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Correlation structure of flow variables for steady flow toward a well with application to highly anisotropic heterogeneous formations

Authors

  • A. Fiori,

  • P. Indelman,

  • G. Dagan


Abstract

The study, a continuation of that of Indelman et al. [1996], aims at deriving the second-order moments of flow variables such as hydraulic head, its gradient, and the specific discharge for steady flow toward a fully penetrating well in a confined heterogeneous aquifer. The log conductivity Y=ln K is modeled as a three-dimensional stationary function of Gaussian correlation of anisotropy ratio e. By using first-order approximations in σ2Y and e, we derive the variance and the vertical integral scale of the piezometric head H, of its radial gradient Er and of the radial component of the specific discharge qr. Owing to the nonuniformity of the average flow, these quantities are functions of the distance from the well. It is shown that the variances of the head σ2H and of its gradient σ2Er, as well as the crossvariance σE,Y between Er and Y vanish at the well, whereas the discharge variance σ2qr tends to the product between the log conductivity variance σ2Y and the squared mean discharge 〈qr2. This behavior pertains to a stratified formation surrounding the well. Far from the well (≈75 horizontal Y integral scales I) the head variance approaches a constant value. For r ≥ 10I the moments σ2Er, σ2qr and σErY tend to the corresponding values for uniform flow but with the local mean head gradient replacing the constant one. The head vertical integral scale grows indefinitely with r, whereas the vertical integral scale of the flux is larger by one log conductivity vertical scale than the one prevailing in uniform flow. This latter property is explained by the presence of the source line, which increases the correlations in the vertical direction. The present results may be used in identifying the log conductivity statistical parameters from flowmeter velocity measurements in piezometers surrounding pumping or injecting wells.

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