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Three-dimensional analysis of variably-saturated flow and solute transport in discretely-fractured porous media

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JO”ANAL OF
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Contaminant Hydrology
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ELSEVIER
Journal of Contaminant Hydrology 23 (1996) l-44
Three-dimensional analysis of variably-saturated flow and solute transport in discretely-fractured porous media
R. Therrien
*,
E.A. Sudicky
Received 20 September 1994; accepted 24 August 1995
Abstract
A discrete fracture, saturated-unsaturated numerical model is developed where the porous matrix is represented in three dimensions and fractures are represented by two-dimensional planes. This allows a fully three-dimensional description of the fracture network connectivity. Solute advection and diffusion in the porous matrix are also directly accounted for. The variably-saturated flow equation is discretized in space using a control volume finite-element technique which ensures fluid conservation both locally and globally. Because the relative permeability and saturation curves for fractures may be highly nonlinear, and in strong contrast to those of the matrix, the robust Newton-Raphson iteration method is implemented according to the efficient procedure of Kropinski (1990) and Forsyth and Simpson (1991) to solve the variably-saturated flow equation. Upstream weighting of the relative permeabilities is used to yield a monotone solution that lies in the physical range and adaptive time stepping further enhances the efficiency of the solution process, A time-marching Galerkin finite-element technique is used to discretize the solute transport equation. Although the methodology is developed in a finite-element frame- work, a finite-difference discretization for both groundwater flow and solute transport can be mimicked through a manipulation of the influence coefficient technique. The use of an ILU-pre- conditioned ORTHOMIN solver permits the fast solution of matrix equations having tens to hundreds of thousands of unknowns. Verification examples are presented along with illustrative problems that demonstrate the complexity of variably-saturated flow and solute transport in fractured systems. _ Corresponding author. Present address: DCpartement de Geologic et Genie GCologique, UniversitC Laval, Quebec, Qut. G
I
K 7P4, Canada. Elsevier Science B.V.
SSDI 0
166.3542(95)00088-7
1 Introduction
The safe disposal of toxic contaminants in the subsurface requires that their migration to the biosphere be prevented for an indefinitely long period of time. Isolation of the contaminants in low-permeability geologic media. where groundwater velocities are typically low and molecular diffusion is the primary solute migration process of importance. is the logical choice for a suitable repository; however, the presence of fractures in such media can greatly influence the mass transport process because they might represent preferential pathways for rapid contaminant migration. Because of the disparity between the rate of advective contaminant migration along the fractures and the slow, but persistent, advance in the adjacent porous matrix, vastly different time scales for transport can exist in fractured porous media. The matrix diffusion process in fractured porous media is also commonly regarded as a transient attenuation mechanism which gradually reduces the solute flux and rate of contaminant advance in the fractures (e.g., Neretnieks, 1980). A realistic description of solute migration in systems where fractures are located in a matrix having a finite porosity should therefore consider mass exchange between the fractures and the surrounding matrix. Significant contributions to the understanding of groundwater Bow and solute trans- port in fractured porous media have evolved in the past decade from the urgent need to safely dispose of radioactive waste (Neretnieks, 1980: Schwartz et al., 1983; Wang and Narasimhan, 1985). While much of the effort has involved the study of fractured. low-porosity crystalline rock, the seemingly ubiquitous presence of fractures in non- lithified near-surface clayey deposits has recently focussed much attention on the problem of contaminant migration in fractured media that are relatively porous. These deposits form aquitards commonly occurrin g at the surface and often overly aquifers comprised of sand, gravel or bedrock (Cherry. 1989). Recent field studies have shown that vertical fractures can be present to significant depths in these clayey materials and create a hydraulic connection between the surface and the underlying aquifer (e.g.. Keller et al., 1986; McKay, 1991). A number of mathematical models describing groundwater flow and solute transport in fractured porous media have been developed in the past. One classical approach is to view a fractured porous medium as a single continuum or equivalent porous medium in which the point-to-point spatial variations in the hydrogeological properties of the rock mass are averaged over a representative elementary volume (REV) in order to define bulk macroscopic values (Bear, 1972). Long et al. (19821, Berkowitz et al. (1988). and
Schwartz and Smith (198X), among others. have studied the applicability of this approach in the context of groundwater tlow and dissolved solute transport in fractured geologic materials under saturated conditions. The double-porosity concept introduced by Barcnblatt et al. (1960) to represent tlow in fractured rock relies on the assumption that the fractured rock mass can be represented by two interacting continua: the primary porosity blocks of low permeability and high storage capacity and the secondary porosity fractures of high permeability and negligible storage capacity. The two systems are linked via a leakage tcrtn representing the exchange of fluid between them. A similar conceptual approach has also been used to represent dissolved solute transport in nonfractured materials containing macropores
R. Therrien, E.A. Sudicky/Journal of Contaminant Hydrology 23 (1996) l-44 3
(e.g., Coats and Smith, 1964). Numerical models that employ the double-porosity approach to describe transport in porous and fractured porous media include those of Bibby (1981) Huyakorn et al. (1983) and Sudicky (1990). In contrast to the two approaches mentioned above, models based on a discretely- fractured conceptualization of the rock mass require that the geometry and the hydraulic properties of each fracture be specified explicitly. Schwartz et al. (1983) Smith and Schwartz (1984) and Cacas et al. (1990) are examples of studies that have examined groundwater flow and solute transport in discrete fracture networks, but without consideration of matrix diffusion. The numerical models proposed by Huyakorn et al. (1987) Berkowitz et al. (1988) and Sudicky and McLaren (1992) incorporate matrix diffusion by using the principle of superposition of one-dimensional fracture elements onto two-dimensional porous matrix elements to solve the coupled fracture-matrix saturated groundwater flow and solute transport equations. Advective transport in the matrix is also accounted for because the two-dimensional matrix blocks are themselves discretized. Variably-saturated flow and solute transport in a fractured medium was investigated by Rasmussen and Evans (1989) who developed a three-dimensional model based on the boundary element method; however, they did not account for contaminant advection in the matrix and matrix diffusion was only approximated by using an arbitrary attenuation coefficient in the fracture transport equation. Wang and Narasimhan (1985) also developed a modelling approach to simulate variably-saturated flow in discretely-frac- tured geologic material but did not extend their analysis to include solute transport. They used the integrated finite-difference approach to solve for the drainage of a regularly fractured matrix block in three dimensions. They assumed that the fractures are rough-walled, with variable apertures characterized by a gamma probability distribution. The fracture walls were taken to be in contact when the aperture at any point in the fracture plane was smaller than a defined cutoff aperture. Effective, macroscale constitu- tive relationships for fracture saturation, permeability and contact area as functions of pressure head were then developed based on the aperture probability distribution function and the fact that the portions of the fractures with apertures larger than the saturation cutoff aperture will be dry. Flow along the fractures was described by a generalized cubic law and a phase-separation constriction factor was used to represent the resistance to flow caused by entrapped air. Richards’ equation and van Genuchten (1980) relationships were used to describe the flow in the porous matrix. Other studies that considered variably-saturated flow include those of Dykhuizen (1987) and Peters and Klavetter (1988) who developed numerical models based on the double-continuum approach, and Nitao and Buscheck (199 1
),
who presented an analytical solution where matrix imbibition is treated as a one-dimensional process. The objective of this work is to present an efficient and robust numerical algorithm for the solution of the three-dimensional variably-saturated groundwater flow and solute transport equations in discretely-fractured media. The inclusion of the third dimension allows for a more realistic representation of the fracture connectivity. Transport pro- cesses including advection, mechanical dispersion, molecular diffusion and sorption in the porous matrix will be fully accounted for because these processes have been shown to be non-negligible for many groundwater flow and solute transport scenarios (Sudicky
and McLaren, 1992). Moreover, under unsaturated conditions, the fractures can become barriers to flow such that solute transport may occur primarily by advection, dispersion and diffusion in the porous matrix. The control volume finite-element method is selected here to solve the variably-saturated flow equation, thus permitting an efficient imple- mentation of the robust Newton-Raphson linearization technique and a straightforward use of upstream weighting of the relative permeabilities. A standard time-marching Galerkin finite-element approach is used to solve the solute transport equation. It will be shown that a fast and robust ILU-preconditioned ORTHOMIN solver permits the solution of the discretized equations with tens to hundreds of thousand of unknowns on modern workstations. Several verification examples will be presented which compare results against those obtained by other numerical simulators for problems involving variably-saturated flow and solute transport in fractured porous media. Example prob- lems designed to illustrate the effects of variably-saturated tlow and the coupled migration of a dissolved contaminant will then be presented. 2.
Physical system
Fig. 1 illustrates a commonly occurring geological scenario that consists of a fractured clay aquitard overlying a sand and gravel aquifer that might be used for water supply. Hazardous wastes that might be buried in the aquitard, close to the ground surface, represent a potential source of contamination if the emplaced contaminants readily dissolve in the groundwater and migrate downward towards the aquifer. If vertical fractures exist in the aquitard and some extend from the bottom of the waste zone downward to the more permeable aquifer, it is then possible that a contaminant plume will form in the aquifer because of the rapid solute advection along the fractures (Harrison et al., 1992). The effect of such fractures on downward groundwater flow and solute transport under variably-saturated conditions has, however, been unexplored to date. For example, how would the shape of the water table be affected by the presence
Infiltration Fig. I. Contaminant
transport
hrough a fractured aqumrd into an underlying aquifer.
R. Therrien, E.A.
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udicky /
Journal of Contaminant Hydrology 23 (I 996) 1-44 5
of the fractures if spatially variable recharge conditions exist at the surface or if the aquifer-aquitard system was stressed by placing a pumping well in the aquifer for remediation purposes. The following section provides a description of the governing equations used to represent variably-saturated flow and solute transport in a discretely-fractured porous medium. Although the scenario illustrated in Fig. 1 is a common one, the theoretical framework and numerical solution methodology presented here are also applicable to other types of fractured geologic materials such as crystalline rocks. 3.
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overning equations
3.1.
Variably-saturatedflow
The description of variably-saturated groundwater flow in a discretely-fractured porous medium requires governing equations for both the porous matrix and the fracture system. The following assumptions are made: the fluid is essentially incompressible, the fractured porous medium is nondeformable and the system is under isothermal condi- tions. Furthermore, the air phase is assumed to be infinitely mobile. A modified form of Richards’ equation is used to describe three-dimensional transient groundwater flow in the variably-saturated porous matrix. The equation has the follow- ing general form in three dimensions, using an indicial notation and the summation convention (Cooley, 1983; Huyakorn et al., 1984):
(1)
where
Kij
is the saturated hydraulic conductivity tensor;
k,, = k,,(S,)
represents the relative permeability of the medium with respect to the degree of water saturation S,; G=
+(x,,t>
is th e pressure head; z is the elevation head; and 0, is the saturated water content, which is equal to the porosity. The water saturation is related to the water content 8 according to: s, =
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/e, (2)
The effect of any source or sink on the flow in the matrix such as a fluid exchange with the fractures or extraction by pumping can be represented in Eq. 1 by Q. In order to solve the nonlinear flow equation (I), constitutive relations must be established that relate the primary unknowns t, ~ and S,. Here, the equation will be solved in terms of the pressure head, with S, = SW($) given by a prescribed functional relationship that is usually determined experimentally. Similarly, the relative permeabil- ity is assumed to be expressible in terms of either the pressure head or the water saturation. A commonly used functional relation is that presented by van Genuchten (19801, based on earlier work by Mualem (1976), where the saturation-pressure relation is expressed by:
s,=[
+ lJ> ??+;;
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