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=lnT is normal and its covariance can be characterized by three parameters: the variance σ_Yc² and the integral scale I_Y of correlated residuals and a nugget σ _Yn², representing variability of small support. The equation of flow is stochastic and the head H is also random. The head-logtransmissivity cross-covariance C_HY and the head variogram Γ_H 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 σ_Yc², pertinent to weak heterogeneity. The aim of the present study is to derive C_HY and Γ_H for highly heterogeneous aquifers of total variance σ_Y²≤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 σ_Y², submerged in a matrix of effective transmissivity T_G (geometric mean). The system is characterized by σ_Y², the integral scale I_Y=8R/(3π) and the volume fraction of inclusions n, which are simply related to the aquifer parameters σ_Yc², σ_Yn² and I_Y. 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 σ_Y²=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 σ_Y²). Comparison between the numerical and analytical solutions shows that C_YH 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.