On Higher‐Order Corrections to the Flow Velocity Covariance Tensor

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Abstract
Second‐order log fluctuating conductivity variance (σƒ2) corrections to the head and velocity covariance functions are derived for a lognormal, stationary hydraulic conductivity field. The Fourier transform method proposed by Deng et al. (1993) is used extensively to obtain numerical estimates of these functions for an exponential log fluctuating conductivity covariance. It is shown that the velocity covariance is insensitive to second‐order corrections in the head field. The velocity covariance, on the other hand, is highly sensitive to second‐order corrections in the velocity when the log fluctuating conductivity variance approaches unity. A closed expression is derived for a second‐order correction to the velocity variance when there is no second‐order correction to the head field. The longitudinal second‐order correction to the velocity variance is 0.4σƒ2 different from the first‐order approximation in isotropic media, 1.5σƒ2 different in a highly stratified formation, and no different when the ratio of vertical to horizontal integral scales approaches infinity. The second‐order corrections to the horizontal and vertical transverse velocity variances are 2σƒ2 different from the first‐order approximations for both isotropic and anisotropic systems.