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

Steady flow through a heterogeneous porous medium in a bounded domain is in vestigated using a recursive perturbation scheme. The effect df boundary co nditions on a two-dimensional flow with arbitrary Variation of the mean flo w (allowing large gradients) is investigated using analytical expressions f or the head and velocity covariance functions. Boundary conditions were dec omposed into deterministic and stochastic components. Two flow cases with t he same zero-order and different first-order boundary conditions were analy zed. Boundary conditions are deterministic for the first case and random fo r 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 bounde d rectangular domain are derived. The resulting integral kernels involve Gr eens functions, and they are evaluated by numerical integration. Boundary c onditions for higher-order problems influence the absolute value and shape of the kernels (and therefore of the head and velocity variance and covaria nce 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 si gma(f)(2) much less than 1; where sigma(f) is the variance of the log hydra ulic conductivity, assumed Gaussian and weakly homogeneous in space. The ty pe of boundary conditions affects' the values of the kernels and therefore determine the convergence limits for the problem. Milder head gradients als o allow larger Values of sigma(f)(2). Nonlocality is also contingent on bou ndary type; Weakly homogeneous log-fluctuating conductivity fields give ris e to head and velocity covariances which are not weakly homogeneous. The in homogeneous 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, nonloca l effects are most important away from the boundaries ("center of the domai n"). Small local head gradients result in small head and velocity gradients , and therefore where observations are made in stagnation regions, data sho uld 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 bound aries influence the flow and therefore the mixing of contaminants.