Calculating The Pseudo-Skin Factor Due To Partial Well Completion
Authors: Vrbik J. 
Published: March 2010

Abstract
A simple formula for the pseudo-skin factor due to partial well completion is derived using equations for steady-stale flow of incompressible fluid and the assumption of uniform flow across the well-bore opening. This is compared with numerical results of the same problem under the (more rigorous) assumption of constant bottom-hole pressure to show that the difference between the two sets of results is insubstantial, thus justifying the practical value of the former simplified approach.

Introduction
The pseudo-skin factor due to a partial well completion is one of the most important quantities when determining the flow of oil into a well-bore. Several papers on this subject have been published in the past, the most notable by Brons and Marting(2) and Odeh(6). Unfortunately, their final results have been presented in the form of tables or charts, thus giving the impression that evaluation of pseudo-skin factor requires extensive use of a computer.
It is shown in [his paper that the findings of previous authors can be reduced by analytical means into numerically simple form of Eq. (5). The advantage of this approach is not just numerical; Eq. (5) explicitly demonstrates the dependence of the pseudo-skin factor on only three basic parameters (reflecting the length and position of the completion interval within the pay zone, the effective well-bore radius, and the ratio of vertical to horizontal permeability), which by itself clearly advances our understanding of the problem. It is worth noting that no knowledge of the outer size or shape of the reservoir is required.
It has been pointed out in the past (see, for example, Odeh's footnote), that a rigorous formulation of the problem leads to a set of mixed boundary conditions, instead of those used in deriving Eq. (5); unfortunately, there is no technique for solving this analytically, and one has to resort to a very elaborate and expensive numerical computation, as done in the second part of this paper. The two sets of results are compared to demonstrate that Eq. (5) is sufficiently accurate for any routine application.
Substantial differences between the two techniques become apparent only when comparing the profiles of oil flow into the well opening. Then, the rigorous approach should provide a more realistic picture of the amount of oil entering the well-bore at different locations along the perforation interval (a typical result is presented in Figure 2).

Derivation of Simplified Formula
Consider the following radial flow of incompressible fluid in a layer of porous media (Fig. 1):
FIGURE 1 (Available in full paper)
Substituting Darcy's equation, in SI units, (i.e. velocity expressed in metre/sec., viscosity in Pascal × sec., permeability in m2 and pressure gradient in Pascal/metre), namely:
Equation (1) (Available in full paper)
into the equation of continuity, yields;
Equation (2) (Available in full paper)
where p is the modified pressure, (i.e. p = P + pgz, where P is the actual pressure, p is fluid density, and z is the vertical coordinate) and kh and kv is the horizontal and vertical permeability of the porous medium, respectively.