1 INTRODUCTION
Saturated hydraulic conductivity and the parameters of constitutive relations between relative conductivi-ty and pressure head in unsaturated soils vary spa-tially in a manner that cannot be described with cer-tainty. Therefore, they are often modeled as correlated random fields, rendering the correspond-ing unsaturated flow equations stochastic. If the (geo)statistical properties of these fields can be in-ferred from measurements, the stochastic flow equa-tions can be solved numerically by conditional Mon-te Carlo simulation. The corresponding first moments constitute optimum unbiased predictors of quantities such as pressure head and flux. Condi-tional second moments constitute measures of asso-ciated prediction errors. The Monte Carlo method is conceptually straight forward but computationally demanding. It lacks well-established convergence criteria and requires that one specify the probability distribution of the parameter fields.
We present a deterministic alternative to condi-tional Monte Carlo simulation which allows predict-ing steady state unsaturated flow under uncertainty, and assess the latter, without having to generate ran-dom fields or variables, without upscaling, and without linearizing the constitutive characteristics of the soil. Neuman et al. (1999) and Tartakovsky et al. (1999) have shown that such prediction is possible
when soil properties scale according to the linearly separable model of Vogel et al. (1991). They have demonstrated that when the scaling parameter of pressure head is a random variable independent of location, the steady state unsaturated flow equation can be linearized by means of the Kirchhoff trans-formation for gravity-free flow. Linearization is also possible in the presence of gravity when hydraulic conductivity varies exponentially with pressure head according to the exponential model of Gardner (1958). This allowed Tartakovsky et al. (1999) to develop exact conditional first- and second-moment equations for unsaturated flow which are nonlocal (integro-differential) and therefore non-Darcian. The authors solved their equations analytically by per-turbation for unconditional vertical infiltration. Their solution treats as a nonrandom constant and is otherwise valid to second order in the standard deviation, Y, of natural log saturated hydraulic
conductivity, Y = ln Ks.
Lu (2000) developed perturbation approximations for the nonlocal conditional moment equations of Tartakovsky et al. (1999), valid to second order in both Y and the standard deviation of = ln, . In this paper, we present only such equations for condi-tional expectations of pressure head and flux. The equations for conditional second moments can be found in Lu (2000). Based on these approximations, Lu developed a finite element algorithm for flow in the vertical plane when Y and are mutually uncor-related. We show some of his computational results
Direct solution of unsaturated flow in randomly heterogeneous soils
Zhiming Lu & Shlomo P. Neuman
Department of Hydrology and Water Resources, The University of Arizona, Tucson, Arizona 85721, USA Alberto Guadagnini
Dipartimento di Ingegneria Idraulica Ambientale e del Rilevamento, Politecnico di Milano, Piazza L. Da Vinci 32, 20133 Milano, Italy
Daniel M. Tartakovsky
Group CIC-19, MS B256, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
ABSTRACT: We consider steady state unsaturated flow in bounded randomly heterogeneous soils under the influence of random forcing terms. Our aim is to predict pressure heads and fluxes without resorting to Monte Carlo simulation, upscaling or linearization of the constitutive relationship between unsaturated hydraulic conductivity and pressure head. We represent this relationship through Gardner's exponential model, treating its exponent as a random constant and saturated hydraulic conductivity, Ks, as a spatially correlated random
field. This allows us to linearize the steady state unsaturated flow equations by means of the Kirchhoff trans-formation, integrate them in probability space, and obtain exact integro-differential equations for the condi-tional mean and variance-covariance of transformed pressure head and flux. We solve the latter for flow in the vertical plane, with a point source, by finite elements to second-order of approximation. Our solution compares favorably with conditional Monte Carlo simulations, even for soils that are strongly heterogeneous.
for = 0 in the presence of a point source, and compare them with those of conditional Monte Car-lo simulations.
2 STATEMENT OF PROBLEM
We start from the steady state Richards' equation
3
( , ) ( ) ( ) 0 K x x gx f x x (1) subject to
D x x x (2)
Q
q x n x x xN (3)where q is flux, K is unsaturated hydraulic conduc-tivity, is pressure head, g is an indicator showing whether the flow is gravity-free or not, being one for flow with gravity and zero for the gravity-free flow, x3 is the vertical coordinate, f is a source term, is
prescribed head on the Dirchlet boundary D, Q is
prescribed flux on the Neumann boundary N, and n
is a unit outward vector normal to the boundary , the union of D and N. All quantities are defined,
and measurable, on a bulk support volume that is small compared to the flow domain The forcing terms f, , Q are random and mutually uncorrelated. This and the fact that K is a random field renders (1) - (3) stochastic.
According to Gardner's (1958) model
( ) ( )
( , ) ( ) ( , ) ( , ) x x
x s r x r x
K K x K K e (4)
where Kr is relative conductivity and is a positive
exponent. Setting = constant allows defining the Kirchhoff transformation (Tartakovsky et al., 1999)
( ) ( ) ( ) 1 ( ) ( ) e
x
x x x Kr d d e (5)which transforms (1) - (3) into
( ) ( ) ( ) ( ) 0 Ks x x g x e3 f x x (6) ( ) 1 ( ) H( ), ( )H e D x x x x x (7)
( )Ks( ) ( ) g ( ) 3 Q( ) N n x x x x e x x (8) where e3=(0, 0, 1)T.3 EXACT CONDITIONAL MEAN EQUATIONS
3.1 Exact equations for (x) We write s( ) s( ) s( ) s( ) 0 ( ) ( ) ( ) ( ) 0 0 K x K x K x K x x x x x (9)
where indicates conditional mean and primed quantities define zero mean random fluctuations about the mean. Substituting (9) into (6) - (8) and taking their conditional mean yields
3 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + ( ) 0 ( ) ( ) ( ) ( ) x x r x x x x x x x e x x x x x n x x x s s K s K D N K g K R K R R f H Q (10) where [*] is identical to the other term in [ ] and( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) r x x x x x x x x R x x x s K s K K R K R K (11)
Substituting (9) into (6) - (8) and subtracting (10) gives
s 3 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 ( ) ( ) x x x x r x x x x x x x x x x x e x x x x s s s s K s K K K g K K K R K R R f H
( ) ( ) x n x x x D N Q (12) Let G(y,x) be an auxiliary function that satisfies3 D N ( ) ( ) ( ) ( , ) ( ) 0 , G( , ) 0 , ( ) ( , ) ( ) 0 , T s s s K G g K G K G y y y y y y, x e y y x x y x y y x x y y y x n y x y (13)
Rewriting (12) in terms of y, multiplying by G and integrating with respect to y over yields
N D s 3 ( ) ( , ) ( ) ( ) ( ) + ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) d ( ) ( , ) d G( , ) ( ) d ( ) ( ) G( , ) ( ) d
y x y x y y r y y y y y y y y y e y y x y x y y y y x n y T y s s K s K s G K g K K R K R R f G Q H K (14) This allows developing explicit integral expressions for all four terms in (11), for example3 3 ( ) ( ) ( ) ( ) ( , ) ( ) ( ) d + ( ) ( , ) ( )d + ( ) ( , ) ( ) ( ) d +g ( ) ( , ) ( ) ( ) d ( ) ( , )
r x x x x y x y y x y x r y x y x y y e x y x y y e x y x s T s x y s T s x y T s y y s T s x y s T s x y K K G K K G g K G K K G K g K G
3 * ( ) ( ) ( ) ( ) d ( ) ( ) ( , ) ( ) ( )
y y y y e x y y x y n y D K s K T s x y s R K R R K H G K d (15) The integral over N and that containing f' have beenomitted because K's and G are independent of Q and
f. The integral over D remains because both H and
G' depend on '.
3.2 Exact expression for q(x)
By virtue of (6) and (7), Darcy's law transforms into
s
( ) ( )[( ) g ( ) ] 3
q x K x x x e (16)
Writing q(x) = q(x) + q(x), substituting (9) into (16), and taking conditional mean, we obtain an exact expression for the conditional mean flux q(x),
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) s s K K K g R g R R 3 3 q x x x r x K x x x e x x e (17)where R(x), RK(x), RK(x) are defined in (11).
4 RECURSIVE CONDITIONAL MEAN FLOW APPROXIMATIONS
The above conditional mean equations are exact but not workable because they include a number of un-known moments. To render them workable, we ex-pand them in powers of Y and , which represent
measures of the standard deviation of Y’(x) and , respectively. For example,
( , ) , 0 0 0 ( ) ( ) ( ) ( ) ! !
x x x x n m n m n Y Y S G n m G m Y K e K n e m (18)where (n,m) designates terms including nth power of
Y and mth power of , and KG and G are
geomet-ric mean values of K and , respectively. The ex-pansion is not guaranteed to be valid for strongly heterogeneous soils with Y 1 and 1. As we
shall see, it actually works well for relatively large values of Y as long as remains small.
4.1 Recursive approximations for (x)
Expansion of (10) to second order in Y and
yields the following set of recursive equations,
(0,0) (0,0) (0,0) (0,0) ( ) ( ) ( ) ( ) 0 ( ) ( ) ( ) * ( ) 3 x x x e x x x x n x x x G G D N K g f H x Q (19)
(0,2) (0,2) 2 (0,0) (0,2) 3 (0,2) (0,2) ( ) ( ) ( ) ( ) 0.5 ( ) ( ) 0 ( ) ( ) ( ) * 0 G G G G D N K gK R H x x x x x x e x x x x n x x (20)
(2,0) 2 (0,0) (2,0) (2,0) 2 (0,0) (2,0) 3 (2,0) ( ) ( ) 0.5 ( ) ( ) ( ) ( ) ( ) 0.5 ( ) ( ) ( ) 0 ( ) 0 ( ) * 0 G Y G G Y G K D K g K R x x x x r x x x x x x e x x x n x N x (21)2 (2,2) (0,2) (2,2) 2 (2,2) (2,0) 2 2 2 (0,2) (0,0) 2 (2,2) (2,0) ( ) ( ) ( ) ( ) 2 ( ) ( ) ( ) ( ) 2 ( ) ( ) ( ) ( ) 2 4 ( ) ( ) 2 x x x x r x x x x x x x x x x Y G G G Y Y G K K K K g K R R R
(2,2) 2 (2,2) (0,2) 3 (2,2) ( ) ( ) + ( ) ( ) ( ) 0 2 ( ) 0 ( ) * 0 x x x x x e x x x n x Y G D K R R x N (22) where H(0,0) and H(0,2) can be derived fromper-turbation expansion of (7). All terms (n,m)(x) with n or m equal to 1 are zero. To illustrate how terms are evaluated recursively to second order, we pre-sent the corresponding nonzero expressions for r,
(2,0) (0,0) Y x (0,0) (0,0) G 3 ( ) ( ) C ( ) ( , ) ( ) * ( ) g ( )
r x x x, y y x y y y e T G y G K G K d (23)
(2,2) (0,2) Y x (0,0) (0,0) G 3 (0,0) (0,2) Y x (0,2) 2 (0,0) G 3 G Y ( ) ( ) C ( ) ( , ) ( ) * ( ) g ( ) ( ) C ( ) ( , ) ( ) ( ) g ( ) 0.5 ( ) g ( ) C (
r x x x, y y x y y y e x x, y y x y y y y e x x T G y G T G y G G K G K d K G K d K x (0,1) (1,1) (0,0) x 3 (1,0) (0,2) x ) ( , ) ( ) ( , ) ( ) ( ) g ( ) ( ) ( , ) ( ) ( )
, y y x x y x y y e x x y x y y T y T y G T G y G G Y G K d K Y G K R d (24)Evaluating to second order requires 1) solving (19) for (0,0) and (13) for G(0,0) (upon replacing K
s
and in (13) by KG and G); 2) evaluating moments
that involve G up to first order, such as G(0,1) and
YG(1,0); 3) evaluating the moments in (11) up to
second order in one of the expansion parameters, for example r(2,0), R
(0,2), RK(2,0); 4) solving (20) -
(21) for (0,2) and (2,0), respectively; 5)
evaluating r(2,2), R
(2,2), RK(2,2), RK(2,2); and 6)
solving (22) for (2,2). A detailed derivation of all
the requisite euqations is given by Lu (2000).
4.2 Recursive approximations of q(x)
Expansion of (17) leads to the following recursive approximations of q(x) to second order,
(0,0) (0,0) (0,0) 3 ( ) ( ) ( ) ( ) q x x x x e x G G K g (25) 2 0 (2,0) (0,0) (2,0) (0,0) 3 (2,0) (2,0) 3 ( ) ( ) ( ) ( ) ( ) 2 ( ) ( ) ( ) 2 ( ) ( ) 2 2 q x x x x x x x x e r x x e x ( , ) Y G Y G G K K g g R (26)
(0,2) (0,2) (0,2) 3 2 (0,2) (0,2) 3 ( ) ( ) ( ) ( ) 0.5 ( ) ( ) q x x x x e x e G G G K g g R x (27) (2,2) 2 (2,2) (0,2) 2 (2,2) (2,0) G 2 2 (0,2) (0,0) 3 2 (2,2) (0,2) 3 ( ) ( ) ( ) ( ) ( ) 2 g ( ) ( ) 2 ( ) + ( ) ( ) 2 2 ( ) ( ) ( ) 2 q x x x x x x x x x x e x x x e Y G Y Y K g R R (2,2) 2 (2,2) (0,2) (2,2) G 3 ( ) ( ) ( ) ( ) ( ) 2 r x x x Y x x e K K K g R R R (28)5 FINITE ELEMENT ALGORITHM
In most cases of practical interest, conditioning points are sparse enough to ensure that conditional mean quantities vary more slowly in space than do their random counterparts. Hence one can resolve the former (by an algorithm such as we propose) on a coarser grid than is required to resolve the latter (by Monte Carlo simulation). Here we nevertheless use a fine grid to allow comparing our direct finite element solution of the recursive moment equations with a finite element Monte Carlo solution of the original stochastic flow equations.
We solve the recursive conditional moment equa-tions by a Galerkin finite element scheme on a rec-tangular vertical grid with square elements, using bi-linear weight functions n(x). For simplicity, we
consider only deterministic forcing terms. To illus-trate our approach we consider the Galerkin orthog-onalization of (19) which, following the application of Green's first identity, yields
(0,0) (0,0)
(0,0) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + ( ) ( ) ( ) ( )
3 x x x e x x x x n x x x x x x D N G G n G n n n K g d K d Q d f d (29) where n = 1, 2, …NN, NN being the number of nodes. Let (0,0) (0,0) 1 ( ) NN m ( )m m x
x (30)where m(0,0) is (0,0)(x) evaluated at node m.
Substituting (30) into the leftmost integral in (29), and defining the matrix components
( ) ( ) 3 ( ) ( ) ( ) ( ) ( ) ( ) e nm nm G n m e e T nm nm G n m e A A d B B d
K x x x e K x x x (31) (29) becomes
(0,0) 1 (0,0) A B ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1,2,... ; D N NN nm G mn m m G n n n g d Q d f d n NN
K x x x n x x x x x x (32)Equations corresponding to n D can be dropped;
at all other nodes, the integral over D vanishes. As
all terms on the right-hand-side are known, the cor-responding integrals can be evaluated numerically and (32) solved for (0,0)(x) at all nodes.
To illustrate how the moments in (11) are evalu-ated numerically, consider for example r(2,0)(x). For
any xe and ye' in elements e and e, let
(0,0)( , ) (0,0)( , ) ( ) ( ) 1 1 ( ) ( ) ( ) N N e e e e e e jk j k j k G G
y, x y x (33) (0,0)( ) (0,0)( ) ( ) 1 ( ) N ( ) e e e p p p y
y (34)where Gjk(0,0)(e,e) is G(0,0) associated with local node j
in element e and node k in element e, p(0,0)(e) is
(0,0)(x) at local node p in element e, N is the
number of nodes in an element, k(e) is associated
with local node k in element e, and j(e) is
associated with local node j in element e. Substituting (33) and (34) into (23), and writing the integral over as a sum of integrals over elements,
(2,0)( ) ( ) (0,0)(e ,e) (0,0)( ) 1 1 1 ( ) ( ) ( ) G ( ) (e,e ) ( ) * ( ) e N N N e e Y G jk p e j k p e e e jp pj k C K G A g B
r x x x (38)where CY(e,e) is autocovariance of Y between
elenents e and e, and
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 ( ) ( ) ( ) , 1, ( ) ( ) ( ) , 1, e e e e e e jp j G p e e e e pj j G p A d j p N B d j p N
T T y K y y y K y y e (39)6 MEAN PRESSURE HEAD
Once the problem has been solved in terms of Kirchhoff-transformed pressure head, it can be back-transformed to yield, with the aid of (5),
(0,0) (0,0) G G 1 ( ) ln ( ) x x (40)
2 2 (0,2) (0,0) G G G (0,2) (0,2) (0,2) 2 (0,0) (0,0) (0,0) G G ( ) ln ( ) 2 ( ) R ( ) C ( , ) 1 + ( ) ( ) 2 ( ) x x x x x x x x x (41) (2,0) (2,0) (2,0) 2 (0,0) (0,0) G ( ) C ( , ) 1 1 ( ) 2 ( ) ( ) x x x x x x (42) (2,2) (2,2) (0,0) G (2,0) (0,2) (2,2) (0,0) (0,0) 2 (2,0) (2,0) (0,0) (0,0) G (2,2) 2 (0,0) G 1 ( ) ( ) ( ) ( ) ( ) C ( , ) ( ) 2 ( ) C ( , ) ( ) 2 ( ) 2 ( ) 1 R ( ( ) x x x x x x x x x x x x x x x x (0,2) (2,0) (0,0) R ( ) ( ) ) ( ) x x x (43)where C is the covariance of . The mean
2,2 0,0 0,2 2,0 2,2 ( ) ( ) ( ) ( ) ( ) x x x x x (44)
and the corresponding mean flux vector by
2,2 0,0 0,2 2,0 2,2 ( ) ( ) ( ) ( ) ( ) x x x x x q q q q q (45) 7 EXAMPLE
Consider a rectangular grid of 20 x 40 square ele-ments in the vertical plane (Fig. 1) having a width L1 = 4, height L2 = 8, and elements with sides
0.2, where is the autocorrelation scale of Y. A water table boundary condition is imposed on the bottom of the domain. A constant deterministic flux Q = 0.5 (all terms are given in arbitrary consistent units) is prescribed at the top boundary and zero pressure head at the bottom. The side boundaries are impermeable. A point source of magnitude QS = 1 is placed inside the domain to render the mean flow locally divergent. The saturated hydraulic conduc-tivity field is statistically nonhomogeneous through conditioning at three points, two above and one be-low the source.
Figure 1. Problem definition and associated grid.
Our moment equations are free of any distribu-tional assumptions. To solve the original stochastic flow equations by Monte Carlo simulation on the same grid (using standard finite elements), we as-sume that Y is multivariate Gaussian. Prior to condi-tioning, Y is statistically homogeneous and isotropic with exponential autocovariance
/ 2
( )
Y Y
C e (46)
where is separation distance, Y2 is the variance of
Y, and is its autocorrelation scale. We started by generating an unconditional random Y field on the grid by using a Gaussian sequential simulator, GCOSIM (Gómez-Hernández 1991), with Y = 1, Y2 = 2 and =1. We took its values at the
condition-ing points to represent exact "measurements" and generated NMC = 3,000 realizations of a corre-sponding nonhomogeneous Y field by the same method. We solved (1) - (4) for each realization with a constant by standard finite elements, and calcu-lated sample mean pressure head and flux at each node, as well as sample variance and covariance of head and flux across the grid. This completed our conditional Monte Carlo simulation of flow in the example.
To render our direct solution of the conditional moment equations consistent with the Monte Carlo solution, we based it on the same conditional mean and autocovariance of Y as generated earlier by GCOSIM (in practical applications of our solution method, one would normally infer them geostatisti-cally from measurements). Figures 2 - 7 compare various moments as obtained by these two methods of solution. Each of these figures includes a contour map and a vertical profile along the center line of the grid (at x1/ = 2.0). Whereas the second order (in
Y) mean pressure head virtually coincides with
Monte Carlo (MCS) results (Fig. 2), the zero-order solution deviates from them slightly, especially near the upper flux boundary. Second-order horizontal (Fig. 3) and vertical (Fig. 4) mean fluxes correspond closely to their MCS counterparts, except for slight discrepancies near conditioning points and the point source. The zero-order results are also reasonably good, but somewhat less so.
It is not possible to obtain zero-order values of variance and covariance. Figure 5 shows a noticea-ble difference between contours of pressure head variance obtained directly to second order and by MCs. These differences are much smaller when viewed in profile. Variances of horizontal (Fig. 6) and vertical (Fig. 7) flux show very good agreement with MCS results, with some exceptions near condi-tioning points and the point source. Considering that our example concerns a strongly heterogeneous me-dium with Y2=2, our direct second-order finite
ele-ment algorithm for solving the moele-ment equations appears to work very well.
q (x ) n (x )=0 2 4 0 2 4 6 8 q (x ) n (x )=0 x1/ x2 / =0 Q Condition-ing point Point source Parameters: Y = 1.0 Y2 = 2.0 = -1.0 2 = 0.0 = 1.0 Q = 0.5 QS = 1.0 NMC = 3000 x1 = 0.2 x2 = 0.2 QS
-0.5 -1.0 -1.5 -2.0 -2.5 -3.0 -3.5 -4.0 x1/ x2 / 0.0 1.0 2.0 3.0 4.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 -3.0 -3.5 -4.0 <> x2 / -4.0 -3.0 -2.0 -1.0 0.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 MC 0thorder 2ndorder
Figure 2. Contour map of mean pressure head and a vertical
profile along x1/=2.0. 0.050.10 0.00 0.00 -0.0 5 -0.1 0 0.05 0.10 -0.10 -0.05 0.000.0 5 0.00 -0 .05 -0 .10 x1/ x2 / 0.0 1.0 2.0 3.0 4.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 <q1> x2 / -0.2 -0.1 0.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 MC 0th 2nd
Figure 3. Contour map and vertical profile along x1 /=2.0 of
horizontal mean flux q1.
-0.7 -0.9 -1.1 -0 .7 -0 .5 -0.3 -0 .5 -0.7 -0.4 -0.6 x1/ x2 / 0.0 1.0 2.0 3.0 4.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 <q2> x2 / -2.0 -1.5 -1.0 -0.5 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 MC 0thorder 2ndorder
Figure 4. Contour map and vertical profile along x1/=2.0 of
horizontal mean flux q2.
0.05 0.10 0.20 0.40 0.60 0.80 0.30 0.50 x1/ x2 / 0.0 1.0 2.0 3.0 4.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 C(x, x) x2 / 0.0 0.5 1.0 1.5 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 MC 2nd 0.05 0.10 0.20 0.30 0.40 0.60 0.50 0.50
Figure 5. Contour map and vertical profile along x1/=2.0 of
variance of pressure head.
Cq1q1(x, x) x2 / 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 MC 2nd 0.05 0.05 0.10 0.05 0.15 0.20 0.2 5 0.10 x1/ x2 / 0.0 1.0 2.0 3.0 4.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0
Figure 6. Contour map and vertical profile along x1/=2.0 of
variance of horizontal flux.
0.20 0.40 0.20 0. 40 0.10 0.20 0.30 0.40 0.20 0.30 0.20 x1/ x2 / 0.0 1.0 2.0 3.0 4.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Cq2q2(x, x) x2 / 0.0 0.2 0.4 0.6 0.8 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 MCS 2ndorder
Figure 7. Contour map and vertical profile along x1/=2.0 of
8 CONCLUSION
It is possible to linearize the steady state stochastic unsaturated flow equations by means of the Kirch-hoff transformation, integrate them in probability space, and obtain exact integro-differential equa-tions for the conditional mean and variance-covariance of transformed pressure head and flux. Approximating the latter equations to second order and solving them by finite elements for conditional mean pressure head, flux, and associated variance-covariance leads to results that compare favorably with those obtained by conditional Monte Carlo simulation, even under divergent flow in a soil that is strongly heterogeneous.
9 ACKNOWLEDGEMENTS
This work was supported in part by the U.S. Nation-al Science Foundation under Grant EAR-9628133, and by the U.S. Army Research Office under Grant DAAD 19-99-1-0251.
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