Effective conductivity of heterogeneous aquifers in unsteady periodic flow

We consider transient flow in 3D and 2D (regional) confined aquifers with spatially variable random hydraulic conductivity K(x) (replaced by the transmissivity T(x) for regional flow). The latter is considered a function of lognormal univariate distribution, characterized by _{K G} (the geometric mean) and σY2, the variance of Y = lnK. The aquifer is modeled as a layer/plane composed of densely distributed spherical/circular inclusions of different K, with time periodic head of frequency ω at the inlet and constant head at the outlet. The self consistent approximation is used to derive the effective conductivity _Kef and the average head 〈H〉 and flux 〈q〉 fields are subsequently arrived at. In the common quasi-steady approximation, _Kef is equal to the steady state effective property Kefst. We derive an expression for the frequency dependent _Kef, which is generally complex, i.e., dynamic. The main result is the delineation of the ranges of the parameters ω,σY2 for which _Kef, 〈H〉 and 〈q〉 show a significant dynamic effect. We examine specific applications to show that generally the quasi-steady approximation is sufficiently accurate while delimiting the cases in which the dynamic effective conductivity is significant. It is also shown that the derived _Kef applies to 2D phreatic flow with time periodic recharge.