Pseudosteady-State Flow Capacity of Oil Wells With Limited Entry and an Altered Zone Around the Wellbore
Authors: Odeh A.S. 
Published: December 2007

Abstract
An analytic solution for pseudosteady-state flow of an oil well with limited entry and with an altered zone is derived. Wells that are open to flow along a traction of their productive interval, and that have a wellbore surrounded by a zone whose physical characteristics have been altered because of damage or improvement, are termed wells with limited entry and an altered zone.
The finite cosine transform was used to solve the partial differential equations that describe the behavior of the physical system. The solution was programmed for the CDC 6400 computer, and many programmed for the CDC 6400 computer, and many computer runs covering a wide range of variables were made. The results were used to reduce the complicated analytical solution to a simple approximate solution that permits the engineer to account easily for the combined effect of limited entry and altered zone on the pseudosteady-state flow behavior of an oil well.

Introduction
This paper reports the derivation of an analytical relation between drawdown and flow rate for a well producing under pseudosteady-state flow with producing under pseudosteady-state flow with restricted entry to flow and with an altered zone around the wellbore. "Altered zone" refers to a zone whose permeability and/or porosity values have been changed because of mud damage, acidizing, or sand consolidation, for example.
The effect of an altered zone around the wellbore on the calculation of the productive capacity of a well is normally accounted for by introducing a skin factor, s, in the proper flow equations. Hawkins developed an algebraic expression relating the wellbore radius, the altered-zone radius and permeability, and the reservoir permeability to s. Hawkins' equation assumes radial flow into the wellbore. This implies that the well is open to flow along the total length of the productive interval.
Flow into the wellbore when only a fraction of the productive interval is open to flow, with or without an altered zone, is not radial. The common practice of using Hawkins' equation in such cases practice of using Hawkins' equation in such cases to calculate s could lead to highly erroneous results. For example, the flow rate calculated for a well producing under pseudosteady state, with restricted producing under pseudosteady state, with restricted entry and with an altered zone and using a skin factor calculated by Hawkins' equation, can be too high by more than 100 percent.
Rowland and Jones and Watts introduced a correction factor into Hawkins' equation to account for the effect of nonradial flow behavior around the wellbore. Hawkins' skin value is multiplied by ht/hp, where ht is the total thickness of the productive interval and hp is the length of the productive interval and hp is the length of the interval open to flow at the wellbore. The ht/hp correction factor is based on the assumption that the flow into the wellbore is radial in the altered zone opposite the open interval. This implies that the nonradial components of flow into the wellbore are negligible. Using model studies, Jones and Watts showed that the error introduced in the calculated skin values by neglecting the nonradial components of flow is less than 20 percent in the worst case. Furthermore, they proposed a modified equation to eliminate this error.
The suitability of the radial flow assumption upon which the factor ht/hp is based depends on the values of many variables such as the radius of the altered zone and its permeability, and the length of the open interval. Because of their discretized nature, numerical models at times do not accurately represent the physical flow pattern around the wellbore. Most models assume the flow to be radial between the wellbore and the adjacent cells, regardless of whether part or all of the productive interval is open to flow. Therefore, results from model studies may not be adequate to define the limits of applicability of the modification to the radial flow formula, such as the ht/hp correction factor. One purpose of the work reported in this paper is to define these limits. paper is to define these limits. The analytic solution is very complex and is not suited for engineering work. Because of this, it was programmed for the CDC 6400 computer, and many computer runs were made.
SPEJ
P. 271