FEMTEC 2013 (4 th International Congress on Computational Engineering and Sciences)

1. FEMTEC 2013 (4th International Congress on Computational Engineering and Sciences) 24thMay 2013 A Fully Coupled Multiphase Flow and Geomechanics Solver for Highly Heterogeneous Porous Media Daegil Yang Texas A&M University George J. Moridis Lawrence Berkeley National Laboratory Thomas A. Blasingame Texas A&M University

2. Reservoir is Deformable 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 • Evidence of land subsidence in Long Beach, California due to oil and gas production (1940s) • Fracture propagation during hydraulic fracturing

3. Reservoir is Heterogeneous • SPE-10a model: • Young’s modulus in a reservoir: 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. • Pre-Khuff Reservoir • Unit of Young’s modulus is psi • Z-direction permeability range: 0.1~1000 md

4. 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 geomechanics equations.

5. 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).

6. 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 single-phase formulation.

7. Coupled Flow and Geomechanics Solver: Mathematical Models • Multiphase (Gas and Water) • Single Phase Replaced with co for oil and water system

8. Coupled Flow and GeomechanicsSolver: Nonlinearity Factors • Porosity Change • where p is total pressure in multiphase flow • Porosity Dependent Permeability • Compressible Flow • Mobility

9. Coupled Flow and Geomechanics Solver: Finite Element Formulation (Single Phase) • where The equations are solved in a fully coupled fashion!

10. Coupled Flow and Geomechanics Solver: Finite Element Formulation (Multiphase) The equations are solved in a fully coupled fashion!

11. Conpled Flow and Geomechanics Solver: Locations of Solution Variables (DOF) and Basis Functions p = pressure, u= displacement, S= saturation v= velocity pressure (piecewise constant) velocity displacement, saturation

12. Coupled Flow and Geomechanics Solver: Verification Discussion: • Numerical solution matched well with analytical solution.

13. Coupled Flow and Geomechanics Solver: Our Method vs Discontinuous Galerkin(DG) Method for Saturation Equation (Single Crack Model) (m/s) DG method Our method Discussion: • Better resolution of saturation solution with our method.

14. Coupled Flow and Geomechanics Solver: Effect of Full Tensor Permeability and Elastic Stiffness • 30o Rotation of permeability: (md) 10m • 30o Rotation of elastic stiffness: (MPa) 10m

15. Effect of Full Tensor Permeability and Elastic Stiffness: Velocity and Saturation after 23 days of Simulation (m/s) After rotation Before rotation Discussion: • Diagonally strong flow pattern with the rotated model.

16. Effect of Full Tensor Permeability and Elastic Stiffness: Pressure after 23 days of Simulation (Pa) After rotation Before rotation Discussion: • Full tensor model needs less pressure gradient than the other model due to diagonally favorable permeability (less resistance to the flow).

17. Effect of Full Tensor Permeability and Elastic Stiffness: Displacement (m) X-direction displacement (m) Y-direction displacement Before rotation After rotation Discussion: • Displacement solution was affected by the pressure solution.

18. Waterflooding in Highly Heterogeneous Reservoir: Reservoir Model

19. Waterflooding in Highly Heterogeneous Reservoir: Rock Properties (md) Permeability (kx=ky) Porosity (Pa) Shear Modulus Lame’s first constant

20. Waterflooding in Highly Heterogeneous Reservoir: Pressure, Saturation, Velocity After 27 days (Pa) (m/s) After 176 days (Pa) Discussion: • Pressure gradient occurred along the x-direction.

21. Waterflooding in Highly Heterogeneous Reservoir: Displacements After 27 days (m) (m) After 176 days (m) (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.

22. Single Fracture Model in Tight Gas System: Reservoir Model (md) Porosity Permeability (kx=ky) Elastic stiffness follows where n=1.5

23. Single Fracture Model in Tight Gas System: Pressure (Pa) (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).

24. Single Fracture Model in Tight Gas System: Saturation 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.

25. Single Fracture Model in Tight Gas System: Y-direction Displacement (m) (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.

26. 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 solverwith 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 solverwith full tensor permeability and elastic stiffness and showed the effect of the full tensor coefficients. • We applied the solverto 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.

27. Acknowledgement: • This work was supported by RPSEA (Contract No. 08122-45) through 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.

28. FEMTEC 2013 (4th International Congress on Computational Engineering and Sciences) 24thMay 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 Lawrence Berkeley National Laboratory Thomas A. Blasingame Texas A&M University