Page 206 - Fundamentals of Reservoir Engineering

STABILIZED INFLOW EQUATIONS 144

2k h
π
PI = e
r 1
µ ln e − + S
r w 2

the Pl ratio increase is

660 1
ln − + 254.02 261.11
= .333 2 1 =
660
ln − − 3.09 4.00
.333 2
= 65.3

6.5 GENERALIZED FORM OF INFLOW EQUATION UNDER SEMI-STEADY STATE
CONDITIONS

The semi-steady state inflow equation developed in sec. 6.2 appears to be restrictive in
that it only applies for a well producing from the centre of a circular shaped drainage
area. When a reservoir is producing under semi-steady state conditions each well will
assume its own fixed drainage boundary, as shown in fig. 5.3, and the shapes of these
may be far from circular. The inflow equation will therefore require some modification to
account for this lack of symmetry. Equation (6.12) can be expressed in a generalized
3
form by introducing the so-called Dietz shape factors denoted by C A, which are
presented for a variety of different geometrical configurations in fig. 6.4. Precisely how
these shape factors were generated, in the first place, will be explained in the
appropriate place, Chapter 7, sec. 6. For the moment the reader is asked to accept the
following tenuous argument for the generalization of the inflow equation. Excluding the
mechanical skin factor, equ. (6.12) can be expressed as

qµ 1 r π 2
pp wf = 2kh 2 ln π r e e 3/ 2 (6.20)

−
2
π

w
in which the argument of the natural log can be modified as
4r e 2 4A 4A (6.21)
π
3/2 2
4 e r w = 56.32r w 2 = γ 31.6r w 2
π
in which A is the area being drained, γ is the exponential of Euler's constant and is
equal to 1.781, and 31.6 is the Dietz shape factor for a well at the centre of a circle,
refer fig. 6.4. Therefore, equ. (6.20) can be expressed in the general form, including the
skin factor, as

qµ 1 4A
pp wf = 2kh 2 ln γ C r 2 + S (6.22)

−
π

Aw
For a reservoir which is producing under semi-steady state conditions, then as already
noted, the volume drained by each well is directly proportional to the well's production
rate. Therefore, it is a fairly straightforward matter to estimate the volume being drained
by each well and, using the average thickness in the vicinity of the well, the area. If