Role of boundary conditions in convergence and nonlocality of solutions to stochastic flow problems in bounded domains

Abstract

Steady flow through a heterogeneous porous medium in a bounded domain is investigated using a recursive perturbation scheme. The effect of boundary conditions on a two‐dimensional flow with arbitrary variation of the mean flow (allowing large gradients) is investigated using analytical expressions for the head and velocity covariance functions. Boundary conditions were decomposed into deterministic and stochastic components. Two flow cases with the same zero‐order and different first‐order boundary conditions were analyzed. Boundary conditions are deterministic for the first case and random for the second. Significant differences between the two cases indicate random (or absence of randomness) processes must be modeled at boundaries. First‐order solutions for the head and velocity covariance functions for a bounded rectangular domain are derived. The resulting integral kernels involve Greens functions, and they are evaluated by numerical integration. Boundary conditions for higher‐order problems influence the absolute value and shape of the kernels (and therefore of the head and velocity variance and covariance functions). It is found that the validity of the perturbation scheme is dependent on the magnitude of the kernels and not only on the condition σf2 ≪ 1, where σf is the variance of the log hydraulic conductivity, assumed Gaussian and weakly homogeneous in space. The type of boundary conditions affects the values of the kernels and therefore determine the convergence limits for the problem. Milder head gradients also allow larger values of σf2. Nonlocality is also contingent on boundary type. Weakly homogeneous log‐fluctuating conductivity fields give rise to head and velocity covariances which are not weakly homogeneous. The inhomogeneous fields are obtained from a linear filter of the solution to the problem without stochasticity. Higher‐gradient regions induce higher head and velocity variances. In the presence of space‐varying gradients, nonlocal effects are most important away from the boundaries lpar;“center of the domain”). Small local head gradients result in small head and velocity gradients, and therefore where observations are made in stagnation regions, data should be analyzed taking into account this effect. The results may be used to interpret experimental data for columns or data taken where conditions do not fit the average uniform flow assumption and when processes at the boundaries influence the flow and therefore the mixing of contaminants.