Understanding the Micro to Macro Behaviour of Rock-Fluid Systems
Geological Society Special Publications Society Book Editors R. J. PANKHURST (CHIEF EDITOR) P. DOYLE F. J. GREGORY J. S. GRIFFITHS A. J. HARTLEY R. E. HOLDSWORTH J. A. HOWE P. T. LEAT A. C. MORTON N. S. ROBINS J. P. TURNER
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It is recommended that reference to all or part of this book should be made in one of the following ways: SHAW, R. P. (ed.) 2005. Understanding the Micro to Macro Behaviour of Rock-Fluid Systems. Geological Society, London, Special Publications, 249. BLOOMFIELD, J. P. & BARKER, J. A. 2005. MOPOD: a generic model of porosity development. In: SHAW, R. P. (ed.) 2005. Understanding the Micro to Macro Behaviour of Rock-Fluid Systems. Geological Society, London, Special Publications, 249, 73-77.
GEOLOGICAL SOCIETY SPECIAL PUBLICATION NO. 249
Understanding the Micro to Macro Behaviour of Rock-Fluid Systems EDITED BY
R. P. SHAW British Geological Survey, UK
2005 Published by The Geological Society London
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Contents
Preface
SHAW,R. P. Understanding the Micro to Macro Behaviour of Rock-Fluid Systems:
vii 1
introduction
HEFFER,K. J. The NERC Micro to Macro Programme: implications for fluid resource management LIu, E., CHAPMAN,M., HUDSON, J. A., TOD, S. R., MAULTZSCH, S. & Li, X-Y. Quantitative determination of hydraulic properties of fractured rock using seismic techniques
29
ODLING, N. E., HARRIS,S. D., VASZI,A. Z. & KNIPE,R. J. Properties of fault damage zones in siliclastic rocks: a modelling approach
43
XIE, Z., MACKAY,R. & CLIFFE,K. A. Precise numerical modelling of physical
61
transport in strongly heterogeneous porous media
BLOOMFIELD,J. P. & BARKER,J. A. MOPOD: a generic model of porosity development
73
SELLERS,S. & BARKER,J. A. Anomalous diffusion in simulations of pumping tests on fractal lattices
79
JOHNSTON,P. B., ATKINSON,T. C., ODLING,N. E. & BARKER,J. A. Models of tracer breakthrough and permeability in simple fractured porous media
91
WORDEN, R. H., CHARPENTIER,D., FISHER, Q. J. & APLIN,A. C. Fabric
103
development and the smectite to illite transition in Upper Cretaceous mudstones from the North Sea: an image Analysis Approach
CASSIDY, R., MCCLOSKEY,J. & MORROW,P. Fluid velocity fields in
115
2D heterogeneous porous media: empirical measurement and validation of numerical prediction BRYDIE, J. R., WOGELIUS, R. A., MERRIFIELD, C. M., BOULT, S., GILBERT, P., ALLISON, D. & VAUGHAN,D. J. The ix2M project on quantifying the effects of biofilm growth on hydraulic properties of natural porous media and on sorption equilibria: an overview
131
SHAW, R. P. Overview of the NERC 'Understanding the Micro to Macro Behaviour of Rock-Fluid Systems'
145
Index
163
Preface
Understanding how fluids flow through rocks is very important in a number of fields. Almost all of the world's oil and gas are produced from underground reservoirs and knowledge of how these energy resources got where they are, what keeps them there and how they migrate through the rock, is very important in the search for new resources as well as for extracting as much of the contained oil/gas as possible. Similar understanding is important for managing groundwater resources and also for predicting how hazardous or radioactive wastes and carbon dioxide will behave if they are stored or disposed of underground. Unravelling the complex behaviour of fluids as they flow through rock is difficult. We can't see through rock, so we need to predict how and where fluids flow and at what rates. This requires an understanding of the type of rock, its porosity, and the character and pattern of fractures within it. Fluid flow can vary with time and over a range of scales, from microscopic pores and cracks to major fault zones. Some of Micro to Macro researchers have been studying rocks from boreholes, excavations and elsewhere, and gathering information from seismic surveys, in an attempt to understand how fluids flow in real rocks in real situations. Others have been working on computer models and laboratory simulations of fluid flow through porous and/or fractured rocks. Put together, these approaches have yielded very useful results, many of which are discussed in this volume. Industries whose resources lie in the subsurface, base most of their planning and investment decisions on models of their sites that require numerical descriptions of the geology. The commercial consequences of poor geological modelling can be particularly severe where fluid flow is involved because fluid flow is governed by the spatial arrangement of extremes in the range of permeabilities. The Micro to Macro Programme has been focused on developing our understanding of the relationships between measured and modelled sub-surface fluid flows spanning the range of spatial and temporal scales relevant to fluid resource management. The programme was motivated by observations and emerging theories of how geological heterogeneities vary
across these ranges in scale, and the consequences for extrapolating fluid behaviour both in time and space; the aim was to provide a clearer physical understanding on which to base more effective geofluid management, and to allow better integration of data for reservoir characterization and improved models for fluid flow. The scope of the programme necessarily involved workers with backgrounds in the hydrocarbon, water, radioactive waste, mining, and geothermal industries and a major objective was to foster communication between disciplines and communities to their mutual benefit. As a result many of the projects funded by the Programme will be of considerable interest to those looking at upscaling issues in the hydrocarbon, groundwater resource and waste disposal (including radioactive waste) industries. In order to highlight some of the results of the Programme to industry, the Steering Committee commissioned Kes Heifer to provide a review of the results of the Programme with implications for the management of fluid resources which forms the basis of Chapter 1 of this volume. While this review is focused on the hydrocarbon industry, it is equally applicable to other sectors where understanding fluid flow is important. One of the purposes of this volume is to disseminate the principal results of the Natural Environment Research Council's (NERC) thematic programme 'Understanding the Micro to Macro Behaviour of Rock Fluid Systems', commonly referred to as 'p~2M', and it forms part of the dissemination strategy of the Programme. This s programme ran from 1998 to 2004 and provided funding to 17 projects following two calls for proposals. In common with other NERC thematic programmes, this Programme was overseen by a steering committee with representatives from industry and academia with expertise and experience in the topics covered by the Programme and knowledge of their potential application. An overview of the Micro to Macro Programme is provided in the last paper of this volume. As well as this book a principal means of disseminating information arising from the Micro to Macro Programme is via a web site,
vii
viii
PREFACE
maintained by the data managers, the British Geological Survey, at http://www.bgs.ac.uk/ micromacro/about.html (or linked from http:// www.nerc.ac.uk/funding/thematics/m2m/) where project updates on most individual projects and links to some of the research
departments can be found). This site will be accessible for at least three years after publication of this volume. Richard Shaw British Geological Survey, Nottingham
Understanding the Micro to Macro Behaviour of R o c k - Fluid Systems: introduction RICHARD SHAW Scientific Co-ordinator, Micro to Macro, British Geological Survey, Keyworth, Nottingham NG12 5GG, UK
The purpose of this volume is to disseminate the principal results of the Natural Environment Research Council's (NERC) thematic programme 'Understanding the Micro to Macro Behaviour of Rock-Fluid Systems', commonly referred to as 'tx2M', and it forms part of the dissemination strategy of the programme. This s programme ran from 1998 to 2004 and provided funding to 17 projects following two calls for proposals. In common with other NERC thematic programmes, this programme was overseen by a steering committee with representatives from industry and academia with expertise and experience in the topics covered by the programme and knowledge of their potential application. An overview of the Micro to Macro Programme is provided in the last paper in this volume. Understanding how fluids flow through though rocks is very important in a number of fields. Almost all of the world's oil and gas are produced from underground reservoirs and knowledge of how these energy resources got where they are, what keeps them there and how they migrate through the rock is very important in the search for new resources as well as for extracting as much of the contained oil/gas as possible. Similar understanding is important for managing groundwater resources and also for predicting how hazardous or radioactive wastes and carbon dioxide will behave if they are stored or disposed of underground. Unravelling the complex behaviour of fluids as they flow through rock is difficult. We cannot see through rock, so we need to predict how and where fluids flow and at what rates. This requires an understanding of the type of rock, its porosity and the character and pattern of fractures within it. Fluid flow can vary with time and over a range of scales, from microscopic pores and cracks to major fault zones. Some of the researchers in the Micro to Macro Programme have been studying rocks from boreholes, excavations and elsewhere, and gathering information from seismic surveys, in an attempt to understand how fluids flow in real rocks in real situations.
Others have been working on computer models and laboratory simulations of fluid flow through porous and/or fractured rocks. Put together, these approaches have yielded very useful results, many of which are discussed in this volume. Industries whose resources lie in the subsurface base most of their planning and investment decisions on models of their sites that require numerical descriptions of the geology. The commercial consequences of poor geological modelling can be particularly severe where fluid flow is involved because fluid flow is governed by the spatial arrangement of extremes in the range of permeabilities. The Micro to Macro Programme has been focused on developing our understanding of the relationships between measured and modelled subsurface fluid flows, spanning the range of spatial and temporal scales relevant to fluid resource management. The programme was motivated by observations and emerging theories of how geological heterogeneities vary across these ranges in scale and the consequences of extrapolating fluid behaviour both in time and space; the aim was to provide a clearer physical understanding on which to base more effective geofluid management and to allow better integration of data for reservoir characterization and improved models for fluid flow. The scope of the programme necessarily involved workers with backgrounds in the hydrocarbon, water, radioactive waste, mining and geothermal industries and a major objective was to foster communication between disciplines and communities to their mutual benefit. As a result, many of the projects funded by the programme will be of considerable interest to those interested in upscaling issues in the hydrocarbon, groundwater resource and waste disposal (including radioactive waste) industries. As well as this book, a principal means of disseminating information arising from the Micro to Macro Programme is via a website, maintained by the data managers - the British Geological Survey - at http://www.bgs.ac.uk/micromacro/ about.html (or linked from http://www.nerc. ac.uk/funding/thematics/mZm/) where project
From: SHAW,R. P. (ed.) 2005. Understanding the Micro to Macro Behaviour of Rock-Fluid Systems. Geological Society, London, Special Publications, 249, 1-3. 0305-8719/05/$15.00 9 The Geological Society of London 2005.
2
R. SHAW
updates on most individual projects and links to some of the research departments can be found). This site will be accessible for at least three years after publication of this volume. The first paper by Heifer provides a review of the results of the programme, with implications for the management of fluid resources. While this review is focused on the hydrocarbon industry, it is equally applicable to other sectors where understanding of fluid flow is important. The remaining papers are ordered approximately in decreasing scale of the main focus of the project from large (macro) to small (micro) scales. Fractures and fracture systems control much of the mechanical strength and fluid transport properties of rocks and are crucial for hydrocarbon production, control and manipulation of water supplies and the dispersal of pollutants. Liu et al. propose the use of seismic methods, based on the phenomenon of shear-wave splitting, for the quantitative determination of open fractures that may form flow pathways, and cemented fractures that may form significant bartiers to flow within a rock mass. Oldling et al. describe a modelling approach to understanding fluid flow through fault damage zones in siliclastic rocks using parameters for fault length and orientation distributions, fault aspect ratio, length-thickness relations both for a single fault and for fault populations, and the fault spatial distribution to generate geologically realistic stochastic models of fault damage zones. These models can then be used to model fluid flow through fault zones. Xie et al. examine several promising upscaling approaches and carry out spatial and temporal analysis of the modelling results to quantify the accuracy and bias of each alternative upscaling method. From this analysis they have determined the limits of applicability of existing upscaling laws and identified improved laws. An important output from the research has been the development of a suite of publicly available, high-resolution, accurate flow and transport simulation datasets comprised of a large number of realizations possessing the large variance and strong textures observed in geological systems. Bloomfield & Barker develop a model of coupled flow and porosity development in heterogeneous porous (fractured) media and use the model to investigate porosity growth phenomena. In order to gain some insight into the range of possible behaviours to be expected from pumping tests, as well as the type of theoretical models needed, Sellers & Barker perform extensive simulations of pressure diffusion for transient groundwater flow, modelled by random walks
on both deterministic and random fractal lattices. For simplicity, this work focused on measurements of the random-walk dimension for generalized Sierpinski carpets, a proposed model for porous and fractured media. Johnston et al. explore, within a simplified modelling framework, the prospects for understanding characteristics of the internal heterogeneities in a medium from evidence provided by tracer experiments. Tracers are harmless marker liquids introduced into an aquifer and their breakthrough is when they are detected at a sampling point some distance away. Field tracer experiments give rise to a variety of tracer breakthrough curves showing distinct characteristics which can be classified into four general types: Fickian; backward tailed; bimodal and multimodal. The Fickian-type curve is typical of a homogeneous and isotropic formation. The other types are thought to arise from flow in more heterogeneous formations. This study demonstrates that different types of breakthrough nfight be characteristic of particular sets of conceptual models for heterogeneities and, as such, may provide a useful pointer in the application and interpretation of tracer tests. Using X-ray diffraction, mercury porosimetry and electron microscopy, Worden et aL have studied the small-scale textures of Upper Cretaceous Shetland Group mudstone cuttings from a range of depths in the Northern North Sea. Relatively shallow samples (1615 m) have an anisotropic mudstone fabric dominated by smectite and have porosity values of approximately 35%. In contrast, more deeply buried samples (3300 m) have developed an isotropic fabric and are dominated by illite and have porosity values of approximately 22%. Image analysis of differentially buried mudstones has proved to be a rapid, flexible and quantitative method for characterizing mudstone textures. The coincidence of mineralogical evolution with textural development and compaction implies that the transformation of smectite to illite occurs by dissolution and precipitation and that chemically facilitated compaction may contribute to porosity loss. Cassidy et al. have developed physical models of complex 2D media with fractal heterogeneity which they use to measure fluid velocity fields. The scale invariance of geological material, and the consequent absence of a length scale on which to base the upscaling of measurements made on geological samples, represents a serious challenge to the prediction of fluid behaviour in rock at economically interesting scales. Numerical simulation is an important tool for understanding constraints in this problem and current
INTRODUCTION discrete fluid models in which complex boundary conditions can be represented have the potential for testing many possible upscaling schemes. At present, however, there are no accurate empirical data on the distributions of fluid velocities in complex, scale-invariant geometries. Their work has started to address this issue. The physical and chemical effects of bacterial biofilm formation upon hydraulic conductivity, mineral-solution interactions and the formation of biogenic mineral precipitates are studied by Brydie et al. over a wide range of scales, from microscopic to macroscopic. In the laboratory, biofilm formation within quartz sand in artificial groundwater resulted in a two orders of magnitude reduction in hydraulic conductivity under constant head conditions. However, under quasi-environmental conditions within macroscopic centrifuge experiments, a reduction of 21% was measured. Evaluation of biofilms within simulated quartz rock fractures and in porous media reveals only a small percentage
3
of the biomass to be in direct contact with the mineral surface, allowing mineral chemistry to be predominantly controlled by mineral surface reactivity. The alteration of mineral surface drastically increases the kinetics of surfacecoordinated trace metal precipitate formation by providing nucleation sites upon extracellular biopolymers (EPS) and cell wall polymers. Over geological time-scales, these processes, particularly the formation of thermodynamically stable pore-blocking mineral precipitates, are envisaged to change markedly the flow paths, flow rates and interaction of migrating geofluids, including water, petroleum, ore-forming solutions, with minerals and rocks. The editor gratefully acknowledges the contribution of all authors who have provided papers for this volume and is indebted to members of the steering committee, many colleagues and specialists for their help in reviewing the papers and for their helpful comments resulting from the reviews.
The NERC Micro to Macro Programme: implications for fluid resource management K. J. HEFFER
Institute o f Petroleum Engineering, Heriot Watt University, Edinburgh EH14 4AS, UK
Abstract: The Micro to Macro (I~2M) Programme has been focused on developing understanding of subsurface fluid flows within geological heterogeneities spanning wide ranges of spatial and temporal scales. This paper highlights the opportunities for industries to incorporate recent observations and emerging theories in this field towards improved fluid resource management. The background to, and objectives of, the 1~2M Programme are reviewed. Selected results from the projects in the programme are discussed and, where possible, compared with evidence from industrial field data. Some conclusions and recommendations for future practice in reservoir characterization are made. For example, there is currently very little recognition of modern theories that point to the likelihood of prevailing criticality in the mechanical state of the Earth's crust and its implication for coherent large-scale collective behaviour emerging from small-scale interactions. Also associated with criticality are long-range spatial correlations and the likelihood that flow properties change during the life of commercial developments: such changes, for example, to absolute permeability, should be looked for and analysed for spatial and temporal patterns. Allied with these features is the importance of coupled processes, principally geomechanics, fluid flow, heat flow and chemistry. Knowing that local faults and fractures play a strong role in fluid flow mechanisms in a potentially time-varying, rather than just a static, fashion, gives even more motivation for acquiring detailed information on micro- and macro-structure over a range of scales.
Industries whose resources lie in the subsurface base most of their planning and investment decisions on models of their sites that require numerical description of the geology. Such modelling has often turned out to be inadequate. The commercial consequences of poor geological modelling can be particularly severe where fluid flow is involved because fluid flow is governed by the spatial arrangement of extremes in the range of permeabilities. The Micro to Macro (p~2M) Programme has been focused on developing understanding of the relationships between measured and modelled subsurface fluid flows, spanning the range of spatial and temporal scales relevant to fluid resource management. The programme was motivated by observations and emerging theories of how geological heterogeneities vary across these ranges in scales, and the consequences for extrapolating fluid behaviour both in time and space; the aim was to provide a clearer physical understanding on which to base more effective geofluid management and to allow better integration of data for reservoir characterization and improved models for fluid flow. The scope of the project involved workers with backgrounds in hydrocarbon, water, radioactive waste,
mining, and geothermal industries and a major objective was to foster communication between disciplines and communities to mutual benefit. In order to place the aims and achievements of the ~2M Programme into context, it is worth first outlining the current standard practice in exercises of characterizing the geology of subsurface commercial resources. Of course, this outline can only be of a general norm, about which there will be, in any one industry, examples of greater or less sophistication.
Current standard practice in characterization of geology and its shortfalls Efforts to improve the realism of spatial distributions of heterogeneity in exercises of reservoir characterization in the oil industry began in the late 1970s and early 1980s, essentially with liaison between sedimentologists, geostatisticians and reservoir engineers. Parallel developments began in the groundwater industry. Models of spatial covariance in heterogeneities were dominated by the statistics of sedimentological data, gleaned mostly from outcrop studies.
From: SHAW,R. P. (ed.) 2005. Understanding the Micro to Macro Behaviour of Rock-Fluid Systems. Geological Society, London, Special Publications, 249, 5-27. 0305-8719/05/$15.00 9 The Geological Society of London 2005.
6
K.J. HEFFER
Most early applications employed limited range variograms and Gaussian frequency distributions. Alternatively, geological bodies were modelled as 'objects' distributed in space, with correlated internal heterogeneities. Later, methods were developed to incorporate so-called 'soft' information on heterogeneities from seismic data. The pioneering work of Hewett (1986) in using fractal interpolation functions (fractional Brownian motion and fractional Gaussian noise) has been applied to many reservoirs since (e.g. Hardy & Beier 1994). However, such modelling has lacked a detailed geoscientific basis, and is, therefore, probably incomplete, for example in anisotropy or relationship to other known structural features. Treatment of structural discontinuities in characterization models was led by the geothermal, mining and radioactive waste industries. Initially, in the hydrocarbon industry, only large, seismically 'visible' faults were included in reservoir models, mainly as disruptions to the geometric continuity of beds and possibly as 'sealing' membranes. Only recently have characterizations begun to incorporate statistical models of fractures and 'sub-seismic' faults, including variability and anisotropy in their properties. However, a notable exception is conductivity of the faults or fractures, which is often assumed to be uniform and uncorrelated with other properties. Also, many 'realizations' of fracture or fault patterns in stochastic modelling exercises do not appear very realistic to the eyes of structural geologists. More fundamental amongst the deficiencies of current practice in any of the industries is that there is very little recognition of modem theories that point to the likelihood of prevailing criticality in the mechanical state of the Earth's crust and its implication for coherent large-scale collective behaviour emerging from small-scale interactions. This is analogous to the critical point phenomena that occur in continuous phase transitions (in liquid-gas mixtures, metallurgy, magnetism, (super-) conductors, etc.) in thermodynamic equilibrium and on which there is a rich literature. The word 'critical' appears in several contexts in this paper, which, although related and in common use, can cause some confusion; Appendix A attempts to distinguish and clarify those contexts. Concomitant with criticality are long-range correlations, power-law distributions, strong susceptibility to perturbation and the magnification of anisotropies. Allied with these features is the importance of coupled processes, principally geomechanics, fluid flow, heat flow and chemistry. The field evidence for criticality
and its application to hydrocarbon reservoirs are given in Appendix B. Omission of these issues in resource characterization can have many practical consequences. Crampin (1999) outlines some implications, and others are implicit in the results of the individual projects of the ix2M Programme. Two of the key implications will be manifest in both 'static' and 'dynamic' aspects of characterization. 9
9
In 'static' modelling, for example, as well as the immediate implication to use variograms with long-range correlation, there is also the consequence that conditioning of stochastic geostatistical models should incorporate distant measured data points as well as more local measurements. More importantly, there is a need to understand the full 3D nature of the scaling that has been observed in many 1D well-log sequences. One possibility is that such scaling has an origin associated with coupled processes at a critical point as outlined above, either modern-day or ancient. If so, there may well be structural patterns to the heterogeneities, implying lineations, strong anisotropy and possible association with older structural trends. In 'dynamic' modelling, the strong stresssensitivity of fault and fracture properties, imply that system permeabilities are likely to change over the development life of a field and that those changes may also exhibit long-range correlations (see also Crampin 1999, 2000).
Currently, time-lapse seismic surveys are showing good promise as a direct means to monitor changing inter-well properties. However, in order to be able to invert the seismic responses with a model containing the complete physics it will be important to incorporate the influence of geomechanical changes in not only the reservoir, but also the over-, under- and side-burdens, on (a) the seismic responses themselves and (b) the reservoir permeability, compressibility and flow behaviour. The prospect of making significant progress with understanding and predicting these complex characteristics of heterogeneities that cover many orders of magnitude in scale was a prime incentive for the ix2M Programme.
Scaling in well-log measurements An allied stimulus for the ~2M Programme was the pre-existing set of observations of spatial correlation in the fluctuations of well-log measurements. Spatial correlation can be described
MICRO TO MACRO PROGRAMME: IMPLICATIONS through its Fourier transform, the powerspectrum, which provides the amount of 'power' in the fluctuations at each spatial frequency, or wavenumber, k. Many researchers (e.g. Hewett 1986; Bean & McCloskey 1993; Bean 1996; Holliger 1996; Dolan et al. 1998; Leary 1998, 2002; A1-Kindy 1999; Marsan & Bean 1999; Leary & A1-Kindy 2002) found that fluctuations in heterogeneities in well logs show scaling of a type that is often described as '1/f', 'flicker' or 'pink' noise. In contrast with 'white' noise, in which the power is distributed evenly over all frequencies, the power in 'pink' noise is distributed evenly in logarithm of frequency. For example, there is as much noise power in the octave 2 0 0 - 4 0 0 Hz as there is in the octave 2000-4000 Hz. 'Pink' noise is the most natural sound to human ears. In terms of wavenumber, k, the spectral power densities of the heterogeneities show power-law behaviour:
S(k) ~ 1/k t~
(1)
where/3 ~ 1.0 to 1.6 (see Fig. 1a). For example, A1-Kindy (1999) found average scaling exponent values/3 = 1.02 ___ 0.1 for 245 logs in both sedimentary and crystalline rocks. The power-law behaviour implies that there is no natural scale to the fluctuations. It is worth examining some of the issues and previous work surrounding
7
these scaling relationships in more detail, although it is fair to say that understanding of the origin for the case of natural rock heterogeneities is still limited and that there is a need for further validation in some aspects.
Potential causes o f 1 / k scaling in heterogeneities The 1/k scaling in well logs has been interpreted as symptomatic of the involvement of self-organized criticality (SOC - see Appendix A) in structural deformation, for which there exists many other indications (e.g. Crampin 1994, 2000; Main 1996; Grasso & Sornette 1998; Leary 1998, 2002; Crampin & Chastin 2000). There are, however, several issues surrounding this interpretation that requh'e further investigation. One problem is that the observed 1/k scaling in well logs, although of a power-law nature, is not consistent with power spectra calculated for usual models of critical phenomena in equilibrium thermodynamics, in which exponent /3 ~ 0 (e.g. Binney et aL 1992); nor with the analyses to date of far-from-equilibrium SOC (Somette et aL 1990; Tang & Bak 1988; Somette 2000). This issue has received some attention (Leary 1998; Heifer in press), but still requires resolution.
Fig. 1. (a) Typical power spectra of well logs showing N 1/k behaviour (Marsan & Bean 1999). Copyright (1999) American Geophysical Union. Reproduced by permission of American Geophysical Union). (b) Spatial correlation functions corresponding to fluctuations described by fractional Brownian motion with various values of the Hurst exponent, H; compared with a more common correlation function used in reservoir description (corresponding to an exponential variograrn) with a finite range (indicated by double headed arrow). Note that the fractional Brownian motion correlations have infinite range but with a significant 'nugget' effect.
8 2.
K.J. HEFFER Anisotropy may exist in the scaling: Somette et al. (1990) developed field equations for a
3.
4.
scalar order parameter representing strain in a SOC model of the lithosphere that scales with distance differently for directions either parallel or orthogonal to the main direction of strain transport. Might, for example, the sensitivity of scaling in well logs with deviation be due to horizontal wells sampling across faults/fractures formed in extensional or strike-slip regimes, whilst the vertical wells are sampling sub-parallel to them? Another remaining puzzle is that the spectral densities of well-logs imply antipersistence (i.e. any two consecutive intervals of log, of any length scale above that resolved by the instrument, are anticorrelated: a positive increment of the log is followed, on average, by a negative increment). The heterogeneity distributions in well logs can be modelled with fractional Brownian motion (fBm) with a Hurst coefficient (Hurst et al. 1965), H = ( / 3 - 1)/2. This implies that H < 0.5 and usually ~0. This is in contrast to the persistenee (i.e. a positive increment is followed, on average, by another positive increment) (H > 0.5) found in the long-run behaviour of other geophysical records related to the weather and climate (e.g. Mandelbrot & Wallis 1969; Feder 1988). Leary (2002) has pointed out that well logs are better fitted with fractional Gaussian noise (fGn), such that the fBm that forms the integral of the fGn will show persistence, with H ~ 1. If the scaling of well-log heterogeneities is attributable to strain fluctuation, then its integral will correspond to fluctuations in displacement (the vector joining the initial and final positions of a point in deforming rock). Intuitively, the latter are, indeed, expected to be persistent. Behaviour of a 1/k nature is found in sedimentary rocks as much as crystalline (Leary & A1-Kindy 2002). Although the origin of scaling is often attributed to the scaling of the fracture set along the borehole (Leary 1991; Holliger 1996), Bean (1996) showed that scaling in the lithology distribution can also be taken as a contributing cause. Bean (1996) has examined this scaling carefully in wells penetrating both volcanic and sandstone facies. There is a slight difference in the scaling exponents between these facies. Dolan et al. (1998) concluded that the fractal dimension obtained from well logs does vary with lithology, but the difference is slight and not detectable
by rescaled range or power-spectral techniques for the available data. Dolan et al. (1998) also stated that the fractal dimensions are different because the controlling mechanisms are different: primary porosity in the clastics and fracture porosity in the volcanics. However, both produce antipersistence. Walden & Hosken (1985) also noted anti-correlations in reflection coefficients at small lags in sedimentary sequences, and cited the importance of this property to the viability of the seismic reflection method. Heifer (in press) has pointed out that the scaling of stiffness modulus at the critical point of failure, as determined in several investigations (e.g. Chakrabarti & Benguigui 1997), is consistent with exponent/3 taking a value ~ 1 in the power spectrum of strain: this supports the role of strain in the fluctuations demonstrated by heterogeneities in well logs, particularly in crystalline rocks where fractures are the main heterogeneity. Dolan et al. (1998) appealed to the fractal dimensions of pore-space distributions in sedimentary rock (Krohn 1988), reporting Hurst coefficients for sandstones similar to those from the porosity tools. However, it is difficult to imagine that the geometry of pore space at grain scales and below would be continued to the larger scales investigated by logging tools if the original depositional process were entirely responsible. It is more likely that the similarity of the fractal dimensions of porosity in unfractured rock with those of rock whose porosity does derive mainly from fractures, is due to tectonic/deformational influences on diagenetic processes (compaction, dissolution, cementation, pressure solution) which over-write the statistics of porosity derived from the original depositional process. The influence of tectonism on deposition (e.g. in controlling avulsions of fluvial systems or the accommodation space available for sedimentation) is also probably significant. Practical factors of measurement need to be considered; in particular, the influence of the stress field surrounding the wellbore on measurements by wellbore tools. There are other causes of 1 / k scaling than SOC. Somette (2000, Chapter 14), examines various mechanisms for power laws. Hooge et al. (1994) have argued that seismic processes are scaling tensor multifractal fields (of e.g. strain or stress) in both space and time. In addition, Li (1991) noted that scale invariance usually derives from balance
MICRO TO MACRO PROGRAMME: IMPLICATIONS between two opposing tendencies. In the context of fracturing, the complex patterns surrounding each fracture of positive and negative stress changes, which act to encourage and inhibit further fractures in the vicinity, are potential candidates to fill the role of opposing tendencies.
Implications of 1/k scaling of heterogeneities for stochastic modelling Irrespective of the origin of 1/k scaling, what is
9
the issues discussed above. Reference to the worker(s) on a tx2M project (a list of these appears separately at the beginning of the 'References' section below) is made in the usual manner, but with the acronym '(ix2M)' replacing a year. It is emphasized that the selected results represent only a small proportion of the overall outcome of the programme; other papers in this volume provide more detail of a fuller scope.
Scaling of diagenetic overprint
a partial loss of predictability from well data even in the immediate surroundings; (b) a long-range correlation that has much more widespread influence than 'usual' variograms with finite ranges (see Fig. lb). Crampin (1999), following Leary (1996), has given example realizations of heterogeneities modelled with 1/k spectral densities in contrast with white noise (constant spectral density for all wavenumbers). Crampin (1999) noted 1/k noise implies that fluctuations at long wavelengths are greater than at short wavelengths, implying strong clustering in the distributions of physical properties. However, the degree of difficulty that 1/k noise poses for reservoir geostatistics is still to be evaluated fully: the property of long-range correlation, in that it 'projects' the spatial influence of measurements, may aid the task of interpolation, as long as anisotropy in correlations is catered for. Heifer (2002, in press) is engaged in developing a methodology for interpolating strain and associated indicators, as illustrated further in Figure 10 and its associated text.
What is the influence of diagenesis on the scaling of heterogeneities seen in well logs? Haszeldine et al. (Ix2M) have validated a new non-destructive screening tool, based on measuring the magnetic susceptibility of the sample, for measuring the content of certain clays, in particular illite, quickly and cheaply. By examining samples from a shoreface facies at different depths in a North Sea reservoir, the co-workers have shown that permeability is correlated strongly with percentage illite content as measured with the new tool, with the interpretation that the illite is filling the remanent pore space left by quartz overgrowths from a previous diagenetic episode. The measurement technique has also been applied to, and is helping to explain the diagenetic histories of, other North Sea reservoirs, including the interpretation of cementation of faults in the Moray Firth through hot fluids advecting cement from the deeper basin. An additional investigation which is highly relevant to the explanation of the 1/k spectral densities of well logs was collection of values of illite % from foot-by-foot core samples, so that the power spectrum could be calculated from this larger bandwidth. A strong spatial correlation between porosity and permeability has been reported in Brae oil field sediments, together with a systematic power-law scaling of log (permeability) over spatial frequencies from 5 km-1 to 3000 km-1 (Leary & A1-Kindy 2002). This was interpreted to result from longrange correlated fracture-permeability networks. The power spectrum of the illite data ostensibly indicated a spatial correlation exponent of 0.54, in line with the porosity and permeability correlation. However, the interpretation is not definitive: the errors involved in the transform from the magnetic data to illite % may have interfered with the interpretation of correlation.
Selected results of the Micro to Macro Programme
Evolution of fracture systems through diagenesis
The following results of the ~2M Programme have been selected on the basis that they illustrate
Diagenetic changes to a pre-existing fracture system can alter its properties significantly. Full
the practical significance of this in reservoir characterization, particularly stochastic modelling exercises? The spatial correlation that is equivalent to 1/k spectral densities (strictly generalized autocovariance function, GACV) is ~log(r), where r is the lag distance (Greenhall 1999). For spectral densities of the form 1/k t~, where/3 r 1, the GACV varies with lag distance as r (t~- 1~.These covariance functions are obviously long-range in nature, although they have a sharp drop-off at small lag distance (Fig. lb). These forms of correlation imply:
(a)
10
K.J. HEFFER
coupling of chemistry with thermal, hydraulic and mechanical processes can be involved, because permeability is often associated with periods of tectonism. However, a lower degree of coupling can arise from the passage of groundwaters through mechanically stable rock, changing the permeability by erosional and/or chemical processes. Such restricted coupling may be applicable to sedimentary aquifers, particularly fractured sandy aquifers or fractured carbonate aquifers, such as the Chalk aquifer of NW Europe, which may be modified significantly over relatively short geological time-scales. Bloomfield & Barker (tx2M) have developed a 2D model (MOPOD) to investigate general relationships in fracture aperture growth and the geometry of evolved fracture apertures using generic growth laws and simple fracture geometries. The work is intended as a precursor to future systematic studies of the emergent behaviour of dynamic fractured aquifer systems. Basic features of the evolved fracture aperture arrays were summarized by Bloomfield et al. (2005). Most pertinent to this discussion of scaling is that the effective permeabilities of the arrays increase as power-law functions of time; the exponent decreases with increase in the erosion parameter (Fig. 2). Effective permeabilities are also lower at the higher values of
20 18 16 14 r I.--
12 10
9 0
e=0.2 e=0.3
&
e=0.4
A 9 []
e=0.5 e=0.6 e=0.7
8
o
6
o
9 9
o o i
0
20
40
60
80
i
100
Time Fig. 2. Fracture porosity development modelled with a generic law for aperture growth (from Bloomfield et al. 2005). Effective transmissivities (TEFF) of the arrays increase as power-law functions of time; note that the exponent decreases with increase in the erosion parameter, e. (Reprinted from Ground Water, copyright (2005), with permission from Blackwell Publishing).
erosion rate: a single flowpath, albeit wider, is apparently less effective than the dispersed flowpaths. However, it is recognized that parameterization of such arrays and prediction of their evolution in terms of the initial boundary conditions are not trivial tasks. One possibility is to investigate multifractal properties of the spatial distributions of the fracture apertures at various stages of their development, in analogy to the analysis of Zhang & Sanderson (2002, Chapter 7). The modelling has some similarities with that of development of drainage networks by Hergarten & Neugebauer (2001), who argued that stationary patterns arising from fixed boundary conditions cannot reproduce the fluctuations characteristic of SOC; however, SOC characteristics were produced when boundary conditions were periodically changed. This might be another consideration to add to the list of future developments outlined by Bloomfield et al. (2005).
P e r m e a b i l i t y o f individual f r a c t u r e s
The characteristics of flow in an individual fracture have never been satisfactorily defined. The roughness of the fracture surfaces cause significant departures from the cubic law for flow-aperture relationship that is often deployed. Ogilvie et al. (Ix2M) have developed a new capability of non-destructive high-resolution profiling of fracture surfaces that avoids alignment problems of previous methods. From the results of such profiling new software is able to derive statistical parameters of the profiles of fracture surfaces and of the aperture between pairs of surfaces, in order to relate these to fluid flow. From the statistical parameters, synthetic fractures can be modelled with more software developed under the p~2M project. Flow experiments on High Fidelity Polymer Models (HFPM) in association with numerical FEMLAB T M modelling of the Navier-Stokes equation within suites of synthetic fractures have the potential to improve the characteristics of fluid flow modelling in rough fractures. An important influence on fluid and electrical transport within a rough fracture is the anisotropy of the fabric. Ogilvie et al. (lx2M) have demonstrated in an HFPM experiment the different characteristics of flow parallel to, and orthogonal to the fabric of the surface roughness. The anisotropy will, of course, be related to the geometry of deformation that created the fracture. Even more interesting will be two-phase flow experiments with these tools, especially perhaps the stress-sensitivity of
MICRO TO MACRO PROGRAMME: IMPLICATIONS two-phase properties of fractures, which are commonly just assumed at present.
Effective permeability o f fractured or faulted rock In deriving effective permeabilities for fractured rock with non-zero background matrix permeability, it is nearly always assumed that the fracture permeability can be locally added to the matrix permeability. On the contrary, using lattice Boltzmann simulation of flow in simplified 2D porous media over a range of solid fractions, Dardis & McCloskey (1998) illustrated the importance of matrix-fracture flow interactions. Figure 4 from Dardis & McCloskey (1998), reproduced here as Figure 3, indicates that the system permeability of fracture and matrix minus the fracture permeability is well in excess of the matrix permeability. That trend reproduces the similar laboratory results of Mattison et al. (1997). Permeabilities of fractures and matrix rock are non-additive. Fluid coupling seems to multiply (in fact by almost an order of magnitude) the effect of fractures on bulk permeability. This large field of influence of flow in a fracture on flow in the surrounding porous medium has also been demonstrated by further lattice Boltzmann modelling of the effect of a relatively sparse population of fractures, not connected, within a porous matrix; the fractures are modelled to cause increase in permeability much more than the nominal calculation of upscaled bulk permeability from, say, effective medium
11
calculations or direct fine-scale modelling of the system as a macroscale continuum. It seems that the pore-scale feedback from fracture to matrix combines with a feedback from matrix to fracture (J. McCloskey, pers. comm. 2002). The spatial extent of the influence of the fracture flow is widespread across the matrix domain. There are potential implications from this finding for many aspects of fluid flow in fractured rock, including influence on relative as well as absolute permeabilities and even on the attenuation factors for seismic waves. Further investigation of these effects is warranted, including perhaps the influence of viscosity (the modelling was, of necessity, run with a relatively high viscosity). Also vital, however, is a means of validating the numerical modelling with a physical model: this was the task of Cassidy et aL (lx2M), who have developed a particle imaging apparatus with which fluid velocities throughout a complex 2D medium can be measured accurately. The velocity fields measured with this apparatus compare visually very well with those predicted by lattice Boltzmann modelling on the same pattern of heterogeneity. However, although the validation has been very successful semi-quantitatively, the lattice Boltzmann modelling is, as yet, unable to simulate the low viscosities of the physical modelling, which remains a task for a future project. An implication that is potentially very important to fluid resource management is that conductive fractures, even before they become connected, can significantly increase the bulk permeability. As well as investigating
Fig. 3. Non-additive influence of fracture and matrix permeabilities - from lattice Boltzmann modelling (after Dardis & McCloskey 1998, copyright (1998) American GeophysicalUnion. Modifiedby permission of American Geophysical Union): (a) configuration of fracture in host rock and typical velocity profiles; (b) modelled effective permeability of fractured media, Kfm,reduced by the fracture permeability, Kf, is much greater than the unfractured matrix permeability, Kin.
12
K.J. HEFFER
non-linear interactions between fracture and matrix flows, Cassidy et al. (lx2M) have developed the ability to examine scaling laws near a percolation threshold (with fractal fracture population and matrix permeability) and scaling of the velocity flow field in comparison with scaling of the material geometry. Harris et al. (Ix2M) have modelled the effect of complex fault structure on fluid flow, to date for the case where faults have lower permeabilities than the host rock. Their methodology can cope with conductive faults but such have not been studied within the Ix2M project. The work has assumed configurations of fault damage zones based on a large background of observational data. A hierarchical clustering model has been developed to give the most realistic realizations smaller faults cluster around larger ones, which cluster around even larger etc. The project has used finite difference, constant volume finite element, and Green element modelling with a variety of sample configurations of faults. The group has also developed a new methodology which both derives the minimum value of the fault rock thickness along flow paths traversing the fault zone, and predicts areas of reduced fault zone connectivity for matrix host rock (Km) and fault rock (Kf) of varying permeabilities. In this method, path tortuosity is controlled by a trade-off between pathway length and net fault-rock thickness crossed. Although it is strictly only applicable to a binary permeability distribution between fault rock and host rock, the method is very quick to apply to very complex geometrical situations. Preliminary results indicate that the geometrical method gives path lengths very similar to those determined by the discrete fracture flow modelling technique of Odling & Webman (1991). A critical threshold value of the ratio in permeabilities is observed to exist at which the flow characteristics transfer from long, tortuous pathways (high Km/Kf) to shorter, direct pathways (low Km/Kf) which encounter an increased thickness of fault rock. An interesting question is whether the permeability distributions of such realizations are consistent with the observations of 1/k spectral densities observed in well logs. One practical outcome for stochastic fault modelling that has been suggested by the findings (Harris et al. 2003) is that clustering tends to degrade the theoretical relationships between exponents for fault-length frequency distributions (1D sample exponent = 2D sample exponent - 1 = 3D sample exponent - 2). Odling et aL (2004) have taken several sets of 2D areal samples from regularly spaced intervals
throughout a large, stochastically modelled hierarchical fault damage zone. For an individual set the size of the samples was uniform, but the size changes between sets from 5 m to 50 m. The effective permeability of each of these samples has been calculated using the 2D, finite difference, discrete fracture flow model. Amongst other findings, the one-point frequency distributions of effective permeability are interesting. The distributions are closer to log-normal than
2.0
frequency distribution ............... frequency curve slope , ~ 9 best-fit log-normal
1.5 ,~i ~'~~''~(~ 0 3 2.0
-2
1.5
~
-I
A
0.5
0-3 2'0i
-~ o,5i o_"3
-2
-t
/
0
0
Increasing size of sample
:: -2
-I
2.0
0 10
5
~
l,O"
00~
~
0.51
"-5
~
-2
_ -t
-10 0
tog k
Fig. 4. Frequency distributions of the effective permeabilities of samples of various sizes from a simulated fault zone calculated by Odling et al. (2004). Each double logarithmic plot shows the frequency distribution (bold line), its local slope (thin line) and a fitted log-normal distribution (triangles). Size of sample increases down through figures. Only at the small sample size (5 m) is the distribution possibly power law; larger samples give distributions which ale closer to lognormal. Reprinted from Journal of Structural Geology, copyright (2004), with permission from Elsevier.
MICRO TO MACRO PROGRAMME: IMPLICATIONS to power law, except possibly at the lower sample size of 5 m. The frequency distributions are shown in Figure 4 and can be contrasted with those produced by coupled modelling at the critical point (see section entitled 'criticality and coupled modelling'). The origin of the power law in the work of Odling et al. (2004) must be a consequence of the statistics of the geometrical variables input to the 'static' fault damage model. However, in the case of coupled modelling, the power-law distribution of permeabilities can arise spontaneously from the interactions of the different processes at the critical point; given that the spatial distribution of permeabilities in this latter case is multifractal, it is unlikely that the univariate distributions are only power-law at certain scales of permeability measurement. There is ample scope for further study of such statistics from both modelling and field data, with the key objective of understanding whether well-test permeabilities measured in the field are dynamic (arising from coupled processes at a critical point) or static (arising from geological heterogeneities unresponsive to production). Criticality in f r a c t u r e p e r m e a b i l i t y
Rather than modelling fractured rock with discrete fractures, a more convenient way is with a continuum model in which effective properties take into account the presence of fractures. Spatial variations in fracture densities, apertures and orientations can be incorporated through
strain modelling in the continuum. One of the most important relationships for such an approach is that between the effective bulk permeability and strain. Various theoretical and laboratory investigations of this relationship have been made and the most common form has been a power law, but with a large range of exponents depending mainly upon the assumed configuration of the fractures (Walsh & Brace 1984; Yale 1984; Charlaix et al. 1987; Bernab6 1988, 1995). Charlaix et al. (1987) indicated that the exponent, s, is larger if the aperture distribution of individual elements which are needed to establish the percolation path at threshold extends continuously to zero with a finite density. One of the difficulties in calibrating these theoretical relationships with laboratory experiments has been in obtaining rock samples that are essentially undamaged prior to testing, and introducing in a controlled manner a characterized fracture set. Meredith et aI. (ix2M) have been able to do this through thermal cracking of a microgranite (the Ailsa Craig microgranite actually used in the tests is commonly used for making curling stones because of its essentially unflawed nature). Both permeability and porosity were measured despite difficulties caused by the extremely low connected fracture density and the essentially zero matrix permeability. Figure 5 shows the crossplot of measurements from one set of tests on the same sample, heated to increasing temperatures (cooled before flow measurements made), and corresponding increasing
Results from Bernabe (1995) incorporating those from Yale (1984) ..... b m e a s u r e m e n ~ ........... m
Bernabe (1995) 2d network model: cracks only] . . . . . J
15
~9
abe (1988) & Walsh & Brace (1984)] easurement crystalline rocks
~
9
2d network modelling
,~Nmnl 9
combined
Log. (lab measurements sst) . . . . . . . Log, (2d network modelling', - -
lo
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9
9 "'""
f 5 10 2d 3d percolation theory I
,t
15
13
20
value of exponent,
25
30
s
Fig. 5. Values of the exponent, s, in the percolation equation for permeability k = a(p - po)S, p > Pc. The range of values of s measured by Meredith et al. (Ix2M) is compared with the values, or frequencies of values, measured or calculated by Bernab6 (1988, 1995), Walsh & Brace (1984) and Yale (1984).
14
K.J. HEFFER
densities of fractures. Fitting to the data the percolation equation: k = a ( p - pc) s, p > Pc
-- 0, p < pc
(2)
where p is the porosity, Pc is the porosity at the percolation threshold, k is the permeability, and a, s are constants, with increasing assumed values of the percolation threshold, ostensibly yields a 'best fit' (highest value of correlation coefficient) when Pc is 0.0075 and s = 4.92. It is interesting that this (very preliminary) interpretation of threshold porosity is just below the actual natural porosity of the starting material (mostly due to isolated altered phenocrysts) of 1% or so (I.G. Main, pers. comm.). The value of the exponent is well in excess of the theoretical conductivity scaling implied by percolation theory (1.3 for 2D; 2.0 for 3D) and lies in the middle of the large range for mixed cracks and pores analysed by Bernab6 (1995) (Fig. 5). It is possible that the large value of the exponent, s, is attributable to heterogeneity in the apertures of the thermally-induced cracks. Whatever the final analysis of these data yields, the data
themselves provide a valuable benchmark against which to compare other values derived from theory or from laboratory measurements made under different conditions. The work has also been another illustration of the extreme sensitivity of permeability to fracture density - a highly non-linear relationship that can act as a threshold in critical behaviour and play a large role in coupled systems of fluid flow and geomechanics. Criticality and coupled modelling
Sanderson et al. (p~2M) (see also Zhang & Sanderson 2002; X. Zhang et al. 2002) looked at the critical point associated with the connectivity of fractures with a 2D distinct element model (UDEC), which couples deformation and fluid flow. The changes in deformability and permeability in the model with increasing input densities of fractures have been calculated (note that, in contrast with later studies described below, the fracture patterns were input into the model rather than induced by failure during deformation). The fracture connectivity is posed as a power-law function of fracture density above a threshold value, as with permeability vs.
Fig. 6. Critical point in coupled mechanics and fluid flow. (a) Fluid flow velocities modelled in loaded domain with three pre-existing fractures: (i) below; (ii) just below; and (iii) at the critical point when new percolating pathways subparallel to Shmaxare created. Reprinted from Zhang & Sanderson, copyright (2002), with permission from Elsevier. (b) Field data confirm that directionalities of flooding axe sub-parallel to the local orientation of Shmax,rotated to align with the modelling of X. Zhang et aL (2002) (adapted from Heifer & Lean 1993).
MICRO TO MACRO PROGRAMME: IMPLICATIONS
porosity described above. Sharp increases in both deformability and permeability are observed at the critical (threshold) fracture density. Four groups of simulated fracture patterns and 15 natural fracture patterns were studied. Exponents of permeability increase above the threshold were found in the range 1.05 to 1.37, in line with 2D percolation theory (exponent of 1.3). When the models were loaded, the stress-strain curves showed softening above the critical fracture density, but then an even greater deformability was observed above a second threshold of fragmentation. Exponents of the relationship between deformability and fracture density above this higher threshold were found to be 0.64 for a zero confining stress and 0.91 for an applied confining stress of 0.3 MPa: it would be a useful exercise to rationalize these values with experimental observations (Chakrabati & Benguigui, 1997, Section 3.4) of the scaling of modulus in a bond percolation model in which increasing densities of bonds incrementally stiffen the model (exponent close to 4 in 2D). The modelling of Sanderson et al. (/x2M) has also provided guidelines for estimating the effective failure variables (friction coefficient and cohesion) for a fractured rock mass. Based on these models they have defined an indicator for criticality in stress state, termed the 'driving
(a)
15
stress ratio' and given by: R=
(fluid pressure - mean stress)
(3)
(~ • differential stress) Instability occurs when the R-ratio exceeds some critical value Rc in the range - 1 to - 2 . These limits respectively represent failure by hydraulic fracturing and by shear failure in a cohesionless material with friction angle of 30 ~. Criticality can occur with shear failure with the fluid pressure still below the minimum principal stress. Sanderson et al. (o.2M) studied the statistics of fracture apertures arising from their modelling in relation to progress of the model to and through a state of criticality (see Fig. 6a). Apertures were actually examined in terms of the fluid flow 'vertically through' the 2D areal model, using essentially a cube law between flow rate and aperture. One-point cumulative frequency distributions of flow rate showed a dependency on degree of criticality: below criticality, the distribution is approximately log-normal; however, at and above the point of criticality, the distribution is better described as a power law. At the critical point the exponent of the power law is 1.1 (Fig. 7a). This modelled distribution can be
(b) 1000
|
',, '1 ,,
SIope~l.1
slope-~1 , t ~
,
.o m 100 "6 r E Z
, vvvv
9~
256
IOOC
ivv
10
N
1 0.001
0.01
0.1
1
10
Vertical flow-rates (x 10 -6 m s -1)
100
1.E-07 1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 I.E+00
Well rate, PI, cum prod, or permeability relative to maximum Im'='Giant field ~
Composite of several smaller fields I
Fig. 7. Frequency distribution of flow rates is power law at and near the critical point. (a) Results of modelling coupled mechanics and fluid flow; reprinted from Zhang & Sanderson, copyright (2002), with permission from Elsevier. Cumulative frequency distribution of flow rates (A) below, (B) and (C) just below the critical point. (b) Field data: cumulative frequency distributions of well rates, cumulative well production or permeability (each divided by the maximum for the field). Data are from one giant field, and also aggregated from a number of smaller naturally fractured fields.
16
K.J. HEFFER
compared with that from field data on the flow productivities from individual wells. Figure 7b shows that power-law distributions also apply to two examples of the latter: one from a giant field; the other as a composite from several fields. The exponent of the fitted power law common to both sets of data is also 1.1. The existence of the power law in the field data combined with the implication from modelling that powerlaw behaviour is expected only at or above the critical point is consistent with the concept of criticality in field behaviour. The equality of the exponent of the power law may not be so significant and further study would be necessary to demonstrate that it is not coincidental. Since, in the modelling of Sanderson et al. (lz2M), flow rate is calculated as the cube of the fracture aperture, the cumulative frequency distribution of fracture aperture is also a power law, with exponent - 3 . 3 . There are few direct datasets on fracture apertures from the field with which to validate these one-point statistics; fracture apertures in recovered core are under relaxed stress conditions. One exception is the large dataset measured downhole with a borehole televiewer log in the Cajon Pass well; from this, Barton & Zoback (1990) calculated a powerlaw frequency density distribution of fracture apertures, with an exponent of - 3 . 0 (equivalent to an exponent of - 2 . 0 for the cumulative frequency distribution). Converting that 1D sample basis to 2D would alter the cumulative distribution to ~ a -3, in good agreement with the distribution of flow rates calculated by Sanderson et al. (tx2M). Sanderson et al. (p~2M) have also investigated multifractal statistics in the distribution of apertures/vertical flow rates arising from their coupled geomechanics-flow modelling. They have found that below the critical point, the spectrum of generalized fractal dimensions Dq(q) varies only weakly with the order q of the moment, indicating an approximate monofractal. The common dimension is equal to 2.0, the space-filling dimension of the underlying input fracture set. However, when the critical point is reached, the multifractal spectrum shows a strong variation, with a sharp decline from negative to positive values of q. No known studies have been made of whether flow rates in a densely drilled field follow a multifractal distribution: such study might lead to further support for criticality in field behaviour. Another example of modelling which produced similar forms of multifractal spectra was the investigation by Cowie et al. (1995) of development of fault patterns by antiplane shear deformation of a 2D plate (in which the displacements
are out of the plane of the plate). No fluid flow was involved in that modelling. Distributions of displacements on the faults were found to evolve with model time from monofractal and space-filling to multifractal. One must be careful not to make too strong a deduction from these model studies: power-law distributions can occur in many ways (Sornette 2000, Chapter 14). Also, interpretation of a power law can be made falsely if the range of data is inadequate, for example extending over only one order of magnitude. However, there are strong indications that geomechanical-flow criticality is a sufficient, if not necessary, condition for power-law and multifractal distributions of flow properties. ls there more direct evidence to support the concept of criticality in oil field developments? Good demonstrations of its applicability are to be found in the North Sea chalk fields, Ekofisk and Valhall. These fields have received intense geomechanical study, mainly because of their strong compaction and its associated, very noticeable, effects of subsidence and casing failures, but it is unlikely that the fields are a special case. Zoback & Zinke (2002) have shown that the stress states in the crests of both fields were consistent with incipient normal faulting at the onset of oil production, and that the subsequent pressure reductions during primary production caused those critically stressed areas to spread downdip to the flanks of the structures (see also Chan et al. 2002). The effective stress states tracked down the Coulomb failure line (with a friction coefficient ~0.6) on a Molar diagram. Passive seismic monitoring in both Ekofisk (Maxwell et al. 1998) and Valhall (Zoback & Zinke 2002) has detected microseismic events, mainly in lower porosity reservoir layers or in the overburden. In Valhall, microseismic events have focal mechanism solutions, also indicating normal faulting. Furthermore, the anisotropy of the detected shear waves has shown evidence of temporal changes. Coleman (p,2M) sought change in fracture characteristics in the Valhall Field, which could be a further indication of criticality. That project has developed a possible diagnostic of fracture activation during reservoir development. In laboratory tests of fluid flow through chalk under stress, it was found that the concentration of the isotope 637C1of the collected fluid was correlated positively with the flux of the fluid through the chalk, this flux being controlled by the fracturing of the rock. Coleman (p~2M) sampled trace waters found in produced oil from several wells in the Valhall Field. No change over time has been observed to date in
MICRO TO MACRO PROGRAMME: IMPLICATIONS the geochemistry of these samples, but the average 637C1 compositions of trace waters varied significantly between wells, always different from that of sea water. It is very interesting that the 837C1compositions indicated more fracture permeability from the crest of the structure than from the flanks (M. Coleman pers. comm. 2002) consistent with the other observations of fracture activity progression. Further modelling by Sanderson et al. (ix2M) also suggests the basis for a reconciliation of the current disagreements in the industry of the importance of critical stressing as a criterion for conductivity of individual fractures. Recent work has shown the strong influence of modern-day stress state on fracture conductivity: fractures which are in a state of incipient shear failure in the modern-day stress field, termed 'critically stressed', will generally be conductive; whilst those fractures stable in the modern-day stress state will generally be nonconductive (Barton et al. 1998; Barton 2000; Chan et al. 2002). An exception to this might be a fracture set that was formed under a palaeo-stress state shortly before, or contemporaneously with, hydrocarbon fill, which inhibited fracture healing when the stress state altered to its modern-day configuration (e.g. Stowell et al. 2001; Gauthier et al. 2002). This scenario is more likely if the original deformation was associated with diagenetic alteration, either dissolution, or partial cementation, such that, when the stress state was altered, bridges between vugs along the fracture path helped to prop open a conductive path. The model of Sanderson et al. (pu2M) of the fluid flow in a granular medium also contained some macro-fractures, with the maximum principal horizontal stress (Shmax) at a large angle (c.60 ~ to the fracture strike (see Fig. 6a). At, or just below, the critical point, smaller-scale fractures formed that were sub-parallel to Shmax, at the same time as the macro-fractures are open. Under conditions of low mean effective stress (as would pertain in waterflooding recovery schemes), the secondary fractures are conductive and form a percolating path for flow. To transpose this to field experience, observations might be made early in the life of a field development of conductive fractures which were formed under some palaeo-stress; whilst, if a secondary recovery scheme is implemented which reduces effective stresses close to a critical point, then coherent fracture trends striking close to the azimuth of Shmax might be equally or even more, influential in governing the directionality of the flooding. This is consistent with the statistics of directionality
17
observed in oil field operations (Heifer & Lean 1993) and in geothermal projects (e.g. WillisRichards et al. 1996). With regard to indicating stress-induced directionality, the modelling complements that of Heifer & Koutsabeloulis (1995) (see Fig. 6b). The semi-quantitative scale invariance of some deforrnational geometries is demonstrated by comparing the results of Sanderson et al. (ix2M), whose model contains overlapping macrofractures at the grain scale, with those of much largerscale modelling of the geomechanical and flow characteristics of a fault relay zone conducted by Y. Zhang et al. (2003), linked to the ix2M project of Yardley et al. (ix2M). In addition to coupled modelling of geomechanics and fluid flow using the explicit finite difference code FLAC in 2D, the modeUing is also explicitly coupled to the finite element code FIDAP, which models chemical reactions. The model has been used to track the mixing of reduced and oxidized fluids, both gold saturated, in the dilatant zones resulting from the geomechanical model. The patterns of fluid mixing are seen to be very similar to the aperture distributions produced by the Sanderson et aL (po2M) model (see Fig. 8). Yardley et al. (Ix2M) are utilizing geochemical methods to investigate palaeo-fluid flow in and around the Navan mine in Eire. A strong control on the flow has been shown to be the density contrast between cooler waters of evaporitic origin overlying hotter hydrothermal waters from the Lower Paleozoic basement. The concentration of lead sulphide mineralization is focused in a ramp zone between two NNE-SSW-trending Caledonian faults, which were activated under a more E-W-directed stress field during Carboniferous-Permian times. The dilatation of this extensional step gave rise to vertical flow to concentrate mixing of the two waters and deposition of lead sulphide. More extensive E - W lineations also hinged upon this focus.
D y n a m i c t r a n s p o r t e q u a t i o n s on fractal structures
If heterogeneous porous media can be described with fractal functions (even if they are uncoupled from geomechanical or chemical changes), is there an effective differential equation which can be applied to describe transport through them? Such an application would have potential for more efficient flow simulations. However, although there have been a wide variety of equations devised in the past to describe flow and transport on a fractal structure, Sellers &
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K.J. HEFFER
Fig. 8. Similarity across scales of coupled modelling of relay zones loaded under anisotropic stresses. (a) Coupled modelling of mechanics and fluid flow at the grain scale reprinted from Zhang & Sanderson, copyright (2002), with permission from Elsevier. The dark lines indicate the largest induced crack apertures. (b) Coupled modelling of mechanics, fluid flow and chemistry at the scale of a mine by CSIRO linked to the project of Yardley et al. (Ix2M); reprinted from Y. Zhang et al. (2003) with permission from Elsevier. Dark regions indicate high flow velocities where mixing of two fluids occurs in dilated zones.
Barker (~2M) showed that there is a lack of justification for those equations. Their project has supplemented previous study of the so-called anomalous diffusion equations which found that none of those yet devised could successfully match the results of random walks on a standard Sierpinski gasket over the full range of times/ distances (Schulzky e t al. 2000). Sierpinski gaskets and carpets are triangle- and squarebased fractal 2D objects; see, for example, http://astronomy.swin.edu.au/~pbourke/fracta Is/gasket/. Sellers & Barker (lx2M), with careful simulations of random walks on Sierpinski ca rpets, also found the following: 9
9
9
boundary effects can be extremely significant and lead to wrong estimates for the dimensions when an insufficient number of time steps is used; log-periodic oscillations can appear superposed onto the asymptotic response, arising from internal boundaries due to a hierarchy of length scales in the fractal; flow dimension is a local quantity that can vary with origin and direction, and is not a global property of the fractal.
Sellers & Barker (ix2M) conclude that the effective differential equation approach can, in some specific cases, provide reasonable solutions, but it is not clear a priori which equations are appropriate to a given fractal. The authors demonstrate the need for better models of transport on fractals. These findings throw further doubt on whether the fractal geometry of fractures can be interpreted
from well tests. Previous work (e.g. Barker 1988) has identified fractional dimensional behaviour during hydraulic tests (Barker's Generalized Radial Flow, GRF, model). By the analysis of synthetic fracture networks with well-known geometric properties, Jourde e t al. (2002) showed that a fractal-like pressure transient response is not necessarily tied to a fractal geometric arrangement of flow paths (fractures or channels). This result agrees with the analytical study by Doe (1991) who stated that a fractional dimension in a well test only requires the change in conduit area with distance from the source point to scale, and does not require the reservoir to have other fractal properties. Consequently the interpretation of a non-integral flow dimension from well tests is, at the present time, questionable, though it may provide qualitative information on the fracture connectivity. Only if the fractal structure is radially symmetric from the source point will the fractional flow dimension be related to the fractal dimension; this is not a very likely natural situation, given the anisotropic character of sedimentary layers, fracture sets and stress fields, etc. Nevertheless, by simulating diffusion on a set of given fractal structures with random walk simulations, an idea of possible types of fracture patterns investigated by a test might be gleaned. Once the assumption (often of convenience) that geomechanical coupling can be ignored is removed, then the practical problem for well tests in fractured rock arises that the fracture storage can be much larger than usual wellbore storage values and, through the
MICRO TO MACRO PROGRAMME: IMPLICATIONS feedback of port-elastic stresses, can also be changing throughout most of the duration of a well test. The practical implication for fluid flow engineering at this stage, therefore, seems to be that there is no 'shortcut' to simulating conditioned reservoir heterogeneities and performing 'conventional' flow simulations on them, preferably in most circumstances with coupling to getmechanical changes.
Reservoir surveillance to monitor changes in properties Seismic m o n i t o r i n g
One of the most attractive consequences of criticality is the association with pervasive, stress-aligned (micro-) fractures and low aspect-ratio pores throughout the crust that are close to failure at the percolation threshold (Crampin 1994, 1999, 2000). Azimuthally aligned shear-wave splitting with very similar characteristics is observed in almost all igneous, metamorphic and sedimentary rocks of all porosities and permeabilities. The anisotropic poro-elasticity (APE) model of Zatsepin & Crampin (1997), in which the mechanism for deformation is fluid movement by flow or dispersion along pressure gradients between neighbouring cracks at different orientations to the stress field, has been very successful in matching a large range of observations of shear-wave splitting, associated with earthquakes, eruptions and other phenomena. These properties imply a basis for various methods of seismic monitoring of reservoir developments. There are various degrees of coupling between stress, pressure and permeability. At the critical point, where the permeability is a strong function of effective stress, all three variables are interdependent. However, in the case of seismic waves passing through rock, the dependence of permeability on small changes in effective stress during the passage of the wave (apart from an average permeability determined by an average effective stress for the process) might be ignored. The reverse coupling, the influence of permeability on the properties of the seismic waves, is still, however, important in the presence of fractures. The coupling arises because of the phenomenon of 'squirt-flow', due to the seismic wave inducing, in cracks at different orientations, or between cracks and spherical pores, pressure gradients that are not parallel to the propagation direction. Squirt-flow and, therefore, its influence on seismic properties,
19
is frequency dependent; the characteristic frequency depends upon factors including the size of the fractures. Various studies have been made of this influence, assuming various geometries of the pore space. Chapman et al. (2002) have developed an approach that allows the introduction of greater generality in geometries (see also Chapman 2003; Chapman et al. 2003; Maultzsch et al. 2003). Using a network model that comprises spherical pores, randomly orientated microcracks and aligned larger fractures, they have determined a strong dependence of seismic properties (velocities, attenuation, dispersion and anisotropy of P- and S-waves) on frequency, pressure (effective stress), viscosity, permeability and fluid characteristics. Their modelling is consistent with the conventional Gassmann model at low frequencies; the existence of the Biot slow P-wave; the dispersion characteristics predicted by previous modelling for mixtures of cracks and pores. With such a mixture, dispersion of S-waves increases linearly with crack density; whilst dispersion of P-waves is zero for the cases of no cracks and all pore space comprising cracks, and reaches a maximum at some intermediate crack density mixed with spherical pores. Although the model is not fully coupled with stress, pressure dependence can be introduced with an externally derived relationship between crack density and effective stress. This correspondence implies strong dependencies of the velocity dispersion and the attenuation on the effective stress. Parameters of the model can be expressed in terms of macroscale, measurable quantities. An exception is the crucial parameter of the relaxation time of fluid ftow (r, typically in the range 10 - 7 to 10 - 4 s), which although linked to permeability, viscosity etc. would generally have to be considered as an unknown parameter to be used for calibration of the model to field data in applications. Calibration has already been performed for laboratory data (for which r values which best fit velocity and attenuation data are reasonably consistent); and also for field data from a gas reservoir where permeability is controlled by pre-existing fractures. Figure 9 shows the calibration of the model against the observed frequency dependence of S-wave anisotropy. An outstanding item of 'ground-truthing' at the moment is being able to associate the radius of macrofracture that is also derived from this calibration with an independent observation of fractures in the reservoir. Further reassurance will be given by a test that matches a larger range of seismic frequencies as well as the relatively limited bandwidth over which anisotropy is predicted to decline. Future
20
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Frequency Fig. 9. Comparison of the frequency dependence of shear-wave anisotropy predicted by the effective medium model of aligned macrofractures, cracks and spherical pores (Maultzsch et al. 2003) with observations from a near-offset multicomponent vertical seismic profile. Reprinted from Geophysical Prospecting, copyright (2003), with permission from Blackwell Publishing. work will include the incorporation of distributions of macrofracture sizes. Work has also progressed under the l-t2M Programme on an alternative model of fluid flow influence on seismic properties by Liu et al. (o~2M) (also Tod 2001). The model incorporates just one set of spheroidally shaped pores of arbitrary aspect ratios. The study has looked at the effects of changing crack shape on wave speeds. A particular application is interpreting seismic data in the case of fractures which are limited in height to bed thickness, whilst varying in lateral length, as commonly occurs with mechanical stratigraphy. Again, strong dependence of the seismic velocities on the seismic frequency has been found, as well as the expected dependence of velocities on the aspect ratios of the cracks. Finite difference modelling has also been conducted by Liu et al. (I~2M) (also Vlastos et al. 2002) to study the effects of fracture spatial distribution and size on the wavefields (not incorporating the influence of squirt-flow). This has shown that when the fractures are smaller than the wavelength, each fracture is a single scatterer resulting in a secondary wavefield independently of the distribution. When the fracture size is larger than the wavelength, the features depend on the distributions. With an almost regular distribution such that the fractures are very close to each other they form clusters which act as
large interfaces; whereas with a completely random distribution the clustering is insignificant and the fractures again act as individual scatterers. It will be interesting to extend this modelling to continuous distributions of fracture size (e.g. power law). The strong dependencies of seismic properties on fracture characteristics, such as size, aspect ratio, orientation and clustering, together with the ambient effective stress, give rise to a range of potential applications in areas such as the interpretation of time-lapse multi-component seismic surveys, pore pressure prediction and the frequency scaling of laboratory measurements.
Temporal and spatial correlations in well-rate fluctuations
Other data that have provided support for the concept of near-criticality in field operations have been derived from fluctuations in production and injection rates at wells over the life of field developments (Heifer et al. 1997). Correlation coefficients have been calculated for temporal series of rate fluctuations at pairs of wells, using the standard Spearman rank correlation (non-parametric) technique. The aggregated results from applications to several fields indicated that rate correlations have the following general properties:
MICRO TO MACRO PROGRAMME: IMPLICATIONS (a)
(b)
(c) (d)
highest positive correlations for well pairs aligned along a direction close to the local orientation of maximum horizontal principal stress; lowest, and on average negative, correlations for well pairs aligned sub-parallel to the local orientation of minimum horizontal principal stress; many of the high correlations are at long range; trends of similar orientation to faulting trends appear in the correlations.
These properties are best explained by the interpretation that the rate fluctuations in a field are (at least partially) due to geomechanical changes in the reservoir causing strain and, therefore, permeability changes. The appearance of long range in the correlations is another indication that the system is close to a critical point. Figure 10 shows the interpolated map of maximum horizontal principal strain corresponding to the principal component with the highest eigenvalue of the matrix of rate correlations between well pairs in one field application. It 'explains' nearly 20% of the variance in the rate fluctuations for this field. The interpolation has been made on the basis that the principal component value corresponds with the local volumetric strain. The lineations that appear in high values of the principal component are even
more plausible when compared with the fault map for the reservoir that is overlaid: there is a good correlation between the trends and locations of faults and zones of high fluctuation. A question that arises is whether the lineations in correlated rate are due to conductive fracture/ fault zones (as, for example, described in Sibson 1996) or to the focusing of fluid flow along sealing faults. It should be borne in mind that the correlations of rate fluctuations imply timevarying properties. Whilst permeability of a set of fractures close to a critical point can be readily time-varying and correlated over long range, it is less easy to imagine the sealing properties of several, spatially separate faults varying in unison. A more sophisticated potential mechanism is that of fluid flow in stress-sensitive flow properties propagating along faults as solitary waves (e.g. Rice 1992; Revil & Cathles 2002).
Conclusions The Ix2M Programme has helped to underpin the concepts of criticality and scaling in reservoir behaviour, with modelling results broadly consistent with observations in the field. There are a number of implications that the lx2M Programme carries for reservoir characterization exercises. 9
9
9
Fig. 10. Map of fault traces on top surface of a reservoir superimposed upon the map of maximum horizontal principal strain as interpreted from the first principal component of the well rate correlation matrix and interpolated using a long-range correlation function appropriate for strain (hotter colours are higher magnitudes of strain). There is a strong association of the trends in high strain with the faulting trends.
21
9
It gives further impetus to deploy long-range spatial correlations in stochastic modelling exercises. The 1/k spectral densities are associated with ~log (lag distance) correlations in real space. It has provided support for the general applicability of reservoir criticality and stress-related anisotropy. Field-specific demonstration and greater understanding of these will require measurement of in situ stress states, magnitudes and orientations, as a matter of course in data acquisition programmes. Associated with criticality is the recognition that flow properties are likely to change during the life of commercial developments. Therefore, for example, in interpreting repeat flow-tests on the same well, changes to absolute permeability should be sought, rather than being considered as aberrational, or force-fit into uniformity, as is the current tendency. Any such changes should be analysed for spatial and temporal patterns, particularly with respect to the local structural and geomechanical characteristics. In order to understand changes in flow properties and provide better predictions for reservoir planning and management, coupled modelling, particularly of geomechanics and fluid flow, is desirable. Although such
22
K.J. HEFFER coupled modelling adds an overhead in computer time and resources to conventional flow modelling, the potential benefits are very large in terms of providing a more representative model of the overall physics of the system. For example, the large operational and commercial influence of horizontal anisotropy in permeabilities on recovery from flooding schemes has been well known in oil reservoir engineering for many decades; additional benefits will surely accrue from modelling the time dependency of such anisotropy and its detailed relationship to local structure and geomechanics. Knowing that local faults and fractures play a strong role in fluid flow mechanisms in a potentially time-varying, rather than just a static, fashion, gives even more motivation for acquiring detailed information on microand macro-structure over a range of scales, from core-logging, borehole image logs, vertical seismic profile and surface seismic surveys. The strong seismic responses that are predicted - especially of anisotropy imply the applicability of a range of seismic techniques such as the interpretation of time-lapse multi-component surveys, particularly of shear-wave signals. The more general association of shearing on pre-existing discontinuities with the progression of fluid flow gives added impetus to deploy passive seismic monitoring of these events. The successes of previous applications can be built upon by validating or calibrating such surveys with coupled geomechanical-flow modelling.
9 9 9 9 9
There are 'new' technologies in which these concepts will be even more pertinent. 9
9
Future developments Whilst the ~2M Programme has helped to underpin new concepts of scaling, criticality, susceptibility to perturbation, long-range correlation etc., there are still many issues outstanding. Many have been outlined above or are mentioned in reports or papers from individual projects. The following are significant examples: 9 9 9 9
Is the applicability of these concepts field specific or ubiquitous? How 'near' is 'near-criticality' in general commercial cases? How best to incorporate power-law spatial correlations and structure-related anisotropy into stochastic modelling? Further searches for multifractal scaling in field data, and development of means of incorporating in modelling.
More detailed understanding of the involvement of sedimentary and diagenetic influences in observed scaling in heterogeneities. Further understanding of the viscous coupling between fracture and matrix flow. Development of faster, more flexible, coupled geomechanical-flow models that can cope with uncertainties in input parameters. Development of acquisition, processing and interpretation techniques for time-lapse shear-wave splitting surveys. Development of equipment for more permanent monitoring of passive seismic events and methodology for incorporating into predictive models of fluid flow and deformation.
9
9
9
CO2 sequestration schemes. One of the key unknowns for projects which seek to 'lockup' CO2 emissions in the subsurface is whether the traps will leak. That places more emphasis upon knowledge of whether potential leakage pathways via faults or fractures are conductive under either original or perturbed conditions. In addition, a sequestration project in an oil reservoir can often only be commercially viable if it assists in enhancing oil recovery: the resolution of issues of heterogeneity patterns, anisotropy and time-variability therefore become even more important for such projects. Geothermal schemes. Much of the awareness of geomechanical influence, including passive seismic monitoring, was pioneered in geothermal projects. However, there is plenty of scope for application of recent technologies and development along the lines listed above. Radioactive waste disposal schemes. The considerations of time variability and longrange correlation become even more acute when applied to schemes that require extreme reliability of prediction for thousands of years. Groundwater. The improved understanding gained by the lx2M Programme, particularly with respect to fractured rocks, will be invaluable in the efficient exploitation of groundwater and in the remediation of contaminated aquifers. Mining industry. Many mineral deposits are closely related to fault/fracture networks and the flow of mineralizing fluids through them. The advances made on modelling fluid flow through fracture networks, at several scales, could be developed
MICRO TO MACRO PROGRAMME: IMPLICATIONS and used, in conjunction with information on such aspects as host-rock type, fluid geochemistry, temperature, stress fields, etc., to help predict favourable sites for mineralization and exploration strategies.
The author thanks Dr Robert Cuss, Professor Rob Knipe, Dr Richard Shaw and Dr Sue Raikes for providing improvements to this paper. Much benefit was also derived from correspondence with Dr Peter Leary and Professor Stuart Crampin whilst writing the paper, without implying that either necessarily agrees with the interpretations given here. Finally, acknowledgement is due to the Natural Environment Research Council for a small grant towards the task of integrating results from the ~2M Programme.
Appendix A: Various uses of the term 'critical' There are several contexts for the term 'critical' in this paper, following common useage in recent literature. Although related in the mechanisms involved, the meanings in the different contexts vary; those meanings are given very brief outlines below.
Rupture as a critical phenomenon The process of faulting or fracturing of rock has been described as a critical phenomenon analogous to those of continuous phase transitions in equilibrium thermodynamics (e.g. in liquid-gas mixtures, metallurgy, magnetism, (super) conductors etc.). As the system stress state approaches the critical point, failures occur, initially at the small scale, and then coalescing to larger scales; the spatial correlation of stresses, strains and earthquakes increases correspondingly. The fact that at the critical point there are no characteristic scales gives rise to power laws in frequency distributions of variables, and relationships between them. The critical point marks the transition from 'intact' to 'fractured' phases of the rock. Fracture criticality is a corresponding term introduced by Crampin (1994), focused towards the implication from widespread observations of shear-wave splitting that there is a very narrow range (a factor of only ~1.5) in average fracture densities between the smallest observed and the threshold for percolation: only a small change in stress state is, therefore, generally required to bring the average fracture density up to the point which will give throughgoing failure.
23
Self-organized criticality (SOC) Critical phenomena are observed in equilibrium thermodynamics only when the system is tuned to the critical point (e.g. by varying temperature). The concept of 'self-organized criticality' was introduced (e.g. Bak 1997) as an explanation of how a system which is far from equilibrium can reach a critical point by self-organization without external tuning, and particularly as an explanation of the origin of 1If noise which is observed in many natural systems. SOC behaviour is found in systems dominated by interactions between many degrees of freedom (rather than the intrinsic dynamics of the individual degrees of freedom) and with thresholds (e.g. for failure) that allow a large number of static metastable configurations (Jensen 1998). It is also required that the system be slowly driven in relation to the time characteristic of the process whereby the threshold in dynamics is crossed (e.g. the build-up of stress on a fault is over much longer time periods than the earthquake that eventually occurs on it). A SOC process is characterized by avalanches of threshold-crossing interactions (e.g. earthquakes) that occur at all sizes, maintaining a significant proportion of the domain in a state close to the threshold (e.g. fractures which are on the verge of further failure, especially in shear, to which the term 'critically stressed' has been applied see below). Several indicators suggest that SOC is a valid model for deformation in the lithosphere (Bak 1997; Sornette 2000), but there is not universal acceptance of this concept (Jensen 1998; Main & A1-Kindy 2002). This use of the term critical in SOC applies to the dynamic state of the whole system which maintains itself in that critical state over an extended time period. At the critical state a significant proportion of the system is close to the threshold at any one time. This concept is congrnous with that of the brittle crust being in a state of failure equilibrium (Zoback & Townend 2001).
Intermittent criticality In contrast to a global state of SOC, the concept of intermittent criticality has been proposed in which the crust is predominantly in a subcritical state, and only approaches criticality during periods of high earthquake activity.
Critical density of fractures The dependence of the permeability of a fractured rock (with insignificant matrix permeability)
24
K.J. HEFFER
on the density of fractures has been treated in terms of percolation theory. There is a critical density of fractures below which there is no connected path of fractures across the rock sample, and the system permeability is negligible: this corresponds to the percolation threshold. Above the critical density, the permeability increases in power-law fashion.
Critically stressed fractures When a fracture has normal and shear stresses (tractions) acting on its surface such that it is in a state of incipient shear failure, it is known as critically stressed (Barton et al. 1995). From field data, it is proposed by Barton et al. (1995) that only fractures which are critically stressed are conductive; more stable fractures are generally non-conductive. This notion, allied to that of a general state of failure equilibrium (see SOC above), implies widespread high bulk permeability through the crust from fractures and, therefore, higher strength than would exist if pore pressures were high due to trapped fluids.
Appendix B: Field evidence for criticality in the Earth's crust, and hydrocarbon reservoirs in particular Some of the following items repeat the lists given by Crampin (1999) and Grasso & Sornette (1998). (1)
(2) (3)
(4)
Direct measurement of in situ stress states in wells: stresses generally lie on the Coulomb frictional failure line and follow this during perturbation of a site (e.g. Zoback & Townend 2001; Zoback & Zinke 2002). Observations of shear-wave splitting that imply fracture criticality, including changes during perturbation (e.g. Crampin, 1999). Observations of induced seismicity caused by commercial perturbations of the subsurface (see www.nyx.net/~dcypser/ induceq/induceq.bib.html for bibliographies which list over 400 references concerned with induced seismicity from fluid injection, oil and gas production, impoundment of water reservoirs, geothermal energy extraction, mining and quarrying and underground gas storage). Triggering of aftershocks by small stress changes (of the order of 1 bar or less) due to the displacements on a main earthquake (e.g. Grasso & Sornette 1998; Stein 1999).
(5)
Power-law frequency distributions of earthquake events and fractal geometries to structures. (6) By considering the relationships between entropy and energy variations calculated from a global earthquake catalogue, Main & A1-Kindy (2002) qualitatively confirmed anticipated criteria for a near-critical state, but conjectured that the degree of variability in entropy was more consistent with intermittent criticality. (7) The observations of 1/k scaling in heterogeneities measured by well logs (see main text). (8) Observed spatio-temporal correlations between rate fluctuations in production and injection from wells in hydrocarbon reservoirs (Heifer et al. 1997), which demonstrate characteristics of (a) longrange and (b) anisotropy related to modernday stress axes. (9) Observations of directionality in fluid injection schemes in oil reservoirs which are strongly correlated to modern-day stress axes (Heifer & Lean 1993) and are reproduced with modelling that involves induced shearing on pre-existing fractures (Heifer & Koutsabeloulis 1995). Microseismicity records during injection in geothermal projects have also revealed a small angle between microseismic clouds and modern-day stress axes occasioned by variable combinations of shear slip and extension on existing fractures (e.g. Cornet & Jones 1994). (10) Power-law frequency density distribution of permeabilities in fractured reservoirs (see main text and Fig. 7).
References Projects under the Micro to Macro (Ix2M) Programme that are referenced in this paper Summaries of these projects may be found in the final chapter of this volume and these are referred to in brackets after each entry in the list below. Bloomfield, J.P. (British Geological Survey) & Barker, J.A. (University College London), 'Modelling porosity development in heterogeneous fracture networks' (A10). Cassidy, R., McCloskey, J. & Morrow, P. (Ulster University, Coleraine), 'Measurement of complete fluid velocity fields in 2D heterogeneous porous media' (A15).
MICRO TO MACRO PROGRAMME: IMPLICATIONS Coleman, M. (Reading University), 'Quantifying contributions from matrix or fracture flow by geochemical analysis of produced oil' (A4). Harris, S.D., Pecher, R., Odling, N.E., Knipe, R.J., Ellis, J.A., Elliott, L. & Ingham, D.B. (Leeds University), 'Scaling of Fluid Behaviour Associated with Flow Through Complex Geological Structures' (A6). Haszeldine, R.S., England, G.L., Quinn, O., Bhullar, A.G., AI-Kindy, F., Barclay, S.A., Graham, C.M. (Edinburgh University); Corbett, P.W.C., Lewis, H., Potter, D. (Heriot-Watt University); Yardley, B.W.D., Cleverly, J., Fisher, Q. (Leeds University); Aplin, A.C. (Newcastle University) & Fallick, A.E. (Scottish Universities Environmental Research Centre), 'Cementation of oilfield sandstones: Micron cementation reveals effects of kilometre-sized hydrogeology, with porosity and permeability scaling' (A2). Liu, E. (British Geological Survey), Hudson, J.A. (Cambridge University), Chapman, M., Vlastos, S., Li, X.Y. (British Geological Survey), Tod, S.R. (DAMTP and British Geological Survey) & Main, I.G. (University of Edinburgh), 'Determination of hydraulic properties of distributed fractures using seismic techniques' (A7). Meredith, P.G., Clint, O.C., Ngwenya, B. (University College London); Main, I.G., Odling, N.W.A. & Elphick, S.C. (University of Edinburgh), 'Crack damage and permeability evolution near the percolation threshold in a near-perfect crystalline rock' (A16). Ogilvie, S., Isakov, E. &Glover, P. (Aberdeen University), 'The Scaling Behaviour of Fluid Flow in Rough Rock Fractures' (A14). Sanderson, D.J., Zhang, X. (Imperial College) & Barker, A.J. (Southampton University), 'Localized flow in fractured rock masses: mechanisms, modelling and characterisation' (A11). Sellers, S. & Barker, J. (University College London), 'Novel flow and transport models for systems exhibiting non-integer flow dimensions' (A9). Yardley, B.W., Barnicoat, A.C., (Leeds University); Wilkinson, J.J. (Edinburgh University); Graham, C.M. (Edinburgh University) & Boyce, A.J. (SURRC) 'Multi-scale fluid-flow path analysis: calibration and modelling using mineralisation systems' (A3). Published references
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study using stochastic models. Journal of Structural Geology, 25, 1281-1299. HEFFER, K.J. 2002. Reservoirs at a critical point - a useful concept in fracture characterization. Extended abstract presented at Petex, 2002, London, December 10-12. HEFFER, K.J. in press. Spatial scaling of effective modulus and correlation of deformation near the critical point of fracturing. Submitted to Pure & Applied Geophysics. HEFFER, K.J. & KOUTSABELOULIS,N.C. 1995. Stress Effects on Reservoir Flow - Numerical Modelling Used to Reproduce Field Data. In: DE HAAN, H.J. (ed.) New Developments in Improved Oil Recovery, Geological Society, London, Special Publications 84, 81-88. HEFFER, K.J. & LEAN, J.C. 1993. Earth stress orientation - a control on, and guide to, flooding directionality in a majority of reservoirs. In: LINVILLE, B. (ed.) Reservoir Characterization IlL PennWell Books, Tulsa, 799-822. HEFFER, K.J., FOX, R.J., MCGILL, C.A. & KOUTSABELOULIS, N.C. 1997. Novel Techniques Show Links between Reservoir Flow Directionality, Earth Stress, Fault Structure and Geomechanical Changes in Mature Waterfloods. SPE Journal, 2, 91-98 (SPE 30711). HERGARTEN, S. & NEUGEBAUER, H.J. 2001. SelfOrganized Critical Drainage Networks, Physics Review Letters, 86, 2689. HEWETT, T.A. 1986. Fractal distributions of reservoir heterogeneity and their influence on fluid transport. Paper SPE 15386. HOLLIGER, K. 1996. Upper-crustal seismic velocity heterogeneity as derived from a variety of P-wave sonic logs. Geophysical Journal International, 125, 813- 829. HOOGE, C., LOVEJOY, S., SCHERTZER,D., PECKNOLD, S., MALOUIN, J.-F. & SCHMITT, F. 1994. Multifractal phase transitions: the origin of selforganized criticality in earthquakes. Non-linear Processes in Geophysics, 1, 191-197. HURST, H.E., BLACK, R.P., & SIMAIKA, Y.M. 1965. Long-term storage: an experimental study. Constable, London. JENSEN, H.J. 1998. Self-Organized Criticality: emergent complex behaviour in physical and biological systems. Cambridge University Press, Cambridge. JOURDE, H., P1STRE, S., PERROCHET, P. & DROGUE, C. 2002. Origin of fractional flow dimension to a partially penetrating well in stratified fractured reservoirs. New results based on the study of synthetic fracture networks. Advances in Water Resources, 25, 371-387. KROHN, C.E. 1988. Fractal measurements of sandstones, shales and carbonates. Journal of Geophysical Research, 93, 3297-3305. LEARY, P.C. 1991. Deep borehole evidence for fractal distribution of fractures in crystalline rock. Geophysical Journal International, 107, 615-627. LEARY, P.C. 1996. Rock heterogeneity and fluid flow. Extended Abstracts L028, presented at the 58th EAGE Conference Amsterdam.
MICRO TO MACRO PROGRAMME: IMPLICATIONS LEARY, P.C. 1998. Relating microscale rock-fluid interactions to macroscale fluid flow structures. In: JONES, G., FISHER, Q.J. & KNIPE, R. (eds) Faulting, Fault Sealing and Fluid Flow in Hydrocarbon Reservoirs. Geological Society, Special Publications, 147, London, 242-269. LEARY, P.C. 2002. Fractures and physical heterogeneity in crustal rock. In: GOFF, J.A. & HOLLIGER, K. (eds) Heterogeneity in the crust and upper mantle: nature, scaling and seismic properties. Kluwer Academic, New York, Chpt 8. LEARY, P.C. & AL-KINDY, F. 2002. Power-law scaling of spatially correlated porosity and log(permeability) sequences from north-central North Sea Brae oilfield well core. Geophysical Journal International, 148, 426-442. LI, W. 1991. Expansion-modification systems: A model for spatial 1/f spectra. Physics Review A, 43, 5242-5260. MAIN, I.G. 1996. Statistical Physics, Seismogenesis and Seismic Hazard, Reviews of Geophysics, 34, 433 -462. MAIN, I.G. & AL-KtNDY, F.H. 2002. Entropy, energy and proximity to criticality in global earthquake populations. Geophysical Research Letters, 29, 7. MANDELBROT, B.B. & WALLIS, J.R. 1969. Some longrun properties of geophysical records. Water Resources Research, 5, 321-340. MARSAN, D. & BEAN, C.J. 1999. Multiscaling nature of sonic velocities and lithology in the upper crystalline crust: evidence from the KTB Main Borehole. Geophysical Research Letters, 26, 275-278. MATT1SON, C., KNACKSTEDT, M.A. & SENDEN, T.J. 1997. Transport in fractured porous solids, Geophysical Research Letters, 24, 495-498. MAULTZSCH, S., CHAPMAN, M., Ltu, E. & LI, X.-Y. 2003. Modelling frequency dependent seismic anisotropy in fluid-saturated rock with aligned fractures: Implications of fracture size estimation from anisotropic measurements. Geophysical Prospecting, 51, 381-392. MAXWELL, S.C., YOUNG, R.P., Bossu, R., JUPE, A. & DANGERFIELD, A. 1998. Microseismic logging of the Ekofisk Reservoir. Paper SPE/ISRM 47276 presented at Eurock 98 SPE/ISRM Rock Mechanics in Petroleum Engineering Conference. 8-10 July, Trondheim. ODLING, N.E. & WEBMAN, I. 1991. A conductance mesh approach to the permeability of natural and simulated fracture patterns. Water Resources Research, 27, 2633-2643. ODLING, N.E., HARRIS, S.D. & KNIPE, R.J. 2004. Permeability scaling properties of fault damage zones in siliclastic rocks. Journal of Structural Geology, 26, 1727-1747. REVIL, A. & CATHLES, L.M. III 2002. Fluid transport by solitary waves along growing faults. A field example from the South Eugene Island Basin, Gulf of Mexico. Earth & Planetary Science Letters, 202, 321-335. RICE, J.R. 1992. Fault stress states, pore pressure distribution, and the weakness of the San Andreas Fault. In: EVANS, B. & WONG, T.-F. (eds) Fault
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Mechanics and Transport Properties in Rocks. Academic Press, San Diego, 472-503. SCHULZKY, C., ESSEX, C., DAVISON, M., FRANZ, A. & HOFFMANN, K.J. 2000. A Comparison of Anomalous Diffusion Equations. Journal of Physics A: Math Gen, 33, 5501-5511 SIBSON, R.H. 1996. Structural permeability of fluiddriven fault-fracture meshes. Journal of Structural Geology, 18, 1031-1042. SORNETTE, D. 2000. Critical Phenomena in Natural Sciences: chaos, fractals, self-organization, and disorder: concepts and tools. Springer-Verlag, Berlin. SORNETTE, D., DAVY, P. & SORNETTE, A. 1990. Structuration of the lithosphere in plate tectonics as a self-organized phenomenon. Journal of Geophysical Research, 95, 17 353-17 361. STEIN, R.S., 1999. The role of stress transfer in earthquake occurrence. Nature, 402, 605-609. STOWELL, J.F.W., LAUBACH,S.E. & OLSON, J.E. 2001. Effect of modern state of stress on flow-controlling fractures: a misleading paradigm in need of revision. Paper presented at DC Rocks', the American Rock Mechanics Association's 38th US Rock Mechanics Symposium, Washington, D.C., July 9. TANG, C. & BAK, P. 1988. Critical Exponents and Scaling Relations for Self-Organized Critical Phenomena. Physics Review Letters, 60, 23. Too, S.R. 2001. The effects on seismic waves of interconnected nearly aligned cracks. Geophysical Journal International, 146, 249-263. VLASTOS, S., LIU, E., MAIN, I.G. & LL X.Y. 2002. Numerical simulation of wave propagation in media with discrete distributions of fractures: effects of fracture sizes and spatial distributions. Geophysical Journal International, 152, 649-668. WALDEN, A.T. & HOSKEN, J.W.J. 1985. An investigation of the spectral properties of primary reflection coefficients. Geophysical Prospecting, 33, 400-435. WALSH, J.B. & BRACE, W.F. 1984. The effect of pressure on porosity and the transport properties of rocks. Journal of Geophysical Research, 89, 9425-9431. WILLIS-RICHARDS, J., WATANABE, K. • TAKAHASHI,H. 1996. Progress towards a stochastic rock mechanics model of engineered geothermal systems. Journal of Geophysical Research, 101(B8), 17 481-17 496. YALE, D.P. 1984. Network modelling of flow, storage and deformation in porous rocks. PhD thesis, Stanford University. ZATSEPIN, S.V. & CRAMPIN, S. 1997. Modelling the compliance of crustal rock: 1. Response of shearwave splitting to differential stress. Geophysical Journal International, 129, 477-494. ZHANG, X. & SANDERSON,D. 2002. Numerical Modelling and Analysis of Fluid Flow and Deformation of Fractured Rock Masses. Elsevier Science, Oxford. ZHANG, X., SANDERSON, D. & BARKER, A.J. 2002. Numerical study of fluid flow of deforming fractured rocks using dual permeability model. Geophysical Journal International, 151, 452-468. ZHANG, Y., HOBBS, B.E., ORD, A., BARNICOAT, A., ZHAO, C., WALSHE, J.L. & LIN, G. 2003. The influence of faulting on host-rock permeability, fluid flow and mineral precipitation: a conceptual 2-d
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Geochemical
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Quantitative determination of hydraulic properties of fractured rock using seismic techniques ENRU L I U 1, M A R K C H A P M A N 1, JOHN A. HUDSON 2, SIMON R. TOD 1'2'3, SONJA M A U L T Z S C H x & X I A N G - Y A N G 1 LI 1
1British Geological Survey, Murchison House, West Mains Road, Edinburgh EH9 3LA, UK 2Department of Applied Mathematics & Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Cambridge CB3 0WA, UK 3present address: BP, UTG Geophysics, Chertsey Road, Sunbury on Thames, Middlesex TW16 7LN, UK
Abstract: There have been significant advances over the last ten years in the use of the seismic anisotropy concept to characterize subsurface fracture systems. Measurements of seismic anisotropy are now used to deduce quantitative information about the fracture orientation and the spatial distribution of fracture intensity. Analysis of the data is based upon various equivalent medium theories that describe the elastic response of a rock containing cracks or fractures in the long wavelength limit. Conventional models assume scale/ frequency independence and hence cannot distinguish between micro-cracks and macrofractures. The latter, however, control the fluid flow in many oil/gas reservoirs, as the fracture size and spacing (hence fracture storability) are essential parameters for reservoir engineers. Recently, a new equivalent medium theory for modelling of wave propagation in media with multi-scale fractures has been presented. The model predicts velocity dispersion and attenuation due to a squirt-flow mechanism at two different scales: the grain scale (micro-cracks and equant matrix porosity) and formation-scale fractures. Application of this model to field data shows that fracture density and fracture size can be inverted successfully from the frequency dependence of the time delay between split shear waves. The derived fracture length matches independent observations from borehole data. This paper presents the results of the latest development in the seismic characterization of natural fractures, with an emphasis on the quantitative determination of fracture sizes.
Fractures and fracture systems control much of the mechanical strength and transport properties of the solid structure and are crucial for hydrocarbon production, control and manipulation of water supplies, and dispersal of pollutants. Open fractures may form flow pathways, but cemented fractures may form significant barriers to flow. Therefore, it is important to distinguish between open and cemented fractures. One of the most promising methods for the detection and characterization of open fractures and prediction of fluid flow directions is undeniably the use of seismic methods, based on the phenomenon of shear-wave splitting (Crampin 1985; Queen & Rizer 1990; Liu et al. 1991, 1993, Li 1997; Potters et al. 1999) and, more recently, on the azimuthal variation of P-wave amplitude versus offset (AVO) (e.g. Lynn et al. 1999; Gray et al. 2002; Li et al. 2003). The success of seismic anisotropy is due to its ability to provide spatial distribution of
subsurface fracture orientations and fracture density. The polarization of fast shear-waves gives the fracture orientation, and fracture intensity can be inferred from time delays between fast and slow shear-waves. As for P-waves, in the presence of aligned vertical fractures in the subsurface, most P-wave attributes (e.g. travel time, velocity, amplitudes) have approximately elliptical variations, where the long axis of the ellipse gives the orientation of fractures, and the relative ratio of short and long axes is proportional to the fracture density. There have been many successful examples in the literature. Note that the majority of the successful applications, particularly those involving surface seismic data, c o m e from areas with simple low relief structures and in areas with limited surface topography. In the geologically complex areas, seismic data processing is harder and interpretation will, in general, be ambiguous. Readers are directed to the papers
From: SHAW,R. P. (ed.) 2005. Understandingthe Micro to Macro Behaviour of Rock-Fluid Systems. Geological Society, London, Special Publications, 249, 29-42. 0305-8719/05/$15.00 9 The Geological Society of London 2005.
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by Li (1997), Li et al. (2003) and Gray et al. (2002) for discussions about the data processing and acquisition design/requirements for successful application of seismic characterization of fractures. Despite this success, reservoir engineers have yet to be convinced to accept seismic anisotropy as a routine technique for fracture characterization because of its failure to provide information about sizes and spacing (hence, volume and storability of fracture network). Though it has been thought that the presence of microscale (grain-scale) cracks and/or macro-scale (metre-scale) fractures are both considered to be the dominant causes of observed anisotropy in hydrocarbon reservoirs (Liu et al. 1993), reservoir engineers are more interested in the latter because fluid flow in reservoirs is believed to be dominated by large-scale fractures. Therefore, a quantitative characterization of natural fracture systems in the subsurface from seismic data would potentially provide essential information for the prediction of permeability and flow patterns within reservoirs. Recent observational evidence suggests that the measured seismic anisotropy as inferred from time delays of split shear-waves actually depends on frequency (Marson-Pidgeon & Savage 1997; Chesnokov et al. 2001; Tod & Liu 2002; Liu et al. 2003a). These observations can be explained adequately and quantitatively modelled using newly developed multi-scale fracture models (Chapman 2003; Chapman et al. 2003). In the past, a range of models which predict frequency dependence of elastic stiffness in fractured rock has been proposed, e.g. Hudson's theories (e.g. Hudson 1981, 1988; Hudson et al. 1996). However, the frequency dependence of seismic anisotropy has not been measured properly until recently. It is now believed that the observation of frequency-dependent seismic anisotropy and the interpretation in terms of fluid flow in multiscale fractured porous media have important implications for an understanding of the causes of seismic anisotropy. In particular, it is shown for the first time that it is possible to extract quantitative information about fracture sizes and spacing from seismic data (Liu et al. 2003a; Maultzsch et al. 2003). Finally, it is suggested that seismic fracture attribute maps can be used to constrain reservoir fracture models using the concept of a discrete fracture network (DFN) model (Rogers et aI. 2003; Vlastos et al. 2003). The majority of our results under the auspices of the NERC-supported micro to macro project have been published. This paper provides an
overview of the achievements, focusing on the current status of seismic techniques for fracture detection, including the latest method for the determination of fracture sizes.
Fracture systems: parameterizations Open fracture systems in outcrops and subsurface reservoirs, such as the one shown in Figure 1, usually have very complex patterns depending on stress distributions (Liu et al. 2000; Rogers 2003). The detailed description of fracture patterns requires many parameters and it is certainly not practical to describe each individual fracture in the fracture network in great detail. However, what are of interest are the parameters controlling the elastic response and hydraulic (fluid flow) response and the aim here is to establish a link between elastic response and flow response of the same fracture systems. For this purpose, the parameters describing fracture systems are classified broadly in the following manner (see Fig. 2). 9
9 9
Fracture density distribution - measure of spatial distribution of the strength or intensity of fracture systems. Statistical distribution - reference to the spatial distributions of fracture orientations, lengths, apertures, surface roughness, etc. Transport properties - controlling parameters of fluid communication in the fracture network, such as fracture permeability (anisotropic), matrix porosity and matrix permeability (isotropic).
Note that specific reference is made to vertical or near-vertical fractures in the context of this paper. The elastic response of fracture systems can be described using various equivalent medium theories (forward modelling indicated by down arrows in Fig. 2) and inversions can then be performed to extract fracture information from seismic data (up arrows in Fig. 2). Not all
20cm I
Fig. 1. A typical fracture pattern from outcrops.
t
DETERMINATION OF FRACTURE PARAMETERS
Fig. 2. Parameterizations of the fractured network in terms of fracture density, statistical properties (length, aperture, roughness) and transport properties (fluid properties, permeability). Down arrows indicate forward modelling (equivalent medium representation) of fracture systems; up arrows indicate inversion process (i.e. estimation of fracture properties from seismic data).
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31
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these parameters can be estimated directly from seismic data. It can be seen in the next section that the most c o m m o n parameters that can be extracted from seismic data are the fracture orientation and fracture density. A c o m m o n parameter in all theories that describe the seismic wave propagation in fractured rock, which is related to the magnitude of anisotropy, is the fracture density e. It is defined as the number density y of cracks multiplied by the crack radius a cubed: e = 3 / a 3 (where y = N/V, N is the number of fractures, V is the volume concerned). This definition o f a fracture has no specific reason and is introduced purely for mathematical convenience. The elastic response and fluid flow response are controlled by different parameters of the fracture systems. Therefore, there are some differences in the parameterization. The fracture density defined above is not the same as the definition used in geological and engineering literature. Geologists and engineers define the fracture density as the n u m b e r of fractures per length. It is difficult to reconcile the two definitions because the seismologists' definition involves two scales (fracture radius and volume), while the engineers' definition (Sue Raikes, pers. comm.) has only one scale (length in which the number of fractures is measured). One possible way to reconcile the two definitions may be as follows (as suggested by one of the referees of this paper). Assuming a standard crack geometry with N = 250 aligned vertical fractures of diameter D = 2a = 0 . 2 m (a is crack radius) and separation perpendicular to the fractures S = I / C = 0 . 1 m in v o l u m e
....
Illlllll
__
_.
Jlllllll
IIII III
II
II
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_
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I III
Fig. 3. Fracture density e is defined as the number density y of cracks multiplied by the crack radius a cubed: e = y a 3 (where y = N/V, N is the number of fractures, V is the volume concerned). The same fracture density can be caused by a few large fractures as shown on the left or many small cracks as shown on the right. V = 1 m 3 (C = 10 is the number of fractures per metre), the fracture density is e = Na3/ V = 250 x 0.13/1 = 0.25. If, instead, a cube is considered with side D, then e = CDa3/ (2D) 3 = CD4/8D 3 : 10 • 0.2/8 = 0.25. It can be seen from the definition of fracture density that a material with only a few large fractures can have the same fracture density as a material with many small cracks, which is illustrated schematically in Figure 3. Most conventional theories do not distinguish the effects of small cracks and large fractures and cannot determine whether the anisotropy is caused by microcracks or macro-fractures. This is regarded as one of the serious limitations of conventional theories.
Modelling cracked and fractured media The interpretation of anisotropic measurements made from seismic data requires theoretical
32
ENRU LIU ETAL.
models that relate measurable seismic parameters to macroscopically determined rock properties (e.g. fracture density and orientations). Based on the assumption that the scale lengths associated with the cracks and fractures are considerably smaller than the seismic wavelength, a description of the average properties of a medium will be sufficient, i.e. an equivalent (or effective) medium description. Various equivalent medium theories have been proposed (Schoenberg 1980; Hudson 1981, 1988; Sheng 1995; Thomsen 1995; Hudson et al. 1996, 2001; Liu et al. 2000; Pointer et al. 2000; Tod 2003a,b). These theories have provided the foundation for extracting fracture information from seismic anisotropy analysis. There is agreement between the models for dry rock, but differences occur in the case of fluid fill and fluid flow between cracks and pores. Also note that these theories are all developed for fractured media with one scale length. For applications to seismic data the Thomsen equant porosity model (Thomsen 1995) and the Hudson crack model (Hudson 1981) are used most widely. Thomsen's model assumes perfect pressure equalization between cracks and equant pores in the surrounding rock matrix. It is, therefore, limited to low frequencies, where the period of the wave is much longer than the time it takes for the pressure to equalize. The flow of fluid from cracks into equant pores can increase the anisotropy significantly (Thomsen 1995). In contrast, Hudson' s ( 1981) model assumes that cracks are isolated and that there is no fluid communication between elements of pore space. It can thus be regarded as a high-frequency theory, bearing in mind that it is still only valid when wavelengths are much longer than the length scale associated with the cracks. Dynamic equivalent medium theories have been proposed by Hudson et al. (1996; interconnected crack model and equant porosity model) and van der Kolk et al. (2001; BOSK model). Tod (2001) considered the Hudson interconnected crack model in the case of nearly aligned cracks, and Tod (2003a) further extended Hudson's model to media with cracks and fractures bounded by layers (called bed-limited or layer-bounded cracks). Tod & Liu (2002) used this later theory to model the frequency-dependent anisotropy observed in earthquake data (Fig. 4). In a separate approach, Tod (2003b) argued that conventional theories are adequate for describing properties of low matrix porosity materials such as carbonates; they provide a poor approximation once the matrix porosity has increased to an extent such that it plays a
significant role in determining the matrix properties, as with sandstones. Due to the significant difference in the behaviour of wave propagation in poroelastic media compared with that in elastic media, an alternative theory is required to describe the full range of porosities encountered in crustal rock adequately. Tod (2003b) started from the basic assumption that a saturated uncracked matrix can be described using Biot theories (Biot 1962) and then used the method of smooth, as used by Hudson (1981) and Hudson et aL (1996) to develop an effective medium theory. The final effective elastic stiffness was given by Tod (2003b) as
Cipjq = C~
0
-Jr- ~c[C~okl- Cipki]Eklpq,
(1)
where C o is the isotropic elastic tensor of the matrix with Lam6 parameters A and/x; C I is the elastic tensor of the materials in the inclusions. ~bc is the crack porosity. E is given in terms of the Eshelby tensor S by Berryman (1997) E = [S S~
I - C~
+ I)] -1,
(2)
where S o is the elastic compliance tensor of the matrix, i.e. the inverse of C ~ and I is the fourth rank identity tensor. Note that the second term in equation (1) is a function of the Lam~ parameters, fluid and fracture properties and frequency. The resulting theory may be used to describe the properties of a material containing a storage porosity associated with the background pore structure of the matrix and a transport porosity associated with the presence of ellipsoidal cracks or inclusions. Due to the presence of the ellipsoidal inclusions, the resulting effective medium exhibits orthorhombic symmetry and, hence, wave velocities will vary with offset and azimuths - the angles to the vertical and horizontal symmetry planes, respectively. An example of this variation is provided in Figure 5, where the two shear-wave speeds are shown to vary with offset angles at a fixed azimuth inline with the narrowest dimension of the ellipsoids, using typical matrix inclusions properties. A considerable degree of shearwave splitting is observed. Modelling multi-scale fractures
Chapman (2003) proposed a poroelastic model based on a squirt-flow mechanism in fractured porous rock. The model considers an isotropic collection of spherical pores and ellipsoidal micro-cracks (either aligned or randomly distributed), the size of which is identified with the
DETERMINATION OF FRACTURE PARAMETERS
33
Fig. 4. (a) Delay time as a function of frequency for three SKS (recorded shear-waves converted from P-waves through Earth core) and two ScS (shear-waves reflected from mantle/core boundary) events recorded at a broadband station in Wellington, New Zealand (Marson-Pidgeon & Savage 1997). (b) Modelling the change in shear-wave anisotropy with frequency.
grain scale and the presence of aligned fractures, which can be larger than the grain scale, but still smaller than the seismic wavelength. Thus, the theory accounts for two different length scales. The resulting medium is transversely isotropic. The model agrees with the results of Brown & Korringa (1975) and Hudson (1981) in the low and high frequency limits, respectively. In the absence of fractures it returns to the earlier squirt-flow model of Chapman et al. (2002). The model has been calibrated by Maultzsch et al. (2003) using the laboratory data of Rathore et al. (1995).
The expressions for the elements of the effective stiffness tensor are given in Chapman (2003). The stiffness tensor is of the form
c0,, : c ~
(3)
where C o is the isotropic elastic tensor of the matrix with Lam6 parameters h and /z; C 1, C a and C 3 are the additional contributions from pores, micro-cracks and fractures, respectively, multiplied by the porosity ~bp, the crack density ec, and the fracture density ef. The corrections
34
ENRU LIU E T AL.
Fig. 5. (a) Schematic of an aggregate in which the misfit between the particles creates a porous system and (b) schematic of a possible distribution of nearly-aligned cracks in an aggregate; the inset shows a reduced version of (a), representing the structure of the material in which the cracks lie. (c) The variation of shear-wave velocities with offset angles for a volume density 0.2 of inclusions.
are functions of the Lam6 parameters, fluid and fracture properties, frequency and a time-scale parameter r, which is related to the squirt-flow (the explicit expressions are given in Chapman et al. 2003). The fact that fluid flow in the model takes place at two scales, the grain scale (microcracks and pores) and the fracture scale, leads to the existence of two characteristic frequencies and associated relaxation times (Fig. 6). The grain-scale fluid flow is related to the traditional squirt-flow frequency (or relaxation time rm), which experiments suggest lies somewhere between the sonic and ultrasonic range (Thomsen 1995). The flow in and out of fractures is associated with a lower characteristic frequency or larger time-scale constant rf, which depends on the size of the fractures. With increasing fracture radius the ratio of surface area to volume decreases. Therefore, more volume of fluid has to move through an element of surface area to equalize the pressure, which requires more time. The two time-scale
parameters are related to each other by the following expression: af Tf = ~ T m ,
(4)
where af is the fracture radius and ~ is the grain size (the scale of pores and micro-cracks), rm is given by Tin =
Cvn(1 + Kc) O-cK~Ct
,
(5)
where Cv is the volume of an individual crack, cl is the number of connections to other elements of pore space, K is matrix permeability and rt is fluid viscosity. O'c = r q x r / [ 2 ( 1 - v)] is the critical stress or equivalently, the inverse of the crack space compressibility and Kc = O-c/kf, with r being the aspect ratio of the cracks, v the Poisson ratio and kf the fluid bulk modulus. The model with two scales (grain-scale pores and meso-scale fractures) described above has
DETERMINATION OF FRACTURE PARAMETERS
Fig. 6. Variation of P-wave velocities with frequency for propagation normal to the fractures (0~ and parallel to the fractures (90~ P-waves do not sense the scale of fractures when they propagate along the fractures, but will show strong dependence on fracture size when they propagate normal to the fractures. Two characteristic frequencies exist: the low characteristic frequency is associated with meso-scale fractures, while the high characteristic frequency is related to the micro-cracks. been extended to accommodate a range of fracture sizes, including a distribution of fracture sizes and orientations (Liu et aL 2003b). The theory models velocity dispersion and velocity anisotropy and, thus, the anisotropy is frequency dependent. The effect is also sensitive to the fracture size. In Figure 7 one can see the change in shear-wave anisotropy with frequency
.4 ~
micro-cracks .
.
.
.
.
.
~-~
.=_
~2 tO')
0
. . . . . . . .
0.1
i
t
. . . . . . . .
i
10
. . . . . . . .
i
100
,
,'~
.....
t
1000
. . . . . . . .
10000
Frequency (Hz)
Fig. 7. Percent shear-wave anisotropy as a function of frequency for different fracture sizes. The waves are propagating at an angle of 60~ measured from the fracture normal. For a given fracture size there is a characteristic frequency range, where anisotropy decreases with increasing frequency. For smaller fractures the change in anisotropy occurs at higher frequencies.
35
as a function of fracture radius. For any given fracture size the anisotropy decreases as frequency increases. This behaviour is consistent with observations from earthquake data (Marson-Pidgeon & Savage 1997; K. Liu et al. 2001). The larger the size of the fractures, the lower the frequency range where velocity dispersion and frequency dependence of anisotropy occurs. The effect has also been observed in vertical seismic profile (VSP) data (Liu et al. 2003a,b). Furthermore, the model can explain a large change in anisotropy due to fluid substitution for frequencies other than the static limit (Chapman et al. 2003). Such an effect has been found by van der Kolk et al. (2001) in shearwave data from a fractured carbonate reservoir.
Estimation of fracture orientation and fracture density Estimating fracture orientation and density f r o m shear-wave splitting Over the past two decades, particularly in the late 1980s and early 1990s, shear-wave data were used in the oil and gas industry to evaluate fractured reservoirs. Field examples that demonstrate the values of shear-wave applications were given by Li (1997), Mueller (1992) and Potters et al. (1999), amongst others. The idea is based on the phenomenon of shear-wave splitting or hirefringence (similar to the birefringence of light in crystal). A shear-wave will split into two waves travelling with different speeds with orthogonal polarizations when entering an anisotropic medium containing aligned vertical fractures. For near-vertical propagation, the fast split shear-wave polarizes parallel to the fracture strike and the slow wave polarizes nearly orthogonal to the fast wave. The time delay between two split shear-waves is proportional to the number density or intensity of fractures. Thus, in theory, one can obtain fracture information of the underlying medium from shear-wave data recorded on the surface or in borehole. With different configurations of sources and receivers, up to nine-component data (called full-wave data) can be recorded consisting of three polarized sources and three component receivers. Ideally, a full nine-component geometry is needed to describe the vector wavefield accurately (hence, called multicomponent seismology). However, in practice, to minimize the cost of acquisition, several configurations of sources and receivers have been used, depending on the purpose of the surveys (see Li 1997).
36
E N R U LIU E T A L .
An example from the Bluebell-Altamont gas field, Uinta Basin in northeastern Utah (readers are referred to the papers by Liu et al. (2003a,b) and Maultzsch et al. (2003) for details) is given in Figure 8. Figure 8a shows the polarization angles obtained through successive rotations after the data were band-pass filtered into five frequency bands. Except for the very low frequency band between 0 H2 and 10 Hz, the polarizations are generally constant over the whole depth interval at 4 0 - 4 5 ~ from the inline direction, which agrees with the direction of predominant fracture orientations of N43~ in the study area (Lynn et al. 1999), and there is no apparent dependence of polarization on frequency. Figure 8b shows the variation in time delays between fast and slow shearwaves. One can identify three distinct intervals. In Interval I (850 m to about 1210 m), as receiver depth increases, time delays increase linearly, indicating this interval is seismically anisotropic. In Interval II (between the depths of 1210 m and 2070 m), the time delays remain almost constant, implying this interval is isotropic for the propagating waves as there is no further shear-wave splitting in this interval. Below a depth of about 2070 m, the time delays begin to increase abruptly. This interval (Interval III), which is also the target reservoir in the BluebellAltamont Field, thus shows strong anisotropy (about 3-4%), which is attributed to the presence of intense fracturing in the reservoir. The shearwave anisotropy is interpreted as being due to the presence of open and aligned vertical fractures, striking northwest in the Upper Green River Formation. If one assumes that the magnitude of shear-wave anisotropy (time delays
between split shear-waves) is proportional to the fracture density, then the highest density of open, gas-filled fractures is interpreted to be in the interval between 2070 m and 2640 m.
Estimating fracture orientation and density from P-wave azimuthal A VO analysis Shear-wave data, though very valuable in providing information about subsurface fractures, are not commonly available. In particular, it is not possible to record shear-wave data in a marine environment (except at seafloors where converted PS waves can be recorded). As a result, there has been a consistent increase in the last few years in the use of 3D P-wave data to characterize fractures (e.g. Li et al. 2003). If it is assumed that the fracture population consists of predominantly one major orientation, the azimuthal variation of P-wave seismic attributes, such as travel time, velocity, reflected wave amplitudes, impedance, etc. can be described approximately by an ellipse. The long axis of the ellipse indicates the fracture orientation and the relative ratio of the short to long axes of this ellipse is proportional to the fracture density or intensity of the rock concerned. It is known that at least three data points are required to define an ellipse in azimuthal planes. Thus, fracture orientation and intensity maps can be built from 3D P-wave data if there is sufficient azimuthal coverage. In the practical application of the azimuthal P-wave AVO analysis, two methods are often employed to extract the fracture information: full-azimuth surface fitting and narrow-azimuth
Fig. 8. Variation of (a) polarization of fast split shear-waves and (b) time delays of split shear-waves with depth after the data have been band-pass filtered into five frequency bands. The angles are relative to the in-line component. The results were taken from the analysis of near-offset VSP data at Bluebell-Altamont Field, Utah by Liu et al. (2003).
DETERMINATION OF FRACTURE PARAMETERS stacking. The first method fits an elliptical surface to data from all available azimuths and offsets by a least-squares fitting technique. The second method divides the data into a number of narrow-azimuth volumes, for example, six azimuths can be chosen with 30 ~ azimuthal bins. Corresponding to these two methods, there are mainly four seismic attributes which may be used to extract the fracture information, including velocity, travel times/interval travel times, amplitude and AVO gradient. The surface fitting method is applicable to the amplitude and travel-time attributes, whilst the narrowazimuth stacking method is applicable to the velocity and AVO gradient attributes. An example is given in Figure 9, which shows the fracture orientation and density maps from an onshore oil field in the Yellow River Delta, China (alter Li et al. 2003). The major faults are overlaid on the seismic fracture attribute maps, from which further fracture porosity and permeability maps may be inferred for drilling planning and for input to reservoir simulations.
Estimating fracture sizes from frequencydependent anisotropy One striking feature in Figure 8 is the dependence of time delays (anisotropy) on frequency. This can be explained by two mechanisms: seismic scattering by heterogeneities and fluid flow in fractured porous rock (discussed in Liu et al. 2003a,b). The polarization angles in Figure 8 are consistent around 43 ~ for all frequency bands. The time delays, in contrast, show a systematic variation with frequency. As frequency increases, the change in time delay with depth decreases, i.e. the magnitude of anisotropy decreases. This behaviour agrees with the theoretical prediction in Figure 7 and has been used to invert for fracture density and fracture radius by Maultzsch et aL (2003), who have presented a detailed study demonstrating the dependence of seismic anisotropy on fracture sizes using the multi-scale fracture model developed by Chapman (2003). This model has been used to invert fracture sizes from field multicomponent shear-wave VSP data (given in Fig. 8). Fracture orientations measured from polarization of fast shear-waves (Fig. 8b) are consistent with borehole, outcrop and core data between N30~ and N45~ Observed fractures are believed to be vertical to sub-vertical. The time delay between the fast and the slow shear-wave shows a sharp increase with depth at the reservoir level, indicating the presence of fractures.
37
From the polarization angles obtained from the field data it is inferred that fractures have an average strike of N43~ which is input into the model. The only unknowns in the model are fracture density and fracture radius. These parameters are estimated by matching the change in time delay with frequency. For each pair of fracture density and fracture radius values, the root mean square (rms) error is computed between the measured and predicted increase in time delay with depth as a function of frequency. Figure 10 displays the error function, i.e. the error between measured and computed time delays as a function of frequency for a wide range of fracture densities and fracture sizes. There is a well-defined minimum at a fracture radius of about 3 m and a fracture density of approximately 4%. Interesting are also the bottom and top end of the diagram are also interesting. They represent approximately what would be obtained using Thomsen's (1995) low frequency and Hudson's (1981) models, respectively. As stated earlier, neither of the models is sensitive to the fracture size, which can be seen clearly in Figure 10. Furthermore, a fracture density of 5% would be inferred from the data by using Hudson's model, while the model of Thomsen yields a value of about 2.5%. However, by incorporating the frequency dependent effects and modelling the data with Chapman's (2003) model, a more tightly constrained estimate of the fracture density is obtained, and fracture size can also be deduced from the data. Figure 11 shows the modelled percentage of anisotropy as a function of frequency in comparison with the real data results. There is a good agreement between the two curves. [Note that Figure 11 is obtained by applying successive short-window band-pass filtering to fast and slow shear-wave components and then subtracting the two components to obtain the time delays. In a previous paper by Liu et aL (2003a,b), the effects of short-window band-pass filtering on the results have been investigated carefully, including synthetic tests and it was concluded that the band-pass filtering technique does not introduce frequency dependency as long as zero-phase band-pass filtering is used.] The error bars represent the error between the measured time delays as a function of depth and the best-fitting straight line. The deduced fracture radius of about 3 m (or fracture length of 6 m) was compared with independent borehole data. There is evidence from borehole images and cores that lengths of fractures in the reservoir lie in the range of 2 - 3 m (Lynn et al. 1999). The inferred average length matches these independent observations quite closely.
38
ENRU LIU E T AL.
Fig. 9. Full-field results for Target T2 for information: (a) fracture orientation and (b) intensity estimated from azimuthal analysis of 3D P-wave amplitudes from an onshore oil field in the Yellow River Delta, China (after Li et al. 2003).
DETERMINATION OF FRACTURE PARAMETERS
39
Fig. 10. Relative error between measured and computed time delay as a function of frequency for a wide range of fracture densities and fracture sizes. There is a clear minimum at a fracture density of 0.04 and a fracture radius of about 3 m.
From seismic data to reservoir simulation: discrete fracture network model Once the seismic fracture attribute maps (orientation, density and possibly fracture size distribution) have been produced, the next logical thing will be to constrain these attributes to build proper reservoir fracture models. The approach is to use the DFN model and the detailed procedure is given in Figure 12, following Rogers e t al. (2003). The seismic fracture intensity and orientation maps are first converted
~
3.6
i 9, 9, . , . ,
3.4
]"
, . , . , 9
Data
3.2
.~ 3.0 0
.~_ t-- 2.8 '~ 2.6
~ 2.4
9 2.2 2.0 I
I
10
12
14
16
18
20
22
24
26
Frequency
Fig. 11. The percentage of anisotropy as a function of centre frequency measured from the VSP data in comparison with the modelled results for the BluebellAltamont Field, Utah (after Maultzsch et al. 2003). The errors represent the uncertainty of the fit to the time delays. The maximum frequency is about 50 Hz (see Liu et al. 2003a,b).
Fig. 12. General workflow for seismically constrained fracture generation based on the DFN technique (after Rogers et al. 2003).
into appropriate fracture parameters using fundamental rock physics relations, preferably calibrated by laboratory measurement, such as the models described in the earlies sections of this paper. If available, vertically zoned seismic impedance tied to individual well locations may also be used to help condition any vertical distribution of fracturing. The geological context must be defined (faults, folds, and stratigraphy). After set up, fractures are generated, constrained to the seismic data in a leastsquares sense. These seismic fracture attributes can then be used as the primary input for advanced fracture modelling tools and for fluid flow simulation (Rogers e t al. 2003; Vlastos e t al. 2003; Liu e t al. 2004). Whilst the determination of meaningful fracture attributes from seismic data is not a trivial process and the route is still, as yet, uncertain, the seismic-fracture model methodology has provided a way of addressing the interwell uncertainty present in most fractured reservoir models. This means that there is the potential for different well configurations and completion strategies to be modelled to improve development planning before drilling. Other geometric issues that could be resolved using the DFN approach are injector-producer short circuits, prediction of early water breakthroughs and also the planning of enhanced recovery methods. It should be emphasized that for the DFN technique to have the maximum impact, the reservoir conceptual model should suggest that the seismic attributes are strongly linked to the features that dominate reservoir permeability. Thus, this
40
ENRU LIU ET AL.
technique lends itself best to low porosity or carbonate reservoirs with a sufficient thickness, for example c. 40 m or greater (which is about a quarter of a wavelength if one assumes that the wave speed is 3000 ms-1 and the frequency is 20 Hz). (It should be noted that in high porosity reservoirs, fractures as well as matrix porosity are likely to affect the seismic response (e.g. Thomsen 1995). Ideally, they would be single layers or multiple layers with similar mechanical properties. There are other items that require attention before this workflow can be established with confidence. The conversion of seismic anisotropy intensity and orientation into true fracture intensity and orientation represents a major theoretical hurdle not addressed to date. Also required is an assessment of the balance between seismic aerial content and borehole 1D data.
Discussion and conclusions Over the last ten years, a wide range of innovative techniques have been developed for mapping the intensity and orientation of fractures using 3D P-wave, converted-wave and S-wave data. Investigation is also taking place to estimate fracture sizes and spacing from frequency-dependent seismic anisotropy (note that fracture spacing is not an independent parameter and, once fracture density and fracture sizes are estimated, it is straightforward to calculate fracture spacing assuming certain types of fracture distribution). Figure 13 summarizes the uncertainty and reliability of fracture parameters that may be estimated from seismic methods. The parameters that can be estimated with least uncertainty are fracture orientations and density. Fluid properties (fluid types and Determination of fracture parameters 9 Fracture orientation 5"
9 Fracture density
o
9 Fluid properties (type,Sw and Pe) 0" ,<
r ,
m
1= or
9 Fracture size
ffJ
,, Fracture spacing
oe--
9 Fracture apertures Fig. 13. Reliability and uncertainty o f seismic characterization o f fracture systems.
saturations) may be inferred from P-wave and converted wave AVO analysis (E. Liu et al. 2001). Progress has also been made, largely due to the effort of the authors of this paper, in predicting fracture sizes and spacing using frequency-dependent seismic anisotropy. Further work is certainly needed to develop this technology for routine use. It will be difficult to estimate fracture aperture, which is the key parameter controlling fluid flow in fracture rock. The reason is that for a given fracture, the effective mechanical aperture controlling elastic response is not necessarily the same as the effective hydraulic aperture determining fluid flow (Renshaw 1995). The authors believe that one of the most important contributions in the last few years in the seismic characterization of natural fractures has been the observation and subsequently quantitative interpretation of frequency/scaledependent anisotropy. Heterogeneous and fractured porous rock may be characterized by observations in different critical wavelength ranges, each reflecting different physical mechanisms. The scale length associated with the heterogeneities or the fracturing has to be much smaller than the seismic wavelength to cause effective anisotropy instead of scattering. This paper has presented results demonstrating the dependence of seismic anisotropic parameters on frequency using a recently developed dynamic equivalent medium theory by Chapman (2003) and Chapman et al. (2003). This model is based on a squirt-flow mechanism and suggests that frequency dependence of anisotropy is sensitive to the length scale of fractures. The model has been tested and calibrated against published laboratory data, which provides the basis for application to field data. A methodology has been developed to invert for fracture density and fracture size from frequency-dependent shear-wave splitting in a near-offset VSP. The derived average fracture length matches geological evidence very well. The study demonstrates that the frequency dependence of shear-wave splitting can be extracted from seismic data and interpreted in terms of an average length scale of fractures. The most important result is the successful discrimination between the effect of micro-cracks at the grain scale or millimetre scale and the effect of formation-scale fractures. Fluid flow and permeability in a reservoir are believed to be controlled much more strongly by formation-scale open fractures than microcracks. Finally, it is suggested that the seismic fracture attributes can be used to constrain optimal reservoir fracture models using the approach of a
DETERMINATION OF FRACTURE PARAMETERS discrete fracture n e t w o r k m o d e l and, thus, can be used as direct inputs for reservoir flow simulations. This will bring seismic characterization of fracture permeability closer to reality (Pride et al. 2003, 2004). H o w e v e r , it is a c k n o w l e d g e d that problems r e m a i n to be solved in characterizing multi-phase fluid flow systems using seismic techniques, as most models describing the seismic response of fractures currently assume single fluid phase. It seems that attenuation anisotropy and associated f r e q u e n c y - d e p e n d e n t azimuthal response of P - w a v e attributes hold promise in characterizing fluid properties and scale of fractures ( C h a p m a n & Liu 2003). The authors are grateful to John H. Queen (Hi-Q Geophysical Inc.), Heloise Lynn (Lynn Inc.) and Evgeni Chesnokov (Oklahoma University) for many useful discussions about frequency-dependent anisotropy and for providing the field VSP data (HL). Thanks go to Steve Rogers (Golder Associates) and Wenjie Dong (ExxonMobil) for useful discussions about the DFN model on various occasions. The authors also want to thank Sue Raikes, Richard Shaw (editors) and anonymous referees for the constructive comments including suggestions for clarification in the definition of fracture density that led to significant improvements in this paper, and the authors have benefited from many discussions with Sue and Richard during project meetings on various occasions. This work was supported mainly by the Natural Environment Research Council (UK) as part of the Thematic microto-Macro (/x2M) Programme (Project No. GST22305) and, in part, by the sponsors of the Edinburgh Anisotropy Project (EAP). This paper is published with the approval of the Executive Director of the British Geological Survey (NERC) and the EAP sponsors.
References BERRYMAN, J.G. 1997. Generalization of Eshelby's formula for a single ellipsoidal elastic inclusion to poroelasticity and thermoelasticity. Physical Research Letters, 79, 1142-1145. BIOT, M.A. 1962. Mechanics of deformation and acoustic propagation in porous media. Journal of Applied Physics, 33, 1482-1498. BROWN, R. & KORRINGA,J. 1975. On the dependence of the elastic properties of a porous rock on the compressibility of the pore fluid. Geophysics, 40, 608-616. CHAPMAN, M. 2003. Frequency dependent anisotropy due to meso-scale fractures in the presence of equant porosity. Geophysical Prospecting, 51, 369-379. CHAPMAN, M. ~z LIU, E. 2003. The frequency dependent azimuthal AVO response of fractured rock. 73rd Annual International Meeting of the Society of Exploration Geophysicists, Tulsa, Oklahoma, USA, 105-108. CHAPMAN, M., MAULTZSCH, S., LIU, E. & Ll, X.Y. 2003. The effect of fluid saturation in an
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anisotropic, multi-scale equant porosity model.
Journal of Applied Geophysics, 54, 101-202. CHAPMAN, M., ZATSEPIN, S.V. & CRAMPIN, S. 2002. Derivation of a microstructural poroelastic model. Geophysical Journal International, 151, 427-451. CHESNOKOV, E.M., QUEEN, J.H., V~CHOREV,A. et al. 2001. Frequency dependent anisotropy, 71st Annual International Meeting of the Society of Exploration Geophysicists, Expanded Abstracts, Tulsa, Oklahoma, USA, 2120-2123. CRAMPIN, S. 1985. Evaluation of anisotropy by shearwave splitting. Geophysics, 50, 142-152. GRAY, F.D., ROBERTS, G. & HEAD, K.J. 2002. Recent advances in determination of fracture strike and crack density from P-wave seismic data. The Leading Edge, 21, 280-285. HUDSON, J.A. 1981. Wave speeds and attenuation of elastic waves in material containing cracks. Geo-
physical Journal of the Royal Astronomical Society, 64, 133-150. HUDSON, J.A. 1988. Seismic wave propagation through materials containing partially saturated rocks. Geophysical Journal International, 92, 33-37. HUDSON, J.A., LIu, E. & CRAMPIN, S. 1996. The mechanical properties of materials with interconnected cracks and pores. Geophysical Journal International, 124, 105-112. HUDSON, J.A., POINTER, T. & LIu, E. 2001. Effective medium theories for fluid saturated materials with aligned cracks. Geophysical Prospecting, 49, 509-522. VAN DER KOLK, C.M., GUEST, W.S. & POTTERS, J.H.H.M. 2001. The 3D shear experiment over the Natih field in Oman: the effect of fracturefilling fluids on shear propagation. Geophysical Prospecting, 49, 179-197. LI, X.Y. 1997. Fractured reservoir delineation using multicomponent seismic data. Geophysical Prospecting, 45, 39-64. LI, X.Y., LIU, Y.J., LIU, E., SHEN, F., LI, Q. & Qu, S. 2003. Fracture detection using land 3D seismic data from the Yellow River Delta, China. The Leading Edge, 22, 680-683. L~u, E., CRAMPIN, S. & QUEEN, J.H. 1991. Fracture detection using reverse vertical seismic profiles and cross-hole surveys at the Conoco Borehole Test Facility, Oklahoma. Geophysical Journal International, 107, 449-463. LIU, E., CRAMPIN, S., QUEEN, J.H. & RIZER, W.D. 1993. Velocity and attenuation anisotropy caused by micro-cracks and macro-fractures in a multiazimuthal reverse VSP. Canadian Journal of Exploration Geophysics, 29, 177-188. LIU, E., HUDSON,J.A. & PO1NTER,T. 2000. Equivalent medium representation of fractured rock. Journal of Geophysical Research, 105, 2981-3000. LIU, E., LI, X.Y. & QUEEN, J.H. 2001. Discrimination of pore fluids from P- and converted shear-wave AVO analysis. In: Ikelle, L. & Gangi, A. (eds.) Ani-
sotropy 2000: Fractures. Converted Waves and Case Studies. Proceedings of 9th International Workshop on Seismic Anisotropy, Society of
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Exploration Geophysicists, Tulsa, Oklahoma, USA, 203-221. LIu, E., QUEEN, J.H., LI, X.Y., CHAPMAN, M., MAULTZSCH, S., LYNN, H.B. & CHESNOKOV, E.M. 2003a. Observation and analysis of frequency-dependent anisotropy from a multicomponent VSP at Bluebell-Altamont Field, Utah. Journal of Applied Geophysics, 54, 319-333. Liu, E., CHAPMAN, M., MAULTZSCH, S., LI, X.Y., QUEEN, J.H. & ZHANG, Z. 2003b. Frequencydependent anisotropy: effects of multi-fracture sets on shear-wave polarizations. 73rd Annual Intemational Meeting of the Society of Exploration Geophysicists, Expanded Abstracts, Tulsa, Oklahoma, USA, 101-104. LIU, E., VLASTOS, S., LI, X.Y., MAIN, I.G. & SCHOENBERG, M. 2004. Modelling seismic wave propagation during fluid injection in a fractured network: Effects of pore fluid pressure on timelapse seismic signatures. The Leading Edge, 23, 778 -783. LIu, K., ZHANG,Z., HU, J. & TENG, J. 2001. Frequency band-dependence of S-wave splitting in China mainland and its implications. Science in China (Series D), 44, 659-665. LYNN, H.B., BECKHAM,W.E., SIMON, K.M., BATES, C.R., LAYMAN, M. 8~; JONES, M. 1999. P-wave and S-wave azimuthal anisotropy at a naturally fractured gas reservoir, Bluebell-Altamont field, Utah. Geophysics, 64, 1312-1328. MARSON-PIDGEON, K. & SAVAGE, M.K. 1997. Frequency-dependent anisotropy in Wellington, New Zealand. Geophysical Research Letters, 24, 3297-3300. MAULTZSCH, S., CHAPMAN, S., LIU, E. & LI, X.Y. 2003. Modelling frequency dependent seismic anisotropy in fluid-saturated rock with aligned fractures: implication of fracture size estimation from anisotropic measurements. Geophysical Prospecting, 51, 381-392. MUELLER, M.C. 1992. Using shear waves to predict lateral variability in vertical fracture intensity. The Leading Edge, 11, 29-35. PRIDE, S., HARRIS, J.M., JOHNSON, D.L. et al., 2003, Permeability dependence of seismic amplitudes. The Leading Edge, 22, 518-524. PRIDE, S., BERRYMAN, J.G. & HARRIS, J.M. 2004. Seismic attenuation due to wave-induced flow. Journal of Geophysical Research, 109 (B01201), 10.1029/2003JB002639. POINTER, T., LIU, E. & HUDSON, J.A. 2000. Seismic wave propagation in cracked porous media. Geophysical Journal International, 142, 199-231. POTTERS, J.H.H.M, GROENENDAAL,H.J.J., OATES, S.J., HAKE, J.H. & KALDEN, A.B. 1999. The 3D shear experiment over the Natih field in Oman: reservoir
geology, data acquisition and anisotropy analysis. Geophysical Prospecting, 47, 637-662. QUEEN, J.H. & RIZER, W.D. 1990. An integrated study of seismic anisotropy and the natural fracture systems at the Conoco Borehole Test Facility, Kay County, Oklahoma. Journal of Geophysical Research, 95, 11255-11273. RATHORE, J.S., FJAER, E., HOLT, R.M. & RENLIE, L. 1995. P- and S-wave anisotropy of a synthetic sandstone with controlled crack geometry. Geophysical Prospecting, 43, 711-728. RENSHAW, C.E. 1995. On the relationship between mechanical and hydraulic aperture in roughwalled fracture. Journal of Geophysical Research, 100, 24 629-24 636. ROGERS, S. 2003. Critical stress-related permeability in fractured rocks. In: AMEEN,M.S. (ed.) Fracture
and In-Situ Stress Characterisation of Hydrocarbon Reservoirs. Geological Society, London, Special Publications, 209, 7-16. ROGERS, S., MACBETH, C., LIU, E. & ANGERER, E. 2003. Constraining models of fractured reservoirs using seismic anisotropy maps, for improved reservoir performance and prediction. 73rd Annual International Meeting of the Society of Exploration Geophysicists, Expanded Abstracts, Tulsa, Oklahoma, USA, 1549-1552. SCHOENBERG, M. 1980. Elastic Wave Behaviour Across Linear Slip Interfaces. Journal of the Acoustical Society of America, 68, 1516-1521. SHENG, P. 1995. Introduction to Wave Scattering, Localization, and Mesoscopic Phenomenon. Academic Press, Inc., London. THOMSEN, L. 1995. Elastic anisotropy due to aligned cracks in porous rock. Geophysical Prospecting, 43, 805-829. TOD, S.R. 2001. The effects on seismic waves of interconnected nearly aligned cracks. Geophysical Journal International, 146, 249-263. TOD, S.R., 2003a. Bed-limited cracks in effective medium theory. Geophysical Journal International, 152, 244-352. TOD, S.R. 2003b. An anisotropic fractured poroelastic effective medium theory. Geophysical Journal International, 153, 1006-1020. Top, S.R. & LIU, E. 2002. Frequency-dependent anisotropy due to fluid flow in bed limited cracks. Geophysical Research Letters, 29, 10.1029/ 2002GL015369. VLASTOS, S., SCHOENBERG,M., MAILLOT, B., MAIN, I.G., LIU, E. & LI, X.Y. 2003. Dual simulations of fluid flow and seismic wave propagation in a fractured network: Effects of changes in pore pressure on signatures of seismic waves. 73rd Annual International Meeting of the Society of Exploration Geophysicists, Expanded Abstracts, Tulsa, Oklahoma, USA, 1382-1385.
Properties of fault damage zones in siliclastic rocks: a modelling approach N. E. O D L I N G , S. D. H A R R I S , A. Z. V A S Z I & R. J. K N I P E
Rock Deformation Research, School of Earth Sciences, University of Leeds, Leeds LS2 9JT, UK (e-mail:
[email protected]) Abstract: Major faults are surrounded by damage zones of minor faults that, in siliclastic
rocks, can form barriers to flow in their own right. Reservoir flow simulation - now a routine part of reservoir management - requires equivalent hydraulic parameters on the scale of the whole fault, while reservoir geological models, from which flow simulator grids are generated, require information on the 3D characteristics of fault populations. Here, a stochastic model of fault damage zone architecture is generated and used to explore the impact of damage zone architecture on extrapolation from 1D (fault throw) and 2D (fault length) to 3D fault population characteristics. Sampling of the simulated damage zone models shows that clustering of faults causes deviations from simple laws relating particularly 1D samples to 3D population power-law exponents, with differences between expected and observed values of up to 0.25. The stochastic model is used to generate input for a 2D discrete fracture flow model for the case where minor (isotropic) fault permeability is four orders of magnitude lower than that of the host rock and, thus, forms partial barriers to flow. The flow model is used to explore the impact of fault damage zones on bulk fault permeability. The damage zone is shown to be around 50% efficient, i.e. a simple estimate of bulk permeability can be made using the harmonic average of fault rock and hostrock permeability weighted by thickness in 1D traverses (e.g. core, well logs), where only half the observed thickness of fault rock in the fault damage zone is assumed. Considering the contributions of the damage zone and the major slip zone, the fault damage zone is likely to make a significant contribution to the bulk permeability of the fault as a whole when the permeability of minor faults in the damage zone is similar to, or at most, one order of magnitude greater than that of the slip zone fault rocks.
Numerous studies of fault zone architecture over the last ten years have shown that, in general, major seismic-scale faults consist of a major slip zone, along which the majority of the displacement occurs, surrounded by a damage zone comprising a complex network of low-throw faults (e.g. Sibson 1992). This article focuses on faults in siliclastic sedimentary rocks where faults within the damage zone take the form of deformation bands along which grain size and porosity are reduced to form partial barriers to fluid flow (e.g. Gabrielsen 1990; Antonellini & Aydin 1994, 1995; Fisher & Knipe 1998). Such faults have been the focus of research in the oil industry for some years and there is now increasing interest in their impact on flow direction and contaminant transport in sandstone aquifers (Wealthall et al. 2001). Flow simulation at the reservoir scale is now a routine task for reservoir management. Due to the limited resolution of flow simulation grids, faults are included as equivalent hydraulic parameters, such as bulk permeability. The often difficult
job of the geologist and/or reservoir engineer is to provide these parameters. Since much of the detail of minor fault architecture within fault damage zones is presently below seismic resolution, the parameters describing the effective hydraulic properties of faults and their damage zones must be deduced from outcrop or core observations on fault architecture combined with flow modelling studies. Generating 3D models of fault damage architecture requires information on many parameters including the size distribution of the fault system. In practice, 3D data on fault geometry are seldom available and the frequency distribution of length and throw must be deduced from 2D maps and sections or 1D togs. Previous studies (Cowie & Scholz 1992b; Bour & Davy 1999) have suggested that simple conversions, valid for spatially random systems, are not necessarily valid in natural systems. Here, the relationship between 1D, 2D and 3D samples of fault length and throw, and the properties of bulk permeability in fault damage zones are demonstrated
From: SHAW,R. P. (ed.) 2005. Understandingthe Micro to Macro Behaviour of Rock-Fluid Systems. Geological Society, London, Special Publications, 249, 43-59. 0305-8719/05/$15.00 9 The Geological Society of London 2005.
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using a stochastic model of fault damage zone architecture together with discrete fracture flow modelling.
Fault damage zones in siliclastic rocks The parameters required to generate geologically realistic stochastic models of fault damage zones are the fault length and orientation distributions, fault aspect ratio, length-thickness relations both for a single fault and for fault populations, and the fault spatial distribution. A number of studies on faults in siliclastic rocks and their damage zones (Antonellini & Aydin 1994, 1995; Fowles & Burley 1994; Knott et al. 1996; Foxford et al. 1998; Beach et al. 1999; Hesthammer et al. 2000; Flodin et aL 2001; Shipton & Cowie 2001; Jourde et al. 2002; Shipton et al. 2002) have outlined their main characteristics. Other parameters required are not readily available from the literature directly but can be inferred from general observations on the properties of fault systems. The evidence that can be used to quantify these parameters is summarised here. M i n o r f a u l t orientations
Fault damage zones consist of a dense network of minor faults and deformation bands with a range of dips and strike orientations that ensure good connectivity (Balberg & Binenbaum 1983; Robinson 1983; Antonellini & Aydin 1994; Shipton & Cowie 2001). Deformation bands dominate and faults with slip planes tend to be segmented and unconnected (Antonellini & Aydin 1994; Shipton & Cowie 2001). The majority of these features trend sub-parallel to the major fault, with a scatter of 25 ~ to 30 ~ about the main fault trend (Shipton & Cowie 2001). Their dips are synthetic and antithetic (i.e. bimodal) to the main fault in approximately equal abundance for faults with displacement of 3 0 m and more (Antonellini & Aydin 1994; Shipton & Cowie 2001). For faults with smaller throws, synthetic dips dominate (unimodal), suggesting that antithetic structures are formed late in order to accommodate increasing deformation in the rock surrounding the major slip plane (Hesthammer et al. 2000). M i n o r f a u l t density a n d f a u l t d a m a g e zone width
Damage zone width and deformation band density generally correlate with fault throw (Beach et al. 1999). A fault of 30 m throw has
a damage zone of around 75 m in width (Shipton & Cowie 2001). The highest density of deformation bands and minor faults occurs close to the fault plane but frequency profiles across fault damage zones show a degree of clustering, particularly around the larger of the minor faults within the fault damage zone (Antonellini & Aydin 1994; Fowles & Burley 1994; Knott et al. 1996; Knipe et al. 1997; Beach et al. 1999; Hesthammer et al. 2000; Shipton et al. 2002). Deformation band densities of up to 20 to 40 per metre for faults with around 30 m throw have been reported (Hesthammer et al. 2000; Shipton & Cowie 2001). M i n o r f a u l t t h r o w a n d length distributions
Power-law fault throw populations within fault damage zones have been observed (Knott et al. 1996; Harris et al. 2003) which, since fault length and throw are generally linearly related (see Cowie et al. 1996 for a review), suggests that fault length distributions are also likely to follow a power law. Fault length distributions from map information are frequently power law with exponents ranging from 0.3 to 2.3 for the cumulative frequency distribution (Bonnet et al. 2001). Although some of this variation probably arises from problems in determining the power-law exponent, such as in the removal of truncation and censoring effects (e.g. Bour et al. 2002), it is thought that this reflects a natural occurring range in power-law exponents. It seems likely that such a range of exponents can also be expected in minor faults and deformation band populations within fault damage zones. F a u l t shape and f a u l t rock thickness
The thickness of fault rock on individual faults is related to fault length, displacement and lithology. Based upon observations from seismic surveys, isolated normal faults are typically planar, approximately elliptical in shape, with an average aspect ratio of around 2, and have a sub-horizontal long axis (Rippon 1985; Nicol et al. 1996). The throw distribution for isolated faults is generally maximum at the fault centre and decreases linearly towards the fault tip line (Childs et al. 1995). Restricted faults, which intersect with other faults, show similar overall patterns, although throw distributions are complicated by interaction with other faults. The relationship between fault length and throw is linear in a linear elastic medium (Pollard & Segall 1987; Cowie & Scholz 1992a), but can be power law in natural systems (Childs et al.
MODELLING FAULT DAMAGE ZONE PROPERTIES 1990; Jackson & Sanderson 1992; Picketing et al. 1996; Steen & Andresen 1999) due to the varying mechanical properties of lithological layers and the linkage and interactions of faults. A review by Gillespie et al. (1992) indicated that fault displacement:length ratios for high porosity sandstones lie in the range of 1:30 to 1:500, centring on a ratio of around 1:100. Manzocchi et al. (1999), from field observations, give the ratio between fault rock thickness and fault displacement for major faults as 1:66 with effective ratios of 1:170 suggested for flow modelling. Assuming, as a first approximation, a linear relationship between fault throw and fault length and using an average thickness: displacement ratio of 1:100 together with a displacement:length ratio of 1:100 suggests a thickness:length ratio for individual faults of around 1:104 at fault centres. The above information on the architecture of fault damage zones has been used to create a stochastic model of a fault damage zone that incorporates geologically realistic length and orientation distributions coupled with lengththickness correlations and a geologically realistic clustered spatial distribution.
45
to 2.8 have been used to represent the range of most commonly occurring power-law length distributions. This corresponds to the core of the frequency distribution of power-law exponents for fault length distributions found in the literature (Bonnet et al. 2001). Variations in both the strike and dip of small faults located around larger structures are expressed through Gaussian distributions, each with standard deviation 10 ~ following field observations, and with a mean strike parallel to the major slip plane and a mean vertical dip.
Describing f a u l t spatial distributions
The spatial distribution of faults is one of the most challenging characteristics to quantify and simulate and attempts to locate faults spatially rely on geometrical rules tested against natural patterns. Many simulation techniques have used a Poisson point process, based on a spatially random and independent distribution of every fault (Baecher et al. 1977; Long et al. 1982). Non-random spatial distributions in models of fault networks include: 9
A stochastic model of a fault damage zone Fault a n d f a u l t system p a r a m e t e r s
A stochastic model of fault damage zones has been created using the parameters summarized in the previous section. The fault models presented here are developed and described fully in Harris et al. (2003) and only a brief description is given here. Each fault within the fault damage zone model is represented by a simple elliptical surface whose aspect ratio follows a Gaussian frequency distribution with mean 2 and standard deviation 0.05. This represents the variation in fault shape caused by interference between faults. The model presented here assumes a linear relationship between length and displacement for each modelled fault, with the displacement increasing from zero at the fault tip line to a maximum value at the fault centre. The ratio of maximum throw to length is set to 1:100, as is the fault rock thickness:throw ratio, giving a length:thickness ratio of 104:1. Fault plane major axes (fault length) are assumed to follow a power law in which the cumulative number of faults of length at least l is F(l)oc 1-D3, where higher values of the 3D power-law exponent D3 indicate the increasing dominance of small faults within the population. The fault population is constrained to lie within a length range lmin < l < Imax. Exponents from 1.6
the 'parent-daughter' model (Hestir et al. 1987); 9 the 'FracMan' software (Golder Associates, http://fracman.golder.com) which is a 'nearest-neighbour' model generating clusters of faults around larger faults; 9 the 'war zone' model (Black 1993; Dershowitz et al. 1998); 9 a multiplicative cascade technique with random input from a L6vy-stable distribution (Belfield 1998).
In papers by Harris et al. (2003) and Odling et al. (2004), the effect of spatial clustering on the geometrical and hydraulic properties of fault damage zone models is explored in detail. Here, the focus is on the most geologically realistic of the models in these articles, in which clustering is hierarchical in nature. In this model, all faults are clustered around larger faults, producing clustering at all scales. The spatial distribution is, therefore, correlated and the choice of location for every fault is influenced by the location of all larger faults. The resultant pattern contains sub-clusters of faults over several fault length scales. A comparison of this frequency distribution with those observed for the Moab Fault, Utah and the Ninety Fathom fault, NE England (see Fig. 1) shows that the hierarchical model produces density variations that most closely resemble these natural examples (Harris et al. 2003). The properties of fault damage zone models created
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Fig. 1. (a) Fault frequency profiles across the fault damage zone model incorporating hierarchical clustering, showing a general increase towards the main fault but also sub-clusters of faults within the damage zone. (b) This profile is compared with the frequency profile across a natural fault (the Ninety Fathom Fault, Northumberland, UK) which shows similar features. using this spatial model are compared to one in which fault locations are random and uncorrelated. From the descriptions above, fault parameters representing a normal fault in semi-lithified siliclastic rocks, with a length of 3 km and a throw of 30 m, have been determined and are listed in Table 1. These parameters have been used to generate a suite of fault damage zone models. Each of the 3D models represents a cuboidal region centred on the major fault plane. This major fault plane is represented by an ellipse with a horizontal long axis of 3 km and a vertical short axis of 1.5 km. The simulated fault damage zone volume measures 1 km horizontally parallel to the fault trend, 150 m in the vertical direction and is 80 m thick. This sub-section of the damage zone was simulated as to simulate the whole damage zone would be computationally prohibitive. Faults were initially generated in a volume
20% bigger in all directions than the final model volume and then trimmed, to avoid edge effects in the simulation. These fault damage zone models have been used to investigate subsampling issues and the hydraulic properties of fault damage zones. Examples of 2D sections through models with different values of the fault length exponent are shown in Figure 2, which clearly shows the spatial clustering of faults.
Sub-sampling 3D fault damage zones Extrapolating f r o m 1D and 2D to 3D - the problem
In practice, 3D data are seldom available and fault characteristics, such as the length and throw distributions, must be deduced from 2D sections (maps of outcrops or seismic horizons)
Table 1. Fault damage zone model parameters Fault attributes Maximum fault length, lmax Minimum fault length, Imin Power law exponent, D3 Aspect ratio Major axis plunge angle Fault throw:length ratio Orientation distribution
Value 10 000 m 2.5m 1.6 to 2.8 Gaussian, mean 2, standard deviation 0.05 0o 1:100 Strike: mean 0 ~ standard deviation 10~ Dip: mean 90 ~ standard deviation 10~
MODELLING FAULT DAMAGE ZONE PROPERTIES
47
Fig. 2. Two-dimensionalvertical sections (150 x 80 m) through 3D fault damage zone simulations for three values of length exponent, D3. When D3 = 1.6, long faults are abundant, creating a denser network of faults. When D3 ~- 2.4, there are many more small faults and the fault network is less dense and occupies a narrower region. In all cases the effects of the hierarchical clustering can be seen in the variation in fault density within the sections.
or 1D sections (line samples in outcrop, cores, borehole logs, or horizon displacements in seismic sections). The correct interpretation of 1D and 2D information and its use to infer 3D fault zone characteristics is of crucial importance for modelling fault network geometries. One-dimensional samples of fault throw populations and 2D samples of fault length populations are frequently observed to be power law (e.g. Childs et al. 1990; Scholz & Cowie 1990; Gillespie et al. 1993; Nicol et al. 1996; Picketing et al. 1996; Watterson et al. 1996; Steen & Andresen 1999), with 1D power-law exponents (cumulative distribution) within the range 0.4-1 and 2D exponents of 0.3-2.3 (Bonnet et al. 2001). If fault throw and length are uncorrelated with respect to location, then the 3D to 2D, and 2D to 1D exponents should each differ by 1 (Marrett & Allmendinger 1991), but such simple rules do not necessarily apply for natural fault patterns (Cowie & Scholz 1992b; Bour & Davy 1999) due to fault clustering (Borgos et al. 2000). A simple example which shows the potential effects of clustering is the case when every fault is vertical, with its centre lying along a single straight line. Twodimensional sub-samples within planes containing the line, and 1D line samples along the line
can, in this case, have the same power-law behaviour as the 3D population.
S u b - s a m p l i n g f r o m the f a u l t d a m a g e zone m o d e l s
Fault damage zone simulations have been generated over a range of size-frequency power-law exponents based on the parameters summarized in Table 1, using spatially random distributions and the hierarchical clustering model. To ensure that the number of sub-sampled faults at each scale are sufficient to deduce population characteristics, 3D domains have been populated with large samples of N fault lengths (2 to 6.5 million faults), drawn randomly from the appropriate power-law cumulative frequency function and they have been distributed spatially according to a range of models from hierarchical to random. The 2D sub-samples are taken using horizontal (x, y)-planes (perpendicular to the major slip plane) at different z locations within the 3D models, and the 1D line samples are taken in the x-direction (normal to the major slip plane) at different positions within these (y, z)-planes. From these samples, the cumulative frequency
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distributions of fault lengths, l, in 2D sections and fault throws, t, in 1D sample lines are determined. The results for the case of D 3 = 2.4 are shown in Figure 3. It is clear that the results for the spatially random model are relatively independent of location for both 2D samples of fault lengths and 1D samples of fault throws. For the hierarchical clustering model, the 2D samples of fault lengths show similar results to the spatially random model, but for the 1D sample of fault throws, the numbers of faults sampled and the slope of the graph can vary significantly according to the sample location.
In addition, the 1D samples do not always show a significant straight line segment from which a slope can be measured. Similar plots to Figure 3 were made for D3 power-law length exponents from 1.6 to 2.8 in steps of 0.2. For each plot, the segment of the cumulative frequency curve where a straight line is appropriate was determined graphically, which is a commonly used method in the literature. In doing this, the limitations imposed by the 3D model have been taken into account, and the robustness of the graphical technique has been verified in Harris et al. (2003).
Fig. 3. Log-log plots of the cumulative frequency function against fault length, 1 (m) and fault throw, t (cm) for 2D and 1D sub-samples, respectively, of the 3D fault damage zone models in the case of D3 = 2.4: models incorporating (a) random spatial location of faults, and (b) hierarchically clustered faults. The dotted lines for the 2D and 1D plots indicate the gradients - 1.4 and - 0.4, respectively, correspondingto a reduction of 1 or 2 with respect to the D3 powerlaw exponent. Multiple lines correspond to the three 2D samples and the (a) seven and (b) nine 1D samples.
MODELLING FAULT DAMAGE ZONE PROPERTIES The straight line segments identified range over one (considered to be the m i n i m u m acceptable range) to over two orders of magnitude, the range generally being greater for the hierarchically clustered model. The results are plotted in Figure 4, w h i c h shows the range of observed D1 and D2 exponents of the sampled populations and their d e p e n d e n c e on the parent D 3 value.
49
The sampled populations have shown that, in the 2D sampling case, the simple rule where D2 = D3 - 1 (Marrett & Allmendinger 1991) is generally obeyed by both spatial models. However, despite the power-law nature of the parent 3D fault s i z e - f r e q u e n c y distribution in all cases, 1D sub-samples do not always show power-law characteristics, particularly w h e n the D3 exponent is low (D 3 = 1.6 to 2.2). Some of these samples
Fig. 4. (a,b) Upper limit to the range over which the D1 and D2 exponents can be estimated. The lower limit is 2.5 m for 2D subsamples and 2.5 cm for 1D subsamples. The scale ranges over which exponents are estimated vary from one to two orders of magnitude. For 1D samples, no straight line segment was identified for values of the D 3 exponent up to 2.4. (c,d) Variation in the D1 and De exponents with the D 3 exponent for the two models. The pale shaded area represents the range of relationships between the D1, D2 and D 3 exponents that are possible. D2 = D 3 - 1 is expected for a spatially random system and D2 =- D3 can occur when all minor fault centres are placed on a single line (see text for further explanation). The darker shaded area indicates a range of exponents found for the hierarchical clustering model.
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contained large numbers of sampled faults (up to 600), so that these observations are unlikely to be due purely to small sample sizes. Generally, one observes a shallowing of the slope at small scales due to limited resolution where small faults are under-represented (termed truncation), and a steepening of the slope at large scales due to the finite size of the samples, causing large faults to be under-sampled (termed censoring) (Lindsey & Rothrock 1995; Picketing et al. 1995; Bour & Davy 1999; Odling et al. 1999). These effects reduce the segment of the graph over which a power-law behaviour can be estimated reliably. It seems that, in some of the 1D cases, the effects of truncation and censoring have dominated, masking the power-law nature of the underlying population. Thus, the presence of a parent powerlaw length distribution need not necessarily be apparent in 1D samples; and, conversely, the absence of a straight line segment does not necessarily imply that the 3D fault population does not have power-law characteristics. In some of the 2D sub-samples of fault lengths, the cumulative distribution appears to show two slopes, separated by a kink, with a steeper slope at larger scales (Fig. 3). There has been a tendency for such distributions to be interpreted as physically meaningful. However, here these cumulative distributions are derived from parent 3D populations that have power-law size distributions with a single exponent. The most pronounced kinks are found for the clustered models with low D3 values (D3 = 1.6 to 1.8) and may, therefore, be the result of uneven sampling of the population due to clustering. This illustrates how it may be misleading to interpret changes in the slope of the subsampled frequency distribution as physically meaningful without other supporting evidence, such as a change in the deformation mechanism (Fossen & Hesthammer 1997; Shipton & Cowie 2001) or control by lithological layering. These results suggest that spatial clustering influences the fault population characteristics of particularly 1D samples and show that the use of 1D samples of fault throw to predict the 3D size-frequency distribution of fault throw should be undertaken with care. In addition, interpretation of kinks in the trend of such subsamples as being physically meaningful should be made with caution.
Two-dimensional flow modelling with faults as partial flow barriers Flow through 2D sections of the stochastic fault damage zone model is simulated using a 2D
finite-difference, discrete fracture, steady-state flow model for flow in porous rocks with fractures (faults) as flow barriers. The model is described in detail in Odling et al. (2004) and only a brief description is given here. In this flow model, both the faults and the rock matrix are discretized onto a regular square grid (Fig. 5a) and the use of large grids up to 400 x 400 in size allows the fault networks to be reproduced faithfully in the model. In discretizing the fault network onto the grid, automatic checks are made to ensure that the connectivity of the network is preserved. Each element within this grid is assigned a permeability representative of the rock matrix together with any fault rock. Both elements that contain a fault and those that abut onto a fault must include fault rock material in the calculation of element permeability in order to model faults as flow bartiers effectively. For flow in both the rock matrix and fault rock, Darcy's law is assumed to apply (non-inertial, laminar flow), which is valid as long as the flow rates are sufficiently small. Elements which represent faults comprise a matrix block containing a thin sheet of low (isotropic) permeability, fault rock material (Fig. 5b). The appropriate permeability of these blocks is calculated using (Fig. 5b) the arithmetic average of permeabilities in parallel and the harmonic average of permeabilities in series (Muskat 1937; Pickup et al. 1995). Corrections to these element penneabilities are made for the orientation of the fault with respect to the grid and for the effect of gridding on fault length, as faults oblique to the grid directions are represented by a 'staircase' of grid elements (Fig. 5c). These corrections ensure that the characteristics of the fault network are represented correctly and that the model results are independent of the grid size. A global pressure gradient is enforced across the model and the pressure at each node is determined by assuming local conservation of flow, i.e. that the net flow towards each node equals zero. Having the pressure field, the flow field can be determined using local permeabilities. An example of a modelled flow field, where flow was perpendicular to the fault trend, in a 2D region (5 x 5 m) within a simulated fault damage zone model is shown in Figure 6. This shows areas of concentrated flow through narrow passages between the faults and at selected points along the faults. The effect of many faults in the region is to 'compartmentalize' the flow field into regions of higher and lower flow with large contrasts in flow rate across faults. The arrows in Figure 6 show how flow directions range through almost 180 ~, which indicates that
MODELLING FAULT DAMAGE ZONE PROPERTIES
51
Fig. 6. An example of a simulated flow field for a 2D, 5 x 5 m region through the fault damage zone model with flow perpendicular to the fault. White denotes low flow speed; black, high flow speed; arrows show flow velocity direction. High flow rates are generated through narrow gaps between the faults and where two or more faults intersect. Flow directions range through a wide angle + 90 ~from the applied pressure gradient as fluid is deflected around the faults. Fig. 5. Discretization of faults onto a regular grid for input to the flow model. (a) In the case of faults as flow barriers the faults are represented by thin plates of low permeability material. The presence of a fault influences the permeability assigned to the grid elements that lie along, or abut against, the discretized trace of the fault. Modifying the permeability of grid elements that abut onto faults is necessary because, without this, fluid can pass along two grid elements on either side of the fault without recognizing the presence of the fault. (b) Grid element permeabilities are given by the harmonic average of matrix and fault permeabilities when the fault is perpendicular to the flow direction, and by the arithmetic average of these permeabilities when the fault is parallel to the flow direction. (c) Faults oblique to the grid directions are represented by a staircase geometry and a correction is made for the resulting increase in length.
flow pathways are tortuous. Harris et al. (1999, 2001) have shown that such pathways can also be simulated using purely geometrical rules by balancing pathway length and fault rock thickness crossed. The shortest pathways involve crossing a large thickness of fault rock, whereas pathways
that minimize fault rock thickness are very tortuous and, therefore, long. A n algorithm that controls the balance between these two opposing tendencies has shown similar pathway geometries to the flow models and so, with calibration, could be used to estimate bulk permeability.
Scaling of bulk permeability in fault damage zones Flow modelling and upscaling permeability Flow modelling is routinely used today to predict h y d r o c a r b o n reservoir and aquifer behaviour and to plan h y d r o c a r b o n production and water abstraction strategies. Due to computational limitations, flow simulator grids cells are typically tens to hundreds of metres across and, therefore, geological details on a finer scale than this cannot be represented (Pickup et al. 1995). The influence of this fine-scale structure on flow must be upscaled to provide representative
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N.E. ODLING ETAL.
permeabilities on the scale of the flow simulator grid cells, and methodologies to do this are a topic of intensive research today. The aim of upscaling is to reproduce the global behaviour of the reservoir while representing the local behaviour as well as possible. There exists a wide range of upscaling techniques in the literature which vary in accuracy, applicability and speed. A number of good reviews of these exist in the literature (e.g. Sanchez-Vila et al. 1995; Kumar et al. 1997; Renard & de Marsily 1997). The most widely applied method that can be used for any heterogeneity field is the solution of the flow field based on Darcy's law. Strictly, the upscaled 'block' permeability is dependent on the boundary conditions of the cell in situ which are influenced by its surroundings, but the accurate determination of these (Almeida et al. 1996) is computationally very intensive. The classical method is to apply no-flow conditions on the boundaries parallel to the flow direction and a constant pressure difference between the other boundaries. This gives the bulk permeability in the direction of flow and considers the block in isolation from its surroundings. This method of upscaling has been used for faults and fault zones by Caine & Forster (1999), Flodin et al. (2001) and Jourde et al. (2002). It is this classical method of upscaling that has been applied here to determine the upscaled permeability properties of fault damage zones.
faults. A ratio of rock matrix to fault rock permeability of 1 0 4 , representative of the permeability contrast commonly found between deformation bands and their host rock, was assumed (Antonellini & Aydin 1994; Taylor & Pollard 2000; Fisher & Knipe 2001). Each 50 x 50 m sub-region was discretized onto a regular square grid with dimensions ranging from 150 to 400, as required to give a reasonable representation of the fault network geometry. Classic boundary conditions comprising noflow top and bottom and a pressure gradient applied right to left were applied. The pressure gradient was applied parallel and perpendicular to the major fault for each sub-region, giving bulk permeability for these two directions. This scheme generated a total of four datasets, each of which contains 260 values of bulk permeability. Bulk permeabilities were estimated using 2D flow modelling and 2D sections through the 3D fault damage zone model. In 3D volumes, there are more possible flow pathways and, therefore, more chances to find high permeability routes through the fault network. Thus, it might therefore be expected that the 2D flow modelling will tend to underestimate bulk permeability, i.e. overestimate the influence of the fault network. However, recent preliminary 3D modelling shows that the errors involved are around 5 - 2 5 % of the 2D bulk permeability and, therefore, minor compared with the variations in permeability with direction and model type.
Two-dimensional sampling o f the 3D models
Bulk permeability distribution parameters
f o r input to the f l o w model
The frequency distributions of bulk permeability for 50 x 50 m regions, for a given flow direction and length exponent, were found to show acceptable fits to a log-normal distribution (Levenberg-Marquardt method, Press et al. 1992) using the Kolmogorov-Smirnov test (e.g. Cheeney 1983). Odling et al. (2004) also show that at small sub-region sizes (5 x 5 m), the frequency distribution becomes asymmetric, with a tendency toward power-law behaviour. This is due to the dominance of the matrix permeability when the fault system is unconnected, which is more likely for smaller sub-regions. The means of log bulk permeability (geometric mean of permeability) are between one and two orders of magnitude lower than that of rock matrix perpendicular to the main fault, and up to one and a half orders of magnitude lower parallel to the main fault. Individual samples show a permeability anisotropy of around an order of magnitude. Models with a length exponent of 1.8 show permeabilities around one half of an order of
Fault damage zone models have been sampled to provide input to the discrete fracture flow model. For the purposes of this paper, the focus is on the bulk permeability of 50 x 50 m regions sampled from two models with hierarchical clustering and power-law length distribution exponents of 1.8 and 2.2, representing the range of the most commonly occurring exponents found in the literature (Bonnet et al. 2001). Odling et al. (2004) give more details of the influence of power-law exponent, sample size and spatial distribution on the statistics of bulk permeability. From each simulated fault damage zone volume, a total of 13 equally-spaced horizontal 2D sections were selected and, from these, a total of 260 subregions, each 50 x 50 m in size, were selected. The major fault has been omitted from each sub-sample so that the effects of the fault damage zone alone can be investigated. These sub-regions provide the fault system geometry and fault rock thickness along the
MODELLING FAULT DAMAGE ZONE PROPERTIES magnitude lower than models with a length exponent of 2.2, for a given flow direction. This occurs for three reasons. First, since the number of faults in all models is the same (around 1.5 million), models with the lower absolute exponent value (and, therefore, a greater proportion of long faults) contain a greater total fault trace length. Secondly, in the model, the thickness of minor fault rock is correlated with fault length so that fault rock thicknesses are generally greater for models with a lower absolute exponent value. Thirdly, models with a great proportion of long faults (lower absolute exponent values) are better connected and there are, therefore, fewer chances for fluid to flow around faults, than in the case where small faults dominate (higher absolute exponent value). Thus, the larger fault trace length, greater thickness of minor fault rock and the greater connectivity for the case of length exponent 1.8 compared with exponent 2.2, all contribute towards lower bulk permeabilities. The variances of the bulk permeability distributions characterize the inherent variation in bulk permeability. Odling et al. (2004) showed that the variance decreases dramatically with an increase in sub-sample region size, for a given flow direction and exponent. At a sub-sample size of 50 x 50m, both models and flow directions show similar variances in the range 0.01 to 0.07.
Bulk permeability in terms of fault damage zone efficiency F a u l t d a m a g e zone efficiency
The bulk permeability of a faulted region is the result of a complex interplay between the fault geometry and fault rock thickness. One of the simplest methods of estimating bulk permeability (e.g. Antonellini & Aydin 1994, 1995; Shipton et al. 2002) from line sample data consists of calculating the hm'monic average of fault rock and matrix permeabilities (Muskat 1937; Pickup et al. 1995), weighted by their relative thicknesses along the sample line:
a a], ~:=kmm--
1-A+Arr
'
(1)
where a is the total fault rock thickness, A is the total line length, ~:' is the bulk permeability and r is the ratio of the fault permeability to the matrix permeability. The 2D equivalent of this 1D approach can be thought of as replacing all of the fault rock in an area with a single fault of uniform thickness that spans the region and is orientated perpendicular to the flow (see Fig. 7).
FLOW
53 FLOW
ii,
It
/
equivalent single fault
fault network
Fig. 7. Definition of fault network efficiency as a barrier to flow. The fault network (left) is replaced by a single spanning fault of uniform thickness (right). Both regions contain the same proportion of fault rock. The single fault represents the configuration of fault rock that provides the maximum barrier to flow across the region and is defined as being 100% efficient. The bulk permeability of this region can be determined analytically from the harmonic average of host rock and fault rock permeabilities. Lower levels of efficiency are defined as single faults of proportionally lower thickness.
In this configuration, the fault rock is at its most efficient as a barrier for flow perpendicular to the fault. Equation (1) underestimates the bulk permeability in this direction as it assumes straight-line flow paths for fluid along the sample line, whereas flow paths are tortuous and strike a balance between the minimum fault rock traversed and the minimum overall path length. Equation (1), giving the bulk permeability of the equivalent single fault system, represents 100% efficiency, as it assumes that all fault rock contributes as much as possible to inhibit flow. Lower levels of efficiency can then be defined by taking proportions of this single fault thickness. The proportion of fault rock can also be determined from line samples such as cores and borehole logs, information which it is possible to obtain from hydrocarbon reservoirs and aquifers (e.g. Hesthammer et al. 2000; Shipton et al. 2002). In the simulated 2D sample regions from the statistical fault damage zone model, the proportion of fault rock has been determined for all samples for which the bulk permeability has been estimated. Manipulating equation (1) gives:
1
~-1=
a(l_l )
x
(2)
Thus, in a plot of log(1/~:- 1) against the log of the fault rock proportion, a/A, the case of an equivalent single fault that spans the entire region is given by a straight line with a
54
N.E. ODLING ET AL.
slope of 1 intercepting the vertical axis (where log(1/k - 5) = 0) at - l o g ( i / r - 1). Different degrees of efficiency are represented by lines of slope 1 with different intercepts, so that the (100c~)% efficient line has an intercept of -log[a(1/r-l)], where 0 < c ~ < 5. Such a plot allows easy evaluation of the fault network efficiency for the fault damage zone models. A plot of these quantities for the two power-law exponents of 2.2 and 1.8, and two directions perpendicular and parallel to the main fault, are shown in Figure 8. Figure 8 shows that bulk permeabilities of the fault damage zone alone perpendicular and parallel to the main fault are separated by a little over an order of magnitude in both models, the bulk permeability parallel to the main fault being the larger and corresponding to lower values of 1 / k - 5. In Odling et al. (2004) it is also shown that, as the sample size increases from 5 m to 50 m, the spread of both the fault rock proportion and the bulk permeability decrease from around three to one order of magnitude. For regions of 50 x 50 m, bulk permeabilities cluster close to the 50% efficiency contour for permeability perpendicular to the main fault, and between the 10% and 1% efficiency contours for permeability parallel to the main fault. Generally, the clouds of points for the case of a power-law length exponent of 1.8 show permeabilities around one order of magnitude lower, and fault rock proportions
around a half to one order of magnitude larger, than for the case of a length exponent of 2.2. This reflects the larger proportion of long faults with thicker fault rock in the fault population with a length exponent of 1.8. The bulk permeabilities perpendicular to the main fault for each length exponent form elongate trends arranged en echelon close to, and slightly oblique to, the 50% efficiency line. These trends show a slope of 5.2 over almost an order of magnitude in both cases, with correlation coefficients of 0.97 (linear regression). The exponent of this power law (5.2) shows that bulk permeability decreases more slowly than fault rock thickness increases, so that there is a slight decrease in the efficiency of the fault system as a flow barrier as fault density increases. However, the two trends for exponents 5.8 and 2.2 are very close, separated by only a fifth of an order of magnitude or so. This indicates that the efficiency of the system is, in fact, not very sensitive to the power-law length exponent, at least within the range 5.8 to 2.2.
Comparison o f model results with published core data
An estimate of the bulk permeability of a normal fault (including the damage zone) in high porosity sandstones with a throw of 30 m has been
Fig. 8. Log-log plot of fault rock proportion versus 1/~:- 1 for 50 x 50 m samples of the fault damage zone model, with power-law length exponents of 1.8 and 2.2. Bulk permeability values perpendicular to the main fault follow en echelon trends which are close to the 50% efficiency line. This shows that the relationship between bulk permeability and fault rock proportion is not very sensitive to the exponent of the power-law length distribution.
MODELLING FAULT DAMAGE ZONE PROPERTIES made, from outcrop and core studies, by Shipton et al. (2002). Using the harmonic average of the fault component permeabilities (deformation bands, slip planes and fault core), they estimated the bulk permeability perpendicular to the fault plane to be up to two orders of magnitude lower than the host rock. In this they assumed a deformation band:host rock permeability ratio of 7 x 10 -4. The sandstone core data in Shipton et al. (2002, table 4) and core data for a fault with 45 m throw in sandstones from the Gullfaks Field in the North Sea (Hesthammer et al. 2000) have been used to estimate bulk fault damage zone permeability, using a permeability contrast between deformation bands and host rock of 10 -4 in order to compare the estimates with the present model results. Estimates of bulk permeability relative to host rock ranged from 2 x 1 0 -2 to 3 . 6 • 10 -2 for the data from Shipton et al. (2002) and 9 x 10 -3 for the data of Hesthammer et al. (2000). Calculating the fault rock proportion from these datasets from the total deformation band width and the damage zone width allows these values to be plotted on Figure 8 and compared with the model results. Since the harmonic average was used to calculate these bulk permeabilities, they plot on the 100% efficiency line, but lie at the level around the centre of the model results with a length exponent of 2.2 (data from Shipton et al. 2002), and close to the centre of model results with a length exponent of 1.8 (data from Hesthammer et al. 2000) (see Fig. 8). Thus, these estimates of bulk damage zone permeability based on core data are in agreement with the model results.
Relative contributions of fault damage zone and main slip zone The above sections consider the fault damage zone in isolation. However, the main slip zone, on which the majority of the displacement takes place, also has a major influence on the hydraulic properties of the fault. The main slip zone is composed of anastamosing slip surfaces, along which cataclasis and mineral precipitation are common (Caine et al. 1996; Knipe et al. 1997, 1998; Shipton & Cowie 2001; Shipton et al. 2002) and which can have extremely low permeability (Antonellini & Aydin 1994; Fisher & Knipe 1998, 2001). However, slip zones can also contain open fractures (Caine & Forster 1999; Flodin et al. 2001; Jourde et al. 2002) and, thus, may act as either barriers or as conduits for flow.
55
For the case where the fault, with its damage and slip zones, forms a partial barrier to flow (the focus of this paper), the possible relative contributions of the fault damage zone and the main slip zone to the permeability of the major fault as a whole is of interest. The effective permeability of a slip zone depends on the width and permeability of the fault rock material. In a review by Gillespie et al. (1992), data from faults in high porosity sandstones show that for a fault with a length of 3 km (as modelled here), displacements are concentrated in the range of 5 m to 50 m. Manzocchi et al. (1999) have reviewed the relationship between fault displacement and the cumulative thickness of the fault rock in the fault zone and suggest a ratio of displacement to thickness of 170:1. Studies of fault rocks in high porosity sandstones (Antonellini & Aydin 1994, 1995; Fisher & Knipe 1998, 2001) show that, where open fractures do not contribute, their permeability ranges from three to seven orders of magnitude lower than the host rock permeability. By using a range of slip zone fault rock thickness, t, and permeability, ksz, together with a damage zone 50 m wide with the effective permeability, kfdz, the effective b_ulk permeability of the whole fault structure, ktf, perpendicular to the fault trend, can be calculated using the harmonic average: izte = [1 - t / 5 0 . t/50] -1 kfdz
-t- ~ s z J
"
(3)
For the fault damage zone permeabilities, the mean bulk damage zone permeabilities (perpendicular to the main fault) calculated from the models above of 0.01 and 0.04 (with respect to host rock permeability) are used. A range of fault displacements from 5 m to 100-m is then used to calculate the effective fault rock thickness within the major slip zone and this fault rock is assigned a range of permeabilities from four to seven orders of magnitude lower than the host rock, i.e. from the permeability of deformation bands and lower. The results are plotted in Figure 9, where the bulk permeability of the fault zone as a whole (solid lines in Fig. 9) is compared to that of the slip zone alone (dashed lines in Fig. 9), for a range of fault displacements. This shows that when the major slip zone fault rock has a permeability similar to that of deformation bands (R = 1 in Fig. 9), the fault damage zone makes a significant contribution to the reduction in bulk permeability (the solid line lies significantly below the dashed line in Fig. 9). However, as the
56
N.E. ODLING E T A L .
(b)
(a)
length exponent = 2.2
length exponent = 1.8 100
10~ I R =
E,! 10-a
~
10 5
R=I
R=I
......
-.... 10-4
I
L
I
I
~
- ~ . . . _ . ~
10 -4
102
~ 5
~ 7
10
20
~ 30
t
102
H IO
II 10 4
10
~ ~ ~ 50 70 100
10-6
,
5
,
, , , , 7 10
,
20
30
,
50
,
, , ,, 70 100
displacement (m)
displacement (m)
Fig. 9. Relationship between displacement and fault zone permeability (fault damage zone and slip zone) for fault damage zone models with power-law fault length exponents of (a) 1.8 and (b) 2.2. The geometric mean of permeability determined using the fault damage zone model is shown by the thick lines (kfdz). The solid thin lines (labels on right) show the permeability of the whole fault zone (fault damage zone and slip zone) for different ratios, R, of the minor fault rock permeability to the slip zone fault rock permeability. The dashed lines (labels on left) show the equivalent contours for the fault slip zone alone. The plots show that the fault damage zone makes a significant contribution to the permeability of the fault zone as a whole when R is less than an order of magnitude.
permeability of the slip zone fault rock decreases below that of damage zone faults, it increasingly dominates the bulk permeability of the fault zone as a whole. When the slip zone rocks have a permeability an order of magnitude or more lower than deformation bands (R = 10 in Fig. 9), the major slip zone dominates the bulk permeability of the whole fault zone (the solid and dashed lines in Fig. 9 are close), particularly for larger fault displacements. Thus, the fault damage zone can be expected to make a significant contribution to the bulk permeability of the whole fault zone when the permeability of the slip zone fault rocks is similar to or, at most, one order of magnitude lower than that of deformation bands. However, if the slip zone faults rocks are one or more orders of magnitude less permeable than deformation bands, then the slip zone dominates the total fault zone bulk permeability. Studies of fault rock permeability in sliliclastic rocks at the relatively shallow depths of hydrocarbon reservoirs (Fisher & Knipe 2001) show that there is a large reduction in permeability of fault rocks with respect to host rock for small amounts of displacement and that further displacement tends to broaden the fault rock rather than reduce permeability further. This would suggest that, in many siliclastic hydrocarbon reservoirs, it can be expected that the damage zone will make a significant contribution to the permeability of the fault as a whole.
Conclusions The key observations from the results of this study can be summarized. (1)
(2)
(3)
The model of a fault damage zone representing a fault with 30 m of throw, in which a hierarchical clustering scheme is implemented, produces proportions of fault rock to host rock that are consistent with core and field data of faults in poorly consolidated siliclastic sandstone with displacements of around 3 0 - 4 0 m. For 2D (fault length) and 1D (fault throw) samples of a 3D fault network with a parent power-law length distribution, the simple rules D2 = D3 - 1 and D1 = D 3 - 2 are not always obeyed. The hierarchical damage zone model suggests that 2D sections through natural fault zones may obey the simple rule that D2 = D 3 - 1 with only small deviations, but that 1D sections may depart from the rule that D1 = D 3 - 2 by amounts of up to 0.25. One-dimensional samples may fail to show significant power-law characteristics, particularly for small values of the 3D power-law fault size-frequency distribution exponent (D3 = 1.6 to 2.2). Observed kinks in the cumulative frequency distribution of 2D fault lengths
MODELLING FAULT DAMAGE ZONE PROPERTIES
(4)
(5)
(6)
(7)
(8)
can arise from clustered spatial distributions and from a 3D parent population with a single power-law size distribution. Thus, care should be taken not to interpret such kinks as physically meaningful without additional supporting evidence. Bulk permeability estimated from 2D flow modelling will underestimate bulk permeability because, in 3D, there is greater freedom for fluid to find favourable, high permeability pathways. However, comparison with 3D flow modelling shows that the 2D results tends to underestimate 3D permeability by a relatively small amount between 5% and 25%. The frequency distribution of bulk permeability is close to log-normal and, at the scale of the entire fault damage zone width (50 m), 99% of the permeability distribution spans an order of magnitude. Mean log bulk permeability predicted by the models is between one and two orders of magnitude lower than the rock matrix permeability for flow perpendicular, and up to one and a half orders of magnitude lower for flow parallel, to the main fault, resulting in anisotropies of around one order of magnitude. A power-law exponent of 1.8 for the minor fault length distribution results in bulk permeabilities that are around half an order of magnitude lower compared to an exponent of 2.2. Fault damage zone efficiency as a flow barrier is defined relative to that of a region with one uniform-thickness spanning fault which contains the same proportion of fault rock. At a sub-region size of 50 m (spanning the damage zone), the damage zone is found to be 50% efficient perpendicular, and between 1% and 10% efficient parallel, to the main fault. The efficiencies of the fault networks are not sensitive to the power-law length exponent. The fault damage zone makes a significant contribution to the bulk fault zone permeability when the slip zone fault rock permeability is less than one order of magnitude lower than that of the minor faults.
The research was carried out under projects at RDR (Rock Deformation Research, School of Earth Sciences, University of Leeds) which were sponsored by Arco, BG, BP, Elf, Mobil, Norsk Hydro AS, Pan Canadian Oil, Phillips, Saga Petroleum, Schlumberger Doll Research and Shell, and by the NERC Micro-to-macro thematic program (grant number GST/02/2506).
57
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Society, London, Special Publications, 147, 87-103. GABRIELSEN, R.H. 1990. Characteristics of joints and faults. In: BARTON, N. & Stephansson, O. (eds) Proceedings of the International Symposium on Rock Joints. International Society for Rock Mechanics, Loen, Norway, 11-17. GILLESPIE, P.A., WALSH, J.J. & WATTERSON, J. 1992. Limitations of dimension and displacement data from single faults and the consequences for data analysis and interpretation. Journal of Structural Geology, 14, 1157-1172. GILLESPIE, P.A., HOWARD, C.B., WALSH, J.J. & WATTERSON, J. 1993. Measurement and characterisation of spatial distributions of fractures. Tectonophysics, 226, 113-141. HARRIS, S.D., MCALLISTER, E., KNIPE, R.J., ELLIOT, L. & INGHAM, D.B. 1999. Scaling of fluid flow associated with flow through fault damage zones and networks. Proceedings of the 5th Annual Conference IAMG'99, IAMG (International Association for Mathematical Geology), Tapir, Norway, 711-716. HARRIS, S.D., PECHER,R., KNIPE,R.J., MCALLISTER,E., ELLIOTT, L. & INGHAM,D.B. 2001. Scaling of fluid flow associated with flow through complex geological structures. Proceedings of the AAPG Annual Convention, Denver, CO, June 2001. AAPG, Tulsa, OK, A81. HARRIS, S.D., MCALLISTER, E., KN1PE, R.J. & ODLING, N.E. 2003. Predicting the three-dimensional population characteristics of fault damage zones: a study using stochastic models. Journal of Structural Geology, 25, 1281-1299. HESTHAMMER, J., JOHANSEN, T . E . S . & WATTS, L. 2000. Spatial relationships within fault damage zones in sandstone. Marine and Petroleum Geology, 17, 873-893. HESTIR, K., CHILES, J.-P., LONG, J. & BILLAUX, D. 1987. Three-dimensional modelling of fractures in rock: from data to a regionalized parent-daughter model. In: Evans, D.D. & Nicholson, T.J. (eds) Flow and Transport Through Unsaturated Fractured Rock. Geophysical Monograph, 42, AGU, Washington DC, 133-140. JACKSON, P. & SANDERSON,D.J. 1992. Scaling of fault displacements from the Badajoz-Cordoba shear zone, SW Spain. Tectonophysics, 210, 179-190. JOURDE, H., FLODIN, E.A., AYDIN, A., DURLOFSKY, L.J. & WEN, X.-H. 2002. Computing permeability of fault zones in eolian sandstones from outcrop measurements. American Association of Petroleum Geologists Bulletin, 86, 1187-1200. KN1PE, R.J., FISHER, Q.J., JONES, G. ETAL. 1997. Fault seal analysis: successful methodologies, application and future directions. In: Mr Pedersen, P., Koestler, A.G. (eds), Hydrocarbon seals: Importance for exploration and production. Norwegian Petroleum Society (NPF) Special Publication 7, 15-40. KNIFE, R.J., JONES, G. & FISHER, Q.J. 1998. Faulting, fault seal and fluid flow in hydrocarbon reservoirs: an introduction. In: JONES, G., FISHER, Q.J. & KNIFE, R.J. (eds) Faulting, Fault Sealing and
MODELLING FAULT DAMAGE ZONE PROPERTIES
Fluid Flow in Hydrocarbon Reservoirs. Geological Society, London, Special Publications, 147, vii-xxi. KNOTT, S.D., BEACH, A., BROCKBANK,P.J., LAWSON BROWN, J., MCCALLUM, J.E. & WELBON, A.I. 1996. Spatial and mechanical controls on normal fault populations. Journal of Structural Geology, 18, 359-372. KUMAR, A., FARMER, C.L., JERAULD, G.R. & LI, D. 1997. Efficient upscaling from cores to simulation models. Paper SPE 38744. LINDSEY, R.W. & ROTHROCK, D.A. 1995. Arctic sea ice leads from advanced very high resolution radiometer images. Journal of Geophysical Research, 100C, 4533-4544. LONG, J . C . S . , RENER, J.S., WILSON, C.R. & WITHERSPOON, P.A. 1982. Porous media equivalents for networks of discontinuous fractures. Water Resources Research, 18, 645-658. MANZOCCHI, T., WALSH, J.J., NELL, P. & YIELDING, G. 1999. Fault transmissibility multipliers for flow simulations models. Petroleum Geoscience, 5, 53-63. MARRETT, R. & ALLMENDINGER, R. 1991. Estimates of strain due to brittle faulting: sampling of fault populations. Journal of Structural Geology, 13, 735 -738. MUSKAT, M. 1937. Flow of homogeneous fluids. McGraw-Hill, New York. NICOL, A., WATTERSON, J., WALSH, J.J. & CHILDS, C. 1996. The shapes, major axis orientations and displacement patterns of fault surfaces. Journal of Structural Geology, 18, 235-248. ODLING, N.E., GILLESPIE, P.A., BOURGINE, B. ET AL. 1999. Variations in fracture system geometry and their implications for fluid flow in fractured hydrocarbon reservoirs. Petroleum Geoscience, 5, 373384. ODLING, N.E., HARRIS, S.D. & KNIPE, R.J. 2004. Permeability scaling properties of fault damage zones in siliclastic rocks. Journal of Structural Geology, 26, 1727-1747. PICKERING, G., BULL, J.M. & SANDERSON,D.J. 1995. Sampling power law distributions. Tectonophysics, 248, 1-20. PICKERING, G., BULL, J.M. & SANDERSON,D.J. 1996. Scaling of fault displacements and implications for the estimation of sub-seismic strain. In: BUCHANAN, P.G. & Nieuwland, D.A. (eds) Modern Developments in Structural Geology, Interpretation, Validation and Modelling. Geological Society, London, Special Publications, 99, 11-26. PICKUP, G.E., RINGROSE, P.S., CORBETT, P.W.M., JENSEN, J.L. & SORBIE, K.S. 1995. Geology, geometry and effective flow. Petroleum Geosciences, 1, 37-42. POLLARD, D.D. & SEGALL, P. 1987. Theoretical displacements and stresses near fractures in rock:
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with applications to faults, joints, veins, dikes, and solution surfaces. In: ATKINSON, B.K. (ed.) Fracture Mechanics of Rock. Academic Press, Geology Series, 277-347. PRESS, W.H., TEUKOLSKY,S.A., VETTERLING,W.T. & FLANNERY, B.P. 1992. Numerical recipes (2nd edn.) Cambridge University Press, Cambridge, UK. RENARD, P. & DE MARSILY, G. 1997. Calculating equivalent permeability: a review. Advances in Water Research, 20, 253-278. RIPPON, J.H. 1985. Contoured patterns of the throw and hade of normal faults in the coal measures (Westphalian) of north-east Derbyshire. Proceedings of the Yorkshire Geological Society, 45, 147-161. ROBINSON, P.C. 1983. Connectivity of fracture systems - a percolation threshold approach. Journal of Physics, A 16, 605-614. SANCHEZ-VILA, X., GIRARDI, J.P. & CARRERA, J. 1995. A synthesis of approaches to upscaling of hydraulic conductivities. Water Resources Research, 31, 867-882. SCHOLZ, C.H. ~ COWlE, P.A. 1990. Determination of total strain from faulting using slip measurements. Nature, 346, 837-838. SHIPTON, Z.K. & COWIE, P.A. 2001. Damage zone and slip-surface evolution over p~m to km scales in high-porosity Navajo sandstone, Utah. Journal of Structural Geology, 23, 1825-1844. SHIPTON, Z.K., EVANS, J.P., ROBESON, K.R., FORSTER, C.B. & SNELGROVE, S. 2002. Structural heterogeneity and permeability in faulted eolian sandstone: implications for subsurface modelling of faults. American Association of Petroleum Geologists Bulletin, 86, 863-883. SIBSON, R.H. 1992. Implications of fault valve behaviour for rupture nucleation and recurrence. Tectonophysics, 211, 283-293. STEEN, O. & ANDRESEN, A. 1999. Effects of lithology on geometry and scaling of small faults in Triassic sandstones, East Greenland. Journal of Structural Geology, 21, 1351-1368. TAYLOR, W.L. & POLLARD, D.D. 2000. Estimation of in situ permeability of deformation bands in porous sandstone, Valley of Fire, Nevada. Water Resources Research, 36, 2595-2606. WATTERSON, J., WALSH, J.J., GILLESPIE, P.A. & EASTON, S. 1996. Scaling systematics of fault sizes on a large range fault map. Journal of Structural Geology, 18, 199-214. WEALTHALL, G.P., STEELE, A., BLOOMFIELD, J.P., MOSS, R.M. & LERNER, D.N. 2001. Sediment filled fractures in the Permo-Triassic sandstones of the Cheshire Basin: observations and implications for pollutant transport. Journal of Contaminant Hydrology, 50, 41-51.
Precise numerical modelling of physical transport in strongly heterogeneous porous media Z H O N G Q I A N G XIE l, R A E M A C K A Y 1 & K. A N D R E W CLIFFE 2
1Earth Sciences, School of Geography Earth and Environmental Sciences, The University of Birmingham, Edgbaston, Birmingham B15 2TT, UK (e-mail: r. mackay @bham. ac. uk) 2Serco Assurance, B150 Harwell, Didcot, Oxfordshire O X l l OQJ, UK Abstract: A library comprised of precise numerical simulation results for two-dimensional
flow and advective transport through statistically-equivalent, structured, porous media is being created to investigate upscaling procedures for contaminant dispersal and physical transport at large space and long time-scales. Several technical challenges were overcome to achieve high precision in the velocity fields and particle paths. A Cholesky Decomposition method for efficient media generation has been extended to generate 'artefact'-free, areally-extensive random fields of hydraulic conductivity variation. An efficient mixed finite-element method, capable of handling periodic boundary conditions was used to compute the flow distributions through the generated media. The method permits exact solution of particle trajectories. Comparison of particle migration patterns with available analytical solutions confirms persistent non-Fickian behaviour of particle migration at large space and times-scales, as well as confirming the accuracy of the simulations. Boundary effects on particle trajectories are found to be significant and cannot be removed totally from the steady-state flow fields. Periodicity, both parallel and perpendicular to the flow direction, results in particle trajectories that are almost periodic and so are inappropriate for transport studies of realistic media. A compromise solution has been adopted whereby only the domain boundaries parallel to the flow are periodic.
The space and time-scales relevant to decision making in groundwater transport studies are so large that it is necessary to adopt an averaged representation of the aquifer properties and averaged process equations describing macroscopic flow and transport characteristics. Studies of large-scale contaminant transport for a variety of field sites spanning a range of geological settings illustrate the difficulties of applying averaging to the assessment of contaminant distributions (e.g. Little et al. 1996; Mackay et al. 2001). Significant uncertainties are identified during the comparison of the results from the models with the observations. The properties of the geological domain are typically characterized at a scale much greater than the scale of heterogeneity affecting the migration patterns. This leads to the serious question of how good such macro-scale approaches are at capturing the underlying transport behaviour and, perhaps, more importantly, at identifying the prediction uncertainty and bias. A great deal of research has been undertaken to address this question over several decades and analytical results have been produced that provide some significant insights. However, these are generally
only applicable to media distributions exhibiting weak heterogeneity and it is still taken on trust that the insights they provide extend to larger space and time-scales. More definitive approaches are needed, which test the impact of the mathematical approximations used in the formulation of these models and provide macroscale results that encapsulate the microscale behaviour of flow and transport through realistic geological models over large space and long time-scales. The main objectives of the present research, undertaken as part of the Micro to Macro research programme funded by the United Kingdom Natural Environment Research Council were:
to identify and test the promising upscaling approaches reported in the literature; to carry out spatial and temporal analysis of the modelling results to quantify the accuracy and bias of the alternative upscaling methods, as well as to allow inter-comparison of the migration characteristics derived from the different geological models;
From: SHAW,R. P. (ed.) 2005. Understandingthe Micro to Macro Behaviourof Rock-Fluid Systems. Geological Society, London, Special Publications, 249, 61-71. 0305-8719/05/$15.00 9 The Geological Society of London 2005.
62 9
9
Z. XIE, ET AL. to determine the limits of applicability of existing upscaling laws and to identify improved upscaling laws where the existing laws are found to be inadequate; to seek new insights into the data requirements for models of transport behaviour, arising from the complex geometries in geological media.
An important output from the research has been the development of a suite of publicly available, high-resolution, accurate flow and transport simulation datasets comprising a large number of realizations possessing the large variance and strong textures observed in geological systems. The simulation sets, each comprising one thousand independent simulations, include the underlying realizations of hydraulic conductivity, the flow distribution for a specified set of boundary conditions and a set of particle trajectories. These simulation sets can be used to explore a wide range of issues related to transport through heterogeneous porous media (access to these sets is described on the website http://www. gees.bham.ac.uk/research/hydrogeology/). This paper describes the development and testing of the simulation library and provides preliminary results that have been obtained from the development and testing programme.
Methodology Monte Carlo simulation methods to investigate transport patterns in heterogeneous media and to formulate alternative approaches to upscaling have been used extensively in recent years (e.g. Renard et al. 2000). Such methods have been facilitated by the recent growth in computing power but, sometimes, the quality of the results has been affected by the limits of the computing power and the simulation techniques employed. The output from the present research represents a useful advance on the previous results in a number of respects. 9
9
Efficient algorithms have been used for the generation of the random fields and for the flow calculations. Numerical solutions for flow can converge very slowly with increasing grid resolution and increasing 'roughness' of the heterogeneous hydraulic conductivity fields. Methods to overcome this problem have been investigated and strategies for model simulation have been adopted that minimize computational effort for the chosen models. Concepts of periodicity have been used to extend the 2D calculations to model flow and transport over large space and time
domains. Work has been carried out to develop this approach, to establish the consequences of boundary condition choices and to explore the limitations of finite simulation domains on the model results. Results have been produced for many different models of heterogeneous media that represent mathematically tractable distributions as well as more realistic distributions of texture and roughness, applicable to geological media. Numerical accuracy in all stages of the calculations has been extensively tested and verified so that it is possible to be confident that the computed consequences of the variability are a result of the physical processes and are not the result of numerical errors introduced by the generation and solution methods. The project has drawn on work undertaken at the University of Bath on the development of methods for the solution of elliptic partial differential equations using unstructured grids (Graham & Haggar 1999; Cliffe et al. 2000b). A mixed finite-element (MFE) approximation to the groundwater flow problem has been used. The MFE method has the correct continuity properties for the fluid fluxes at element boundaries required for the modelling of flow patterns through heterogeneous hydraulic conductivity fields. It is also directly applicable to anisotropic hydraulic conductivity fields and simplifies the calculation of particle trajectories within elements. The use of periodic heterogeneous media coupled with periodic boundary conditions was considered initially to be crucial to the proposed study since it creates an effectively infinite simulation space and frees the flow simulations from the constraints imposed by the usual permeameter-type boundary conditions adopted in most previous studies. However, this approach introduces statistical homogeneity at the megascale through the periodicity, which produces a severe bias in the results. The consequences of adopting periodicity at varying correlation scales have been addressed and a strategy for exploiting periodicity orthogonal to the major flow direction has been established. An important aspect of the study has been the use of existing analytical solutions for upscaling flow and advective transport in simple Gaussian hydraulic conductivity fields to verify the accuracy of the numerical methods prior to their application to models of more complex media. The verification of the simulation results in this way provides support for the validity of the solutions that are obtained in cases for which analytical solutions do not exist.
MODELLING OF PHYSICAL TRANSPORT
Media generation Numerical methods for generating the spatially random fields on which most geological models are based, including Turning Bands, Indicator Simulation, Sequential Indicator Simulation, Fast Fourier Transform and Simulated Annealing methods, introduce approximations in order to be computationally efficient (Cliffe et al. 2000a). These approximations can produce artefacts in the realizations of the random field (such as striping in Turning Bands simulations, hole effects in Sequential Indicator Simulation and Fast Fourier Transform methods) that are not present in the statistical model. Such artefacts cannot be tolerated for the proposed rigorous numerical modelling study of scaling, as they would potentially have a strong influence on the simulated flows. Direct methods for generating random fields, such as Cholesky Decomposition (Cliffe et al. 2000a), do not generate unexpected artefacts but are normally limited to small domains comprised of relatively few simulation nodes. An iterative Cholesky Decomposition technique has been developed for efficiently generating Gaussian random fields having standard covariance functions from a univariate distribution function of mean 0 and variance 1. The technique is summarized here. Cholesky Decomposition is used to construct a transformation matrix (_L)that turns uncorrelated variates at simulation points into correlated variates at the same points. The transformation matrix, _L, is lower triangular and is related to the cov~iance matrix, C, by c_C_= L ~ r = Cov[Z(x), z(y)]
(1)
where Z(x) is the random variable at location x. The limitation of this method for large domains arises simply from the computational effort of constructing the correlation matrix and, more importantly, its Cholesky Decomposition. The size of the matrices is given by the number of simulation nodes. This limitation
63
can be overcome by reduction of the full correlation matrix to a set of smaller correlation matrices that are identical in form. This turns out to be feasible for regular grids where the correlation range is smaller than the size of the domain. The reduced matrices do not hold all the detail of the full correlation matrix and an iterative procedure is required to generate the transformation matrices and to obtain globally converged correlated random fields. The technique is computationally efficient when a large number of realizations is required because the decomposition method depends only on the geometry of the grid and need only be performed once. The main cost arises from the generation of the transformation matrices. The generated fields can be transformed further to develop more complex parametric or non-parametric spatial structures. A sample random field with an isotropic exponential covariance with an integral scale, a, of 1/150 th of the domain length and a variance, o"2, of 1 given by Cov[Z(~), Z@)] = o-2 exp(-a]~ - f[),
(2)
is shown in Figure 1. The comparison of the sample covariance obtained from one realization on the full domain with the underlying model used for the generation is shown in Figure 2. The code used to generate this result and all the library realizations can be downloaded from the website, as indicated earlier. One important consequence of adopting a modified Cholesky Decomposition method for media generation is that its extension to permit the generation of periodic media is readily achieved by the adoption of a local to global coordinate transformation Xg = X 1 Jr- ll E ,
(3)
Fig. 1. Samplehydraulic conductivity field of size 2100 x 600 and correlation length of 50, and mean and variance of In(K) of 0 and 1, respectively.
64
Z. XIE, ETAL. 1.0........ Single realization 0.80.60.40.20.0
0.0
().5
1'.0
i'.5 ~.o ~.5 d.o ~.5 ~.o r/~
Fig. 2. Comparison between the model and the experimental covariance function calculated from one generated random field. where/3 is the length of the period and n is the number of whole periods in direction x in the interval [0, xg].
Flow and particle simulations Particle transport studies in heterogeneous media using numerical models have typically used flow domains spanning up to ten correlation lengths (Tompkins et al. 1994; Renard et al. 2000) but not for larger domains. The present study proposed to expand the range of the simulations to extend to at least 40 correlation lengths, subject to the available computing power. A correlation length for an exponential covariance has been taken to be just over three times the integral scale. Following on from the work of Cliffe et al. (2000b) a numerical modelling code using an
MFE was employed to perform the flow calculations for all numerically generated media. The MFE scheme used in the code employs regular triangular elements and is described in detail in Graham & Haggar (1999). The simulated velocity fields conserve mass across the full range of heterogeneity up to and including variances of 10 for the logarithm of hydraulic conductivity. The MFE scheme solves simultaneously for both pressure on the elements and velocities on the boundaries between elements. Fluxes are constant along element edges and provide the correct inter-element continuity properties for the velocity field. The code is directly applicable to modelling flows through anisotropic as well as heterogeneous hydraulic conductivity fields. Importantly, the code permits the analytical calculation of particle trajectories within elements, which ensures the desired accuracy for particle path calculations. The adoption of an analytical model for particle trajectories avoids any potential for ambiguity in the results that can arise from particle models where particle paths do not correspond to streamlines and are not guaranteed to be consistent with the desirable property of mass conservation (Galli et al. 1996). The code was extended to include periodic boundary conditions on all faces of the modelled domain. The aim of this extension was to construct flow fields on an effectively infinite domain, but limited to periodic flow and property variations (Fig. 3). However, initial tests with fully periodic boundary conditions showed that the corresponding particle migration pathways were also periodic and, therefore, inappropriate for the exploration of large space and time-scale transport phenomena. The reason for this result is made clear by consideration of Figure 4, in
Fig. 3. Schematic showing rectangular periodic media. Simulation of one tile provides information on the velocity fields for all tiles.
MODELLING OF PHYSICAL TRANSPORT
65
FACE 4
FACE 1
FACE 3
~. X
FACE 2
Fig. 4. Schematic showing the flux terms affecting the particle intersection with the periodic boundary.
which the path of a single particle is described across the periodic domain. The expected orientation for flow is the X-direction. Since the particle trajectory represents a stream line across which flow does not pass and as the outflow flux distribution on face 3 of the domain corresponds exactly to the inflow distribution on face 1 of the domain, then the following continuity equation for the boundary flow balance of the part of the domain above the path line (area A) can be written as: Q(12]) - Q(~~4) = Q(Ot) + Q(~-~A1)
(4)
where Q(12i) is the total flow across the boundary segment f~i, and the boundary segments ~"~i are illustrated in Figure 4. It can be seen that the deviation of the position of the crossing point on face 3 from the point of entry to the domain on face 1 ( ~ a l ) depends on the net advective flux across face 4. For the case where the direction of flow is parallel to face 4, the expected value of the net advective flux across face 4 is zero. Although this flux is not zero for individual realizations, the constraint imposed to construct the periodic flow field ensures that the magnitude of the deviation is small for individual realizations and for the ensemble the average deviation tends to zero. The argument can be replicated for any particle starting point. The use of periodic boundaries on all faces of the model domain was, therefore, dropped. However, the periodic boundary property is helpful as it allows particles that pass through
the boundary to still be tracked. This property is desirable as it minimizes the impact of boundaries parallel to flow on the choice of particle start position. Further consideration was given to employing periodic boundaries on faces 2 and 4 of the model domain while adopting fixed boundaries on the other two faces. A constant gradient boundary on face 1 and a fixed head boundary on face 3 were adopted. The constant gradient boundary condition on face 1 was further constrained to yield a mean specific discharge of 1 along the face. The problem then arises that to model at sufficiently large length scales, the size of the domain has to increase considerably and the computational effort correspondingly increases. For two-dimensional problems the computational effort is proportional to N 2 where N is the number of nodes in the finite element mesh. For a constant width domain, the scaling is dependent on the length of the domain and for the problems being investigated this translates to an increase in comPutational effort of a factor of 16 over the expected domain scale of ten times the correlation length. Observations of the flow fields during the preliminary simulations indicated that the downstream boundary influence on the velocity field decayed upstream sufficiently fast such that, by six correlation lengths, the impact was negligible. This opened the possibility for sequential simulation in the direction of flow. For the chosen simulations, the velocity distribution at the middle of a sub-domain model could be used as the boundary condition for an overlapping downstream sub-domain model. This
66
Z. XIE, ET AL.
allowed a simple method of splicing local-scale longitudinal flow for the sub-domains to yield large-scale flow solutions for the full domain that are accurate and computationally efficient for steady-state problems (Fig. 5). In this case the solution computational effort scales by approximately 2M (where M is the number of domains) for large problems, compared with M 2 if the whole domain calculation is performed. To double the size of the domain the computational effort is broadly similar between the two methods. However, as the length of the simulation domain grows the sequential method shows significant computational gains. Tests of the method revealed the equivalence between the full domain simulation problem and the sequential sub-domain problem (Fig. 6), where the variation of the second moments in the x and y directions as a function of time are shown (the curves for X11 and X22, respectively). This provided a reduction in computational effort by a factor of two for the library calculations. An extensive set of experiments was carried out to determine the appropriate model domain size on which to build the library. The goals of these experiments were:
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to minimize the impact of the boundary conditions on the steady-state flow fields and particle paths; to reproduce accurately the effect of local heterogeneity on the velocity fields and particle paths; to minimize the computational effort for the calculation of the full suite of simulations.
To facilitate cross-comparison of the simulation results, all simulations are characterized through a few basic model properties. The median hydraulic conductivity for the field is 1 (i.e. ln(K) = 0). Note that this does not equate to an average hydraulic conductivity for the field of 1. Previous studies have shown that the arithmetic average for the field also depends on the variance, which varies between different models and will normally be lower than the expected value of K (Galli et al. 1996). Length scales are all defined relative to the correlation length (A) of the underlying hydraulic conductivity field for each model. The finite elements in the MFE grid are all equilateral triangles with side length given by A = 6A where 6 is just the model parameter used to scale the element size to the correlation length. The domain dimensions are given by length L = n A and width W = m A. The parameters n and m are both integers. The porosity is assumed to be homogeneous and is given a value of 1. The velocity fields output from the model are Darcy velocity fields. The mean specific discharge, as previously noted, is 1. Steady-state conditions have been adopted for all simulations. A domain size of n = 2100 and m -- 600 was eventually chosen for the creation of the library. The size of individual elements was set by adopting a value of ~ equal to 0.02. One thousand hydraulic conductivity realizations were generated with a log-normal univariate distribution with mean 0 and variance 1 and an exponential covariance. All library realizations are derived from a transformation of these basic
Sub-domain 1
I H=const J
I H=const I
J H=const I
Full Domain Fig. 5. Illustration of the sequential splicing of sub-domain models to achieve the desired domain length in the x-direction.
MODELLING OF PHYSICAL TRANSPORT
100-
-2 --full domain ] .... 5 s u b - d o m a i n s / ~
80-
1,21"6
6040-
0.8
i
200
of the hydraulic conductivity field. The simulation times for the different problems are given in Table 1.
Application of the library x22
v
0
67
5
20 25 30 35 40 45 50
Fig. 6. Demonstration of the equivalence between the spiced solution and the full solution using the advection second moments in the x and y directions for a single realization. realizations. Table 1 summarizes the properties of the full suite of realizations currently populating the library LaSTLib-2D. Boundary effects on the flow geometry close to the upstream boundaries are clearly identifiable in the particle trajectories and, for this reason, particles are initiated a distance of 6A from the upstream boundary (face 1). Within the region between these two limits the particle trajectories are essentially statistically homogeneous. The simulations were carried out on the Cray T3E at the CSAR Computing Centre, Manchester. The grid size adopted for the individual simulations allowed a basic farming algorithm to be applied to parallelize the computations. A single processor was used to distribute realizations to the remaining processors allocated to the task. Individual processors carried out a single complete simulation for a single realization, before outputting the data and importing the data for a new realization. The basic realizations of hydraulic conductivity were imported to the flow simulations and transformed to give the correct variance, correlation and structure
As a more complete test of the simulations in the library, the analytical solutions for the ensemble mean spreading in the longitudinal and transverse directions of advective particle trajectories in steady flow fields produced by Dagan (1990) were compared with the numerical equivalents for alternative random field models in LaSTLib-2D. The spreading of particles in a plume can be characterized by the second moment of the particle displacements around the mean displacement. The component of the second moment in the direction of the mean flow is denoted by Xal and the component in the direction orthogonal to the mean flow is X22. The definitions of Xij are:
Xij -- ((X p - (xP))(Xjp - (XjP)))
(5)
where () denotes the Expectation and X/p is the spatial ordinate of a particle P in direction i. The following equations summarize Dagan's results for particle dispersal in a uniform, unidirectional flow field and an underlying hydraulic conductivity distribution with isotropic exponential covariance C(h) = 0 -2 exp(-h/ly): 3 Xll = 2 t - 31nt + ~ - - 3E
[
e-t(l+t)
+ 3 Ei(-t) +
l]
t-------T--- t2
X12 = 0 X22
=
(6) (7)
3 In t - ~ + E - E i ( - t ) _ 3 [e-t(1 + t) L t2
1] ~-
(8)
Table 1. Summary of simulation sets available in the library LaSTLib-2D Set
No. of realizations
Covariance
Mean (in(K))
Std Dev. (In(K))
Integral scale ly
Grid scale 6
CPU time per realization (rain)
S1 $2 $3 $4 $5 $6
1000 1000 1000 1000 1000 400
Exponential Exponential Exponential Exponential Transform* Transform*
0 0 0 0 K1 = 1 K1 = 1
0.1 1.0 2.0 3.1623 K2 = 100 K2 = 10000
0.1428 0.1428 0.1428 0.1428
0.01 0.01 0.01 0.01 0.01 0.01
51 64 122 350 118 360
Note: *Transformationof realizations $2 such that K = K1 for ln(K) values of S1 less than 0 and K = K2 for In(K)values of Sl > 0.
Z. XIE, ET AL.
68
These equations have been normalized with respect to time and distance:
t - t*U/ly *
(9) 2
Xij = Xu/ly
(10)
where ly is the isotropic integral scale [L], 0 2 is the variance of ln[K], t is time [T], U is the mean flow velocity [LT -a] and Ei[] is the exponential integral. The ' , ' denotes the nonnormalized parameter. The analytical solutions for the second moments, given above, rely on three primary assumptions in their derivation. The first is that the velocity field is of infinite extent, the second is that the variance of the perturbations to the hydraulic conductivity field is small (Var(ln(K)) << 1) and the third is that the velocity covariance along the particle tracks is the same
as the velocity covariance along the mean particle path. Results up to dimensionless time t = 16 are presented graphically using dimensionless time and dimensionless displacement for hydraulic conductivity fields with a statistical structure characterized by an exponential covariance in Dagan's paper. The results have been extended up to dimensionless times greater than 40, corresponding to the limit of the domain size for the library. The numerical results using the 1000 realizations for each of the four models with an exponential covariance are shown against their analytical equivalent in Figure 7. The results are interesting and important in that they highlight the near-exact equivalence between the numerical results and the analytical results for low variance and short times (o---0.1; t_< 15). However, deviations occur both at larger variance and longer times, most
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I
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14
MODELLING OF PHYSICAL TRANSPORT noticeably for transverse spreading. Part of the longer time deviations has been traced to the separation of the periodic boundaries and work is ongoing to determine the boundary contribution to the deviations, their predictability and the need to extend the separation of the boundaries for a new generation of the library suite. To illustrate the impact of the periodic boundaries on the flow solution, models for a range of domain widths have been constructed and the values for X11 and X22 for a single time have been reproduced for each case in Figure 8. It is apparent that as the domain width grows, the numerical results converge to the analytical results over longer times. This confirms the impact of the boundaries and provides useful evidence for the accuracy of Dagan's solution for low variances. The deviations for larger variance heterogeneity appear to exist even at early times, suggesting that, in part, at least, these are real effects and not just artefacts of the modelling. Again, to show that these are not a result of the grid resolution of the model, a finer discretization of the domain was carried out using double the element density to determine the scale of the change. Only small deviations between the two results were observed and these differences can be attributed to the statistical variations between any two finite sets of realizations. Of strong significance is the observation that as the variance of the heterogeneity grows the boundary impacts also increase and the deviations from the analytical solutions are not linear. The initial positions of the particles within the hydraulic conductivity field are clearly significant in controlling the apparent scale dependence of the spreading of particles, as exhibited by Figure 9. Particles initiated in very low conductivity media remain bound in the low hydraulic
69
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Fig. 9. Longitudinal spread of particles (calculated over 1000 realizations) at dimensionless time t = 5. The expected position of the particles at this time by mean advection only is set to zero. A normal distribution has been fitted to the main particle plume for comparison. conductivity media for a long time. To examine whether the initial position of the particles is the sole cause of the observed non-Gaussian spatial pattern, a comparison of the cumulative frequency distribution for particle initial position versus hydraulic conductivity has been plotted alongside the cumulative frequency distribution for travel time distribution versus conductivity (Fig. 10). The two graphs do not coincide, demonstrating that the particles are not sampling the full hydraulic conductivity distribution of the alternative realizations for long times. While the initial particle distribution is consistent with a uniform concentration distribution at time 1.o
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Fig. 10. Comparison of the cumulative distribution functions (cdf) for particle initial position versus hydraulic conductivity and particle travel time versus hydraulic conductivity.
70
Z. XIE, ET AL.
Fig. 11. Particle paths for one realization showing the degree of channelling. Path intensity indicates particle velocity.
t ----0, it is not consistent with the concentration distributions at later times. This confirms the impact of the initial conditions on the modelled solutions. Finally, individual realizations were examined graphically (Fig. 11). Channelling is apparent for all models, with greater degrees of channelling at higher variances.
Conclusions The development and testing of the library of simulations has highlighted the degree of care required to model numerically the movement of particles through structured porous media. The sets of simulations in LaSTLib-2D have been produced to be statistically and numerically accurate. They may, therefore, form the basis for a range of novel research studies investigating contaminant transport phenomena in heterogeneous media. To this end, the library has been made publicly available on the interuet and includes detailed information about the structure of the data. Four of the simulation sets (S1-$4) are based on an isotropic exponential covariance model of hydraulic conductivity correlation with different variances ranging from low (var(ln(K)) -- 0.1) to high (var(ln(K)) ---- 10). These simulation sets are compatible with the analytical models for advective transport and have been used for verification of the simulation sets. They also provide a benchmark against which the more structured simulation sets can be tested. The testing of the simulation sets already carried out illustrates their accuracy but
also characterizes the role of the boundary conditions on the steady-state velocity fields. The remaining two simulation sets ($5-$6) adopt binary fields for the heterogeneity of the hydraulic conductivity field but still maintain isotropy of the correlation structure. The transformation of an isotropic to anisotropic system for physical transport can be performed by a suitable spatial transformation without the requirement to re-simulate the flow field. Additional simulation sets are being considered with more complex hydraulic property fields. The developments that have arisen from the research can be summarized as: 1
2
3
4
a new algorithm for generating large-domain, high resolution random fields using Cholesky Decomposition; a modification of a mixed finite-element flow code to handle periodic boundaries in two dimensions that can produce accurate, mass conservative velocity fields for the study of strongly heterogeneous media; a sequential method to reduce the computational effort for generating flow distributions using the MFE code; a precise set of simulation sets each comprising 1000 realizations that can be used for further study of transport in strongly heterogeneous porous media.
Work is ongoing to use the simulation sets to explore upscaling and prediction methods in flow and transport modelling.
MODELLING OF PHYSICAL TRANSPORT The research team would like to acknowledge the financial support for this work by the Natural Environment Research Council under the Micro to Macro thematic programme and also the support of the United Kingdom Nirex Limited, who provided funding for the inputs to the programme by Andrew Cliffe. The major simulation work has been carried out using the extensive parallel computing facilities at CSAR, Manchester, without which this study could not have been attempted.
References CLIFFE, K.A., FRANKLIN, D.J. & MACLEOD, E.J. 2000a. A review of methods for generating heterogeneous hydrogeological property fields. Report No. AEAT/R/ENV/0190 for United Kingdom Nirex produced by AEA Technology. CLIFFE, K.A., GRAHAM, I.G., SCHIECHL, R. & STALS, L. 2000b. Parallel computation of flow in heterogeneous media modelled by mixed finite elements. Journal of Computational Physics, 164, 258-282. DAGAN, G. 1990. Transport in heterogeneous porous formations: spatial moments, ergodicity and effective dispersion. Water Resources Research, 26, 1281-1290. GALLI, A., GOBLET, P., GRIFFIN, D., LEDOUX, E., LE Loc'n, G., MACKAY, R. & RENARD, P. 1996.
71
Quick upscaling of flow and transport related parameters. Final report of work completed under the Geoscience 2 Reservoir Engineering Project, Topic 4, JOU2-CT92-0182. Commission of the European Communities, p. 167. GRAHAM, I.G. & HAGGAR, M.J. 1999. Unstructured additive Schwarz conjugate gradient method for elliptic problems with highly discontinuous coefficients. SlAM Journal of Scientific Computing, 20, 2041-2066. LITTLE, R., MULLER, E. & MACKAY, R. 1996. Modelling of contaminant migration in a chalk aquifer. Journal of Hydrology, 175, 473-509. MACKAY, R., RILEY, M.S. & WILLIAMS, G.M. 2001. Simulating groundwater contaminant migration at Villa Farm lagoons. Quarterly Journal of Engineering Geology and Hydrogeology, 34, 215-224. RENARD, P., LE LOC'H, G., LEDOUX, E., MARSILY, G. DE & MACKAY, R. 2000. A fast algorithm for the estimation of the equivalent hydraulic conductivity of heterogeneous media. Water Resources Research, 36, 3567-3580. TOMPKINS, J.A., GAN, K.C., WHEATER, H.S. &. HIRANO, F. 1994. Prediction of solute dispersion in heterogeneous porous-media-effects of ergodicity and hydraulic conductivity discretisation. Journal of Hydrology, 159, 105-123.
MOPOD: a generic model of porosity development J. P. B L O O M F I E L D 1 & J. A. B A R K E R 2
1British Geological Survey, Maclean Building, Crowmarsh Gifford, Wallingford, Oxfordshire OXIO 8BB, UK (e-mail: j. bloomfield@ bgs.ac, uk) 2Hydrogeology Group, Department of Geological Sciences, University College London, Gower Street, London WC1E 6BT, UK (e-mail:
[email protected]) Abstract: A code, MOPOD, has been developed to investigate general relationships between simple porosity growth laws and pore growth phenomena. MOPOD has been formulated as an 'initial value problem' and to date, investigations have focused on a very simple porosity growth law of the form dai(t)/dt : vT, where e is the aperture growth rate exponent. A range of qualitatively distinct evolved geometries have been described for porosity growth on 2D and 3D arrays of varying geometries and connectivities as a function of the exponent, e, of the aperture growth-rate law, and the width of the initial aperture distribution, o z. At low growth-rate exponents and moderate values of o-z over time there is a homogenization of apertures oriented sub-parallel to the head gradient. At moderate growth-rate exponents these apertures become increasingly heterogeneous in evolved arrays, with planar heterogeneities developing sub-parallel to the head gradient for low values of o-z while anastomosing structures develop at higher values of o z. For larger growth-rate exponents preferentially enlarged array-spanning paths develop. No selforganization phenomena have been observed because periodic or cyclic behaviour is not inherent in the simple growth laws investigated to date.
The objective of this Micro to Macro project has been to produce a MOdel of coupled flow and POrosity Development (MOPOD) in heterogeneous porous (fractured) media and to use the model to investigate porosity growth phenomena, including scaling phenomena. Some of the preliminary results from the study have already been published (Bloomfield & Barker 1999, 2003; Bloomfield et al. 2000, 2001) and a full description of the model with observations on porosity growth phenomena is given in Bloomfield et al. (2005). Consequently, this short paper is limited to a brief description of the model and of additional model functionality that has been developed since that described in Bloomfield et al. (in press) and to a note on some new observations related to porosity growth phenomena. The paper is concluded with a discussion of potential areas of application of the code and a note on how it may be developed in the future.
Context of the MOPOD study The approach taken to modelling fractured aquifers is commonly that of stochastic network modelling but such models may not exhibit the
long-range preferential pathways, associated with channelling of flow through fracture networks, which are important for contaminant transport (Berkowitz & Balberg 1993). One approach to the generation of more realistic spatial correlations in fracture networks is to explicitly model the development of fractures according to fundamental physical and chemical laws (Groves & Howard 1994; Dreybrodt 1996; Bekri et al. 1997; Djik & Berkowitz 1998; Siemers & Dreybrodt 1998; Kaufman & Braun 1999). However, the idealized processes that are modelled, such as mineral dissolution and precipitation, can grossly oversimplify the more complex field situation where hydromechanical plucking, abrasion and microbially mediated processes may be significant. MOPOD has been developed as an alternative middle-way between stochastic and process-based construction of fracture networks for flow and transport modelling, especially where a range of fracture modification processes are active, simultaneously. Using MOPOD, patterns of fracture aperture growth can be described based on simple growth laws applied to fracture arrays with simple initial aperture distributions. The approach is, therefore, in the spirit of statistical mechanical studies of simple systems such as
From: SHAW,R. P. (ed.) 2005. Understanding the Micro to Macro Behaviour of Rock-Fluid Systems. Geological Society, London, Special Publications, 249, 73-77. 0305-8719/05/$15.00 9 The Geological Society of London 2005.
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J.P. BLOOMFIELD & J. A. BARKER
percolation models (Berkowitz & Balberg 1993; Sahimi 1994; Stauffer & Aharony 1994). The objective of this type of study is to reveal general characteristics of behaviour, such as geometrical phase transitions, ahead of studies of behaviour in relation to specific processes with their individual magnitudes and time-scales.
MOPOD code and functionality The MOPOD model is described in detail in Bloomfield et al. (2005). The following is intended only as a brief summary. The MOPOD model has been formulated as an 'initial value problem' expressed by a set of N first-order equations of the form dai(t)/dt = G(ai,vi,p), where ai(t) is the aperture of pore (or fracture) i (i = 1. . . . . N) at time t, subject to the initial aperture values, ai(O) = ai0. G is a user-defined growth function, p is a set of parameters describing the growth of the apertures and vi is the magnitude of the volumetric flow rate in pore i. To solve the first-order equations one must evaluate the function G. This involves the evaluation of volume flow rates within pores and this, in turn, requires the solution of the upstream and downstream heads in each individual pore. The solution of the firstorder equations has been implemented using a Runge-Kutta method and heads have been found using a variety of methods line successive over-relaxation (LSOR), preconditioned conjugate gradient (PCG) and Gaussian elimination), depending on the geometry and size of the problem. The model uses an array of pores or fractures that have been seeded with an initial aperture distribution. The apertures that are calculated using the first-order equations are then used subsequently in the next iteration of the model and the model is run through as many iterations as are desired. MOPOD is flexible in the form of the initial geometry of the porosity elements, the boundary conditions, and in the form of the fracture growth laws that can be modelled. MOPOD can model 2D and 3D arrays with both regular (orthogonal symmetry) and random structures, where the random arrays can be modelled as fully or partially connected systems. The size of the array that can be modelled is limited primarily by the computational effort required to solve the head distribution in the array. For practical purposes arrays are generally limited to a maximum size of 20 x 20 x 20; however, larger arrays may be modelled. A wide range of initial aperture distributions is available (including uniform, normal, log-normal, negative exponential, gamma and triangular) and the code can be modified to
accept user-defined initial aperture distributions. The boundary conditions are flexible and include constant head or constant flow conditions. Flow can either be across the array or radially towards the centre of the array. Flow rates are taken to be proportional to the cube of the aperture ('cubic law') and to the head gradient (Darcy' s law). However, this too can be modified as required (for example to take account of turbulent flow). The MOPOD code models aperture growth as a polynomial function of aperture and flow rate (as defined by the set of parameters, p). This allows a reasonable amount of flexibility, but the form of growth law is readily replaced and there is little restriction on the form or complexity of the growth law that could be used.
Porosity growth p h e n o m e n a To date, investigations have focused mainly on a very simple porosity growth law of the form dai(t)/dt = v~, where e is the aperture growth rate exponent. In addition, all the model runs described here and in Bloomfield et al. (2005) used initial aperture distributions that were spatially uncorrelated and log-normal. The study worked with the statistics of the logarithms of the apertures, zi = loglo ai, where the initial mean of the zi values is denoted by ~z and standard deviation is denoted by o-z, respectively. The geometric mean of the initial, aio, values was set to unity (equivalent t o / z z = 0). Bloomfield & Barker (2003) and Bloomfield et al. (2005) have described a range of qualitatively distinct evolved geometries for porosity growth on fully connected 2D orthogonal arrays as a function of the exponent, e, of the aperture growth-rate law and the width of the initial aperture distribution, o z . They noted that at low growth-rate exponents (e < 0.3) and moderate values of o-z (o-z < 0.4) there is a homogenization of apertures orientated parallel to the head gradient with time. At moderate growth-rate exponents (0.3 < e < 0.6) row apertures become increasingly heterogeneous in evolved arrays, with planar heterogeneities developing parallel to the head gradient for low values of o-z, while anastomosing structures develop at higher values of o-z. For growth-rate exponents in the range 0.6 < e < 0.8, preferentially enlarged array-spanning paths develop. When both e and O'z are large the array-spanning structures no longer develop; instead, isolated enlarged apertures develop. Figure 1 (from Bloomfield & Barker 2003) summarizes these phenomena. Additional work on irregular 2D, and regular (orthogonal) and irregular (fully and partially
GENERIC MODEL OF POROSITY DEVELOPMENT
75
Increasing e 0.2 to 0.8
r~ o o
o
I
I
c--
I
c~ L.
o c-
I
m I
m
I
m
I I
I
Fig. 1. Schematic illustration of the structure of evolved fully connected 2D orthogonal fracture arrays as a function of e and o z. Only the significantly enlarged components of the arrays are illustrated.
connected) 3D arrays has shown similar qualitative trends in array development. For example, preferentially enlarged array spanning paths (similar to those schematically illustrated in the top right-hand corner and centre right of Fig. 1) develop at relatively high values of e and o-z for each of the model geometries. However, the absolute thresholds at which these structures begin to appear vary, depending on the model geometry. Figure 2 shows three examples of preferentially enlarged array spanning paths that develop on a range of array geometries and under different boundary conditions. Based on the observation that qualitatively similar evolved structures develop regardless of the geometry of the fracture array or the boundary conditions, it is inferred that the form of the evolved structures and the associated phase transitions are inherently associated with the form of the simple growth law. Parameterization
of the evolved arrays
A number of methods have been explored in an attempt to parameterize the arrays. Basic statistics provide quantitative descriptions of the arrays. A
limited study of geometrical percolation in the arrays has been undertaken (Bloomfield & Barker 1999); however, because the aperture fields become highly structured the assumption of a random aperture field required for analysis using percolation methods becomes invalid as soon as the arrays evolve. There is no evidence for scaling of either the porosity or flow fields in the evolved arrays for the growth laws investigated. Variograms can be used to characterize the development of spatial correlation in the aperture field. Although spatial correlations develop in the porosity distribution, pore structures evolve towards a final state and self-organization phenomena are not observed because periodic or cyclic behaviour is not inherent in the simple growth laws investigated to date (Bloomfield et al. 2005). The development of correlations in flow rates between adjacent fractures can also be tracked by comparing the maximum flow into and out of a fracture intersection. Increasing correlation in maximum inflow and outflow with time is a characteristic feature of array development. However, more rigorous parameterization of the evolving arrays is more problematic and it has not been possible to identify parameters
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J.P. BLOOMFIELD & J. A. BARKER
Fig. 2. Three examples of preferentially enlarged array spanning structures developed on irregular 2D and 3D arrays. The boundary conditions for the arrays are as follows: (a) through flow under a constant head gradient (left to right) with no flow at top and bottom boundaries; (b) radial flow out of the centre of the array with a constant head gradient from the margins to the centre of the array; (c) through flow under a constant head gradient (left to right) with no flow at the top, bottom, front and back boundaries.
that can be used to predict the form of the e v o l v e d arrays on the basis of the initial aperture distributions or growth-rate laws.
Areas of application for the code and future development The initial motivation for the study was to develop a m o d e l to simulate the geometry of preferentially enlarged fractures in the Chalk of N W
E u r o p e in order to understand the d e v e l o p m e n t of the aquifer better. H o w e v e r , it is envisaged that M O P O D m a y have useful applications in a n u m b e r of areas w h e r e there is a n e e d to m o d e l porosity development. For example, M O P O D could be used to m o d e l borehole acidization of carbonate reservoirs and aquifers or borehole d e v e l o p m e n t in friable sandstone formations due to sand pumping. In addition, M O P O D could be used to generate realistic templates of natural systems (i.e. porosity fields that show
GENERIC MODEL OF POROSITY DEVELOPMENT long-range preferential pathways, such as those associated with channelling of flow through fracture networks) for use in flow or transport models. For example, it could be used to generate evolved fracture arrays prior to the modelling of pumping and tracer tests in simulated fractured aquifers, or it could be used for the modelling of breakthrough phenomena, or as part of core flooding, immiscible phase pore-entry or residual saturation studies. It could also be used as the basis for numerical models of remediation in fractured rocks. Future work will focus on extending the functionality of the MOPOD code. At present, MOPOD is configured so that it performs a single growth realization. However, because of the nature of the approach that has been adopted, the MOPOD methodology is well suited to Monte Carlo-type simulations and this functionality will be developed. In addition, work will be undertaken to study breakthrough phenomena on the evolved arrays. This work was undertaken as part of the NERC-funded Micro-to-Macro Thematic Research Programme (grants GST/02/2505 and GST/03/2505) and as part of the National Groundwater Survey of the British Geological Survey (BGS). Published with permission of the Executive Director, BGS (NERC).
References BEKRI, S., THOVERT,J.-F. & ADLER, P.M. 1997. Dissolution and deposition in fractures. Engineering Geology, 48, 283-308. BERKOWlTZ, B. & BALBERG, I. 1993. Percolation theory and its applications to groundwater hydrogeology. Water Resources Research, 29, 775 -794. BLOOMFIELD, J.P. & BARKER, J.A. 1999. Modelling development of fracture aperture distributions using a simple aperture growth law. British
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Geological Survey Technical Report No. WC/ 99/18. British Geological Survey, Keyworth, UK. BLOOMFmLD, J.P. & BARKER, J.A. 2003. MOPOD: a model to investigate fracture porosity development. In: KRASNY, J., HRKAL, Z. & BRUTHANS,J. (eds) Groundwater in Fractured Rocks. IHP-VI Series on Groundwater No. 7, IAH, Prague, Czech Republic, 397-398. BLOOMFIELD, J.P., GAUS, I., BARKER, J.A. & ROmNSON, N. 2000. Modelling porosity development and flow in heterogeneous media. Geoscience 2000 (abstract), 175. BLOOMFIELD, J.P., BARKER, J.A., GAUS, I., GRIFFtTHS, K. & ROmNSON, N. 2001. Modeling porosity development and flow in heterogeneous porous media. Geophysics Research Abstracts, 3, 335. BLOOMFIELD, J.P., BARKER, J.A. & ROBINSON, N. 2005. Modelling fracture porosity development using simple growth laws. Ground Water, 43, 314-326. DJIK, P. & BERKOWITZ,B. 1998. Precipitation and dissolution of reactive solutes in fractures. Water Resources Research, 34, 457-470. DREYBRODT, W. 1996. Principals of early development of karst conduits under natural and manmade conditions revealed by mathematical analysis of numerical models. Water Resources Research, 32, 2923-2935. GROVES, C.G. & HOWARD,A.D. 1994. Early development of karst systems. 1. Preferential flow path enlargement under laminar flow. Water Resources Research, 30, 2837-2846. KAUFMAN, G. & BRAUN, J. 1999. Karst aquifer evolution in fractured rocks. Water Resources Research, 35, 3223-3238. SAHIMI, M. 1994. Applications of percolation theory. Taylor & Francis, London. SIEMERS, J. & DREYBRODT,W. 1998. Early development of karst aquifers on percolation networks in fractures in limestones. Water Resources Research, 34, 409-419. STAUFFER, D. & AHARONY,A. 1994. Introduction to percolation theory. Taylor & Francis, London.
Anomalous diffusion in simulations of pumping tests on fractal lattices S H A U N S E L L E R S & J O H N A. B A R K E R Department of Earth Sciences, University College London, Gower Street, London WC1E 6BT, UK (e-mail:
[email protected]) Abstract: The recent interest in fractals in the geosciences literature has led to several
proposed theoretical models for the hydraulic testing of fractured-rock systems that exhibit a fractal-like geometric structure. There is, however, no agreement on the correct form of the resulting model equations. In order to gain some insight into the range of possible behaviours to be expected from pumping tests on such systems, as well as the type of theoretical models needed, extensive simulations of pressure diffusion for transient groundwater flow, modelled by random walks on both deterministic and random fractal lattices were performed. For simplicity, the focus was on measurements of the random-walk dimension for generalized Sierpinski carpets, a proposed model for porous and fractured media. In addition to the expected anomalous slow down in diffusion in fractals as measured by the random-walk dimension, the simulations show further novel and unexpected anomalous behaviour due to the presence of internal boundaries at all scales. None of the proposed theoretical models for pumping tests on fractals appears consistent with all of the observed anomalous behaviours. The simulations suggest that interpretation of experimental pumping tests in terms of well-defined non-integer dimensions can be difficult, even when finite-size effects are negligible.
Power laws characterizing the geometry of fracture systems have become fairly common in the geosciences literature where they have been used, for example, to describe measurements of pore-geometry, surface roughness and the relationship between fault number, lengths and widths (see Bonnet et al. 2000 for a review). Since a l o g - l o g plot of a power law yields a straight line with the slope being the scaling exponent, approximate fits of straight lines to the experimental data are typically used to justify power-law relationships. Often, such straight lines on l o g - l o g plots are also interpreted as evidence of some type of fractal structure. Although there is still some discussion about the interpretation of these data and the relevance of power laws and fractals to fracture systems (e.g. Bonnet et al. 2000), it seems reasonable that there are some fracture systems that can be described as approximately fractal, at least over a reasonable range of scales. Clearly a fractal structure should have some effect on the hydraulic properties of the fracture system. Pumping tests, in particular, are a standard way of determining hydraulic properties at the field scale. Thus, there exists a need to understand the response of a pumping test in fractal systems that have non-integer dimensions. In standard Euclidean systems with 2D or 3D radial flow, pumping tests are usually modelled
with a diffusion equation for the pressure head, where the form of the equation depends explicitly on the dimension (Barker 1988). For fractal systems, one might expect some generalization of the diffusion equation with, perhaps, appropriate non-integer dimensions appearing in the governing equation. In fact, two such models have already been proposed in the hydrogeological literature: the generalized radial flow model of Barker (1988) and the fractal model of Chang & Yortsos (1990). In addition, it is now common to see data reported in terms of the parameters of these models (such as the flow dimension). To cite two recent examples from well tests, Le Borgne et al. (2003, 2004) report flow dimensions ranging from 1.4 to 1.7 for an aquifer in Brittany, and Riemann et aL (2002) report flow dimensions of 1.75-1.85 for an aquifer in Bloemfontein, South Africa. Although the physical meaning of the flow dimension is not yet clear, it is often interpreted as a measure of the geometric fractal dimension of the fracture network. Le Borgne et al. (2003, 2004) also find anomalously slow diffusion of pressure given by a random-walk dimension of 2.8. (Note that the random-walk dimension provides a measure of the anomalous slow down in diffusion and is defined in the results section. In an n-dimensional Euclidean lattice the random-walk dimension is exactly 2
From: SHAW,R. P. (ed.) 2005. Understanding the Micro to Macro Behaviour of Rock-Fluid Systems. Geological Society, London, Special Publications, 249, 79-89. 0305-8719/05/$15.00 9 The Geological Society of London 2005.
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(ben-Avraham & Havlin 2000).) This anomalous value of 2.8 was obtained by fitting a straight line to a log-log plot of characteristic time versus distance from the pumping well. There are also several additional models in the physics literature purporting to model diffusion on fractals: O'Shaughnessy & Procaccia (1985), Metzler et aI. (1994), Compte & Jou (1996) and Giona & Roman (1992). Interestingly, they all lead to different generalizations of the diffusion equation. One could then ask which model is the 'correct one' for diffusion on a fractal. A recent comparison (Schulzky et al. 2000) has shown that for suitable choices of parameters, they all can provide a reasonable fit over a certain range of data. But none fits over the entire range. Thus, there seems to be no agreement at this point as to the 'correct' model, if indeed there is one. The uncertainty about the correct theoretical model clearly raises doubts about the meaning and usefulness of reported values of non-integer dimensions for pumping tests. In order to get a better understanding of how to model pumping tests on fractal systems, as well as the possible types of behaviour to expect, extensive Monte Carlo simulations of diffusion in various fractal lattices were performed. This work is intended to illustrate possible behaviour, as well as to provide examples that can be compared with results from theoretical models. Previous simulations have either tended to use complicated, random geometries or a very simplified geometry to verify certain predictions (e.g. Acuna & Yortsos 1995; Doughty & Karasaki 2002). It can be difficult, however, to interpret the results from complicated geometries, and very simplified geometries do not always provide a broad range of behaviour. In order to understand and relate the diffusive response to the underlying geometry better, generalized Sierpinski carpets were chosen as a simple computational model for fractal fracture systems. Sierpinski carpets have been already proposed as a model for constructing porous and fractured media and they can be constructed with a given geometric fractal dimension (Tarafdar et al. 2001). In order to simplify the problem further, focus was placed on measurements of the random-walk dimension associated with the pressure diffusion on the Sierpinski carpets. These 'numerical experiments' correspond to the measurement of characteristic time and distance from the well as in the field experiments of Le Borgne et al. (2003, 2004). They provide a direct way of determining the random-walk dimension and are computationally simpler than solving flow in fractal networks, as carried
out by Acuna & Yortsos (1995) and Doughty & Karasaki (2002). Note that the random-walk dimension does not characterize diffusion on fractals fully. Nor is it always the most relevant quantity. But it does provide a good starting point in making comparisons to predictions from proposed theoretical models as well as to experimental data. In the discussion, the generation of the fractal lattices is discussed first, followed by the use of random walks to simulate diffusion on the lattices and then the results from the simulations. Diffusion in fractals is known to be anomalous (e.g. Ben-Avraham & Havlin 2000) but the results from this study illustrate additional behaviours that appear to be novel and unexpected. Further, some of these results are inconsistent with common assumptions used to obtain the above-cited generalizations of the diffusion equation, thus suggesting even more limitations on the various proposed generalizations of the diffusion equation for fractals, as well as the interpretation of the resulting non-integer dimensions from experimental data. Clearly, more theoretical work needs to be done in order to improve the understanding of pumping tests on fracture systems having fractal properties.
Generation of Sierpinski carpets Although a special class of fractals, Sierpinski carpets provide a simple and versatile tool for investigating fractals. They are completely specified by their generators, thus computationally easy to implement (Tarafdar et al. 2001). In addition, by choosing an appropriate generator, one can build carpets with a specified geometric fractal dimension and shape. For these reasons, Sierpinski capets have already been proposed as models for porous media (Tarafdar et al. 2001) and for fractured rock (Doughty & Karasaki 2002). A deterministic iteration procedure generates the Sierpinski carpets from the generators. Four different examples of 3 x 3 generators are shown in Figure 1. The shaded squares represent the pore space. The iteration procedure is to replace each shaded square by the generator. Figure 2 shows the second and fourth iterations of the generators in Figures la and b. More sophisticated generators are given in Tarafdar et al. (2001), where more realistic shapes can be obtained by choosing larger generators. Here, the focus is on the 3 x 3 generators in order to simplify the picture as much as possible. For the simulations, the generators were iterated nine times to obtain a 19 683 x 19 683 lattice. This size is more than sufficient to allow for
ANOMALOUS DIFFUSION IN PUMPING TESTS
81
Fig. 1. Four 3 • 3 generators for Sierpinski carpets: (a) produces a fractal dimension of log 8/log 3; (b)-(fl) all produce the same fractal dimension of log 7/log 3. The shaded squares represent the void space. random walkers to sample the fractal properties and also avoid finite-size end effects. Note that the reported data for real fracture systems rarely exceed three orders of magnitude of selfsimilarity. It is also straightforward to generate Sierpinski carpets in three-dimensional space, though this
can entail additional significant computational time and m e m o r y requirements for the simulated r a n d o m walks. R a n d o m carpets were also generated by randomly choosing the generator at each level of iteration. There are several ways in which this randomness can be introduced. For example,
Fig. 2. Second and fourth iterations of the generators illustrated in Figure la and d. The fractal dimensions are log 8/ log 3 and log 7/log 3, respectively. The shaded squares represent the void space. At each time step during a random walk, a walker either moves to a neighbouring shaded square or remains stationary.
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S. SELLERS & J. A. BARKER
one can simply choose the generators at each iteration level and apply the same generators at every point. Figure 3 shows two such examples constructed by randomly removing, respectively, one and two elements at each iteration level. Alternatively, one can randomly choose the generators for each point. Figure 4 shows two such examples. Here, the resulting carpets are not always well connected - in fact, one sees many isolated islands. In addition, depending upon the choice of generators, the distribution of islands can be highly non-uniform.
Random walks The specific problem under investigation is diffusion with an initial, instantaneous point source at a given location of a fractal. The solution of this problem corresponds to the fundamental solution of a differential equation as it can be integrated to yield solutions to more general boundary data. Discrete time-discrete space random walks are used to simulate diffusion. In this case, a large collection of independent and identical walkers is assigned an initial location and the positions are randomly incremented by one spatial step at each time step. Two common algorithms for the updating procedure are the 'blind' and 'myopic ant' procedures (Ben-Avraham & Havlin 2000). This paper reports only results for the blind ant procedure, as there were no measurable numerical differences in the resulting dimensions. In the blind ant procedure, one chooses with equal probability among the four possible directions. If the neighbouring site is available (i.e. a shaded square in the figures), the walker moves to this site; if not, then it stays at the current position. For each fixed origin, an ensemble of walkers is created (typically 30 0 0 0 - 5 0 000 particles) with total time
(a)
steps ranging from 1 million to 100 million. In order to simulate a point source, all walkers in the ensemble have the same origin or starting point. For more information on simulating random walks on Sierpinski carpets, see Schubert (1999) and Seeger et al. (2001). Typical trajectories of a single walker are given in Figures 5 & 6 for two different lattices. While individual walkers are not meaningful statistically, the figures do illustrate that the scaling need not be the same in all directions. In order to make this statement meaningful, however, ensembles of random, identical walkers must be considered.
Results The property of interest is the scaling of distance with time, which is determined by the randomwalk dimension dw. It is defined by t ~ (Ir - rol2) aw/2 = (IAr[2) &/2,
(1)
where r0 is the origin of the walker and () the ensemble average over independent, identical walkers. For Euclidean systems, dw = 2, regardless of the geometric or topological dimension. The case dw ~ 2 is called anomalous (BenAvraham & Havlin 2000) and arises due to the presence, for example, of obstacles and dead ends at all scales, which are typical of fractals. In particular, for dw > 2 the characteristic time for diffusion is larger, which indicates that the diffusion process is slower. Figure 7 shows a typical plot of characteristic time versus distance for an unbounded 2D lattice. The data fall neatly on the indicated line with slope two, which corresponds to dw = 2. All diffusion models for fractals assume radial symmetry. To test this assumption, the following
(b)
Fig. 3. Two examples of random carpets after four iterations, in which the generator is randomly selected at each iteration. In (a) one of the nine elements is randomly removed, and in (b) two elements are randomly removed. The fractal dimensions are log 8/log 3 and log 7/log 3, respectively.
83
ANOMALOUS DIFFUSION IN PUMPING TESTS
~ ,!,:,~:,,:,,',,:,,',,,',,',,,~:,,',',c~t~t
(a)
Fig. 4. Alternative random carpets can be obtained by randomly choosing generators at each point: (a), (b) fourth and fifth iterations constructed by randomly removing a corner element at each point; (c), (d) fourth and fifth iterations constructed with a slight bias for removing the lower corners. Both fractals have a dimension of log 8/log 3, the same as that of Figures 2a and 3a. Note that this procedure can result in carpets that are not well connected.
scalings were introduced t ~
dx/2
and
t ~ ~ / 2 ,
(2)
where &x and Ay are, respectively, the displacements in the x and y directions from the origin of the walkers. The exponents d~v and dY w are anisotropic generalizations of the r a n d o m - w a l k dimension. If diffusion in fractals is isotropic or radially symmetric, then d x = d y = dw. Note that for E u c l i d e a n systems, d~w = d r = dw = 2 (although diffusion can be anisotropic with a non-spherical diffusion tensor). The next part of the discussion n o w turns to the r a n d o m - w a l k simulation results for fractal lattices. For space reasons, results for a few select carpets, w h i c h we chosen to be representative of the extensive simulations are presented. The data are plotted to illustrate the above three scalings and, thereby, to extract the r a n d o m - w a l k dimensions. A least-squares fit of logarithmically equally spaced points provides the p o w e r - l a w relationship. Figure 8 shows the resulting characteristic times versus displacement for two specific origins of the point source on the classical Sierpinski carpet of Figure 2c. In Figure 8a, the data fall neatly onto well-defined straight lines,
(b) Fig. 5. Typical trajectories of 30 000 time-steps of a single random walker on the two lattices of Figure 2. Note that the walker in (a) spends substantially more time moving in one direction than in the other, due to the limited number of upwards connections. This can lead to different scaling properties, hence different dimensions, for the two directions. each with slope ~ 2 . 1 . The response is symmetric with the same scaling in the x, y and radial directions: dw =d~v = d y = 2.1. Note that this r a n d o m - w a l k dimension differs from the geometric fractal dimension, w h i c h is approximately
Fig. 6. Typical trajectory of 30 000 steps of a single random walker on the random lattice of Figure 4a. Due to a lack of connectivity, not all of the void space is sampled.
84
S. SELLERS & J. A. BARKER 1e+06 le+05
.le5
.le4~ .leg; .le2~
,i
.le2
.le3
.le4
distance . . . .
. . . . . .
t=r 2
unbounded
Fig. 7. Results for an unbounded 2D Euclidean lattice. 1.89. Also, the radial component (green) always lies below the x (red) and y (blue) components, as the x and y components include only information in the respective directions, whereas the radial component includes both (At 2 = &x2+ Ay2). Figure 8b shows that a different location for the point source can yield a slightly different response where the data can deviate from straight lines. Asymptotically, however, the slopes are close to the previous value of 2.1. Presumably, if the simulation times were extended further the data would converge to a straight line with this slope. Other origins showed similar deviations from linearity in the data, with the tendency to converge asymptotically. Figure 9 shows the results for four different choices of origin of the point source on the
fractal in Figure 2d. Figures 9a and b illustrate similar behaviour, with similar though slightly different slopes. Small but definite oscillations in the data make accurate determination of the slopes difficult. Perhaps for longer simulation times the slopes of the two figures would approach each other. Importantly, there is a significant difference in the scalings for the x and y components of the displacement. For Figure 9a, a least-squares fit of logarithmically equally spaced points yields dw -- 2.16, d~v = 2.10, dY w = 2.59. Thus, not only is there anisotropy in the diffusion, but the apparent diffusion coefficients scale differently in the different directions. This anisotropy of the scalings is surprising since it is commonly assumed that diffusion is isotropic on fractals (Barlow et al. 1995). In fact, differential equations for diffusion on fractals assume radial symmetry. Figure 9c illustrates similar, though still slightly different, behaviour for yet another choice of origin for the point source. Finally, Figure 9d shows a completely different response for the same lattice. This time the response appears to be isotropic but with large oscillations in the data. Measurement of the dimensions yields dw = d~v -- dY w ~ 2.5. Also, the data for the x component (red) lie above those of the y component (blue), in contrast to Figures 9a-c. As this example surprisingly shows, the same fractat lattice can exhibit both isotropic and anisotropic behavior. Clearly, the governing power laws depend strongly on the origin of the point source for this lattice. This result further suggests that the random-walk dimension can depend on
Fig. 8. Results for two specific origins in the classical Sierpinski carpet of Figure 2c. The blue curve is the displacement in the y direction, the red curve is the displacement in the x direction, and the green curve is radial displacement. The best fitting power law is given in the figure for each curve. Note that the geometric fractal dimension ~1.89, which differs from the random-walk dimension.
ANOMALOUS DIFFUSION IN PUMPING TESTS
85
Fig. 9. Results for four different origins in the fractal of Figure 2b. The best fitting power law is given for the y, x and r components.
the choice of origin and hence, the particular problem. Thus, a single, well-defined random-walk dimension need not characterize the entire fractal. In fact, many dimensions may be necessary. Figure 10 illustrates the data for two origins on the random lattice of Figure 4b. In Figure 10a, the data are approximately isotropic, with small oscillations in the data possibly reducing the accuracy of the dimension measurements. However, Figure 10b shows that another choice of origin of the same lattice does not yield any
apparent scaling behaviour, even after 107 time steps. Perhaps longer simulations would indicate scaling. Note, however, that in the previous figures it took less than 103 time steps for the data to converge to the asymptotic properties. The simulation results for the random fractal in Figure 4d also appear strongly to depend on the starting point of the walkers. Figure 11 shows the results for a specific origin. Initially, the curves appear to be straight, but then begin to deviate from straight lines. For the x component, there is no identifiable power-law
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S. SELLERS & J. A. BARKER
Fig. 10. Results for two origins of the random fractal of Figure 4b: (a) the data scale well and appear to be approximately isotropic; (b) no clear power-law behaviour can be observed for these time-scales. description. The y and radial components deviate less from linearity but still do not appear to follow a clear power law for the simulated time-scales. Again, it is not clear if longer simulation times would provide a single power law, or
if there is no single power-law description. One possible explanation is the significant lack of connectivity in this carpet, which may result in walkers actually sampling paths that have a fractal dimension less than the geometric
Fig. 11. Results for the random fractal of Figure 4d. No single power-law behaviour can be observed for these time-scales.
ANOMALOUS DIFFUSION IN PUMPING TESTS fractal dimension. Thus, a walker may sample a region that appears to have certain dimension and then, at a later time, it may sample another region that appears to have a different dimension, resulting in an apparent time-evolution of the dimension. One of the characteristic features of fractals is the presence of internal boundaries at all scales. In order to understand their effects as well as their significance on diffusion better, random walks on 2D Euclidean lattices with boundaries were also simulated. Figure 12 illustrates the effects of a boundary on the random walks. In Figure 12a, the origin of the walkers lies next to an edge boundary. The boundary initially reduces the mobility of the walkers, so that the initial data points lie above the red curve. As the walkers gradually move away from the boundary, their mobility returns to normal and the data points quickly converge to the red line. The asymptotic slope is 2; consequently the random-walk dimension remains unchanged by the external boundary. Figure 12b illustrates the effect of a boundary located 15 units from the origin. Initially, the walkers do not feel any boundary and the initial slope of the data is 2. However, as the walkers approach the internal boundary, their mobility decreases; hence, the data points lie above the red curve. In addition, as the walkers eventually move away from the boundary, their mobility increases, so that the slope gradually returns to 2. Thus, the effect of
87
an internal boundary on the diffusive process is to slow it down in the vicinity of the boundary. The result is a 'bulge' in the plot. A closer examination of the previous figures for Sierpinski carpets reveals that the apparent straight lines actually consist of small log-periodic oscillations (see, for example, Fig. 13). And these oscillations themselves consist of finer-scale oscillations. The simulations of random walks in Euclidean lattices with boundaries allow one to easily interpret these oscillations as effects from internal boundaries at many scales. In fact, the walkers are constantly sampling paths that contain many different length scales. As seen in Figure 12, each internal boundary contributes to observed bulges in slope with a magnitude characteristic of the particular length scale of the boundary. Since a power law relates the length scales for Sierpinski carpets, the resulting bulges are log-periodic and appear as many superposed oscillations. By superposing the bulges in various ways, the resulting shape and slope can be controlled. Thus, the distribution of the internal boundaries, which determines the bulges, appears to determine the shape and slope of the data and, hence, the random-walk dimension. The inhomogeneity in the distribution of internal boundaries in fractals then provides a possible explanation for the apparent dependence of the dimensions on the initial and boundary conditions.
Fig. 12. Results for 2D Euclidean lattices with boundaries: (a) boundary at the origin; (b) boundary 15 units from the origin. Near the boundaries the diffusion process slows down, thereby increasing the slope of the data points. In both cases the slope graduallyreturns to two as the boundary effects become negligible, hence dw = 2. The boundary results in 'bulges' in the data.
88
S. SELLERS & J. A. BARKER 1e+05 -
.9e5 distance ......
detail of Fig 9 a
Fig. 13. Detail of Figure 9a. The data appear to consist of bulges of the same shape as those in Figure 12b.
Discussion The results presented here show that diffusive behaviour in both deterministic and random Sierpinski carpets can be quite complex. It has been shown that a single scaling law with a unique random-walk dimension does not always appear to characterize diffusive data on Sierpinski carpets. In fact, many dimensions may be needed. Further, the dimensions appear to vary significantly with direction and with the location of the source. In the examples given above, the dimensions ranged from about 2.1 to 2.6. Naturally any variation will depend on the particular type of fractal. Note that these random-walk dimensions are significantly different from the geometric fractal dimension and, unfortunately, it is not yet clear how to relate these dimensions. By contrast, all the generalized diffusion models for fractals have assumed radial symmetry as well as a single, well-defined dimension that characterizes the entire fractal. A pumping test is usually modelled by pressure head diffusion, so that the results hold in particular for pumping tests on Sierpinski carpets. An anisotropic response means that one set of piezometers for a fixed pumping well could give one apparent power law and corresponding random-walk dimension while another set of piezometers in different locations, but with the same pumping well and fracture system, could give another power law with a completely different scaling. As shown in the previous examples, measurements made in one direction can differ significantly from those made in another direction. Further, a pumping test with the pumping well located in one place
of the fracture system could yield a completely different power law from another well located in a different place of the same fracture system. Current theoretical models for pumping tests on fractals do not allow for such behaviour, which raises questions about the limitations, if not the correctness, of current models for general fractals. One could argue that an ensemble average over a sufficient number of origins would provide a single random-walk dimension for the fractal. In fact, this is commonly done in simulations. But this approach does not seem physically relevant to a pumping test, where a well has a fixed location. Standard practice does not involve averaging data for many different well locations. Also Davison et al. (2001) have already criticized averaging methods, showing that they mask interesting properties. Admittedly, one could also argue that Sierpinski carpets are very special fractals and not necessarily representative of fractures likely to be found in aquifers. However, they certainly indicate possible behaviour in more complicated fractals. Further, one can create much more complex fractals by choosing the appropriate generator. Thus, this work puts into question the assumption that a single or even a small number of non-integer dimensions will characterize pumping tests on fractals. Whether or not it is possible to represent such a pumping test with a small number of non-integer dimensions will, of course, depend on the particular fracture system. More work is needed to determine if the anomalous behaviour observed in the simulations of this study is actually relevant to real fracture systems. In summary, simulations have been presented that illustrate novel and surprising behaviour for diffusion on generalized Sierpinski carpets. They include:
9
9
9
9 9
no apparent power-law scaling over the simulated times for some fractals, so that no dimension can be identified; an apparent change in the scaling over time, leading to an apparent time-varying dimension; small oscillations in the scaling due to internal boundaries, leading to errors in estimates for the dimension; anisotropic scaling contradicting the common assumption of radial flow; a dimension that varies with the location of the source, so that no single random-walk dimension can be identified for fractals.
ANOMALOUS DIFFUSION IN PUMPING TESTS These results suggest that there may be nontrivial difficulties in interpreting reports of noninteger dimensions from pumping well tests. This work was undertaken as part of the NERC-funded Micro to Macro Thematic Research Programme (grant GST/02/2664).
References ACUNA, J.A. & YORTSOS, Y.C. 1995. Application of fractal geometry to the study of networks of fractures and their pressure transient. Water Resources Research, 31, 527-540. BARKER, J.A. 1988. A generalized radial flow model for hydraulic tests in fractured rock. Water Resources Research, 24, 1796-1804. BARt.OW, M.T., HATTORI, K., HATTORI, T. & WATTANABE, H. 1995. Restoration of isotropy on fractals. Physical Review Letters, 75, 3042-3045. BEN-AVRAHAM,D. & HAVLIN, S. 2000. Diffusion and Reactions in Fractals and Disordered Systems. Cambridge University Press, UK. BONNET, E., BOUR, O., ODLING, N.E., DAVY, P., MAIN, I., COWIE, P. & BERKOWITZ, B. 2000. Scaling of fracture systems in geological media. Reviews of Geophysics, 39, 347-383. CHANG, J. & YORTSOS, Y.C. 1990. Pressure-transient analysis of fractal reservoirs. Society of Petroleum Engineers Formation Evaluation, 5, 31-38. COMPTE, A. & Jou, D. 1996. Non-equilibrium thermodynamics and anomalous diffusion. Journal of Physics A: Mathematical and General, 29, 43214329. DAVISON, M., ESSEX, C., SCHULZKY,C., FRANZ, A. & HOFFMANN, K.H. 2001. Clouds, fibres and echoes: a new approach to studying random walks on fractals. Journal of Physics A: Mathematical and General, 34, L289-L296. DOUGHTY, C. & KARASAKI, K. 2002. Flow and transport in hierarchically fractured rock. Journal of Hydrology, 263, 1-22.
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GIONA, M. & ROMAN, H.E. 1992. Fractional diffusion equation for transport phenomena in random media. Physica A, 185, 87-97. LE BORGNE, T., BOUR, O., DE DREUZY, J.-R. & DAVY, P. 2003. Flow model relevant to fractured crystalline aquifers: insights from a scaling interpretation of pumping tests. In: KRAZNY, J., HRKAL, Z. & BRUTHANS, J. (eds) Groundwater in Fractured Rocks. Proceedings of the IAH International Conference, Prague, Series on groundwater 7, UNESCO, 263-264. LE BORGNE, T., BOUR, O., DE DREUZY, J.R., DAVY, P. & TOUCHARD, F. 2004. Equivalent mean flow models for fractured aquifers: Insights from a pumping tests scaling interpretation. Water Resources Research, 40, W03512 (doi: 10.1029/2003WR002436). METZLER, R., GLOCKLE,W.G. & NONNENMACHER,T.F. 1994. Fractional model equation for anomalous diffusion. Physica A, 211, 13-24. O'SHAUGHNESSY, B. & PROCACCIA, I. 1985. Analytical solutions for diffusion on fractal objects. Physical Review Letters, 54, 455-458. RIEMANN, K., VAN TONDER, G. & DZANGA, P. 2002. Interpretation of single-well tracer tests using fractional-flow dimensions. Part 2: A case study. Hydrogeology Journal, 10, 357-367. SCHUBERT, S. 1999. Random walks in complex systems - anomalous relaxation. PhD thesis, TU Chemnitz, Germany. SCHULZKY, C., ESSEX, C., DAVISON, M., FRANZ, A. & HOFFMANN, K.H. 2000. The similarity group and anomalous diffusion equations. Journal of Physics A: Mathematical and General, 33, 55015511. SEEGER, S., FRANZ, A., SCHULZKY,C. & HOFFMANN, K.H. 2001. Random walks on finitely ramified Sierpinski carpets. Computer Physics Communications, 134, 307-316. TARAFDAR, S., FRANZ, A., SCHULZKY, C. & HOFFMANN, K.H. 2001. Modelling porous structures by repeated Sierpinski carpets. Physica A, 292, 1-8.
Models of tracer breakthrough and permeability in simple fractured porous media P. B. J O H N S T O N 1'4, T. C. A T K I N S O N 1'3, N. E. O D L I N G 2 & J. A. B A R K E R ~
IDepartment of Earth Sciences, University College London, London WC1E 6BT, UK 2School of Earth Sciences, The University of Leeds, Leeds LS2 9JT, UK 3School of Environmental Sciences, University of East Anglia, Norwich NR4 7TJ, UK 4Enviros, 20-23 Grenville Street, Farringdon, London EC1N 8SS (e-mail: [email protected]) Abstract: Tailing and bimodal behaviour of tracer breakthrough curves from tracer tests
conducted in fractured porous media are commonly presented as 'deviations' from the Fickian model for homogeneous porous media. Tailing is mainly attributed to: (1) tracer storage brought about by diffusion between mobile and static regions of fluid; (2) a concentration of flow towards the wider (aperture) and, thus, more permeable fracture zones; and (3) the high variance in fracture conductivity and consequent mixing of tracer. Bi- (or multi-) modality has been attributed to solute following a few highly permeable flow paths. Systematic numerical simulations of flow and transport in geometrically simple fractured porous media were conducted using a 2D finite difference flow code and a particle tracking transport model. As a simplification only advective dispersion was considered. The modelling study produced a large variety of tracer breakthrough curves, including two patterns commonly seen in field data - the backward tailed uni-modal type and the Gaussian type. The study demonstrates that different types of breakthrough might be characteristic of particular sets of conceptual models for heterogeneities and, as such, may provide a useful pointer in the application and interpretation of tracer tests.
The role of this paper is to explore, within a simplified modelling framework, the prospects for understanding characteristics of the internal heterogeneities in a medium from evidence provided by tracer experiments. Field tracer experiments give rise to a variety of tracer breakthrough curves showing distinct characteristics which can be classified into four general types 9 9 9 9
Gaussian backward tailed bimodal multimodal.
The Gaussian-type curve is typical of a homogeneous and isotropic formation. The other types are thought to arise from flow in more heterogeneous formations. How do non-Gaussian breakthrough curves arise? All of these curves are frequently presented as 'deviations' from the Fickian model for homogeneous porous media, as described by the 1D Advection Dispersion Equation (ADE). This assumes that the centre of the mass of the tracer plume moves with the
average fluid velocity, and that dispersion behaves macroscopically as a Fickian diffusive process, with the dispersivity being assumed constant in space and time (Kosakowski et al. 2001). The Fickian approach uses dispersivity as a parameterization of local heterogeneities in the flow field that arise due to such factors as spatial variations in pore sizes and viscous shear within pores. It is intrinsic in this 'classical' approach that the heterogeneities be small with respect to the region of interest and that the integrated effect of the local deviations for migration be normal with zero mean. The variety of nonGaussian breakthrough curve types results from situations in which this is not true, for example where heterogeneities in permeability fields may be similar in spatial scale to the region of interest. Here, fractures are a common example. In fact, the rate at which a tracer plume spreads is rarely constant and dispersive behaviour often changes with time and/or distance travelled by the plume (Gelhar et al. 1992). This is especially true in fractured porous media, where flow can take place both through the fine-scale pore network (matrix) and through
From: SHAW,R. P. (ed.) 2005. Understanding the Micro to Macro Behaviour of Rock-Fluid Systems. Geological Society, London, Special Publications, 249, 91-102. 0305-8719/05/$15.00 ~) The Geological Society of London 2005.
92
P.B. JOHNSTON E T AL. test. The breakthrough curve is an example of multiple peaks and backward tailing behaviour. Figure lb is a breakthrough curve from a weak-dipole gradient tracer test conducted in the crystalline bedrock at the Mirror Lake site, New Hampshire, USA (Becker & Shapiro 2000), consisting of relatively less densely fractured pelitic schist which is intruded by more densely fractured granite. The test was conducted over a distance of 36 m. There is one broad peak and the breakthrough curve is skewed at late time (or large pumped volume). For reasons explained later, Becker concludes that it is unlikely that the tailing was caused by a purely diffusive transport
the fractures, which if greatly more permeable than the matrix will tend to act as channels. The following examples of tracer tests from the published literature show some typical breakthrough curve forms. Figure 1a is a breakthrough curve from a forced gradient tracer test conducted in the Triassic sandstone aquifer of Liverpool, UK (Streetly et al. 2002). The test was conducted over a distance of 5 m. There are two distinct peaks, at approximately 0.1 relative time units (corresponding to 2 days) and at 1 relative time units (corresponding to 23 days), with smaller multiple peaks occurring throughout the duration of the
(a) Fractured Sandstone 5 metres Radial Flow
2.~"d ~., Ld j~l~ ~,
--Bactcwardtailing
peaks
Multiple 9
,
r
2 3 4 RelativeTime: tl(~m~ to max]
1 (b)
i 1.4d /,Uni~. m~al,~.~.~Bac~
Fractured Crystalline metamorphic rock
Weak dipole flow 36 metres d tailing
3
5
,
~
,'1
~
~
Re~atlveTime: tf(11meto maxl
(c)
.04 d ~~'-" ~\ -modal
,~
~
\..~_..~_ 0
KarsticChalk 2000 metres
Natural Gradient
~ ~3d
....
o
2
3
4
5
6
R e ~ v e 1~e: t~t~me to max]
Fig. 1. Examples of tracer tests from the published literature: (a) multiple peaks; (a), (b) backward tailing behaviour; (c) bimodal behaviour.
MODELS OF TRACER BREAKTHROUGH mechanism and is likely to be a result of fracture flow channelling. This breakthrough curve is an example of backward tailing behaviour. Figure lc is a breakthrough curve from a natural gradient tracer test conducted in the Cretaceous Chalk aquifer, Northern Ireland (Barnes 1999). The test was conducted over a distance of 1540 m, with a head gradient of 45.5 m km -~. There are two distinct peaks, a large peak at 1 relative time units (corresponding to 25 hours) and a smaller peak at 3.7 relative time units (corresponding to 80 hours). The curve shows that little dispersion occurs within the aquifer, suggesting to the author that flow may be channelled through a limited number of fractures, which represent a subset of the entire network. The breakthrough curve is an example of bi-modal behaviour. Many authors have attributed backward tailing, as seen in Figures l a, b, to the diffusion of mass from the relatively mobile fluid in the fractures into the relatively immobile fluid in the rock matrix, known as matrix diffusion, (Grisak & Pickens 1980; Tang et al. 1981; Sudicky & Frind 1982; Moench 1995; Park et al. 1997). Others have attributed the tailing to the diffusion of mass into stagnant zones along the fracture wall (Raven et al. 1988), or into low flow regions associated with smallaperture spaces within the fracture channel (Abelin et al. 1991; Kunstmann et al. 1997). Bi-modal or multi-modal peaks, often in combination with backward tailing, as seen in Figures 1a, c, have been attributed to multiple flow channels (Neretnieks et al. 1982; Cacas et al. 1990; Neretnieks 1993; Nordqvist et al. 1996; Park et al. 1997; Tsang & Neretnieks 1998). The early time bi-modal peaks are likely to be caused by solute following a few highly permeable flow paths. Tailing is likely to be caused by a concentration of flow towards the wider and, thus, more permeable fractures and by the high variance in fracture aperture and, thus, fracture conductivity. Kosakowski et al. (2001) pointed out that it is the interaction (or mixing) of solute between high velocity flow paths and low velocity regions that often leads to non-Fickian transport behaviour. The question of the relative importance of matrix-diffusion-dominated- to flowchannelling-dominated transport arises and the importance of the relative velocities between the high permeability fractures and the low permeability matrix. To investigate the importance of matrix diffusion, multiple tracers with varying rates of molecular diffusion have been used by, for example, Garcia-Gutierrez et al. (1997), Jardine et aL (1999) and Becker &
93
Shapiro (2000). The separation of the breakthrough curves suggests that the transport is dependent upon molecular diffusion. According to Becker & Shapiro (2000) the most commonly cited field evidence of matrix diffusion is an extended tail on a single tracer breakthrough curve. In a rock such as chalk, where the matrix porosity is typically between 20% and 45%, matrix diffusion plays an important role in the dispersion of a tracer, as shown by Gamier et al. (1985), who conducted field experiments in a fractured chalk and found a clear separation of the breakthrough curves for different tracers with different diffusion coefficients. It is well known that molecular diffusion does occur in crystalline rock, as shown by Birgersson & Neretnieks (1990), who conducted an in situ diffusion experiment in a granitic rock. The porosity of a crystalline rock, however, is so small it is considered negligible. As a result, it is unlikely to play such a major role, compared with the Chalk, in mass transfer on the time-scale of tracer tests. Becker & Shapiro (2000) conducted a multi-tracer experiment in a fractured crystalline bedrock and found that there was no clem" separation of the breakthrough curves especially at late time, implying that the backward tailing was not caused by any purely diffusive transport mechanism. Becker concluded that it is more likely that channelling is the controlling mechanism in tracer behaviour. Garcia-Gutierrez et al. (1997) also conducted multi-tracer experiments in fractured crystalline bedrock but found a clear separation of the breakthrough curves and concluded that diffusion into the crystalline rock played an important role. Becker & Shapiro (2000) pumped tracer for approximately 23 days over a distance of 36 m. Garcia-Gutierrez et al. (1997), on the other hand, pumped tracer for 50 days over a distance of 20 m. The flow rate was nearly five times greater for Becker & Shapiro than for Garcia-Gutierrez et at. The flow rate controls the tracer travel time and, thus, the time available for tracer to diffuse into the matrix or other low permeability zones. Thus, Becker & Shapiro (2000) may not have observed diffusion because of the shorter test time. In addition, the matrix porosity and fracture aperture affect the mass of tracer that can be diffused into low permeability zones. Large fracture apertures and low matrix porosities will tend to increase the characteristic time required for diffusion to affect tracer migration significantly. This article aims to show that 9
simulating the introduction of simple fracture patterns into a statistically homogeneous
94
P.B. JOHNSTON E T AL. porous medium (the spatial variation of the hydraulic conductivity of the medium being insignificant on the scale of the region of interest) can produce a variety of tracer breakthrough curves which encompass the commonly observed curve forms; the different types of breakthrough are characteristic of particular sets of conceptual models for heterogeneities.
Under certain conditions advective processes can dominate groundwater flow and mass transport. As a simplification, mass transport is assumed to be via advective processes only. In addition, the fracture aperture is assumed constant for each simulation.
Modelling approach A 2D finite difference, discrete fracture model (Odling & Webman 1991) for flow in fractured rocks was used to simulate the flow field and tracer transport (Odling & Roden 1997). This used an advection-biased, random-walk technique, through fracture networks in which dispersion due to advection was the only process giving rise to the tracer dispersion. In this flow model, both the fractures and the rock matrix are discretized on a regular square grid. Grid elements are assigned permeabilities representative of either fractures or rock matrix. The rock matrix can be thought of as a porous medium, such as sandstone or chalk, or a highly dense network of connected fractures in a crystalline rock, such as a granite. The flow code is capable of coping with very high permeability contrasts that arise between fractures and embedding porous medium. The reader is referred to Odling & Webman (1991) for a full description of the model. Odling & Roden (1997) conducted a series of modelled transport tests on a simple en e c h e l o n fracture pattern, and on an unconnected and a connected natural fracture pattern, measured from outcrop from a shale and sandstone formation, respectively. Odling & Roden (1997) concluded that, for a fractured porous medium, connectivity of the fracture network may play a secondary role to fracture orientation and density in terms of flow and transport. In this article, the work by Odling & Roden (1997) has been developed further by systematically altering the fracture network and hydraulic properties of simple simulated fracture patterns in order to investigate the relationship(s) between fracture aperture, spacing (or density) and angle to the direction of hydraulic gradient on unconnected fracture patterns.
A series of experiments was conducted across a square region, with a tracer injected evenly along the cells of the up-gradient boundary. The breakthrough was monitored across all the cells of the down-gradient boundary. The setup is analogous to either a natural gradient twowell tracer test in the vertical plane or flow between two canals or adits in the horizontal plane (Fig. 2). Tracer tests used to characterize the transport properties at sites for proposed underground nuclear waste repositories, such as in the Stripa mine, Sweden (Olsson & Gale 1995), frequently involve tracer migration in the horizontal plane. An equal number of particles was injected at each node of the up-gradient boundary, irrespective of the type of node (fracture or matrix node). If it is assumed, as here, that flow parallel to the inflow boundary is negligible, a constant mass injection boundary is considered appropriate. If flow parallel to the boundary is not considered negligible, a constant concentration injection boundary is more appropriate. In fact, the constant head condition imposed on the boundary ensures that locally all flow is perpendicular to the boundary. The top and bottom boundaries were either periodic or impermeable. A periodic boundary was used to reduce the influence of impermeable upper and lower boundaries on the flow field for the continuous and e n e c h e l o n fracture patterns (see below). For the continuous and en e c h e l o n cases the fractures were wrapped. This means that the fracture ends were positioned so that the fracture end leaving the model on the upper boundary was at the same x-coordinate as the fracture end entering the model on the lower boundary. This ensured that fracture continuity
Fig. 2. Boundaryconditionsand analogyto vertical section between two boreholes.
MODELS OF TRACER BREAKTHROUGH was maintained and particles leaving the upper boundary reappeared at the same horizontal position on the lower boundary and vice versa. The models were constructed in a series of steps. An uncorrelated and isotropic statistically homogeneous porous medium was discretized onto a grid of 200 • 200 elements. This was the minimum size at which the bulk permeability of the model was equal to the geometric sample mean of the distribution of matrix values. The deviations from the mean are negligible when compared with the fracture to matrix permeability contrast. An average matrix permeability was chosen - in this case a permeability of 10-15m2 was used, a typical value for the Chalk matrix (Allen et al. 1997). The matrix permeability for each node was chosen from a lognormal distribution, with a standard deviation of In permeabilities of 0.57, a value close to that for the Chalk matrix (J. Bloomfield, British Geological Survey, pers. comm.). Granite, due to the presence of microfractures, and sandstone permeabilities are slightly hi~her and lie within the range 10 -13 to 10-14m "~ (Freeze & Cherry 1979; Odling & Roden 1997). A constant porosity of 20% is assumed throughout. The porosity causes the breakthrough curve to shift along the time axis without changing the form of the curve. Although the porosity used here is atypical of the Chalk it does not detract from the validity of the findings. Simple fracture patterns (for an explanation of their 'real' representation see below) were superimposed on the porous medium, consisting of: 9 a uniformly spaced continuous single set (Pattern A, Fig. 3a); 9 a uniformly spaced en e c h e l o n single set with small overlap (Pattern B, Fig. 3b); 9 a uniformly spaced en e c h e l o n single set with large overlap and lower fracture density (Pattern C, Fig. 3c); 9 a uniformly spaced varying coverage single set (Pattern D, Fig. 3d). The flow direction was allowed to vary according to the boundary conditions. The varying coverage was simulated by randomly choosing locations along the fracture length using a linklist technique, to disallow overlap and replication of the fracture locations. A fracture coverage of 75% was simulated, with 100% representing a fully percolating fracture. The fracture patterns were generated initially over the entire grid area and, in order to avoid a lower fracture density close to the boundaries of the flow domain, the largest rotatable square was chosen for the flow and transport simulations.
95
Patterns A and D might represent either bedding plane fractures or joints, depending on whether the plane of the 2D model is considered as being perpendicular to bedding (fractures are bedding plane fractures) or in the plane of the bedding (fractures are joints). Patterns B and C have greatest similarity to single sets of vertical joints so that the 2D model plane may be considered as horizontal for these two patterns. The simulated patterns most resemble simplified bedding plane fractures and joints in stratabound formations. For each pattern the aperture, spacing and orientation with respect to the direction of hydraulic gradient were altered systematically. The apertures used were 0.01, 0.04, 0.1 and 1 mm (typical for near-surface conditions), which, for a unit pressure gradient, correspond to fracture-element-to-matrix-element permeabilitys ratios of 8.3 • 10 4, 1.3 x 105, 8.3 • 10- and 8.3 • 10 7. Below 0.01 mm the model behaves as if no fracture network is present. An aperture of 1 mm is considered somewhat less than the upper limit of fracture apertures as observed in Chalk, granite and sandstone outcrops and borehole logs. Reeves (1979) reports Chalk apertures of 0 . 5 - 4 0 m m and Bloomfield (1996) reports Chalk apertures of 0 . 5 - 2 0 m m but that only 10-20% of the enlarged bedding plane fractures have high flow rates. These large apertures have been enhanced by dissolution of the Chalk matrix. Patsoules & Cripps (1989) report Chalk microfractures with apertures between 0.1 mm and 0.6 mm and Bahat (1989) reports Chalk apertures of a few millimetre or less. Johnson & Dunstan (1998) provide detailed logs of 40 boreholes. The boreholes penetrate igneous and metamorphic rock, granite and schist. The majority of the fractures have apertures of 5 mm or less. It is clear that even though large ( > 1 ram) apertures are present, they are not ubiquitous. To take into account the effects of the partial contact of fracture walls, due to compressive stresses, a fracture porosity of 70% was assumed. This corresponds to the contact area found under relatively low normal stresses (Vickers et al. 1992), as expected for shallow aquifers. The permeability of a fracture node is described using the cubic law. Thus, by varying the fracture aperture, the bulk permeability of the fracture system is also varied systematically. The fracture angles used were parallel to, or rotated by 22.5 ~, 45 ~ and 67.5 ~ with respect to the direction of the hydraulic gradient. The spacing was varied from 2.5m (Spacing 1) to approximately 0.2m (Spacing 4), with Spacing 2 half the spacing of Spacing
96
P.B. JOHNSTON E T A L .
(a)
Pattern A
III 84 II
..... I
{b)
Pattern B
(d)
Pattern D
I[HI
A
Flew~
(c)
Pattern C
,qDl~ m
2~ALL
I
A
Mlmix
tl~mll~ow~ Fig. 3. Examples of rotated fracture patterns, (a) and (d) with 0 ~ and (b) and (c) with 45 ~ rotations. (a) Pattern A: a uniformly spaced continuous single set; (b) Pattern B: a uniformly spaced en echelon single set with small overlap; (c) Pattern C: a uniformly spaced en echelon single set with large overlap; (d) Pattern D: a uniformly spaced varying coverage single set.
1 and Spacing 3 half the spacing of Spacing 2. The crystalline rock at Stripa has fracture spacings on the order of 1 - 3 m (Nordqvist e t al. 1996), a range which overlaps the fracture
spacings of 0 . 2 - 2 . 5 m used in this study. The grid size for the background h o m o g e n e o u s permeability is 0.025 m. For each realization the fractures have a constant aperture.
MODELS OF TRACER BREAKTHROUGH The flow code was used to calculate the flow field for each fracture pattern and to determine the bulk permeability in the general direction of the hydraulic gradient. One thousand particles per node were released (sufficient to ensure the identification of the main peaks and reproducible results) and monitored at their exit, that is, at the whole outflow boundary, to provide breakthrough curves in the form of numbers of particles arriving within short, predetermined time intervals. One hundred time bins were used in all of the experiments, with duration chosen so that the peak of the Fickian breakthrough curve from the matrix always occurred in the same time bin.
Results Approximately 1000 patterns were simulated. The great majority of breakthrough curves fell into one of five distinct classes, although transitional cases between the classes could also be recognized. Figure 4a shows the generalized form of the five distinct classes. Figure 4b
97
provides examples of simulated breakthrough curves with some transitional cases: 9 9
9
9
9
Type 1 is a matrix-like curve; Type 2 is a forward tailed matrix-like peak; Type 3 is a bi-modal curve with early breakthrough and peak, plus a matrix-like peak at later time; Type 4 is an L-shaped curve with very early breakthrough, but with a rise to the peak and backward tailing; Type 5 is an example of an L-shaped curve with very early breakthrough and peak and backward tailing.
Figure 5 a - d shows how the breakthrough curve type varies across an orthogonal variable space defined by the fracture aperture, spacing and angle. The data are arranged into a three-dimensional array, representing aperture, spacing and angle, and each dimension contains four elements, each containing a number to represent the type curve. The cube has been separated into vertical slices to emphasize how the type curve
(a)
Type 1 Type 2 Type 3 Type4 Type5 (b)
35000 30000 03
!l /
CD
25000 -
O~ Q. O c~
E z
--Type,I
Vi
,,,,,,--Type 21
20000 15000
,
|
---Ty .....
4 t
10000 5000 0 0
10
20
30 40 Time (Dimensionless)
50
60
Fig. 4. (a) Form of the five distinct classes of breakthrough curve. (b) Examples of simulated breakthrough curves produced by the modelling exercise, including some transitional cases.
P, B. JOHNSTON E T AL.
98
(a) Pattern A
Angk~67~$ d e g l ~
/~eing,
4
1
<=
1
2
Paean.
An~le~15 d e ~ r ~
4
Angk~22.5 degre~
.~ .~
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v*,etne
,i
Angle-~t deglx~es
(b) Pattern B A~tgle=6%5degrees
Anglr
tlk.~rees
Type 1 Type2
Type3 Type4
Type5
Fig. 5. Type curve positions for fracture patterns (a) A, (b) B, (c) C and (d) D 75% in spacing, aperture and angle space.
MODELS OF TRACER B R E A K T H R O U G H
99
(c) Pattern
C
AngL~=67.5degrees
"~1 , T " ' , ~ / ' - " ' ~
' ~N,',I.,
.AJagle=22.5d egrt~
}, (d) Pattern D 7 5 %
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4 Angle~ degrees
Type I Fig. 5. Continued.
Type 2 Type3 Type4 Type5
100
P.B. JOHNSTON ET AL. Pattern D (75% coverage) (Fig. 5d): The
changes throughout the space. The results may be summarized in two ways: (1)
(2)
majority of the curves are either of Type 1 or are transitional forms. Many of the transitional curves have an initial rapid rise, indicative of fracture-dominated transport, but the first arrival occurs at relatively late time when compared with Patterns A and B, which is due to the discontinuous nature of the fractures. All of the curves show a matrix-dominated aspect to transport, and the degree to which the matrix controls the transport increases as the angle to the direction of the hydraulic gradient and the fracture spacing increases.
by taking each fracture pattern in turn and examining the relationships between breakthrough curves and geometric parameters; by looking at these same relationships from the perspective of each type of breakthrough curve in turn.
Fracture patterns 1.
2.
3.
Pattern A (uniformly spaced continuous
single set) (Fig. 5a): Pattern A is capable of producing the complete range of type curves. Generally, high apertures produce Type 4 and 5 curves and low apertures produce Type 1 and Type 3 curves. There is some variation with angle - as the angle is increased from zero to 67.5 ~ the type curves change from Type 3 to Type 4 or 5. This effect is more pronounced for large fracture spacings. Pattern B (uniformly spaced, en echelon, single set, small overlap) (Fig. 5b): Pattern B is similar to Pattern A, except when the angle is at 67.5 ~. At this high angle the direction of the en echelon overlap is at the greatest angle to the direction of the hydraulic gradient and the pressure drop across the overlap is not as pronounced when compared with the models with smaller angles of rotation. For the smaller angles the majority of the high aperture curves are of Type 5. The large pressure drop across the fracture overlap causes rapid transport through the intervening matrix and allows Pattern B models to produce Type 4 and Type 5 curves. In a natural system this effect will only be observed if the fracture network is not connected in the third dimension. A set of en echelon vertical joints bounded by horizontal (closed) bedding planes is an example of such a network. Pattern C (uniformly spaced, en echelon, single set, large overlap) (Fig. 5c): The overlap is large and the pressure drop across the overlap is smaller in comparison with the pressure drop across the en echelon fractures in Pattern B. The majority of the curves have 'diffuse flow' characteristics (i.e. Type 1 or Type 3), which result from the higher proportion of transit time spent in the regions between the en echelon fractures in this pattern. For the smallest and intermediate fracture spacing Type 4 curves can be produced.
Breakthrough curves
A second way to summarize the results is to take each type of breakthrough curve in turn and itemize the conditions which may give rise to it. 1.
2.
3.
4.
Type 1: Type 1 curves are produced when the bulk permeability of a fracture pattern is similar to the bulk permeability of the purely matrix case, for example, due to the fractures being small in length and of small aperture, or at high angle to the direction of the hydraulic gradient. Type 2: Type 2 curves are rare, in both the field and the simulations. They are similar to Type 3 curves in that they are an intermediate between Type 1 and Types 4 and 5. Type 3: This bi-modal curve type arises when a substantial proportion of the particles travel mainly in the matrix. Those that travel mainly within a fracture arrive in the first peak and those particles that travel mainly within the matrix arrive in the later peak, with a transit time similar to that of the pure matrix case. Type 3 curves are intermediate in their shape between Type 1 ('matrix-dominated') and Types 4 and 5 ('fracture-dominated'). In terms of the geometric properties of fractures, this transition from curve Type 1, though 3 to 4 and 5 can be produced by progressively changing critical parameters such as spacing and aperture. Although not simulated here, Type 3 curves could also be produced by fractures of varying mean aperture, as seen in natural fracture systems. Type 4: Type 4 curves can be produced by near-percolating fracture patterns or by non-percolating patterns that have extensive fracture coverage. Pattern D produces Type 4 curves, but the first arrival may not occur in the first time bin, due to flow being
MODELS OF TRACER BREAKTHROUGH forced through the matrix between fractures because of their non-percolating arrangement. As discussed previously, this type of effect will only be observed if the fracture network is not connected, in terms of flow, in the third dimension. Type 5: Type 5 curves can be produced by percolating or near-percolating fracture patterns, for example, continuous fractures or an en echelon pattern with small overlap. Type 5 curves are typical of transport within a fracture-dominated system, represented here by a constant fracture aperture within and between the fractures.
Discussion Tailing behaviour was observed by Becker & Shapiro (2000) and could be shown not to be due to diffusion because different tracers, with large differences in molecular weight and, therefore, diffusion coefficients, showed the same tailing. Becker & Shapiro (2000) speculated that this might be due to two different scales of fractures being present - fine fractures forming a 'matrix' and wider-spaced fractures. They showed that under certain conditions advective processes can dominate groundwater flow and mass transport. Therefore, as a simplification, mass transport was assumed in this study to be via advective processes only. In the simulations the degree and form of the non-Gaussian behaviour for a particular curve is produced by three mechanisms of dispersion: the dispersion caused by particles either moving mainly within the matrix; and/or particles interacting continually between the fracture and matrix, a mixing process; and/or particles moving mainly within the fractures. The small-scale dispersion - the spread within any particular peak - is due to the frequent fracture-to-matrix interaction and this can be observed in all of the curve types. The largescale dispersion - the presence of more than one major peak - is due to particles taking different routes through the fracture system. The example curve labelled Type 3 in Figure 4b is an example of large-scale dispersion. Type 4 and 5 curves are commonly observed in tracer tests conducted in fractured porous media (cf. Fig. l a - c ) . It has been shown that these curves can be produced using very simple fracture patterns by varying a range of hydraulic and geometric properties - fracture spacing and aperture, fracture continuity, bulk conductivity and fracture angle.
101
Conclusions The modelling study shows that 2D fractured porous media can produce a large variety of tracer breakthrough curves, including two patterns rather commonly seen in field data - the backward tailed uni-modal, non-Gaussian type and the Gaussian. While field examples may arise from other processes, such as double-porosity diffusion, this study shows that in 2D constructs it is the combination of the fracture spacing, aperture, continuity and angle to the direction of the hydraulic gradient which dictates the type of curve produced and not necessarily the type of fracture pattern. Moreover, the subjective classification scheme shows that the different types of breakthrough may be characteristic of particular sets of conceptual models for heterogeneities. However, only a restricted range of patterns has been investigated and the authors are presently in the process of simulating the effects of more complex patterns.
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FREEZE, R.A. & CHERRY, J.A. 1979. Groundwater. Prentice-Hall, Englewood Cliffs, NJ. GARCIA-GUTIERREZ,M., GUIMERA,J., LLANO, A.Y.D., HERNANDEZ, A., HUMM, J. & SALTINK, M. 1997. Tracer test at E1 Berrocal site. Journal of Contaminant Hydrology, 26, 179-188. GARNIER, J.M., CRAMPON, C., PREAUX, G., POREL, G. & VREULX, M. 1985. Tracage par 12-C, 2-H, I- et uranin dans la nappe de la craie senonienne en encoulement radial convergent. Journal of Hydrology, 78, 379-392. GELHAR, L.W., WELTY, C. & REHFELDT, K.R. 1992. A critical review of data on field-scale dispersion in aquifers. Water Resources Research, 28, 1955-1974. GRISAK, G.E. & PICKENS, J.F. 1980. Solute transport through fractured media, 1. The effect of matrix diffusion. Water Resources Research, 16, 719730. JARDINE, P.M., SANFORD, W.E., Gwo, J.P., REEDY, O.C., HICKS, D.S., RIGGS, J.S. & BAILEY, W.B. 1999. Quantifying diffusive mass transfer in fractured shale bedrock. Water Resources Research, 35, 2015-2030. JOHNSON, C.D. & DUNSTAN,A.H. 1998. Lithology and fracture characterization from drilling investigations in the Mirror Lake area, Grafton County, New Hampshire. USGS Toxic Substance Hydrology Program - Water Resources Investigation Report 98-4183. KOSAKOWSKI,G., BERKOWITZ,B. & SCHER, H. 2001. Analysis of field observations of tracer transport in a fractured till. Journal of Contaminant Hydrology, 47, 29-51. KUNSTMANN, H., KINZELBACH, W., MARSCHALL, P. & LL G. 1997. Joint inversion of tracer tests using reversed flow fields. Journal of Contaminant Hydrology, 26, 215-226. MOENCH, A.F. 1995. Convergent radial dispersion in a double-porosity aquifer with fracture skin: Analytical solution and application to a field experiment in fractured chalk. Water Resources Research, 31, 1823-1835. NERETNIEKS, I. 1993. Solute transport in fracture rock Applications to radionuclide waste repositories. In: BEAR, J., TSANG, C.F. & MARSILY, G. de (eds) Flow and contaminant transport in fractured rock. Academic Press, San Diego, 39-127. NERETNIEKS, I., ERIKSEN, T. & TAHTINEN, P. 1982. Tracer movement in a single fissure in granitic rock: some experimental results and their interpretation. Water Resources Research, 18, 849-858. NORDQVIST, A.W., TSANG, Y.W., TSANG, C.F., DVERSTORP, B. & ANDERSSON, J. 1996. Effects of high variance of fracture transmissivity on -
transport and sorption at different scales in a discrete model for fractured rocks. Journal of Contaminant Hydrology, 22, 39-66. ODLING, N.E. & RODEN, J.E. 1997. Contaminant transport in fractured rocks with significant matrix permeability, using natural fracture geometries. Journal of Contaminant Hydrology, 27, 263-283. ODLING, N.E. & WEBMAN, I. 1991. A 'conductance' mesh approach to the permeability of natural and simulated fracture patterns. Water Resources Research, 27, 2633-2643. OLSSON, O. & GALE, J.E. 1995. Site assessment and characterization for high-level nuclear waste disposal: results from the Stripa Project, Sweden. Quarterly Journal of Engineering Geology, 28, S17-$30. PARK, C., VANDERGRAAF, T.T., DREW, D.J. & HAHN, P. 1997. Analysis of the migration of nonsorbing tracers in a natural fracture in granite using a variable aperture channel model. Journal of Contaminant Hydrology, 26, 97-108. PATSOULES, M.G. & CRIPPS, J.C. 1989. Survey of macro and micro-fracturing in Yorkshire chalk. In: Proceedings of the International Chalk Symposium. Brighton Polytechnic. Thomas Telford Ltd, London, 87-93. RAVEN, K.G., NOVAKOWSK1,K.S. & LAPCEVIC, P.A. 1988. Interpretation of field tracer tests of a single fracture using a transient solute storage model. Water Resources Research, 24, 20192032. REEVES, M.J. 1979. Recharge and pollution of the English Chalk: some possible mechanisms. Engineering Geology, 14, 231-240. STREETLY, H.R., HAMILTON, A.C.L., BETTS, C., TELLAM, J.H. & HERBERT, A.W. 2002. Reconnaissance tracer tests in the Triassic sandstone aquifer north of Liverpool, UK. Quarterly Journal of Engineering Geology, 35, 167-178. SUDICKY, E.A. & FRIND, E.O. 1982. Contaminant transport in fractured porous media: analytical solutions for a system of parallel fractures. Water Resources Research, 18, 1634-1642. TANG, D.H., FRIND, E.O. & SUDICKY,E.A. 1981. Contarninant transport in fractured porous media: Analytical solution for a single fracture. Water Resources Research, 17, 555-564. TSANG, C. & NERETNIEKS,I. 1998. Flow channeling in heterogeneous fractured rocks. Review of Geophysics, 36, 275-298. VICKERS, B.C., NEUMAN, S.P., SULLY, M.J. & EVANS, D.D. 1992. Reconstruction and geostatistical analysis of multiscale fracture apertures in a large block of welded tuff. Geophysical Research Letters, 19, 1029-1032.
Fabric development and the smectite to illite transition in Upper Cretaceous mudstones from the North Sea: an image Analysis Approach R. H. WORDEN 1, D. CHARPENTIER 1'2, Q. J. FISHER 3 & A. C. A P L I N 4 ~Jane Herdman Laboratories, Department of Earth and Ocean Sciences, University of Liverpool, 4 Brownlow Street, Liverpool L69 3GP, UK (e-mail: [email protected]) 2UMR G2R - CREGU, BP239, 54506 Vandoeuvre les Nancy, Cedex, France 3School of Earth and Environment, University of Leeds, Leeds LS2 9JT, UK 4NRG, School of Civil Engineering and Geosciences, University of Newcastle, Newcastle NE1 7RU, UK Abstract: In this study, Upper Cretaceous Shetland Group mudstone cuttings from a range of depths in the Northern North Sea, have been studied using X-ray diffraction, mercury porosimetry and electron microscopy. Millimetre to micrometre mudstone textures have been quantified using image analysis of backscattered electron microscope images. Relatively shallow samples (1615 m) have isotropic mudstone fabric (no alignment of clay minerals), are dominated by smectite and have porosity values of approximately 35%. In contrast, more deeply buried samples (3300 m) have developed an anisotropic fabric (distinct alignment of clay minerals), are dominated by illite and have porosity values of approximately 22%. The change in mineralogy is due to smectite replacement by illite, which occurs simultaneously with porosity-loss and fabric development during progressive burial. Image analysis of differentially buried mudstones has proved to be a rapid, flexible and quantitative method for characterizing mudstone textures. The coincidence of mineralogical evolution with textural development and compaction implies that the transformation of smectite to illite occurs by dissolution and precipitation and that chemically facilitated compaction may contribute to porosity loss.
The migration and storage of both aqueous and petroleum fluids in sedimentary basins are controlled significantly by the presence of mudstones, due to their low permeability and high capillarity. However, the aquitard and sealing capacity of mudstones are not predicted simply since a variable combination of detrital and diagenetic controls influence porosity, permeability and capillary pressure characteristics. Muds have very high porosity at the time of deposition (60-90%), decreasing to 1 0 - 2 0 % at depths of several thousand metres. Compaction appears to proceed by the preferential collapse of larger pores, with seemingly little impact on smaller pores (Katsube & Williamson 1994; Dewhurst et al. 1998; Yang & Aplin 1998). Increasing effective stress forces grains and clay aggregates together, resulting in porosity loss and a decrease in pore size (Aplin et al. 1995; Vasseur et al. 1995; Dewhurst et al. 1998). However, the compressibility of
mudstones is also affected by grain size (Aplin et al. 1995; Yang & Aplin 2004), so that porosity loss is also strongly controlled by facies and provenance, as well as clay content and burial depth. It is well known that mineral assemblages in mudstones change progressively as temperature increases during burial and diagenesis. Progressive burial diagenetic mineral reactions in mudstones have been summarized by Hower et al. (1976) and a critical component of the sequence is the transformation of detrital or eogenetic smectite to illite. This reaction is typically thought to occur over the temperature range of 70-100~ although time, temperature, porefluid geochemistry and the rock-fluid ratio can all influence the way in which the reaction occurs (e.g. Freed & Peacor 1989; Huang et al. 1993; Sachsenhofer et al. 1998). Considerable controversy exists about the effective reaction mechanism in natural systems. Reaction mechanisms for smectite illitization Call be classified
From: SHAW,R. P. (ed.) 2005. Understandingthe Micro to Macro Behaviourof Rock-Fluid Systems. Geological Society, London, Special Publications, 249, 103-114. 0305-8719/05/$15.00 9 The Geological Society of London 2005.
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into two major categories (Altaner & Ylagan 1997): (1) (2)
solid-state transformation; dissolution and crystallization.
Solid-state transformation involves the gradual layer-by-layer replacement of smectite by illite and implies that the illite daughter crystals should have similar grain sizes, shapes and fabrics as the parent smectite crystals. Dissolution and crystallization involves dissolution of the parent smectite, followed by nucleation and growth of the daughter illite. This mechanism implies that loss of the morphological characteristics of the primary smectite is likely. The degree of alignment of clay minerals in mudstones can be quantified by determining the anisotropy of the fabric. Anisotropy is determined ideally perpendicular to the maximum stress direction. For mudstones that have not undergone either soft-sediment (plastic) deformation or compressive tectonic deformation, the maximum stress direction typically is approximately parallel to the primary bedding laminations. Primary detrital minerals are typically poorly aligned at the time of deposition (Bennett et al. 1991). Many depositional muds have no distinct clay mineral fabric but it is well known that very high-grade diagenetic to low-grade metamorphic mudstones have welldeveloped fabric anisotropy (Merriman & Peacor 1999; Jacob et al. 2000). The degree of alignment and anisotropy must, therefore, increase during burial, heating and compaction, whatever the initial fabric state. A link has been suggested between smectite illitization and the development of preferred orientation of phyllosilicates in mudstones (e.g. Bjcrlykke 1998; Ho et al. 1999). Because physical properties are related to mineralogy and crystallite size, the smectite-to-illite transition may be accompanied by changes in porosity and permeability. Moreover, fluid flow can be expected to be easier along, rather than across, the direction of aligned phyllosilicate grains. Since highly compacted clay should have a greater degree of grain orientation than non-compacted clay, an anisotropic fabric is expected to result from burial. Under the effect of uniaxial consolidation, clay fabric should result in a relatively increased ratio of horizontal to vertical permeability (Vasseur et al. 1995). Despite the likely importance of the smectiteto-illite transformation, there are few studies that report quantitative data on the relationships between mineralogy, degree of preferred orientation and porosity (but see Ho et al. 1999;
Aplin et al. 2003). Consequently, this study examines Upper Cretaceous Shetland Group mudstones (Hancock 1990) from the Northern North Sea (Fig. 1) to determine the interrelations of mineralogy, fabric and porosity. Mineralogy of the mudstones was determined by X-ray diffraction (XRD) analysis and backscattered scanning electron microscope observations. Porosity was determined using mercury injection porosimetry. The degree of anisotropy of rock fabric in terms of the relative alignment of clay minerals was quantified by analysis of backscattered electron images. Finally, these results were compared with porosity data. One other objective of this work was to assess the feasibility of image analysis of mudstones as a method of quantifying microfabrics.
Background geology and key assumptions The Shetland Group sediments accumulated in the North Viking Graben in the Upper Cretaceous (Fig. 1). They are approximate time-equivalents of the Chalk that is dominant in the Central and Southern North Sea (Hancock 1990; and well data reported here). The sediments of the Shetland Group are dominated by mudstone and are up to 1800 m thick at the present day. The sediments are quite uniform and were deposited below storm wave base at rates of a few tens of metres per million years. The wells sampled in this study were 211 / 13-1 and 211/13-3. The Shetland Group rocks were not cored in these wells so that the study had to utilize cutting samples. Washed and dried cutting samples were collected, with the largest rock fragments being less than 1 cm in diameter. The shallowest sample (1650m) is at about 60~ while the deepest sample at 3316 m is at about 110~ at the present day. Subsidence curves based on stratigraphic analysis show that there have been no major phases of uplift and cooling; the thermal history shows that heating has thus been broadly continuous. Present-day temperatures are the maximum that these rocks have experienced. Clay mineral studies, including the transformation of smectite to illite, have been routinely employed to help ascertain thermal and burial histories of basins. Historically, these studies have relied on the assumption, in some cases explicit although in more cases implicit, that the starting sedimentary material was broadly uniform for mudstones in a given formation in terms of the primary depositional clay mineralogy. In essence, the critical assumption is that the variation of the ratio of smectite to illite is
N O R T H SEA M U D S T O N E FABRIC
63~
5ow
105
7~
Well Location
J
East Shetland Basin
r162
g
60~
~p
,g
Scotland Forties Approach Basin
100 km Fig. 1. Map of the well location in the Northern North Sea including the coastal outlines of Scotland and Norway and the main structural elements of the North Sea basins. Well from 211/13-3 is marked for reference.
a function of present-day depth of burial and is not a result of primary differences in the mineralogy of the clay minerals fed into the basin. The assumption seems to be reasonable in general since the depth-related variation of smectite and illite is consistent in many basins around the world. Moreover, such a uniformity of mineral variation would not be expected if it were a result of primary sedimentological variations since there is no a priori reason why old (deeper) sediments should always have a greater illite/smectite ratio than young (shallow) sediments. As in many other studies, the assumption of uniform sediment input is employed in the analysis of the Shetland Group mudstones. Although there are no published studies of the sedimentology or provenance of this formation to help either prove or disprove this assumption, geophysical logs indicate broadly uniform lithological characteristics.
It is also assumed that these Shetland Group mudstones had similar initial clay mineral fabrics at the time of deposition. This is the only way that one can begin to interpret the variation of clay fabrics as a function of depth of burial. Methods
Physical properties and mineralogy Sub-centimetre-sized washed cuttings samples were disaggregated by freezing and thawing (Yang & Aplin 1997). Porosity was calculated for freeze-dried samples from measured grain densities, plus the mass and bulk volume of samples placed in a mercury injection porosimeter. Clay fractions ( < 2 Ixm grain size) were separated in settling tubes and orientated slides were prepared for XRD using a procedure
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similar to the 'Millipore ':~:' Filter Transfer Method' described by Moore & Reynolds (1997). X-ray diffraction was undertaken using Cu-Kot radiation, with a Hiltonbrooks DG2 Xray generator connected to a Phillips PW1050 goniometer and a Phillips PW1752 single crystal graphite monochromator. The step size was 0.04~ with a counting time of 0.02 ~ s-1 Air-dried, glycolated and 550~ samples were analysed, and the expandability of mixed-layer illite-smectite was estimated from the position of the I/S 002/003 reflection of glycolated samples using a version of the NEWMOD ~ computer program (Reynolds 1985).
Backscattered electron image acquisition Mudstone samples were made into polished sections for textural and mineralogical characterization. The mudstone sections were mounted on glass slides and cut perpendicular to primary bedding structures, as defined by subtle colour changes, assumed to be indicative of minor mineralogical and grain-size differences. Highresolution backscattered electron microscopy (BSEM) was used for the observation of the mudstones, using a Phillips XL30 electron microscope operating at an accelerating voltage of 15 kV with a spot size of 1 txm diameter. Images of the mudstone samples were captured as greyscale backscattered electron images (BEI) at magnifications of x600 and x 1500 (Fig. 2a). The x600 images routinely included silt- and mica-rich layers, as well phyllosilicate-rich layers. The x 1500 images recorded clay-rich layers (i.e. mudstones) away from silt and mica.
Image processing NIH freeware, Scion Image software, was employed to perform image analysis. Untreated images, recorded with scale bars, were imported as bitmap files. As is typical of backscattered electron images (BEI), pores are dark and mineral grains tend to be much brighter. The BEI were inverted to produce greyscale negative images (Fig. 2b). The treated images thus had pore spaces represented by bright pixels and discrete solid grains represented by darker pixels. The first step for image processing was the conversion of greyscale BEI into binary images by taking a density slice whereby the software recognized a user-defined pixel intensity as a cut-off. If all pixels with any brightness greater than black were selected then the original grains in contact with each other simply appeared to be a large irregularly shaped mass using the
Fig. 2. Image analysis method of microfabric determination based on backscattered electron microscope images (in this case, high resolution) and the determination of frequency of alignment and orientation of clays in any particular orientation.
NORTH SEA MUDSTONE FABRIC density slice method. If too low a density were selected then the image was composed of masses of connected objects (clay mineral grains or aggregates) that would be impossible to analyse in terms of orientation and size. If a very high density was selected then the image would hold few objects of ill-defined shape. The selection of pixel-intensity cut-off was thus made so as to optimize the separation of grains, while preserving as much as possible of the grain shape. The same optimized pixel-intensity cut-off was applied to all images in the study to ensure comparability of results. At this stage any grains in contact appeared to form complex aggregates. The second step was to optimize the separation of grains into discrete objects. Thus, the grain aggregates were further separated from each other by using the object erode tool to remove the outer layer of pixels of all aggregates until discrete clay objects could be discerned. Obvious artefact holes in the slides, an inevitable result of working with cuttings of mudstones (especially the most porous shallow samples), were removed from the pore inventory since they are of no relevance to an analysis of grain fabrics in natural mudstones. The result of this process was a modified binary image in which discrete packets of white pixels represented discrete clay mineral grains and black pixels represented the artificially exaggerated pore spaces in the rock. The third step was to use the Scion Image object analyse option to determine the shape, length and orientation of each artificially defined, discrete clay mineral grain. The orientation angle thus corresponded to the angle between the arbitrary horizontal for the image and the main axis of the particle. The orientation data have no absolute Cartesian significance since the original cuttings samples were not orientated. However, the polished sections were cut perpendicular to primary bedding so that orientation relative to bedding could be determined. The number of object-orientation analyses obtained from each image in all directions allowed the production of a rose diagram, using standard graphing software and, thus, the determination of an anisotropy ratio of the sample. The number of objects aligned at 0 ~ 45 ~ 90 ~ and 135 ~ relative to the image was determined, although greater angular resolution could be employed if desired. The anisotropy ratio can be defined as a measure of the degree of fabric anisotropy and was determined by dividing the maximum angular frequency (defined as the orientation with greatest number of objects) by the minimum angular frequency (defined as the
107
orientation with lowest number of objects; Fig. 2c). By definition, a rock with an isotropic fabric, such as a perfectly-sorted sandstone composed of closely-packed spheres, has an anisotropy ratio of one (i.e. it is isotropic). Using the image analysis approach outlined here, a lowgrade metamorphic slate, with very wellaligned phyllosilicate minerals, typically has an anisotropy ratio of about 25. The degree of anisotropy can be visualized further by drawing the best-fitting ellipsoid around the rose diagram which allows definition of preferred orientation in between the specified measurement angles of 0 ~ 45 ~ 90 ~ and 135 ~ From the Scion Image-generated list of grain orientations, it was found to be important to discard the very smallest objects composed of four or fewer pixels, since these introduced artefacts into the dataset. The artefacts resulted from simple geometric rules of nearestneighbour alignment and led to false alignment preferences at 0 ~ 45 ~ 90 ~ and 135 ~ This essential step prevented the generation of spurious orientation preferences. By treating a number of BEI from the same sample, it was possible to assess the natural heterogeneity of the sample and the degree of variability within immediately adjacent portions of rock (Fig. 3). This image analysis approach could be used at a variety of scales from 100 ~m 2 to 4 mm 2 from BEI to several cm 2 from optical images collected from petrographic thin sections.
Results
Porosity, petrography and mineralogy Porosity in Shetland Group mudstones decreases from approximately 35% at 1615 m to 22% at 33i6 m. The change is not entirely smooth with depth but there is a distinct pattern of decreasing porosity with increasing depth of burial (Fig. 4). A porosity of 35% is relatively high for 1615 m but is by no means exceptional for fine-grained, smectite-rich mudstones (Yang & Aplin 2004). All SEM images were obtained in backscattered electron mode on thin sections of cuttings sliced perpendicular to the bedding. The initial conclusion from the petrography is that Upper Cretaceous Shetland Group rocks from the North Sea are mudstones with detrital grains predominantly less than 20 pLm in size and with a large proportion of clay particles. Primary bedding can be discerned from the alignment of localized silt-rich layers. After clay minerals, the most common detrital minerals are quartz, micas and feldspar. Quartz is
108
R.H. WORDEN ETAL. Porosity (%)
0 1500
10
20
30
I
I
I
40
/-
2000E
D 2500
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Fig. 4. Porosity-depth data for the Shetland Group mudstones from well 211/13-3. commonly present in a variety of grain sizes (from 1 txm to more than 50 Ixm). Detrital micas consist of elongated crystals of biotite and muscovite between 10 I~m and 40 Ixm long.
Mineralogy of the < 2 txm fraction
Fig. 3. Image analysis method of the determination of fabric heterogeneity. Images, at any magnification (scale), can be split into discrete areas, each of which can be determined for fabric degree and orientation of clay alignment.
The 001 diffraction peak of illite is present in all samples (Fig. 5), but is most intense in the deepest samples. In the case of the shallowest samples, the smectite 001 trace is located at 5.3 ~ for the glycol-treated preparation. The maximum-intensity smectite peak (001) in the shallowest sample is symmetrical and has a very high intensity relative to the diminutive illite 001 peak. The shallowest sample thus has smectite with a very low proportion of illite layers. In the case of the deepest samples the smectite 001 peak is very small relative to the illite 001 peak. In the deepest sample, the illite 001 peak is broad probably due to the presence of a minor quantity of illite-dominated mixed-layered illite/ smectite. Chlorite and kaolinite are present in all the samples (peaks at about 12 ~ 2-0) but based on their relative X-ray trace intensities are most abundant in the deepest samples.
Quantification of the illite/smectite ratio The proportions of illite in illite/smectite (l/S) are reported in Figure 6. In the shallowest
NORTH SEA MUDSTONE FABRIC
109
12000 smectit~
1615m
10000 8000
6000 4000 2000 ,
,
10
15
,
,
20 25 Degrees~ 0
,
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8000 ,.,., o (D
6000
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E
c O
o 4000
2000
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5
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15
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35
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Degrees ~ 0
Fig. 5. Comparison of X-ray diffraction spectra obtained on the <2 p~mfraction of samples located at 1615 m and 3316 m (shallowest and deepest Shetland Group mudstones in the series).
sample, at a depth of 1615 p~m, illite corresponds to c.25% of the I/S. Between 1771m and 3021 m the illite proportion varies from 40% to 55%, whereas at 3316 m, illite forms 80% of the illite/smectite. The data are insufficient to draw out the detail of the reactions by which smectite is lost and illite gained in these samples, but the general trend from smectiterich to illite-rich is clear.
Clay matrix fabric Clay matrix fabrics from different samples studied by BSEM and image analyses are shown in Figure 7. The image analysis process converted multi-toned backscattered electron micrographs into inverted binary images. Each image was carefully and individually processed
to optimize the separation of mineral grains into discrete objects. It is possible that an image could give different results depending on the integrity of the human intervention. Like any analysis, there are issues of reproducibility (precision). To estimate the reproducibility of the anisotropy ratio data, some images were processed and analysed repeatedly at different times. A~fisotropy ratios determined from the same image (n = 10, mean = 1.4) fall within a range of 1.1 to 1.7, resulting in an uncertainty of approximately ___0.3. The heterogeneity of the mudstone clay matrix fabric was assessed by taking multiple images (e.g. Fig. 3). Each mudstone sample had standard deviations of the anisotropy values from sub-samples of approximately +0.35 representing the natural heterogeneity. Comparison of the intrinsic uncertainty
R.H. WORDEN ETAL.
110
1500
0
Illite in Illite-smectite (%) 20 40 60 80 I
I
I
100
I
2000
E v r-
~_ 2500 D
3000
3500
Fig. 6. Estimation of the illite percentage in mixedlayered illite/smectite (I in I/S) in the North Sea Shetland Group samples as a function of sample depth. The proportions of illite and smectite layers in the mixed-layered I/S mineral have been determined by comparing the diffractogram patterns of the ethylene glycol-solvated samples with patterns obtained on standards. The values given have an error of approximately + 5%.
following image analysis and the natural heterogeneity suggests that they have a similar magnitude. The error bars on the graphic display of anisotropy ratio vs. depth (Fig. 8) thus represent the uncertainty of the anisotropy ratio. Clay minerals can be wrapped locally around relatively rigid silt grains, thus distorting the matrix fabric. The distortions in the vicinity of silt grains owe more to the relative proximity of the silt grain rather than any fabric that might have developed due to realignment or recrystallization in an effective stress field. To illustrate the clay matrix fabric variation with depth, areas of l l001xm 2 (35 ixm x 35 txm) were selected because they were dominated by clay minerals and contained few silt grains (Fig. 7). All of the mudstone samples contained clay matrix. The clay matrix must have been compacted in comparison to the primary depositional fabric since the density-derived porosity values are much lower than typically reported mudstone depositional porosity values of 60-90%. The micrographs in Figures 7a and b correspond to the clay matrix of samples located at 1615 m and 1771 m. The matrix in these samples has
no clear preferred orientation of clay minerals. The micrographs in Figures 7c, d and e correspond to the clay matrix of the samples located at 2073 m, 2868 m and 3021 m. The matrix has minor preferred orientation of clay particles. The micrograph in Figure 7f corresponds to the clay matrix of the sample located 3316m, which shows that the matrix has a reasonably well-developed, preferred orientation of clay particles. The anisotropy ratios of the clay matrix (illustrated in Fig. 7) are presented as a function of depth in Figure 8. They vary from 1.4 to 3.2, thus quantitatively confirming the observed variable degrees of preferred orientation of the clay particles in the previous section and evident in Figure 7. The values confirm the approximately isotropic microscale fabric of the shallowest samples and the distinctly anisotropic fabric of the deepest sample. They also indicate that the clay mineral fabric has evolved from isotropic to significantly anisotropic due to burial from 1615 m to 3316 m.
Discussion Primary mineralogy and fabric It is not possible to study samples that are completely unaffected by diagenesis and burial, but the observation of the shallowest samples indicates that the initial mineralogical composition might have been dominated by smectite-rich clay minerals, with lesser quartz, albite, K-feldspar, biotite and muscovite, carbonate bioclasts, with minor Ti-oxides and apatite. The fabric of the clay-dominated matrix was most likely isotropic at the time of deposition since the samples following burial to 1615 m are effectively isotropic.
Mineralogical diagenesis By integrating SEM observations with XRD data, it is possible to infer the variations of mineralogy and bulk rock geochemistry with increasing depth of burial. The main reaction was the illitization of smectite. This is important in these Cretaceous mudstones since between 75% and 80% of the initial smectite has been transformed to illite by the time the rocks have been buried to 3316 m. Consideration of the stoichiometry of this reaction suggests that there was probably concomitant loss of silica, iron, magnesium and sodium from smectite, probably resulting in chlorite formation (Hower et al. 1976; Lee et al. 1985; Lindgreen et al. 1991).
NORTH SEA MUDSTONE FABRIC
111
Fig. 7. BSEM micrographs of North Sea samples located at (a) 1615 m; (b) 1771 m; (c) 2073 m; (d) 2868 m; (e) 3021 m; (f) 3316 m. The high magnification photographs illustrate the clay matrix (areas of c. 1100 ixm2). The ellipses correspond to the distribution of the particle orientation obtained from image analyses.
112
R.H. WORDEN ETAL.
1 1500
Anisotropy ratio 2 3 I
40 a
4
I
35
"~_ 30 o
2000
25
20
E
v
10
20
i0
50
Illite in I/S %
r"
"" 2500 c~.
i
i
J
i
i
i
i
i
i
(9
a
b
+ + +
O
3000
o "E 2 <
3500 Fig. 8. Estimated anisotropy ratio in the North Sea series as a function of the samples, depth, calculated from the ellipses of orientation distribution (Fig. 7). The values are given + 0.3. F a b r i c evolution
There is a good inverse correlation between porosity and the illite/smectite ratio (Fig. 9a), indicating that the transformation of smectite into illite occurred simultaneously with the loss of porosity. Since both processes occur as a result of increased burial, it is impossible to simply determine if illitization has led to porosity loss. However, chemical compaction is well described in both sandstones and carbonates, occurring as a result of the increase in the rate of mineral reactions as a function of increasing temperature and effective stress. It is at least possible that the loss of porosity during burial in these Shetland Group mudstones is at least partly a result of the diagenetic transformation of smectite to illite. By using image analyses of BSEM observations, it has been possible to quantify partially the evolution of clay mineral fabric with depth. The image analysis work was focused on clayrich areas, since these should show the greatest response to increasing effective stress during burial. At the time of deposition, clay minerals are here assumed to have been aligned randomly and thus have an isotropic fabric. After burial to
+ 10
30
50
60
Illite in I/S %
Fig. 9. Cross-correlation of the derived illite-smectite ratio (from XRD data) with (a) porosity (derived from density measurements) and (b) the anisotropy ratio derived using electron microscope image analysis.
1615 m, the clay minerals show no clear preferred orientation and the fabric is very close to being isotropic (anisotropy ratio c.1.4-t-0.3, where a value of 1.0 represents a totally isotropic fabric). During burial from the surface to 1615 m, the matrix clay fabric has undergone porosity loss with almost no development of an aligned fabric. Thus, even though the principal effective stress will have increased progressively during burial and its direction will have remained constant, the clay minerals have not responded by developing a preferred orientation. Increase in burial depth (and, thus, presumably effective stress) did not lead to a preferred fabric development prior to a substantial change in the proportion of illite in mixed-layer I/S. This is consistent with observations made previously by Aplin et al. (2003) and Charpentier et al. (2003) in smectite-rich deep-water Gulf of Mexico mudstones, in which there was very little fabric realignment at burial depths of 5000 m and porosities of 15-20%. In samples deeper than 1615 m, mineralogical variations imply a series of linked reactions dominated by
NORTH SEA MUDSTONE FABRIC the loss of smectite and growth of illite. In samples from 2070 m to 3020 m, the anisotropy ratio increased slightly (between 1.7 and 2.2), indicating a systematic reorganization of clay minerals coincident with an increased illite/ smectite ratio (Fig. 9b). At 3315 m, the illite percentage in mixed-layered I/S is very high (80%) and the anisotropy ratio is about 3.2. The deepest sample is, thus, both illite rich and has substantially aligned clay minerals. Although the clay matrix fabric evolved from isotropic to moderately anisotropic, the evolution is not continuous with depth. There is a clear increase in the anisotropy ratio from 1.4 at 1650m to 2.2 in less than 4 5 0 m of extra burial. The values remain approximately constant for the next 1000 m, before another increase from 2.2 to 3.2 over a 300 m interval. The anisotropy ratio is correlated broadly to the illite percentage in illite/smectite (Figs 6 and 8), so that the anisotropy ratio increases as the illite percentage increases and remains constant as the illite percentage remains constant. Given the synchronicity of the mineralogical and fabric changes (Fig. 9), smectite illitization is probably connected intimately with the reorientation of clay minerals. The two principal mechanisms proposed for reorientation of clay particles are mechanical rotation of pre-existing grains and dissolution and recrystallization of neoformed mineral perpendicular to the principle effective stress (e.g. Ho et al. 1995). In this case, as in the Gulf of Mexico (Ho et aL 1995; Aplin et al. 2003; Charpentier et al. 2003), it appears that dissolution and recrystallization of netformed minerals is the primary driver for fabric realignment. Continued recrystallization presumably results in the high levels of anisotropy (up to 25) observed in low-grade metamorphic mudstones. A key unknown, which is not resolved in the present study, is the extent to which clay mineral recrystallization reduces porosity beyond the level that can be achieved by mechanical compaction alone. Chillingarian & Knight (1960) suggested that smectitic mudstone can mechanically compact to only c.30-40% porosity, although more recent work has shown that mudstones which have not undergone substantial diagenetic mineral transformations can exhibit porosities of 15% (Aplin et aL 2003). Although the results presented here suggest that mineral transformation may well influence the pattern of porosity loss in mudstones as a function of depth or temperature, more detailed studies are required to confirm and quantify the process. Whilst bulk rock mineralogy (percentage of silty and sandy minerals, and of clay particles)
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undoubtedly has a major impact on mudstone properties (e.g. Dewhurst et al. 1998), this study of North Sea Cretaceous Shetland Group mudstones underlines the potential role of clay mineral diagenesis and chemical compaction in the reorientation of clays (Fig. 9). It is possible that mineralogical reactions - and smectite illitization in particular - can reorientate clay minerals and induce a significant porosity decrease.
Conclusions (1) Upper Cretaceous Shetland Group mudstones from the North Sea show mineralogical variations that have resulted from diagenetic reactions. Relatively shallow samples (1615 m) have approximately 35% porosity, isotropic mudstone fabrics (minimal alignment of clay minerals) and are dominated by smectite. In contrast, more deeply buried samples (3300m) have developed an anisotropic fabric (visually distinct alignment of clay minerals), are dominated by illite and have porosity values of approximately 22%. (2) The dominant geochemical process was the illitization of smectite. At least 60-80% of the smectite has been transformed into illite by burial to 3316 m. (3) Image analysis of backscattered electron micrographs collected using polished thin sections prepared from cuttings samples, has proved to be a useful, rapid and flexible method to gain microstructural data from mudstones. This approach can be used at a range of scales to determine the effects of sedimentary grain alignment and claymatrix fabric evolution during burial and diagenesis. This approach can be used to assess heterogeneity in mudstones, also at a wide range of scales (g~m to cm). (4) In clay-dominated layers, the clay mineral fabric of Shetland Group mudstones changes from almost isotropic at 1615 m to significantly anisotropic at 3316m. Anisotropy only begins to develop when the proportion of illite in illite/smectite increases. Mineralogy and anisotropy track each other closely. (5) The reorientation of clay minerals is probably due to the dissolution of smectite and consequent precipitation of illite aligned in the effective stress field rather than mechanical rotation of pre-existing grains. (6) The transformation of smectite to illite is responsible for the increasing anisotropy. The realignment of clays is concomitant with a substantial loss of porosity.
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Compactional porosity loss has thus been facilitated by g e o c h e m i c a l reactions in the mudstones. This work was supported by NERC grant number NER:T:S:2001:00682. The authors would like to thank Shell for providing samples, Cornelius Veltkamp for assistance with SEM work and Lisa Worrall for advice on the use of the image analysis software. Reviewers Stuart Haszeldine and Dick Merriman are warmly thanked for helping to improve the clarity of the paper, including the explanation of critical assumptions, finer details of the methods and the discussion.
References ALTANER, S.P. & YLAGAN, R.F. 1997. Comparison of structural models of mixed-layer illite/smectite and reaction mechanims of smectite illitisation. Clays and Clay Minerals, 45, 517-533. APLIN, A.C., YANG, Y. & HANSEN, S. 1995. Assessment of [3, the compression coefficient of mudstones and its relationship to detailed lithology. Marine and Petroleum Geology, 12, 955-963. APLIN, A.C., MATENAAR, I.F. & VAN DER PLUIJM, B. 2003. Influence of mechanical compaction and chemical diagenesis on the microfabric and fluid flow properties of Gulf of Mexico mudstones. Journal of Geochemical Exploration, 78-79, 449-451. BENNETT, R.H., O'BREIN, N.R. & HULBERT, M.H. 1991. Determinants of clay and shale microfabric signatures: processes and mechanisms. In: BENNETT, R.H., O'BRIEN, N.R. & HULBERT, M.H. (eds) Microstructure of Fine-Grained Sediments. Springer-Verlag, New York, 5-32. BJORLYKKE, K. 1998. Clay mineral diagenesis in sedimentary basins - a key to the prediction of rock properties. Examples from the North Sea Basin. Clay Minerals, 33, 15-33. CHARPENTIER, D., WORDEN, R.H., DILLON, C.G. & APLIN, A.C. 2003. Fabric development and the smectite to illite transition in Gulf of Mexico mudstones: an image analysis approach. Journal of Geochemical Exploration, 78-79, 459-463. CH1LLINGARIAN, G.V. & KNIGHT, L. 1960. Relationship between pressure and moisture content of kaolinite, illite and montmorillonite clays. American Association of Petroleum Geologists Bulletin, 44, 101-106. DEWHURST,D.N., APLIN, A.C., SARDA,J.P. & YANG, Y. 1998. Compaction-driven evolution of porosity and permeability in natural mustones: An experimental study. Journal of Geophysical Research, 103, 651-661. FREED, R.L. & PEACOR, D.R. 1989. Variability I temperature of the smectite/illite reaction in Gulf Coast sediments. Clay Minerals, 24, 171-180. HANCOCK, J.M. 1990. The Cretaceous. In: GLENNIE, K.W. (ed.) Introduction to the Petroleum geology of the North Sea. Blackwell Scientific Publications, Oxford, 254-272. Ho, N.C., PEACOR, D.R. & VAN DER PLUIJM, B.A. 1999. Preferred orientation of phyllosilicates in
Gulf Coast mudstones and relation to the smectite-illite transition. Clays and Clay Minerals, 47, 495-504. HOWER, J., ESLINGER, E.V., HOWER, M.E. & PERRY, E.A. 1976. Mechanism of burial and metamorphism of argillaceous sediment: 1. Mineralogical and chemical evidence. Geological Society of America Bulletin, 87, 725-737. HUANG, W.L., LONGO, J.M. & PEVEAR,D.R. 1993. An experimentally derived kinetic model for smectiteto-illite conversion and its use as a geothermometer. Clays and Clay Minerals, 41, 162-177. JACOB, G., KISCH, H.J. & VAN DER PLUIJM, B.A. 2000. The relationship of phyllosilicate orientation, X-ray diffraction intensity ratios, and c/b fissility ratios in metasedimentary rocks of the Helvetic zone of the Swiss Alps and the Caledonides of Jaemtland, central western Sweden. Journal of Structural Geology, 22, 245-258. KATSUBE, T.J. & WILLtAMSON, M.A. 1994. Effects of diagenesis on shale nano-pore structure and implications for sealing capacity. Clay Minerals, 29, 451-461. LEE, J.H., AHN, J.H. & PEACOR, D.R. 1985. Textures in layered silicates: Progressive changes through diagenesis and low-temperature metamorphism. Journal of Sedimentary Petrology, 55, 532-540. L1NDGREEN, H., JACOBSEN, H. & JAKOBSEN, H.J. 1991. Diagenetic structural transformations in North Sea Jurassic illite/smectite. Clays and Clay Minerals, 39, 54-69. MERRIMAN, R.J. & PEACOR, D.R. 1999. Very lowgrade metapelites: mineralogy, microfabrics and measuring reaction progress. In: FREY, M. & ROBINSON, D. (eds) Low-Grade Metamorphism. Blackwell, Oxford, 10-60. MOORE, D.M. & REYNOLDS,R.C. 1997. X-Ray diffrac-
tion and the identification and analysis of clay minerals (2nd ed). Oxford University Press, Oxford. REYNOLDS, R.C. 1985. Thermal transformation of smectite to illite. American Association of Petroleum Geologists Bulletin, 69, 300-301. SACHSENHOFER, R.F., RANT1TSCIa, G., HASEMHI3TTL, C., RUSSEGGER, B. & JELEN, B. 1998. Smectite to illite diagenesis in early Miocene sediments from the hyperthermal western Pannonian Basin. Clay Minerals, 33, 523-537. VASSEUR, G., DJERAN-MAIGRE, I., GRUNBERGER, D., ROUSSET, G., TESSIER,D. & VELDE, B. 1995. Evolution of the structural and physical parameters of clays during experimental compaction. Marine and Petroleum Geology, 12, 941-954. YANG, Y.L. & APLIN, A.C. 1997. A method for the disaggregation of mudstones. Sedimentology, 44, 559-562. YANG, Y.L. & APLIN, A.C. 1998. Influence of lithology and compaction on the pore size distribution and modelled permeability of some mudstones from the Norwegian margin. Marine and Petroleum Geology, 15, 163-175. YANG, Y.L. & APLIN, A.C. 2004. Definition and practical application of mudstone porosity - effective stress relationships. Petroleum Geoscience, 10, 153-162.
Fluid velocity fields in 2D heterogeneous porous media: empirical measurement and validation of numerical prediction R A C H E L C A S S I D Y 1, J O H N M C C L O S K E Y 1 & P H I L I P M O R R O W 2
l Geophysics Research Group, School of Environmental Sciences, University of Ulster, Coleraine, Co. Derry BT52 1SA, N. Ireland (e-mail: ri. cassidy @ulster, ac. uk; j.mccloskey @ulster, ac. uk) 2School of Software Engineering, University of Ulster, Coleraine, Co. Derry BT52 1SA, N. Ireland Abstract: The scale invariance of geological material and the consequent absence of a
length scale on which to base the upscaling of measurements made on geological samples represent a serious challenge to the prediction of fluid behaviour in rock at economically interesting scales. Numerical simulation is an important tool for understanding constraints in this problem and current discrete fluid models in which complex boundary conditions can be represented have the potential for testing many possible upscaling schemes. At present, however, there are no accurate empirical data on the distributions of fluid velocities in complex, scale-invariant geometries, with which to validate such models. To address this, fluid velocity fields in complex 2D media with fractal heterogeneity were measured. Digital models of rock geometries were created and translated into physical form using electric discharge machining and stereolithography. These physical models were then enclosed between parallel sheets of glass and Perspex forming a Hele-Shaw cell which was permeated with water, doped with small neutrally buoyant spheres and pumped at accurately steady and reproducible velocities. Local velocity vectors were estimated by the analysis of sequential images captured using high-resolution video. Precision digital control systems were used to move the cell relative to the camera and repeated measurements allowed the construction of full 2D velocity fields. The accuracy of the technique was assessed by comparison between automated and manual measurements, confirming the accuracy over approaching three orders of magnitude in velocity. The results have been compared to the output of lattice Boltzmann (LB) simulations of flow in identical geometries, showing that the correlation between simulated and measured velocity fields is strongly dependent on the viscosity used in the numerical simulation. In particular, the LB scheme used in these tests is incapable of simulating correct viscosities for complex geometries. Some important effects are shown to be strongly viscosity dependent and it is concluded that some simulations may be able to predict the behaviour of high viscosity fluids only. Nonlinear effects between fracture and matrix flow are likely to be more important in these cases.
Though still the subject of some debate, it is widely accepted that, at least in many important situations, heterogeneity in the Earth's crust is scale invariant. In particular, matrix porosity (e.g. Katz & Thompson 1985; Jacquin & Adler 1987), fracture length distributions (e.g. Davy 1993; Somette et al. 1993), the spatial distribution of fracture networks (e.g. Barton 1995; Odling 1997), fracture roughness (e.g. Brown 1987; Power & Tullis 1991; Power & Durham 1997) and aperture (e.g. Belfield 1994; M~heust & Schmittbuhl 2001) have been shown to conform to power-law or fractal frequency-size statistics (see, for example, Bonnet et al. 2001). These properties together largely
control the movement of fluids through the crust and understanding of this control is of primary importance in several economically and socially important areas of earth science, perhaps the subject of most current study being the efficient recovery of hydrocarbon from fractured porous reservoirs. The central aim of many such studies is the development of schemes for the extrapolation or interpolation of material properties across scales. Thus it would be extremely useful if, for example, accurate predictions of the permeability of a reservoir at a length scale of 10 km could made on the basis of samples taken from boreholes (e.g. with a length scale of 10 cm). It would be equally
From: SHAW,R. P. (ed.) 2005. Understandingthe Micro to Macro Behaviour of Rock-Fluid Systems. Geological Society, London, Special Publications, 249, 115-130. 0305-8719/05/$15.00 9 The Geological Society of London 2005.
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useful if the fracture connectivity through a reservoir, often controlled by relatively small structures, could be robustly interpolated from seismic sections which have very limited smallscale resolution. This scale-changing problem is seriously complicated by the scale invariance of material properties. Basically, so that it is possible to model the large-scale properties of a rock body without measuring every feature of that body, it is vital to assume a homogenization scale above which the rock mass properties can be assumed to be uniform. Thus, the average of the property under consideration must converge with scale. In fractal distributions, however, no representative sample size can be defined, there is no convergence of average properties with scale and simple extrapolation is impossible. This lack of a homogenization scale has profound implications for understanding and predicting the behaviour of fluids in the crust. Indeed, attempts to calculate accurately the expected yield from oil reservoirs, to predict and remediate groundwater pollution and to identify permanent, safe repositories for nuclear waste are inhibited by the inherent scale-invariance of many flow-relevant fields. Approaches to the solution of this problem, to finding a method for upscaling in complex, possibly scale-invariant geometries have been varied and field studies, laboratory investigations and numerical modelling have all been used. This paper concentrates on the latter and, in particular, on the validation of numerical simulations which are being used to test upscaling schemes. Numerical modelling of fluid transport in geological materials has become an important method of assessing the importance of complexity and in testing theoretical methods for dealing with the upscaling problem. Unfortunately, models based on continuum methods, such as traditional finite element simulations, rely on the definition of spatial averages and, therefore, must assume the existence of a homogenization scale. Relatively recently, lattice gas (Frisch et al. 1986) and lattice Boltzmann (Benzi et al. 1992) methods, in which a fluid is represented by the movement of particles on a regular grid in two or three dimensions, have been shown to yield solutions to the Navier Stokes equations. The ease with which complex boundaries and boundary conditions can be incorporated in such a lattice make them particularly suitable for solving flow problems in complex materials and it can be argued that they offer the greatest potential for progress in the upscaling problem since, at least in principle, they are not reliant on defining a representative sample size. The validation of
simulations in complex geometries is, however, not straightforward and, while it has been shown that these models do make accurate predictions for flow in simple geometries, their predictions in complex and scale-invariant geometries has not been adequately demonstrated (see Dardis (1998) for a review). Models of flow in complex geometries are, generally, reliant on comparison with existing empirical data for their validation and such data, when available, are generally limited to non-unique, bulk measurements of properties such as permeability and are not adequate to validate complex numeric schemes fully. Here, an experimental system is described which has been developed to measure detailed 2D velocity fields in precisely defined complex physical geometries and an attempt is made to validate a numerical simulation in an identical system. Specifically, the following sections describe a procedure for the production of 2D physical models with precisely defined geometries and a method through which measurements of fluid velocity are made at precise locations in these geometries. The first stage involves the generation of 2D digital models of rock. These digital models are translated into a physical format using stereolithography or wire electric discharge machining (EDM) and encapsulated between transparent sheets to form a HeleShaw cell. The physical models are then placed in an experimental rig and permeated with water, pumped at steady and reproducible rates using a high pressure liquid chromatography (HPLC) pump. The experimental system includes a fully automated positioning system which moves the flowcell relative to a camera and illumination system to analyse small subareas. Velocity measurements are taken at each node of a regular grid using particle image velocimetry (PIV), a technique which uses the sequential imaging and tracking of neutrally buoyant particles to quantify the fluid velocity over a small volume. Schemes are also described which have been used to assess the accuracy of the velocity measurements. Some sample results are then presented and detail provided on the formats in which information about the geometry of the void space is stored, together with the corresponding components of the velocity vectors. The results are then discussed, with particular reference to their limitations and work is described which is presently being carried out with this system. Finally, a velocity field, measured for a fracture with self-affine fractal walls, is used to validate a lattice Boltzmann simulation of flow in the same fracture.
FLUID VELOCITY FIELDS IN 2D POROUS MEDIA
The models Each physical model, whose geometry is defined in accordance with the objectives of a specific investigation, is first represented as a binary image of solid and void space. This Boolean image acts as the basis for numerical simulations while simultaneously providing a template for the production of the physical model using computerized machining and rapid prototyping techniques. The results allow the comparison of predictions of any particular numerical scheme with measurements of the velocity field in an analogous physical model. The basis of the physical model is a HeleShaw cell, a pair of rigid, transparent plates separated by a narrow gap, used to examine fluid behaviour under quasi-two-dimensional conditions. In this application a 2 mm thick representation of the digital image is placed between the plates, allowing fluid motion throughout the void space to be quantified by saturating the model with a fluid containing reflective micro-particles and measuring the temporal displacement of particles. The physical model is generated in two ways. Simpler geometries with fewer component parts (e.g. Fig. la) are cut from aluminium plate by wire EDM. Geometries consisting of many unconnected elements, such as a model of rock matrix where numerous individual grains must be represented (e.g. Fig. lb), are produced in plastic directly
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onto a glass plate by stereolithography. Both techniques ensure reproduction of the numerical model in a physical format as a 200 x 200 mm transparent flowcell containing a 2 mm thick layer of either resin or anodized aluminium matching the numerical geometry defined at the outset.
Model production using wire EDM The geometry of a profile, taken across a fracture surface is well described by a self-affine fractal trace. The log-log plot of the power spectral density against spatial frequency for a selfaffine trace has the slope/3, which relates to the fractal dimension D for the trace as (5 - / 3 ) / 2 . Similar power-law relationships have been quantified in outcrop roughness with relationships spanning 10 - 3 - 1 m for field measurements and 1 0 - 5 - 1 0 - 2 m for laboratory measurements (Brown & Scholz 1985). Power et aL (1987) reported fractal scaling of roughness extending over 11 orders of magnitude by combining measurements of the power spectral density of fault surfaces over the wavelength of 10 - 5 1 m and measurements of the roughness of the San Andreas Fault. A self-affine fractal geometry such as this was chosen as one of the models of complexity. It should be pointed out, however, that the detail of choice of rule for producing heterogeneous models is not a central concern of
Fig. 1. Numerical geometries: (a) a digital model of three fracture apertures with different offsets, used as the basis for production of the physical models using wire EDM; (b) a porous-fractured geometry, which contains many small sections, is produced using stereolithography.
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this work. This study need only guarantee a complex structure so that tests are made of the predictive capability of any numerical prediction. A self-affine fractal satisfies this fundamental criterion. The trace is produced following the method described in Turcotte (1993). First n elements of a one-dimensional array are filled with Gaussian-distributed random numbers (hn). A discrete Fourier transform is then taken of these values to map the n real numbers (hn) into n complex numbers (14,,). The coefficients are given by N--I
Hm -- AT ~ hn e2~nm/N n=O
(1)
The transform of the set produces a flat spectrum, where all amplitudes are equal within the statistical scatter. This flat spectrum must then be modulated by a given power law by filtering the resulting Fourier coefficients H,, using the relation /3
Fig. 2. Physical models consisting of large segments. These are cut in aluminium sheet using wire EDM and assembled manually between transparent plates. The six aluminium sections, which form this model, are displaced and offset to match the digital model shown in Figure la, cut to 180 mm lengths and assembled between Perspex plates.
''m---An inverse discrete Fourier transform is then taken of the filtered Fourier coefficient to generate the trace. The sequence of points is given by
Model production using stereolithography
N-1
z h'n
1 ~H1me_2~nm/N NATm=o
comparison with numerical simulations of the flow fields.
(3)
These points constitute the fractional Brownian noise and must then be normalized to the scale of interest and a 300 mm section extracted and output as a script file to AutoCAD where the model is constructed as a 2D image and prepared for cutting in 2 mm aluminium using wire EDM. This technique uses a copper electrode to generate an electric discharge and erode a trace (0.1 mm wide) through the 2 mm aluminium sheet. The resolution of the resulting model is on the order of 0.1 mm (c. 2% of the minimum amplitude). When complete, the aluminium sheet is removed from the machine, trimmed to size and anodized to prevent the aluminium oxidizing when saturated with fluid. The fracture sections are then bolted in place between transparent plates (Fig. 2) to match the digital image of the model. Where any disparities result from the manufacturing process, these are corrected in the numerical geometry to ensure complete correspondence and allow point-to-point
A complex, dual porosity digital model (Fig. lb) is defined, comprising a porous matrix superposed with a fracture set, that permits the nonlinear interaction between matrix and fracture flow to be investigated (Mattisson et al. 1997; Dardis & McCloskey 1998a). Again, any debate as to the importance of the particular geometry employed here is irrelevant; it is simply complexity that must be guaranteed. Matrix porosity is represented by a distribution of nonoverlapping discs of different radii that are packed through a simulated compression using a discrete element method (Cundall & Strack 1979) until a specified porosity is attained. The strength of interaction among discs is controlled by elastic constants whose value is defined locally as a fractally correlated field which is mapped onto the distribution of discs, allocating an elastic constant to each disc based on its position within the field. The model causes discs with larger elastic constants to repel each other more strongly, producing looser packing and, therefore, a higher porosity in that area. Conversely, where the elastic constant is small, the discs
FLUID VELOCITY FIELDS IN 2D POROUS MEDIA contact and reduce porosity. The discrete element model compresses the discs both vertically and horizontally until the required porosity is achieved. Once the digital model of the matrix is completed using this technique, an area of 200 x 200 nun is removed and the coordinates of each disc are written to a formatted script file as a point with a given radius. A fracture network is then defined consisting of a distribution of line segments positioned randomly but with specified length, orientation and thickness. The precise rules employed are completely controlled by the modeller. Since the number-length distribution of natural fractures is, in general, well described by a power law (e.g. Bonnet et al. 2001), here such a fracture set is generated by defining lengths 1,, for random fractures according to: l,, = 10- l l ~
(4)
where x,, is a random number between 0.0 and 1.0 and D is the fractal dimension of the length distribution. Here, three preferred fracture orientations were also chosen, with allowance for a random variation of 5 ~ on the orientation of each. Fracture aperture is defined as a linear function of length. All fractures are then written to a script file which is used to reconstruct the pattern in AutoCAD. Initially, in AutoCAD, the completed matrix and fracture pattern are treated as separate objects. These are superposed, the fractures cut from the matrix and the resulting form extruded to a 2 mm thick layer in stereolithography (STL) format for production using stereolithography (Fig. 3). The basis of stereolithography is the use of a computer-controlled laser beam to polymerize a liquid photosensitive monomer
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formulation, layer by layer, according to a 3D representation of the object in a CAD system (Bernhard et al. 1994). To generate a model a glass base-plate (200 x 200 mm) is first placed on a platform and submerged in liquid resin to a depth of 0.1 ram. This provides a base on which the medium is built and subsequently forms one side of the flowcell. Following the digital map of the medium a laser then moves across the area, selectively curing areas corresponding to the solid areas of the model. Once a layer is complete, the model is again submerged in the resin to coat the previous layer and the process is repeated, continuing until the layer is 2 m m thick. The model is then removed from the vat, rinsed with alcohol to remove uncured resin and fully cured in a UV oven. It is estimated that the accuracy of the physical model produced using this technique is approximately 0.1 mm (c. 2% of the smallest diameter disc used). Prior to encapsulation by bonding a Perspex top plate, the physical model is compared with the digital image of the medium to identify flaws and then cleaned thoroughly to ensure no residue remains on the glass base-plate that would impair measurement. To bond the top plate, epoxy resin is prepared and applied across a 200 x 200 mm square in the centre of a frame covered with tensioned nylon mesh. This is repeatedly screeded using a spreader, both on the top and from beneath, so that every pore in the mesh is filled and all excess removed. The frame is then placed so the glued mesh rests 5 mm above the surface of the base plate and is then repeatedly rolled and depressed to apply small deposits of epoxy to the model thi'ough the pores of the mesh. When the top plate is applied and weighted these deposits flatten to cover each section with a thin layer which bonds to the top plate. The model is then left to cure for 48 hours before use.
The experimental system
Fig. 3. Stereolithography produces a precise copy of a digital model by use of a laser to selectively harden photosensitive resin on a glass base-plate to a depth of 2 ram. A 30 x 40 mm section of such a model is shown.
The transparency of the flowcell allows fluid movement in the void space of each physical model to be visualized. Measurements of fluid velocity are made throughout each model by seeding the fluid with neutrally buoyant microparticles and applying an imaging technique which exploits the relationship between the shift in particle positions in temporally separate images and the velocity of the fluid. A strong steel rig houses the components of the measurement system (Fig. 4). The flowcell is attached to a pair of moveable trolleys on the central support
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Fig. 4. The experimental system. Numbers refer to components as follows: (1) flowcell unit; (2) camera; (3) horizontal and vertical trolleys and stepper motor drivers; (4) lighting source directed through branched fibre optic cable to two cross-focused condensers at the rear of the unit; (5) fluid lines; (6) HPLC pump; (7) reservoir and pressurized eleunt flask containing marker particle suspension; (8) BNC (Bayonet-Nut-Connector) cable to frame grabber and CPU.
of the rig which allow the position of the flowcell to be adjusted relative to the camera and lighting, which are m o u n t e d on stands to the front and rear. Moving the camera and lighting systems relative to the flowcell was unfeasible due to the requirement for two linked positioning systems to co-align the camera and lighting and the increased potential for damage to the
camera charged-coupled device (CCD) sensor from persistent movement. The fluid system is arranged around the flowcell supplying a continuous, controlled flow for the duration of the experiment. These components ensure the systematic m e a s u r e m e n t of f u i d velocity in small sub-areas to produce a full, high-resolution map of velocity throughout each model.
FLUID VELOCITY FIELDS IN 2D POROUS MEDIA
The fluid system A reinforced, clamped unit seals the flowcell and provides inlets and outlets for the fluid, which is circulated through the flowcell at a constant rate for the duration of each analysis. A 10 mm Perspex back plate forms the base of the unit to which foam-lined, brass clamps are fitted that compress against the edges of the flowcell and seal it. The clamps at the top and base of the unit are fitted with connectors to the inlet and outlet pipes. The clamped unit bolts vertically to trolleys at the front of the rig and fluid is pumped from the base of the system through the flowcell in a constant, controlled flow with a HPLC pump. The volumetric flow rate through the rig is kept low (c. 2 - 4 ml min -~, depending on model porosity) to ensure flow is globally laminar with a parabolic velocity profile between the plates. The fluid is seeded with 10 ixm neutrally buoyant, hollow, glass spheres that follow the movement of the fluid and allow fluid motion to be visualized. The particles contrast strongly when backlit in water and are large enough to be visible within the measurement limits of the camera (for an 8 mm field of view a pixel is c. 7 ~m wide). The particles are added to 100 ml of water along with 1 ml of weak detergent solution to inhibit flocculation of the marker particles, producing a well-mixed suspension which is gradually added to another 1.5 1 of water in the system, monitoring particle concentration to ensure particle density is optimal for analysis (a density of c. 20 particles per mm2).
The imaging system Particle image velocimetry, a non-intrusive, image-based measurement technique, has been widely applied in engineering for 20 years (for a review see Grant (1997)). The technique involves seeding a fluid with neutrally buoyant particles which have a strong contrast in refractive index when compared to the permeant so that fluid motion can be visualized by following their progress. Using a camera and illumination system to record the displacement of the moving particles in temporally separated images, the 2D velocity of the fluid is determined remotely, without interfering with its motion. In this system, fluid velocities are low (the maximum measured velocity is c. 15 mm s -1) making it feasible to capture single exposures of the particle field in each image frame using a continuous light source and a precisely triggered camera. This simplifies the process of
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image analysis used to determine the displacement of the moving particles, as the order in which a sequence of images is taken indicates the direction of flow. This is not possible in high-speed flows where, due to limitations in camera speed, multiple particle images are captured in each image frame using a pulsed light source. The camera and illumination set-up involves indirect cross-focused backlighting of a subarea of the flowcell, which is imaged by a mega-pixel camera with a minimum field of view of 8 mm, focused on the plane central to the flowcell and recording the positions of the flow marker particles at intervals. To maximize the contrast between the micro-particles and the fluid, a custom-designed back-lighting system is used, which provides two intense, collimated light sources that are cross-focused precisely on the measurement plane of the camera, illuminating the marker particles moving through that area. The region of interest for measurements is a narrow (c. 50 lxm thick) layer centred on the midpoint of the 2 mm space between the plates of the flowcell. Under laminar flow conditions this midpoint represents the peak of the parabolic velocity profile which develops between the plates and, thus, measurements are confined to a zone across this peak that is 2.5% the width of the aperture between plates. By restricting the camera and lighting to ensure that only particles in this narrow region of maximum velocity are recorded, a quasi-two-dimensional measure of velocity is possible. Particles beyond this region are out of focus, allowing their exclusion using image-processing techniques. Any out-ofplane motion of particles at the boundaries of the 50 Ixm layer has minimal impact on the measured velocity, as the cross-correlation algorithm returns an averaged value based on the best fit to the shift of a population of c. 20 marker particles in a 1 mm 2 interrogation area. Any particles not present in both of the image pairs will add noise to the returned correlation field, but repeated measurement and stacking (described in the next section) negate this effect. The camera, a mega-pixel, analogue Adimec MX12P, is fixed at the front of the flowcell unit and focused on the central plane. The CCD array comprises 1024 • 1024 pixels and delivers up to 30 frames per second through a composite video output signal to the frame grabber memory. A telecentric lens with a 2X-extender sets the minimum field of view to 8 • 8 mm and ensures a constant magnification across this area, eliminating the viewing angle error (fisheye effect) inherent with conventional lenses. The minimum field of view determines the
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measurement resolution of the system, as the flow marker particles must be clearly visible within the 1024 x 1024 pixel image frame (a pixel resolution of 7.8125 Ixm). Image acquisition is in a controlled mode, where an externally generated trigger input signal supplied from the controlling software triggers the camera to acquire each image. This ensures that velocity measurements, made on the basis of comparing particle positions in temporally separate images, relate precisely to a velocity in mm s -1.
The drive system The flowcell is measured sequentially by defining 8 mm square sub-areas and positioning the flowcell relative to the fixed imaging system so that each lies within the field of view of the camera. Each sub-area is further subdivided into 1 mm interrogation areas (IA) within which particle displacements will be integrated to determine the local velocity, spaced at 0.5 mm intervals. The drive system controls the analysis of the entire flowcell by systematically moving each sub-area into the field of view of the camera and engaging position correction software to ensure the location corresponds to that section of the digital image within tolerance. The tolerance is set at 0.1 mm throughout these experiments to correspond with the resolution of the model construction techniques. The flowcell unit is attached to a pair of trolleys resting on linear guide ways and is driven horizontally and vertically by geared stepper motors, bringing the flowcell to the correct position relative to the camera. The stepper motors turn a prescribed amount producing an u p - d o w n , left-right shift of the flowcell unit. All movement is controlled by generating pulsed commands to the stepper motors from within the software using a digital I / O (input/output) and counter timer, each output pulse producing a shift of 9.26 ixm in the position of either trolley. To initiate the analysis of a flowcell, an 8 mm reference area is located in the digital model and the same area is located in the physical model. The flowcell is then moved to bring this area within the field of view of the camera. The coordinates of this area are then set within the software as a reference and all movements to other sub-areas throughout the analysis are relative to this. On each move to a new sub-area, the accuracy of the position relative to the digital model is assessed. A software routine assesses the goodness of fit between an image of the field of view after each movement of the flowcell and the
corresponding section of the digital model and ensures that each sub-area is located, again within tolerance, and automatically adjusted where required.
Measurement of 21) velocity fields Cross-correlation operates on pairs of images, separated by a time interval that is adjusted iteratively to ensure maximum measurement accuracy. The technique relies on a particle population captured in the first of a pair of images, captured at time t, also being present in the second image captured at t + At, so that comparing images allows the offset corresponding to particle displacement during the interval to be quantified. One of the primary considerations in the analysis was, therefore, to devise a means of optimizing the separation time, At, between images to suit the range of velocities in each model. The definition of the size of the IA and the separation time are crucial to the success of this approach and must be optimized to achieve a balance between maximizing the number of particles common to both images while ensuring a significant particle displacement between images. The IA was restricted to 1 mm 2 and velocity variations were accounted for by applying an adaptive separation time. Within any 8 mm sub-area, there can be a significant variation in fluid velocity; for example a slow moving pore throat with velocities < 1 mm s - i can occur in proximity to a high velocity fracture with velocities of c. 15 mm s -1. This precludes using a single separation time per sub-area and requires, instead, that each IA be dealt with individually to ensure the displacement between images for any IA is within a range of 1 0 - 4 0 pixels in each direction. The returned velocity vectors in each IA are assessed to locate those with minimal displacement and these are resampled at separation times up-scaled to ensure a significant displacement. In sub-areas exhibiting a broad range of velocities, up to six different separation times may be necessary to analyse the velocity field accurately.
Clarity of the IA Image quality is critical to accurate measurement and all models are affected to some degree by areas that cannot be measured due to poor contrast between the marker particles and the fluid. This is generally due to impurities such as residual resin from the stereolithographic process on the transparent plates or diffraction of light on some edges of the physical model from back illumination. Identifying these areas
FLUID VELOCITY FIELDS IN 2D POROUS MEDIA and excluding them from analysis prevents meaningless attempts at velocity measurement and provides a measure of clarity that can be used in post-processing to eliminate erroneous measurements. Prior to a measurement run, all IA within each model are examined automatically and assigned a DIRT (Degraded IA Removal Threshold) value that refers to the fraction of each image that is over-illuminated. The basis of determining the DIRT for each IA is that all clean, fluid-filled areas of the model should have a low greyscale value. By assessing the fraction of the void space in each IA with a high greyscale value, out of a worst case where the entire area is obscured, problem areas can be excluded from analysis. Primary processing
The initial step in extracting velocity information from an image pair involves subtraction and thresholding to remove noise and isolate the flow marker particles. Each pixel has a greyscale value from 0 (black) to 255 (white). Under illumination, the flow marker particles appear white (high-greyscale values) while the fluid appears grey and the solid areas either black or white (dependent on whether a resin or aluminium material layer is used). Direct subtraction of the image pair isolates all altered pixels between images and eliminates stationary values. The main disparity between the images is caused by the moving particles, creating a strong signal by their presence at a location in the first image and their absence in the next. Those particles which are in focus (within the depth of field) generate the strongest signal and changes due to noise are excluded by careful optimization of a threshold. The threshold assigns a unit value to all subtracted pixel values above a limit and a zero to those below. This is performed for both the positive and negative returns from subtraction, the former corresponding to the first image in the pair and the latter to the second image. The threshold is carefully optimized to ensure that only particles within the depth of field, which have the highest greyscale levels, are included in the analysis. Following this process, two binary images are output containing the isolated flow marker particles corresponding to the first and second images captured. Cross-correlation
Following primary processing, individual IA (corresponding to an area of 128 • 128 pixels) in each of the processed image pairs are extracted
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and analysed using a frequency domain crosscorrelation algorithm. The maximum value in the resultant 2D correlation function represents the physical displacement, in pixels, between image sections producing the best fit and relates directly to fluid velocity in the area. A single analysis is not, in general, sufficient to obtain an accurate determination of velocity for all IA in any sub-area. The density of marker particles in an IA may be too high or low during capture, resulting in a spurious peak being returned. In addition, the effect of resin residue in narrow pore spaces will affect the quality of the image for analysis. In order to strengthen the signal-to-noise ratio of the cross-correlation and obtain a more accurate velocity measurement, multiple image pairs are analysed and the successive correlation fields for each IA are stacked by addition. After the correlation fields for five image pairs are stacked, the vector field for the sub-area is assessed, first to determine areas requiring analysis at longer separation times to account for lower velocity regions and, secondly, to identify areas requiring further analysis to improve the signal peak due, in general, to contamination of the field of view. After the required iterations, the accumulated peak for each IA is interpolated to a sub-pixel scale to provide a higher resolution estimate of the location of the cross-correlation maximum. The x and y-components of velocity are converted from the displacement in pixels, dp, to velocity, v, (in mm s -1) by the relation 1000
v = -=--pdo Lkt - -
(5)
where At is the separation time and p is the size of a pixel in millimetres (for an 8 mm field of view captured on a 10242 CCD, this corresponds to 0.0078125 mm). The PIV method is the cornerstone of the system described here and its accuracy is clearly crucial to the success of the project. This accuracy was tested thoroughly by comparison of the velocities measured by the automated cross-correlation technique for different locations and velocities in the flowcell, with measurements taken by hand from a live video of flow in the same areas for a wide range of flow rates (Fig. 5). This exercise was the real test of the quality of the experimental system and demonstrates excellent agreement over almost three orders of magnitude, confirming confidence in the robustness and accuracy of the technique.
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y = +1.010X 1-0.01334, r2=0.9984 a8=+0.004848, ab=+0.008704
0.1
0.01 0.01
0.1
1
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Manual Measurement (ram s "t)
Fig. 5. Comparison of the automated measurements of fluid velocity obtained from cross-correlation with those measured manually in the image pairs for different velocities. The goodness of fit confirms the accuracy of the crosscorrelation-based measurement algorithm.
Each measurement run takes up to 48 hours and produces an output data file containing on the order of 80 000 velocity measurements each referenced to the centre of the corresponding IA.
Measured velocity fields A range of digital models and equivalent physical models has been produced and the fluid velocity fields measured. To disseminate these data as widely as possible, all results from the project, along with full details of the model geometries, are archived on the web at http://www.errigal. ulst.ac.uk/NERC_m2M/index.htm. Formatting standards have been applied to facilitate the use of the measured velocity fields and the numerical geometries and all are freely available to download. This paper confines a description of the measured geometries to those described in the preceding section. Data are displayed as vector plots overlain on the digital image of the medium; each vector is centred on the corresponding IA and is scaled according to colour and length. Areas for which accurate measurements do not exist are left vacant; interpolation is never used.
Velocity fields in fractured-porous models The completed velocity field for the fracturedporous medium is shown in Figure 6a. The
measurement resolution of this approach allows the strong interaction between matrix and fracture flow to be observed clearly and measured. The channelling of flow is apparent, as are the intricacies of fluid interaction at pore junctions where fractures act as preferential flow paths into which the matrix fluid is fed, enhancing flow in some areas while retarding it in others. While the fractures dominate flow, the matrix contribution, connecting areas of high permeability is significant. Observations such as this highlight the inadequacy of modelling techniques that consider matrix and fracture flow separately and simply sum the flow fields to obtain a global measure of permeability. They support the findings of Matfisson et al. (1997), who investigated the effect of a discrete fracture on a porous matrix composed of sintered glass spheres and found the combined permeability to differ by one order of magnitude from that of the matrix and fracture considered separately. This model also exemplifies some of the problems with resin residue that have emerged with recent models produced using stereolithography. This is manifested in the areas of the model for which there are no plotted vectors, predominantly in the narrow interstices of the matrix where accumulated hardened resin could not be removed from the glass base-plate. This is attributed to a change in the resin and equipment used by the stereolithographers and is a problem
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Fig. 6. Measured velocity fields: (a) (i) the velocity field for the porous-fractured geometry shown in Figure lb and beneath (ii) an enlarged section showing the complex interaction between fracture and matrix pore space; (b) (i) velocity field for the model of three fracture apertures (Fig. I a) and beneath (ii) enlarged sections of a fracture with and without offset.
currently being worked on. As will be seen later, these deviations in detail between the digital models and the physical models produced by stereolithography mean that comparisons between observed velocity fields in these models and those predicted by numerical simulations for the same geometry were not carried out. For these comparisons to be meaningful and informative it is vital that the flow paths in the physical and simulated media are identical. In fact, the predictions of the numerical model employed here were very significantly different than the observed velocity fields for the stereolithographic models. This strongly suggests that the small variations in pore throat geometries produced by weaknesses in the production process may have unexpected and significant effects on fluid transport. This interesting phenomenon is the subject of more detailed study at present and will be reported elsewhere.
Velocity fields in self-affine fractures The models produced using wire EDM, consisting of larger sections that are emplaced manually in the flowcell, are not affected by contamination to the same extent as those produced by stereolithography. A typical velocity field is shown in Figure 6b. The absence of measurements along the edges of the fractures here are the result of flow stagnation close to the fracture wall. While the velocity measuring techniques employed here, by increasing the image separation times, are able to measure extremely low flow speeds, the time taken for the measurements increases rapidly as the minimum measurable velocity is decreased. Lowering the minimum measurable velocity is also subject to diminishing returns since more and more time is spent on measuring smaller and smaller areas near the fracture walls. For the purposes of this study the lowest velocity measured here was
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thought to be an acceptable compromise between data volume and areal coverage, and the increased time required to measure slow speeds. Two enlarged sub-areas are shown of fractures with and without offset between the traces. With a constant actual aperture at zero offset, the effective aperture and, thus, the flow field varies significantly with the angle of the fracture trace relative to flow direction. The quality and resolution of this model were high, as wire EDM ensures very accurate reproduction of the fracture traces. In addition, any errors in the positioning of the aluminium sections between the transparent sheets can be corrected in the digital model to ensure correspondence. The authors are confident that the error in the position of any point between the numerical and digital model is not greater than 0.1 mm. As the physical model corresponds very closely with the digital model of the medium, a comparison of the data with the simulated velocity field is possible and work on the validation of a numerical model is presented in the next section. Validation of a lattice B o l t z m a n n simulation This section investigates the effectiveness of a 2D lattice Boltzmann (LB) fluid model of the fluid velocity field in a complex fracture whose walls consist of self-affine traces with variable offset (Fig. 2). The LB scheme is a development of the earlier lattice gas models in which the fluid is represented by discrete particles moving between the nodes of regular grid. Particle interactions take place using a set of rules which are designed to conserve mass and fluid momentum. Lattice Gas (LG) models have been shown correctly to reproduce macroscopic fluid behaviour for large grids. In the LB scheme, the discrete particles of the LG scheme are replaced by the probability of the presence of a particle at any node, a technique which significantly reduces the computation time required for any simulation. In this study, the lattice Bhatagnar-Gross-Krook (BGK) scheme (Qian et al. 1992) is tested, in which the fluid is represented by a real valued particle density Ni(x, t) at location x, time t, and moving in direction i, on a regular hexagonal lattice. The evolution of Ni at each time step is given by: Ni(x, t + A t ) -- N i ( x - ei, t) + w(N~i (x - el, t) - N i ( x - el, t))
(6)
where ei is the lattice unit vector in direction i. The first term on the right represents the advection of
fluid and the second term results in the relaxation of highs or lows in fluid density towards the equilibrium density, N~/q, at a rate which is governed by the relaxation parameter to. Thus, o9 is a surrogate for the viscosity of the fluid, large values resulting in rapid decay of density anomalies (low viscosity). The kinematic shear viscosity, v, is related to 09, by:
v = ~
-
1
(7)
Increasing w increases the rate at which momentum is transferred on the lattice, lowering viscosity, which vanishes at o9 = 2. Conversely, decreasing o9 reduces the transfer of momentum and increases viscosity. This scheme has been tested extensively (see for example Dardis (1998) and Dardis & McCloskey (1998a,b)) and validated by comparison with analytical solutions to the flow equations in simple geometries for which solutions are possible, and with available bulk, experimental measurements of permeability from rock samples. In these tests, the lattice BGK scheme has given accurate predictions. Validation of predicted fluid velocities in media with geologically feasible complexity is addressed here. Figure 7 shows two velocity fields for an enlarged sub-section of the model shown in Figure l a. The velocity field for the measured physical model is shown in Figure 7a and the predicted velocity field for the same section of the digital model is shown in Figure 7b. Qualitatively, the correspondence is excellent; the vectors coincide for all areas, signifying that the LB simulates the behaviour of the fluid realistically. The colour scale differs, however, indicating a difference in the structure of the flow field, with simulated velocities in areas of the flow field significantly higher than the equivalent measurements. As the measured velocities have been extensively validated, this is clearly due to inaccuracies in the numerical simulation and, as the difference is systematic and the flow field is qualitatively similar, the difference may be attributable to incorrect definition of the model parameters. In the measured velocity field, the area of fracture over which the majority of flow takes place is noticeably smaller than that of the simulated velocity field. The correlation length in the simulated flow field is longer; the effect of the constricted part of the fracture extends over a larger area. This figure is the best possible match between predicted and observed velocities; to obtain this fit the viscosity of the fluid in the simulation has been systematically decreased, bringing the predicted field
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Fig. 7. Enlarged sections of the EDM model in Figure la showing (a) the measured velocity field and (b) the predicted velocities at the same locations. The correspondence is good in as far as the fluid accelerates in more constricted parts of the channel but the correlation length is greater in the predicted field, with the fluid velocities relatively greater at the edges of the fractures compared with the measured field. This is consistent with a viscosity difference, the viscosity being higher in the simulated field.
closer to the observed. For this model, however, the simulation becomes unstable for o9 > 1.85 and the simulation is not able to reproduce the correct velocity field accurately. To quantify this mismatch, the measured ycomponent of velocity for the fracture section is plotted against the predicted y-component of velocity in Figure 8a. The main points made here are equally valid for the x-direction. While the plot of predicted against observed velocity (Fig. 8a), is poor, it is not random; there is a
clear, underlying structure. It was concluded above that the failure to model the velocity field in the complex geometry correctly is due to inadequate resolution of viscosity in the numerical simulation. The structure in Figure 8a must then be due to a systematic effect of the incorrect value of viscosity, probably reflecting the geometry of the area under study. If this were so, it should be possible to reproduce this non-random structure by deliberately simulating the field with the wrong
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Fig. 8. (a) Graph of the measured against predicted y-component of velocity for the fracture section shown in Figure 7. (b) Predicted y-component of velocity simulated at the lowest viscosity possible (to = 1.85) plotted against a predicted y-component of velocity at higher viscosity (to = 1.2). There is a marked similarity between these graphs.
viscosity. Consider the situation in which a fluid flowing in the geometry of Figure 7 is simulated at a low viscosity but one which is manageable with the LB scheme. Let this be called the 'observed' field. Let this field be simulated using an to value which is deliberately set too low and then the results plotted as above (Fig. 8b). There is quite a remarkable similarity between this plot and Figure 8a. This exercise illustrates a couple of important points. First, the lattice BGK model is unable to simulate low viscosity flows in complex media correctly. Secondly, and of more importance, the inaccuracies produced by the viscosity mismatch are non-random and are related to the geometry of the medium. This becomes clearer when the relationship of the mis-prediction is exposed by identifying the location of the worst points. Each lobe corresponds to a particular area of the fracture, the greatest deviations relating to the narrower parts of the channel and relating to the boundaries of the fluid with the medium. Chen et al. (1991) showed that the lattice BGK scheme incorrectly modelled the fluid-solid interaction for to # 1.0. Here, it is shown that the limitations on the available viscosities severely degrade the prediction accuracy and, what is more, this degradation is dependent on material heterogeneity. These results suggest strongly that the promise of accurate prediction of f u i d transport in complex materials using lattice Boltzmann simulations will require a more physically sound control on fluid viscosity.
Discussion The core objective of this project was to provide reliable, high-resolution data on fluid velocity within known 2D geometries as a resource for the validation of existing numerical models. In
addition, a first attempt was made to validate a numerical simulation using these data. A methodology has been realised to define a numerical model as a binary image and translate it to a physical model, a transparent flowcell, containing a precise copy of the digital model generated using automated machining techniques. This flowcell is then incorporated in a purpose-built experimental rig, which controls analysis by maintaining a constant, controlled flow of seeded fluid through the flowcell and sequentially positioning the flowcell relative to a camera and illumination system to analyse velocities in sub-areas of the model, using an algorithm based on cross-correlation. A positioning system controls the systematic location and measurement of all sub-areas to produce a complete, accurate map of velocity. Checks imposed at all stages ensure measurements are as accurate as possible and correspond to precisely defined points within each model. The system is now fully operational and capable of measuring detailed velocity fields throughout a range of complex geometries, incorporating the realistic characteristics of rock. Considerable time has been spent checking and optimizing the system and ensuring the accuracy of the technique and the quality of the output datasets. It has been shown that the system produces a detailed, high quality assessment of the distribution of fluid velocities in a complex medium with scale-invariant heterogeneity. These datasets consist of a map of up to 80 000 points across the void space of each medium, each with an associated velocity vector. Quality control in the form of careful checks of the correlation between automated and manual measurements have been undertaken. The results of these checks show that the method returns very accurate velocity measurements that are reliable over a wide range of velocities.
FLUID VELOCITY FIELDS IN 2D POROUS MEDIA Furthermore, it is believed that many of the areas excluded in the analysis because of their slow velocities can be measured accurately if longer image separations are allowed. It is only high velocity measurements that are limited by the speed of the camera. It is believed that this system has produced accurate data measurements of fluid velocities in complex media for which the material geometry is known accurately. The results should be of real benefit to the numerical modelling community. Preliminary comparisons with a modified lattice Boltzmann model have highlighted problems with the scheme and identified several issues for further consideration. Based on the results of the comparison of the simulated velocity fields with real data, the efficacy of the relaxation parameter, o~ (which controls the transfer of momentum on the lattice), in controlling the viscosity of the modelled fluid has been shown to be limited and, particularly in deviations from unity, o) has been shown systematically to distort the predicted velocity fields in proximity to the solid boundaries. This is consistent with the findings of Li et al. (2003) who showed the BGK scheme to predict permeability wrongly for oJ ~ 1.0. This study emphasizes the strong and nonlinear control of viscosity on the flow field and the need, in real applications, to model the fluid viscosity properly. Several limitations of the system are currently being resolved. A recent change in the resin and equipment used by the manufacturers has resulted in deposits of hardened resin residue lodged among the solid elements in certain models, which cannot be removed in all instances and thus make comparison of measurements with predictions in the originally defined geometry impossible. Blocking a pore throat significantly alters the velocity field, especially in complex geometries close to the percolation threshold and, as it is impossible to alter the digital model file to match, the use of these data for validation is limited. This problem has, however, produced the interesting result that small differences in model geometry produce very significant differences in the flow pattern, resulting in changes in global permeability. This suggests that, in general, the problem of predicting bulk permeability by reductionist methods, such as LB modelling of accurate representations of rock geometry, are unlikely to be successful. Comparisons of measurement with predictions for the models analysed to date are consequently restricted to those models produced using wire EDM. As accurate copies of the media were obtained before the recent deterioration in
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quality, this issue should be resolved either through negotiations with the company or sourcing a new supplier. It is believed, however, that stereolithography is a potentially powerful technique for addressing many important problems in the earth sciences. Further work in fluid transport in complex media includes the examination of the effects of the growth of biofilms on fluid flow through pore throats and how this affects total medium permeability with time. The possible construction of 3D models whose material properties and internal geometry are known in detail, also offers great possibilities for the examination of the effect of stress and pore fluid pressure on permeability. Both of these issues are currently the subject of much research. In conclusion, accurate measurements of fluid velocity fields for complex media with fractal heterogeneity have been produced. These data, with the appropriate meta-data, are stored on the web and are available for general use. Results to date have shown, first, that very small changes in material geometry can produce very significant changes in global flow patterns and significantly alter global permeability. This nonlinear control of flow at large scales by changes at small scales further compounds the upscaling problem. Secondly, it has been shown that a standard LB simulation for flow in a complex medium fails to reproduce the detail of the fluid velocity field. The source of the error has been identified as unphysical representation of fluid viscosity, which incorrectly models the fluid-solid interaction. This indicates the need for development of more physically sound techniques for viscosity representation in the LB models. This work was supported by the Natural Environment Research Council (NERC) under the Micro to Macro 0x2M) Thematic Research Programme. The authors thank Harry Smyth and Terry Griffin of the University of Ulster Mechanical and Electronic Workshops for assistance in developing the experimental equipment. The contributions of Rae Mackay and John Bloomfield in reviewing this paper are acknowledged.
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JACQUIN, C.G. & ADLER, P.M. 1987. Fractal Porous Media II: Geometry of porous geological structures. Transport in Porous Media, 2, 571-596. KATZ, A.J. & THOMPSON, A.H. 1985. Fractal sandstone pores: implications for conductivity and pore formation. Physical Review Letters, 54, 1325-1328. LI, H., PAN, C. & MILLER, C.T. 2003. Viscous coupling effects of two-phase flow in porous media. Paper HllG-0934, presented at the AGU 2003, San Francisco, 12th- 18th December Meeting. MATTISSON, C., KNACKSTEDT,M.A. & SENDEN, T.J. 1997. Transport in Fractured Porous Solids. Geophysical Research Letters', 24, 495-498. MI~HEUST, Y. & SCHMITTBUHL, J. 2001. Geometrical heterogeneities and permeability anisotropy of rough fractures. Journal of Geophysical Research, 106, 2089-2102. ODLING, N.E. 1997. Scaling and connectivity of joint systems in sandstones from western Norway. Journal of Structural Geology, 19, 1257-1271. POWER, W.L. & DURHAM, W.B. 1997. Topography of Natural and Artificial Fractures in Granitic Rocks: Implications for Studies of Rock Friction and Fluid Migration. International Journal of Rock Mechanics and Mining Sciences, 34, 979-989. POWER, W.L. & TULLIS, T.E. 1991. Euclidean and fractal models for the description of rock surface roughness. Journal of Geophysical Research, 96, 415-424. POWER, W.L., TULLIS, T.E., BROWN, S.R., BOITNOTT, G.N. & SCHOLZ, C.H. 1987. Roughness of natural fault surfaces. Geophysical Research Letters, 14, 29-32. QIAN, Y., D'HUMIERES, H., & LALLEMAND,P. 1992. Lattice BGK models for Navier-Stokes Equation. Europhysics Letters, 17, 479-484. SORNETTE, A., DAVY, P. & SORNETTE, D. 1993. Fault growth in brittle-ductile experiments and the mechanics of continental collisions. Journal of Geophysical Research, 98, 12111 - 12139. TURCOTTE, D.L. 1993. Fractals and chaos in geology and geophysics. Cambridge University Press, UK, 73 -94.
The 1~2M project on quantifying the effects of biofilm growth on hydraulic properties of natural porous media and on sorption equilibria: an overview J. R. B R Y D I E 1, R. A. W O G E L I U S 2, C. M. M E R R I F I E L D 3, S. B O U L T 2, P. G I L B E R T 4, D. A L L I S O N 4 & D. J. V A U G H A N 2
1British Nuclear Fuels plc, Sellafield, Cumbria CA20 1AH, UK 2Department of Earth Sciences and Williamson Research Centre for Molecular Environmental Science, University of Manchester, Oxford Road, Manchester M13 9PL, UK (e-mail: [email protected]) 3School of Engineering, University of Manchester, Oxford Road, Manchester M13 9PL, UK 4School of Pharmacy and Pharmaceutical Sciences, University of Manchester, Oxford Road, Manchester M13 9PL, UK Abstract: The physical and chemical effects of bacterial biofilm formation upon hydraulic conductivity, mineral-solution interactions and the formation of biogenic mineral precipitates have been studied over a wide range of scales, from microscopic to macroscopic. Several novel pieces of equipment have been designed, constructed and commissioned in order to measure the physical effects of biofilms upon fluid flow through fractures and porous media, the overall effects of biofilm formation upon mineral surface reactivity, and the imaging and identification of mineral precipitates formed due to the presence of biofilm and bacterial cell surface polymers on a quartz surface. This paper presents an overview of key experimental methods and selected results; further experimental information is being published elsewhere. Biofilm formation within quartz sand in artificial groundwater resulted in a two orders of magnitude reduction in hydraulic conductivity under bench-scale constant head conditions. However, under quasi-environmental conditions within macroscopic centrifuge experiments, a reduction of 21% was measured, revealing differences in measurements and, hence, the value of the macroscopic experimental work in scaling from micro to macro. ln-situ microscopic evaluation of biofilms within simulated quartz rock fractures and in porous media reveal only a small percentage of the biomass to be in direct contact with the mineral surface, allowing mineral chemistry to be predominantly controlled by mineral surface reactivity, rather than by a diffusion-limited mineral-biofilm-solution interface. This is true even when a mineral surface is apparently completely covered by biofilm. The alteration of mineral surface drastically increases the kinetics of surface-coordinated trace metal precipitate formation by providing nucleation sites upon extracellular biopolymers and cell wall polymers. Over geological time-scales, these processes, particularly the formation of thermodynamically stable pore-blocking mineral precipitates, are envisaged to markedly change the flow paths, flow rates and interaction of migrating geofluids (water, petroleum, ore-forming solutions) with minerals and rocks.
Bacteria and bacterial biofilms are virtually ubiquitous within pore spaces and fractures of soils, sediments and rocks at and near the Earth's surface (Neu & Lawrence 1999). Some bacterial communities have even been reported from depths of several hundred metres (Beveridge et al. 1997; Cunningham et al. 1997). The effect that these biofilms have upon fluid flow through aquifer rocks and sediments, and on the reactions between such rocks and dissolved aqueous chemical species within these fluids, is
not understood fully. It is known that bacteria colonize the solid-fluid interface and often produce extracellular polymers (EPS) which invariably coat mineral grain surfaces and fractures, resulting in the formation of a biofilm (Taylor & Jaffe 1990). This biofilm not only alters the physical and chemical characteristics of mineral surfaces, but results in an overall reduction in pore space, leading to the phenomenon known as bioclogging. This problem is a major concern where abstraction or introduction
From: SHAW,R. P. (ed.) 2005. Understandingthe Micro to MacroBehaviourof Rock-Fluid Systems. Geological Society, London, Special Publications, 249, 131-144. 0305-8719/05/$15.00 9 The Geological Society of London 2005.
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of fluids within aquifers (or petroleum reservoirs) is required. EPS is thought to strengthen the attachment of cells to a surface, forms links between cells and provides a protective matrix which coats the bacterial cell clusters (Gilbert et al. 2002). The bulk of the volume of natural bacterial biofilms is often provided by the EPS which, although highly hydrated (approximately 99% fluid), constitutes >90% of the dry weight of the biomass. These matrix polymers retain nutrients and extracellular products which condition the microenvironment of the cells and provide an interface between bulk fluids and the underlying inorganic (mineral) substratum. The architecture of each biofilm is highly dependent upon several key variables, including hydrodynamic environment, nutrient conditions and physico-chemical properties of the substratum (Massol-Dey~i et al. 1995; Paulsen et al. 1997; Stoodley et al. 1999). The specific colonization behaviour of individual microorganisms also has an effect upon biofilm structure. Any bacterial biofilm is metastable and will adapt its structure, and relative cell distribution, to accommodate changing environmental conditions. Bacterial biofilms may also result in the precipitation of biogenic minerals from dissolved trace metals within migrating fluids. Over geological timescales, bioclogging - the production of biogenic mineral precipitates and biologically mediated mineral-fluid interactions - is likely to have a profound effect upon processes which involve the movement of fluids and their interaction with mineral surfaces, e.g. ore-fluid flow, meteoric water circulation (and associated water-rock interactions), hydrocarbon migration and the transport of contaminant plumes through aquifers. It is, therefore, essential to be able to image and quantify bacterial biofilm distribution in terms of mineral surface coverage, and to determine the extent to which a biofilm-modified mineral surface continues to interact directly with bulk solutions. This paper presents an overarching review of research carried out as part of the University of Manchester's Micro to Macro ('lx2M') programme. The results and conclusions of this research are being reviewed presently and individual foci of research are being published in specific journals, depending upon the nature of the study (Brydie et al. pets. comm.) (Further details of these publications are available from the authors of the present paper.) Essentially, the objectives of this research programme have been to undertake experimental work relevant to microscopic, mesoscopic and macroscopic scales on the growth of bacterial biofilms within artificial groundwaters, and to study
their influence upon both the hydraulic properties of geological systems (porous sediments and fractured rocks) and on the sorption of major and trace metals from the bulk fluids. In the microscopic-scale experiments, biofilms have been grown predominantly in artificial groundwater in simulated rock fractures and within quartz sand-filled flowcells. For the most part, biofilms have been studied in situ and in the presence of bulk solutions using advanced imaging techniques, including environmental scanning electron microscopy (ESEM) and confocal scanning laser microscopy (CSLM). Mesoseopiescale experiments, involving custom-built columns filled with natural quartz sand, have enabled the effects of biofilm growth upon hydraulic conductivity and the nature of biofilm formation within pore spaces to be determined at centimetre scale. Macroscopic studies involved the design, construction and commissioning of novel porous medium testing equipment to be used in conjunction with a 500 g tonne geotechnical centrifuge (radius 2.7 m), based in the Manchester School of Engineering. This equipment enables hydraulic conductivity tests on porous media to be carried out under an environmental stress regime equivalent to a natural depth of 6 0 - 1 0 0 m . This technique essentially bridges the gap between laboratory bench-scale experiments and field-scale studies, but with the advantage of carrying out tests under highly controlled conditions.
Experimental methodology M i c r o s c o p i c - s c a l e research
Experiments have been carried out in order to understand physico-chemical processes controlling biofilm structure and to understand chemical interaction between bulk solutions, biofilm and the mineral substrate at the biofilm-mineral interface. Miniature flowcells have been used to grow biofilm samples within a simulated rock fracture environment, using artificial groundwater as the bulk solution and manufactured silica tiles as fracture walls. Initially, microscopic experiments to study bacterial biofilm formation were carried out using a 50% nutrient broth as the growth substrate, in order to test the experimental equipment. All subsequent experimentation (including flowcell studies, bench-scale columns and centrifuge column work) was carried out using an artificial groundwater containing the following constituents per litre of deionized water: 2 . 5 7 m g l -a MgClz.6H20, 10.40 mg 1-I MgSO4.7H20, 0.53 mg 1- t Mg(NO3)z.6H20, 9.32 mg 1-1 NaHCO3, 3.65 mg 1-1 CaClz.2H20, 1.13 mg 1-1 KH2PO4,
BIOFILM GROWTH EFFECTS ON POROUS MEDIA 4.10 mg.1-1 NaNO 3 and 138.96 mg 1-1 CH3COONa. A schematic experimental set-up, shown in Figure 1, involves two chemo-mechanically polished quartz plates forming the fracture walls into which artificial groundwater, inoculated with a model bacterium ( P s e u d o m o n a s aeruginosa PA01), was introduced. Further details of experimental conditions are outlined in Figure 1. With this set-up, all aspects of the biological and geochemical environment within the cell can be controlled by varying fracture width, the chemical environment, fluid flow rate and so on. The fracture may also be filled with granular materials to produce a porous medium experiment. Fluorescent probes can also be introduced to help characterize biological components of the biofilm (including bacterial cells, polymers and even trace metal distribution) in situ and without introducing sample preparation artefacts. Techniques such as fluorescence microscopy and CLSM may then be used to determine spatial relationships between biofilm components, mineral surfaces and bulk solutions. Until recently, the examination of bacterial biofilms developed within natural (or simulated) geological systems was performed using techniques including scanning electron microscopy (SEM), transmission electron microscopy (TEM) and fluorescence microscopy (Naomi et al. 1994; Beveridge et al. 1997). Sample preparation techniques used during these early studies (air drying, critical point drying and freeze drying)
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typically introduce artifacts, such as diminished sample integrity, shrinkage of cells and EPS and an overall collapse of the biofilm structure. More recently developed techniques, such as ESEM, allow the imaging and analysis of hydrated samples under low vacuum conditions and with minimal sample preparation. However, even this technique may result in the partial disruption of the sample matrix (and associated biofilm) through the progressive removal of bulk fluids and the imposition of surface tension forces during ESEM investigation of the sample. CLSM, on the other hand, is an ideal method for the imaging and quantitative spatial analysis of bacterial biofilms due to its non-invasive nature and its ability to image live, structurally unaltered and fully hydrated biofilms (Neu & Lawrence 1999; Davies et al. 1998; Whitely et al. 2001). CLSM, a technique not previously used for such studies by earth scientists, results in a series of sequential two-dimensional fluorescence digital images, each representing an optical section of a given volume and containing pixels on a 512 x 512 grid. When contoured and stacked in sequence, the images may be reconstructed as a grid of volumetric pixels (voxels). Correlation of voxels between layers allows isosurfaces (3D surfaces) to be rendered, which are then identifiable as components of the bacterial biofilm, such as bacteria or exopolymers (Castleman 1996; Xavier et al. 2001). Once generated, images are fully 3D and can be rotated for view in any
Fig. 1. The experimental set-up for growth of biofilms in a controlled environment (simulated fracture) and in-line staining of biofilm components using multiple fluorescent dyes. The entire model was then sealed, and imaged using a confocal laser scanning microscope (CLSM). The simulated fracture comprises two glass slide surfaces of dimensions 10 x 10 • 0.1 mm separated by a shaped silicone spacer of 1 mm thickness.
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user-defined orientation in order to understand better the spatial relationships of the biofilm in relation to the mineral surface. Furthermore, the sequential application of molecule-selective fluorescent dyes allows quantification of specific biofilm components. For example, the determination of numbers and spatial distribution of bacteria relative to their supporting matrix becomes straightforward (Lawrence et al. 1998). Mesoscopic-scale research Research at the mesoscopic scale has involved experiments in which bacterial biofilms have been grown in a 70 mm long acrylic column (internal diameter 25 mm) which, for most of the experiments, has been packed with a naturally occurring quartz sand (Congleton sand). Using a custom-designed column system (Fig. 2), the chemically sterilized (2% Viton solution rinsed six times using autoclave sterilized
deioinized water) quartz sand-filled column was operated under constant hydraulic head conditions for 150 hours, using an artificial groundwater as eluant. The rate of effluent discharge may be measured using pressure transducers and an automatic logging system which allows the rate of eluant solution passing through the column to be measured directly. Once a stable reduced effluent rate is reached, following biofilm growth, the column may be dismantled and examined using ESEM, TEM or fluorescence microscopy. Additional analysis of cells, polymers and proteins may also be carried out in order to determine bacteria and biofilm distribution within the porous medium. Trace metal breakthrough curves can also be determined by direct sampling and analysis of column porewater solutions. The analysis of biofilm samples, and any biogenic minerals precipitated, may then be performed at the end of each experiment.
Fig. 2. The experimentalarrangementfor hydraulicconductivitymeasurementsin a sand-filledcolumn.The reservoir contains artificialgroundwater containingper litre of deionizedwater: 2.57 mg 1-1 MgC12.6H20, 10.40nag 1-1 MgSO4.7H20, 0.53 mg 1-1 Mg(NO3)z.6H20, 9.32 mg 1-1 NaHCO3, 3.65 mg 1-1 CaC12.2H20, 1.13 mg 1-1 KH2PO4, 4.10 mg 1-I NaNO3 and 138.96mg 1-I CH3COONa.
BIOFILM GROWTH EFFECTS ON POROUS MEDIA
Macroscopic-scale research In order to simulate fluid flow phenomena at field scale (tens to hundreds of metres) within geological porous media, as affected by the presence of biofilm and biogenic mineral precipitates, a 500 g Tonne geotechnical centrifuge was employed (Fig. 3). This machine has a radial arm length of 2.7 m and can achieve equivalent physical lithostatic conditions to approximately 60 m depth, within a 600 mm long experimental quartz sand column when under a centrifugal acceleration of 100 g. Put another way, applying the universally accepted scaling laws associated with multigravity modelling, i.e. seepage flow conforming to Darcy law flow conditions generated under N gravities reduces the equivalent flow time by N 2, use of the centrifuge in this way simulates a flow regime within a porous medium column at mesoscale equivalent to approximately 27 years of fluid flow at field scale within a single day of experimentation (when operating at 100g). The conditions which prevail in the centrifuge test at N gravities
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therefore simulate both pressure magnitudes and gradients found at field scale. This ensures that the integrity of the soil column is maintained whilst appropriate flow conditions such as reflected in a suitable Reynolds number for laminar flow, are adhered to. The alternative method of applying an equivalent hydraulic gradient across the column length at unit gravity to create an accelerated flow may result in local hydrofracture or the development of preferential flow paths within the soil mass. In Figure 3, a schematic view of the geotechnical centrifuge is shown, highlights including the centrifuge arm (housed underground adjacent to the Manchester School of Engineering), the centrifuge payload and the control room from where centrifuge operations are directed and data acquired. As in the mesoscopic-scale research, there is a column, in this case 600 mm high, which may be packed with sand or other porous material. A novel insert has also been constructed as part of this research which accurately simulates a fracture of controlled width within a quartzite block. This insert fits neatly inside the cylindrical
Fig. 3. Schematic diagram of the geotechnical centrifuge and payload used for experiments to determine hydraulic conductivity of a sand-filled column under different conditions: sterile; containing biofilm, and containing mineral precipitates.
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BIOFILM GROWTH EFFECTS ON POROUS MEDIA column body and comprises two parallel polished quartzite blocks held at a fixed distance apart within a stainless steel framework. This system may be used to study hydraulic properties of a fracture during, and following, the formation of bacterial biofilms and the generation of biogenic mineral precipitates under simulated aquifer conditions. As shown in Figure 3, alongside the analytical column are two solution (and/or contaminant solution) source reservoirs and an effluent reservoir. Solutions from each source reservoir may be passed through the porous medium column/simulated fracture under a precisely controlled hydraulic gradient and the effective hydraulic conductivity measured as a function of effluent solution accumulation rate. This procedure is carried out for aseptic equipment (control experiments) as well as upon fractures and sand columns in which bacterial biofilms and mineral precipitates have formed. All of these measurements occur while the whole experimental rig is spun around at up to 100 gravities. During the centrifuge 'flight', pressure transducers are used to monitor fluid flow rates within the column. Typical results of a test run are provided in Figure 4. Here, a test was run using chemically sterilized quartz sand (again Congleton sand) as column fill. The results illustrated here show the stepwise increase of the centrifuge from rest to 80 g at intervals of 10g. The centrifuge speed is increased incrementally to allow stabilization of all equipment and the column tests rig at each g-level. A stepwise increase in pressure is noted within the source reservoirs as the centrifuge is accelerated. At 80 g, a valve is opened beneath the source reservoir allowing fluid to flow through the sand-containing column. Pressure naturally drops off rapidly within the source reservoir and correspondingly increases in the effluent reservoir. Due to the fact that the system is under constant hydraulic head conditions, as hydraulic equilibrium is reached using the concept of a Marriotte bottle, which ensures a remotely controlled constant head of influent on the column, effluent accumulates in a linear manner over time. The rate of effluent increase may then be used directly to calculate effective hydraulic conductivity at a particular g-level. Comparison and extrapolation of hydraulic conductivities at a series of g-levels (or effective field-scale depths) then allows predictions of hydraulic conductivities at greater g-levels, and analogously, at greater depths. In typical test runs, effluent accumulation is monitored as the centrifuge speed is incrementally decreased. Figure 4 shows decreases from 80 g down to 20 g.
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Experiments were conducted to determine the effect of biologically mediated mineral precipitation of hydrated iron oxides onto biofilmaltered mineral surfaces (the alteration having been previously physically and chemically characterized by spectroscopic and ESEM examination of biologically colonized mineral surfaces under experimental conditions); the effects that such mineral precipitates have upon hydraulic conductivity may also be measured directly in real time. Biofilm-conditioned columns were subjected to high g-level centrifugation to remove excess biomass and to allow the discrimination of hydraulic conductivity as a function of mineral precipitation as opposed to mere bioclogging.
Results and discussion Microscopic scale The results of the in-situ CLSM studies of biofilm development upon a silica surface reveal structural features within developing bacterial bitfilms resulting directly from the hydrodynamic environment within the simulated fracture. During the initial stages of bacterial biofilm formation, primary colonizing bacteria are seen to align themselves within the predominant bulk fluid flow direction (Fig. 5). Bacterial cells seen in Figure 5 are approximately 1 &m long and 0.5 p~m in diameter. This alignment is apparently used as an initial template for subsequent biofilm architecture and may even be imaged in much more developed biofilms. Images shown in Figure 5 are of only one of the two parallel simulated fracture surfaces; a similar biofilm develops on the opposing surface. A biofilm developed under relatively elevated nutrient conditions (50% nutrient broth, Oxoid Chemicals) (Fig. 6) occupies approximately 3.15% of the fracture void space on either side of the fracture, resulting in a combined reduction in fracture width of 6.30%. Typical biofilms imaged during this study consisted of a raised umbrella-like canopy of cells and EPS, covering an interconnected network of fluid channels, anchored to the surface by cells and EPS. Despite the fact that, under visible light, the biofilm appears to coat large areas of the simulated fracture surface, CLSM imaging reveals that only a small percentage of cells and related EPS contribute to the area of biofilm attachment (biofilm footprint) at the surface. Measured surface area coverages in this study range from 6 - 8 % surface area and are consistent with those of Lawrence et al. (1998) who reported between 2.5% and 9% coverage at the biofilm colonized
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Fig. 5. Three-dimensional isosurface model of a developing biofilm within a simulated fracture. This composite image was generated from confocal laser scanning microscopy (CLSM) images and shows alignment of bacterial cells consistent with fluid flow direction during biofilm formation.
surface and a bimodal distribution of biofilm EPS. Figure 6, again showing only a single side of the fracture, clearly demonstrates the bimodal distribution of the EPS and bacteria, from the surface out towards the bulk solution. The biofilm footprint layer is composed predominantly of EPS (shown in yellow), but with clusters of nucleic acids (shown in red) that are contiguous with columnar populations of the canopy. Inelastic displacement (breakage of biofilm components as opposed to plastic deformation) and translocation of these colunms, which include both cells and EPS, in a direction consistent with bulk fluid flow during growth (Fig. 6), is strongly indicative of EPS migration due to shear forces rather than direct deposition of displaced colonies during growth. An overall canopy displacement of 32 ixm is evident within this image, implying a purely physical inelastic disruption of biofilm matrix. These features have not been reported previously for relatively thick, matrix-supported, laterally continuous monocultural bacterial biofilms, and are in direct contrast to the conclusions of Stoodley et aL (1999) who described biofilm
detachment as a result of the elastic passive migration of both cells and matrix. The biofilm features described here were observed as a direct consequence of being able to use CLSM images to produce in situ 3D isosurfaces capable of being manipulated to any user-defined orientation, allowing entire datasets to be viewed and interpreted in real-time. For example, manipulation of the entire CLSM dataset as a single isosurface model reveals in situ evidence of the mechanics of biofilm deformation which would not have been seen easily by visual inspection of the CLSM images or reconstituted 3D models available within the CLSM acquisition software. As a result of this, cells and biofilm components were imaged in 3D throughout the biofilm formation process, from initial bacterial attachment at the surface through to the development of a complex biofilm. Mesoscopic scale
Throughout the sterile experiment, effluent discharge rate was constant with time. Following
BIOFILM G R O W T H EFFECTS ON P O R O U S M E D I A
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Fig. 6. Three-dimensional isosurface model of a developing biofilm within a simulated fracture (dimensions of box shown: width 321.4 Ixm; depth 321.4 txm; height 35.1 ~m). This composite image was generated from confocal laser scanning microscopy (CLSM) images and shows a thick biofilm containing distinct bacterial communities and polysaccharides within the EPS.
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the inoculation of the column, by entraining a cell suspension (cultured in the same artificial groundwater as used for eluant solution) for four hours, the column feed was switched back to sterilized artificial groundwater and the effluent discharge rate monitored. At approximately 30 hours of operation, the rate of effluent discharge decreased rapidly until the rate stabilized at a reduction of 70% after approximately 100 hours of column operation. This experiment
was repeated several times and typical results are shown in Figure 7. The change in hydraulic conductivity caused the effluent discharge flow rate to decrease from 6 . 3 m l m i n - m to 1.5 ml min-1. The decreased discharge rate was associated directly with previously determined biofilm accumulation rates under identical conditions. ESEM images of the biofilm obtained subsequent to the column study (Fig. 8a) shows cross-linked streamers of EPS between grains,
Fig. 8. ESEM images obtained from biofilm materials grown in the mesoscopic-scale sand column: (a) streamers of extracellular polysaccharides (EPS) plus dense biofilm on the quartz sand substratum; (b) single bacterium (Pseudomonas aeruginosa PA01) attached to the mineral surface and surrounded by EPS.
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Fig. 11. (a) ESEM image showing iron oxyhydroxide precipitates and their nucleation point; a single bacterium and associated EPS. (b) The results of PIXE analysis reveal the composition of such precipitates to be iron oxyhydroxide and additional phosphorous and sulphur.
BIOFILM GROWTH EFFECTS ON POROUS MEDIA and denser polymers at the surface of the quartz grains, with a layered structure parallel to the surface. The images also reveal occasional single bacterial cells attached to the mineral surface (Fig. 8b).
Macroscopic scale When the results of centrifuge experiments using sterilized large quartz sand columns to extend the scale of the mesoscopic experiments (see Fig. 9) are examined, it can be seen that hydraulic conductivity (k) of the quartz sand is similar to that measured at the mesoscopic scale (7.5 x 1 0 - S m s - a ) . When a fully developed biofilm is present, k drops to 0.5 x 10 -5 m s -1 as a direct result of bioclogging. This was measured in a constant head flow test at 15 g, with the biofilm intact. By increasing g values in the same flow test, bulk pore fluid shear forces cause the majority of the biofilm superstructure to slough (detach). This sloughed biomass is then carried through the column and is eluted. The resulting k value measured following biomass detachment was 5.3 x 10 -5 m s -~. The remaining biofilm footprint at the solid-solution interface is capable of interacting with dissolved trace metal species within pore solutions, often resulting in the nucleation and precipitation of trace metals. The most recent experiments explored the process that is critically important to trace metal transport, whereby a dissolved trace metal, in this case iron at a concentration of 100 mg 1-1, was continuously passed through the centrifuge column during flight (and hence simulated field conditions) under aseptic conditions. As seen from Figure 10, this causes a further substantial decrease in hydraulic conductivity (4.8 x 1 0 - S m s -1) due to the precipitation of hydrated iron oxides within the pore space. Imaging of precipitates from these experiments, and other experiments performed under identical conditions but using quartz plates as the mineral surface, shows the morphology of precipitates along with their association with bacteria and EPS (Fig. 11). In this case, ovoid bacteria and mineral precipitates are imaged using ESEM. Chemical characterization of these precipitates using the high spatial resolution analysis of the proton probe (PIXE or proton-induced X-ray analysis) confirms that one is dealing with an iron oxyhydroxide precipitate (Fig. 1 lb) and Xray diffraction analysis revealed the mineral to be lepidocrocite. Minor peaks within the proton probe analysis show the presence of 1 wt% phosphorous and sulphur, which is attributed to phosphate and sulphate ions incorporated into the
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rapidly precipitated oxyhydroxide matrix. An important observation from these experiments is that iron precipitation from relatively dilute solutions is enhanced by the presence of a small quantity of bacterial biomass on the mineral surface. This effect is noted for live and dead microbial matter, indicating surfacemediated precipitation as opposed to metabolic formation of minerals within the environs of the bacterial cells.
Concluding remarks A number of general points can be made from the various studies to date. First, even under the low nutrient conditions characteristic of most natural environments, prevalent in the bench-scale and centrifuge column experiments, the presence of biofilms developed within porous sediments or fractured rocks may significantly decrease hydraulic conductivity; in a particular case studied, by over 70%. Secondly, even when a biofilm apparently covers an entire mineral surface, a very substantial amount of the mineral surface is not directly covered by cells or EPS and so remains in direct contact, and able to readily react, with bulk solutions. Thirdly, fluid shearing and sloughing of biofilm within the elevated g-level environment typical of a centrifuge mean that experiments involving biofilms must be limited to relatively low glevels (<25 g). However, 'biofilm-conditioned' columns and mineral precipitation experiments may be carried out over a much larger range of g-values, hence substantially reducing experimentation time. Mineral precipitates formed within pore spaces are thermodynamically stable and may, therefore, persist and cause hydraulic conductivity reduction over geological time-scales. As stated earlier, this paper provides a summary overview of experimental work and research already carried out. Further research using the methods outlined above will include the use of multiconsortia biofilms, simulations carried out under anoxic conditions, and detailed studies on the chemical and structural controls imposed by mineral surfaces upon the inception and growth of biofilms under natural environmental conditions. Dr Sarah Schooling and Mr Tony Wade are thanked for their help with biofilm growth and imaging studies. The centrifuge experience of Mr Martin Cruikshank was invaluable and technical support from colleagues in the Earth Sciences Department at Manchester University, particularly Dr Paul Wincott and Steve Caldwell, is also acknowledged. Advice and support were also received
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biofilms of fixed film reactors treating contaminated groundwater. Applied and Environmental Microbiology, 61, 769-777. NAOMI, A., HUGHES, S.N. & HANDLEY,P.S. 1994. A comparison of conventional SEM techniques, low temperature SEM and the electroscan wet scanning electron microscope to study the structure of a References biofilm of Streptococcus crista CR3. Journal of Applied Bacteriology, 76, 448-454. BEVERIDGE, T.J., MAKIN, S.A., KADURUGAMUWA, J.L. & LI, Z. 1997. Interactions between biofilms NEU, T.R. & LAWRENCE,J.R. 1999. In situ characterisation of extracellular polymeric substances (EPS) and the environment. FEMS Microbiology in biofilm systems. In: WINGENDER, J., NEU, T.R. Reviews, 20, 291-303. CASTLEMAN, K.R. 1996. Digital Image Processing. & FLEMMING, H.C. (eds) Microbial Extracellular Prentice Hall, New Jersey. Polymeric Substances; Characterisation, StrucCUNNINGHAM, A., WARWOOD, A.B., STURMAN, P., ture, Function. Springer Verlag, Berlin, 21-47. HORRIGAN, K., JAMES, G., COSTERTON, J.W. & PAULSEN, J.E., OPPEN, E. & BAKKE, R. 1997. Biofilm HIEBERT, R. 1997. Biofilm Process in Porous morphology in porous media, a study with microMedia - Practical Applications. In: AMY, P.S. & scope and image techniques. Water Science and HALDEMAN, D.L. (eds) The Microbiology of the Technology, 36, 1-9. Terrestrial Deep Subsurface. Lewis Publishers, STOODLEY, P.J., BOYLE, D., DE BEER, D. & Boca Raton, Florida, 325-344. LAPPIN-SCOTT, H.M. 1999. Evolving perspectives DAVIES, D.G., PARSEK, M.R., PEARSON, J.P., of biofilm structure. Biofouling, 14, 75-90. IGLEWSKI, B.H., COSTERTON, J.W. t% GREENBERG, TAYLOR, S.W. & JAFFE, P.R. 1990. Biofilm growth E.P. 1998. The involvement of cell-to-cell signals and the related changes in the physical properties in the development of a bacterial biofilm. of a porous medium. 1. Experimental Investigation. Science, 280, 295-298. Water Resources Research, 26, 2153-2159. GILBERT, P., MAIR-LITRAN, T., MCBAIN, A.J., WHITELY, M., GITA BANGERA, M., BUMGARNER,R., RICKARD, A.H. & WHYTE, F. 2002. The physiPARSEK, M.R., TEITZEL, G.M., LORY, S. 8z ology and collective recalcitrance of microbial GREENBERG, E.P. 2001. Gene expression in biofilm communities. Advances in Microbial Pseudomonas aeruginosa biofilms. Nature, 413, Physiology, 46, 203-256. 860-864. LAWRENCE, J.R., NEU, T.R. & SWERHONE,G.D. 1998. Application of multiple parameter imaging for the XAVIER, J.B., SCHNELL, A., WUERTZ, S., PALMER, R., WHITE, D.C. ~z ALMEIDA, J.S. 2001. Objective quantification of algal, bacterial and exopolyrner threshold selection procedure (OTS) for segmentacomponents of microbial biofilms. Journal of tion of scanning laser confocal microscope images. Microbiological Methods, 32, 253-261. Journal of Microbiological Methods, 47, 169-180. MASSOL-DEYA, A.A., WHALLON, J., HICKEY, R.F. & TIEDJE, J.M. 1995. Channel structures in aerobic
from industrial collaborators (Drs P. Humphreys and I. Beadle, BNFL) and staff at the PIXE facility of the Vrije Universit~it, Amsterdam. NERC are thanked for financial support via the Micro to Macro Special Topic Programme.
Overview of the NERC 'Understanding the Micro to Macro Behaviour of Rock-Fluid Systems' R. P. S H A W Scientific Co-ordinator, Micro to Macro, British Geological Survey, Keyworth, Nottingham, NG12 5GG, UK Abstract: The NERC Ix2M Programme funded 17 projects over a period of six years. A brief outline of the programme and of all the component projects is provided here.
This paper provides an overview of the objectives of each of the four themes of the NERC Micro to Macro Thematic Programme and provides a brief summary of all 17 projects funded by the programme. The programme focused on developing an understanding of the relationships between measured and modelled subsurface fluid flows spanning the range of spatial and temporal scales relevant to fluid resource management. The programme was motivated by the growing recognition that assumptions of uniformity at certain scales are inadequate for extrapolating fluid behaviour both in time and space. A clear example is the power-law behaviour that characterizes many aspects of observed rock properties seemingly independently of rock type. Developments in the areas of geological and geofluid observations and modelling that focus on scaling relationships in generic rock help to provide a clearer physical understanding on which to base more effective geofluid management. To make progress towards this objective, research spanning a wide spectrum of observation and simulation scales was undertaken by the programme which can be divided into four themes: 9 9 9
9
understanding the natural processes which lead to scaling relationships between size and magnitude of rock and flow heterogeneity; quantification of essential fluid flow properties and their spatial pattern from measurements; identification of appropriate statistical models and scaling laws describing rock property heterogeneity and fluid-rock interactions in geological media; understanding the relationships between rock property distributions and flow model parameter distributions.
The aims and objectives of these themes are discussed in more detail below.
Programme themes
Theme 1: Understanding the processes leading to rock property/fluid flow scaling Depositional, climatic, tectonic, structural and geochemical processes are all involved in the development of pathways for fluid flow within a geological medium. Areas of interest include: the evolution of fracture and porous media permeability distribution from micro- and macroscale studies of rock heterogeneity; geochemical sealing and re-sealing of pores under changing stress fields and chemical environments; chemical kinetics of rock dissolution in actively flowing systems; the coupling between fluids and stress in actively deforming rock; mineral and chemical signatures as indicators of permeability/porosity development; and patterns of self-organization and associated power-law behaviour resulting from coupled non-linear geological processes. The development of conceptual or generic models of the evolution of scaling rock properties that control flow provide valuable insights into heterogeneity and structure distributions that may not otherwise be attainable.
Theme 2: Quantification of essential fluid flow properties and their spatial variation The integrated interpretation of the full range of observations, including geophysical data, to extract information on rock geometry and property distributions that control fluid flow needs further development. Geophysical inversion
From: SHAW,R. P. (ed.) 2005. Understanding the Micro to Macro Behaviour of Rock-Fluid Systems. Geological Society, London, Special Publications, 249, 145-161. 0305-8719/05/$15.00 9 The Geological Society of London 2005.
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currently provides valuable constraints on rock geometries and may provide viable information on the geomechanical and hydraulic behaviour of a rock mass. Further research may enable these techniques to be refined and validated. Increasing data resolution and bandwidth of non-invasive measurements, together with new insights from rock evolution modelling (Theme 1), open new possibilities for defining both geometric and hydraulic data patterns in rocks at all depths. Combining measurement methods with insights into the natural processes governing property distributions may ultimately produce efficient and accurate data inversion methods for permeability and multi-scale flow predictions. In addition to geophysical data, geochemical datasets are important indicators of fluid migration patterns. New insights into the reliability of these data obtained across a range of space scales improve the use of this data resource. Note that the development of new measurement tools was outside the scope of the programme. The output of this theme feeds directly into Theme 3.
Theme 3: Statistical models and scaling laws for rock heterogeneity and fluid-rock interactions Geostatistical models encompassing fractal, multi-Gaussian, Boolean, Markovian as well as non-parametric, non-Gaussian approaches have all been adapted for the statistical characterization of the parameter distributions of fractured and porous media at a range of spatial scales. For robust application of these methods, only low order statistical moments have been extracted from the available observation data. While the expectation values derived for intermediate points between observation values have been found to be well conditioned, the textural patterns in real geological media controlling fluid flow have not been well reproduced. Biases in the fluid flow realizations generated using low order statistical models of rock properties are recognized but the magnitude of prediction errors are not characterized satisfactorily. While work undertaken in the/x2M Programme has improved understanding greatly, further research is needed to understand more completely the application of such models to fluid flow patterns in geological media. Integration of new measurement methods with new statistical models reduces the bias in the output of such models. Errors in prediction are as much related to uncertainties in the conceptual models describing the behaviour of the rock mass
as they are to uncertainties in the parameterization of these models. There is, therefore, a need to improve the conceptual models as well as the parameterization and the addressing of parameter uncertainty. Developments in self-similar systems in which power-law scaling are found across a wide spectrum of the spatial scales from the microscopic to the macroscopic. Such laws may allow better inference of behaviour at large or small scales to be made from data obtained from limited scale ranges and using current laboratory and field measurement methods. Research on the mathematics of heterogeneous 'noisy' (e.g. 1 / f noise) systems previously developed in unrelated disciplines provides new knowledge when applied to geological media.
Theme 4: Relationship between rock property and flow parameter distribution Simulators, whether used for data inversion from field experiments or for analysis of field-scale flow and transport patterns, require consistent unbiased input datasets derived from the available field data. Simulators needed for flow modelling at one scale demand upscaled property distributions derived from models of finer-scale variations in the geological media. Effort is also justified in analysing the level of detail to which modelling of geology should be taken prior to the upscaling procednre, which, to a certain extent, averages out much of the detail. Useful analogues are likely to be found in the fields of statistical mechanics and (non-equilibrium) thermodynamics. These may suggest how the statistical parameters of the underlying geology and (equally important in a commercial development) their uncertainties may be incorporated directly in fluid flow simulation. Mechanical, thermal and chemical processes can influence rock transport properties significantly, even during exploitation lifetimes. Hydraulic modelling may not be valid unless these additional effects are incorporated, particularly in fractured rock. Understanding the characteristics of the coupled non-linear dynamic system, particularly spatial patterns and scaling relationships, enables efficient modelling of the pertinent physics compatible with the small amount of information usually available in commercial resource development.
The Programme The broad scope of the programme' s themes outlined above, and the various constraints on the programme, did not allow all issues to be
OVERVIEW OF THE NERC ix2M PROGRAMME addressed. However, the funded projects all focused on fundamental elements of these issues and the majority of projects covered aspects from more than one, if not all four, of the themes. Brief summaries of all projects in the Micro to Macro Programme are provided below. The summaries have been compiled from information provided by each project team, mainly for the end-of-programme meeting held in November 2003, and their contributions are acknowledged gratefully. Many of the projects are covered in greater detail by other papers in this volume. The summaries are approximately ordered in decreasing scale from - basin (macro-) to small (micro-) scales. They have been numbered so that they can be cross-referenced easily from the introductory paper by Heifer.
A1 New micro-geochemical traces of fluid and solute transport in sedimentary basins (Graham, C.M. & Haszeldine, R.S. (Edinburgh University)) This pilot project aimed to develop and apply new analytical techniques to advance understanding of the sources of quartz cements in oil field sandstones. Quartz cements clog up pore space that could otherwise be filled with oil. In previous projects, the oxygen isotopic composition (180/160) of quartz cements has been used to constrain the temperature and timing of quartz cementation and the source of the fluid from which the cement precipitated. This pilot project used a multi-element tracer approach (silicon isotopes (63~ and lithium (Li), boron (B) and aluminium (A1) trace element concentrations) to help constrain the sources of quartz cement. Constraining the sources and distribution of quartz cement in the subsurface is important for helping oil companies to drill oil wells in the best quality reservoirs and to maximize production. Analyses were performed using the NERC Ion Microprobe Facility at Edinburgh University, which allows quartz cements to be analysed in situ in polished samples of sandstone. Two types of sample were analysed: 9
experimentally precipitated quartz overgrowths, grown at various temperatures in the laboratory from fluids containing Li, B and A1. The results show that the uptake of these elements into the quartz cements is temperature dependent and, therefore, that they can now be used to attempt to calibrate
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their use as natural geothermometers in quartz cements; quartz cements in North Sea reservoir sandstones in which the sources of quartz cements and the trace elements within them are more complex. The analytical technique for measuring silicon isotopes was refined to give precisions of _+ 1%o and quartz overgrowths and the remains of siliceous sponges were analysed to determine if sponge dissolution is linked to overgrowth precipitation. The similarity of the silicon isotopes in sponges and overgrowths from the various reservoirs examined suggests that this process is a contributor to quartz cementation in some reservoirs, which has not previously been considered to be important. The results of this study provide support for the adoption of a new, flexible multi-tracer microanalysis approach to determining reservoir hydrogeology and quality, and for constraining sources of silica for sandstone cementation. This work is not described separately in this volume but is discussed in Macaulay et al. (2000a,b). This pilot project led to the next project summarized.
A2 Cementation of oil field sandstones: Micron cementation reveals effects of kiiometre-sized hydrogeology, with porosity and permeability scaling (Haszeldine, R.S., England, G.L., Quinn, O., Bhullar, A.G., A1-Kindy, F., Barclay, S.A., Graham, C.M. (Edinburgh University); Corbett, P.W.C., Lewis, H., Potter, D. (Heriot-Watt University); Yardley, B.W.D., Cleverly, J., Fisher, Q. (Leeds University); Aplin, A.C. (Newcastle University) & Fallick, A.E. (Scottish Universities Environmental Research Centre)) The four principal objectives tackled by this project were: 9
9 9
to establish statistically valid and defensible sampling procedures to enable small size analyses to be scaled up to represent large size features; examination of sandstone cements with new micro-analysis techniques; to integrate small-scale mineral analysis with fluid inclusion chemistries, palaeo-pressures and with large-scale basin modelling;
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R.P. SHAW to undertake four case studies from the UK to develop, illustrate and apply the new techniques.
Using shoreface sands of the Scott Field in the Outer Moray Firth, statistical methods were developed to enable accurate sampling of the Coefficient of Variation (Cv) and the statistically appropriate number of samples required (No). The number of samples differs for different properties, and the spacing of samples along a core is also dictated by the rate of change of a property. For example, in the field studied illite cement requires one sample per 250 mm to constrain permeability within 100mD; Quartz cement and •180 ion microprobe values require one sample per 6 m, whereas permeability requires one sample per 50 mm. Using the Scott shoreface and Magnus submarine fan oil fields, porosity profiles were compared on a hundreds-metre scale with cementation effects on a micron scale. The micro-zonation of quartz cement in Cathode Luminescence, and the 8180 profile differs for the two fields: cements in the Magnus Field were forming during oil filling under hydrostatic pressure, whereas the cements in the Scott Field formed before oil charge with basin-derived saline fluids at a time of fracturing. The Magnus Field has a rapid porosity decline with depth, while the Scott Field has a normal porosity decline. One 100mm 3 sandstone sample may now be adequate to discriminate these two major oil field types. Studies of 3D exposed sand-sand faults in the onshore Moray Firth area at seismic scale and sub-seismic scale revealed that although these sediments have not been buried deeper than two kilometres, asymmetric quartz cementation has greatly reduced permeability across fault planes. Cementation was not caused by cataclasis or granulation, but by hot fluids advecting from the deeper basin. A strong spatial correlation between porosity and penrteability has been reported in Brae oil field sediments, together with a systematic power-law scaling of log-permeability. This is interpreted to result from connected large-scale fracture networks at spacings of over 5 km -1. Non-destructive magnetic susceptibility measurement of illite content in core plugs from the same borehole, has produced powerspectrum data showing correlation with a scaling exponent of 0.54. This may indicate that fracture networks control the distribution of diagenetic illite cement. In summary, micron-scale cementation effects do record macro-scale (km) hydrogeological
features and small samples can now be used to represent genetic units of sandbodies and accurately scale-up to reservoir and basin-sized models.
A3 Multi-scale fluid-flow path analysis: calibration and modelling using mineralization systems (Yardley, B.W., Barnicoat, A.C., Freeman, S., Banks, D., Gleeson, S. (Leeds University); Wilkinson, J.J. (Edinburgh University); Graham, C.M. (Edinburgh University); Boyce, A.J. (SURRC); Blakeman, R. (Glasgow University and SURRC); Everett, K. (Imperial College) & Ashton, J. (Tara Mines, Navan, Eire)) The aim of this project was to integrate microand meso-scale structural and geochemical data from the Irish Midlands in order to understand the fluid flow and fluid mixing responsible for the development of world-class zinc-lead orebodies, in particular the Navan deposit.
Flow path distribution A detailed analysis of grade data from Tara Mine, together with underground mapping, has shown that fracture networks play a fundamental role in controlling ore distribution (and, by inference, fluid flow paths) in the Navan zinc-lead deposit. Detailed SEM and CL petrography indicated that only very limited gangue mineral deposition, attributable to low-temperature, very saline brines, has occurred away from the orebody. This suggests strongly that, despite significant porosity and inferred permeability at the time of ore deposition in surrounding lithologies, there was little flow of either fluid necessary for ore formation through the stratigraphy other than via fractures. Faults and fractures formed a mesh (sensu Sibson 1994) developed at very low strains by the reactivation of Caledonian structures in the basement in response to the early stages of Carboniferous extension. Fluid flow, mixing and ore formation were much more effective when this mesh was active, prior to the formation of through-going faults now dominating the area's structure.
Fluid types Clear evidence has been collected from sphalerite-hosted fluid inclusions showing that two fluids, one hotter and of moderate salinity, the other cooler and much more saline, mixed during ore formation in the Navan orebody.
OVERVIEW OF THE NERC p~2MPROGRAMME Chlorine and bromine analyses of inclusions of the cooler fluid show that it originated by evaporation of sea water beyond the point of halite saturation at the Earth' s surface, and that the sulphur it carried has a negative 634S signal, indicating that bacteriogenic sulphate reduction had been operative in the near-surface environment.
Integrated view of fluid flow paths Petrographic evidence indicates that little or none of the low-temperature brine migrated along stratigraphy, so, when combined with the recently published results of Blakeman et al. (2002), these data suggest that the two fluids found at ore formation stage both migrated along fault/fracture permeability. Underground and petrographic observations show that normal processes of chemical and, to a lesser extent, mechanical compaction lead to reductions in porosity to between 10% and 20% (depending on lithology) prior to fracturing associated with the early stages of extensional faulting. This led to extremely large increases in system permeability along a mesh of small faults, which ceased at greater displacements when a small number of through-going structures developed.
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Wall- rock interaction The hot, deep fluid has reacted with host rocks in the basement as shown by the isotopic composition of lead and the deep-sourced sulphur shown by both the data from this study and those of Blakeman et al. (2002) and others. The cool, surface-derived brines reacted with carbonates on their way to the site of ore formation, and the dolomite 'plume' overlying the Navan orebody is potentially the result of brines migrating down faults and, to a limited extent, along the most permeable stratigraphy.
Flow directions and driving forces The data collected in this project show clearly that one of the fluids was migrating downwards driven by its remarkably high density caused by its salinity, and the other was moving upwards. The numerical modelling reveals that the density-driven down flows are so strong that thermal effects alone are insufficient to allow for its upward migration. Overpressures (limited to 10% of the difference between hydrostatic and lithostatic pressures) were essential to allow its upwelling. Either breaching of lithostatically pressured compartments or transient, deformation-induced overpressures were necessary to allow the deep fluid's upflow.
Modelling of chemical and physical behaviour The chemistry of the two fluids inferred to have been responsible for ore formation have been modelled using: 9 a combination of fluid inclusion and mineral saturation constraints (hot, less saline fluid); 9 forward modelling of sea water evaporation and heating and reaction with limestone (cool, saline, surface-derived fluid). Reaction of these two fluids together yields, depending on which is mixed into a reservoir of the other, ore and gangue minerals that match quite well with observed distributions. The physical process of fluid mixing has been examined in single- and multiple-fault systems with dense saline fluids introduced at the surface and hot, less saline fluids at the base of the model. With a rock-package permeability structure similar to that at Navan, both fluids migrate to more permeable parts of the stratigraphy along faults. Reaction, the distribution of which has been tracked by computing salinity gradients with time, occurs at differing places within the most permeable units. These features model very well the distribution of ore found at Tara Mine.
A4 Quantifying contributions from matrix or fracture flow by geochemical analysis of produced oil (Coleman, M. (Reading University)) Oil in a chalk reservoir is hard to produce because, although the porosity may be very high, the permeability is usually very low. The porosity of the chalk matrix may contain most of the oil but it is usually extracted via fractures in the rock. To plan for optimized production of oil it would be valuable to monitor the contributions from the matrix and the fractures. This project successfully combined a number of novel and potentially risky approaches to attack the problem. It was postulated that brines from fracture porosity and matrix should have different stable isotope compositions of the chemically inert element chlorine, and the project tested this concept in the laboratory and the field (oil field). In high-pressure compaction and brine flow experiments, where axial load and confining pressure could be varied independently, flow was started and permeability determined by measuring the flux of brine through chalk
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cores. Control of the confining pressure allowed fractures to be opened and closed, with corresponding changes in permeability. Isotopic compositions of the brines used in the experiments were measured. Variations in chloride isotope composition occurred as expected. ~37C1 (the measure of C1 isotope composition) correlated positively with mean permeability for each pressure stage of the experiment, that is the more positive values were associated with fracture permeability and the negative ones with lower permeability in matrix. Attempts to characterize chalk porosity were made by impregnating samples with resin and dissolving the chalk to leave a pore-cast for SEM examination. The method was only partially successful and was not satisfactory for identifying fractures unless they were near to the surface and within range of the shallow penetration depth of the resin. SEM images of the chalk itself are prone to artefacts because subsamples either break along fractures or may have fractures induced by the strain of taking a sub-sample. However, a novel method of 3D imaging of 5 mm diameter chalk cores using X-ray micro-tomography was developed. Internal fractures can be seen clearly in samples in which they might be expected, and the method could be developed to give quantitative estimates of porosity and permeability. Produced oil samples were specially taken for this project monthly over two years from a selection of 14 wells in the Valhall Field, with minimum co-produced water. The method, developed under a previous NERC project, to extract trace water from the 'dry' oil was used to obtain samples of the aqueous phase that the oil carries from the oil zone of the reservoir. Chemical analysis of the extracted trace waters showed that they were sodium chloride brines. Overall salinity and trace solutes (e.g. potassium, magnesium, sulphate and barium) defined six end-member compositions, some mixtures between them and three hydrologically isolated areas. The compositions related to chemical zonation of the field with some controls on composition apparent: for example, the most saline was adjacent to a salt dome. ~37C1 of extracted trace brines showed considerable variation. The oil field is a dome structure with highest permeability from fractures, generally found in the crest area, where the strain of flexure was greatest. The crest samples had the most positive ~37C1 values, which the experiments showed was associated with fracture permeability. Conversely, the lowest values were associated with the samples from flanks of the field with lowest measured permeability, from
matrix. The one exception to this was from one off-crest well to the west of the field that had high 637C1 values, but this was associated with a fault where fractures might be expected. During the two years of monitoring chemical and isotopic compositions, each well retained its distinctive values and showed no significant temporal variation. This indicates that there was no compaction of fractures measurable in this period. The most significant finding is that the chlorine isotope variations are independent of the chemical compositions of the brines and their zonation are thus a function solely of reservoir permeability. The approach developed offers a method of monitoring variations in reservoir permeability and, especially, compaction of fractures, during extended periods of production. However, it can be performed cheaply by routine analysis of produced fluids rather than by intervention, which would interrupt production and also be expensive and, therefore, unlikely to be undertaken.
A5 Mudstone microstructure evolution during burial and diagenesis and effect on cap-rock sealing capacity (Worden, R., Charpentier, D. (Liverpool University), Aplin, A. (Newcastle University) & Fisher, Q. (Leeds University)) Natural mudstone properties are important because they control how much petroleum can be trapped beneath a mudstone cap rock deep in sedimentary basins and the isolating capacity of aquicludes in shallow aquifers. This project was planned as a pilot project to investigate the controls and scale-effects of mudstone rock properties. Two basins that had contrasting thermal and burial histories were selected as natural laboratories: Miocene mudstones from the rapidly buried Gulf of Mexico; and Cretaceous Shetland Group mudstones from the more slowly buried Northern North Sea. Mudstones from a wide range of depths from each basin were chosen to capture changes in properties during burial. These mudstone samples were examined with a wide range of techniques, including core analysis, scanning and transmission electron microscopy, X-ray texture goniometry and X-ray diffraction. It is well known that mudstones, typically rich in the clay mineral smectite at the time of deposition, become progressively dominated by the micaceous clay mineral illite during burial and heating - a form of incipient metamorphism.
OVERVIEW OF THE NERC ix2M PROGRAMME The samples from the Gulf of Mexico and North Sea conformed to this pattern. More unusual was the conclusion that the compaction experienced in the mudstones occurred over the same depth interval as the transformation of smectite into illite. Microstructural studies at a range of scales revealed that mudstones without identifiable fabric (isotropic microstructure) after minimal burial developed a distinct fabric over the same depth interval that saw the evolution to low porosity and the change from smectite to illite. Image analysis of backscattered electron micrographs using Scionlmage proved capable of quantifying the fabric change at a 10 Ixm scale. Transmission electron microscope analysis, tricky with such low strength samples, revealed that shallow-buffed smectite-rich mudstones are dominated by high-angle boundaries with attendant microporosity (on the nanometre scale), while deeper-buried, illite-rich mudstones are dominated by low-angle boundaries (subparallel crystals) with lower microporosity. The study has revealed that porosity loss and smectite-illite transformation in mudstones are likely to be interrelated and that they occur at the same time as the formation of well-developed fabrics. These simultaneous changes are unlikely to be coincidental. It is possible that the master variable is the time-temperature-dependent mineralogy change that facilitates grain rearrangement and porosity loss. This novel conclusion will be investigated further to help refine understanding of the ultimate controls on mudstone rock properties. The long-term goal is to be able to predict mudstone permeability for flow modelling at a range of scales in a range of basinal settings from aquifers to oil fields.
A6 Scaling of fluid behaviour associated with flow through complex geological structures (Harris, S.D., Pecher, R., Odling, N.E., Knipe, R.J., Ellis, J.A., Elliott, L. & Ingham, D.B. (Leeds University)) This project undertook the development of new methods for integrating the geological structural characterization of fault zones with the numerical modelling of fluid flow behaviour at different scales.
3D stochastic model The new three-dimensional fault zone flow model (Harris et al. 2003) developed by the project allows the analysis of tortuosity and connectivity through arrays of faults using
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percolation techniques and pathway searching algorithms. The spatial clustering of minor faults about major faults is a critical element of the model described here, with the principal aim being to account for the 'sub-seismic'-scale damage zones which exist around seismically mappable faults (Antonellini & Aydin 1994; Knipe et aL 1997; Shipton & Cowie 2001). A new methodology has been developed which both derives the minimum value of the fault rock thickness along flow paths traversing the fault zone and predicts areas of reduced fault zone connectivity for matrix host rock and fault rock of varying permeabilities.
Discrete fracture flow model (DFFM) A 2D, DFFM approach for flow in fractured rocks has been extended to handle fractures (faults) as partial to complete flow barriers in 2D and 3D (Odling et al. 2004). The newly developed DFFM has been applied to 2D slices through the new stochastic models of fault damage zones described above using a permeability contrast of four orders of magnitude, typical of minor faults in permeable sandstones (Odling et al. 2004). With respect to the whole fault zone, the fault damage zone is found to dominate the hydraulic behaviour when the major slip zone fault rock permeability is at most one order of magnitude lower than that of the minor faults. The new fault damage zone efficiency concept provides a method of estimating bulk rock permeability from the proportion of fault rock which can be measured from core or borehole logs and provide input into reservoirscale flow models, such as Eclipse. Control volume finite element (CVFE) model The CVFE scheme (Fung et al. 1992; Verma 1996) can be used with unstructured grids in two- and three-dimensional domains. New methods for incorporating faults within the CVFE approach have been developed. The permeability of the faults relative to the matrix was varied over a wide range and a critical transition was observed at which the faults influence the flow properties. The CVFE model allows for the matrix permeability to be inhomogeneous and an anisotropic tensor, while the faults can have varying thickness and permeability over their surface.
The Green element model (GEM) GEM (Taigbenu 1996) is an exciting new technique derived from the standard boundary
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element method that combines the high accuracy of the boundary integral techniques with the broad versatility of the so-called domain techniques. In this project, significant progress has been made in tackling the practical problems of the application of the new GEM approach to flow in faulted rocks (Pecher et al. 2001) and has highlighted the future potential of the method.
Key achievements/developments New sophisticated models of flow (control volume finite element and Green element approaches) have been developed together with a new finite-difference model (2D/3D) for flow through rocks with variably permeable faults. Application of a 2D/3D flow model to the new 3D stochastic model of fault damage zone geometry allows comparisons of flow properties predicted by the flow model and pathways through the geometric model. These models have been used to establish scaling laws for the bulk permeability of the fault damage zones and methods to estimate fault damage zone bulk permeability for reservoir-scale models.
A7 Determination of hydraulic properties of distributed fractures using seismic techniques (Liu, E. (British Geological Survey) & Hudson, J.A. (Cambridge University)) The research in this project has focused primarily on the following tasks: 9 9 9 9
scaling; fluid-rock response; statistical analysis; data analysis.
The project has been very successful and has made significant advances in all these areas. There is now a growing interest in monitoring dynamic reservoirs using time-lapse seismic methods; this research topic has been very timely and clearly provided new understanding of the dynamic behaviour of the rock mass (i.e. stress-sensitive elastic deformation of porous rock) and the seismic characterization of fluid transport to meet the industry need. Significant progress has also been made on the understanding of dynamic characteristics of fluid-solid interactions through frequency-dependent wave propagation in cracked porous rock. The main achievements are summarized below.
Development of a new fluid dynamic model arising due to the interaction of micro- and macro-scale fractures and pores at the time period of the seismic wave. In contrast to the predictions of more simplified models, it is shown that the scale length of the fractures plays a key role in the analysis, leading to frequencydependent anisotropy at seismic frequencies. The fluid dynamics are considered with the aid of a lattice model. The model allows for the fractures to be of a different size to the micro-cracks and pores. The advantage of this arrangement is that the length scale of the fractures enters the analysis. In conventional models of anisotropic fracture distributions, behaviour depends only on the crack density. This means that it is impossible to discriminate between the cases where anisotropy is caused by a few large cracks or many small cracks. Such a distinction is critical in the attempt to link seismic anisotropy to the large-scale fluid flow properties of the rock. The commonly used Hudson theory for modelling wave propagation in cracked media has been extended in a number of ways, including interconnected nearly-aligned cracks, fluid flow in bed-limited or layer-bounded media and cracks in poroelastic media. The later extension allows cracks to be inserted in porous media (Biot's porous media), in contrast with the earlier Hudson's model, which only considers cracks in elastic solids. The earlier model of fracture models with micro-crack alignment on fracture surfaces is now extended to include arbitrary crack orientations. In particular, the bed-limited crack model can now handle spheroidal cracks of arbitrary shape. As a limiting value of the general result valid for arbitrary shape, simple expressions are derived for the elastic stiffnesses in the case of flat elliptical cracks that take the same form as the original expressions. These then reduce to those original expressions in the further limit in which the two parameters controlling the shape of the ellipse are identical. In addition, the theory has been extended to incorporate a model of fluid flow that is dominated by the flow from one crack to another and is valid at all frequencies from seismic through logging to ultrasonic. A numerical method using 2D pseudo-spectral methods to model spatial and scale distribution of discrete fractures has been developed. The fractures are treated as planes of weakness using the concept of the linear-slip deformation or displacement discontinuity model. The implementation of fractures with a vanishing width in the finite difference grids is done using the method introduced by Vlastos et al. (2003).
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Numerical results have been performed to investigate the effects of scale length (sizes) and spatial distributions of fractures on the characteristics of propagating waves. These show that the wavefronts can be affected significantly by the presence of fractures with different scales or lengths relative to the wavelength, and also show that different spatial distributions of fractures can give characteristic features on the wavefields, implying that the information about the fracture distributions in natural rock may be obtained from seismic data.
to the direction of the hydraulic gradient that dictates the type of curve produced and not necessarily the type of fracture pattern. Two-set patterns, on the other hand, produce a more restricted range of curve types. A breakthrough curve classification scheme, using principal component analysis (PCA), was devised and linked to fracture pattern properties. There are two approaches to classifying additional breakthrough curve data using PCA.
A8 Tracer tests, hydraulic properties and fracture networks at sub-continuum scales in aquifers (Johnston, P. (University College
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London)) Fractures spaced at c.0.1 m to c. 10 m impart continuum scales to fractured aquifers which may exceed the size of many practical problems. Sub-continuum flow models often use statistical simulations of explicit fracture networks, which can lead to serious problems of system identification and uniqueness. To address this problem, the aim of the research was to determine whether tracer breakthrough curves could be used to characterize fracture topology at subcontinuum scales. The approach used in this project was one of forward modelling, starting with a prescribed fracture network in a homogeneous permeable medium. The flow field and tracer transport were simulated for each network in turn, using a 2D finite difference numerical code (Odling & Webman 1991; Odling & Roden 1997) in which advection was the only dispersive process. Both one-set and two-set patterns were simulated. The output from the model - the tracer breakthrough curve - is a record of the fracture geometry, flow field and bulk equivalent permeability of the model and of the chosen injection pattern. Linear and radial flow fields were simulated, allowing the results to be compared with natural and forced gradient tracer tests conducted in field experiments. By repeating the modelling procedure for different networks, the combinations of bulk permeability and breakthrough curves that each produced were explored. Over 3000 patterns were simulated. Results indicate that 2D fractured porous media can produce a large variety of tracer breakthrough curves, including two patterns rather commonly seen in field data: the backward-tailed unimodal, non-Fickian type; and the Fickian. For the one-set case, it is the combination of the spacing, aperture and angle
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The breakthrough curves are added to the raw dataset and a new covariance matrix is determined. The raw dataset is considered to be definitive, giving a fixed set of orthogonal functions in terms of which any further function can be determined as a linear sum with coefficients representing the principal components.
In this study the latter was assumed. Given a breakthrough curve, produced by a previously un-modelled pattern, it is shown that, using PCA and permeability as constraints, one can reduce the number of possibilities of fracture spacing, aperture, angle of rotation to the hydraulic gradient and pattem responsible for producing that curve.
A9 Novel flow and transport models for systems exhibiting non-integer flow dimensions (Sellers, S. & Barker, J. (University College London)) The original project aim was to investigate the possibility of developing novel methods for modelling transport in heterogeneous systems. This was based partly on the increasing number of hydraulic tests, mainly in fractured rocks, that exhibit non-integer dimensional flow (i.e. not linear, cylindrical or spherical flow). Further motivation came from the fact that observations of fractal dimensions have become quite common: typical measurements refer to pore geometry, surface roughness, fault traces and relationship between fault number, lengths and widths. An underlying fractal structure appears consistent with non-integer flow dimensions and, as a pumping test is modelled by a diffusion equation (pressure diffusion), the work came to focus on (generic) diffusion on fractals. An initial literature search indicated that there are numerous models for diffusion on fractals, with no agreement as to the correct one. The approach taken here was to construct simple fractals with known characteristics (such as fractal dimension) and then simulate diffusion with random walks. Sierpinski carpets were chosen
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as they are flexible enough to generate arbitrary fractal dimensions. Although rather special cases, they have been proposed as a means of constructing porous media. Random walks of up to ten million time steps were simulated on ninth generation Sierpinski carpets discretized with a 19 683 x 19 683 lattice. The resulting displacements as a function of time were averaged over up to 100000 particles with the same initial starting point. The standard procedure in the literature is to average the results over initial conditions to obtain results independent of the starting conditions. The goal, however, was to simulate a boundary value problem on a specific fractal as might occur in an experimental situation.
Main results The presence of intemal boundaries at all scales significantly affects the behaviour of the random walks. In fact, the graph of displacement squared versus time often shows significant deviations from a straight line on the standard log-log plot for the investigated time-scales. The deviations due to the intemal boundaries are oscillations about a straight line, with periods that can last several decades, thus making difficult or impossible the determination of the asymptotic slope and, hence, the random-walk dimension. The random walks can show anisotropic behaviour in that random-walk dimensions for horizontal displacement and vertical displacements can be unequal. The dimension results are strongly dependent on the origin. In fact, one origin can yield isotropic response whereas a nearby origin of the same fractal yields anisotropic response. Equivalent points corresponding to different generation levels of the fractal, however, yield identical dimensions. Thus, the random-walk dimensions have an anisotropic, multifractal structure with respect to origin. The dependence of the random walks on the origin does not decay in the simulated timescales, indicating a long-term memory effect. This result differs from that in Euclidean lattices where the effect of the origin decays exponentially in time.
Conclusions and significance Random walks have been used to show that very simple fractals such as Sierpinski carpets can exhibit complex and surprising behaviour. In particular, the characteristic linear graph on a log-log plot of displacement against time may not be observable for data collected on a
limited time-scale. Any underlying fractal structure may, therefore, not be readily observable through pumping tests. All known diffusion models based on differential equations assume isotropy and attribute a single random-walk dimension to a fractal. The results from the simulations are inconsistent with these models and cast doubt on the applicability of a single differential equation for the entire fractal. Further, the dependence on the initial conditions indicates that fractals are heterogeneous, so that data collected at one point of the fractal may not be valid for another point of the fractal. If the differential equations were modified to allow for anisotropy, then they would be expected to hold at most at a fixed point. Also, it is not clear how to relate the observed random-walk dimensions with the underlying fractal dimension.
A10 Modelling porosity development in heterogeneous fracture networks (Bloomfield, J.P. (British Geological Survey) & Barker, J.A. (University College London)) The objective of the project has been to produce a model of coupled flow and porosity development in heterogeneous porous media and to use the model to investigate scaling phenomena. A code, MOPOD, has been produced and the task of computing porosity development has been formulated as an 'initial value problem'. The model is highly flexible: it is capable of modelling 2D and 3D arrays with both regular and random structures, a wide range of initial aperture distributions and flexible boundary conditions, including constant head or flow conditions. In addition, codes for visualizing the arrays and for simulating particle transport, which reveals the effects of filtering in relation to particle size, have also been developed. Even though the model is formulated in terms of a simple system, evolved arrays can be highly complex and parameterization and prediction of their evolution is not trivial. Consequently, investigation has focused on a very simple porosity growth law of the form dai/dt = v~, where ai(t ) is the aperture of pore i at time t, vi is the magnitude of the volumetric flow rate in pore i, and e is the aperture growth rate exponent. The evolved structures are highly sensitive to initial porosity distributions, growth-rate exponents and hydraulic boundary conditions; they range from relatively uniform porosity development, through complex anastomosing geometries, to the development of preferentially enlarged
OVERVIEW OF THE NERC [z2M PROGRAMME array-spanning paths with long-range correlations. A limited study of percolation in the arrays has been undertaken; however, because the porosity fields become highly structured as they evolve, the assumption of a random porosity field required for analysis using percolation methods - becomes invalid. There is no evidence for scaling of either the porosity or flow fields in the evolved arrays for the growth laws investigated. Although spatial correlations develop in the porosity distribution, pore structures evolve towards a final state and selforganization phenomena are not observed because periodic or cyclic behaviour is not inherent in the simple growth laws investigated. The MOPOD code has been used to characterize variations in transmissivity and to investigate the breakthrough characteristics as the porosity field develops. The effective transmissivity of the arrays is sensitive to the growth-rate exponent and is a power-law-like function of time. The form of the tracer breakthrough is also dependent on the evolved pore structure and, hence, the growth-rate exponent. For example, porosity fields with preferentially enlarged array-spanning paths are associated with high concentrations of tracer breakthrough at relatively early times. The principal outcomes of the work and their significance are: 9 a flexible, generic, model of coupled flow and porosity development in heterogeneous media has been developed that produces porosity structures with long-range correlations without relying on process-specific assumptions. The model provides an alternative to current methods: 9 stochastic methods that do not readily reproduce such long-range correlations or pathways; and 9 process-specific approaches that may grossly oversimplify the field situation and that can require significant computational effort. New insights have been gained into the development of secondary porosity. It has been demonstrated that complex porosity fields can develop from even simple porosity growth laws, that critical exponents are associated with phase changes in the geometry of the evolved structures and that porosity fields develop dynamically stable configurations during an initial phase of growth followed by simple amplification of porosity at later times. This work provides a better understanding of how secondary porosity
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is likely to develop in aquifers and hydrocarbon reservoirs and will enable more realistic prediction of flow and transport.
A l l Localized flow in fractured rock masses: mechanisms, modelling and characterization (Sanderson, DJ., Zhang, X. (Imperial College) & Barker, A.J. (Southampton University)) This project mainly involved coupled mechanical/hydraulic modelling of the effects of stress on fracture networks using distinct element numerical methods. The main achievements of the research are noted below. Numerical modelling has demonstrated that deformation and flow exhibit critical behaviour in terms of stress, with both increasing nonlinearly to become highly localized (Zhang & Sanderson 2001, 2002a; Sanderson & Zhang 2004). At the critical stress state, rock deformation is enhanced by slip on fractures, with rock friction being the main controlling factor. The material properties (e.g. stiffness) control the geometry of the resulting structures. Flow becomes highly localized at the critical stress state and can be described in terms of multifractals. The critical stress state may be described in terms of a driving stress ratio R = (fluid p r e s s u r e - m e a n stress)/l(differential stress). Instability occurs where the R-ratio exceeds some critical value, Re, in the range - 1 to - 2 . This result may be used to evaluate in situ stress in the crust, which appears to be close to the critical state throughout much of the upper crust. Models with fractures and smaller polygonal discontinuities (grains) have demonstrated the importance of dilational shearing on deformation and flow, providing discrete dual porosity/dual permeability simulations of flow (Zhang & Sanderson 2002b; Zhang et al. 2002). Models demonstrate sensitivity to fracture geometry, grain texture and stress, and to cyclic loading. Simulated well tests have been conducted, including slug tests (Zhang et al. 2002) and changes in flow during excavation of a shiplock in China have been simulated (Zhang & Sanderson 2002c). Permeability/depth variations in fractured rock have been simulated and compared with borehole data. Results indicate heterogeneous permeability and localized flow similar to that modelled at the critical stress state. Near-borehole effects have been modelled and a new type of 'block loosening' deformation identified (Zhang & Sanderson 2002d), which is quite different to traditional 'wellbore breakout'.
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Fieldwork has allowed quantification of damage around faults and has established a power-law distribution of vein opening from decimetre to micro- (10lxm) scales. At the micro-scale, grain processes dominate; at the macro-scale, reactivation of slip on fractures is most important.
range of heterogeneous media types. For this library to be useful, several factors had to be addressed. These included 9 9
A12 Quantifying fluid movements in heterogeneous formations at different scales and their contribution to physical transport (Xie, Z., Mackay, R. (Birmingham University) & Cliffe, K.A. (Serco Assurance)) The use of numerical models to predict subsurface flow and transport has received much attention in the fields of water resources, petroleum engineering and waste disposal. While modelling is important, the predictions made with the available models are often highly inaccurate, particularly for transport problems. The cause of this inaccuracy is attributed primarily to the unknown fine-scale heterogeneity of the flow paths and secondarily to weak determination of the fluid stresses on the system. New data collection strategies have made little progress in improving accuracy, but significant progress has been made towards bounding the errors by exploring the prediction uncertainty using Monte Carlo simulation methods. These methods employ stochastic descriptions of aquifer heterogeneity. However, to model at the space- and time-scales of relevance to decision making, averaged process equations are adopted and the stochastic properties of the geological domain are typically characterized at a scale much greater than the scale of heterogeneity affecting the migration patterns. This leads to a serious question about how good the large-scale equations and the supporting property distributions are at capturing the transport behaviour and the prediction uncertainty. Many papers have been produced that address this question but most adopt either simplified models of geological heterogeneity or consider only short space- and time-scales. Results that are relevant to more realistic geological models and large space- and time-scales are needed. A rigorous numerical modelling study is presently being undertaken that aims to bridge the apparent gap between practice and theory. The foundation for this work has been the preparation of a library of reliable, large space and time simulations for two-dimensional domains encapsulating detailed descriptions of the finescale flow paths and flow velocities through a
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the minimization of numerical artefacts introduced to generate the media and simulate fluid movement; the production of highly accurate flow and transport simulations; the control of boundary influences on flow geometry and transport behaviour; and, importantly; validation against analytical solutions for known heterogeneity.
An iterative Choleski decomposition technique has been developed for generating Gaussian random fields obeying standard covariance functions. This technique is computationally efficient for generating large numbers of alternative realizations with the same spatial statistical model. Importantly, ensemble statistical behaviour is well reproduced over a large number of realizations generated by this method. Nonparametric transformations of the Gaussian fields permit the production of more geologically realistic structures. One thousand realizations have been produced for several spatial statistical models. A mixed finite element discretization of the steady-state groundwater flow equation has been used for the numerical simulation of the flow paths. This formulation allows exact calculation of particle trajectories considering advection only. Initial work to implement periodic boundary conditions perpendicular to the major flow direction was abandoned when particle migration patterns were found to also degrade to a periodic form. Periodicity parallel to the major flow direction proved not to suffer from the same restriction. Exploitation of this condition has allowed the domain width perpendicular to flow to be considerably less than the domain length. Sensitivity analyses have been performed to determine the optimal rectangular domain size for the random fields. Domains corresponding to 40 correlation lengths parallel to flow by 12 correlation lengths perpendicular to flow employing a grid resolution of not less than 50 elements per correlation length were eventually adopted as optimal for the library. Validation of the combined suite of media generation model, flow model, particle migration model and chosen domain size has been undertaken by comparison of the numerical results against the analytical results for ensemble spreading parallel and perpendicular to the flow direction developed by Dagan (1990).
OVERVIEW OF THE NERC I,z2M PROGRAMME The results are accurate for low variance media for short to intermediate times but deviate significantly for larger times. The causes of this deviation are currently being explored but are partly contributed by boundary interference, which seems unavoidable. However, not all of the deviation can be explained in this way and a re-examination of the flow model approximations as well as the approximations exploited in the development of Dagan's analytical model is underway.
A13 Analysis of reaction and flow in stochastically heterogeneous porous media (Huppert, H.E. (Cambridge University) & Bonnecaze, R.T. (University of Texas at Austin)) Reaction and flow in porous media occur in a variety of situations. These include orebody mineralization, use of acids in oil wells to enhance oil recovery and the movement of non-aqueous phase liquids and aqueous contaminants in rocks and soils. In homogeneous porous media, these systems lead to an instability that, in turn, leads to the development of fingering of the reaction and flow front created by the positive feedback between flow and reaction. Real porous media are not homogeneous and it is unclear how their heterogeneities affect the instability. Details of the heterogeneities are usually unknown but they can be described statistically. This pilot study has examined the stability of a reactive front in a sinusoidal and stochastically heterogeneous porous medium by both analytical and computational methods. It has the aim of predicting the characteristics of the instability from the statistical description of the heterogeneous porous media. A model of reaction and flow for two types of heterogeneous porous media has been analysed. The first model considered flow in a porous media with two-dimensional sinusoidal variation in permeability that is altered by advected solute with the rock. This model system can be analysed to understand flow and reaction in nonhomogeneous porous media. This heterogeneity generally enhances the instability and rate of growth of the fingers and creates a preferred mode for growth roughly corresponding to the frequency of the heterogeneity. Based on the understanding gained from this model, a model for reaction and flow in a stochastically heterogeneous porous media was developed. This model provides a means to predict the statistics of the instability in such porous media.
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AI4 The scaling behaviour of fluid flow in rough rock fractures (Ogilvie, S., Isakov, E. & Glover, P. (Aberdeen University)) Fluid flow through natural fractures depends upon the scaling behaviour of their fractually rough surfaces. Techniques have been developed for imaging fluid flow in natural rock fractures themselves, by using high fidelity physical models (HFPM) and high-resolution numerical simulations. These advances have been used to examine the scaling behaviour of miscible/nonmiscible fluid flow in fractures, paying particular attention to: 9 9 9 9
channelling, fingering, dispersion and flow stability; surface wettability; normal and shear deformation; and nature and development of fractural fracture matching.
Digital optical imaging techniques Techniques have been developed fully to either observe a HFPM fracture pair during fluid flow through the rough fracture or, on a single fracture surface covered with dyed fluid, to enable pointwise determination of fracture surface topography. The result is a high resolution optical determination of fracture surfaces and apertures with topographies of the two fracture surfaces for all project samples now determined to within 15 txm in the fracture plane (640 x 480 pixel image) and to within 15 ~xm vertically. This method provides a faster, higher resolution and cheaper technique than the more commonly used stylus profilometry. A further advance in optical profiling is the robust use of devices to calibrate the dyed fluids and equipment used in the imaging process. Previous studies arbitrarily scaled the resulting surface heights. Profiles of fracture surfaces and aperture maps have been determined for all samples.
Fracture parameterization and creation of synthetic fractures This experimental work has led to the in-house development of software (SynFracTM), which creates suites of 'digital' fractures whose characteristics are finely tuned to those of the measured fractures. It is important to have such a program for generating synthetic fractures because fluid flow modelling is best carried out on a suite of fracture data that share the same characteristics to enable the derived parameters to be a reasonable average rather than depending upon one
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implementation that may not be representative as a result of its specific structure. Clearly, such a dataset is impossible to obtain from real rocks because it would be too time consuming and different rock fractures may differ in their basic structural parameters. Using SynFracT M it is possible to create synthetic fractures at five different resolutions, from 64 x 664 to 1024 • 1024. Three different synthesis methods can be used, the simple Brown (1995) method, the Glover et aL (1998) method and a new, improved method (Ogilvie et al. 2003; Isakov et al. 2001) that allows high quality partial correlation of random deviates. There is also a choice of three different extremely high quality random number generation methods. The synthetic fractures can be simply analysed for mean fracture aperture and surface height information in the program, which also enables the operator to view the resulting synthetic rock fractures and apertures both in greyscale plan view and along any mouse-click chosen perpendicular profiles. The program outputs in jpeg, tiff, usgs, table text and triple column text formats for use in documents or in fluid flow modelling programs. Fluid flow experiments
Fluid flow experiments have been carried out on the HFPMs of all samples at a wide range of flow rates using a flow rig and imaging equipment. This involves releasing water, then dyed water from fluid reservoirs on the rig, moving the output position which controls flow velocity and recording pressure difference on manometers. For each measurement, a Reynold's number is calculated and a series of flow images versus time are captured using Adobe Premier 5.1TM video capture software. 2D flow models
All data from previously described stages of the project provide boundary condition input into 2D fluid flow models in the plane of the fracture surface. The modelling was performed in a FemLabTMenvironment, fracture geometries from SynFracT M 'digital' fractures and Reynold' s numbers from experimental fluid flow. This combined experimental-numerical approach is very successful in enabling the physical constraints upon fluid flow in rough rock fractures to be well characterized. The Reynold's equation does not account for fracture surface roughnesses, therefore the Femlab modelling involved the solution of the Navier-Stokes, Diffusion and other partial differential equations for an incompressible fluid confined by the
complex boundary conditions set by the rough fracture walls. The preliminary stages of this work have already been published (Ogilvie et al. 2001) and are about to be published (Ogilvie et aL in press).
A15 Measurement of complete fluid velocity fields in 2D heterogeneous porous media (Cassidy, R., McCloskey, J. & Morrow, P. (Ulster University, Coleraine)) The ubiquitous scale invariance of geological material and the consequent absence of a length scale on which to base the upscaling of measurements made on geological samples, represent a serious challenge to the understanding of fluid behaviour in rock. Numerical simulation is an important tool for understanding and predicting the movement of fluid in geological materials and current discrete fluid models, in which complex boundary conditions present no serious challenge to the modeller, have the potential for testing many possible upscaling schemes. At present, however, there is no accurate empirical data on the distributions of fluid velocities in complex, scale-invariant geometries and models can only be tested against analytical solutions in simple geometries or against bulk experimental measurements. The assumption that a model that successfully passes these two tests is able to solve accurately the flow problem in more realistic media remains untested. The project set out to measure fluid velocities everywhere in complex 2D media with fractal heterogeneity. First, digital models with scale-invariant geometries were created and then translated into physical form. The two models described here are: 9
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rough fracture, defined by two self-affine fractal surfaces, which is cut from a sheet of aluminium using wire erosion machining and; digital combination of rock matrix and fracture porosity produced by the superposition of the void space from a discrete-element model and from a fracture model containing a fracture set with a fractal length distribution. This model is cast in resin using selective hardening of photosensitive polymer; a technique known as stereolithography.
These models were then enclosed between parallel sheets of glass and Perspex forming a Hele-Shaw cell. The cells were permeated with water, doped with small neutrally buoyant spheres and pumped at accurately steady and
OVERVIEW OF THE NERC ~2M PROGRAMME reproducible velocities using a high precision HPLC pump. Local velocity vectors were estimated by the analysis of sequential images of the spheres over areas of about 0.25 mm z using a high-resolution video camera. Precision digital control systems were used to move the cell and measure the fluid velocity; repeated measurements allow the construction of full 2D velocity fields. The accuracy of the technique was assessed by comparison between automated and manual measurements, confirming the accuracy over approaching three orders of magnitude in velocity. Results for a variety of media and formats for the storage of the void space geometry and measured velocity fields have been obtained. These results have been compared to the output of lattice Boltzmann (LB) simulations of flow in identical geometries. The results show that: 9 9
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the LB model successfully predicts the velocity field for simple geometries; the correlation between predicted and observed velocity fields is strongly dependent on optimization of the simulation viscosity (incorrect viscosities result in incorrect predictions); this LB scheme is incapable of simulating correct viscosities for complex geometries a systematic de-correlation is observed for increasing viscosity mismatch; some important effects relating to interactions between matrix and fracture flow are strongly viscosity dependent.
Some simulations may be able to predict successfully the behaviour of high viscosity fluids only. Non-linear effects between fracture and matrix flow are likely to be more important in these cases.
A16 Crack damage and permeability evolution near the percolation threshold in a near-perfect crystalline rock (Meredith, P.G., Clint, O.C., Ngwenya, B. (University College London); Main, I.G., Odling, N.W.A. (Leeds University) & Elphick, S.C. (Edinburgh University)) The importance of the evolution of crack networks during deformation is now becoming widely recognized as one of the key factors that control important processes involving fluid flow in rocks of low permeability, for example, hydrothermal circulation at mid-ocean ridges, energy recovery from geothermal reservoirs and accelerating deformation preceding volcanic eruptions. Hence, this project has been investigating the scaling properties of crack
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populations near the percolation thresholds for fracture and fluid flow in Ailsa Craig Microgranite (ACM) with increasing levels of damage. The work has shown the ACM to be a nearideal material for this investigation: it is almost perfectly isotropic, with P- and S-wave velocity anisotropy much less than 1%; and the fluid permeability of the undeformed rock is remarkably low at 1.5 • 10 -23 m 2, determined at an effective pressure of 10 MPa. No pre-existing microcracking could be observed either by optical or scanning electron microscopy. In the first phase of the study, thermal stressing at temperatures up to 800~ was used to induce crack damage in cores of ACM. Elastic wave velocities and fluid permeability were measured at room temperature before and after each thermal treatment cycle. A reduction of 48% in P-wave velocity and 32% in S-wave velocity was found with thermal stressing up to 800~ indicative of the high level of induced microcracking. However, the velocity measurements showed that this new damage was isotropically distributed at all temperatures investigated. Fluid permeability was found to have increased by seven orders of magnitude over the same treatment range. The increase in permeability is very non-linear, as induced cracks increasingly link up to create more pathways for fluid flow. In the second phase, the study was extended by examining the relationship between slow mechanical deformation, microcrack growth and permeability in ACM close to the fracture interconnection percolation threshold. To examine these interrelationships, creep experiments were undertaken on cores of ACM during which acoustic emissions, solute breakthrough curves and electrical impedance (EI) were measured at a strain rate of 10- 6 s- 1 . The initial network of distributed damage was generated by heating samples at 1~ min-1 to 900~ then cooling to room temperature at the same rate. The treated core was then placed in a Hasler cell and initially loaded to 20 MPa. To measure breakthrough curves, solutions of deionized/distilled/ degassed water were alternated with a degassed 1 M NaC1 solution. Initial 'breakthrough' tests under hydrostatic conditions showed that the EI response detects first a linear change in impedance as the solute front advances through the core followed by a more gradual decrease as the front 'breaks out' of the core end. The El response, therefore, describes both the advecfion and dispersion terms associated with the solute front in the core. The data allow the interrelationship between crack generation/growth, effective cumulative aperture, permeability and hydraulic dispersion to be examined during the
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process of creep deformation. Of particular interest is the behaviour of the system close to failure, where rapid changes in fracture network characteristics occur.
A17 Quantifying the effects of biofilm growth on hydraulic properties and on sorption equilibria: micro to macro measurements (Vaughan, D.J., Wogelius, R.A., Boult, S., Merrifield, C., Brydie, J.R. (Manchester University) & Large, D. (Nottingham University)) The objectives of this project were to undertake experiments, relevant to microscopic, mesoscopic and macroscopic scales, on the growth of bacterial biofilms, and to study their influence on both the hydraulic properties of geological systems (porous sediments, fractured rocks) and on the sorption of major and trace metals in introduced aqueous fluids. In the microscopic experiments, biofilms have been grown in various media using artificial groundwaters under both oxic and anoxic conditions; the rates of biofilm growth have been measured by studying the rate of reduction in discharge through a porous medium, and the biofilms themselves characterized using advanced imaging methods (environmental scanning electron microscopy, confocal laser scanning microscopy). Important information on the structure of immature and mature biofilms has been obtained from the study of model systems, notably the incomplete coverage of the substrate surface, even in mature biofilms, and the influence of fluid flow, seen in imbricate and in shear structures. Sorption and precipitation phenomena were studied by the addition of iron and lead to biofilm-conditioned surfaces, with rates of iron hydroxide precipitation being studied using imaging and spectroscopic methods (atomic force microscopy, X-ray photoelectron spectroscopy, etc.). Mesoscopic experiments (e.g. involving columns packed with quartz sand) have enabled the effects of biofilm growth on hydraulic conductivity at the centimetre scale to be determined. Continuous logging of discharge rate from a column as a function of biomass accumulation has demonstrated reductions in hydraulic conductivity of two to three orders of magnitude during the course of experiments. When iron was introduced, the optimum conditions of hydroxide precipitation were found to be when the mineral surfaces are conditioned by biofilm. The macroscopic studies have involved design, construction and commissioning of novel
equipment to mount in a 500 g geotechnical centrifuge (radius 3.2 m) in the Manchester School of Engineering. This enables the scaling up of hydraulic conductivity tests to the 6 0 - 1 0 0 m range, bridging the gap between laboratory and field measurements and enabling field-scale studies in a highly controlled environment. Centrifuge experiments involving 600 mm high columns of a porous medium (Congleton Sand) encountered problems due to sloughing of the biofilm at higher g-levels. Greater success was achieved following addition of iron in solution with subsequent hydroxide precipitation. Changes in hydraulic conductivity observed in the scaled-up centrifuge system experiments again show potential reductions of two to three orders of magnitude following biofilm development, and comparable further reductions following iron hydroxide precipitation. A model fracture system has also been constructed for use in centrifuge experiments and is still being commissioned. Overall, a great deal of new information has been acquired concerning biofilms, their structures and related characteristics, their importance in controlling the hydraulic properties of sediments and fractured rocks, and their role in the migration of minor/trace impurities in waters. Indeed, the importance of biofilms has been demonstrated and quantified over a wide range of scales for the first time. The compiler acknowledges the contributions from all the Principal Investigators of the NERC Micro to Macro Programme and their teams that have been used to compile this introduction and summary of the programme. This paper is published with the permission of the Executive Director of the British Geological Survey (NERC).
References ANI"ONELLIN~,M. & AYDIN,A. 1994. Effect of faulting on fluid flow in porous sandstones: Petrophysical properties, American Association of Petroleum Geologists Bulletin, 78, 355-377. BLAKEMAN, R.J., ASHTON, J.H., BOYCE, A.J., FALLICK, A.E. & RUSSELL,M.J. 2002. Timing of interplay between hydrothermal and surface fluids in the Navan Zn + Pb orebody, Ireland; evidence from metal distribution trends, mineral textures, and delta (super 34) S analyses. Economic Geology, 97, 73-91. BROWN, S. 1995. Simple mathematical model of a rough fracture. Journal of Geophysical Research, 100, 5941-5952. DAGAN, G. 1990. Transport in heterogeneous porous formations: spatial moments, ergodicity and effective dispersion. Water Resources Research, 26, 1281 - 1290. FUNG, L.S., HIEBERT, A.D. & NGHIEM, L. 1992.
Reservoir simulation with a control volume finite-
OVERVIEW OF THE NERC ix2M PROGRAMME element method. SPE Reservoir Engineering, 7, 349-357. GLOVER,P., MATSUKI,K., HIKIMA,R. & HAYASHI,K. 1998. Synthetic rough fractures in rocks. Journal of Geophysical Research, 103, 9609-9620. HARRIS, S.D., McALHSTER, E., KNIFE, R.J. & ODLtNG, N . E . 2003. Predicting the threedimensional population characteristics of fault zones: a study using stochastic models. Journal of Structural Geology, 25, 1281-1299. ISAKOV, E., OGILVIE,S.R., TAYLOR,C.W. & GLOVER, P.W.J. 2001. Fluid flow through rough fractures in rocks. I: High resolution aperture determinations. Earth and Planetary Science Letters, 191, 267-282. KNIPE, R.J., FISHER, Q.J., JONES, G. et aL 1997. Fault seal analysis: successful methodologies,application and future directions. In: MOLLER-PEDERSEN,P. KOESTLER, A.G. (eds) Hydrocarbon Seals: Importance for Exploration and Production. Norwegian Petroleum Society (NPF) Special Publication, 7, 15-40. MACAULAY,C., GRAHAM,C.M. & HASZELDINE,R.S. 2000a. Palaeo-hydrogeology in oil reservoir sandstones from multi-tracer micro-analysis of mineral cements. Paper presented at Geoscience 2000, Manchester, 175, Geological Society, London. MACAULAY,C., HASZELDINE,R.S., GRAHAM,C.M. & FALLICK, A.E. 2000b. Silicon isotopes identify sponge spicules as source of silica cement in North Sea oilfield sandstones. Paper presented at the American Association of Petroleum Geologists Annual Meeting, New Orleans. ODLING, N.E. & RODEN, J. 1997. Contaminant transport in fractured rocks with significant matrix permeability, using natural fracture geometries. Journal of Contaminant Hydrology, 27, 263 -283. ODL1NG, N.E. & WEBMAN, I. 1991. A 'conductance' mesh approach to the permeability of natural and simulated fracture patterns. Water Resources Research, 27, 2633-2643. ODLING, N.E., HARRIS, S.D. & KNIPE, R.J. 2004. Permeability scaling properties of fault damage zones in siliclastic rocks. Journal of Structural Geology, 26, 1727-1747. OGILVlE, S.R., ISAKOV, E., GLOVER, P.W.J. & TAYLOR, C.W. 2001. Use of Image Analysis and Finite Element Analysis to Characterise Fluid Flow in Rough Rock Fractures and their Synthetic Analogues. Proceedings of the 8th European Congress for Stereology and Image Analysis. Image Analysis and Stereology, 20(2) suppl. 1,504-509. OGmVIE, S.R., ISAKOV,E., TAYLOR,C.W. 8r GLOVER, P.W.J. 2003. Characterisation of rough fractures in crystalline rocks. In: PETFORD, N. & MCCAFEREY (eds) Hydrocarbons in Crystalline Rocks. Geological Society, London, Special Publications, 214, 125-141. OGILVIE, S.R., ISAKOV,E., TAYLOR,C.W. & GLOVER, P.W.J. in press. Fluid flow through rough fractures in rocks. III: (Single phase) flow experimentation and modelling. Earth and Planetary Science Letters. PECHER, R., HARRIS, S.D., KNIPE, R.J., ELLIOTT, L. & INGHAM,D.B. 2001. New formulation of the Green
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Index Page numbers in italic refer to figures advection dispersion equation 91 Ailsa Craig microgranite 13, 159 anisotropy 6 fracture surface 10 mudstone 104, 109-110, 112, 113 seismic 29-40 frequency dependence 32, 37 vertical seismic profile 35, 36, 37, 38 shear-wave 19, 20, 33, 36 well-logs 8, 9 antipersistence 8 apertures fracture 15-16, 95-96, 97, 100, 119 growth rate 10, 74-75
bacteria see biofilms, bacterial barriers, flow 43, 50-51 bioclogging 131, 132, 137, 143 biofilms, bacterial effect on fluid flow 131-143, 160 birefringence, shear-wave see splitting, shear-wave Bluebell-Altamont gas field, shear-wave anisotropy 36, 38 Boltzmann simulation, lattice 11, 116, 126-129, 159 Brae oil field cementation of sandstone 148 porosity/permeability correlation 9 breakthrough curves see tracer experiments, breakthrough curves Brownian motion, fractional 6, 7, 8, 118
Cajon Pass well, fracture apertures 16 carpet, Sierpinski see Sierpinski carpets carpet, random 81-82, 82, 83 cementation, quartz 9, 147-148 chalk Cretaceous, tracer test 92, 93, 95 MOPOD model 76
as oil reservoir, porosity/permeability 149-150 channelling, flow 93 Cholesky Decomposition 63, 156 clay minerals fabric evolution 112-113 matrix 109-110, 111 reaction mechanisms 103-104 see also illite, transformation; smectite, illitization CO2 sequestration 22 conductivity hydraulic 62, 66, 67, 134, 135-136, 136, 141, 143 see also fractures, conductivity connectivity, fracture 14, 44 contaminants see transport, contaminants control volume finite element model 151 coupling, fluid 6, 10, 11, 14 crack damage, in crystalline rock 159 crack model 32, 152 Cretaceous, Upper Shetland Group mudstones 104-114, 150151 see also mudstone, Upper Cretaceous Shetland Group critical point 6, 12, 13, 14, 20, 23 criticality and coupled modelling 14-18 in fracture permeability 13-18 in hydrocarbon development 16 intermittent 23 self-organized 7-8, 23 terminology 23-24
Darcy's Law 50, 52, 74, 135 deformability 14-18 deformation, and fluid flow 14-15, 155-156 deformation bands 44, 52 density crack 19 fractures 14-18, 23-24, 29, 30-31, 31
164 minor faults 44 diagenesis effect on fracture systems 9 - 1 0 effect on scaling 9 mudstone 103-104, 110, 112 diffusion anomalous, fractal lattices 79-89 matrix 93 molecular 93 directionality 17 discrete fracture flow model 151 discrete fracture network model 38-39 dispersivity 91 displacement 8
Ekofisk field, criticality 16 elastic response 31, 32 equant pore model 32
failure equilibrium 23 faults damage zones 3D extrapolation 46-47 barriers to fluid-flow 43-57 efficiency 53-54 flow modelling 12, 50-57, 151 minor fault characteristics 44-45 in siliclastic rock 43-45 stochastic model 45-46 sub-sampling 46-50 upscaled bulk permeability 51-57 hydraulic properties 43-57 main slip zone 55-57 minor, characteristics 6, 44-45 spatial distribution 45-46 sub-seismic 6 FEMLAB modelling 10 flow channelling 73, 93 flow, fluid barriers 43, 50-51 complex geology, scaling 151 - 152 effect of biofilms 131 - 143 in fault damage zone 12, 50-57, 151 in fractures 10, 15, 31, 151 in heterogeneous porous media 11, 18, 61-70, 115-129, 157, 158-159
INDEX matrix-fracture interaction 11, 93, 118, 125 modelling 5, 10, 18, 19, 116 in fault damage zones 50-57 generalized radial flow 18, 79 geomechanical 14-17 in multiscale fractures 32-35 Navan zinc-lead deposit 17-18, 148-149 path analysis 148-149 velocity field measurement 116 - 129, 158-159 flowcell 116-117, 119-122, 128, 158 numerical model 116 flowcell, Hele-Shaw 116-117, 119-122, 120, 128, 158 fractals 10 dynamic transport equations 18 effect on hydraulic properties 79 generalized radial flow model 18, 79 lattices anomalous diffusion models 80-89, 153-154 internal boundaries 87 random walk simulation 83-88, 154 sedimentary versus crystalline rock 8 well-logs 8 see also Sierpinski carpets fracture patterns 97, 98, 99, 100 fractures apertures 10, 15-16, 74-75, 95-96, 97, 100, 119 biofilm development 137-138, 139 cemented 29 characterization 30 conductive/non-conductive 17 connectivity 14, 18 density 14-18, 30-31, 31, 36-38 critical 24 P-wave azimuthal AVO analysis 36-37 shear-wave splitting 35- 36 discrete flow model 151 discrete network model 38-39 effect of diagenesis 9 - 1 0 en echelon, tracer tests 94-95, 96, 100 fluid flow matrix-fracture interaction 11 scaling 151-152, 157-158 localized flow in rock masses 155-156 multiscale 32- 35
INDEX open 29 orientation estimation 35- 37 parameterization 30, 31, 157-158 permeability 10-18 roughness 10, 117, 157-158 size estimation 37-38 synthetic, modelling 10, 157-158 theoretical models 31-35 frequency dependence, anisotropy 37-38
Gassmann model 19 Gaussian noise 6, 8 geomechanics, flow modelling 14-16 geothermal power 6, 17, 22 granite tracer test 93, 95 unflawed 13, 159 Green element model 151-152 groundwater 5, 22 artificial, biofilm growth 132, 133, 160 transport 61 Gulf of Mexico, Miocene mudstone 150-151
head gradient 74, 76 heterogeneity, geological 115 modelling 5 - 9 physical transport 61-71, 156 tracer experiments 91 - 101 well-logs 6 - 9 High Fidelity Polymer Model 10 hydrocarbons criticality 16 reservoir modelling 6 scaling problems 115-116
illite cement, North Sea oil fields 148 correlation with permeability/porosity 9, 112 illite/smectite ratio 104-105, 108-109, 110, 112, 112, 113 transformation 103-105, 150-151 iron oxyhydroxide precipitate 137, 142, 143
165
LaSTLib-2D 67, 70 lattice Boltzmann simulation 11, 1 I6, 126-129, 159 lattices see fractals, lattices lead see zinc-lead orebody
p.2M see Micro to Macro Programme Magnus oil field, cementation of sandstone 148 matrix chalk 149-150 clay 109-110 diffusion 93 permeability 11, 91-92, 93 porosity 32, 115, 118 matrix-fracture flow 11, 118, 125 Micro to Macro Programme 5-24, 145-147 mining 6, 22 Miocene, Gulf of Mexico mudstone 150-151 modelling Boltzmann 11, 116, 126-129, 159 coupled 14-18 dynamic 6, 13 fluid velocity fields 116, 129 geological heterogeneity current practice 5 - 6 scaling 9 particle transport mixed finite element approximation 62, 64-67 velocity field measurement 116, 119-129 static 6, 13 monitoring, seismic 18-20 Monte Carlo simulation diffusion in fractal lattices 80 MOPOD 77 transport in heterogeneous porous media 62, 156 MOPOD model of porosity development 10, 73-77, 154-155 areas of application 75-77 MOPOD code 74 Moray Firth, fault cementation 9, 148 mud diagenesis 103 porosity 103
166
INDEX
mudstone Miocene, Gulf of Mexico, microstructure evolution 150-151 Upper Cretaceous Shetland Group 104-114 anisotropy 104, 109-110, 112, 113 clay matrix 109-110, 111, 112-113 illite/smectite ratio 104-105, 108-109, 110, 112, 113 image analysis 105-107 microstructure evolution 150-151 mineralogy 107-108, 110 petrography 107-108 porosity 104, 107, 108, 112-113 Navan mine, palaeo-fluid flow 17-18, 148-149 Ninety Fathom fault, fault frequency distribution 45, 46 North Sea sandstone cementation 147-148 Upper Cretaceous Shetland Group mudstones 104-114, 105, 150-151
oil, chalk reservoirs 149-150 orientation fractures 35-37 minor faults 44
P-waves, azimuthal AVO analysis 29, 35, 36-37 particle transport see transport, particles percolation 13-14, 15, 17, 24, 75, 155, 159 threshold 11, 13-14, 19, 159 permeability bulk upscaled 147 fault damage zones 51-57 effect of diagenesis 9-10 effective 11 - 13 evolution in crystalline rock 159 fractures 10-18 matrix 11, 23, 30, 50, 91-92, 93 non-additive 11 persistence 8 phyllosilicates, in mudstone 104 polymers, extracellular 131 - 132, 137-138, 140
poroelastic model 32-35 porosity development in fractures 10 development model (MOPOD) 73-77, 154-155 fractal dimension 8 growth phenomena 74-75 matrix 32 mudstone 103-104, 107, 108, 112, 113 oil field sandstones 147-148 Pseudomonas aeruginosa PA01 133, 140, 142 pumping tests, Sierpinski carpets 79, 88, 153-154
quartz cementation 9, 147 see also sand, quartz
random carpets 81-82, 82, 83 random field generation 62, 63, 156 random walks diffusion modelling 18, 79, 80, 81, 82, 83-88, 83, 154 effect of internal boundaries 87 Sierpinski carpets 82-88, 84, 85, 86, 154 reservoirs hydrocarbon chalk 149-150 criticality 24 modelling 6, 7 discrete fracture network model 38-39 flow simulation 43, 51-57 monitoring 18- 21 sandstone cementation 147-148
sand, quartz, biofilm growth experiments 134, 135, 137 sandstone contaminant transport 43 quartz cementation 9, 147-148 Triassic, tracer test 92, 92, 93 scaling bulk permeability, fault damage zones 51-57 effect of diagenesis 9
INDEX fluid flow in rough fractures 157-158 through complex geology 151-152 problems of extrapolation 115-116 well-log measurements 6-9 see also upscaling schist, fractured, tracer test 92, 92 Scott oil field, cementation of sandstone 148 seismic surveys 6, 152-153 see also anisotropy, seismic shear-wave splitting see splitting, shear wave Shetland Group mudstones 104-114, 105, 150-151 see also mudstone, Upper Cretaceous Shetland Group Sierpinski carpets 18, 80-88, 153-154 generation 80, 81 random walks 82-88, 84, 85, 86 smectite illite/smectite ratio 104-105, 108-109, 110, 112, 113 illitization 103-105, 113, 150-151 splitting, shear-wave 19, 29, 32, 35-36 squirt-flow 19, 32-34 stereolithography 119 stiffness, elastic 8, 32-33 strain, and effective bulk permeability 13 stress critical 15, 17, 24 effective 19
tailing behaviour 92-93, 101 Tara Mine, Navan, fluid flow paths 148-149 thermodynamics, equilibrium 7 throw, fault damage zone 44, 47, 48, 50 tracer experiments 91,153 breakthrough curves 91-101, 97, 98, 99, 100-101, 153
167
transport contaminants 43, 73 dynamic 18 fluid, numerical modelling 61-70, 116 mass, advective 94 molecular diffusion 93 particles heterogeneous porous media 64-67 velocity fields 115-129 trace metal 143
upscaling bulk permeability 51-57 physical transport 61-70
Valhall field criticality 16 oil analysis 150 trace water analysis 16 velocity fields 2D measurement 122-129, 158-159 flowcell 116-117, 119-122 fractured-porous models 118, 124-125 self-affine fractures 117, 125-126 viscosity, fluid flow in fractured rock 11
waste, radioactive 6, 22 well tests see pumping tests well-log measurements, scaling 6 - 9 well-rate, fluctuation 20- 21
Yellow River Delta oil field, P-wave azimuthal AVO analysis 37
zinc-lead orebody, Navan, flow path analysis 148-149
Understanding the Micro to Macro Behaviour of Rock-Fluid Systems Edited by R. P. Shaw
Understanding how fluids flow through though rocks is very important in a number of fields. Almost all of the world's oil and gas are produced from underground reservoirs. Knowledge of how they got where they are, what keeps them there and how they migrate through the rock is very important in the search for new resources, as well as for maximising the extraction of as much of the contained oil/gas as possible. Similar understanding is important for managing groundwater resources and for predicting how hazardous or radioactive wastes or carbon dioxide will behave if stored or disposed of underground. Unravelling the complex behaviour of fluids as they flow through rock is difficult, but important. We cannot see through rock, so we need to predict how and where fluids flow. Understanding the type of rock, its porosity, the character and pattern of fractures within it and how fluid flows through it are important. Some contributors to this volume have been trying to understand real rocks in real situations and others have been working on computer models and laboratory simulations. Put together, these approaches have yielded very useful results, many of which are discussed in this volume.
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Cover illustration: Background image is a section through a septarian nodule from the Oxford Clay, Latton, Wiltshire (Richard Shaw), top left image is calcite veins in LiassicLimestoneat Kilve, Somerset(Dave Sanderson), centre image is the cliffs at Burton Bradstock, Dorset (Neville Hollingworth) and lower right is an extract of a 3D seismic image of Sleipner, Norwegian sector of the North Sea (Andy Chadwick).
