Localization of mean flow and apparent transmissivity tensor for bounded randomly heterogeneous aquifers

We explore the concept of apparent transmissivity for bounded randomly heterogeneous media under steady-state flow regime. The novelty of our study consists of investigating a tensorial nature of apparent transmissivity. We demonstrate that apparent transmissivity of bounded domains is anisotropic even though an underlying local transmissivity field is statistically isotropic. For rectangular flow domains, we derive an analytical expression for the apparent transmissivity tensor via localization and perturbation expansion of the nonlocal mean flow equations in the variance of log-transmissivity. In this expression, almost everywhere the off-diagonal terms are several orders of magnitude smaller than the diagonal terms. When the domain size relative to the log-transmissivity correlation scale is large, the longitudinal and transverse components of the apparent transmissivity tensor approach the geometric mean of local transmissivity. While rigorously valid for mean uniform flows only, our expression for the apparent transmissivity tensor leads to mean hydraulic head distributions that compare favorably with those obtained through Monte-Carlo simulations and the nonlocal mean flow equations even in the presence of pumping wells. This agreement deteriorates in the vicinity of wells and as pumping rates increase.