Download ^ dt dx [^ dx) + dz \ dz + *v

ModelCARE 90: Calibration and Reliability in Groundwater Modelling (Proceedings
of the conference held in The Hague, September 1990). IAHS Publ. no. 195, 1990.
HOW EFFECTIVE ARE EFFECTIVE MEDIUM PROPERTIES?
S. MISHRA
INTERA Inc., 6850 Austin Center Boulevard, Suite 300, Austin,
Texas 78731, USA
J. L. ZHU & J. C. PARKER
Virginia Polytechnic Institute & State University, 241 Smyth Hall,
Blacksburg, Virginia 24061-0404, USA
ABSTRACT Effective medium properties describing unsaturated
flow in heterogeneous media are derived using a perturbation
approach and are numerically evaluated from fine-scale transient
simulations. Application of these parameters with coarser grids
does not appear to cause any significant difference between
hetereogeneous and equivalent homogeneous system responses.
INTRODUCTION
Spatial variability in natural soil materials is often an impediment to field-scale
modelling of unsaturated flow and transport processes because of increased data
requirements and computational burden. An alternative modelling approach
which has received growing attention in recent years is based on treating the
actual heterogeneous medium as an equivalent homogeneous system with a set of
effective properties.
Several studies dealing with unsaturated flow in
heterogeneous media have appeared in the recent literature. These include
analytical studies using spectral-perturbation methods (Yeh et aJL, 1985;
Mantoglou & Gelhar, 1987) as well as numerical simulations in single- or
multiple realizations of distributed unsaturated flow parameter fields (El-Kadi,
1987; Ababou & Gelhar, 1988; Hopmans et ah, 1988; Binley et a l , 1989).
Our objectives in this paper are twofold. First, we present a derivation of
effective medium properties from a perturbation approximation of Richards'
equation, and show how these can be obtained from fine-scale simulations of
transient unsaturated flow in a heterogeneous system. Next, we investigate how
the behavior of the effective homogeneous system deviates from that of the
actual heterogeneous system, when the mesh is progressively coarsened beginning
with the grid at which the effective properties were computed.
PERTURBATION ANALYSIS OF RICHARDS' EQUATION
Transient unsaturated flow in a 2-D vertical domain is described by Richards'
equation under the assumption of a passive air phase, viz.:
c
Ôh
^ dt
=
d_ (K dh)+
dx [^ dx)
+
d_ (K 9h , K\
dz \ dz
+
*v
(1)
W
where h is pressure head, K is unsaturated conductivity, C=d0/dh is moisture
capacity with 8 the water content, t is time, and x and z are horizontal and
vertical space coordinates, respectively. At the local (grid) scale, we assume the
K(h) function to be isotropic and the C(h) function to be non-hysteretic, with
the C-K-h relationship characterized via Van Genuchten's (1980) model:
521
522
SMishra et al.
e = 9T + h + {ah)n\
-m
(e, - er)
(2)
m 2
-m 2
/
11 1
n
K = K s {1 - (ah) " [l + ( a h f ] } / {[l + (ah) ]
}
(3)
where 9, is the saturated water content, 6r is a 'residual' water content, K3 is the
saturated hydraulic conductivity, a and n are Van Genuchten (VG) model
parameters, and the exponent m = 1-1/n. Differentiation of (2) readily yields
the moisture capacity, C = dfl/dh.
For heterogeneous soil materials, it is assumed that the spatial variability of
C and K can be treated in the context of stationary random functions, viz:
K(h) = K,(h) e ^
C(h) -
,
5(h) + C'(h),
E[lnK\ = ln(K9),
E[C] = C
,
E[f ] = 0
(4)
E[C'] = 0
(5)
where Kg is the geometric mean conductivity, C is the mean capacity, f is the
fluctuation in InK, C' is the fluctuation of C, and E[ ] denotes the expectation
operator. Assuming further that stochastic hydraulic conductivity and water
capacity functions result in the hydraulic head, h, being a stationary stochastic
process, we have:
h = h + h' , E[h] = h , E[h'] = E[ôh'/Ôt] = E[dh!/dXi] = 0
(6)
with Xj=x,z. Substituting for C, K and h in (1), taking the expectation of both
sides and rearranging, we obtain a stochastic mean flow equation having the
same form as the Richards equation:
C
e |
= & ( K e x §)
+ & (Ke z f
+ K6z)
(7)
with an "effective" moisture capacity defined as:
C e = C + E[C f
] / (g)
(8)
and "effective" hydraulic conductivities defined as:
K
ex,.
K,
{Ele', + E [ e « ,
/ (?&H))}
(9)
The effective properties as defined by (8)-(9) exhibit added nonlinearity due to
dependence on spatial and temporal derivatives of the mean head, h. Yeh et ah
(1985) and Mantoglou & Gelhar (1987) have used spectral representation
techniques to evaluate expectations of perturbation derivative terms similar to
those occuring in (8)-(9) in order to derive closed-form analytical expressions for
the effective parameters. In this work, such terms are evaluated numerically
from fine-scale simulation results using a Monte-Carlo methodology. Details of
the two-dimensional finite element model used to solve (1) for this purpose may
be found in Kuo et ah (1989).
RANDOM FIELDS OF SOIL HYDRAULIC PROPERTIES
The stochastic parameters of interest (9,, 9r, K*, a, n), which completely
characterize the C(h) and K(h) functions, are assumed to be second-order
stationary random functions. Distributed fields of these parameters, for use in
the fine-scale simulations, are generated using an indirect protocol suggested by
Mishra et aL (1990). Their procedure, based on the assumption that variability
How effective are effective medium properties?
523
in soil properties is essentially due to spatial variations in particle size
distributions, can be summarized as follows:
(a) Generate spatially variable autocorrelated fields of saturated water content,
6S, and particle size distribution moments, ^[Znd] and cr[Znd], assuming these
variables to be normally distributed with known mean and auto-covariance,
and assuming particle diameter, d, to be lognormally distributed.
(b) Compute the full particle size CDF at each node of the simulation domain
from these randomly generated variables.
(c) Estimate the stochastic VG model parameters (a, n) and saturated
conductivity (K*) from particle size distribution data via physically based
pore-structure models, with the residual water content, 6r, taken to be equal
to zero for convenience.
Table 1 shows the statistics of the input and the output variables in the random
field generation process for a hypothetical 30x30 porous medium used in this
study. Distributions of a, n and K» appear to be lognormal, as observed in
recent field measurements of soil spatial variability (Hopmans et aL, 1988). Note
that d is expressed in units of cm, a in cm"1, and K* in cm h"1.
TABLE 1 Statistics of random fields in hypothetical medium.
Input
Output
Std Dev.
Corr. Length
6,
-5.50
0.55
0.35
0.25
0.25
0.02
4 blocks
4 blocks
4 blocks
ln(n
hi(K. )
-4.32
0.58
1.01
0.313
0.018
0.405
3.8 blocks
3.5 blocks
3.8 blocks
p
a
Ind
Ind
EVALUATION AND PARAMETERIZATION OF EFFECTIVE PROPERTIES
Effective soil hydraulic properties (i.e., the effective C-K-h relationship) are
evaluated in a manner similar to that presented by Zhu et ah (1990). Figure 1
depicts the system geometry and applied boundary and initial conditions used in
this study. The governing equation for transient unsaturated flow, (1), is first
solved numerically to simulate the infiltration event (i.e., t < 4.7 h) for 10
realizations of the heterogeneous system. At each time step, the following
quantities are calculated for every element by taking expectations over Jthe total
number of simulations: (a) mean pressure head, h, and its gradients, ôh/ôt and
dh/dz, (b) mean water saturation, S=E[0/0.], (c) variance of log-hydraulic
conductivity, rf /ky a n d (d) the fluctuation term E[exp{f'}(dh'/i9z)]. The mean
soil moisture capacity and the geometric mean conductivity are evaluated from
(2) and (3), respectively, as functions of the mean pressure head and the
geometric means of K5, a and n. The choice of the geometric mean as an
averaging operator is motivated by the lognormality of these parameters, and is
also based on results from stochastic analyses of steady-state unsaturated flow in
heterogeneous media (Mishra & Parker, 1989).
S.Mishra et al.
524
h = 0
or
q = 0
= 0
-150 cm
FIG. 1 Schematic of hypothetical system and boundary conditions.
Since the mean flow is vertical, averaging is also carried out horizontally
over all 30 elements for each simulation, otherwise 10 realizations would be
insufficient for the averaging to produce reasonable results. This procedure
produces 30 sets of the above variables for any given time step, with each set
representing averaged values at a given depth. Note that K e will not affect the
computation of mean flow in the vertical direction as uniform pressure heads are
prescribed along the top and bottom boundaries. This eliminates the need for
evaluating the expectation term E[exp{f'}(dh'/dx)}. Furthermore, the effective
capacity, C e , is taken to be equal to the mean capacity, C, since numerical
evaluations have indicated that E[C'(Sh'/#0] is at least one order of magnitude
smaller than C at all times (Zhu et ah, 1990).
Based on an examination of the spatial and temporal distribution of
E[exp{f'}(cm'/<9z)], this term was found to be reasonably well approximated by:
1
&% i - "!f>bi t - )
(10)
where b={dh/<9z}, b m = M a x | b | with the maximum taken over the spatial
domain. S u b s t i t u t i o n ^ (10) in (9) then yields the required expression for Kg .
Effective C e -K e -h relationships were computed via the procedure described
above for the hypothetical heterogeneous system. These were then used to
predict the response of the equivalent homogeneous system by solving (7) for the
same initial and boundary conditions used in the base simulations. Excellent
agreement was obtained between the response of the hetereogeneous system and
that of the equivalent homogeneous system - both at early times when flow is
highly transient as well as at late times when steady state is approached.
Similar results were obtained by Zhu et aL (1990), who also concluded that the
time-dependence of effective properties must be taken into account to obtain an
accurate large-scale representation of transient flow phenomena in heterogeneous
media. Simple geometric averaging of hydraulic properties (e.g., K», a, n) might
yield reasonably accurate results for steady-state flow scenarios, but will be
inadequate for reproducing the effects of transient unsaturated flow.
How effective are effective medium properties?
525
OPTIMUM LEVEL OF DISCRETIZATION
If unsaturated flow is to be simulated by substituting large-scale effective
parameters for the distributed parameter field in the original mesh, the primary
savings in computation will be due to reductions in problem nonlinearity and
data requirements.
The major burden of computation, i.e., solution of a
nonlinear system of equations, will be unchanged as it depends on the total
number of grid blocks. Substantial savings in computing can only be achieved if
the number of grid blocks in the system is reduced, albeit at the risk of
increasing truncation error. It is therefore useful to investigate the performance
of the equivalent homogeneous system as a function of mesh size.
The approach taken in this work for determining an optimum level of
discretization involves progressively coarsening the mesh (beginning with the grid
at which the effective parameters were computed) to examine how the response
of the effective homogeneous system deviates from that of the actual
heterogeneous system. The system considered is the same as that shown in Fig.
1, but with both infiltration and redistribution periods taken into account. Since
the effective parameters were determined only from fine-scale simulations of
infiltration, adding the redistribution period provides a further test of the
applicability and robustness of these effective parameters under a different set of
boundary conditions. Four levels of discretization are considered: (a) 30x30, (b)
25x25, (c) 20x20, and (d) 15x15. For each of these cases, effective parameters
determined from simulations of infiltration into the 30x30 heterogeneous system
were used to compute the response of the equivalent homogeneous system with
the same initial and boundary conditions.
Figure 2 shows the calculated rate of infiltration into the system across the
top boundary.
The base case corresponds to the response of the 30x30
heterogeneous system. The responses of the equivalent homogeneous systems
generally agree well with the base case results, although the infiltration rate
tends to be overpredicted when effective parameters are used.
1200 -m
Y
F
o
Y
"
"
800-
H)
F
vo
-'
<i>
•+->
"
"
_
ooooo
a DODO
" " »
• »o«o
Heterogeneous system
15x15 equiv homogeneous
20x20 equiv homogeneous
25x25 equiv homogeneous
30x30 equiv homogeneous
u
cc
r
o 400-
-*-<
o
L.
-4->
-
M—
_c
0 | I I I I I I I I I | I M I M I I I | I I I I I I M I | I 1 M I I M I | II I I À \ I I \
0
1
2
3
4
5
Time (h)
FIG. 2 Comparison of infiltration rate for heterogeneous and
equivalent homogeneous systems.
526
S.Mishra et al.
A comparison of the head distributions for the base case and the four
equivalent homogeneous systems are shown in Fig. 3. Pressure profiles at two
different times during infiltration (t < 4.7 h) and redistribution (t > 4.7 h) are
depicted in these figures. There is good agreement between the two sets of
responses in general, although the infiltration data (top) are much better
matched than the redistribution data (bottom) when using effective parameters.
The worst agreement occurs consistently for the 15x15 mesh equivalent
homogeneous system, which is easily explained by the higher truncation error
associated with this coarse discretization.
Infiltration
150-
E
o
.91
I
50-
t = 4.6 h
0 I ! I I
-200
I [ I I I I
-150
-100
i i i i |
-50
0
150-
t = 18 h
100-
E
o
_D1
50-
Redistribution
0
] I 1 I ! I I I I I | I t I 1 I I I I I | I I I I 1 I I I I | I I I 1 I I | I I j
-200
-150
-100
-50
Pressure Head ( c m )
FIG. 3 Comparison of head distributions for heterogeneous and
equivalent homogeneous systems.
How effective are effective medium properties?
527
Two other interesting observations can be made from the data presented in
Fig. 3. The maximum deviation between the heterogeneous and the equivalent
system responses appears to occur at the point of inflection on the pressure
profiles. Since this corresponds to the location where pressure (and saturation)
gradients are changing the most, it is apparent that effective parameters are
unable to reproduce large gradients at the local scale. The second observation
concerns the tendency of the equivalent homogeneous system pressure profiles to
converge to the hetereogeneous system profile at large redistribution times. This
implies that the use of effective properties will be even more appropriate when
the effects of a step change at a boundary (i.e., from infiltration to
redistribution) have dissipated throughout the system.
DISCUSSION O F RESULTS
The computations reported here indicate that effective medium properties can be
effectively used to model unsaturated flow in heterogeneous media, at least under
the conditions of this study. The variances in soil properties considered in this
study are not large, but they are in the normal range for field soils. However,
our critical assumptions were: (i) 2-D flow with 1-D boundary conditions, and (ii)
absence of any long-range correlation (relative to the scale of simulations). It is
conceivable that the use of effective properties may not be appropriate when
variances and/or correlation length scales are large, when initial conditions are
non-uniform, or when more complex boundary conditions are imposed (i.e.,
several cycles of infiltration-redistribution, presence of a strip source, etc.). On
the other hand, addition of a second (or third) active dimension to the flow
problem might help relax some of the constraints on averaging.
In summary, the results from this study suggest that for heterogeneous
media with small spatial variability and correlation lengths, effective properties
can be useful for modelling transient unsaturated flow. Furthermore, reasonably
accurate predictions of flow can be made with the effective properties even when
these are used in conjunction with coarser meshes where the number of grid
blocks is reduced from the base case by as much as 50%.
ACKNOWLEDGEMENTS Financial support for this study was provided by the
Appalachian Soil and Water Conservation Laboratory of USDA-ARS under
contract 58-3615-7-003.
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Binley, A., Beven, K. k Elgy, J. (1989) A physically based model of
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El-Kadi, A. (1987) Variability of infiltration under uncertainty in unsaturated
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