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...for simulating ground water flow fractured aquifers. The most common approach is to represent the system with an equivalent continuum model, often referred to as an equivalent porous media (EPM) model. With this approach, equivalent continuum properties assigned to model cells represent the combined effects of individual fractures and the rock matrix. The discrete fracture model, or discrete fracture network model, is the second approach used to simulate ground water flow in fractured rocks. With this approach, flow is explicitly simulated in each fracture using, for example, solutions to the Navier-Stokes equation (Bear 1993), Kirchoff's laws for electrical circuits (Kraemer and Haitjema 1989), or hydraulically connected circular discs (Cacas et al. 1990a, 1990b). The third approach is a hybrid method that uses discrete fracture models to estimate effective properties for continuum approximations. Regardless of the approach, accurate representation of dominant fractures probably is more important than model selection (NRC 1996; Selroos et al. 2002).
The use of continuum models to directly simulate flow within individual fractures may be considered as another approach for representing fracture flow. While these continuum models are not considered true discrete fracture flow models, they can be used to explicitly simulate flow within individual fractures. Fractures and densely fractured flow zones can be represented in continuum models by adding zones of increased hydraulic conductivity in the appropriate orientation. Rayne et al. (2001) demonstrate that subhorizontal fracture zones can be represented with a continuum model by using thin, highly permeable layers that follow the depths of the mapped fracture zones. Eaton et al. (2001) show that simulated head gradients are more accurate with explicitly incorporated fracture zones than with equivalent continuum properties. Svensson (2001a) shows how individual fractures can be directly incorporated into continuum models to simulate flow in a fractured aquifer in Sweden (Svensson 2001b). Selroos et al. (2002) conclude that similar estimates of travel times can be obtained with a discrete fracture model or with a continuum model that explicitly incorporates transmissive features. These are a few recent examples of what will be referred to herein as fracture zone continuum models.
This paper describes a stochastic method for developing two-dimensional fracture zone continuum models. The method is demonstrated by quantifying uncertainties in travel time estimates that result from uncertainties regarding the exact locations and orientations of vertical fracture zones. Development of the method was motivated by the common occurrence of hydraulically significant vertical fracture zones, such as those found in west-central Florida. This paper also describes a procedure for converting a calibrated steady-state EPM model into a fracture zone continuum model.
Method for Incorporating Individual Fracture Zones into Continuum Models
Ground Water Flow
Svensson (2001a) presents a method for incorporating fractures or fracture zones into continuum ground water flow models. The approach presented here is similar to Svensson (2001a) except it is intended to work with commonly used finite-difference and particle-tracking programs, such as MODFLOW (McDonald and Harbaugh 1988) and MODPATH (Pollock 1994). This paper focuses on the incorporation of fully penetrating vertical fracture zones in a two-dimensional aquifer; Svensson (2001a) shows how the method can be applied in three dimensions.
For a fracture zone at a 45[degrees] angle to the model grid (Figure 1), Svensson (2001a) shows that the exact flux, [Q.sub.e], from the center of cell (3,1) to the center of cell (2,2) is
[Q.sub.e] = -[K.sub.f] * W * b[[[DELTA]h]/[[DELTA]x[square root of 2]]] (1)
where [K.sub.f] is hydraulic conductivity of the fracture zone, W is fracture zone width, b is thickness of the aquifer, [DELTA]h is head difference, and [DELTA]x is grid spacing. In the Svensson (2001a) formulation, flow from cell (3,1) to (2,2) takes two pathways (Figure 1). One pathway leads through cell (2,1) and the other through cell (3,2). In contrast, the method used herein restricts flow through a fracture zone to a single pathway as shown by the flow vectors in Figure 1. By specifying that flow is through cell (3,2), the equation for flow from cell (3,1) to (2,2) is
[Q.sub.e] = -[K.sub.c] * [DELTA]x * b[[[DELTA]h]/[2 * [DELTA]x]] (2)
where [K.sub.c] is the cell hydraulic conductivity. By setting Equations 1 and 2 equal, [K.sub.c] for a 45[degrees] angle fracture zone becomes
[K.sub.c] = [K.sub.f][W/[[DELTA]x]][square root of 2] (3)
The general equation for [K.sub.c], determined by evaluating various fracture zone orientations, is
[K.sub.c] = [K.sub.f][W/[[DELTA]x]][sin ([theta]) + cos ([theta])] (4)
where [theta] is the angle measured from the x or y axis, whichever is less.
Flow between matrix blocks and a fracture zone, as calculated with the methods presented here, may be slightly affected by fracture zone orientation. For fracture zones in a permeable matrix, small flow errors may result because MODFLOW uses [K.sub.c] (an adjusted value only valid for fracture zone flow) and harmonic averaging to calculate internodal conductance values with the four adjacent cells. Future applications of this method would benefit from a fracture zone conductance package for MODFLOW that uses the adjusted [K.sub.c] value to calculate conductances only between fracture zone cells. Harmonic averaging would use [K.sub.f], instead of [K.sub.c], to calculate internodal conductances between fracture zone cells and matrix cells.
The spatial extent of a fracture zone is only one cell wide in the model grid, but use of Equation 4 allows the true width of fracture zones to be specified. Therefore, the hydraulic effect of fracture zone width is indirectly included in the ground water flow calculations through adjustment of cell hydraulic conductivity. This approach limits detailed evaluation of particle traces in and directly adjacent to fracture zones, because true widths and spatial extents of fracture zones are not explicitly included in the model grid. In the example presented in this paper, widths of fracture zones are assumed equal to cell spacing.
Adjustment for Particle Tracking
With this method for incorporating fracture zones, an adjustment is also required for particle-tracking routines, such as MODPATH (Pollock 1994). MODPATH's semi-analytical particle-tracking method approximates velocities at particle locations using a piecewise linear interpolation scheme. For fracture zones that align with rows or columns, MODPATH will calculate accurate travel paths and travel times. For fracture zones at an angle to the model grid, however, particle paths and travel times calculated by standard MODPATH would not be accurate unless an adjustment is made. Figure 2a shows the travel paths for three particles within a transmissive fracture zone surrounded by impermeable rock matrix. None of the paths in Figure 2a are straight because of the velocity interpolation scheme. Thus, for certain fracture zones, standard MODPATH calculates travel times that are too long. Moreover, travel times through the fracture zone are not uniform; each path should be straight and particles should travel at the same velocity.
[FIGURE 1 OMITTED]
A simple solution to this problem can be obtained by noting that particle paths calculated by...
NOTE: All illustrations and photos
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