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Advances in Water Resources Vol. 22, No. 7, pp 729±740, 1999
Ó 1999 Elsevier Science Ltd
Printed in Great Britain. All rights reserved
0309-1708/99/$ ± see front matter

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 estimate of the ¯ux covariances, but do not signi®cantly a€ect the estimate of the
head covariances. Ó 1999 Elsevier Science Ltd. All rights reserved

domains. Numerical solutions for head covariances and
head-conductivity cross-covariances in bounded domains have also been reported by McLaughlin and
Wood,18,19 Sun and Yeh,35 Li and McLaughlin,17 and
Van Lent and Kitanidis.36 Typically, when a numerical
approach is taken, the covariances and cross-covariances 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 Sun32 point out that, in general, accurate numerical
calculations of gradients of unknown functions in coupled 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

1 INTRODUCTION
Over the past two decades, stochastic methods have
been widely used to describe the variability of groundwater systems. Of long standing interest is the determination 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 Bakr et al.,3 Mizell et al.,20 Graham
and Mclaughlin,13 Rubin,27 Rubin and Dagan,30 and
Zhang and Neumann.37,38 Osnes,24 Rubin and Dagan28,29 and Na€ and Vecchia22 have used semi-analytical methods to solve for head covariances in bounded
domains. Osnes25 used semi-analytical methods to solve
for velocity covariances in two dimensional bounded

*

Corresponding author.
729

730

A. I. James, W. D. Graham

mixed ®nite element method. Over the past three decades, mixed ®nite element methods have been applied
to a wide range of problems.2,5±7,12,23,26 When applied to
groundwater ¯ow problems the mixed ®nite element
method has the advantage of simultaneously approximating 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 documented.1,4,8,11,21 In this paper we show that these advantages
extend to the determination of covariances and crosscovariances involving ¯ux terms. The outline of this
paper is as follows: In Section 2, we give a brief introduction to the mixed ®nite element method. In Section 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

h…x† ˆ gD ;

x2X

x 2 CD

ÿ …K…x†
rh…x††
n ˆ gN ;

…1†
…2†

x 2 CN

…3†

Here, K is the hydraulic conductivity tensor (an n  n
symmetric positive de®nite matrix which is uniformly
bounded below and above in X), h the piezometric head,
f a source term, gD is the value on the ®xed head
(Dirichlet) boundaries, gN is the value on the ®xed ¯ow
(Neumann) boundaries, and n is the outward unit normal of @X. In what follows, we assume that gN ˆ 0 on
CN . In typical groundwater problems it is of interest to
determine the Darcy ¯ux q, given by q ˆ ÿK
rh.
Rather than solve the above system eqns (1)±(3) and
determine q from di€erentiating h, we introduce q into
the above system and derive the equivalent ®rst order
system of PDEs:1,26
q…x† ‡ K…x†
rh…x† ˆ 0;
r
q…x† ˆ f ;
h…x† ˆ gD ;
q…x†
n ˆ 0;

x2X
x 2 CD
x 2 CN

x2X

H …div; X† consists of vector-valued functions that have
divergences that lie in L2 …X†. Our approximating functions come from two spaces W and V which are de®ned
by:
V ˆ fv 2 H …div; X†jv
n ˆ 0 on CN g


W ˆ w 2 L2 …X†jw is piecewise constant

…4†
…5†
…6†
…7†

The mixed ®nite element method can be applied to a

system of the form of eqns (4)±(7). In this method, the

…8†
…9†

W is 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 @X are zero.7
We multiply eqns (4) and (5) by functions v 2 V and
w 2 W , respectively, and integrate over X to obtain the
mixed weak form of the system,
ÿ ÿ1

K q; v ‡ …rh; v† ˆ 0; v 2 V
…r
q; w† ˆ … f ; w†;

Let X be a bounded domain in Rn (n ˆ 2 or 3) with
boundary @X, with the boundary decomposed so that
@X ˆ CD [ CN , with CD \ CN ˆ 0. We consider the
Dirichlet±Neumann problem for groundwater ¯ow:

ÿ r
…K…x†
rh…x†† ˆ f ;

two variables (here the head h and the Darcy ¯ux q) are
simultaneously approximated using two di€erent approximating spaces. Let L2 …X† denote the space of
square integrable functions on X, and de®ne the space
H …div; X† by

ÿ

n

H …div; X†  v 2 L2 …X† jr
v 2 L2 …X†

w2W

where …
;
† represents the L2 inner product. Applying
Green's formula to the ®rst equation and using the fact
that v
n ˆ 0 on CN , the weak solution to the system
eqns (4) and (5) is obtained by seeking a pair …q; h† 2
V  W such that
ÿ ÿ1

…10†
K q; v ÿ …h; r
v† ˆ hgD ; v
niCD ; v 2 V
…r
q; w† ˆ … f ; w†;

w2W

…11†

To discretize the domain X, we choose a grid Tr over X.
For comparison purposes, we will use both the lowestorder Raviart±Thomas±Nedelec23,26 (RTN0 ) and Brezzi±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 , respectively. We denote the two mixed ®nite element
1
0
 W  V and Wr  VBDM
W 
spaces by Wr  VRTN
r

r
V for RTN0 and BDM1 , respectively. These ®nite element subspaces are de®ned further in Appendix A. The
discrete, mixed form of our problem is then to seek a
pair …qr ; hr † 2 Wr  Vr such that
ÿ ÿ1


K qr ; vr ÿ …hr ; r
vr † ˆ hgD ; vr
niCD ; vr 2 Vr ;
…12†

…r
qr ; wr †; ˆ … f ; wr †;

wr 2 Wr
0
VRTN
r

1
VBDM
.
r

…13†

or
The discretized
where Vr refers to either
version of eqns (12) and (13) is, unfortunately, a saddle
point problem. By introducing a hybrid space of

Numerical approximation of head and ¯ux covariances in three dimensions using mixed ®nite elements
Lagrange multipliers restricted to the face of each element, Kr  L2 …@E† for an element E, 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 hybridized, discrete form of our problem is then to seek
…qr ; hr ; kr † 2 Vr  Wr  Kr such that
ÿ ÿ1


K qr ; vr ÿ …hr ; r
vr † ‡ c…kr ; v† ˆ 0; vr 2 Vr

…14†

…r
qr ; wr † ˆ … f ; wr †;
c…lr ; qr † ˆ 0;

wr 2 Wr

lr 2 Kr

…16†

The inner product c…
;
† is de®ned as:7
XZ
c…lr ; vr † ˆ
lr v
n dr
E

…15†

…17†

@E

This results in a discrete system of equations:
 ÿ BH ‡ C K
 ˆG

Kÿ1 AQ

…18†

 ˆ F
BT Q

…19†

ˆ0
CT Q

…20†

where the matrices A, B and C can be evaluated once a
particular approximating space is chosen. Note that this
 and H
system is block diagonal and so the unknowns Q
can be eliminated at the element level using static condensation, resulting in an equivalent symmetric positive
de®nite system (e.g., Brezzi and Fortin,7 p. 181).

3 APPLICATION TO PERTURBATION EQUATIONS

h…x† ˆ H …x† ‡ h0 …x†
q…x† ˆ Q…x† ‡ q0 …x†
ln K…x† ˆ Y …x† ‡ y 0 …x†
and substitute these quantities into eqns (4)±(7) (from
this point onward we assume for simplicity that K is a
scalar function of x). We assume that the source term f
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 values Q and H :

r
Q…x† ˆ f ;

x2X

x2X

x 2 CD

Q…x†
n ˆ 0;

…21†
…22†

…23†

x 2 CN

…24†

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


q0 … x† ‡ eY y 0 … x†rH ‡ rh0 … x† ‡ y 0 …x†rh0 … x† ˆ 0;
x2X

…25†

r
q0 … x† ˆ 0;
h0 … x† ˆ 0;

x2X

…26†

x 2 CD

q0 … x†
n ˆ 0;

…27†

x 2 CN

…28†

Several di€erent methods may be used to derive the
head and Darcy ¯ux covariances, as well as the crosscovariances between ¯ux, head, and log-conductivity.
One method is to multiply eqns (25)±(28) by perturbations in log-conductivity, head, and velocity at independent points x0 and then take the expected value
(neglecting third and higher order products or perturbations) 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 perturbation ¯ow problem we start by taking the ®rst-order
variations in eqns (25)±(28) (neglecting products of ®rstorder perturbation terms):32,33
dq…x† ‡ eY ‰ dy…x†rH ‡ rdh…x†Š ˆ 0;

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:

Q…x† ‡ eY …x†rH …x† ˆ 0;

H …x† ˆ gD ;

731

r
dq…x† ˆ 0;
dh…x† ˆ 0;

x2X

x 2 CD

dq…x†
n ˆ 0;

x 2 CN

x2X

…29†
…30†
…31†
…32†

where we have dropped the primed subscripts for perturbation quantities.
We then multiply eqn (29) by an arbitrary vectorvalued function w1 …x† and eqn (30) by a scalar function
w2 …x† (assuming the requisite smoothness of both functions) and integrate over the domain X:


ÿ

ÿ
…dq; w1 † ‡ eY dyrH ; w1 ‡ eY rdh; w1 ˆ 0; x 2 X
…r
dq; w2 † ˆ 0

x2X

where we have suppressed the explicit dependence on x.
Again, …
;
† is the L2 …X† inner product, de®ned for two
vector-valued functions u, v as
Z
…u; v† ˆ ui vi dX
X

732

A. I. James, W. D. Graham

where summation over i is implied. Adjoint operations33
yield:
ÿ

ÿ


…w1 ; dq† ‡ eY rH w1 ; dy ÿ eY r
w1 ; dh
‡ hdh; w1
ni@X ˆ 0;

x2X

ÿ …rw2 ; dq† ‡ hdq
n; w2 i@X ˆ 0;

…33†

x2X

…34†

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 eliminated by choosing w1 and w2 such that
w1 …x†
n ˆ 0;
w2 …x† ˆ 0;

x 2 CN

…35†

x 2 CD

…36†

Thus, eqns (33) and (34) become
ÿ

ÿ


…w1 ; dq† ‡ eY rH w1 ; dy ÿ eY r
w1 ; dh ˆ 0;
x 2 X;

ÿ …rw2 ; dq; † ˆ 0;

x2X

…38†

where R…q; h; y; x† is a user de®ned function.31,3 The ®rst
order variation in J is32,33 (in inner product form):

 
 

@R
@R
@R
; dq ‡
; dh ‡
; dy
…40†
dJ ˆ
@q
@h
@y
Adding eqns (37), (38) and (40) gives

 

@R
@R
‡ w1 ÿ w2 ; dq ‡
ÿ eY r
w1 ; dh
dJ ˆ
@q
@h


@R Y
…41†
‡e rH w1 ; dy
‡
@y
The ®rst two inner products on the right hand side of
eqn (41) can be eliminated if we choose w1 and w2 to
satisfy
x2X

…42†

x2X

…43†

Once w1 and w2 have been determined by solving
these equations we can calculate the partial functional
derivative of J with respect to the hydraulic conductivity
in the jth grid block yj by calculating the last inner
product in eqn (41)32,33
Z 
Xj

@R
‡ eY rH w1
@y

…45†
x2X

w2 …x† ˆ 0;

x 2 CD



…46†

w1 …x†
n ˆ 0;

…47†

x 2 CN

…44†

…48†

Using the solution w1 to the system of eqns (45)±(48) in
eqn (44) we have:
Z
Z
@hi
ÿ Q
w1
…49†
ˆ eY rH w1 ˆ
@yj
Xj

Xj

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€erentiate w2 , the
adjoint state of the head. Furthermore, if Q and w1 are
approximated using the same basis functions, the integral 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 conductivities. If we choose R ˆ qk …x†d…x ÿ xi †, where qk is
the ¯ux component in the xk direction, we have
~ …x† ÿ rw
~ …x† ˆ d…x ÿ xi †nk ;
w
1
2

@R
ÿ e r
w1 …x† ‡
ˆ 0;
@h
Y

@J
ˆ
@yj

x2X

with the boundary conditions:
…37†

X

@R
ˆ 0;
@q

w1 …x† ÿ rw2 …x† ˆ 0;

ÿ eY r
w1 …x† ˆ d…x ÿ xi †;

Now we assume a general performance criterion of
the form
Z
…39†
J ˆ R…q; h; y; x†

w1 …x† ÿ w2 …x† ‡

where the integration over Xj is performed over the
support of w1j (i.e., where the jth basis function of w1 is
non-zero). In order to perform the integration an appropriate function R must be chosen. By choosing R ˆ
h…x†d…x ÿ xi † (where xi denotes the ith measurement
point), we can determine @J =@yj ˆ @hi =@yj by solving
eqns (42) and (43) for w1 …x†. The partial derivative
@J =@yj ˆ @hi =@yj represents the sensitivities of a head
measurement at location xi to 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

~ …x† ˆ 0;
eY r
w
1
~ …x† ˆ 0;
w
2

x2X

x2X

x 2 CD

~ …x†
n ˆ 0;
w
1

x 2 CN

where nk is a unit vector in the xk direction. The tilde
indicates that the solutions to the second system of
equation will in general be di€erent from the ®rst set. We
can then determine the entries of JQk , the Jacobian
matrix of sensitivities of ¯ux parallel to xk with respect

Numerical approximation of head and ¯ux covariances in three dimensions using mixed ®nite elements
to conductivity, given by @qki =@yj in exactly the same
~ …x†:
way as for JH using the solution w
1
Z
@qki
~
ˆ
…50†
ÿQ
w
i1
@yj
Xj

Once the entries for the Jacobian matrices are determined, the Jacobian matrix can be used to determine the
®rst-order covariance matrices for head and ¯ux:32
Phh …x; x0 † ˆ JH Pyy …x; x0 †JTH

…51†

Phy …x; x0 † ˆ JH Pyy …x; x0 †

…52†

Pqk qk …x; x0 † ˆ JQk Pyy …x; x0 †JTQk

…53†

Pqk y …x; x0 † ˆ JQk Pyy …x; x0 †

…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 ˆ Lx2 ˆ 1, 101  101  1 elements) while the second
is a three dimensional cube (i.e., Lx1 ˆ Lx2 ˆ Lx3 ˆ L ˆ 1,
21  21  21 elements) shown in Fig. 1. We use the twodimensional domain to determine the head and ¯ux
variances at the center of the domain as a function of ky ,
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 to ky than we can
readily obtain in three dimensions. Furthermore, we can
compare these results with two-dimensional analytical24,25,28,29 and numerical36 ®nite domain results. In

Fig. 1. The ¯ow domain. All boundaries parallel to x1 are no¯ow, and the boundaries at x1 ˆ 0; L are ®xed head.

733

both the two- and three-dimensional cases, head is ®xed
at x1 ˆ 0 and x1 ˆ L so that the gradient across the
domain is 1. The mean hydraulic conductivity Kg in both
cases is also 1. No-¯ow boundaries parallel to the x1 -axis
and a stationary conductivity ®eld are imposed to create
a uniform head gradient. We use an isotropic exponential log conductivity correlation function,
Pyy …x; x0 † ˆ exp …ÿjx ÿ x0 j=ky †.16 Note that extending our
analysis to incorporate an anisotropic exponential conductivity function or other log conductivity correlation
functions (such as the separated exponential used by
Osnes24,25 or a ``hole-type'' function16) can be accomplished by simply incorporating the appropriate function 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 RTN0 at the price of three times as
many unknowns.
Fig. 2a shows the normalized head standard deviation rh =J ry ky at the center of the domain …L=2† as a
function of distance to the boundary normalized by
the hydraulic conductivity correlation scale …ky †. Results for RTN0 elements (open diamonds) and BDM1
elements (®lled squares) are both shown, and give almost exactly the same results. Fig. 2(a) shows that the
normalized standard deviation of head increases (approximately 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 J 2 r2y k2 .16 Fig. 2(a) shows that
the two-dimensional ®nite domain head standard deviation is larger than the in®nite domain results obtained 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 covariances and head-conductivity cross-covariances in
bounded two-dimensional domains. His results (his
Figures 1,10 and 11) also indicate that the head covariance at the center of a square domain is substantially higher than the 2-D in®nite domain result for
L=kY ˆ 15 and increase as the width of the domain
decreases. Van Lent and Kitanidis,36 using a two-dimensional Monte Carlo numerical spectral approach,
also showed that head variance increases as the domain size increases and that head variances in large
domains are signi®cantly underpredicted by ®rst-order

734

A. I. James, W. D. Graham
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 towards 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 deviation for values of L=2ky > 12:5.
Fig. 3 shows normalized head standard deviation
rh =J ky ry calculated for the three dimensional case
using RTN0 elements (open diamonds) and BDM1 elements (®lled squares) parallel to the x1 -axis for
0 6 x1 6 L=2 at x2 ˆ x3 ˆ L=2 for ky =L ˆ 0:1. The head
standard deviation is zero at the ®xed head boundaries,
and increases to a maximum at the midpoint x1 ˆ L=2.
Also shown in Fig. 3 are normalized head standard
deviation using RTN0 elements (open right triangles)
and BDM1 elements (®lled triangles) parallel to the x2 axis for 0 6 x2 6 L=2 at x1 ˆ x3 ˆ L=2 also for
ky =L ˆ 0:1. In this case, the head standard deviation is
a maximum on the no-¯ow boundaries, and decreases
to the midpoint value at x2 ˆ L=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.17,22 Fig. 2(b) and 3
show that when no-¯ux boundaries exist in both the
vertical and transverse directions to mean ¯ow the

Fig. 2. (A) Normalized head standard deviations rh at the
center of the two-dimensional domain as a function of log
conductivity correlation scale ky . Shown for comparison is the
two-dimensional in®nite domain result for head standard deviation for a ``hole-type'' covariance function. (B) Normalized
head standard deviations at the center of the three-dimensional
domain as a function of log conductivity correlation scale.

in®nite domain results. It should be noted that Van
Lent and Kitanidis36 imposed periodic boundary conditions, rather than ®xed head or no ¯ux boundary
conditions, in order to avoid non-stationarities associated 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 BDM1 elements 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

Fig. 3. Normalized head standard deviations in the three-dimensional domain as a function of xi =ky , for i ˆ 1; 2. Results
parallel to x1 are located along the line x2 ˆ x3 ˆ L=2, while
those parallel to x2 are located along the line x1 ˆ x3 ˆ L=2.
In®nite domain results are shown by the dotted line.

Numerical approximation of head and ¯ux covariances in three dimensions using mixed ®nite elements

Fig. 4. Head correlations in the three-dimensional domain as a
function of xi =ky , for i ˆ 1; 2. Results parallel to x1 are located
along the line x2 ˆ x3 ˆ L=2, while those parallel to x2 are located along the line x1 ˆ x3 ˆ L=2. The point x0 is located at
the center of the domain at x1 ˆ x2 ˆ x3 ˆ L=2.

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
qhh …x; x0 † ˆ Phh …x; x0 †=Phh …x0 ; x0 † parallel to the x1 -axis
from 0 6 x1 6 L=2 at x2 ˆ x3 ˆ L=2 for RTN0 elements
(open diamonds) and BDM1 elements (®lled squares);
and parallel to the x2 -axis from 0 6 x2 6 L=2 at x1 ˆ
x3 ˆ L=2 for RTN0 elements (open right triangles) and
BDM1 elements (®lled triangles). The point x0 is located
at the center of the domain, x1 ˆ x2 ˆ x3 ˆ L=2. Also
shown in Fig. 4 are the three-dimensional in®nite domain 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 the x2 (and x3 ) directions, however, 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 variance r2q1 =Kg2 J 2 r2y
and r2q2 =Kg2 J 2 r2y at the center of the two-dimensional
domain as a function of distance to the boundary normalized by ky . It is apparent that better accuracy is
obtained using BDM1 elements rather than RTN0 elements 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

735

Fig. 5. Normalized ¯ux variances r2q1 and r2q2 at the center of
the two-dimensional domain as a function of log conductivity
correlation scale ky . In®nite domain results are shown by the
dotted lines.

equations for ¯ux) the additional approximation terms
provide greater accuracy. Unlike the head variance, the
approximation to r2q1 using BDM1 elements approaches
the in®nite domain result as the domain size increases
(note that the approximation using RTN0 elements diverges slightly). However, there remains a small di€erence between our results and the in®nite domain result16
of 3/8 for even the largest domain tested …L=2 ˆ 25ky †.
The results for r2q2 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,25 who derived analytical solutions for velocity in bounded two-dimensional domains with boundary conditions similar to ours. His
results show (his Fig. 2) that for bounded domains the
normalized velocity variance r2u1 =U 2 r2y (the ¯ux and
velocity results are comparable since both are normalized) is somewhat larger than the in®nite domain value,
while ru2 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 r2q1 with L=2ky > 15 (0.39) are somewhat less than his (0.43). This slight discrepancy may
be due to the fact that Osnes used a separated exponential function for log transmissivity correlation rather
than the isotropic exponential correlation function used
here, although Gelhar16 has shown that the ¯ux covariance 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 domain results were robust predictors of r2q1 ; but slightly

736

A. I. James, W. D. Graham

Fig. 6. Normalized ¯ux standard deviations in the three-dimensional domain as a function of xi =ky , for i ˆ 1; 2. Results
parallel to x1 are located along the line x2 ˆ x3 ˆ L=2, while
those parallel to x2 are located along the line x1 ˆ x3 ˆ L=2.
Note that rq2 is the same parallel to either the x1 or x2 axis.

Fig. 7. Flux correlations qq1 q1 …x; x0 † in the three-dimensional
domain as a function of xi =ky , for i ˆ 1; 2. Results parallel to x1
are located along the line x2 ˆ x3 ˆ L=2, while those parallel to
x2 are located along the line x1 ˆ x3 ˆ L=2. The point x0 is
located at the center of the domain at x1 ˆ x2 ˆ x3 ˆ L=2.

underpredicted r2q2 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 approximate the three-dimensional ¯ux covariances and standard deviations. In Fig. 6 the normalized standard
deviation of ¯ux rq1 =Kg J ry parallel to the x1 -axis for
0 6 x1 6 L=2 at x2 ˆ x3 ˆ L=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 deviation rq1 =Kg J ry
parallel to the x2 -axis for 0 6 x2 6 L=2 at x1 ˆ x3 ˆ L=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 twodimensional variances whereas ours are for three dimensional standard deviations). One di€erence is that
Osnes'25 results 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 =ky ˆ 0:2 from the boundary, not directly on the
boundary (the forcing term for the adjoint state equations for ¯ux is equivalent to speci®cation of unit ¯ux at
a point, which cannot be imposed on a no-¯ow boundary). Also shown in Fig. 6 is the normalized ¯ux
standard deviation rq2 =Kg J ry (®lled triangles) parallel to
the x1 -axis (x2 -axis) for 0 6 x1 6 L=2 (0 6 x2 6 L=2) at
x2 ˆ x3 ˆ L=2 (x1 ˆ x3 ˆ L=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 results16 and are
similar to those of Osnes.25
Fig. 7 shows the ¯ux correlation qq1 q1 …x; x0 † ˆ
Pq1 q1 …x; x0 †=Pq1 q1 …x0 ; x0 † parallel to the x1 -axis for
0 6 x1 6 L=2 at x2 ˆ x3 ˆ L=2 (®lled squares) and parallel
to the x2 -axis for 0 6 x2 6 L=2 at x1 ˆ x3 ˆ L=2 (open
diamonds). Again, x0 is located at the center of the domain, x1 ˆ x2 ˆ x3 ˆ L=2. The correlation decreases
more rapidly parallel to the x2 axis than parallel to the x1
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 covariance 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 correlations qq2 q2 …x; x0 † ˆ
Pq2 q2 …x; x0 †=Pq2 q2 …x0 ; x0 † parallel to the x1 -axis for
0 6 x1 6 L=2 at x2 ˆ x3 ˆ L=2 (®lled squares) and parallel to the line x1 ˆ x2 (45 to the mean ¯ow direction)
(®lled diamonds). The results parallel to the x2 -axis are
equal to those parallel to the x1 axis. The correlation
parallel to x1 (or x2 ) decreases from the maximum at the
center of the domain, becoming slightly negative before

Numerical approximation of head and ¯ux covariances in three dimensions using mixed ®nite elements

Fig. 8. Flux correlations qq2 q2 …x; x0 † in the three-dimensional
domain as a function of xi =ky , for i ˆ 1; 2. Results parallel to x1
and x2 are equal. Results parallel to the line x1 ˆ x2 (45 to the
mean ¯ow direction) are also shown. The point x0 is located at
the center of the domain at x1 ˆ x2 ˆ x3 ˆ L=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 increases. 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 at x0 . However, this is in contrast to Van Lent
and Kitanidis,36 who reported results for the transverse
component of speci®c discharge that had higher variance 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 assumptions, and imposed periodic boundary conditions
to avoid non-stationary e€ects introduced by ®xed
boundary conditions.

737

covariances in ®nite two- and three-dimensional domains using the mixed ®nite element method. This
method is useful for determining head and ¯ux covariances for non-stationary ¯ow ®elds where in®nite domain results are not applicable, such as those induced
by injection or extraction wells, impermeable subsurface barriers, or non-stationary hydraulic conductivity
®elds. Furthermore, 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 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±Nedelec23,26 (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 predictions 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 Kitanidis have shown underestimates head and transverse
¯ux variances in large domains. However, if these ®rstorder covariances are used together with ®eld observations in iterative inverse modeling algorithms, errors
associated with the small perturbation technique will
become less important as more ®eld observations become available.

5 CONCLUSIONS

ACKNOWLEDGEMENTS

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

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

738

A. I. James, W. D. Graham

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
0

paper, RTN0 and BDM1 , are denoted by Wr  VRTN
r
BDM1
 W  V, respectively, where
W  V and Wr  Vr
Wh …E† ˆ faja 2 Rg
n
0
…E†
ˆ
… a1 x ‡ b1 ; a2 y ‡ b2 ; a3 z
VRTN
r
o
T
‡b3 † jai ; bi 2 R

…55†

…56†

VrBDM1 …E† ˆ
80
19
a1 ‡ b1 x ‡ c1 y ‡ d1 z ‡ r0 zx ‡ 2r1 xy ÿ t0 xy ÿ t1 x2 >
>
>
>
>
>
>
B
C>
>
=
< B a2 ‡ b2 x ‡ c2 y ‡ d2 z ÿ r0 yz ÿ r1 y 2 ‡ s0 yx ‡ 2s1 yz C >
@
A
>
> a3 ‡ b3 x ‡ c3 y ‡ d3 z ÿ s0 zx ÿ s1 z2 ‡ t0 zy ‡ 2t1 zx >
>
>
>
>
>
>
>
;
:
jai ; bi ; ci ; ri ; si ; ti 2 R
…57†

7

for each element E 2 Tr for RTN0 and BDM1 . The
0
1
or vr 2 VBDM
degrees of freedom of a vector vr 2 VRTN
r
r
are determined by the moments of the ¯ux through the
faces of each element:7
Z
v
npk dr 8pk 2 Pk …@Ei †
@Ei

where k ˆ 0 or 1 for RTN0 or BDM1 elements, respectively. Also, Pk …E† is the space of polynomials on an
element E of total degree 6 k, and Pk …@Ei † is the space of
polynomials on the ith face of E of total degree 6 k.
Thus, any vector vr 2 Vr can be represented as
vr ˆ U
V

…58†

where V is an n  1 vector of coecients and U is the
3  n matrix of basis functions, where n ˆ 6 for RTN0
and 18 for BDM1 . For RTN0 , on a regular rectangular
grid where d1 , d2 , and d3 are the dimensions of an element in the x1 , x2 , and x3 directions, respectively, we
have:
2 d ÿ2x
3
1
1
0
0
2d1
6 d ‡2x
7
6 1 1
0
0 7
6 2d1
7
6
7
d2 ÿ2x2
6 0
7
0
2d2
6
7
Uˆ6
7
d
‡2x
2
2
6 0
7
0
2d2
6
7
6
d3 ÿ2x3 7
6 0
7
0
2d3 5
4
0

0

d3 ‡2x3
2d3

while for BDM1 we have:

2

ÿ dx11

1
2

6
6 6x2 12x1 x2
6 d 2 ÿ d1 d 2
6 2
2
6
6 6x3 12x1 x3
6 d2 ÿ d d2
1 3
6 3
6 1 x
1
6
6 2 ‡ d1
6
6 6x2 12x1 x2
6 d2 ‡ d d2
1 2
6 2
6
6 6x3 12x1 x3
6 d2 ‡ d d2
6 3
1 3
6
6
0
6
6
6 ÿ3 6x21
6
6 2d2 ‡ d12 d2
6
6
6
0
6
Uˆ6
6
0
6
6
6
2
6 3 ÿ 6x2 1
6 2d2 d1 d2
6
6
6
0
6
6
6
0
6
6
6
6 ÿ3 ‡ 6x21
6 2d3 d12 d3
6
6
6
0
6
6
6
6
0
6
6
6x2
6 3
6 2d3 ÿ d 2 d12
6
1
4
0

0
ÿ3
2d1

6x2

‡ d d22
1 2

0
0
3
2d1

6x2

ÿ d d22
1 2

0
1
2

ÿ dx22

6x1
d12

1 x2
ÿ 12x
d2d

6x3
d32

12x2 x3
d2 d32

1 2

ÿ
1
2

‡ dx22

6x1
d12

1 x2
‡ 12x
d2d

6x3
d32

2 x3
‡ 12x
d d2

1 2

2 3

0
0
ÿ3
2d3

6x2

‡ d 2 d2

2 3

0
0
3
2d3

3

0

6x2

ÿ d 2 d2

2 3

7
7
7
7
7
6x23 7
ÿ3
‡
7
2d1
d1 d32 7
7
7
0
7
7
7
7
0
7
7
2 7
6x
3
3 7
ÿ
2
2d1
d1 d3 7
7
7
0
7
7
7
7
0
7
7
2
6x3 7
ÿ3
7
‡
2d2
d2 d32 7
7
7
0
7
7
7
7
0
7
7
2 7
6x
3
3 7
ÿ d d2 7
2d2
2 3
7
7
x3
1
ÿ
7
d3
2
7
7
12x1 x3 7
6x1
‡ d2d 7
d12
1 3 7
7
12x2 x3 7
6x2
ÿ d2d 7
d22
2 3 7
7
x3
1
7
‡
d3
2
7
7
12x1 x3 7
6x1
‡
2
2
d1
d1 d3 7
7
5
12x2 x3
6x2
‡
2
2
d
d d
0

2

2 3

Now, using eqns (17) and (58) the matrices A, B and C
in eqns (18)±(20) are given by:
XZ ÿ T

U
U dx
Aˆ
E2T

Bˆ

E2T

Cˆ

E

XZ

r
U dx

E

XZ ÿ

@Ei 2T

@Ei



UT
ni dr

Note that if the mean ¯ux Q and the adjoint state w1 (or
~ ) are both approximated using the same mixed ®nite
w
1
element method, the integrals used to calculate the
sensitivities eqns (49) and (50) have the same form as the
integral in A appearing above. This makes determining
the entries of the Jacobian matrices straightforward:
@hi
 j
Aj
W
 1j
ˆ ÿQ
@yj

…59†

@qi
~
 j
Aj
W
…60†
ˆ ÿQ
1j
@yj
~ are the coecients of the mean
 1j , and W
j , W
where Q
1j
¯ux vector and the adjoint states of head and ¯ux on

Numerical approximation of head and ¯ux covariances in three dimensions using mixed ®nite elements
element j, respectively, and Aj is the elementary matrix
of the jth element.

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