Page 277 - Fundamentals of Gas Shale Reservoirs

PULSE‐DECAY PERMEABILITY MEASUREMENT TEST 257
The major steps of a pulse‐decay experiment for permeability model that may describe the tight gas and shale gas perme-
measurement follow: ability. Kaczmarek (2008) and Wu et al. (1998) proposed
analytical interpretations of the pressure‐decay response,
1. Valves 1, 2, and 3 are opened to allow the gas flow assuming the Klinkenberg effect and a constant mass flow
through the sample and through upstream and rate for linear and radial gas flow, respectively. Jannot et al.
downstream vessels. (2007) investigated the experimental conditions that affect
2. The entire system reaches the equilibrium pressure the determination of Klinkenberg parameters and showed
p ; valves 2 and 3 are closed. that better estimates of k and b are achieved with an infinite
d0 D
3. The upstream vessel pressure is increased by a few downstream vessel volume. They also showed that the
percent and allowed to reach the new equilibrium upstream vessel volume, core diameter, and length do not
state; valve 2 is then opened and the pressure difference significantly affect the k and b estimations. Darabi et al.
D
(∆p) is measured with respect to time. (2012) showed that an analytical pseudo‐pressure solution
may better estimate the permeability, mainly because of the
strong sensitivity of the permeability parameter estimates
The pressure difference ∆p = p − p across the core. The to the variation of fluid properties. In addition, they show
u
d
pressure‐decay response ∆p is analyzed to determine the the pseudo‐pressure partial differential equation may be
permeability. numerically solved with accurate estimations of gas density,
Several studies investigated the pressure‐decay partial viscosity, and volumetric compressibility.
differential equation to determine an analytical solution and The following presents the pulse‐decay equation along
subsequently estimate the core permeability. Hsieh et al. with an analytical solution for the constant permeability and
(1981) derived the late‐transient solution for the pressure constant µc assumptions, and a numerical algorithm with all
g
difference across the core sample. Jones (1997) used the pressure‐dependent parameters. The pulse‐decay material
analytical solution originally derived by Hsieh et al. (1981), balance equation for gas flow in a one‐dimensional core
and investigated the conditions under which the late‐transient sample with gas adsorption is
solution falls into a single exponential decline. Other studies
proposed different solution forms such as series (Brace et al., 1
1968; Chen and Stagg, 1984; Dicker and Smits, 1988; (1 ) q n r n kp , (11.24)
Haskett et al., 1988; Hsieh et al., 1981; Neuzil et al., 1981; t t r x x
Wang and Hart, 1993), error function (Bourbie and Walls,
1982), or exponential decay (Dana and Skoczylas, 1999; where q is the adsorbate density per unit sample volume, k is
Ivanov et al., 2000). Other studies expand the scope of pulse‐ permeability (either Darcy permeability or a pressure‐
decay experiment applications to partially water saturated dependent permeability function), and n = 0,1,2, respectively,
samples (Homand et al., 2004; Newberg and Arastoopour, present one‐dimensional flow in Cartesian, Radial, and
1986) and permeability measurement with incompressible Spherical coordinates.
fluids (Amaefule et al., 1986; Trimmer, 1982). Equation 11.24 accounts for the gas desorption. The shale
adsorption isotherms are commonly described by the
Langmuir isotherms (Cui et al., 2009; Lancaster and Hill,
11.8.1 Pulse‐Decay Pressure Analysis 1993; Ross and Bustin, 2007; Saulsberry et al., 1996)
The main body of proposed analytical solutions to the pulse‐
decay equation assumes constant gas density, viscosity, and q qp , (11.25)
L
volumetric compressibility (defined as c = 1/ρ × ∂ρ/∂p). As a a p L p
g
result, the gas‐flow rate is linearly proportional to the local
q
pressure gradient. Another common assumption is that q sa , (11.26)
Darcy’s law is valid for the gas flow in the core. With these V std
assumptions the intrinsic permeability is best estimated
when the upstream and downstream vessels have equal vol- where q and p are the Langmuir volume and pressure,
L
L
umes V = V (Dicker and Smits, 1988; Jones, 1997). If the respectively, ρ is the bulk density of the core, and V is the
std
s
d
u
core porosity is not known, it’s best to have the upstream and gas molar volume at standard pressure (101,325 Pa) and
downstream volumes set close to a first‐order estimate of standard temperature (273.15 K).
sample pore volume (Wang and Hart, 1993). The choice of appropriate coordinates depends on the
These assumptions of linear proportionality of flow rate core sample shape and the boundary conditions. We select
and pressure gradient and the Darcy’s law may not be valid Cartesian coordinates through the rest of the analysis, that is,
for gas flow in shales. The Klinkenberg equation (Klinkenberg, n = 0. Hsieh et al. (1981) presented the analytical solution
1941) is an alternative pressure‐dependent permeability assuming Darcy permeability (i.e., k = k ), and constant gas
D