Transmissivity and head covariances for flow in highly heterogeneous aquifers

Spatially variable transmissivity T of aquifers is modeled as random. Analysis of field data [Water Resour. Res. 21 (1985) 563] indicate that the logtransmissivity Y=ln T is normal and its covariance can be characterized by three parameters: the variance σYc2 and the integral scale IY of correlated residuals and a nugget σYn2, representing variability of small support. The equation of flow is stochastic and the head H is also random. The head-logtransmissivity cross-covariance CHY and the head variogram ΓH can be used conveniently to solve the direct and inverse problems. These covariances are derived for an unbounded domain and for mean uniform flow of constant head gradient −J. Under these conditions, analytical expressions were determined in the past by first-order approximation in σYc2, pertinent to weak heterogeneity. The aim of the present study is to derive CHY and ΓH for highly heterogeneous aquifers of total variance σY2≤4. This goal is achieved by adopting a multi-indicator model of the aquifer consisting of circular inclusions of radius R and of normal logtransmissivity of variance σY2, submerged in a matrix of effective transmissivity TG (geometric mean). The system is characterized by σY2, the integral scale IY=8R/(3π) and the volume fraction of inclusions n, which are simply related to the aquifer parameters σYc2, σYn2 and IY. The flow problem is solved numerically at high accuracy by the analytic element method. The medium is modeled by 50,000 inclusions and parameters values are σY2=0.1, 1, 2, 4 and n=0.4, 0.65, 0.9. Analytical solutions are derived by the effective medium approximation (EMA), in which each inclusion is regarded as submerged in a medium of effective transmissivity, and by first-order approximation (FAO in σY2). Comparison between the numerical and analytical solutions shows that CYH is overestimated by FOA and is in agreement with the EMA. The head variogram is in agreement with EMA for n≤0.65, but underestimated for n=0.9, when it is close to the FOA. The latter effect results from cancellation of errors. An outline of application of results concludes the study.