Second‐order expressions for velocity moments in two‐ and three‐dimensional statistically anisotropic media

Abstract

Second‐order expressions are derived for the mean and covariance of steady state seepage velocity under mean uniform flows in infinite two‐ and three‐dimensional domains. The order of approximation is defined in terms of the variance σ2 of a statistically homogeneous and anisotropic natural log hydraulic conductivity field Y with a Gaussian spatial autocorrelation function. Results show that second‐order mean velocity either exceeds or is close to its first‐order counterpart, depending on anisotropy. Head fluctuations of order larger than σ affect second‐order velocity moments to the same extent as do head fluctuations of order σ in virtually all cases, hence neglecting the former renders the results nonasymptotic. Velocity variances are generally larger when approximated consistently to second than to first order, The ratio between second‐ and first‐order variance approximations is larger in three than in two dimensions, larger for transverse than for longitudinal velocity, and increases with σ2. Anisotropy has a significant effect on second‐order velocity variance. Second‐order effects have the greatest influence on longitudinal velocity variance at extreme anisotropy ratios and on transverse velocity variance in isotropic domains.