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)K(x) (replaced by the transmissivity T(x)T(x) for regional flow). The latter is considered a function of lognormal univariate distribution, characterized by KGKG (the geometric mean) and View the MathML sourceσY2, the variance of Y=lnKY=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 ωω at the inlet and constant head at the outlet. The self consistent approximation is used to derive the effective conductivity KefKef and the average head 〈H〉〈H〉 and flux 〈q〉〈q〉 fields are subsequently arrived at. In the common quasi-steady approximation, KefKef is equal to the steady state effective property View the MathML sourceKefst. We derive an expression for the frequency dependent KefKef, which is generally complex, i.e., dynamic. The main result is the delineation of the ranges of the parameters View the MathML sourceω,σY2 for which Kef,〈H〉Kef,〈H〉 and 〈q〉〈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 KefKef applies to 2D phreatic flow with time periodic recharge.