Page 357 - Fundamentals of Reservoir Engineering

REAL GAS FLOW: GAS WELL TESTING 292

for a real gas β = m(p)

and for gas-oil (two phase) β = m(p)
′
Although the term linearization has been applied to the conversion of equ. (5.1) to
equ. (8.66), it should be remembered that linearization is only achieved for the case of
liquid flow (undersaturated oil) for which the coefficient k/φµc is a constant. For both
real gas and two phase (gas-oil) systems, the µc product is pressure dependent,
meaning that equ. (8.66) is still non-linear.

The basic building block in well test analysis is the constant terminal rate solution of
equ. (8.66), which predicts the pressure or pseudo pressure response at the wellbore,
resulting from the production of a well at constant rate from a state of equilibrium
pressure. Expressing equ. (8.66) in dimensionless form

1 ∂ r ∂ β = ∂ β D (8.67)
D
r ∂ r D D r ∂ D t ∂ D
D
2
2
where r D = r/r W and t D = kt/φµcr (= 0.000264 kt/φµcr in field units- t in hours), the
w
w
general constant terminal rate solution, for r D = 1, can be expressed as
α
f(p) = β D (t ) S (8.68)
+
D
q
In this equation the various component parts are as listed in table 8.14, (in field units),
again, dependent on the nature of the fluid.

To interpret the majority of practical well tests requires the superposition of constant
terminal rate solutions, for different constant production rates acting for different
th
periods of time, to give the value of f(p) n at time t n during the n flow period, as
n
n
+
α f(p) = ∆ q β D (t D − t D ) q S (8.69)
n
j
−
j1 n j 1
=
in which
Undersaturated Real Two phase
oil gas gas-oil
α (fieldunits) 7.08 10 − 3 kh kh 7.08 10 − 3 kh
×
×
q q µ o B o 1422QT q o
o
f(p) p i − p wf m (p i) − m(p wf) m(p ) m(p )
′
′
−
i
wf
β D (t D) p D(t D) m D (t D) m´ D(t D)
S S S + DQ S
TABLE 8.14

4t′
′
β o (t D n − t D j 1 ) = β D (t ) ′ = 2π t′ + 1 2 ln γ D − 1 2 β D(MBH) (t ) (8.70)
D
DA
DA
−