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    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


    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.


    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


  • [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


  • 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


    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

  • 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)

  • G(y)>0


    TANGENT AT y *

    Figure lb. Standard space representation of the performance function of Figure la, illustrating the first-order approximation for P[Z

  • Figure 2. Illustration of the margin against exceedance, Z, and probability of ex- ceedance, p. fz(z) is the (unknown) PDF of Z.


  • 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

  • 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 decomposit