On perturbative expansions to the stochastic flow problem

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 tracta ble, the log conductivity fluctuation, f, about the mean log conductivity, lnK(G), is assumed to have finite variance, sigma (2)(f). Historically, per turbation schemes have involved the assumption that sigma (2)(f) < 1. Here it is shown that sigma (f) may not be the most judicious choice of perturba tion parameters for steady flow. Instead, we posit that the variance of the gradient of the conductivity fluctuation, sigma (2)(del)(f), is more appro priate choice. By solving the problem with this parameter and studying the solution, this conjecture can be refined and an even more appropriate pertu rbation parameter, epsilon, defined. Since the processes f and delf can oft en be considered independent, further assumptions on delf are necessary. In particular, when the two point correlation function for the conductivity i s assumed to be exponential or Gaussian, it is possible to estimate the mag nitude of sigma (delf) in terms of sigma (f) and various length scales. The ratio of the integral scale in the main direction of flow (lambda (x)) to the total domain length (L*), rho (2)(x)=lambda (x)/L*, plays an important role in the convergence of the perturbation scheme. For rho (x) smaller tha n a critical value rho (c), rho (x) < rho (c), the scheme's perturbation pa rameter is epsilon=sigma (f)/rho (x) for one- dimensional flow, and epsilon =sigma (f)/rho (2)(x) for two- dimensional flow with mean flow in the x dir ection. For rho (x) > rho (c), the parameter epsilon=sigma (f)/rho (3)(x) m ay be thought as the perturbation parameter for two-dimensional flow. The s hape of the log conductivity fluctuation two point correlation function, an d boundary conditions influence the convergence of the perturbation scheme.