On the second-order correction to velocity covariance for two-dimensional statistically isotropic porous media

Second-order perturbation theory has recently been applied to flow and transport in statistically isotropic porous media. The second-order head correction has been found to be important in the velocity covariance and thus also in transport. This result was obtained by using numerical integration on the second-order correction for velocity covariance with the higher-order head fluctuation. The present paper is a continuation of previous studies and aims at deriving an analytical, closed-form expression for the second-order correction term involving higher-order head fluctuation in lognormal statistically isotropic media with exponential log conductivity correlation. The higher-order correction for velocity covariance is found to be most significant in the transverse direction, a result that is consistent with previous analysis.