On certain singular integral equations arising in the analysis of wellbore recharge in anisotropic formations

• The solution can be factorized into a singular and a non-singular part.
• Polynomial approximation is practical for determining the density functions.
• Density functions for large similarity parameter c exhibit similarity.
• The solution at the similarity limit converges to a single function of z/c.
• For large c  , the density functions scale as logc-1.

The problem of determining the steady state pressure field for single and multi-well configurations with non-trivial wellbore boundary conditions is considered in this work as an integral equation problem. The aquifer, where the well configuration resides, is assumed to have different vertical and horizontal intrinsic permeabilities and it is bounded above and below by impermeable geological settings. The solutions of the integral equation, known as density functions, are studied from two points of view. First, the singular behavior of the density function is investigated by studying the singular part of the kernel of the integral equation; on this basis the density function is suitably expressed in terms of a non-singular counterpart, the reduced density function, for which a polynomial approximation is formulated and constructed numerically. The convergence of the approximation is studied with respect to the order of the polynomial and shown to be adequately fast. Second, the density functions for large depth to radius ratio and/or large horizontal to vertical permeability ratio exhibit similarity. The density functions depend on the parameters of the single-well problem through a single similarity parameter c. For large values of c  , corresponding to the physical limits just mentioned, the density functions are reduced to essentially a single function, modulo a factor (logc)-1. This property simplifies considerably the analysis of all the large c cases. Considering the case of two wells, as an illustrative example, we also show that the properties of the single well case are also exhibited by the multi-well density functions.