area of research, but field-validated models that are directly

applicable to the remediation of contaminated sites are not yet

available. Nonetheless, when used with appropriate site

characterization data, the available models can be helpful in

developing a qualitative understanding of the behavior of the

fractured rock system and the interactions of contaminants with

the system.

Selection of a suitable model requires the model user to have a

specific objective in mind. For example, is the purpose of the

modeling effort to determine the location of additional monitoring

wells, to design optimal well placement for hydraulic control

during site remediation, to interpret existing data, or to determine

sources or predict the fate of a pollutant? No single model will

serve all purposes. The choice of model will depend on the

system conditions, the decision to be made, and the extent and

availability of site characterization data.

Before ground-water models can be applied to any system,

fractured or not, it is necessary to have extensive data about that

system. Data are needed to (1) select or develop an appropriate

model based on the processes acting on the system; (2) define

boundaries in space and time of the domain in which these

processes are acting; (3) determine the state of the system at

some point in time from which predictions, either forward or

backward in time, can be made; and (4) estimate the effects of

future stresses or inputs to the system (Konikow and Mercer,

1988). Modeling and data collection are complementary activities,

neither being a substitute for the other. Not only are data

necessary for successful modeling, but modeling results may be

used to guide data collection efforts.

Because of the heterogeneous and anisotropic nature of fractures

in the subsurface, the data requirements for modeling the

movement of water and contaminants in fractured media are

somewhat different than the requirements for modeling more

Stephen G. Schmelling and Randall R. Ross

Contaminant Transport in Fractured Media: Models for Decision Makers

EPA Superfund

Ground Water Issue

United States

Environmental Protection

Agency

Superfund Technology Support Centers for Ground Water

Robert S. Kerr Environmental

Research Laboratory

Ada, OK

Office of Solid Waste

and Emergency

Response

Office of

Research and

Development

EPA/540/4-89/004

The Regional Superfund Ground Water Forum is a group of

ground-water technical specialists, representing EPA’s Regional

Superfund Offices, organized to exchange up-to-date information

related to ground-water remediation at Superfund sites. Site

characterization of fractured rock sites, and modeling groundwater flow and contaminant transport in fractured media, have

been identified by the EPA Regional Superfund Ground Water

Forum as major issues of concern for decision-makers at many

Superfund sites. The ability to reliably predict the rate and

direction of ground-water flow and contaminant transport in

fractured rock systems would be of great value in planning and

implementing the remediation of contaminated aquifers. This

paper summarizes the current status of modeling ground-water

flow and contaminant transport in fractured rock systems. A

companion paper summarizing the status of site characterization

at fractured rock sites is in preparation.

For further information contact Stephen G. Schmelling, RSKERLAda, FTS 743-2434; Randall R. Ross, RSKERL-Ada, FTS 743-

2355.

Summary

Mathematical models have a potentially useful role to play in

arriving at a decision on the remedial action to be taken at a

contaminated site. Where there is a need for a quantitative

estimate of the threat to public health resulting from a particular

course of action, of the estimated cost and time of clean-up for

a particular remediation strategy, or of the results of other

actions to be taken at a contaminated site, mathematical models

have a greater potential to provide the needed information than

any other approach to the problem. For contaminated sites in

fractured rock, however, this potential has yet to be realized.

The development of predictive models for ground-water flow

and contaminant transport in fractured rock systems is an active
2

homogeneous unconsolidated porous media. The development

of techniques to characterize the hydrogeologic properties of

fractured rock systems has proceeded in parallel with the

development of models. These techniques are often more

complex and more difficult to interpret than analogous techniques

used in unconsolidated media.

There are at least four categories of models to describe flow and

transport in fractured rock systems. (1) Models developed for

use in unconsolidated porous media have been successfully

applied to certain fractured rock systems. These models consider

the fractured rock system as an equivalent porous medium

(EPM). EPM models are more likely to accurately predict

ground-water flux than to correctly predict solute transport

(Endo, et al. 1984). (2) A second category of fractured rock

models explicitly considers discrete fractures. The extensive

data required by discrete fracture models for fracture system

characterization limit their use to sites with a relatively small

number of well defined fractures. (3) A third category of models

represents the fractured system by a set of matrix blocks of well

defined geometry. Such models are largely research tools that

are useful for enhancing our conceptual understanding of

pollutant transport in fractured rock. (4) A fourth category of

models uses a stochastic approach to describe the fracture

distribution. At the present time these are research models. It

is important to note again that the choice of an appropriate

model depends on the conditions present at the site, and on the

decision to be reached by using the model.

Before discussing models in more detail, Section II provides

some basic information on ground-water flow and contaminant

transport in fractured rock systems. Section III briefly discusses

methods of characterizing contaminated fractured rock sites.

Site characterization and modeling are complementary activities

and each is essential to the other. The last part of the paper

discusses the conceptual basis of ground-water models and

gives some specific examples of the types of models that are

publicly available.

Fractured Rock Systems

Development of the theory of flow through porous media began

with experimental work by Henri Darcy, published in 1857. The

study of fluid flow through fractured rock was first developed in

the petroleum industry. These studies resulted from observations

that oil and gas production could be significantly increased by

fracturing the producing formations near the well bore (Duguid

and Lee, 1977). Gale (1982) notes that the first comprehensive

experiments on flow through artificial fractures were conducted

in the early 1950’s. During the past decades, the amount of

research on flow and transport in unconsolidated porous media

has greatly exceeded that devoted to fractured media. This is

due in part to the complexity of fractured rock systems and the

lack of economic incentives.

Most fractured rock systems consist of rock blocks bounded by

discrete discontinuities comprised of fractures, joints, and shear

zones, usually occurring in sets with similar geometries

(Witherspoon, et al., 1987). Fractures may be open, mineralfilled, deformed, or any combination thereof (Nelson, 1985).

Open fractures may provide conduits for the movement of

ground water and contaminants through an otherwise relatively

impermeable rock mass. Mineral-filled fractures are filled either

partially or completely by secondary cementing materials such

as quartz or carbonate minerals, thereby reducing or eliminating

fracture porosity and permeability. Deformed fractures may be

infilled with permeability-reducing gouge, a finely abraded

material produced by the cataclasis of grains in contact across

a fault plane during displacement of the rock masses. Other

deformed-fracture features include slickensides, which are

striated surfaces formed by frictional sliding along a fault plane.

Slickensides reduce permeability perpendicular to the fracture

plane, but the mismatch of fracture surfaces may increase

permeability along the fracture plane. Very little displacement is

necessary to produce gouge or slickensides. The deposition of

a thin layer of low permeability material, fracture skin, may

prevent the free exchange of fluids between the rock matrix and

fracture (Moench, 1984).

Major factors affecting ground-water flow through fractured rock

include fracture density, orientation, effective aperture width,

and the nature of the rock matrix. Fracture density (number of

fractures per unit volume of rock) and orientation are important

determinants of the degree of interconnection of fracture sets,

which is a critical feature contributing to the hydraulic conductivity

of a fractured rock system (Witherspoon, et al., 1987). Only

interconnected fractures provide pathways for ground-water

flow and contaminant transport. Fractures oriented parallel to

the hydraulic gradient are more likely to provide effective

pathways than fractures oriented perpendicular to the hydraulic

gradient. Fractured rock systems simulate equivalent porous

media when the fracture apertures are constant, the fracture

orientations are randomly distributed and the fracture spacing is

small relative to the scale of the system (Long, et al., 1982).

The cross-sectional area of a fracture will have an important

effect on flow through the fracture. Fracture-flux is proportional

to the cube of the fracture aperture (distance between rock

blocks). The relationship between flux and aperture appears to

be true for fractures with apertures greater than 10 microns

(Witherspoon, et al., 1987). Fracture apertures, and therefore

flow through fractures, are highly stress-dependant, and generally

decrease with depth (Gale, 1982).

The nature of the rock matrix plays an important role in the

movement of water and contaminants through fractured rock

systems. Metamorphic and igneous rocks generally have very

low primary porosity and permeability. Fractures may account

for most of the permeability in such systems and the movement

of water and contaminants into and out of the rock matrix may

be minimal. Sedimentary rocks generally have higher primary

porosity and varying permeability. Coarse-grained materials

such as sandstone have relatively high primary porosity and

significant matrix permeability. Fine-grained materials such as

shale have high primary porosity and low permeability. Fractures

may enhance the permeability of all types of materials. High

porosity allows significant storage of water and contaminants in

the rock matrix. Authigenic clays formed during the weathering

of certain rock-forming minerals may significantly reduce the

porosity and permeability of the fractures and rock matrix. Rates

of contaminant migration into and out of the rock matrix will

depend on the permeability of the matrix, the presence of lowpermeability fracture skins, and the matrix diffusion coefficient of

the contaminant.

A complete description of a contaminated fractured rock system

would include data on the dimensions of the system, the length,

aperture width, location, and orientation of each fracture, the

hydraulic head throughout the system, the porosity and
3

required. Hsieh, et al. conclude that failure to fit data to an

ellipsoidal representation indicates that the rock cannot be

represented by an equivalent, continuous, uniform, anisotropic

medium on the scale of the test. Depending on the application

to be made, the test may be repeated on a larger scale, or the

data may be interpreted in terms of discrete fractures of the

system.

Aquifer tests can provide information on aquifer anisotropy,

heterogeneity and boundary conditions, but do not provide

information on the range of fracture apertures or surface

roughness. One of the major drawbacks associated with longterm aquifer testing is storage and treatment of the large volume

of water discharged during the test.

Tracer Tests

Tracer tests can provide information on effective porosity,

dispersion and matrix diffusion, generally unobtainable from

other hydrogeologic methods. Tracer tests can either be

conducted under natural gradient or forced gradient conditions.

The primary disadvantages of tracer tests include the time,

expense, number of necessary sampling points, and difficulties

associated with data interpretation. However, the important

information provided by tracer tests is difficult to obtain by any

other means. Davis, et al. (1985) provide a good introduction to

the use of tracers in ground-water investigations.

Geophysical Tools

Both surface and borehole geophysical methods can be used to

characterize fractured rock systems. Application of surface

geophysical methods such as ground-penetrating radar,

magnetometer surveys, and seismic and remote sensing

techniques should be evaluated before a drilling program is

initiated. These techniques may provide insight to potential

monitoring well locations by revealing the orientation of major

fracture systems. However, the correlation of major surface

geophysical features with contaminant transport processes in

fractured media has yet to be thoroughly characterized.

Borehole walls are usually less susceptible to fractures induced

during drilling operations than cores. Borehole geophysical

techniques can usually provide a more reliable estimate of

fracture density than cores. However, as indicated by Nelson

(1985) in a review of down-hole techniques, responses used to

detect fractures on well logs are non-unique and require detailed

knowledge of the tool and the various rock property effects,

which could cause fracture-like responses. Borehole geophysical

methods include acoustic, electrical resistivity, caliper, gamma

and other high energy borehole logging techniques. The

acoustic televiewer presents a continuous image of the acoustic

response of the borehole face, and can detect fracture apertures

as small as one millimeter. This oriented tool also allows the

determination of fracture orientations. Caliper logs are best

suited for determining relative fracture intensity in continuous,

competent rock. Advances in electronic miniaturization have

led to the development of down-hole cameras, capable of

providing in-situ viewing of fractures in the subsurface.

Borehole Flowmeters

Flowmeters have been used for many years in industry. However,

only recently has instrumentation been developed capable of

accurately measuring very low flow rates. Borehole flowmeters

permeability of the rock matrix, the sources of water and

contaminants, the nature and concentrations of the contaminants

throughout the system, and the chemical interactions between

the contaminants and rock matrix. Presently, collection of such

detailed information is neither technically possible, nor

economically feasible on the scale of most contaminated sites.

However, most ground-water models, especially those describing

contaminant transport, require this type of information as input.

In general, the more detailed the site characterization, the

greater the probability of success in modeling the site. The

accuracy of flow and transport modeling in fractured rock

systems is highly dependent on the accuracy and extent of site

characterization data.

Hydrogeologic Characterization Methods

Hydrogeologic characterization methods are usually most

successful when used in conjunction with one another. These

methods may include coring, aquifer tests, tracer tests, surface

and borehole geophysical techniques, borehole flowmeters,

and other tools. Important information may be gathered before,

during, and after drilling operations.

Coring

Core material obtained during drilling operations can yield

information on the density, location, and orientation of fractures,

and provide samples for physical and chemical testing.

Information concerning fracture roughness and mineral

precipitation on fracture surfaces can also be obtained from

core samples. Information collected during air hammer drilling

operations with open hole completions includes the location of

major water bearing fractures, changes in hydraulic head with

depth, and changes in the ground-water geochemistry. In certain

instances, cores may be taken diagonally to intercept near

vertical fractures and determine fracture azimuth. A major

drawback of coring is the relatively high cost. However, the

information obtained from coring operations often makes this

characterization technique cost-effective.

Aquifer Tests

Aquifer tests, including constant rate pumping tests and slug

tests, can provide hydraulic conductivity and anisotropy

information for fractured formations. These tests also allow the

estimation of average fracture apertures of a medium. The

same tests that are commonly used for unconsolidated porous

media can be used for fractured media. The test results,

however, will generally be more difficult to interpret. Barker and

Black (1983) note that transmissivity values will always be

overestimated by applying standard type curve analysis to

fissured aquifers. Other more complex tests, such as cross-hole

packer tests, are particularly applicable to fractured media.

Hsieh, et al. (1985a,b) describe a method of determining the

three-dimensional hydraulic conductivity tensor. The method

consists of injecting fluid into, or withdrawing fluid out of,

selected intervals isolated by inflatable packers and monitoring

the transient response in isolated intervals of neighboring wells.

This method is applicable to situations where the principal

directions of the hydraulic conductivity tensor are not necessarily vertical and horizontal. A minimum of six cross-hole tests

is required to determine the six independent components of the

hydraulic conductivity tensor. In practice, scatter in the data is

likely to be such that more than six cross-hole tests will be
measure the incremental discharge along screened or openhole portions of wells during small-scale pumping tests. The

three major types of flowmeters currently being developed

include impeller, heat-pulse, and electromagnetic. Heat-pulse

and electromagnetic flowmeters have no moving parts that may

deteriorate over time; they also have greater sensitivity than

impeller flowmeters (Young and Waldrop, 1989). The greater

sensitivity may allow the detection of the vertical movement of

water within the borehole under non-pumping conditions. Under

pumping conditions, fracture zones contributing ground water to

a borehole may be identified. Currently only prototype heatpulse and electromagnetic flowmeters have been developed.

However, commercial models should be available in the near

future.

Other Characterization Methods

Fracture traces, fault planes and other lineaments are often

identifiable on aerial photographs, but must be field verified to

distinguish anthropogenic features such as fences and buried

pipelines from geologic features. The orientation of all fractures

identified from aerial photographs and field observations (e.g.,

outcrops and excavations) should be measured and plotted on

rose diagrams to identify major fracture trends. Such trends are

usually related to the geologic (tectonic) history of a site. A basic

understanding of a site's tectonic history and subsequent fracture

orientations should allow a better understanding of potential

contaminant pathways.

Graphical geochemical techniques commonly used in porous

media may provide valuable information at fractured rock sites.

Hem (1985) and Lloyd and Heathcote (1985) provide overviews

of methods used to identify the sources and extent of mixing of

ground waters.

Models

Geometric Concepts

A model is a simplified representation of a physical system. The

focus of this paper is on mathematical models which may be

appropriately described as “a mathematical description of the

processes active in a ground-water system, coded in a

programming language, together with a quantification of the

ground-water system it simulates in the form of boundary

conditions and parameters” (van der Heijde, et al., 1988).

Preceding and underlying the mathematical model is a largely

qualitative description of the structure of the system under study

and the physical, chemical, and biological processes to be

included in the model. This qualitative description is called a

conceptual model. Several different conceptual models have

been used to describe flow and transport in fractured media.

The most common conceptual picture of flow and contaminant

transport in a fractured porous medium is that the advective flow

of water and transport of pollutants is largely, or entirely, through

the fractures. Water and contaminants may diffuse into and out

of the porous rock matrix. This diffusion can act to spread out the

contaminant plume in space and time, and to retard it. In

situations where transient water flow is involved, water may also

be stored in, and released from, the rock matrix. To the extent

that there is sufficient primary porosity in the matrix to allow

advective flow and transport, as might be the case for a sandstone,

this basic conceptual picture will be in error, as will any model

that is based on it. If the rock matrix has very low porosity, such

as would be the case for granite, then the role of the rock matrix

can often be neglected.

Several different approaches, or concepts, have been used to

model flow and transport in fractured media. Models can be

roughly classified as equivalent porous media models, discrete

fracture models, geometrically based models, and stochastic

fracture distribution models (van der Heijde, et al., 1988). One

could also develop models that overlap these categories.

 The EPM approach is to treat the fractured rock system as if it

were an unconsolidated porous medium. This approach is most

likely to be successful when the spacing of the fractures is small

compared to the scale of the system being studied, and the

fractures are interconnected. Good results in modeling groundwater flow have been obtained when these conditions were met

(Pankow, et al., 1986). The validity of using the EPM approach

to model pollutant transport in a fractured system is less well

established.

A modification of this approach is to model the system as if it

were composed of two overlapping continua with different

porosities and permeabilities. Low porosity and high permeability

are associated with the fractures and high porosity and low

permeability are associated with the rock matrix. The model

allows for transfer of contaminants between the fractures and

the rock matrix. This multiple interacting continua (MINC)

approach (Preuss and Narasimhan, 1985) requires that the

fractures be closely spaced relative to the size of the system and

that the fractures be frequently interconnected.

In the discrete fracture approach, the fracture geometry is

explicitly included (e.g. Grisak and Pickens, 1980; Sudicky and

Frind, 1982). Fractures are most often represented as channels

with parallel sides, and the individual fractures are combined

into fracture networks. The simplest network has a set of parallel

fractures in what is basically a one-dimensional problem. A

more complex network has two sets of parallel fractures oriented

at some angle to each other in a two-dimensional array (Smith

and Schwartz, 1984). Another increase in complexity, and one

step closer to reality, is to allow the fractures to have varying

lengths, locations, and orientations relative to one another (Long

and Billaux, 1987). These models can have either two- or threedimensional fracture arrays. Some of the discrete fracture

models only account for solute transport by advection, and

others include both advection and dispersion. Essentially all of

the discrete fracture models are research models. One obvious

problem in the practical application of discrete fracture models

is that it is almost impossible to define the fracture system at a

site in fine enough detail to apply the model. The best possibility

for this approach seems to be through some sort of statistical

modeling of the fracture system to duplicate the measured

hydrology at the site. Most of the work on complex discrete

fracture networks has been done in connection with the disposal

of nuclear waste in crystalline rocks and has not included

diffusion into the rock matrix.

Another approach to modeling flow in fractured media is to

represent the fractured system by a set of porous matrix blocks

of well defined geometry. The most common examples are

parallel prismatic blocks (e.g., cubes) or spheres arranged in a

regular array. The spaces between the blocks are the fracture

channels. The blocks are assumed to be porous so that solutes

4
can diffuse into and out of the matrix. This approach combines

dual porosity with the discrete fracture approach. While no real

aquifer has such a well defined geometry, the model can provide

insight into the important factors in solute transport in fractured

porous media. A recent review article by van Genuchten and

Dalton (1986) provides an excellent summary of work using this

approach.

Real rock fractures may have rough surfaces that are not parallel

to each other, and the fracture may be partially blocked by

translocated or precipitated filling material. Research models

have been developed that attempt to account for these effects.

A recent paper by Tsang and Tsang (1987) describes a fracture

system with flow through a series of tortuous intersecting

channels.

None of the above conceptual pictures is “best” in an absolute

sense. Rather, each may be appropriate for a particular situation.

Models that are conceptually simpler have the advantage of

being easier to implement as a rule, but they may also oversimplify

the situation and miss important phenomena that are taking

place. More complex models have the potential to provide a

more detailed description of what is happening at the site being

modeled, but they are also likely to be more difficult to implement

and may require data that cannot be collected with currently

available techniques.

Pankow, et al. (1986) compared two contaminated fractured

rock sites which differed in regard to fracture aperture, fracture

spacing, matrix porosity, and matrix diffusion coefficient. They

concluded that the EPM approach would work well in describing

contaminant transport for the system with small interfracture

spacing and high enough matrix porosity and diffusion coefficient to rapidly establish matrix/fracture equilibrium. They also

concluded that the EPM approach would not be appropriate for

the other system where matrix/fracture equilibrium was not

rapidly established. Pankow’s paper presents an excellent

summary of attempts to model contamination at real fractured

rock sites.

Model Interactions and Processes

A complete model for flow and transport of contaminants in a

fractured rock system would need to include all of the interactions

and processes that one has in a model for flow and transport in

unconsolidated porous media. For any given situation, some of

these interactions and processes will be important, and some

can be neglected. The existing models that describe flow and

transport in fractured systems are as a rule less complete than

models for unconsolidated porous media. The appropriateness

of a model for a particular circumstance will obviously depend on

how well the assumptions and processes built into the model

match the conditions of the system to which the model is being

applied.

Flow Conditions

Models assume that ground-water flow is laminar. It is apparent

from observations of flow issuing from fractured systems that

this assumption is not always valid. However, at the present

time, the assumption of laminar flow is not a major limitation on

the use of models to describe flow and transport in fractured

systems.

5

Virtually all the models describing flow and transport in fractured

formations assume that the flow of water and the advective

transport of contaminants is only through the fractures. In many

models, water and contaminants diffuse in and out of the rock

matrix in a direction perpendicular to the direction of flow in the

fracture, but there is no flow or advective transport of contaminants

through the matrix. There are some situations where this

assumption will not hold, and it will not be appropriate to make

it.

The discussion in this paper is largely concerned with models

that describe flow and transport in the saturated zone. Flow

through fractures in the unsaturated zone can be exceedingly

complex and is more difficult to model than it is in the saturated

zone. For example, the distribution of water between the

fractures and the matrix in fractured porous rock depends on the

water content. As the water content decreases, a greater and

greater proportion of the water is found in the matrix. Likewise,

other parameters controlling flow are also dependent on the

water content. While flow and transport through unsaturated

fractured rock will be important at some sites, the prospects for

modeling these phenomena at the present time are less good

than they are in saturated fractured rock. Good summaries of

modeling in unsaturated fractured crystalline rock have recently

been published by Dykysien (1987) and Evans and Nicholson

(1987).

Diffusion and Dispersion

As a rule, the models assume that the concentration of the

contaminant is constant across the narrow dimension of the

fracture. Mechanical dispersion consists of longitudinal dispersion only. The geometry of the fracture system is a major

determinant of variation in the flow velocity and consequently of

the mechanical dispersion.

Molecular diffusion within the fracture is usually considered to be

unimportant relative to mechanical dispersion. However, as

mentioned above, diffusion is the main process by which

contaminants are assumed to move within the rock matrix.

Diffusion in and out of the rock matrix will be significantly affected

by the nature of the surface of the fracture. A thin layer of low

permeability, or “fracture skin,” can impede the interchange of

water and contaminants between the fracture and the matrix

(Moench, 1984). Experimental work on the diffusion of

contaminants through the rock matrix and between the matrix

and the fracture is very limited, and more research is needed in

this area.

Adsorption and Desorption

Models for solute transport generally account for adsorption and

desorption of contaminants on the surface of the fractures and

within the rock matrix. To date, all modeling of adsorption and

desorption assumes that the processes are described by linear

isotherms and ignores sorption kinetics. Some models account

separately for sorption processes in the fractures and within the

rock matrix. In porous rocks, the available surface area within

the matrix is likely to be so much greater than that in the fractures

that sorption within the matrix will probably be more important

than sorption in the fractures, assuming that the contaminants

have time to diffuse into the matrix.
None of the available models appear to explicitly account for ionexchange processes. While many of the organic chemicals of

concern are not ionized in solution, other contaminants such as

heavy metals could be, depending on the pH of the system.

Little experimental or theoretical work has been done in this

area; more work is needed.

Radioactive Decay

Much of the work on modeling flow and transport in fractured

systems has been motivated by concerns about the disposal of

radioactive waste. Consequently, there are a number of models

that account for the radioactive decay of single radionuclides

and radionuclide chains.

Chemical Reactions and Biological Processes

Very little work has been done that deals specifically with

chemical reactions and biological processes in fractures. None

of the models account for these processes. This is an important

topic, and more work needs to be done on it.

Multiphase Flow

Models for flow and transport in fractured systems appear to be

limited to single-phase flow. That is, they can simulate the flow

of water alone, or the transport of contaminants that are dissolved

in water. The models cannot simulate the flow of a system of

water and an immiscible phase such as an oily waste, nor the

transport of a contaminant dissolved in an immiscible phase. At

the present time neither the capability of modeling multiphase

flow in homogeneous media, nor the capability of modeling

solute transport in fractured systems is advanced enough to

implement a practical code for modeling multiphase flow in

fractured systems (Streile and Simmons, 1986). Schwille (1988)

has studied the qualitative behavior of dense non-aqueous

liquids in laboratory models. His book contains a number of

interesting photographs. This is another area where more

research is needed.

Available Models

This section describes types of models that are available to

describe flow and transport in fractured rock systems. Specific

models will not be mentioned by name because the information

is likely to become outdated in a short period of time. A good

starting place for obtaining information on publicly available

models is the International Ground Water Modeling Center,

located at Butler University in Indianapolis, Indiana. They can

provide a list of available models and information on specific

models on the list.

Freeze and Cherry (1979) describe the development and use of

a mathematical ground-water model as “a four-step process,

involving (1) examination of the physical problem, (2) replacement of the physical problem by an equivalent mathematical

problem, (3) solution of the mathematical problem with accepted

techniques of mathematics, and (4) interpretation of the mathematical results in terms of the physical problem.” Successful

modeling of a ground-water contamination problem, whether in

fractured rock or not, requires that all four of these steps be

carried out correctly. While step (3), solution of the mathematical problem, could conceivably be carried out by a person

6

who knew almost nothing about ground water, successful

completion of the other three steps requires a thorough knowledge

of hydrogeology in general, along with specific knowledge of the

hydrogeology of the site to which the model is to be applied.

Fractured systems present several difficulties which hinder the

replacement of the physical problem by an equivalent mathematical problem — step (2). One difficulty is that the spatial

distribution of the fractures and the way in which they control the

flow is usually not known, nor is it even knowable in a practical

sense. A second difficulty is that even if this information were

available, including it in the mathematical problem would make

the mathematical problem so complex that a solution could not

be found. As a result, all mathematical models include certain

assumptions about, and simplifications of, the actual physical

problem. For example, the model may assume that the fractures

are of uniform width with parallel sides, and/or that they form a

regular geometric pattern. Comparable assumptions are also

made about other physical, chemical, and biological processes.

Consequently, the mathematical model provides only an

approximate description of the physical system under study.

The model can still be very useful, but anyone using a

mathematical model should be fully aware of these assumptions

and simplifications and their effect on the appropriateness of the

model for the problem of interest.

Mathematical models that describe ground-water systems are

usually written in terms of partial differential equations. If the

equations are simple enough, the solution to the equations can

be expressed in a closed mathematical form (e.g., a formula).

This type of solution is called an analytical solution. More

generally, the equations in the model are too complex to find an

analytical solution, and a numerical method must be used to

solve the problem. Models for fractured systems include those

with analytical solutions as well as those with numerical solutions.

For either type of solution, the product available to the model

user is a computer code in a high level language such as

FORTRAN. The hardware requirements to run the code will vary

from a personal computer (PC) to a main-frame type of computer.

As expected, the more complex models usually have greater

hardware requirements.

Models with Analytical Solutions

Models for which analytical solutions have been obtained are

basically one dimensional. An example of this type of solution

is a model describing solute transport in a single fracture or a set

of parallel fractures. Processes that are included in the model

include diffusion in and out of the rock matrix along a direction

perpendicular to the plane of the fracture, adsorption on the

fracture face and in the rock matrix, and radioactive decay. The

solution to this problem has been published in the open literature

(Sudicky and Frind, 1982) and is simple enough that the computer

code can run on a personal computer.

To arrive at an analytical solution for this problem, the parallel

fracture model assumes that the fracture or set of fractures has

a uniform aperture and is in a homogeneous rock matrix, and that

the fractures in a set of parallel fractures are uniformly spaced.

To use this model, one must have an estimate or measurement

of the following parameters: the flow velocity in the fracture, the

longitudinal dispersivity in the fracture, the fracture aperture or

width, the fracture spacing for a set of fractures, the matrix
models to field problems is a qualitative procedure, a combination

of science and art.”

There are a number of numerical models that have been written

to model flow and transport through systems of randomly oriented

fractures and compare the results with experimental data. The

results of this work have been of use in understanding the nature

of flow and transport in fractured media, but this type of modeling

is a research effort at this time.

Summary Remarks

The development of models to enhance understanding of and to

predict contaminant transport in fractured rock systems continues

to be an active area of research. Reliable, field-validated models

that can be used to predict the results of clean-up scenarios at

contaminated sites are not yet available. The models that are

available help in developing an understanding of the behavior of

the fractured rock system.

The application of mathematical models to contamination in

fractured rock systems is hampered by difficulties in at least two

major areas. The first problem is site characterization — the

collection of the necessary data to adequately describe the

geologic and hydrologic properties of the system. Mathematical

modeling is not a substitute for collecting data. In fact, data

collection is an essential part of modeling the behavior of a site.

Collecting the data required by existing models is difficult and

expensive, and in many cases not possible with present

techniques. More research is needed to find better ways to

measure the properties of fractured rock systems. Conversely,

there may be value in developing models that require data that

can be collected.

The second problem is model validation—comparing the results

of modeling to results obtained in the field. Validation of models

is necessary if decision makers are to have confidence in them

and be able to use them in planning and carrying out remedial

work. Research is being done to validate models of contaminant

transport in fractured rock systems, but more work is needed.

One problem with model validation studies is the shortage of

data sets for sites with a variety of geological and hydrogeological

characteristics. This is where cooperation and coordination

between those in the research community and those charged

with remediating contaminated sites could prove mutually

beneficial.

REFERENCES

Barker, J.A., and J.H. Black. 1983. Slug Tests in Fissured

Aquifers. Water Resour. Res. 19:1558-1564.

Davis, S.N., Campbell, D.J., Bently, H.W., and T.J. Flynn. 1985.

Ground-water Tracers. National Water Well Association,

Worthington, Ohio. pp.200.

Duguid, J.O., and P.C.Y. Lee. 1977. Flow in Fractured Porous

Media. Water Resour. Res. 13:558-566.

7

porosity, the matrix tortuosity, the diffusion coefficient of the

solute in water, the fracture retardation factor or partition

coefficient, the matrix retardation factor or partition coefficient,

and the half-life if the solute is a radioactive species.

This model obviously describes a highly idealized situation and

would not be a suitable predictive tool for dealing with a real

contamination problem. However, it could have some use in

building an understanding of the system and the interactions of

the pollutants within the system. It is easy to vary the effect of

the fracture apertures, fracture spacing, matrix porosity, and so

forth, and see the effect that each of these parameters has on

the rate at which pollutants move through the system. Used with

the proper degree of professional judgement, the results of the

model could provide guidance in making a decision. One

possible application would be in comparing solute transport in

two fractured systems from which cores had been collected to

provide information on the parameters that are included in the

model.

Analytical solutions have also been published in the open

literature for dual-porosity, or “two-region” models of onedimensional flow through systems composed of porous blocks

with well-defined geometry (van Genuchten and Dalton, 1986).

Those geometries for which solutions are available include

close-packed spheres, hollow cylindrical macropores, closepacked solid cylinders, and rectangular blocks. Data and

hardware requirements for running these codes are similar to

those for the parallel crack model.

Models with Numerical Solutions

Models for which only numerical solutions are available are

often referred to as numerical models. Codes for these models

are likely to require more input data and greater computer power

than those for analytical models. The reward for this extra work

is that one can investigate more complicated problems using

numerical models than one can using only analytical models.

One example of a numerical model for fractured rock systems

is a two-dimensional model that can simulate both ground-water

flow and contaminant transport in a fractured aquifer. The step

up from a one-dimensional system to a two-dimensional system

allows one to model a more complex situation, but it also

requires a more complex set of differential equations and a more

complex code. The model uses a dual porosity approach and

allows for some specific fracture geometries. Processes that

are included in the model include advective-dispersive transport

in the fractures, diffusion in the matrix blocks, sorption in the

fractures and in the matrix, and radionuclide decay chains.

This two-dimensional code requires data on the system

dimensions, the transmissivity, the storage coefficient, and the

fracture aperture and spacing if it is to be used to predict groundwater flow. The input data requirements for predicting solute

transport include all of those listed for the one-dimensional

analytical model described above plus values for the solute

concentrations at the system boundaries.

While codes like this have utility for understanding system

behavior, they should not be used as predictive tools. Like all

ground-water models, they will be most useful when applied by

a person with the necessary training and experience--typically,

a professional hydrogeologist. To quote a recent report (van der

Heijde, et al., 1988), “The application of computer simulation
8

Dykysien, R.C. 1987. Transport of Solutes through Unsaturated

Fractured Media. Water Res. 12:1531-1539.

Endo, H.K., J.C.S. Long, C.R. Wilson, and P.A. Witherspoon.

1984. A Model for Investigating Transport in Fracture Networks.

Water Resour. Res. 20:1390-1400.

Evans, D., and T. Nicholson (eds.). 1987. Flow and Transport

Through Unsaturated Fractured Rock, Geophys. Monograph

42, American Geophysical Union, Washington, D.C.

Freeze, R.A., and J.A. Cherry. 1979. Groundwater, PrenticeHall, Inc., Englewood Cliffs, N.J.

Gale, J.E. 1982. Assessing the Permeability Characteristics of

Fractured Rock. Geol. Soc. Amer., Spec. Paper 189, 1982, pp.

163-181.

Grisak, G.E., and J.F. Pickens. 1980. Solute Transport Through

Fractured Media. Water Resour. Res. 16:719-730.

Hem, J.D. 1985. Study and Interpretation of the Chemical

Characteristics of Natural Water, 3rd edition: U.S. Geological

Survey Water-Supply Paper 2254, 263p.

Hsieh, P.A., and S.P. Neuman. 1985a. Field Determination of

the Three-Dimensional Hydraulic Conductivity Tensor of

Anisotropic Media, 1. Theory. Water Resour. Res. 21:1655-

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Hsieh, P.A., S.P. Neuman, G.K. Stiles, and E.S. Simpson.

1985b. Field Determination of the Three-Dimensional Hydraulic

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Konikow, L.F., and J.W. Mercer. 1988. Groundwater Flow and

Transport Modeling. J. of Hydrology, 100:379-409.

Lloyd, J.W., and J.A. Heathcote. 1985. Natural Inorganic

Hydrochemistry in Relation to Groundwater. Clarendon Press,

Oxford. 296p.

Long, J.C.S., and D.M. Billaux. 1987. From Field Data to

Fracture Network Modeling, Water Resor. Res. 23:1201-1216.

Long, J.C.S., J.S. Remer, C.R. Wilson, and P.A. Witherspoon.

1982. Porous Media Equivalents for Networks of Discontinuous

Fractures. Water Resour. Res. 18:645-658.

Moench, A.F. 1984. Double-Porosity Models for a Fissured

Groundwater Reservoir with Fracture Skin. Water Resour. Res.

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Nelson, R.A. 1985. Geologic Analysis of Naturally Fractured

Reservoirs. Contributions in Petroleum Geology and

Engineering, v.1, Gulf Publishing Company, Houston, Texas,

pp. 320.

Pankow, J.F., R.L. Johnson, J.P. Hewetson, and J.A. Cherry.

1986. An Evaluation of Contaminant Migration Patterns at Two

Waste Disposal Sites on Fractured Porous Media in Terms of

the Equivalent Porous Medium (EPM) Model. J. of Cont.

Hydrology, 1:65-76.

Preuss, K., and T.N. Narasimhan. 1985. A Practical Method for

Modeling Fluid and Heat Flow in Fractured Porous Media. Soc.

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Schwille, F. 1988. Dense Chlorinated Solvents in Porous and

Fractured Media (translated by J. Pankow). Lewis Publishers,

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Influence of Fracture Geometry on Mass Transport in Fractured

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Transport of Organic Chemicals: An Assessment of Current

Modeling Capability and Priority for Future Research Directions

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Fractured Porous Media: Analytical Solutions for a System of

Parallel Fractures. Water Resour. Res. 18:1634-1642.

Tsang, Y.W., and C.F. Tsang. 1987. Channel Model of Flow

Through Fractured Media. Water Resour. Res. 23:467-479.

van der Heijde, P.K.M., A.I. El-Kadi, and S.A. Williams. 1988.

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