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 σ2 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 σ2 = 1 and σ2 = 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 σ2. 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 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.
ASJC Scopus subject areas
- Water Science and Technology