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STOCHASTIC FINITE ELEMENT ANALYSIS OF
GROUNDWATER FLOW USING THE FIRST-
ORDER RELIABILITY METHOD
Jeffrey D. Cawlfield Dept. of Geological Engineering
University of Missouri-Rolla Rolla, MO 65401
and Nicholas Sitar
Dept. of Civil Engineering University of California, Berkeley
Berkeley, CA 94720
Abstract
An approach for stochastic analysis of groundwater flow based on the first-order reliability method is presented. The method can fully utilize any level of probabilistic information from the minimum of second-moments to the complete full joint distribution. It is well suited for problems in which the statistical information is incomplete, as is common when considering groundwater flow. The results of a first-order reliability analysis include an estimate of the probability of exceeding a specified performance criteria and the sensitivity of the stochastic solution to changes in the uncertain variables and their moments. First-order reliability results from an analysis of one-dimensional flow with uncertain hydraulic conductivity compare well with results from a previously presented Monte Carlo analysis. In our two-dimensional stochastic finite element model, the uncertain variables are the element hydraulic conductivities and the values of constant head or point flux boundary conditions. Example two-dimensional applications are presented for confined flow between two constant head boundaries. The effect of spatial correlation between element hydraulic conductivities on the probability estimate and the sensitivity information is investigated.
Introduction
In recent years much attention has been given to the incorporation of uncertainty into the analysis of subsurface flow in order to explicitly account for porous media variability. Sources of uncertainty which arise during analysis of subsurface flow were listed by Tang
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[1984] as follows: inherent spatial variability, random measurement errors, systematic sampling bias, calibration errors and estimation errors from insufficient sampling. In addition, the solution model may contribute uncertainty due to model simplifications and assumptions. The analyst must attempt to account for each source of uncertainty, and a solution model should incorporate as much of the resulting statistical information as possible.
In this paper we assume that the statistical characteristics of the uncertain parameters are available from an appropriate analysis and we present a method for utilizing that statistical information in a stochastic model using the first-order reliability method. The first-order reliability method is an attractive stochastic modeling technique for a number of reasons:
(1) as a minimum, the method requires only second-moment information concerning the uncertain variables; however, additional information (such as marginal distributions) can be incorporated into the solution,
(2) the method directly yields an estimate of probability associated with a particular uncertain event; for example, exceedance of fluid head at a given location in a flow domain,
(3) the first-order reliability calculations directly yield sensitivity information,
(4) the method can be used with numerical solution models,
(5) the method is efficient even for very low probability events.
After a brief review of previously used stochastic models, we present the theoretical background of the first-order reliability method and then show the basic implementation of the method to specific groundwater flow problems. In addition, example one-dimensional analytical flow solutions and two-dimensional stochastic finite element analyses are presented. A more extensive treatment can be found in Sitar et al. [1987] and Cawlfield and Sitar [1987].
Previous Stochastic Modeling
Early groundwater flow models which incorporated uncertainty assumed statistical independence between hydraulic conductivity values (e.g., Warren and Price [1961], McMillan [1966] and Freeze [1975]). However, hydraulic conductivity (or transmissivity) usually exhibits spatial correlation over some distance. Gelhar [1976] incorporated spatial correlation through an application of stochastic continuum theory, where the spatial covariance function of hydraulic conductivity was related to the covariance function of the head random field through spectral analysis. The stochastic continuum approach relies upon small perturbations of the random variables and, in order to obtain closed-form functional relationships between hydraulic conductivity uncertainty and head uncertainty, the method is usually only applied to tabular aquifers of infinite extent. Additional work using the stochastic continuum approach has been carried out using spectral analysis (e.g, Bakr et al. [1978], Gutjhar et al. [1978], Gutjahr and Gelhar [1981] and Mizel at al. [1982]). Other solution methods have also been used to analyze the stochastic continuum (e.g., a Green's function approach by Dagan [1982], and classical small perturbation theory by Tang and Pinder [1977]).
Smith and Freeze [1979a and 1979b] used a nearest-neighbor technique to model spatial correlation in conjunction with Monte Carlo simulation. Their approach accounted for more complex boundary conditions and problem geometry than were considered with
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stochastic continuum theory. Others have used geostatistical techniques to incorporate spatial variability into Monte Carlo analyses (e.g., Delhomme [1979], Clifton and Neuman [1982] and Clifton [1985]).
These previous methods of stochastic modeling have concentrated on analysis of the uncertain moments, usually the mean and variance of the random head. Although these moment calculations are important, decisions are usually based on a quantitative evaluation of risk. For use in formal decision analysis, results must be expressed as a measure of probability, and the first-order reliability method directly yields such measures by estimating the probability associated with a particular flow event. The method also directly provides information concerning the sensitivity of the stochastic solution to changes in the uncertain variables and their moments. Therefore, the first-order reliability method provides an attractive alternative to the available stochastic continuum and Monte Carlo techniques.
First-Order Reliability Theory
Introduction
The first-order reliability method has been used in structural stochastic analysis for some time. The method can, in principle, incorporate any amount of probabilistic information, from the minimum requirement of second moments to the full joint probability distribution function (PDF). In groundwater flow problems the statistical information is typically incomplete, but often the marginal distributions are known or can be reasonably assumed (i.e., hydraulic conductivity is commonly assumed to be lognormally distributed). Therefore, the first-order reliability method is particularly well suited for stochastic groundwater flow analysis.
The method is based on two main concepts which will be discussed in some detail:
(1) formulation of a performance function which describes the flow behavior of interest in terms of the random variables, and
(2) transformation of the problem into standard normal space, where an estimate of probability is obtained.
The performance function is symbolized as Z=g(X), where X is a vector of the n uncertain parameters in a problem. For example, X may contain hydraulic conductivities, boundary conditions and problem geometry. For convenience, the limit state is defined as Z=g(X) = 0. The limit state is, in general, an n-dimensional hypersurface which divides the performance into two regions: a region where Z=g(X) < 0, and a region where Z=g(X) > 0. We will refer to the region where Z <. 0 as the region of exceedance, and the objective is to estimate the probability of exceeding an arbitrary target value (symbolized as P[Z <_ 0]).
To illustrate these concepts, consider two-dimensional space and a function q = f(X„X2). The objective is to evaluate the probability that q is greater than some value of interest, say qt. The performance function is formulated as follows:
Z = g(X) = qt - q = qt - f(X„ X2)
Figure la illustrates this performance function, showing the limit state surface as a line for this case. The probability that q J> qt is equivalent to P[Z <. 0], which is obtained by integrating the joint PDF in the region where g(X) <. 0 as follows:
P[Z <. 0] = FZ(0) = / fX(x)dx
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where Fz(0) is the cumulative distribution function (CDF) for Z evaluated at 0, fx(x)dx is the joint PDF of X and the integral is taken over the region where g(X) <. 0. Figure 2 gives a graphical interpretation of Z and its relation to the limit state. Z is a margin
Figure la. Two-dimensional example performance function, Z = g(X) = qt - f(X„ X2).
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G(y)>0
HYPERPLANE^
TANGENT AT y *
Figure lb. Standard space representation of the performance function of Figure la, illustrating the first-order approximation for P[Z <_ 0]. /3, a and y* are the reliability index, vector of direction cosines and design point as described in the text.
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Figure 2. Illustration of the margin against exceedance, Z, and probability of ex-ceedance, p. fz(z) is the (unknown) PDF of Z.
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against exceedance (i.e., a safety margin), and the probability content which we seek to evaluate is shown cross-hatched in Figure 2.
The exact evaluation of P[Z <_ 0] requires an n-fold integration of the joint PDF. This exact evaluation is difficult for practical problems because the joint PDF is unknown, the n-fold integration is over a generally complex region where Z = g(X) <. 0 and the integration is virtually impossible for n greater than about 2 or 3.
Reliability Index
Because of the difficulties in evaluating the n-fold probability integral, approximate methods for evaluating the probability content defined by Z <. 0 have been developed based on a measure of reliability known as the reliability index. Within the context of second moment information, Cornell [1969] defined the reliability index Q3) as follows:
P = Mz / °z
where / ^ and CTZ are the mean and standard deviation of the performance function, Z, respectively. Figure 2 illustrates the concept of Cornell's reliability index, which is shown to represent the distance between the mean performance and the limit state (measured in units of the standard deviation of Z). Therefore, /3 can be thought of as a measure of the margin against exceedance (or safety margin), and a high /3 indicates a low probability of exceedance.
If the performance function is linear, the moments JJL^ and o~z may be computed exactly in terms of the means and covariances of the random variables X. However, when g(X) is a nonlinear function of X, the full joint PDF is required to exactly evaluate ^ and CTZ. Because the full joint PDF is seldom known, early approaches used a first-order mean centered Taylor series approximation as follows [Ang and Cornell, 1974]:
Mz *> g(M)
a z2 = 2 S ôg/3Xj dg/ôXj pjj CTJCTJ
i J
where the double summation is over all combinations of i and j , and the partial derivatives are evaluated at the point M (the vector of mean values of X). The approximation requires only second moment information concerning the X random vari- ables, and the performance function must be differentiable with respect to each X variable. The reliability index obtained using these approximations is known as a mean value first-order second moment reliability index (/?MVFOSM)-
The MVFOSM index has been shown to have a major shortcoming: the value of J^MVFOSM ' s n o t invariant with respect to mutually consistent formulations of Z = g(X) [Hasofer and Lind, 1974]. Consider, for example, a performance function Z = qt-X,X2, which could be equivalently formulated as Z = (qt / X,)-X2. In both cases, Z will be zero, less than zero and greater than zero for the same values of X„ X2 and qj, so the performance functions are mutually consistent with respect to the limit state. However, the value of /^MVFOSM 'S n o t necessarily equal for these two performance functions. This result is unsatisfactory, because consistency requires that both the linearization point and a unique performance function must then be specified for each problem.
Hasofer and Lind [1974] showed that the invariance problem can be overcome by linearization on the limit state surface (rather than at the mean point), because mutually consistent performance functions all have identical surfaces defined by Z = 0. The conceptualization of the reliability index was illustrated in Figure 2 as a measure of the distance between mean performance and the limit state. Following that logic, Hasofer and Lind [1974] defined an invariant reliability index in terms of the minimum distance
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between the limit state and the mean point in standard space. The standard space consists of uncorrelated variâtes with standard deviations equal to one and means equal to zero (therefore, the mean point (M) in standard space is conveniently the origin). Figure lb is the standard space representation of the original space shown in Figure la.
The standard space exhibits rotational symmetry; that is (refer to Figure lb), for all hyperplanes of equal distance from the origin, the probability content P is constant within a second moment representation (i.e., upper Tchebycheff bound), regardless of the orientation of the hyperplane [Der Kiureghian and Liu, 1986].
Evaluation of the reliability index using the Hasofer-Lind definition requires that the original random variables (X) be transformed into uncorrelated standard variâtes (Y). This transformation is conveniently expressed as
Y = L"1 D"1 (X - M)
where D is a diagonal matrix of standard deviations of X, L is a lower-triangular decomposition of the correlation matrix
R = [ Pij ]
and M is the vector of means of X. The transformation matrix L"1 may be evaluated through Cholesky decomposition.
The point on the limit-state surface nearest to the origin in standard space is known as the design point (y*), and its counterpart in the original space is symbolized as x*. The reliability index, /3, can be expressed in terms of the design point as
fi = a* y*
where a* is the unit normal at the design point directed towards the region where Z <_ 0. An appropriate optimization algorithm for determining the design point is given by Madsen et al. [1986], as
Yi+i = [<*m + G(yi)/|VyG(y i)|]a iT
here g(X)= g(DLY + M) = G(Y), VyG(y) signifies the gradient of G(y) with respect to the vector yj and | VvG(y) | signifies the magnitude of the gradient vector. The design point (y*) can be thought of as the most likely set of circumstances leading to exceedance. In standard normal space the contours of probability are concentric spheres, and the probability decays with distance from the origin. Therefore, the design point, which is the closest point to the origin on the limit state surface, has the highest likelihood of occurrence of any point in the region where Z <_ 0. Without complete joint PDF information, the actual form of probability decay is unknown and, strictly speaking, the design point is not necessarily the most likely set of circumstances leading to exceedance. However, such an interpretation is plausible as a general concept.
Probability Estimate
With only second moment information, in a strict sense, P[Z <. 0] cannot be estimated. However, an ad hoc approximation given by
P[Z <. 0] = *(- /3)
is often used, where *( ) is the standard cumulative normal probability function. This approximation assumes that Z is normally distributed.
Now consider the case where the joint PDF is known. Then it is convenient to
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transform the X random variables into standard normal variâtes Y. The transformation
Y = T(X)
is nonlinear for nonnormal X, and is given in terms of the joint PDF of X [Hohenbichler and Rackwitz, 1981; Der Kiureghian and Liu, 1986].
Standard normal space possesses rotational symmetry in a stricter sense than was mentioned previously for standard space. Referring to Figure lb, the probability P defined by the hyperplane is given exactly by *(- jS). Therefore, for a general limit state surface, a first-order approximation of P[Z <. 0] is defined by
P[Z < 0] - *(- j8)
This approximation is obtained by replacing the actual limit state surface with a tangent hyperplane at the design point y*. The approximation works well because, in standard normal space, the probability density decays exponentially as a function of distance squared from the origin; therefore, most of the probability content in the region Z <. 0 is concentrated in a small area near the design point, where the tangent hyperplane is a very good approximation to the limit state surface (Figure lb). The approximation is good in most practical problems, but more refined approximations that account for the curvature of the limit state surface near the design point are discussed by Madsen et al. [1986] and Der Kiureghian et al. [1987].
In groundwater flow problems, the analyst usually operates with information that falls between the two extremes just discussed; that is, between only second moment information (where the probability estimate is ad hoc) and full joint distribution information (where the properties of standard normal space can be used to obtain an exact first-order estimate of probability). Usually, the groundwater analyst will be able to assign marginal distributions to most, if not all, the uncertain variables in the problem (i.e., lognormal hydraulic conductivities). For such problems with incomplete probability information, Der Kiureghian and Liu [1986] have developed a general joint distribution model which is consistent with all available statistical information and which greatly simplifies the transformation to standard normal space. The joint PDF according to this model is given by
fxW = 4>n(z,R0) [fXiW- fXn(xn)l / [*W - <P^n)]
where zj = *"'(Fxi(xj)), <£( ) is the standard normal PDF and *n(z, RQ) is the n-dimensional normal density of zero means, unit standard deviations and correlation matrix RQ-
The correlation matrix RQ is obtained using the integral relation [Der Kiureghian and Liu, 1986]:
Pij = / / [ (xi-mi) / o-j] [(xj-mj) / o-j] 02(z;,zj,poij)dzidzj
where the double integration is carried out from - °o to + oo. For each pair of variables Xj and X; with known marginal distributions and correlation coefficient pjj, the integral relation can be solved for the coefficient pQij, which is an element of Roij- Liu and Der Kiureghian [1986] have developed semi-empirical formulas which allow convenient calculation of selected marginal distribution types.
The transformation to the standard normal space for the general joint distribution model is given by
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^(FxiOq)) Y = LQ"1
*"1(FXn(%))
where LQ is the lower-triangular decomposition of RQ. This transformation is convenient, because it requires only marginal distributions for each X.
The reliability index is evaluated as before, by iteratively solving for the minimum distance between the limit state surface and the origin in standard space. If marginal distribution information is known, the general model is used and the transformation is carried out at each trial linearization point. The reliability index resulting from the calculations can then be used to give a first-order estimate of P[Z <. 0]. The accuracy of the estimate is a function of the curvature of the limit-state surface near the design point. In addition, the probability estimate assumes that the general joint PDF model is appropriate for the problem.
Sensitivity Information
As mentioned previously, the first-order reliability method directly yields sensitivity information. The most elementary measure of sensitivity is the partial derivative of 0 with respect to the values of the random variables at the design point (y*). Such a sensitivity measure gives an indication of how /3 will change when a particular random variable is slightly perturbed about its design point value. The sensitivity measure is given by
a* = y*/1 y* | = Vy* j8
where Vv*/3 signifies a vector of gradients of /3 with respect to the components of the vector y*. The values of a* result directly from solving for the design point.
For practical applications, a sensitivity measure in the space of the original random variables (X) is required. Using the chain rule of differentiation, Der Kiureghian and Liu [1985] suggest a scaled sensitivity measure in original space given by
7 = (Vx*/3) D / | (VX*J3)D |
The gamma vector indicates the relative importance of each X variable, and the use of the standard deviation matrix D as a scaling factor provides a measure of the sensitivity of /3 with respect to equally likely changes in the random variables X. Sensitivity vectors can also be derived to measure sensitivity with respect to deterministic parameters in the performance function. If the performance function is given by
g(X) = g(X, -n)
where T) is a vector of deterministic parameters, Madsen et al. [1986] have shown that
\ P = (V-n (g(x*,Ti)) / |VyG(y*)|
In groundwater flow applications, where either a target head or target flux is usually used as a deterministic parameter in the performance function, this formulation can be used to measure the sensitivity of )8 to the target value.
Madsen et al. [1986] also give a formulation for evaluating the sensitivity of /3 with respect to distribution parameters for the random variables X ( i.e., means and variances). Suppose fx(x) = fx(x, 6), where 0 is a vector of distribution parameters. Then
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V0 jS = a* J T 0
where JJQ is a matrix containing the partial derivatives of the transformation T with respect to the parameters 6, evaluated at the design point.
Example Applications
One Dimensional Problems With Analytical Flow Solutions
Sitar et al. [1987] used the first-order reliability method to evaluate the uncertain head in a one-dimensional flow region; the problem had been previously analyzed by Freeze [1975] using Monte Carlo simulation. The flow region is shown in Figure 3, and it is divided into ten equally dimensioned blocks which are assumed to have hydraulic conductivities (K) described by independent lognormal random variables with means of Y equal to 1.0 logjo cm/sec and standard deviations of Y equal to 0.5 log^o cm/sec (where Y=logioK). An analytical solution for the fluid potential ($) in any section of the flow region was given by Freeze [1975]. For comparison with the Monte Carlo results of Freeze [1975], the first-order reliability performance function is formulated to compare a target fluid potential at a specific location in the flow region with the fluid potential calculated using the analytical solution (cp), as follows:
z = g ( X ) = <pt-4>
An estimate of P[Z <_ 0] = P[ <p > 4>t ] *s obtained using the first-order reliability analysis, and a PDF for the fluid potential can be developed from the individual results over a large range of target values.
Consider the PDF for <p at x=45.0 cm in the flow region. Figure 4 compares the first-order reliability PDF to a normal curve fit to Monte Carlo results of Freeze [1975]. The curves agree quite well, although there is some deviation at either end of the PDF. This discrepancy is an artifact of fitting a normal curve to the Monte Carlo results, whereas the curve fit to the first-order reliability results simply connects the individually calculated PDF points. Had a normal curve been fit to the first-order reliability PDF points, the two curves would have been practically identical.
Figure 5 shows a comparison of PDF forms for <p near the constant head boundary at x=85.0 cm in the flow region (near the high potential boundary), where a skewed PDF is obtained. In this case, the two curves are nearly identical.
We should note that using the first-order reliability technique to develop a complete PDF form is somewhat inefficient, because the first-order reliability analysis yields only one estimate of probability for each calculation. However, for an analysis of a specific probability associated with one (or a few) target values, the first-order reliability approach will usually be much more efficient than Monte Carlo simulation, particularly for low probability events. Additionally, the first-order reliability evaluation provides sensitivity information, which is inefficient to obtain using Monte Carlo simulation.
Stochastic Finite Element Analysis of Two-Dimensional Flow
We have recently implemented the first-order reliability method in conjunction with a finite element model of two-dimensional steady state groundwater flow [Cawlfield and Sitar, 1987]. The performance function is formulated either in terms of nodal heads (hjj or flux at constant head nodes (qj) as follows:
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x=o X=L
s y
>
i=I K.
i=2
K2
i=3 K3
i=m^
K„H
i=m
Km * L
Figure 3. One-dimensional flow region of Freeze [1975]. L = 100.0 cm.
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First-Order Reliability
Monte Carlo
2 0 40 60 8 0 FLUID POTENTIAL (cm)
100
Figure 4. Comparison of First-Order Reliability PDF and Monte Carlo PDF for <p at x 45.0 cm.
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i
0.05
0.04
0.03
0.0Z
0.01
0
i
-
-
F i r s t — — ~Monte
1 — —-
-Order
Carlo
Reliab
1
Llity /\\
A i
k h *<t> - 8 5 . 0
m
1 L 20 40 60 80 FLUID POTENTIAL (cm)
100
Figure 5. Comparison of First-Order Reliability PDF and Monte Carlo PDF for <p at x •• 85.0 cm.
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g(X) = h t + 2 akhk
or
g(X) = qt + S a ^
where h t and qt are the target values, ak or a\ are deterministic œnstants and the summations are over all values k=l to m or 1=1 to p (m is the number of nodes and p is the number of constant head nodes). The solution then requires evaluation of the partial derivatives 3g/dXj for all random variables Xj. These can be explicitly calculated by direct differentiation of the numerical equations resulting from the finite element discretization.
For confined steady state flow, the finite element discretization results in the following general matrix equation:
Kh = q
where K is the global conductance matrix, h is the vector of nodal heads and q is the vector of fluxes. If the random variables of concern are the element hydraulic conductivities, direct differentiation of the two types of performance functions given previously leads to the following:
dg/dKj = -aT K"1 (ÔK/dKi) h
for the first type of performance function and
dg/diq = aT3Q/dKi
for the second type, where each entry of the ôQ/dK; vector is given by
oCty 3Kj = SKjj / ÔK; h + Ky (-K^ÔK / dKjh)
In the last equation, Kj; is a row vector with j taking on values from 1 to n and n is the number of nodes. IC* and ôK/dKj can be explicitly calculated from the global conductance matrix. The inverse matrix must be calculated only once during each iteration. The ôK/dK{ matrix must be calculated once for each uncertain Kj during each iteration, but the computational requirements are relatively minor because only a few terms in the conductance matrix are involved. In a similar fashion, explicit calculations of the partial derivative of the performance function with respect to other random variables can be formulated (for example, the vector of random variables (X) might include constant head or flux boundary values). The full derivations are not given here, but it should be apparent that explicit calculation of these partial derivatives increases the efficiency of the computations, as opposed to using, for example, a finite divided difference approximation of the derivatives.
In our model, each element hydraulic conductivity, values of constant boundary heads and values of externally applied point fluxes may be considered random variables. Problem geometry is assumed to be deterministic. Any of the random variables may be correlated to each other, and the model is capable of generating correlation coefficients between element hydraulic conductivities, based on an assumed autocorrelation functional form. The autocorrelation functional form may be assigned as a point function, in which case the element properties are derived using the spatial averaging techniques of Vanmarke [1983]. In the examples discussed in this section, we have assumed that the point hydraulic conductivity spatial correlation can be described by an exponential decay function, and the x and y direction spatial correlation functions are independent (separa-
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ble). Consider the finite element mesh shown in Figure 6, containing 66 nodes and 50
elements, with confined flow between two constant head boundaries. Suppose we are interested in estimating the probability that the head at node 33 ($33) is greater than some target head (<£t). The performance function is then formulated as
Z=g(X)=0 t - 033
Assume the constant head boundaries are deterministic and fixed, so that the only uncertain parameters are the element hydraulic conductivities, which are lognormally distributed and have the following point statistics:
yxK = 0.0001 and org; = 0.0001
In addition, the K field is isotropic, with isotropic exponential autocorrelation (i.e., the x and y autocorrelation functional forms are identical) as follows:
(-1 d I / 0.58) p = e
where d is the distance of separation between two K values and 8 is the scale of fluctuation. The scale of fluctuation is equal to two times the correlation integral scale, which is also often used to express correlation strength.
The steady state head at node 33 for a deterministic analysis of confined flow in a uniform porous media would be 100.0. Now consider two target heads at node 33 for our stochastic analysis: the relatively likely event of <p=110.0, and the much lower probability event of <p=140.0. The probabilities estimated using the first-order reliability method and the given statistical information are shown for these two cases in Figures 7 and 8 as a function of scale of fluctuation. First note that P[<p >. 110.0] is much higher than P[(p 140.0], as expected. Perhaps the most important result illustrated by this example is the sensitivity of the probability estimate to the scale of fluctuation. It appears that, for this type of flow problem, the scale of fluctuation is extremely important with respect to probability estimates, but only if the scale of fluctuation is less than about one-half the distance between the two constant head boundaries.
The gamma sensitivity results are shown in Figures 9 and 10 for the two target heads. The sensitivity results are not surprising, as it is expected that the stochastic solution would be most sensitive to the K values in elements immediately surrounding node 33. For the low probability event (<pl = 140.0), the solution is very sensitive to the K values in the elements immediately to the left of node 33, and also surprisingly sensitive to the element K values along the right constant head boundary.
To demonstrate potential applications to more complex problems, consider a layered flow region with a low conductivity material surrounded by a high conductivity material, such as shown in Figure 11. Both materials have coefficients of variation equal to 1.0, and autocorrelation occurs only between element K values in the same material (with isotropic exponential autocorrelation as before). If the same performance function is used to evaluate probabilities associated with the head at node 33, we find that the probability estimates are not affected too much by the introduction of the low conductivity layers (i.e., the probability estimates are within about 10% of the estimates for the one material case considered previously). However, the sensitivity information is much different for the two material case, as shown for the gamma sensitivity in Figure 12. Note that the solution is now most sensitive to values of K in the elements along the bottom row of the mesh (in the high conductivity zone closest to node 33). These results illustrate the potential utility of the first-order reliability method for more complex analyses.
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NO FLOW BOUNDARY
• = 50.0
NO FLOW BOUNDARY
Figure 6. Finite element mesh for example stochastic finite element analysis of confined flow region.
= 150.0
100. Q
100.0
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o d
< Ld
cc CD Q < LU
O l-O-
o 6
O
o-oO
MA- = O A - = 0 . 0 0 0 1 5, = 6 y
200 400 600 SCALE OF FLUCTUATION
800 1000
Figure 7. P[033 > 110.0] at node 33 as a function of scale of fluctuation (S).
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o
O"
O"
<£ to
<
o*
o ^
^ = 0 , . =0.0001 6, =6,
200 400 600 SCALE OF FLUCTUATION
800 1000
Figure 8. P[</>33 > 140.0] at node 33 as a function of scale of fluctuation (8).
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GAMMA SENSITIVITIES
0.1 2
0.13
0.13
0 J 4
0.13
0.11
0.12
0.13
0.13
0.13
0.1
0.12
0.14
0.14
0.13
0.08
0,12
0.16
0.17
0.14
0.04
0.07
0.27
0.28
0.08
0.02
0.04
0.24
0.24
0.04
0.08
0.11
0.16
0.16
0.13
0.12
0.13
0.14
0.15
0.15
0.13
0.13
0.14
0.15
0.15
0.13
0.13
0.13
0.14
0.15
Figure 9. Gamma sensitivity for P[#33 > 110.0].
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GAMMA SENSITIVITIES
0.09
0.12
0.14
0.14
0.11
0.08
0.1
0.12
0.12
0.1
0.07
0.1
0.12
0.12
0.1
0.07
0.11
0.16
0.16
0.11
0.06
0.09
0.32
0.33
0.11
0.03
0.02
0.16
0.15
0.03
0.04
0.08
0.12
0.12
0.09
0.13
0.13
0.1 +
0.15
0.17
0.19
0.16
0.16
0.18
0.21
0.18
0.15
0.15
0.15
0.19
Figure 10. Gamma sensitivity for P[#33 > 140.0].
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• . •
» • • » « • • • • ' t • .
S • ff •
• * a « « • • • * * H
Mat. 1:
' . M E 0
6
1 • • •
* • * it. 2 :
* • * * %•
Kx -Kyy\LK-
x = 8 y =
• * " .
KX=K
• * » . • •
= 500.0
,>!% = 200.0
• • .
= l x l C
t * •
1x10
a • * •
^
-6 , •
• • • -
• • .
* * .
v a o
• . * .
• « « « «
Figure 11. Two-dimensional confined flow region with two materials.
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GAMMA SENSITIVITIES
0.13
0.14
0
0
0.26
0.13
0.14
0
0
0.25
0.12
0.14
0
0.01
0.25
0.11
0.15
0.02
0.03
0.25
0.06
0.1
0.14
0.15
0.15
0.02
0.05 f
0.15
0.14
0.07
0.1
0.14
0.03
0.03
0.23
0,14
0.15
0.01
0.01
0.28
0.15
0.16
0
0
0.29
0.15
0.15
0
0
0.29
Figure 12. Gamma sensitivity for P[033 > 110.0] for two material case.
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Conclusions
The first-order reliability method provides an attractive technique for stochastic analysis because it is general and flexible, it is capable of incorporating a wide range of statistical information (including marginal PDF forms), it provides an estimate of probability associated with a particular subsurface flow event and it directly yields information concerning the sensitivity of the stochastic solution to changes in the random variables and their moments. The method can be used with either analytical or numerical solutions of the direct flow problem, and the performance function can be generally formulated such that the probability associated with a wide variety of flow events may be estimated.
Results from application of the first-order reliability analysis to one-dimensional problems with analytical flow solutions indicate that the first-order reliability estimates of probability compare very well to Monte Carlo simulation estimates. Our results using a stochastic finite element model of two-dimensional steady state flow illustrate the potential utility of the first-order reliability method in the analysis and interpretation of more complex groundwater flow problems.
ACKNOWLEDGMENTS
The work presented in this report has been supported in part by the National Science Foundation under PYI Award No. CEE-8352147. The opinions, findings and conclusions expressed in this report are those of the authors and do not necessarily reflect the views of the National Science Foundation. The authors wish to express their appreciation to Professors Armen Der Kiureghian and T. N. Narasimhan for their helpful review and suggestions throughout the development of this work.
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