Fully coupled simulation for fluid flow and geomechanical properties in oilfield simulation operations
The invention relates to a computer system for modeling an oilfield having a subterranean formation and an underground reservoir therein. The computer system includes a repository storing a geomechanical model for representing at least a portion of the subterranean formation and the reservoir, the geomechanical model comprising a stress equation and a fluid flow equation, a processor and memory storing instructions when executed by the processor comprising functionalities for combining the stress equation and the fluid flow equation into a matrix equation, and modeling the oilfield by solving the stress equation and the fluid flow equation simultaneously.
1. A computer system for modeling an oilfield having a subterranean formation and an underground reservoir therein, comprising:
a repository storing a geomechanical model for representing at least a portion of the subterranean formation and the reservoir, the geomechanical model comprising a stress equation and a fluid flow equation;
a processor and memory storing instructions which when executed by the processor cause execution of functionalities for:
combining the stress equation and the fluid flow equation into a matrix equation by:
converting a combination of the stress equation and the fluid flow equation into an integral form; and
converting the integral form into a discrete form based on a finite element method;
solving the stress equation and the fluid flow equation simultaneously;
configuring a neural network associated with the geomechanical model; and
calculating stress and elastic and plastic strains at various locations using the neural network.
2. The computer system of claim 1 ,
wherein the fluid flow equation is formulated in a hyperbolic form;
wherein the matrix equation comprises a plurality of coupled equations; and
wherein the plurality of coupled equations are solved simultaneously.
3. The computer system of claim 2 , wherein the hyperbolic form comprises:
A
i
*
∂
s
w
∂
t
+
B
i
*
∂
s
g
∂
t
+
C
i
*
∂
p
w
∂
t
+
D
i
*
∂
ɛ
v
∂
t
+
E
i
*
v
t
·
∇
s
w
+
F
i
*
v
t
·
∇
s
g
=
Q
i
*
,
where t represents time, S w represents saturation of water, S g represents saturation of gas, p w represents water phase pressure, ε v represents volumetric strain, v i represents phase velocity, Q i represents flow rate, and at least one selected from a group consisting of A i *, B i *, C i *, D ii *, E i * and F i * represents a parametric function of at least one selected from a group consisting of fluid compressibility, volume formation factor, gas-oil ratio, Biot's coefficient, capillary pressure, porosity, pressure and saturation.
4. The computer system of claim 2 , wherein the fluid flow equation is formulated in a hyperbolic form based on a substitution equation V i =f i v y +g i ∇s w +h i ∇s g , where at least one selected from a group consisting of f i , g i and h i represents a parametric function of at least one selected from a group consisting of fluid compressibility, volume formation factor, gas-oil ratio, Biot's coefficient, capillary pressure, porosity, pressure and saturation.
5. The computer system of claim 1 , the instructions when executed by the processor further cause execution of functionalities for:
configuring the neural network for at least one selected from a group consisting of:
processing input data, by assigning data related to stress and material properties as inputs to the neural network and generating a set of values to be assigned to an array of individual grid blocks representing a portion of the oilfield modeled by the geomechanical model;
calculating strain and plasticity, by assigning loads and elastic stiffness matrix for stress equations at a given time as inputs to the neural network and generating plastic and elastic strain throughout the array of grid blocks representing the portion of the oilfield modeled by the geomechanical model;
analyzing well placement and schedule, by assigning various choices of well trajectory and production and injection schedules as inputs to the neural network and generating simulation outputs for a revised set of parameters; and
resimulating based on revised input data by assigning changes in material properties or observed stress and strain as inputs to the neural network and generating simulation outputs with revised properties.
6. The computer system of claim 1 , the instructions when executed by the processor further cause execution of functionalities for:
modeling the oilfield based on the geomechanical model comprising a viscoplastic model for calculating plastic strain based on a current yield surface;
obtaining a current stress using the viscoplastic model;
using a plurality of iterations for achieving convergence; and
determining an incremental change in the plastic strain for at least one of the plurality of iterations, based on a distance of the current stress from the current yield surface;
providing a matrix for representing a relationship between the plastic strain and change in stress; and
determine a pseudo-time-step of the plurality of iterations based on a maximum eigenvalue of the matrix,
wherein the matrix is represented by
A
=
(
∂
ɛ
.
yp
∂
σ
)
=
γ
{
(
∂
F
∂
σ
)
T
(
∂
Q
∂
σ
)
+
F
(
∂
2
Q
∂
σ
2
)
}
,
where A represents the matrix, and
wherein the pseudo-time-step is determined as
Δ
t
c
=
2
Ξ
Λ
,
where Λ represents the maximum eigenvalue of the matrix and Ξ is a parameter for achieving convergence.
7. The computer system of claim 6 , wherein the incremental change in the plastic strain is assigned according to
δɛ
vp
=
γ
F
(
∂
Q
∂
σ
)
Δ
t
c
,
where δε vp represents the incremental change in the plastic strain, F represents a yield function, γ represents a fluidity parameter, Q represents plastic potential, σ represents stress and Δt c represents the pseudo-time-step.
8. A computer implemented method for modeling an oilfield having a subterranean formation and an underground reservoir therein, the method comprising:
modeling, using a computer, the oilfield based on a geomechanical model comprising a viscoplastic model for calculating plastic strain based on a current yield surface;
obtaining, using the computer, a current stress using the viscoplastic model;
using a plurality of iterations for achieving convergence;
determining, using the computer, an incremental change in the plastic strain for at least one of the plurality of iterations, based on a distance of the current stress from the current yield surface.
9. The method of claim 8 , further comprising:
providing a matrix for representing a relationship between the plastic strain and change in stress; and
determine a pseudo-time-step for the plurality of iterations based on a maximum eigenvalue of the matrix,
wherein the matrix is represented by
A
=
(
∂
ɛ
.
vp
∂
σ
)
=
γ
{
(
∂
F
∂
σ
)
T
(
∂
Q
∂
σ
)
+
F
(
∂
2
Q
∂
σ
2
)
}
,
where A represents the matrix, and
wherein the pseudo-time-step is determined as
Δ
t
c
=
2
Ξ
Λ
,
where Λ represents the maximum eigenvalue of the matrix and Ξ is a parameter for achieving convergence.
10. The method of claim 9 , wherein Ξ is less than 0.5 for achieving stable and non-oscillatory convergence.
11. The method of claim 9 , wherein the incremental change in the plastic strain is assigned according to
δ
ɛ
vp
=
γ
F
(
∂
Q
∂
σ
)
Δ
t
c
,
where δε vp represents the incremental change in the plastic strain, F represents a yield function, γ represents a fluidity parameter, Q represents plastic potential, σ represents stress and Δt c represents the pseudo-time-step.
12. A computer implemented method for modeling an oilfield having a subterranean formation and an underground reservoir therein, comprising:
modeling, using a computer, the oilfield based on a geomechanical model comprising a viscoplastic model for calculating plastic strain based on a current yield surface;
calculating, using the computer, damage to the subterranean formation based on plastic strain;
modeling sand generation based on damage;
calculating, using the computer, permeability based on plastic strain and damage;
modeling sand transportation based on the permeability;
forecasting, using the computer, production based on the sand generation and the sand transportation.
13. The method of claim 12 ,
wherein the damage is calculated according to d=ε vp + +ψε eqp , where d represents the damage, ε vp + represents dilitant plastic volumetric strain, ε eqp represents equivalent plastic strain tensor, ψ represents a rock material property associated with the subterranean formation, and
wherein the permeability is calculated according to A=η(1−d)A 0 and
η
=
η
0
+
(
1
-
η
0
)
ɛ
eqp
B
+
ɛ
eqp
,
where A represents a current friction parameter, A 0 represents an initial friction parameter, η represents a current hardening parameter, η 0 represents an initial hardening parameter, and B represents a second rock material parameter associated with the subterranean formation.
14. The method of claim 12 , further comprising:
obtaining a current stress using the viscoplastic model;
using a plurality of iterations for achieving convergence; and
determining an incremental change in the plastic strain for at least one of the plurality of iterations, based on a distance of the current stress from the current yield surface.
15. The method of claim 14 , further comprising:
providing a matrix for representing a relationship between the plastic strain and change in stress; and
determine a pseudo-time-step of the plurality of iterations based on a maximum eigenvalue of the matrix,
wherein the matrix is represented by
A
=
(
∂
ɛ
.
vp
∂
σ
)
=
γ
{
(
∂
F
∂
σ
)
T
(
∂
Q
∂
σ
)
+
F
(
∂
2
Q
∂
σ
2
)
}
,
where A represents the matrix, and
wherein the pseudo-time-step is determined as
Δ
t
c
=
2
Ξ
Λ
,
where Λ represents the maximum eigenvalue of the matrix and Ξ is a parameter for achieving convergence.
16. The method of claim 15 , wherein the incremental change is assigned according to
δɛ
vp
=
γ
F
(
∂
Q
∂
σ
)
Δ
t
c
,
where δε yp represents the incremental change in the plastic strain, F represents a yield function, γ represents a fluidity parameter, Q represents plastic potential, σ represents stress and Δt c represents the pseudo-time-step.
17. A computer implemented method for modeling an oilfield having a subterranean formation and an underground reservoir therein, comprising:
obtaining, using a computer, a statistical size distribution of a plurality of pores located in the subterranean formation;
establishing a fluid flow model based on an interconnected pipe network for representing the plurality of pores, the interconnected pipe network having network geometry determined based on the statistical size distribution;
adjusting, using the computer, the interconnected pipe network based on a stress/strain change in the subterranean formation to generate an adjusted network geometry; and
updating, using the computer, the fluid flow model based on the adjusted network geometry.
18. The method of claim 17 , the interconnected pipe network comprising a pipe having a pipe diameter and a pipe length, the method further comprising:
determining the pipe diameter and an average porosity based on the statistical size distribution;
determining a pipe length based on the average porosity; and
calculating permeability for the fluid flow model based on the pipe diameter and pipe length.
19. A computer implemented method for modeling an oilfield having a subterranean formation and an underground reservoir therein, comprising:
modeling, using a computer, the oilfield based on a geomechanical model for representing at least a portion of the subterranean formation and the reservoir, the geomechanical model comprising a stress equation and a fluid flow equation;
configuring, using the computer, a neural network associated with the geomechanical model, the neural network being configured for at least one selected from a group consisting of:
processing input data, by assigning data related to stress and material properties as inputs to the neural network and generating a set of values to be assigned to an array of individual grid blocks representing a portion of the oilfield modeled by the geomechanical model;
calculating strain and plasticity, by assigning loads and elastic stiffness matrix for stress equations at a given time as inputs to the neural network and generating plastic and elastic strain throughout the array of grid blocks representing the portion of the oilfield modeled by the geomechanical model;
analyzing well placement and schedule, by assigning various choices of well trajectory and production and injection schedules as inputs to the neural network and generating simulation outputs for a revised set of parameters; and
resimulating based on revised input data by assigning changes in material properties or observed stress and strain as inputs to the neural network and generating simulation outputs with revised properties.