CORRELATION STRUCTURE OF FLOW VARIABLES FOR STEADY FLOW TOWARD A WELLWITH APPLICATION TO HIGHLY ANISOTROPIC HETEROGENEOUS FORMATIONS

The study, a continuation of that of Indelman et at. [1996], aims at d eriving the second-order moments of flow variables such as hydraulic h ead, its gradient,and the specific discharge for steady flow toward a fully penetrating well in a confined heterogeneous aquifer. The lag co nductivity Y = In K is modeled as a three-dimensional stationary funct ion of Gaussian correlation of anisotropy ratio e. By using first-orde r approximations in sigma(Y)(2) and e, we derive the variance and the vertical integral scale of the piezometric head H, of its radial gradi ent E-r and of the radial component of the specific discharge q(r). Ow ing to the nonuniformity of the average flow, these quantities are fun ctions of the distance from the well. It is shown that the variances o f the head sigma(H)(2), and of its gradient sigma(E)(2), as well as th e crossvariance sigma(E),(Y) between E-r and Y, vanish at the well, wh ereas the discharge variance sigma(qr)(2) tends to the product between the log conductivity variance sigma(Y)(2) and the squared mean discha rge < q(r) >(2). This behavior pertains to a stratified formation surr ounding the well. Far from the well (approximate to 75 horizontal Y in tegral 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 ver tical integral scale grows indefinitely with r, whereas the vertical i ntegral scale of the flux is larger by one log conductivity vertical s cale than the one prevailing in uniform flow. This latter property is explained by the presence of the source line, which increases the corr elations in the vertical direction. The present results may be used in identifying the log conductivity statistical parameters from flowmete r velocity measurements in piezometers surrounding pumping or injectin g wells.