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Advances in Ware-r Resources, Vol. 18, No. 4, pp. 191-201, 1995 Copyright 0 1995 E1smie.r Science Limited Printed in Great Britain. AU rights reserved

0309-1708(95)00013-5 0309-1708/95$09.50 + 0.00

Darcy velocity computations in the finite element method for multidimensional randomly

heterogeneous porous media

Rajesh Srivastava & Mark L. Brusseau Soil and Water Science Department, University of Arizona, Tucson, Arizona 85721, USA

(Received 26 July 1994; accepted 15 March 1995)

For numerical modeling of transport of contaminants through porous media, an accurate determination of the velocity field is a prerequisite. Some previous schemes of obtaining the Darcian velocity field through numerical solution of the flow equation result in a physically inconsistent velocity distribution in a heterogeneous medium. A scheme that is consistent with the physics of velocity variation near material interfaces is examined and compared with previous schemes. Numerical simulations are used to demonstrate the capability and accuracy of the proposed scheme for randomly heterogeneous porous media.

Kev work Darcy velocity, heterogeneous porous media, numerical simulation, material interface. _

INTRODUCTION

Efficient and accurate evaluation of the flow velocity is a prerequisite for any nutnerical model of multi- dimensional transport through porous media. The requirement of efficiency becomes more desirable for simulations of a randomly heterogeneous medium, as typically hundreds of reabzations are required to obtain a meaningful description of the transport process. An accurate representation of the velocity field is especially critical to the performance of numerical models based on the particle tracking technique.2P6P139’6 With the recognition of the complexity of transport and in the absence of usable analytical expressions, numerical modeling has become the method of choice for field- scale transport simulation. The Finite Difference Method (FDM) and the Finite Element Method (FEM) are the two most commonly used numerical techniques for groundwater flow and transport. Tradi- tionally, the numerical methods first solve for the hydraulic head (potential) at the grid points and then obtain the velocity by numerical differentiation. This results in a loss of one order of accuracy in the velocity computations and also gives rise to discontinuities in velocity components at the element boundaries. Goode’ presents an excellent dkussion of this aspect for the FDM and Cordes and Kinzelbach3 provide an overview for the FEM.

At the interface of two different materials, the velocity component normal to the interface should be contin- uous while the tangential component will, in general, have a discontinuity. These conditions are not satisfied in the existing numerical methods. For example, when using numerical differentiation of the (linear) head values (referred to as Scheme 1 in this paper), the x-component of the velocity would be constant in the x-direction and linear in y- and z-directions and, in general, would have discontinuities at all the material interfaces. Using a separate application of the FEM to the velocity components (Yeh15) leads to velocity components which are linear within an element and continuous on the element boundaries (Scheme 2) but requires more computational effort compared to Scheme 1. It is clear that Scheme 1 does not preserve the continuity normal to the material interface and Scheme 2 does not preserve the discontinuity tangential to the interface direction. The mixed FEM2,” attempts to overcome the loss of accuracy in velocity computation by simultaneously solving for the head and the velocity components. The computational effort however is much larger than that for either Scheme 1 or Scheme 2, especially for unsaturated flow simulations where iterations need to be performed over the whole system of equations, with three or four degrees of freedom (for 2- and 3-dimensional problems, respectively) at nodal points. Also, the required continuity and discontinuity

191

192 R. Srivastava, M. L. Brusseau

l’rl’ AY

111, 4 AX W

Fig. 1. Element size and local node numbering of the finite elements.

conditions for the velocity components are satisfied only for the lowest-order formulation. The higher-order formulations do not preserve the velocity component discontinuity tangential to the interface.

In this paper, we examine a scheme (Scheme 3) which combines Scheme 2 and the mixed FEM in a way that maintains the advantage of both techniques. This scheme preserves the interface conditions in situations where all the element boundaries are material interfaces. For the element boundaries that are not material interfaces, the continuity of the velocity component tangential to the boundary is not preserved. Also, for element boundaries which are not parallel to a coordinate axis, the required continuity conditions are not met. These may not be serious limitations for simulations of randomly heterogeneous porous media, as typically the medium is conceptualized as consisting of blocks of different conductivity and all the element boundaries are material interfaces located parallel to the coordinate axes.

NUMERICAL SCHEMES

In all the schemes discussed above, the head distribution is obtained by applying the Galerkin technique (cf. Pinder & Gray’D) to the governing equation for flow. Linear shape functions are used for the approximation of head within an element. For this study, we consider only steady-state saturated flow conditions but all the schemes are equally applicable to variably saturated transient flow conditions with little or no modification. Also, for ease of presentation, we consider only two- dimensional isotropic flow conditions; the extension to three dimensional anisotropic flow is straightforward.

The governing equation for steady-state saturated flow in porous media in absence of any source/sink is

V.(K.VH) = 0 (1)

where K is the hydraulic conductivity tensor (LT-‘) and His the hydraulic head (L). Application of the Galerkin FEM to eqn (1) leads to the following system of linear

algebraic equations

[AI{W = IQ1 (2)

where the coefficient matrix A and the flux vector Q are given by

(3)

QI = Jr N,qbdI’

in which R is the flow domain, r is the boundary with qb being the specified normal flux on the boundary and N1 are the shape functions. This set of equations is solved using the Jacobi Conjugate Gradient iterative solver” to obtain the nodal values of hydraulic head. The various schemes for the computation of velocity components are then applied with these head values as described below.

Scheme 1

In this scheme, numerical differentiation of the hydraulic head values within each element is performed to obtain the velocity components. Since the head is assumed to vary linearly within each element, the x-component of the velocity will be constant along the x-direction and linear along the y-direction and is given by

q,=-+K @Y-Y)(HI - H2)+_@4-H3)

Ay Ax

(4) The element size and node numbering are presented in Fig. 1. A similar expression for the y-component can be written, indicating that it is linear along the x-direction and constant in the y-direction.

Scheme 2

In this scheme, a separate application of the Galerkin FEM to Darcy’s law is utilized to obtain continuous values of the velocity components. Using the linear shape functions to describe the intraelement variation of the velocity components and applying the Galerkin technique, we obtain

IA41 {rli

in which

for i=x,y (5)

A qZJ =

Thus, a set of N linear simultaneous equations has to be solved for each velocity component. The resulting components are linear in both x- and y-directions and are continuous at all element boundaries.

l3arcy velocity computations in randomly heterogeneous porous media 193

Fig. 2. Nodal point locations for the x-component of velocity for Scheme 3.

Scheme 3

In this scheme, we describe the intraelement variation of the velocity components in such a way that the component in any direction is linear and continuous along that particular di.rection and linear with jump discontinuity in other directions. Thus, the x-component of the velocity will be linear and continuous in the x-direction and linear with jump discontinuities at the element boundaries in the y- (and z-) direction. This necessitates a different set of nodes than those used for the heads. Figure 2 shows the nodal point locations for the x-component of the velocity with double nodes placed along all the rows inside the domain. Application of the Galerkin FEM in this case leads to

[A’l{d = {&I ~A’l~qJ = PJ

(7)

where the coefficient matrix and the flux vector are given as

4 2 1 2

[A’] = c i ” 4 2 ’ 1 e

A;?

HI H2

H3

H4

HI H2

H3

H4

the e under the summation implying a sum over all elements.

The assumption that the x-component of velocity has jump discontinuity in the y-direction leads to uncoupled symmetric banded systems of equations for each row, with a semi-bandwidth of four. These systems can be solved much more efficiently than the set of N simultaneous equations obtained in Scheme 2, especially for a three-dimensional problem. Scheme 1 will, of course, require the least CPU time as all computations are performed at the element level and no simultaneous equations have to be solved.

Another commonly used scheme, the mixed FEM (cf. Allen et al.‘), is not considered in this study as it is very CPU intensive and the possible increase in accuracy may not be worth the additional effort. Also, for variably saturated transient flow problems, where iterations are needed at each time step due to the dependence of conductivity on the pressure head, the mixed FEM is extremely inefficient as it involves iterations over all variables (the hydraulic head and the velocity components). An analysis of the performance of the mixed FEM for randomly heterogeneous media was performed by Durlofsky.4

An additional complication of the division of the porous medium into hypothetical conductivity blocks is that all the corner points of these blocks are points of singularity where the velocity becomes infinite. The two prevalent approaches towards the problem of singularity are: (i) use of special elements near the points of singularity, as is commonly done in solid mechanics for problems involving crack formation and in the boundary element method (cf. Li;8 Lafe et aL9) and (ii) use of the argument that Darcy’s law will not be valid close to the singular points as the velocity becomes very large and hence capturing the singularity in the numerical model is not essential. The second approach is utilized in the present study as there is some evidence that singular functions are not absolutely necessary for problems similar to those under con- sideration here. l4

An alternative formulation of Scheme 3 can be done similar to the lowest-order Mixed Finite Element method by placing the nodes for the x-component midway between the rows. This amounts to assuming that the x-component is linear and continuous along the x-direction and constant and discontinuous along the y- (and z-) direction. Application of the Galerkin FEM in this case leads to

[A’lk) = {&I ~A’lkQ = {ByI

(9)

where the coefficient matrix and the flux vector are

194 R. Srivastava, M. L. Brusseau

NoFlow

24 6

24 6

b-1

0 II

I

No Flow

Fig. 3. Flow geometry and boundary conditions for the first set of numerical experiments.

given as

[A’] = xy : : e [ 1

This leads to a tridiagonal system of equations for each row which can be efficiently solved. Also, by not placing the nodes at the corners, the singularity in velocity components is avoided. This scheme will be less accurate than Scheme 3 but will take less computational time. The results of this procedure are not discussed in the subsequent sections since its performance, in terms of both accuracy and execution time, lies between those of Schemes 1 and 3.

NUMERICAL EXPERIMENTS

To compare the accuracy of different schemes, we simulate two flow situations. The first one (Fig. 3) involves flow around a square block of different conductivity in an otherwise homogeneous medium. The boundary conditions are no flow across the top and bottom boundaries, specified head at the right boundary and specified flux at the left boundary. The conductivity of the medium is fixed at 1 unit and that of the block is taken as 0.1 units for the less-conductive block case and 10.0 units for the more-conductive block case. The size of the block is taken as 1 unit x 1 unit and the domain size is taken as 11 units x 11 units. Six different finite element grids are used with 1, 2, 3,4, 20 and 40 elements per unit length in each direction. The two finest mesh

results were compared and were found to be almost identical to each other and for all the schemes. There- fore, these are assumed to represent the ‘true’ solution and the other results are compared with them to ascertain the accuracy. The two sources of approximation in the velocity components are (i) the approximation of linear variation of head within an element and (ii) the approximation for the velocity component itself. Since the same head distribution is used for all the schemes, the difference in accuracy of different schemes for a particular discretization will be due only to the velocity approximation in that scheme. Following the usual practice, the discretiz- ation for the hydraulic head and the velocity compo- nents was kept the same although some previous investigators (e.g. Zhang et al.‘*) have used an analytical solution for the hydraulic head to compare different schemes.

The velocity components are compared along lines l-l through 6-6 (Fig. 3) which are 3 units long. Lines l- 1 and 2-2 are a natural choice due to the symmetry of the geometry, lines 3-3 and 4-4 are chosen close to the material interfaces to analyze the effects of the corner singularity and lines 5-5 and 6-6 are located to ascertain the effects of not preserving the continuity of tangential velocity components in Scheme 3 at element boundaries that are not material interfaces. For odd numbers of elements per unit length, these lines (with the exception of 3-3 and 4-4) pass through the center of finite elements and the appropriate interpolation scheme is used to obtain the values of the velocity components for each scheme. For even numbers of elements per unit length, all the lines (again except 3-3 and 4-4) coincide with element boundaries. In this case, the lines are assumed to approach the element from below or from the left and the values for that element are used. The lines 3-3 and 4-4 are located slightly away from the material interfaces (at the center of the second row or column for the finest grid) to reduce the effect of the corner singularity on the results.

The second flow situation (boundary conditions similar to Fig. 3) involves the flow through a randomly heterogeneous media. The hydraulic conductivity dis- tribution is generated by using the Turning Bands Method’ assuming a mean of natural log conductivity equal to 0 and variances of 1 (low variance case) and 3 (high variance case). The correlation lengths in both horizontal and vertical directions are taken as 5 units and the domain size is taken as 11 correlation lengths in each direction. The mesh for generation of the con- ductivity values is chosen with 5 blocks/correlation length, giving rise to 55 unit-length conductivity blocks in each direction. Each block is then divided into 1 x 1, 2 x 2, 3 x 3, 4 x 4 and 15 x 15 finite elements respectively, again assuming that the finest grid solution represents the ‘actual’ values (computer memory limita- tions dictated the choice of the finest grid). The results

Marcy velocity computations in randomly heterogeneous porous media 195

6.5 I I I

4.54- I I 5’ 6 7

Y

5.0 r--q

4.o4wj

I I 6 7

Y Fig. 4. Hydraulic head distribution around the less-conductive block along lines (a) l-l (b) 4-4 and (c) 6-6. The legend represents the number of elements per unit length in each direction. (FG indicates the finest grid with 40 elements per

unit length.)

are compared along the set of lines l-l, 2-2, 3-3 and 4-4 as for the previous case (with the line lengths being 3 conductivity blocks) for a single realization of the conductivity field with the assump- tion that the relative accuracies of various schemes will be similar for the ‘average’ over a number of realizations.

- (a) FO- .

0.0: I I I 5 6 7

X

-0.4: I 5 6 7

X Fit. 5. Variation of the x-component of velocity along the line l-l for the less-conductive block for (a) one element and (b) two elements per unit length. The legend represents the scheme

number.

RESULTS AND DISCUSSION

The head distribution along the line l-l for the flow through the homogeneous medium with a less con- ductive block is shown in Fig. 4(a), from which it is observed that the coarsest discretization of a single element per unit length is not very accurate, especially near the interface (x = 5 and x = 6). Using as little as 2 elements per unit length, however, we are able to obtain a considerably better head distribution. Similar results are obtained for lines 2-2,3-3 and 5-5 (not shown) but the numerical results were more in error along lines 4-4 and 6-6 (Fig. 4(b) and (c), respectively).

Figure 5(a) shows the variation of the x-component of velocity along the line 1- 1 for the coarsest discretization for the less conductive block case. As epected, the linear variation from Scheme 3 is closer to the actual solution than the piecewise constant variation of Scheme 1. Scheme 2, however, is seen to be much worse and is even less accurate for the finer discretization of two elements per unit length (Fig. 5(b)). All such schemes which obtain a single nodal value of velocity based upon the average conductivity of surrounding elements (e.g.

196 R. Srivastava. M. L. Brusseau

2.4

2.2

2.0

9, 1.8

1.8

1.4

1.2

1.04 I I 5 8 7

X Fig. 6. Variation of the x-component of velocity along the line l-l for the more-conductive block for two elements per unit

length.

Zhang et al. ‘*) are inherently inadequate to represent the real velocity distribution in heterogeneous media. This can easily be seen in the simple case of saturated unit gradient flow through layered soils in which the nodes at the interface should have different values in the two layers (equal to the saturated conductivity of the respective material). Thus the schemes which obtain a

0.0; I I I

5 8 7

Y

2.5) I I , .

2 _ . _ _ _ _ _

2.0 - 3 --VW ____.I. FG -

0.0: I I 5 6 7

Y Fig. 7. Variation of the x-component of velocity along the line

2-2 for (a) less- and (b) more-conductive block.

0.8

0.6 a

0.4

X

X

Fig. 8. Variation of the (a) x-component and (b) y-component of velocity along the line 3-3 for the less-conductive block.

nodal velocity value tend to cause ‘numerical accelera- tion’ of flow in low conductivity areas and a corre- sponding deceleration in the high conductivity areas. The implications for simulation of an aquifer remedia- tion event, e.g. by pump and treat, using such schemes are then quite obvious in that the remediation time may be severely underestimated as it is usually governed by the low conductivity areas. Also, the diffusive process, which is an important factor in determining the remediation time,12 may be erroneously deemed to be insignificant. Scheme 2, although more accurate for homogeneous or continually varying conductivity fields (such as those considered by Zhang et al.18), should not be used for randomly heterogeneous fields. Subsequent results clearly show the inability of Scheme 2 to obtain accurate velocity components in heterogeneous fields. Most of the discussion from this point onward is therefore limited to a comparison of Schemes 1 and 3.

Figure 6 shows the velocity distribution obtained along the same line as reported in Fig. 5 (l-l) but for the case when the heterogeneous block is 10 times more conductive than the surrounding medium. Again, Scheme 3 is seen to be better but none of the schemes reproduce accurately the parabolic variation of velocity within the block.

Darcy velocity computations in randomly heterogeneous porous media

o.04-’ I

5 6

Y

Y Fig. 9. Variation of the (a) x-component and (b) y-component of velocity along the line 44 for the less-conductive block.

Figure 7 (a) and (b) show the velocity distributions obtained along line 2-2 from the coarsest discretization with the less-conductive and the more-conductive block, respectively. Particularly noticeable is the almost oppo- site trend of variation obtained from Scheme 2. The other two schemes provide more representative approx- imations to the fine grid solution with neither scheme being superior.

Figure 8 (a) and (b) depict the variation of the x-component and y-component of velocity, respectively, along the line 3-3 for the flow with the less-conductive block. The singularity near the corner (x = 5 and x = 6) has a pronounced effect on the velocity distribution. As the corner is approached, the velocity in the medium approaches infinity and that in the block approaches minus infinity. This singularity is manifested in a discontinuity in the x-component along the direc- tion. For eample, the x-component is about O-8 in the medium and about O-2 in the block. As a result, Scheme 3 (which assumes a continuity of the x-component along the x-direction) is not able to obtain accurate results (Fig. 8(a)). Scheme 1, on the other hand, is very accurate as it arssumes the components to be discontinuous at all boundaries. As previously stated,

1.2

1.1

1.0

I I 4 5 6 7

X Fig. 10. Variation of the x-component of velocity along the line 5-5 for the less-conductive block for (a) one element and (b)

four elements per unit length in each direction.

the corner singularity (and the resulting discontinuity in the velocity component) is an artifact of assuming the validity of Darcy’s law at very high velocities and in most cases Scheme 3 should work satisfactorily. Also, the conductivity contrast between two adjacent elements may not be as high as that considered in this example and the severity of the singularity may be quite small.

0.2 t _.." .? ':,. ., , ,,,,..,,.,,..,,,., ,..' I 2 .______ 3 ---

FQ - 0.1 ,...... /“. %> F \ ,..‘ ,..’ ‘..

qy 0.0

X Fig. 11. Variation of the y-component of velocity along the line 5-5 for the less-conductive block. (Schemes 2 and 3 have

identical results.)

198 R. Srivastava, M. L. Brusseau

3 ---

____.____-_---

.’ ‘. t I I Y

4 5 6 7

Y

Y

Fig. 12. Variation of the (a) x-component and (b) y-component of velocity along the line 6-6 for the less-conductive block.

These aspects will be further discussed for the second set of simulations for a randomly heterogeneous porous medium.

Figure 9 (a) and (b) are similar to Fig. 8, but along the line 4-4. In this case, Scheme 3 does consider the discontinuity in the x-component and is therefore quite accurate. Scheme 1 is also seen to be of comparable accuracy. For the y-component, however, none of these schemes is satisfactory. One of the reasons for this is the inaccuracy of the head distribution (Fig. 4(b)) and the other reason is the corner singularity.

Figure 10 (a) and (b) show the variation of the x-component of velocity for 1 and 4 elements per unit length, respectively, for the case of the less-conductive block. The superiority of Scheme 3 over the other two schemes is clearly demonstrated. Also worth noting is the fact that even for the finer discretization, Scheme 2 produces large errors, specially within the block and the elements adjacent to it (Fig. 10(b)). Figure 11 shows the variation of the y-component along line 5-5. The assumed discontinuity of the y-component across the element boundaries for Scheme 3 is seen to be incon- sequential. The figure shows a continuous profile for Scheme 3, which is closer to the fine grid solution. Along

(a) 2726

1

27 26 26 X

26.5 -

, . . , , 2 - --_-__. .

(b) FQ -

26.b 27 26 26 Y

Fig. 13. Hydraulic head distribution for the random con- ductivity field with low variance along lines (a) 1 - 1 and (b) 2- 2. The legend represents the number of elements per unit

length in each direction.

line 6-6, however, the assumed discontinuity of x-component has a pronounced effect on the results of Scheme 3 (Fig. 12(a)) and Scheme 1 is much closer to the fine grid solution. For the y-component, none of the schemes is satisfactory (Fig. 12(b)). As mentioned previously, though, for a random field simulation all the element boundaries will be material interfaces and lines similar to 5-5 and 6-6, which pass through the same material, will not exist for the usual discretization of one element/conductivity block.

Next we describe the results for the randomly heterogeneous field. The head distribution along lines l-l and 2-2 are shown in Fig. 13 (a) and (b), respectively, for the low variance case. It is observed that even the coarsest discretization of one element/ conductivity block reproduces the head distribution quite accurately. In fact along all lines (3-3 and 4-4 not shown) the numerical results are parallel to the fine grid solution indicating that the gradient is almost the same and therefore the effect on velocity computations should be minimal.

Figure 14 (a) and (b) show the variation of the x-component of the velocity along line 1-l for the

llarcy velocity computations in randomly heterogeneous porous media 199

X

Fig. 14. Variation of the x-component of velocity along the line l-l for the random conductivity field for (a) low and (b) high

variance. The legend r’epresents the scheme number.

low-variance and high-variance case, respectively. The fine grid solutions have similar profiles with sharper variations for the high-variance case and similar trends are observed for the numerical schemes. For the sake of brevity, only the low-variance case is described in the rest of this section. Scheme 3 is much better than the other two schemes and Scheme 2 is surprisingly

Fig. 15. Variation of the y-component of velocity along the line l-l for the random conductivity field with low variance.

1.41 1 1 , , , . . , .

2 _._____ s--

‘. ‘. M-

‘. 1.0 -

% ‘\ 0.8 - ‘. ‘. ‘. .

0.8 -

, (a) I

‘-48 1 I I

27 28 29 Y

----I I

-0.08

cc, -0.08

-O.l& I 27 28 28

Y Fig. 16. Variation of the (a) x-component and (b) y-component of velocity along the line 2-2 for the random conductivity field

with low variance.

satisfactory. The latter fact may be due to the reduced contrast of conductivity between adjacent elements for this simulation as compared to the previous one. Figure 15 shotis the variation of the y-component of velocity along line l-l. The discontinuity at the element boundaries is clearly seen for both Schemes 1 and 3.

Figure 16 (a) and (b) show the variation of velocity components along line 2-2. Both Schemes 1 and 3 are quite accurate for the x-component, whereas Scheme 2 performs poorly. The profile obtained from Scheme 2 clearly shows the detrimental effect of assuming a continuous variation of the velocity component across the element boundaries where it is actually discontin- uous. For the y-component however both Schemes 2 and 3 are close to the fine grid solution (Fig. 16(b)).

Figure 17 (a) and (b) show the variation of the x-component along line 3-3 using one and four elements/conductivity block, respectively and Fig. 18 shows the variation of the y-component. For the finer discretization (Fig. 17(b)), both Schemes 1 and 3 are sticiently accurate but there is not much improvement in the accuracy of Scheme 2. The comer singularities appear to be less strong in this case compared to the first

200 R. Srivastava, M. L. Brusseau

.-----

___---- ____/---- , . I . . ,

* _. _ _ _ . .

3 ---_,

FQ- I

o3’6 27 26

X

1 .oo

0.90

% 0.60

0.70

6.W

1 __-- .____.-- _.--

_.-I

.’ ,.--

, 2 -. - _ -. _ 3 ---_

(b) FQ-

27 26

X

29

Fig. 17. Variation of the x-component of velocity along the line 3-3 for the random conductivity field with low variance for (a) one and (b) four elements per unit length in each direction.

set of simulations because of the reduced contrast in conductivities of adjacent elements. The fine grid profiles in these figures show a pronounced jump at the right interface (x=28) but a smoother variation at the left interface (x = 27). This may be due to the fact that for this particular realization of the random field, the conductivity contrast at the left interface is small compared to that at the right interface. Consequently,

0.1 , 1 . ..I . . . . . . . . . . . . . . . . . . . . . . .

0.0

% LL _,,...’

_o_, ,,,,((,,._,,..,............ ~~~--““’

, .I . ,.

* _. _ _ _ _,

s---- FQ-

26 27 26 29

X

Fig. 18. Variation of the y-component of velocity along the line 3-3 for the random conductivity field with low variance.

1.4. 1 . . 2 ___---m 2 w_-

‘\ FQ-

1.0 -

q* ??.

I ‘??S

I

27 26

Y

I I 26 27 26 29

Y

Fig. 19. Variation of the (a) x-component and (b) y-component of velocity along the line 4-4 for the random conductivity field

with low variance.

even for the coarsest discretization, Scheme 3 works better near the left interface. Figure 19 (a) and (b) show the variation of the x- and y-components, respectively, along line 4-4. Scheme 3 can be said to be marginally better than Scheme 1 along both lines 3-3 and 4-4.

SUMMARY

A method of obtaining a physically consistent velocity distribution in a randomly heterogeneous soil is exam- ined in detail and is compared to some often-used techniques. The performance of various schemes are compared under two flow situations. The results from the first flow situation provide some insight into the nature of velocity variations near interfaces of different materials. The second flow situation is a representative of the usual Monte Carlo technique of simulation of a randomly heterogeneous porous medium. The results for this situation confirm the applicability and accuracy of the proposed method. Both Schemes 1 and 3 work satisfactorily in most situations. Near a singular point in the presence of strong singularities, Scheme 1 performs

Darcy velocity computations in randomly heterogeneous porous media 201

better than Scheme 3 dule to its ability to reproduce the discontinuity in the velocity component normal to the interface. Similarly along lines which pass through the same material, Scheme :I gives better results due to its ability to preserve the continuity in the velocity component tangential t’o the interface. In almost all other domain locations, however, Scheme 3 seems to be better suited for computation of velocity components. For simulations of randomly heterogeneous porous media, where the conductivity contrast in adjacent elements is not very large and where all element boundaries are material interfaces, Scheme 3 is more accurate. Ultimately, the: choice of which scheme to use rests upon a balance of programming effort and CPU time versus the level of desired accuracy. An alternative formulation of Scheme 3 is also suggested which lies in between Schemes 1 and 3 in both these aspects. It should be mentioned that the ,accuracy criterion used in this study is based on the local accuracy of the velocity field. Inclusion of the global flow field and mass conservation criteria was not attempted but can be performed along similar lines.

ACKNOWLEDGEMENTS

This study is funded in part by the DOE Env. Rest./ Waste Management Program and in part by the NIEHS Superfund Program.

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