Steady-State Flow Capacity of Wells With Limited Entry to Flow
Authors: Odeh A.S. 
Published: December 2007

Abstract
This paper analyzes the effect of limited entry to flow at the wellbore on the steady-state productivity of a well. Wells that have been opened to flow along a fraction of their productive interval are termed wells with limited entry. Previous work treated the cases of a partially penetrating well, a well producing from the central Portion of the productive interval and a well in which several intervals equally spaced were open to flow. In this paper the open interval can be located anywhere within the productive interval. Thus, in a sense, it generalizes previous work.
The finite cosine transform was used to arrive at a solution for steady-state flow of a slightly compressible fluid. The solution was programmed for a CDC 1604 computer. Numerical values for rd = 660 ft, rw = 1/4 ft, and range of sand thickness of 20 to 200 ft are presented in graphical form. The effect of rd and rw values on the result is shown in a table. The correct calculation of skin and damage ratio in the presence of limited entry to flow is explained and illustrated by examples. Moreover, the paper shows bow to calculate the net decrease in productivity due to the combined effect of limited entry and perforations.

Introduction
In some wells only a fraction of the productive interval is open to flow. Location of this fraction is usually dictated by formation characteristics and reservoir behavior. For instance, if a gas cap exists, the open interval is located away from the gas-oil contact to prevent any possible gas coning. Wells that intentionally have been opened to flow along a fraction of their productive formation are termed wells with limited entry. Obviously, unintentional completions of this type also exist. Limited entry to flow decreases well productivity. Magnitude of the loss depends on the fraction of the formation open to flow, on the thickness of the sand, on the location of the open interval and on the ratio of rd/rw, where rw is well radius and rd is the drainage radius of the well.
The use of pressure buildup data on producing wells to calculate the condition of the formation around the wellbore is an accepted practice. van Everdingen and Hurst introduced the concept of the skin factor s considered to be due to a thin layer of different permeability immediately around the wellbore. These authors dealt with the case of a well of complete radial geometry, i.e., a well with open-hole completion that completely penetrates the formation. The presence of a low-permeability skin results in a loss of productivity, as does limited entry. Therefore, if pressure buildup data obtained on a well with limited entry are used to establish the presence or absence of skin (i.e., formation damage), and a correction is not made for this loss of productivity, the calculations would result in an erroneous skin value. They might indicate the presence of formation damage when in reality there is none, or they might indicate a value larger than the true value. This could lead to an incorrect basis for planning remedial measures.
Muskat studied the problem of partially penetrating wells for the case of incompressible flow. He presented equations and figures which allow estimation of loss in productivity. Brons and Marting, using equations based on Nisle's work, studied the loss of productivity for three cases. The first was for a partially penetrating well; the second was for a well producing from only the central portion of a productive interval; and the third was for a well in which several intervals equally spaced were open to flow. Their work was for steady-state depletion-type reservoirs wherein the well radius of drainage is established and the fluid is considered to be slightly compressible.
Considered in this paper is the problem of wells with limited entry in which the open intervals are located anywhere within the productive sand.
SPEJ
P. 43ˆ