Average Pressure From Buildup Data For A Rectangle With Constant Pressure Boundary
Authors: Dykstra H. 
Published: December 2007

Abstract
This paper shows the relation between the average pressure in a rectangle with a constant pressure boundary at the instant the well is shut in and the buildup pressure behavior of the well. The paper includes a set of tables giving this relation for a producing well at various locations in a producing well at various locations in a rectangle with one or more constant pressure boundaries. The data in the table are for producing times greater than the time producing times greater than the time required to reach steady state.
Several methods are described that can be used to obtain the average pressure at the instant the well is shut in. The principle of superposition is used to get the principle of superposition is used to get the pressure buildup behavior of the well. Two pressure buildup behavior of the well. Two examples are given to show the effect of a constant pressure boundary as opposed to a no flow boundary.

Introduction
In 1937 Muskat presented an equation for the pressure buildup of a well producing from a radial system with constant pressure at the outer boundary. Miller, Dyes, and Hutchinson (MDH) extended the analysis to systems with several different outer radii and to systems with closed outer boundary. The dimensionless buildup pressure was plotted against the log of the dimensionless plotted against the log of the dimensionless shut-in time. For these solutions the well was at steady state prior to shut in for constant pressure boundary and at pseudo steady state for closed outer boundary. Horner considered the case for an infinite outer boundary and showed that the press build-up can be plotted against the log ((t + delta t) /t) and extrapolated to infinite shut-in time. Matthews, Brons, and Hazebroek used the method of superposition to determine the pressure behavior of a well producing from a bounded system for a producing from a bounded system for a variety of well locations and boundary shapes. They presented a set of curves giving the difference between a "false pressure" obtained from a Horner plot and the average pressure of the bounded system as a function pressure of the bounded system as a function of producing time. Perrine took the MDH results and presented a chart showing how the wellbore pressure built up as a function of dimensionless shut-in time to the outer boundary pressure for a constant pressure boundary, and to the average pressure at the instant of shut-in for a closed outer boundary. Earlougher, et al presented a set of MDH curves similar to Perrine for wells producing from rectangular reservoirs with producing from rectangular reservoirs with constant pressure boundaries.
The method of presenting the MDH curves for constant pressure boundaries along with curves for closed outer boundaries was somewhat deceiving. It has been the experience of this author that many engineers have incorrectly used such curves for determining the average pressure for drainage systems having one or more constant pressure boundaries. Normally an engineer is interested in the average pressure in the drainage area at the instant the well is shut in for use in material balance calculations and the curves for a constant pressure boundary as presented by Perrine pressure boundary as presented by Perrine and by Earlougher, et al cannot be used for this purpose.
Buildup curves for the special case of a well in the center of a constant pressure square were calculated by Kumar pressure square were calculated by Kumar and Ramey. They presented MDH curves, Horner plots and an MBH curve giving the difference between the "false pressure and the average drainage pressure at the instant the well was shut in.