Page 322 - Fundamentals of Reservoir Engineering

REAL GAS FLOW: GAS WELL TESTING 257

magnitude greater than for typical sandstone samples. So far, the experiments have
not been repeated on sandstone but in using equ. (8.29) it is assumed that the same
physical principles will apply. Although correlation charts giving β as a function of the
9
permeability exist in the literature the reader should be aware that they are not always
applicable. Irregularities in the pores can greatly modify the β versus k relationship
making it advisable, in many cases, to experimentally derive a relationship of the form
given by equ. (8.28).

8.8 THE CONSTANT TERMINAL RATE SOLUTION FOR THE FLOW OF A REAL GAS

The constant terminal rate solution of the radial diffusivity equation

()
()
1 ∂ r ∂ mp = φµ c ∂ mp (8.11)
r ∂ r r ∂ k t ∂

for the flow of a real gas, describes the change in real gas pseudo pressure at the
wellbore due to production at a constant rate from time t = 0. Equation (8.11) is
identical in form with equ. (5.20) except that pseudo pressure replaces real pressure as
the dependent variable. Therefore, the constant terminal rate solution of equ. (8.11)
must, by analogy, have the same form as the solution presented for liquid flow in
Chapter 7.
For small flowing times, the transient, constant terminal rate solution of equ. (8.11), in
Darcy units, is similar to equ. (7.10), i.e.

4 kt
mp i mp wf ) = constant × ln ⋅ 2 + 2S′
(
() −
γ φ µ cr w
in which S' = S + DQ. The constant can be evaluated using the relationship

2p
∆ mp ∆ p
() =
µΖ

therefore,

2p qµ 4 kt
() mp
mp − ( wf ) = ln 2 + 2S′
i
µΖ 4 π kh γ φ µ cr w
and converting to field units, using the fact that pq/Z = (p sc q sc)T/T sc, gives

711QT 4t
() mp
mp − ( wf ) = ln D + 2S′ (8.30)
i
kh γ
Taking the analogy with the liquid flow equations a stage further, equ. (8.30) can be
expressed in dimensionless form as

kh () mp m () S′ (8.31)
t
(
i
1422QT (mp − wf )) = D D +