A Study of the Behavior of Partially Penetrating Wells
Authors: Clegg M.W. , Mills M. 
Published: December 2007

Abstract
This paper presents an approximate analytical solution to the problem of the pressure distributions arising from the production of a compressible liquid in a partially penetrating well. The limits of validity of the approximation are discussed and it is shown that the basic approximation is the same as that used in previous numerical studies of the problem. The variation in pressure at the production problem. The variation in pressure at the production well with time for different penetration ratios is investigated and is shown to agree with previous work.
The solutions are used to assess the effects of partial penetration at large distances from the well partial penetration at large distances from the well and at the wellbore itself. In the latter case the effects of anisotropy are also studied for constant, but unequal, horizontal and vertical permeabilities. It is suggested that measurements of the pressure drawdown below the producing interval in a well should give some indication of a thickness-vertical permeability product for the formation; this would permeability product for the formation; this would characterize the effectiveness of the formation below the producing interval.

Introduction
In many oil and gas reservoirs the producing wells are completed as partially penetrating wells; that is, only a portion of the pay zone is perforated. This may be done for a variety of reasons, but the most common one is to prevent or delay the intrusion of unwanted fluids into the wellbore. This paper studies the effects of partial penetration on the pressure distribution resulting penetration on the pressure distribution resulting from the production of a compressible liquid at a constant rate. To simplify the mathematical analysis the case of a single well located in an infinite reservoir of constant thickness is considered, and it is assumed that this well is perforated over an interval adjacent to the upper perforated over an interval adjacent to the upper (or lower) impermeable boundary of the formation. A well producing from an arbitrary interval within the bulk of the formation could be treated in a similar way, but this action would merely add to the complexities of the solution and contribute little to the appreciation for the over-all effects of partial penetration. partial penetration. In a recently published paper by Odeh, the more general problem of producing from an arbitrary interval within the pay zone was studied for the quasi-steady state flow of a compressible liquid. This is the case in which there is a constant potential at the outer boundary and a constant flux potential at the outer boundary and a constant flux across the same boundary. This contrasts with the problem studied here, which is concerned with the problem studied here, which is concerned with the nonsteady-state behavior in an infinite reservoir.
The exact solution of the partial penetration problem presents great analytical problems because problem presents great analytical problems because the boundary conditions that solutions of the partial differential equations must satisfy are partial differential equations must satisfy are mixed; i.e., on one of the boundaries the pressure is specified on one portion and the flux on the other. This difficulty occurs at the wellbore, for the flux over the nonproductive section of the well is zero, and the potential over the perforated interval must be constant. In the case of constant rate production from the well, this uniform potential is time dependent and unknown, and the additional condition that
where h1 is the thickness of the perforated interval, must also be satisfied.
This problem may be overcome in the case of constant rate production by making the assumption that the flux into the well is uniform over the entire perforated interval, so that on the wellbore the flux perforated interval, so that on the wellbore the flux is specified over the total formation thickness.
SPEJ
P. 189