Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol22.Issue7.1999:

Numerical approximation of head and ¯ux

covariances in three dimensions using mixed ®nite

elements

Andrew I. James

*

_{& Wendy D. Graham}

Interdisciplinary Program in Hydrologic Sciences, University of Florida, Gainesville, FL 32611, USA

(Received 14 April 1998; revised 27 September 1998; accepted 8 October 1998)

A numerical method is developed for accurately approximating head and ¯ux covariances and cross-covariances in ®nite two- and three-dimensional domains using the mixed ®nite element method. The method is useful for determining head and ¯ux covariances for non-stationary ¯ow ®elds, for example those induced by injection or extraction wells, impermeable subsurface barriers, or non-stationary hydraulic conductivity ®elds. Because the numerical approximations to the ¯ux covariances are obtained directly from the solution to the coupled problem rather than having to di€erentiate head covariances, the approximations are in general more accurate than those obtained from conventional ®nite di€erence or ®nite element methods. Results for uniform ¯ow example problems are consistent with results from previously published ®nite domain analyses and demonstrate that head variances and covariances are quite sensitive to boundary conditions and the size of the bounded domain. Flux variances and covariances are less sensitive to boundary conditions and domain size. Results comparing approximations from lower-order Raviart±Thomas±Nedelec and higher order Brezzi±Douglas±Marini9 ®nite element spaces indicate that higher order element space improve the esti-mate of the ¯ux covariances, but do not signi®cantly a€ect the estiesti-mate of the head covariances. Ó _{1999 Elsevier Science Ltd. All rights reserved}

1 INTRODUCTION

Over the past two decades, stochastic methods have been widely used to describe the variability of ground-water systems. Of long standing interest is the determi-nation of head and velocity covariances, and head/ conductivity, head/velocity, and velocity/conductivity cross-covariances. These covariance functions have been determined analytically for two- and three-dimensional in®nite domains by Bakret al.,3Mizellet al.,20Graham and Mclaughlin,13 Rubin,27 Rubin and Dagan,30 and Zhang and Neumann.37,38 Osnes,24 Rubin and Da-gan28,29and Na€ and Vecchia22have used semi-analyt-ical methods to solve for head covariances in bounded domains. Osnes25used semi-analytical methods to solve for velocity covariances in two dimensional bounded

domains. Numerical solutions for head covariances and head-conductivity cross-covariances in bounded do-mains have also been reported by McLaughlin and Wood,18,19 Sun and Yeh,35 Li and McLaughlin,17 and Van Lent and Kitanidis.36Typically, when a numerical approach is taken, the covariances and cross-covari-ances involving velocity are obtained by numerically di€erentiating the covariance functions of head. This can lead to loss of accuracy in the resulting functions if head covariances change rapidly over small distances within the domain, as may occur if conductivity is highly variable, if complex boundary conditions are present, and/or source and sink terms are present. Sun and Yeh33 and Sun32point out that, in general, accurate numerical calculations of gradients of unknown functions in cou-pled inverse problems can be dicult.

In this paper we examine a method for numerically evaluating covariances and cross-covariances between log hydraulic conductivity, head, and Darcy ¯ux using a Printed in Great Britain. All rights reserved 0309-1708/99/$ ± see front matter PII: S 0 3 0 9 - 1 7 0 8 ( 9 8 ) 0 0 0 4 4 - X

*

Corresponding author.

mixed ®nite element method. Over the past three de-cades, mixed ®nite element methods have been applied to a wide range of problems.2,5±7,12,23,26When applied to groundwater ¯ow problems the mixed ®nite element method has the advantage of simultaneously approxi-mating both head and ¯ux, rather than having to obtain the ¯ux from di€erentiation of the head. This results in greater accuracy of the ¯ux approximations. This and other advantages in using a mixed ®nite element method for groundwater ¯ow problems are well document-ed.1,4,8,11,21In this paper we show that these advantages extend to the determination of covariances and cross-covariances involving ¯ux terms. The outline of this paper is as follows: In Section 2, we give a brief intro-duction to the mixed ®nite element method. In Sec-tion 3, we use an adjoint state method to determine the covariances and cross-covariances of head and Darcy ¯ux and discuss application of the mixed ®nite element method to the resulting equations. In Section 4 we present numerical results comparing the accuracy of two di€erent mixed ®nite element spaces with analytically derived results.

2 MIXED FINITE ELEMENT SPACES

Let X be a bounded domain in Rn (nˆ2 or 3) with boundary @X, with the boundary decomposed so that

@XˆCD[CN, with CD\CNˆ0. We consider the Dirichlet±Neumann problem for groundwater ¯ow:

ÿ r K_{x† rh…x†† ˆf; x2X …1†}

h…x† ˆgD; x2CD …2†

ÿK_{x† rh…x†† nˆgN; x2CN}3†

Here, K _{is the hydraulic conductivity tensor (an nn} symmetric positive de®nite matrix which is uniformly bounded below and above inX),hthe piezometric head, f a source term, gD is the value on the ®xed head (Dirichlet) boundaries, gNis the value on the ®xed ¯ow (Neumann) boundaries, and nis the outward unit nor-mal of @X. In what follows, we assume that gNˆ0 on

CN. In typical groundwater problems it is of interest to determine the Darcy ¯ux q, given by qˆ ÿK _rh. Rather than solve the above system eqns (1)±(3) and determine q from di€erentiating h, we introduceq into the above system and derive the equivalent ®rst order system of PDEs:1,26

q…x† ‡K_{x† rh…x† ˆ}0; x2X …4†

r q…x† ˆf; x2X …5†

h…x† ˆgD; x2CD …6†

q…x† nˆ0; x2CN …7†

The mixed ®nite element method can be applied to a system of the form of eqns (4)±(7). In this method, the

two variables (here the headhand the Darcy ¯uxq) are simultaneously approximated using two di€erent ap-proximating spaces. Let L2_{X† denote the space of} square integrable functions onX, and de®ne the space

H…div;X†by

H…div;X† v2ÿL2_X†n

jr v2L2_X†

H…div;X†consists of vector-valued functions that have divergences that lie inL2_{X†. Our approximating} func-tions come from two spacesWandVwhich are de®ned by:

Vˆfv2H…div;X†jvnˆ0 on CNg …8†

W ˆw2L2…X†jwis piecewise constant

…9†

Wis the space of all piecewise constant square integrable functions on the domain X, while V is the space of functions lying in H…div;X†whose components normal to the boundary on@Xare zero.7

We multiply eqns (4) and (5) by functionsv2Vand

w2W, respectively, and integrate overXto obtain the mixed weak form of the system,

Kÿ1_q;v

ÿ

‡ r… h;v† ˆ0; v2V

r q;w

… † ˆ…f;w†; w2W

where …;† represents the L2 _{inner product. Applying} Green's formula to the ®rst equation and using the fact that vnˆ0 on CN, the weak solution to the system eqns (4) and (5) is obtained by seeking a pair …q;h† 2 VW such that

Kÿ1_q;v

ÿ

ÿ…h;r v† ˆhgD;vniCD; v2V …10†

r q;w

… † ˆ…f;w†; w2W …11†

To discretize the domainX, we choose a gridT_roverX. For comparison purposes, we will use both the lowest-order Raviart±Thomas±Nedelec23,26 (RTN0) and Brez-zi±Douglas±Marini5 (BDM1) ®nite element spaces in our numerical examples, and we assume that they can be de®ned over the grid. Here, we assume that the grid will be a regular rectangular grid, with the grid spacing in the

x1, x2, and x3 directions denoted by d1, d2, and d3, re-spectively. We denote the two mixed ®nite element spaces by WrVRTNr 0W V andWrVBDMr 1 W

V for RTN0 and BDM1, respectively. These ®nite ele-ment subspaces are de®ned further in Appendix A. The discrete, mixed form of our problem is then to seek a pair…qr;hr† 2WrVrsuch that

Kÿ1_qr;vr

ÿ

ÿ…hr;r vr† ˆhgD;vrniCD; vr2Vr;

…12†

r qr;wr

… †;ˆ…f;wr†; wr2Wr …13†

whereVrrefers to eitherVRTNr 0orV BDM1

Lagrange multipliers restricted to the face of each ele-ment,KrL2…@E†for an elementE, this problem can be converted into a problem with a positive de®nite coef-®cient matrix.2,4,7,8 The problem can then be easily solved by conjugate-gradient type methods. The hy-bridized, discrete form of our problem is then to seek

…qr;hr;kr† 2VrWrKr such that Kÿ1_qr;vr

ÿ

ÿ…hr;r vr† ‡c…kr;v† ˆ0; vr2Vr

…14†

r qr;wr

… † ˆ…f;wr†; wr2Wr …15†

c…lr;qr† ˆ0; lr2Kr …16†

The inner productc…;†is de®ned as:7

c…lr;vr† ˆ X

E

Z

@E

lrvndr …17†

This results in a discrete system of equations:

Kÿ1_{AQÿBH‡CK ˆG …18†}

BTQˆ_{F …19†}

CTQˆ0 …20†

where the matricesA,BandCcan be evaluated once a particular approximating space is chosen. Note that this system is block diagonal and so the unknowns_{Q andH}

can be eliminated at the element level using static con-densation, resulting in an equivalent symmetric positive de®nite system (e.g., Brezzi and Fortin,7p. 181).

3 APPLICATION TO PERTURBATION EQUA-TIONS

We now apply this same mixed ®nite element method to equations describing the mean and covariance of head and Darcy ¯ux. To derive these equations, we expand the head, velocity, and log hydraulic conductivity into the sum of a mean and a zero-mean perturbation:

h…x† ˆH…x† ‡h0…x†

q…x† ˆQ…x† ‡q0…x†

ln K_{x† ˆY…x† ‡y}0_x†

and substitute these quantities into eqns (4)±(7) (from this point onward we assume for simplicity thatK_{is a} scalar function ofx). We assume that the source termf

and the Dirichlet boundary value gD are deterministic and known. Expanding the resulting equations and taking the expected value gives the equations for the mean valuesQandH:

Q…x† ‡eY…x†rH…x† ˆ0; x2X …21†

r Q…x† ˆf; x2X …22†

H…x† ˆgD; x2CD …23†

Q…x† nˆ0; x2CN …24†

Subtracting equations eqns (21)±(24) from the expanded equations16,19 gives the exact equations for the pertur-bation quantities:

q0… † ‡x eYy0… †rx H‡ rh0… † ‡x y0…x†rh0… †x

ˆ0;

x2X …25†

r q0… † ˆx 0; x2X …26†

h0… † ˆx 0; x2CD …27†

q0… † x nˆ0; x2CN …28†

Several di€erent methods may be used to derive the head and Darcy ¯ux covariances, as well as the cross-covariances between ¯ux, head, and log-conductivity. One method is to multiply eqns (25)±(28) by perturba-tions in log-conductivity, head, and velocity at inde-pendent points x0 _{and then take the expected value} (neglecting third and higher order products or pertur-bations) to obtain ®rst-order PDEs for Phh…x;x0†; Pyy…x;x0†; Pqq…x;x0†; Phy…x;x0†; Phq…x;x0†; and Pqy…x;x0†

(e.g., McLaughlin and Wood,19 Graham and McLaughlin13). Here, we use the adjoint state method to determine state sensitivities of the system, from which covariances are easily obtained. We follow closely the methods of Sun and Yeh33,35 and Sun32 for coupled inverse problems.

To ®nd the adjoint state for the ®rst-order pertur-bation ¯ow problem we start by taking the ®rst-order variations in eqns (25)±(28) (neglecting products of ®rst-order perturbation terms):32,33

dq…x† ‡eY_{‰dy…x†rH‡ rdh…x†Š ˆ0; x2X …29†}

r dq…x† ˆ0; x2X …30†

dh…x† ˆ0; x2CD …31†

dq…x† nˆ0; x2CN …32† where we have dropped the primed subscripts for per-turbation quantities.

We then multiply eqn (29) by an arbitrary vector-valued functionw1…x†and eqn (30) by a scalar function

w2…x†(assuming the requisite smoothness of both func-tions) and integrate over the domainX:

dq;w₁

… † ‡ÿeYdyrH;w₁

‡ÿeYrdh;w₁

ˆ0; x2X

r dq;w2

… † ˆ0 x2X

where we have suppressed the explicit dependence onx. Again,…;†is theL2_{X†inner product, de®ned for two} vector-valued functionsu,vas

…u;v† ˆ

Z

where summation overiis implied. Adjoint operations33

The last terms on the left hand sides of eqns (33) and (34) represent boundary integral terms resulting from the adjoint operations. These two terms can be elimi-nated by choosingw1 andw2 such that

w1…x† nˆ0; x2CN …35†

w₂…x† ˆ0; x2CD …36† Thus, eqns (33) and (34) become

w1;dq

Now we assume a general performance criterion of the form order variation inJ is32,33 (in inner product form):

dJˆ @R

Adding eqns (37), (38) and (40) gives

dJˆ @R

The ®rst two inner products on the right hand side of eqn (41) can be eliminated if we choose w₁ and w₂ to

Once w₁ and w₂ have been determined by solving these equations we can calculate the partial functional derivative ofJwith respect to the hydraulic conductivity in the jth _{grid block y}

j by calculating the last inner

product in eqn (41)32,33

@J

where the integration over Xj is performed over the

support ofw1j (i.e., where thej

th_{basis function ofw} 1 is non-zero). In order to perform the integration an ap-propriate function Rmust be chosen. By choosing Rˆ h…x†d…xÿxi† (where xi denotes the ith measurement

point), we can determine @J=@yjˆ@hi=@yj by solving

eqns (42) and (43) for w1…x†. The partial derivative

@J=@yjˆ@hi=@yj represents the sensitivities of a head

measurement at locationxito perturbations in hydraulic

conductivities at location xj. These partial derivatives

are the entries of the Jacobian matrix of sensitivities of head with respect to conductivity JH.31,33 Explicitly writing this out, we have

w1…x† ÿ rw2…x† ˆ0; x2X …45†

ÿeYr w₁…x† ˆd…xÿxi†; x2X …46†

with the boundary conditions:

w2…x† ˆ0; x2CD …47†

w1…x† nˆ0; x2CN …48† Using the solutionw1to the system of eqns (45)±(48) in eqn (44) we have:

where Q is obtained from the solution to the mean equation.

The system eqns (45)±(48) has the same form as the original equation, and can be solved using the same mixed ®nite element method as the mean equation. As discussed above, the advantage of this approach is that

w₁ can be directly (and accurately) approximated using this method, eliminating the need to di€erentiatew₂, the adjoint state of the head. Furthermore, ifQandw1 are approximated using the same basis functions, the inte-gral in eqn (49) is easily calculated (see Appendix A for details).

Alternatively, we can determine the sensitivities of ¯ux in a given direction with respect to hydraulic con-ductivities. If we choose Rˆqk…x†d…xÿxi†, whereqk is

the ¯ux component in thexk direction, we have

~

where nk is a unit vector in the xk direction. The tilde

to conductivity, given by @qki=@yj in exactly the same way as forJH using the solutionw~1…x†:

@qki

@yj ˆ

Z

Xj

ÿQ_w~

i1 …50†

Once the entries for the Jacobian matrices are deter-mined, the Jacobian matrix can be used to determine the ®rst-order covariance matrices for head and ¯ux:32

Phh…x;x0† ˆJHPyy…x;x0†JTH …51†

Phy…x;x0† ˆJHPyy…x;x0† …52†

Pqkqk…x;x 0_{† ˆJQ}

kPyy…x;x 0_†JT

Qk …53†

Pqky…x;x 0_{† ˆJQ}

kPyy…x;x

0_{† …54†}

4 NUMERICAL EXAMPLES

In this section we present some numerical examples to illustrate the performance of this method in determining head and velocity covariances in two di€erent domains. The ®rst is a square two-dimensional domain (Lx1 ˆLx2ˆ1, 1011011 elements) while the second

is a three dimensional cube (i.e.,Lx1 ˆLx2ˆLx3ˆLˆ1,

212121 elements) shown in Fig. 1. We use the two-dimensional domain to determine the head and ¯ux variances at the center of the domain as a function ofky,

which is a measure of the distance to a boundary. We use a two-dimensional domain for this case in order to use a much ®ner grid spacing relative toky than we can

readily obtain in three dimensions. Furthermore, we can compare these results with two-dimensional analyti-cal24,25,28,29 and numerical36 ®nite domain results. In

both the two- and three-dimensional cases, head is ®xed at x1ˆ0 and x1ˆL so that the gradient across the domain is 1. The mean hydraulic conductivityKgin both

cases is also 1. No-¯ow boundaries parallel to thex1-axis and a stationary conductivity ®eld are imposed to create a uniform head gradient. We use an isotropic expo-nential log conductivity correlation function,

Pyy…x;x0† ˆ exp…ÿjxÿx0j=ky†.

16

Note that extending our analysis to incorporate an anisotropic exponential con-ductivity function or other log concon-ductivity correlation functions (such as the separated exponential used by Osnes24,25 or a ``hole-type'' function16) can be accom-plished by simply incorporating the appropriate func-tion into the matrix eqns (51)±(54). As discussed previously, for comparison purposes we use two di€erent types of ®nite elements, the Raviart±Thomas± Nedelec elements of order zero (RTN0) and the Brezzi± Douglas±Marini elements of order one (BDM1). The BDM1 elements give higher order accuracy in the ¯ux approximation than RTN0at the price of three times as many unknowns.

Fig. 2a shows the normalized head standard devia-tion rh=Jryky at the center of the domain …L=2† as a

function of distance to the boundary normalized by the hydraulic conductivity correlation scale …ky†.

Re-sults for RTN0 elements (open diamonds) and BDM1 elements (®lled squares) are both shown, and give al-most exactly the same results. Fig. 2(a) shows that the normalized standard deviation of head increases (ap-proximately logarithmically) as the normalized distance to the boundary increases, as found by Rubin and Dagan28 and Osnes.24 In two dimensions the in®nite domain head variance is very sensitive to the shape of the covariance function at large separation distances. Analytical results using an exponential correlation scale give an in®nite head variance, however; results using a ``hole-type'' covariance function yield a ®nite variance equal to 8=p2_J2_r2

yk

2

.16 Fig. 2(a) shows that the two-dimensional ®nite domain head standard de-viation is larger than the in®nite domain results ob-tained using a hole-type function20 when the distance to the boundary is more than two correlation scales. Since we use an exponential correlation function for log conductivity rather than a hole-type, it would be expected that our results increase as L=ky increases.

Osnes24 derived analytical solutions for head covari-ances and head-conductivity cross-covariances in bounded two-dimensional domains. His results (his Figures 1,10 and 11) also indicate that the head co-variance at the center of a square domain is substan-tially higher than the 2-D in®nite domain result for

L=kY ˆ15 and increase as the width of the domain

decreases. Van Lent and Kitanidis,36 using a two-di-mensional Monte Carlo numerical spectral approach, also showed that head variance increases as the do-main size increases and that head variances in large domains are signi®cantly underpredicted by ®rst-order

Fig. 1.The ¯ow domain. All boundaries parallel tox1 are

in®nite domain results. It should be noted that Van Lent and Kitanidis36 imposed periodic boundary con-ditions, rather than ®xed head or no ¯ux boundary conditions, in order to avoid non-stationarities asso-ciated with boundary e€ects.

Fig. 2(b) shows a similar case, but this time in the three-dimensional domain. Since RTN0 elements give virtually the same results as the computationally more expensive BDM1elements for head covariances, we used the only lower-order elements in this case. Again, the normalized head standard deviation is larger than the in®nite domain result (which is relatively insensitive to

the shape of the covariance function at large separation distances) when the distance to the boundary is greater than two correlation scales. However, for values of

L=2ky>6 the normalized head variance decreases

to-wards the in®nite domain value. Note that to maintain approximately one element per correlation scale in three dimensions we did not evaluate the head standard de-viation for values ofL=2ky>12:5.

Fig. 3 shows normalized head standard deviation

rh=Jkyry calculated for the three dimensional case

using RTN0 elements (open diamonds) and BDM1 el-ements (®lled squares) parallel to the x1-axis for 06x16L=2 at x2ˆx3ˆL=2 for ky=Lˆ0:1. The head

standard deviation is zero at the ®xed head boundaries, and increases to a maximum at the midpointx1ˆL=2. Also shown in Fig. 3 are normalized head standard deviation using RTN0 elements (open right triangles) and BDM1elements (®lled triangles) parallel to thex 2-axis for 06x26L=2 at x1ˆx3ˆL=2 also for

ky=Lˆ0:1. In this case, the head standard deviation is

a maximum on the no-¯ow boundaries, and decreases to the midpoint value at x2ˆL=2. Note that the maximum is slightly above the in®nite domain value (shown as the dotted line) consistent with the results shown in Fig. 2(b). The fact that the ®nite domain head variance exceeds the in®nite domain results is due to the no-¯ux boundaries. No ¯ux boundaries have been previously shown to increase the head variability in the two-dimensional semi-in®nite case29 and in the three-dimensional semi-in®nite case.17,22Fig. 2(b) and 3 show that when no-¯ux boundaries exist in both the vertical and transverse directions to mean ¯ow the

Fig. 3.Normalized head standard deviations in the three-di-mensional domain as a function ofxi=ky, foriˆ1;2. Results parallel to x1 are located along the linex2ˆx3ˆL=2, while

those parallel to x2 are located along the line x1ˆx3ˆL=2.

In®nite domain results are shown by the dotted line. Fig. 2. (A) Normalized head standard deviations rh at the

region of higher head variability propagates well into the interior of the domain. This is most likely due to the spatial persistence of the head ®eld in directions transverse to mean ¯ow.

Fig. 4 shows the head correlation function

q_hh…x;x0† ˆPhh…x;x0†=Phh…x0;x0† parallel to the x1-axis

from 06x16L=2 at x2ˆx3ˆL=2 for RTN0 elements (open diamonds) and BDM1 elements (®lled squares); and parallel to the x2-axis from 06x26L=2 at x1ˆ

x3ˆL=2 for RTN0 elements (open right triangles) and BDM1elements (®lled triangles). The pointx0_{is located} at the center of the domain, x1ˆx2ˆx3ˆL=2. Also shown in Fig. 4 are the three-dimensional in®nite do-main head correlations.3 Our numerical results for the head correlation parallel to the x1 axis decrease faster as separation increases than do the in®nite domain results, and approach zero at the front and back boundaries as required by the ®xed head boundary conditions. Parallel to thex2 (and x3) directions, how-ever, our results decrease less than the in®nite domain results due to the no-¯ux boundary conditions in these directions.

Fig. 5 shows the normalized ¯ux variancer2

q1=K

2

gJ

2_r2

y

and r2

q2=K

2

gJ

2_r2

y at the center of the two-dimensional

domain as a function of distance to the boundary nor-malized by ky. It is apparent that better accuracy is

obtained using BDM1 elements rather than RTN0 ele-ments in larger domains. This is due to the fact that BDM1 elements approximate both the zeroth and ®rst moments of ¯ux on each face of an element, while RTN0 elements approximate only the zeroth moment. Thus, in areas where there are sharp gradients in ¯ow (such as arise from the forcing terms in the adjoint state

equations for ¯ux) the additional approximation terms provide greater accuracy. Unlike the head variance, the approximation tor2

q1using BDM1elements approaches

the in®nite domain result as the domain size increases (note that the approximation using RTN0 elements di-verges slightly). However, there remains a small di€er-ence between our results and the in®nite domain result16 of 3/8 for even the largest domain tested…L=2ˆ25ky†.

The results forr2

q2 are also close to the in®nite domain

results, but show a larger deviation from the in®nite domain result as L=2k increases. Both of these results agree with those of Osnes,25who derived analytical so-lutions for velocity in bounded two-dimensional do-mains with boundary conditions similar to ours. His results show (his Fig. 2) that for bounded domains the normalized velocity variance r2

u1=U

2_r2

y (the ¯ux and

velocity results are comparable since both are normal-ized) is somewhat larger than the in®nite domain value, whileru2 is smaller than the in®nite domain value, even

for quite large values of L=2ky. Comparing our Fig. 5

with Fig. 2 of Osnes25 shows similar results, although our results for r2

q1 with L=2ky>15 (0.39) are

some-what less than his (0.43). This slight discrepancy may be due to the fact that Osnes used a separated expo-nential function for log transmissivity correlation rather than the isotropic exponential correlation function used here, although Gelhar16 has shown that the ¯ux co-variance functions are relatively insensitive to the log conductivity covariance function used. Our results are also consistent with those of Van Lent and Kitanidis36 who found that the Darcy ¯ux variance was not as sensitive to domain size as the head variance. Van Lent and Kitanidis36 also found that ®rst-order in®nite do-main results were robust predictors of r2

q1; but slightly Fig. 5.Normalized ¯ux variancesr2

q1 andr2q2 at the center of the two-dimensional domain as a function of log conductivity correlation scaleky. In®nite domain results are shown by the

dotted lines. Fig. 4.Head correlations in the three-dimensional domain as a

function ofxi=ky, foriˆ1;2. Results parallel tox1are located

along the linex2ˆx3ˆL=2, while those parallel tox2are

lo-cated along the linex1ˆx3ˆL=2. The point x0 is located at

underpredicted r2

q2 for the approximately stationary

¯ow-®eld they considered.

Fig. 6 shows the results for normalized ¯ux standard deviations in the three-dimensional domain. Given the higher accuracy of the BDM1 elements in determining ¯ux variances, we only used these elements to approxi-mate the three-dimensional ¯ux covariances and stan-dard deviations. In Fig. 6 the normalized stanstan-dard deviation of ¯ux rq1=KgJry parallel to the x1-axis for

06x16L=2 at x2ˆx3ˆL=2 is shown by the ®lled squares. It decreases from the maximum at the ®xed head boundaries to a minimum at the center of the domain (though still slightly above the in®nite domain result16). The normalized standard deviationrq1=KgJry

parallel to the x2-axis for 06x26L=2 atx1ˆx3ˆL=2 (open diamonds) decreases from the center value to a minimum near the no-¯ow boundaries. Comparing these to Fig. 1 of Osnes25 again shows that our results are qualitatively similar to his (his results are for two-dimensional variances whereas ours are for three di-mensional standard deviations). One di€erence is that Osnes'25results show a distinct upturn in variance near the no-¯ow boundaries (his Fig. 1(b)). Our results do not show this, since the nearest point to the boundary for which we could obtain results is at a distance of

x2=kyˆ0:2 from the boundary, not directly on the

boundary (the forcing term for the adjoint state equa-tions for ¯ux is equivalent to speci®cation of unit ¯ux at a point, which cannot be imposed on a no-¯ow boun-dary). Also shown in Fig. 6 is the normalized ¯ux standard deviationrq2=KgJry(®lled triangles) parallel to

the x1-axis (x2-axis) for 06x16L=2 (06x26L=2) at

x2ˆx3ˆL=2 (x1ˆx3ˆL=2) (the results are equal

parallel to either axis). The standard deviation is a maximum at the center of the domain, and decreases as either type of boundary is approached. Again, these results are less than the in®nite domain results16and are similar to those of Osnes.25

Fig. 7 shows the ¯ux correlation q_q1q1…x;x0† ˆ Pq1q1…x;x

0_†=P

q1q1…x

0_;x0_{† parallel to the x1-axis for} 06x16L=2 atx2ˆx3ˆL=2 (®lled squares) and parallel to the x2-axis for 06x26L=2 at x1ˆx3ˆL=2 (open diamonds). Again,x0_{is located at the center of the} do-main, x1ˆx2ˆx3ˆL=2. The correlation decreases more rapidly parallel to thex2axis than parallel to thex1 axis, indicating that ¯ux variations are more strongly correlated in the mean ¯ow direction. The correlations in both directions agree closely with the in®nite domain results, diverging only slightly near the boundaries. This is in qualitative agreement with the results of Osnes25 who found that while the two-dimensional velocity co-variance was somewhat higher than the in®nite domain velocity covariance,27 the correlations have similar shapes. This is also in agreement with Van Lent and Kitanidis36 who found that the variogram of the longitudinal component of speci®c discharge closely matches the (®rst-order) in®nite domain result in two dimensions.

Shown in Fig. 8 are the ¯ux correlationsqq2q2…x;x

0_{† ˆ}

Pq2q2…x;x

0_†=P

q2q2…x

0_;x0_{† parallel to the x1-axis for} 06x16L=2 at x2ˆx3ˆL=2 (®lled squares) and par-allel to the linex1ˆx2(45 to the mean ¯ow direction) (®lled diamonds). The results parallel to thex2-axis are equal to those parallel to the x1 axis. The correlation parallel tox1(orx2) decreases from the maximum at the center of the domain, becoming slightly negative before

Fig. 7. Flux correlations q_q1q1…x;x0_{† in the three-dimensional}

domain as a function ofxi=ky, foriˆ1;2. Results parallel tox1

are located along the linex2ˆx3ˆL=2, while those parallel to

x2 are located along the line x1ˆx3ˆL=2. The point x0 is

located at the center of the domain atx1ˆx2ˆx3ˆL=2.

Fig. 6. Normalized ¯ux standard deviations in the three-di-mensional domain as a function ofxi=ky, foriˆ1;2. Results parallel to x1 are located along the line x2ˆx3ˆL=2, while

those parallel to x2 are located along the linex1ˆx3ˆL=2.

increasing to zero at the boundaries. The correlation at 45_{to the mean ¯ow direction decreases away from the} center, but not as rapidly as the results parallel to x1. The correlations in both directions di€er from the in-®nite domain solutions, showing greater correlation than the in®nite domain solutions as separation in-creases. Note that the variance at x0 is lower than the in®nite domain result and normalizing by this value increases the correlation for separations greater than zero relative to the in®nite domain values. This agrees qualitatively with the two-dimensional results of Osnes25 (his Fig. 4), taking into account the lower variance atx0_{. However, this is in contrast to Van Lent} and Kitanidis,36who reported results for the transverse component of speci®c discharge that had higher vari-ance and showed less correlation than ®rst-order in®-nite domain results. Note, however, that both our analysis and that of Osnes25 use the ®rst-order small perturbation assumption, and calculate covariance functions in bounded domains. Van Lent and Kitanidis on the other hand used a numerical spectral Monte Carlo approach to avoid ®rst-order perturbation as-sumptions, and imposed periodic boundary conditions to avoid non-stationary e€ects introduced by ®xed boundary conditions.

5 CONCLUSIONS

We have developed a numerical method for accurately approximating head and ¯ux covariances and

cross-covariances in ®nite two- and three-dimensional do-mains using the mixed ®nite element method. This method is useful for determining head and ¯ux covar-iances for non-stationary ¯ow ®elds where in®nite do-main results are not applicable, such as those induced by injection or extraction wells, impermeable subsur-face barriers, or non-stationary hydraulic conductivity ®elds. Furthermore, because the numerical approxi-mations to the ¯ux covariances are obtained directly from the solution to the coupled problem rather than having to di€erentiate head covariances, the approxi-mations will be more accurate than those obtained from conventional ®nite di€erence or ®nite element methods, particularly if injection and/or extraction wells are present or the mean conductivity ®eld is variable.

Results for uniform ¯ow example problems are consistent with results from previously published ®nite domain analyses17,22,24,25,28,29,36 and demonstrate that head variances and covariances are quite sensitive to boundary conditions and the size of the bounded domain. Flux variances and covariances, however, are less sensitive to boundary conditions and domain size. Results comparing lower-order Raviart±Thomas±Ned-elec23,26 (RTN0) and higher order Brezzi±Douglas± Marini5 (BDM1) ®nite element spaces indicate the higher order element space improves the estimate of the ¯ux covariances, at a cost of roughly three times as many unknowns. The higher order elements were not found to signi®cantly a€ect the estimate of the head covariances for the examples investigated here.

The head and ¯ux covariances predicted using this methodology should be useful for quantifying the uncertainty of groundwater ¯ow predictions in heterogeneous aquifers.3,9,10,16,20 Furthermore they provide a logical means of improving model predic-tions using site speci®c ®eld measurements through inverse modeling techniques such as non-linear least squares and maximum likelihood methods, Bayesian conditioning, and Kalman ®ltering.10,14,15,19,32±35 The methodology developed here is limited by the small perturbation assumption which Van Lent and Kit-anidis have shown underestimates head and transverse ¯ux variances in large domains. However, if these ®rst-order covariances are used together with ®eld obser-vations in iterative inverse modeling algorithms, errors associated with the small perturbation technique will become less important as more ®eld observations be-come available.

ACKNOWLEDGEMENTS

The authors would like to thank two anonymous reviewers whose comments improved this paper. This

Fig. 8. Flux correlations qq2q2…x;x0† in the three-dimensional domain as a function ofxi=ky, foriˆ1;2. Results parallel tox1

andx2are equal. Results parallel to the linex1ˆx2(45to the

mean ¯ow direction) are also shown. The pointx0_{is located at}

material is based upon work sponsored by the U.S. Air Force under grant F08637-97-C-6018.

APPENDIX A

The two mixed ®nite element spaces we utilize in this paper, RTN0 and BDM1, are denoted by WrVRTNr 0 are determined by the moments of the ¯ux through the faces of each element:7

Z

@Ei

vnpk dr 8pk 2Pk…@Ei†

wherekˆ0 or 1 for RTN0or BDM1elements, respec-tively. Also, Pk…E† is the space of polynomials on an

elementEof total degree 6k, andPk…@Ei†is the space of

polynomials on the ith face of E of total degree 6k. Thus, any vectorvr2Vr can be represented as

vrˆUV …58†

where _{V is an n1 vector of coecients and U is the}

3n matrix of basis functions, where nˆ6 for RTN0 and 18 for BDM1. For RTN0, on a regular rectangular grid where d1, d2, and d3 are the dimensions of an ele-ment in the x1, x2, and x3 directions, respectively, we have:

while for BDM1 we have:

Uˆ

Now, using eqns (17) and (58) the matricesA, BandC in eqns (18)±(20) are given by:

AˆX

Note that if the mean ¯uxQand the adjoint statew1(or

~

w1) are both approximated using the same mixed ®nite element method, the integrals used to calculate the sensitivities eqns (49) and (50) have the same form as the integral in Aappearing above. This makes determining the entries of the Jacobian matrices straightforward:

@hi

elementj, respectively, andAj is the elementary matrix

of thejth element.

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