Wellbore Pressures in Reservoirs With Constant-Pressure or Mixed No-Flow/Constant-Pressure Outer Boundary
Authors: Larsen Leif 
Published: December 2007

Abstract
This paper reports On a general relationship existing between wellbore pressures in reservoirs with constant pressure or mixed no-flow/constant-pressure outer pressure or mixed no-flow/constant-pressure outer boundary and in related closed reservoirs, and uses this to derive expressions for shape factors, wellbore pressures, and Matthews-Brons-Hazebroek (MBH) functions for static reservoir pressure.

Introduction
Effects of reservoir shape and well location on wellbore pressures have been studied extensively for closed pressures have been studied extensively for closed reservoirs, but only a few cases have been presented for reservoirs with constant pressure or mixed no-flow/ constant pressure outer boundary. However, the latter group can be increased considerably by the methods presented in this paper, since the wellbore pressures for a wide variety of such reservoirs can be expressed by differences of wellbore pressures from related closed reservoirs. As a consequence, the shape factor, CA, and the dimensionless wellbore pressure in the steady-state period, can be determined readily from existing tables for a large number of cases. Moreover, the end of the infinite-acting period, tDAeia, can be estimated, and a rough upper period, tDAeia, can be estimated, and a rough upper bound can be given for the start of the steady-state period, tDASS. Modifications of standard methods to period, tDASS. Modifications of standard methods to determine the static reservoir pressure from buildup data also are obtained readily.
The results of this paper are related closely to those of Earlougher et al. and Ramey et al. Tables from those papers also are used as a basis for this paper. All cases papers also are used as a basis for this paper. All cases that can be handled by direct use of these tables are included in Fig. 1. Tables presented by Earlougher also are used. By using the approach of Ramey and Cobb, it is possible to work with either dimensionless wellbore pressures or with MBH functions. These functions also pressures or with MBH functions. These functions also are used to determine the static pressure from Horner analysis.

Theory
Only single-well reservoirs that can be generated by regular patterns of producing and injecting (image) wells, with each well having the same rate, q, in absolute value, are considered. Storage and skin effects are not included. The dimensionless pressure drop for the actual well,
.....................................(1)
can therefore be expressed as a sum of exponential integrals. Here, A denotes drainage area, pi initial fully static pressure, pi flowing wellbore pressure, and
.....................................(2)
dimensionless time based on the wellbore radius. Dimensionless time based on drainage area, tDA= r2wtD/A, also is used.
Consider PwD for such a reservoir with constant pressure or mixed no-flow/constant-pressure outer pressure or mixed no-flow/constant-pressure outer boundary. Since the reservoir can be generated by a regular pattern of producing and injecting wells, a related closed pattern of producing and injecting wells, a related closed reservoir is obtained by letting all wells be producers. By superimposing the two patterns, each injecting well cancels a producing well, and a new pattern of only producing wells is obtained, with each well having rate 2q producing wells is obtained, with each well having rate 2q and draining an area 2A. The superscript "new" refers to the new closed reservoir and "old" refers to the first closed reservoir. It follows that
PwD(tD, A) 2P new WD (tD, 2A) -P old wD (tD, A)......(3)
Moreover, since
.....................................(4)
where PDMBH denotes the MBH function, and tD(2A)= (1/2)tDA, it follows that .....................................(5)
JPT
P. 1613