Modeling the large-scale dynamics of saturated groundwater flow using spatial-filtering theory: 1. Theoretical development

We develop a model and a framework for understanding the explicitly resolved, large-scale dynamics of groundwater flow in fully saturated, heterogeneous porous media. The approach is based upon the large eddy simulation (LES) methodology from geophysical fluid dynamics. The idea behind the LES approach is to explicitly resolve the larger scales of flow and provide a closure model for the effect of subgrid scales upon resolved scales. The technical foundation of the method rests on a spatial filtering operation, which allows one to precisely define the scale of the model. We spatially filter Darcy's law and the continuity equation to produce a model for the resolved-scale dynamics. We develop LES approximations for the resulting flow equations and a closure model for unresolved, subgrid-scale terms. We model the subgrid terms using resolved-scale quantities only and thus do not require subgrid-scale information. The accuracy of these closure models depends upon the scale at which hydraulic conductivity is 'measured' by field tests. We characterize the approximations used to model subgrid- scale effects asymptotically with a small parameter, the dimensionless filter width ε, defined as the ratio of the filter width to the dominant length scale of the explicitly resolved variables.