A note on head and velocity covariances in three‐dimensional flow through heterogeneous anisotropic porous media

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Abstract
A procedure for the characterization of the head and velocity fields for saturated flow in heterogeneous, statistically anisotropic formations is developed. The fields are characterized through a series of spatial covariances as well as the velocity‐head, velocity‐log conductivity and head‐log conductivity cross covariances. All these functions are given in terms of the parameters which characterize the input log conductivity field and the average head garadient. A general procedure is outlined, followed by explicit derivation of head and velocity covariances for heterogeneous, statistically anisotropic formations. Closed form analytic results are obtained for the statistically isotropic formations, while the results for the anisotropic ones required taking quadrature over analytic expressions. The results may be used for transport predictions and for conditioning on measurements, and may be particularly useful for testing of numerical codes.