Stochastic analysis of well-type flows in randomly heterogeneous porous formations

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Abstract

[1] Well-type flow takes place in a heterogeneous porous formation where the transmissivity is modeled as a stationary random space function (RSF). General expressions for the covariances of the head and flux are obtained and analyzed. The second-order approximation of the mean radial flux is represented as the product between the solution equation image valid in a homogeneous domain and a distortion term equation image, which adjusts equation image according to the medium heterogeneity. The spatial dependence of the function equation image is studied. In view of the formation identification problem, the equivalent T(eq) and apparent T(ap) transmissivity are computed. The important result is the relationship equation image (equation image may be either “eq” or “ap”), where TH and TG represent the harmonic and the geometric means of the transmissivity, respectively. The position-dependent weight equation image is explicitly calculated. Indeed, close to the well, it yields equation image, which is understandable in view of the fact that the limit equation image is equivalent to I → ∞, which is the heterogeneity structure of a stratified formation. Nevertheless, the effective transmissivity of a stratified formation is precisely TH. In contrast, far from the well, one has equation image, with the flow being slowly varying in the mean there. It is shown that equation image grows with increasing equation image. In the case of T(eq), the rate of growing is found (similar to Dagan and Lessoff (2007)) to be strongly dependent upon the position in the flow domain, whereas T(ap) is a more robust property. Finally, it is shown how the general results can be used for practical applications.

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