Estimating permeability in unconventional subterranean reservoirs using diagnostic fracture injection tests
In an example diagnostic fracture injection test (DFIT), or a “minifrac,” a fracturing fluid is pumped at a relatively constant rate and high pressure to achieve a fracture pressure of a subterranean formation. Sometime after achieving formation fracture pressure, the pump is shut off and the well is shut in, such that the pressure within the sealed portion of the well equilibrates with the pressure of the subterranean formation. As the pressure declines, the pressure of the injection fluid is monitored. This collected pressure data is then used to determine information regarding the permeability of the subterranean formation. In some implementations, a model for before closure analysis (BCA) can be used to estimate the permeability of a formation based on before closure pressure data (i.e., data collected before the fracture closure pressure is reached) based on a solution to the rigorous flow equation of fracturing fluid, which is leaking off into the formation during fracture closure.
1. A method for determining a permeability of a subterranean formation, the method comprising:
injecting a fluid through a well into a subterranean formation at an injection pressure sufficient to cause a fracture of the subterranean formation;
shutting in the well after injecting the fluid;
before closure of the fracture, monitoring a pressure of the fluid injected into the subterranean formation after shutting in the well to provide before closure pressure data;
determining information about the permeability of the subterranean formation based on the before closure pressure data using a mathematical formula relating a measured pressure, P i , at the time of shutting in, t i , and the measured pressure, P n , at a later time prior to closure of the fracture, t n , the mathematical formula corresponding to a solution of a flow equation of the form:
∂
2
P
D
∂
x
D
2
=
∂
P
D
∂
t
D
,
where P D is pressure, x D is a spatial dimension of the fracture, and t D is time.
2. The method of claim 1 , wherein the mathematical formula is of the form:
A
·
t
i
·
k
1
2
(
1
t
n
-
1
t
i
)
+
P
i
=
P
n
where k is the permeability of the formation and A is a proportionality factor.
3. The method of claim 2 , wherein
A
=
B
r
p
Δ
P
inj
C
D
f
c
f
(
∅
c
t
μ
)
1
2
,
in which P is a constant, r p , is a ratio of permeable area to fracture area of the formation, ΔP inj is a change in pressure at the end of injection, C Df is a fracture storage coefficient, c f is a fracture compliance parameter, Ø is a porosity of the formation, c t is a total reservoir compressibility parameter, and μ is a viscosity of the formation.
4. The method of claim 3 , wherein c f corresponds to a Perkins-Kern-Nordgren geometry, a Kristonovich-Geertsma-Daneshy geometry, or a Radial geometry.
5. The method of claim 1 , wherein the subterranean formation is comprised of a porous medium.
6. The method of claim 1 , wherein the formation is an unconventional formation.
7. The method of claim 1 , wherein the before closure pressure data corresponds to the pressure of the fluid injected into the subterranean formation over a timespan of between approximately 10-3000 minutes, and wherein the pressure of the fluid injected into the subterranean formation is between approximately 5000-10000 psi.
8. A non-transitory computer readable medium storing instructions that are operable when executed by a data processing apparatus to perform operations for determining a permeability of a subterranean formation, the operations comprising:
obtaining before closure pressure data, the before closure pressure data corresponding to a pressure of fluid injected into a subterranean formation from a well measured after the well is shut in and before a fracture of the subterranean formation is closed;
determining information about the permeability of the subterranean formation based on the pressure measurement data, the information determined using a mathematical formula relating a measured pressure, P i , at the time of shutting in, t i , and the measured pressure, P n , at a later time prior to closure of the fracture, t n , the mathematical formula corresponding to a solution of a flow equation of the form:
∂
2
P
D
∂
x
D
2
=
∂
P
D
∂
t
D
,
where P D is pressure, x D is a spatial dimension of the fracture, and t D is time.
9. The computer readable medium of claim 8 , wherein the mathematical formula is of the form:
A
·
t
i
·
k
1
2
(
1
t
n
-
1
t
i
)
+
P
i
=
P
n
where k is the permeability of the formation and A is a proportionality factor.
10. The computer readable medium of claim 9 , wherein
A
=
B
r
p
Δ
P
inj
C
Df
c
f
(
∅
c
t
μ
)
1
2
,
in which B is a constant, r p is a ratio of permeable area to fracture area of the formation, ΔP inj is a change in pressure at the end of injection, C Df is a fracture storage coefficient, c f is a fracture compliance parameter, Ø is a porosity of the formation, c t is a total reservoir compressibility parameter, and μ is a viscosity of the formation.
11. The computer readable medium of claim 10 , wherein c f corresponds to a Perkins-Kern-Nordgren geometry, a Kristonovich-Geertsma-Daneshy geometry, or a Radial geometry.
12. The computer readable medium of claim 8 , wherein the subterranean formation is comprised of a porous medium.
13. The computer readable medium of claim 8 , wherein the formation is an unconventional formation.
14. The computer readable medium of claim 8 , wherein the before closure pressure data corresponds to the pressure of the fluid injected into the subterranean formation over a timespan of between approximately 10-3000 minutes, and wherein the pressure of the fluid injected into the subterranean formation is between approximately 5000-10000 psi.
15. A system for determining a permeability of a subterranean formation, the system comprising:
an injection module adapted to inject a fluid through a well into a subterranean formation at an injection pressure sufficient to cause a fracture of the subterranean formation, and to shut in the formation after injection the fluid;
an instrumentation module adapted to, before closure of the fracture, monitor a pressure of the fluid injected into the subterranean formation after shutting in the well to provide before closure pressure data;
a data processing apparatus operable to determine information about the permeability of the subterranean formation based on the before closure pressure data using a mathematical formula relating a measured pressure, P i , at the time of shutting in, t i , and the measured pressure, P n , at a later time prior to closure of the fracture, t n , the mathematical formula corresponding to a solution of a flow equation of the form:
∂
2
P
D
∂
x
D
2
=
∂
P
D
∂
t
D
,
where P D is pressure, x D is a spatial dimension of the fracture, and t D is time.
16. The system of 15 , wherein the mathematical formula is of the form:
A
·
t
i
·
k
1
2
(
1
t
n
-
1
t
i
)
+
P
i
=
P
n
where k is the permeability of the formation and A is a proportionality factor.
17. The system of claim 16 , wherein
A
=
B
r
p
Δ
P
inj
C
Df
c
f
(
∅cC
t
μ
)
1
2
,
in which B is a constant, r p is a ratio of permeable area to fracture area of the formation, ΔP inj is a change in pressure at the end of injection, C Df is a fracture storage coefficient, c f is a fracture compliance parameter, Ø is a porosity of the formation, c t is a total reservoir compressibility parameter, and μ is a viscosity of the formation.
18. The system of claim 17 , wherein c f corresponds to a Perkins-Kern-Nordgren geometry, a Kristonovich-Geertsma-Daneshy geometry, or a Radial geometry.
19. The system of claim 15 , wherein the subterranean formation is comprised of a porous medium.
20. The system of claim 15 , wherein the formation is an unconventional formation.
21. The system of claim 15 , wherein the before closure pressure data corresponds to the pressure of the fluid injected into the subterranean formation over a timespan of between approximately 10-3000 minutes, and wherein the pressure of the fluid injected into the subterranean formation is between 5000-10000 psi.