Higher‐Order Effects on Flow and Transport in Randomly Heterogeneous Porous Media

A higher‐order theory is presented for steady state, mean uniform saturated flow and nonreactive solute transport in a random, statistically homogeneous natural log hydraulic conductivity field Y . General integral expressions are derived for the spatial covariance of fluid velocity to second order in the variance σ 2 of Y in two and three dimensions. Integrals involving first‐order (in σ) fluctuations in hydraulic head are evaluated analytically for a statistically isotropic two‐dimensional Y field with an exponential autocovariance. Integrals involving higher‐order head fluctuations are evaluated numerically for this same field. Complete second‐order results are presented graphically for σ 2 =1 and σ 2 =2. They show that terms involving higher‐order head fluctuations are as important as those involving lower‐order ones. The velocity variance is larger when approximated to second than to first order in σ 2 . Discrepancies between second‐ and first‐order approximations of the velocity autocovariance diminish rapidly with separation distance and are very small beyond two integral scales. Transport requires approximation at two levels: the flow level at which velocity statistics are related to those of Y , and the advection level at which macrodispersivities are related to velocity fluctuations. Our results show that a second‐order flow correction affects transport to a greater extent than does a second‐order correction to advection. Asymptotically, the second‐order transverse macrodispersivity tends to zero as does its first‐order counterpart. An approximation of advection alone based on Corrsin's conjecture, coupled with either a first‐ or a second‐order flow approximation, leads to a transverse macrodispersivity which is significantly larger than that obtained by standard perturbation and tends to a nonzero asymptote. Published Monte Carlo results 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 σ 2 < 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 σ 2 ≥ 1.