Page 278 - Fundamentals of Gas Shale Reservoirs

258 GAS TRANSPORT PROCESSES IN SHALE

(a) property values for ρ, μ, and c . Darabi et al. (2012) showed
g
1 that the pressure solutions under the typical pulse‐decay
Late transient analytical solution conditions (i.e., upstream and downstream pressures,
Numerical solution
0.9 standard temperature, large pressure gradient across core
sample) with constant fluid properties and pressure‐dependent
0.8 properties are significantly different. Figure 11.14a presents
a sensitivity analysis on ρ, μ, and c pressure‐dependency
g
p D 0.7 effect and showed that under the typical pulse‐decay condi-
tions, the variations of μ and c have a negligible effect on
g
the pressure solution, whereas the pressure dependency of ρ
0.6
has a significant impact.
The pseudo‐pressure function m(p) is defined as follows:
0.5
p k
0.4 mp 2 pdp . (11.27)
0 0.2 0.4 0.6 0.8 1 0 z
t D
(b) Equation 11.24 is reformulated using Equation 11.27 and
1
Late transient analytical solution the real gas law (ρ = pM/zRT) that accounts for pressure
Numerical solution dependence of density is
0.9

m k 2 m
0.8 , 0 xL, t 0 (11.28)
.
t c 1 K x 2
g a
n D 0.7 Equation 11.28 accounts for the gas desorption in the
term K , defined as the derivative of adsorbate density with
0.6 a
respect to gas density, K = ∂q/∂ρ. Equation 11.28 has two
a
boundary conditions and one initial condition,
0.5
t
t
m
BC-1:( , 0 ) m () , t , 0 (11.29)
0.4 u
0 0.2 0.4 0.6 0.8 1
t
t
t D BC-2 :(mL , ) m d () , t , 0 (11.30)
(c)
, )
0
1 IC :(mx , ) m di , x 0 , m (00 m ui . (11.31)
Late transient analytical solution
Numerical solution
0.9 Three parameters, μ, c , and k are assumed to be constant
g
to derive an analytical solution. A constant permeability
assumption is a reasonable assumption because the pressure
0.8
variation across the core is fairly small (e.g., 500 kPa).
n D 0.7 Darabi et al. (2012) showed that the assumption of constant
viscosity‐volumetric compressibility product introduce an
acceptable error.
0.6 Three dimensionless groups, ∆m = (m − m )/(m − m ),
d
u
u0
d0
D
t = kt/(µc φL ), and x = x/L are defined and substituted into
2
D
g
D
0.5 Equations 11.28–11.31. Using an analogy to the Hsieh et al.
(1981) method, the analytical solution to Equation 11.28 is
0.4 derived for the late‐transient time as follows:
0 0.2 0.4 0.6 0.8 1
t D
ln( m ) ln() st, (11.32)
f
FIGURE 11.14 Comparison of the late‐transient analytical solu- D 0 1
tion with numerical solution. The numerical solution assumes that
all parameters are pressure dependent. The analytical solution f 0 2 a b 2 1 2 b a 2 1 2 b 2 1 2 /
assumes that (a) Darcy permeability is used and ρ, μ, and c are 2 2 2 2
g
(
constant, (b) Darcy permeability is used, μ and c are constant, and 1 1 aa bb ab a bbab), (11.33)
g
ρ is pressure‐dependent, (c) APF is used, μ and c are constant, and
g
ρ is pressure dependent.