Page 251 - Introduction to Petroleum Engineering

238 TRANSIENT WELL TESTING
Gas well deliverability tests are discussed in the next section, and PTT for charac-
terizing gas flow in the reservoir is discussed here. Conversion factors correspond
to standard temperature 60°F = 520°R and standard pressure 14.7 psia.

12.3.1 Diffusivity Equation
The starting point of gas well PTT is a diffusivity equation. Pressure was the
dependent function in the diffusivity equation for oil well PTT. By contrast, the
diffusivity equation for gas well PTT must account for nonlinear gas properties.
This can be done by expressing the diffusivity equation for gas flow in terms of real
2
gas pseudopressure m(p) (psia /cp), which is defined as
p p
m p 2 dp (12.18)
p ref Z
where p is the reference pressure (psia), p′ is the dummy variable (psia), Z is the
ref
real gas compressibility factor (fraction), and μ is the gas viscosity (cp). Real gas
pseudopressure m(p) accounts for the pressure dependence of Z and μ.
The diffusivity equation for single‐phase gas flow in the radial direction is
2 mp 1 mp mp (12.19)

r D 2 r D r D t D
Dimensionless radius and dimensionless time are defined as
r kt
r D t ; D . 0 000264 2 (12.20)
r
r w c T i w
where r is the radial distance from the well (ft), r is the well radius (ft), k is the
w
permeability (md), t is the time (hr), ϕ is the porosity (fraction), μ is the gas viscosity
(cp), and c is the total compressibility (psia ). The subscript i shows that gas vis-
−1
T
cosity μ and total compressibility c are evaluated at initial pressure. Total system
T
compressibility is
c T c r c S o c S w c S g (12.21)
g
o
w
where c is the rock compressibility and fluid‐phase compressibility is the saturation‐
r
weighted average of oil‐, water‐, and gas‐phase compressibilities. If all three phases
are present, the diffusivity equation for single‐phase gas flow is based on the assump-
tion that gas is the only mobile phase.
12.3.2 Pressure buildup Test in a Gas Well
We illustrate analysis of gas well PTT by analyzing a PBU test in a gas well. A gas
flow rate is maintained at a stabilized rate for a duration t . The well is shut in after
F
the stabilized flow period, and the pressure response at the well is recorded as a
function of shut‐in time Δt. The superposition principle is combined with the