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OCR for page 79
3
Flow Processes
INTRODUCTION
As suggested by the discussion in the previous chapter, most ap-
plications of ground water models to aid decisionmaking begin with
the potential energy and flow of water alone. First of all, ground
water flow models, with their focus on the prediction of head, vol-
umes, and velocity of flow, can be important tools in the assessment
and development of water resources. For example, predictions of the
economic yield of an aquifer, or of the impacts of new or increased
pumping on existing wells, or of ground water recharge below ir-
rigated agriculture, all require an understanding and prediction of
ground water head and flow. Second, ground water flow models are a
crucial component of all analyses of contaminant transport because
of the need to define the ground water velocity field. As noted in
a number of sections of this report, advection with the flow field
is often the dominant process controlling the direction, if not the
rate, of transport. In the absence of significant density differences
caused by contaminant concentration differences, the velocity field
is independent of chemical and biological transport processes. Thus
transport modeling studies usually begin with a prediction of the
velocity field based on a ground water flow model.
Historically, the earliest ground water models were developed to
79
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80
GROUND WATER MODELS
predict head and volumetric flow in fully saturated, porous (nonfrac-
tured) geologic environments. Because of this relatively long history,
saturated continuum flow models have been investigated extensively
and are quite well understood in the context of a wide variety of prob-
lems. The modeling of flow in unsaturated, nonfractured settings has
a shorter history and is largely dominated by problems of understand-
ing and predicting infiltration from rainfall, irrigation, rivers, canals,
and ponds. Our understanding of such models is less sophisticated
than our understanding of saturated flow models. Least well devel-
oped, and indeed only in its infancy, is our understanding of recent
attempts to mode! head and flow in both saturated and unsaturated
fractured environments. For all three cases saturated continuum
flow, unsaturated continuum flow, and fracture flow most of our
understanding of flow modeling has been gained for problems requir-
ing predictions of head and volumetric flow rates. The demands of
contaminant transport prediction, in which the critical flow variable
is velocity, are more challenging and have only recently become the
focus of attention in ground water flow modeling.
Although ground water flow modeling ~ older and more ad-
vanced than ground water transport modeling, many issues and un-
certainties remain in the application of flow modem to decisionmaking
problems, particularly those involving transport. This chapter sum-
marizes the committee's sense of the state of the art of ground water
flow modeling and of the issues related to the current and future use
of flow models in decisionmaking.
SATURATED CONTINUUM FLOW
As mentioned in the introduction to this chapter, the earli-
est models of ground water were saturated continuum flow models.
There are several reasons for this early interest in fully saturated flow
First, the dominant problems 30 years ago were problems of water
resources development. Attention was focused on questions of avail-
able ground} water resources ant! on the impacts of the installation of
wells on these resources (and surface water resources). Variables of
interest were head and volumetric flow rate. Relevant spatial scales
were large aquifers and aquifer systems. Ground water quality was
assumed to be high, except in areas of saltwater intrusion.
A second impetus for the early focus on fully saturated flow
was the relative ease with which the physical processes of greatest
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FLOW PROCESSES
81
importance to water resources questions can be represented. Gov-
erning equations are linear or nearly so; assumptions of spatial ho-
mogeneity and temporal steady state are often justified; one- and
two-dimensional formulations yield relevant, useful information; and
relatively unsophisticated numerical approximation techniques are
adequate. Put simply, the easiest types of ground water problems to
mode! are fully saturated flow problems.
This ease is only relative, however. fully saturated ground water
flow modeling remains quite challenging. This is especially true
for problems of contaminant transport below the water table. For
such problems, the role of ground water flow modeling is to provide
an estimate of the flow velocities. Head predictions are of little
direct interest. Velocity estimates, however, are usually based on
hydraulic head differences and therefore are much more sensitive
to modeling errors than are estimates of hydraulic head alone. In
addition, satisfactory predictions of transport often require that the
velocity field be well predicted on fine spatial grids. The use of
large-scale average velocities, which are usually very adequate for
water supply problems, can place high demands on the dispersive
component of a transport model, demands that we are only just
beginning to understand (see Chapters 2 and 4~. This need for high
spatial resolution presents formidable challenges for data collection,
parameter estimation, mode} formulation, numerical methods, and
computational power and speed.
State of the Art
The physical processes controlling the flow of water through fully
saturated porous rock or soil are well understood, both theoretically
and experimentally. The mathematical statements of the funda-
mental physical laws governing general fluid motion-conservation
of mass, momentum, and energy which are collectively known as
the Navier-Stokes equations, are universally accepted (White, 1974~.
More important, the simplifications of these equations, which lead to
Darcy's law for fully saturated flow through porous media (equation
t2.43), have been investigated both in the laboratory and in theory.
The Darcy equation is known to yield good predictions of head and
flow under a wide range of conditions encountered in the subsurface
(cf. Freeze and Cherry, 1979~.
The conditions under which the Darcy equation is not adequate
for prediction have been reasonably well delineated. Darcy's law
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82
GROUND WATER MODELS
is known to fad! for high-velocity flows, which might occur in very
porous grave! or boulder deposits, karat terrain, or the immediate
vicinity of pumping wells (Bear, 1972, pp. 125-127~. Darcy's law is
suspected to fail for flow through extremely small pores under low-
pressure head gradients (Freeze and Cherry, 1979, p. 72), conditions
that might occur at great depth, for example, in the vicinity of
potential radioactive waste repositories. Considerable controversy
about this behavior exits. In neither case of failure is there adequate
support for universally accepted alternative formulations (short of
the Navier-Stokes equations), although a number of models have
been proposed (Bear, 1972,pp. 170184~. Prediction uncertainty for
these flows must be considered larger than for the vast majority of
flows for which Darcy's law is a valid approximation.
The mathematical properties of the governing partial differential
equation for fully saturated continuum ground water flow (Equation
t2.5~) are well understood. The form of the equation is typical of a
wide variety of physical problems and so has been studied extensively
in many contexts. Because the equation is linear, many powerful
tools of mathematical analysis are applicable. Exact, analytical so-
{utions are available for a wide variety of problems characterized by
very simple geometries, boundary conditions, initial conditions, and
parameter fields (usually homogeneous). These analytical solutions
are essential in testing and verifying approximate numerical solution
techniques, and they often provide considerable insight into more
complex problems. In cases where prediction of detailed velocity or
concentration fields is unnecessary, analytical solutions often provide
adequate precision for certain problems and goals.
For those problems in which the simplifications necessary for at-
taining analytical solutions are inappropriate (there are many), nu-
merical approximation techniques are highly developed and widely
available. A sophisticated literature exits, including several texts
devoted exclusively to ground water flow problems (Huyakorn and
Pinder, 1983; Remson et al., 1971; Wang and Anderson, 1982~. Nu-
merical accuracy and its control are well understood. A number
of well-documented, robust, and flexible computer codes are readily
available (Bachmat et al., 1980; see also information from the Inter-
national Ground Water Modeling Center, Indianapolis, Indiana). In
addition, in the last few years a variety of computational and graph-
ical tools have been introduced, such as pre- and post-processors
and expert systems, designed to aid in the application of such codes.
Proper use of numerical codes, however, still requires considerable
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FLOW PROCESSES
83
training and experience, and it is unlikely that solution procedures
will ever be fully automated.
Until recently, most numerical methods have focused on efficient
and accurate computation of ground water heads for one- and two-
dimensional problems. However, in response to the increased avail-
ability of affordable computational power, the last few years have
seen significant progress in three-dimensional solution techniques.
Such techniques are no longer experimental and are beginning to be
used in practice (e.g., Ward et al., 1987~. In addition, researchers
are now focusing attention on the accurate computation of head
gradients (velocities), a task much more challenging than accurate
computation of heads (Bear and Verrujt, 1987~.
The nature of the parameters appearing in the various forms
of the fully saturated ground water flow equation ~ reasonably well
understood. Both the hydraulic conductivity and the specific storage
are empirical parameters that arise from the simplifications leading to
Darcy's law and a workable statement of continuity (see discussion
on ground water flow in Chapter 2~. While they are not directly
measurable, theoretical and experimental studies have clarified how
these parameters depend on the properties of the rock and of the fluid
when used to predict flow in laboratory columns and boxes (Bear,
1972, pp. 132-136~. Less well understood are the natures of these
parameters when used to predict average flows over large distances
through heterogeneous geologic deposits. Theoretical studies have
explored the relationship between large-scale conductivity (and/or
transm~ssivity) and the variability of local conductivity (cf. Dagan,
1986; Gelhar, 1986), but our understanding remains limited.
The state of the art of fully saturated continuum flow modeling
is least well developed in the area of hydrologic characterization. The
magnitude of flow parameters and their spatial variability currently
remain unpredictable a priori. As discussed in detail in Chapter 6,
this unpredictability is a major source of uncertainty in ground water
flow modeling today.
It is of course impractical to fully characterize an aquifer's perme-
ability distribution via small-scale permeameter testing, since such
tests are conducted on disturbed samples of material and are ex-
pensive. In addition, simple correlations between more readily mea-
surable geophysical and soil physical parameters have proven elusive
(e.g., Lake and Carroll, 1986, pp. 181-221~. Several investigators
have carried out detailed studies of the spatial structure of per-
meability and porosity in ground water environments (e.g., Byers
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84
GROUND WATER MODELS
and Stephens, 1983; Hoeksema and Kitanidis, 1985; Smith, 1981;
Sudicky, 1986~. Petroleum engineers have devoted considerable at-
tention to of! and gas reservoir characterization, developing both
techniques and insight that are useful to hydrogeologists. These
studies have shown that ground water geologic environments are
highly variable, but in general, the quantitative knowledge remains
very limited.
Parameter values must in general be inferred from field obser-
vations of head response to stress. Well tests are the most obvious
example. In most applications of ground water flow models, param-
eter values are obtained via calibration using some type of "inverse
technique," leavened by well test estimates and geologic knowledge.
Parameter values are chosen that yield satisfactory predictions of
observed head at selected observation points (usually few in number)
under known conditions.
A large body of theoretical literature has grown up around the
ground water "inverse problem" (Yeh, 1986~. It is known from these
studies that parameter values estimated in this way are nonunique
and are very sensitive to errors in measured head data. A number
of automated techniques have been suggested for dealing with these
problems and for quantifying the resulting uncertainties. There is,
however, no agreement on the best approach to this problem, nor
is there reason to expect such agreement. Automated techniques
remain experimental in practice, and most calibrations proceed by
trial-ancI-error fitting procedures with no quantification of uncer-
tainty. In the hancis of an experienced, knowledgeable hydrogeologist,
trial-and-error techniques can yield satisfactory results for problems
requiring modest spatial resolution. Parameter estimation remains,
however, one of the crucial challenges in successful saturated contin-
uum flow modeling, especially for problems of contaminant transport.
Because of the inherent uncertainty in defining the parameter
fields for ground water models, a number of investigators have begun
exploring the uncertainty in head and velocity predictions that results
from parameter uncertainty (e.g., Dagan, 1982; Smith and Freeze,
1979a,b). While limited by a number of restrictive assumptions, these
studies are showing how predictive uncertainties may be related to
uncertainties in parameters and boundary conditions. They also
suggest that prediction uncertainties can be large, especially for
velocities. While not yet common in practice, techniques for the
quantification of uncertainty are gradually becoming more accepted
and accessible.
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FLOW PROCESSES
85
Implications for the Use of Saturated
Continuum Flow Models in Decisionmak~ng
Because of the relatively long history of development and use
of saturated continuum flow models, the issues surrounding their
application to modern decisionmaking problems are generally not
conceptual or theoretical, but are practical. As noted in the previ-
ous section, the physical processes controlling saturated flow are well
understood, and the mathematical models describing these processes
have been studied extensively. The challenges posed by practical
application arise in situations where it is not feasible to mode} a
flow system at the spatial and/or temporal scale appropriate to con-
ceptual and mathematical understanding. Saturated continuum flow
models rest on fluid mechanical principles and laboratory column val-
idation of Darcy's law. Field application at this scale is not possible;
we lack complete data sets, and if we had such data sets, modeling
costs in time and computer resources would be extraordinarily high.
Therefore, successful application of ground water flow models rests
on the skill and art of the hydrogeologist in understanding when,
where, and how to simplify and respond to a lack of information.
The next few paragraphs summarize the most important issues that
must be addressed in applying saturated continuum flow models to
practical ground water problems, given the current state of the art.
Spatial Dimensionality
Many ground water flow problems may be successfully addressed
by assuming that flow occurs in only one or two dimensions, i.e., in a
single direction or in a plane. Several texts provide careful discussions
of the implications of such an assumption (e.g., Bear, 1972; Freeze
ant] Cherry, 1979~. The computational savings are obvious. The cost
of such a simplification is that mode} parameters are defined as spatial
averages of the "fundamental" parameters (e.g., transmissivity is
a depth-average of hydraulic conductivity) and that the predicted
responses (head and/or velocity) are similarly averaged. The utility
of a reduced dimension mode! (from a three-dimensional reality)
is generally greatest for problems focusing on spatially averaged
predictions (volumetric flow rates and/or heads in wells with long
screens) and away from boundaries and stresses (welIs, for example).
E`ully three-dimensional flow models typically justify their expense
only for problems requiring significant resolution in the vicinity of
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86
GROUND WATER MODELS
complex geometries (of boundaries or heterogeneities) or physically
small sources or sinks.
Boundary Conditions
The uniqueness of any particular ground water flow problem
is expressed in a mode} in part by locating mode! boundaries and
defining conditions along those boundaries (see Chapter 2~. Often
the boundaries correspond to physical boundaries in the environment
along which conditions are known or can be estimated by the use of
data. In many other situations, mode} boundaries must be defined on
the basis of practicality-physical boundaries are unknown or are at
great distance from the region of interest. In either case, boundary
condition specification ~ extremely important in many problems
and requires a thorough understanding of the mathematical role
of boundary conditions as well as the hydrogeolog~c environment.
Boundary condition Specification is an often overlooked source of
significant error (Franke et al., 1987~. No matter how complex the
model, proper application will always depend on a knowledgeable,
trained user working with data that have been collected in such a
way as to shed light on boundary conditions.
Transient Versus Steady State
Another valuable assumption in the application of saturated
continuum flow models ~ that of steady state, i.e., that conditions
remain constant over time. Because the stresses that drive ground
water flow often vary only slowly in time much more slowly than
the system requires to respond steady state assumptions are often
justified. However, there are situations where transients must not
be ignored. For example, Sykes et al. (1982) have suggested that
ignoring seasonal periodicities in ground water flow direction can
lead to otherwise unexpected dispersion during transport. Pulsed-
pumping remedial schemes, which are becoming more common, also
may demand explicit consideration of transient effects if accurate
prediction of contaminant breakthrough is required.
Discretization
The accuracy of numerical approximations to the mathematical
equations used to represent ground water flow depends on the size
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FLOW PROCESSES
87
of the discretization used relative to the rate at which the gradi-
ent of hydraulic head changes. In other words, if the gradient of
head varies rapidly because of hydraulic property heterogeneity or
boundary conditions, then discretization must be fine to achieve com-
parable accuracy to coarse discretization in regions of slowly varying
gradient. The choice of grid discretization (in space and/or in time)
is further complicated by the averaging incorporated into numerical
models. Most models make some very simple assumption about how
parameter values vary between computational nodes. For example,
many models assume parameters to be constant over a grid block.
This averaging, in the face of geologic heterogeneity, requires careful
consideration of the resolution required to answer a particular ques-
tion. Details of flow behavior below the scale of the discretization
are usually lost. A general rule of thumb is that the discretization
must be at least as dense as the data available for defining param-
eter heterogeneity. Because data are usually scarce, considerations
of numerical accuracy will often determine the grid discretization
necessary in many practical problems.
Velocity Computation
More and more problems of saturated continuum flow focus on
the prediction of ground water flow velocities. As noted earlier, the
accurate prediction of head gradients, on which velocities directly
depend, is much more difficult than the accurate prediction of head
alone. Computed head gradients are more sensitive to numerical
errors of approximation. In addition, head gradients are more sensi-
tive to parameter values. Thus problems requiring accurate velocity
prediction, e.g., transport problems, may require more sophisticated
numerical methods and may require more careful specification of
parameter values for sufficient accuracy.
Parameter Values
The dominant problem in the application of saturated continuum
flow models, given today's state of the art, is the specification of pa-
rameter values, i.e., the characterization of the geologic environment.
Direct measurement is at best costly. Complete characterization is in
any case impossible because nondestructive measurement methods
are not available. Parameter identification via inversing techniques is
computationally difficult. However, when appropriate data are avail-
able, the approach provides a practical way to characterize large-scale
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88
GROUND WATER MODELS
distributions, for example, hydraulic conductivity distributions from
hydraulic head data. While a tremendous amount of current research
is focused on these problems, simple panaceas do not seem to be on
the horizon. Parameter evaluation will remain a challenging task
demanding education, experience, skill, and wisdom.
FLOW IN THE: UNSATURATED ZONE
This report is primarily devoted to a discussion of various is-
sues related to modeling water flow and contaminant transport in
the saturated zone. A detailed discussion of flow and transport in
the unsaturated zone would seem out of place and not necessary.
However, the unsaturated zone is the region through which contam-
inants must pass to reach the saturated zone. The various processes
occurring within this region, therefore, play a major role in deter-
mining both the quality and the quantity of water recharging into
the saturated zone. It is necessary to understand the role played by
the unsaturated zone in ground water contamination and how the
processes in this zone are either similar to or different from those in
deeper flow systems.
Characterization of the Unsaturated Zone
Of course, the major feature of the unsaturated zone that dis-
tinguishes it from the saturated zone, as the terms clearly indicate,
is the degree of saturation of the pore spaces, in this case by wa-
ter. In the saturated zone, all of the pores are filled with water (or
other water-miscible or immiscible liquids) and the volumetric water
content (~) is equal to porosity Ail. In contrast, the fluid phase
occupying the pore spaces in the unsaturated zone may be liquids
(mostly water and sometimes nonaqueous-phase liquids iNAP Es])
and gases. The degree of liquid saturation at a given time varies
considerably depending on the soil's physical properties (primarily
pore-size distribution, which is related to soil texture and structure),
and the pattern of inputs and losses of water at the soil surface.
A brief examination of what happens as a completely saturated
soil gradually becomes unsaturated is necessary. It is the largest
pores that become air-fi~led first, and removal of water from the
smaller pores becomes increasingly more difficult (i.e., requires more
work or energy). This phenomenon may be explained by considering
the capillary forces that are responsible for water retention in pores.
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FLOW PROCESSES
r2r3
.__
a J
1 l -120
h1 ~ ~ ~ -100
h ~ ~- ~-80
~-~: ~I
::::::::::::::: ::::::::::::::::::::~:::
::::::::::::::: ::::::::::::::::::::::::
-60
.~-~-~- ~
~- ~- ~In
-:- h Oh
- 3 LL -40
-20
r1 < r2 ' r3
h1 > h2 ~ h3
89
Capillary Rise Equation
\
0 50 100 150 200
DIAMETER OF TUBULAR PORE, r (,um)
FIGURE 3.1 Relationship between pore size (r) on capillary rise and pressure
head (h).
When capillaries of different sizes are placed in water in a beaker,
water will rise to different levels above the free water surface. If these
capillary tubes are then lifted out of the water, they will not drain
unless external pressure is applied. The capillary (or suction) forces
that hold water inside the capillary against the gravitational forces
arise from the attraction of water molecules for each other (cohesion)
and the attraction of water molecules to the walls of the capillaries
(adhesion).
The height to which water rises in a capillary is indeed a measure
of the capillary forces. These forces are stronger in the smaller capil-
laries, as reflected by the higher rise of water there than in the larger
capillaries. In fact, the capillary forces are inversely proportional
to the radius (r) of the capillary (see Figure 3.1~. Water inside a
capillary tube (and, by analogy, in soil pores) is under suction and is
further illustrated by the concave curvature of the water-air interface
at both ends of the capillary tube when it has been taken out of
the water. By convention, free water is taken as the reference and is
assigned a value of zero for the capillary forces; thus capillary forces
are assigned a value less than zero, which is why they are referred to
as "suction" forces.
If a porous medium can be thought of as a random network of
capillary tubes of varying sizes, it can be seen that the suction force
with which water is held in different pore sequences varies inversely
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102
GROUND WATER MODELS
So far nothing has been said about the flow of ground water
in conduit systems. In general, the character of flow is so very
different often turbulent and partially saturated that most texts
do not treat the basic theory. A few books and papers (LeGrand and
Stringfield, 1973; Milanovic, 1981; White, 1969) treat ground water
flow in karst and provide a starting point for readers interested in
this fascinating topic.
Strategies for Modelmg
To date, single-phase or multiphase flow in fractured media has
been modeled using one of three possible conceptualizations: (1) an
equivalent porous continuum, (2) a discrete fracture network, and
(3) a dual-porosity medium. With the first of these approaches, it is
assumed that the medium is fractured to the extent that it behaves
hydraulically as a porous medium. Under this condition, the con-
tinuum equations for porous medium flow developed in Chapter 2
describe the problem mathematically. The actual existence of frac-
tures is reflected in the choice of values for the material coefficients
(e.g., hydraulic conductivity, storativity, or relative permeability).
Often these parameters take on values significantly different from
those used for modeling a porous medium (Shapiro, 1987~. Exam-
ples of this approach an cited by Shapiro (1987) include Elkins (1953),
Elkins and Skov (1960), and Grisak and Cherry (1975~.
With the discrete fracture approach, most or all of the ground
water moves through a network of fractures. This approach assumes
that the geometric character of each fracture (e.g., position in space,
length, width, and aperture) is known exactly as well as the pattern
of connection among fractures. In the simplest theoretical treatment,
the blocks are considered to be impermeable. Figure 3.9a is an ide-
alization of a two-dimensional network of fractures consisting of two
different sets. Note how each fracture, represented on the figure by a
line segment, has a definite position in space, length, and aperture.
The hydraulic characteristics of the fracture system develop as a
consequence of the intersection of the individual fractures. In three
dimensions, the network can be described in terms of intersecting
planes that could be rectangular (Figure 3.9b) or circular in shape
(Figure 3.9c). Examples of the discrete fracture treatment of flow in
networks include Long et al. (1982, 1985), Robinson (1984), Schwartz
et al. (1983), and Smith and Schwartz (1984~.
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FLOW PROCESSES
a
~ ~~3;
4;Z~
~1
~. .
FIGURE 3.9 Three different
conceptualizations of fracture
networks: (a) a two-dimensional
system of line segments (from
Shimo and Long, 1987~; (b) a
three-dimensional system of
rectangular fractures (from Smith
et al., 1985~; and (c) a three-
dimensional system of `'penny-
shaped~ cracks (from Long, 1985~.
103
b
c
'··· .,
A::::::::::::
;;; ;'
:::::::::: ..::
· ~
:~:~:~:~:~:~:~
·...~.....
a::::.:.:.:: :; __
............... -
:::: A:: ::::::
.._ .....~.....
. ~ ...........
.. an.... V
:::::::::'
· · · · ~
·e ~ ~e ~
:~:~:~:~:~:~:~:
· ~ ~ ....
· ~
::::::>
. 4:
r
l ~
.
~.
The dual-porosity conceptualization of a fractured medium con-
siders the fluid in the fractures and the fluid in the blocks as separate
continua. Unlike the discrete approaches, no account is taken of the
specific arrangement of fractures with respect to each other there is
simply a mixing of fluids in interacting continua (Shapiro, 1987~. In
the most general formulation of the dual-porosity model, the possi-
bility exists for flow through both the blocks and the fractures with a
transfer function describing the exchange between the two continua.
Mathematically then, one flow equation is written for the fractures
and one for the blocks, with the equations coupled by the source-sink
terms. Thus a loss in fluid from the fracture represents a gain in
fluids in the blocks (Shapiro, 1987~.
In most applications involving multiphase flow, a more restrictive
approach is followed. Fluids are assumed to flow in the fracture
network, with the blocks acting as sources or sinks to the fractures
(Torsaeter et al., 1987~.
Examining the mathematical details of the dual-porosity for-
mulation is beyond the scope of this overview. To understand the
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104
GROUND WATER MODELS
equation development or to locate the available references, readers
can refer to Huyakorn and Pinder (1983), Shapiro (1987), and Tor-
saeter et al. (1987~.
ISSUES IN MODELING
The modeling approaches just described are subject to significant
complexities of both a theoretical and a practical nature that affect
the modeling process. Three main issues are discussed here: (1)
whether fractured media can even be approximated as continua, (2)
computational constraints on discrete network models, and (3) the
uncertainty in establishing the network geometry.
A Fractured Medium as a Continuum
For a porous medium, it is not difficult to believe in the existence
of what is termed a representative elemental volume for various con-
trolling parameters. Consider a parameter like hydraulic conductiv-
ity as an example. The representative elemental volume is a sample
volume for which the hydraulic conductivity is independent of sample
volume or averaging volume. In other words, the representative ele-
mental volume exists when a small change in the sample volume does
not result in a change in hydraulic conductivity. This concept can
be demonstrated using Figure 3.10. When the volume of a porous
medium is small (e.g., a few pores), even a slight change in the sam-
ple or averaging volume can cause appreciable changes in hydraulic
conductivity. As the sample size increases, there comes a point when
the hydraulic conductivity is not sensitive to the averaging volume.
In modeling a porous medium as a continuum, it is assumed
implicitly that the domain or individual cells within the domain for
which the flow equation is written satisfy a representative elemental
volume condition. For most cases, this assumption is reasonable.
In modeling a fractured medium using continuum approaches' the
sample assumption is required. However, in the case of a fractured
medium there is much less certainty in the assumption of a represen-
tative elemental volume being valid (Schwartz and Smith, 1987~.
The first major problem with fractured media is that a represen-
tative elemental volume can only be defined when fracture densities
are above some critical density. The critical density is defined as that
density of fractures that provides connectivity of the network (Fig-
ure 3.11~. Below the critical density, the network is not connected
OCR for page 105
FLOW PROCESSES
o
C)
At,
I
o
105
air
REV
I I ~
AVERAGING VOLUME
FIGURE 3.10 Variation in hydraulic conductivity as a function of the av-
eraging volume. The dashed lines point to volume where the assumption of
a representative elemental volume (REV) is valid. SOURCE: Modified from
Shapiro, 1987.
(nonpercolating) and the mean hydraulic conductivity will be zero no
matter how large the averaging volume (Schwartz and Smith, 1987~.
Thus in modeling a fractured system, simply choosing a large volume
of rock for a cell will not necessarily guarantee that the assumption
of a representative elemental volume is met.
Situations also exist in which the concept of a representative ele-
mental volume is either impractical or invalid. Consider the following
two examples. By assuming a network to be connected but sparsely
fractured, the averaging volume necessary to obtain a representative
elemental volume of the medium could be much larger than the scale
of interest. For example, the minimum averaging volume might be
a block of rock 200 m on a side, while the scale of interest is 100
m, which makes the concept impractical. Further, a network could
be connected, but with a hierarchy of fracture types. In this case,
as the sampling volume expands, hydraulic conductivity might keep
increasing without necessarily becoming constant.
The concept of a representative elemental volume probably does
not hold for some fractured media. Thus there are going to be
systems that cannot be modeled by using continuum approaches.
Without relatively detailed analyses, it will probably be difficult to
identify these systems in advance. When the continuum approaches
to modeling are not appropriate, one must turn to discrete modeling
approaches, which are clearly not without their own problems, as the
next two sections show.
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GROUND WATER MODELS
i//
/ _
Nonpercolating
Critical Density
Percolating
FIGURE 3.11 Examples of percolating and nonpercolating networks in two
dimensions. The critical density is the point where infinite clusters of fractures
appear and connectivity is achieved. SOURCE: Schwartz and Smith, 1987.
Computational Constraints on Discrete Network Models
Modeling fluid flow in a network of discrete fractures does not
require that the fractures behave as a continuum. All of the problem
cases discussed in the previous section can be modeled without the-
oretical constraints. Unfortunately, there are practical constraints
that severely limit the capability of modeling discrete fracture net-
works. The most serious is the number of fracture intersections,
because describing flow in the network requires that hydraulic head
be calculated at each intersection. In two dimensions, a network
with 50,000 intersections will require a major computational effort
and will be expensive. Yet the size of a network would in many
cases be much smaller than the size of the region of interest. One
probably cannot create a discrete fracture system in two dimensions
that is large enough to solve intermediate and regional problems.
The situation is even more pessimistic when one tries to account
for the presence of the rock blocks, or the three-dimensionality of
the fracture network. With these more complex flow conditions, one
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FLOW PROCESSES
107
has to further reduce the number of discrete fractures that can be
incorporated in the model.
Discrete models are of limited practical value, although they
are potentially a theoretically more powerful approach to modeling
fractured systems. Their main use to date has been to explore the
fundamentals of flow and mass transport in fractured media.
Uncertainty In Establishing the Network Geometry
One further limitation in the use of discrete fracture models is the
requirement to specify the exact geometry of the network. There will
never be a situation where the geometry of a natural fracture network
is exactly known. At best, hydraulic testing can provide estimates
of apertures, and fracture mapping in tunnels or on the surface may
provide indications of fracture orientation, fracture lengths, and the
pattern of connection. However, no tests can provide a definitive
description of the network within a rock or sediment mass.
Uncertainty in describing a network ultimately translates into
uncertainty in mode] predictions made for flow in the system. Es-
sentially, the less one knows about a system, the less confident one
can be in predicting system response. Stochastic modeling methods
(e.g., Smith and Freeze, 1979a) offer a possible approach to making
predictions and establishing the potential range of uncertainty in the
face of uncertain data.
The complexity of natural fracture networks and the difficulty in
making measurements practically guarantee that predictions made
by using discrete fracture models will be relatively uncertain. The
same situation will probably hold for continuum models as well.
However, so little work has been conducted on natural systems that
it will require years to fully assess how uncertain predictions in
fractured rock systems might be.
Adequacy of Modeling Technology
In most practical problems involving saturated flow in fractured
media, there has never been much hesitation in applying continuum-
type models. For example, many would argue that the classical
methods of well hydraulics appear to mode! the response of fractured
systems to the extent necessary for design. The question remains,
however, as to how realistically such models account for the fractured
flow processes. Experience from the petroleum industry does suggest
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108
GROUND WATER MODELS
that in some cases more sophisticated flow formulations (e.g., dual-
porosity models) will be required. Further, issues of stress coupling
and the validity of the cubic law will require further study. In addition
to the theoretical questions that remain to be resolved, there is a
significant gap in practical knowledge about flow in fractured media.
For example, only very limited testing has been carried out with
well-characterized media, except on the laboratory scale.
The modeling tools exist to deal with fractured media, but at
present, results should be interpreted with caution. Systems are often
complex and extraordinarily difficult to characterize, especially with
the level of effort considered normal for most site investigations. The
state of the art in field testing provides a relatively rudimentary esti-
mate of values for some parameters like hydraulic conductivity, while
other parameters, like storativity, must be established through fitting
simple theoretical models (usually of the porous medium type).
Unsaturated flow modeling of fractured systems is a subject of
increasing interest, particularly in light of work at Yucca Mountain
in Nevada to assess the feasibility of disposing of high-level nuclear
waste in an engineered repository. Most of the same theoretical
and practical concerns that were discussed for flow in saturated and
fractured media hold for unsaturated media as well. The greatest ad-
ditional problem lies with the increased difficulty in measuring perti-
nent parameters down boreholes. According to Evans and Nicholson
(1987), a lack of data has restricted the validation of models to a
few very simple systems. This problem of the unavailability of data
or a large range in variability of existing data was also identified by
Pruess and Wang (1987) as an impediment to progress in modeling.
Thus the capability in modeling again exceeds the ability to fully
establish the validity of the model.
Not much has been written here concerning the multiphase
transport of fluids in ground water systems. Notwithstanding the
significant capability of solving these problems in petroleum-related
applications, there is much less research and overall experience with
contaminant-related, multiphase modeling. In the case of dense non-
aqueous-phase liquids (DNAPEs), especially the contaminant and
petroleum types, problems are sufficiently different that not all of
the of! field capabilities are directly transferable. Again, the commit-
tee would consider the capability of modeling to exist but without the
theoretical and practical experience with the models to consider these
applications in any sense routine. As was the case with unsaturated
flow modeling in fractured media, limitations in the data provide
OCR for page 109
FLOW PROCESSES
109
a further impediment to progress. However, because a variety of
organic liquids could be involved, even fewer data are available.
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Representative terms from entire chapter:
water flow
