Quasi-Monte Carlo methods for computing flow in random porous media

We devise and implement quasi-Monte Carlo methods for computing the expectations of nonlinear functionals of random fields arising in the modeling of fluid flow in random porous media. Specific examples include the effective permeability of a block of rock, the pressure head at a chosen point and the breakthrough time of a pollution plume being convected by the velocity field. The mathematical model is a system of first order partial differential equations in space with a random field describing the permeability. Our emphasis is on situations where a very large number of dimensions are necessary to obtain a reasonable accuracy in probability space, and where classical Monte Carlo methods with random sampling are currently the method of choice. As an alternative we introduce quasi-Monte Carlo methods which use deterministically chosen sample points in probability space. Our algorithm performs finite element approximation for each realization of the field, and approximates the necessary element integrals by using values of the field computed only on a suitable regular grid, with the help of FFT techniques. In this way we avoid the use of a truncated Karhunen-Loève expansion, but introduce high nominal dimension in probability space. Numerical experiments with 2-dimensional rough random fields, high variance and small length scale are reported, showing that quasi-Monte Carlo method consistently outperforms the Monte Carlo method, with a noticeably better than O(N−1/2) convergence rate and a smaller implied constant, where N is the number of samples. Moreover, the rate of convergence of quasi-Monte Carlo methods does not appear to degrade as the nominal dimension increases. Examples with dimension as high as 10 are reported.