three-dimensional analysis of variably-saturated flow and solute transport in discretely-fractured...

$: Three-dimensional analysis of variably-saturated flow and solute transport in discretely-fractured porous media$
JO”ANAL OF

Contaminant Hydrology

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 framework, 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-preconditioned 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 transport 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-fractured 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 constitutive 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 processes 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 implementation 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 problems 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 through a fractured aqumrd into an underlying aquifer.

R. Therrien, E.A. Sudicky / 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. Governing 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 conditions. 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 following 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, = e/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 permeability 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,=[ l+(lJ> ??+;; O<m<l

and the relative permeability is obtained from:

k,, = S,i/‘[ I - (1 - $/“‘)“‘]’ (4)

where (Y and m are parameters obtained from a fit of Eqs. 3 and 4 to experimental results; f, is the capillary pressure (F’, = - $I>; and S, = (S, - S,,)/( 1 - S,,), S,, being the residual saturation. Although other fundamental relations exist (Nielsen et al., 19861, any arbitrary, but physically realistic, function for S,(4) and k,, (S,) can be handled through the use of tabular data as input to a variably-saturated numerical model. Although it has been observed experimentally, the hysteresis phenomenon will not be considered for this work. The implications of neglecting hysteresis in the constitutive relations for fractured media will be examined in the future.

The fractures are idealized as two-dimensional parallel plates. This implies that the total head is uniform across the fracture width. The equation for variably-saturated flow in a fracture of aperture 2h can be written by extending the saturated fracture flow equations (Berkowitz et al., 1988: Sudicky and McLaren, 1992) and using the analogy of Richards’ equation (I) for the porous matrix. With the extension. the governing equation has the form:

;(“‘)K,k,,, d”;A;r,, as,, - 4,Ll/ + q,,ir f Q, = (2h) ~ at ’ i,,j= I,2 (5)

I /

where k,.,, is the relative permeability of a fracture; $, = t,bf(x,,t> and zr are the pressure and the elevation heads within the fracture, respectively; S,., is the water saturation for the fracture; and Q, represents externally applied sources or sinks. The saturated hydraulic conductivity of a fracture, K, . is given by (Bear, 1972):

(6)

where p and p are the fluid density and viscosity, respectively; and g is the acceleration due to gravity.

Eqs. I and 5 are linked via the fluid leakage tluxes L[,,,, and Y,(,,- across the two surfaces I and I ‘. respectively, of a fracture. Along surfaces comprising a fracture- matrix interface, the tacit assumption will be made that the pressure head in the matrix along such surfaces equals the pressure head in the adjoining fractures.

Initial and boundary conditions are needed to solve Eqs. 1 and 5 describing fluid flow in the porous matrix and in the fractures. Boundary conditions can be of the first- and second-type and are defined in the usual way (Huyakorn and Pinder, 1983). A seepage face constitutes another type of boundary and its representation will be discussed in the section describing the numerical solution of Eq. I.

Constitutive relations must be also available to describe variably-saturated flow in the fractures. There is a very limited number of studies where these relationships have been derived experimentally (e.g., Reitsma and Kueper, 1994). Several theoretical studies have, nevertheless, been performed in order to characterize the nature of the relationships. Wang and Narasimhan (1985) and Rasmussen and Evans (I 989) generated

R. Therrien, E.A. Sudicky / Journnl of Cmtaminant Hydrology 23 (1996) I-44 I

synthetic relations between pressure, saturation and relative permeability for a single fracture plane containing a distribution of apertures. Their results were based on capillary theory and they used the well-known cubic law to represent flow in the fracture. Pruess and Tsang (1990) considered the problem of two-phase flow in a rough-walled fracture plane. They subdivided the fracture plane into sub-elements and assigned a spatially-correlated aperture to each. The occupancy of each element by either the wetting or the non-wetting phase fluid was based on a prescribed entry pressure relationship. Once the fluid occupancy was assigned to each element, the relative permeabilities were obtained by performing two single-phase flow simulations: one for the region occupied by the wetting phase and the other for the non-wetting phase.

In this study, relative permeability and saturation-pressure head relationships are based on the work of Mendoza and Sudicky (1991), which follows the general conceptual framework of Pruess and Tsang (1990). However, in contrast to Pruess and Tsang (19901, Mendoza and Sudicky (199 1) rigorously account for local accessibility in assigning the phase occupancy of each fracture sub-element. They also consider fluid trapping and they calculate uncertainty in the constitutive relations by using the Monte Carlo simulation method.

Another process that needs to be considered when describing flow in a variably- saturated fractured porous media is the reduction of the area available for flow across a fracture-matrix interface. Some portions of a fracture, as it desaturates, cannot transmit any water and thus reduce the area available for matrix flow across the fracture. We use the approach of Wang and Narasimhan (1985) who represented this phenomenon with a function describing the change in the effective fracture-matrix area as a function of pressure. This effective area is only applied to those matrix nodes that are also fracture nodes. Further details on the numerical implementation of the effective area concept will be discussed in the section on numerical methods.

The validity of the assumption of a passive air phase, which is implicit in the use of Richards’ equation, was examined by Forsyth (1988) who used a cell-centered finite-difference solution describing two-phase flow in a porous medium and compared results to those obtained from a single-phase solution based on Richards’ equation. He found that the agreement between the single-phase and the two-phase formulation was good except when there was a highly nonlinear dependence of relative permeability of the air phase with respect to the air saturation. Fractures were not considered by Forsyth (1988) and hence the need to account for a finite air phase mobility in fractured porous media cannot be assessed. Moreover, air entrapment and air compressibility effects could be important in cases where infiltration occurs into a complex three-dimensional fracture network that is initially dry.

In order to describe groundwater flow in the saturated zone, the storage terms forming the right-hand side of Eqs. 1 and 5 are expanded in a way similar to that presented by Cooley (1971) and Neuman (1973). They relate a change in storage in the saturated zone to a change in fluid pressure through compressibility terms as is conventionally done in hydrogeological applications. They also assume that the bulk compressibility of the medium is constant for saturated and nearly-saturated conditions. For unsaturated conditions, it is assumed that the compressibility effects on the storage

of water are negligible compared to the effect of changes in saturation. The following expression is obtained for the storage term in Eq. 1 (Cooley, 197 I ; Neuman, 1973):

;fV, =S&; + 0,% (7)

where S, is the specific storage coefficient of the porous medium (Freeze and Cherry, 1979).

Using similar arguments, the storage term in Eq. 5 describing variably-saturated flow in the fractures becomes:

where S,r is the specific storage coefficient for the fractures. Because it is assumed here that the fractures are non-deformable and fluid-filled, there is no contribution to the storage term from fracture compressibility. Thus, the specific storage coefficient for a fracture under saturated conditions is related to the water compressibility, cy,, according to:

s\,- = P.!TQ, (9) The validity of assuming non-deformable fractures is likely to be reasonable if the fractures have a high normal stiffness or if changes in the effective stress field within the system due to pumping, for example, are small.

3.2. Solute trunsport

In order to describe solute transport in a discretely-fractured porous medium, two equations are needed, one for the porous matrix and one for the fractures in a manner analogous to that for the flow problem. It is assumed that the solute can adsorb onto the fracture walls and onto the solid phase comprising the porous matrix and it may also be subject to first-order decay. Continuity of concentration and of solute mass flux at the matrix-fracture interface provides a coupling between the fracture and the porous medium equations.

Three-dimensional transport in a variably-saturated porous matrix is described by the following equation (Bear, 1972):

where c = c(x,,t) is the solute concentration; Dli is the hydrodynamic dispersion coefficient; and h is a first-order decay constant. The assumption of fluid incompress- ibility is made in Eq. 10. The retardation factor. R, is given by (Freeze and Cherry, 1979):

R. Therrien, E.A. Sudicky/ Journal oj’Cmtaminant Hydrology 23 (1996) l-44 9

where pb is the bulk density of the matrix; and K, is the equilibrium distribution coefficient describing ‘a linear Freundlich adsorption isotherm. Note that for variably- saturated conditions, the water saturation appears in the definition of R.

The fluid flux in Eq. 10 is obtained using the Darcy equation:

The hydrodynamic dispersion coefficient Dii is given by (Bear, 1972):

(12)

(13)

where cr, and LY~ are the longitudinal and transverse dispersivities, respectively; 141 is the magnitude of the Darcy flux; T is the matrix tortuosity; Dd is the free solution diffusion coefficient; and 6;, is the Kronecker delta. The product ~0~ represents an effective diffusion coefficient for the matrix. Other expressions for an effective diffusion coefficient in unsaturated media, such as that described by Millington (1959), can be substituted.

The equation for two-dimensional solute transport in a variably-saturated fracture can also be deduced from the equation describing solute transport in a fully-saturated fracture (e.g., Tang et al., 1981; Sudicky and McLaren, 1992). Its form is:

i,j= 1,2 (14)

where c, = c,( ~{,t) is the concentration in a fracture; and Dfij is the hydrodynamic dispersion coefficient of the fracture. An expression similar to Eq. 13 can be used to represent Dfij, where dispersivities and fluxes correspond to those of the fracture and the fracture porosity is assumed to be unity. The retardation factor Rs is defined according to (Freeze and Cherry, 1979):

R,- = 1 + 2 K;/(2b)

* where K, is a fracture-surface distribution coefficient. The Darcy equation is again used to define the fluid flux in the fracture:

( 15)

qf, = -K&v, a( IcIf + zt>

dXj (16)

The terms involving 0, in Eq. 14 represent the advective-dispersive loss (or gain) of solute mass across the fracture-matrix interfaces I and I+ due to fluid leakage and hydrodynamic dispersion. They are discussed in detail elsewhere (Berkowitz et al., 1988; Sudicky and McLaren, 1992).

The boundary conditions for Eqs. 10 and 14 describing solute transport in the porous

matrix and in the fractures, respectively, can be of the first-. second- or third-type and are defined in the conventional way (Huyakorn and Pinder, 1983).

4. Numerical formulation

4.1. Vclriab!v-satctrarecE~-,s~lturut~~l,floM.

Eqs. 1 and 5 describing variably-saturated flow in the porous medium and the fractures are nonlinear and their solution is further complicated because strong contrasts in material properties are likely to exist between the fractures and the porous matrix. In addition to these complexities, a fully three-dimensional discrete-fracture representation can easily involve many tens to hundreds of thousands of unknowns. It becomes imperative, therefore. that an efficient and robust numerical technique be used to solve the governing equations.

The proposed method of solution for the flow problem is based on the control volume finite-element approach (e.g., Kropinski, 1990; Forsyth, 199 I ) which has been shown to be particularly well-suited for a fast and efficient implementation of the Newton-Raph- son linearization technique (Forsyth and Simpson, 1991). Although the Newton-Raph- son technique requires a larger amount of work for each solution step compared to other linearization methods such as Picard iteration, the robustness and higher order of convergence of the Newton method make it attractive. The resulting linearized matrix equation is solved using an efficient preconditioned ORTHOMIN solver. Details concerning the numerical algorithm and its implementation are provided below.

The method of solution of the variably-saturated flow equation for the porous matrix and the fracture equation is identical except for the difference in dimensionality; hence only the porous medium Eq. 1 will be treated here. The basic idea of the control volume finite-element approach is to obtain a discretized equation that mimics the governing mass conservation equation locally. A volume of influence, referred to as a control volume, is assigned to each node. The discretized equation for a given node then consists of a term describing the change in fluid mass storage for that volume and another term representing the divergence of the fluid mass flux in the volume. The fluid mass flux will depend on the physical properties associated with the volume and the difference in the value of the primary variable (i.e. $) between the node in question and its neighbours. The control volume finite-element approach is described in detail by Kropinski ( 1990) and Forsyth (199 I ).

First, an approximating function is defined in the usual way:

( ‘7)

where n, (i = I, 2, 3) are the spatial coordinates; and J is a nodal index ranging from I to ~1, where y1 is the total number of nodes. Although Eq. 17 describes the approximation for the pressure head, the same procedure is used for the water saturation. In the following, the dependency of the primary variables on X, and t in Eq. 17 are dropped for clarity.

R. Therrien, E.A. Sudicky/ Journul qf Contaminant Hydrology 23 (1996) 1-44 11

The standard Galerkin technique (see, e.g., Huyakorn and Pinder, 1983) is used to discretize Eq. 1 over the domain of interest, V, leading to:

a(C+z) 1

aij A

% + Q - SS&a, - f/+$ I= 1, . ..) n

(18) We now only consider the equation applying to node I. Upon approximating the time derivative by a finite-difference representation and using a lumped mass approach to treat the storage terms in Eq. 18, we can write:

(19) where u is the region or control volume associated with node I; L is the time level; and At is the time-step size. Here, a fully implicit discretization in time is used. Likewise, for the region u associated with node I, the source/sink term in Eq. 18 becomes:

Q, = /QN& I’ (20)

It is now desired to express the first term on the left-hand side of Eq. 18 as a function of the total head difference between node I and each of its neighbours. By first applying the divergence theorem to this term in Eq. 18 and, using the definition of the Darcy flux in the porous matrix given by Eq. 12, one obtains:

I N,du = - / K,ik,, a(C+z) aN,

/’ Zd” f /y’N,dB

ax, I B

(21) The last term on the right-hand side represents the fluid flux normal to the boundary, B, of volume L’. Let us assume for clarity that this quantity is zero.

Considering only the pressure head term, I,$, in Eq. 21 and disregarding the elevation head term for the moment, use can be made of Eq. 17 to get:

where the summation is carried over all the nodes. Using the fact that

CN,=l

(22)

we have:

N, = 1 - c NJ .I#1

(23)

such that:

(24)

From the results Eqs. 23 and 24. the right-hand side of Eq. 22 can now be rewritten in the following way:

The relationship (25) is used to obtain:

(25)

(26)

where

Y/J = ( C!, + -7.1) - ( +, + :,) (27)

The discretization of the domain into finite elements will create a nodal connectivity (i.e. a table of nodal incidences for the elements). Defining q, as being the set of nodes connected to node I, it is obvious that the nodes not included in q, will not contribute to the change in storage or fluid flow at node I. Using this and the fact that the y,,‘s are nodal quantities and that the summation and integration operations are interchangeable, the right-hand side of Eq. 26 becomes:

(28)

The hydraulic conductivity term, K,,, is taken to be an elemental value. The right-hand side of Eq. 28 therefore becomes:

K k ‘N”N’d,, = c y,,,h,,l-,,, ” IM ?I.\-, ax,

(29) .JE ‘I,

where

r,,, = K,,j,-2 zd,> I 1

(30)

and where the h,,,‘s represent a weighted value of the relative permeabilities for nodes I and J. If upstream weighting is applied, the h,,,‘s are given by:

k TX’1

if rJ,,( $J + z,, - *, - z,) > 0 A,, =

k if r,,,($,,+t,,-*,-z,)<O (31)

r\\,

R. Therrien, EA. Sudicky/Journal of Contaminant Hydrology 23 ~1996) 1-44 13

Upstream weighting of the relative permeability is highly recommended in order to ensure the monotonicity of the solution (Forsyth, 1991). A monotone solution will yield saturations that will always remain in the physical range (i.e. between 0.0 and 1.0). Kropinski (1990) provides a vivid illustration of the importance of the type of relative permeability weighting on the quality and stability of the solution of Richards’ equation. She considered a two-dimensional unsaturated flow problem in which the air entry pressure was low, which makes the governing equation more hyperbolic in nature. Results showed that the use of central weighting produced significant negative saturations as the wetting front advanced in a relatively dry soil; however, the solution was stable and remained in the physical range when upstream weighting was used. Although it is well-known that the use of upstream weighting for advection-dispersion problems can lead to excessive smearing of a concentration front, its use in conjunction with hyperbolic-type equations, such as the one for variably-saturated flow, does not because the solution is self-sharpening. It has also been shown that, for purely hyperbolic equations, the use of central weighting can cause complete failure of the solution (Sammon, 1988).

By combining Eqs. 19, 20 and 29, the final form of the discretized equation for node I becomes:

,,L+w+‘-~1) +8 w-w s w At s At

*

IJ If )L+lr

I.1

+,,+,

1

(32) where the volume of influence for node I

u, = /

N,du (33) 1’

The reduction of the area available for flow across a fracture-matrix interface can be easily incorporated in Eq. 32 in a manner suggested by Wang and Narasimhan (198.5). The area between nodes I and J, which are both located in the matrix, is imbedded in the Z,, term, defined by Eq. 30. For cases where Z or J also coincide with a fracture node, this area between the two nodes is multiplied by a factor representing the new effective area, accounting for a matrix-matrix flow area reduction when the fracture desaturates.

The source/sink term in Eq. 32 can be manipulated in order to specify imposed boundary conditions (Forsyth, 1988; Kropinski, 1990). To assign a prescribed pressure head, I+IJ *, to a portion of the domain, the source/sink term becomes:

&I = K;jkrwW,( do * - +I) (34) where WI is a number large enough (e.g., 102’) to ensure that t,!~, = 4 * when the assembled system of equations for all nodes is solved. This can be viewed as injecting or withdrawing sufficient fluid at node I to maintain the prescribed pressure head.

Seepage faces represent boundaries that require special treatment and a variety of methods have been suggested to implement such boundaries (Neuman, 1973; Cooley, 1983). The method used&here is from Forsyth (1988) and requires that the approximate

location of the seepage face be known a priori. The appropriate form of the source/sink term at the nodes forming the seepage face will be:

where K ’ is the component of the hydraulic conductivity tensor normal to the seepage face. The above expression allows seepage only when the pressure in the medium is greater than the atmospheric pressure, Ic;,,,,. For cases where the flux normal to the boundary is known, use can be made of the boundary term involving y ’ in Eq. 21 to impose this prescribed boundary condition.

Another special treatment of the source/sink term, Q,, is employed when an extraction or an injection well is present in the computational domain. It is sometimes assumed that the flux distribution with depth at the well nodes is uniform; however, this assumption is likely to be incorrect when the well is located in a heterogeneous material or if it intersects a fracture. The method proposed by Cooley (1971) has been implemented in the numerical algorithm and it consists of an iterative procedure to determine the nodal flux distribution along the well screen that will produce hydraulic heads at the well nodes that are uniform. The reader is referred to Cooley (197 1) for further implementation details. The use of one-dimensional elements to represent wells (Sudicky et al., 1995) is presently being extended to variably-saturated conditions and will be reported on in a future paper.

The Newton-Raphson technique is used to linearize Eq. 32. The method is described in Huyakorn and Pinder (1983) and is reproduced here to demonstrate the advantage of using the control volume finite-element approach over a conventional Galerkin method. If Eq. 32 is rewritten in the following way:

f;‘= s,s: i

(v - rcIL) + * (Sk - Sk> *r ‘, Ar i c/ - ~~.,A,JL,)’ - Q; (36)

I J%

where Y represents the iteration level, application of the Newton method to Eq. 36 produces the following matrix equation:

F/I A$;+ ’ = -,f,’

In Eq. 37, the Jacobian matrix, F,‘,, is defined as:

(37)

The iteration process is carried out repeatedly until the change in the pressure head, A$;+ ‘> becomes less than a specified tolerance at all the nodes. The evolution of the residual, f;‘, is also monitored during iteration to ensure proper convergence.

Full Newton iteration can be computationally expensive mainly because of the need to evaluate the Jacobian matrix. It is therefore highly desirable to implement a scheme capable of evaluating it in an efficient manner. One option is to evaluate the Jacobian

R. Therrirn, E.A. Sudic!q/ Journal qf Contaminant Hydrology 23 (1996) 1-44 1.5

numerically (Kropinski, 1990; Forsyth and Simpson, 1991). Term-by-term evaluation of the Jacobian can be represented by (Kropinski, 1990):

V; f/7 +; + ??> -f;( Icl;>

z= E (39)

where E represents a small numerical shift in the pressure head value. Obviously, from Eq. 39, numerical differentiation requires more than one function evaluation for each term; however, it can be shown that not all terms in the Jacobian will require two function evaluations. The form of the discretized equation (36) also makes it intuitively easy to evaluate the Jacobian numerically. This is because the fluid flow terms appearing in the summation in Eq. 36 depend only on nodes I and J. Kropinski (1990) and Forsyth and Simpson (1991) give a detailed procedure for the Jacobian evaluation and it is reproduced below for completeness.

The diagonal term of the Jacobian for node I can be determined from Eqs. 36 and 38 as:

aQ; -- W,

The entries in column I of the Jacobian will be, excluding the diagonal:

(40)

(41)

where J E 7,. As stated previously, the only term in f,’ depending on t/t,’ will be the one representing flow from node I to node J. Therefore:

?f; -= a*; - ~(YJ,Mr,)r

I (42)

Because of local conservation of mass, the fluid flow term between I and J appearing in the equation for node I will be similar to the one appearing in equation for node J. Therefore we have:

YIJ = - YJI

4, = A,, (4)

r,J = &,

Using Eq. 43, Eq. 42 becomes:

The right-hand side of Eq. 44 is also found in the summation appearing in Eq. 40. The expression for the diagonal term (40) can therefore be expressed as:

-=- s,s’ (v- dJ”> + 8 (s:, -w “,_ c afi aQ; ’ w At ’ At (45)

1 JET, a*; a*:

which shows that the evaluation of the diagonal term for node I incorporates all the terms appearing in column I of the Jacobian. The Jacobian can therefore be constructed by only evaluating the diagonal terms and subsequently filling in the off-diagonal terms in a column-wise fashion.

Kropinski (1990) and Forsyth and Simpson (I 991) show that for n unknowns and with the summation in Eq. 36 extending from unity to 7,. the building of the Jacobian requires n(2 + 27,) function evaluations. The Picard iteration scheme, on the other hand, requires at least n(1 + q,) function evaluations to compute the residual, which is seen to be only a factor of 2 less than the more robust Newton method. Numerical differentiation is also attractive because it allows easy use of tabular data to represent arbitrary constitutive relations should analytical expressions be unavailable.

A variable time-stepping procedure similar to the one outlined by Forsyth and Sammon (1986) and Kropinski (1990) has been incorporated in the solution procedure. After obtaining the solution at time level L, the next time step is selected according to:

AtL+’ _ s;

max IS;,+ ’ - .S,“,l AtL

where S, is the maximum change in water saturation allowed during a single time step. This implementation of variable time stepping therefore allows the use of increasingly larger time steps if the dependent variable does not experience drastic changes. The time-step selection can also be based on changes in pressure head by using an expression identical to Eq. 46. It is also recognized that the number of Newton iterations taken to achieve convergence is a good indicator of the suitability of the current time-step size. If the number of iterations exceeds a specified maximum, IT,,,,, at time level L + 1 in which the time step is currently At, the solution is restarted at time level L and the time step is reduced. typically by a factor of 2. Overall, the procedure can lead to a very significant reduction in computational effort, especially when the solution is desired only at a few widely-spaced target times.

The discretized equations presented above are independent of the choice of element type. Of the numerous types of three-dimensional elements that can be used to discretize the porous blocks, rectangular prism elements (Huyakorn et al.. 1986) are implemented here. The two-dimensional fracture planes are discretized using rectangular elements (Huyakorn et al., 1984). This choice of simple elements allows use of the influence coefficient technique (Frind, 1982; Huyakorn et al., 1984) to analytically evaluate the integrals appearing in Eq. 28 in an efficient manner. A restriction of the use of these simple elements is that irregular domain geometries and random fracture orientations cannot be naturally accommodated. However, the numerical model is presently being modified to allow the subdivision of rectangular prisms into tetrahedral elements, which will permit discretization of irregular domains. Because the fracture elements are generated such that they correspond to sides of the three-dimensional rectangular prisms, the nodes comprising the fracture elements are therefore common to nodes comprising the porous matrix elements. The commonality of these nodes thus ensures the continuity of hydraulic head at the fracture matrix interface. Also, by superimposing the contributions at each node from both element types, there is no need to explicitly calculate the

R. Therrien, E.A. Sudicky/ Journal of Contaminant Hydrology 23 (I 996) 1-44 11

fluid leakage terms appearing in Eq. 5. The final system of equations for the superim- posed fracture- and matrix-element system has the following form:

+Q,‘?+’ I ) I=l, . ..) n; Z’=l, . ..) nf (47)

where nf represents the number of nodes associated with fracture elements; a, is the area associated with each fracture node; rl/* represents the set of nodes connected to node I’ through the two-dimensional fracture elements; and the r,r,, A,:, and r,,J terms, which are quantities associated with the nodes belonging to the fracture elements, are defined in a manner similar to that described for the three-dimensional matrix.

An option that has been implemented in the numerical formulation is a choice between a finite-element or a finite-difference representation using a methodology identical to that described by Therrien (1992) and Panday et al. (1993). The problem and boundary conditions are defined in terms of finite elements upon input and new nodal connectivities are established when a finite-difference formulation is chosen. The influence coefficient matrices for the finite-element method, presented by Huyakorn et al. (1986), are manipulated in order to mimic a finite-difference discretization. The reader is referred to Therrien (1992) and Panday et al. (1993) for details on the implementation. The finite-difference form of the needed influence coefficient matrices are provided in the Appendix. Because of the different number of nodal connections (i.e. 7 for finite differences, 27 for finite elements), the finite-element method requires nearly four times as much memory to store the coefficient matrix compared to that for the finite-difference method. The size of the assembled coefficient matrix from Eq. 47 is n X 27 for finite elements and n X 7 for finite differences. A variety of groundwater flow simulations that we have performed for fractured porous media under either fully- or variably-saturated conditions, have indicated that the finite-difference and finite-element representations yield similar results. Also, experience has indicated that the CPU time required when using the finite-difference method is also nearly a factor of 4 less.

It should be noted that in the discretized equation (321, the saturation term is represented exactly and no use is made of the water capacity term which is known to induce severe mass-balance errors (Milly. 1985; Celia et al., 1990). Various schemes ranging in complexity have therefore been derived to resolve this problematic mass-balance error (e.g., Cooley, 1983; Milly, 1985; Celia et al., 1990). In this work, the use of the Newton-Raphson procedure allows a direct representation of the saturation term, thereby completely avoiding the difficulties arising from the use of the water capacity term.

The resulting system of Eqs. 47 can be very large for a fully three-dimensional field-scale problem involving numerous fractures. A fast and efficient matrix solver is therefore critical in order to reduce core memory requirements and CPU time. The preconditioned ORTHOMIN solver has been shown to be very efficient and robust for solving large systems of equations (Behie and Forsyth, 1984) and it has therefore been implemented to solve the system of Eq. 47. The preconditioning chosen consists of performing an ILU decomposition of the assembled coefficient matrix without altering its original sparsity pattern (Behie and Forsyth, 1984). The solver has the capability, however, to perform a higher-order ILU decomposition of the coefficient matrix, where additional steps of a Gaussian elimination are performed, resulting in the addition of extra bands to the original matrix. Performing a higher-order decomposition results in a better-conditioned decomposed matrix. which can improve the convergence rate of the solver, but at the expense of increased storage requirements and additional computation time.

4.2. Solute transport

The transport equation is linear and its solution is therefore not as involved as that for the variably-saturated flow equation. A standard time-marching procedure based on the Galerkin method has been implemented (see, e.g., Huyakorn and Pinder, 1983). The type of elements used for transport are identical to those used for the variably-saturated flow problem and the superposition of the two-dimensional rectangular elements representing the fractures onto the three-dimensional rectangular prism elements representing the porous matrix is also performed. This ensures the continuity of concentration at the fracture-matrix interface and avoids the need to explicitly determine the solute mass exchange terms involving a,, in Eq. 14.

The final form of the discretized equations obtained by assembling the porous matrix and fracture elements can be expressed as:

R. Therrien, E.A. Sudicky/ Journal of Contaminant Hydrology 23 (1996) I-44 19

+(w- 1) i9ng* aN,* cYN,*

4f- N,* + SwfDfu- - dA” I dx!i dx, dx,

I

+ e w ~&S,D,,I,~N,N,dBp +~~2b)S,,I),,ln;~N~*dB: i J=l e e’ I Li I

n

+ c (1 - w) x0$, D,&;N,N,dB’ J= I i c

+~(2b)S,,D,,,,~;&*dB; Cf

i,j = 1, 2, 3; k,l=1,2; 1=1,2 ,..., n (48)

where X_, and C,I refer to the summation over the three-dimensional porous matrix and two-dimensional fracture elements, respectively, that are connected to node J. The volume of a three-dimensional matrix element is represented by V’ and the area of a two-dimensional fracture element is represented by A’. The boundaries of the three-dimensional and the two-dimensional elements are represented by B’ and B,f, respectively. The N * are interpolation functions defined on the two-dimensional rectangular elements. The parameter w (0 < w I 1) is a time-weighting factor, L represents the time level at which the solution is known and L + 1 represents the time level for which a solution is desired.

An option to use either a finite-difference or finite-element discretization, as described earlier, has also been implemented for the transport solution. The finite-difference form of the influence coefficient matrices related to the advection terms are given in the appendix. Experience has indicated that for discretely-fractured porous media in which the matrix has low permeability such that mechanical dispersion in the matrix is weak relative to molecular diffusion, the finite-element and finite-difference representations give essentially identical results. This suggests that the cross-derivative terms in the transport equation are small compared to the terms that are retained in the finite-difference approach. In Eq. 48, a consistent representation of the mass matrix is used for the finite-element discretization. A lumped representation, which is naturally given by the finite-difference formulation, could also be used if desired.

As in the case of the flow problem, a mass balance is performed to assess the accuracy of the solution. A procedure similar to that described by Huyakom and Pinder (1983) is used.

The transport matrix equations are solved using the same ILU-preconditioned OR- THOMIN solver (Behie and Forsyth, 1984) as is used for the flow problem. The description of the numerical solution to the problem of variably-saturated flow and dissolved contaminant in discretely-fractured porous media is now complete. In order to verify the proposed solution algorithms, some test problems have been designed for

20 R. Therriun. E.A. Sudick?: / Journal qf Contuminanf Hydrology 23 (I 9%) I-44

which comparisons to numerical solutions obtained with other simulators are available The results of the comparisons for these problems and some illustrative examples designed to investigate variably-saturated flow and solute transport for a simple fracture geometry and for a complex fractured aquitard-aquifer system will be described next.

5. Verification and illustrative examples

Some examples are now presented to verify the flow and transport solutions and also illustrate the capabilities of the numerical model to help understand variably-saturated flow and transport processes in fractured porous media. The first verification example consists of two-dimensional, saturated groundwater flow and solute transport in a complex fracture network imbedded in a porous clay matrix. Comparisons are made with results obtained using the two-dimensional model described by Sudicky and McLaren (1992). The second example is intended to verify the variably-saturated flow simulator. It involves the vertical drainage of a three-dimensional fractured tuff column and was presented by Wang and Narasimhan (198.5). Results obtained with the present simulator will be compared to results published in Wang and Narasimhan (1985). Some illustrative examples will then presented. The first series will examine the case of a single fracture located in a porous matrix, in order to investigate the impact of different matrix hydraulic properties on variably-saturated flow and solute transport in the fractured system. The second series of examples will focus on three-dimensional variably-saturated flow and solute transport in a complex fractured aquitard-aquifer system.

5.1. Flow and trunsport in u complrx ,fracture network

The transient migration of a solute in a two-dimensional saturated porous matrix containing a complex fracture network is used as the first verification example. Because the problem is two-dimensional, a direct comparison with the model described by Sudicky and McLaren (1992) can be made. Fig. 2 shows the geometry of the problem and the distribution of the fractures. The dimensions are 7.0 m in the x-direction and 3.0 m in the y-direction. A unit thickness has been assigned in the z-direction for simulation purposes. The porous matrix has properties typical of clays found in southern Ontario, Canada (hydraulic conductivity and porosity equal to 8 . lop6 m day-’ and 0.45, respectively). The longitudinal and lateral dispersivities of the clay matrix are both 0.0 m and the effective diffusion coefficient for the matrix is 6.91 . lo-’ m2 day ‘. The solute is conservative, having a retardation factor of 1.0 and the decay constant is zero.

The fractures are generated randomly in the two-dimensional plane. Two orthogonal sets of fractures are considered, one set being parallel to the x-axis and the other parallel to the y-axis. The fractures have a constant aperture equal to 30 pm and constant longitudinal and lateral dispersivities of 0.1 m. The magnitude of the lateral dispersivity is irrelevant because transport in each fracture plane becomes one-dimensional for the problem at hand. A uniform distribution, with maximum and minimum lengths tanging from 0.5 to 2.0 m is assumed for the fracture lengths. The mean fracture density within

R. Therrien, E.A. Sudicky/ Journal of Contaminant Hydrology 23 (1996) 1-44 21

3.0

2.7

2.4

2.1

1.8

1.5

1.2

.9

.6

.3

.0 II I I II I Ill1 III I .0 .7 1.4 2.1 2.8 3.5 4.2 4.9 5.6 6.3 7.0

X-Distance Cm)

Fig. 2. Physical system, location of fractures and steady-state hydraulic heads for fracture network problem. The heads vary from 3.5 m on the left-hand boundary to 0.0 m on the right-hand boundary. The contour interval is 0.35 m.

the domain is - 5.0 fractures/m2. Once the fractures are generated, the mesh is refined where necessary to ensure that there is a minimum of three nodes between adjacent fractures in each of the orthogonal sets. The generated mesh contains 185 nodes in the x-direction, 158 nodes in the y-direction and 2 nodes in the z-direction for a total of 58,460 nodes. The number of porous medium elements in the grid is 57,776. The total number of individual fractures generated is 180 of which 85 are parallel to the x-axis and 95 are parallel to the y-axis. For the discretization used, the total number of rectangular plane elements representing all fractures is 6591.

The boundary conditions for the flow problem consist of a specified hydraulic head equal to 3.5 m along the left-hand boundary at x = 0.0 m, and a value equal to 0.0 m along the right-hand boundary at x = 7.0 m. The remaining flow boundaries are impermeable. A total of 144 ORTHOMIN iterations were necessary to achieve conver- .gence for the steady-state flow solution using an absolute convergence criterion equal to 1 . lop5 m. The hydraulic head distribution is shown in Fig. 2.

The velocities computed from the flow simulation results were then used for the transport problem. The solute is initially absent from the domain and a first-type boundary condition given by c = c, was imposed along the left-hand side of the domain. The other boundaries for transport are zero dispersive flux. The convergence criterion used in the ORTHOMIN solver is 1 . 10-5. Variable time stepping was used to march the solution to 200 and 500 days.

Fig. 3a and b shows the concentrations in the two-dimensional plane for the two specified times values. A comparison of the computed concentrations with those obtained using the two-dimensional model described by Sudicky and McLaren (1992) indicated that both models yielded nearly identical results. Therefore only the results

22

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R. Therrien, E.A. Sudicky/Journal of Contaminant Hydrology 23 (19961 1-44 23

obtained with the present model are shown. The cumulative solute mass-balance error, defined as the difference between the mass flux at the boundaries and the change in mass stored for a specified time, was 0.2% for the first time (200 days) and 0.5% for the last time (500 days). The total CPU time for the flow simulation was 0.5 min and that for the transport solution was 3.5 min on the IBM” RS/6000 Model 590. These CPU times include input/output, element assembly and ORTHOMIN solve time.

The more rapid advance of the solute front in the fractures and the smoothing effect of diffusion in the matrix are apparent at both times. The smoothing caused by matrix diffusion is more pronounced near the source because it is the region where the solute contact time between the fractures and the matrix blocks is largest. Although groundwater velocities in the fractures are on the order of several tens of metres per day, concentrations at the right-hand boundary located 7.0 m away from the source are still very low after 500 days, indicating that for this case matrix diffusion acts as a strong retardation mechanism for solute movement. Also, although the fractures control the rate of advance of the leading edge of the plume, the amount of contaminant mass stored in the fractures is < 0.07% of the total mass stored in the system at 200 and 500 days. The contaminant mass located in the low-mobility porous matrix represents a long-term source of groundwater contamination. This becomes an important factor to consider when dealing with fractured porous media should a remediation scheme be implemented.

5.2. Drainage of a fractured tuff column

An example is now presented to verify the variably-saturated flow solution in discretely-fractured porous media. Because the variably-saturated flow equation is nonlinear and analytical solutions are at best approximate, the numerical formulation was verified by comparison to the numerical solution presented by Wang and Narasimhan (1985) for an example problem which involves the vertical drainage of a three-dimensional fractured tuff column. Fig. 4 illustrates the geometry of the physical system. The porous tuff matrix contains three fracture sets, two sets are vertical with a constant fracture aperture equal to 240 pm and one set is horizontal, with a fracture aperture equal to 3 10 pm. It should be noted that the fractures have not been drawn to scale in Fig. 4. The fractures partition the matrix into blocks, with each block having dimensions equal to 0.22 m X 0.22 m X 0.48 m. A total of 27 such blocks are represented in Fig. 4. The saturated hydraulic conductivity of the tuff matrix is 3.2 . IO-’ cm s-‘, its porosity is 0.09 and its specific storage equals 1 lop6 m-’ The constitutive relations describing the matrix saturation and relative permeability are represented by the van Genuchten relations (Eqs. 3 and 4) with S,, = 9.6. 10-4, LY = 7.027 lo-’ m-l, m = 0.45 and IZ = 1.818. The specific storage of the fractures is equal to 4.4 . 10ph mm ‘. The theoretical expressions developed by Wang and Narasimhan (1985) to describe the saturation, relative permeability and effective fracture-matrix flow area for the fractures, as functions of the fluid pressure, were implemented in this model and used for the simulation. The comparison was made for the case where the phase-separation constriction factor, which was used by Wang and Narasimhan (1985) to represent the effects of the air phase on the flow of water, was neglected.

24

0.48 m

Fig. 4. Verification example involving fractured porous tuff, from Wang and Narasimhan (1985).

Due to the symmetry of the system, Wang and Narasimhan (1985) only considered one vertical column bounded by four vertical fractures. For this comparison, only one quarter of this column, i.e. total dimensions equal to 0.11 m X 0.1 1 m X 1.44 m, need be discretized, taking advantage of the horizontal symmetry of the drainage process. The column was discretized using 7 nodes in each of the horizontal directions and 31 nodes in the vertical direction, for a total of I5 19 nodes (1080 three-dimensional elements). The nodal spacing used was identical to the one reported by Wang and Narasimhan (1985). Each vertical fracture was represented by 180 two-dimensional elements and 36 elements were used to discretize each horizontal fracture.

The column was initially saturated, the fluid was static and its potential was everywhere zero. Drainage was performed by applying a suction equal to - 1 12.0 m at the bottom of the column, all other boundaries being impermeable. This suction caused the fractures and the matrix to desaturate with time. Time-stepping control was used to move the solution through time. A maximum change in pressure head of 1 .O m for each time step was used in con.junction with Eq. 46. The total CPU time for the Bow simulation was 6 min on the IBM’ RS/6000 Model 590 and a total of 51 I variable time steps were necessary to reach the final simulation time of 10’ yr. It should be noted that convergence of the Newton procedure occurred typically after the first iteration for most time steps.

The comparison of the results obtained with this model and those of Wang and Narasimhan (1985) is presented in Fig. 5, which shows the change in fluid pressure with time for four different locations in the column. Location A represents the middle of the porous block, location B is the middle of a vertical fracture bounding the matrix block and points C and D are in the middle of the horizontal fractures. It can be seen from Fig. 5 that there is a very good agreement between the results obtained and those reported by Wang and Narasimhan (1985). The pressure head is seen to decrease

R. Therrien, E.A. Sudicky/ Journal of Contaminant Hydrology 23 (1996) l-44

-60.0

-120.0 ’ I , I I I I I I I I I 1o-5 1o.3 10-l 10' IO3 lo5

Time(year)

Fig. 5. Pressure drop at selected points during the drainage of a fractured porous tuff. Lines indicate results from this simulator, the swhols are results from Wang and Narasimhan (1985).

gradually at early times for all observation points. The decrease is more rapid at point D, which is closer to the drainage boundary. The pressure at points A and B is identical, revealing that the drainage process for this case is mainly influenced by the porous matrix when unsaturated conditions prevail.

The drainage simulation was repeated using both the finite-element and the finite-difference representations. Both representations produced identical results, although the finite-difference representation required one quarter of the CPU time compared to the finite-element scheme.

5.3. Vuriablv-saturuted~ow and transport in a single fracture

This series of simulations involves variably-saturated transient flow and solute transport in a porous matrix containing a single fracture. It is intended to illustrate some important features of the coupled variably-saturated flow and transport process that are contrary to the behaviour expected in completely saturated systems. The simulations involve the injection of water containing a dissolved solute into the initially unsaturated fracture-matrix system. Different contrasting porous matrix and fracture flow properties will be used in order to observe their effect on the fluid and solute behaviour. In order to better grasp the sensitivity, only a single one-dimensional fracture is considered, thus

26

Table I Tabular data describing the material conhtitutivc relationships for the hinglc fracture example

Sandstone - IOO.0 0.02 - 12.0 0. I - 3.5 0.2 - 3.0 0.205 ~ I .o 0.35

Clay ~ IWO 0.26 -7.x 0.6

- 6.5 0.65

-3.5 0.7

~ 2.3 0.8

- 0. I 0.9 I 0.0 I .o

Fracture ~ I.5 0.29 ~ 0.225 0.29 - 0. I5 0.3

0.0 I .o

0.0 0.2 0.42 0.64

I .o

0.0 0.37

0.6 I 0.0 I .o

0.0 0.000 I 0.0 I 0. I I .o

0.0 0.0000 I 0.00 I 0. I I .o

0.0 0.0 0.2399 0.0 0.24 0.00 I 7 0.33 0.025 I I .o I .o

eliminating geometrical effects caused by the presence of numerous intersecting fractures.

We consider a two-dimensional vertical soil column having a length equal to 3.2 m in the x-direction, a height of 5.0 m in the z-direction and a unit thickness in the y-direction. A single vertical fracture is located in the center of the domain, at x = I A m, and extends from the top of the domain, at : = 5.0 m, to an elevation equal to 2.0 m. Variable spacing is used to discretize the domain, with a finer nodal spacing being used near the fracture. The discretization is such that 39 nodes are used in the x-direction, 49 nodes in the z-direction and 2 nodes in the v-direction. This yields a total of 3744 nodes and 1786 rectangular prism elements. The fracture is subdivided into 36 rectangular plane elements.

In the first simulation, the porous matrix consists of a relatively permeable material having properties typical of a permeable sandstone. Its hydraulic conductivity and porosity are equal to 1.08 m day-’ and 0.25, respectively, and its specific storage coefficient is equal to 0.000 I m ’ The constitutive relations used for the porous matrix were estimated from graphical information for example problem three of Huyakorn et al. (1984). The data describing the constitutive relationships are shown in Table I. The solute transport parameters used for the sandstone are a longitudinal disperaivity equal to 0.5 m, a transverse dispersivity equal to 0.1 m and an effective diffusion coefficient equal to 1.28. IO ’ m’ day-‘. The solute is assumed to be conservative.

For this example, the fracture has a unifortn aperture equal to 75 pm, which, according to the parallel plate theory, corresponds to a saturated hydraulic conductivity of 397.3 m day ’ The value of the specific storage coefficient of the fracture, assuming

R. Therrien, E.A. Sudicky / Journal of Contaminant Hydrology 23 (I 996) I-44 27

that the fracture is fluid-filled and is rigid, is equal to 4.4 . lo-6 rn-‘. The constitutive relations describing variably-saturated flow in the fracture were obtained from simulations performed by Mendoza (1992). The relationships are an ensemble average result obtained from Monte Carlo simulation of two-phase flow (i.e. air and water) in a rough-walled fracture having a mean aperture equal to 75 pm. Although Mendoza (1992) considered the drainage of a fracture and subsequent imbibition causing the trapping of the air phase, only the primary drainage curve which neglects air entrapment is used for this scenario. The constitutive relations for the fracture are presented in Table 1. The longitudinal and transverse dispersivities of the fracture are equal to 0.5 and 0.1 m, respectively. The effective diffusion coefficient of the solute in the fracture equals 1.8. 10s5 m* day-‘.

A steady-state simulation was performed to establish initial conditions involving downward flow in the column such that approximately the upper one-half of the column was unsaturated. The boundary conditions used for this steady-state simulation involved a constant hydraulic head equal to 3.0 m at the top of the domain (Z = 5.0 m) and an outward normal Darcy flux equal to 0.001 m day-’ imposed at the bottom of the domain at (Z = 0.0 m). The lateral boundary were assigned a zero fluid flux condition. The outflow boundary condition produced a hydraulic head value equal to 2.5 m at the bottom of the domain. Upstream weighting of the relative permeabilities was applied for this example. Upon solution, saturation values at the nodal points were computed from the input soil-moisture characteristic curves for the fracture and the matrix using the computed nodal pressure-head data.

After establishing the steady-state initial condition for flow, a coupled flow and transport simulation was performed. The boundary conditions used for the transient infiltration problem consist of a constant hydraulic head equal to 5.0 m at the fracture inlet (i.e. at x = 1.6 m and z = 5.0 m) while maintaining the same outward normal Darcy flux along the bottom boundary that was used to establish the initial flow regime. Again, the lateral boundaries were treated as being impermeable. At the onset of infiltration in the fracture, a first-type boundary condition for contaminant transport (i.e. c = c,) was imposed at the fracture inlet. A zero-dispersive flux was imposed at all the other boundaries.

The simulation was carried out up to a time value equal to 2.5 days by using the variable time-stepping procedure described previously for the unsaturated flow problem and a Crank-Nicolson time discretization for the transport problem. A total of 150 time steps were required to arrive at the final target time. The total CPU time was 3.2 min on the IBM” RS/6000 Model 590. Fig. 6a-c shows the evolution of the wetting front and the concentration plume over the 2.5-day simulation period. For clarity, the fracture location is not shown on the saturation results presented in Fig. 6a-c. Fig. 6a shows the degree of saturation and solute concentration values at a time equal to 0.1 day. It can be seen that the water is imbibed primarily by the matrix and that the fracture remains unsaturated along the majority of its length. Inspection of the shape of the solute plume at this time reveals that there is no evidence of preferential solute transport along the fracture. On the contrary, the plume pattern is similar to that which would be expected if no fracture was present. The saturation values at a time equal to 0.5 day (Fig. 6b) indicate that the infiltration front is reaching the lower saturated region of the domain.

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.90

-00

.70

.60

"50

.40

-30

"20

.10

.00

1.00

.90

.!30

070

.60

.50

‘40

.30

.20

.10

.00

R. Therrien, E.A. Sudicky/ Journal of Contaminant Hydrology 23 (1996) l-44 29

This observation becomes more noticeable at a time equal to 1.5 days (Fig. 6c), at which point the fracture and the porous matrix both become almost saturated on account of the progressive rise in the water table. The corresponding concentration patterns at this time indicate that the leading edge of the plume, where the fracture exists, is beginning to advance at a slightly faster rate. This is caused by the fact that the leading edge of the plume is located in the fully-saturated portion of the fracture, which has a higher saturated hydraulic conductivity than the porous matrix. Although not shown here, the entire domain becomes fully-saturated after 2.5 days and the shape of the solute plume indicates that the solute migration rate along the saturated fracture is enhanced.

The fluid flux at the fracture inlet was also computed and its behaviour with time is shown in Fig. 7a. It can be seen that the fluid flux into the domain rapidly decreases at early time due to the increase in saturation that occurs in the immediate vicinity of the fracture inlet. After a time equal to - 0.1 day, the fluid flux attains a value of 0.0225 m3 day-’ and remains essentially constant until a time value of - 1.75 days, which corresponds to the time when the entire domain attains full saturation.

This example shows that the contrasts in the saturation-pressure head relationships, hydraulic conductivities, and the relative permeability between the fracture and the matrix can cause flow to occur primarily in the porous matrix rather than the fracture. The concurrent solute transport simulation also demonstrates that, in this case, the hydraulic properties of the porous matrix control the shape of the contaminant plume and that there is no marked preferential migration along the fracture until it becomes fully saturated.

Another simulation was conducted whereby the porous matrix was assigned a much lower hydraulic conductivity, equal to 8.2 . 10m6 m dayy’, which is typical of that of a clayey material. Its specific storage coefficient and porosity equal 0.001 m-’ and 0.45, respectively. The constitutive relations needed to define the variably-saturated flow behaviour are again taken from Huyakorn et al. (1984) and are shown in Table 1. The following parameters are used to describe transport in the clay matrix: a longitudinal dispersivity and a transverse dispersivity equal to 0.1 m and an effective diffusion coefficient equal to 6.22 . IO-" m2 day-‘.

The initial conditions for flow are such that there is static equilibrium in the domain and a uniform hydraulic head equal to 3.0 m is prescribed at all the nodes. This causes the upper half of the domain to be unsaturated and again, because of the nature of the constitutive relations for the matrix and the fracture, the saturation values in the fracture are initially lower than those in the surrounding matrix. At time t = 0, a hydraulic head equal to 5.0 m was imposed at the fracture inlet, and a zero-flux boundary condition was applied to all the other boundaries. This causes infiltration to occur into the domain and, as was used in the previous example, a solute concentration equal to c, was imposed at the fracture inlet. Initial concentrations were again equal to 0.0 everywhere in the domain. For this example, upstream weighting of relative permeabilities was also used.

Fig. 6. High-permeability matrix, saturations on the left and concentrations on the right at: (a) 0.1 day; (b) 0.5 day; and Cc) 1.5 days.

30

0.20 r

a)

0.15 -

0.05

0.00 0.0 0.5 1 .o 1.5 2.0 2.5

Time (day)

0.25 b)

-

0.20

% s k 0.15 -

3 LL

0.10

0.05 -

J

0.00 1 0.000 0.002 0.004 0.006 0.008 0.010

Time (day)

Fig. 7. Fluid flux at the fracture inlet for: (a) high-permeability matrix: and (b) low-permeability matrix

The combined transient flow and solute transport simulation was performed until a maximum time value equal to 900 days was reached. Fig. 8a shows the concentration and saturation patterns at a time equal to 50.0 days. It can be seen that, for this example, the injection of the tluid at the fracture inlet causes flow to occur primarily in the fracture, which becomes fully-saturated at an early time. The infiltration front begins to

R. Therrien, E.A. Sudic@/Joumal c~Contaminant Hydrology 23 (1996) 1-44 31

_;

50

4.5

4.0

3.5

3.0

2.5

2.0

I .5

I 0

.5

.0

10

1.5

4.0

3,s

3.0

2.5

?.0

I .5

I .0

.5

.0

X-Distance Cm)

I .00

.95

.90

,QS

.Q0

I ..a0

“95

.90

“85

.00

5.0

4.5

4.0

3.5

3.0

2.s

2,0

1.5

1.0

.5

-0 .0 .6 1.3 r.9 7.6 3.2

S,0

4.5

4.0

3.5

3.0

2.5

2.0

1 .s

1.0

.5

.0 .0 .6 1;3 1;9 2;6 3.2

X-Distance (mi

1.00

,90

.00

,70

.60

.50

.40

.30

,210

*I0

.00

I .00

.90

-80

.70

.60

.50

.40

.30

.20

.10

.00

Fig. 8. Low-permeability matrix, saturations on the left and concentrations on the right at: (a) 50 days; and (b)

900 days.

propagate in the matrix from the lower end of the fracture rather than from the fracture inlet as was the case in the previous example. Fig. 8b shows the distribution of saturations and concentrations at the final time value. Fig. 8a and Fig. 8b clearly illustrates that there is preferential fluid flow and solute movement along the fracture. The fluid flux at the fracture inlet was also computed during the course of the simulation (Fig. 7b). It can be seen that the fluid flux rapidly declines to a constant value equal to - 0.005 m3 day-’ until a time equal to - 0.0005 day. This time corresponds to a value when the fracture becomes fully saturated. For larger time values, the fluid flux becomes increasingly dependent on the hydraulic properties of the matrix rather than those of the fracture. It should be noted that, for the purpose of clarity, fluid flux values are only shown up to a time value equal to 0.01 day in Fig. 7b. It can be seen by comparing Fig.

7a and Fig. 7b that the fluid flux for this problem is much less than for the previous one, reflecting the contrasting values of the hydraulic properties of the porous matrix between the two examples.

The simulations described above have demonstrated that, under variably-saturated flow conditions, a fracture will not necessarily represent a preferential pathway for fluid flow and solute transport. The combination of saturated hydraulic conductivity, relative permeability and degree of saturation of the porous matrix and the fracture will ultimately control the behaviour of water infiltration and solute migration in fractured porous media. Field or laboratory determination of these properties for both the porous matrix and the fractures will therefore be critical in order to perform realistic field-scale simulations. Perhaps equally as important will be the need to properly characterize the geometry and interconnectivity of the fracture network which is likely to have a major influence on fluid movement and solute migration in a field setting. This last aspect is examined in the next example.

5.4. Three-dimensional field-scale example

The last example is a fully three-dimensional illustration of variably-saturated groundwater flow and contaminant transport in a system comprised of a low-permeability aquitard overlying a sandy aquifer, The aquitard contains a complex fracture network that is connected in all three dimensions. The downward migration of a contaminant from a buried source located near the surface and into the underlying aquifer will be examined. We will also investigate, in a groundwater remediation framework, the influence of pumping the aquifer on variably-saturated flow conditions near the surface of the aquitard and the contaminant recovery at the well.

The physical system considered consists of a IO-m-thick fractured aquitard overlying an aquifer having a thickness equal to 3.0 m (Fig. 9). The horizontal extent of the domain is 90 m in the x-direction and 90 m in the v-direction. It is assumed that a buried waste is located in the aquitard, with the waste zone extending from 6.0 to 40.0 m in the x-direction, 20.0 to 70.0 m in the v-direction and 10.0 to 12.0 m in the

Waste Facility

f 10m

L90mL

Fig. 9. Physical system for the three-dimensional illustrative example

R. Therrien, E.A. Sudicky/Journal of Contaminant Hydrology 23 (1996) 1-44 33

Table 2 Tabular data describing the material constitutive relationships for the fractured aquitard-aquifer example

Material k nv

Aquifer - 100.0 0.02 0.0 0.0 - 12.0 0.1 0.2 0.0001

-3.5 0.2 0.42 0.01 -3.0 0.205 0.64 0.1 - 1.0 0.35 1.0 I .o

Aquitard - 100.0 0.26 0.0 0.0 -7.8 0.6 0.37 0.0000 1 -6.5 0.65 0.61 0.00 1 - 4.5 0.7 0.9 0.1 -2.3 0.8 1 .o I .o -0.1 0.91

0.0 1.0

Fracture - 1.5 0.29 0.0 0.0 - 0.225 0.29 0.2399 0.0 -0.15 0.3 0.24 0.0017

0.0 I.0 0.33 0.025 1 I .o I .o

z-direction. The dissolved solute contained in the waste zone is assumed to be nonreactive. This particular problem closely resembles the scenario examined by Harrison et al. (I 992), where a fractured clay aquitard overlies an aquifer; however, the third dimension is explicitly accounted for here. Also, for this example, a fully-penetrating pumping well is located in the centre of the aquifer (i.e. at x = 45.0 m and y = 45.0 m) and will be pumped once a significant plume forms in the underlying aquifer.

The hydraulic and dispersive properties of the aquitard are similar to those used by Harrison et al. (1992). The hydraulic conductivity of the aquitard equals 1.0. lo-‘” m S -I, its specific storage coefficient is 0.001 m-‘, its porosity equals 0.40 and the constitutive relations describing variably-saturated flow in the aquitard matrix are presented in Table 2. The longitudinal and transverse dispersivities of the aquitard are both equal to 0.1 m and the effective diffusion coefficient equals 1.47 . 1O-'o m’/s. The hydraulic and dispersive properties of the waste zone are identical to those of the aquitard matrix, except that its hydraulic conductivity is two orders of magnitude higher.

The hydraulic conductivity of the sandy aquifer underlying the aquitard is equal to 1.0 . lo-” m SC’ which is six orders of magnitude higher than that of the aquitard. Its porosity and specific storage coefficient equal 0.35 and 0.0001 m- ‘, respectively. The constitutive relations used to describe variably-saturated flow in the aquifer are presented in Table 2. The longitudinal dispersivity of the aquifer is equal to I.0 m and its transverse value equals 0.02 m. These values are typical of moderately heterogeneous sandy aquifers (Sudicky, 1986; Garabedian et al., 199 1). The effective diffusion coefficient of the solute in the saturated aquifer material is equal to 5.22 . lo- ‘” m2 s- ’ .

Three mutually orthogonal sets of fractures, two vertical and one horizontal, were randomly generated within the aquitard. A uniform distribution having minimum and maximum values equal to 0.0 and 90.0 m, respectively, was used to assign the location of the vertical fractures in the horizontal plane and all of the vertical fractures were assumed to originate at the top of the aquitard. A log-normal distribution having a geometric mean equal to 8.0 m and a geometric variance of 2.0 was then used to determine the horizontal extent of all the fractures in the x- and y-direction. The lengths of the vertical fractures were prescribed according to an exponential distribution, reflecting the fact that the probability of encountering a fracture decreases with increasing depth. The fracture apertures were randomly generated from a uniform distribution having minimum and maximum apertures equal to 25 and 35 pm, respectively. This range is similar to that estimated by McKay (199 I> for a surficial clayey till in southwestern Ontario. An exponential decay function was applied to the randomly- generated apertures causing the values to decrease with depth in a manner not unlike that predicted by the geomechanical theory presented by Mase et al. (1990). The form of the applied function is:

2b-2b* exp( -d<) (49)

where 2 b is the depth-varying aperture assigned to a fracture element; 2 b is the randomly-generated aperture; d is the depth below the surface of centroid of the fracture element considered; and 5 is an aperture decay constant. A value equal to 0.03 was used for <. Using these statistical parameters, a total of 337 fractures were generated in the aquitard, of which 127 were horizontal, and 108 and 102 vertical fractures were generated parallel to the X- and the y-axis, respectively. Of all the vertical fractures present, a total of 57 fully penetrated the aquitard. Because it was assumed that the waste zone was unfractured, the fracture generation process excluded the presence of fractures in this zone. The dispersive properties of the fractures were constant everywhere. The longitudinal and transverse dispersivities were 0.5 and 0.1 m, respectively, and the effective solute diffusion coefficient in the fractures was 1.27 . lo-’ m2 s ’ The specific storage coefficient of the fluid-filled fractures was 4.4 10mh mm ’ and the constitutive relationships describing variably-saturated flow in the fractures were obtained from Mendoza (1992) and are shown in Table 2.

The grid and the fracture generation were performed simultaneously in order to produce an adequate discretization. The three-dimensional mesh contains 79 nodes in the s-direction, 74 nodes in the v-direction and 26 nodes in the z-direction, yielding a total of 151,996 nodes. The total number of rectangular prism elements representing the porous non-fractured material equals 142,350, and 20,205 individual rectangular cle- ments representing the fractures were generated. The nodal spacing was variable in each direction, with a smaller spacing being used in the vertical direction near the aquitard surface and at the aquitard-aquifer interface. Also, a finer mesh discretization was used in the vicinity of the pumping well.

A steady-state saturated flow simulation was performed first prior to the start of pumping in the aquifer; unsaturated flow behaviour will be considered later after the plume forms in the aquifer and pumping begins. The boundary conditions consisted of a prescribed recharge flux equal to 4 lo- ’ m day- ’ at the aquitard surface, thus causing

R. Therrien, E.A. Sudicb / Journal of Contuminant Hydrology 23 (1996) I-44 35

the water table to be located slightly above the upper surface of the domain. Hydraulic head values equal to 9.05 and 8.85 m were assigned over the left-hand (x = 0.0 m) and right-hand (X = 90.0 m) sides of the aquifer, respectively. All the other boundaries are assumed to be impermeable. This combination of boundary conditions produces downward vertical flow through the aquitard and left-to-right horizontal flow in the aquifer.

Assigning recharge in a low-permeability medium represents a major problem from a numerical modeling point of view. For the case examined here, the top surface of the domain consists of clay in which fractures are present. If the medium is fully-saturated, any recharge that occurs should preferentially flow into the fractures. To allow for this behaviour, we use what will be referred to as redistribution elements on the top surface of the domain (R. Therrien and E.A. Sudicky, in prep.). These redistribution elements are two-dimensional horizontal elements which are handled similarly to fractures in the numerical code. It is assumed that these elements possess a sufficiently high hydraulic conductivity such that the fluid entering as recharge redistributes itself by flowing laterally on the surface in regions where the vertical hydraulic conductivity is low. A hydraulic conductivity value equal to 1.25 lop5 m s-’ was assigned to these redistribution elements and their vertical thickness equalled 0.05 m. This combination of hydraulic conductivity and thickness produces a transmissivity that allows optimal redistribution for the system at hand. It was found that a further increase in the transmissivity value of the redistribution elements did not affect the results. From a physical point of view, these redistribution elements can be viewed as either a zone at the top of the aquitard where fracturing is more intense and where the hydraulic conductivity of the medium will be high, or they could be viewed as a mechanism to simulate overland-type flow.

A total of 419 ORTHOMIN iterations were necessary to attain convergence of the flow solution, using a relative convergence tolerance equal to 1 10p7. The total CPU time required to compute the steady-state flow solution on an IBM” RS/6000 Model 590 was 3.25 min, including input, matrix assembly, matrix solution and output. A 7-point finite-difference spatial discretization was used for the flow simulation.

The steady-state hydraulic head distribution along a vertical cross-section parallel to the x-axis and located at y = 45 m is shown in Fig. 10. The hydraulic head pattern is seen to be irregular in the aquitard due to the presence of the fractures. The same type of erratic hydraulic head pattern was found by Harrison et al. (1992) who performed

10.4

Fig. 10. Steady-state hydraulic heads along a vertical cross-section located at .v = 45 m (bottom contour = 8.85 m, interval = 0.5 m).

Fig. I I. Relative concentration imsurface for c’/ c’,) equal to 0.005 after 30 yr of transport

steady-state flow simulations in a two-dimensional fractured aquitard-aquifer system. It can also be seen that groundwater will generally flow from the fractures to the matrix except in the case of those fractures that extend downward to intersect the more permeable aquifer.

Contaminant migration from the waste zone under saturated conditions will now be considered. The initial concentration was assigned a value equal to c, in the waste zone and 0.0 elsewhere. A first-type boundary condition consisting of c = 0.0 was imposed at the inflow (i.e. left) end of the aquifer in order to represent the influx of uncontaminated groundwater. A third-type boundary condition is used at the top of the system where it is assumed that the concentration of the contaminant in the recharging water is 0.0. A zero dispersive flux was assigned to all the other boundaries. The transport simulation was performed up to a target time value equal to 30 yr by using the standard time-marching Galerkin finite-element scheme. As was the case for the steady-state flow problem, the finite-difference option was also used for the solute transport simulation. A Crank- Nicolson discretization and adaptive time stepping was used to march the solution through time.

A three-dimensional isosurface of the contaminant plume after 30 yr of transport is shown in Fig. 11. The isosurface corresponds to a relative concentration value (c/c<,> equal to 0.005. They gray plane shown in Fig. 11 (and Fig. 12) represents the fractured aquitard-aquifer interface. It can be seen that the solute has migrated down the fractures and has formed a contaminant plume that extends laterally beyond the outflow boundary in the aquifer. Although the relative concentrations in the aquifer are low, absolute concentrations could greatly exceed the drinking water standards for many organic and inorganic contaminants, depending on the value of c’,, in the source. The erratic three-dimensional nature of the plume is particularly evident at the aquitard-aquifer interface. The shape of the plume at this interface bears little resemblance to a plume that would be formed in an aquifer overlain by a non-fractured aquitard. Also, in the aquitard below the contaminant source zone, the presence of sparse vertical fractures has led to discontinuous zones of contamination that would be difficult to interpret through a field monitoring program unless information was gathered concerning the pattern and frequency of fractures in the aquitard. Even if this information was at hand, it is clear that a prediction based on a simplified two-dimensional model would be inappropriate.


Fig. 12. Location of the water table after 500 days of pumping.

For the results shown in Fig. 11 at 30 yr, only N 0.01% of the total mass stored in the domain is contained in the fractures. The remainder of the contaminant is stored in the source zone, aquitard matrix in zones adjacent to the fractures, and in the aquifer. The consequence of having a large amount of contaminant stored in the low-permeability, low-porosity aquitard matrix on account of matrix diffusion is that pump-and-treat remediation of the aquifer is likely to be problematic.

The total CPU time taken to perform the transport simulation on an IBM” RS/6000 Model 590 was 8.1 min. Variable time stepping was used and a total of 32 time steps were necessary to attain the target time value of 30 yr. The criterion used to select the time-step size was constrained by a maximum allowable change in relative concentration equal to 0.1 during any particular time step. This allowed the use of larger time-step sizes in the later stages of the simulation when matrix diffusion tended to smooth out any early-time plume irregularities caused by the presence of fractures.

The transient response of the system under variably-saturated conditions will be examined next, where the recharge at the top is decreased to lop5 m day-’ and a pumping well that fully penetrates the aquifer is located at x = 4.5 m and y = 45 m. Except for the reduction in the recharge rate, the boundary conditions for the flow problem are identical to those used to compute the steady-state saturated hydraulic heads described previously. The boundary conditions for transport are also the same as those used for the fully-saturated simulation. The initial head distribution and solute concentrations were specified by using the values calculated at a time equal to 10,000 days under fully-saturated flow conditions. The pumping rate in the well was specified to equal 175 m3 day-‘. The groundwater flux distribution at nodes along the well screen was evaluated at each time step by the iteration method of Cooley (1971). It was found that there was no need to introduce an extra iteration level to compute the nodal flux distribution along the well screen, because the updating of hydraulic heads at the well nodes was performed prior to each Newton iteration.

The simulation was performed until a time equal to 500 days was attained after the start of pumping. The total CPU time on the IBM” RS/6000 Model 590 for the coupled transient flow and transport problems was 6.41 h. A total of 52 variably-spaced time steps were required to arrive at the target time of 500 days. On average. four Newton iterations were required during each time step. The fluid and solute mass-balance error never exceeded t_ lo-“% and +O.l%, respectively.

A notable feature of the simulation results is the location of the water table after 500 days which is shown in Fig. 12. The reader is reminded that prior to pumping, the system was fully saturated and that the water table was located near the top of the aquitard. It can be seen that the combined effect of pumping and the reduced recharge rate has caused desaturation of the upper portion of the fractures that intersect the top of the aquitard, with greater perturbations in the water table occurring near those fractures that fully-penetrate the aquitard. This enhanced decline in the water table is due to the drawdown in the aquifer from pumping which propagates along the fully-penetrating fractures. The desaturation occurs first in the fractures and then extends laterally to the surrounding porous matrix. Because the fully-penetrating fractures desaturate, they become barriers to water flow and are no longer acting as preferential recharge features. The applied recharge is therefore redistributed to the fractures intersecting the surface which in turns affects the movement of dissolved solute in this portion of the domain.

The flux-averaged concentration at nodes along the well screen was computed and the results are shown in Fig. 13 for the first 200 days of pumping. Because contamina-

50.0 100.0 150.0 Time (day)

Fig. 13. Flux-averaged concentration at the pumping well.


tion existed at the well location prior to pumping, the concentration in the effluent is initially high as the well captures parts of the plume in the aquifer; however, the concentration rapidly decreases from its initial peak value until it reaches an almost constant value at a time equal to N 100 days. The well concentration in the effluent remains nearly constant until the final simulation time of 500 days. Because of slow diffusion of solute from the matrix to the fractures in the aquitard, it is to be expected that the concentration in the discharge of the pumping well will exhibit an exceedingly long tail should pumping continue. The strongly-tailed results therefore raises serious concerns about the utility of pump-and-treat technologies implemented for the purpose of remediating aquifers receiving contamination from an overlying fractured aquitard.

The hydraulic response of the system, combined with the erratic behaviour of the water table, demonstrates that variably-saturated flow and solute transport in fractured porous media is a highly complex process. Indeed, it is likely that even a dense array of observation wells located in a fractured medium that are monitored during periods of changing recharge conditions will yield seemingly anomalous data. Rigorous, physically-based mathematical simulation tools such as that developed here will therefore be of much benefit for unravelling at least some of this complexity.

6. Conclusions

A comprehensive three-dimensional numerical model has been developed for the rigorous solution of variably-saturated groundwater flow and solute transport problems in complex discretely-fractured porous media. Because of the large grids that are expected when solving field-scale problems and because of the high contrast in material properties between the fractures and the matrix, the numerical techniques were carefully chosen to achieve both efficiency and robustness. The control volume finite-element technique has been shown to be well-suited for the solution of the variably-saturated groundwater flow equation in discretely-fractured porous media. It also enables a highly efficient implementation of the robust Newton-Raphson linearization technique which is crucial for handling situations where hydraulic properties vary by many orders of magnitude. For large problems, use of the influence coefficient approach with the option to switch from finite-element to finite-difference type interconnectivity is highly advan- tageous to reduce computer memory requirements and increase execution speed. Up- stream weighting of the relative permeability, the exact treatment of the saturation term and variable time stepping represent other features that further enhance solution efficiency and robustness. The solute transport equation was solved by using a standard time-marching Galerkin method. Advection, dispersion and diffusion are accounted for in both the fractures and the porous matrix.

Test problems involving two-dimensional infiltration and advective-dispersive mass transport in a fractured porous matrix that was initially unsaturated demonstrated that the complex interactions caused by contrasts between the saturation, relative permeability and pressure head relationships for the fractures and the matrix will control the redistribution of water in the system and will thus strongly affect the shape of a contaminant plume. Depending on the nature of these relationships, groundwater flow

and solute transport can be controlled by either the porous matrix or the fractures, thus illustrating the necessity for an explicit and coupled description of the relevant processes occurring in both the fractures and the matrix.

A field-scale simulation of contaminant transport through a fractured aquitard and into an underlying aquifer illustrated the importance of the three-dimensional nature of fracture interconnectivity. This fracture interconnectivity will create erratic hydraulic head patterns in the aquitard and will produce an irregularly-shaped contaminant plume in the underlying aquifer. The interpretation of data obtained by monitoring the aquitard and the underlying aquifer for hydraulic head and solute concentration will therefore be very problematic if the nature of fracturing is unknown.

It was also shown that the water-table configuration in fractured porous media is likely to be a complex dynamic surface that changes with time. This raises questions concerning the applicability of conventional field technologies for recharge estimation based on water-table responses and natural isotope distributions that evolved from studies of nonfractured geologic materials. It is also suggested that a standard pump- and-treat methodology for the purpose of remediating an aquifer receiving contamination from an overlying fractured aquitard may be of limited utility due to the extreme tailing of contaminant arrival-time curves at the well caused by the slow diffusion of contaminants from the matrix to the fractures in the aquitard.

Several assumptions were made in order to solve the governing equations. The implications of some of these assumptions, for example neglecting the flow of the air phase and the deformation of the fracture-matrix system, are still unclear. Extension of the present work through a relaxation of these assumptions would further enhance our understanding of variably-saturated flow and solute transport processes in discretely- fractured porous media. This work is in progress and will be reported on later.

Acknowledgements

Funding for this research was provided by an NSERC operating grant and a grant from the Ontario Ministry of the Environment and Energy awarded to E.A.S. Additional funding was in the form of an NSERC postgraduate scholarship and a scholarship from Le Fonds pour la Formation de Chercheurs et 1’Aide a la Recherche du Quebec awarded to R.T. We acknowledge funding originating from the University of Waterloo Consor- tium Solvents-in-Groundwater Research Program which is sponsored by: The Boeing Company, Ciba-Geigy Corporation, General Electric Co., Eastman Kodak Co., Laidlaw Environmental Systems Ltd., Mitre Corporation, PPG Industries Inc., NSERC as well as the University Research Incentive Fund of Ontario. The assistance of S. Shikaze in producing the colour output has been greatly appreciated.

Appendix A

This Appendix presents the influence coefficient matrices for a seven-point finite-difference representation. The reader is referred to Huyakorn et al. (1986, 1987) for the

R. Therrien, E.A. Sudicky / Journal of Contaminant Hydrology 23 (1996) I-44 41

influence coefficient matrices required by the finite-element technique in three dimensions using rectangular prisms elements.

The following integrals are defined over a three-dimensional rectangular prism element having a volume equal to V’:

dV’ (A-1)

(A-2)

CTJ = j,eN,N,dV’ (A-3)

If 1 represents the mean dimension of the element along the x,-axis, m represents the mean dimension of the element along the x,-axis and H represents the mean dimension of the element along the x,-axis, then the use of the influence coefficient technique to evaluate the above integrals leads to the following results:

(A-4)

(A-5)

where the modified 8 X 8 elemental matrices representing a seven-point finite-difference template are given by:

(A-7)

(A-8)

(A-9)

(A-10)

(A-l 1)

(A-12)

(A-13)

42

and

I

Lk - -1 u -

/ 0 0

I

a“’ zzz L 0 0

-I

u:.- = b: = c,

-I 0 I 0 0 I 0 -1

0 0 I -1

-I I 0 0

-I 0 I 0 0 I 0 -I

0 0 -I -I

I I 0 0

0 0 0 0 1 0 0 I I

0 0

-I I !

-I 0 0 1 1

0 0 I

-1 I

-1 0 0 I

(A-14)

(A-15)

(~-16)

(A-17)

(A-18)

(A-19)

Here, y1 represents a 4 X 4 null matrix.

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