Numerical simulation of the water saturation at the interface between homogeneous porous medium
Simulación numérica de la saturación del agua sobre la interface entre medios porosos homogéneos
E. Cariaga1, A. Vergara-Fernández2*, M. Lévano3 y N. Vergaray3
1 Department of Mathematical and Physical Sciences-Universidad Católica de Temuco Casilla 15D-Temuco-Chile.
2 Núcleo de Investigación en Energías Renovables-Universidad Católica de Temuco Casilla 15D- Temuco-Chile. * Corresponding author. E-mail: avergara@uctemuco.cl Tel.: +56 45 20 56 84, Fax +56 45 20 56 30.
]]> 3 School of Informatic Engineering- Universidad Católica de Temuco Casilla 15D- Temuco-Chile.
Received December 3, 2012
Accepted October 20, 2013
Abstract
The oil-water flow in a heterogeneous porous medium was studied numerically, with special emphasis on the interface between two homogeneous layers of the porous matrix. The heterogeneity considered consists of a discontinuous capillary pressure on the interface. The differential equation was solved using a fully implicit scheme based on the upwind finite volume method. The unknown was the water saturation. This study evaluated the impact of changes in: the porosity of trie entire domain, the initial water saturation, the water injection rate, the gravitational force, and the material grain size, on the water saturation at the interface. Experiments have improved the understanding of hydrodynamics on the interface. A full characterization of the porous matrix is an essential condition before defining conditions of oil extraction. The studied algorithm has great potential for use in earlier stages of design and planning for oil extraction.
Keywords: discontinuous capillary, porous medium, interface, finite volume, oil-water flow.
Resumen
]]> En este trabajo se estudió numéricamente el flujo agua-petróleo en una medio poroso heterogeneo, con énfasis en la interface entre dos capas homogeneas de la matriz porosa. La heterogeneidad considerada consistió en una función discontinua de capilaridad sobre la interface. La ecuación diferencial fue resuelta utilizando un esquema tipo upwind completamente implícito basado en el método de volúmenes finitos. La incognita principal fue la saturación del agua. Se evaluó el impacto sobre la saturación de la interface de diversos parámetros tales como: la porosidad, la saturación inicial de agua, la razón de inyección de agua, la fuerza gravitacional, y el tamaño del grano del material. Se concluye que una completa caracterización de la matriz porosa, junto con una adecuada combinación de parámetros de control y modelado, son condiciones fundamentales para definir condiciones de extraccioín adecuadas. El algoritmo estudiado posee un gran potencial para ser utilizado en etapas tempranas de diseño y planeación para la extracción del petróleo.Palabras clave: capilaridad discontinua, medio poroso, interface, volúmenes finitos, flujo agua-petróleo.
DESCARGAR ARTÍCULO EN FORMATO PDF
Acknowledgements
EC has been supported by Fondecyt project 11100358.
References
Amini, A. and Schleiss, A. (2009). Numerical modeling of oil-water multiphase flow contained by an oil spill barrier. Engineering Applications of Computational Fluid Mechanics 3, 207-219. [ Links ]
Aziz, K. and Settari, A. (1979). Petroleum Reservoir Simulation. Elsevier Applied Science Publishers. London. [ Links ]
Bastian, P. and Helmig, R. (1999). Efficient fully-coupled solution techniques for two-phase flow in porous media-Parallel multigrid solution and large scale computations. Advances in Water Resources 23, 199-216. [ Links ]
Bear, J. (1988). Dynamics of Fluids in Porous Media. Dover. New York. [ Links ]
Bertsch, M., Passo, R. and van Duijn, C. (2003). Analysis of oil trapping in porous media flow. SIAM Journal on Mathematical Analysis 35, 245-267. [ Links ]
Cancès, C. (2008). Écoulements diphasiques en milieux poreaux hétérogènes: modelisation et analyse. Thèse Docteur de L'Université de Provence. Universite de Provence. [ Links ]
Cancès, C. (2009). Finite volume scheme for two-phase flow in heterogeneous porous media involving capillary pressure discontinuities. M2AN Mathematical Modelling and Numerical Analysis 43, 973-1001. [ Links ]
Cancès, C., Gallouët, T. and Porretta, A. (2009). Two-phase flows involving capillary barriers in heterogeneous porous media. Interfaces and Free Bound 11, 239-258. [ Links ]
Cancès, C (2010). On the effects of discontinuous capillaries for immiscible two-phase flows in porous media made of several rock-types. Networks and Heterogeneous Media 5,635-647. [ Links ]
Cariaga, E., Concha, F. and Sepuílveda, M. (2005). Flow through porous media with applications to heap leaching of copper ores. Chemical Engineering Journal 111, 151-165. [ Links ]
Chen, Z., Huan, G. and Ma, Y. (2006). Computational Methods for Multiphase Flows in Porous Media. Society for Industrial and Applied Mathematics (SIAM). [ Links ]
Correa, A. and Firoozabadi, A. (1996). Concept of gravity drainage in layered porous media. SPE Journal March, 101-111. [ Links ]
Enchery, G., Eymard, R. and Michel, A. (2006). Numerical approximation of a two-phase flow problem in a porous medium with discontinuous capillary forces. SIAM Journal on Numerical Analysis 43, 2402-2422. [ Links ]
Ern, A., Mozolevski, I. and Schuh, L. (2010). Discontinuous Galerkin approximation of two-phase flows in heterogeneous porous media with discontinuous capillary pressures. Computer Methods in Applied Mechanics and Engineering 199, 1491-1501. [ Links ]
Ersland, B., Espedal, M. and Nybo, R. (1998). Numerical methods for flows in a porous medium with internal boundary. Computers & Geosciences 2, 217-240. [ Links ]
Eymard, R., Gallouet, T. and Herbin, R. (2000). Finite Volume Methods. Handbook of Numerical Analysis. Vol. VII. Edited by P. G. Ciarlet and J. L. Lions. Elsevier Science B. V. [ Links ]
Fucik, R., Mikyška, J., Sakaki, T., Beneš, M. and Illangasekare, T. (2010). Significance of dynamic effect in capillarity during drainage experiments in layered porous media. Vadose Zone Journal 9, 697-708. [ Links ]
Gerritsen, M. and Durlofsky, L. (2005). Modeling Fluid Flow in Oil Reservoirs. Annual Review of Fluid Mechanics 37, 211-238. [ Links ]
Helmig, R. (1997). Multiphase Flow and Transport Processes in the Subsurface: a Contribution to the Modeling of Hydrosystems. Springer. [ Links ]
Helmig, R. and Huber, R. (1998). Comparison of galerkin-type discretization techniques for two-phase flow in heterogeneous porous media. Advances in Water Resources 21, 697-711. [ Links ]
Hérard, J. and Hurisse, O. (2009). Some recent numerical advances for two-phase flow modeling in NEPTUNE project. International Journal for Numerical Methods in Fluids 59, 285-307. [ Links ]
Hoteit, H. and Firoozabadi, A. (2008). Numerical modeling of two-phase flow in heterogeneous permeable media with different capillary pressures. Advances in Water Resources 31, 56-73. [ Links ]
Jiménez-Islas, H., Calderón-Ramírez, M., Navarrete-Bolaños, J.L., Botello-Álvarez, J.E., Martínez-González, G.M. and López-Isunza, F. (2009). Numerical study of natural convection in a 2-D square cavity with fluid-porous medium interface and heat generation. Revista Mexicana de Ingeniería Química 8, 169-185. [ Links ]
Lee, K. (2010). Simulation on the surfactant-polymer flushing of heterogeneous aquifers contaminated with non aqueous phase liquids. Engineering Applications of Computational Fluid Mechanics 4, 558-568. [ Links ]
Mikyska, J., Beneš, M. and Illangasekare, T. (2009). Numerical investigation of NAPL behavior at heterogeneous sand layers using VODA multiphase flow code. Journal of Porous Media 12, 685-694. [ Links ]
Niessner, J., Helmig, R., Jakobs, H. and Roberts, J. (2005). Interface condition and linearization schemes in the Newton iterations for two-phase flow in heterogeneous porous media. Advances in Water Resources 28, 671-687. [ Links ]
Kinjal, R., Manoj, N. and Twinkle, R. (2012). A mathematical model of imbibitions phenomenon in heterogeneous porous media during secondary oil recovery process. Applied Mathematical Modelling. doi: http://dx.doi.org/10.1016/j.apm.2012.06.015. [ Links ]
Salazar-Mendoza, R., Vázquez-Rodríguez, A., Ramos-Alcántara, J. R., Espinosa-Martínez, E. G. and Espinosa-Paredes, G. (2004). A two-region averaging model for solid-liquid flow with a moving bed in horizontal pipes. Revista Mexicana de Ingenieríía Quíímica 3, 273-286. [ Links ]
Schweizer, B (2008). Homogenization of degenerate two-phase flow equations with oil trapping. SIAM Journal on Mathematical Analysis 39. 1740-1763. [ Links ]
van Duijn, C., Molenaar, J. and de Neef, M. (1995). The effect of capillary forces on inmiscible two-phase flow in heterogeneous porous media. Transport in Porous Media 21, 71-93. [ Links ]
]]>