Abstract
Water injection in petroleum reservoirs is the most used technique to improve the oil recovery factor. To design a water flooding process, it is necessary to precisely understand and predict the saturation and pressure behavior in porous media. Simultaneous flow of water and oil is governed by a system of non-linear partial differential equations representing the phases conservation laws, in which the water injection flow rate is modeled by a boundary condition (BC) at the inlet face. This work presents an approximate solution for the variable internal boundary condition problem uncoupling the saturation and pressure equations for a step-change injection rate. The proposed solution is compared to numerical results for typical rock and fluid properties found in one-dimensional core flooding experiments. We also investigate the effect of phase compressibility on the approximate solution. The proposed analytical solution might be useful to design laboratory core flooding experiments or to estimate parameters of relative permeability curves.
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References
Abbaszadeh, M., & Kamal, M.: Pressure-transient testing of water-injection wells. SPE Reserv. Eng. 4(01), 115–124 (1989). Society of Petroleum Engineers (SPE). https://doi.org/10.2118/16744-pa
Azevedo, A.V., Souza, A.J., Furtado, F., Marchesin, D., Plohr, B.: The solution by the wave curve method of three-phase flow in virgin reservoirs. Trans. Porous Media 83(01), 99–125 (2009). Springer Science and Business Media LLC. https://doi.org/10.1007/s11242-009-9508-9
Bratvold, R.B., Horne, R.N.: Analysis of pressure-falloff tests following cold-water injection. SPE Form. Evaluation 5(03), 293–302 (1990). Society of Petroleum Engineers (SPE). https://doi.org/10.2118/18111-pa
Buckley, S., Leverett, M.: Mechanism of fluid displacement in sands. Trans. AIME 146(01), 107–116 (1942). Society of Petroleum Engineers (SPE). https://doi.org/10.2118/942107-g
Banerjee, R., Thompson, L.G., Reynolds, A.C.: Injection/falloff testing in heterogeneous reservoirs. SPE Reser. Evaluation Eng. 1(06), 519–527 (1998). Society of Petroleum Engineers (SPE). https://doi.org/10.2118/52670-pa
Castañeda P., Abreu, E., Furtado, F., Marchesin, D.: On a universal structure for immiscible three-phase flow in virgin reservoirs, Comput. Geosci. 20(01), 171–185 (2016). Springer Science and Business Media LLC. https://doi.org/10.1007/s10596-016-9556-5
Corey, A.T., Rathjens, C.H., Henderson, J.H., Wyllie, M.R.J.: Three-phase relative permeability. J. Petroleum Technol. 8(11), 63–65 (1956). Society of Petroleum Engineers (SPE). https://doi.org/10.2118/737-g
Fayers, F.J., Sheldon, J.W.: The effect of capillary pressure and gravity on two-phase fluid flow in a porous medium. Trans. AIME 216(01), 147–155 (1959). Society of Petroleum Engineers (SPE). https://doi.org/10.2118/1089-g
Guzmán, R.E., Fayers, F.J.: Mathematical properties of three-phase flow equations. SPE J. 2(03), 291–300 (1997). Society of Petroleum Engineers (SPE). https://doi.org/10.2118/35154-pa
Stehfest, H.: Algorithm 368: numerical inversion of laplace transforms. Commun. ACM 13, 47–49 (1970)
Hazebroek, P., Rainbow, H., Matthews, C.S.: Pressure fall-off in water injection wells. Trans. AIME 213(01), 250–260 (1958). Society of Petroleum Engineers (SPE). https://doi.org/10.2118/925-g
Isaacson, E.L., Marchesin, D., Plohr, B.J., Temple, J.B.: Multiphase flow models with singular Riemann problems. Mat. Apl. Comput. 11(02), 147–166 (1992). Sociedade Brasileira de Matemática Aplicada e Computacional
Johnson, E.F., Bossler, D.P., Neumann, V.O.: Calculation of relative permeability from displacement experiments. Trans. AIME 216, 370–372 (1959). Society of Petroleum Engineers (SPE), SPE-1023G
Kazemi, H., Merrill, L.S., Jargon, J.R.: Problems in interpretation of pressure fall-off tests in reservoirs with and without fluid banks. J. Petrol. Technol. 24(09), 1147–1156 (1972). Society of Petroleum Engineers (SPE). https://doi.org/10.2118/3696-pa
Liu, T.: Hyperbolic and Viscous Conservation Laws. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 01. SIAM, Philadelphia (2007)
Lax, P.D.: Hyperbolic systems of conservation laws II. Commun. Pure Appl. Math. 10(04), 537–566 (1957). Wiley. https://doi.org/10.1002/cpa.3160100406
Oleinik, O.A.: On the uniqueness of the generalized solution of the Cauchy problem for a non-linear system of equations occurring in mechanics. Uspekhi Mat. Nauk 12, 169–176 (1957)
Peres, A.M.M., Boughrara, A.A., Reynolds, A.C.: Rate superposition for generating pressure falloff solutions for vertical and horizontal wells. SPE Annual Technical Conference and Exhibition, Society of Petroleum Engineers (SPE) (2004). https://doi.org/10.2118/90907-ms
Peres, A.M.M., Boughrara, A.A., Reynolds, A.C.: Rate superposition for generating pressure falloff solutions. SPE J. 11(03), 364–374 (2006). Society of Petroleum Engineers (SPE). https://doi.org/10.2118/90907-pa
Peres, A.M.M., Reynolds, A.C.: Theory and analysis of injectivity tests on horizontal wells. SPE J. 8(02), 147–159 (2003). Society of Petroleum Engineers (SPE). https://doi.org/10.2118/84957-pa
Sheldon, J.W., Cardwell, W.T.: One-dimensional, incompressible, noncapillary, two-phase fluid flow in a porous medium. Trans. AIME 216(01), 290–296 (1959). Society of Petroleum Engineers (SPE). https://doi.org/10.2118/978-g
Thompson, L., Reynolds, A.C.: Well testing for radially heterogeneous reservoirs under single and multiphase flow conditions. SPE Format. Evaluation 12(01), 57–64 (1997). Society of Petroleum Engineers (SPE). https://doi.org/10.2118/30577-pa
Welge, H.J.: A simplified method for computing oil recovery by gas or water drive. J. Petrol. Technol. 4(04), 91–98 (1952). Society of Petroleum Engineers (SPE). https://doi.org/10.2118/124-g
Acknowledgements
The authors acknowledge Universidade Estadual do Norte Fluminense (UENF) for financial support. This study was also financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—Brasil (CAPES)—Finance Code 001.
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Barros, W.Q., Pires, A.P., Peres, Á.M.M. (2022). Approximate Solution for One-Dimensional Compressible Two-Phase Immiscible Flow in Porous Media for Variable Boundary Conditions. In: Constanda, C., Bodmann, B.E., Harris, P.J. (eds) Integral Methods in Science and Engineering. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-07171-3_1
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