Models of groundwater flow in statistically homogeneous porous formations

Abstract

Most natural porous formations display significant variations in space of permeability K and storativity S. Such formations are regarded as random structures characterized by the permeability frequency distribution and its autocorrelation. With the aid of three basic length scales (l, microscale; £, integral scale of K autocorrelation; and L, length scale of space change of average K) a classification of aquifers is suggested. A similar classification is proposed for the flow regimes. The study is limited to statistically homogeneous or slowly varying formations (£ ≪ L) and to uniform or slowly varying (in space and time) flows. In the section dealing with steady flows, effective permeability, as well as variances of head gradient, specific discharge, and head, is computed for one‐, two‐, and three‐dimensional flows. Bounds and an estimate of the effective permeability in terms of a log normal permeability distribution are given. The computations are based on physical, simplified models of formation structure. It is shown that the head variance is grossly overestimated for one‐dimensional flow through blocks ‘in series,’ and the same is true for specific discharge for layers ‘in parallel.’ The two‐ and three‐dimensional variances are much lower and are close to each other. The unsteady flow is analyzed with the aid of the relaxation time needed for blocks of different K, S to adapt to the environment. For flows which change slowly and uniformly the effective permeability is that derived for steady uniform flows, and the effective storativity is equal to the S arithmetic mean. The head gradient variance computed with the aid of some physical models is compared with that of steady uniform flow, and it is shown that for sufficiently slow time changes the flow field can be considered momentarily uniform. The various results are employed to estimate effective properties, as well as fluctuations of the head and specific discharge, in aquifers, with possible applications to prediction, the inverse problem, and hydrodynamic dispersion.