Adaptive time-splitting scheme for two-phase flow in heterogeneous porous media

Mohamed F. El-Amin, Jisheng Kou, Shuyu Sun, Amgad Salama

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In the present paper, an adaptive time-splitting scheme is introduced to investigate the problem of two-phase flow in heterogeneous porous media. The pressure and saturation equations are coupled by the capillary pressure which is linearized in terms of saturation. An IMplicit Pressure Explicit Saturation scheme is used to solve the problem under consideration. We use the time schemes for the pressure and saturation equations. The external time interval is divided into two levels, the first level is for the pressure, the second one is for the saturation. This method can reduce the computational cost arisen from the implicit solution of the pressure equation and the rapid changes in saturation. The time-step size for saturation equation is adaptive under computing and satisfying the Courant–Friedrichs–Lewy (CFL<1) condition. In order to show the well performance of the suggested scheme, we introduce a numerical example of a highly heterogeneous porous medium. The adaptive time step-size is shown in graphs as well as the water saturation is shown in contours.

Cited as: El-Amin, M., Kou, J., Sun, S., et al. Adaptive time-splitting scheme for two-phase flow in heterogeneous porous media. Advances in Geo-Energy Research, 2017, 1(3): 182-189, doi: 10.26804/ager.2017.03.05


Time-splitting, IMPES, two-phase flow, porous media, reservoir simulation

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Al-Dhafeeri, A.M., Nasr-El-Din, H.A. Characteristics of high-permeability zones using core analysis, and production logging data. J. Pet. Sci. Eng. 2007, 55: 18-36.

Arbogast, T., Yotov, I., Wheeler, M.F. Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences. SIAM J. Numer. Anal. 1997, 34(2): 828-852.

Belytschko, T., Lu,Y.Y. Convergence and stability analyses of multi-time step algorithm for parabolic systems. Appl. Mech. Eng. 1993, 102(2): 179-198.

Bhallamudi, S.M., Panday, S., Huyakorn, P.S. Sub-timing in fluid flow and transport simulations. Adv. Water Resour. 2003, 26(5): 477-489.

Chen, Z., Huan, G., Ma, Y. Computational Methods for Multiphase Flows in Porous Media. Philadelphia, USA, SIAM Computational Science and Engineering, 2006.

Coats, K.H. Impes stability: Selection of stable timesteps. Paper SPE 84924 Presented at SPE ReservoirSimulation Symposium, Houston, TX, June, 2001.

El-Amin, M.F., Kou, J., Salama, A., et al. An iterative implicit scheme for nanoparticles transport with two-phase flow in porous media. Procedia Comput. Sci. 2016, 80: 1344-1353.

El-Amin, M.F., Kou, J., Sun, S. Convergence analysis of the nonlinear iterative method for two-phase flow in porous media associated with nanoparticle injection. Int. J. Numer. Meth. Heat Fluid Flow 2017a, 27(10): 2289-2317.

El-Amin, M.F., Kou, J., Sun, S. A multiscale time-splitting discrete fracture model of nanoparticles transport in fractured porous media. Paper SPE 188001 Presented at SPE Kingdom of Saudi Arabia Technical Symposium and Exhibition, Dammam, Saudi Arabia, 24-27 April, 2017b.

El-Amin, M.F., Kou, J., Sun, S. Multiscale adapted time-splitting technique for nonisothermal two-phase flow and nanoparticles transport in heterogenous porous media. Paper SPE 186047 Presented at SPE Reservoir Char-acterisation and Simulation Conference and Exhibition, Abu Dhabi, UAE, 8-10 May, 2017c.

El-Amin, M.F., Kou, J., Sun, S. Discrete-fracture-model of multi-scale time-splitting two-phase flow including nanoparticles transport in fractured porous media. J. Comput. Appl. Math. 2017d, 333: 327-349.

Gravouil, A., Combescure, A. Multi-time-step explicit-implicit method for non-linear structural dynamics. Int. J. Numer. Meth. Eng. 2000, 50(1): 199-225.

Hoteit, H., Firoozabadi, A. Numerical modeling of two-phase flow in heterogeneous permeable media with different capillarity pressures. Adv. Water Resour. 2008, 31: 56-73.

Klisi ´nski, M. Inconsistency errors of constant velocity multi-time step integration algorithms. Comput. Assisted Mech. Eng. Sci. 2001, 8(1): 121-139.

Kou, J., Sun, S. A new treatment of capillarity to improve the stability of impes two-phase flow formulation. Comput. Fluids 2010, 39(10): 1923-1931.

Kou, J., Sun, S., Yu, B. Multiscale time-splitting strategy for multiscale multiphysics processes of two-phase flow in fractured media. J. Appl. Math. 2011, 4: 1-21.

Lu, Q. A parallel multi-block/multi-physics approach for multi-phase flow in porous media. Austin, The University of Texas, 2000.

Park, Y.J., Sudicky, E.A., Panday, S., et al. Application of implicit sub-time stepping to simulate flow and transport in fractured porous media. Adv. Water Resour. 2008, 31(7): 995-1003.

Singh, V., Bhallamudi, S.M. Complete hydrodynamic border-strip irrigation model. J. Irrig. Drain. Div. 1996, 122(4): 189-197.

Singh, V., Bhallamudi, S.M. Hydrodynamic modeling of basin irrigation. J. Irrig. Drain. Eng. 1997, 123(6): 407-414.

Smolinski, P., Belytschko, T., Neal, M. Multi-time-step integration using nodal partitioning. Int. J. Numer. Meth. Eng. 1988, 26(2): 349-359.

Smolinski, P., Sleith, S., Belytschko, T. Stability of an explicit multi-time step integration algorithm for linear structural dynamics equations. Comput. Mech. 1996, 18(3): 236-243.

Sun, S., Geiser, J. Multi-scale discontinuous Galerkin and operator-splitting methods for modeling subsurface flow and transport. Int. J. Multiscale Comput. Eng. 2008, 6(1): 87-101.

Telytschko, T., Lu, Y.Y. Convergence and stability analyses of multi-time step algorithm for parabolic systems. Appl. Mech. Eng. 1993, 102(2): 179-198.

Vanderkwaak, J.E. Numerical simulation of flow and chemical transport in integrated surface-subsurface hydrologic systems. Canada, University of Waterloo, 1999.

Wang, Y., Sun, S., Yu, B. Acceleration of gas flow simulations in dual-continuum porous media based on the mass-conservation POD method. Energies 2017, 10(9): 1380.

Wang, Y., Yu, B., Cao, Z., et al. A comparative study of POD interpolation and POD projection methods for fast and accurate prediction of heat transfer problems. Int. J. Heat Mass Transf. 2012, 55(17): 4827-4836.

Young, L.C., Stephenson, R. A generalized compositional approach for reservoir simulation. SPE J. 1983, 23(23): 727-742.


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