Modeling and numerical calculation of piston-like oil displacement for doubly-periodic systems of field development
DOI:
https://doi.org/10.7242/1999-6691/2015.8.1.7Keywords:
mathematical modeling, piston-like displacement of oil by water, oil-water boundary, Weierstrass elliptic functions, singular integral equationAbstract
Prediction of the oil-water boundary movement has great importance to the design problems of oilfield development by waterflooding: knowledge of the nature of the coupled motion between the displaced oil and the displacing water in the reservoir allows us to optimize the system of oil field development. The simplest model of joint oil/water filtering is the model of “multicolored” liquids, which assumes that oil and water have the same or similar physical properties (density and viscosity). In this paper, we consider a more complex model “piston-like” oil-water displacement, which takes into account differences in the viscosity and density of two fluids. An oil reservoir is assumed to be homogeneous and infinite, of fixed thickness, and with constant values of porosity and permeability coefficients. The reservoir is developed by a group of a finite number of production and injection wells recurrent in two directions (doubly-periodic cluster). Filtration of liquids is described by Darcy's law. It is assumed that liquids are weakly compressible, and the pressure in the reservoir satisfies the quasi-stationary diffusion equation. The model of piston-like displacement leads to the discontinuity in the tangential component of the velocity vector at the boundary of oil-water contact. Use of the theory of elliptic functions in conjunction with the generalized Cauchy integrals reduces the problem of finding the current boundaries of oil-water contact to the system of singular integral equations for the tangential and normal components of the velocity vector and the Cauchy problem for the integration of the differential equations of motion of the oil-water contact boundary. An algorithm for the numerical solution of this problem is developed. The monitoring of oil-water boundary motion for different schemes of waterflooding (linear row, four-point, five-point, seven-point, nine-point, etc.) is carried out.
Downloads
References
Zeltov U.P. Razrabotka neftanyh mestorozdenij. - M.: Nedra, 1998. - 365 s.
2. Uillhajt G.P. Zavodnenie plastov. - M.-Izevsk: Institut komp’uternyh issledovanij, 2009. - 788 s.
3. Basniev K.S., Kocina I.N., Maksimov V.M. Podzemnaa gidromehanika. - M.: Nedra, 1993. - 416 s.
4. Gerol’d S.P. Analiticeskie osnovy dobyci nefti, gaza i vody iz skvazin. - M.-L.: Nefteizdat, 1932.
5. Kasatkin A.E. Sravnitel’nyj analiz shem rasstanovki skvazin pri zavodnenii // Vestnik SamGU. - 2013. - No 9-2 (110). - S. 196-207.
6. Masket M. Tecenie odnorodnyh zidkostej v poristoj srede. - M.-Izevsk: Institut komp’uternyh issledovanij, 2004. - 640 s.
7. Carnyj I.A. Podzemnaa gidrogazodinamika. - M.-Izevsk: Institut komp’uternyh issledovanij, 2006. - 436 s.
8. Danilov B.L., Kac P.M. Gidrodinamiceskie rascety vzaimnogo vytesnenia zidkostej v poristoj srede. - M.: Nedra, 1980. - 264 s.
9. Fazlyev R.T. Plosadnoe zavodnenie neftanyh mestorozdenij. - M.-Izevsk: Institut komp’uternyh issledovanij, 2008. - 256 s.
10. Hajrullin M.H. Issledovania po problemam teorii fil’tracii v KNC RAN // Obzory issledovanij po mehanike splosnoj sredy. K 50-letiu KNC RAN. - Kazan’, 1995. - S. 60-78.
11. Astaf’ev V.I., Roters P.V. Modelirovanie dvoakoperiodiceskih sistem dobyvausih skvazin // Vestnik SamGU. - 2010. - No 4 (78). - S. 5-11.
12. Astaf’ev V.I., Roters P.V. Modelirovanie dvoakoperiodiceskih sistem dobyvausih skvazin. 2. Koefficient produktivnosti // Vestnik SamGU. - 2011. - No 8 (89). - S. 118-127.
13. Astaf’ev V.I., Roters P.V. Modelirovanie i optimizacia razrabotki mestorozdenij mnogoskvazinnymi dvoakoperiodiceskimi klasterami // Vestnik SamGU. - 2013. - No 9/2 (110). - S. 170-183.
14. Koiter W.T. Some general theorems on doubly-periodic and quasi-periodic functions // Proc. Kon. Ned. Akad. Wt. Amsterdam. - 1959. - Vol. A62, no. 2. - P. 120-128.
15. Lavrent’ev M.A., Sabat B.V. Metody teorii funkcij kompleksnogo peremennogo. - M.: Nauka, 1973. -749 s.
16. Gromadka II T., Lej C. Kompleksnyj metod granicnyh elementov v inzenernyh zadacah. - M: Mir, 1990. - 303 s.
17. Krejg F.F. Razrabotka neftanyh mestorozdenij pri zavodnenii. - M.: Nedra, 1974. - 192 s.
18. Saffman P.G. Viscous fingering in Hele-Shaw cells // J. Fluid Mech. - 1986. - Vol. 173. - P. 73-84. DOI
19. Stone H.A. Philip Saffman and viscous flow theory // J. Fluid Mech. - 2000. - Vol. 409. - P. 165-183. DOI
20. Zaslavskij M.U., Pergament A.H. Issledovanie neustojcivosti tipa "fingers" v fil’tracionnyh teceniah: Prepr. / Institut prikladnoj matematiki im. M.V. Keldysa RAN. - M., 2002. (URL: http://keldysh.ru/papers/2002/prep31/prep2002_31.html).
21. Chen C., Meiburg E. Miscible porous media displacements in the quarter five-spot configuration. Part 1. The homogeneous case // J. Fluid Mech. - 1998. - Vol. 371. - P. 233-268. DOI
22. Daripa P., Glimm J., Lindquist B., McBryan O. Polymer floods: A case study of nonlinear wave analysis and of instability control in tertiary oil recovery // SIAM J. Appl. Math. - 1988. - Vol. 48, no. 2. - P. 353-373. DOI
23. Kornilina M.A., Samarskaa E.A., Cetveruskin B.N., Curbanova N.G., Akobovskij M.V. Modelirovanie razrabotki neftanyh mestorozdenij na parallel’nyh vycislitel’nyh sistemah // Matem. modelirovanie. - 1995. - T. 7, No 2. - S. 35-48.
Downloads
Published
Issue
Section
License
Copyright (c) 2015 Computational Continuum Mechanics
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.