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, 1nK _G , is assumed to have finite variance, σ ² _f . Historically, perturbation schemes have involved the assumption that σ ² _f <1. Here it is shown that of 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, a ² _Δf , is a more appropriate choice. By solving the problem with this parameter and studying the solution, this conjecture can be refined and an even more appropriate perturbation parameter, ε, defined. Since the processes f and V f can often be considered independent, further assumptions on V 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 avf in terms of o f and various length scales. The ratio of the integral scale in the main direction of flow (A,) to the total domain length (L*), px = a. _x /L*, plays an important role in the convergence of the perturbation scheme. For p _x smaller than a critical value p _c , px < p _c , the scheme’s perturbation parameter is e = o f/p _x for one-dimensional flow, and s = o f/p,² for two-dimensional flow with mean flow in the x direction. For p _x p _c , the parameter e = of/p _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.