Understanding the micro to macro behaviour of rock fluid systems

Understanding the Micro to Macro Behaviour of
Rock-Fluid Systems


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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 />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 />about.html (or linked from c.
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

- -

Log. (combined}
9 "''..
9

9

9
"'""

f
5
10
2d 3d percolation theory I

,t

15

13

20

value of exponent,

25

30

s

Fig. 5. Values of the exponent, s, in the percolation equation for permeability k = a(p - po)S, p > Pc. The range of
values of s measured by Meredith et al. (Ix2M) is compared with the values, or frequencies of values, measured or
calculated by Bernab6 (1988, 1995), Walsh & Brace (1984) and Yale (1984).


14

K.J. HEFFER

densities of fractures. Fitting to the data the percolation equation:
k = a ( p - pc) s, p > Pc

-- 0, p < pc

(2)

where p is the porosity, Pc is the porosity at the
percolation threshold, k is the permeability, and
a, s are constants, with increasing assumed
values of the percolation threshold, ostensibly
yields a 'best fit' (highest value of correlation
coefficient) when Pc is 0.0075 and s = 4.92. It
is interesting that this (very preliminary) interpretation of threshold porosity is just below the
actual natural porosity of the starting material

(mostly due to isolated altered phenocrysts) of
1% or so (I.G. Main, pers. comm.). The value
of the exponent is well in excess of the theoretical conductivity scaling implied by percolation
theory (1.3 for 2D; 2.0 for 3D) and lies in the
middle of the large range for mixed cracks and
pores analysed by Bernab6 (1995) (Fig. 5). It is
possible that the large value of the exponent, s,
is attributable to heterogeneity in the apertures
of the thermally-induced cracks. Whatever the
final analysis of these data yields, the data

themselves provide a valuable benchmark
against which to compare other values derived
from theory or from laboratory measurements
made under different conditions.
The work has also been another illustration of
the extreme sensitivity of permeability to fracture density - a highly non-linear relationship
that can act as a threshold in critical behaviour
and play a large role in coupled systems of
fluid flow and geomechanics.
Criticality and coupled modelling

Sanderson et al. (p~2M) (see also Zhang &
Sanderson 2002; X. Zhang et al. 2002) looked
at the critical point associated with the connectivity of fractures with a 2D distinct element
model (UDEC), which couples deformation and
fluid flow. The changes in deformability and
permeability in the model with increasing input
densities of fractures have been calculated
(note that, in contrast with later studies described

below, the fracture patterns were input into the
model rather than induced by failure during
deformation). The fracture connectivity is
posed as a power-law function of fracture density
above a threshold value, as with permeability vs.

Fig. 6. Critical point in coupled mechanics and fluid flow. (a) Fluid flow velocities modelled in loaded domain with
three pre-existing fractures: (i) below; (ii) just below; and (iii) at the critical point when new percolating pathways subparallel to Shmaxare created. Reprinted from Zhang & Sanderson, copyright (2002), with permission from Elsevier.
(b) Field data confirm that directionalities of flooding axe sub-parallel to the local orientation of Shmax,rotated to align
with the modelling of X. Zhang et aL (2002) (adapted from Heifer & Lean 1993).


MICRO TO MACRO PROGRAMME: IMPLICATIONS

porosity described above. Sharp increases in both
deformability and permeability are observed
at the critical (threshold) fracture density. Four
groups of simulated fracture patterns and 15
natural fracture patterns were studied. Exponents
of permeability increase above the threshold
were found in the range 1.05 to 1.37, in line with
2D percolation theory (exponent of 1.3). When
the models were loaded, the stress-strain
curves showed softening above the critical fracture density, but then an even greater deformability was observed above a second threshold of
fragmentation. Exponents of the relationship
between deformability and fracture density
above this higher threshold were found to be
0.64 for a zero confining stress and 0.91 for an
applied confining stress of 0.3 MPa: it would be
a useful exercise to rationalize these values

with experimental observations (Chakrabati &
Benguigui, 1997, Section 3.4) of the scaling of
modulus in a bond percolation model in which
increasing densities of bonds incrementally
stiffen the model (exponent close to 4 in 2D).
The modelling of Sanderson et al. (/x2M) has
also provided guidelines for estimating the effective failure variables (friction coefficient and
cohesion) for a fractured rock mass. Based on
these models they have defined an indicator for
criticality in stress state, termed the 'driving

(a)

15

stress ratio' and given by:
R=

(fluid pressure - mean stress)

(3)

(~ • differential stress)
Instability occurs when the R-ratio exceeds some
critical value Rc in the range - 1 to - 2 . These
limits respectively represent failure by hydraulic
fracturing and by shear failure in a cohesionless
material with friction angle of 30 ~. Criticality
can occur with shear failure with the fluid
pressure still below the minimum principal stress.

Sanderson et al. (o.2M) studied the statistics of
fracture apertures arising from their modelling in
relation to progress of the model to and through a
state of criticality (see Fig. 6a). Apertures were
actually examined in terms of the fluid flow 'vertically through' the 2D areal model, using essentially a cube law between flow rate and aperture.
One-point cumulative frequency distributions of
flow rate showed a dependency on degree of
criticality: below criticality, the distribution is
approximately log-normal; however, at and
above the point of criticality, the distribution is
better described as a power law. At the critical
point the exponent of the power law is 1.1
(Fig. 7a). This modelled distribution can be

(b)
1000

|

',,
'1
,,

SIope~l.1

slope-~1 , t ~

,

.o

m 100
"6
r
E
Z

, vvvv

9~

256

IOOC

ivv

10

N

1
0.001

0.01

0.1

1

10

Vertical flow-rates (x 10 -6 m s -1)

100

1.E-07 1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 I.E+00

Well rate, PI, cum prod, or permeability relative to
maximum
Im'='Giant field ~

Composite of several smaller fields I

Fig. 7. Frequency distribution of flow rates is power law at and near the critical point. (a) Results of modelling coupled
mechanics and fluid flow; reprinted from Zhang & Sanderson, copyright (2002), with permission from Elsevier.
Cumulative frequency distribution of flow rates (A) below, (B) and (C) just below the critical point. (b) Field data:
cumulative frequency distributions of well rates, cumulative well production or permeability (each divided by the
maximum for the field). Data are from one giant field, and also aggregated from a number of smaller naturally fractured
fields.


16

K.J. HEFFER

compared with that from field data on the flow
productivities from individual wells. Figure 7b
shows that power-law distributions also apply to
two examples of the latter: one from a giant
field; the other as a composite from several

fields. The exponent of the fitted power law
common to both sets of data is also 1.1. The existence of the power law in the field data combined
with the implication from modelling that powerlaw behaviour is expected only at or above the
critical point is consistent with the concept of criticality in field behaviour. The equality of the
exponent of the power law may not be so significant and further study would be necessary to
demonstrate that it is not coincidental.
Since, in the modelling of Sanderson et al.
(lz2M), flow rate is calculated as the cube of
the fracture aperture, the cumulative frequency
distribution of fracture aperture is also a power
law, with exponent - 3 . 3 . There are few direct
datasets on fracture apertures from the field
with which to validate these one-point statistics;
fracture apertures in recovered core are under
relaxed stress conditions. One exception is the
large dataset measured downhole with a borehole
televiewer log in the Cajon Pass well; from this,
Barton & Zoback (1990) calculated a powerlaw frequency density distribution of fracture
apertures, with an exponent of - 3 . 0 (equivalent
to an exponent of - 2 . 0 for the cumulative
frequency distribution). Converting that 1D
sample basis to 2D would alter the cumulative
distribution to ~ a -3, in good agreement with
the distribution of flow rates calculated by
Sanderson et al. (tx2M).
Sanderson et al. (p~2M) have also investigated
multifractal statistics in the distribution of
apertures/vertical flow rates arising from their
coupled geomechanics-flow modelling. They
have found that below the critical point, the

spectrum of generalized fractal dimensions
Dq(q) varies only weakly with the order q of
the moment, indicating an approximate monofractal. The common dimension is equal to 2.0,
the space-filling dimension of the underlying
input fracture set. However, when the critical
point is reached, the multifractal spectrum shows
a strong variation, with a sharp decline from
negative to positive values of q. No known
studies have been made of whether flow rates
in a densely drilled field follow a multifractal
distribution: such study might lead to further
support for criticality in field behaviour.
Another example of modelling which produced similar forms of multifractal spectra was
the investigation by Cowie et al. (1995) of development of fault patterns by antiplane shear deformation of a 2D plate (in which the displacements

are out of the plane of the plate). No fluid flow
was involved in that modelling. Distributions of
displacements on the faults were found to
evolve with model time from monofractal and
space-filling to multifractal.
One must be careful not to make too strong a
deduction from these model studies: power-law
distributions can occur in many ways (Sornette
2000, Chapter 14). Also, interpretation of a
power law can be made falsely if the range of
data is inadequate, for example extending over
only one order of magnitude. However, there
are strong indications that geomechanical-flow
criticality is a sufficient, if not necessary, condition for power-law and multifractal distributions of flow properties.
ls there more direct evidence to support the

concept of criticality in oil field developments?
Good demonstrations of its applicability are to be
found in the North Sea chalk fields, Ekofisk and
Valhall. These fields have received intense geomechanical study, mainly because of their strong
compaction and its associated, very noticeable,
effects of subsidence and casing failures, but it is
unlikely that the fields are a special case. Zoback
& Zinke (2002) have shown that the stress states
in the crests of both fields were consistent
with incipient normal faulting at the onset of oil
production, and that the subsequent pressure
reductions during primary production caused
those critically stressed areas to spread downdip
to the flanks of the structures (see also Chan
et al. 2002). The effective stress states tracked
down the Coulomb failure line (with a friction
coefficient ~0.6) on a Molar diagram. Passive
seismic monitoring in both Ekofisk (Maxwell
et al. 1998) and Valhall (Zoback & Zinke 2002)
has detected microseismic events, mainly in
lower porosity reservoir layers or in the overburden. In Valhall, microseismic events have focal
mechanism solutions, also indicating normal faulting. Furthermore, the anisotropy of the detected
shear waves has shown evidence of temporal
changes.
Coleman (p,2M) sought change in fracture
characteristics in the Valhall Field, which could
be a further indication of criticality. That
project has developed a possible diagnostic of
fracture activation during reservoir development.
In laboratory tests of fluid flow through chalk

under stress, it was found that the concentration
of the isotope 637C1of the collected fluid was
correlated positively with the flux of the fluid
through the chalk, this flux being controlled by
the fracturing of the rock. Coleman (p~2M)
sampled trace waters found in produced oil
from several wells in the Valhall Field. No
change over time has been observed to date in


MICRO TO MACRO PROGRAMME: IMPLICATIONS
the geochemistry of these samples, but the
average 637C1 compositions of trace waters
varied significantly between wells, always different from that of sea water. It is very interesting
that the 837C1compositions indicated more fracture permeability from the crest of the structure
than from the flanks (M. Coleman pers. comm.
2002) consistent with the other observations of
fracture activity progression.
Further modelling by Sanderson et al. (ix2M)
also suggests the basis for a reconciliation of
the current disagreements in the industry of the
importance of critical stressing as a criterion
for conductivity of individual fractures. Recent
work has shown the strong influence of
modern-day stress state on fracture conductivity:
fractures which are in a state of incipient shear
failure in the modern-day stress field, termed
'critically stressed', will generally be conductive; whilst those fractures stable in the
modern-day stress state will generally be nonconductive (Barton et al. 1998; Barton 2000;
Chan et al. 2002). An exception to this might

be a fracture set that was formed under a
palaeo-stress state shortly before, or contemporaneously with, hydrocarbon fill, which inhibited
fracture healing when the stress state altered to
its modern-day configuration (e.g. Stowell et al.
2001; Gauthier et al. 2002). This scenario is
more likely if the original deformation was
associated with diagenetic alteration, either dissolution, or partial cementation, such that,
when the stress state was altered, bridges
between vugs along the fracture path helped to
prop open a conductive path.
The model of Sanderson et al. (pu2M) of the
fluid flow in a granular medium also contained
some macro-fractures, with the maximum principal horizontal stress (Shmax) at a large angle
(c.60 ~ to the fracture strike (see Fig. 6a). At,
or just below, the critical point, smaller-scale
fractures formed that were sub-parallel to
Shmax, at the same time as the macro-fractures
are open. Under conditions of low mean effective
stress (as would pertain in waterflooding recovery schemes), the secondary fractures are conductive and form a percolating path for flow.
To transpose this to field experience, observations might be made early in the life of a
field development of conductive fractures
which were formed under some palaeo-stress;
whilst, if a secondary recovery scheme is
implemented which reduces effective stresses
close to a critical point, then coherent fracture
trends striking close to the azimuth of Shmax
might be equally or even more, influential in governing the directionality of the flooding. This is
consistent with the statistics of directionality

17

observed in oil field operations (Heifer & Lean
1993) and in geothermal projects (e.g. WillisRichards et al. 1996). With regard to indicating
stress-induced directionality, the modelling
complements that of Heifer & Koutsabeloulis
(1995) (see Fig. 6b).
The semi-quantitative scale invariance of some
deforrnational geometries is demonstrated by
comparing the results of Sanderson et al. (ix2M),
whose model contains overlapping macrofractures
at the grain scale, with those of much largerscale modelling of the geomechanical and flow
characteristics of a fault relay zone conducted by
Y. Zhang et al. (2003), linked to the ix2M
project of Yardley et al. (ix2M). In addition to
coupled modelling of geomechanics and fluid
flow using the explicit finite difference code
FLAC in 2D, the modeUing is also explicitly
coupled to the finite element code FIDAP, which
models chemical reactions. The model has been
used to track the mixing of reduced and oxidized
fluids, both gold saturated, in the dilatant zones
resulting from the geomechanical model. The
patterns of fluid mixing are seen to be very
similar to the aperture distributions produced by
the Sanderson et aL (po2M) model (see Fig. 8).
Yardley et al. (Ix2M) are utilizing geochemical
methods to investigate palaeo-fluid flow in and
around the Navan mine in Eire. A strong
control on the flow has been shown to be the
density contrast between cooler waters of evaporitic origin overlying hotter hydrothermal

waters from the Lower Paleozoic basement.
The concentration of lead sulphide mineralization is focused in a ramp zone between two
NNE-SSW-trending Caledonian faults, which
were activated under a more E-W-directed
stress field during Carboniferous-Permian
times. The dilatation of this extensional step
gave rise to vertical flow to concentrate mixing
of the two waters and deposition of lead sulphide.
More extensive E - W lineations also hinged
upon this focus.

D y n a m i c t r a n s p o r t e q u a t i o n s on
fractal structures

If heterogeneous porous media can be described
with fractal functions (even if they are uncoupled
from geomechanical or chemical changes), is
there an effective differential equation which
can be applied to describe transport through
them? Such an application would have potential
for more efficient flow simulations. However,
although there have been a wide variety of
equations devised in the past to describe flow
and transport on a fractal structure, Sellers &