Multiscale finite-volume formulation for multiphase flow in porous media: black oil formulation of compressible, three-phase flow with gravity.(English)Zbl 1259.76049
Summary: Most practical reservoir simulation studies are performed using the so-called black oil model, in which the phase behavior is represented using solubilities and formation volume factors. We extend the multiscale finite-volume (MSFV) method to deal with nonlinear immiscible three-phase compressible flow in the presence of gravity and capillary forces (i.e., black oil model). Consistent with the MSFV framework, flow and transport are treated separately and differently using a sequential implicit algorithm. A multiscale operator splitting strategy is used to solve the overall mass balance (i.e., the pressure equation). The black-oil pressure equation, which is nonlinear and parabolic, is decomposed into three parts. The first is a homogeneous elliptic equation, for which the original MSFV method is used to compute the dual basis functions and the coarse-scale transmissibilities. The second equation accounts for gravity and capillary effects; the third equation accounts for mass accumulation and sources/ sinks (wells). With the basis functions of the elliptic part, the coarse-scale operator can be assembled. The gravity/capillary pressure part is made up of an elliptic part and a correction term, which is computed using solutions of gravity-driven local problems. A particular solution represents accumulation and wells. The reconstructed fine-scale pressure is used to compute the fine-scale phase fluxes, which are then used to solve the nonlinear saturation equations. For this purpose, a Schwarz iterative scheme is used on the primal coarse grid. The framework is demonstrated using challenging black-oil examples of nonlinear compressible multiphase flow in strongly heterogeneous formations.
MSC:
| 76M25 | Other numerical methods (fluid mechanics) (MSC2010) |
| 76S05 | Flows in porous media; filtration; seepage |
| 76M12 | Finite volume methods applied to problems in fluid mechanics |
Keywords:
multiscale finite volume;Black oil;compressibility;gravity;operator split;reservoir simulationSoftware:
GSLIBCite
Full Text:DOI
References:
| [1] | Aarnes, J.E.: On the use of a mixed multiscale finite element method for greater flexibility and increased speed or improved accuracy in reservoir simulation. Multiscale Model. Simul. 2(3), 421–439 (2004) ·Zbl 1181.76125 ·doi:10.1137/030600655 |
| [2] | Arbogast, T.: Implementation of a locally conservative numerical subgrid upscaling scheme for two-phase darcy flow. Comput. Geosci. 6, 453–481 (2002) ·Zbl 1094.76532 ·doi:10.1023/A:1021295215383 |
| [3] | Arbogast, T., Bryant, S.L.: A two-scale numerical subgrid technique for waterflood simulations. SPE J. 7, 446–457 (2002) |
| [4] | Aziz, K., Settari, A.: Petroleum Reservoir Simulation. Elsevier, Amsterdam (1979) |
| [5] | Chen, Y., Durlofsky, L.J., Gerritsen, M., Wen, X.H.: A coupled local-global upscaling approach for simulating flow in highly heterogeneous formations. Adv. Water Resour. 26, 1041–1060 (2003) ·doi:10.1016/S0309-1708(03)00101-5 |
| [6] | Chen, Z., Hou, T.Y.: A mixed finite element method for elliptic problems with rapidly oscillating coefficients. Math. Comput. 72, 541–576 (2002) ·Zbl 1017.65088 ·doi:10.1090/S0025-5718-02-01441-2 |
| [7] | Chen, Z., Yue, X.: Numerical homogenization of well singularities in the flow transport through heterogeneous porous media. Multiscale Model. Simul. 1(2), 260–303 (2003) ·Zbl 1107.76073 ·doi:10.1137/S1540345902413322 |
| [8] | Christie, M.A., Blunt, M.J.: Tenth SPE Comparative solution project: a comparison of upscaling techniques. SPE Reserv. Eng. 4(4), 308–317 (2001) |
| [9] | Craft, B., Hawkins, M.F.: Petroleum Reservoir Engineering. Prentice-Hall, Englewood Cliffs (1959) |
| [10] | Dagan, G.: Flow and Transport in Porous Formations. Springer, New York (1989) ·Zbl 0673.76026 |
| [11] | Deutsch, C.V., Journel, A.G.: GSLIB: Geostatistical Software Library and User’s Guide. Oxford University Press, New York (1998) |
| [12] | Efendiev, Y., Wu, X.: Multiscale finite element methods for the problems with highly oscillatory coefficients. Numer. Math. 90, 459–486 (2001) ·Zbl 0997.65129 ·doi:10.1007/s002110100274 |
| [13] | Gautier, Y., Blunt, M.J., Christie, M.A.: Nested gridding and streamline-based simulation for fast reservoir performance predicition. Comput. Geosci. 4, 295–320 (1999) ·Zbl 0968.76059 ·doi:10.1023/A:1011535210857 |
| [14] | Hou, T., Wu, X.H.: A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comp. Phys. 134, 169–189 (1997) ·Zbl 0880.73065 ·doi:10.1006/jcph.1997.5682 |
| [15] | Jenny, P., Lee, S.H., Tchelepi, H.A.: Multi-scale finite-volume method for elliptic problems in subsurface flow simulation. J. Comput. Phys. 187, 47–67 (2003) ·Zbl 1047.76538 ·doi:10.1016/S0021-9991(03)00075-5 |
| [16] | Jenny, P., Lee, S.H., Tchelepi, H.A.: Adaptive multiscale finite volume method for multi-phase flow and transport. Multiscale Model. Simul. 3, 50–64 (2004) ·Zbl 1160.76372 ·doi:10.1137/030600795 |
| [17] | Jenny, P., Lee, S.H., Tchelepi, H.A.: Adaptive fully implicit multi-scale finite-volume method for multi-phase flow and transport in heterogeneous porous media. J. Comput. Phys. 217, 627–641 (2006) ·Zbl 1160.76373 ·doi:10.1016/j.jcp.2006.01.028 |
| [18] | Juanes, R., Dub, F.-X.: A locally-conservative variational multiscale method for the simulation of porous media flow with multiscale source terms. Comput. Geosci. (2008) doi: 10.1007/s10596-007-9070-x ·Zbl 1259.76067 |
| [19] | Lee, S.H., Tchelepi, H.A., Jenny, P., DeChant, L.J.: Implementation of a flux-continuous finite difference method for stratigraphic, hexahedron grids. SPE J. 7, 267–277 (2002) |
| [20] | Lunati, I., Jenny, P.: Multi-scale finite-volume method for multi-phase flow with gravity. Comput. Geosci. (2008) doi: 10.1007/s10596-007-9071-9 ·Zbl 1259.76051 |
| [21] | Lunati, I., Jenny, P.: Multi-scale finite-volume method for compressible multi-phase flow in porous media. J. Comput. Phys. 216, 616–636 (2006) ·Zbl 1220.76049 ·doi:10.1016/j.jcp.2006.01.001 |
| [22] | Peaceman, D.W.: Fundamentals of Numerical Reservoir Simulation. Elsevier, Amsterdam (1977) |
| [23] | Tchelepi, H.A., Jenny, P., Lee, S.H., Wolfsteiner, C.: Adaptive multiscale finite volume framework for reservoir simulation. SPE J. 12, 188–195 (2007) |
| [24] | Watts, J.W.: A compositional formulation of the pressure and saturation equations. In: Proceedings of 7th SPE Symposium on Reservoir Simulation: SPE 12244, pp. 113–122, San Francisco, CA, 15–18 November 1983 |
| [25] | Wolfsteiner, C., Lee, S.H., Tchelepi, H.A.: Modeling of wells in the multiscale finite volume method for subsurface flow simultion. Multiscale Model. Simul. 5(3), 900–917 (2006) ·Zbl 1205.76175 ·doi:10.1137/050640771 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
