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

The study, a continuation of that of Indelman et at. [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 lag conductivity Y = In K is modeled as a three-dimensional stationary function of Gaussian correlation of anisotropy ratio e. By using first-order approximations in sigma(Y)(2) and e, we derive the variance and the vertical integral scale of the piezometric head H, of its radial gradient E-r and of the radial component of the specific discharge q(r). 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 sigma(H)(2), and of its gradient sigma(E)(2), as well as the crossvariance sigma(E),(Y) between E-r and Y, vanish at the well, whereas the discharge variance sigma(qr)(2) tends to the product between the log conductivity variance sigma(Y)(2) and the squared mean discharge < q(r) >(2). This behavior pertains to a stratified formation surrounding the well. Far from the well (approximate to 75 horizontal Y integral scales I) the head variance approaches a constant value. For r greater than or equal to 10I the moments sigma(E)(2), sigma(qr)(2) and sigma(E,Y) 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.