**P**revious chapters have addressed fracture geometry, physical and geomechanical properties of single fractures, fracture detection, and the interpretation of hydraulic and tracer tests. All of these elements must be integrated when developing a mathematical model to represent fluid flow and solute transport in fractured media. The focus of this chapter is mathematical models and the model-building process. The process of building a hydrogeological model crosses discipline boundaries, demanding the combined expertise of geologists, geophysicists, and hydrologists.

The hydraulic properties of rock masses are likely to be highly heterogeneous even within a single lithological unit if the rock is fractured. The main difficulty in modeling fluid flow in fractured rock is to describe this heterogeneity. Flow paths are controlled by the geometry of fractures and their open void spaces. Fracture conductance is dependent, in part, on the distribution of fracture fillings and the state of stress. Flow paths may be erratic and highly localized. Local measurements of geometric and hydraulic properties cannot easily be interpolated between measurement points. In contrast, granular porous media, although also heterogeneous, commonly exhibit smoothly varying flow fields that are amenable to treatment as equivalent continua. Model studies of fluid flow and solute transport in fractured media that do not address heterogeneity may be doomed to failure from the outset.

The formulation of a hydrogeological simulation model is an iterative process. It begins with the development of a conceptual model describing the main features of the system and proceeds through sequential steps of data collection and model synthesis to update and refine the approximations embodied in the

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6
Field-Scale Flow and Transport Models
Previous chapters have addressed fracture geometry, physical and geomechanical properties of single fractures, fracture detection, and the interpretation of hydraulic and tracer tests. All of these elements must be integrated when developing a mathematical model to represent fluid flow and solute transport in fractured media. The focus of this chapter is mathematical models and the model-building process. The process of building a hydrogeological model crosses discipline boundaries, demanding the combined expertise of geologists, geophysicists, and hydrologists.
The hydraulic properties of rock masses are likely to be highly heterogeneous even within a single lithological unit if the rock is fractured. The main difficulty in modeling fluid flow in fractured rock is to describe this heterogeneity. Flow paths are controlled by the geometry of fractures and their open void spaces. Fracture conductance is dependent, in part, on the distribution of fracture fillings and the state of stress. Flow paths may be erratic and highly localized. Local measurements of geometric and hydraulic properties cannot easily be interpolated between measurement points. In contrast, granular porous media, although also heterogeneous, commonly exhibit smoothly varying flow fields that are amenable to treatment as equivalent continua. Model studies of fluid flow and solute transport in fractured media that do not address heterogeneity may be doomed to failure from the outset.
The formulation of a hydrogeological simulation model is an iterative process. It begins with the development of a conceptual model describing the main features of the system and proceeds through sequential steps of data collection and model synthesis to update and refine the approximations embodied in the

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conceptual model. The conceptual model is a hypothesis describing the main features of the geology, hydrological setting, and site-specific relationships between geological structure and patterns of fluid flow. Mathematical modeling can be thought of as a process of hypothesis testing, leading to refinement of the conceptual model and its expression in the quantitative framework of a hydrogeological simulation model.
The relationship between the conceptual model, laboratory and field measurements, and the hydrogeological simulation model is illustrated in the flow chart in Figure 6.1. Simulation models are usually used as tools for enhancing understanding of flow systems as an aid in reaching management or design decisions. Three basic questions must be addressed before a model can be used as a site-specific design tool. First, does the conceptual model provide an adequate
FIGURE 6.1 Flow chart identifying steps involved in the development of a hydrogeological simulation model for the purpose of reaching a management or design decision.

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characterization of the hydrological system? If not, it should be revised and reevaluated. Second, how well does the model perform in comparison with competing models? Model results are nonunique. Consequently, field measurements may be successfully matched using fundamentally different conceptual and mathematical models. A model that successfully reproduces field measurements is not necessarily correct or validated. An important test of any model is to see how well it predicts behavior under changed hydrological conditions. Once the conceptual model is accepted, a third question must be asked: is the data base adequate to estimate the model parameters with sufficient reliability that the associated prediction uncertainties are acceptable in light of the intended application of the model in the decision process? If not, additional data collection is justified. In most instances this latter question is best addressed in an economic framework (Freeze et al., 1990).
The flow chart in Figure 6.1 is not unique to fractured geological media. It outlines the modeling process for any hydrogeological system. It is commonly the case, however, that these three questions are considerably more difficult to resolve when dealing with fractured media in comparison to problems involving fluid flow through granular porous media.
The purpose of this chapter is twofold: (1) to summarize recent views on the development of conceptual models of fluid flow and transport in fractured porous media and (2) to discuss and assess the state of the art in mathematical modeling and to identify research needs to advance modeling capabilities. The chapter focuses on general issues of model design. Mathematical formulations, numerical techniques, and specific computer codes are not included in the discussion. The chapter reviews the ways different types of simulation models incorporate the heterogeneity of a rock mass.
Much of the experience involving the simulation of fluid flow and solute transport in fracture systems has developed through the application of dual-porosity models in reservoir analyses. An extensive literature exists on the use of these models (e.g., Warren and Root, 1963; Kazemi, 1969; Duguid and Lee, 1977; Gilman and Kazemi, 1983; Huyakorn et al., 1983a,b; to name but a few). Although dual-porosity models are included in this discussion, greater emphasis is placed on more recent models that accommodate more complex fracture geometries than normally assumed in dual-porosity models. In addition, the discussion in this chapter is limited to fracture systems and model applications where it is not necessary to consider the effects of fracture deformation on fluid flow. This topic is addressed in more detail in Chapter 7.
DEVELOPMENT OF CONCEPTUAL AND MATHEMATICAL MODELS
Overview
Figure 6.1 highlights the central role of the conceptual model in the modeling process. When formulating a conceptual model to describe conductive fractures

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and their permeabilities, three factors come into play: (1) the geology of the fractured rock, (2) the scale of interest, and (3) the purpose for which the model is being developed. These three factors determine the kinds of features that should be included in the conceptual model to capture the most important elements of the hydrogeology.
Geology of the Fractured Rock
A geological investigation seeks to identify and describe fracture pathways. These pathways are determined by material properties, geometry, stress, and geological history of the rock. There are two end members that describe the distribution of fracture pathways: (1) a system dominated by a few relatively major features in a relatively impermeable matrix, as commonly observed in massive crystalline rocks, and (2) a system dominated by a network of ubiquitous, highly interconnected fractures in a relatively permeable matrix, as might be found in an extensively jointed, layered rock with strata-bound fractures. Fracture systems exist at many levels between these two extremes in many different rock types. A geological investigation attempts to identify which features of a fracture system, if any, have the potential to dominate the hydrology.
Scale of Interest
A fracture system may be highly connected on a large scale, but it may be dominated by a few, relatively large features when viewed on a smaller scale. Classical thinking holds that the larger the scale of interest, the more appropriate it is to represent fractured rock in terms of large regions of uniform properties. The major problem then becomes one of estimating the large-scale properties from small-scale measurements. More recently, some workers have suggested that there are fracture features at all scales of interest. For these cases the traditional strategy of representing the rock as having large regions of uniform properties is less likely to be adequate. A series of new approaches are being developed to address fracture representation at large scales.
Purpose for Which the Model Is Being Developed
The level of detail required in the conceptual model depends on the purpose for which the model is being developed—for example, whether it will be used to predict fluid flow or solute transport. Experience suggests that, for average volumetric flow behavior, predictions can be made with a relatively coarse conceptual model provided data are available to calibrate the simulation model. Thus, a continuum approximation may be used to predict well yields with sufficient accuracy, even if a fracture network is poorly connected. For solute transport a considerably more refined conceptual model is needed to develop reliable predic-

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tions of travel times or solute concentrations because of the sensitivity of these variables to the heterogeneity of fractured systems (Chapter 5). A more specific description of fracture flow paths may also be necessary to identify recharge areas for a well field located in a fractured medium.
The importance of identifying fracture pathways in contaminant transport problems can be visually demonstrated by observations from a laboratory model. Figure 6.2 illustrates the development of a solute plume in a fractured porous medium constructed from blocks of porous polyethylene. In this experiment a tracer was injected into a permeable matrix block for a period of 75 minutes, and its distribution was mapped. The pattern of spreading is complex and strongly dependent on the local structure and hydraulic properties of the network.
To this point, the conceptual model has been expressed in terms of a hypothesis describing the heterogeneity of the rock mass. In a more general sense, a conceptual model encompasses all of the assumptions that go into writing the mathematical equations that describe the flow system. Decisions are required about whether the problem at hand involves saturated or unsaturated flow, isothermal or nonisothermal conditions, single-phase or multiphase flow, and reactive or nonreactive solutes. Issues related to the development of conceptual models for these more complex processes are explored in the final section of this chapter.
The development of an appropriate conceptual model is the key process in understanding fluid flow in a fractured rock. Given a robust conceptual model, different mathematical formulations of the hydrogeological simulation model will likely give similar results. However, an inappropriate conceptual model can easily lead to predictions that are orders of magnitude in error.
Development of a Conceptual Model
The steps in building a conceptual model of the flow system include (1) identification of the most important features of a fracture system, (2) identification of the locations of the most important fractures in the rock mass, and (3) determination of whether and to what extent the identified structures conduct water. The important fractures are significantly conductive and connected to a network of other conductive fractures. All fractures are not of equal importance. Identification of preferred fluid pathways is crucial in the development of a conceptual model. So too is an understanding of the orientation of the fractures (or fracture sets) that contribute to flow, because these orientations provide insight into the anisotropic hydraulic properties of the rock mass.
Inferences about the relative importance of fractures can be made in a variety of ways. As described in Chapter 2, geological observation of fracture style is a powerful tool. Fractures have been studied in many locations and in many rock types, and several patterns have been identified (e.g., La Pointe and Hudson, 1985). For example, fractures may be evenly distributed in the rock mass or may occur in concentrated swarms or zones. There may be polygonal joints or

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FIGURE 6.2 Laboratory experiment of tracer injection in a fractured porous medium. The model domain is 80 cm in length. Tracer was injected for 75 minutes and allowed to migrate under the influence of a uniform hydraulic gradient. The lines in the boxes represent the fracture networks. From Hull and Clemo (1987).

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en-echelon features. Understandings gained through investigations of fracture style provide insights for predicting the character of fractures in unexposed parts of the rock mass.
Understanding the genesis of fractures can provide insights into the hydrological properties of the rock mass. For example, subvertical shear zones near Grimsel Pass in the Swiss Alps were studied in outcrop by Martel and Peterson (1991) and shown to have distinctly different fracture patterns depending on the orientation of the zone with respect to the fabric of the rock. Shear zones at an angle to the fabric (so-called K zones) tend to have parallel groups of fractures, whereas those parallel to the fabric (so-called S zones) have more braided, anastomosing fractures (Figure 6.3). The hydrological characteristics of these zones are likely to be quite different. K zones may have high vertical permeabilities in the vicinity of the fractures because they formed by extension; in contrast, S zones may be more uniformly, if anisotropically, permeable.
The Stripa mine in Sweden provides an example of another way to make inferences about fracture hydrology. A 50-m-long drift was excavated in the granitic rock mass, and every fracture with a trace longer than 20 cm was mapped (Figure 6.4). The rock appears to be ubiquitously fractured, with a concentration of fractures in the central 10 or 15 m of the drift (the H zone). Careful measurement has shown that essentially all the inflow comes from this zone; 80 percent of this inflow is from a single fracture (Olsson, 1992). Outside of this zone, the myriad other fractures are unimportant in contributing to the fluid influx. The zone itself does not behave as a homogeneous porous slab. It can be surmised that the flow systems in these zones are likely to be only partly connected. Fracture zones also dominate the hydrology at the Underground Research Laboratory in
FIGURE 6.3 Schematic diagrams comparing the arrangement of fractures in K zones and S zones from the Grimsel Pass in the Swiss Alps. From Martel and Peterson (1990).

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FIGURE 6.4 Fracture map from the Stripa validation drift, Sweden, showing the H fracture zone. Arrow shows the fracture that carries 80 percent of the flow into the drift. From Olsson (1992).

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Canada (Davison et al., 1993) and at the Grimsel Rock Laboratory in Switzerland (Long et al., 1990).
The state of stress can be a controlling factor in determining which fractures or parts of fracture systems are important. An example is provided by a study in a fractured oil reservoir in the Ekofisk field in the North Sea (Teufel et al., 1991). Fractures with two distinct orientations were logged from boreholes; one of the sets was far more abundant than the other. Subsequent hydraulic testing showed that the direction of maximum permeability was aligned with the less abundant fracture set and that this direction was parallel to the maximum compressive stress. Apparently, fractures parallel to the maximum compressive stress tend to be open, whereas those perpendicular to this direction tend to be closed.
Once the flow-controlling fractures are identified, the next step is to define and locate them in the rock mass. Ideally, geophysical and geological data can be combined to determine the three-dimensional geometry of the major structures that might conduct water. Geological and geophysical investigations are clearly complementary. Geological investigations are well suited to identifying, locating, and characterizing exposed fractures and to determine how the fractures were formed; they are limited in their ability to determine how far to project known fractures into the rock and to detect unexposed fractures. Geophysical investigations can locate unexposed fractures, as discussed in Chapter 4, but are limited in their ability to uniquely determine the geometry of the detected fractures. Geological and geophysical methods do not generally yield quantitative information about the hydraulic and transport properties of fractures. They do provide information about the structure of the rock mass that can be used to organize hydrological investigations and interpretations.
There are two types of geological information available for integration with geophysical data: (1) geological maps of outcrops and (2) underground exposures and borehole data. The interpretation of borehole data has certain limitations. Individual borehole records typically do not allow the shapes and dimensions of fractures to be determined, nor do they provide information about fracture connectivity. In addition, the orientation of fracture zones, which can be determined from borehole measurements, can differ significantly from the orientation of individual fractures in the zone (e.g., the K zone in Figure 6.3). Methods of collecting and analyzing statistical data on fracture systems based on borehole observations are discussed in greater detail later in this chapter.
Finally, it must be determined if and how the identified fractures conduct water. There is probably no better way to see if a fracture conducts water than to pump water through it while monitoring the hydraulic response in the rock. A conceptual model of the fracture geometry provides a framework for designing this hydrological investigation. Instead of simply measuring injectivities at random, the tests can be focused on determining the hydrological properties of specific features. If the conceptual model includes a major feature, the well testing program can be designed to investigate the permeability of this feature. If there

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are a number of features, the well tests can be designed to see if they are hydraulically connected. This diagnostic testing is described in Chapter 5. Geophysical tools described in Chapter 4, such as borehole flowmeters or borehole televiewers, also provide indications of the location of hydraulically conductive fractures. Under favorable circumstances, examination of the chemical composition of waters from different locations can provide an indication of water sources and whether or not the sources are mixing.
The result of this process is a model of those features of the fracture system that control the hydrology and some understanding of the nature of flow in the system. The process is by its very nature iterative. As data are collected and analyzed, the conceptual model is updated, and this may influence subsequent decisions on the types of data to be collected and on measurement techniques or locations. Revision of the conceptual model or selection of a competing model is driven by comparison of model predictions with new observations. An example of this process that emphasizes the role of geophysics is the conceptual modeling of the Site Characterization and Validation Experiment at the Stripa mine, which is discussed in Chapter 8.
Mathematical Models
Mathematical models fall into one of three broad classes: (1) equivalent continuum models, (2) discrete network simulation models, and (3) hybrid techniques. The models differ in their representation of the heterogeneity of the fractured medium. They may be cast in either a deterministic or stochastic framework. The scale at which heterogeneity is resolved in a continuum model can be quite variable, from the scale of individual packer tests in single boreholes to effective permeabilities averaged over large volumes of the rock mass. Discrete network models explicitly include populations of individual fracture features or equivalent fracture features in the model structure. They can represent the heterogeneity on a smaller scale than is normally considered in a continuum model. Some of the more recent innovations in mathematical simulation are best classified as hybrid techniques, which combine elements of both discrete network simulation and continuum approximations.
Table 6.1 presents a summary of the main classes of simulation models for fractured geological media. Included in this table are key parameters that distinguish the models. Also listed are references that illustrate recent applications of each modeling approach. The few references were chosen simply to point the direction toward the relevant literature; it is not meant to imply they are necessarily viewed as the ''best" papers on a given topic. Subsequent sections of this chapter explore these model types in greater detail. Additional coverage of some of these modeling concepts can be found in a recent text by Bear et al. (1993) and in another National Research Council (1990) report on groundwater models.

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TABLE 6.1 Classification of Single-Phase Flow and Transport Models Based on the Representation of Heterogeneity in the Model Structure
Representation of Heterogeneity
Key Parameters that Distinguish Models
Recent Examples
Equivalent Continuum Models
Single porosity
Effective permeability tensor
Carrera et al. (1990)
Effective porosity
Davison (1985)
Hsieh et al. (1985)
Multiple continuum
Network permeability and porosity
Reeves et al. (1991)
(double porosity, dual permeability, and multiple interacting continuum)
Matrix permeability and porosity
Matrix block geometry
Nonequilibrium matrix/fracture interaction
Pruess and Narasimhan (1988)
Stochastic continuum
Geostatistical parameters for log permeability: mean, variance, spatial correlation scale
Neuman and Depner (1988)
Discrete Network Models
Network models with simple structures
Network geometry statistics
Fracture conductance distribution
Herbert et al. (1991)
Network models with significant matrix porosity
Network geometry statistics
Fracture conductance distribution
Matrix porosity and permeability
Sudicky and McLaren (1992)
Network models incorporating spatial relationships between fractures
Parameters controlling clustering of fractures, fracture growth, or fractal properties of networks
Dershowitz et al. (1991a)
Long and Billaux (1987)
Equivalent discontinuum
Equivalent conductors on a lattice
Long et al. (1992b)
Hybrid Models
Continuum approximations based on discrete network analysis
Network geometry statistics
Fracture transmissivity distribution
Cacas et al. (1990)
Oda et al. (1987)
Statistical continuum transport
Network geometry statistics
Fracture transmissivity distribution
Smith et al. (1990)
Fractal Models
Equivalent discontinuum
Fractal generator parameters
Long et al. (1992)
Chang and Yortsos (1990)

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largest length scale when the density is above the critical value. For simple lattice percolation the critical density corresponds to a value for p called pc. In the continuum percolation case the controlling parameters are the spatial density, size, and orientation of the random sets. In some cases one can find a way to map the parameters of the continuum percolation model into an effective p and then apply the results predicted by simple lattice percolation (Hestir and Long, 1990).
The second basic property of percolation networks is that universal scaling laws appear when the fracture density is near the critical value. To explain this, suppose p is near pc and M represents a particular physical property of the percolation network, such as average permeability. Then M will be related to p in the following way:
where the exponent is referred to as a universal scaling exponent. The term universal is used because numerical studies suggest that the exponent depends only on the dimension of the percolation model (e.g., two or three dimensions) and not the details of the model, such as the kind of lattice used or the types of random sets in the continuum percolation case.
It is reasonable to suggest that most random network models will have two basic properties: critical density and scaling laws with universal exponents. Hence, these properties are also expected to apply in the case of a network of fractures. However, there is no general way to apply these concepts in a field situation, so percolation theory can only be used as a general framework for thinking about fracture and fluid flow problems. For some example applications of percolation theory to flow in a single fracture, see Chapter 3.

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APPENDIX 6.C
CONNECTIVITY
A flow system with many fractures can have a high or low permeability depending on the connectivity of the fracture network (Figure 6.B1). Although the term connectivity has no generally accepted definition, it is normally used to describe the subjective appearance of a fracture network. Highly connected networks are more permeable. In some cases, where there is a specific mathematical model for the fractures, connectivity can be rigorously defined.
One of the best-known cases where connectivity has been studied is the percolation model. As noted in Appendix 6.B on percolation theory, percolation networks are constructed by creating a lattice of conductors and randomly turning some of them off. Several quantities can be defined that describe connectivity in percolation networks; each is based on assigning an arbitrary point on the lattice to be the origin. Connectivity measures include (1) the expected number of conductors connected to the origin, and (2) the probability that there is a path from the origin to a point x in the lattice. This latter probability is called the connectivity function. The higher the expected number of conductors, or the greater the probability of a path between the origin and point x, the greater the connectivity, .
A model that is similar to percolation networks is the continuum percolation model. One method for quantifying connectivity in these models involves choosing an arbitrary fracture and labeling it level one (Billaux et al., 1989). The fractures intersecting the level one fracture are then counted and labeled as level two fractures. Next, the number of fractures intersecting level two fractures that were not previously labeled are counted and labeled as level three fractures, and so on. The connectivity measure is given by a count of the number of fractures at each level; more connected systems have an increasing number of fractures at each level, whereas less connected systems have fewer fractures or no fractures at each level.
A third measure of connectivity is the average number of intersections per fracture (Figure 6.B1). This measure was used by Robinson (1984) and Long and Hestir (1990) for Poisson line systems. In this simplified system an analytical expression can be derived for the average number of intersections per fracture. This number is then taken to be the connectivity measure. In this case the permeability of the network is a definable function of connectivity.
Except for the Poisson line model, the other measures of connectivity described above have no known analytical description but can only be determined by numerical methods. To the extent that fracture systems can be reduced to statistical models characterizing the hydraulically active fracture system, the geometric analysis of connectivity will be indicative of permeability.

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