Method and apparatus for measuring stress dependency of shale permeability with steady-state flow
A method for determining stress-dependent permeability includes providing a core sample in a pressurized core container of a testing apparatus and generating a steady-state flow of gas from an upstream reservoir in the testing apparatus along an axial direction through the pressurized core container into a downstream reservoir in the testing apparatus. During the steady-state flow, an inlet pressure at an inlet to the core sample, an outlet pressure at an outlet of the core sample, and a midpoint pressure at a midpoint of the core sample are measured. The stress-dependent permeability is calculated from a flow rate of the gas through the core sample and the measurements of the inlet pressure, the outlet pressure, and the midpoint pressure.
1. A testing apparatus, comprising:
a core container containing a core sample comprising:
an inlet;
an outlet; and
a pressure sensor positioned at a midpoint of the core sample;
an upstream reservoir fluidly connected to the inlet of the core sample;
an upstream pump connected to the upstream reservoir;
a downstream reservoir fluidly connected to the outlet of the core sample;
a downstream pump connected to the downstream reservoir;
a confining pump connected to the core container;
at least one flow meter;
an inlet pressure sensor positioned proximate the inlet of the core sample; and
an outlet pressure sensor positioned proximate the outlet of the core sample.
2. The apparatus of claim 1 , further comprising a temperature sensor positioned proximate the inlet and the outlet of the core sample.
3. The apparatus of claim 1 , wherein the at least one flow meter is provided on at least one of the upstream pump and the downstream pump.
4. The apparatus of claim 1 , wherein a temperature sensor and a pressure sensor is provided on each of the upstream pump, the downstream pump, and the confining pump.
5. The apparatus of claim 1 , wherein the at least one flow meter is positioned downstream of the outlet of the core sample.
6. A method for determining stress-dependent permeability, comprising:
providing a core sample in a pressurized core container of a testing apparatus;
generating a steady-state flow of gas from an upstream reservoir in the testing apparatus along an axial direction through the core sample into a downstream reservoir in the testing apparatus;
during the steady-state flow, measuring an inlet pressure at an inlet to the core container, an outlet pressure at an outlet of the core container, and a midpoint pressure at a midpoint of the core sample; and
calculating the stress-dependent permeability from a flow rate of the gas through the core sample and the measurements of the inlet pressure, the outlet pressure, and the midpoint pressure.
7. The method of claim 6 , wherein providing the core sample in the pressurized core container comprises:
wrapping at least one sleeve around the core sample;
loading the sleeved core sample into a sample cell in the core container;
using a confining pump to introduce a confining fluid into the core container;
continuing to flow confining fluid through the sample cell after the sample cell containing the core sample is filled with confining fluid;
locking the confining fluid and the core sample in the filled sample cell; and
applying a confining pressure to the sample cell using the confining pump.
8. The method of claim 6 , wherein an upstream pump in the testing apparatus pumps the gas from the upstream reservoir into the inlet of the core sample and a downstream pump in the testing apparatus pumps the gas from the outlet of the core sample to the downstream reservoir.
9. The method of claim 6 , wherein the testing apparatus is in an oven at a constant temperature during generating the steady-state flow and measuring.
10. The method of claim 6 , further comprising using the stress-dependent permeability to predict a formation permeability of a formation under a formation pressure different from a confining pressure in the pressurized core container.
11. A method for determining stress-dependent permeability, comprising:
loading a core sample into a core container;
generating a steady-state flow of gas through the core sample under a confining pressure;
taking a plurality of measurements during the steady-state flow of gas through the core container, comprising:
measuring a flow rate through the core sample;
measuring an inlet pressure at an inlet to the core sample;
measuring an outlet pressure at an outlet of the core sample; and
measuring a midpoint pressure at a midpoint of the core sample; and
calculating the stress-dependent permeability from the measurements of the inlet flow rate, the inlet pressure, the outlet pressure, and the midpoint pressure.
12. The method of claim 11 , wherein generating the steady-state flow of gas comprises:
using an upstream pump to pump gas from an upstream reservoir into the inlet of the core sample;
using a downstream pump to control the outlet pressure of the gas out of the outlet of the core sample into a downstream reservoir; and
wherein a relative difference between an inlet flow rate and an outlet flow rate is less than 1 percent.
13. The method of claim 12 , further comprising increasing the inlet pressure of the gas to an initial inlet pressure using the upstream pump.
14. The method of claim 13 , further comprising:
applying the confining pressure within the core container using a confining pump fluidly connected to the core container; and
increasing the confining pressure while increasing the inlet pressure to maintain a constant pressure differential between the inlet pressure at the inlet of the core sample and the confining pressure in the core container.
15. The method of claim 11 , wherein after generating the steady-state flow of gas, further comprising:
increasing the inlet pressure to a final inlet pressure, wherein the difference between the confining pressure and final inlet pressure is greater than a selected range of effective stress pressures of interest; and
maintaining the final inlet pressure while taking the plurality of measurements.
16. The method of claim 11 , further comprising determining a stress sensitivity parameter, a, using a stress sensitivity equation:
L
L
x
=
1
+
∫
p
o
p
x
exp
[
α
(
p
-
p
o
)
]
ρ
d
p
μ
∫
p
x
p
i
exp
[
α
(
p
-
p
o
)
]
ρ
d
p
μ
where p i is the inlet pressure, p x is the midpoint pressure, p o is the outlet pressure, ρ is a density of the gas, μ is a viscosity of the gas, L is a length of the core sample, and L x is a midpoint distance from the inlet to the midpoint.
17. The method of claim 16 , further comprising determining an outlet permeability, k o , of the core sample at the outlet pressure using an outlet permeability equation:
k
o
=
Q
L
A
∫
p
o
p
i
exp
[
α
(
p
-
p
o
)
]
ρ
d
p
μ
where Q is the flow rate measured during the steady-state flow of gas, and A is a cross-sectional area of the core sample.
18. The method of claim 17 , further comprising calculating a permeability, k, of the core sample at the confining pressure, p, using a permeability equation:
k=k o exp[α( p−p o )].
19. The method of claim 18 , further comprising predicting a formation permeability k f of a formation under a formation pressure different from the confining pressure, using a formation permeability equation:
k f =k o exp[−α(σ−σ o )]
wherein σ is a formation effective stress, the formation effective stress being equal to the formation pressure minus a pore pressure of the formation; and
wherein σ o is an effective stress of the core sample proximate to the outlet, the effective stress being equal to the confining pressure minus the outlet pressure.
20. The method of claim 18 , further comprising inputting the permeability equation into a reservoir simulator to predict a hydrocarbon production from a reservoir, the reservoir simulator comprising at least one computer implemented software application with instructions to simulate the reservoir on a computer.