FEMTEC 2013 (4th International Congress on Computational Engineering
and Sciences)
24th May 2013
A Fully Coupled Multiphase Flow and
Geomechanics Solver for Highly
Heterogeneous Porous Media
Daegil Yang
Texas A&M University
George J. Moridis
Thomas A. Blasingame
Texas A&M University
D. Yang, G.J. Moridis, T.A. Blasingame FEMTEC 2013
A Fully Coupled Multiphase Flow and Geomechanics Solver for Highly Heterogeneous Porous
Media
Slide — 1/28
Lawrence Berkeley National Laboratory
Evidence of land subsidence
in Long Beach, California due
to oil and gas production
(1940s)
■
Fracture propagation during
hydraulic fracturing
D. Yang, G.J. Moridis, T.A. Blasingame FEMTEC 2013
A Fully Coupled Multiphase Flow and Geomechanics Solver for Highly Heterogeneous Porous
Media
Slide — 2/28
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From:Oilfield Wiki. Hydraulic Fracturing.
http://www.oilfieldwiki.com/wiki/Hydraulic_fracturing
From: City of Long Beach Website. www.longbeach.gov/oil/subsidence/story.asp
Reservoir is Deformable
■
● Young’s modulus in a
reservoir:
Z-direction permeability
range: 0.1~1000 md
■
■
Pre-Khuff Reservoir
Unit of Young’s modulus is psi
D. Yang, G.J. Moridis, T.A. Blasingame FEMTEC 2013
A Fully Coupled Multiphase Flow and Geomechanics Solver for Highly Heterogeneous Porous
Media
Slide — 3/28
● SPE-10a model:
Al-Qahtani, M.Y. and Rahim, Z.. 2001 A Mathematical Algorithm for Modeling
Geomechanical Rock Properties of the Khuff and Pre-Khuff Reservoirs. Paper SPE
68194.
Reservoir is Heterogeneous
Need for New Coupled Flow and Geomechanics Solver for a
Heterogeneous System:
●Conventional FEM based solvers do not satisfy local mass
conservation of the flow equation.
●Want to model complex geometry .
■ i.e., Discrete fracture, complex shape of cracks, fracture network.
●Want to obtain more accurate numerical solution.
■ Better velocity and saturation solution of the flow equation.
●Heterogeneous media contain anisotropic properties.
■ Full tensor coefficients should be included in the flow and
D. Yang, G.J. Moridis, T.A. Blasingame FEMTEC 2013
A Fully Coupled Multiphase Flow and Geomechanics Solver for Highly Heterogeneous Porous
Media
Slide — 4/28
geomechanics equations.
Literature Review and Motivation:
●Fully Coupled vs Iteratively Coupled
■
■
■
Guitierrez et al. (2001) mentioned that the iterative approach is not robust
since there is no proof that the iterative algorithm guarantee the unique
solution.
Wan (2002) indicated that a certain mapping of solution might be required
for the iterative method since the discretizations of two separate modules
might be different.
Dean et al (2006) stated that the fully coupled approach is the most stable
one of the three approaches (explicitly coupled, iteratively coupled, and
fully coupled).
●Mixed Finite Element Method (Mixed FEM)
■
Mixed FEM known to satisfy local mass conservation and provide more
accurate and continuous description of velocity solution by solving
coupled mass and Darcy’s equations simultaneously (Chavent and Robert
1991; Durlofsky 1994; Hoteit and Firrozabadi 2006a, 2006b).
Mixed FEM can deal with discontinuous full tensor permeability and
unstructured mesh (Durlofsky and Chien 1993; Wheeler and Peszynska
2002; Younes et al. 2004; Klausen and Winther 2006; Wheer et al. 2012).
D. Yang, G.J. Moridis, T.A. Blasingame FEMTEC 2013
A Fully Coupled Multiphase Flow and Geomechanics Solver for Highly Heterogeneous Porous
Media
Slide — 5/28
■
Literature Review and Motivation:
●Mixed FEM in Coupled Flow and Geomechanics
■
■
Gai (2004) used the mixed FEM formulation to model multiphase flow
equations that are coupled with the equilibrium equation. The formulation
ended up with the cell centered finite difference.
Jha and Rubens (2007) used the mixed FEM formulation to model coupled
flow and geomechanics problems. Limited to only single-phase
formulation.
Ferronato et al (2010) used the mixed FEM formulation to model a 3D
coupled flow and geomechanics problem. They showed that Mixed FEM is
numerically more stable than the standard FEM. Limited to only singlephase formulation.
D. Yang, G.J. Moridis, T.A. Blasingame FEMTEC 2013
A Fully Coupled Multiphase Flow and Geomechanics Solver for Highly Heterogeneous Porous
Media
Slide — 6/28
■
Coupled Flow and Geomechanics Solver: Mathematical
Models
●Multiphase (Gas and Water)
Replaced with co for
oil and water system
D. Yang, G.J. Moridis, T.A. Blasingame FEMTEC 2013
A Fully Coupled Multiphase Flow and Geomechanics Solver for Highly Heterogeneous Porous
Media
Slide — 7/28
●Single Phase
Coupled Flow and Geomechanics Solver: Nonlinearity
Factors
● Porosity Change
where p is total pressure in multiphase flow
● Porosity Dependent Permeability
● Compressible Flow
D. Yang, G.J. Moridis, T.A. Blasingame FEMTEC 2013
A Fully Coupled Multiphase Flow and Geomechanics Solver for Highly Heterogeneous Porous
Media
Slide — 8/28
● Mobility
Coupled Flow and Geomechanics Solver: Finite Element
Formulation (Single Phase)
where
D. Yang, G.J. Moridis, T.A. Blasingame FEMTEC 2013
A Fully Coupled Multiphase Flow and Geomechanics Solver for Highly Heterogeneous Porous
Media
Slide — 9/28
The equations are solved
in a fully coupled fashion!
Coupled Flow and Geomechanics Solver: Finite Element
Formulation (Multiphase)
D. Yang, G.J. Moridis, T.A. Blasingame FEMTEC 2013
A Fully Coupled Multiphase Flow and Geomechanics Solver for Highly Heterogeneous Porous
Media
Slide — 10/28
The equations are solved in a fully
coupled fashion!
Conpled Flow and Geomechanics Solver: Locations of
Solution Variables (DOF) and Basis Functions
displacement, saturation
velocity
pressure (piecewise constant)
D. Yang, G.J. Moridis, T.A. Blasingame FEMTEC 2013
A Fully Coupled Multiphase Flow and Geomechanics Solver for Highly Heterogeneous Porous
Media
Slide — 11/28
p = pressure, u= displacement, S= saturation v= velocity
Discussion:
● Numerical solution matched well with analytical solution.
D. Yang, G.J. Moridis, T.A. Blasingame FEMTEC 2013
A Fully Coupled Multiphase Flow and Geomechanics Solver for Highly Heterogeneous Porous
Media
Slide — 12/28
Coupled Flow and Geomechanics Solver: Verification
Coupled Flow and Geomechanics Solver: Our Method vs
Discontinuous Galerkin (DG) Method for Saturation Equation
(Single Crack Model)
(m/s)
Our method
Discussion:
● Better resolution of saturation solution with our method.
D. Yang, G.J. Moridis, T.A. Blasingame FEMTEC 2013
A Fully Coupled Multiphase Flow and Geomechanics Solver for Highly Heterogeneous Porous
Media
Slide — 13/28
DG method
Coupled Flow and Geomechanics Solver: Effect of Full
Tensor Permeability and Elastic Stiffness
● 30o Rotation of permeability:
10m
(md)
● 30o Rotation of elastic stiffness:
(MPa)
D. Yang, G.J. Moridis, T.A. Blasingame FEMTEC 2013
A Fully Coupled Multiphase Flow and Geomechanics Solver for Highly Heterogeneous Porous
Media
Slide — 14/28
10m
Effect of Full Tensor Permeability and Elastic Stiffness:
Velocity and Saturation after 23 days of Simulation
(m/s)
After rotation
Discussion:
● Diagonally strong flow pattern with the rotated model.
D. Yang, G.J. Moridis, T.A. Blasingame FEMTEC 2013
A Fully Coupled Multiphase Flow and Geomechanics Solver for Highly Heterogeneous Porous
Media
Slide — 15/28
Before rotation
Effect of Full Tensor Permeability and Elastic Stiffness:
Pressure after 23 days of Simulation
(Pa)
After rotation
Discussion:
● Full tensor model needs less pressure gradient than the other model due
to diagonally favorable permeability (less resistance to the flow).
D. Yang, G.J. Moridis, T.A. Blasingame FEMTEC 2013
A Fully Coupled Multiphase Flow and Geomechanics Solver for Highly Heterogeneous Porous
Media
Slide — 16/28
Before rotation
Effect of Full Tensor Permeability and Elastic Stiffness:
Displacement
(m)
X-direction displacement
(m)
Y-direction displacement
After rotation
Discussion:
● Displacement solution was affected by the pressure solution.
D. Yang, G.J. Moridis, T.A. Blasingame FEMTEC 2013
A Fully Coupled Multiphase Flow and Geomechanics Solver for Highly Heterogeneous Porous
Media
Slide — 17/28
Before rotation
D. Yang, G.J. Moridis, T.A. Blasingame FEMTEC 2013
A Fully Coupled Multiphase Flow and Geomechanics Solver for Highly Heterogeneous Porous
Media
Slide — 18/28
Waterflooding in Highly Heterogeneous Reservoir: Reservoir
Model
Waterflooding in Highly Heterogeneous Reservoir: Rock
Properties
(md)
Permeability (kx=ky)
Porosity
Lame’s first constant
Shear Modulus
D. Yang, G.J. Moridis, T.A. Blasingame FEMTEC 2013
A Fully Coupled Multiphase Flow and Geomechanics Solver for Highly Heterogeneous Porous
Media
Slide — 19/28
(Pa)
Waterflooding in Highly Heterogeneous Reservoir: Pressure,
Saturation, Velocity
After 27 days
(Pa)
(m/s)
After 176 days
Discussion:
● Pressure gradient occurred along the x-direction.
D. Yang, G.J. Moridis, T.A. Blasingame FEMTEC 2013
A Fully Coupled Multiphase Flow and Geomechanics Solver for Highly Heterogeneous Porous
Media
Slide — 20/28
(Pa)
Waterflooding in Highly Heterogeneous Reservoir:
Displacements
After 27 days
(m)
(m)
After 176 days
(m)
Discussion:
● X-direction deformation (left) is 3.5 times higher than y-direction deformation (right).
● Larger deformation occurred in the upper region due to lower mechanical properties.
D. Yang, G.J. Moridis, T.A. Blasingame FEMTEC 2013
A Fully Coupled Multiphase Flow and Geomechanics Solver for Highly Heterogeneous Porous
Media
Slide — 21/28
(m)
Single Fracture Model in Tight Gas System: Reservoir Model
(md)
Porosity
Permeability (kx=ky)
Initial gas saturation
0.8
Initial water saturation
0.2
Initial pressure (MPa)
20
Temperature (oC)
30
Elastic stiffness follows
P-R
where n=1.5
D. Yang, G.J. Moridis, T.A. Blasingame FEMTEC 2013
A Fully Coupled Multiphase Flow and Geomechanics Solver for Highly Heterogeneous Porous
Media
Slide — 22/28
EOS
Single Fracture Model in Tight Gas System: Pressure
(Pa)
After 4.6 hour
After 12 days
Discussion:
● At the beginning of the simulation (left), reservoir pressure became higher than
initial pressure (20 MPa).
D. Yang, G.J. Moridis, T.A. Blasingame FEMTEC 2013
A Fully Coupled Multiphase Flow and Geomechanics Solver for Highly Heterogeneous Porous
Media
Slide — 23/28
(Pa)
Discussion:
● High mobility of gas and low compressibility of water increased the water
saturation inside and near the fracture.
● Rock compression reduced pore space which increased water saturation.
D. Yang, G.J. Moridis, T.A. Blasingame FEMTEC 2013
A Fully Coupled Multiphase Flow and Geomechanics Solver for Highly Heterogeneous Porous
Media
Slide — 24/28
Single Fracture Model in Tight Gas System: Saturation
Single Fracture Model in Tight Gas System: Y-direction
Displacement
(m)
Discussion:
● The decrease in pore pressure inside the fracture increased the magnitude of the
effective stress, which compressed the rock .
● Above the production area compression occurred and below the production area
dilation occurred.
D. Yang, G.J. Moridis, T.A. Blasingame FEMTEC 2013
A Fully Coupled Multiphase Flow and Geomechanics Solver for Highly Heterogeneous Porous
Media
Slide — 25/28
(m)
Conclusions:
●Formulation
■
■
We derived a fully coupled system of nonlinear equations to describe
multiphase flow in deformable porous media.
We presented a finite element formulation to solve for pressure, saturation,
velocity and displacement.
●Use of the Solver
■
■
■
■
We verified the solver with analytical solution and showed a good
agreement between the numerical solution and the analytical solution.
We showed that our formulation provides better resolution of saturation
than the DG formulation.
We tested the solver with full tensor permeability and elastic stiffness and
showed the effect of the full tensor coefficients.
We applied the solver to model waterflooding simulation of highly
heterogeneous reservoir systems and observed the strong impact of
heterogeneity on fluid flow and geomechanical deformation.
We modeled a tight gas reservoir system, saturated with water and gas,
with a single fracture and found that the water saturation increases when
pore space in the rock was reduced under deformation .
D. Yang, G.J. Moridis, T.A. Blasingame FEMTEC 2013
A Fully Coupled Multiphase Flow and Geomechanics Solver for Highly Heterogeneous Porous
Media
Slide — 26/28
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Acknowledgement:
●This work was supported by RPSEA (Contract No. 08122-45) through
D. Yang, G.J. Moridis, T.A. Blasingame FEMTEC 2013
A Fully Coupled Multiphase Flow and Geomechanics Solver for Highly Heterogeneous Porous
Media
Slide — 27/28
the Ultra-Deepwater and Unconventional Natural Gas and Other
Petroleum Resources Research and Development Program as
authorized by the United States Energy Policy Act of 2005.
FEMTEC 2013 (4th International Congress on Computational Engineering
and Sciences)
24th May 2013
A Fully Coupled Multiphase Flow and
Geomechanics Solver for Highly
Heterogeneous Porous Media
End of Presentation
Daegil Yang
Texas A&M University
George J. Moridis
Thomas A. Blasingame
Texas A&M University
D. Yang, G.J. Moridis, T.A. Blasingame FEMTEC 2013
A Fully Coupled Multiphase Flow and Geomechanics Solver for Highly Heterogeneous Porous
Media
Slide — 28/28
Lawrence Berkeley National Laboratory