Higher-order effects on flow and transport in randomly heterogeneous, statistically anisotropic porous media

• Main Author: Hsu, Kuo-Chin, 1963-
• Other Authors: Neuman, Shlomo P.
• Language: en_US
• Published: The University of Arizona. 1996
• Subjects: Hydrology.
• Online Access: http://hdl.handle.net/10150/282175

A higher-order theory is presented for steady state, mean uniform saturated flow and nonreactive solute transport in randomly homogeneous, statistically anisotropy natural log hydraulic conductivity fields Y. General integral expressions are derived for the spatial covariance of fluid velocity to second order in the variance σ² of Y in two and three dimensions. Analytical expressions are evaluated for integrals involving first-order (in σ) fluctuations in hydraulic head in two- and three-dimensional Y fields with exponential and Gaussian correlation functions. Integrals involving higher-order fluctuations of hydraulic head are evaluated numerically. Our results show that corrections involving higher-order head fluctuations are as important as those without them; neither should be disregarded. 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. The variance of longitudinal and transverse two- and three-dimensional velocity is larger in anisotropic than in isotropic fields due to higher-order effects. Second-order mean velocity is larger than first order in three dimensions and for most anisotropy ratios (except 1-6) in two dimensions. Transport requires approximations at flow and advection levels. Our results show that, in two dimensions, the combined effects of second-order correction to flow and to advection impact transport to a greater extent than does the sum of their individual effects. The choice of Y correlation function has greater effects on macrodispersivity than does second-order correction in flow. Published two-dimensional results of Monte Carlo simulations yield macrodispersivities that lie significantly below those predicted by first- and second-order theories. Considering that Monte Carlo simulations often suffer from sampling and computational errors, that standard perturbation approximations are theoretically valid only for σ² < 1, and that Corrsin's conjecture represents the leading term in a renormalization group perturbation which contains contributions from an infinite number of high-order terms, we find it difficult to tell which of these approximations is closest to representing transport in strongly heterogeneous media with σ² ≥ 1.