The problem of flow through a porous medium, with the permeability treated as a Gaussian random field, can be thought of as a high-dimensional problem: the dimensionality might be the number of terms in a truncated Karhunen-Loève expansion; or (as we prefer) the number of points in a discrete sampling of the porous medium. In this paper, describing recent joint work with F Kuo, I Graham, D. Nuyens and R Scheichl, we explore the use of quasi-Monte Carlo methods to study various expected values of the flow through the medium, and to compare the results with the Monte Carlo method. The problem is computationally difficult if the permeability changes markedly from point to point, but the numerical results (obtained by evaluating integrals with as many as one million dimensions) are encouraging.