Quantifying flow uncertainty in heterogeneous porous media using random walk simulations

Elliptic and parabolic partial differential equations (PDEs) model conservation laws of many physical applications, such as heat transfer and flow in porous media. Often these PDEs have an input parameter field (conductivity) that is both heterogeneous and uncertain, and that poses significant computational challenges for quantifying the uncertainty in the flow performance. Based on the relation between stochastic diffusion processes and PDEs, Monte Carlo (MC) methods have been proposed to solve elliptic and parabolic PDEs for problems where the input conductivity field is homogeneous. However, in many practical problems with highly heterogeneous conductivity, these MC solution strategies are not directly applicable.

Here, we describe a MC method to solve conservation laws with heterogeneous conductivity fields using general diffusion processes. The stochastic representation of the conservation law makes it possible to compute the solution at any point independently of the solution at other locations in the domain without solving a global linear system. It is shown that the method provides an efficient alternative for computing the statistical moments of the solution to a stochastic PDE at any point in the domain.