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).


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.


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


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,


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-


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


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.


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,


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


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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on the importance of geological heterogeneity for flow simulation

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