An Eecient Probabilistic Finite Element Method for Stochastic Groundwater Flow

We present an eecient numerical method for solving stochastic porous media ow problems. Single-phase ow with a random conductivity eld is considered in a standard rst-order perturbation expansion framework. The numerical scheme, based on nite element techniques , is computationally more eecient than traditional approaches, because one can work with a much coarser nite element mesh. This is achieved by avoiding the common nite element representation of the conductivity eld. Computations with the random conductivity eld only arise in integrals of the log conductivity covariance function. The method is demonstrated in several two-and three-dimensional ow situations and compared to analytical solutions and Monte Carlo simulations. Provided that the integrals involving the covariance of the log conductivity are computed by higher-order Gaussian quadrature rules, excellent results can be obtained with characteristic element sizes equal to about ve correlation lengths of the log conductivity eld. Investigations of the validity of the proposed rst-order method are performed by comparing nonlinear Monte Carlo results with linear solutions. In box shaped domains the log conductivity standard deviation Y may be as large as 1:5, while the head variance is considerably innuenced by nonlinear eeects as Y approaches unity in more general domains.