HIGHER-ORDER EFFECTS ON FLOW AND TRANSPORT IN RANDOMLY HETEROGENEOUS POROUS-MEDIA

A higher-order theory is presented for steady state, mean uniform satu rated flow and nonreactive solute transport in a random, statistically homogeneous natural log hydraulic conductivity field Y. General integ ral expressions are derived for the spatial covariance of fluid veloci ty to second order in the variance sigma(2) of Y in two and three dime nsions. Integrals involving first-order (in sigma) fluctuations in hyd raulic head are evaluated analytically for a statistically isotropic t wo-dimensional Y field with an exponential autocovariance, Integrals i nvolving higher-order head fluctuations are evaluated numerically for this same field. Complete second-order results are presented graphical ly for sigma(2) = 1 and sigma(2) = 2. They show that terms involving h igher-order head fluctuations are as important as those involving lowe r-order ones. The velocity variance is larger when approximated to sec ond than to first order in sigma(2). Discrepancies between second- and first-order approximations of the velocity autocovariance diminish ra pidly 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 advec tion level at which macrodispersivities are related to velocity fluctu ations. Our results show that a second-order flow correction affects t ransport to a greater extent than does a second-order correction to ad vection. Asymptotically, the second-order transverse macrodispersivity tends to zero as does its first-order counterpart. An approximation o f advection alone based on Corrsin's conjecture, coupled with either a first- or a second-order flow approximation, leads to a transverse ma crodispersivity which is significantly larger than that obtained by st andard perturbation and tends to a nonzero asymptote. Published Monte Carlo results yield macrodispersivities that lie significantly below t hose predicted by first- and second-order theories. Considering that M onte Carlo simulations often suffer from sampling and computational er rors, that standard perturbation approximations are theoretically vali d only for sigma(2) < 1, and that Corrsin's conjecture represents the leading term in a renormalization group perturbation which contains co ntributions from an infinite number of high-order terms, we find it di fficult to tell which of these approximations is closest to representi ng transport in strongly heterogeneous media with sigma(2) greater tha n or equal to 1.