POD-based Monte Carlo approach for the solution of regional scale groundwater flow driven by distributed recharge

We present a methodology conducive to the application of a Galerkin model order reduction technique, Proper Orthogonal Decomposition (POD), to solve a groundwater flow problem driven by spatially distributed stochastic forcing terms. Typical applications of POD to reducing time-dependent deterministic partial differential equations (PDEs) involve solving the governing PDE at some observation times (termed snapshots), which are then used in the order reduction of the problem. Here, the application of POD to solve the stochastic flow problem relies on selecting the snapshots in the probability space of the random quantity of interest. This allows casting a standard Monte Carlo (MC) solution of the groundwater flow field into a Reduced Order Monte Carlo (ROMC) framework. We explore the robustness of the ROMC methodology by way of a set of numerical examples involving two-dimensional steady-state groundwater flow taking place within an aquifer of uniform hydraulic properties and subject to a randomly distributed recharge. We analyze the impact of (i) the number of snapshots selected from the hydraulic heads probability space, (ii) the associated number of principal components, and (iii) the key geostatistical parameters describing the heterogeneity of the distributed recharge on the performance of the method. We find that our ROMC scheme can improve significantly the computational efficiency of a standard MC framework while keeping the same degree of accuracy in providing the leading statistical moments (i.e. mean and covariance) as well as the sample probability density of the state variable of interest.