NDLI: On Perturbative Expansions to the Stochastic Flow Problem

Content Provider	Springer Nature Link
Author	Bonilla, F. Alejandro Cushman, John H.
Copyright Year	2001
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.
Starting Page	3
Ending Page	35
Page Count	33
File Format	PDF
ISSN	01693913
Journal	Transport in Porous Media
Volume Number	42
Issue Number	1-2
e-ISSN	15731634
Language	English
Publisher	Kluwer Academic Publishers
Publisher Date	2001-01-01
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Industrial Chemistry/Chemical Engineering Hydrogeology Geotechnical Engineering Mechanics, Fluids, Thermodynamics Civil Engineering
Content Type	Text
Resource Type	Article
Subject	Catalysis

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