PERTURBATION AND GAUSSIAN METHODS FOR STOCHASTIC FLOW PROBLEMS

Most applications of stochastic ground water modeling rely on a perturbation assumption; the magnitude of the randomly distributed parameters in the governing equation should be small to obtain a valid solution. A Gaussian method was developed to solve the ground water flow equation with random hydraulic conductivity distribution. In the Gaussian method, the moment generating function for multivariate normally distributed random variables was used to express joint high order moments of hydraulic head and conductivity in terms of low order moments. A set of coupled partial differential equations for the moments of the hydraulic head were formulated. The number of unknowns in the Gaussian method was the same as in the perturbation method, however the 'closure problem' was solved without the truncation of high order terms. Comparison with an exact analytical solution suggested that the Gaussian method can be more appropriate in the case of short correlation length. The Gaussian and the perturbations methods can be equivalent for the case of steady-state flow. For transient ground water flow, the two methods are different. The Gaussian method preserves the known lognormal spatial frequency distribution of the conductivity in the field.