On Perturbative Expansions to the Stochastic Flow Problem |
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Authors: | Bonilla F Alejandro Cushman John H |
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Institution: | (1) Center for Applied Math and Department of Civiland Environmental Engineering, USA;(2) Departments of Mathematics and Agronomy, Center for Applied Mathematics, Purdue University, Math Sciences Building, West Lafayette, IN, 47907, U.S.A |
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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,
f
2. Historically, perturbation schemes have involved the assumption that
f
2<1. Here it is shown that
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, f
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, , 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 f in terms of f and various length scales. The ratio of the integral scale in the main direction of flow (
x
) to the total domain length (L*), x
2=x/L*, plays an important role in the convergence of the perturbation scheme. For
x
smaller than a critical value c, x < c, the scheme's perturbation parameter is =f/x for one- dimensional flow, and =f/x
2 for two-dimensional flow with mean flow in the x direction. For x > c, the parameter =f/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. |
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Keywords: | Flow stochastic perturbation velocity head gradient |
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