ream function i- is to be dctennined, \·orticCS' must be simulated. Actually, vortices bnve no analogy in the electric resistance model, but this presents no difficulties because the inverted model must be used for tencing the streamlines. With the inversion, where streamlines are simulated by electric potential Jines, the vortices are in,'ertcd to Irou recs or sink!. For the gr:tdual transit ion wne the strrnll:lh of the source in an areil dA is - (k/I') (<1),/')z) dA, secording to (12). F or the abnlpt interface the streng th for a line ~mcnt d& ~nrresflOnding to an clel'ation incrcm~nt dyo is, according to (Z2), r'dyo = (k/",)b. - 'YI) dyo

(59)

The correct streamlines would be obtaine in tbe region outside the zone containing the singularities, and since 4>

Selected Works of G. de Josselin de Jong

3755

and v satisfy the Cauchy_Riemann equations (see equation 14) , the e and if lines form an orthogonal, equilateral grid in tbis outside region. Inside the gradual transition wne, e and if no longer form squares because e is unequal to 4>, and therefore the Cauchy-Riemann relations no longer hold between 0 and i-. For completeness it may be mentioned that vortices are encounte red in t he tbeory of magnetiffiIl, because any wire conducting an electric current creates a magnetic field around it, wbieh, according to the law of Biot and s.wart, obeys the same formulas 118 the potential around the vortex filament. To make a model in which the magnetic analogy iB uscd is difficult from a practical standpoint. The application of boundary conditions, the determination of the flux, and the direct current necessary for the elim ination of eddy cur rent losses aU create unattractive complications. The elaboration of t his type of model ill therefore omitted in this study.

Verification with an electric relistance anaW<J1J model. I n the previous section it was demonB~rated bow streamlines and potenti:tl line!! c~n be determined by means of an electric resistance analogy model. A test by this means wrur conducted for the verti ca l interface situa tion inl'estigated previously by analytic and pa rallel plate model procedures. Conductin~ l~~per was used to si mul ate the porous medium. Instead of applying the continuous si nk ami doublet dist ribution as required theo reticall y for an abnlpt interfrtcc, a limitC{1 number (10 for the haH height c) was used. For the stre!lmline determination ill the ill' ·erted model, each sink, acco rd ing to (59), requ ires a discharge of 0' -

~'/(c/l0) -

"lb. -

'Y 1)c/ IO",

(60)

where I is the thickness of the aquifer. T he electric current / flowing th rough the electric model of thickness I. is considered representative of the discharge flowing througb the .~quifer of thickn!!SS f. Ohm's IllW fo r the current flowing through a stream chrtnnel of breadth b, perpendicular to the potential gradient (dEjds), with p the speei fi c electric resistivity, gives / = - (bl.!p ) (dE/dl ). With (5) the discharge througb a breadth b in the hydraulic case is Q = - bl (ili-/iln). Because of the inversion d/ds is simil ~r to d/dn,

183

.~

. . I~ ~

O. Do: JOSSELIN us JONG

3756

and the relation between model a.nd aquifer ia

o

•o,

• •"

~ ~

.~

(f.Ejp)[amperee] ..... /'I'[em' /_ee)

(61)

The intervals for the streamlines .0.'1' were chosen at tl.e arbitrary nine of Ai< - th. - 'r,le/w" (62) By (Ill), these were simulated by ele<:tric por (cnlial intervals of magnitude fl.H, according to

CJ./ p)

t. E ..... / ft.• .,. [k6-. - y,}cl/ W,,] (63) A SOO-volt a-c source (S-) was used w supply

cu rrent to the iX>int oourCila. With a OCKI-kfl resistance, the current at each ooucce WiIJ! E/500,000 [amp} which, acoording to (60), simulates k (y, - y,) cl/IO 1'-_ The resistance of the

(65) Q" - 2kl(-r, - ..,.,)<:/1' The required current for the sources wsa obtained by replacing the 500-kn resistance by a smaller one of 250 kn. For the sinlu! & counterphase potential E· of equal magnitude with a 250-kn _istance wsa used. So 8*/250,000 [amp] WAS equivalent \Q 2kf (y. - l") e/I<. To ohtain potential lines at !imihr intefl'als :1<1

in (62)

a9

~

k(.y, - ..,.,)420"

the eleetric potenti,.] intervals af: had 10 be !:Iken as

dE - (p/I.)(f dO/Q ' ,)(E· /250,OOO) = 1.625 X IO- 'E· In io'igure 15 the full lines are streamlinC8 and the dotted lines are the potenti,.] lines obtained with the eleetrie resistance model. This Cllblet.! of equal strength. Since, crete IIOUI"OO!! is shown to the left; the potential aceording to (59 ), the doublet slrenglh in- distortion by the discrete doublets is ahown in crt'1I$I)ll with height 1/., the strenglh distribution Figure 15 to the right. Outside this region the lines fonn squam. is triangular. The ten doublet.!! were the refore DiDert:ru:e in vi'lC~tl/. If, besides specific placed at height.!! weigb~ dilfcrenCCII, ,·anauon! of viscosity also [2c/3V1OJl[(m - 1)'" mil'] ltaye to be C
+

+

L'

- (kl/20p)(..,.. - ..,.,)<:' (64) The doublet.!! were represented by a souree and a sink wit.h discharge Q" at a mutual distance ~" = .:/40, and their strength was AN :=: Q"e/40. To meet the requirement (64), tbe discharges of the sources and sinks were gi"cn by

184

6. - kp "9. - - (iJ6./fJ,)

+ k-ry

+ ky{fty/fJ,) -

-(iN'./fJIlj

+

M. - -(fJ9.JatI) ky(fty/fJIl) _ +(fNt./iJl ) E~abors.tion of the conditions of equilibrium Ihen gives

,, -It • ..

- k(irr/fJz)

Soil Mechanics and Transport in Porous Media

MULTIPLE-FLUID

now

T HROUCH POROUS MEDIA

Hg. 15. Result of eleclric rel!;,,(ancc model teIJt, showing lines for constant fI (dolled lines). an expression devoid of the /L term. The oondition of continuity remains div q = O. Tn the electric resistance model Ohm's law giVCll

pi .,. -grad E

where 1 IS the current deIl!lity vector, and p, the specific resistsnce, ia now a variable function of position. The relation between hydraulic and electric qUfln~ities may DOW be !1.8 follows: For the tracing of potential lines

'I'

3757

(full lines) and constaDt

proportional. AP, the totsl l"C1Ii!ltance in the electric model consiate of p multiplied by the thickness, it ia sometimes mote convenient to UlIIl a fluid of oonatsnt resistivity 811 conducting material and to vary the layer thickness of the fluid to meet the diffcrencell in viscOIlity. CoNCLUSION

These equatiollS show that E is a single-valued function. The valuC1l of div i indicate a source and sink dist ributioD which difteN! but Slightly from the previous resulta. In the first case the speeific resistance must be directly proportional to the viseo6i.ty, in the second case inversely

This investigation hIlS shown that it is possible to determine the flow pattern in a porous medium which is !!Ittumted by several fluids of different specific weight by the use of singularities. All the fluids are repla.ced. by one hypotheticaJ. homogeneous fluid. The motion of the hypothetical fluid will be identical to the movements of the different fluids, for which it is I1Ilbtltituted, if singularites ate applied at all the pointa where the original tluide show change in apeeifie weight. h is proved to be convenient to give the singularities the character of vortices, if the motion of the fluids ia to be determined. If the emphasis is on the pressure deklrmination, it is better to use a distribu tion of sources and si nks. I n the case of the vorticell, a stream funetion is the basic variable; in the case of the sources and sinks, it is a mUltiple-fluid potentia!. Outside the region of the singularities the potential and stream functions obey the CauchyRiemann equations; inside they are not orthogonal. The magnitude of the si ngul arities is related to the gradienta in specific weight. If the initial position of the interface is known, the action of the singularities on the fluids contained within boundaries with known conditions can be determined from potential theory. The solution is given in the fonn of an integral for which

Selected Works of G. de Josselin de Jong

185

e~ --+

E;

("11/1') grad 'Y] --+ I;

[q -

/L

--+ p

For the tracing of streamlines grad

w. --+

-I;

(aE/&); --+ (-aE/an); f}.

f}.

--+

p--+ l / p

A[>[>lication of the conditions of continuity and equilibrium to the hyd raul ic system gives the following conditions for thc electric quantities: For the case of potential lines

-k div [(yip) grad 'Y) --+ d iv i {o' e/O~

an - a"e/an a8] ... --+

[a" B/ a.

il~ -

0

;fE/an a.] -

0

For the case of stream lines div q

=

0 --+ [o"E/o$ an - a"ElOn 08] = 0

-"(o-,/a::) --+ div i

G.

3758

I)!

JOSSELIN D! JONG

the region of integration is that part of the aquifer where weight diffe rences are present. In most practical eases the interrace or the mixing zone will have a fonn which prohibita a mathematicu.lly convenient caleulation of the integral in dosed form. There are, however, methods available for approximation of the illtegl'8Is involved to any desired degree.

dynamic balance of fresh water and salt water in a coastal aquifer, J. GeOphll" Reoearch, 64, ~61-467. 1959. Diet~, D. N., A

theoretical approach to the problem of encroaching lind bypa8!!ing edgewater. Kooinkl. Ned. Akad. lVetClUChap~ Proc., Ser. n, 66, 83--92. HISS. Edelman, J . B_. Slrooming van Zoot en Zout Ws ler, Rapp/. 194f) in take de too/ertl()OTticninu till ... A~lerdam, Byillge 2, 1940. Glover. R. E~ T he pattern of frosh_water flow ;n II coastal aquifer, J. OeOpll"ll3. Re~o,ch. 8,. 457_

T he method requi res a knowledge of the position of the interface at a certain moment, and, from tbat and the known boundary conditions, the subsequent motion of the boundll ry C&II be computed from the specific discharge perpendicular to the interface. The displacement of the interface creates a new position of the singularities, with spedfic discharge changed accordingly, from which a gradual or step-wise movement of the interface can be computed. A special feature of the singularity distribu-

J . A ~ T. R. SilUpIIOn, I,. K. [..,u, Jo'. I•. Dotes, and P. B. McCauhey. Laboratory research On sen water intru.sion into fresh ground-water ""urrea and methods of it.s prevention. Final Rept., Sanita".. 8ng. Re3eorch LAb., Univ . Calif.. Berkeley, 68 pages. 1953. Henry. B . R .. Salt intrusion into fresh_water aqui_ fers, J. Geophll', ReMarch, 64, 19tt- 1919, 1959. He",berg, B.. Die WlI89Crversorgung einiger Nord_ l!eebader, J. Casbeleuchtung lind Wl\8!!Ierversor-

tion is the possibility it offers for the use of the elect ric resistance model, which is especially useful because it makes possible the direct t racing of potentials and streamlines _ Aclllally, weight differences of fluids have no analogy in electric resistanee models, but the hypothetic~l fluid with homogeneous properties and the singulllrities applied to this fluid can be simulated by the use of sources and sinks with different distributions for the determination of both potential lines and streamlines. By introducing II !mlall change in the definition of tbe potential and stream function, the influeoce of viseosity differeneell may also be studi ed with an electric resist. \oce analogy. T hen similar sources and sinks take care of the ijpecific weight variations, but differences in the specific resistance of the model are needed to simulate the viscosity variations.

Hubbert. M . K., The theory of ground-water motion. J. 01.'01., 47, 785-944. 19·10. Hubbert. 1\1. K .• Entrapment of petroleUm under hydrodynamic conditions. BuU. Am. A"tN:. Petrol. Gc%(1i.
A cknaw/edtlmen/. Thie st.udy was "I'onsored by the Waler Resources Center of the Univcraity of Califarnia and was conducted in the Hydrau lic I.Aboratory at Berkeley. For the apportunity to "tlldy this IUbject, far vfiriaus lJIlggestians in the courllc of the invcetigAti''". Rnd for valunble help in the proparation of the manuscript I am especially indebted to Profcssor David K . T odd. Univeraity of Cnlifornia. I n the laboratory ph.sc~ of the study Dr. J . Bear, Technion, Israel, offered his experienced assistance.

Malavsrd. L. C., The use of rheoelectrical anll!_ ogies in aerodynamica. AGARDov,aph 18, NATO Advisory Group of AerOllaut. Researx:h I\.nd Develop., Parie, 175 pp., 1956. Muskat, M .• 7'he Flo", 0/ U r>mogeneotu P /t,id. thrr>ugh Por01J.~ Media, McCra w-Hill Book Co_, Ne w York. 763 pages, 1937. Santing. G., Modele pour ]'~tude dell prob!cmes de ncoulement simultane dC1l eaux sollterrains douees et sal~. AuembUe GbtA,o/e de Brw:eller, A4WC .• 1.. tem. d.'Hvdr%r;ie &ienlifiq!MI, t,

REFERENCES

Todd. D. K., and L. HuismMl, Cround water !lo,,' in t.he Netherlands ooaata.l dUDes, J. 0/ the Iflldral. Div., Proc. Am. Soc. Civil Eng"., 8f;, no. B Y 7. 63-81, ]llS9. ( Manu",-,ript reo:eived June 28, 1960.)

459. 1959. H ~rder.

gung. 44. 81S-S19. 842-844 , 1901.

1!l37).

1&1- 193. 1%1.

&don, Chijben, W .• Nota in verband met de voorgenomen putboring nabij Anuterd!lJD, Ingen~u,. Utrecht, p. 21, 1888-1889. Cooper, ll. B ., Jr., A hypothesis concerning the

186

Soil Mechanics and Transport in Porous Media

VOLu ,n

66, No. 10

0""... 1961

Moire Patterns of the Membrane Analogy for Ground- Wa ter Movement Applied to Multiple Fluid Flow

G. DE

JOSSELIN DE JO!fO

Civil Enginuring lk",,'[1mII1, TtdMlogical Un;"en;lv Delft, 1M N.thtTlan~

InlrOOuclWn. In the membrane analogy for flow through poroU8 media, contou r line8 of a deflected membr9.ne repl"('8Cnt either a(.reamlinell or equipowntial lines. l1u~ membrane WII.S used by Prandll [1003] to solve torsion problem9. lIan3e71 [1952] gllve a deseription of the application of the analogy to soh-e the flow patterns re!!ulting from systems of sourecs lind ainks. Multiple fluid flow through porous media may be treated by considering a 8uitllble distribution of 80ureea and sinh [de J oudin de Jrmg, 1960] and therefore the membrane analogy is also applicable to this problcm. The obje.;t of this letter is to point out that such problems may be 80I ved by a moi re method, since thi8 technique pro\'idcs a com'cnient procedure for cstabliBhing contour lincs of a deflected membrane. The proeedurfl is a slight modification of the method initiated by Lig/lmberg [1955], who superimposed two photographic expoaurCll of a loaded and an unloaded modeL Ligtenberg used the reflection of the grid Ill! seen in the mirrored model surface to obtnin a moire pattern which is a meaaUrfl of surface rotation. The modified procedure diiJcflI in that a grid pattern is projected on the nonreflecting model and the moir~ pattern obtained by the lIuperimpollCd imagcs is a mCBBure of the nonnal displaooment. OIU) [1954] mentioned this kind of experiment in connection "ith the study of hanging roofs. MttllbraM-mqjr~ 1e3l setup_ The membrane i8 a thin rubber sheet placed vertically to eliminate ita own weight and stretched uniformly by a unit foroo S. Sinks or llOurOO8 arfl simulated by concentrated point loads P normlll to the plane of the memhranoo as shown in Figure I. A horizontal bellm of pllrallel light ray., lit an IIngie of incidence a with the normal to the unloaded mcmbranc, is projected through a grid IlO aa to throw a shadow imuge on the membrane. The grid consists of a ruling of parallel blllck

lines on a glll.S8 plate, oriented in such a lI"ay that the Jinell are vcrticlIl. When the membrane is given a smllll displacement, w, normsl to it!! plane, the shadow image of the grid is displaced. If the IIXUt of the camcra is placed norma!!y to the unloaded membrane the apparent normlll displacement, u, of the proje.;ted grid line!! is

u - wtana With b the pitch of the shadow linea in the unloaded condition, a displacement of n epacingB corre.ponda to a deflection w. equal to

w. = nb cota If the splICing of the grid linea is such that the pitch is t"ice the width of the lines, photographic aupcrpoeition will produce complete eX]l06ure in those regions where w' is given by

w' =- (n

+ t)b cota

(l)

therefore the8e regioTl!l lire black in the negative and show up 811 white bands in the poaiti ve prints. These bands indicate the dCllirod contonr lines for the defle.;ted mcmbrane. An example of the result!! which can be obtnincd by this method is gil'en in Figure 2. This figure shows the streamlinCfl for II confined &Quifer fined with two fluids of different specifi c weigh t at the moment the fluid motion atart;!! from an abrupt vertiCil! interface . Since this is the I\&me problem Il8 treated previously by the author (1960), the upper half of Figure 2 can be compared with the full li nes of Figure 15 of the article quoted, wherein the results of the electric rcsistnnoo model are represented. Clmlputation 0/ analogous quantitie.. The dimell$iOIll! of the test setup were determined 811 follows. The relation between the deflection w and the normu.1 10lld per unit area p of a mem_ brane uni formly stretched by a unit force S is

3625

187

I,b'TTERS TO THB BDlTOR

3626

.

w

Fig. I. Top view of {'-"!II. arrangement and det ..il of dil!phu:ed mcmbrRnn. given by the well -known equation

'\7'10= -piS

(2)

fe.g. see Tim08htn/ro and Goodier, p. 269, illS!.] T his equation i8 analogous to the basic equation for multiple fluid flow given by the author in the referenoo quoted. Equating (1 0) to (11) from this reference giVe!!

For the electric resista.noo model the streamline interval WI\8 arbi trarily lIC!cetcd 1\8 (eq. 62) /l'lr = (k/p)h. - 1',)(c/20)

To pennit comparison of test results the contour interval for the moire teat was therefore chose n

to'" /l1O = /l'i'

(3)

where 'Ir is the specific discharge stream function and <.0 is the vorticity. Therefore, a vortex of strongth wdA acting in the region dA ean be represented by a distribution (-p/S)dA over the region dA of the membranoo. T he foree distribution p may be approximated by a conoontrated foree, P = -pdA, applied at the centroid of t he region oA . The vortex strength for an abrupt interfaee is given by equation 22 of the article quoted in corrected form:

w oA

-

<.0

d8

on ... (k/J..)(1'. - 1',) dy.

From (2) and (3) the relationship between analogollll quantities is therefore elltabliahed :

~=~= -pd A/8 'I' W <.od A

"'"

188

P/8

(k/p)c"y.

""1',) 0/1.

P/S I'e ""1',) dy. = 20S dy. (k/ p)(1'.

Since nine point loads were used, for a half aquifer height c in the membrane, dYe equall! c/9, and /l1O = 9P/ 20S

(.)

In order that /lw corrcapond to one interval, is 1 in (1), and tow of (4) should be equal to b cot a. Thllll it followa that

JI

P = (20/9)8b cot a

Using II stretching forec 8 = 200 ycm, a projeeted grid width b - 0.22 em and an angle of incidence a "" 4{;0, point loads of magnitude p = 98 g wcre required. Since the theory is only valid for 6mall vnlucs of 10 a correction had to be made for regiollll of deviation. Figul"1l 3 gives the pattern for the illlltsntancous streamlinCB corresponding to a subsequent poIIi. tion of the interface occurring at a time' - !-1fl.1C/

Soil Mechanics and Transport in Porous Media

, ,

'1,1

11 '"

LE'ITERS TO TH E EDITOR

Selected Works of G. de Josselin de Jong

3627

189

3628

LETTERS TO THE EDITOR

k(,,(. - -y,) aee, after the initial vertical poaition of the interface given in Fig"", 2 . 'l'he l""'itinn of the interface at a subsequent instant may be

approximnWd from the st~mline pattern for the previous instant, assuming stell.dy-etate flow during the time interval. In the membrane analogy it is then ne<:essary to shift the point loads to the new position of the interfll.Ce. F or a oomplete investigation of the movement of the fluids, a succession of photographs representing the instantaneQus motion in the entire field at each of the time intervilis selecWd is requited. The membrane moir!! results of Figures 2 and 3 are oompfU"llble to the parallel plate tel!t rcsulUl given in FiguI'CB 13 and 14 of the article quoted. Their agreement is demonstrated by the superposition shown in Figurel! 4 and 5. Equipotential linCII m&y IIllIO be obtained by the moi r!! method. Point load moment couples are then applied to the membrane to represent the doublets in the analogy. The principal advantage of the membrane

190

moir!! method is the convenient way of obtaining a photograph of stre"mlin..~ !Inri ~q"ipot"'"ti ... IR_ Thill procedure e1imin&tes the need for point to point plotting of contou r lines, which ill required with the electrical resistance analogy. REI'~IlENer.s

Ha!>!en, V. E., Complicated well problema I!Olved by membrane (L'u..togy, Tra M. Am. GeJTp1l1l8. Union, " , 912-916, 1952. de JOS8elin de Jong, G., Singul&rity distributions for the anal)'llUs of multiple-fiuid flow through porous moois, J . G(:(>JJlIlI~ . &U
Soil Mechanics and Transport in Porous Media

WATt k

VOl •. '. NO . •

RESOURC~S

It t: St:A It CH

FOU ItTtI OUARTEIt

',,$

A Many-Valued Hodograph in an Interface Problem C. DE JOSSELIN DE JONG

Civil E''9ineering DepoT/.ment, Technt;;/DfJico.l Univenity Ihlft, the

N~ther/on,lf

Ao.truot. The hodograph method fOT determining ~ttern. of aow of VOUnd ..... tc. in .. cM$lal aquifer with a drain ""'lU;"'" the treatment of double-vh~ted hodOjll'apb planell. Tbl, many_valuedneM d<>ell not prohibit the ule of Schwal"l-Ch. iatoJI'el aualyai. ;f the hodosraph domain ;. aimply ""nnected. In tbe t ..o-lluid Cal. of fresh water aOwitl4\ over etation"ry .,.It the hodosrapb is ';mply connected, and the bodoernph method is $ho .... to give .. IOlution. This 101ution was verified by to teat thM ,hoa'ed " $lAbl. interfa<:e in the predicted position.

".,.te.

' NTROm.1 CT10;>l

III BOlving several problems of twated in tbat study "'ere all !ingle-valued. There are, however, eases of practical importance where the hodogrnph is ",,,-ny_valued. An enmplc is a drain age "'ell 10eated in the vicinity of the l\I1l1COast to recover part of t he fJ'l!:'lh water t.ht floW!! out to the sen. In their progress report Bear and Dagan [ 1062] suggested the po&'libility of many-valuedness of the hodoglltpb in that case. This dminage problem will be treated bere to show how to deal with hodographs wben they are many'·allled. Many-valuedncss occurs when the s:tme specific dischrge vector is encountered in se"eral different places of the aquifer. PolubarinooaKochina [1952J shows tbe shape of a double"alued Itodograph in the case of seepage through '" dam, without (unher mathematical eltobora_ tion (see 1962 traHalation, p. 46). To 8how the characte r of II double-valued hodograph finn in a sim plified form, the one fluid C8!1t: of ...·ate r flowing arolmd a .. ~1I while a drain interrepis part of that water will be treated hem (Figure 10). Double-valuedness is a ""nsequence of t.he inflection points of the streamlines in the region CSGPD. The two-Huid cue of fresh waler flowing Over a stationuy salt wale r region with 11 drain

shO"'! a !imilar pattern of the streamlines (BCtl Figure 2<1). The intcrf~cc has inflection poinl$ in Ihe rcgiull of upconing created by Ihe drain. ONI!:-t·LUIO CAllI:

The one-Huid case (Figure la) can be solved "'ithout the uac of a hodograph, because it is pO/lSible to map the eomple.: potential n = 4> + iv (Figure Ie) dire<.:tly on the % = % iy plane, whiCh is II acetion of the aquifer (Figure

+

l a).

'The potential of> is ~ero along the seabottom, i.e., the positive z axis, the line GE up to infinity. The strellm function 't' is zero along the line CS, which beyend S !ep:lraWB in the two linC-9 SA. and SG. The discharge
Since the flow region is bounded in the n plano by straight lines, mapping un be affecled by the Schwan-Ch ristoffel formula . BecaUl!e t his fonnul ... maps on the upper hal f of a plane, the % plane is tumed into the t ptane, repre8CI!tcd in Figure Ib, hy multiplying z with e" := - I . The relation belol'ccn (l and t can

'" 191

I£J , •

•

"

"

0) "

01"-. 0>- •

,

" co..,..... "O'f'~ ""

• •

•

Fig. 1. One-Huid c.... or a. drain at A intercepting "·ate, Howing underneath the impermeable line HG toward the ""a, bot\c>m. All Sgum. are drawn {or the cue II = e, Ow= 2Q,.

192

Soil Mechanics and Transport in Porous Media

545

A i\1wI !I- Vulll(d J/ udogrupll 111(>1l

be written as

in the colllplex ,~ pinH". Hero w is defined as

1O = u+w "' q.-iq. By this definition the relation between

- or,

'" J' ~ "" J' >.,,,+a,

(A_a) ~ +fJ

2a I'" I

til

+ a, (" a'" - ,)

To obtain the complex specific disdmrge is applied to (1)

• Q,

10 = -\ ..- (a

1m (nCt)] ... -Q,

Re (I)

for

at permits

G:

n(I)'" 0

ev~lull.tion

>

for

a

1m (I) -

0

t = 0

of the constanta

<1:,

Q,a ' 12/,,(1}.

P,

-iQ,

- 8)

Turning t into n" finally gives the complex potential RS & fUllction of t, Q Il(z) _ -...l 'II"

= -dfl/Ih

(5)

L-q uaUOIlS.

Introduction of the bound.'lry conditions

A,B:

n and til is

beclluse q = -grad !Iond the hannonic funetioM and if obey the Cauchy-Riemann

''') + p, log ("" I"' ~ :lIi on

(4)

[~ '" + "/2 "')] 2i(~ )+101( (.~a'7li a- , 10 Z (I)

In (1) only r ill unknown. If the dischnrge of the drain Q, is kept constant, the value of I changes, if the strength of the How field from infinity to the boundary GE is varied. Without drain the flow around a wall is given by the complex potential (l(t) = iQ . (z/e)'"

(2)

where Q. is the diseharge through the line GE with length e. For large values of z the two potential ficld9 (1 ) an d (2) will tend to coincide if

(a - 3) = 2Q,(M)'''/(..-Q.,.)

(3)

This defines !. In Figure 1 aU drawings correspond to the case a = ennd Q. 2Q,.

=

lI0d0graph in lJne·ftuid oou.

The hodogJ'll.ph

ill represented in Figure ld, where the endpoint!! of the specific discharge vectors q are mapped

Selected Works of G. de Josselin de Jong

(z

+ .)+ a) [0]' " -;

I)(Z

10 (5)

(6)

The 10 plane is drawn in Figure ld with its positive imaginary axis ~ downward, because then n.ecording to (4) the speeific discharge vector q in the hodograpb plant has the same direction as the oon-esponding vector in the z plane. In the representation of Figure Id, the map turns out to be double-liheeted, as can be seen by following a path that encircles the lower half of the z pbne once, by starting in A, to the left of the drllin encircling the drain counterdockwise to At, then guing over SGF' to E, encircling the aquifer at infini ty clockwise from E.' to H, and returning to A" The endpoints of the corresponding specific discharge vectors then trace the following path. In A, of z this vector points to the right, 80 that A, is mapped on the positive II axis. Encircling the drain the vecto r endpoint turn9 over a large circle counterclockwise to A •. The point S being a stagnation point, the velocity is zero there and its map in the til plane is the origin. 10 the point G the velocity is infinitely large and changes direction pointing to the right in G, and upward in G., eo that the veetor endpoint traces 171" of a large ci rcle. Since all along G.E the velocity is upward and decreasing in strength, in the hodogrnph the endpoint of tho vector follows the negative v axis toward the origin. T he origin of the j'odograph corresponds to the infinity of the z plane EDeH, because the velocity reduces to zero there. From B to A, the velocity vector points to the right, eo that SA, is mapped along the po6itive u axis of the hodograph pl.:llle. If the path BA,A"sGFEDCB is traveled in the aquife r (Figure 10), the total Bow region is encircled onCt and lies at the right-hand side

193

546

O. l.>J.: JOSSlU,!!," DE J<):
of tlat path. In the hodocrapb (FiCUre Id ) the relevant part of the plAne lies to the. left of the travele r, ~aUl!e. there is a reflection by d rawine the positive v nis downward. Figu re Id ~hows that the hodog raph path makes a double lOOIJ, &lid therefore the cncircled region is d oubleabeeted. i3ecaulit the ~ingJe-\"~lued :: pl~ne is turned into a double-abeeted w plaue, tbe w plane must con~ in a branch point of order I. Thit point M can be found becausc, as will be shown prescntly, a point 0: 10 is a bmnch point of order 1 if the lim derivative vani.shes there lind the seeond derivative ill not u ro. In the symbols { "'..... - (dlll/d::) ••• " -

0

(7)

",..," .. (rfw! tb')' .'Jt ;0<1 0 T he \·alidity of this atatcment can be shown by expanding w in a T:'lVlor series a round w. as

w ..

Wit

+ w..,'(. - ,..,) + +W..,"(I -

• ..,)'

+

Beclluse of (7) this reduces to

w-

"'......

+",..,"(. - ' . . ). +

(8)

If (, - ::.) = ~ •• with p a small, constant length, the point, cberibes a small circle with radiull p around : •. A com plete revolution is traced when" incfC:w'!& from 0 to 2.... The point tD then traces approximately a small circle with radi Ull r = 'iilll."'jp· (_ Figure Ie), which is a deuil of Figure 1d. If we call tD - w. = re'·, then a«:o rding to (8) .; it equal to 2, + arg (w ...... ). If tberefore , incl"t:flSCll from 0 to 2:.-, the angle ¥ increases by 4... , indicating thllt the poInt w. is encircled twice if the point t. it encircled once. A branch of higher order, say n, is 0btained if aU deri\·ath,t!I up to w."" vanish and tD ....... it not uro. For more information about branch points the reader is referrod. to the textbooks on function theory. A dctl\iled deIICription of branch points is, for instance, gIVen by Nehori {l952). pages 78 and 152, Thit deIICription will be helpful in undenrtandin, the repreeentation in Figure Id, which may be difficult to visualize since it is imJlOl!llible to build a co rrect paper model of such surfaces. From (6), w.' = 0 if

194

w..,' -

Thi~ i~

. Q,

I, . . • + (a.

l ...

2(a

the

e~se

+

for

,. . ' + (3J -

+

a)' .... .)(' .... a)'

(I),..,

+

(IIJ

[;-;.r

(II . .

- 0

0

l!;i\"iu(I; the roots

'.w

+ (I

-

Ho -

~) ' "

·[3(0 - ~)'" ± (98 -

a)'" /

(9)

It can be \·crified by direct computatKIIIII that Ill.." is not uro for t .. givcn by (9 ). Therefore I . u defined by (9) it mapped into the branch point of the w plane. If DF is the strellmline passing through the point lIf in thc : plane given by:., thcn only the 61reamlines between DF lind have inRec tion points. In the hodograph (Figure ld) these IItreamlines ma ke loops around the bra nch point (_, for inab nce, the stre:unline for '" = !fa Q.) . The % sign in (9) reRect. the symmetry with f CllptCt to the z axis. Since the lower half of the :: plane it conside red hcre, only the - sign is nc«led. Beeause of the br:inch point the hodograpb is not only double-vlllued but also doubleconnected. The location of the branch point in 111 cannot be infcrred beforehand, and therefore it is imJlO!!8ible to indicate the cut (as shown in "'jgore I d) that makes the hodograph lrimply connected. In the t .....o-fluid case thi!: difficulty it avoided ~Il!le the hodograph is i!.llelf simply connected.

esc

TwO-n.UID

CASE

1n the two-fluid callC {he presence of a drain that intercepts pan of the fresh wate r that flo"Q toward the _ will cause upconing of the interface between the freah water and wt water. The flow pattern in the aquifcr for thit ~ is shown in Figure Za. All picture3 in figure 2 represent tbc valuea of the test described in the last seetion. In FigUre 2£, however, the points F, M., }.f" and S are shifted fIOmcwhat in the hori:wntal dire-ction in order to make them discernible. Figure Za shoWl! that in tbe aquifer the drain is located at a distance tJo from the 8Cuhore. The interface between the moving fnlllh water

Soil Mechanics and Transport in Porous Media

o ,

•

Fig. 2. Two-lluid e&IIe of I. dn.iD. ..t A ictercepting fresh ..ater lIowing underneath the impem>eable line BG toward t he .ea bottom over et.ationary ...tt ....ter. All figu..", are df1l.wu fOl" the c.... Q,jK',.. = O.3CH, Q./K'a. = 0.00:116 except 2e, which i8 out of .c.le.

Selected Works of G. de Josselin de Jong

195

548

Q . DE .JOSSEL1N DE lONG

IIlId the underlying st.{ltionary salt wate r (horiionllli shading) is the line DM,M,P. This liDe has two inflection points, !of, and M •. T he hump between these two points can be considered as the upeoning. If the salt water is stationary , the interfrtee is kntlwn to map in the hodogrnph plane as a circle with diameter

Complex Potentwl in th e Two-Fluid C
The mapping of 0 on the upper half of the

t plane is obtained by the Schwnrz-Christoffel

K ' = k('Y. - "(,);-',

formulrt as follows:

with k = hydraulic conducth'ity, Y. = speeific weight of salt \ ater, and y, = specific weight of fresh water. The bodograph is representoo in Fi gure 2b. It is left to the render to verify that the path BA ,A,SCPlII,M,DCB that encircles the region of th(! aquife r occupied by the fresh water is mapped in the bodogTaph by the path BA,A,SO,G,FM ,lIf,DB. The arelUl A,A , and G,G, in t he hodographs are cireles with large radii. In the same way a5 in Figu re 1 the re!cvant region of the nquifer lies to tile righ~ of the traveler and in the hodograph to the left of the traveler. The points FM,MJ) from the interrace lie on the circle with dinmeter K'. Since M, and }.t. a re inflection points on the interface in the aquifer, these point! arc turning points on the circle of the hodograph. This makes, for example, the line FM,M . a slot in the sheet that is shaded vertiC:llIy in Figure 2b. T here is a narrow bridge between the points SM, that connects the region PG,G, to A•. F rom A. ov~ r the large circle to A, and farther toward B there is an ovcrlap encircling the slot at M •. This o"e rlap ends in the white beak that contains the points Band D. T he !"(llevant region in the hodogr:lph is double-sheeted in such a. way that it cannot be cut out of one piece of paper. By gluing two pieces togethe r it is possible to make a model that is simply connected. It is therefore possible to map the hodograph on a half infinite plane, for eJ[ample, the t plane (Figure 2e), which is also simply connected. T he complex potential is shown in F igure 'M. The discharges Q, and Q. through the drain and the sea bottom PG, respectivel y, are considered known. The location of the stagnation point S is lLS yet unknown. The shape of the interfaee in the ~ plane can be computed by mapping the pertinent domaillll in both the nand w planes ( Figures 2b and U ) on the upper half of the same t plane (Figure

196

2c). By using rebtion (5) , % can then be expressed as n function of t he parameter t, which forms ,.. solution because fo r every point of z the value of n lind w can be computed through the intermc
(x _ a) X''"(A "" + 1)1/1 + fJ J '~

fI(l) = a. = a,

J' ""+ ' + J AI'"(A

a.

(X

fI(t) ... a,[log l" - (n -

'(0

I)"t {a-rid>. a)A "'(A I)"i

+

+ fJ

~)a-II'

+ rr'''£(I, I, a)1 + fJ.

( 10)

where

+ (I + 1)'l'l'Il + I)"l + (I + 1)"'a'I' ] - log I(a -

l" - [t'lO {

.e(I, I, a) = 2 log [t' /'(o

(11)

t)

The value of the constants a., p, snd a remtion betwccn the lIS yet unknown values of 0, f, ~ can be obtnined from the boundary condition!. From (II) it can 00 verified t hat for j au the reB.! axis the imaginary parts of log r a nd £(t, I, 0) have the following values : 1m [log(j'" 0

for

0<1<""

1m liog r] .. ir

for

-

=

<»

""e"

< ,< -/ - Ie" and

(12)

rm (£(/, /, a)l = 0

for

0

<

I,

for

a


1m (£(/,

a)J "" i1f

t


•• d

-<» <1 < -1 The bou ndary couditiollll for 0 can be taken

Soil Mechanics and Transport in Porous Media

A

Mony-Vulu~d

549

llodlJgrl).)J"

from Figure 2d to be for t on thc real axis on A,B:

r" on PD:

- Q,.£(I, /, 0)1

1m [O(t)[ = -iQ,

<

I).

lIodO<Jraph in Two·Fluid Case

< en

t

1m [O(t) J - +;Q,

r" at G:

-.

ott) -

for


(13)

+ Q,)/Q,

+ {)'''/(a

- ,)

because the bounda ries then become straight lines. As is seen in Figure 2c particularly, the circle DM,M,F maps into II line pa ra llel to the te"l uis, at a. distance 1/K' abol'e it. T hc mapping of w' on t is then given by tbe expression

_ J'

( 14)

The final expression for Ott ) is then OCt) -

.'

...-'[(Q,

- a.

[r, og

+ Q,) -

log

(16)

W* = I/w

I - 0

- 0"' (0

To apply the Schwarz·Christoffel formula fo r the mapping of the hodograph on the t plane i ~ is convenient to trnnsfonn the W plnne into 1he w' plane by the invcrsion

0

Introduction of (12) lind (13) ill (10 ) pe rmits evaluation of a. and fl. lind doouctioH of thc foUowing relation between a, I, 8 1111<1 the discha rgE'!! :

(Q,

(15)

1<>* - a ,

r

(A (A

+

+ ....) dA + f)' It + {J

tn ,H A a/,\'i'(,\

which afte r integrlltion becomes

em, + .Hm, + ,)1'''(1 + D"' s(s + D(t s)

+ (m, + ,)(m, + 3)(S + to , ' ''(8

- (m , +

(17)

+ /loh

r. =

The boundar)' oonditions are now from Figure 2c:

[a'"

+ + D"t]'/I (I).

( 20)

The condition in A is fultilled if on

G8 and SB : 1m [w·(t)] - 0 for

t

< ""

t = 0

find

0

<

log

(m ,

r. '"

inAandG: fot on PD:

(2 1) I = 0

( IS)

1m [W*( I)] _ I/K'

With (1 4), (19), (21) it is possible to express 6, tn" 114 11.8 functions of Q" Qt, a, and 1 a$ follows:

(or -..,
( m,

+ a)(»I. + 3)a"'(a + D'" ,(, + f)(a 8)

+ a)(m. + a)(, + ~D - (m, + m, + ~),(, + D

(19)

Since both log' and t'" vanish for t = 0, the condition in G indicates that 11. = O. In the point A, t is equal to I). and the value of , is

Selected Works of G. de Josselin de Jong

8 =

m,

I). -

[Q,/(Q,

+ if)

+a=

iAl(a

+

+ DA-'J'lt l i,\l(, + if) + [(a + if)' - 48(a + {)A- ']' It , -

m.

+ Q-J][I).(a + DI"'

8

-

- [(a

+ if)'

~8(a

(22)

where

A

~

(Q,/(Q,

+ Q,)J log r.

(23)

Finally the condition along FD makes ...

=

197

550

O. OF. JOSS£I.IN DE .lONG

10 9

l/"K'. Therefore the expression for

'0

10· - ( I/, K'{ log

reduces

t (24)

Rewtiqn between z ami t, The two expressions (15) and (24) permit computing Z &8 a (unction of t. Writing (5) in a slightly different way and introducing (16) gives

z -

-

r

[Q'+.

+D

I/ ']

(t - 0)

1

1'1I(t

+ I)"!

Insertion of Ih is rC8ult and (24) in (25) gives with (1 4)

~

-[(Q,

2J~ . (log t.)' +

a" -

+ Q.lI.-' K']

r

[(!±1)"'(r .. - I)] logt. a tI-

+ (r ' / 6) -

+ {1

w·(dO!dt) dt

Q, o"'(a

z

the condition that A coincides in both planes is = - a. for t = a. This oondition gives

t

rtQ,K' { -log

Q,

+ -;-

oomes 1 and log' vlU1iahea fo r ' = O. Let the distance GA in the aquifer be a.; then

(25)

From (15) we obtain dO ;U-

The expression (27) can be further adjusted 110 that the poinla A and G in I oo rrespond to these pointa in t. In order that G lie in the origin of I, /3. should be zero, because' be·

log r d(1og r)

-r

lata

2Li,( I/ t .)

+ Li,(l / t :)}

(29)

If for the aquifer the distante a., the discharges Q" Q., and the value for K' are given, it is possible to determine all with (29), sinte it is the only unknown there, It is not Ieaaible to give an explicit expression for all except when it is very large, (All the cases mentioned by Bear and DIl{IBn .( 1962] have very large all valnes). From (29) it follows that for all» 1 therE" r('""h ~

+ [Q,/'-'K'l{log r . + 1)/1(1 + 1))'''(1

r

(! -

0)-' dl

+P

(26)

+ {J.

(27)

~ r!(2K'ao ' - iQ,)/Q.J'"

(30)

-

at'

log r dl}

Integration of thii! exprc&ion gives

z -

-Q~/K?'l log~J' + i~}{JOga ~ - Li,(UrJ

t logr.

+

where [.i.{.,.) st8nds for the dilogarithm defined

by Li,(.,.) _ - [~_l log (1 -~)

d).

(28)

T he terms eonsisting of dilogarithlll1l in (27) are derived from the last integral of (26). Since the integration of this term is not 60 easily ve rified, a further justification of the result is given in the appendix. The other terms of (26) have a more element8ry charaeter.

198

+

L"(l / r.)

log

IOg[J 1_ df;J[ logr + L"(n.)

- L..(r.)}

r. ~ log (40/1)

This relation is represented in Figure 3. The region of applicability of F igure 3 is limited, becaU!le the analysis developed is only valid jf the interface in the aquifer tontains inflection pointa M, and M•. From (22) it is seen that the expressiollB for,", and !no tontAin roots". T hese roots- are only real if

Soil Mechanics and Transport in Porous Media

.A. M!PIy·Valued llooograph

••,

'I:t/It'a~

551

1~(t.

r-

1

"

//

r

L

•

.1

" I.

I

"

"u

I

~

~~

••

Fig. 3. Plot of log

(. +;n' >

46(6

r.

as

•••

/I.

+ /)11.-'

< [(~ + W"IO 4a(~+{)

rJ -

g .

.

(31)

,.

left of the vertiCAl . This is 11nderstood if it is

realized that a velocity vector pointing to the upper left means an interface with fresh water below salt water. When the dot (M.M,) moves over the top of the circle, M. is the first to pass the vertical. For illlltability not to occur the criterion is therefore "that the real part of w at M, remains positive. This implies in the inverted hodograph that

Rc [w·(M.))

I

,• ' .

function of Q, alld Q. accordil1& to equation 30.

If (31) ill not satisfied, the roota are imagiuary and there are -no inflection points. Then the hodograph contains a branch point as shown in the one-fluid use. If (31) ill not satisfied, the fonnula$ are not valid becaU8e tbe Schwarz.. Chrilltoffe! analysis as used here is not appliuble if a branch point is involved. The condition (31) sets an upper bound to the ratio Q.,tQ, which hat the fonn Q, Q,

~

(32)

This line limits the zone of validity in Figure 3 to the left. The I"(lgion of applicability in Figure 3 is also limited at the bottom. This condition is visualized by considering the ease that the discharge through the dmin Q, is kept constant while the !!Upply from infinity decreases, 80 that Q. diminishes and eventually reduces to zero. Du ring this process a moment will come when the interface beeomes unstable and the drain starts to dillcharge salt water. By diminishing Q. the points M, and M, will move along tbe circle in the hodograpb in a counterclockwise direction. Instability OCCUI"!I for that pr!.rt of the interface that IIlIIpIl in the hodograph on pointa of the circle that Jie to the

Selected Works of G. de Josselin de Jong

>

0

(33)

The cri terion (33) can only be represented in a simplified fonn if a :» I, K'a. :» Q" and Q, :» Q., a8 ill true for the region of Figure 3. In that ease the requirement for the lower bound is

1

+ ... q + log ... q -

[Q,/12K'Oo)

>

0

(34)

v.ith IJ - (2Q.K'00)'''/ Q,

On the scale of Figure 3 the lower limit is 80 dose to the horizontal axis that it is not to be distinguished from it. Shape 01 the inlerlau lor a/I» J. For large values of all the fonnulas describing the inter~ faee can be given in a more compact form. The interface is the line DM,ltf,F in the z plane (eee Figure 2a) , which will be represented by z. =

199

552

G. DE JOlIS!:UX DE .faNG

+

iy,. T his line corresponds to a part of the neglltive re.... l axis in the t plano. Let to be defined by t = t"e"; then t. is a posith'c quantity

the fluids was ~ = 0.34 crn'/see. From these values K' can be deduced to have been K' = [ (Yo -"(,IJy,)(gb'/12v) ::: 2.54 cm/ aee. The

along DM,M,fi'.

duin distaDC{!

For point~ to the left of M , the value of t. is large with respeet to I (~allse m. i~ of the

through the drain (per unit width) was Q. = 29.0 cn)'/ see. The amou nt of fluid flowing through PO per unit width was Q. = 3.97 cm'/!!eC. Wi th these specifications it pm be deduced tbM all» I and log C. = 21.2 (see Figure 3).

r.

order 0). Therefore the quantity '- defined by , := C.e" is then approxima tely C. - 4t.1f· Inserting these quantities in (27) and separating TC:l.1 :md imaginary p.1rts of t . yields

a.

was 37.5 em. The discharge

'/f' K' Re (zo) "" r'K'zo

+ t.)/t .] + t Li,(t:lto":l - lA.(t .lto) + (,..'/6)1 + Q.{ - !(log to]' + !,..'] (35) -,..{Q, log ( t. + t .)/t.l + Q. log to] (36)

Q, lHlog (to/t.»)' - log (to/t .) log [(to

,..'K' 1m (:0) - ,..· K'yo cz

be ~impliflcd for C. » , .. J" the fonnula for 7 , thll dilogarithms can be approxima ted by I. i,{l/t) _ Ill' for z » 1 and they then drop out. Th e f"{>m~ining tenns penni~ elimination of C, by squaring y. und compa ring the terms obtained with 2(Q, + Q,)z./K'. The following relat ion between r. and Y. is the result:

Th~ .~xpTt."lS>lions ~an

Yo' ... -2(Q,

+ QJzo/K' - Q,Q.[ log r, ]'/(,..K')'

+ (Q, + QJ (3Q. + Q,)/ 3K" By mring the ap proximation (30) for log C. this reduct!s to yo· ... -2(Q,

+ Q,)Zo/ K' + [2Q,' + 4Q,Q.

+ 3Q.'

- 6Q,K'a.,1I3K'· (37) This is the shape of the interface for the quantity of water (Q, + Q.) coming {rom infinity and streaming around a wall with its edge located nt 11. d istance

+

+

+

(Q, /( Q, Q.)J[a., (Q. 2Q.)/oK'] (38) to the left from the actual edge. Tu t remlt. T o verify the above formulas a test was run in a parallel plate model, with a slot width of b ;0;; 2.1 mm. Figure 4 is a photograph of the resulting Bow pattern. The two fluids were re:cin oil (")" := 0.864 gf/em' ) and glycerine (y. = 1.071 gf/P.Ill· )' The viscosity of

200

From (22) it then follows that 3 = 0.120, m, = OJJ0(l45a, m, = 2.0a. Inserting these values in (27) gives the loclttion of poillts S, M" lind M , in the t plane:

z( M ,)

- 20.02 - 8.05i

em

(39)

-36.44 - 14.9i cm re M,) From (24) the inclination of the interface at M, lind M , can be computed, because the angle of the interface with the horizontal is equal to the angles M,AB and MoAB in the inverted hodog raph (Figure 2c) . These angles can be obtained from the complex values for W- at ,u, and ilJ., which afe

+ i)I K ' (l.28 + t}J K'

w*( M,) - (4.85 w'"(M.) ""

(40)

The streamlines obtained from the photog raph of Figure 4 are reproduced in Figure 5. In thi.! drawing the pointa surrounded by a square give the computed location of the points 5, M" and M , llceording to (39) . Th e computed inclination of the interface at M, and M. aeeording to (40) is represented by the dashed lines through these points. The agreernellt between test result and theory can be considered S1.tisfactory. T he point S coiricides with the stagnation point that separates the water flowing to the well, A, and the wate r flowing to the !!Ca, GF. The point! M,

Soil Mechanics and Transport in Porous Media

A

_M_

-.

-.

-"

Manli·Val~

Hodograph

,

A

-"

M.

-"

, •, • -"

M.

-"

Fig. 6. Plot of iDtetfaee aDd tltretunlinell or test. in parallel plAte model. Verilieat;on between theory and e:rpenments eolllisul of tile stagnation point 8 and the ;olleetioo pOints M., M. with the looal inclinatiollll of the ;otetf.....

Selected Works of G. de Josselin de Jong

201

554

G. DE JOSS£LIN DE JONG

and M. coincide with the ioHcetion points, and the dashed lines are practically tangent to the observed interface. The interface to the left of M. is approximated by the parabola (37) which by application of the parametel'll becomes

+ 25 .9K(, + 740 =

0 The deviation between the shape oomputed with 110'

this fonnula and the observed interface is too !lIlIan to be reproducible in Figure 5. From the agreement between test reIluita and theory, it mfY be deduced that the equations developed do represent the upeoning of the salt water-fresh water interface under a drain in a ooastal area.. The test Bho"'ed that the shape of the interrace W!IIl stable, in the sense that small disturbances disappeared without appreciably changing the location of the interfaoo.

Consider J, and !lIlb!;titute integral then becomes

J I -

-

·l

l

+ log r .

l

I

f [

dt

a)

(AI)

I[(r - 1}'/4rj

- I[(f - 1)/4rl

4aa+ _ (2

d

'/"10

+ 4(a/f))r + 11

(A2)

Using the constant {, introduced by the relation (20), [2 + 4(a/f)] is equal to [C. + ({,)_I]. The denominator of the integrand in (A2) can therefore be written IL!I

(, - ,.)1, - (,.)"'1

r{. -

=

J,

10grIOg(l-t)

L f

l, 202

(r - r.J

1.•

p.-' log (I - IA) dp.

log r log [I - (rlrJ I

- Li,(I/tJ

similar operation applied to T. gives llS a final result for (.'1.3)

J "" log

iogrdr [r (rJ- 1]

[1 1- «(ft>]

log

r + [,i,«(/t.)

+

- Li.(l/rJ - Li,(rr. ) Li,(rJ (A4) This result is used in (27). In these expressions the notation Li,(a) stands for the dilogarithm l1li indicated in (28). This fUllction is tabulated for the real po!litive value.. of " in the region 0 < OJ < + 1 (".,., for instance uwin [1958J), which also indicates how to deal with comple!O: values of
- Li,(l/a)

- i[[og .,.j'

+ (...'/3) + i". log a

for

1

<

a

<

(I)

(AS)

u.( -a)

for Li,(a)

I

'I' p.-' log ( I - p.) dp.

"1,

Li,(a) = !Li,(a"j -

+

+ J.

r~ - ll

log ( I - ,,) d"

III,

Deeause ({.t'] 4(a{a 1)/t]"", the integral J falls apart into two pieces J -

l ,-,

+ Li,(t!t.l

The domain of t. corresponding to the domain of t is gi,"en in Figures 2/ and 2". The result of substituting, in (AI) is J =

log (I - p.) I:;f:

A

dt - I[(f - 1)/4!'j dr

+ f)]'/J

[t(t

gives

f/l •

-

J, -

Consider the variable' introduced in (11). The fol!owing relatiODll between t and, exist: j ...

p.)

integr~l

!ogp.log( l - p.)lf~~:

J, -

+

+O J'" /.o ' [ "(" t(t + f) log !It'" + {t + D'''J'1tJ (I

(I

Partial integration of this

APPENDIX

Equation 26 requires the integration

+ log r.) dp.

,r. (log p.

',I.

= {Ie.; this

}J

-iLi,(l/a"j

-1
(A6)

+ Lit( -1/a)

- (...·/6) - ![log (-a)]" (A3)

for-""
(A7)

Soil Mechanics and Transport in Porous Media

50' The expression (AS) diffel'll from the value gh'en b)' Lewin with rtIlpeet to the sign of tbe term i .. log
+

arg, = O. Ackl'LQltll,dgmml •. For verification ud cxpall!lion of the mathematiCII I am indebted to A. Verruyt for hia invaluable aaistanee. The verir~atio" !.est "'lIS lIkillru lly run by K. Vcrvoort an d O . Z&nd8tra in the Uoivereity LAboratory Dr Soil Meehania at Delft.

fre.m and salt ..."ten ;1\ couta.l aquifers, ~ Report I , The lJteady interface between t .. o immiecible Buidr in • two dimensiOUllI aqui fer. 123 pp., Tu/utioll-lrrDel ifulilvle 01 TtchllologV, HlIdrau1ic Lob~ P.¥.I$/fJ!, December 1962. Btu, J ., and G. Dapo, Some exact eoIul.ions of illterfaee problems by means of Ute hodosraph method, J. G'OpAVI. Re•.• 89, 1 $63- 1~T2, 1964. !.tWin, L., DikJgarith_ QlIa AA~ioltd FUlII:liDnI, 353 pp~ MacDonsld, LoDdoo , 1958. ~ehari, Z., Con/cnmGl MaP1ring, 396 pp ., MeG ..... D ill Book Company, ht ed., t ~2. l>olubarinov,,-Kochin", P. I ., 1'he~v 0/ GrolilldtOOter Movement, t ran!lat.ed from the Russian by ROf!;cr J . M. Dc.... ic!lt, 613 pp., l'rincetoD t~n

Univeroity P re,,", 1962.

RXnRltNCI!:S

Bear, J ., and O. Dapn, The transition looe be-

Selected Works of G. de Josselin de Jong

( Manuecr,pt reeeived October 23, 1004.)

203

IfP • •NTm fllOM flOW TH.OUGH POtoUS MEOlA

@ 1919 ACAOEMIC PUSS INC.. NEW '1'011{

9

Generating Functions in the Theory of Flow through Porous Media G. de Josselill de j O"g Civil E'lgi..u ring Dt partmcot !klft Umw.rtity oj Tuhnology Delft, the Ndherlanth

I. 2. 3. 4. 5. 6. 7. 8.

Imroduc' ion l1a sic PropertiL'S of !he Flow Single.Valuedncss of ' he Functions Multiple Fluid Functions for /' and k Constant, p Variable Velocity of the Flow Field Sol utions with Sinb'i..ll.riti~, Multipl e Fluid Potentia l for Variabl ~ p, p, and k. ElecTric An alogy for V3riable /'. p, and k . 8.1. POIen!;al Lines. 8.2. Streamlines 9. Variable D ensity Flow in Th rt(: Dimensions References .

377

379 382 383 387 390 394 395 395 396 397 400

I. Introduction In this stud y the Row of fluid s through porous media is considered for the case in which Ruid and porous med ium arc inhomogen~ou s. The properties that may differ from place to place in the field are the density and viscosity of the flu id and the intrinsic permeability of the medium. These inhomogeneities are responsible for rota tions in the fl ow pattern. Since the Darcian fl ow of fluids through porous media is usually irrotational, il was considered instructive in this presentation to elaborate especial1y the character of these rota tions. For the determination of the distribution of th~ p ressure and the specific discha rge vector it is convenient to .make use of potential and streamfunctions. These functions may be called generating fun ctions, because their gradicnts gcnerate the specific discharge vector. For incomp ressible homogeneous flui ds these functions obey Laplace equations of the type

204

378

G. DE J OSSEL1N DE JONG

where:F is the relevant functio n. The solution of a problem is obtained by selecting from the functions that obey th is basic equation those that satisfy the boundary cond it ions. I n this study, function s will be determined that in a similar way generate t he specific discharge vector by differentiation if inhomogeneities are present. The basic equation s obeyed by these generating functions will turn out to be Poisson equations. These are equations of the type

where C!J is a func tion of place. If the value of C!J is known everywhere, it is possible to solve :F in a region enclosed by boundaries along which either the value of F or its derivative normal to the boundary is known. The solution of the Poisson equation will be given in the form of an integral which has to be extended over all points where C§ has a nonvanishing value. This method of solving may be called the solution by singularities. In addition, solutions of t he Laplace equation have to be applied to satisfy t he remaining boundary cond itions. If <§ is an unknown function (for instance, if <§ depends on F ), it is not possible to use the solution by means of the singularities mentioned. It may be possible that other kinds of singularities are able to cope with Poisson equations where <§ is a linear function of F. This possibility is not investigated here and only fun ctions obeying a Poisson equation with known values for <§ are considered. The generating functio ns appropriate for flow of fluids with varying properties may be called multiple fluid JUlletions . It will be required that the multiple Auid functions are si ngle valued in the entire field irrespective of how many fluids are involved. The multiple fluid function s will be chosen such that they resemble as closely as possible the potential and streamfunctions for the single Auid case. The multiple fluid functions developed here are essentially the same as those described in a previous article (de Josselin de Jong, 1960). In that paper Poisson's equations were established for the streamfunction 'I' and a multiple fluid potential, 8. Yih (1961 ) arrived at t he same result with respect to '1'. Knudsen (1962) showed that the pressure obeys a Poisson equation of a kind very similar to that for IP. Lusczynski (1961) suggested several types of heads for multiple Auid Aow. De Wiest (1964, 1965) reviewed this work and selected H i", the type called "point water head" by Lusczynski, as the most suitable physical quantity to use. It will be shown here that Lusczynski's "point water head" H iP is directly related to 9. By elaborating the

Selected Works of G. de Josselin de Jong

205

9.

379

GENERATiNG FUNCTI ONS

e,

second~o rder

differential equation governing H j p and it turns ou t, however. that HiP obeys a Poisson equation t hat is less convenient to solve than the relation governing The name chosen by de Josselin de Jong (1960) to ind icate €J may be confusing because (1 is not a poten ti al in the sense that grad 8 is proportional to the specific discharge vector. Because t here is a rotation in the fluid with varying density, the specific discharge ca nnot be derived from a potentia l. Until a better name is suggested, the term multiple fluid potential will be used here, with the stipulation t hat it is not a velocity potential in the usual sense. I n Sections 2- 8, the case of a two·dimensional aqui fer is treated. In Section 9, formulas are given for t he three· dimensional case. The density is considered variable throughout the entire chapter. The viscosity of the fluid s and the permeability of the porous medium are taken constant, except in Section 8. I n that section, an electric analogy is described which provides a means of treating variability of these other properties. AJI fluids will be considered to be incompressible. This means t hat fluid elements will preserve their density during displacement. The variability of density considered is the difference in density between fluid elements at differen t places in the field. No surface tension will be assumed to exist between fluids of different properties. The porous medium is isotropic. The multiple fluid functions generate the flu id displacement at a given instant. It is assumed in t his analysis that the distribution of the fluids and their properties is known at t hat moment. From the mUltiple fluid functions it is then possible to determine t he displacement of fluid particles during a small time interval. By taking into account the manner in which the properties of the fluid s are convected by this displacement, the distribution of the properties after that time interval can be determined. This creates a new situation which requires new multiple fluid functions. It is not the objective of this study to pursue what is happening at successive time intervals, but only to describe the instantaneous flow situation.

e.

2. Basic Properties of the Flow Since the fl uids are assumed to be incompressible, conservation of mass requires that t he specific discharge vector q obeys the continuity equation div q =

206

aq.. + aq. ax az

=

0

(I)

Soil Mechanics and Transport in Porous Media

380

G. DE J OSSELI N DE JONG

The second requ irement is Darcy's law which can be written in terms of an equilibrium of forces. When the forces acting on the fluid per unit volume of the fluid are eq uated, the equilibrium equat ion results: (p.{k) q = - grad p - p grad F

(2)

The left side of (2) expresses the force per unit volume of flu id necessary to drive a specific discharge q th rough the porous medium accord ing to Darcy's law. In (2) p- is the dynamic viscosity, k is the intrinsic permeability, and p is the density of the fluid. The two terms on the right of (2) represent the two forces that can work on t he fluid and produce the driving action. The first term is obtained from the gradient of the pressure p. The second term represents a body force which can be derived from a potential F acting per unit mass of the fluid. If only gravi ty works, F is equal to gz, in which z is measured vert ically upwards. T he equilibrium equation becomes in this case (I'-/k) q = -grad p - pg grad z

(3)

Equations ( I) and (2) are the basic relationships sufficient for the esta blishment of the multiple flu id functions. They are relations describing the flu id behavior in a point. Because (3) describes a point relationship it is irrelevant for the last term whether the density is a constant or a variable in the field . Since this poi nt was questioned, it may be helpful to derive this term by integrating the forces acting on the fluid in an infinitesimally small circular disc with radius r 1 ( Fig. I ). Let t be a coordinate running in the direction of grad P and let ~ be a coordinate perpendicu lar to Further, let the subscript zero in the expressions Po, (op/ot)o ,... , etc., indicate the value of these quantities in the center of the circle . Then the density in the circle is approximated by its T aylor series:

r

P = Po

+ {(8pj8{)o + H2(82p/8{2)o + a(82pj8{ aoo + g2(82pj a{2)o + ...

(4)

The force acting on a unit of mass in the gravity field is - g grad z. In order to find the total force on the fluid element, the mass of that fluid amount has to be evaluated by integration over the volume of the disc. Iff is the thickness of the disc and 1l the porosity, this gives

Jl1 =f• I• r,

2~

Selected Works of G. de Josselin de Jong

p · ljf·dr·rd8

(5)

207

9.

l81

GENERATING FUNCTIONS

p

M,

z

M,

FiG. I.

where p can be replaced by the series in (4). Let the series be split in three parts, giving for the integral three contributions Mil M 2 , M 3 , respectively, such that M, is obtained fro m the first term in the series, M t from the second term, and Ms from the third, fourth, and fifth terms together. 208

Soil Mechanics and Transport in Porous Media

382

G. DE J OSSELIN DE JONG

The result is (6)

Since M3 is of the order '\4, its value may be disregarded with respect to Ml which is of order '1 2 , because in the limit " tends to zero. To obtain the force per unit volume of the fluid, (6) has to be multiplied by g grad z and divided by nf· 'I1T 12 , the volume of the fluid in the disc. Thus, the force on the infinitesimal element finally becomes - p~ grad z as indicated in (3). The second term from the series expression for p does not contribute to a force on the element because M2 is zero. In the diagram for p in Fig. I, ( apla ~)o is the inclination of the line EFG, the tangent to the curve for p . The second term, t(8p/a g)o. covers the triangles BFE and CFG, which being similar and opposite give cancelling forces. However, they contribute a clockwise torque which can be written as r,

T =g cosa

f, J, f ·£(opfo f)onfdr ·rdO b

where a is the angle between f and x. By evaluating this integral it is found that

Since cos a = (O{/ox), the torque ca n be written as (7)

This torque is responsible for the rotational properties of the fl ow fie lds, as will be shown in Section 6. The expressions ( I) and (2) are therefore point relationships which only require Po, the value of p at the point considered. The variability of p with location only contributes if higher order differential equations are established. This actually happens in the subsequent treatment.

3. Single-Valuedness of the Functions

For the multiple Auid functions to be applicable in the analysis it is of importance that they be single-valued because otherwise they have no physical meaning and may become misleading in computations.

Selected Works of G. de Josselin de Jong

209

9.

GENERATING FUNCTI ONS

383

The single-valued ness of a function
(8)

This theorem can be proved by starting from the contour integral expreSSiOn

fc «()clJf8s) ds = 0

(9)

which indicates that for every closed contour C, t he value of
fc

(&
fc

[«()clJf8x ) dx

+ (&cPf8z) dz]

= fc(Mdx + Ndz)

fL[(aNlax) - (aM lax)] dx d. ~ ff [(0'<1>10' ax) - (O'lax ax)] dA ~ 0 • ~

(10)

Since (9) applies for every contour C in the fie ld , (10) applies for every area A enclosed by that contour. By reducing the contour to as small a circle as one wishes, the last integral of ( 10) indicates that (8) applies to every point in the field. Wherever expression (8) will occur in the sequel, it will be concluded that single-valuedness of the relevant fun ction is established .

4. Multiple Fluid F unctions for p. and k Constant. p Variable Of the multiple flu id functions, the stream function 1Jf resembles most t he conventional stream function. It will be defined in twodimensional flow as q.

~

- (O'Plax).

q.

~

+ (O'Pla.)

(I I)

This means that the specific discharge vector has the magnitude of the gradient of 1Jf and runs perpendicular to grad IJI such that the angle ~ from q to grad IJI is traced clock wise (Fig. 2) .

210

Soil Mechanics and Transport in Porous Media

384

C. DE ]OSSE LI N DE JONG

, q x

:~~

,

T

,- ---

.". of Flc. 2.

Application of the continuity Eg. ( 1) gives - ((J2Y'j8z ax)

+ (fP'P18x 8z) =

0

(12)

which shows that '¥ is a single-valued function irrespective of the properties of the Ruids involved . More information about 'f' may be obtained with the equ ilibrium Eq. (2). This equation written in its ,\', z components gives (el k) q. ~ - (lJplax),

(el k) q, ~ -(lJplax) - pg

(13)

From the physical viewpoint the pressure p must be single-valued because it is a physical scalar quantity which has one value in every point of the field. Therefore, the relation - (Otp/ax oz)

+ (fJ2p/8z ax) =

0

(14)

is satisfi ed everywhere in the fie ld. The identity (14) can be used to eliminate p from (13). Since (p-lk ) is a constant, differentiating Eqs. ( 13) with respect to z and X, respectively, and subtracting gives (Pl k)[(ilq.laz) - (ilq,l ax)J ~ - (fI'pl a.< ax) ~

+ (a'plaz ax) + [iJ(pg)l axJ

iJ(pg)lax

(IS)

Applying expressions (I I) to this result gives

0'

I V''P ~ - (kle)[iJ(pg)laxJ I Selected Works of G. de Josselin de Jong

(16)

211

9.

3S5

GENERA TI NG FUNCTIONS

T his relation, obtained previously (de Jossclin de Jong, 1960), permits computation of IJI wit h the aid of singularities of rotat ional type if the distribution of p is known. This will be shown in Section 5. Applying the continuity Eq. ( I) to (13) gives

v./k) div q

~

- (8'p/8x') - (8'p/8,') - [«,,)/8'J

~

0

0'

VIp

=

- O(pg)/oz

I

(17)

This relation for p was obtained by Knudsen (1961). The two Poisson Eqs. (16) and ( 17) are very similar because they contain the derivatives of the density in the x and z directions, respectively. For p, solutions of bou ndary value problems can be obtained by use of singularities in a sim ilar way as used for IJI. I n a previous article (de Josselin de Jong, 1960), a functio n was introduced which can be considered the equivalent of the potential fun ction for the flow of homogeneous flu ids. This fun ction 8 was called the multiple fluid potential, although it is not a potential in the sense that grad 8 is proportional to t he discharge vector. This fun ction 8 is related to pressure p and elevation z by the expression

v./k) (;

~

P + pgz

(IS)

H ere p is the density of the fluid at the point of consideration. Since

p, p, and z are single valued, 8 is also single valued. From ( 18) it follows t hat the relation between 8 and /JiP' the point water hMd as defined by Lusczynski ( 196 1), is (19)

Taking the grad ient of (18) gives, since (p./k) is constant, (p./k) grad 8 = grad P + a' grad(pg)

+ (pg) grad a'

(20)

A comparison with (3) shows that t his expression contains one term more than the expression for (p./k)q. Therefore, 8 is not a regular potential. By use of the equilibrium Eq. (3), the pressure can be eliminated to give (po/k) q = - (POlk) grad 8

212

+ % grad(pg)

(2 1)

Soil Mechanics and Transport in Porous Media

386

G. DE J OSSEL IN DE JONG

Taking the divergence of the expression in order to establish continuity gives (po/k) div q = 0 = - (P/k) \7 2 8 + div[z grad(pg)] (22)

This finally gives for

e the Poisson equation = + (kfp.) div[z grad(pg)]

\7 t 8

(23)

The Poisson equation (23) shows that B can be solved by use of singularities. Since H iP' the "point water head," is so closely related to 8 it might be expected that H iP also obeys some convenient Poisson equation. Elaboration along the same lines as above gives ViH jf) = (pg}- l{div[z grad(pg)] - 2 g rad Hlp' grad(pg) - Hi" V2(pg)}

Since the right-hand side contains H iP ' this equation is inconvenient in the analysis. H~p is therefore less suitable then 8 . The results of this section are summarized in Table I. TABLE I Multiple Fluid Functions Multiple Iluid functions:

,

,q

k

• -

,q

k

p

~

,

- -k -a",+ - -ax-

+ J: _

'p - - - pg-

- -k -a",+ - -8",Phys ical neo::es,sity

divq _ 0

_

, a-

"

divq _ 0

Phys ical necessity

Poisson equation established by:

Elimination of p

divq _ 0

Type of singularity

+ pg"')

-",-

,~

Single-valued ness estahl ished by:

Gives;

(k /I')(p

--,- ,~

· -

8 -

k 8(pg)

, ,Vortex

Vip _

a(pg)

,-

Soun:e or sink

Selected Works of G. de Josselin de Jong

V' 8 _

l'

a8

(}(Pg)

l'

a8

&(pg)

,.

,

+ - d ,v[z grad(pg»)

Source-sink dipole

213

9.

387

GENERATING FUNCTIONS

5. Velocity of the Flow Field The Poisson equation (16) indicates that the flow contains rotation in those points where the gradient of the fluid density has a horizontal component. Although reasons can be found in textbooks why this equation represents vorticity, it is helpful for the subsequent treatment to show this here. Consider the case that in a circular area A2 with radius "2 (Fig. 3), the following relation holds 'V2tp =

2w,

o < ,. < ,.~

(24)

r

FiG. 3.

214

Soil Mechanics and Transport in Porous Media

388 with w

C. DE JOSSELIN DE JONG

=

constant, and that outside A2 it is V21J1 = 0,

(25)

Let S be a circular contour, concentric with A2 and with a rad ius r that can be either larger or smaller than'2 ' Consider first the case where T is smaller than ' 2 > T hen only (24) is relevant. Integration of (24) over the area A gives

If. O"PdA ~ If. 2wdA ~ 2w If. dA ~ 2=,'

(26)

By Gauss' theorem the first integral can be written (27)

In this expression 11 is the ollter normal to the circle S, which is related to the radius r in such a manner t hat (alJljoll) = (oV'lor ). Because the system is axial symmetric, (e'Pj or) is constant over the circle S, and (27) becomes

fLV 1J1 dA = 2

f

(olf/lor ) s dS = (o'f'/or) brT

(28)

Combination of (26) and (28) gives (29)

(a'f'/ar) = wT,

From Fig. 2 it is secn that (alP/or) represents a discharge q$ tangential to the circle S. H ence, (29) can be written

0 < , < ':

(30)

and this indicates that w in the region A creates a tangential flow circulating around this region with a discharge velocity proportional to the distance to the center. w is called the 'c:o,ticity . The second case to consider is , larger than '2 . Since according to (25), the value of V2tp is zero outside A 2 , integration of V21Jf over the area A now only consists of a contribution over the area A 2 . This gives (31)

Selected Works of G. de Josselin de Jong

215

9.

GENERATING FUNCTIONS

389

Again, the fi rst integral of (3 1) can be elaborated by the Gauss theorem as (27) to yield a result (28) wi th, now the radius larger than ' 2 ' By combining (28) and (3 1), there results BlJIlBr = w('2 2j,)

which shows that the discharge circulat ing around A2 is (32)

This expression indicates that the discharge outside the region A 2 , which contai ns the vorticity, circulates around that region with a velocity which is inversely proportional to the dislance from the center. The d istribution of the specific discharge as given by the Eqs. (30) and (32) is represented in Fig. 3. In that figure it is seen that on the boundary of the region A2 containing the vorticity, the circulating discharge q6 is continuous. This follows from (30) and (32) which give the same value for q~ if, is made equal to ' 2 ' The continuity of q. is a direct consequence of the above analysis, wh ich shows that a discontinuity can only exist in a place where w is infinite. From (16) and (24) it follows that the vort icity is equal to (33)

This expression s hows that w is only infinite if op/8x is infinite. This is the case if p changes ab ruptly, as is encountered at a sharp interface between two different fluids. Since this aspect was treated extensively by de Josselin de Jong (1960), its treatment will be omitted here. Only gradually changing densities (grad p = finite) will be taken into account in the sequel. A further conclusion from (33) is that every small region, where the density varies in the horizontal x direction, contributes a vorticity similar to the region A2 in the example. Arou nd every elementary region LlA2 containing density variation, a circu lating discharge Llqs is created which obeys Eq. (32). The influence of the density variations in the entire field is thus obtained by adding vectorially all elementary discharges Llqs created by all elementary areas LlA 2 . By choosing the elementary areas LlA2 infinitesimally small, the discharge Llqs in the area itself obeying Eq. (30) can be disregarded . If this reasoning is correct it is also necessary t hat the discharge (30) be obtained in this way by integrating discharges of the character (32) as generated by infinitesimally small vortices over the region A 2 • This may seem contradictory because it requires that the result

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obtained for the region outside A2 (which differs from the solution inside), when appl ied to the region inside, actually gives that different inside solution. A lengthy but straightforward evaluation of the integral involved shows, however, that the solution (30) is indeed obtained. The verification is an entertaining exercise in elliptic integrals. It may seem to be a difficulty that the discharge according to (32) becomes infinite for r = O. This means that the vortex at the point of consideration contributes an infinitely large discharge. However, this difficulty disappears since the magnitude of t he vortex at a point, being proportional to the area, becomes infinitely small of the same order. Therefore, the integral remains finit e. This exorbitant behavior at a point containing vorticity is the reason we apply the word singularity. Usi ng the solution (32) for the discharges around points containing vorticity an d adding those in an integral is called the integral solution with singularities. Instead of adding vectorially elementary discharges, it is mathematically more convenient to add the elementary streamfunctions which generate them. T his gives an integral expression for IJI which is explained in Section 6.

6 . Solutions with Singularities If a small region of the size dxo dzo around the point xo • %0 contains vorticity w(.\'o , %0)' t he discharge is rotating around that point according to (32) with a magnitude

qs = 2w(xo' %0) d.>:o d::ro/21TT

(34)

where T is the distance from the point of vorticity. Written in terms of the streamfunction IJI, this is

which by integration gives (35)

The constant is not a function of 0, because qr = 0 since t he discharge only rotates around the point of vorticity. Thus, 81J1180 = O. Since we are only interested in the derivative of IJI, the constant is irrelevant and can be taken as zero without loss of generality.

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217

9.

GENERATING FUNCTIONS

391

Because all t he regions containing vorticity contribute to 'P in this manner, the total expression for 'P as created by all the vortices is the integral (36)

where In the multiple flu id case where density variations create vorticity according to (33), the expression fo r 'P becomes lJ'1(X, z) = ( - k/hp.)

ff [O(pg)/ox](Iln r dX(I dZ(I

(38)

T he surface integral for 'PI covers the entire region between the boundaries containing fluids. This expression for 'PI forms t he vortex part of the streamfunction. I n general, the values of IJI, on the boundaries differ from the values required by the boundary conditions. These differences form a new boundary value problem to be satisfied by a harmonic streamfunction IJI II obeying the Laplace equation V21J111 = O. The function 'Pn forms the second part of the solution. T he fi nal solution to t he boundary value problem is IJI = IJI I + 1f'1I . T his way of solving the problem is known in potential t heory as the solution of Poisson's equation by singularities. I n this case t he Poisson equation fo r IJI) is (16) which has (38) as its solution. In a similar way, a solution fo r the pressure p or the multiple fluid potential 9 can be given because they obey similar Poisson equations i.e., (17) and (23). This gives PI

= (- 1/271-)

fJ[o(pg)/oz](Iln r dxo dzo

9 1 = (+ k/bTP.) JJ [z V2(pg)

+ O(pg)/oz]o In r dxo dzo

(39) (40)

Also in these cases, add itional functions Pn and 8" obeying Laplace's equation must be introduced to complete the solution and to satisfy all boundary conditions. An alternative way of decomposi ng the solution of Poisson's equation fo r boundary conditions is pointed out by De W iest (1969) . I nstead of In r which can be considered as a Green's funct ion for the infinite domain, Greens functions can be introduced which have zero conditions

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G. DE ]OSSELIN DE JONG

along t he boundaries of the domain considered in each specific problem. (The function or its normal derivative is zero along the boundary.) The differences wit h t he required bou ndary values are then those boundary values themselves, creating a boundary value problem with a solution satisfying Laplace's equation. Since, however, a boundary value problem has to be solved anyway, the method proposed here seems less laborious because it circumvents the construction of a Green's function, which in contradistinction to In r is different for every bou ndary geometry and for every point in the field. We are now ready to compare the torque, caused by density variation in a circular area as determined in Section 4, to t he driving force necessary for the circu lating discharge caused by the vorticity. If, however, the vorticity is limited to the circu lar area, the available torque is not sufficient to support the discharge outside the area. This outside discharge requires a torque of infinite magnitude. T his discrepancy is a consequence of the vorticity, being caused by density variations. In other instances vortices exist which can be limited to isolated regions. In this case it is impossible to visualize an isolated region containing a density gradient. Isolation means that it is surrounded by a field which, up to infinity, has the same density. The grad ient actually means that the field contains fl uids of different density. If th is difference exists up to infinity, this entails also a torque which is infinitely large. In order to circumvent this difficulty and to obtain a realistic situation, the case of a circular aquifer of finite dimensions an d surrounded by an impermeable boundary will be considered (see Fig. 4). The radius is ' 3, the thicknessJ. and t he porosity n. For the analysis to remain simple the density will be taken to vary linearly over t he field accord ing to (41)

where Po' is a constant. For this situation, (aplax) is constant over the fiel d and equal to 8p 8p 8~ , 8x =8g· 8x = Po

COS(I

(42)

From (33), the vorticity is

0 < , < 'a

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219

9.

GENERATING FUNCTIONS

393

p

~

'~ .

. F! c. 4.

which is also constant. As explained above, an integration of the influence accord ing to (32) of all the vortices in the entire region gives the solution similar to (30) 9s

=

wr = - !(klp)[8(pg)j8.t] r

(43)

Since this gives directly a discharge perpendicular to the radius, this solution satisfies the boundary condi tions which have actually been chosen in this example in such a way that the discharge obeys the simple formula (43). Therefore, the second part of the solution is zero. Since in Fig. 3, qs is positive for counter clockwise rotation, (43) gives a clockwise rotation for positive p~. I n order to show that the clockwise torque (7) is apt to create this circulating flow, the torque will be computed here by considering the required forces.

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G.

DE JOSSEL I N DE JONG

T he force necessary to drive fluid t hrough the ring between rand r + dr is

Hence the total torque clockwise in the circular region is

(44)

A com parison wit h (7) shows t hat if r = r3 , the two torques are equal. From this analysis it follows t hat the rotational flow within t he circular area is possible because a torque created by density variations acts on the flui d of th is region. This is comparable with t he general concept t hat forces of the boundaries cannot create rotations in the interior of t he fl uid . Vorticity can only occur in those places where body forces act in t he form of torques on the fl uid. The geometry of Fig. 4 gives a fl ow pattern, which by its simplicity permits us to predict t he rotation of the fl uid at all subsequent t imes. The discharge is such that t he Au id rotates as a rigid body. Therefore, regardless of dispersion and diffusion, t he Auid conserves its density distribution (41 ) if t he coordinate g is aHowed to rotate clockwise with t he fl uid. This means that the angular velocity is equal to

Integration gives fo r

01

=

0, t = 0, (45)

7. Multiple Fluid Potential for Variable 1-'. P. and k Equ~ t1 on ( 1,6) I ~ no t co rre et.

Sl neo tY h

e· ~~ "

-+

defi ned by (1 8 ) t o be

~~ E z

"

(46)

.. 0 lin d with ( 21)

(47)

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221

9.

395

GENERATING FUNCT IO NS

ther efore the contin.,it)' equation

~ ho"ld

be

(48) It is l.mpo"s1bl" to b uild " P"i~non equation tn

e

f ro," thh ""pfe n t on and

t herefore .. l..<> it 15 impos,dble to c'Ons~ruct an 1nte (, r~1 e~prcsRlon for@ ·"f the f or,"

(~e).

which represents the rotational part of the solution. For 'P and p, the equations are

~ B
V2tp = -

'"

V2p = -

(49)

k

ox

'"

a<.;;) -

grad

(~ )

, [grad p

+ pg grad .3']

(50)

Both these expressions contain 'P and p, respectively, on the right sides. Therefore, the solution cannot be evaluated with t he integral expression because the integrand still contains unknown terms, A possible way to obtain a solution is by electric analogy. [See Malavard (1956).) Although this manner of solving a multiple fluid problem may on ly be of academic value, it will be given here for the sake of completeness.

8. Electric Analogy for Variable "'. p. and k 8.1.

POTEN TIAL

LINES

If the objective is to determine the pressure in the fluid . the electric potential can be identified with the multiple Ruid potential multiplied by (p. jk). This slightly revised funct ion will be called 8*, The analogy between flu id and electric quantities is 8*

= (p

+ pgz)_ E

(",Ik) q - z grad(pg) H Pe ie

(P/') -

222

Po

(51)

(52) (53)

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396

C. DE jOSSELIN DE JONG

in which the electric quantities are: E. the electric potential, p ~ , the specifi c electric resistance, and i e • the specific electric current vector. Because p, z, 1-'-, p, and k are single-valued (unctions, 8* is singlevalued and therefore, the single-valued ness of E is assu red . I n the electrical model the current will flo w according to Ohm's law, wh ich can be wr itten as Pel" = - grad E

(54)

which holds also if p .. is variable in the field. For the flu ids t his means that the fo llowing relation is supposed to hold (p.fk) q - z grad(pg) = - grad p - z grad(pg) - (pg) grad z

(55)

A comparison with (3) shows that this is indeed required. Application of the continuity equation for the fluids can be effected after divid ing (52) by (jL/k) at the left side and by Pe at the right side . Taking the divergence then gives div q - div(kz/p.) grad(pg) _ div ie

(56)

Requ irement (I) indicates that div i e must be made equal to the electric analogy of - div(kz/jL) grad(pg) . Since div i e is the amount of electric current that has to be injected into the electric fl ow field, it means that in those p laces where p or k or jL and P vary. current has to be fed into the model. I n an electric model, only electric potential can be observed or for that matter, equal potential lines may be drawn . From these lines the values of the pressure can be found by means of (51).

8.2.

S TREAMLI NES

If the objective is to determine the streamlines in the fl uid, the electric model must be used in an inverse way such that electric potential lines are related to the streamlines of the fluid. The inversion of streamImes and potential lines is obtained by inverting the boundary conditions. But besides that, in the interior, the inversion also has its consequences because the analogy of the resistance m ust be made reciprocal. T he analogy betwecn fluid and electric p roperties will now be taken as

q., _

- (oE/8z),

q. _ + (oE/ox)

P./kH I/Pe

Selected Works of G. de Josselin de Jong

(57) (58)

223

9.

397

GEN ERATI NG FUNCT IONS

Continuity of the fluids requires that - (f;~ E/oz

ox)

+ «(}2E/o.'( az)

=0

which indicates t hat E is single-valued. Ohm's law applied to the electric system gives with (57), (58), and (13) the fo llowing relation between the specific electric current and the specific discharge of the fluids .

le~

I aE

= - -Pe -(J.'( -

jJ.

-k q,

8p

= + -0%+

pg (59)

. Ie

l aE

p.

8p

= -Pe-az--+ -k q~ = -ax •

Taking the divergence of ie and using the single-valuedness of p then gives di v i.e _ [8(pg)/ox] (60) From (60) it fo llows that also in this case an injection in the electric fl ow field is necessary with a magnitu de of the electric current equivalent to [8(pg)/8x]. Because of the inversion, injection in this case has the effect of vorticity. From (57) it follows directly that lincs of equal electric potential will trace streamlines.

9. Variable Density Flow in Three Dimensions For three dimensions, Poisson equations similar to the two dimensional case can be derived. For the pressure p, this was done by Knudsen (1962). Writing the equilibrium equation (3) in vector notation gives ("Ik) q ~ - Vp - pgk

(6 1)

where the vector k indicates the unit vector in the z direction, vertically upward. Continuity applied to t his expression if /J.. and k are constan t, leads to the follo wing result in view of (I), (p./k) V . q = _ 'V2p - 'Vp"gk = 0 0'

(62)

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G. DJ,; j OSSEL I N DE J ONG

For e, t he Poisson equation is already stated in general coordinates by (23) giving '17 28 = + (k/p.) V . [z Vp] (63) For the streamfunction it is more complicated since the t ransit ion to t hree dimensions requires the introduction of two strcamfunctions. How this can be done was described by Yih (1957), who showed that the discharge can be written as q ~ (V'!') X (Vx)

(64)

Since the establishment of the appropriate Poisson equations for lJI and X was too difficult to ach ieve, it will be shown here that the discharge itself obeys a Poisson's equation which permits a direct computation with t he solution by singularities. Application of the cu rl operation to (61) gives, if p. and k are constant, (p.{k) V X q =

-v

X

'Vp - V X pgk

(65)

Expansion of the first term to the right side gives

v X Vp

~

i (B'play ox) - ("'plox oy )]

+ j(",p/ox ox) + k((o'p/ox ay) -

(o'plox ox)] (o'play ox)]

For p hysical reasons p is single-valued, hence by use of (10), aU three vectors are zero and the term vanishes. T hus, (65) becomes (Plk) V X q

~

- V X pgk _Wplay) - j(oplox)]

(66)

Since \7 X q represents a vortex, this expression shows that the axis of the vortex lies in the horizontal plane and has the direction of lines fo r p constant in that plane. Further, si nce i (op{ox) j (op/oy) is the horizontal component of t he density gradient, the expression (66) shows that the vortex strength is proportional to this horizontal component of g rad p. T he curl operation o n (66) gives

+

(.Ik) V X (V X q)

~

- V X (V X pgk)

The fi rst term can be written as " X (\7 X q ) = V(V . q ) - V2q

Continuity requires t hat \7 • q be zero so there remains V~q

= (kifi-) V X (V X pgk)

Selected Works of G. de Josselin de Jong

(67)

225

9.

GENERATI NG FU NCTIO NS

399

This is Poi sson's equation for q . Solutions of this equation can be given in the form of the integral qr = + (k/41TJ.L)

f,

V X (V X pgk ) r - 1 dV

(68)

This integral describes the part of the discharge vector that is created by the vortices caused by density variations. A second part, qu , that is irrotational has to be added to sat isfy the bounda ry conditions. The integrand of (68) contai ns the density variations because the development of the curls gives V X (V X pgk )

=

+ (02p/OX oz) i + <02p{ay oz) j .,. [(02p/ox 2) ...,. (82p/ayZ)] k

FIG. 5.

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Soil Mechanics and Transport in Porous Media

400

G. DE JOSSELIN DE JONG

The behavior of the discharge is represented in Fig. 5 where a threedimensional picture is given of surfaces of equal density. In the region where the density varies continuously, many of these sheets can be visualized as lying perpend icular to the density gradient. According to (66). the discharge vortex created by the density gradient has its axis in the d irection of horizontal lines on these sheets. The vortex axis, which is horizontal, is perpendicu lar to the vertical direction of gravity; it lies in the isopicnic (of equal density) sheet and is perpendicu lar to the gradient of density. T herefore, the vortex axis is perpendicular to the plane through the vertical and the density gradient (shaded area in Fig. 5). The discharge from the vortex rotates in a plane perpendicular to the vortex axis. Hence, finally, the discharge vortex in a point is rotating in the plane through the vertical and the density gradient in that point. T he strength of the vortex is proportional to the horizontal component of the density gradient.

REFERENCES de Josselin de l ong, G. (1960). Singularity distributions (or the analysis of multiple fluid flow through porous media, J. GeophYJ. ReI. 65, 3739-3758. D e W iest, R. }. M. (1964). Dispersion and Salt water Intrusion, Groundwatt1', }. 0/ NWWA 2, 39-40. De Wiest, R. J. 1\1. (1965). "Geohydrology," pp. 305-3 13. Wiley, New York. De Wies!, R. }. M. ( 1969). Green's fu nctions in the flow th rough porous media, this volume, Chapter X. Knudsen, W. C. (1962). Equations of fluid flow through porous media-incompressible fluids of varying density, J. GeophYI. ReI. 67, 733-737. Lusczynski, N . J. (196 1). Head and Flow of Groundwater of variable D ensity, J. GeophYI. ReI. 66, 4247-4256. Malavard, L. C. ( 1956). The use of rheoelectrical analogies in aerodynamics, AGARDograph 18, NATO Advisory Group of Ae ronautical Res. and D evel., Paris. Yih, C. S. (1957). Fonctions de courant dans les ecoulements a trois dimensions, La Houille Blanche 12,439-444. Yih, C. S. (196 1). Flow of a nonhomoge neous fluid in a porous medium, j. Ffllid Mech. 10, pp. 13 3- 141.

Selected Works of G. de Josselin de Jong

227

GEOSCIENCES

87

Vortex Theory for Multiple Fluid in Three Dimensions G. de Josselin de Jonq

Geot6chnical Labozoatoz-y Departmsnt of Civil Enginsering Delft Uniwrsity of Technology K$lVBrling Bui8/ll/VVB!i 1

2628 CL Ds lft, The Netherlands Del~ ~r.

Rep., 4

(1979) pp. 87-102

Received, 1 IV 79

The specific discharge in a pOl'e fluid !<1ith specific wight, y, tllat is

variable in space, is a rotational flotJ possessing V01'ticity in tllose 1'6gions, wM1'6 y varies in 1I0"';sontal direction. Tile vortioos have lIorisontal = s

parallel to lines of COIIBtant y. They create a cireulating jlol,J in the aquifer a1'OW1d them. On. 0 sharp interface the con.tou1'lines of equal height f01'fll !1Ortez tines , that enclose l'f/entl'ant vortez ribbOrlB of constant strength. F01'1llUlas are given for the specific discharge in an arbitrary point of the aquifer, created by the vortices in 0 triangular part of an interface. These relations are suitab le for detemrining the displacement of an intel'fooo in time. FinalZy the specific discharge in a point of the intel"face , as created by the VOl'ticss in a s ma H eil"cular region of the int6l"face around that point, is demonstrated to con.sist of a shear fl= only, similQI' to the shear fl= ocaurring in the dimenaional case .

tIJo

Introduction

The vOTtex theoTY. tOT the s i multaneous flow of fluids with different specific "eiqh t t hrou9h POTOU S media , ... as revi e wed in this journal (Tet. 3) . for the two d imensional ca s e. In an adjoinin9 aTticle

Haitjema (Ref. I) published the

JX>ssi bi lities o tte T"d by a computer progra m based on the two dimensional voT t ex theory. The extension towards thTee dimensio n s of this work turned out to be invo lved a nd to Tequ i re IDOTe than tTivial el a boTa tions. TheTefore it was consideTed ltPPTopr i atel:O present the thTee dimen s ional vortex theory h"re In so .. " detaiL s UT Unq f To .. the ba s i c equations, a llthouqh the basis of the t h eo r y w,,- s al r eady p ublish ed p re'liou 9 ly (Ref. 2) . Fo r collateral r "'uli n g , c hapte r 5 of the book "llydrodyna m1cs·· by l4r.b i s reco """,n
228

88

4 (1979) DELFT PROGRESS REPORT

1. Basic equations for pressure and s tream functions Daruy ' 8 Law and C!OI1tinuity The fluid flow through the pol(es of an aquifer is gove"ned by Darcy ' s law and the requireoIlent of continuity. Darcy ' s

'laI.l "an

be obtained by considering

equilibrium of forces acting on the pore fluid. The specific weight of the fluid, y, and the gradient of the pressure, p , produce forces . that drive the fluid through the pores. 'these dr lvlnq forces equalize the resistance f orce of

•

•

!llll.gnltude ()I/.,)q , ecountered by the fluid when a specific discharge q 1" flowing through the pores. ~.

Expressed in ter",,. of the components

' \ ' qz ' of the vector •q the

pertinent relations al(e (see f.i. Vel'l'Uiit. 1970)

(0:/111 (3ptax)

1'u,

(.:hl (3p/3yl

(O:/I1){(3p/az l + y )

where r 1s the intrinsic permeability of the a quifer and 1I ill dynamic viilCOsity of the fluid. The x. y coordinates are in the horizontal plane and z is vertical upwards. In the following treatment.

0:

and 1I will be con .. idered to be

constants throughout the aquifer. but'( is v a riable in space . The requir ement of

continuity is expressed by

o

(1.2)

Poisson equation foro pY'e88U1'e 8y

introducin<} Darcy's law (1.1) into (1 . 2), the specific di scharge

component s can be eliminated and there results, if ("Ill) is a constant,

0..3) This relation is due to Knuda .."

(1962) and shows that the fluid pressure p

obeys a Poisson type equation. Such an equation describes the distribution of the potential c reated by sources of s trength,

(3y/3z). Every volume dV, where '(

varies with height, con tributes to the distrib.ltion of p. This contr ibution h"s in a n infinite medium the magnitude (l/4I1R) (
poin~,

where p is considered. The total

result is an integral Over the entire region, where '( is v"riable, p .. (l/h)

IJJ (3,(/3z)

R-l <.IV

Selected Works of G. de Josselin de Jong

( 1.41

229

"

GEOSe I ENeES Since this value of p occurs in an infinite medium,adjustments PI nece ssary t.o comply wit.h boundary cond ition s adjustment.s obey t.he

us~al

have to be added. These

Laplace equat.ion

The solution (1.4) .... y be of interest in p r o b lems . where t.he emphasis 1.& On pre ssu r e . The specific discharge vector can then he obtained fr01ll the gradient o f t.he p r es sure by use of (1.1). In the case , ",here the d + sPlaCetllen t.s of sal1.ne ppr .. wat. .. r or o t.h .. r cont.amminant .. Ar<> of pri""ipal lmport""" ... i t 's .-nnv<,nfpnl:

1\...,,-,, :1'41

t

to(o: l spos .. of " met.hod to det.ermine t.he specific d i scha rge directly . Thi s is possible. since the solu tion for discharge can be writ.t.en in the fOnD of integclI ls similar to (1. 4). These in t.egrals are obtai ned by considering stream functio n s that obey Poisson equations of t.he t.ype ( 1. 3) .

Stream jUnctions In t.he t.wo dimensional case only One s tream funct.ion is sufficient. for describing t.he flow behaviour . In the three dimensiona l CIlSe it is necessary t.o introduce t.hree st.rellm funct.io n s, called P , G, a here. These st.ream functions generate the s pecific dischllrge

'. " '.

(aa/ay)

(aG/a",)

(aF/a",)

(3a/a ,, )

(aG/ax)

(aF/ayl

cOlllponent ~

liS follows

The partial derivat.ives of F. G . a not. mentioned in (1 .6 ) are still tree and are as s umed to satisfy the IIdditional r equirement.

(aFIa,,) + (3G/ay) + (aa/az) • 0 The functions F, G,

a

(1 . 1)

lire mat.hemat i cally II ccepta ble . only i t they are single

valued in the entire region. Then their mi xed derivllt.ives li re cOlllllll.l t a t i v e. In pa rticular we l'Wve

a2 Fnya", a2G/Oza" a2 a/a,,3y

~lF/aza y

3 'G/ax3 z ~2H/aya x

'l:!'te equaLit ies (1 . 8 ) are sufficient. to e nsure continui t.y. liS can be verif ied by l utrodu Cing (1 . 6) into (1 . 2) . For the equivalence o f singleva l uedness lind t\.(. comnrul..lltive proper t y of partilll derivatives see f. L previous arti c le (dlt Josule/.il'! de Jong,

230

the appendi" o f the

1911) .

Soil Mechanics and Transport in Porous Media

90

4 (1979) DELFT PROGRESS REPORT

Vorticity vector At thie point i t Ie app ropri~t c to introduce the vorticity v o ctOr

0 with

components Ox, n y • Oz in x, Y. z directions. These components are rel a ted to the specific discharge components q", qy, qz by

" ,• "

"•

(3q"l;;y)

(3~/h)

(ag,,!;;,,)

(3q,/3kJ

(a~/ax)

(3q Jay)

•

(In Lamb's notation: The COlDponents

r. •

•

20y - n,

2f1 z • t ·)

fly ' Oz con::espond to a r ota tion around the r espective

coordinate a xi s in the direction of a right handed screw. The components are drawn as double pointed arrows in fig.

1 and their

modI! of rotation is shown

by the circular arrows.

T"

components of vorticity 0" be e xpr ess ed in ter",s of the stream .functions by eliminating from (1.10) by use of (1.6) • Th" gives

,." ,• 20

,

'. ", '.

(a 2 G/hc3y)

(a 2 FJay2)

(a 2 F/h2)

(a 2 Hny3z)

(;;2G/az 2 )

(a 2 GJa ", Z)

(3 2 Fnzh)

(a 2 Hn",2)

(3 2 H/3y2)

• (3 2u/3dz) • (a 2f'nyh) •

(3 2C/3z3y)

l"'"

These expressions can be simplified by use of the a dditional requirement (1.7). By taking the ",-derivative of (1 . 7) it follows that (1.12)

Since the st ream functions r, G, Hare s1ng l e v a lued everywhere 1n the aquifer, their mixed derivatives a':e commutative. So (1.12) can be used in the first of (1. 11) to elinminate G and H. By similar operations on the othc .: two of (I. 11) there .:esults

Pig. 1. Conpone>its of VOI'ticity and their mode of r otation .

Selected Works of G. de Josselin de Jong

231

"

GEOSCIENCES

Poisson equation f01' the strecn fimctions Thf,

equations (1.13) have the character of Poillson equations. Their so lution

can be given in the form o f integrals if the vorticity componentll Ox' 0y ' Oz have a known maqnitude. Actually this is the case, because their value can be obtained by

e l ~inatinq~,

Since the pressure p

i~

qy'

qz from (1.IO) by use of Darcy's la w (I.l).

a phy sical quantity, that hall only one value in a

point, p is single valued and it s mixed derivative s are commutative. So we have

a2p/ayax a2 p/azay

a 2 p/ax3y aZp/ayaz a2 p/hb

a2 p/axaz

Therefore introducing (I .I ) in (1.10) also reBUlts i n el1minatinq p and there remains (.::/11)

(3y/3y)

+ ('::/11) (ay/b)

o Therefo r e the stream funct i ons F, G, H obey Poi "...,n type equations of the fo rm

+

('::/11) (ay/ay) (.::/11)

(ay/ax)

o Since the right hand

side~

are k nown quantities , the solution of F, G, R can

be written in the fo r lD of integ-nls similar to the e xpress ion ( 1.4 ) for p.

These sol.utions a re

, . 0/4 .. ) o•

0/ 4.. )

, . ( 1/4 .. )

III III III

211 x ' 2fl y ' 20", '

,-, ,-,

dV ' .. dV ' "

•

( .::/ 4"11 ) 1.::/ 4 "11)

III III

(3y/ay')

,-'

(3y/a x' )

,-, d" '


R- 1 dV ' - 0

)" '"

2. Physical interpretation Local coordinate system .o.ccordlng to (I. I S) the specific discha rge has vor tidty (a nd therefore r otates) in t hose points of the aquifer, Whe r e the specific weight y varie s in " and/or y direction . EverY"'here else , where y is a constant , the flow is irr<>t·'tionaL In order to speci f y the charac ter o f tI,e ""'tion i t is a ppropria t e t~

232

Introduce a loca l orthogonal, right handed coordinate system n, S , t,

Soil Mechanics and Transport in Porous Media

4 (1979) DELFT PROGRESS REPORT

92

system oriented 1'I0ffl1al and tangent to the y-const-ant plane .

tangent to the p l ane of constant y fig. 2. The o r 191n of the n, s, t coordinates 1s in the centre of an infinitesimal volume dV conta in! "", '(variation and t heref ore vorticity. Let n be normal to the plane of constant y.

poi nting upwards. Then n has

the direction of g rad i en t y. Since ho_vcr the lighter fluid wi ll be above the he a vier , the positive direction of n corresp:mds to y decrease and therefore

with

-Vy _ '\"he coo.-d lna t es S ,

t

are orthogonal to n and ly both in the '(-

constant plane. We then have ay/an • -

(2. I)

Vy

The coordinate s ", t are chosen In such " "",nner, that t

is horizontal and

s is in the direction of steepest ascent. The angle between 5 a nd its p ro jection on the ho r i z ontal pillne is 0, whil e the lingle between this p r ojection and t.he x- a x is i s S, see fig . 3. With x', y ', z ' the coordinat.es o f the centre of dV , the transfor .... t ion r e l ations be tween the coordinates x, y, z and n, S , tare

(.

x')-

(,

y ')-

(.

z ' )-

"

• ,

,. ,. • ,.

" " •"

-'" -'"

0 0

00_ 0

, •• •• --'" , ,• 00.

)

00. 0

0

0

00,

00. 0

000

)

),

"

,

sin S

1

000

j(2.2l

-'" , • , ,

00_ 0 00.

d"

.'.' -'" , • .' ) d "

S

S

, ' ) sin , ')

0

0"' 0

(y - y') cos S

, • ,. -'" .," , • "

.'.' .," ) )

00. 0 0

Di rectir:m of vorticity Using the chain rul e (~y/~xl

.

(aynn) (an/. x)

. M s i mIlarly fo r (ay/ax)

(ayl3:r-l (aylaz)

• 0, •"

0,

"sin " " 0

0"'

"in

"

00.

"

"'"

•

( ~ ylasl

(a s /ax)

•

(ay/at) (at/ax)

(2.4)

found by use of (2 .1 ){2.3) that

S

,

Selected Works of G. de Josselin de Jong

233

GEOSC I ENCES

9J

Fig. J . OPientation of n,

8.

t

COOl'dinates by angl.es a and 8.

VOl'ticity vee-tor paraHel to t .

t

Application of this result to (1.15) gives for the components of the vo rti ci ty

+

(K/~)

Vy sin a sin II

( .j~)

Vy sin a cos II

o The coponents of vorticity On Os 0 t in the n . s . t direction s can be found by

..0.," .0. •• ,, =. 0"'" ••• • .0• .," • ,• "'" •

application of silrlilar transformation formulas as (2.3) giving

0

0

• 0" 0 • • • 0 0, • "

0

0

0

• 0

and this results Zfl - 0

0 .0'

d "

0

0

0

cos 8

•

sin

0

}12 "

0

wIth (Z.O) in

,

20 -

:lOs- 0 ;

(KIll)

(2 . 8)

Vy sin o.

ThiS r esu lt indicates . that the vorticity in a point created by a s pecific weight gradient in the pore fluid. has its a d s

t..lnqel'lt. to tl\e intu5edlon ]inc

of a horizontal plane and the plane of constant y. This vorticity is represented by the d oub l e poin ted arrOW in fig. 3. Its

mo.l.e of rotation iB

such as to r e v olve the y _consta nt plane towards a horizontal position . a physically pla USible action .

Sharp interface A sharp interface between two fluids . of specific veiqht

n.

1'2 respectively.

can be considered to be a zone of thickness h of gradually decreasin9 l' fro .. 1'\ to 1'2. in the li..it that h is reduced to zero . The zone contains sheets o f planes for y _con stant, that a re locally parallel and in the li.. i t are sqeezed toqether i n to o n e pla ne. Consid er a n infinitesimal area dA of the interface . This area corresponds to a volume hdA of the zone of thickness . h. In this volume the vorti c i ty is accord i ng to (2.8) and the tota l strength o f the vorte" actinq is 2fl

234

t

dV -

(KIll)

V'( sin

(1

hilA

(2 . 9)

Soil Mechanics and Transport in Porous Media

4 (1979) DELfT PROGRESS REPORT Since y is a",sUllled to deereaSe gra dually within the zone we ha v e

"

(2 . 10)

and 80 the vortex acting in an area dA of a sharp interface has the magnitude (2. II) The direct-ion of the vor tex corresponds to the positiv e t-coordinate, which i. locally tangent to the line of intersection between planes of constant y and the horizontal plane. In this case the planes of constant y form a staCk, that is sqeezed into t.he "'harp int.erface plan". So the vortex ha s its axis parallel to the horizontal contour lines of the interface plane. A sharp interfa c .. plan .. bet.ween t.wo fluids of different density actually is a vortex sh .. et. . The horizontal contour lines are the vortex line .. On that .. heet fi9. 4. The ribbon cut out by t.wo adjacent. horizontal c ontour lin .... form" a reentrant vortex tube . Let. the height. distance bet.ween the contour line s be Az. Then 6z is an constant for these two contour lines and the width of the ribbon

is

~/sin
dz/9in 0, and

therefore the total strength of all vortices per unit length of ribbon i9 with (2.11) (2.121

This is a constant and so (2.12) shows t.hat. the ribbon cut out from a sharp int .. r face by two horizontal contour line s with co n s tant he i ght d i ff e rence, is a reentrant vortex tube wit.h constant. st.rength .

-- ...... - - -- ' ---fIf- -- - - - 4.

Tlw. contOW'Unes

0 11

between

Fig.

cQntOlU'~ines ar'e

vorlex tubes loti t h cons tant strengt h.

a s harp i n terface are vort u

Selected Works of G. de Josselin de Jong

lines. TI,e ribbons

235

GEOSC IEN CES

95

3. Ci r culation of the pore fluid

The vortice .. in the lOeqions,

.me.. y

valOies. plOoouce a lOotation of the fluid

in those re<;lions . Besides that, each volOtez induce s a motion of the f luid arQUDd

its re<;lion of action up to infinity. Th.1s 1DOtion consist s of

Ii

c1r=lat1.on of

the fluid anmnd the axis of thfl vortex. The character of this IDOtion can be. shown by deducing the specific discharge created in a point x, y, z outside the region of vorticity freo the solution s (1.17) for the stream fw'lctions . The solutions (I. 17) consist of intaqrals over the entire lOeqion of the

aquifer, where y variations occur. Consider the contributions dl", dG, d R to the stream functions P, G, H produced by the infinitesimal volw.e dV ', only. These contributions an,takinq account of (1. 1 5) which specifies dF

(l/h)

dG

0/4,.)

d.

0,

.'

",ny ' ,-' ,-.

that~,

· 0,

dV' dV'

with

R. {Ix - .,)2 • (y _ y , )2 + (z _ z ' )2}1.

(3.21

The quantity R is the distance between the point

ti

x ', y ', z ' ,wIlere the.

VOitel<

is l ocated, and the point x, y. z ,where the specific discbarge is to be

computed . Introducing (3.1) into the lOelatJDns (1.61 , the contribut1.ons d
to the specific d.ischarge components


qz in the point x, y.

Z

d.~

&re

found to be dqx·

""'" d'\. -

II/h) {+ 211 ' y

(.

(I/h)

{- 211.,

(.

( I/h)

{- 211y • (.

,-, dV ' z') } ,-, dV' z· )}

x ' ) • 2(lx ' ( Y- y')} R-3 dV'.

)",3)

These relations are simplified by transforming them into n, s , t directions.

By

substituting the relation (2.8) for (It into the equations (2.6 ) i t is found

",," Furt.her the terms (z - z') etc. can be transformed by use of (2.2) .

236

Soil Mechanics and Transport in Porous Media

"

'"

~

(3 . 3) becomes

d"x"

"' -.... Y

(dV ' /hR'j (
",

", '"'" ", '"

00. 00 •

d"

, =. , • •

,• • ., . •

• •

.in

.in .in

O~

(1979) DELFT PROGRESS REPO RT

• • .in 00 •

S!

"

From these the components of •q in n, s , t direction s are found

by

us1llg

relation s derived fC()(II (2.3)

d",dq,,-

•

d"c-

• =. , -""v "" • .," , • dq, • • =. , + d q y • .," , • dq, .," • , . in • d'Y s

d"x .in d q , 00.

....

00.

00.

COs

which give with (3.5)

d",-

(dV ' !4:rR]j

( -2U '&)

dQs "

(dV ' /41IR 3 )

( +20 · nJ

t

t

,

dq- 0

Form of fZow pattem The result (3.7) represent s the specific discharge created around a vortex

of strength 2fl

•

t

dV ' with its allh in t-dh'ection . The specific discharge vector

dq lies in a plane perpendicular

;0

the axi s of the vortex , bec:t.use dq

t

is '!'ero

see fig. S. Purther, the vector dq is perpendicular to the line, that projects the point of consideration x, y. z on the t-axis. So the flow is a circulating

motion around the t-axis as executed by points on the rim o f a wheel . Actually all points on a sphere around the vortex have specific discharge s , that have their direction and magnitude coinciding with the motions of points on the sphere, when i t rotates as a rigid body around the axis of the vortex. Thi" applie s to all con centric spheres revo lving .. ith angular velocities that decrease proportional to R- 3 .

Fig . S. An iGo l
!lol't~;r

C1'eates a cil'aulating flO\,)

per>pendiculal' to ita aria.

Selected Works of G. de Josselin de Jong

237

GEOSC I ENCES Carp~tion

97

of the soh
The result (3.7) is derived from the relations IJ.1) , that are so l utions of the relevant Poisson equations (1.16), which for the case of an isolated vortex 20 dV' have the form ..,2 (dF)

-f~ 2Qx' dV'

within dV ' outBide dV' within dV' outside dV'

V2 (dH) .

0

(3.B)

everywhere

In order to constitute the solution to the problem also the equation (1.7) has to be satisfied, which in this case has the form H1F) /ax ... a (dG) lay'" a (daJfoz .. 0

(3.9)

It can be shown (see f.i. Lamb, pg. 209) that in order to satisfy alBa \3.9), it is necessary to combine isolated vortices into reentrant vortex tubes , whose surfaces consist of clos ed vortex lines. In fact this situation is encountered in the c a se of a sharp interface, that is subdivided into vortex ri bbons by horizontal contour lines . So the solutions (3.7) are complete on l y after inteqration over an entire interface. It is possible to show, that also in the case of gra dua lly changing y the entire region of y variations can be subdivided into reentrant vortex tubes, with constant vortex strength in their cross sections. So also in that case the solution is COIIIp lete only after integration over the entire region o f varying y.

4. Motion of a sharp interface SUbdivision in triangles with. (2..11) is

The solution (3.7)

suitable to determine the motion of a

sharp interface. Let the interface

position be given at a certain instant.

Then a surface element dA contain s

a vortex of known strength and dir ection.

The direction is parallel to the contour line in dA and it s strenqth is proportional to sin" , with II the local inclination angle of the Interface . The displacement of the i n terface during a time step is obta ined by determinin9 the specific disch.arge in points of the interface surface a s an inte<jration of the individual contribut.ion of all area's dA. For an arbit.rary Shape of the interface, this 1s executed by choosing a nWllber of nodal points a nd SChem..tlzing the gradually cu.-ved interface by a

238

Soil Mechanics and Transport in Porous Media

"

t

(1979) OELFT PROGRESS REPORT

diamolllbhapecl surface consbting of flat trianql"s between nodal po1nts.. Yrom the coordinates of the nadal point.II, t.ru. angle s CI and IS

"1''10

deUnoined t.bat

specify the local coordinate BYBte .. n, II. t. Sinc" the triangles ar e flat, B axe conlltants for each tria ngle. Let 9 be the vortex strength per unit

0. and

area, suc;:h that accoroin<) to (2.11) 9 •

'nIen

(oc/~) (Yz -

10180

q 1s

II.

(4. J)

Yl) sin o .

constant for each trianqle.

Each trianqle contributes" specific d ischarge

in a point P,whose coor dinates In the lecal n. B,

q .,ith components t


coordinates of the triangle

are c ' . p' tp' The crigin of the local coordinates is in the centre of p q", .. vity of the txi"ngle. By use of (3.7) and (2.11) the specific discluJ,rqe

components created by " triangle are ('l/h)

(q/4TT)

JJ , JJ ,

(-5 +11.') 11.- 3 d,,'de' (+0. )

11.- 3 ds 'de'

0 where integration 1s over s ' lind c ' throughout the a rea of the triangle and R _ {n 2 + {s_s , )2 + {t_t,)2)1 p

p

p

,

(4.3)

the distance between the point of inteqration (0, s ' , t') and P, the fixed point of interest, see fi9 6. The components

~,

qs'

'Ie

are to be converted subsequently i n x, y, z

directions by use of transformation rela tions with the coefficients of (2 .2). Summation of the contribution of aU triangles gives the specific di scha r'Je in every de sired point P of the aqui fer. EspeCially the specific discharge in the nodal points are of interest. Dividing these discharges by the porosity give . the advancing velocity of the interface.

Fig. 6. 1'r>iang 14 abc on

inter/ace with local n, 6, t coordinates . P if! projected on the s , t plane in point P' .

Selected Works of G. de Josselin de Jong

239

GEOSCIENCES

99

When the distance Ro bet.... en the centre of gravity of a triangle and the point P is larger than 2S tlloes the smallest triangle side, an errOr s",,",ller than 0,02\ is introduced by concentrating vorticity

ot

the triangle in its

cent.re of gravit.y . 'l1>en the int"",ral. of (4.2) slloplify. because s· . 0 and Ro. The result is

R ·

, ,

(-s )

1""

(+n )

J

with A - !ldS' dt' _

the area o f the triangle. However, if P is closer to the

triangle, lnteqration has to be performed with s'and t'v&riable. This MIOunts to & cumbersome analysis with the following re s ult.

Re8Ult of integration OlJer triangl€ Let the cornerpoint s a . b, c of the triangle be located Such, that viewed frOlll above the tri&ngle is at the left, when tr,,"versing the sides from a to b , b to c. c to a. see fig. 6. Let. P' be t.he project.ion of the point P of consideration on the s . t plane (the plane of the triangle). The distance frOlll P to P' is equal to np the coordinate of P normal to the s, t plane, np is positive for P above the plane, negative below. The projection of P' on the Hne ij 18 P

. 1j The specific dischar ge components qn qs created by the triangle abe then are

(4. S)

(4 .6)

Summat i on is over the sides of the triangle, such t.hat 1j are subsequent.ly ab, be, ca . The line pieces bet.ween

I I

are always positive. The line iPi:l

is postive, if the direction of going frO P

to i is in the arrow directions ij as indicated in fig. 6. In fig. 6 the line aP 1s therefore negative. ab The line P ' P is positve,if P' 1s Located to t.he left whe n going from 1j towards j. In fig. 6, P ' P is positive, but P ' P is negat.ive. The factor f ab be equals 1, if all P ' P are posit.ive. Then P' is located within tile triangle ij abe. The factor f is l-ero . it one of the lines P'P is negative. Then P ' is ij located outside the tr iang le. The factol' f is introduced in order to limit the arc t an () to values between -~" and + ~" .

240

Soil Mechanics and Transport in Porous Media

100

4 (1979) DELFT PROGRESS REPORT

SingwZar behaviour in a nodaZ point The expr es sion (4. 5) becane's i nfinite , when the point P approaches One of the sides of the trianql~, because either """,,,rator 0 " denomiMotor in the loqarit~ becomes ~ero,when p and P conincid e. This in ter fe r es unacceptably ij with the determination of the speci fic disc~rge in a noda l point. since all

a d jacen t t"tangles then contribute infinite normal components . In order to establish the influenc e of these tria",:!L"s .. specia l procedure has to be

f ollowed, based on

II.

detailed &nalys i s of the s pecific discM rge s encountered

in the vicinity of the interface. At a noda l point severa l tria ngl e s meet, in " diamondshllped interface

schematisation . Therefore the substi tut i on by tri&n91es is not appropriate a t " nodal point . because instead of the top of " pyr.u>i
Specific dioch=ge in centre o f dioc Consider a cir cular fl a t disc o f radius r at an IlIIgle

(J

wi th the horizontal p l ane.

The vortex strength per unit area is g from (4, 1) . The question is to determine the specific discharge at a point P on the llOnl>al in the centre of the disc, when P a pproaches the disc from above Or below, The coordinates of P are taken np. A, "p.O, t p.O, wi th A a small poSitive or negative quantity for P apprOo'1chlng the disc centre froo> above or below,respectively , !le
t ---"\\

r

Fig . 7, CiI'cuw pW't of inte,.f<1.ce idth Vor tic,>s ,

Specific diBCharoge in P on n01'm<27..cl!I'OU{!"- CB"tZ'3 .

Selected Works of G. de Josselin de Jong

241

GEOSC I ENCES

'"

Integration is OVer a ring shaped area with radius p a~width dp . Th~n in equations (4.2) (4.]) we have s' - pcos9; t ' -psin9; ds 'dt' '' pdpde, R" (A 2 +p2 ) ' and the specific discharge in P has compone nt.

(q/4 .. )

qs -

(g/4 .. )

I, I" I, I" o

0

o

0

(p cos 9) (A 2 + p2)-3/2 pdpd9,

AtA2 + p2)-3/2 pdpd9

Integration with respect to 9 gives

o ig l

r

(l2 + p2) - l/2 pdp •

o

So that finally only the component qs remaines.

giving

,, .

(4 .9)

FOr ],.-+±O this reduces to (4.10) with in the denominator the absol ute va lue of A, because the root of l2 is alwa ys po . itive . For l .. +0 the point P is a bove the disc in the lighter fluid with y .. "Y\. For l

~

-0 the point is in the fluid with '12' Let the discharges in the two

fluids be indicated a s ql and q2 respectively. Then reintroducing (4 .1 ) the r esult (4,10) Can be written

",, • "

.,

l (~/\I) (Y2

nisin
~ (~/u) {Y2

'II) sin
, ,

just

""'.,

Further according

q~ '.

".

1 q~

...

•

.. ~.... ...

..

interfac e

just below interface ~

'" '"

fluid fluid

,

(4 . 2) . 00 (4.8)

q~_O;

1"'" (4.12)

The se relations are identical to thOse found in the two dimensional case as presented previously (de J088elin de Jong .

1977). They show,

that the vortices, in a flat, disc shaped ar ea of

> ..

q,J ....

a sharp interface between two fluids, only create a shear tlow parallel to the interface. Above the interface , the flow is in the direction of steepest ascent, and below it in opposite

Fig . 8.

242

direction , as delllOnstrated in the a dJacent fig. 8 .

Soil Mechanics and Transport in Porous Media

4 (1979) DELFT PROGRESS REPORT

102

Conclusion An int.erface between two pore fluids of different specific weiqbt 1.lI

accorcl.inq to the three dimensional extension of the vortex theory

II.

vortex:

sheet. The vortices have horizontal axes a nd the contourlines of equal beLgllt on the interface are vortex lines . The ribbons between contourlbte ll" are

r eentra.nt vortex tube"_ Every vortex on the interface creates .. clrculating flow around it :tnd the speCific dischar91!. in an arbitrary p)lnt of the .. quifer, 1s obtained by integration of the influences of ,,11 vortic es oVer the

entire interface using equations (l.ll ,

(3.4) and (2.S).

The three dimensional vortex theory wa s applied to produce

II.

computer

pr ogram for determining the movement of a n interface wi th time . This work was initiated by Haitjema as an extension of the computer program he developped for the two di .. ensiofUlol c"se a nd reported in this journ"l

{Haitjema. 19771.

'nle three di ..ensioMI elaooration w"s continued by Luger , who elaborated the use o f equations (4.5),

(4 .6) "fter they became "v"llablc.

In the computer program the re"l , &IDOOthly curved sh"pe of an interface is repl"ce4 by " di"lDOndshaped pl"ne consisting of fht trhngles. E"ch trhnqlc i s " vortex sheet with vortices homogeneously distributed over it. The COrnCr points of the tri"ngles "re the no(b1 points, whose displacements with cOur8C of time are determined by comput1nq the specific discharge cre"ted hy ..II vortices in all trhnqLes and dividing th is discharge by the porosity. Only the triangles adjacent to the nodal points are e x .... pted because their influence becomes infinite according to (4. 5) . These adjacent triangles are replaced partly by a little circular disc , r..mgen t to the interf"ce a t the nodal point , whose contribution to the specific disch.l!.rge in its centre is sbo"", in section 4 to consist of a shearflow par"lleL to the disc, on ly. This shear flow is disregarded in the computation, because it does not contribute to the displacement of the interface .

Hai tjcma (19/"1) , "Nul!Ierical applic"tion of vortices to multiple fluid flow in porous media". Delft Proqr.Rep. 2, PI> 237-24B . G. de Jos8eUI1 de Jong (1969), HGenera ti nq funetio rls i n the theory of flo',· through porou s media" Chapter 9 from "Plow throuqh porous medi " " (R.l.M. de Wiest, editor), Academ ic Press, tie " York. pp 377-4 00 G. de JosseUn de Jong (1977),HReview of vortex theory fo r multipl", tl"ld flow H Delft Proqr.Rep. 2, pp. 225-236. w. e . Knudsen (1962), "Equations of fluid flow t hrough porous ",eeli" Incompressible fluids of varyinq densityH, J .Geophys . Re s . 6 7, pp 7J~-737. H. Lamb ( 19J2),"Hydnxly"amics ". 6th ed.DOver Publ., New'{o l'k. A. ~'e1'J'l.lijt (197ni , ·'Theory of gcoundwa t er How", Macmillan , London.

I . H.M.

2. 3. 4. S.

6.

Selected Works of G. de Josselin de Jong

243

Pr~ifl9$o'Euromech ,43 / Delfr / 2-4 ~tember 1981

The simultaneous flo w offresh and salt water in

aquife~

ofl arge

horiw ntal extension detennined by shear flow and ' vortex theory C.DE JOSSE ll N DE JON G Reti,ed from Unl.eTlify of Dtlff. Ne lherlands

Edel""",',. IIOU haa eacapelete Edel""'" interf""e . Ilis treatis" st a rts with a she"," nO\l conditions ",,01 the Di~t~_Dupuit clari fying d"",cription of the "",ch81>i~m of .. pproxi .... tions. A solution is verified with ground"o.ter now ",,01 its relation to pres_ " ,..,ault fro .. exact vortex t heory. sures , expressed in terms of e'l.uilibrium o f forces. From pase 642 on , hO'JCver , the INTRODUCTION analysis is contused by the presupposi t ion, that groundw&ter flow i. alwa,ys s~je ct to \>0. Shear flow tenti .. IR and the treatllllnt continues wi th ~b e n tvo fluids (1 , 2) o~ di fferent spe d fic (J. fo r'" of Do.rcy'a 1 .... , t." .. t is uncapable to d~8cribe variable density now , correctly . weight rl' Y2 resp ectively, occupying .. n Applying ...~ appa.rently rigourous r easoning &d that ther"Core a d~ftcription with the specifi" discho. r ll!' corcpct"Jents para1le l potenti .. ~ is i<:possible . L.. czyns ki (1961) to the inter~e , cu;t occur in orl1er to r.uara.ntee that along the inte rrace the pres _ attempted to formulste heads, that could s e rve as potentiah . Here, heado .. r e not re_ Sures in the two rIuids at either side is commendable r<)r practic&l use , because the y e'l. .... l . A snail error , unfo r tunately , crept into the end o f the derivsti oo , le o.ding to are pIIeudo ·potentio.ls "hos e gradi ent. are rel& te d to the noor only in particular di _ the inoo,...ect e xpres6ioo .... ctions. This is .holm hy Bear (1915) in l q" 1- (Y2!Yl )'1..,2 j-(.:/l') (Y2 - Yl)aino; ...here,': i. in~rir... ic per:nesbi li ty his e q .... tion (9 . 5.19) • u i. d/fn"";c viscoai ty . ' Using the formulation of Darcy ' s 1&" in l.o .. h.le .. I'~Hc .. tion , Eone nts 'I. I ' 'I.n2 in shown , that rotation e xist . propOrtior.a.l to the two fluids at eithe r side ~t be "'1. ....1 th e hor;'.ootal component of the density gra_ in order to satisry continuity . In forl:lula dient and hO\l the sped ric diseh..,.,.., can be corcputed ;n th e {::tire "'lui fer by locat ing ( 1.3) vortices in the r egion" of rot .. tion . By this theor)' it is pOsdl.l e to tre,,' any

75

244

distriblltion ot densiti e~ , abo ot tit" gra.d""l tr",,~i tioo U>I'Ies , th!l.t exi~t beween miscible rlll;d. beca ... e o f difrusioo ..... d di s persion , "n>e theory i. veriried in the pap"rs by redlleing a !l:radual tr ...... ition ~one to • sharp intert..:e ard ""to-blishi"g the ..... gnitude ot th e shearfl"" , 'lhe &olu_ t ion~ then become .; nglll"r .:,d pro<\\ICe discontinllities oC 1M.gnitudea "s given by the "'I.uationa (1 , 2) llIld (1 , 3) , he re ,

.li~tly .

Hori wntally extended aqui n.rs Dieh (1953) tre"t.. the col!lbined flo ... of fluids .. ith diffe...,nt properties in elon _ gated , con f ined "'I.uifers , "n>e "" ..1ysi. &tarts .. ith the con""pt of shearn"" crea _ t ed by th" e'l.""li t y "r prenW'e On either . ide of a aharp interface, "n>en the COr_ r ect values of the s h.,.,.flow cooponents , parallel to the a "lIi fer boWldarin , "re at_ t r ibute d "ith a Dupuit ... aumption to Ule oped fic diach arSl' co"",onente parallel to the aqui fer in every pl ..... e normal to th e aQui f e r . Finally the partial di fCerentio.l e'l.uation (21.,,) i. obteined for tile !DOtion o f the interf"c .. in coW'se of ti"" , Be ar (1975) pg 535 i""to""" the Dieu _ Dupuit "".. .lysi a by reJD:>ving a .. inac cu"""y mentioned by Dietz (1953) PI> 88 . "n>i. lead. to Bear'. e'l..(9.5.6~) which i. similar to Dieu ' a (2 1,a) ettept for a com_ mutation of a nUlli>er of ter ms with the Second ./ax oC .'n/3x· , eo..,,,riaor. of the tim pape,.. is so"",vh,,t i",eded, bec",,,,e in a.ear ' s (9 .5.6b) the nu""rator beween bra""" is mlspr.inted . The reader c"n rea<1ily p"rform the correction by ,"""orking the anal .. " i • . Such" correction CM be verified .. ith Bear ' s (9 . 1 . 20) , .mioh ia Obt aiMd with .. "illlil&r an&lysis, Th" vork of Dietz and "bo e'l. . (9 . 5 .6b) of Bear describes"" aquifer , that ia tilted with respect to the hOrhon , and the coor_ dinate system i. parallel ..... <1. normal to the aquifer .Other , horizontal eoordin &teft are used in Bear ' s treat ... nt leading to (9 . 7 . 20) and that e qUlltion is therefore better sui_ ted for co_rison vi th relations developed in this paper . In the an&l ..sh of Dieu (195]) ""d BMr '. (1975) "",..ion of it , the discont i nw,," ch"racter of the n"" "t the inte .... tIIce ia only "",coWlte'" f or in s directioo parallel to the aquife r . It is an objec _ tive of thi. paper to ~hOlf , how the an&1 .. .ia i s changed by considering relations of the kind (1 .2) (1 . 3) , s""h th&t the charac _ ter of the discontinui ties in the flow , both par&11el ",,<1 "" rnU to the inter !'see , is tBken into "",,count, It appears, Ulat f . i. introduction of ""uif"r ani$Otropy change l the inter!'ace ..,tion eqUlltion onlv

,,,11

All exupl.ry , For" coo fined aquifer this theory re'l.uires the introduction of an inf'inite .e ri es of i ... !1' vortic;oes , which i. "o""'t;".... beli,, _ ""d. to ;""ede pr..:tical applic"tion, whe n the &quic" r is elongated ""d the interface inclination an,uc is s:ne.11 , beea""e oC the integr&tiona required in the analysis . I t ... ill be shown here that for the c .... e of parallel , i""ervioua boundaries, the sol,, t im i. si""lified , bec ..uae the integrals ha"" ~losed sollliiona. '!he USe of vortex theory i. required , .men the interface inclin"tion angl e i . not s""'11, .... C., . in the vi a cinity of wells , ""d the Diet~-Dupuit approxi"",tion ia in _ sufficient . In that c&se a coo:p"ter i. in dis""n."ble for es t "bli.hing tile integ .... _ tions n"""ric&lly, A description of the applic"ti-on of vortex theory to cO"'Puter anaJ.y~is i~ given by Hoitjelll'l. (1~71), Other program; "no in co,,",e of develop"",nt. 'Itle /ll'neral objective of thia papcr i s to r "cond Ie the Diet~_D1..'Pui t approxi "",tion with ~he """"le~e Edel""", " hear no .. cond.i tions ..... d vortex theory .

2 FLO" CONDITIONS AT

A.~

INTERFACE

"n>e c!I..e is considered here of " .harp inter_ face bet"""n fresb and aalt .. .. toer, fluid 1,2 "".pectively, '!he specific .....,igl1ts are then ""1<)'2 and the viscosities are ~1<~2 ' The dirference in both prop e rties is of the o r der of "everal proc:ent8, 'Itle intrinsic pcr!l1Cabilities of the anisotropic aquifer are Kh ' "'v in hori""ntal and .... rtical dir"c t iQl.S respectivel y , and these are considered to be the prindpo.l dire ctions. '!he fluids .... e miscible and r epl""e "",nt in the pore .p see is co"",lete sllCh that Kh and "'v are the lame Cor both fluids, Shea ... ~ flo"" "

,

qn l \

,

-.-

r l. ;'" I ,

\

~

.-

. - .- . -.~~'

1 '

",~,._-" d ('

q ~2

,"

:: fC":':'C':c::c:::c:c:ccc:cc:::::c:::-:--'; Fig,l, bterface line .. ith coord".aeea n , 8 and specific discharges


;

'I.s 1 >'I. s 2·

76

Selected Works of G. de Josselin de Jong

245

The :>.-eoo.. din .. t.e i8 " e rti " .. l, th e x,y coo,,_ din .. te a ....e h ori WIlt. .. I . 'nl e hei gilt 0 r th e interr"" .. i . c ...lIed l; . So UI .. e
t;quilibriwn In order to ~ .. tis ry "q"iZib...:w. the fluid pre90urea P I , P2 in th e nuic1s Ilt either side of the interface a re e
tma*(a ~/ h)

. i,,(1*( aclh ) II ,. ( a~l3x)' J Ii" <'080*11 [,...( 3clh) > l~ ':

(2.2 ) (2.3)

At the in te rrac e l<x:al , or t.hogonal coordin .. tea n,_ respectively nor "",1 .. nd aJ.ong the interf .. ce line ar .. considered (rig . l) . The coordin .. te. n,' .... e tllken positi~ , vIlen pointing in pos iti'" ,. ,x direc ti on. res pectivel y. The cOlIPooents onents
(2.4) (2 . 5)

and the inversiw
[ (ap l'e x ) - ( "P2/ 3>:) Jc08 a+

(2 •• )

In order to a ..tisry cOl1t';~uitlf , Ule nor .... l specit'ic disch a rge co""on ents q,,1 .
In order to expre .. thi s r~lati on (2 . 12) in ter~ o f sp ecif'ic discha.rl§' CO"Ponents , the r el!l. tiooB bet\l~en floy Brld pressure devclop<> d Crolll e quilibriulD of forces are w.ed . Th ~se are for the ith fluid :

(2 . 8)

... ith ~1' ~2 respectivel:y

~~

(~I -II:2)'l" *-(Ul<1Xl-ll:!
(2 .13)

l OCh )qxi * - ( aP i I ax) i (IIi loc )<1,..; *-O Pi lad -Yi

(U

F,limi" ..ting P h ( 2 . 14) gives

(2 . 14)

P2 1'''0111 (2 . 12) , (2 . 13) and

(1I 1<1 •. , - u ,
ne"

(2 .1 5)

COq>ooenUl

'!he reh.tions ( 2 .9) , ( 2 .15) can be solve d to

"W

(u l q x l -U2<1,;2) «(cos '«lOCh) . (ain ·o.loc)) " *( y 2- Y1)B in ncC6a _( U - 1l:?) 'l" (.inal,,) (2 . 16) 1 (U <1 I ~ U2<1:a) 1 z

[(cos'a/"t, ) . (8i n'al,,) )(2 .17)

q" '"q" l *-qx 1· ina -oq. l c osa • :.!~tiplic 8ti(:O:l

(2 . 12)

[ (ap 113,,) - ( eP2/" z) Jsino*O

Diacmtinuiti es in the

''l,,2- - q ,.zcinoo ''qz2''''''Q.

gi"'.

( U 1<1 x 1-l'2 q JC2 ) ( casCli' ''h ). (2 . 6)

Con ti nui t.y

246

(2 .11 ) in X," di,..,ctions

( 2 .1 )

(2.9)

TIlese rel!l.ti oo. gi ..... an i""reni on o f the discoot.i n u'ties in th e specific di sch .... ge coq>ooenu at the interfac e . The norral .peci fie disch!Ll"ge eO"floncnt
Soil Mechanics and Transport in Porous Media

\./hen the interf..,e "",v," up>f ... rd~ with a "'lo~ity ~~/at, a surfac e of a""", A ot the interfa"", plane ..,""" durin!'; '" ti"", inter_ val dt , UlrouW! a vol~ o.f the aquifer o.f ""'l!7'itude c08a(3~!3t)dt . A . (aee fi!,; . 2) . 'llle ",,,,,,unt of fluid in thi s &qui~rvo l urne is c times the volume, wh"n C is t.he ..,ro._ sit;( Of the aquifer . So. the a"",unt o.f fluid displac e d is ec""a(3~/h)dt . A . The speci fic discharge 'In i$ det'i.ned "" the w1"",, of nuid p""sing IIor"",l t.o t.lLc iat." .. Cace . throu;.i' .. u , it. ",.cs d urin r. u"it ti ... . So. the a""'\lOt cf fluid p&.'!sing th~ !.he ar"a A dur in!'; a ti ... dt i s '!nAdt. CCqlaring the w o vcluroo" gives ~", £ccSCl(3~/3t)

Fi!,; . 2 _ I n tA! rfaee "ovin!,; upwardS

3 n.TERrACE )OTICU EQUATION

Continuity in x -d irection

Let th e a quifer have iu upperboundary at z-Z) and its lQller boundary at z-Z2' '!h e heigllt occupied by fluid 1 is (2 ,-~) and for fluid 2 it is (I; - Z:!). s e e fig. 3 . 'llle hori ""ntal .speci ric dischar~ components 4 x 1, 4,&""<; "" . an "pp ro~iD'... t ion ass ume d to b e constant In .. vertIcal plane . Per unit .. idth the disch..,.ges Q"" Qx2 of t'luicls 1, 2 resp . ...... then

(2 . 18)

Usin!'; this value for Q. in (2 . 16),(2 . n) ",,01 appJ.«in!,; (2 . 2!,( 2 . )~ result.s in

Q x1 -t"

('I . 1)

, (7. I ·C)

Qx2-q:a(~ -Z2 ) '

(2 . 19)

(3 . 2 )

'llle total discha rg>o of both nuido tog>ot.he r over unit width is called, Q". 'lben I/e h"ve Q"oQ"I+Q:a"' -Q.x I( Z I - ~) .... :a( t - 2 2 )· Let there be in fil tration of n uid Ulroul,$l the upperboundary a.ddi ng a volu",", $,. per unit ho.rizontal area a nd unit tim to the di~charge Qx ) and simirlarly S2 to Q,,2 fro., bel"" . Continuit.y reqmrC'5 the n. that th e to.tal dis~har~ Qx " a ti" fie" the relatio.n

At this point i t rNIII be "" .... rked , thllt the relaticns (2 , '9),(2.20) a .... valid for any value Of a in the interval _11/2'0"'/2 . Beyond tha t interval the interface i s un stable . becaus e heavier fluid is abo"" liW>ter . ¥/he" t he viscosi t ie. are equal , su:;h that ~)"'l-'2"'~ . eQ.uatiOf1S(2 . 16) , (2 . n) redu:;", to

. (Y2- Y, )Slnoc-csa (q", -
J

() . 4) When the in t erfac e "'veS upwards vi th a velocit.y (al;/3t), the wlu"", of nuid ) ""ded to the disch..,.!!" Q~1 over a uni t h oriwntal area ",,01 unit u .... is c(al;!3t). wi th E the aQ.ui fer perOlli t,y . 'nle sa"", VO_ lu::e i$ s\lbt r acted fro., Q;4' Continuity or e ach fluid sep .... o.t.ely .... Q.u'r""

(2.21)

(Y -Y ,)sin'0 2 Uain!,; the tensor character of pel'Dl'lIbility, it is possibl1o to s hc l/ that th'" 'te r", [(cOS'Cl/Kn)~(.in'a/"v)) .... pre.entll the .... dprov.u of the intrinsic per"" ..bility . 1< , in s _direc'tioo . Uaing f'urther (2.5) it is~ fOUl-d , that. (qs (-qs2)-(K a /ll) (YrY1 )sina. ,

(
() . 5)

( 3Qx2/ax)-S2-€:(3~!3t)

(3 . 6)

Equation in

(2 . 23)

~

alone

When Q ~, 51 ' 62 a .... knovn f ro .. boundary condi tions , "relation in the unknow n varia_ ble I; "lone i. obtained by eliminating
a .... l .. tion f or the shea r now remindin!,; of

( 1.2).

78

Selected Works of G. de Josselin de Jong

247

,z~

'.

lUI'

./ In this last expres.ion A is not rN."""d. si""e fro., (3.7) it .ppe.... th.t the difference [1-(1'2/111)1 Q(I be ove rrul ed. by the .ni so tropy pr~rt!on ("'h"'v) , ""ieh eln be quit .. hrge in hcdzontdly d.e~i _ ted aquifer • . Hovever , linee A eontsi .... (3Clit) , it i. insppropri.U! to i""orp;l r .te it in. t i,.,.t d.et .. r.,in.tion o r (3'/ ~ t) . The proce dure i ~ to eonside r A u . eor _ reetion ter .. , set e qUIIl to ri .... t and .d.justeel i t er.t ively . fl; .. rvo.rd • • 'nle r...,tor in (3 . 13) ..1. 0 depend.1 on MiIOtT'OPY • .., .. ordlnll to (3.8) . He", the proportion ( Kh/K"l ~ b$ lu/llt enoush to b.l.nee the inte rr..... inclination fow:tor ( i r;/ h )' ",,<1 to r<"
Ll-L1-______________

,

~

By ",in, tile abbrniationa

['ah2 -y 1 )O:~h,ll ~·c [1. ( ~2/\1,) J (~/o<) (3l;/at)

() . 6)

"'(~/ ot;v )(3l;tax)'

().6)

"''1'0"

(l.ll

"'&tlon [2.19) bt'eollll!!l

( 1l1<1",,~q:.:2)"lI I(r-")~f( 1 ~)

(3 .9 )

.,<1 .obinS with ().3) far q ~l' '1,,::0 givetl

tn the ".ntral t hree
qxl l \11(t-~)~(Zl -( ) I -

-Il:!Qx"U'(

r_A)«(_~)*J( '++)

'1:.;:2[\1, (l;-'Z:!)"'2[Z ,-t)

%- .

().IO)

\-

"1I}Q,,-I/ , (r-M (z l-t1*/(

h~ )

* 5 1"6 2 ' vhere V st ands tor (a/a ~l . (31ay). Th e . he.r flo" eqUllt i ons are

(3 . 11)

I t i, nov PQllib l e to obtain .. r elat ion in I; alone , eiUle r by eliMino.ting 'I x} ILnd '1,,1 by ().lO) in (3 .1) and using (3.5), or by el!IOinatinil '112 and Qx2 by ().11) in (3.2) and .. ins (3.6). The two expreuiona are _ '';':Lar but not ident i eal. Avel"&gin, the. gi. vn

is the hO)'"i2.01l1.&1 <1i.-.etion on the {nte rraee. The ..... an.lysis . . .bove p ....... <1\ICU the tbll"",in, inte r r ..... .,tion .. quation tor \J 1"\'2"')1 , vile", t

datlatl · HS,-6 2 )· '\

r;:

I7(Z, - t) - Il,(t- Z2 }

c( i~l3t) <; (S 1-5 )2

1'\\J:;!{Zj -l; )+U,(l; -Z2) + (1, ,-t)(t - Z2 )(at/ax )

V[Q('Z.1 + z2-2~)12 (Zl -ZA )

+u ,
5 (Zl -~)(~ -Z2)9r;

().12)

..

1


'Jbia i. th i"~.l'rac. I>IOti"" ,.quatiOl1 . eo ... parable with Bear ' . (9 .7. 20) . \/hen the lo_ ca ti on o f U\e interr., .. i . Itn""", aL • p.rti cular ..... n t" il Itn (Jlm • • • tuneti"", or :r. Ntd the "P'.rds .ation 3r;13t o f th .. inter_ fac .. e ... W dt'terllined with (;3.12) for e"",,.,. point o f it . In the Cue of t"",h, aalt groundw.ter tbe ditference bet.... en II I ..~d ~ ;. . .. ll and introd "dn, 111"')12'"11 gives

(3 .

,~ )

In the axid .y ..... trie eue , "ith .. the ho r-'ial coor1iinate, th e inl ern..,e IDltion equation , v ith S,-5 i, 2 ri~lal

-o ,

((a~nt).(Q/211rlf; 1 (Z' ''2.:! -2~1/2(Zl -Zz )j . • ' { (Zl -I;)(~-~)(a~/a r ) ) .. ~ ar r (Z _z H l e..., .nd thoBe ~veloped in liUrat"..., is in the t e r., eon taining ""isotropy: ("III" .. ) .

" 248

Soil Mechanics and Transport in Porous Media

Ch&r&<:t.~ r

or

t.h~ int~r t'&e~

..:>t ion

~Q.uation

'!tI~ ~q u&tions

(3 . 12), (3 . 13) , (3.14) , (3.15) "", t ion or th~ int~rr.c~ in dirr~r~nt cu~" , hsve th~ ch.r&cter of ""n _ line .. r di r tus i on "
~ £XA~LS

th~

or

A roLlJ'l'ION

Solving by "h

r

I

~ep"r ation

I

or vsriables gi"CB

r

u-lnr -rr> · 4r:T- ~t.-to) ' "0 0

(~

.8)

qXIH - r< rx+lH)

rl [ 1+("h/ ,,) rl l

'1;aH . r( rx- JII)

O I1+ ("h/'') rl l

(b . 10)

'!tine vallles .. re "",su..,d to be constants o""r eve ry vertical. In order to rind the co-..ponents '1.1 , '1z2 at the interfoooe , (2 . 18) i. cootoinedwith (2 . 8) to give c(a~/at) .. _tq ·+hl· an d USe or (4 . ~) gi"u vith (4 . 91 , N . IO • for ~.~:

(4 . 2)

'!tI .. "",tion is considered or "" interralin~ with ;ncHnu i on angle o.-sre t ~ n f , . " "h thu i t ;a r epresented by the r ehtion C(t)"/x+!1I (4 . 3) It.s intersection poi nts Sl , S2 ,,; th the boundar ies ha"" coordiM t es "S 1-- (H/2t) ; >>
ce , th&t &t time t is a 8trai@llt

"

(b . 1)

vhe r e integration constant8 ar e &dded , such thst st ti.., ta , r eqUIIIs fo' 'I11 .. result (b . B) s hows, that the inclin&t;on angle .. deerm.ses in COUTSe of ti..,. Using (3.10) , (3 . 11 ) gi",," , for "1 ~"2.~ and C1 x-Aa() , the values or 'lxi, qx2 at th~ in t e rface

In order to oho" .. uolution or the inter_ t'&<:e motion equ&tion (3 . 12) , this e to di.r~g .. rd " throughout . Then (3 . 12) is

£H( 3,/O t) - - 2 Tf'x/( I"';(r'l

(4 . 3) r or the inte rr""e , the time dependen _ ce is onl y in r, b .. ing (t) a runeti on Of t alone . So dir!er~nti at; n" (b . 3) " ith res _ pect to t, gi"". (a~/at).(dr/dtJx (b . 6) Ccrnbin;ng l
q~ IH-r(-f x. }H) r'/t 1+("h/") f' ].

(I, . 11)

1+("h/") r' J

(4 . 12)

q~H _ r( -!x-JHI t' II

In orde r to satisfY continuity in the !'resh and s a lt " ate r repons , t he compon~nts 'lz p 'l z2 a re dist ri buted linearly ow,. the hel@llta : ,
(4.5)

'1. 1

"

::~q~I-r¥r'/ [1+( "h/")r' l

(4 . 13)

r

'!his res ult . how' , that t he i nc r e""e Of the 'lr2'ftC2 .. r~r' 1{1+('), / " ) 1 (4 . I ~) interf'scc he i ght, i. liMar in x and the r e _ The relatioos (4 . 9) , (4 . 10) and (b . I ) )(4 .1 4) tore the int~r !'ace apparentiy turns as .. lie,... "".. d to eon8t r ue t tig. 4 . ri gid line. Thi. means , that in equation

80

Selected Works of G. de Josselin de Jong

249

~~~~~~~:~~~~~~~;~:~::~:;'~~~1l~·'·!·~~:-~~:M \

~

\

f

..

dioeharll!

~ ~

FiI!distribution .5. Speci fie

~

•

~

•

deter!!l.ine4 b)' e ,...., t vertn thecl"}' . ... ina mult (&.2)

•

•

E'XAC1' 5OWl'IO~ BY 'o'01m:X THEORY

Vonice. in confined aquifer Accordin, to vorux thllOl"}' the sped fie di .cha ... "" dilt ri but.ion , .... soci.ted vitI! II section .u at. .. point. S ot' an incUr-eel in _ t.erfa:-e , can be dete"';'n"d b)' locating. varUx or tt.rengtn r4~ in 1:; , \/ lUl t dirrer:nce spatmed by ds. The vor tex h"" • horiUlntal "xi~ p ..... Uel int"rr~ and create ... coution, thOlt turn. the inter!'..,,, 1.00< ...4. Ute hori _

t o the

..,ntal. Each YOrt"" contribute. to • nrel'" func _ tion ' . deriV1.tiwl " .... the s pecific dileharlP' e"mponente

who."

'b.. · (n/h) ; qz· ..(a'/h)

(5 . 1)

'!be eontributim o f the V<'r t.ex in d. to" if in t v o r pi i. the dist"""" , be_ We .. n P .. nel S , defined ~Y rp. " !("'~Xs)· . (.,,~ ... )·h, ..nth ", . .. the coordin.t ... Of the point S , wh e ... t he Yor_ te~ i . located • ••,,1 Olp . zt, the coor-din.1.etI o f . point p. where the ili.cllarae i. consi . dered. 'nIe contribution (5 . 2) to 'f io "" _ llindina; O f the contribo.ztion to .. potential j by. '"".di -.:en tional ",,11 of .,..~itude rd ... . In • conrined Ilquife r. vith i..,.r....ble bo".,dariu . t ,,-0 and ~ , an infinite rail of i~ 'IOrti"",. h . . to be introd"""d in the """,ner of i m 8" ~lllo. Usin, .. "",ult of """'kat (1931) . nct 9 . 8 =nccming an i n _ tinit e rov of velh , "..,h "IO rte~ con tribu_ te. to If' by an • .,.",t dlf' fti ""n by

A:I interr..,.., e).l,,,,,nt between SI .... d 32 T""'Iui .... s integr ..tion Of (5.3) o""r ... fro", ....1 .... ~... 2. l";ne:r... l .. ~ · ..... 1.. t;"n t n ...... i . ry (5 . 3) , give. ror'" the value (,1. . 1) in table A. SinM "" i. a IUlction of ' .. inte _ /lration of (A . 1) i . di r ticult, In order to ,i""Ii!)" inte«ration , the di.ch..,.I!'"" in P ..... cod>ined as l"ollovs


- i
tII'nl!

(5.5)

A. l) thi. gi"". (A . :» . 5wtracting

1 1"1"0," the fi nit te .... betll;:en braces and sdding 1 to the ..,,,,,n4 term , (A .)) i. prodUCfld .

Let the interr."", interVal be tween $1 WId h."" the inclination angle o.a r etan! . then the inflniteai1<&l di.t&nee d"" alonl! the ;Merf""" equals d ~/ r and using (5 .~ ) it i. pouible to lIrite d ... in (A.3) as 4z,.(H/ ... ) (d~ ddZa) ( r/ H i r ) (5.6) U.. "& tlli. in (A . 3) pe"'; ... ir>tel!.... tion prod"",inl'. ~he "",,,It (A .Io) . The 61!ht ftide of (A . ~) i •• ep.,.,.ted in • ""al ""d an part by .pplyi"l! Eule ... · s equation tv;"",. TIli . 1I;""'S (B . 1) in Table w;"g abbreviations 6., • • ~a. lips derined , there. The reaul (B . ~) cCO"ltains the vill...ea for 'l~ ..d
52

.""'&i" .....

B.

Rl'"

" 250

Soil Mechanics and Transport in Porous Media

ITABLE III R _Ii: 1+'1 'jl . ps

wIth

I;

pEl

pEl

P"

'

i1*

p"

-I e ' +I\.'J~' e ."r etan(" ps

va

•

pEl

ps

If.

) . a* ...... "tan(ll. It

pB '

+wI '/f' ){_ (i .. rllnRps +( i - f) InRp.. +( 1_; fle ps - ( 1+; rJep,.}~~

q

a(r/lI}(r/, +r'li _r ln(R "iR

q

_iq -iq

p!I

pa

)

.. (e XPcosz _ex,.cosz ) ; !l a("Xps inz _" XSsinZ ); 1\* ~(eXp!linZ ",XS&inZ ) p .. ps p s pEl p s


XV

I'"

zp

p6~

pi!

,I _ire 2+;1"11 p.

pa

(B . 1)

,.«(I_in,,>1

.(r/H)(r/l + r) i- ~H +fxp "ilZ +«(l-irlH)} p

xp~p

Miaotropy, such t hat Kh ;':v ' the vorte x xsl ' zsl 8Il d ,,<;2 . '"s2 the ....sult (B . 1) i s theory is 5i",;lar , ooly the ~ - coordinate collPa r"b l" to Haitje= ' s (1977) eqUlltion (9) deri""d t or .. s tr aight interface .. 1.. has t o be ..u1tiplied by a factor (Kh/ ';v)l ".,nt AIl in an infinite a'lui l "r . Th ~ pl"OCe throughout . The ooly consequence i. , that dw-e for CO!l'binin g such el1! "" ots to analy _ the factor (l +r ) in (B . 3) i s rep laced by se a "","\led inter!'ace . desc ribed. by ( 1+( >;h / >; ,,,)f·j . Haitje .... , ean be appli e d similar l y to (B . 1), In (B,l) angles involved , defined REFERENC£S by arcs '1f tangents . These angl"s h,ne to be cho~en in th e i nterval 0<6< .. , when 9~s aesr,J. 1915 , I¥na.ucB of fiuids in por ous is conC<'!rne d and Bps . "h~n P ia in fluiil 2 . ... ~ia , Bod< 2nd ed .,IUn. Elaevie r Publ . Wh~" P i 5 in fluid 1, ",,01 Y i. beween DieU , D. II . 1953, A theoretic" l approo.ch to Y,,1 ""dYs2 , th~ value o f (~82-ep8d h 5s to be increased vi th 211 . For .. justil"ic .. the proble .. of e ncroo.chi ng and byp""inl! e dge" ater . Proe . Kon . l!ed . Aeade";e Ylln tion see !); Jouel i n de JOOI! (1960) , p31~ lI"eteMchappen , series B, 56 , no. 1, 83-92 . 3153 . Van ~ , C . J. & L .A . Peleti c r, 1911 , A cl",," of similari ty solutions of the non _ Co"l'ari~on vith eXIUIflle of section II liMar d.iffusion equation . Noolinear Ana_ lysis , The ory , Mct hocls ""d Applications , In th e ex"""le the interface is a st r ai!'1lt line with linPt points 5 15 2 on the bounda1, 223--233 . VM D~ , C . J . 1979, Nonli near di:rusion ri es , s uch that z , ><0, :~2.H and xsl , x.2 given by (1, . 4) boundary values f or problems , Dr . Thes is, Lelden . Edelman , J.H. 1940 , Stroo",;nF\ """ wet en /;po Md Ill'" aN t-ben : f wut grood "'ater , Rappor t 19~O in ....k e de ~.1 .... '1>cosZp-e-inl ", .. tet"\IQorzieni ng vM Amterdam . Bijla", 2 <"s2aexPcoo"lp+ehl f (5 . 7) 8_ 14 . EdelllWl ,J.H. 19"1 , Over de berekening van llpa'O< -l'Ip s Xpsin"lp s grondwatentrolningen , Dr . Thesis, ~ltt . Becausc of t he identity of the ,, ' s , (B . 1) lIaitjema,H .M., 1977 . Ku"",rical application reducea to (B . 2) , vhere the term be tveen of vortices :" rouitiple f~uid nov in po _ h&s to be added , i r fluid 1 i. concerr o,," "",<'Iia . Delrt Progr . Rep . 2 ,231-248 . ned . The values of (B .2) were us e d to con Hwbert,M .II . 1940 , The theory o f groundw"a_ struot the spedf'ic disch"" ", <'Iistributioo ter fiow , vol XLViII , 8 , 185-94 4 . of rig . 5 . 0.. J05selin dc Jong , G. 1960 , Singul""ity Whe n the point of coo s iderat ion P is lo di n ribution8 for the """l,ysi. o f mul _ cated in th e r e gioo "a1 <"1>< ",,2 such thst. t iple fluid flow through porous ",edia , J . Cephys . Res ., '101. 65 , 11, 3139 - 3158 . (5.8) De JO$$elin de Jong , G. 1911 , R~ vie w of 'IOrtex theory inr .rout. tiple fluid flO'ol , the v:u-i ables can be approxi"",ted by o..lft Progr . Rep. 2 , 225--236 . R :,.""h/ f . R t "eXp· "1) ' de J05selin de Jong , G. 1919, Vorte x theory p" ' pa ' ps " P S P for .."Hiple fiuid flow in three di",,",, _ ""d using (5 . 4) , (B . 2) reduoes to (B . 3) . s i <:ns , Delft Progr . Rep . ~ , 81 - 102 . Sep"" . tion of real snd i .... ginary p""ts Muskat, M. 1931 , The flow of homgeneous "1>:)",, , that the resu lt (B . 3) gives the s""'" values as (k .91, (4 . 10) Md (4 . 13) , (4 . 14) , nuicls through porous me <1l a , Book , ~ Gr ..", Hill . "hen Kh~v and the rooanin g of the b rllCk etB is taken into account. In the case or

e .....

'I"h:

n;"l

2·np 2....

<>

e ...

e , .,

<>

" Selected Works of G. de Josselin de Jong

251

&Iroil tk /" PubliC(Uwn nO 4 1 de {' AuocWJiQII I nlemaliollale d'Il.lfdrologi f (de I'U.G.G.P.) . Sylnpotria Darty ( D ijtm, 1056)

l..' ENTRAINEMENT D E PART ICUl..fS PAR l..E CO U RANT INTERSTIc rEL pn Ir. G. de J OSSE LI N de J ONG

(Laboratoire de la M6caniq ue

d~

Sols. Delft

S ummary The pore system of a packed bed is schematizcd to a syslem of canals in order 10 permil pro.babilily-<::omI?Ulations. for a Sl~an,e particle carried by llie pore "(iller movement 10 arrive at a certaIn place In a 'ertam lime. Thecomp ulalions lead to explicit values for the coefficient of longitudinal and transversal di!fusivity. A lest device is des<.:Tibcd which permits determ inat ion of longitudin al diffusivity. Relation hetw~n test result and t heory is di$l;u!lnd.

I. Introduction Dansla loi de Darcy tous les phenomenes ph ysiqu es qui d~termincnl Ie mouvcment du liquide intersticiel dans un tassement de grains, sonl rassemblts d ·une fa~n ingenicusement simp le et pratique permetlant Ie caleul ra pide de nombreuscs applications Icchn iques. II exi ste n ~anmoins des probl~mes re1 at ifs aux mouveme nts de ees liquides qui demandent a ce que I'on consid~re d'une facon rigoureuse les phenomenes micro·structurels qui COnlililUentla lo i de Darcy. Un de ces prob~mes miCrO.Slructurcls sc pose quand on vcUI etudier Ie mou,·ement d·une particu le et ran~re entrain6e par Ie courant interstieiel. Par exemple. I'in t roduction d'cau sal~ dans un massifsatur~ d· cau douce peut ameller it Se poser la question de la r~r­ tilfon de salinite Qui s·en suit aprts Que ceUe quan t it~ sal~ ail ~e entrainl!<: pendant un cer· lain temps It travers Ie lassemenl de grains. D ans un pays comme la Hollan de qui est bord~ par la mer ct doil prol¢ger reau douce neccssaire II la vie de sa pop ulation, soo cheptel et ~s vtgetau~. conlre "infiltration involontaiTC d·ea u salt<:, ce probl~me sc rencontre main tcs fols. D· aulre .part, tes probt~mes Qui sc prtscntent dans les optralions ehimiques Quand un liquide Qui sc Irouve dans une colonne it remplissage eSI remplac~ par un autre produit. ne sc rtsolvenl Qu'en considtran l les mouvements micro-struclurels ex~UI~ par les parlieu]cs individuelles. U.s c~perieoees on t dtmonlre qu·une quantitt de particules elranstres introduite se disperse en loute direction relativemenl au mouvement moyen du liquide interslieie! pa r leur mouvemeot ent re les obstacles des grains. Par analogi e avec la diffusion mol«:u!airc q ui engendre une d ispersio n semblable I'on a introdui l les conceptions de d iffusion 10ng'lu_ dinale et de diffusion t ransversale pour indiquer ce ph~o m~ne. ( Klinkenherg (I), Baron (~). Nous voulonl demontrer d·a ns eet article commenl on peut determiner Ics coefficien ts de diffusion relatifs II oes deux di ff~rents modes de dispersion en oonsid~ranl Ie mecanisme du mouvement du liquide It t ravers les pores. Lc ealeul camplel etant l .op lo ng pour ~tre publ ie lei, no us nous limitcroos II la presen· ta tion des points de d¢part . du mode de caleul ct des r~ul tats obtcnus en r¢serva nt la demonstrat ion rigo ureu!IC ~ une autre publ ication. La description d' un essai avec leq uel nOlls avons determine Ie coefficient de di ffusion longitud inale, el des rtsulta ls obtenus qui d¢monlrcnl I'applicabilil~ de la IMor ie campl~era ces consideratio ns. 2. Le mou\"t'ment d'une parlieult' t tf1lngere Nous vou lons considerer une particule etrangtre Qui, en trai n&: par Ie cou rant inlcrst icid, '
252

La quest ion qui se pose est la atiermination du d~placement de celie Ilarticule apres qu'eUe ait ~t6 entrain~ un cerlain temps. Comme dans un systcme de grains tass6s arbitrairement les interstices ont une forme arbitraire, Ie lNII"Wurs que la plrticule suivra n'est pas ~ fixer. On ne peUI pn!dire que la probabilit~ pour la partlcule d'arriver ~ un cenain endroit. Le calcul de cette probabil;l~ est Ie but de notre recherche, car Ie coefficient de diffusion peut 5C d~duire de II d~viltion standard de la n!partition de la probabilit~. Dans ce calcul la variable stochastique sera d~termin~ pa r Ie passage d'un interstice fI un autre, car la longueur du parcours total se compose de la somme des longueurs individuellcs de ccs passalles INIrcourus et Ie total du temps passt est 18 somme des temps de s~jour passts dans les d ivers in terstices. De cette fa~on des calculs on t ~t~ etfcctu6s par Danckwerts e), Klinkenberg (I) qui, en consid~rant I'elfel de la probabilit~, ne se basent pas sur un mttanisme pMnom~nologique et ont reeours fila d(:termination d'un coefficient num~rique qui ne s'obtient que par I'exp~­ rience (essais de !(ramers ~), Klinkellberg (I). Les mttanismes introduilll par Baron (dilf. transv. (I) et Kramcrs (diff. Longt. (S) donnent d~jfl une impression de ce coeffic ielll mai s sont d'ulle autre natu re trop si mple, illtroduisant une distribution discrete pour Ie clioi~ d'un d~placement + f ou - f. Nous voulons trailer ici Ie cas ou la variable stochastique aura unc distribution conti nue, d~pendant d'un mttanisme idtalis(:. 11 y a troi s effets principaux qui d~terminent la disttibut ion du variable-. I. Dans 18 section d'un pore, la vilesse du liquide n 'est pas conSlan!e mais plus grande vers Ie cent re. 2. Les interstices sont de diff~rentes grandeurs. 3. La dire<:tion du courant dans un interstice peut avoir un angle quelconque a vec la direction moyenne du courant principal.

ad. J) Perpendiculairement if. la direction du courant dans un pore, les vile.sses du iquide ne sont pas les memes et il semble nttessaire de consid~rcr fllaquelle de ces vi tesses la panicule prendrait part. A cause du mouvemen t Brownien et l grande vitesse fI cause de la turbulence, il existe dans Ie pore unediffusion radial e qui rend la vitesse de la particule ~gale it. la vitesse moyenn e du liquide dans certaines conditions. i.e ca1cul numtrique de cel dIet est possible avec les multats de van Deemter tt). Nous avons suppoS\! dans les calculs suivanl$ que cette diffusion radiale soil complete et que Ie mouve mcnt de la particule soi l ~gale l la vilCSse moyenne da ns Ull pore. ad. 2) En obse rvant la Forme d'un interstice qui se produit entre quatre grains de forme fI peu prt s spMroldale (fig. I) , on voit que Ie COUTant entre par un canal triangulai re assez

ttroit, arri ve dans un ample t ~traM re, se partage en troi s et quine Ie pore par des canaux ~galeme nt tria ngulaires. en clUlaux triangulaires se retrouvent partou i dan~ Ie tassement de grains et Formellt la rlkistance principale dans tOUles directions. Si les grains sonl taus des sphtres et de meme grandeut, ces canau~ Iriangulaires ne seront pas tgaux l cause du tassemenl irrt gulier. Nous avons, en premitre approximat ion, suppos~ que 1a oiffusion provcnant de l'in~gali tt de ces r6sistances serait petite en comparaison du troisitme effct CI nous avons schtmatist notre systemc par deS r:anaux conini~ indiquts' plll"' IB fig 2. ad. 3) Nous vou lons suppoliCr que les deux effets prttMents sont ntgligeablcs en comparaison avec les diff~rences de vitesses qui s'c n suivent de la direction des cana" x tl t mentaires. Plus I" vitcsse moyenne dans un canal sera grande, plus sa direction correspondra avec la direction principale Z. Pour 18 distribution de ces directions, on supposera que Ie tassement des grains a ~tt tel qu'il n'y a pas de direct ions de prtftrence.

140 Selected Works of G. de Josselin de Jong

253

~....

...

,

! z

, Fig. I Pore minimal ent re quatre spheres.

Fig. 2 I'o rc schematiS4! par des canau~.

Fig. 3 Route arbitmire pr ise par une particule iI t ra vers les canaux.

3. Deduction de la probabilite ~Ion ces J considerations prk~dentes. Ie systeme de canau~ dan~ leqacl la particulc est emrain6e sera schtmatise de la faton indiquCe par la figure J. II consiste en des canaux de meme longueur - I - de meme resistance au courant. orientes uniformement dans toutes directions. La particule suit une route tortuouse :i travers cei> divers canaux 11 uno vitessc egale 11 la vitesse moycmne du liquide dans los canaux differents . Pla~ons la coordonntc Z dans la direction principalc du COurant intcrsticiel total, et moyen sur ~aucoup de canaux. Nomons l"all81e entre cette dir,::clion ella direction du canal elementaire (j ) : OJ 01 'h rangle de sa projec tion aVec la coordonnee X (fig. J).

Supposons que Ie temps de sejour dans co canal soit tj ~ to/cos OJ ou to est Ie temps de sejour minimal possible el qui est obtcilu dans un canal o ri en te dallS la direction Z. Quand la partieule a passe par N de ces canaux , elle eSI arrive<: :i un endroi t X Y Z el dans un temps T donne<; par X _ k.Yj = L/sin OJ cos ofj

Y ~ L)'j ~ L Isin OJ sin (I)

'h

Z ~ LZj - LleosOj

T ~

L Ij

~

L Io/COS OJ

ou les sommes sont effcctuCes pou r j _ I it N.

·141 254

Soil Mechanics and Transport in Porous Media

O~ns ces expressions les angles OJ et IjIj sont maimenanl les variables slochastiljues Nous voulons suppOscr que la p robabi l i t~ que la part icule prenne un e direction 6j. IjIj au moment, qu'arrivte II une bifurcation. eHe dOlt choisir un nouveau canal II. suivre. cs t don n~ pa r la fracti on du liq uide total qui cou le dans ceu e direct ion : 6j. Ij.-j. Un coleul simple indique que ceUe probabi lit~ est do nn~ par

fC6j.

(2)

ou I'on

v~ritie

(/Ii) dO

d<.\l

=

(l l r:) sinO cose de d<.\l

que

I" I~ de

t" dt\lf(O.

<.\I) - I

Pour ]a d~tcrmination de la probabilit~. W" CX Y Z nd X dY d'ZdT que la partkule arriverait apres Ie passage de N Cana U)( arbitra ires dans I' endroit X Y Z eI dans un temps T . nous appliquons maintenant Ie thwremc de Markoff (voir p. 1')(. Chand rasc kha r (1». En considerant T comm<: l'un des composanlS du veeteu r stochastiquc tquivalent au~ ~utres composants (X Y Z). on oblicnt alon;

0) _

x

e~p { - j (~X

"

_

00

_

"

_

'"

+ ') Y + ~Z + ":Tj)

x

A"

avec

.

"

Toules 11'5 inl~gralions son t faisablei par la m~thode du col et nous arrivons enfin II une expression pour la proba bil il~ W " ( R Z T) ou R est egale a ()(2 + y2)1~. dtplaccm<:n t dans une direction perpendieulai re a la direction principalc Z. Toutefois ceue probabilit~ W" n'cst que la probabili te que cel clfe t soi t oblenu en N passages. Mais il esl bien pOss ible que la panicule arr ivera a R. Z, dans un temps T avec differcn ts nombn.:s de passagC5 dependant de la roule qu·eH., a prise. I I nous faut done. pour obten; r la probabi litt lota le. que la partieul., arr ive dans l'endroi t voul u el dans Ie lemps pos~, inlegrer W" pOur tou tes lcs valeu rs possibles de N. c.-a-d. W( R ,Z, T )dRd'ZdT ~ d R d'ZdT

/l CSt possible d'cffectuer encore eeue demitre nous obtenons alon Ires approximativemen t (4)

f"

W,.dN.

inl~gration

Rd R. d'Z.dT [ W ( R, Z, T ) d R dZdT - 8fh::(ac bt)JI,! (A

C

+

par la methode du col el

J'.

A')3

x exp {l- 2 (A

+ A')t '2 C' ,2 + 8 }

dans laquclle nous 3"ons inlrodui t Ics notations suiva ntes:

,, _ 0._5/4)/2. b _ -

(5)

c _

1/6 t. f _ 1/ 2 (Ij.

+ 1/3 6 f2

_ I f 2 (;"2j

Zj

- Ij.

-

ill

Zj)

(1 _ 21. _ lj

e _ 2/3f _ Zj

I - 1/8 (l

_ 1!2 (;ij -

'/)

142

Selected Works of G. de Josselin de Jong

255

+ eTiJ/4 (ae _ b 2) B _ [at' Z _ b (dZ + "T) + edT}/4 (ue _ C ... {ae 2 _ 2b dt' + ed 2114 (ue _ bJ2 A = [uZ 2 -

2bZT

b 2)

A' _ R2/4f. Un trait indique la valeur moyenne d'une vari~ble qui se d,!Lermine ~is(:me"t avec (2). Par sym~trie on atlend r~it pour II une expression conten~ntl2j, mais parce que ccl(e valeur est infinie elle cst evit~ dans Ie resullat du calcul, ou I'on reneontre A qui est delin;

P"

,(e'·' O, + I) ~ 4 Zf/ ou y cst la consta nte de Euler. L'expression pour W ( R , Z, T) n'eSI ras. rnai s ressemble bcaueoup a une repartition normale de Gausz. Le mal
(S')

O"ll ..,

(6)

crT _ 0.-; ".

VZ j;2,2 _ rJ'"1"1f31" = 1

2z1 -.- ;:'";-·; z2

ii-"-;3iil]iia ... IQ \ !3ZC):':': i /4 )/1

y"[j3z----V:=Zi4"iti

De ce resultat se d&1uisent les coefficients de diffusion longitudinale (0,) et transversale (OM) qui d&rivenl la diffusion par rapport au mouvement moyen de vites; V Z IT ~ 1f~IQ' car scion Einstein 0 _ 02/ 2T . Done Q

(1)

Dz - 0",.212T .... 3 "i·q 1 (A -

DJ! - 0""2/ 2T ... 3/16

...

1/4)

'·0 I.

5i les canaul< elemen taires ne sont pas Cll'LUX, comme nous I"avons suppose au debUl de celte derivation, les valcurs de ;:-2. z2-: /2 ct ;.t" differe ront de ce que nous avons donn~ "Ins haut. t. a r~partition de r j, Zj ct Ij connue, il n'csI pas difficile d·en lenir compte dans Ie resulla t (6) ou I"on rem art/ue que les deviations standard augmenteront. L'£ssai

Un essai de contrOle fut effcctue avec un d ispos itif de percolation (tlg. 41. Un qlindre (06 em, hauteur ZOcm) fut rcmpli de houles de ,"erre. se nsible ment spMroldalcs et d·un diam~tre de 0,2 mm (Mvia tion standard 0,05 mm). Le remplissage contrO]~ soigneusement procura une pOTosite uniforme de 38.5 %. D'abord Ie syst~me fut sature d·un e eau di still&: dllns !aqueUe o n ajoutait une quanti te de NaC] co rrespondant it. une conc~ntration de 0,02 normal. Le liquide sous Ie tlltre fUI remplace par de I'eau ayant une eon~ntration de ],0 normal NaCI et on fll ensuite percoler e syst~me de bas en haut rar ~ ]iquid.::. it une vilcsse de 6 cma/ minute. Tcn~nt comptc de a porosit~, la vitesse moyenn e dans les grains fut I

•

(6/ 471:.6' ) x (]/0,385) - 0,55 em/ minute.

A differentcs hau teurs 2.5-7,5-]2,5-17.5 cm du fond, ]a p~roi etait pere~ pa r de~ ele<::. trodes qui permettaient de mesurer ]a concentration de salinitc du liquide intersticieL Ccs ~Iectrodes eonsistaient en des tlls de rlatine couverts de noir en platine (diamelre 0.7 Illm. ) mm long). La rtsistance ~Iectrique que c<:s pointes epro"verent e n appliquant un courant alt ernatif (1000 Hz, 50 micro amp.) fUI mesuree ra r Ie pont de Wheat stone indique dans 13

256

Soil Mechanics and Transport in Porous Media

OEl41L

HECIRoO(

, oSCIUAfE CIl 1000

~l .

! Fig. 4 Disposilif de I"cssai.

figure 4. Un elalonnagc dans Ics liquidcs de divc=~ eonccntralions de NaCI avail donne la relalion entre eeue res;s!ance ele<:trique et la salinilt. L'tlalonnage fu! cx6cu!t aussi bien dans Ie liquide seul qU'Hprts remplissage avcc des buules de verre, donnan! des valeuts qui correspondaient ~,'ec Ie calcul !htorique. Au moyen des lectures de ces res istances on pouvait done $u;vre les changements de la saliniu! dans I"cn!ouragc de ~""Cs tlcctrodes au fur ct It mesure Que la fronticre des coneen_ Ira!ion~ rassaL! de 0,02 normal It 1,0 norma l.

'144

Selected Works of G. de Josselin de Jong

257

"---;;= = =;;;---,--, I ..,"0' e, ••• 1 / MJSQuI . . . .. . . . . .

+ -{

.,.

., 1.-,r..------,.c--------__~

,, ,

. r-----------------~~,

•

• . .-.-.---.--- ....

~

------ ~• L-""' ...-:::: --~ -. . - .

,

_ C ...... , •• " ...

Sa1injt~

.

'0 "' ...

Fi". !i cn rt&is (rfe par le$ electrodes.

145

258

Soil Mechanics and Transport in Porous Media

Cctte frorl\i~re ttait brusque au d~but de la percolation. elle se dispersa en montant dans les grains. Dans la figure 5 sont pr6sen!~ les r~ultalS des observa~ions avec les t le<:t rodes. 1"lt,,,riIUlin"$ des f t SU/lulS d'usn;

Dans la figure 5 nous avons dessint la sali nitt enregistrtc par les ele<:trodes au moment ou une parlicule. voyageant A 13 vitesse moyenne. eSI a"iv~ A la hauteur indiqute par I·abscisse. Pour demonirer I'influence de l"epaisseur de rtlectrode, nous avons dessine au..dessus ce qU'une elect rode enregl5trerai t 5i une frontitre brusque passait. Celie ligne obtenue th ~riq uement montre que la frontitre brUSque n'cst dUormte que tres faiblcment et que les electrodes sont asse~ minces pour negliger celle influence. l es lignes obtenues par les ele<:trodes demontrent sensiblement une distribution normale-. On observe que leur pente augmente a mcsure que leur emplacement est plus cleve. Celie pente est une mcsure pou r la dtviation standard. comme nous ravons calculk. et doit donc suivre une loi comme indiq ute par o. de la formule 6. Dans la figure 6 est pr6sentte la relation entre y'Z \a racine de la hauteur du electrodes et la deviatlon standard o. observ6e. Pour obtenir la dtviation standard o. qui ne depend que de la diffusion longitudinale, i1 faut encore soustraire I'influence de la diffusion moh!culaire. Celle influence a ete determinre par I'ell~ricnce. Dans un essai, Ie front passa d'abord par Jes eJe<:trodes inferieures, puis la percolation fut arrette pendant 24 hcures et fut rt tablie ensuite pour faire passer Ie frOnl des ele<:lrodes reltantes. les lignes observc!es, pointi llc!es dans la figure S, monlTent un basculement additif qui correspond a un coefficient de diffusion moleculairc de 0", _ 0,7 X IQ-s cm2fs«.

les essais normaUll ne prenant que 30 minu tes, ['influence de la diffusion mole<:ulairc fUI asscz petite comme l'indlque 13 fill. 6 ou nous "avons rapPoriCe.

it" ,

e e

U'

~

.,."

t,

~.

, ,

, ,, D~vlatlons

"

lscm

Fill. 6 standa rd s observtes compar6es avo<:: la Ih60rie.

146

Selected Works of G. de Josselin de Jong

259

Dans celie figure sont donnks egalemern les lignes thtoriques obtenues ave<: la formule 6 pour I
difli<;ult~s

REFERENCES (') KU NKEN BERG . A. , SJENITZER. F. Holding time dist ri butions of the Gaussian type. Chem. Eng. S~i. Vol. 5. 1956 (Faraitra). (I) BARON, T. Generalized trafhica metllods for the des ign of fixed bed <;alalyti<; reactors. Chem . Eng. Progress. o. 48. 19S2. pp. 119. (') DANCK WERTS, ... V. Continuous now systems. DiSlribution of residence times. eke"" Eng. Sri.. 1953, VoL 2. pp. I to 13. (.) KRAMERS, H., ALB ERDA, G. Frequency response analysis of <;ontinuous flow systems. Chem. Eng. Sri .• 1953, vol. 2, pp. 173 to 181. (0) KR AME RS, H. Communication priv~. (0) VAN DEEMT ER. BROEDER, LA UWE RIER. Fluid displacement in capillaries. Appl. Sci. Res. Section A , vol. 5, Nr 5, 19S5. (') CHA NDRASE KHAR . StOChastic problems in Physics and Astronomy. Rev. Mod. Pkys. voL IS, Nr 1. 1943.

147

260

Soil Mechanics and Transport in Porous Media

h.""". ",.

Vol, JO, No,1

Longitudinal and Transverse Diffusion in Granular Deposits G.

Ot: J OSSEUN DE

Jose

Aln/,... /- The pore system 01 ... packed bed i. rtpr .... nted. by a oyotem 01 caul. in order to permit prot>.biUty computations lor a lordgn partkle carried. by the 1"'''' liquid move· ment to arrive at • ccrt.un place in a cortain time. The computations lead to explicit valu .. lor the c... fficient 01 longitudinal and transverv.l diffusion . ,\ test device is d=ribed. which Jl<:rmhs the determination 0/ tbe longiludinal diffu";vit}". The relatiorush ip ~tw""n ,..t r.. ult and Ihoor}" i. discussed.. Inlroduclio ..- The no",' of liquids through porous media is defined by Darcy's law when bulk movement is considered. I n several case'l,lwwcvcr, it is of interest to know how elements of "olume Or discrete particles carried by the liquid will tmvel. For in.tance, in the .tudy of ground-water mO,·e· ment, radio active salts are injected into the soil. The salt will travel through different porcs and after a given interval of time will arrive at dif· ferent places, thei r distance from the sta rting point being dependent upon how tortU(}U$ was the path they fallowed. T his re5ults in a dispersion of the injected !>alt which is addi tional to the molecular diffusion. This dispersion caused by Ihe geometry of the pore ranal system also has the character of a diffusion with a greater value in the mean direction of flow titan perpendicular to thi$ dirc<;tion. Th~rcfore the lwo different concepts longitudinal diffusion IKliI.k",tbt1-, and Sjfflilu. , 1956] and transverse diffusion [Bar,,", 1952] have been introduced in order to indicate this phenomenon. The passage of the fluid bIotwe<:n the grlLins con$ists of the su mmation of a great number of random phenomena, every single phenomenon being represented by the Iransi tion from One pore 10 another. The randomness is originated by the geometry of the pore (;anals arQUnd the grain, whicll is delcnninl-d by the adjacent grains whose positions have" random character. This suggests the applicalion of probability calculus in order 10 obtain an impr..,sion of the diffusion coefficients. Man}' workers have undertaken such probability computations. o,'IJdu-vls [195J], Sc/rrilk" .. [1954], Klidrnbc.g and Sjmilu. [19561 obtain rliffusion coefficients by application of probability, wi thout defining the micrOSl ructural nlechanism of the particle movement. Therefore their n>sull conlains a numerical constant, describing the granulomctric propcr t iCll of the porous medium which can only be dctermined by experiment (Iests of K.amers and A/~oo [1953] anrl Klink",,_ bt'g and Sjt"il"" [1956]). Also Day (1956) de_

scribes such an approach, using the results of SchriJkUet' 11954]. T he latter introduces an equal magnitude of tran.verse and longitudinal diffusion. Tests, however, pointed out that there is a marked difference (6 to 8 times). In Ihe detcrmination of the diffusion coefficients prcscoted here Ihe slarting point is a schemal ized pore canal system and the movement of the liquid through lhe canals is accounted for in detail. Previously, others have applied mechanisms which in troduced the grain dimensions (for lrans" erse dillusion, &von [1952]; for longitudinal diffusion, Kramers in a private communication, and RiJai and others [1957]). These mechanisms show a probability distribution of discontinuous character, the Iraveling parlicle being able to I, 0, Or - I. choose on ly between movem ents of We preferred an approach where the probability distribution is continuous with respect to tbe direction of the pa lh and proporlional to the dis· charg~ of all the canals orienled in the direction considered . This approach enable,; derivation of an cxpr"""ion, which conlains both lransverse and longitudinal diHus.ity combined in one formula, and shows the observed difference of the diffusion coefficients in transverse and longitudinal direction. Thi~ resul t was published by the author fdt hmdin d~ hlt/g, 195-6] in Dijon [or brevity's sake without rletailed mathematical explanat ion. Beca us.:. of a regrettable computational mistake and some printing erron the formulas shown there are incorrect; so this article serves as a rectification and a jusliflution of the mathematical treatment. NDta/ion- The following nota tion ill used. l length of elementary canal x, y, Z,' coordinate$ of the exit of an elementary canal, when the enlrance i$ placed in the origin residence time for parlicle in elemcolary

261

+

~o.J

u residence lime f(lr demcotary canal in prin_ cipal directioll of flow

68

G. DE JOSSELIN DE l ONG

\, ! i Z i

,

\,

•

y

,

X

Fro. I - DiaB;ram of pore space,
angles d~ribing direction of elementary =01 q~ ,q., Q liquid discharge

(J, '"

mean velocity of liquid How

~.

l~

"" g~

distributiQJI

po"" P probability function

lunttiQ!I~

for choke of

X, Y, Z, R coordina tes of arrival point of a T

p&rticie, starting in the origin time of travel of the particle

N

X"

number of canals covered by the particle y., Z . , Reo coordinates of the maximum

number of particles traveling during the time T. T , arrival time of the ma:rimum number 01 particles covering II. distance Z., R. N. number of eanals oovcred by the maximum number of particles ,,~,


tT r standard deviations of proba·

bility function with , .. peel to Z, R, T D~ longitudinal diffusion coefficient D~ tran.versal diffusion coefficient ~,,!, (, T integration variables

262

0· , 0, b, " d, e, /

functions of the lint and second moments of the stochastic variables given by (9) and ( 12) A, A', B, C functioM given by (15) ). function of distance Z. gi\"en by (13) and (23), represented graphically in Fig. 4 In 'Y - Euler's constan t .. 0.571 Sckm.lli.at;on 0/ «mal system-The pores between grains of spherical form consist of nearly polyhedral cavities connected by Iriangut.r shaped canals (Fig. l a). Tho resistance to the waterflow is principally de termined by the narrow passages, these being wionted at random. We wm schematizc this pore system by a net· work of canals linked together as indicated in Figure l b. The orientation of the canals is at random, but unifonnly distributed in all directions. We will further suppose that the pressure gradient in each canal is proportional to COl! 8, 8 being the angle between its direction and the direction of principal flow, Z (Fig . Ie). If for a lint approximation the lengths and the conductivity of all elementary canals are taken to be equal, the

Soil Mechanics and Transport in Porous Media

69

Dl Ft'US10X' IN GRANULAR DEl'OStTS me&n velocity of the liquid in each callal is propOrtional to cos (J. The velocity of IIow in the canal varies over its aection, 100 that if a louign particle mO'"e!I a t a vdocityequal to that of the lIuid in its immediate vicinity, ill velocity wiD be depended on its radial distance Irom the cenl<:r. Brownian motion, however, caU!e$ a radial diffusion in the canal 1_ pa'" Dumk, and otbers, 1955J, so that if the canal section is small enougb "'c may assume tbat the "c!ocityof tbe foreign partide i. equal to the mean veloci ty in tbe canal. With tbese assumption~ tbe rcsideno: time 'i in tbe canal j is gI "cn by

'0

" - .. foot. I,

where tI i5 tbe shor test residence time po$Sible, occurring in a canal oriented in the Z direction. II the length 01 an elementary canal is l, the d istance cove~ in the r , y, ~ directions by a canal

"

r ,_ l lln"cos""

"_llin"Iin",,

'"

"1_ l oot.'/

Cbiu of J>i:illo- A I.,...,ian particle carried by the t;Urren t through the canals has to ehoose a new dire<:lion of motion every time it Ilrri\"tS Il t a junction. We will assu me that tbe choice of direc· tion beu.cen (J, 0/- and (J + dI,o/- + do/- isdistribute
a canal in di'e(tion I, of lor a lime t.J «II). 'rite ,'eloc:ily in Ihis ca.naJ is (lI") em (J, so that the devia tions lu and &- are ~ .. 1lI·(l/II) -."1" j(41 ' (/ II)(1 ~

_ Ill· (t f ll)

"n , cos, .. i/41-t/tI) "n 201

Thl'S(: deviat ions lie on a sphere wilh radius At 'lI2" and center a l l - t.J,t/2u. The d istribution function t~ - (2-a')- '. s.in 2(J d(28) dofalso shows that the arrival IlOin ts on the sphere are uniformly distribut ed ovcr its surface. This sca.tl<:ring therefore bean a resemblance to Brownian motion super-imposed on Il transla tion, and the problem has become one of random lIight. However, there is a difference in that Ihe residence time in the free patlos lor Br ownian motion is independen t of the orientation , ~-or the problcm considered here, the residence time is inversely proportional to em (J, so th.tt its value ill infinite in !he n treme cue (J - "./2, For the determination of the diffmion coeffirien t ..-e can, with lOme al terations, make use of the probr.bility calculus developed lor Bro ..",ian motion ICiallli'raukw, 19-13, pp. 8- 16]. IniroduclUm o ( trtXJobi/ily-Wht n the particle has pused through a large number, N , 01 canals, il has arrived at a point .'t, Y, Z "'here

L ~' - Ll ..n'/OOIPI} 1:,/- 1:l ..n./ .... ",'

X .. y o.

"J

L tCOl',

Z L I/ The summa tion being framj _ I to N , The time of travel i5

1: I, - :L: .. foot. 'I

T -

do/< canals

oriented in a direction be t ween (J, t and (J + d8, t + dt. The discharge 01 tbese canal s is q~ .. A q,(2·. r ' cO! 8 sin 8 d8 dof where q. is the discharge of an elementary canal oriented in the Z d irection, The total d~ha'8C ollhe A canals is

+ OOIl1)

(5)

With the aid of Markoffs theorem IClumdrasd,Iuu-, 1943, I', 9J we can determine the probahility P,. (X, Y, Z, T) dX ·d Y ·dZ ·dT, that the particle after N pas$.tI!CI has arrived in the volume be lween (X, Y, Z) and (X + dX, Y + dY, Z + dZ) duri", the lime inteNal T to T + dT, as

p,,(.r , I', Z,T/ dXd YtlZdT

So the frarlion of diso:hargt: in the 8 - (J of - of + dt direct ion is

+

d(J,

,»

The path foUo,,'ed by !he particle is determined by the ..ubscqucn t choke;, c:ach being distributed :l(O)I'ding to thc distribution function ,~. In order to show the scattering effect let uS ("QnJidtr how far a particle is d",~atcd whC1l it fof lO'lo'S

Selected Works of G. de Josselin de Jong

dXdYUdTL-

-

"~r J'

..

x Up 1where

A ... -

lit

L-

J(! X + . Y

[L'" L" dI

dp

..

d~

L..

<4

L..

+ t Z + .nl x A .•

dr (6)

'Of

e.ql 1+ j(.t~i +

.'1 + roi + " 1>1 ]'

263

70

G. DE ]OSSELIN DE JONG

By introduction of T as th~ f(lllr(h component 01 the v~lor whose probability is det~rmined, the dependenCl: of the residence times on the stochastic variaWcs is accounted lor. The result P", however, is not yet sufficient beeau"" the particle may arrive atX, Y, Z in a time T in differen t ways with diJIcreot nurnbl.", of pasISIljj:<:S N. J'or instanre, the pa.rticle may choose a path which, being oriented in the Z direction, brings it quickly very near to XYZ and then tah a (:anal nearly pe'llCndicular to Z with such a long residence time that the felt (If T is <;OlIlIumed there. In that case the number of choices, is vcry small in comparison to the mean number of paMagCS necesary. I t is therefore neu:ssary 10 integrate PI' over all possible values of N to obtain the required probability. P(X, Y,

z, 't)

..

J".

P,,(X, 1', Z, T, N) dN (7)

"' On the Z axis (X .. y .. 0) the extreme

For poin!.!! values for Narc N, .. ZI t and N , .. (ZTluf)i . St<JndJml ~'<>i;qm-The probability distribution P(X, Y, Z, T) for the arrival of one panicle is a measure for the concentration distribution when a great number of particles i$ injected. The di!persion may be described by diffusion coefficients which according to Einstein equal D _ ,.>/2T

(OJ

where IT is the standard deviation of Ihe probability distribution. As P(X, Y , Z , T ) is a function of X, Y, Z, and T we may dcri"e for a gi"en T, the place X, Y , Z where the probability is a manmum, and the standard deviation ax, IT" a z in the directions perpendicular and paranel to the principal .tream around this mnimum. But also we can take a fixed point in space and determine the standard deviation IT-r dcso;ribing the changes of concentration in course 01 time at the moment when the mnimum concentration passes. In order to determine tbese standard de\"iations from the probability described by (6), se\"eral integrations h\"e to be executed. I lIIegr<>i,'!m oj I/u aprtuirm JIlT Ihe probabililyFirst the value 01 AA' has to be determined, which may be effected in II. ",II.y similar to that shown by Chandra.ukha, 11943). We encounter here, howe\"er, the difficulty that 1/ becomes infinite for 9 _ r/2 acrording to ( I) . We will not enter into the mathematical details, which can be obtained by request to the author, but infer, that 10 the neglect of small terms for N» I, the following result is obtainl-d.

264

+ Jr + /"'111

(9)

with Q.0

II'"

[In (;flu) -

_

h - ll(lI:) - (I)(s)J - - ul/6

+ (J/M

ll(") - (_)') -

c -

. . . (0) _

2i/3

(.>:) -I,y)-(, ) - O / -

t(zO) ..

tU-) ..

j{rl) -

fJja

where ( ) indicates the mean value. I ntroduction 01 tha value lor ..lA' into (6) for PI< sh~ that nut in tegrations vcrsus or, \, ~, and 'I ha"e to be executed. Wi th ""i'«t to \, ~ and 'I this is readily performed by the u"" of the well known result

L~ up (i..., -

~I ~' up (-"'/411')

,.'fJ') d" -

(10)

For the f, 'I part of the integral thill gives, using polar coordinates, a result which yields the probability, that the partide should arrive in the ring between Rand R + dR Cl~R dRlbN/)

upl -R'/4NJ/

(It)

From the,., I pa rt of the integral only the integra_ tion wrsus \ may be treated according to (10) aiving (2 ..).... dT dZ(r/,,\'j1

1~ d.

+ i{bM.lz -

X

t:
1- i.IT -

,.\1 - {IZ -

dN]

.N)t/k\'j

- N,.t{d· - (61MII

The subsequent integration with respect to r shows the difficulty that Q. * is a function of or. Here again we cannot extend upon the mathematical details, (which can be obtained by reque~ t) by which may be proved that a very good approxima_ tion is obtained by substitution of the constant Q. instead of a ~ according to G OO ().. -

t - In,..)"'

(12)

with;>. the root of N .. j .. ~/(~-t-In,..)

(Il)

T he relationship between;>' and N according to ( 13) is represented in Figure 2. Again the larger N, the bettcr the approximation. With Ihis substitution, the coefficient of,' in the integrant is independent of ,., and (10) may be used. Then finally the following resull ill obtained for PN •

Soil Mechanics and Transport in Porous Media

D I FFUSIO~

,

From (1(;) il is ob,·ious that the standard deviat;nn of P(JI) approaches the value

•

•• - 1-

r"

/

'"

/

•

_

• •

•

••

•\

, . "1"11'1, ,.. .. IOV Z./ J

FIG. 2 - Graphico.l ~pr_ntation 01 Eq . 13 combillCd with F.q. 21; ~ as a lunction 0174 1b~ diSlance of the maximum """«nlrati",,. and f Ihe eI~m"ntar)' canallenglh Z, R) dT dZ dH - (dT dZ HdH / S.-N'j(a< - b')1)

Xe>q>I - (A+A ')N- '+8-CNj

(14)

,,·herein

.z _

leT' - 2bTZ

B _ 21ar - be,T

c _

PI!') d" -

1«1' -

2~

d.

C'

."

L~ explFl" J

,

(I i)

exp

WW!

dl'

c.,n by ,·;rtue of the substitution (16) be 31'· proximatl~l by

•

f--/ , , •

F"(}..JII

The evaluation oi an integral of the lonn

/

•

P.~( T,

7\

I:': GRAN UL\ I/. DEPOSITS

+ aZ'l/4(a< - ~ + dZ) + ~'ZJ/4(a.:

+ ....1/4(..,;

)

- b') (15)

- b')

A' _ R'/4J

-

U- ,...(...l](p. - ,..,)']

IZ ~/

- f··(..,))t up If·('''H

N . _ I(A

(18)

+

.~ '}/C)1

• - INo/C)1

'J

+C A')'

dTdZRdR [ PIT. Z. H) dT dX dH - 8}1,.{a< b'»)i (A

X exp 18 -1[1

+ C(A + A')~J

Fv..l

+ tv. -

~ ,)'fI"r...)

(16)

where jJ.. is the value of jJ. for whi(h F( /Io ) has its maximum. Be<"au se the diwibution of P(JI) is approximately normal the higher order terms in the Tay]or expansion of F(/Io) may be neglecled.

Selected Works of G. de Josselin de Jong

(20)

whcrein A , A', B, C arc given by (]S) and <1, h, e, d, e,} by (9) and (12). CIuJ'iK/a oj/he p,oooUi/ily I'(T, Z, R)-From (20) ior lhe probability !>Cvcral conclusions may lIC drawn with respect 10 standard deviations. The expre,;sion is a iUllction uf T , Z and R because thc factors "' , A' and B contain Ihem. The maximum of peT, Z, N) lil"!! in T • . 70. , R.givcn by "

F(..) -

(19)

and the wobability for Ihe particle to arrive in time T at a point with coor<.linates Z, R is

we can replace the function F oi "in the exponent

b,·

l

The error intrormed on (14) ac.:ording to (7), the valu,," of N, . N" and F(N) are such that in Rgrl"Cment wilh this condition the extension 01 th e hound" ri~.. to + .. and -.., is IICrm~ible. T herefore Ihe result of the integration ve",us N ca.n be ob tained by dNermining the value of N. for which F (N) i'l (16) is a maxin'um and sub· ~equentJy F O(N .) , These d"lerminations necessi. tate on ly differen tialions on F(N) being operaliOll~ which arc always feasible. Neglecting terms 01 small order, there ,.',.'sults

In order to obtain the diffusion coefficicnts as Ihe final result of our considerations, the sum.,· quent treatment 01 (14 ) for Ihe probability 1'.,., that a part ide after N choi""" Rrrives at a point Z, H. in a lime T, involvc.. an integration with rl'Spect to N. and Ihe determination 01 the stand· ard deviations wilh r~pect to Z, R, Or T. Th~ operations become easy to perform if it is realiu..J that acrording to Ihe cenlral.limi t th"",,,m the different probabiliti.,;; in,·oked will approach a nonnal law for N » I. Therefore. if a result is obtained for t he probability distribution of a va.iable jJ. given by

d(p. - ""

-

0

(21\

This meanS thai thc maximum concentratioll 0/ an injected subslance travels at a velocity r. equa l to the mean vdoci ty (= )/(1) _ tt/". T hat the maximum alSQ foliO\\"$ the axis R _ 0 is C"ident. T he , Iandard deviations about the ma~imum may be derived with (17). We obtain Ihen

265

c.

72 _II

-

"r -

117..,/ _)1 - [Z,{")/{z)ll [2Z,(
2M. + ""J/"'1

DE JOSSEUN DE

JO~G

([JZo/S f ll

.,~

Ellcr""""

1

IZ, «")(I)' - 2(zl)(s) (f)

(22)

+ to. + ; - In y)1i' (0)')/(0)'1 1 0 .. - uIJZ. (). + t _In -r)/'P (. /i)" .. - t l[JZ.(). + t - In ")')/ljl "

-

In these cxprL'$1lion~ Z. i, the distance along the z axis, where the maximum is at the ",ornent T •. The maximum numlJer 01 choi«s N . for the muimum at Z , is gi'"cn according to (19) by

N. _ Z,/_ - 37.,/21

(23)

The length of ( is gi\"Cn by the geometry of the pore canal system, while ;>. can be read from Figure 2 as a function of N . or computed by (13). The standard deviation
0" _ ... ~/2T, - J "//t6 O• ... ... ·.I2T.... . ,l{>.

)

+I-

In ..,)/6

(24)

From t his result we ded uce thai the interrelation between these diffusion cocflicien15 is V ziO,. - 8(>'

+t -

In y)/9

(25)

which according to ( 13) i. dependent on the distance, .in<;e X is a function of N , and N , ... 3 Z,;U. The order of magnitude of DzID" would be 6.5 for N , _ 10'. In the foregoing compu tat ions ( the length of the elementary canal s was invariable. The introduction of a statistically distributed canal length in the given analysis, entail. nO essential diJficultics. We did not r~prt'sent its iniluence because we assumed that for the exampl e given by the experiment this influ ence was small in «Impari""n to the scattering effect producOO by the orientation of the canals. E.'?t',imenlt--Sevcral experimen ts have been run with the percolation appa ratus represented

266

~

~I

OSttu., ...

. . . ".

i

~ f

0

! t·tG. 3 - Test device

diagrammatically in Figure J. The cylindcr{6 cm , heigh t 20 cm) was packed with glass spheres of 0.2 mm diameter (standard deviation 0.05 mm) . Th e packing was carefully con trolled to procurt' a uniform poroIlity of 38.5 volume pet. The system was satu rated with distilled water with NaCl to a concentration of 0.02 nonna!. The liquid under the filter was replaced by water With a 1.0 normal NaCI amcentration. Nexl the waterflow was started at a filter velOCity upward~ of

Soil Mechanics and Transport in Porous Media

73

DlfFliSION IN GRANULAR DEPOSITS

,, -, --

-

-

-------------

"r

,

"'+------------------+e-,

,~ ,*, .. t .. t*"ant ([. 1'1<':. ;, - Ot..."ved standard

~via\ion .

a. related 10 theory

cm'!minute. The mean ,-clocity through the pores (.)/(1 ) was

(i

(611.6') X (1/0.385) - 0.55 ~m/min

At different distances from the bottom (2.5, 7.5, 12.5, [7.5 em) ekctroon were inserted inlO the grain material, and permitted salinity measurements of the interstitial liquid. The e1e<;trodes consisted of a I,latinum wire
stone bridge (1000 cps, 50 "A) rcprnented diagrammatically in Figure J. Calibration in dif-

•

------------~

:• N,

--~-~--"' .-.- ........... .

-.- ----;;;:-----::=------~-- --=----------------

j,

I

• FIG. 4 - Salinity as rtgiSlcred b}"

, ....
eI~ tmtles

Selected Works of G. de Josselin de Jong

ferent l\aCI CQllcentrations gave the relation be_ twt<:n salinity and electric resistance in the circuit. Calibration in the liquid alone as weU as after intrIX/uetion of the grains ga,'c values which correspond with thooretical computations based on potential distribution around ellipsoids and reduction of conductivity by the grain material. By measuring the resistidty variations with respect to tim e the break·through curve as the saline front passed by, could be determined. In Figure ~ is given the salinity as registered hy the electrodes at the moment that a particle traveling at the mean velocity ....·ou ld ha,'c arri,·ed at the distance indicated by the ah;ci$&l. The error in(rIX/urecl hy the finite dimension of the electrodes is shown by the curve at the top of Fig. 4 where, based on theoretical calculations of the potential distribution round ellipsoids, the apparent concentration ,·ariations measured by an electrode have been drawn lor the passing 01 an abrupt front. The observed CUrves show a distribution that is neady normal. Their indination increases with the distance fTom the bottom indicating that also the

267

G. OE jOSSEI.iN IllS JONG slal,dard deviations (Tz increase with Z. In f igure 5 Ihe rciation between (Tz and viz i~ shown. In the same figure we in>ertl"c,;t. This value is about t of the main grain diameter _ In a close pa(king this is the diSl ance of Ihc center, of IWO neighbouring tetrahedral pores (!K1:" Fig_ la). Here the packing was looser. but Ihe difference in 5izes of Ihe spheres may accounl for Ihe sma ll value of ( since the number of smaller spheres Prl'Sent was not negligible In ..'igure.5 the moloxular diffusion has also been shown because in the lotal observed diffusion the effecl of Brownian mOl ion is prt"SCnl_ ~ I ole<:u!ar diffus ion was determined in the percolation apparatus as follows. The perrolMion ll"St was cxeculed as Boove for the front passing the two lower electrodes. Then the liquid How was SIOPP'.xt duri,,!: 24 hours and started again in order to let loe Ironl pass the two upper electrodes. T he additional inclinalion of Ihe break -through CUrves correspond to an ae«ptable diffosion coefficien t of 0 .. _

0.7 X 10-' em'I"'--.:.

T he normal runs lasled about 30 miuulCS. SO the iulluence of molC("1Jlar diffusion was not suHicient 10 affect Ihe interpretalion 01 the II'St results (see Fig. 5). C"'IC/usi......-The introdUClion of a continuous probability ,iist ribution for Ihc random choice of tra,·d dire<:tion at a bifurcalion in the port: canal !)'Stem permilS lhe computalion of longitudinal and transvcn;al
268

sc,,·cd. but Ihe effect may be of importance when scale Il"St results have to be interpreted. The It'St result! repre:5cnlt..J by R;fai and olhers {19561 Sl~m nOI to be in <:ontradict ion with this re!'ult, Ihat D% should be a function of Z. . Sin~ Ihc tr:ms,'eTSC diffusivily docs not depend on 7." the ratio belw.... n Dz and DR increases as the mean distantt of travel increases _ From the few experimental data available values of IJz/ DR from 6 10 10 ha"" been reported. According to ou r theory this would correspond to a tra"cling distanCE" of 10' 10 10' times the clemcntary canal length. Ad,..,..d ....&m.~/ s---- \\" ~ a,~ ind~bted to R. Timman for hi. assistance in Ihe mathematical .tudi.. and H . Kramcrs for his sugg"'lions in physical m~lIe"'. Mi •• Hoving carried out I~e e",)~riments. REfERE):CES

BARON, T. , G<:neralizt.'lem! in ph}'.i", and ulTOnonr, 1If;!:. Jfod. I'I,y, .• 16. 1-89, 19-43. DANel< WEus. P. V., Continuous ~ow system s; dist.i. hUlion 01 ...i,lence ,im.,., Cile"" E",. Sci., 2, t-13, 1953. 0",·, P. R .• Di.p".";on of a mo"ing saIl water boundary, TraM. A m". Grophy,. (h,;".., 37, 595-601, 1'156. [IE JOSSELIN oE JO,,-C. G .. l.·ento:rtainemcnt de l>arlicui ... par It courant inle.,;t ici.l. I'ubl. 41 . ,\ ssn , Hydro!,,!!)", VGGI. Sr",~,;a DdTC,!, 139-147, 1956 JAHNKE, E., A"-" F. E""F., Tahl~, Df lum/;m... Do,'e' l'ubl,. I'ew York. 382 pp .. 19-;5. KLl"KKSB ';IlC. ,\... ASO F. SJESIT1.U. H o\din~ time di'lribulions of lite GauS6ian Iy~. Che"'. Eng. Sd., ~. 258-270. 1956. K .".. ERS, 11., ,""" G !"UHD." Frequency response analy.i, of conlinuou. flo\\" ')"~I.m •. Ch.",. E"I. Sd.,

2, 173- 181,1953. RI TAI. M_ N. E., \\. J. KAUI>'''''-' "~.,, D. K. TODD Dist>~r-sion phenomena in laminar flo\\" through [>O.ous media. 1''"8'U' lI~p. l, CoM! StO"'U. media, J_ Appl. Physics, 26, 99-;- 1001, 1954. VAS DEE"TER. L"lJ"·E1U£~. A~pI.

J. J.,

j.

J.

(boEnER. A~f>

H. ,\.

displacement in capittari~s. S
Lllb(ml/uriu", 1'0<" G,~~J~,.~ilui,a. Daf/, TlI~ IY'fl,../~rni,

Pus/bur

60,

(Communi[ateoJ manu)Ccipl <ecei"ed Mar<:h 25, 195 •. and, as ",~ised, Seplember 19, 1957; ~n for fo.mal di",u ..io,n unlil Jul y L 1958.)

Soil Mechanics and Transport in Porous Media

v.... J'I,

1'; •. G

Discussion of "Longitudinal and Transverse Diffusion in Granular Deposits" IIY

G_

Dt: J OSSE I,IN 0],; JONG

!T'~"$.,

39, 61- 14, L958J

J. A. Cole (Dcparlment oj Grodes)' and G<'Ophysics, University of Cambrirlge, Cambridge, England)-The writer finds the paper most valuable, because he [Col. , 1951] hIlS made expcriments with eolumnsol granular material, basically similar 10 tho:;e described by the author, in order to de-lermine the relative importanc~ 0/: Effect (a) radial molecular diffusion in each pore combined with a velocity gradient across the pore, and l~ffcct (b) the geometry 01 the porc system, in I)roducing a longitudinal dispersion of an injected substance. T he author men Lions Eff~t (a), but makes the assumption that it may be neglected if the pore section is small enough. Thouj;h valid, suth an assumption i. not oJlcn real iud, as the following numerical examples illustrate. 1"a."/lI' !l953, 1954ab1 has .hown that the diffusion oocfficicnt Dwoj !-:freet (a) with strean,line flow in straisht cylindrical tubes oj rarlius a is

."

• •

."

•

""If

~."

•

I

d

."

,,

, ,,

,,

,... ._._._.__ ._. _. _. _._ .I1!~:!..~_''::i'."!._ . __

.'

(26)

Ftc. 6 -

I. being the dis!ancc along the lube in which the rencc:nlration of the injected substance has an apllrt.'Ciab!c gradien!, and is appro~imatcly <-qual to 4 ffz. The firsl rendition in (27) can easily be met hy using large cllough values of Z . Figure 6 is a. presentation of D~ versus 1'0 on a log-log graph, for two values 0/ o. Where tbe two line. arc broken the second rendition iu (27 ) does not hold, and (26) is unly an approximation. I n applying (26) 10 granular materials, which do not have cylindrical pores as a rule, one uses a mean pore radius (see abelow) in remputing D •. The random orientation 01 Ihe pores results in one's observing not D$, hut D./r _ o'«J)/ (I»'/48D.. , where (1)/(1) _

•

,

01, ... ;" I ... ,.. ~I

,•

when (27)

OI" .. bl

mat~rial,

.'

~J' .. ~'

~~/"J'

.

Con'pari$On of D. 01 .....·0<1 in th. granular ,n Tahtc I ,,·itb theoretical valuts of

10,10<1

D•.

Ma· .. rioJ

,, " C

Compooi."", of ... l
I

,

•

~

Gtass bead, GI ..... kads

Gl .... bead. etun
0.020 0.047 0.027 0.034

0."" 0 .35

0.33 "' 0 . 33

•

-

0._ 0.018 0.0105 .. 0.01.1

~"d

Clearly a line (otf<'Sponding to D./f', for a gi,'en value of a, plolloo versus «:)/(1»' is reincident with tbat 0/ D. "ersus V. for the same value of a .

269

11 61

Dl SCUSS I O~

The author gi.,es for his measurements in a gLa..s.bQd column, .r~ _ 0.6 em ,,·hen Z _ 17.0 em and (.0) 0.55 em/min _ 0.0092 em/KC. Accordingly Dz - 0.6' X 0 .0092 / .... en,·/KC - 9.6 X 10-' an/ -=. Th is value of Dz is sho"'n (as the point fo r Ma terial Al on F igure 6. Abo shown a re Mmle .·.lues of D z in columns of glass beads a nd of dean quaru sand, which were obtained by thc writer. Materials A, B, C, and D have thc properties ip«ifi.:d in Ta ble I, where d _ mean grain diameter, ; _ coefficient of volume of porosity, and (d/2)[P/(1 - ; )]1 _ the mean radiu s of thepor~, approxima t ing Ihe polyhedra to spheres. These valuCi of ii correspond to values of D~/f' ,,·hieh arc far from negligible. In Material A, the ca!IC quoted by the author. it is likely that Effect (a) wal appreciable (one ten th of the observed D. say) . In the other eramples Effect (a) was of major imporlana., amounting 10 a quarter of thc observed D z in Materials B and D, arK! alm06t all therft)f in material C. In ronl ..SI to thc above laboratory nperirnent.>. Ihe ... rilcr Ie,,", 19571 . 1"" made field uperiment.> n·ilh t .. <:en fton'ing in chall; aquifers. These :
Am -

a-

RU EKEXCt:S

Cou. J. ,\ ., TI~ J= ~/I'M~d \rot." ~1. S. th~ , i,. University of C.mood~. 2(» 1'1'" 1951. DE VACU, 0 .. n.c.:..r 01 chron>atogranhr. J. ,I",.,. C". .... S"'., M, 532-5-10, 19l1. Fosnl. ,\ . G., Po.'~maLief, of c.d,.nst proc_ in find columns. pt. L. ;\bthematical 001 ..· t"'" . nd ..,.ml'totic CIl",nslon5, Pr«. R. SH .• 21eA . 1$1- 185, 1953. "fA"LOl, G.:or,uv. l:lisj.....-.ion of ooIubie m>'ter in

",Innl l'Io""nl JIoo,o'I)' 'hlOUllh 219A. 186-203, t933

270

~

tube. P"". R.5oo< ..

TAfLOl, GW'nt:Y, The d~ of .... Itcr in turbulent !low throu,h a ~, /'roc. R. S«., 223A. 446-168, 195-b. T"'·Wl, GWf"nEY, Ow>t\ition'IIMtr .. hich dispf:roion of a lOlul<: ;n a &L r ~m of iOI.'Wt C&JI be \t6Cod to mtUU'" molecular dift"uNon, P,oc. R. S« .• 215A, .f7J-f17,

1~54b.

THO""', Ib;".. y C., IIcte'."""'" ion tlChan~ in a no .... ing I}"ltcm, J. ,I ..... Ck",. S""., 66 . 166-1t666, I9+! .

G. dt hut/i" d( "'", (Laboratorium Voor Grondmechanica. Postbox 69, Delft. T he 1\cth er· I.nds, Author', closure)- In his di'ICU",ion Cole givell fu rther informat ion on Ihe devia t ion from our theory to IJCl exp>eclcd in CXI>crim cnb for .... hi(h the basic a!;Sumption of piston /low in the elemenlary C1lnal. is unsatisfactory so that Ihe influence 01 holdback can not be neg!ected. According to Ihe conception of 1'..11 DU1II/tr and others 11955), this holdback amount! 10 11 - 0 .02 in the ten dtKribed, becau,,", the faclor (0 .. // ..·)1 _ 'I , where I) .. _ molecular diffuiion _ 0.1 X Hr' an'/sec, / _ t ime to co.·cr the distana. >I v. with mean velocit)' _ 2A cm / O.OO92tm/sec .. 270sec,lnda _ radiw;olpotfS_ O.(J().I em. For the interpretltion of Ihe test rewl t5 in the .rtide this amounl of holdback "'as considered as negl~ible. According to his ' hmr)" Cole obtains a totncwhat larger inlluena. about one tenlh 01 the longitudinal diffusion. '\ 5 Ihe I"incipal point of our consideralions "·U 10 show the in terrelation bct"'ccn longitudinal and lrans\"ene diffu sion and Ihe poSSi bility 10 de. .cri1.>c (h(3C two ph enomena in one mathema t ical expression, the inlluence nf holdback and molecular diffusion both were omitted in Ihe theoretical consideralion§ for simplicity'!! !;like. lIu! we do agree: that Ih~ two eff~"(;ts and the chcmisorption as l1lentio,,~'(1 in the di'ICUS!lion may ha,·c an O,'Crwhdming in ihwnce in certain cireum~tan(es. I " ord~r to show the phenomenon of longitudinal aml tran s.'cf"S(' diffusion we add here some photograllhs of I Chris t;Dn~n f,ltcr (Fig. 7). T his is a dcposi t of crush~'d oplical glass (grain diameter 0.-42 to 1.19 mm) !aluraled wilh a solu t ion of :\ HJ in water of 5u(h a wn~ntralion that looth refraction indicti 01 grains and )lOre li(tuid are rqua! ror sodium litht. ."" Ihe light is not d ispersed by de"iation 01 iu direction at Ihe interlaco of IIraiTUI and lirluid the syslem becon":a I)!.""picace. (Wc arc indcbt~'(1 10 tho: Shcll I.aboratory •. 'm. lt.dam. for thisdC\·ice.l The pure liquid runs ........ ticall)' through the

Soil Mechanics and Transport in Porous Media

1162

,

I)ISC USS ION

!

"n

!

..p.:;;.

~O

,

l.

.,: ,

11 1

• :~ I

..p.:;;.

;to "

'.

0

..

~

, "

.-

0

". '. :

..

', 1#

~

..

~

_

~1IOOI"'k 1

I

I

l

0

_

........ •.coc",

FlO. 7-Chri 5tiansen filter $howin,; longitudinal and transverse diffusion deposit. At! analine dye is introduced with an injection needle. The molecular diffusion of the dye is abou t D • .. 10-' em" sec. The photographs show the dispersion originated at two velod tie!l, 0.(0)8 em / sec and 0.008 em /sec, which induce a holdback over a di~tance 01 0.5 em 01 respectively 0 .025 and 0 . 10 . The standa rd deviations originated by molecular diffusion in combination with longitudinal and transverse diffusion as computed from the theory are added in the figure lor comparison.

Selected Works of G. de Josselin de Jong

At the lower velocity the elonga tion or the standard deviation cllipi!Oid is principally determined by the longitud inal diffusion, while at the higher velocity the influence of the holdback enterS into the observation. RE FERESCE

1. 1., J. J. BaoEDn:, AND H. A. LIIl_ WEau:a, ~luid displacem.nt in capillari~ AppJ. Sd., RoJ., Me. A., 6, 374-388, ]955.

VAN DUllttll,

271

Journal of Hydrology, 84 (1986) 55- 79

55

Elsey ier Science Publ ishers B. V., Amsterdam - Prin ted in The Netherlands [2]

TRANSVERSE DISPERSION FROM AN ORIGINALLY SHARP FRESHSALT INTERFACE CAUSED BY SHEAR FLOW G. DE JOSSELIN DE J ONG and C.J, VAN DUlJN Delft Unillersity of Techn%gy, P.O. Box 356, 2600 AJ Delft (The Netherlands)

(Receiyed January 7, 1985; revised and aceepted Oetober 5, 1985)

ABSTRACT De Josselin de Jong, G. and Van Duiin, C.J., 1986. Transyerse dispers ion (rom an originally

sharp fresh--&alt

interC.~

eaused hy shear now. J . HydroL, 84: 55--79.

In this paper the innuenee of transyers.al dispersion and molecular dirfusion on the distribution of sal t in a plane flow through a homogeneous porous medium is studied. Sinee the dispersion depends on the veloeity and the veloeity on the distribution of s.alt (through the specific weight) this is a non linear phenomenon. In particular for the flow situation considered , this leads to a differential equation wh ich has the character of nonlinear diffusion. The initial situation (at t = 0) is chosen such that the fresh· and s.alt water are separated by an interface, and each fluid has a constant specific weight 7 1 and 71, respeetively. For this initial situat ion, the solution of the nonlinear diffusion equ ation has the form of a similarity solution, depending only on where denoteB the loeal eoordinate nonnal to the original interface plane and t denotes time. Properties of this similarity solution are diBeUssed. In partieular it is shown how to obtain this solution numerieatiy. The interpretation of these mathematieal results in terms of their hydrological significance ill given (or a number of worked out examples. These examples describe the distribution of salt, as a function of t and t, for various now conditions at the boundaries t = ±oe. Also eumples are given where the molecular diffusion can be disregarded with respect to the transvers.al dispel'Sion.

t/Vt,

t

INTRODUCT ION

When fluids of different densities are present in an aquifer and the density varies in ho rizontal directions, the fluid motion contains rotation, In the case of a sharp interface th is rotation results in a shearflow, which is proportional to t he density difference and the interface inclination. Its magnitude was established by Edelman (1940). The rotations and shearflows resulting in t he more general case of gradual d ensity variations were treated by De Josselin de Jong (1960). Specific discharges in an aquifer are accompanied by dispersion. Therefore it can be expected, that at an inclined interface dispersion occurs, which results in changing the abrupt transition from one density to the other, in a 0022·1694/fl6/ $03.50

C 1986 Elsevier Seience Publishera B. v.

272

56

gradual transition zone. Since dispersion can be described in terms of differential equations (see Bear, 1975) it must be possible to express the spreading from an abrupt interface mathematically in terms of the dispersion parameters. The governing relations form a coupled system , consisting of equations describing the specific discharge rotations, due to the gradual density variations, and the dispersion, caused by t he specific discharge distribution. Solving this system analytically in the general case of arbitrary density distributions and additional superimposed specific discharge distributions seems rather tedious. For practical purposes, therefore, Verruijt (1971) proposed to introduce a new parameter to describe the spreading, without specifying its relation to the dispersion parameters and t he density variations. The purpose of this paper is to show, that it is possible to describe the spreading process analytically, when starting from an abrupt interface in certain simple circumstances, such that t he discharge remains parallel and a plane flow situation occurs. For that simplified problem the coupled system can be reduced to one ordinary differential equation of diffusion type. The properties of solutions of this equation are known, because they have been studied extensively from the mathematical standpoint, by Gilding and Peletier (1977), Van Duijn and Peletier (1977) and Van Duijn (1986a). How these solutions are applied to describe the spreading of density variations from an abrupt interface is shown in this paper.

SIMPLIFIED PROBLEM

In this paper the case is considered of a dispersion zone developing from an originally flat, inclined interface, that extends in all directions to infinity. In order to simplify the analysis, the conditions at the boundaries of t he infinite aquifer are assumed to be such, that the flow is constant in planes parallel to the original interface plane. Let coordinates ~, 1'/ be in the original interface with 1'/ horizontal and ~ pointing upwards at an angle 0:' with the horizontal, see Fig. 1. The coordinate r is normal to the original interface plane and points upwards. Flow conditions then are such, that the specific discharge components qt, q", qr satisfy: q" =

qr

= 0

aq t, a~

=

aqt,ihI

= 0

(1)

r

indicating that on ly qt is nonzero and a function of only and the t ime t, i.e. qt = qtCr, t). At time t = 0 the in terface is assumed to be sharp, such that the plane r = 0 separates the aquifer into two regions in which the density of the fluids is a constant. Above it is freshwater with density PI and below it salt water with density P2' In the description below the specific weight 'Y = PC is used instead of density p . So in formula the situation is initially:

Selected Works of G. de Josselin de Jong

273

57

,

,\

,

Fig. 1. Interface plane at angle water below it, at time t = O.

Ci

> 0, t

'Y = 1'1

~

'Y = 'h

r
with horizontal, separati ng freshwater above it from salt

= 0 ~

I (2)

0

Under influence of this specific weight difference, initially an Edelman (1940) shear flow q exists at t he interface of magnitude:

q

= (q t , - qt , ) = (KjP.)('Yl - 1'dsina

(3)

(see e.g. De Josselin de Jong, 1981). In this expression qt " qt , are the specific discharge components parallel to the interface in the fresh· and salt water regions, respectively. K is the intrinsic permeability of the aquifer, considered to be a constant, and IJ. is the dynamic viscosity of the fluids. For fresh- and salt water the viscosity differs by an amount small enough to disregard its influence o n t he results of the analysis below: for a justification see e.g. Verruijt (1980). Superimposed on the shear flow an average specific discharge in ~ direction of magnitude fjq is considered to occur in this paper, with fj a number, that remains constant in time. The initial flow conditions are then in accordance with eqn . (3) given by: q ~,

=

q~ ,

{fj +! )q

for

(fJ- ~ )q

for

t>O,t t
0 = 0

The following situations are to be distinguished, see Fig. 2:

fJ

<- !

fj =

-!

-! < fJ < 1 fJ = ! fj> ;

t

both fresh- and salt water flow downward the fresh water is stationary fresh flows up, salt flows down the salt water is stationary both fresh- and salt water flow upwards.

As time proceeds the transition from"Yl to "Y2, which originally is sharp at 0, will spread by hydraulic dispersion and molecular diffusion. Salt water

=

274

Soil Mechanics and Transport in Porous Media

68

= 0 for different values or II.

Fig. 2. Distribution of qt at time I

will mix wit h freshwater, and the specific weight will become a variable

function of position and time. Because of the plane character of the case ~

considered, "'( will be a function of implies that:

and t only. i.e. 'Y = 'Y(t. t), which (6)

At infinity the specific weights will tend towards the original values 1'1

at t

-jo

+ 00 and 1'2 at t -

- 00,

The mathematical description of the spreading process is developed below from basic equations, describing the changes in specific weight and specific discharge qt in course of time. The boundary conditions at infinity are assumed to be such, t hat the specific discharges remain constant there and

equal to the initial values eqn. (4) . Thus: qt = (f3

q,

+ i)q

(Jl-

foc

t-++OO,t>O

r . . . -oo,t> O

!io

I (7)

The result of the analysis will be a distribution of specific discharge and specific weight as functions of ~ and t. An example is shown in Fig. 3 for the

,

r,

,

._ 1, ._

Fig. 3. Distribution of specific discharge (a) and specific weight (b) for the case m == 0, f 4 at some time, t> O.

il = l

Selected Works of G. de Josselin de Jong

275

,9

a

case ~ = at some later time, t> O. A more detailed description of this figure is given in the sections below.

BASIS EQUATIONS

Flow rule When an aquifer contains flu ids of different spe<::ific weight, the flow rule is obtained by considering equilibrium of forces. Let Kq be the force required to act on a unit volume of pore fluid in order to maintain the specific discharge q through a porous medium. According to Darcy 's experiments this force equals Kq = (P. ! K)q. Let Kp be the force exerted on a unit volume of pore fluid by the gradient in the pressure p and the influence of gravity on the specific weight, "y . This force equals Kp = - grad p + 'Ye, where e is a unit vector in the direction of gravity, i.e. downward . Equating Kq and Kp results in: (J.I. ! K)q = - gradp +-ye

(8)

By taking the curl of this relation, the pressure p is eliminated. This leads

to , (P. /K) curl q = curl ('Ye)

(9)

which, accounting for eqns. (1) and (6), reduces to: (10)

Integration of eqn. (10) is possible, because variable. This gives:

qt = - (K /P.) 'Y sin a

t

is the only independent

+ constant

where the in~gration constant can be determined by use of eqn. (7). This gives, taking account of eqn . (3):

or:

(11)

In this equation qt and 'Y are the variables, which are both functions of and t. It may be remarked here, that eqn. (11) is a linear relationship between these two variables. This is reflected in Fig. 3, where the curves for q ( and 'Y , respectively , are shown to be each others mirror image, when drawn on appropriate scales.

t

276

Soil Mechanics and Transport in Porous Media

60

Continuity of fluids When both fl uid and porous mediure. can be considered to be incompressible, continuity of fluid is satisfied, when div q is zero. Using eqn. (1) it can be verified, that all tenns of div q vanish and so continu ity of fluid is guaranteed, identically .

Continuity of salt In stationary groundwater salt is spread by molecular diffusion. In addition, this spreading is enhanced when the fluid moves, because inhomogeneities in the pore space scatter and recombine fluid elements. In the periods of being adjacent, salt is transmitted by molecular diffusion to neighbouring streamlines and carried off in directions deviating from the average flow paths. This process is called mechanical dispersion (see e.g. Bear, 1975). Averaged over the pore space a salt flux F occurs, which expressed as weight transport per unit time and unit area of the aquifer is: F

~

(12)

- Ograd)'

where 0 is a second rank tensor. It is the dispersion tensor consisting of terms due to molecular diffusion and mechanical dispersion. Continuity of salt is satisfied, when the divergence of the exchange flux is balanced by the local rate of change of the specific weight o'Y/ ot and its convective rate of change div(q'Y). When n denotes the porosity of the porous medium, t his balance is expressed by: n(o'Yl ot)

+ div(q'Y)

~

- div F = div(O grad "1)

Taking account of eqns. (1 ) and (6) this expression reduces to:

n(ar/at)

=

a((nDmo' + .Tlq, 1l arlatJIa,

(13)

where Dmoi is the molecular diffusion coefficient and O::T is the transverse dispersion length in t he direction of (i.e. in the direction perpendicular to qt) . A special feature of the dispersion is, that not the specific discharge itself, but its absolute value has to be taken into account. Equation (13) can be simplified by taking advantage of the linear relation· ship (11) between "1 and qt. This permits to write it in terms of qt only as:

r

n(aq, /at)

=

.T a[(qm + Iq,1l aq,/an la,

(14)

where m is a dimensionless parameter representative for the ratio between the influence of molecular diffusion and mechanical dispersion. It is given by: (15)

Expressed in these terms the salt flux vector F given by eqn. (12) has only one component F~ of magnitude:

Selected Works of G. de Josselin de Jong

277

61

(16)

Equation (14) has to be solved subject to the initial and boundary conditions, eqos. (4) and (7). The mathematical implications of this system of equations is treated in the next sections.

SIMILARITY TRANSFORMATION

The partial differential equation (14) can be converted into an ordinary differential equation with simple bOW1dary conditions, by subjecting it to the Boltzmann (1894) similarity transformation. This means, that the two variables t and t are replaced by the independent similarity variable r according to: r = f /(aTqt /n)112

(17)

alar

Introduction of this variable implies that = (Jrj at)d j dr ... etc., so that eqn. (l4) is transformed into the following ordinary differential equation: (18) A solution of eqo. (18), which only depends on r is called a similarity solution of the original equation (14). Let the new variable w = w(r) be defined by:

w = q(lq

(19)

Then eqn. (18) reduces to:

!

r{dw/dr)

+ d[(m + Iw I) dw /drl /dr

= 0

(20)

This is a no nlinear, o rdinary differential equation in which the relevant coefficient has the form (m + Iwl). By its dependence on the absolute value of w, this coefficient creates a special nonlinear character of the problem. The boundary conditions (4) and (7) reduce by introduction of eqns. (17) and (19) to: [0'

[0'

r-+

I

(21)

-

QO

Problem P(m, 13)

The problem, defined by the differential equation (20) with the boundary conditions (2 1), will be referred to as P {m, 13), indicating that m and 13 are the essential parameters. Since these parameters are related to the molecular diffusion (see eqn . 15) and to the specific discharge at infinity (see eqn. 7),

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Soil Mechanics and Transport in Porous Media

62

it is justified physically to assume that they are real numbers, satisfying: (22)

and

Using the substitutions (17) and (19) it is possible to write the salt flux from eqn. (16) in t he Conn: F t = (,Yl -'Y d(a T Qn /t) If2 [(m

+ Iwl) dw/ dr]

(23)

Application of similarity solu tion

A solution of P(rn ,

/3) is called a similarity solution of eqns. (14), (4)

and (7 ). In Fig. 4 it is shown, how such a solution w as a function of r is related to the curves for qt as a fun ction of ~ and t. The heavy curve in Fig . 4a is the solutio n w(r) of P(D, a), determined in a manner that is explained in example 3-ii in t he section on practical application . This curve intersects the axis r = 0 in Wo = 0.37 2. This means that according t o eqns. (17 ) and (19) q t = 0.372 q at t = 0 for every time, t > O. Thus, all the curves in Fig. 4b, which represent the distribution of qt over the height t at different time t > 0, have the intersection with the axis t = 0 in commo n. The heavy curve in F ig. 4a in tersects the axis w = a in ro = -0.503. This means according to eqns. (17) and (19) that the plane where the discharge qt is zero, is located in to = - 0.503 (O:Tqt j n)1 / 2. As a consequence this plane descends wit h time proportional to t 1 / 2 as sho wn in Fig. 4c. The curves in Fig. 4b all have the same shape as t he heavy curve in Fig. 4a but are stretched with a factor, that is proportional to t 1/2 .

\

\

\

, '

- \ i

\ \ ... '.

® Fig. 4. Relation between a solution w(r) in (a) and the specific discharge qt (~, t)in (b). The plane, where qt = 0 descends in course of lime according to ..;t, see (c). The curves are for the case m ::::: 0, f3 = 1/4.

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63

AUXI LIARY PROBLEM Q

The problem P(m. tJ) is not explicitly solvable, because of its no nlinear character , although much is known about its solutions. A special d ifficulty mentioned already is the occurrence of the absolute value [w i. which causes solutions to consist of combinations of parts, where w > 0 and w < O. This difficulty is solved by considering first the auxiliary problem Q, which is d efined by the differential equation: !s(du lds)

+ d[u(du lds) l /ds

(24)

= 0

with boundary condition:

u

=

1

(25)

and the additional condition:

o ~ u(s) .;;;; 1

(0<

- 00<8<+00

(26)

In this section the solu tion set of this problem Q is considered. In subsequent sections it is shown, that with these solutions it is possible to produce the solutions of P(m, (3) for all m and /3 by application of an appropriate rescaling procedure.

Solutions of Q The solutions required in t his paper are given by the fam ily of curves represented in Fig. 5. Solutions of equations similar to eqn. (24) were studied extensively by Gilding and Peletier (1977 ), Van Duijn and Peletier (1977), Gilding (1980) and Van Duijn (1986a) . They gave rigorous mathematical proofs about existence and uniqueness of solutions and they studied their behaviour. Using ideas developed in these papers, the following basic facts about solutions of Q can be established, see Appendix A. All t he curves from Fig. 5 are strictly increasing, with du lds > 0, at points where u > O. Different curves cannot intersect. Further, all curves approach the upper boundary u = 1 as a compleme ntary error function such that: l - u = O (erfc(+!s)}

(27)

Expressions for t he curves in Fig. 5 cannot be given in closed form. They were constructed with a shooting method, which was executed by using a finite difference approximation of eqn. (24) , described in Appendix B. The curves are indicated by type numbers I and II, such that type I curves are located above the heavy line in Fig. 5, and ty pe II curves below that line. The separat ion line is called separatrix. In the description below a few remarks are inserted on the construction of the curves. These are included for those readers who wish to dispose of more accurate values than can be inferred from Fig. 5.

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Soil Mechanics and Transport in Porous Media

64

Fig. 5. Family of curves representing solutio ns of problem Q. Triangles are for example 3·i, where m = 1{4, t3 = 1/ 4. Dashed lines are for example 3·ii, where m = 0, t3 = 1/ 4.

Type I curues These remain strictly positive in t he entire interval for s, such that u tends to a positive value q, as a lower boundary, i.e.: u = rp

with

0<41< 1

fo,

(28)

This lower boundary is also approached as a complementary error functio n, in this case such that: fa.

s ~

-

00

(29)

The numerical procedure for establishing the type I curves is to start from different values Uo on the vertical axis, s = O. Next u~, the value of (du lds)o in each starting point, is chosen in such a manner t hat constructing the curve to the r ight, it reaches the value u = + 1 asymptotically as a complementary error function according to eqn. (27). Subsequently starting with these values 1.10 and u~ the curves are constructed to the left. The asymptotic value ¢J mentioned in eqn . (28) is established using again the complementary error function approximation, now with eqn. (29). For every value of ¢, t he corresponding starting values are assembled in Fig. 6, 1.10 on the left-hand scale and u~ on the right-hand scale. For example, the value ¢J = 0.580 corresponds to 1.10 = 0.8, u~ = 0.132.

Type II curves These remain zero for all values of s below a value so, specific for each curve, i.e .: 1.1

= 0

fm

At So the curves start on the base line, such that:

(3~) 1.1

Selected Works of G. de Josselin de Jong

= 0, with a vertical tangent 1.1' ~

+

00,

281

65

a. t

i- .-----

~- ---- --- ~- - .7

; - - - - --;-rrr

.• 1--,..... .,

.,

H-+++-+-+H-+~+--- " 'I rH +-+-+H-+-H·' i'

T--kj-N-'·+-H-I-I·'

.,

00

tt'~FR "

rfi

'.0

Fig. 6. Startvalues Uo and u~ = (du /ds) in s"" 0 corresponding to different endva!ues if, for type-I curves.

2u(du /ds) = A

u = 0

(31)

s = So

in

with X a value specific for each curve. The numerical procedure for establishing the type II curves is to start

from different

So

values on the base line, to choose a A and to use the series

expansion in terms of A mentioned in Appendix C for small values of (5 - so). Subsequently, the curves are constructed towards the right, using the finite difference scheme of Appendix B. The asymptotic end value of u is established by use of the complementary error function approximation (eqn. 27 ). The value of A is finally chosen in such a manner that the end value equals u = 1. The A values corresponding to different start values So are assembled in Fig. 7.

" "0 --

·\ · z.. .1~

- 1.0

-·s

.~

0

-- .... , -. -..

... ,_...... ... . ·5

1·0

S"

Fig. 7. Startvalues So and A. = 21,1(d1,l/ds) on 1,1 == 0 for type·II curves ending in u = 1, B

== + 00.

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Soil Mechanics and Transport in Porous Media

66

Separatrix The separatrix is the limiting case between t he two types of curves. It is a type I curve with ¢ reduced to zero, and a type II curve with A reduced to zero. The series expansion for small values of (s - so) of the separatrix differs from the series applicable to the type II curves (see Appendix C).

PRACTICAL APPLICATION

In this section it is shown how curves of Fig. 5, representing the solutions of problem Q, can be used for solving problem P(rn, (1) in practical situations. It is assumed here, that the relevant hydraulic parameters m and /3, defined by eqns. (15), (4) and (7) are known. The situations where Itll;;;"! and !PI < ! are distinguished. When UJI > the boundary conditions (eqn. 21) show that either w(co) >0 and w(- 00) ~ 0 (case 1) or w(oo) ~ 0 and w( - 00) < 0 (case 2). In both cases the corresponding solutio n of P(rn, tl) does not change sign on the entire interval (- 00, (0). It will be shown that in both cases a solution of P(rn, tl) can be obtained from a solution of problem Q (i.e . a curve from Fig. 5 for that matter) by applying an elementary transformation involving rescaling and displacing the curve in u-direction. When jpl 0 (case 3). Consequently, the solution w changes sign on the interval (- 00, (0). This introduces an additional difficulty caused by the absolute value of w in the differential equation (20). Let TO be the value of r where the solution vanishes: i.e. w(ro) = 0 , then w(r) < 0 for r < ro and w(r) > 0 for r > ro. The solution thus consists of two parts (w > 0 and w < 0) . Each part is an appropriate transformed solution of problem Q. They are joined together at ro using the continuity of the concentration (or velocity qt) and the continuity of the saJt flux. T he treatment here is aimed to describe the required procedure of rescaJing and combining the curves appropriately. without too much mathematical details. For more detailed information the reader is referred to Van Duijn (1986b).

t

Casel:{3 ~ !

When (3 is larger than ! , both fresh- and salt water flow upwards. When tl equals!, only the freshwater flows upwards while the salt water is stagnant. Both situations are shown in Fig. 2. Now let u( s) be a solution of problem Q and consider for 0 > 0 the transformation:

r = os

(32)

w(r) = o2u(rl o) - m

(33)

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67

This transformation consists of a rescaling (caused by 0) and a displacement (over m) of a relevant curve from Fig. 5. By the transformation t he rliffp.rp.ntial p,quation (24)

tr dw l dr

hecomp.~:

+ d[(m + w) dw /dr] / dr

= 0

(34)

because the a cancels. Next let u(s) be a solution of problem Q of type I or the separatrix. Then u(oo) = 1 and u(- 00) = ~ ;;a. O. Using this in eqn. (33) gives: w(oo) =

02

-m

(35)

and: w(-oo) =

0 2 1/1

- m

(36)

Thus when choosing a such that:

(3+t = o2 - m

or

o = «(3+!+m)l fl

(37)

and
= o21/! - m

0'

" = (~ -\

+ m)/ (~ + l + m)

(38)

it follows that the funct ion w(r) satisfies the boundary conditions (eqn. 21). Moreover w(r) ;;.. 0 for all - 00 < r < ""', implies w(r) = Iw(r) l. Therefore eqn. (34) is identical to eqn. (20). Thus the function w(r), defined according to eqns. (32), (33) and (37), (38) is a solution of P(m, 13).

Example l·i: m = 1 and 13 = 0.881 Then from eqns. (37) and (38) there results (1 = 1.543 and ~ = 0.580. Using Fig. 6, the value of
=

(12U -

m = 2.381

U -

1

and: r = 1.545

These values form the curve of Fig. 8, which is readily verified to satisfy the boundary conditions (eqn. 21 ). The curve in Fig. 8 resembles a comple· mentary error function. Indeed this kind of function is to be expected as a solution of problem P(m, (3) in the limit·case where m is large with respect to one. Example I·U: m = 0 and 13 = ~ Then the molecular diffusion can be disregarded with respect to the effect of the lateral dispersion and the salt water is stagnant. This situation can be considered as the limit of the case where 13 > , and m > O. From eqns. (37)

284

Soil Mechanics and Transport in Porous Media

68 1. 5 ... .

'l' ~ /q

...... __ ... .......... ........... ,,-

1·'SI •

"

'"

., ... .•.... ......... . _. _. _ · JJI

_• .0

,

-2 ·0

,.,

•

Fig. 8. Similarity solution for m = 1, (j = 0.881.

and (38) it follows that: 0 = 1

.nd

(39)

" ~ 0

As mentioned ahove, this value of.p indicates that the relevant curve is the

separatrix of Fig. 5 and the value of a shows, that the similarity solution is the undefonned separatrix. Since this simple result produces a clarifying example, the corresponding specific weight and specific discharge distributions are shown in Fig. 9. This figure is to be interpreted in the manner as explained for Figs. 3 and 4. In Fig. 5 the separatrix is specified by the intersections with the coordinate axes. These are: s = 0 So

= - 1.23675

uo = 0.5873

u

=

0

I

(40)

These values are reencountered in Fig. 9 in the following manner. Since the scale factor in this case is a = 1, see eqn. (39), the s, u values are directly the r, w values, which are related to physical quantities by eqns. (17) and (19). Using these relations it follows that s = 0 corresponds to t = 0 for all time t which is the original height of the fresh-sa lt interface. The first line of eqn. (40) therefore indicates that the specific discharge at the original interface height is constant for all t and equal to qt = 0.5873 q, see Fig. 9b. The second line of eqn. (40) indicates that the depth to, below which the groundwater is still stationary and the specific weight is not yet reduced, equals - 1.23675 (aT9t l n) !12. This means that this depth increases proportional to root time, see Fig. 9c. Case2: (3';;;;;-~

When (3 is smaller than - ~, both fresh- and salt water move downwards. When (3 equals - L only the salt water flows downward while the freshwater

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69

,

©

Fig. 9. Example t-ii, case m = 0, /3 = 1/ 2. Development of brackish zone, when molecular diffusion can be disregarded a nd the sal t water is statio nar y.

is stagnant. Both situations are shown in Fig. 2. Now the appropriate transfonnation is :

r

=

-as

(41)

- w(r) = Olu(- rio) - m

( 42)

Applying this transformation, eqn. (24) becomes: irdw/dr+d{(m - w)dw /dr] /dr = 0

(43)

Let again u(s) be of type I or the separatrix. Then eqns. (41) and (42) imply that : - w(- oo ) =

0 2

(44)

- m

and: -w(+oo) = o2¢ - m

( 45)

Now a must be chosen such that:

-{3+! = o2 - m

a = (-

tJ + ! + m)1 12

(46)

and ¢ such that:

-,B - !

= o2¢ - m

0'

'"

~ ( - ~ - l +m) / (-~+l +m)

( 47)

in order that w(r) satisfies eqn. (21). Moreover, w(r) '" 0, implies t hat -w(r) = Iw(rll. Thus eqn. (43) is identical to eqn. (20), showing that w(r) in this case is in fact a solution of P(rn, ~). Summarizing, for this case t he solution of P(rn ,~) again consists of a type lor separatrix curve from Fig. 5. Since both eqns. (41) and (42) contain a minus sign the relevant curve from Fig. 5 is rotated over 1800 to produce the similarity solution in the r, w plane.

286

Soil Mechanics and Transport in Porous Media

70 Case

3: -

t < (J < + t

In this case fresh- and salt water flow in opposite directions (see Fig. 2) and so q~ and therefore also w from eqn. (19) change sign in the region of integration. The positive part of the solution is denoted by w· and its

negative part by W - . Then w+ resembles case 1 and w- resembles case 2. Before showing the practical elaboration of two examples in the sub· section "use of Fig. 10" below, a few concepts required in the procedure are mentioned here first. The positive part of w(r) is defined according to: ( 48)

with

and the negative part by: ( 49)

with

In eqns. (48) and (49), u· and u- are parts of two different curves from Fig. 5. Because of the minus signs in eqn. (49), the part u - is rotated over 1800 in the rescaling process. By choosing 0 according to eqns. (48) and (49), it follows that w+(oo) = (3 + , and w -(~ 00) = (3 ~ ,. It remains to organize the solution in such a manner that the two parts w+ and w- match together at the point where w = O. More precisely, it remains to select ro and curves u+(s) and u-(s) from Fig. 5 so that the composite function: w+(r)

fm

r> ro

w(r) =

(50)

r 1, it is shown by Van Duijn (1986a) how to join the solutions on the base line, u = 0 in his case. He points out, that the fitting conditions are to be deduced from additional physical considerations. In the present problem the condition to be satisfied is that the salt flux should be continuous. Using expression (23) this condition can be written as: 2(m

+ w+)dw+ /dr

= 2(m ~ w - )dw -/dr = 1\ w+ =

u; -

= 0

at

for

r = ro

(51)

Elaboration of t his condition is rather involved, since the relation between 1\ and ro cannot be determined directly. In order, however, to provide a first approximation of the solution, Fig. 10 is added here, in which 1\ and ro values are assembled for a relevant region of m, (3 values. Selected Works of G. de Josselin de Jong

287

71

A 11\.

·······r······T"· ·~;;- ··-··· ~········

.\'" .', ..... ' ........ ~..... ,! ....... . i" --_._ -,

- ';,y.~-.,

...• •.•:

.~

_-+_..... ,

~ ... ___.L 4

,\~ - ~

~.

i .... ~ -

• 0/", :

:

---i -

..•..... , ....... .

:

i --!--T--'--------- -.------:-- -j ----,-------,

.. ... .

-·3.·~' -

Fig. 10. Values for

'0

o

-· ·

......,

'.

and A for different m, {3 combinations.

Use of Figure 10

Example 3-;: m = -i and (J = i The similarity solution in r, w coordinates is the dashed line in Fig . 11. This curve is obtained as follows. From Fig. 10 the appropriate values for this example are found to be A = 0.292, ro = - 0.699. The value of ro means that the dashed curve in Fig. 11 intersects the axis w = 0 in the point r = ro = - 0.699. The vaJue of A indicates with eqn. (51) that the inclination of the curve is :

dw ldr = A/2m = 0,585

w = 0,

in

ro = - 0.699

This information is sufficient to construct the two parts of the curve with the finite difference scheme of Appendix B, by extending them from the starting point r = ro, W = 0 in both directions up to infinity. It is also possible to obtain the curve by transforming two curves from Fig. 5. From eqns. (48) and (49) it follows that the scale factors are u+ = 1 for the positive part and u- = - (, )1 12 = - 0 .707 for the negative part, respectively . Again from eqns. (48) and (49) it can be deduced that the starting points of the corresponding curves in Fig. 5 are:

!

and

8~ = ro /u + = -

=!

and

So

u+ =

0.699

and:

u

= ro / u - =

+ 0 .989

The points are indicated by two triangles in Fig. 5. It is readily verified, that the upper part of t he dashed curve in Fig. 11 is identical to the unde· formed curve starting in the left triangle in Fig. 5, undeformed because a+ = 1. The other one is rescaled by a- = - 0.707.

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Soil Mechanics and Transport in Porous Media

72

i

o

................... , _·l' Fig. 11. Similarity solut ions for t he example g-i, case m = 1/4, (j = 1/ 4 (dashed line) and example 3 -i;, case m = 0, /3 = 1/4 (full line ).

Example 3-ii: m = 0 and (3 = ! The solution in r, w coordinates is t he fu ll line in Fig. 11. This curve is obtained as follows. From Fig . 10 the appropriate values for this example are fo und to be 1\ = 0.183 and TO = - 0.503. The value of TO indicates that the full line in F ig. I I intersects the w = 0 axis in the point r = TO = - 0 .503. T he value of J\ being nonzero, i t follows from eqn . (51) in this case where m = 0, that dw+ Idr and dw -{dr are infinite. The two parts of the curve have a vertical tangent in the intersection point with the axis w = O. T hey can be constructed , however, by starting on either side of the intersection point with t he series expansion of Appendix C because 2w·dw·/d r = - 2w - (r) dw -/dr = i\ is finite and known in this case. Using the finite difference scheme of Appendix B the curve can be extended towards r -.. ± 00. It is also possible to obtain the curve by transforming two appropriate curves of Fig. 5. From eqns. (48) and (49) it fo llows that the starting points of t he curves are:

u· = 0

and

56

= r%· = - 0.503/ (3{4)"1 = - 0.581

and

So

= r%

and:

u

= 0

- = - 0 .503/- (1 /4)112 =

+ 1.006

The corresponding curves are indicated by dashed lines in Fig. 5. It may be remarked, that u· intersects the axis s = 0 in u· = 0.496 . Rescaled wit h eqn. (48) this point becomes r = O,wo = 0.372 . This value and ro = - 0.503 mentioned above are reencountered in Fig. 4, which was dicussed in the su bsection "Application of Similarity Solution".

ANISOTROPY

From the practical standpoint the case of anisotropy is of importance. The results of t he analysis presented here are still valid in that case. Only the

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73

coefficients K for the intrinsic permeability and C/(T for the transverse dispersion, have to be adjusted. It was shown (De Josselin de Jong, 1981) t hat for the case of a soil with a horizontal permeability Kh deviating from Kv the vertical permeability. the relation (3) for the shear flow, q, remains valid. Only K has to be replaced by"., the intrinsic permeability in the direction of the interface, such that (1 /".) = (cos 2a / Kh) + (sin1O:/Kv )' With respect to the transverse dispersion it may be remarked here, that the custom to attribute aT a value of one tenth of the longitudinal dispersion length aLI which is common in these days, is not justified in general. It is certainly an overestimation in the case of anisotropy created by lenses of more permeable material, that are elongated in horizontal direction. The longitudinal dispersion length in the direction of the lenses may be of the order of the lens lengths and/or their mutual distance. The longitudinal dispersion length perpendicular to the lenses, may be of the size of the lens thickness and/or spacing. But the transverse dispersion may be much smaller because that effect is due to the possibility for the groundwater to exchange .salt with fluid elements in neighbouring .streamline.s. Since the elaboration of streamline patterns and the ensuing exchange possibilities is rather involved, this point is not pursued in detail here. For practical use it may suffice to mention, though, that it is more realistic to envisage a value much smaller than one tenth for the ratio between CiT and Ci L . RECAPITULATION OF THE RESULTS

In the preceding sections, the mixing process of fresh· and salt ground. water due to molecular diffusion and transverse dispersion was discussed. This was done for plane flow under several different conditions. In all cases, the initial distribution (at time t = 0) of the specific weight is that the fresh and salt fluids have constant specific weights, 'YI and 'Y2 respectively, and are separated by a sharp interface. The difference is in the specific discharges qt of the two unmixed fluids. These are considered to have a constant value in each of the two regions above and below the interface at t = 0, and to keep that same magnitude at infinity both above and below, == ± co, for all later times t ;;;' O. For t > 0 the fluids become more or less mixed and the specific weight 'Y becomes a function of the local height and the time t. The specific discharge qt changes accordingly because it depends linearly on 'Y, see eqn. (11). Because of the plane fl ow and other simplifying assumptions, the system of partial differential equations, that describes the spreading process can be reduced to the single differential equation (20). This is achieved by introducing the similarity variables wand r. From these, w is related to the specific discharge qt by eqn. (19) and to t he specific weight 'Y by using eqn. (11 ). The variable r is related to the height and the time t by eqn. (17). The governing equation (20) has as relevant solutions the family of curves

r

r

r

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Soil Mechanics and Transport in Porous Media

74

shown in Fig. 5. By rescaling, displacing and combining in various manners t he appropriately chosen curves of this family, it is possible to construct t he solution for various values o f molecular diffusion and flow conditions at infinity. Molecular diffusion in comparison to dispersion is described by t he pnrrunet.cr m, see ..qn. (115). the flo w conditio ns o riginally and at infinity by

the parameter (3, see eqos. (4) and (7). The differential equation (20) is a nonlinear diffusion equation with (m + Iwl) as diffusion coefficient. The absolute value of w in this coefficient is unusual and requires a special treatment when w changes sign in the integration interval. By eqn. (1 9), this o ccurs when the fluids flow in

opposite directions. In this paper three cases are considered that differ in the way the two unmixed fluids flow. In the cases 1 and 2 both fluids flow initially and at infinity in the same direction . In case I, the choice fJ ~ i guarantees that both fluids flow upwards or only the freshwater flows u pwards and the salt water is stagnant (qt ~ 0). In case 2 «(3 '" - !). both flow downwards or only t he salt water flows downwards and the freshwater is stagnant (q ~ '" 0). In case 3 (- i
ACKNOWLEDGEMENT

The authors are indebted to J. van Kan for a number of clarifying discussions about t he numerical approach .

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75

APPENDIX A In th is appendix, some elementary p rOJHl rties of sol ution. of problem Q are discussed. For convenience, the notation ,j = dulds is being used here.

Uniquene,. Consider eqn. (24). Suppose that at some point So the values:

,.d

(AI)

prescribed, where C , an d C 2 are given const.anta. Taking C, > 0, it was shown by Atkinson and Peletie. (1971) that for s ny -oo< C 2 there exists a unique soiution of eqn. (24), which satisfies the conditio ... (AI) a nd which exists on the largest possible interval, where it is positive. A consequence of the uniqueness is m o no t onicity of solutions of Q. 1m!

< .... ,

Monotonicity Let II be a solution of problem Q and sup pose that there exisl.$ II point So where: u ('o)

=

C,

with

O
,,'{,o) = 0

and

(A2)

<

Now observe, that the eonst ant function u(.) = C, for - 00> 5 < "" also utisfies eqns. (24) and (A2). Then the uniqueness requires, that the solutions U and must be identical, which implle& that u(.) = C I for aU - "'" < 5 < ""'. Thill contradicts t he bou ndary condition (25). Therefore the only solu tion of Q, which is not str ictly increasing i, the constant U = 1. Al l other sol utions with 0 < u < 1 must $lItisfy u ' 0 in order to satisfy the bou n dary condition (25).

u

>

ln lerleetion Next it i. thown , tbat two solutions of Q cannot intersect. Let UI and Ul be two solu tions of Q and su p pose, t hat there ex ists an intersection point '0, where ud'o) = U1(50). By t he uniqueness, u~{'o)'I":II;('O), because otherwise III and III would be identi ca l. Without loss of generality it is assumed here, that ..,'1('0) > uj(.o). Then two situationacan arise: (1 ) There exists an other intersection point 510 such that:

,.,

,.d (2) UI('»

111(') for ali I > '0 and both solutiona sat isfy eqn. (25).

Ad {lJ

Integnlion of eqn. (24) with respec t to. from '0 to '1 gives for II I and ul, respectively: Uj('j)u'I{'I) - III{'O)U'I{'O)+!'IIII(Bll -!aOIlI('O)-!

J-,

and :

'0

U1('I)U1('I) - 1I1('0)U;('0) + hIUl(SI) - houl('o)

Subtractinll" thlllie equation. lI"ives:

'I

UI('I)[U'I - U1](.d - udso)[u;-uil(so ) - !

J

-,

- !

ul(.)d, = 0

J-,

ul(.jd8

(M)

'0

[ul(') - 1I1(')]& = 0

However, by the above aS6umptionl 1 1I'I - U~]{.d O. This contradicta the equal sign in eqn.

-,

0

(A3)

(A5)

11I', - u;](.o»O and (A5). Therefore case (1)

cannot arise.

292

Soil Mechanics and Transport in Porous Media

76 Ad (2)

A similar argument gives for this ease also a contradiction. Thus both (1) and (2) cannot arise and thus no interseclion point exists.

Asymptolic behaviour Next an argument due to Peletier (1970) is used to obtain eqns. (27) and (29). The starting point is the following observat io n. At points where u > 0, eqn. (24) can be written as: (A6)

which can be integrated to give: (ul)'(s)

:=:

2 (u )'(solexp

f.,•

{- 1

(A7)

[z/u(z» ) dz}

>

Here 80 is an arbitrary chosen point, where u{s ol O. SinC(! 1.1(.)< 1, it follows from eqn. (A7), that: (u2j' (8)";(u2)' ('O)eIp( _ lt.2-8~)J

for

8 > '0

(AB)

Using 1.1'(.) > 0 and thus I/(s) > u{sol for 8 > 'a in eqn. (AS) gives: O
for

' ;;'''0

(A9 )

Integration of eqn. (A9) :ith respect to 8 from II po in t , ;;" '0 to "" gives:

1 - u(i) < 1.1'(80) e"p

(!a~)

where erfc (x) "'" (2 {V1i)

f

exp (-

j"" i exp (•

!a2 )ds Z2

= u'(so) exp (!.hJtrerfc (!i)

(AI0)

)d: .

From eqn. (AlO), condilion (27) follows when i" .... ""'. A similar argument gives eqn. > O.

(29), wheMver .p

APPENDIX B A finite difference melhod to obtain the solutions of problem Q as shown in Fig. 5, is discussed here. The lIOlulions of problem Q were distinguished in type-! curves, lype-II curves and a &eparatrix. In all three cases, first a value of u and du{ds was chosen at a point where u is positive. For curves of type I this was done at s = 0 and the appropriate values for u(O) and du{ds(O) were found by a try and error method. For curves of type II and for the separatrix a series expansion was used to approximate the value of u and du{d8 at a point 8 L which was chosen sufficiently close to t he point 80 where the solution vanishes, see Appendix C. Thus for t he finite-difference approximation one has to solve an in itial value problem of the form:

(u 2 )"+su' = 0

(Bl )

uri)

Ii

(B2)

= U'

(B3)

U'(6)

:=

where eqn. (81) is a rewritten version of eqn. (24) and where ;, Ii and Ii' are three given numbers such that u > 0 and U' O.

>

Selected Works of G. de Josselin de Jong

293

77

Let ~I

=; + i&, wh(!re &

from,/ to

gives:

'i.)

(U 2 )'{'I+d-{u 2 )'{.;l+

.

denotes the discretization interval. Inte gration of eqn. (81 )

' j+ I

J

su'(s)ds

(84)

0

=<

,

Integra ting the third term in eqn. ( B4) by parh and 6f!tting u = p gives: 2Uj+IP/+l

=

'/ + I

2uiP/+'jU/-'I+\UI+I+

,

J

.,

(85)

u(s)ds

where 1.
=

Uj

+

f

(86)

p(.r )ds

., The integral, in eqn •. (B5) and (86) are approximated by a thIFd·degree Hermite polynom ial, see Ralston (1965, p . 60) USing t he notation difference scheme:

It = q

t hIS leads to the finite-

,.'

(S7)

12(Pr-PI + l)

,.

and:

-

2

( B8)

Cpj+Pi+ . )+

This scheme hllll a local truncation error of 0(& 5), see Ralston (1965, p. 212), and it is therefore expected that the global accuracy is of O{&4). The value. of q at 3; and 31+1 are obtained by u sinS- the differential equation ( B l )d • = 'I and 3 = '1+ I. For q l this gives:

pl - 'i Pi -

q; = - III

( B9 )

2UI

Substituting this expreSllion and the corresponding expression for qi+1 into eqn. ( B8) yields: UI+I

=

111 +

,. "'(pl - (Pi+PI+I) - - -+'; 2

12

IIi

Pi P~+I -2111

ul+1

-

PI +I )

8/+1 -2 - u/+ 1

( BI 0)

Thus for given values of '1, 8/ +1> II/ and PI, the two nonlinear equations (B7) and (BI0) have to be solved to obtain ui .. 1 and P i +l - However, because of the nonlinearity this cannot be done directly and the follow ing iteration process was used to obtain approximate values for U/+I and PI+I' Let u1~\ = " i. Substituting this value into e~n_ ( B7) and solving the reSUlting linear equat ion for P I .. I, gives the approximate value PI~{ ' This value in turn is substituted into eqn. (BI0!_ The resu lting quadratic equation in 111+1 can be solved directly to obtain th e value u!l as a next approximat ion for 1.1/+\. Since this process converge!! rapidly only a few of these iteration steps were needed at each value of i = 0 , 1, 2, ...

APPENDIX C In this appendix an approximat io n is considered of curves of type II and the leparatrix from Fig. 5 for small values of u. This approximation has the form of a series expansion

294

Soil Mechanics and Transport in Porous Media

78 which solves the initial value problem defined by the differential equation (24) for 3>'0 and the initial conditions (eqn. 31).

Type-II curves For type-II cUn'es, )., u{s)

=:

> 0 and II seril)$ expansion o f th e form :

[A{s - 'ol ) ll2 + a (S-so)+b(s - sO)311+ c (s - 'O)1

(el)

is chosen as an approximation in 11 sufficiently small neighbourhood of so. Clearly, any a pproximation of the form (el) satisfies eqn. (31). The values of the constants a, band c are obtained by substituti ng eqo. (el) into the differential equation (24). This gives t he follow ing set of equations: from the term with (. - .0 )- IIl:

2o..jX == - 1 'ov'X

hom t he term with (s - solO;

a2

from the term with (. - '0)1 12 :

2ab+ 2cv'X == 1~-fb80 - tv'X

from the term with (. - so):

b 2 +2ac == ~c -~ a

from the term with (. - .0))/2 :

2bc = - ;b

from the term with (8 - '0)1;

c 2 ==

+ 2b..ft

= - ~a.o

- !c

Thus for A > 0, the choice:

a = - ~.o

b = ,~/ 36v'X

c = (s~ / 270~) - (1 / 15)

(C2 )

guarantees, that the approxi ma tion (Cl) satisfies equation (24) up to terms with {s - so)l n . The first point o n the curve (U1' '1) is chosen in such a manner, that ('1 - '0) is a aufficientlr small number for approaching the value of UI by the series expansion. The value of U I , which represents (du /ds) in the fust point, is taken from the derivative of the series ex pansion. The values "l> 51 are the starting values for establishing the rest of the curve upward s and to the right by applyi ng the finite·difference procedure of Appendix B. The parameter to adjust in this case is II from eqn. (31).

U;,

Separalrix For the separat rix, an approximation of the form (C I) is c ho sen with

'0

< O. Then the following two sets of value1l for a, band c occur:

a = 0

~

= 0 and

b = 0

c = 0

(C3)

b = 0

c = - '0/ (38 0 + 1 )

(C4)

and:

a = -

!'O

The set of values (C3) leads to an approximation, whic h is identically zero. This solution corresponds to the part of the separatrix in the region 1/ or;;; '0. In the region s > '0, the set (C4) gives:

ii{s) = - !sO(' - '0) - ' 0(' - '0)2 / (3Io+ 1)

(C5)

and this a pproximation satisfies eqn. (24) up to terms with {' _ '0)112. To obtain n um er ically t he solution, that starts at this value of '0 an identical procedure as for t he case of type·II curves was used. The parameter to adjust in this case is ' 0. It is found, that for.o = - 1.23675 the correspondi ng so lution reaches the value II = + 1 asymptotically, when 1/ -+ + 00.

Selected Works of G. de Josselin de Jong

295

79

REFERENCES Atkirl$On , F.V. and Peletiar, L.A. , 1971. Similarity profiles of now through porous

media. Arch. Ration. Meeh. Anal., 42: 363- 379. Bear, J., 197 5 . Dynamics of F luids in Porous Media. Elsevier, New York, N.Y., 2nd ed. Boltzmann, L., 1894. Zur Integration der Diffusionsgleichung be; variabeln Diffusionscoeffici(!nlen. Ann. Ph ys. Chern., 53 : 959-964 . De Josselin de Jong, G. , 1960. Singularity distribu t ions for the analysis of multiple nuid now through porous media. J . Geopbys. R(ls ., 65 (11): 3739- 3758. De J osselin de Jong, G., 1981. The simultaneous now of fresh and salt water in aquife rs of large horizontal extension determined by shear now and vortex theory. In: A. Verruijt and F.B.J. Barends (Editors), Proc. Euromech. 143, Balkema, Rotterdam, pp.75-82. Edelman, J .H., 1940. Strooming van zoet en zout grondwater. Rapport 1940 inzake de watervoorziening van Amsterdam, Biil. 2, pp. 8-14. Gild ing, B.H., 1980. On a class of similarity solutions of the porous media equation Ill. J. Math. Anal. Appl., 77 : 38 1- 402. Gilding, B,H. and Peletier, L .A., 1977. On a class of similarit y solutions of the porous med ia equation II. J. Math. Anal. Appl. , 57 : 522- 538. Peletier, L.A., 1970. Asymptotic behaviour of temperature profiles o f a dass of nonlinear heat conduction problems. Q. J. Me<:h. Appl. Math., 23 : 441 - 44 7 . Ralston, A., 196 5. A First Course in Numerical Analysis. McGraw·Hill, New York. N. Y. Van Duiin, C. J ., 1986a. On a class of s imilarity so lutions of the porous media equation. ( In prep.) Van Duiin, C.J" 1986b, A mathematical analy sis of density dependent dispersion in fresh-aalt groundwater flow. Proc. 9th Salt Water Intrusion Meeting, Delft. (In prep. ) Van Duiin, C.J . and Peletier, L.A. , 1977 . A class of similarity solutions of the non linear diffus.ion equation. Nonlinear AnaL., Theory, Methods, Appl. , 1: 223 - 2 3 3. Verruijt, A., 1971. Stesdy dis persion across an interface in a porou s medium. J, Hydro!. , 14: 337- 347 , Verruijt, A., 1980. The rotstion of a vertical interface in a porous medium, Water Re$Our . Res., 16: 239- 240.

296

Soil Mechanics and Transport in Porous Media

THE TENSOR CHARACTER OF THE DISPERSION COEFFICIENT IN ANISOTROPIC POROUS MEDIA G . DE JOSSELIN DE l ONG

INTROD UCTION

In their attempt to describe dispersion phenomena in mathematical f.:)rm several authors arrived at the conclusion, tha,t a porous medium possesses a coefficient of dispersion, which has the character of a tensor. This tensor is formulated in such a manner, that it represents the geometrical aspects of the porous mediwn responsible for the scatter of tracer particles, when carried by a fluid flowing through it. It is therefore a property of the porous medium alone, and can be considered to materialize the tortuosity of the particle trails caused by the random arrangement and the interconnectivity of the cnannels constituing the pore space. Nikolaevskii ( 1959) originated the tensor. He constructed it with a dimension of length and predicted the magnitude of this length to be in the order of particle size. He obtained a tensor of the fourth rank for an isotropic porous medium by postulating, that the random modification of the velocity vector, from mean velocity to local velocity, is a tensor of second rank. This postulate requires that the velocity vectors considered possess the property of linear superposition. It will be shown subsequently that this requirement is nOI always fulfilled, and tbat, for instance, Nikolaevskii's result for the isotropic medium cannot be extended to the general case of an anisotropic medium. Nikolaevskii states that the rank of the tensor must be even,l:Iut includes the possibility of any number for that rank . Bear (1961) started with the results of the dispersion computations proposed by the author (De Josselin de Jong, 1958). These computations were executed for an isotropic model material exhibiting a particular manner of scaltering tracer particles. The basic scattering me<:hanism adopted was apparently realistic enough to reproduce mathematically the dispersivity phenomena observed in tests and espe<:ially the difference in longitudinal and transverse dispersion. The specific problem treated was the determination of concentration distribution developing from a point injection of tracer particles jf these were carried away by the fluid flow through the porous medium. The resulting concentration distribution turns out to be almost normal (Gaussian), in all directions, with the points of standard deviation lying on an ellipsoid.

,so

297

260

G. lH Josu/in tk JOIIJ/

This concentration d istribution can be uniquely defined by its mean point and the variances or second central moments with respttt to the three space coordinates. By superposition of point injections and rotation of coordinates, Bear (1961) showed Inat the variances can be considered as the components of a second Tank tensor, and he developed the relation between this tensor and Nikolaevskii's fourth rank tensor. Inspired by the work of Nikolaevskii and Bear, Scheidegger (1961) suggested that the dispersion constant is also a fourth Tank tenso r in the genera l anisotropic case. This assertion was not substantiated by considering the physics of the scattering mechanism underlying the dispersion phenomenon. Therefo re his suggestion is not imperative. It is the purpose of this paper to reestablish the dispersion tensor for the anisotropic medium starting from the basic scattering mechanism. Since the mechanism used by the autho r, (De Jossel in de Jong, 1958) proved rea listic enough for the isotropic case it was considered to be acceptable for the anisotropic case as well. SCATTER INQ MECHANISM

The scattering mechanism adopted previously consists of assigning a choice to every tracer particle when arriving at a junction point in the pore channel system. It was suggested that the probability for a particle to choose a certain direction is proportional to the d ischarge in that di rection considered as a fraction of the total discharge through all junction points and in all directions. Once a particular channel has been chosen in junction A the particle travels a certain distance in a certain direction to the next junction point B. This d istance is described by its space coordinates x, Y. z wilt!. origin in A. The

Fig. I.

298

Soil Mechanics and Transport in Porous Media

TIre Dil perJ;Oll Cot/firiell' ill Anil olropic POrDUS Media

261

travel time required for the particle to travel from A to B is called the residence time , I. The components x, y, Z , I are stochastic variables with a probability distribution determined by the adopted flow mechanism in the channels and in the junction points. In this work use will be made of the mean values of these stochastic variables and their squares. These mean values are indicated by a bar and combinations by a circonfiex.

(J)

j

mean value of x

x2

mean value of x 2

xy "" mean value o f x . !

mean value of square minus square of mean value xy = xy - .t . y.

The summation of the stochastic components x , y, Z, 1 of each individual channe l over th.e many channels covered by a tracer particle form its total travel path. and traveltime. The probability for a particle to arrive at a certain point X , Y, Z , after time To is the combined probability for that particle to travel through a certain co mbinat ion o f channels. The probability distribution P(X, Y, Z , To) in space of the arrival points is the tracer concentration distribution at time To after the mo ment that an amount of particles was injected in the origin of the X, Y, Z coordinate system .

,

n

I I

y

I

---- __ f.

injection point {io'Ji'-------------+--7--------~ X

z Fig. 2. Ellipsoid of standard dev ialion

Selected Works of G. de Josselin de Jong

299

G. De Jone/(" de long

262

By use of Chandrasekhar's analysis of Markoff processes and a special form of saddle point analysis this probability turns out to be given in good approximation by (2)

In this expression

~I

and {J are space coordinates with their origin in the point

Xo = To

~;

Yo = 'To

~;

20 = To:

(see Fig. 2).

This is the point of average displacement of the tracer particles during travel time To. Throughout this paper subscripts can have three values t, 2, 3 corresponding respectively to x, y, z. if tbey are Roman. and if they are Greek they run over 4 values corresponding to x, y, z and t . is the determinant of a matrix f with tbe components hj. These 9 components /IJ are related to the mean squares and square means of the stochastic variables, by the following cllprcssions given here witHout proof.

III

(

(3)

)lj

= -

; (bi/bl" - b,,,bb)il'i. . To b"-.i,,i.

In this formula b". are components of matrix b. which is the inverse of a mattix ex given by

(4)

b- I

=0:=

xx

xy

xy

"

yy

y'

"

y,

"

xl

yl

"

xl , yl ,

",

If

By the inverse of 0:, it is meant that tbe components o f band the expressions

0:

are related by

(S)

The form (2) for the probability distribution is Gaussian in all directions, and the ellipsoid of standard deviation centered in the mean point is given by (6)

Since the distribution is Gaussian it is uniquely defined by the second moments of the concentration around the mean point. These second moments can be combined as the components of a matrix a such that

300

Soil Mechanics and Transport in Porous Media

-

'-

IIf

'" Ct,p(X . Y,Z. To)d{,d{,d{,

III

p(X, Y,Z, To)d{ ,d{,dC•.

By laking II\e partial d.ri~al ive of(2) with ""pectlo i", i1 is possible 10 establi,h, thaI the malrix a is (he inverse of the mal so thaI 'hei, ~ompon.n!S are rdated by

,i. /.

(8) By usc of (J) and (4)

.labor~lion

of a,.. lives the folio win, e"p",ssion

T6 ' _

".. - P

A

".,1" + "',.x,l ]. _ .

[IIX ..... - X.lii -

_

,

By introduction of I I) this can be reducWlo

~Hi'.i.x. - .yi,l- 4 .i.' + >:-;:<,1'].

a .. -

(9)

,h~

'h. u
h""" ~1

c_"",,,, n L' 01 ••• .,.,,,," TOnk

(l~61)

U AR

thu ,I,. rel",\Qn h«v • • n

'h o . ", i. ne.. OM . I,e <""ro""." ' ~!hc or ch. hy.!..~U< l<.di.nt •• D pq

L

pq..

(~.!L I l ~tI.) "

S;nc. th o . . . . . . . i ••

,

".

bet~«n

b • .., . . i • • • • cond

n" ~

t on'''"

, I.. .q
di . !><'rI;.i ,y. e ~n " .

i , • >._ h • • ~" . , to t.h. di . uo<'

t""

" ... " •• by

of I:TOvi" 01 th. inje«.d ."ti
I.

(h • • • i

l,yb "~ Ue

. o""ok . ... tho dc<",nfo""o of the . . done •• on tl..

••• di . nt ve
tho

~, .Jl. nt

, b.

,"nt" (,"" ' .. 0

. i. ple,

< _nc. to ' rp." of ,h. o,d«

bee,u "

.,odocto

of

( 2.02) . wh c
t o inl; nit y. ' h"e •• 04u,... I"r ~

" n'"''

. f even

bot iM lni,. " n ~.

s;"oo 'he ".""on' 01 di ' l"' .. ivi(y •• , . . ... .

!n

tc .....

tI,.

to",."

of

v, d."" • • ' pq ,1,i .1o OTO U,. <=1""'''.'' of •

,,"' ... ~i ",", · •

.:,.""tor ' i ' ,
01 • "'n'~' of i ;Ii.i • • " n~ i" tho ' .'a.".pi.
In .. do' '0 "dry . h. dcpcn' one. of ' pq ." . h e ' ,ydnuli.

,.adi.".

v"",o.

it

i.

""c.. . .. ,.

necc .... ry IOda!>orOlelhe mean valu .. of lh~ !lona!~ni.<>trop;" c~ .. ,hi. wi!! be done in lhe neXl s.<:elin" with i.),. i and Xi. providing lhe JIO"ibility 10 e<>n.idcr lhe form ofa".

,

...'.

..... -. ...- ...... ~~-----------.

\

,-

FiJ. l .

Selected Works of G. de Josselin de Jong

301

264

G. De JO$stlill tU JOIII ELABORATION OF' MEAN VALUES

Let n(O)dO be the amount of channels baving an angle between 0 and 8 + dO

with the positive x.direction, such that the total amount of channels considered : N - I~::::n(O)dO is a large number. The integral only covers half the circle, because every channel has two opposite directions. Let 1.(0) be the conductivity of the channels n(O)dO. Anisotropy is obtained, if n(0)A(8) varies with 6. Let 1(0) be the length of the channels n(O)dO. We will take for 1(0) a constant length I here, because it simplifies the computations without effecting the result essentia lly. Let c(O) be the cross sectional area of the channels n(8)dO. In the fluid a gradient of head, VQ), is assumed to exist which is everywhere equal and makes an angle a with the positive x-direction. This means that the cos (q- 0:). The discharge gradient in each of the channels is equal to of such a channel is tben

IV4>I ·

(10)

0(8) -

'(0)1 .'"1·0.,(0 -

0).

According to the assumption, that the probability for a particle to choose a certain direction is proportional to the amount of fluid flowing in that direction, that probability is for a direction between 0 and 0 + dO equal to (11)

In this expression Q is the lotal amount of Huid passing through all the N channels. Using (10) this becomes (12)

Q _

+ t / h +C

f

n(O)q(O) ' dO -

_1 /1.+c

I V4> 1

f.

t / l.+c

n (O)A(O)CO'(O - 1X)dO.

-1 / 1.+.

Integration is performed over the range - -ill + t:r: < 0 < + -ill + 11, because only channels in those directions carry waler away from the junction points. That only discharge departing from the junction points is counted, is a require-ment for the probability calculus. Thcchoice is made forward in time and not backv..ard. The x-component of one step in the Markoff process is the projection of the channel in x-direction. For each of the n(O)dO channels, trus is IcosO. The mean value x of these x-components is this value leosO multiplied by its probability o f occurrence p(0 _0 + dO) integrated over all possible directions. This gives with (10), ( 11), (12) I / h+.

(13)

302

f x-

n(O)l cos Oq(O)dO

-1/1.+·

Q

If n(O),J.(O)cosOcos(O - ll)dO

-

f

n(O)..1.(O)cos (O - ll)dO

Soil Mechanics and Transport in Porous Media

11u! Dispersion eM/fic/tnt in An/sotropic Porous Mtdia

For the y-component, which is /slnO, this gives _

( 14)

f

IJl •• ~

-1 / 2.+0

y~

f f

n(O)IsinOq(O)dO

'"

/ n(O)).(O)sinOcos(O-Cl)dO -",--------

Q

n(O)J.(O)cos(O - Cl)dO

The residence time I of one step in the Markoff process, is the time a particle stays in one channel. If the discha rge is q(O) and the volume of the channel is / . c(O) then this residence time is I = (I . c(O)/q(O». The mean value i of the residence time is this value multiplied by its probability of occurre nce, P(O ..... 0 + dO), integrated over all possible directions. This gives with (10) ( II ), ( 12),

, f==--111.+.

- 1/ 2.+0 n(O)/·

_

(15)

I

c(8)dO

~

~

-

n(0)c(8)dO

- ' ; 0 - --

I V¢l1

Q

f f

n(O)J.(O)cos(O - Cl)dO.

In the same manner we find

= _ 1'-,f'-c"(:.-o~),,:-(o,:,),",,o_'_'_O'~O_'.:.(O_-~'~)d_O n(0)1(0) cos (0 - Cl)dO The general case of anisotropy is obtained by assigning to the n(O)d{O channels an arbitrary distribution with respect to 0 of the combined conductivities n(O»).{O)dO and the combined cross sectional areas n(O)c(O)dO. Every possible distribution can be expressed as a Fourier series. Let these be Xl

(16)

J

{

(17)

n~) ),(0) = 1:

._0Ahcos 2n(0 -

"N(O)c(O) =

..1:,

Cl2.)

Blocos2n(O-{Jl.)'

Unly the even terms appear, because every channel figures in two oppposite directions. Execution of the integrals in (13), (14), (15) and (16), all between the limits - til: + a and til: + a gives finally . I i = R

(18)

Y=

~

-

I

;

[A ocosa

+ tAl COS(2Cl 2 -

; [A osinCl

+ tA 2 sin(2a 1 -

II:

Cl)] a)]

B

'~RI·" I' with R =

f

1110+.

nCO ) - - J.(O)cos(O - a)dO.

-1 120+0

Selected Works of G. de Josselin de Jong

N

303

G. De Josulill de long

'66 The real mean velocity v, = f ll. With (l8) this is

Vx

= Xjl and

V"

=

~

=

[2~o COS 11 + 4io COS(2IX~ -IX)] -I V 1

v,

=~

=

[2~:SinO:+4~2(1Sin(20:2-a)]· IV41I.

(19) {

has x- and y-components given by

II

Since the gradient of ¢> has the direction IX, the partial derivatives of !II can be V¢lI' CQsa, o sin 11. Then (19) becomes

introduced as 8<1>jox = V"

(2~:

=

(20) {

I

I I

+

4io cos2a

1 ) ::

+

4~lO sin2a1 : ;

AI' 0<1> (Ao A2 )0<1) v, = 480 sm2O: 1 ax + 2B o '" 480 COS21l1 iii"

These expressions form the relation between the real mean velocity vector, v and the gradient vector, V¢'. Because the vector components are obtained by linear superposition of the four coefficients at the right of (20), these coefficients form the components of a second rank tensor. This tensor is the anisotropic permeability tensor divided by the porosity, because specific discharge is equal to real mean velocity multiplied by the porosity. This result is in agreement with the concept developed by Ferrandon (1948). In the permeability tensor expressed by (20) only A o , Bo and A2 appear, showing that the other terms of the infinite series (17) are of no importance to the permeability. In contrast to this simple result the dispersion phenomenon cannot be described without using all the terms in the infinite series. This will be seen by elaborating ap.f for the case that the conductivity distribution is expressed in an infinite series as given by (17). Introduction of v... and v, in (9) gives for the coefficient all ::: au, corresponding to x, = Xq = x, the form (21)

Consider as an example the last term between brackets. Elaboration of the integral (16) with (17) gives the value of Xl. Because of Tol f before brackets, this has to be multiplied by (RTo I V<J) l/nIB o), to give finally (22)

TOx2 "" f

Toll V<J)I' nBo

. I(-l)"A "_0

304

[(4nl+3)cos201+(8nJ - lOn)sin201-4n2+9] 2"

(4n2

9)(4n2

1)

Soil Mechanics and Transport in Porous Media

267

Couidero:l as a polynomc in (coSet) and (sina), tbis tron of G. . contain. (cou)JaU and (aina)1oo+1. The contribution of the other two !ermA in (21) dOCI not rcdo.cc tbe order of the polynome. ThiI means, thattcao be conwcrtd as the eompooents of an even tensor of rank (2Jt + 2). S' n caD run up to Wioity it is &n even tensor of infinite

t~ ~~weto~

nnk.

,tet""'~/lo+OU ~ t1te-

"J "",.,

CONCLU!JON

D)o.

ik. "!j..u-tJ(:. y~ tn'I

~ ~MA.tJic. .fjnvli....t

SiDce tholvarlancc of dispersion for the general anisotropie cascJs a tensor of infinite rank the dispersion constant is also a. tensor of infinite rank. As a coDlCIQucncc of this fact the differential equation for dispcnion developed for the isotropic case can only be 8cncraliscd to the anisotropic case by mtra-

ducina a diJpersioD coefficient, whose dependence on the din:ctiOD. of ftow is expressed by a SCM of infinite terms.

~CES

t. 8&Aa. J. (15161). On the tensor rorm or dispcnioo in porous media, I . c.oplt. k:r., M.1I85- 1197. 1. 01 JoawN D! JeNa, O . (1958). LonaitudlnaJ and lnIlSvcne diffuskm I'll aranulat dopoIiu. TNuu. Anwr. Gtop/!. UnJolI, 39, 61':'74. 3. Nw>t.Uvsin. V. N . (1959). Convcdivc dilfusion in porous media. Prlkl. Mat. Mddt .• fl, J042- J05O. 3. ScH:rm.ooa. A. B. (1961). Ge:ners.l theory of dispersion in poroUI media, J. Chi)p/I. lW., _, 3273-3218. 5. FD.a.umoH. 1. (1948). Les loil ck 1'6eouJemtot de filtralion, u Gi,w Clril, 124,

>4-21.

ENolNualNO D.,......11oCJiNt'. DaLn U~ Of 1'lIcHNowo¥,

Qvn.

00rrJ'1.AHT1OI!W

25, DILl'J', N8'J'HEIll...oUo

Selected Works of G. de Josselin de Jong

305

DISPERSION IN FISSURED ROCK

by

G de Josse1in de Jong and Shao-Chih Way

Geoscience Department New Mexico Institute of Mining and. Technology, Socorro, New Mexico 87801 July 1972

306

1

Introduction It is the purpose of this study to show that the dispersion of particles, transported by a fluid flowing through a system of fissures in rock, can be treated as a special case of the general theory for dispersion, developed by use of the probability theory (DE JOSSELIN DE JONG, 1969). A case study of the dispersion of a large amount of locally injected particles will first be presented computation wise. It will be computed how the particles are partitioned at each intersection of the fissures, how the subgroups of particles are transported through the fissures and uhere the subgroups \'1ill be located at successive time intervals. The result consists of a distribution of particles at a certain time, jr , after injection, with discrete

amoun~of

particles at discrete

points. In the second place the general theory developed on the basis of probability concepts will be applied to the same case. This theory gives as a result a continuous distribution of particles, which is essentially Gaussian. In order to show that the discrete distribution obtained from the computation and the continuous distribution obtained from the probability have the same general form, a figure a (Pig.4 ) Hill be used. i'l'umerically, the blO results ~lill be compared by considering the centre of gravity of the diapersed particles and the second moments of the distribution around that centre. The case study is simplified by considering a system of two intersecting families of fissures. Let the fissure planes be parallel to Z (Pig.l), and let the conditions in all planes perpendicular to Z be identical. Thus the problem to be treated is two-dimensional and only the

Selected Works of G. de Josselin de Jong

307

2

x,y coordinates are essential. Fluid flow through the fissures proceeds parallel to the xy plane. Discharges are per unit deuth in the Z-direction. Sections l, 2, 3, 4 deal \'11 th the computational experiment. Sections 5, 6, 7, 8 deal with the theoretical prediction. The two resulting particle distributions will be compared in Section 9.

308

Soil Mechanics and Transport in Porous Media

1. Particle Behaviour in Fissures and their Intersections The behaviour of particles in passing through the individual fissures and their intersections is partly deterministic, partly random. Let us first consider, v:ha"t happens \d thin a fissure In the follo\"1in0 the di::;charges through the fissures are divided in lamellae. Each lamella carries the same amount of discharge per unit depth. Since the fluid velocity in a fissure cross section is not uniform, the velocity in the different lamellae is not the same.

'I' '

It is assumed here, that the length, of the individual fissure segments between intersections/and the average fluid velocity, \f< ' through them are kno\m. The index, ~ , refers to the particular family of fissures (chct:r.","ct~rized b:,' their direction,

length, CQ"ductivi ty , ... etc~. In the case considered here, there are two different

kinds of fissures, so f-l has the value I , Jl. , sec Fig .1. Furthermore it is assumed that thE: Hidth, Ct'- , of the fissure is small Hith respect to its length, it'-, and molecular diffusion, Pmof , large enough for the quantity .t

"

"

/0",,, Jt' to be small w~ th respect to one. Then Brmvoian motion will force a particle to visit all fluid stream lamellae within a fissure, such that the overall particle velocity corresponds to the average fluid velocity. The time, t f , that a particle will reside in a fissure segment is then equal to Cf< iff"'

<1.1)

This influence of BroHnian motion on the particle behaviour creates a deterministic effect, because the residence time,

tf-< ' has a known value for every particle travelling through the fissure.

Selected Works of G. de Josselin de Jong

309

4

Brownian motion, however, also has a second probabilistic influence on the particle behaviour.

Because a particle

is scattered over all stream lamellae it has an equal probability to be found at any point of the cross section at the exit of a fissure, irrespective of its location in the cross section at the entrance.

The choice of stream

lamellae that carries the particle at the exit is probabilistic and not correlated to the entrance stream lamellae. Let us no\~ consider, \'lhat happens at an intersection. Redistribution of particles at an intersection of fissures is again deterministic, if the fluid and the particles behave in the manner proposed by SNOH (19? ). He assumes, that there is no turbulence and that the time a particle spends lVithin an intersection is too small to permit a particle to switch stream lamellae • . According to SNOH the redistribution of stream lar.-.cllae at an intersection can be determinnd from the discharges in different directions.

Since the particles stay lVith

the stream lamellae and they are equally distributed over the stream lamellae by BrOlmian motion in their preceding travel through the fissure, the redistribution of particles at an intersection is the same as the redistribution of the fluid. And since the fluid distribution is determinable, also the particle distribution is deterministic.

310

Soil Mechanics and Transport in Porous Media

5

2. Regular. Array of Fissures The example considered is a system of cracks, consisting of tHO intersecting families of fissures indicated by subcripts I

and JI

(see Fig.1).

The angles of the fissures ,vi th respect to the x-axis are respectively:

().r.

intersections are:

and

.91t

JI and

•

Their lengths betvleen

lIt.

The projections of the

fissure segments on the x- and y-coordinate axes are

(2.1) The case considered here consists of a system of parallel and equidistant

fissures, such that for each family the

fissure length and direction angle are constant throughout the field. A unif.orm hydranllc gradient

J" at an angle

01..

with

respect to the x-axis is acting throughout the f1m'l field (Pig.l).

Because of the regularity of the crack system,

the gradients in the tHO families of (2.2)

:;; = J (OS (0<- t9I

)

cracl~s

are respectively

and

If the hydraulic conductivity per unit depth of the fissures is .J..z and Ax, the discharges depth of the fissures are

Cfz and CfIt per unit

(2.3) If the vlidth of the b/o kinds of fissures are respectively Cor and C][. , then the average velocities Hill be

(2.4)

"i=

'II CI

and

Selected Works of G. de Josselin de Jong

311

6

The corresponding residence times are using (1.1) (2.5) computations will be performed for the special case, where (see Fig.3)

(2.6)

f)x-

30"

~

=30"

Furthermore all fissures are equal, such that ( 2. 7 )

Jl. =11[

=.f. ;

C'x =

c:v = c

then formula (2.3) becomes

(2.8)

9:z

= AT ("S(OO) =A...J ;

~

= AT Cos (6,,':1

=

:/;..>t:r

and formula (2.5) becoces

(2.9)

312

tI _

.Rc A.."j

Soil Mechanics and Transport in Porous Media

7

3. Continuation Probability In the case considered, the ratio of 2 to I, see (2.D).

9z

to

rJl

is

In Fig.2, the discharges are represented

in a ratio 5 to 2, in order to avoid arabigui ty resulting from the equal subdivisions, ~:~. The discharges are represented by stream lamellae: a, b, c, d, e, f, g, each of \
, that a particle carried through

a fissure of direction

will continue after the intersection

P~'1

~

R in direction Sf. This probability \<1111 be called the continuation probability. The lamellae a, b, carrying the discharge tt-p; through the fissure 1'.C in

tlJ!-direction, continue their path beyond

C in the 0.I-direction. At C, therefore, certainty exists, that a particle arriving along OJ! \<1111 continue along {)z, and not along OIL. 11.s a consequence the continuation probabilities

(3.l)

f

f

and f~lI~JL have the values

c.

1L~Z

t;

]I

-+.r

-

C

fl£~J! =

1

0

The five lamellae c, d, e, i, g, carrying discharge

tJi:

through fissure BC are divided into t\·l0 groups. The lamellae c, d, e, continue along ~z, the lamellae f, g, continue along

/}]!.

As can be seen from the figure this

sU!Jdivision is such that

1+1

=.

rJ[

Particles entering fissure BC at B are evenly distributed over the lamellae by BrO\mian motion when arriving in C. This is independent of their entrance position at B. Therefore, the continuation probability,

Selected Works of G. de Josselin de Jong

313

8

f;~I' for a particle started at B along (J.r to continue G in the tlI-dircction is proportional to c+d+e, and fI~,£ to continue in the B~-direction is proportional to f+g. Since the particles started at B in the 6lz -direction arc certain to continue in some direction, it is necessary, that

combining these results, the following values for the continuation probabilities are obtained (3.2)

In this case where all intersections have the same characterestics, the letter C in the expressions for the

....

continuation probabilities ~:z' etc can be replaced by the letter indicating any other intersection.

=

314

=0

Soil Mechanics and Transport in Porous Media

9

4. Computntional Experiment The computational experiment consists in pursuing the travel paths of a large amount of particles injected along a line inZ-direction. This line intersects the xy plane in a point P, which will be called the injection point. As shown in Section 3, the a~ount of particles passing an intersection is subdivided into subgroups that continue their path in different directions. The probability to continue in a certain direction was expressed as a ratio given by the for~ulae (3.1) and (3.2). Using for the specific discharges '7I and '7;r the values (2.8),

the continuation probabilities become f

(4.1)

C

7I~'I

fJI::-

-- J. .z..

~:1!

=

1][:1£

1

I

L _

0

TiJe "rAtios of 5/tbdirih"C\"v of fI,e iflJtdtd. <{~tJhld- a,~ ,~ T0t11;t-ld M.., fU,3 wl.P-Iethe location of the subgroups is shown after time intervals J

71:~

in {.(" . ."t• .!

of magnitude !Jt, 2.Llt, }Ld, 4Llt, SAt_AAt is equal to the residence time in fissures I , such that according to (2.9)

(4.2)

td

=

t:r

= f t AJ

Since according to (2.9) the residence time in fissures is t,"lice t;r., subdivision of the particle groups only occurs at time intervals, which are integer multiples of

1l

td After continuing the procedure of subdivision, (shown for the first 5 timesteps in Fig.3) ov~r a time interval

Selected Works of G. de Josselin de Jong

315

10

T ,

consisting of 40 timesteps, such that

T

(4.3)

4-0

At =

pc

£f0 .AJ

the result of Pig.4 is obtained.

The particle groups arc

indica1:ed by solid circles, '''hose areas correspond approximately to the sizeS of the subgroup::;. The numbers accompanying the circles are the ratios of subdivision, from total amount of particles to subgroups. The circles do not represent a spatial distribution, because all particles are considered to be located at the centre of each circle. All centres 0 are located on the dash dot line at an angle t~= 60 , see Fig.4. a In Fig.4 the distribution of particles along the line of spread is shmm by the stepHise solid line, \-Thich represents the cumulation curve. Numerically the results of this experiment are expressed _by the coordinates of the centre of gravity of the particles and the second moment around that centre.

The origin of

the coordinates )( , ~ used for the centre of gravity is located in the injection point, P, see pig.4. Using a supusui/>t- eis found that after the coordinates (4.4)

x

JZ

T =

for the experimental values, it 40 1.~ the centre of gravity has

= zs; 93 Y.

;

~

y =

4.91 .R.

The second moment is taken in the direction of the line of spread at 60", the result ,-,ill be called al~. 0

The spread perpendicular to the line at 60 H'ill be called ~ Since there is no spread in that direction, the

Ok

second moment, a~~ is zero.

(4.5)

316

af~ = 3.90 1.2.

So ''Ie find

...

a... ~ =

0

Soil Mechanics and Transport in Porous Media

11 5. General Theory of Dispersion

baSed.

The general theory of dispersion

0)1,

U,Stl'l$ CI-lAI!P;2ASF1~HARS [j('w;,rA~'(5p.Al~~<

DE

JO~G,

r1~\?()J.,Ll<.L 1V

'(]

-Ift.CI,1"v

01 MAi?i
after being trans90rted during a time interval

is

spread in-co a spa ti 0.1 Gaussi'i,1.- d{str~ bl"t~lIlv wi~l t/,e pcn'"fJ corresponding to the standard deviation in any direction loca-ted on an ellipsoid.

If the flOl·l system is

t,~o-dimensional~

points of standard deviation arc located on an ellipse. We present here the two dimensional case only, because extension to 3 dimensions is obtained by adding terms of a similar character as those presented here. The centre of the ellipse has coordinates )( , ~ , (\-lith origin in the injection point) given by

X = T

(5.1 )

<;<>

The quanti tics displace:nent

'tT = :J



{X~,

coordir~ates



T



are the averages of the

Xm ,

Y",

corresponding. to the

passage through elementary conveyor units (defined in Section G) and (t> is the average of the residence time

tm

spent in an elementary conveyor unit.

X,,,,

y", , t

h•

The quantities

are the stochastic variables, subject to a

probabilistic choice, because the elementary conveyor units have different magnitude and orientation and particles ChOOS0 to enter them according to probabilistic rules. The size, shape and orientation of the ellipsoid can be computed from the variances or second moments a"", a,,:! a~~ , ~lhose magnitude according to the general theory is given by

(S.2)

'

:~~ + ~:;:] T [(X,> _ <xt> <~> _ <'Jt7 <X) + at> <.K>(~>]



T [( l1~>



_ 2. (~t>

<0

5.'i!. -r

Selected Works of G. de Josselin de Jong

<0

:]

(t>

317

12 The quantities (xx>,

(X~>,

<:JP, (.x0,

<~t>,

at> are the

averages of the products of the stochastic variables 7"" Yn. , tm as cor.tbined bet\veen the brackets, < >. By introducing the second moments of the stochastic variables and indicating them (5.3)

(Xt>

-<7>

"

.. ,de

the expressions for the variances obtain the form O;c;<

(-5.4)

{

0~ a~~

/\ oy~] '" - : / A' x t(X> = -T [ ;a - -I" t t T [ :I-'J -;xt "<'P A <X) -r tt A = - 'Vt {t> A

'J' - J,!it -

T [/\

= -(t>

A

<~>

/\

-I"

tt

<x;><~>]

"-

<~>z

z]

The relation between the variances and the size, the shape and the direction of the ellipse is as follows (see BEAR, 1961, by 1191). The quantities (1:>.:>., components of a second rank tensor values are a"

(5.5)

,

a~L'

all

=i

0.l.t

-==

{

f

ax~,

0 11 ,

\'Those

are the principal

given by

(au.-t 0:1,)

(O;u l'

a~,)

The roots of these principal values are equal to the magnitudes of the principal axes of the ellipse, ~ and 6", according to (5.6)

6,

=;a;;

The angle, \f , bet\veen the major principal axis, and the x-axis is given by

<S.7)

318

~_.

_ ....

6,

La~¥___ ... _________ _

Soil Mechanics and Transport in Porous Media

13

6. Elementary Conveyor Units The results mentioned in Section 5 are theoretically obtained as a distribution of the probability to find a particle in a certain location at a certain time. A particle can arrive there following different paths. The analysis consists in deriving the probability for a particle to choose a certain path and to select from all possible paths only those, that end up in the desired location at the ·desired time. Mathematically this produces an integral expression with a closed solution. It does not fit in the frame\·;ork of this presentation to expose all the details of this analysis (a separate report deals with this aspect). It is, however, necessary to mention here that the analysis requires,that the particle path be subdivided into elementary steps of a special nature. In the first place the steps must be such that it is possible to determine the magnitude of the probability for a particle to choose a step of a particular kind. Ilv -Ute-- 5CW1~ j,/ftlk it is necessary that the steps are not correlated, the reason for noncorrelation is that the individual steps must be combined into a product. It is

known that according to probability theory the probability for the subsequent occurence of several events consists of the multiplication of the probabilities of the separate occurence of the individual events, only if the events are not correlated. In order to be able to apply the mentioned product, it is therefore necessary that the steps are not correlated. When-particles are transported by a fluid through a porous system, the above mentioned analysis can be applied to predict their dispersion if the paths followed by

Selected Works of G. de Josselin de Jong

319

14 the particles can be subdivided into steps the requirements imposed by the theory.

that satisfy

We will use the

form «probabilistic step» , if the steps do indeed satisfy these requirements. Physically the particle path consists of a number of traverses through pores, cavities or fissures, depending on the nature of the porous system. In general there is not necessarily a coincidence betHeen the probabilistic steps of the particle path and the traverses through individual pores, cavities or fissures. combination of them.

It can be any

In order to indicate the physical counterpart of the mathematical steps, the term«elementwy conveyor unit» is proposed here.

An elementary conveyor unit can be any

part of the porous medium traversed by transported particles. It can be one channel bet\.;een grains or any combination of channels or fissures up to complete lenses or layers of soil. The combinat:ions of connn:i. ts consti tl\tint] elementary conveyor units can be different for different directions and magnitudes of the hydraulic gradient. In order to constitute an elementary conveyor unit the combination of conduits must satisfy the four requirements listed ·beloH. These requirements guarantee that the passage through the elementary conveyor units from their entrances to their exits, coincide with the probabilistic steps.

In the following the parameter, m,

will be used to indicate the properties of an elementary conveyor unit of the mth kind. The reqpirements are

ITI

a particle, travelling from entrance to exit through an elementary conveyor unit of the mth kind, covers a distance, whose coordinate components ?C/h, a known magnitude.

320

!:lin ,

ZIn ,

have

Soil Mechanics and Transport in Porous Media

15

m

a particle, travelling from entrance to 8::i t through

an elementary conveyor unit of the mth kind, remains within the unit during a residence time, magnitude.

tIn, Hi th a knm'.'n

"fl)

the choice being made by a particle to enter the next unit after having completed its travel through a

previous unit has a probab~lity distribution, ://11 , "'hich is a known function of the~arameter, m, of the unit it I Hill enter.

BJ .

the probability distribution of choice, J/II , does not depend on the parameter, m, of the previous unit. This independence guarantees

that subsequent steps are

uncorrelated. ·Because the values of X/11, Yhl , Z", , t/1/ as chosen successively by the particle during its travel through the porous system are subject to a probabilistic choice, these quantities are called the stochastic variables. In the case considered here, there are two kinds of elementary conveyor units. Therefore m has the values 1, 2. In order that the distribution of choice represents a correctly mormalized probability it has to satisfy the condition

(6.1) The averages (x>, , , (xx>, in th~ results obtained from the

etc mentioned theory are defined as follo\\'s by sur:ll:tations of the stochastic variables multiplied by the corresponding probability of choice (;(J> , • • • • • • • • • •

ge~eral

In

(X> =

(6.2)

{

L Xln71l1 In

= L !J",7", Ito

= I..

tin ~J~

Selected Works of G. de Josselin de Jong

hi

(xx>

::=-

(X,>

=

.t

L X", dill Irt

1 X",Y,,,7,n

•••• 4!tc

321

16 If m, instead of consisting of a finite number of discrete values, is continuously distributed, then the requirement on

~/JI

(6•3)

J

is

-1"00

?-/ll till'

:=

f

-ro

In this case the averages are defined as follows by integrals over all possible values of m

<x.'l = (6.4)

322

1m

X",

?1J1

dhl

<~>

=1



=L tin ff'I? dm

In

'!1m '1/11 dm

():x> ==

(XJ>

1 x~ '}/tl din h'

=1m

•...

X".!J)II

7", (hn

~t<

Soil Mechanics and Transport in Porous Media

17 7. Combination of Fissures into

Elementary Conveyor Units In order to assign the status of elementary conveyor unit to fissure segments or combinations of fissure segments, the 4th requirement provides the decisive criterion. Therefore, we have to examine first, where in the particle transportation system the correlation disappears between local decision of subsequent path direction and the preceding path direction. This can be done by considering the stream lamellae of }'ig.2 again. Particles entering the fissure CE at C arc carried by either of the tt-lO lamellae, f or g. Both those lamellae were previously in the fissure BC. Therefore fissures CE and BC are correlated and cannot be considered as tt-lO seperate elementary conveyor units, but have to be combined. Furthermore at E both stream lamellae, f. and g, \·Till continue in the 0l-direction as can be deduced from the behaviour of the streaM lamellae, a and b, as they pass intersection C.

Particles entering the fissure BC at B are transported by one of the lamellae c, d, e, f or g. If entering \dth c or d the particle had previously the

tlJL-direction, if

entering with e, f or g the particle had previously the ~-direction. Molecular diffusion scatters particles randomly over the lamellae such that arriving in C, the location of a particle is no more influenced by which lamellae carried them at B. One could say that the particle has forgotten its past history. At C,part o~ the particles continues along O~ (those carried by c, d, e) and part continues along 01I. The continuation probabilities at C are, because of molecular diffusion, independent of the particle location at B and therefore not correlated to the travel direction of the particle before arriving at B.

Selected Works of G. de Josselin de Jong

323

18

Since the rupture in correlation is produced by molecular diffusion within the ~ fissure (such as BC) the transition points between elementary conveyor units, where exits and entrances of subsequent units meet, must be located halfway along these fissure segments. The discharge of particles is subdivided in C. The lamellae c, d, e continue through CD in the ~-direction and reach an exit of an elementary conveyor unit half Hay along CD. The la,:lcllae f and g con-tinue first along &JL through CE and then, in E, procede along Oz through EF, reaching the exit of an elcmentaty conveyor unit half Hay between E and F. This is represented by the shaded fluid flow in figure Sa. The elementary conveyor units are represented in figure Sb by the s6lid arrOHS overlying the dotted fissures. There are two kinds of elementary conveyor units labeled m=l and m=2. The conveyor unit, m=l, consists of tHO halves of fissure segments in the 1-direction, such as going from a point halhlay betHeen Band C to a point half Nay between C and D. This distance has a length 1x and a direction

Ox The conveyor unit, m=2, consists of one half of a fissure segment in the ~-direction, a full length of fissure segment in the O~-direction, and again a half in the t9x- direction, such as going from half\-lay bet,-,reen BC over C to E upto halfway between E and F. The first two requirements on elementary conveyor units are satisfied, because it is possible to compute the displacements and residence times. These are, using ( 2; 1) and (2. S ) For m=l

(7.1)

324

{

;(1 :=

"i1

t1

:: ::

-J:

1 XI + yXz

= }z

1 !fz +± !I;r. Z" t.z of f tz = 1

((Js

JI Si" .Ix Cr

[}z ())r

1I

Soil Mechanics and Transport in Porous Media

19 For m=2 I

:t;J.

(7.2)

{

XXI -/" 7:11 -r

'1j" = Ly ~ I

t:z..

=

f

1-

t.:r

of·

;}11

of"

(IT

1"

f

Z XJ:

±yz ±t.r

iz

=

Si".9z

Ix Cr 'Iz

-r 1:JL Sin

i'r

t9J[

CJT

T--

'Ix

In order to shoH that finally also the third requirement is satisficd and ~, ~ can be computed, it is necessary to determine the probability that a particle, starting half,·ray betHcen Band C, ''iill end up either half''iay between C and D or half,'iay bebleen E and F. 'fhese probabilities can be obtained by considering the continuation probabilities in C and E. A particle can be considered to traverse an elementary conveyor uni t, m=l, if arriving in C from the .9J: -direction it \Vill continue in the 4- direction. The probability ~

for a particle to do so is IZ-"I as defined in Section 3, and therefore ...Ti th (3.2) we obtain

9z - r.zr

(7.3)

'I.:r A particle can be considered to traverse an eleQentary

.conveyor unit, m=2, if arriving in C from the tlz -direction, it ...Till continue along 011. and subsequently, arriving in E from the 6'JI-direction it Hill continue along OJ:. The probability for a particle to do so consists of two consecutive probabilities, .

Ie

%-+

It

combined with

JE

J:J[-#z

•

According to the rules of probability theory these b10 probabilities have to be multiplied. This gives for the probabili ty, J2, , that a particle chooses an elementary conveyor unit, m:2, Hith (3.2) and (3.1) c

(7.4)

lj:z. =.~ ... ][ ~~ =

911.

r;

£. IIt:x

* ~:z .. f

-

Selected Works of G. de Josselin de Jong

c

71

if

9.zr

'II

325

20 The

t\olO

prob<J.bili tics of choice

71

J:z.

and

satisfy

the requirement (6.1), because (7.5) Because the combination of fissure segments as combined above into the units, m=l and m=2, satisfy the four requirements imposed by probability calculus, these units represent the elementary conveyor units in this case.

computation

of the averages <x:> , , , (xx>, , ••• etc. amounts to the use of (~.3) and (7.4), (7.5) in the expressions (6.2). This gives the following result. for the averages

<x> = ;;<,71

(7.6)

1"

x:t.J.1.

( lz

Y1

~1 +

lJ",7z

( JI sill

JII e][) If;; .. ( II CC$ SZ f.r 1" Z P.r 1.J[


-=

tl

J;

1"

t.. <,J...

<xx>

:=

X/''ll

-r

X;C;" =

<x~>

=

:Xf!it ~1 ",)(.1.

'J.. 9~

=

YI~'~1

1"

?f:~.1.

7I +

iII

6;
'1Ir) 19I

()I 9I 1"

111.

Sin f}1I

9][) I'lz-

( fI Cz -I-

=

..

=

.1.

.

1"

(,Pz {os til.z Sin i9.x

+ J:t


CM 0I

326

Xl

t f 71 -tx.d.:-7:.=

'7.r + h lE

fICd!).r

1"

{CJ &lI

l'Jr.1. C"< f}JI

'II .,. .z lrf:;z Sin()z t

<:xt> =

.lJz cc, OJI 7E) 19;;

IJI {b< IJr Sin /Pr '1Jr

(lr'" $,n't;z

Cto< f}.r Ccs <9$ 'IE

..

.t; C

$,;' ~JI

(JZ<:r;:J1 E)

Sini9ll

'1]/

S,;' IJE '1][} l'1z

Sin t9JI 'IE

'1JI) /

'II

+JJ[CC
Soil Mechanics and Transport in Porous Media

21 for the second moments, as defined by (5.3)

'" - <xx> - <x '1 ~ = xx.

(7.7>

.fII~

<xJ> - o:>- = J:;

./-

1I~

"

=

<:1 ~ > - <~>~

1\

=

<Xi> -

='

<Jt> - <,> =

1][

tt. = -~ =

.1:...

/I

~t

f;..(f.r - ,/,;:)

'1,L.L

..,

"Y" =

;ct

"

(&S {}JI

JJ[

=

/I

Selected Works of G. de Josselin de Jong

..,

$'-," ('1.r CO$lJ1!.

Z

$/" tlJL

)JL C05~1L

Sin Or.

-'1I:)

r:

1J! ('Ix -'II)

r}

eE

~

S/h eJI CJ[

(P,r - 'II)

---;;: (1:-'/;)

9i

{':r~ (9.r - '/;: J

rI'" 'Ix

327

22 Application to the Example

8~

The example, considered for the experil;:ent, had the following values for the direction of the fissures and the direction of the overall hydraulic gradient (Pig.3) (8.1)

All lengths were equal, and all cross sections were equal, so that (8.2)

1:r. = -1:J[

=.,R ,

The discharges are according to (2.8)

(8.3)

~

=AJ

(0.$(0")

= AT

using these values in (7.6) and (7.7) gives

<;L?= 3$ 4-

<J>

j.

1 = 7; J..

<0 =

cl.

Z;;:j

(8.4)

<:;t> = =

328

1

c.(~

7j:

30"

!:i

~....

"'1.":A:J

Soil Mechanics and Transport in Porous Media

23 The ilngle

11

betv;een x-axis and path follo''led by the centre

of gravity is given by (0.5)

Ii

=

ttl)! -

1 <~>

=

<X)

10.9·

Applying the results (8.4) to the formulae of Section 5 developed from the general dispersion theory gives: for the coordinates of the midpoint displacement (8.6) for the second moments

(8.7)

{

Oxx

= /.:1.8 2-

T.R.~~

a~.'J

313 =-

T J.:,./o..(eT

a,~

to?

6

.ie

.JL T - p.8

With. . T

.e- I.e

.).J

the time after injection equal to

T

(8.8)

=

.pc 4-0 ;::j

... g~ ves, IV~ th a

th~s

.

t!'"

(8.9)

{

O;xx. = Il..

aXJ

=

tl..

a~~

L

0·94

2.-.

~

..e

l.

1. GZ

(8.10)

{

If-

o:l!.. -1

3.75..e

.

theoret~cal:

= ~5. 98

X

yll. -:

1.:-

$1

for

j.

k

U.sing these values in (5.5), 011

if.

superscr~pt

k

1.9<'1 ..J.

Selected Works of G. de Josselin de Jong

/f..

5.00

1-

(5.6) and (5.7) gives

0:z.:.

=

O.z,1!..

=

0

'I'll... =

60'

0

329

24

The values (8.9) and (D.10) are used to produce the dotted continuous cumulation curve in Fig.4~, which according to the theory has the form of a Gaussian distribution.

330

Soil Mechanics and Transport in Porous Media

25

9. Discussion of Results, Conclusions

and Further Development The results of the computational experiment and of the theory, expressed in the values of the coordinates of" the center of gravity and the second moments as obtained in (4.5),

(4.6) and (8.8),

(8.10), are summarized as

follo\'1s ~

-:::: ZS.93.R..

y.L

=

4.91 £.

'!j

~

3.90 ;.'"

Q"

0

au.='

~

"a" "-

(9.1)

If.-

X

a,z.2.

toe .e-

61

0:

::

= =

X =z6'.98.£ ;lIf.

=

s. (}(J

f

~

3.75

~

iI.

..,

0

't~= 60'"

60·

ft:.

1.976' .£

61

0

It 6"" =

0::-

1.911 1. 0

Both experimental computation and theory predict a dispersion ellipse, \'1hose major principal axis has a direction of 60° with the X~axis and a minor principal axis equal to zero.

This means, therefore, that particles are spread

in a line at 30° with respect to the hydraulic gradient. (see Fig.4). The values of the centre of gravity coordinates and the standard deviation disagree by less than 2%.

The

experimental centre of gravity lies belo\-1 the theoretical prediction. These errors are largely due to the arbitrariness of the condition of injection. had been chosen at a point

Q

If the injection point

along

Selected Works of G. de Josselin de Jong

f)z, half"1ay bet\-1een

331

26 the fissure intersections, rather than at P (Pig.6), the experiment ,.,.ould have given the sume spread of particles ''lith the same experimental values for the second moments as injection at P. poin~also

By displacing the injection the

)(Y

is desplaced. PR=~

f13

the origin of

coordinates used for locating the centre of gravity Since the displacement consists of components

X ,

and RQ=!4 £, the coordinates

Y

have to be

increased by these amounts respectively.

The result of

the experiment starting from Q therefore is: .e

Xa = 2!>.93

(9.2)

{ ":1;

4-.91

+ 0.'1-)3 + o. z5

.e

6.la = 1.97-' .f.

= 26.36.£

.A!

5.16

~a

~

=

0

The travel time to obtain the values given by Fig.4 starting from Q is longer than starting from P by half the residence time in a

():z

fissure.

So

T

must be taken to be

(9.3) Using this value

~

in (8.6) and (8.7) gives the

theoretical prediction

x-/f.. tit (9.4)

If...

d't a =

= z. 6. 3 i.

('95.R.

If~

Yo, =

(h"'"'a

5.06i.

=

0

The error in centre of gravity location is again less than 2%, but the experimental centre of gravity now lies above the theoretical prediction.

The error in standard deviation

0', is no'''' about 1%. These errors can still be reduced by starting the injection in a more balanced ,.,.ay.

This means that the

distribution of particles over the two kinds of fissures

332

Soil Mechanics and Transport in Porous Media

27

at the injection point is chosen in such a manner that this distribution remainz unaltered, although after every time interval

~t

, particles are redistributed.

Then

the centre of gravity coordinates of the experiment \-.1ill IJot follow the tortuous path as shOvTn in Fig. 6 but a straight line that matches the theoretical values exactly and the error in standard deviation reduces to less than 1%.

He \dll not elaborate this refinement further because

explanation of a balanced injection may confuse more than its contribution warrants. Besides this numerical comparison,

Fig.4

a

permits

a visual comparison of the cumulation curves of the particle distributions from computational experiment (solid, stepwise curve) and theory (dashed, continuous curve).

The two

curves fit everywhere \·11 thin the mesh width of the fissure pattern. Since experiment and theory agree numerically and visually within values that are smaller than the mesh width of the fissure pattern, the conclusion may be drm\1n that the theory is capable of predicting the outcome of the experiment within acceptable limits. The theory is expressed in terms of averages of the stochastic variables.

The prediction formulae can therefore

be used in every case \-There the stochastic variables and their probability distribution is known.

As an example

of further development it \-.1ill be sho\-Tn here how to apply the theory if the lengths of the fissure segments are not equal (as in the previous treatment), but are randomly distributed..

Then the distribution of choice

changed to include the variability of length, t

7",

must be

.

Let the distribution of the lengths for both fissure families be Gaussian \-1i th the same mean value Jf' and the same standard deviation, \dll then contain a term

1~.

Selected Works of G. de Josselin de Jong

The probability of choice

333

28

(9.5) which represents the probability that a fissure segment has a length of magnitude bet\.leen

j

and.1 -rdJ.

!/".

The previously obtained values for and (7.4) then become

(9.6)

{

91

::

RJ./

=

'1:r;

lrfjU

78

f

'II

./rf;:i/i

1---'1",[.i7i;

The factor

1

d

"f,/:('r-

-'"

,£xf { -

.2)«( { -

given by (7.3)

f.f-.J,J"/",l;} d£

O-,1f')/zJ:}
is added in order to normalize the

probabili ty of choice

J-t-{9",d.R =

,

9'-" - '1:

•

properly,

satisfying

IdJ.A.Xr{-a-Jf'f/.?J:J [f.:~5'z I.l'

-.4

-t

the requirement

'tJ

dR -=

f

Besides a summation of the form (6.2), this requirement contains an integration over all possible lengths.

In order

to obtain simple results from the integration, it is common practice to integrate lover the range from -od to "'" This means, that unreleastic negative values of i are included.

However, if 10 is small \Vi th respect to .If',

the error introduced is small \Vith respect to the convenience

a.

ofAs~mp

1 er

.

express~on.

The averages of the stochastic variables are obtained by formalae similar to (6.2) and (6.4). For the average of the x-component of the displacement this gives with ( 7 • l) and (7. 2 )

<x>

334

= J.

~

IJ'A:t·7C

I" .JI.xr { -

(J -If'

)';/.2.R;)

If

_cd

Soil Mechanics and Transport in Porous Media

29 and for the average of the x-component squared

lu

<xx> = ) ...

(q,s)

1:

L)(f { -

0-.1,,)"/:/...1;]

* {f 1% C&< 6.1 ( 9.t - '1JL J

)t

-t f.I Cc.r~.1 + .I C<" ,pI/' 1c) /1.1} tip

and simil~r expressions for the other averages. Considering the example of Section 8 and introducing the following values for the mean length and its standard deviation

the averages become (to be compared with (8.4) )

(9. 10 )

0:::>

=

=

3}3 ;

<:xx) =

t/-

1.0&

f

I~

7i..R.

~

:If ;.7.

"'!-JI

<~J>

=

(t> = 2.£1. >.7

<~:J>

..J... .,(" = 1.3 8

<~:>

4-

1.

<xt> =


(9. 11 )

aXe<

1.3

f.i8

Tj"~ ~c

a"J

1. 1

/.Zs

3/3

T

J.%~

9

T

iZ.'~

a"

1.3

3

/.is

and this gives for

T

f f";l -

l'

7.fi.l~-=-

~

<:it> ::- (f

)l~(-tr~ (;tlli\v"Ji ,\,1

From this the second moments become .cl

(.06

>':1

.z. c

;;; 53>.( . . 1.06'15 .>."r"

'1, J

i [I + J,:)

,.(c.

.It:-

=

~

'fo

Xj

a dispersion ellipse with

major principal axis:

6. =- ~

b', ~ .( .u; I

minor principal axis: 6~ = o. s~ direction of major principal aBs: t 6~.:l·

E).l.

=

=0

Dispersion in this case spreads the particles in two directions because

~ ~ 0

,

instead of along a line as

,~as

the case when all fissure segnents had the same length.

Selected Works of G. de Josselin de Jong

335

30

REFERENCES

J. Bear, <.( On the Tensor Form of Dispersion in Porous Media. J. Geoph. Research, Vol. 66, No.4, Pages 1185-1197, 1961.

»

S. Chandrasekhar, Stochastic Problems in Physics and Astronomy.» Reviews of Modern Physics, Vol. IS, No. I, Pages 1-89, 1943.

«

G. de Josselin de Jong, «Longitudinal and Transverse Diffusion in Granular Deposits.» Trans. Amer. Geoph. Union, 39, Pages 67-74, 1958. G. de Josselin de Jong, The Tensor Character of the Dispersion Coefficient in Anisotropic Porous Medi~ Proc. lAHR Symposium, Haifa, 1969, Pages 259 -267.

«

D. T. Snow, ??

336

Soil Mechanics and Transport in Porous Media

,, ,,

J

y

):t

Orientat'~on of of the hydraul'~cthe , • of f'~ssures and gr t\-lO ad~ent ' f amilies

J

Selected Works of G. de Josselin de Jong

337

_:_:~::~::~f

aIr

---

®

Subdivision of the discharges at intersections of fissures.

1: Fig.

338

2

Soil Mechanics and Transport in Porous Media

I

:.,

,

'.... ,

\

n

°0 <"'.

'K-'.

\<S.

\

I

:'

..,

°0

"

I

~

:----..

:

: .: "

339

Fig. 3

1

---------------------'

.;,'-------~·\:-i

Selected Works of G. de Josselin de Jong

340

~

;

..... ~.

,../

I

,

/

;

....

,.

»

~~

,.

:1/ ~ •

-. , -• 1

Soil Mechanics and Transport in Porous Media

{Fig. Distribut"l.on of started h particles alfway b et\V'een Band

__ -

1>- __

~

I'."Sl",rt

~r

sal c.

;r

I

I

I I I I

I

I

I I

--&--I

I

I

Elementary Conveyor Units, m=l and m=2.

Selected Works of G. de Josselin de Jong

341

342

I'M!. of cuh. of 3r"'~ilj (~pe.li1l1e•.t-) p.,tk

Ii1 c~ '1 iY""'la (.fW1'j )

y

Soil Mechanics and Transport in Porous Media

t/'\

//

. ~

ill

~

Paths followed by centre of gravity of particles.

NIEUW ARCHIEF VOOR WISKUN DE (4), YoU, (1985) 207-208

Cube in Tessaract An Introduction to the FOllowing Article J .H. de Boer & J. van de Craats (Section Editors)

Tessorocf (or jour-dimensional measure polytope) is the four-dime nsional analogue of cube. In the November 1966 issue of "Scientific American", Martin Gardner devoted his column " Mathematical Games" to higher dimensional polytopes ("polytope" is the general term in the sequence: point, segmen t, polygon, polyhedron, ... ). He raised the question of "finding the largest cube, that can be fitted in a unit tessaract". lllis problem was inspired by the three-dimensional phenomenon, that inside a uni t cube a square may be constructed with sides large r than unity, leading to the surprising result, that it is possible to cut a hole in a unit cube such that a cube wi th edges larger than I can pass through it (the hole must be cut in a direction perpendicular to the square). Pieter Nieuwland (1764 - 1794) found the maximum edge length for such a cube (cL Section 3 of the following article). The weaker result that a cube can be perforated in such a way that the second cube of Ihe same size may pass through the hole, is ascribed by Joh n Wallis (1616 - 1703) to Prince Rupen, Count Palatine of the Rhine (1619 - 1683) (d. O.J. Schrek: Prince Rupert's problem and its extension by Pieter Nieuwland, Scripla Malhemalica 16 (1950) pp. 73-80 and pp. 26 1-267). In 197 1, G. de Josselin de Jong (professor of Soil Mechanics, now retired from Delft University of Technology) sent a letter to Gardner containing a description of the construction of a cube with edges b in a tessaract with edges a. where b >0. He also expressed his opi nion that this cube might be the larg. est cube that can be tilled in the tessaracl. He did not have a proof of this conjecture, which he based on visual intuition only. Martin Gardner answered, that he did not feet competent to judge the meTits of this result, and he recommended to send il to a mathematical journal. To find out wether the result was wonh publishing, de Josselin de Jong asked the advice of H.S. M. Coxeter who proposed to formulate it as a problem in

343

208

J.H. de BOER & J. van de CRAATS

the American Mathematical Monthly. It appeared as problem 5886 in 1972 (vol. 79, p. 1140). No reactions were received. However. in 1974 (vol. 8 1, p. 294) the result of Section 6 of the following article was given, but without any explanation how it was obtained. Nine years later, de Josselin de Jong, still welcoming an adequate oppOrlunity for showing the geometrical aspects of the result, came in contact with one of the editors of OUT section "Recreational Mathematics". Upon seeing the material we asked him 10 publish his construction in our journal. TIle result is

the following article. We expect that many will enjoy reading it, and, of cou rse, we also hope thai some will find it a challenge to improve upon the results by constructing a larger cube, or to supply a proof that in fact the cube is the largest possible. Also the general n-dimensional case might be worth considering.

344

Soil Mechanics and Transport in Porous Media

NIEUW ARCHIEF VOOR WISKUNDE (4), VoLl, (1985) 209-217

Cube with Edges Larger than those of the Enclosing Tessaract G. de Josselin de Jong Ary Scheffersrraaf 227

2597 VT Den Haag The Nerher-Iands

I. INTRODUCTION

The purpose of this paper is to show, how inside a tessaracl (jour.dimensional measure polytope, or hyper-cube) a three-dimensional cube can be constructed with edges larger than those of the enclosing tessaract. As an example of ales-

saraet consider in euclidian four-space R4 the polytope T = {(XI,X2,X3,X4)ER 4

. . T has 16 vertices (m

· COOT d mates

I-

I I "2Qo;;;XI,X2,X3,X4"';;; + "2a).

(-+-"2a, I -+-"2a, I +"2a, I -+-"'2a», I 32 edges (0r

length a), 24 two-dimensional faces and 8 three-dimensional cells. To show thaI T contains a cube with edges larger than a, it may suffice to give the coordinates of the eight vertices of the cube (see (10) in Section 7) and leave it as an exercise to the reader to verify the claim by linear algebra. However, it might be of interest to show, how the result was found by rotating a rectangular parallelepiped around an axis, adjusting the lengths of its edges in such a manner, that its vertices are on the bounding surface of the tessaract and its edges are of equal length. The procedure results in producing the largest cube, that can be constructed by this particular rotation. Since all other rotations the author could imagine, did not produce a larger cube, it is conjectured that the maximal cube is found . This conjecture is based on visual intuition only. and still requires a rigourous proof.

345

210

G. de JOSSELIN de JONG

2. A NOTE ON THE FIGURES

lei it first be shown bow a tessaract can be represented visually as in Fig. 2. The construction of this figure is understood by starting with the oblique parallel projection of a three-dimensional cube, as in Fig. I. Two faces of the cube are undistorted and presented as two congruent squares. The four other square faces appear as oblique parallelograms. In an analogous wayan oblique parallel projection of a (four-dimensional) tessaract in three-space may be produced by presenting two boundary cells undistorted as two congruent cubes. From these the front cube is drawn with heavy lines in Fig. 2. The back cube is displaced with respect to the front cube

in an arbitrary direction backwards. Then corresponding vertices are conneeted by 8 parallel edges. The six remaining boundary cubes appear as six oblique parat1elopipeds. Figure 2 gives a plane picture of the three-dimensional projection of a tessaract. TIlls figu re should be visualized as a three-dimensional model. It is instructive to locate on this model all 8 boundary cells, which in R4 are cubes. Before treating the problem of finding a cube with edges larger than those of the enclosing tessaract, it is hclpful to consider first the problem of constructing the largest square in a cube. This problem is treated here in a circumstantial manner, unnecessary for such a trivial case. This is done in order to introduce the elements of the reasoning which, being obvious in three dimensions, can be extended by analogy to the less obvious four-dimensional case.

3. SQUARE IN CUIIE

Visual insight suggests, that the most favourable position for the square is obtained by choosing its plane through the centre M of the cube. This centre lies (Fig. 3) halfway on the line connecting the midpoints nand n' of two opposite vertical edges of the cube. Consider this line to be the axis of rotation for a plane a, that will contain the square. Note that in a cube two edges are opposite when, although being paralleL thev are not in a common boundary face.

/'

/'

/'

/'

Fig. 1.

346

Cube projected in plane of paper

Fig. 2.

Projection of tessaract in Rl, projected in plane of paper

Soil Mechanics and Transport in Porous Media

211

CUBE IN TESSARACT

The plane (I intersects the upper and lower faces of the cube by lines parallel to the diagonals in these faces. The lengths of these lines is limited by the edges of the cube. giving symmetrically located vertices for the square. Let the edges of the cube have lengths a and those of the square b. Intrer d ucing a parameter h according to Fig. 3, gives the relation mm' = b = aV2-2 h.

( I)

The upright edges of the square are located in a plane normal to the axis of rotation. This plane intersects the cube in the dotted rectangle with sides D.. and a, such that wi th Pythagoras'~ theorem

b 2 = (D..I+02.

(2)

Elimination of h gives the relation (3)

with the root b l a = 3/2V2:o:; l.0607, showing that b can be larger than u. This is Pieter Nieuwland-f*i- 's solution men tioned in the editor ~'s introduction. Visual insight in this case clearly indicates that it is impossible to manoeuvre the square in a position which produces a higher value for b 1 u.

" ' , .,

-

a

.

-' - " ' "

/

,

/

~~1 ~- )..-..-...~ ....: Fig. 3.

Largest flat square in cube

Selected Works of G. de Josselin de Jong

347

G. de JOSSELIN de lONG

212

4. CUBE IN TESSARACT

A similar procedure is now applied to the construction of a cube with edge length b, that can be inserted in a tessaract with edge length a, such that b >0. The tessaract is the body represented in Figure 4 by ABC D E F G HA' B' C D'E' P G' H'

The cube, that has to fit in it, is shown in Fig. 6 as P Q R S P' Q' R' S. Visual intuition suggests that the most favourable position for the cube is obtained, when its centre coincides with the centre of the tessaract. Let m and m' be the intersection points of the diagonals of the opposite faces P Q R S and P' Q' R' S' of the cube (Fig. 6). The centre M of the cube lies halfway mm'. For the tessaract its centre M lies halfway the centres n and n' of the two opposite faces B F F' B' and D H H'D' (Fig. 4). These faces are opposite, because, although being parallel, they are not in a common cubic boundary cell. In order to position the cube in the tessaract the choice is now made to superimpose the lines mMm' and nMn' so that the points M coincide. Further, the line nmMm'n' is considered to be the axis of rotation for a plane fJ, that will contain the diagonal plane PRR'P' of the cube. The plane fJ intersects the upper and lower faces of the tessaract (E'F'G'H' and ABC D in Fig. 4) in lines parallel to the diagonals in those faces. The segment R R' in face E' F' G' H' is shown separately in Fig. S. Its position is specified by the parameters A and p.. Its length b then satisfies the relation RR' = PP' = b = a h - 2"-

(4)

The symmetric positions of PP' and R R' in the upper and lower faces guarantee that P R R' P' is a rectangle. The lengths of the upright edges of this rectangle are b V2 and with Pythagoras' theorem it follows from Fig. 4 that (5)

(Note that the distance between the faces ABCD and E' F' G' H' is equal to a h.) Rotation of the plane fJ around the axis n m M m' n' varies the parameter p.. In Fig. 5 the situation P.
348

Soil Mechanics and Transport in Porous Media

CUBE IN TESSARACf

213

'c" '\ •

\ \ \

C

\ Fig. 4.

T~t with axis of rotation venioes I'RR'P' of tbe cube

~'m ' MII1II

r

I •

and

R'

. / "-,,-',- '

t '. ·H· ..·'

"

\,

•

~

•

.;

.";..---- -- - .. --Fig. S.

Location o r RR' as condi tioned by the parameters >. and jl Fig. 6.

Cube with edges of kng!h b, !hat

Selected Works of G. de Josselin de Jong

7

/ 349

214

G. de JOSSELIN de JONG

5. THE REMAINING FOUR VERTICES

The plane P R R' P' is kept fixed in the tessaract during the procedure to locate the other diagonal plane S Q (! S' of the cube. Unlike in three-space, where the position of a cube is completely detennined by one diagonal plane, in four-space it is possible to rotate the other diagonal plane: S Q Q' S' around the axis n m M m' n', while keeping it perpendicular to the plane P R R'r. In order to study the possibilities offered by this second kind of rotation, it is helpful to construct the intersection of the tessaract with the threedimensiona1 space through m normal to n m M m' n'. The line PR is located in trus space and S Q can rotate freely in the plane normal to P R in this space. After choosing the most favourable position for S Q the cube is determined completely. In Fig. 7 the dotted orthogonal parallelepiped is the intersection body of the tessaract with the space through m perpendicular to n m M m' n'. This assertion can be verified by noting that the dotted lines, being mutually orthogonal, are all normal to the diagonals F' H' or BD, which in turn are parallel to n m M m' n'. Moreover, all dotted lines are situated in faces of the tessaract and therefore on its boundary. The dotted intersection body is shown separately in Fig. 8. It has eight edges of length a, one set of four being parallel to B F, the other set parallel to F F'. The remaining four edges, from which two carry P and R, have lengths ']).. This is verified in Fig. 5, where the dotted line through R (perpendicular to R R') is the upper edge of the intersection solid in Fig. 8. The diagonal S Q of the inserted cube is situated in the dotted intersection body of Fig. 8, and more specifically in a plane 'Y through m normal to P R. This plane y intersects the dotted body in the heavy lined hex.agon in Fig. 8. The segment SQ has to remain within this hex.agon which is presented undistortedly and with its dimensions in Fig. 9. The most favourable position for S Q is as shown in Fig. 9. S Q is a diagonal in the cube and therefore its length is b V2. Using the values shown in Fig. 9 gives (SQ)' = 2b' = (aV2 -

350

v..pV2 '1 a

+ 4;"(1 + 2'£). a'

(6)

Soil Mechanics and Transport in Porous Media

CUBE IN TESSARACT

215

~

~

\!\\ , E

i~r\j-~>,

G

'-f-..ll- ". If '

jD""' --H-~ !i.If"---I\-'-c--+i '... J= .····".."·I('· '.

!\\ ", ,

"'. "

'.

:\' '... :. " 1 ...

'.

\,

; ___ \

/""_~'_'

: \.

\,

'.;

\,

: \i\ .:

.;:

,,_ :.

c

\: \.

P

Teuaract inlCneClod by threo-spacc through ,", perpcndjcular 10 !be rotation LUI

l:" t~~1' 2~ 'l _. ,,

.

Q -...!'~/1

-

·

•• ••• I

;

.u

I

•

I---,m~''--l

(" -z~ )(z'

I

1 Fi"

I . Shape of inlerXction body willi plane IIIlrIml to PR

Selected Works of G. de Josselin de Jong

Fi" 9. Hal&""- limitin& lhc: si.l.c of SO in plane T through '" DOmIaI to PR

351

G. de lOSSELIN de lONG

216

6. RESULT

Solving (6) for A, after

1.1.

has been eliminated by means of (5), gives

,

~ = + aV2(b 2 - a 2 )1 / (bV2 + a). Only the upper sign is applicable, since A cannot be negative. Use of (4) then

gives

, (aV2 - bXbV2 - a) = aV2(b' - a')' .

(7)

There is only one root, viz.

(8)

b/a~I.OO7435 .

This result shows. that there exists indeed a cube with edges larger than those of the tessaract. From (4) and (5) it follows that

A/

a~O.2034

, "/ a~O.0864.

(9)

In the solution of the problem in the Am. Math. Monthly, mentioned in the editor prilRe-'s introduction, the ratio (8) was given as a root of an equation of degree eight. instead of the more simple equation (7). This was due to a less adequate elimination of>.. and p. from (4), (5) and (6). Fig. 10 shows how the cube fits into the tessaract. It may be noted that the point Q of the cube is located in the face BeG F of the boundary cell ABC D E F G H. Since both P and P' are situated in the face ABC D of that boundary ceU the entire face P Q Q' P' is located in that cell. Similarly. the face S R R' S' is a subset of the opposite boundary cell A' B' C D' E F' G' H'. Omy the two faces P Q Q' P' and S R R' S' of the cube are located in the boundary space of the tessaract. All other planes of the cube are in the interior. The vertices, however, are all situated on the boundary of the tessaract.

352

Soil Mechanics and Transport in Porous Media

217

CUBE IN TESSARACT

Fig. 10. Cube placed in the tessaract 7. COORDINATES

In orde r to facilitate verification of the resuh, the coordinates of the vertices of the cube ~re given bel~w as located in the tessaract of Section I with vertices (± fa'±"2a, ± fa' ± Ta). The lower signs in these expressions relate to the coordinates of the points R', S', P', (I. I

I

.~

_ I

_

I

I

. ,- _

1

P.R' = ( ± (Ta~"2(A+J..I)v2)' + "2a' + (Ta - "2(A - J..I)v2),+Ta)

Q, S' -- (+ .1a + ('!a _ 2AJ..I) - AV2) ' + .1a) - 2 ,- 2 a ' + (.1a 2 2 11

. t::I

_

11

_'-I

R,P' = (::t(Ta - "2(A - J..I)v2), ± Ta. +(Ta - T (A + J..I) v 2), ±"2a) S , Q' -- ( -+ (.1 2 a -,\ V2) . +- (.1 2 a _ 2AJ..I) a ' -+ .1 2 a' +.1) - 2a . It is possible 10 verify by linear algebra that the conditions guaranteeing P Q R S P' (I R' S' to be a cube, lead to equations (4), (5) and (6).

Selected Works of G. de Josselin de Jong

353

6 Complete Bibliography of Scientific Publications

6.1 List of publications The numbering of the papers is identical to the original numbering made by Gerard de Josselin de Jong, and therefore still completely in line with the actual hardcopy archive of the papers. [1] De spanni ngvcrdcJing rondom vertikale in zandige dekterreinen geboorde holten in stand gchoudcn door zwarc vloeistor. (T he stress distribution around a vertical hole ill oohesionless soil. ), G. de Josselin de Jong & J. Gcertscma, De IngeniclI.f, 1953, pp MI-M5

[2] Consolidation around pore-pressure meters, G. de Josselln de Jong, J. Appl. Phys ., Vol. 24, no. 7, 1953, pp 922-928 , American Ins titute of Physics [3J Wat scbeurt cr in de grond tijdens bet heien? (What happens in the soil during piled-driving?), Dc fngenicur, 1956, pp B77-B88 [4] Discussion, Proc. 3m Int. Conf. Soil Mech. fj Found. Eng_, Zurich, 1953, p 161

[5] L'cnt rainement de particules par Ie courant intcrsticicl, Proc. Symposia Darcy, Dijon, 1956, Publ. nr 41 of the UGGl, pp 139-1 47, UGGI

[6] Spoorbaanafsehniving (Slide in a. ra.i lway embankment) , LGM Mededelillgen, Vol. 1,1956, pp 7·40

[7] Verification of the usc of peak area for the quantitative differential thermal allaly~is, G. de Josselin de JOIIg, J. Am. Cerum. Soc., Vol. 40, no . 2, 1957, pp 42-49 , Ame ri can Cerl).mi c Societ y

354

6.1 Complete list of publications

355

[8] A capacitative cell apparatus, G. de Josselin de Jong & E.C.W.A. Geuze, Proc. ~th Int. Con/. Soil Mech. (1 Found. Eng., London, 1957, Vol. I , pp 52-55, Butterworth Scientific Publie"tiOlI~ {9] Application of stress functioilll to consolidation problcm~ , G. de Jossclin de Jong, Proc. ~th Int. Con/. Soil Mech. (1 Found. Eng., London, 1957, Vol. 1, pp 320-323 [10] Discussion, Proc. ~th Int. ConI- Soil Mech. fj Found. Eng. , London, 1957, Vol. 13, pp148-149

[11] Grafis<.:he bcpaling van glijlijn-patronen in de grondmechallica (Graphical method for the detertnination of slipJine fields in soil meo-:hanics) , Hodograaf zonder rotatie i~ fout. , De Ingenieur, 1957, pp 861·865 [12} Grafische methode tcr bepa-ling van glijlijn-pMronen in grond met weinig samellhang (Graphical method for the det.crmination of slip_line fields in cohcsionlcss soil), LGM Mededelingen, Vol. 2., 1957, pp 61-78 [13] Longitudinal and transverse diffnsion in grauu1ar deposits, G. de Joosclin deJong, Tmm.A.Geophys. Uniofl, Vol. 39, No.1, 1958, pp67-74, AGU 13 a. Discussion of "Longitudinal and transverse diffusion in granular deposits", G. de Josselin de Jong, 'TI-ans. A. Geophys. Union, Vol. 39, No. 6, 1958, pp 1160-1162, AG U

[14J The indefiniteness in kinematics for friction materials, Proc. Brossels Coni. on Earth Pre$sure Prob/eml, Vol. 1, 1958, pp 55-70 [15] Discnssion, Proc. BnJIIsels COllI. on Earth PreJlSUf"e Problems, Vol. 3, 1958, pp 57-58, pp 64-65 [16]

Static~

and ki nematics in the failahle 7,one of granular material, Doctor's Thesis Delft University 0/ Technology, Waltman Delft, 1959

[171 Vortex theory for mnltiple pha.sc flow through paroW! media, Water Resources Center Contribution, 23, Hydr. Lab, Univ. of California, 1959 (intern report)

[IS} Foto-elasti!ICh .onderwek van korrelstapelingen, LGM Mededelingen, Vol. 4,1960, PI> 119-134 [19] Singularity distri bution~ for the analysis of multiple-fluid flow through porous media, G. de Jossclin de Jong, J. Geophys. Res. , Vol. 65, 110.11, 1960, pp 3739-3758, Americall Geophysical Unioll [20] Moire patterns of the membrane analogy for ground-water movement applied to multiple fluid flow, J. Geophys. Res., Vol. 66, no. 10, 1961, pp 3625-3628, American Geophysical Un ion [21) Discussion, Proc. Sthe Int. ConI. Soil Mech. fj Found. EfI9., Paris, 1961, Vol. 3, pp 326-327 21 a. Disc::u!i>lion of paper by J acob Bear Oil the Tensor Form of Dispersion in PoroW! Media, G. de Joslielin de Jong & M.J. Bosscll, J. Geophys. Res., Vol. 66, No. 10, 1961 , pp 3623-3624 21 b. Refraction moire analysis of curved surfaces, Proc. Symposium Shell Research, Delft, 1961, Nort h-Holland Publishin.g Comp., pp 302-308

Selected Works of G. de Josselin de Jong

355

356

6.1 Complete list 01 publications

[22] Vacuum voorgespannen zand als coustructiemateriaal , I (Vacuum prestressed sand as construction material, I), De IngenieuT, 1962, 8127-8134 [23) Vacuum voorgcspannen ?and als oonstructicmateriaal , II (Vacuum prestressed sand as construction material, 11) , De IngenieuT, 1963, 8113-8121

[24) Discussion, p=. EUT. Con/. Soil Mech. 1963, p 35 [25] Discussion, p=. EUT. Con/. Soil Mech. 1963, p 76

e e

Found. Eng. , Wiesbaden, Found. Eng., Wicsbaden,

[26] Consolidatie in drie dimensies, I, LOM Medelingen, 7, 1963, pp 57-73 [21] Consolidatie in drie dimensics, II, LOM Medelingen, 8, 1963, PI> 25-38

[28J Consolidatie in driO! dimO!nsies, 1II, LOM Medelingen, 8, 1964, PI> 53-68 [29] A many-valued hodograph in an interface problem, G. deJooselindeJong WateT ResouT. Res. , Vol. 1, no. 4, 1965, pp 543-555, American Geophys ical Union

[30] Primary and secondary consolidation of a. spherical sample, G. de JosseJin de J ong & A. Verruijt, Proc. 6th lnt Conf. Soil Mech. Montreal, 1965, Vol. 1, pp 254-258

e Found.

Eng. ,

[31) Lower bound collapse theorem and lack of normality of strainrate to yield surface for soils, G. de Josselin de Jong, Proc. l UTAM Symp. on Rheology and Soil Mechanics, edited by G. Kravtchen ko and P~d Siricys Grenoble, 1964, pp 69-78, Springer-Verlag [32) Discussion, G. de Josselin de Jong, Proc. EUT. Conf. Soil Mech. fj Found. Eng., Oslo, 1968, pp 199·200, Norwegia n Inst it ute of Technology [33] Consolidation models oonsisting of all assembly of visoous elements or a cavity channel network , G. de Josscliu de Jong, Geotechniquc, Vol. 18, 1968, Pi> 195-228 , Till.) Institu te of C ivil I~ ng i" ....cn;

[34] Generating functiOIl~ in the theory of flow through porous media, G. de Jossclinde Jong , Chapter 9, Flow Through Porous Media, Acadell1ic Press, 1969, pp 377-100 [35J Etude photo-clastique d 'un empilement de disqucs, G. de J085Ci in de J ong & A. Verrllijt, CahieT Groupe Fmw;ais de R/u;ologie, Vol. 2, No. 1, 1969, pp 73·86

(36J The double slidiug, free rotating model for granular assemblies, G. de .Jossclill de Jong,Giotechnique, 1971 , pp 155-163, The Im;l.itulioll of C ivi l Eligi llccl"s [371 Discussion, Proacding., Roscoc Memorial Symposium, Cambridge, 1971, pp 258-262 (38J Dispersion of a point injection in an anisotropic porOIlS med ium, Socorro Report, New MCJl:ioo 87801 , 1972 38 a. Dispersion in fissured rock, G. de Jossclin de Jong & Shao Chill Way, New Mexico 87801, 1972 [39J Dispersion described by differential CC]uation developed with Lagrangian Correlation Fullction~ , RappoTt Geotechniek Delft, 1973

356

Soil Mechanics Mechanics and and Transport Transport in Porous Porous Media Media Soil

6.1 Complete list of publications

357

[40] The tensor character of the dispersion coefficient in anisotropic poroll8 media, G. de Josselin de Jong , Pmc. JA HR Congress, Haifa Israel, 1972, pp 259-267 [41J Photoel{l.';tic verification of a mechanical model for the flow of a grannlar material, A. Drescher & G. de Jossclin de Jong, J. Meeh. Phl/s. So/ieU, 1972, Vol. 20, pp 337-351, Pergamon Press [42J A limit theorem for material with internal friction, Pmc. Symp. PIMtic· ity, Cambridge, 1973, pp 12-21 [43J Aelotropie van gestrnctureerde materialen, Cement XXVI, No.4, 1974, pp 166- 176 [44J Rowe's stress dilatancy relation based on friction, Geotechniql.le, 178, pp 527-534 [45J Groudmechauische aspecten van korrelstapelingen, Procestechniek, Jaargang 31, No. 12, 1976 45 a. Evaluation of principle strCliS difference in photo-elastic granular mooia by use of a compensator in pure bedding, &pporl. Laborolorium voor Geolechniek Delft, J une 1976 [46] Mathematical elaboration of the double sliding free rotating model, Archi· wun Mechr.miki Stosowr.mej, 29, No.4, Wars1.awa, 1977 46 a. Model for the behaviour of granular materials in progress ive flow, Contribution Conf. in Jab/ona, Poland, 1976, lecture notes [47J Review of vortex theory for multiple fluid flow , Delft Progre&8 Repor/, 2, 1977, pp 225-236 [48J Constitutive relations for the flow of a granular a.ssembly in the limit state of stress, Speciality Session 9, Pmc. Int Conf. Soil Mech . & Found. Eng., Tokyo, 1977, pp 87·95 [49} Lecture notes Int. Center for Mech. Sciences (CISM), Udine, 1974, Model for t he behaviour of granular material in progressive flow [501 Improvement of t he lowerbound solut ion for the vertical cut off in a oohesive, frictionless soil, G. deJosselin de Jong, GeQtechnique , T echnical notes June 1978, pp 197-201, T he Institution of Civil Enginccrs [51) Vortex theory for multi ple fluid in thre(! dimetl.'iions, C.deJosselinde Jong, Delft Progren Report, 4, 1979, pp 87-102 [52J Diskontinuitiiten in Grellzspannungsfelderu, Geotechniek, Jahrgang 2, Heft 1, 125-129 (translated by Smoltczyk) [53) Het verloop van een drukstoot door cen paal, K IVI Publicatie, Vreed.enburghdag 1977, 19 oktober 1977 [54} Application of the calcu l u~ of variations to the vertical cut off in cohesive frictionlCliS soil, G. de Jesselin de Jong, Geotechnique, 30, No.1, 1980, pp 1-16, The institutioll of Civil Enginccrs

Selected Works of G. de Josselin de Jong

357

358

[55)

6. 1 Complete list of publications

11I 11~traties (Illustrations), Uitgcgcven door de vakgrocp GL'OtL'chlliek tel" ge legenllcid \11.1l het afscheid van pTof.dr.ir. G. dc .TOSl:!din de J ong, June

25, 1980 [5 6 J 1'rilmte 1.0 Professor de Jossditl dc Jong, LGM McilcddiJlgcJI, Part XXI, No.2, June 1980 [57J A limit theorem for soils, Delft I'rogl'Css /lCf'O,·t, 5, 1980, PI) 280-291 [58J A variational fallacy, C. de JC>liscJ iu de Joug. Geott:C/wiqu c, JUlie H181 , Vol. :11 , lIu.2, pp 289·290. Tllf" rll~ti t.m i(>1) of Civil Enginf't'1'S [59 J The si multaneous flow of fresh and salt water ill aquifers of large horizontal extension determined by shear flow and \"Ol'tex 1heory, C. de J()Iisclin de .Iong, Proc. Euromech. Colleg., Edited by A. Verruijt & F. B. J. Baremls, Sept. 198 1, pp 75·82, A. A. Dalkema [6 0 J Behavior of an elasto-plastic doubk...sliding frL'(.... rotating material if subjectoo to an ideal sim ple shear test., P''OC. IUTAM Conf. on DeformatiQn alld Failure of Gm nu/Ilr Materials, Edited by Vermeer & Luger, Delft, 1982, P)I 555-562 [61 J Cube with edges larger tlmu lilcl!;e of the endo~i ug te;&'\ract, C. de .Josscl ill de J Ollg, Nieuw An;hicf voor de lViskuude, 4, Vol. :1, I!JM5, Pi> 20!J·2 17 (+ introduction: pp 207.208 ), Wiskulldig GCHootsdoap

[(2) 'l'riHlsversc d is pcr" io\1 from (1 \1 originally sharp frt.",h·salt intcrfao:.'e caused by shear tlow. G.de.los.<;elindcl-long find C ..l . Van Duijn. J. Hytir%.l}Y, 84, 1986, Pi> 55-59, Elscvier

1631 EI'.,"_pI,","i' ""i,,, of "" "",hI, , 1i,Ii"S mod,I i" ,,,,d,,in,,I 'impI' shear tests. C.de JOli.'\Clin de .Jong, Geoleclmique, 38, :'l"o. 4, 1988 , pp 5:1:1-555 (+ 63a DisCIlS~ioll G. de Josseli ll de Jong Geoteclmique, 39, No.3, 1989, pp 565-5(6). T he Institution of Civil Engineers

[64] KC\'erling [luisman Lezing, 1989 [65 J Co-rota tio na l solution in simple shear tests. IYro/1! Memorial Symposium , Oxford, 1992. Proceedings Predictive Soil i....It.'<:h,mics, Thomas Telford, i9!H

358

Soil Mechanics and Transport in Porous Media

7 Acknowledgements

The following articles have been reproduced with the kind permission of the copyright holder. The numbers in brackets refer to the numbering of the articles in the complete bibliography (Chapter 6).

[2] Consolidation around pore pressure meters, G. de Josselin de Jong, J. Appl. Phys., Vol. 24, no. 7, 1953, pp 922-928, American Institute of Physics. Reprinted with the permission of the American Institute of Physics [5] L’entrainement de particules par le courant intersticiel, G. de Josselin de Jong, Proc. Symposia Darcy, Dyon, 1956, Publ. nr 41 of the UGGI, pp 139-147, UGGI. Reprinted with the kind permission of UGGI [7] Verification of the use of peak area for the quantitative differential thermal analysis, G. de Josselin de Jong, J. Am. Ceram. Soc., Vol. 40, no. 2, 1957, pp 42-49, American Ceramic Society. Reprinted with the permission of the American Ceramic Society, Post Office Box 6136, Westerville, Ohio 43086-6136 [8] A capacitative cell apparatus, G. de Josselin de Jong & E.C.W.A. Geuze, Proc. 4th Int. Conf. Soil Mech. & Found. Eng., London, 1957, Vol. 1, pp 52-55, Butterworth Scientific Publications. Reprinted with kind permission of the copyright holder Prof. J. de Josselin de Jong [9] Application of stress functions to consolidation problems, G. de Josselin de Jong, Proc. 4th Int. Conf. Soil Mech. & Found. Eng., London, 1957, Vol. 1, pp 320-323. Reprinted with kind permission of the copyright holder Prof. J. de Josselin de Jong [13] Longitudinal and transverse diffusion in granular deposits, G. de Josselin de Jong, Trans. A. Geophys, Union, Vol. 39, 1958, pp 67-74, AGU. Reprinted with the kind permission of the American Geophysical Union (AGU) 359

360

Acknowledgements

, [13a] Discussion of “ Longitudinal and transverse diffusion in granular deposits’ , G. de Josselin de Jong, Trans. A. Geophys. Union, Vol. 39, No. 6, 1958, pp 1160- 1162, AGU. Reprinted with the kind permission of the American Geophysical Union (AGU)

[19] Singularity distributions for the analysis of multiple-fluid flow through porous media, G. de Josselin de Jong, J. Geophys. Res., Vol. 65, no. 11, 1960, pp 3739-3758, American Geophysical Union. Reprinted with the kind permission of the American Geophysical Union (AGU) [20] Moiré patterns of the membrane analogy for groundwater movement applied to multiple fluid flow, G. de Josselin de Jong, J. Geophys. Res., Vol. 66, no. 10, 1961, pp 3625-3628, American Geophysical Union. Reprinted with the kind permission of the American Geophysical Union (AGU) [29] A many-valued hodograph in an interface problem, G. de Josselin de Jong, Water Resources Research, Vol. 1, no. 4, 1965, pp 543-555, American Geophysical Union. Reprinted with the kind permission of the American Geophysical Union (AGU) [31] Lower bound collapse theorem and lack of normality of strainrate to yield surface for soils, G. de Josselin de Jong, Proc. IUTAM Symp. on Rheology and Soil Mechanics, edited by G. Kravtchenko and PM Sirieys Grenoble, 1964, pp 69-78, Springer-Verlag. Copyright Springer-Verlag [32] Discussion, G. de Josselin de Jong, Proc. Eur. Conf. Soil Mech. & Found. Eng., Oslo, 1968, pp 199-200, Norwegian Institute of Technology. Reprinted by kind permission of the Norwegian Geotechnical Institute [33] Consolidation models consisting of an assembly of viscous elements or a cavity channel network, G. de Josselin de Jong, Géotechnique, Vol. 18, 1968, pp 195-228, The Institute of Civil Engineers. Reprinted by kind permission of the Institution of Civil Engineers [34] Generating functions in the theory of flow through porous media, G. de Josselin de Jong, Chapter 9, Flow Through Porous Media, Academic Press, 1969. Reprinted with kind permission of Academic Press-Elsevier [35] Etude photo-élastique d’un empilement de disques, G. de Josselin de Jong & A. Verruijt, Cahier Groupe Francais de Rhéologie, No. 1, 1969, pp 73-86. Reprinted with the kind permission of the Cahier Groupe Francais de Rhéologie

360

Soil Mechanics and Transport in Porous Media

Acknowledgements

361

[36] The double sliding, free rotating model for granular assemblies, G. de Josselin de Jong, Géotechnique, 1971, pp 155-163, The Institution of Civil Engineers. Reprinted by kind permission of the Institution of Civil Engineers

[38a] Dispersion in fissured rock, G. de Josselin de Jong & Shao-Chih Way, New Mexico 87801, 1972. Unpublished material. Reprinted with kind permission of the copyright holder Prof. J. de Josselin de Jong

[40] The tensor character of the dispersion coefficient in anisotropic porous media, G. de Josselin de Jong, Proc. JAHR Congress, Haifa Israël, 1972, pp 259267. Reprinted with kind permission of the copyright holder Prof. J. de Josselin de Jong [41] Photoelastic verification of a mechanical model for the flow of a granular material, A. Drescher & G. de Josselin de Jong, J. Mech. Phys. Solids, 1972, Vol. 20, pp 337-351, Pergamon Press. Reprinted with kind permission of Elsevier [50] Improvement of the lowerbound solution for the vertical cut off in a cohesive, frictionless soil, G. de Josselin de Jong, Géotechnique, Technical notes June 1978, pp 197-201, The Institution of Civil Engineers. Reprinted by kind permission of the Institution of Civil Engineers

[51] Vortex theory for multiple fluid flow in three dimensions, G. de Josselin de Jong, Delft Progress Report, 4, 1979, pp 87-102. Reprinted by kind permission of Delft University Press [54] Application of the calculus of variations to the vertical cut off in cohesive frictionless soil, G. de Josselin de Jong, Géotechnique, 30, No. 1, 1980, pp 1-16, The Institution of Civil Engineers. Reprinted by kind permission of the Institution of Civil Engineers [58] A variational fallacy, G. de Josselin de Jong, Géotechnique, June 1981, Vol. 31, no. 2, pp 289-290, The Institution of Civil Engineers. Reprinted by kind permission of the Institution of Civil Engineers [59] The simultaneous flow of fresh and salt water in aquifers of large horizontal extension determined by shear flow and vortex theory, G. de Josselin de Jong, Proc. Euromech. Colleg., Edited by A. Verruijt & F. B. J. Barends, Sept. 1981, pp 75-82, A. A. Balkema. Reprinted by kind permission of AA Balkema Publishers-Taylor and Francis The Netherlands

Selected Works of G. de Josselin de Jong

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362

Acknowledgements

[61] Cube with edges larger than those of the enclosing tessaract, G. de Josselin de Jong, Nieuw Archief voor de Wiskunde, 4, Vol. 3, 1985, pp 209-217 (+ introduction: pp 207-208), Wiskundig Genootschap. Reprinted by kind permission of ‘‘Nieuw Archief voor Wiskunde’’

[62] Transverse dispersion from an originally sharp fresh-salt Interface caused by shear flow, G. de Josselin de Jong and C. J. van Duijn, J. of Hydrology, 84, 1986, pp 55-79, Elsevier. Reprinted with kind permission of Elsevier [63] Elasto-plastic version of the double sliding model in undrained simple shear tests, G. de Josselin de Jong, Géotechnique, 38, No. 4, 1988, pp 533-555 (+ 63a discussion G. de Josselin de Jong Géot., 39, No. 3, 1989, pp 565-566). The Institution of Civil Engineers. Reprinted by kind permission of the Institution of Civil Engineers

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Soil Mechanics and Transport in Porous Media

Theory and Applications of Transport in Porous Media Series Editor: Jacob Bear, Technion – Israel Institute of Technology, Haifa, Israel 1.

H.I. Ene and D. Polisˇsevski: Thermal Flow in Porous Media. 1987 ISBN 90-277-2225-0

2.

J. Bear and A. Verruijt: Modeling Groundwater Flow and Pollution. With Computer Programs for Sample Cases. 1987 ISBN 1-55608-014-X; Pb 1-55608-015-8

3.

G.I. Barenblatt, V.M. Entov and V.M. Ryzhik: Theory of Fluid Flows Through Natural Rocks. 1990 ISBN 0-7923-0167-6

4.

J. Bear and Y. Bachmat: Introduction to Modeling of Transport Phenomena in Porous Media. 1990 ISBN 0-7923-0557-4; Pb (1991) 0-7923-1106-X

5.

J. Bear and J-M. Buchlin (eds.): Modelling and Applications of Transport Phenomena in Porous Media. 1991 ISBN 0-7923-1443-3

6.

Ne-Zheng Sun: Inverse Problems in Groundwater Modeling. 1994 ISBN 0-7923-2987-2

7.

A. Verruijt: Computational Geomechanics. 1995

8.

V.N. Nikolaevskiy: Geomechanics and Fluidodynamics. With Applications to Reservoir Engineering. 1996 ISBN 0-7923-3793-X

9.

V.I. Selyakov and V.V. Kadet: Percolation Models for Transport in Porous Media. With Applications to Reservoir Engineering. 1996 ISBN 0-7923-4322-0

10.

J.H. Cushman: The Physics of Fluids in Hierarchical Porous Media: Angstroms to Miles. 1997 ISBN 0-7923-4742-0

11.

J.M. Crolet and M. El Hatri (eds.): Recent Advances in Problems of Flow and Transport in Porous Media. 1998 ISBN 0-7923-4938-5

12.

K.C. Khilar and H.S. Fogler: Migration of Fines in Porous Media. 1998 ISBN 0-7923-5284-X

13.

S. Whitaker: The Method of Volume Averaging. 1999

14.

J. Bear, A.H.-D. Cheng, S. Sorek, D. Ouazar and I. Herrera (eds.): Seawater Intrusion in Coastal Aquifers. Concepts, Methods and Practices. 1999 ISBN 0-7923-5573-3

15.

P.M. Adler and J.-F. Thovert: Fractures and Fracture Networks. 1999 ISBN 0-7923-5647-0

16.

M. Panfilov: Macroscale Models of Flow Through Highly Heterogeneous Porous Media. 2000 ISBN 0-7923-6176-8

17.

J.M. Crolet (ed.): Computational Methods for Flow and Transport in Porous Media. 2000 ISBN 0-7923-6263-2

ISBN 0-7923-3407-8

ISBN 0-7923-5486-9

18.

R. de Boer: Trends in Continuum Mechanics of Porous Media. 2005 ISBN 1-4020-3143-2

19.

R.J.Schotting, H. (C. J.) van Duijn and A.Verruijt (eds.): Soil Mechanics and Transport in Porous Media. Selected Works of G. de Josselin de Jong. 2006 ISBN 1-4020-3536-5

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