Analysis of flow through heterogeneous random aquifers: 2. Unsteady flow in confined formations

The direct problem of determining the head field in a formation of given stationary isotropic random structure is solved approximately for unsteady flow conditions. Under restrictive assumptions of sufficiently small conductivity variance and average head gradient slowly varying in time and space, closed form expressions are derived for the effective conductivity or storativity and for the head variogram. Two types of unsteady flow are considered: transients and periodic flows. In the case of transients, it is assumed that initially the head is constant (no flow) and that ultimately the flow becomes uniform. It is found that the effective conductivity and transmissivity are time‐dependent and drop from the arithmetic mean (initially) to the steady state value during a relaxation time. The latter is much larger for one‐dimensional flows than for two‐ or three‐dimensional ones. The head variogram also relaxes from the one corresponding to the initial lack of correlation to the steady state variogram during a relaxation time which grows with the lag. While in three‐dimensional structure the relaxation time scales are relatively short and steady state analysis can be used accurately even under transient conditions, the tendency to steady state may be very slow in the case of transmissivity because of its large integral scale. Hence steady state analysis for transients might not be warranted for regional transient flows. In the case of periodic flows of seasonal nature, the head gradient is assumed to be made up from a steady component and a periodic one. If the amplitude of the latter is sufficiently small compared to the steady one, steady state value of effective transmissivity and storativity can be adopted. The head variogram has also its steady state form for large lags and quasisteady for small lags. Hence the variogram lies between the steady one (steady component of head gradient) and the quasisteady one (instantaneous head gradient).