The role of scaling laws in upscaling

In this work the process of coarse-graining in complex subsurface hydrologic systems is discussed, with a particular effort made to examine the difference between the mathematical process of averaging as distinct from the process of upscaling. It is possible to show that the process of averaging itself is not sufficient to reduce the number of degrees of freedom required to describe a complex heterogeneous system. The reduction of the number of degrees of freedom is accomplished entirely by the subsequent scaling laws that one assumes are valid filters for eliminating redundant (or low value) information in the system. A specific example is given for the upscaling of the effective dispersion tensor in a randomly heterogeneous porous medium. It is shown that: (1) generally, an averaged description can be developed, but the system does not contain any less information than the original problem; (2) by adopting a number of scaling laws, a nonlocal in time and in space formulation can be obtained that is identical to results obtained previously, and (3) by making one additional assumption, a local macroscale transport equation can be obtained.