Report

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 ■ 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 ■ 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