A computationally practical approach for modeling complex mean flows in mildly heterogeneous media

We present in this paper a critical review of recent research on nonuniform mean flows in heterogeneous porous media, examine why existing stochastic methods are computationally so difficult to implement, and introduce a new and efficient alternative. Specifically, we reformulate the nonstationary spectral method of Li and McLaughlin (1991, 1995) and present a new way for its numerical implementation, combining the best advantages of efficient analytical solutions and flexible numerical techniques. The result is a substantially improved stochastic technique that allows modeling efficiently the nonlinear scale effects for moderately heterogeneous media in the presence of general nonstationarity. In particular, the reformulated approach allows computing the nonlocal and nonstationary mean "closure" flux using a coarse grid without having to resolve numerically the small-scale heterogeneous dynamics. The methodological innovation significantly increases the size and expands the range of groundwater problems that can be analyzed with stochastic methods. The effectiveness of the new spectral approach is illustrated with two concrete examples and a systematic comparison with existing stochastic methods.