Abstract
General and simple algorithms are presented to determine the coordinates of sufficient numbers of image wells to compute pressure distributions and Matthews-Brons-Hazebroek (MBH) functions for rectangles of any shape, triangles with internal angles (30,60,90), (45,45,90), (60,60,60), and (30,30,120) degrees, rhombi with acute angle 60 degrees, and hexagons. A simple extension to compute their slopes also is included. The well must be located at the center of the hexagon, on the short diagonal of the rhombus, and on the corresponding height of the last triangle. For the other shapes there are no restrictions on well location. Any combination of no-flow and constant-pressure outer boundaries can be handled for pressure distributions in rectangles and (45,45,90) degrees triangles, but only special cases are possible for the other shapes.
Introduction
Matthews, Brons, and Hazebrock introduced the special function (p* -P)/(70.6qBu/kh) to determine the average pressure, P, from the Horner false pressure, p*, for closed homogeneous reservoirs of certain basic shapes. The reservoirs were assumed filled with a fluid of small and constant compressibility, and each was produced at a constant surface rate, q, from a single fully penetrating well with zero skin and no wellbore storage.
With these conditions, it follows from results of Ramey and Cobb that the MBH function
(1)
can be expressed in the form
(2)
where
(3)
is dimensionless time based on the drainage area and
(4)
is dimensionless wellbore pressure drop. Note that the dimensionless time based on the wellbore radius is given by tD=AtDA/r2w
The drainage shapes considered in this paper are similar to those in Ref. 1 in the sense that each can be generated by adding a regular infinite pattern of image wells to the actual well in an unbounded homogeneous reservoir, with all wells starting to produce or inject at time zero. If each well is a producer, then a closed drainage area is generated, while at least part of the outer boundary will remain at constant pressure otherwise.
If qi denotes the rate of the ith well (negative for injecting wells) and q = q1>0 is the rate of the actual well, then
(5)
It is assumed here that the response of each well is given by the line-source solution, and that, and (xi, yi) is the location of the ith image well for i>2. In most cases considered in this paper, =.
The dimensionless pressure drop, PD, at an arbitrary observation point ( ) within the drainage area, but outside the wellbore, is given by
(6)
(7)
for the closed area drained by the well at the origin, with the infinite sum negligible for producing times in the infinite-acting period. To determine the linear and semilogarithmic slopes of
PwD and PDMBH, note that
(8)
SpEJ
P. 113^
Pressure Drop Equations for a Partially Penetrating Vertical Well in a Circular Cylinder Drainage Volume
