Galerkin Finite Element Procedure for analyzing flow through random media

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Abstract
A method is presented for analyzing flow through a porous medium whose parameters are random functions. Such a medium is conceptualized as an ensemble of media with an associated probability mass function. The flow problem in each member of this ensemble is deterministic in the usual sense. All the solutions belong to a particular Hilbert space, and hence they can be written in terms of linear combinations of its basis functions. This is similar to the Galerkin formulation except that the coefficients in the linear combination are no longer deterministic quantities but random functions. The finite element method in conjunction with a Taylor series expansion is used to get the first two moments of the solution approximately. The method does not require specification of full probability mass functions of the parameters but only their first two moments, and spatial correlations can be easily accounted for. However, it is assumed that the probability mass functions are peaked at the expected value and are smooth in its vicinity. A sample problem is solved to illustrate the procedure. It is observed that the result is sensitive to the element size in the numerical scheme and the variances and spatial correlations of parameters. The expected value of the hydraulic head is found to differ significantly from the results that would have been obtained if the problem had been solved deterministically.