On the Importance of Geological Heterogeneity for Flow Simulation
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Transcript of On the Importance of Geological Heterogeneity for Flow Simulation
www.elsevier.com/locate/sedgeo
Sedimentary Geology 18
On the importance of geological heterogeneity for flow simulation
Timothy T. Eaton *
School of Earth and Environmental Sciences, Queens College, City University of New York, 65-30 Kissena Blvd Flushing,
NY 11367, United States
Abstract
Geological heterogeneity is recognized as a major control on reservoir production and constraint on many aspects of
quantitative hydrogeology. Hydrogeologists and reservoir geologists need to characterize groundwater flow through many different
types of geological media for different purposes. In this introductory paper, an updated perspective is provided on the current status
of the long effort to understand the effect of geological heterogeneity on flow using numerical simulations. A summary is given of
continuum vs. discrete paradigms, and zonal vs. geostatistical approaches, all of which are used to structure model domains. Using
these methods and modern simulation tools, flow modelers now have greater opportunities to account for the increasingly detailed
understanding of heterogeneous aquifer and reservoir systems.
One way of doing this would be to apply a broader interpretation of the idea of hydrofacies, long used by hydrogeologists.
Simulating flow through heterogeneous geologic media requires that numerical models capture important aspects of the structure of
the flow domain. Hydrofacies are reinterpreted here as scale-dependent hydrogeologic units with a particular representative
elementary volume (REV) or structure of a specific size and shape. As such, they can be delineated in indurated sedimentary or
even fractured aquifer systems, independently of lithofacies, as well as in the unlithified settings in which they have traditionally
been used. This reconsideration of what constitutes hydrofacies, the building blocks for representing geological heterogeneity in
flow models, may be of some use in the types of settings described in this special issue.
D 2005 Elsevier B.V. All rights reserved.
Keywords: Geological heterogeneity; Groundwater flow; Paradigms; Hydrofacies
1. Introduction
Understanding geological heterogeneity is critical
for characterization of flow in the subsurface. Hetero-
geneity includes variations in grain-size, porosity, min-
eralogy, lithologic texture, rock mechanical properties,
structure and diagenetic processes. All these factors
cause variations in hydraulic conductivity, storage,
and porosity, and thus control flow and transport
through these rocks. Since contaminant plumes are
subject to dispersion, the details of sedimentary
0037-0738/$ - see front matter D 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.sedgeo.2005.11.002
* Tel.: +1 718 997 3327; fax: +1 718 997 3299.
E-mail address: [email protected].
sequences are more important for transport simulation
than they are in regional flow simulation for water
supply or reservoir assessment. However, geological
heterogeneity is recognized as a major control on res-
ervoir production (Dutton et al., 2003), and constraint
on many other aspects of quantitative hydrogeology
such as model calibration (Cooley, 2004) and recharge
estimation (McCord et al., 1997). Geological heteroge-
neity is readily apparent in surface outcrops and well
logs, yet these represent only small windows into sub-
surface aquifers or reservoirs, and analogous outcrops
may not even be present for many groundwater sys-
tems. Moreover, in recent years, flow simulation has
benefited from ever-faster and more powerful computer
4 (2006) 187–201
T.T. Eaton / Sedimentary Geology 184 (2006) 187–201188
processors and graphical user interfaces (GUIs). So, the
major obstacle to representation of such geological
heterogeneity in groundwater flow models is now less
computational than informational.
Quantitative hydrogeology originally de-emphasized
geological heterogeneity because its theoretical under-
pinning is based on flow through a homogeneous
bequivalent porous mediumQ. Of course, this is a
gross simplification applied to the heterogeneous real
world. The bias toward an assumption of homogeneity
is not surprising because hydrogeologists were at the
outset primarily focused on problems of water supply in
relatively uniform and highly conductive porous media.
Darcy (1856) conducted his experiments with the goal
of evaluating the size of sand filters needed for munic-
ipal water supply, resulting in the empirical law bearing
his name. Theis (1935) developed his approach to
calculating aquifer properties by mathematical analogy
to heat flow in homogeneous materials. It was only with
the advent of studies of groundwater contamination
(Bredehoeft and Pinder, 1973; Pinder, 1973; Fried,
1975) that the importance of geological heterogeneity
began to be recognized. Still, the limitations of com-
puter simulation capability have long been an important
constraint on quantitative analysis of groundwater flow,
ensuring that the default assumption of relatively ho-
mogeneous media remains entrenched.
The purpose of this paper is to provide an updated
perspective on the current status of the long effort by
hydrogeologists to understand and represent geological
heterogeneity in flow models. A comprehensive re-
view of the vast topic of addressing heterogeneity in
groundwater systems is beyond the scope of this work.
Many reviews (e.g., Koltermann and Gorelick, 1996;
Anderson, 1997) and compilations (Fraser and Davis,
1998; Huggenberger and Aigner, 1999; Faybishenko et
al., 2000; Bridge and Hyndman, 2004) on different
aspects of this subject are now available. Instead,
following some background information, several im-
portant concepts that underlie the understanding of
flow through heterogeneous media will be briefly
summarized. Then, an overview of trends in the anal-
ysis and simulation of flow in heterogeneous aquifer
systems will be presented. Special emphasis will be
placed on limitations of methods, selected develop-
ments that seem most promising, and a few recent
studies of note. The concept of hydrofacies (Poeter
and Gaylord, 1990), which had much influence on the
field of hydrogeology, will be reviewed. An operation-
al broadening of this concept is proposed that may be
of some use in understanding and simulating flow
through heterogeneous media. This provides an intro-
ductory framework for the following papers that illus-
trate some innovative characterization and modeling
approaches which deserve wider application.
2. Ubiquity of geological heterogeneity
Geological heterogeneity that controls flow mani-
fests itself in all rocks, from unlithified surficial sedi-
ments to sedimentary and crystalline bedrock. But most
geological formations are not considered aquifers or
hydrocarbon reservoirs because they have insufficient
porosity and permeability to conduct significant flow to
wells. Many of these are fine-grained mudrocks that
constitute more than 60% of the sedimentary rocks in
the world (Potter et al., 1980). Their hydrogeologic
properties have been rarely studied compared to those
of more permeable formations, and for the most part,
they have been considered boundary conditions: aqui-
tards confining aquifers or bsealsQ for hydrocarbon
reservoirs. As aquitards are of increasing interest for
protecting aquifers from groundwater contamination
(Cherry et al., in press), it is becoming clear that
many are quite heterogeneous and often fractured
(e.g., Eaton and Bradbury, 2003).
Even geologic formations that are used for water
supply or petroleum resources have significant spatial
heterogeneity in hydraulic conductivity, as indicated by
various types of hydraulic testing, and detailed analysis
of well logs and outcrops. Of course, the type of geolog-
ical heterogeneity that needs to be taken into account
depends on the scale of the problem under consideration
(Schulze-Makuch and Cherkauer, 1998; Beliveau, 2002;
Neuman, 2003). But it would seem that significant het-
erogeneity is present everywhere, on all scales down to
centimeters (Allen-King et al., 1998), and even in aqui-
fers originally thought to be classical homogeneous
equivalent porous media, such as the Borden site
(Sudicky, 1986; Allen-King et al., 2003) in Canada.
In unlithified sediments, geological heterogeneity
that controls flow is represented by variations in litho-
facies, whereas in indurated, crystalline bedrock, it is
also represented by fractures. The hydrogeology of
unlithified sediments has received much attention by
hydrogeologists interested in the question of heteroge-
neity (Fraser and Davis, 1998; Huggenberger and
Aigner, 1999; Bridge and Hyndman, 2004), as has the
hydrogeology of fractured rocks (Faybishenko et al.,
2000). The focus on these end-members of heteroge-
neous media has probably been enhanced by the his-
torical development of two classes of methods for
describing flow: continuous and discrete methods (dis-
cussed in more detail later). However, sedimentary
Fig. 1. Example of karstic geological heterogeneity in carbonate bedrock from a mine in Poland: 1, dolomite; 2, breccia; 3, clay; 4, sand; 5, dolomite
fragments in clay and sand (from Motyka, 1998). Reproduced with permission from Springer.
T.T. Eaton / Sedimentary Geology 184 (2006) 187–201 189
bedrock aquifers present a special challenge because
their heterogeneity can be due to variations in lithofa-
cies, or fracturing, structural deformation, diagenesis
and even dissolution (e.g., Fig. 1). While this type of
heterogeneity is well known to petroleum geologists, it
has been less of a focus for hydrogeologists than the
heterogeneity in unlithified material or fractured crys-
talline rock.
The principal goal of this special issue of Sedimen-
tary Geology, therefore, is to stimulate new approaches
to applying the tools of sedimentology, stratigraphy,
structural geology, rock mechanics, and diagenesis to
the problem of characterizing geological heterogeneity
and simulating flow in sedimentary bedrock aquifers.
With the shortage of hydrogeologic data being a typical
challenge for most studies, geophysical methods as well
as geologic reasoning are often used to interpolate
subsurface structures between data points. Hence, the
need has never been greater for cooperative studies
between hydrogeologists and other geoscientists to pro-
vide a robust basis for flow simulation in heterogeneous
groundwater systems.
3. Concepts behind continuum vs. discrete
paradigms
3.1. Equivalent porous medium methods
The development of hydrogeologic theory rests
upon a continuum assumption - that is: flow is consid-
ered on a volumetrically averaged basis at a macroscop-
ic scale in what is assumed to be equivalent to an ideal
porous medium. Potential field theory can then be used
to describe the smooth variation in hydraulic head, for
example the shape of the water table across an area.
Application of the principle of mass conservation to the
averaged volumes used allows the derivation of govern-
ing partial differential equations to represent flow. These
equations are the basis for analytical and numerical
solutions (generally implemented using finite-difference
or finite-element methods) that quantify groundwater
flow.
The principal parameter governing flow: hydraulic
conductivity, is defined at some scale larger than that of
the microscopic pores, in order to predict flow given
some hydraulic gradient according to Darcy’s Law. The
minimal volume over which the governing equations of
flow apply is commonly referred to as a representative
elementary volume (REV) (Bear, 1972). The dimen-
sions of such an REV (Fig. 2) are defined according to
the purpose of the investigation, but it must be of a size
range within which measurable characteristics are sta-
tistically significant and remain more or less constant
(Bachmat and Bear, 1987; Bear and Bachmat, 1991).
Therefore, in a heterogeneous continuum, the REV size
must be smaller than the major variations in hydraulic
conductivity for this approach to quantitative hydro-
geology to be applicable. This REV size range can be
less than the size of a core sample (Brown et al., 2000)
based on computer tomography, or even smaller. Using
Fig. 2. Conceptual position of a representative elementary volume (REV) (V3 or V4) larger than the microscopic scale, and within the macroscopic
size domain (adapted from Bear, 1972; Freeze and Cherry, 1979). Reprinted by permission of Pearson Education, Inc.
Fig. 3. Illustration of fractured geological heterogeneity and its representation in numerical flow models: (a) real fracture network; (b) distribution of
head in real network; (c) discrete fracture (DF) model; (d) distribution of head in DF model; (e) continuum model; (f) distribution of head in
continuum model (from Hsieh, 1998).
T.T. Eaton / Sedimentary Geology 184 (2006) 187–201190
Fig. 4. Representation of (a) three-dimensional fracture geometry as
intersecting disks in space (from Dershowitz and Einstein, 1988); and
(b) inferred flowpaths through fracture planes (from Tsang and Ner-
etnieks, 1998). Copyright 1998 American Geophysical Union. Repro-
duced by permission of American Geophysical Union.
T.T. Eaton / Sedimentary Geology 184 (2006) 187–201 191
pore-scale flow simulation, Zhang et al. (2000) sug-
gested the concept of a statistical REV to overcome
inconsistencies in observed REV scales for different
properties. Extension of the REV concept to hydrogeo-
logic domains in which discrete fractures constitute the
principal heterogeneity is a challenge. Estimates of
REV size in fractured rock based on simulation of
fracture network geometry are on the order of cubic
meters (Wang et al., 2002; Min et al., 2004). From a
flow modeling perspective, the size of an REV can be
defined as a volume across which hydraulic head
changes are not significant (Anderson and Woessner,
1992), which is in effect the size of model grid cells
(Ingebritsen and Sanford, 1998). Decisions about the
resolution of flow model discretization in equivalent
porous media must therefore depend on what REV size
is appropriate to capture the geological heterogeneity
for a given modeling application.
The continuum approach can also be taken to un-
derstanding flow in highly heterogeneous settings such
as fractured rocks (Hsieh, 1998; Selroos et al., 2002;
McKenna et al., 2003; Ando et al., 2003). In this case,
flow through each individual fracture is not considered,
but overall flow through the fracture network is as-
sumed to be reproduced sufficiently well by an equiv-
alent porous medium (Fig. 3). When hydraulic
properties due to flow through pore space and fractures
are considered simultaneously in overlapping continua,
this constitutes what is known as a bdouble-porosityQmodel (NRC Committee on Fracture Characterization
and Fluid Flow, 1996). Some approaches to flow in
carbonate rocks consider btriple-porosityQ models that
account for flow in matrix, fractures and conduits
(Motyka, 1998; Kaufmann, 2003).
3.2. Discrete fracture and other methods
Alternatively, the discrete fracture approach takes
into account flow through each fracture characterized
by a location, orientation, size and transmissivity.
Often, fractures are represented in three dimensions
by intersecting circular or elliptic disks (Fig. 4a). The
major difficulty with this approach is that it is extreme-
ly computationally intensive (Long et al., 1985). It is
generally practical only at very small site-specific
scales or for special applications like siting of national
nuclear waste repositories (Selroos et al., 2002). Slight-
ly simpler approaches are to limit analysis to two
dimensions (Min et al., 2004) or to represent the
three-dimensional network of disks with a network of
channels (Fig. 4b) (Cacas et al., 1990a,b; Moreno and
Neretnieks, 1993; Tsang and Neretnieks, 1998) but
neither of these techniques is in widespread use for
computational reasons. In contrast to the continuum
approach of a hydraulic conductivity field, the discrete
fracture approach considers the structure of the model
domain to be a network of transmissive fractures
(Hsieh, 1998). Both seek to reproduce the basic ele-
ments of the heterogeneous flow field but cannot rep-
licate all the details (Fig. 3).
Another even less common approach is that of per-
colation theory (Berkowitz and Balberg, 1993; Stauffer
and Aharony, 1994), in which statistics about connec-
tions between elements that represent pores or fractures
provide information about the hydraulic conductivity of
larger systems. This connectivity is associated with a
critical threshold (the percolation limit) above which a
system will conduct flow. The effective hydraulic con-
ductivity of systems above the percolation limit
depends on the number of throughgoing flowpaths
(Berkowitz, 1995). Percolation theory has most often
been described in the physics literature using lattices of
T.T. Eaton / Sedimentary Geology 184 (2006) 187–201192
equidimensional, conductive elements (e.g., Sahimi and
Mukhopadhyay, 1996; Bruderer and Bernabe, 2001).
Such a lattice above the percolation limit will behave as
an equivalent continuum if observed on a large enough
scale, but also has aspects of a discrete network. One
method for generating such a percolation lattice is
through simulated annealing such that model results
match hydrologic data (Long et al., 1997).
Relatively few workers have applied percolation
theory to understanding flow through heterogeneous
rocks. Koudina et al. (1998) studied the permeability
of a three-dimensional fracture network relative to the
percolation threshold. Hestir and Long (1990) obtained
an equation for relative hydraulic conductivity based on
the average number of fracture intersections between
fractures for a two-dimensional network. A recent study
of solute dispersion in heterogeneous media (Rivard
and Delay, 2004) used a 2D percolation network. Al-
though this approach has most often been applied to
pore-scale problems or fracture networks, Proce et al.
(2004) related interconnectivity of a hierarchical model
of sedimentary architecture to predictions of percolation
theory. When heterogeneities are strong enough that
flow is controlled by a finite number of preferential
flowpaths, a combination of percolation theory and
critical path analysis (Hunt, 2001) holds promise for
analyzing such systems.
4. Approaches to flow simulation in heterogeneous
media
Simulation of flow through heterogeneous media
requires decisions about structuring the model domain
in light of the treatment of heterogeneity as described
above. Much depends on how the uncertainty of the
model, an inherent property, is perceived. There are
two general approaches to handling model uncertainty
which can both be applied, whether continuum or
discrete fracture conceptualizations are used. They
are commonly referred to as the zonal (usually deter-
ministic) formulation and the geostatistical formulation
(Gorelick, 1997). The first assumes that the hydraulic
conductivity field can be subdivided into zones in
which the properties are constant but uncertain, to be
determined by model calibration. The second assumes
a groundwater domain with continuously variable
properties, in which the spatial distribution of these
properties is uncertain. Recent developments are in-
creasingly blurring the lines between the two as mixed
formulations of zonal and geostatistical approaches to
parameterization of model domains become more
widespread.
4.1. Zonal formulation
The zonal formulation is that which is most widely
used among hydrogeologists, because of the ease with
which fixed parameters, such as hydraulic conductivity,
can be assigned to subsets of model elements (grid
blocks or polygons) into which groundwater flow
model domains are commonly discretized. Model cali-
bration using btrial and errorQ or inverse methods is then
used to adjust these model parameters until simulated
hydraulic heads and flows match observation targets of
water levels in wells and measured streamflows (Ander-
son and Woessner, 1992). In recent years, much prog-
ress has been made in incorporating methods of
nonlinear parameter estimation using inverse methods
(McLaughlin and Townley, 1996; Hill, 1998) into stan-
dard hydrogeologic practice. A number of numerical
codes for this purpose such as UCODE (Poeter and
Hill, 1998) and PEST (Doherty, 2002) are now widely
available.
However, as noted recently by Doherty (2003), most
modelers overlook the uncertainty in the spatial zona-
tion of hydraulic properties, as opposed to that of
parameter values assigned to different zones. The pa-
rameter zonation, which constitutes the model structure,
is usually fixed at the stage of model conceptualization,
and constitutes an important source of uncertainty.
Different conceptual models could involve different
schemes of zonation of parameters in a numerical
model. Failure to test these alternatives using inverse
methods may conceal a major source of error
(McLaughlin and Townley, 1996). A good example of
such conceptual model testing is the study of the Death
Valley regional groundwater flow system in the south-
western USA (D’Agnese et al., 1997, 1999). During the
study, based on inverse modeling analysis, the concep-
tual model evolved with an increase in the number and
significant change in the location of zones of hydraulic
conductivity.
The increase in size and resolution of continuum
model domains makes it increasingly difficult and
time-consuming to test different configurations of
model zonation as well as parameter values assigned
to different zones. Nevertheless, recent developments
integrating the testing of conceptual model structure
into an inverse modeling approach seem quite promis-
ing. In particular, Tsai et al. (2003a,b) describe a com-
binatory optimization scheme using a genetic algorithm
and Voronoi tesselation for parameter structure identi-
fication, in 2D and 3D inverse models. Likewise, Doh-
erty (2003) presents a method of pilot points and
regularization, integrated into the inverse code PEST,
T.T. Eaton / Sedimentary Geology 184 (2006) 187–201 193
which allows the characterization of spatial properties
as well as estimation of parameter values. The incor-
poration of model structure estimation into the inverse
problem can help overcome another difficulty in
groundwater flow modeling – that of overparameter-
ization. Overparameterization occurs where not enough
observations are available to constrain the property
values of interest, or where unrealistic parameter values
are estimated (Doherty, 2003). A related difficulty
commonly encountered is that of parameter correlation
such that hydraulic conductivity and recharge, for ex-
ample, cannot be independently estimated. Hill and
Osterby (2003) have recently proposed a new method:
singular value decomposition, to detect extreme param-
eter correlation.
4.2. Geostatistical formulation
In a geostatistical formulation, if a discrete fracture
network conceptualization is used, it is considered im-
possible to collect enough field data to completely
characterize the locations, orientations, extents and
transmissivities of the discrete fractures. Therefore,
fracture geometric properties and parameters of the
flow model are considered to be random variables
whose probability density functions are estimated
from field data (Hsieh, 1998). Likewise, for a continu-
um conceptualization, the variation of the hydraulic
conductivity in the subsurface is generally considered
to be a random variable from a multivariate Gaussian
distribution. Its statistical properties are described by a
semivariogram fitted to field data. Realizations of syn-
thetic fracture networks or images of hydraulic conduc-
tivity fields with these properties are then taken to be
statistically equivalent to the actual fractured or hetero-
geneous domain.
In such classical stochastic methods (e.g., Gelhar,
1993; de Marsily et al., 1998), hydraulic properties are
considered stationary random variables; in other words,
the mean is constant, and variability is independent of
spatial location (Myers, 1989; Isaaks and Srivastava,
1989). Furthermore, the ergodic hypothesis, i.e. the
ensemble average is statistically equivalent to the spa-
tial average of a variable in one realization, is assumed
to hold (Dagan, 1997). Such a classical stochastic
approach has not been as widely used as it might be,
perhaps due to the apparent non-stationarity of porous
media at most scales (Anderson, 1997). In addition, the
validity of the ergodic hypothesis has been questioned
for highly heterogeneous media (Sposito, 1997; Zhan,
1999). These difficulties with this geostatistical formu-
lation result from the reliance on ensemble moments
described by Neuman (1997) as, btheoretical artifacts ofa convenient mathematical frameworkQ that, while use-ful, requires a leap of faith to accept. It is most often
applied in settings where field data are considered too
limited for a deterministic zonal model to be appropri-
ate, such as extremely large-scale or deep heteroge-
neous environments.
In practice, sequential stochastic simulation is often
used to reproduce images of geological heterogeneity
by means of equiprobable realizations that honor a
semivariogram describing spatial covariance of the
field data (Deutsch and Journel, 1997). A widely used
version of this process is sequential Gaussian simula-
tion, in which the simulated values are drawn from
Gaussian (spatially periodic) distributions with para-
meters based on a kriged solution. However, sequential
Gaussian simulation has the drawback of relying on the
assumption of multi-variate Gaussianity. This leads to
maximum entropy realizations in which the extreme
values are highly disconnected, unlike observations of
common geologic structures (Gomez-Hernandez and
Wen, 1994), with obvious implications for effective
hydraulic conductivity and transport. Conditioning or
constraining simulations, such that interconnectedness
inferred from field data is incorporated into simulations,
can improve the representation of and flow simulation
in realistic geologic structures.
Another common variety of such modeling is se-
quential indicator simulation, which allows for better
representation of discrete heterogeneity. In this case,
the stochastic process of hydraulic conductivity distri-
bution is transformed to a step function defined by
thresholds of categorical variables (Deutsch and Jour-
nel, 1997). Sequential indicator simulation can also be
conditioned to field data and it does not assume statis-
tical homogeneity, unlike sequential Gaussian simula-
tion. A Monte Carlo approach in sequential indicator
simulation is generally used, in which flow and/or
transport is simulated through large numbers of equi-
probable hydraulic conductivity fields (Ritzi et al.,
1994, 2000; Pohlmann et al., 2000), generated from
the semivariograms fitted to available field data. Sta-
tistical analysis of the results provides some confidence
that the range of possible outcomes of such modeling
has been adequately captured.
4.3. Mixed formulations
The idea of architectural elements in sedimentary
lithofacies, popularized by Miall (1985), Miall and
Tyler (1991) for fluvial strata and introduced to hydro-
geologists by Anderson (1989), has been very influen-
T.T. Eaton / Sedimentary Geology 184 (2006) 187–201194
tial in conceptualizing heterogeneity in aquifer systems,
particularly in unlithified sediments. The architectural
elements approach allows the building of a framework,
based on depositional environments and geological
processes, to describe the geometry of large-scale het-
erogeneity. This framework specifies the spatial posi-
tion of bounding surfaces for architectural elements,
usually deterministically or based on log or outcrop
studies. The smaller-scale variation of hydraulic proper-
ties within such architectural elements can be specified
deterministically or with a geostatistical formulation. To
date, the most common application of this approach has
been for unlithified aquifers (Scheibe and Freyberg,
1995; Heinz et al., 2003; Lunt et al., 2004), but similar
techniques have been applied to indurated sedimentary
aquifers and reservoirs (Fisher et al., 1998; Willis and
White, 2000; Trevena et al., 2003). Related work has
relied more heavily on geostatistical formulations to
specify the boundaries of geologic features using petro-
physical criteria (Rossini et al., 1994; Rovellini et al.,
1998).
Others have specified the location and geometries of
architectural elements using object-based processes
(Anderson et al., 1999; Moreton et al., 2002; Tye,
2004; McKenna and Smith, 2004). Genesis methods
have also been described to represent deposition of
alluvial materials of different geometries (Lancaster
and Bras, 2002; Teles et al., 2004), that would be suited
to simulation of flow through these deposits. A com-
bined structure-imitating, geostatistical and geophysical
imaging approach was used to reconstruct the spatial
structure of hydraulic properties in a channel bend
deposit (Cardenas and Zlotnik, 2003). These methods
seem particularly promising to describe the geological
heterogeneity in indurated aquifer and reservoir sys-
tems, particularly if diagenetic processes can be incor-
porated. However, none of them has apparently yet
been applied to indurated aquifers.
An important advance has been the formulation of
transition probability-based indicator geostatistics
(Carle and Fogg, 1996; Carle et al., 1998). The basis
for this approach is the use of multidimensional Markov
chains to represent the variability of sedimentary struc-
tures (Carle and Fogg, 1997). This method allows the
incorporation of easily observable geological informa-
tion, such as asymmetry, proportion, mean length and
juxtaposition of lithofacies into an indicator geostatis-
tical framework for characterizing heterogeneity. It has
already been used to simulate unlithified deposits for a
flow model in alluvial fans (Weissmann and Fogg,
1999; Weissmann et al., 2004) and buried valley aqui-
fers (Ritzi, 2000; Ritzi et al., 2000, 2003). However,
there is no obstacle in principle to its use for character-
izing heterogeneity and distributing hydraulic proper-
ties for flow simulation in sedimentary bedrock
aquifers. The necessary geologic observations can be
made in indurated formations, as described in the fol-
lowing papers in this special issue.
5. New developments in flow simulation in
geologically heterogeneous settings
Recent developments in incorporating geological
heterogeneity in flow simulation can be categorized
as either theoretical advances or notable studies and
flow modeling applications. The former have mainly
occurred in the development of new geostatistical
approaches that illustrate and begin to overcome some
of the limitations of classical stochastic methods de-
scribed above. The latter consists of selected recent
studies addressing aspects of geological heterogeneity,
and notable flow modeling applications that employ
codes designed for equivalent porous media (EPM)
simulation in new ways. These modeling applications,
in conjunction with increased computer-processing ca-
pability, open new directions in representing geological
heterogeneity in sedimentary aquifers.
5.1. Theoretical advances in geostatistical approaches
Neuman and di Federico (1998) and Neuman (2003)
have noted that the constant-sill semivariogram models
used to infer statistical homogeneity may in fact be an
artifact of an infinite hierarchy of mutually uncorrelated
homogeneous fields at increasing scales. It has been
suggested that heterogeneity in geologic media is actu-
ally characterized by evolving scales (Cushman, 1997),
and new methods for stochastic simulation of such
heterogeneous property fields have been proposed
(Rubin and Bellin, 1998). A particular case of evolving
heterogeneity is that of fractal scaling, in which the
correlation structure can be described by a power law
(Neuman, 1994; Molz et al., 2004). These approaches
overcome the limitations of classical multi-Gaussian
geostatistical methods, but remain challenging to put
into practice. They are nevertheless exciting new devel-
opments that may provide more realistic frameworks to
characterize geological heterogeneity for the purpose of
flow simulation.
In response to the limitations of classical geostatis-
tical methods as outlined earlier, recent work has
addressed the problem of flow in highly heterogeneous
media where the assumption of stationarity does not
hold. An example of non-stationary flow and solute
T.T. Eaton / Sedimentary Geology 184 (2006) 187–201 195
flux due to multi-scale geological heterogeneity and
complex boundary conditions was given by Wu et al.
(2003) (Fig. 5). It is likely that such non-stationary flow
conditions are the rule in highly heterogeneous sedi-
mentary bedrock aquifers, as has been illustrated in
unlithified materials (Heinz et al., 2003). Efforts to
quantify flow through such settings have employed
analytical non-stationary spectral methods (using Four-
ier transforms) (e.g., Lu and Zhang, 2002a, Li et al.,
2004). Composite media approaches, in which different
heterogeneous structures of contrasting hydraulic prop-
erties, such as inclusions of different shapes, have also
been used to quantify flow numerically (Winter and
Tartakovsky, 2002; Winter et al., 2002; Dagan et al.,
2003). These numerical methods can also be used to
solve for flow in multi-modal structures of heteroge-
neous porous media, in which the first two statistical
moments are not adequate to characterize properties (Lu
and Zhang, 2002b). As these methods become more
widely understood, and implemented in readily avail-
able modeling codes, their application will allow a
geostatistical approach to even the most heterogeneous
flow systems, a significant advance.
5.2. Recent studies of heterogeneity and notable flow
modeling applications
Characterizing heterogeneity and simulating flow in
fractured sedimentary bedrock aquifers is particularly
Fig. 5. Simulation of flow in a representative cross section at Yucca Mountai
the hydraulic conductivity distribution is non-stationary (from Wu et al., 20
challenging, and some new developments are highlight-
ed here. A facies analysis approach in carbonate set-
tings has usually focused on how texture controls
hydraulic properties (Rovey and Cherkauer, 1994;
Hovorka et al., 1998; Schulze-Makuch and Cherkauer,
1998). While recognizing that properties of the rock
matrix determined by lithofacies are certainly relevant,
particularly in recent carbonates (Budd and Vacher,
2004), other workers have focused on heterogeneity
caused by fracturing. In relatively undeformed rocks,
bedding-plane fractures have been found to constitute
particularly important flowpaths (Novakowski and Lap-
cevic, 1988; Yager, 1997; Michalski and Britton, 1997).
Building on detailed stratigraphic work on the Silu-
rian carbonate aquifer in Wisconsin, USA, Muldoon et
al. (2001) have identified and correlated such planar
high permeability zones at scales of up to 16 km. While
pumping test data indicated that the aquifer responds as
an equivalent porous medium at large (100–1000 m)
scales, a conceptual model was developed in which
horizontal flow and hydraulic conductivity were almost
entirely due to bedding plane fractures. The hydrostra-
tigraphic conceptual model was then incorporated into a
groundwater flow simulation using thin, high-perme-
ability layers in an equivalent porous medium model
(Rayne et al., 2001). This novel approach was needed
to account for the highly transient nature of recharge
and well water fluctuations in the aquifer, and to delin-
eate capture zones for municipal wells. A similar ap-
n, showing flow lines in an extremely heterogeneous aquifer in which
03). Reprinted with permission from Elsevier.
T.T. Eaton / Sedimentary Geology 184 (2006) 187–201196
proach has been used by Swanson et al. (2006—this
issue) to simulate springflow from a sandstone aquifer.
In a numerical flow modeling study of an exhumed
fractured sandstone reservoir (Aztec Sandstone) in
Nevada, USA, Taylor et al. (1999) examined the distri-
bution of fluid flow between a joint set and the rock
matrix. The objective was to identify the proportion of
joints that would have conducted flow in the past, and
understand the pattern of chemical alteration and frac-
ture mineralization on the now exposed rock. Of inter-
est here is the implication that fracture heterogeneity,
not normally considered in flow simulation in sand-
stone bedrock sequences, can be a major control on
flow. Taylor et al. (1999) represented fractures as linear
features with a range of lengths embedded in an equiv-
alent porous medium using a finite-element numerical
simulation. Their results showed that depending on the
hydraulic conductivity contrast between fractures and
rock matrix, flow was either dominated by indirect
connections between fractures (low contrast) or domi-
nated by flow in the interconnected fracture network
(high contrast) (Fig. 6). They concluded based on the
field observations of chemical alteration and joint min-
eralization that the actual joint permeability must have
been ~5 orders of magnitude greater than that of the
rock matrix (Taylor et al., 1999). An extension of this
study (Eichhubl et al., 2004) has examined mixing
between basinal fluids and meteoric water through
Fig. 6. Partitioning of paleoflow between joints and rock matrix as a
function of the contrast in hydraulic conductivity, inferred from
chemical alteration of fractures in a sandstone aquifer (from Taylor
et al., 1999). Copyright 1999 American Geophysical Union. Repro-
duced by permission of American Geophysical Union.
regional and outcrop-scale fluid migration pathways,
demonstrating the importance of focusing flow by
such structural heterogeneity in these typical shoreline
facies.
Many other types of geological heterogeneity have
been recognized to control flow in the subsurface, and
only a few more recent examples will be given here. In
a study on the same Aztec Sandstone described above,
Sternlof et al. (2004) describe the hydraulic properties
of deformation bands: zones resulting from shear and
compaction associated with stresses that cause faulting
(Main et al., 2000). Three different types of deforma-
tion bands were found to have hydraulic conductivity
up to two orders of magnitude lower than undeformed
rock (Sternlof et al., 2004). Such contrasts are equiva-
lent to those normally used by hydrogeologists to dis-
tinguish aquitards from aquifers, and failure to
recognize such features for purposes of flow simulation
could cause significant error.
Various types of diagenetic processes that control
hydraulic properties down to the bed-scale, such as
texture- and grain-size control of porosity and cemen-
tation in sandstones (Milliken, 2001) have been de-
scribed. Davis et al. (2006—this issue) discuss the
effect of carbonate cementation on fluvial aquifer het-
erogeneity. On a larger scale, dolomitization processes,
and burial/compaction leading to fracturing have long
been understood as a principal determinant of carbonate
reservoir properties. However, Westphal et al. (2004)
have recently presented a case where hydrothermal
brecciation as well as dolomitization and calcite cemen-
tation are responsible for reservoir heterogeneity in the
Wind River Basin, Wyoming. Finally, increasing atten-
tion is being paid to biological processes that can be
responsible for geological heterogeneity and ultimately
influence weathering and erosion. In a recent study,
Gingras et al. (2004) analyzed the effect of worm bur-
rows in creating a dual-permeability system, demon-
strating tortuous flow paths in an Ordovician limestone.
6. Revisiting the concept of hydrofacies
It is clear from the trends in understanding and
simulating flow through heterogeneous geologic media
that it is more important than ever to incorporate major
aspects of the structure of the flow domain into numer-
ical simulation models. These may be interconnected
high permeability features such as fractures, or carbon-
ate dissolution channels, or sand stringers in unlithified
or indurated sands. From another perspective, the spatial
distribution of low-permeability structural or diagenetic
features, such as faults, breccia or deformation bands,
T.T. Eaton / Sedimentary Geology 184 (2006) 187–201 197
can channel flow. In any event, in such heterogeneous
aquifers, higher conductivity networks are formed
which have a dominant influence on flow and transport
that must be reflected in the model. The paradigm of
such interconnected networks has long been advanced
by Fogg and others in unlithified settings (Fogg, 1986;
Fogg et al., 2000) as well as in low-permeability envir-
onments (Fogg, 1990). Fractured media represent a clear
example where such networks are critical to flow, but
any type of structure in a heterogeneous medium can
form interconnected networks that control flow. This
idea has been associated with the concept of hydrofacies
since the work of Poeter and Gaylord (1990), building
on the work of Anderson (1989), who defined hydro
(geologic) facies as bhomogeneous but anisotropic
unit(s) that (are) hydrogeologically meaningfulQ.With more widespread interest in the control of
groundwater flow and transport by geological hetero-
geneity, advances are needed in incorporating hetero-
geneous structure in simulation models to reproduce
actual flow observations. Therefore, an operational
broadening of the concept of hydrofacies is suggested
here. Hydrofacies is fundamentally a hydrogeologic
concept, yet it is commonly applied as a label to units
that have already been delineated using geologic (usu-
ally lithofacies) criteria. To accommodate variations in
hydraulic properties due to non-lithofacies structure (for
example fractures or diagenetic variations in bedrock),
an expanded definition of hydrofacies with reference to
the representative elementary volume is proposed
(Eaton, 2002). Specifically, hydrofacies can be regarded
as scale-dependent hydrogeologic units with a particu-
lar representative elementary volume (REV) or struc-
ture of a specific size and shape.
From a flow modeling perspective, hydrofacies are
only hydrogeologically meaningful if the model incor-
porates the structure of the heterogeneity that consti-
tutes the hydrofacies, whatever the geologic origin of
that heterogeneity. So, for instance, in an equivalent
porous medium flow model, the grid cell size (REV)
must be small enough to resolve this hydrofacies
structure, otherwise the simulation results will not be
accurate. Unfractured and fractured zones, or zones of
different diagenetic alteration in the same lithofacies,
could be considered two different hydrofacies. For
example, Low et al. (1994) identified units of similar
hydrogeologic properties in fractured rock called
bHydrogeologic unitsQ that would constitute hydrofa-
cies in this view. In a discrete fracture model, the
major fracture interconnections that make up the geo-
logical heterogeneity must be adequately represented
in the model for the flow simulation to be accurate. In
such cases, the fracture network backbone itself (not
all the fractures) becomes the hydrofacies.
The major advantages of such an expansion of the
concept of hydrofacies is that it allows the description
of heterogeneous flow domains that are not limited to
unlithified materials (as has been mostly the case up to
now), incorporates many other sources of geological
heterogeneity that control flow in indurated rocks, and
imposes a simulation-based constraint on hydrogeolo-
gic characterization. In other words, the delineation and
representation of hydrofacies are adequate if the result-
ing numerical model results in an error smaller than
some acceptable level with respect to field data. Iden-
tification and explicit representation of the structure
represented by different hydrofacies, using an adequate
discretization, becomes an important part of the process
of constructing numerical flow models. It is hoped that
such an operational broadening of the concept of hydro-
facies will be helpful to workers characterizing the
geologic controls on flow in heterogeneous aquifer
systems and hydrocarbon reservoirs.
7. Summary and conclusion
A perspective on the current status of integrating
geological heterogeneity into numerical simulations of
flow is presented. A review of basic concepts behind
such flow simulation describes equivalent porous me-
dium and discrete fractured network paradigms, as well
as zonal, geostatistical and mixed formulations. Recent
developments in incorporating geological heterogeneity
in flow simulation have included theoretical advances
in geostatistical approaches and novel applications of
well-known equivalent porous media approaches to
illustrate preferential flow. Two such applications are
described at some length, and some other recent exam-
ples are reviewed of publications on geological hetero-
geneity as it affects flow.
A common framework would be useful in consider-
ing how to incorporate geological heterogeneity of all
kinds into flow models. Therefore, the concept of
hydrofacies, which has generally been associated with
lithofacies, is revisited. Hydrofacies, or units defined by
unlithified materials with different hydraulic properties,
have long been used by hydrogeologists in constructing
flow models. A new interpretation is suggested which
encourages the idea of applying hydrofacies to indurat-
ed as well as unlithified aquifers. Such an operational
broadening incorporates a larger range of geological
mechanisms that determine variations in hydraulic
properties, and links hydrofacies to a minimum resolu-
tion of detail that is needed to accurately simulate flow
T.T. Eaton / Sedimentary Geology 184 (2006) 187–201198
in a heterogeneous aquifer. Readers are invited to con-
sider the concepts addressed above as they study the
following papers in this special issue of Sedimentary
Geology.
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