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From page 307... ...
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.

From page 308... ...
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.

From page 309... ...
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 dualporosity models in reservoir analyses.

From page 310... ...
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 developedfor 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.

From page 311... ...
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.

From page 312... ...
End test 1500 Time (min) 90 450 FIGURE 6.2 Laboratory experiment of tracer injection in a fractured porous medium.

From page 313... ...
The Stripa mine in Sweden provides an example of another way to make inferences about fracture hydrology. A 50mlong drift was excavated in the granitic rock mass, and every fracture with a trace longer than 20 cm was mapped (Figure 6.4~.

From page 314... ...
314 _ n ~ m ·; U: .~ C3 Ct IS 3 Cal Ct Al hi: U: O ,  ~ 3 so ¢ go 3 ,8 ~ .= 3 no Cal 3 .

From page 315... ...
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.

From page 316... ...
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.

From page 317... ...
317 ._ ._ o ~4 o o ._ Ct Cal o Cal Cal m o o C~ Ct E~ C~ 3 o _1 C~ Ct ._ V)

From page 318... ...
The reality of hydrogeological simulation is that there may be a number of possible models with different combinations of geometry and media parameters that reproduce the observed response at a given point in the rock mass. Most flow and transport data from fractured rock sites are open to more or less equally successful interpretations by means of fundamentally different conceptual and mathematical models.

From page 319... ...
Fluid flow in deformable fractures is discussed in Chapter 7. EQUIVALENT CONTINUUM SIMULATION MODELS The Continuum Approximation In a conventional equivalent continuum model, the heterogeneity of the fractured rock is modeled by using a limited number of regions, each with uniform properties.

From page 320... ...
1 = 0.000 1 [m/sec] l'Z 15 20 FIGURE 6.5 Deriving equivalent continuum properties.

From page 321... ...
Matrix blocks between the conducting fractures can significantly increase the storage properties of the rock mass. Equivalent continuum models for fractured media are of two general types, single and dual porosity.

From page 322... ...
Continuum approaches that make use of discrete network models as a means of estimating fieldscale continuum properties of the fractured medium are discussed later in this chapter, under "Hybrid Methods: Using Discrete Network Models in Building Continuum Approximations." SinglePorosity Models Developed in a Deterministic Framework Fluid Flow Singleporosity models consider flow and transport only in the open, connected fractures of the rock mass. When using a numerical model, such as a finitedifference or finiteelement model, a grid is superimposed on the flow domain and values of permeability are assigned to each grid block.

From page 323... ...
As indicated earlier, reliable prediction of solute transport in a fractured rock mass is considerably more difficult than the corresponding flow problem. Common practice, which is fraught with uncertainty, is to adopt the continuum approximations embodied in the conventional form of the advection dispersion equation.

From page 324... ...
For a rock mass with large porous matrix blocks between the conducting fractures, dualporosity models have been used to account for the release of fluid from storage in the matrix blocks into the fracture network. The geometry of the fracture network is idealized to the extent that it can be represented by a small number of geometric parameters (e.g., the average dimension of a matrix block)

From page 325... ...
The primary advantage of dualporosity flow models is that they provide a mechanism to account for the delay in the hydraulic response of the rock mass caused by fluid that is resident in less permeable matrix blocks. Dualporosity

From page 326... ...
. ~ 106 109 FIGURE 6.7 Interpretation of a pumping test in fractured rock at Yucca Mountain, Nevada, using a dualporosity model.

From page 327... ...
Solute Transport In cases where there is the potential for substantial diffusive transfer of mass into the rock matrix blocks, it is essential to incorporate this process into the mathematical structure of the simulation model. Concentration distributions can be greatly affected in comparison to a case in which mass is restricted to the open fracture network.

From page 328... ...
Stochastic Continuum Models The stochastic representation of fluid flow and solute transport in heterogeneous porous media has led in recent years to a powerful new set of tools for hydrogeological analysis. These tools have also been applied to fractured geological media (e.g., Neuman, 1987~.

From page 329... ...
The stochastic equations also permit estimates to be made of the magnitude of the uncertainties in flow and transport predictions. As an alternative to estimating parameter values in a governing stochastic equation, multiple realizations of the hydraulic properties of the rock mass can be generated from the probability model (e.g., Figure 6.9)

From page 331... ...
Assessment of Continuum Modeling The strength of the continuum approach lies in its simplicity; it reduces the geometric complexity of flow patterns in a fractured rock mass to a mathematical form that is straightforward to implement. For most applications that are encountered in practice, some type of continuum approach remains the preferred alternative.

From page 332... ...
Methods for implementing this information in the framework of a continuum model require further investigation, especially in the case of sparsely fractured rock masses. DISCRETE NETWORK SIMULATION MODELS Why Consider Discrete Network Models?

From page 333... ...
Various issues involved in collecting and analyzing statistical data on fracture networks are reviewed later in this section, followed by the outline of a number of models that have been proposed to represent the statistical characteristics of fracture networks. Discrete network models are closely linked with concepts of stochastic simulation.

From page 334... ...
. Numbers on the right are the mean and standard deviations of travel times for solutes that enter the left side of the systems through a single fracture and travel downstream to the right side of the systems.

From page 335... ...
Conditioning on the bulk permeability provided a stronger constraint on estimates of the rate of advective mass transfer than did geometric data on fracture location without additional information on fracture connectivity, which cannot be obtained from borehole fracture intersections. To apply discrete network models in a field setting, it is essential to have the capability for threedimensional simulation.

From page 336... ...
Favorable conditions for discrete network modeling might best be met, for example, in some jointed rocks. These factors are similar to the conditions suited to the application of continuum models.

From page 337... ...
This lack of interconnection is the main reason that discrete network models are chosen over equivalent continuum models, but it is also the reason that homogeneous network models may fail to capture the distribution of hydraulic conductors and consequently fail to reproduce the hydraulic behavior of the fracture network. Evidence of hydrologically inactive fractures has been found at numerous sites.

From page 338... ...
the conductive properties of fractures; and (6) in some cases the conductive properties of fracture intersections.

From page 339... ...
and for fracture flow modeling by Long et al.

From page 340... ...
(1983) and to the hydrology of fractured rock masses by Rouleau (1984~.

From page 341... ...
· The percentage of fractures that terminate against other fractures as a function of orientation, as obtained from outcrop or underground excavation. Estimates of the transmissivity of individual fractures from hydraulic tests.

From page 342... ...
Then a series of simulated networks is sampled in a manner congruent to that used in the field: simulated boreholes or surfaces are used to collect fracture spacing, orientation, and trace length. The simulated data are compared to the real data, and adjustments are made to the model parameters to improve the fit to the observed data.

From page 343... ...
For example, given the density of fracture intersections along a line (i.e., fracture frequency) and the fracture size and orientation distributions, it is possible to calculate the expected number of fractures per unit volume (volumetric fracture density)

From page 344... ...
On the practical side, there is probably no other effort in the analysis of fracture statistics that deserves a goodquality assurance program as much as the data collection and reduction process for fracture orientation. Fracture Size From available exposures, the distribution of fracture size is obtained from the orientation data and fracture trace data.

From page 345... ...
If only borehole measurements are available, problems associated with obtaining estimates of fracture size can be severe. Transmissivity of Individual Fractures In applying a discrete network model, the characterization of transmissivity values is likely to be a dominant source of uncertainty in a flow or transport simulation.

From page 346... ...
Applications of Discrete Network Models in Media with Significant Matrix Porosity Discrete network models are necessarily more complex in cases where (1) the permeability of the rock mass in which the fracture network is embedded is a significant fraction of the network permeability or (2)

From page 347... ...
The second view is that discrete network models are practical tools for sitespecific simulations (e.g., Dershowitz et al., 1991a; Herbert and Lanyon, 1992~. The advantage of a discrete fracture simulation is that volumeaveraging approximations are avoided at the scale of the fracture network.

From page 348... ...
FIGURE 6.1< Prediction of solute distribution using a discrete network model in a medium where matrix diffusion is important. A fractured lowpermeability aquitard overlies a permeable aquifer (lower unit)

From page 349... ...
349 p~ ///// C ._ lo ._ it .

From page 350... ...
Whether it be for continuum models or discrete network models, too few prediction studies have been checked against subsequent behavior. Perhaps the most often heard comment about discontinuum models is that they are not practical.

From page 351... ...
HYBRID METHODS: USING DISCRETE NETWORK MODELS IN BUILDING CONTINUUM APPROXIMATIONS Fluid Flow Because it is difficult or impossible to measure largescale permeability directly, it would be extremely useful to be able to estimate such values by taking some kind of average of the localscale measurements. In the case of fracture systems, discontinuum models can provide a framework for attempting such an analysis.

From page 352... ...
, among others, have extended the approach of estimating continuum properties from a fracture network analysis by developing models of fluid flow in fracture networks. The fracture networks are specified by giving the location, orientation, extent, and hydraulic aperture of each fracture.

From page 353... ...
(3) The calibrated flow model was then used to generate 17 "intermediatescale" realizations, and a hydraulic conductivity ellipse was

From page 354... ...
(4) These intermediatescale values were averaged, using equations developed for heterogeneous porous media, to estimate a largescale hydraulic conductivity for the rock mass surrounding the edit.

From page 355... ...
. One way to develop expressions for permeability as a function of the stochastic network parameters is to relate a discrete network model to a simple percolation model.

From page 356... ...
Solute Transport As indicated earlier, one of the key difficulties in the formulation of a fieldscale model to represent solute transport in fractured rock is estimation of the parameters that characterize the dispersive properties of the rock mass. Model studies have been used to examine equivalent porosity in much the same way that models have been used to examine equivalent permeability.

From page 357... ...
The discrete network model is a small piece of the much larger domain for which fieldscale simulation is required, but it is intended to be representative. The basic inputs to the model are the geometric properties of each fracture set

From page 358... ...
DISCRETE NETWORK MODELS WITH SCALEDEPENDENT PROPERTIES Basic Issues There are two basic concerns with the stochastic models of fracture networks described earlier in this chapter. First, there are typically no mechanistic underpinnings to the proposed stochastic structure of the fracture network; their primary objective is to reproduce the statistics of fracture maps, not the spatial relationships between fractures.

From page 359... ...
(b) Results using a discrete network simulation.

From page 360... ...
The concept of conditioning discrete network models on hydrological observations usually involves an extension to crosshole hydraulic tests from the singleborehole tests used in estimating fracture transmissivity. Rules describing the scale dependence of the hydraulic and transport properties of geological media are being actively explored by the research community.

From page 361... ...
The evidence suggests that the concept of hierarchical systems is applicable to many fractured rocks. The question remains, in adopting a discrete network model, of how to generate network models that provide a realistic portrayal of these spatial structures.

From page 362... ...
(1989~. The parameters of the parentdaughter process were derived by comparing the variogram derived from the field data with a series of theoretical variograms derived from different sets of parentdaughter parameters.

From page 363... ...
were the first to present a twostage fracture model designed to discriminate between fracture sets in which the fracture traces are extensive and those sets in which the fractures have an irregular pattern and truncate against the extensive fractures. These authors modeled the extensive or primary fractures as unbounded Poisson lines.

From page 364... ...
Geometric models of fracture networks can also be built in a way that mimics the natural genesis of fractures. If this is done alteratively, with each stage of fracturing depending on the results of the previous stage, hierarchical fracture systems can be obtained.

From page 365... ...
(c) Modeling of PA100 hierarchical fracture trace model.

From page 366... ...
In principle, there is no conceptual difficulty in applying discrete network models to fractures with

From page 367... ...
Discrete Network Flow Models Conditioned on Hydraulic Behavior As discussed earlier, geometric information from fracture statistics does not form a sufficient basis to construct a discrete network model. Hydraulic data must be available to define transmissivities at the scale of individual fractures.

From page 368... ...
The equivalent discontinuum model is similar to an equivalent continuum model, except that some lattice elements are removed.

From page 369... ...
It is possible to use these models to reproduce the discontinuous hydraulic behavior of a fracture system without modeling every fracture. An inversion technique called simulated annealing can be used to construct equivalent discontinuum models (Long, 1993~.

From page 370... ...
Emerging Concepts For fracture networks near the percolation threshold, the equivalent discontinuum models described above result in a fractal structure through the development of a percolation lattice. Several attempts have been made to model fracture systems using numerical approximations of fractals directly.

From page 371... ...
.............. ~2 61 FIGURE 6.26 Twodimensional equivalent discontinuum model of the H zone at Stnpa, annealed to C1C2 crosshole hydraulic test.

From page 372... ...
FIGURE 6.27 Comparison of the C1C2 well test response data to the prediction of these test results provided by the equivalent discontinuum model.

From page 373... ...
Figure 6.29 shows preliminary models of the H zone at the Strip a mine constructed through IFS inversion using the same data as the model in Figure 6.26. Recently, interest has been growing in neural network models.

From page 374... ...
Finally, uncertainties about the model parameters can be used to estimate uncertainties about fracture system geometry. The idea of using crosshole hydraulic tests to calibrate discrete network models also appears promising.

From page 375... ...
Nonlinear relationships between the media properties that characterize the conductance of a fractured rock mass and the degree of phase saturation are very difficult to quantify at field scales. Coupling between the thermal, hydrogeological, mechanical, and geochemical systems also is characteristic of these more complicated systems (see de Marsily, 1987~.

From page 376... ...
Fluid saturation and hydraulic conductivity are functions of moisture content and, thus, fluid pressure. The conceptual model of fluid flow that is favored by some workers is illustrated in Figure 6.30.

From page 377... ...
A conceptual model of this system would differ significantly from the conventional view of unsaturated flow in fractured rock, which predicts that the matrix controls the flow and that fractures act as barriers to it. Indeed, this conceptual model would predict a more rapid downward transmission of fluid than the conventional model, where flow is confined to the rock matrix.

From page 378... ...
has developed an efficient threedimensional, discrete fracture model that solves for variably saturated groundwater flow and solute transport. The matrix blocks between the discrete fractures are porous and permeable.

From page 379... ...
To model fluid flow in unsaturated fractured rocks using currently available models, relationships must be developed for moisture content, fluid pressure, and relative permeability. Additional data are needed to develop relationships for fluid exchange between the fractures and the matrix blocks.

From page 380... ...
Modeling Multiphase Flow in Fractured Rocks Models that simulate multiphase flow in fractured rock systems arise in petroleum reservoir engineering, in the analysis and development of some geothermal systems, and in contaminant hydrogeology. Experience with multiphase flow models has been considerably more extensive in the petroleum engineering field than in the field of groundwater hydrology.

From page 381... ...
Multiphase flow models describing NAPL migration in fractured rock could be used in practice to provide muchneeded insights into physical (and chemical) processes for NAPL migration at the network scale.

From page 382... ...
/ &:~ . ONAPL IN // FRACTURED ,~ .~FRACTURE // Ct AY OR Rack ~MATR IX ;'' ' · Hi \ DISSOLVED PLU ~ E // GROUNDWATER LOWER ~FLOW AQU I FER ~N \ N 'I ' ' ' ~ ~ ~ ~ ~ ~ ~ ~ it, ~ \ \ ' FIGURE 6.32 Conceptual model describing movement of a dense nonaqueousphase liquid (DNAPL)

From page 383... ...
As frequently noted in this chapter, it is difficult to characterize flow paths at the field scale in fractured rock masses. Wels and Smith (1994)

From page 384... ...
Heat exchange between the matrix and fracture network depends on rates of fluid flow in each domain and the geometry of the fracture network. The main problem in modeling a geothermal reservoir to assess its production potential is to obtain representative values for the hydrological and thermal properties of the fractured rock mass.

From page 385... ...
The most basic issue facing the modeling community is how to use measurements obtained at one scale to estimate model parameters that represent another scale of heterogeneity of a fractured rock mass. The conceptual model plays a fundamental role in dealing with the issue of scale in flow and transport models.

From page 386... ...
Stochastic continuum models represent the heterogeneity of the fractured rock as a continuous random field, with the scale of heterogeneity tailored to the scale of measurements of permeability made in the field. Discrete network models explicitly include large numbers of individual fractures that conduct fluid in the model structure.

From page 387... ...
Stochastic theories provide equations for estimating largerscale hydraulic and transport properties from the smallerscale field measurements (Neuman and Depner, 1988~. The validity of the stochastic continuum approach rests on the suitability of approximations embodied in the representation of a fractured rock mass as a heterogeneous porous medium and in the suitability of mathematical simplifications that underlie stochastic transport theory.

From page 388... ...
In some circumstances, discrete network models may provide one alternate means of estimating largescale hydraulic and transport properties of the rock mass for use in equivalent continuum models. Two classes of models have been proposed to represent the geometric properties of fracture networks.

From page 389... ...
The concept of conditioning discrete network models on hydrological observations involves an extension to crosshole hydraulic tests from the singleborehole tests used in estimating fracture transmissivity. In this approach an inverse technique is used to construct a representation of the fracture network that is functionally equivalent to the observed system in the constraints imposed by the conceptual model.

From page 390... ...
The laboratory, excavated in crystalline rock of the Lac du Bonnet batholith, consists of a main vertical shaft approximately 450 m deep, with horizontal tunnels at different depths. Many different experiments have been carried out, or are planned, to assess current and emerging site characterization methods, to document rock mass response to tunneling and thermal loading, and to assess the current ability to model groundwater flow and solute transport in a crystalline rock site.

From page 391... ...
. FIGURE 6.A1 Conceptual model of the rock mass at the Underground Research Laboratory (URL)

From page 392... ...
\~ 1984 198S 1986 1987 120 90 C] 60 ': C' LU o FIGURE 6.A2 Comparison of observed and predicted drawdown using a continuum model for the URL drawdown experiment.

From page 393... ...
Conversely, there is a high probability that a cluster will span the ~  0.8 C=0.2 C=~.O ;_8.0 (a FIGURE 6.B1 Examples of fracture networks with different degrees of connectivity. Parts (a)

From page 394... ...
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.

From page 395... ...
One of the bestknown 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.

From page 396... ...
1989. Threedimensional statistical modeling of a fractured rock mass an example from the FanayAugeres Mine.

From page 397... ...
Pp. 331338 in Proceedings of the 2d International Workshop on Scale Effects in Rock Masses, Scale Effects in Rock Masses 93, Pinto du Cusha, ed.

From page 398... ...
1986. A model to evaluate the transient hydraulic response of threedimensional, sparsely fractured rock masses.

From page 399... ...
Pp. 267323 in Flow and Contaminant Transport in Fractured Rocks, J

From page 400... ...
In Fractured and Jointed Rock Masses. Rotterdam: A

From page 401... ...
In Proceedings of the Symposium Conference on Fluid Flow in Fractured Rocks, Georgia State University, May 15 18. Pruess, K., and J

From page 402... ...
1990. New approaches to the simulation of field scale solute transport in fractured rocks.

From page 403... ...
1985. Hydrologic mechanisms governing fluid flow in a partially saturated, fractured porous medium.

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