UC Santa Barbara

In the stochastic analysis of steady aquifer flow, the log hydraulic conductivity is the random “input” and the piezometric potential is the random “output.” Their joint behavior is governed by a differential equation that, in view of the random nature of its dependent (piezometric potential) and independent (log hydraulic conductivity) variables, represents a stochastic differential equation. The analysis of the distributional properties of the piezometric potential field involves the product of the gradients of log hydraulic conductivity and of the piezometric potential. Previous research in this field assumes that such product of gradients is small in some sense. This paper derives a closed‐form expression for the standard deviation, and hence the order of magnitude, of the product of the random gradients of log hydraulic conductivity and of piezometric potential. It was found in this research that for statistically homogeneous log hydraulic conductivity fields (1) the product of the random gradients may or may not have a zero mean, depending on whether the specific discharge is a constant or a random quantity, respectively; (2) under joint normality of the log hydraulic conductivity and the piezometric potential fields, their random gradients are statistically independent if the specific discharge is constant but are dependent when the specific discharge is random; (3) the standard deviation of the product of random gradients is proportional to the variance of log‐hydraulic conductivity times a term involving three quantities: the covariance of the piezometric potential, the covariance of the log hydraulic conductivity, and the cross covariance of the latter two fields; (4) a necessary and sufficient condition for the smallness of the product of random gradients is that the second derivatives of the covariance of the log hydraulic conductivity, the covariance of the piezometric potential, and the cross covariance of the latter two random fields be finite and that the variance of log hydraulic conductivity be much less than one. This paper also reviews some fundamental principles on the stochastic analysis of random fields and their importance to the modeling of log hydraulic conductivity fields and to the analysis of subsurface flow. Specifically, the paper highlights the role of Gaussian distributional assumptions in deriving key results of stochastic groundwater flow via perturbation analysis. Copyright 1990 by the American Geophysical Union.