Random walk path solution to groundwater flow dynamics in highly heterogeneous aquifers

Random walk path methods including walk on spheres and walk on rectangles have been used to solve elliptic and parabolic partial differential equations (PDEs). These methods are able to provide not only the pointwise solutions to the linear PDEs but also contributions of boundaries and all source/sink terms as an analytical solution does. However, due to difficulty in dealing with heterogeneity, these methods cannot be applied to groundwater flow problems in highly heterogeneous aquifers. A novel method called walk on grid (WOG) is proposed based on lattice random walk to overcome the difficulty. WOG algorithm is verified in a 1D homogeneous transient problem, a 1D heterogeneous steady-state problem, a 2D heterogeneous transient problem and a 3D optimization problem. It is demonstrated that WOG is effective in solving groundwater flow problems in highly heterogeneous confined aquifers. Probabilities of walkers arriving at prescribed boundaries (terminal weights) and source counts may be useful for characterization of medium heterogeneity. WOG method sheds a new light on solving the PDEs of complicated groundwater problems in a changing environment and on analyzing medium heterogeneity.