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

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.