Methods and systems for determining gas permeability of a subsurface formation
Methods and systems disclosed here include conducting two pressure-dependent permeability tests having the same range of effective stress but two different values of pore pressure. For the test with the higher pore pressure, the permeability is only impacted by the mechanical deformation of the rock, while for the one with lower pore pressure the permeability is impacted by both mechanical deformation of the rock and the Knudsen diffusion. By using the same range of effective stress, the contribution from the mechanical deformation of the rock should be the same. Therefore, by subtracting the permeability with higher pore pressure from the one with lower pore pressure, the impact of Knudsen diffusion and the mechanical deformation of the rock can be determined.
1. A transient flow method for determining gas permeability of a subsurface formation, comprising:
acquiring a sample of the subsurface formation;
positioning the sample in a pressure vessel comprising a fluid and a pressure gauge and applying a constant confining pressure, P c (1);
running a first test by equilibrating the sample at a predetermined first pore gas pressure, p i ;
applying a predetermined constant second gas pressure, p o , to an inlet of the sample, the second gas pressure being greater than the first pore gas pressure;
measuring a third pore gas pressure, p, as a function of time, t, at a plurality of locations along the axis of the sample in the pressure vessel;
in a computer, determining a gas density or total gas mass per unit volume of the subsurface formation, m; and
in the computer, determining the gas permeability of the subsurface formation as a function of pore pressure, k(p), based at least in part on the first pore gas pressure, the second gas pressure, the third pore gas pressure as a function of time, and the gas density as a function of pore pressure.
2. The method according to claim 1 , further comprising:
determining a transport parameter of the subsurface formation, D(p), using a first formula:
D
(
p
)
=
-
∫
p
i
p
λ
2
d
m
dp
dp
dp
d
λ
where λ is an independent variable calculated using the formula k=xt −1/2 ; and
determining gas permeability k of the subsurface formation from D(p) using
D
(
p
)
=
k
ρ
μ
where μ stands for gas viscosity, and ρ for gas density.
3. The method according to claim 2 , further comprising:
determining the total gas mass per unit volume of the subsurface formation, m, using a second formula:
m =ϕρ+(1−ϕ)ρ a
where ϕ is porosity of the subsurface formation, ρ is gas density of the gas, and ρ a is adsorbed gas mass per unit volume of the subsurface formation.
4. The method according to claim 3 , further comprising:
determining the porosity ϕ of the subsurface formation using a third formula:
ϕ
=
B
-
A
∫
p
i
p
0
λ
d
ρ
a
dp
dp
A
∫
p
i
p
0
λ
d
(
ρ
-
ρ
a
)
dp
dp
where A is a cross-sectional area of the sample, and B is a slope of a curve of the cumulative gas flow into the sample at x=0 versus t 1/2 .
5. The method according to claim 4 , further comprising:
determining the slope of the curve, B, using a fourth formula:
B
=
A
∫
p
i
p
0
λ
d
m
dp
dp
6. The method according to claim 1 , wherein the subsurface formation comprises at least one of shale, limestone, and sandstone.
7. The method according to claim 1 , further comprising:
running a second test with a fourth pore pressure and a fifth pressure with a second constant confining pressure, P c (2), wherein the fourth pore pressure is greater than the first pore pressure, the fifth pressure is greater than the second pressure, and the second constant confining pressure P c (2), is greater than the first constant confining pressure P c (1);
generating a graph plotting permeability versus effective stress for the first test and the second test;
determining impact of mechanical deformation of rock on permeability as a function of the effective stress;
determining the difference between permeability values obtained from the first test and the second test with respect to the effective stress; and
determining impact of Knudsen diffusion on permeability as a function of the pore pressure.
8. A non-transitory computer-readable medium having computer executable instructions that cause a computer to perform the operations of:
reading a measurement of a first pore gas pressure, p i , of a gas after equilibrating the sample in a pressure vessel comprising a fluid and a pressure gauge that reads the constant confining pressure P c (1) after it is applied;
reading a measurement of a predetermined constant second gas pressure, p o , applied to an inlet of the sample, the second gas pressure being greater than the first pore gas pressure;
reading a measurement of a third pore gas pressure, p, at time, t, at a plurality of locations along the axis of the sample;
determining a total gas mass per unit volume of the subsurface formation, m; and
determining gas permeability of the subsurface formation as a function of pore pressure, k(p), based at least in part on the first pore gas pressure, the second pressure, the third pore gas pressure, and the gas density as a function of pore pressure.
9. The non-transitory computer-readable medium of claim 8 , wherein the computer executable instructions further cause the computer to perform the operation of:
determining a transport parameter of the subsurface formation, D(p), using a first formula:
D
(
p
)
=
-
∫
p
i
p
λ
2
d
m
dp
dp
dp
d
λ
where λ is an independent variable calculated using the formula xt −1/2 ; and
determining gas permeability, k, from D(p) using
D
(
p
)
=
k
ρ
μ
where μ stands for gas viscosity, and ρ for gas density.
10. The non-transitory computer-readable medium of claim 9 , wherein the computer executable instructions further cause the computer to perform the operation of:
determining the total gas mass per unit volume of the subsurface formation, m, using a second formula:
m =ϕρ+(1−ϕ)ρ a
where ϕ is porosity of the subsurface formation, ρ is gas density of the gas, and ρ a is adsorbed gas mass per unit volume of the subsurface formation.
11. The non-transitory computer-readable medium of claim 10 , wherein the computer executable instructions further cause the computer to perform the operation of:
determining the porosity ϕ of the subsurface formation using a third formula:
ϕ
=
B
-
A
∫
p
i
p
0
λ
d
ρ
a
dp
dp
A
∫
p
i
p
0
λ
d
(
ρ
-
ρ
a
)
dp
dp
where A is a cross-sectional area of the sample, and B is a slope of a curve of the cumulative gas flow into the sample at x=0 versus t 1/2 .
12. The non-transitory computer-readable medium of claim 11 , wherein the computer executable instructions further cause the computer to perform the operation of:
determining the slope of the curve, B, using a fourth formula:
B
=
A
∫
p
i
p
0
λ
d
m
dp
dp
13. The non-transitory computer-readable medium of claim 8 , wherein the subsurface formation comprises at least one of shale, limestone, and sandstone.
14. The non-transitory computer-readable medium of claim 8 , further comprising
reading a measurement of a fourth pore gas pressure, of a gas after equilibrating the sample in a pressure vessel comprising a fluid and a pressure gauge that reads the constant confining pressure P c (2) after it is applied;
reading a measurement of a predetermined constant fifth gas pressure, applied to an inlet of the sample, the fifth gas pressure being greater than the fourth pore gas pressure; and
reading a measurement of pore gas pressure, p, at time, t, at a plurality of locations along the axis of the sample.
15. The non-transitory computer-readable medium of claim 8 , wherein the computer executable instructions further cause the computer to perform the operation of:
generating a graph plotting permeability versus effective stress for the first test and the second test;
determining impact of mechanical deformation of rock on permeability as a function of the effective stress;
determining the difference between permeability values obtained from the first test and the second test with respect to the effective stress; and
determining impact of Knudsen diffusion on permeability as a function of the pore pressure.