THE UNIVERSAL STRUCTURE OF THE GROUNDWATER-FLOW EQUATIONS

Most groundwater flow models are not sufficiently detailed to allow an explicit representation of all dynamical scales. Instead, most models are constructed at such a coarse scale of resolution that unresolved subgrid variability often exists. It is therefore important to understand the interaction between unresolved dynamics and explicitly resolved dynamics. The concept of universality is central to this interaction and is here related to the problem of rescaling models of groundwater flow. We examine when it is possible to construct an accurate model for the explicitly resolved large scales, without an explicit description of the subgrid scale dynamics. We demonstrate how unresolved subgrid scale dynamics interact with resolved scale dynamics. We show that a universally valid resolved scale model can be constructed if the resolved dynamics are sufficiently independent of the details of the subgrid scale dynamics. In that event, a resolved scale model is composed of a universal structure and accompanying model parameters. The model parameters represent the effect of unresolved dynamics upon resolved dynamics. We examine two possible model structures for groundwater flow. We show theoretically, and numerically, that a local Darcy law is a universally valid resolved scale model if the resolved and subgrid scales of the hydraulic conductivity are separated in scale by a so-called spectral gap. If, however, the hydraulic conductivity possesses many scales of variability, then a more general nonlocal Darcy law is a more appropriate model structure. When the nonlocal Darcy's law is more appropriate, numerical experiments suggest that errors using a local Darcy's law with effective parameters are most significant at the smallest resolving scale of the model, and are minimal at scales between 8 and 16 times the resolving scale.