Pressure Buildup for a Well With Storage and Skin in a Closed Square
Authors: Chen H.K. , Brigham W.E. 
Published: December 2007

This paper investigates the effect of wellbore storage and skin on pressure buildup behavior. The closed square is used as the basic unit of analysis and the principle of desuperposition is used to generate the pressure functions. Skin is found to have an effect for the cases presented.

Introduction
An estimate of effective formation permeability can be obtained from pressure buildup data using several conventional graphing methods. The best known are the Muskat, Miller-Dyes-Hutchinson, and Horner (or Theis) graphs. In general, the Horner (or Theis) graph produces the most reliable results. produces the most reliable results. The theory of conventional pressure buildup assumes that a well is closed at the sand face and no after-production occurs. Actually, however, a well is shut in at the surface and flow continues after shut-in because of the fluid compressibility and/or a change of liquid level in the wellbore. This is known as wellbore storage, which affects the pressure behavior during the early buildup time.Formation damage, or skin effect, also can affect the early-time behavior. Skin alone has no influence on buildup behavior, but skin with wellbore storage causes longer distortion of the pressure data than does wellbore storage alone.When a buildup test is run in a closed system, the late-time data are affected by the boundary. This behavior was studied by Ramey and Cobb for a well in the center of a closed square; however, their study did not include the effects of storage and skin. This paper augments their study to include these important parameters.

Theoretical Approach
Before discussing the approach used to calculate the behavior of a well in a closed square, it might be well to comment on the use of the closed square as a basic calculation parameter. The closed square has broader application than first suspected because it is possible to superpose a number of closed squares to get a large variety of rectangular patterns with nearly any mix of boundary conditions. Of course, the closed square itself has considerable use because it is the geometry in many depletion systems with uniform spacing.To find the pressure behavior of a well with storage and skin in the center of a closed square, we used the idea of desuperposition discussed by Gringarten et al. For example, start with the pressure behavior (PD) of an ideal well in the center of a closed square, subtract the pD, for an ideal well in an infinite reservoir, and add the pD, for a well with storage and skin in an infinite reservoir. The result will be close to what is desired. This concept can be represented in generalized equation form as follows.
(1)
This concept also can be described graphically (Fig. 1). The first term on the right of Eq. 1, pD (CD = 0, s = 0,), signifies a well without storage or skin in a closed square. This is mathematically the same as an infinite-square array of wells without storage or skin. This system is shown diagrammatically in Fig. la. The square of interest is outlined, and the wells are shown as open circles to indicate no storage or skin.
JPT
P. 141