ON HIGHER-ORDER CORRECTIONS TO THE FLOW VELOCITY COVARIANCE TENSOR

Second-order log fluctuating conductivity variance (sigma(f)(2)) corre ctions to the head and velocity covariance functions are derived for a lognormal, stationary hydraulic conductivity field. The Fourier trans form method proposed by Deng et al. (1993) is used extensively to obta in numerical estimates of these functions for an exponential log fluct uating conductivity covariance. It is shown that the velocity covarian ce is insensitive to second-order corrections in the head field. The v elocity covariance, on the other hand, is highly sensitive to second-o rder corrections in the velocity when the log fluctuating conductivity variance approaches unity. A closed expression is derived for a secon d-order correction to the velocity variance when there is no second-or der correction to the head field. The longitudinal second-order correc tion to the velocity variance is 0.4 sigma(f)(2) different from the fi rst-order approximation in isotropic media, 1.5 sigma(f)(2) different in a highly stratified formation, and no different when the ratio of v ertical to horizontal integral scales approaches infinity. The second- order corrections to the horizontal and vertical transverse velocity v ariances are 2 sigma(f)(2) different from the first-order approximatio ns for both isotropic and anisotropic systems.