Page 282 - Fundamentals of Gas Shale Reservoirs

262 GAS TRANSPORT PROCESSES IN SHALE

In Equations 11.49 and 11.50, K is the ratio of the void This may result in the poor quality and a time lag in the
c
volume of the main and sample vessels V to the crushed recorded early‐time pressure data. Cui et al. (2009) recom-
c
sample pore volume, mend that the late‐time pressure data are more reliable for
the permeability estimation.
V
K bc , (11.51)
c
M (1 ) K a 11.9.4 Crushed Sample Permeability Estimation with
Late‐Time Pressure Data
where ρ is the sample bulk density and M is the gas molar
b
mass. Substituting the pseudo‐pressure solution m in The logarithm of cumulative residual gas penetration ratio
Equation 11.45 into 11.50, F is given as follows: F in Equation 11.52 becomes a linear function of time when
R R
the dimensionless time τ > 0.1. If K ≥ 50, then the exact F
cm
R
2
e K n / t R a 2 solution in Equation 11.52 can be approximated with
F 6 K K 1) . (11.52) Equation 11.53. The straight line part of the solution for
(
R c c K 2 2 9 ( K 1)
n 1 c n c τ > 0.1 is approximated as follows:
If the ratio of the sample pore volume to the total void
volume of the main and sample vessels is small (K → ∞), the ln(F R ) f 0 st, (11.56)
1
c
pseudo‐pressure ratio solution F simplifies to
R where s is the slope of the straight line,
1
6 22 2 1
/
F e nKtR n . (11.53) K
R n 2 s 1 , (11.57)
n 1 1 2
R a
Cui et al. (2009) show that for K > 50 Equation 11.53 is
c
an appropriate approximation for the analytical solution, and and α is the first solution of Equation 11.34, and the y‐inter-
1
the early‐time and late‐time pressure history in the void cept of the straight line is
volume may be used to determine the permeability.
6 KK 1)
(
f 0 ln 2 2 c c . (11.58)
11.9.3 Crushed Sample Permeability Estimation with K c 1 9( K c 1)
Early‐Time Pressure Data
The early‐time solution of Equation 11.53 can be approxi- The slope s of the straight‐line part of the solution is
1
mated as (Carslaw and Jaeger, 1947; Do, 1998): related to the permeability as
2
R [ 1 ( ) K ] cs
6 K a g 1
F 1 F t. (11.54) k 2 . (11.59)
U R
R 2 1
a
Equation 11.42 implies that the early‐time cumulative The full scope of the pressure data, including the early
uptake gas penetration ratio F versus the square root of time time and the late‐time periods, may be used in a numerical
U
yields a straight line on a linear scale. The line slope s is estimation method to determine the permeability. The reader
1
related to the permeability, may refer to Section 2.2 for the details of the numerical
workflow.
2
2
sR [ 1 ( ] ) c
k 1 a g . (11.55)
36 11.10 CANISTER DESORPTION TEST
Equation 11.55 is valid for large values of K (>50) The canister desorption test is performed on the drill cores to
c
and relatively short time after the gas expansion into the estimate the rock permeability and diffusion. Core sample,
sample vessel, specifically when the dimensionless time usually in a cylindrical shape are obtained from the produc-
τ = Kt/R < 0.0002 or F < 0.2 (Cui et al., 2009). The large tive zone of the wells and transferred into the canister
2
U
a
differential pressure between the high‐pressure main vessel apparatus, which is schematically shown in Figure 11.17.
and the almost vacuum sample vessel may cause an adia- The drill core is kept at the reservoir temperature and an
batic temperature change in the system. Also, pressure ambient pressure. The cumulative desorbed gas volumes are
measurement at the early times may be affected by the recorded and used to determine the permeability and
kinetic expansion from the main cell into the sample cell. diffusivity.