College, South Hadley, Massachusetts, June 17-21, vol. 295, pp. 49-61. 4. Chen, Z., 2000, Formulations and numerical methods of the black oil model in porous.
1
Numerical Simulation of three-phase flows in heterogeneous petroleum reservoirs E. Abreu,
a
F. Furtado,
b
D. Marchesin,
c
and F. Pereira
a
a
Departamento de Modelagem Computacional, Instituto Polit´ecnico/UERJ, Caixa Postal 97282, 28601-970 Nova Friburgo, RJ, Brazil. b
Department of Mathematics, University of Wyoming, Laramie, 82071-3036 U.S.A. c
Instituto Nacional de Matem´atica Pura e Aplicada, Estrada Castorina, 110, Rio de Janeiro, 22460-320, RJ, Brazil. In immiscible three-phase (oil, gas and water) flow typical of petroleum reservoirs, the leading oil front can split into two, a classical Buckley-Leverett front followed by a new type of shock wave. This new shock wave is related to the existence of an elliptic region or an umbilic point for the system of nonlinear conservation laws describing the convective transport of the fluid phases. Unlike classical shock waves (e.g., Buckley-Leverett fronts), this nonclassical “transitional” shock wave is very sensitive to the form of the parabolic terms in the equations that arise from diffusive effects (see [6] and references therein). It is then imperative that capillary pressure effects be modeled accurately in order to calculate the physically correct transitional waves. The purpose of this work is the presentation of an accurate numerical procedure for the simulation of three-phase flows which includes capillary pressure effects. Our new procedure combines mixed finite element methods for the approximation of parabolic and elliptic problems with a non-oscillatory, second order, conservative central difference scheme to handle a system of conservation laws. Different approaches for solving numerically the three-phase flow equations can be found in [3–5]. This procedure is validated against semi-analytic one-dimensional results [1,2]. Here we focus on two-dimensional, heterogeneous problems. Our procedure is used to verify that transitional waves, identified theoretically for one-dimensional problems, are present in two-dimensional heterogenous reservoirs. REFERENCES 1. Abreu, E., 2003, Numerical simulation of three-phase water-oil-gas flows in petroleum reservoirs, M.Sc Thesis, IPRJ/UERJ, Brazil. (in Portuguese - Available at http://www.labtran.iprj.uerj.br/Orientacoes.html) 2. Abreu, E., Furtado, F., and Pereira, F, 2003, On the numerical simulation of threephase reservoir transport problems. To appear in Transport Theory and Statistical Physics (2005). (Available at http://www.labtran.iprj.uerj.br/Preprints.html)
2 3. Berre, I., Dahle, H. K., Karlson, K. H., and Nordhaug, H. F, 2002, A streamline front tracking method for two- and three-phase flow including capillary forces, In Proceedings of an AMS-IMS-SIAM, Joint Summer Research Conference on Fluid Flow and Transport in Porous Media: Mathematical and Numerical Treatment, Mount Holyoke College, South Hadley, Massachusetts, June 17-21, vol. 295, pp. 49-61. 4. Chen, Z., 2000, Formulations and numerical methods of the black oil model in porous media, SIAM Journal on Numerical Analysis, vol. 38, No. 2, pp. 489-514. 5. Chen, Z., and Ewing, R. E., 1997, Fully-discrete finite element analysis of multiphase flow in ground-water hydrology, SIAM Journal on Numerical Analysis, vol. 34, pp. 2228-2253. 6. Marchesin, D. and Plohr, B. J., SPE 56480, 1999, and 2001, Wave structure in WAG recovery, SPEJ, vol. 6, no. 2, pp. 209-219.
