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
|
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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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Permeability. SPE paper 58993, presented at the 2000 SPE International Petroleum Conference and Exhibition, Villahermosa, Mexico, 1-3 February. BARTON, C.A. & ZOBACK, M.D. 1990. Self-similar distribution of macroscopic fractures at depth in crystalline rock in the Cajon Pass Scientific Drillhole. In: BARTON, N. & STEPHANSSON, O. (eds) Rock Joints, Balkema, Rotterdam, 163-170. BARTON, C.A., ZOBACK, M.D. & MOOS, D. 1995. Fluid flow along potentially active faults in crystalline rock. Geology, 23, 683-686. BARTON, C.A., HICKMAN, S.H., MORIN, R., ZOBACK, M.D. & BENOIT, D. 1998. ReservoirScale Fracture Permeability in the Dixie Valley, Nevada, Geothermal Field. SPE paper 47371, presented at the SPE/ISRM Eurock '98 conference, Trondheim, Norway, 8-10 July. BEAN, C.J. 1996. On the cause of 1/f-power spectral scaling in borehole sonic logs. Geophysical Research Letters, 23, 3119-3122. BEAN, C.J. & MCCLOSKEY, J. 1993. Power-law random behaviour of seismic reflectivity in boreholes and its relationship to crustal deformation models. Earth and Planetary Science Letters, 117, 423-429. BERNABt~, Y. 1988. Comparison of the effective pressure law for permeability and resistivity formation factor in Chelmsford granite. Pure and Applied Geophysics, 127, 607-625. BERNABI~, Y. 1995. The transport properties of networks of cracks and pores. Journal of Geophysical Research, 100, 4231-4241. BINNEY, J.J., DOWRmK, N.J., FISHER, A.J. & NEWMAN, M.E.J. 1992. The Theory of Critical Phenomena - an Introduction to the Renormalization Group. Oxford University Press, Oxford. BLOOMFIELD, J.P., BARKER, J.A. & ROBINSON, N. 2005. Modeling fracture porosity development using simple growth laws. Ground Water, 43, 314-326. CHAKRABARTI, B.K. & BENGUIGUI, L.G. 1997. Statistical Physics of Fracture and Breakdown in Disordered Systems. Oxford University Press, Oxford. CHAN, A.W., ZOBACK, M.D., FINKBEINER, T. & ZINKE, J. 2002. Production Induced Faulting and Fault Leakage in Normal Faulting Regions: Examples from the North Sea and Gulf of Mexico. Abstract presented at the AAPG Annual Meeting, 10-13 March 2002, 'Pathways of Hydrocarbon Migration, Faults as Conduits or Seals'. CHAPMAN, M. 2003. Frequency dependent anisotropy due to meso-scale fractures in the presence of equant porosity. Geophysical Prospecting, $1, 369-379. CHAPMAN, M., ZATSEPIN, S.V. & CRAMPIN, S. 2002. Derivation of a microstructural poroelastic model Geophysical Journal International, 151, 427-451. CHAPMAN, M., MAULTZSCH, S., LIU, E. & LI, X.Y. 2003. The effect of fluid saturation in an anisotropic multi-scale equant porosity model. Journal of Applied Geophysics, 54, 191-202. CHARLAIX, E., GUYON, E. & ROUX, S. 1987. Permeability of a random an'ay of fractures of
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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
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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" ,
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
86
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.
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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
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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
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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
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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)
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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
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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 (