On Perturbative Expansions to the Stochastic Flow Problem

Abstract

When analyzing stochastic steady flow, the hydraulic conductivity naturally appears logarithmically. Often the log conductivity is represented as the sum of an average plus a stochastic fluctuation. To make the problem tractable, the log conductivity fluctuation, f, about the mean log conductivity, lnK _G, is assumed to have finite variance, σ _f ². Historically, perturbation schemes have involved the assumption that σ _f ²<1. Here it is shown that σ _f may not be the most judicious choice of perturbation parameters for steady flow. Instead, we posit that the variance of the gradient of the conductivity fluctuation, σ_Δf ², is more appropriate hoice. By solving the problem withthis parameter and studying the solution, this conjecture can be refined and an even more appropriate perturbation parameter, ε, defined. Since the processes f and ∇f can often be considered independent, further assumptions on ∇f are necessary. In particular, when the two point correlation function for the conductivity is assumed to be exponential or Gaussian, it is possible to estimate the magnitude of σ_∇f in terms of σ_f and various length scales. The ratio of the integral scale in the main direction of flow (λ _x ) to the total domain length (L^*), ρ_x ²=λ_x/L^*, plays an important role in the convergence of the perturbation scheme. For ρ _x smaller than a critical value ρ_c, ρ_x < ρ_c, the scheme's perturbation parameter is ε=σ_f/ρ_x for one- dimensional flow, and ε=σ_f/ρ_x ² for two-dimensional flow with mean flow in the x direction. For ρ_x > ρ_c, the parameter ε=σ_f/ρ_x ³ may be thought as the perturbation parameter for two-dimensional flow. The shape of the log conductivity fluctuation two point correlation function, and boundary conditions influence the convergence of the perturbation scheme.