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Scaling of relative permeability functions as a method of regularizing numerical solution of the water-oil displacement problem

Authors: V.V. Ovcharov (TNNC LLC, RF, Tyumen)

Key words: scaling of functions, relative permeability, method of regularization, numerical diffusion, reservoir simulation.

The article dwells on the problem of simulating discontinuous solutions in numerical models as exemplified by the oil-water displacement problem in porous medium with neglected capillary pressure (Buckley-Leverett problem). It has been shown that coarse sampling in numerical models enhances the process of numerical diffusion, leading to “blurring” of the displacement front. The work presents a new method to regularize the numerical solution of the water-oil displacement problem in porous medium by scaling the relative permeability functions, which enables reconstruction of the displacement front’s movement using coarse grid with no changes in the numerical scheme. The carried out test calculations have confirmed the applicability of this method.

References
1. Bakhvalov NS., Zhidkov N.P., Kobel'kov G.N., Chislennye metody (Numerical methods), Moscow: BINOM Publ., 2003, 632 p.
2. Petrov I.B., Kholodov A.S., Zhurnal vychislitel'noy matematiki i matematicheskoy fiziki – Computational Mathematics and Mathematical Physics, 1984, V. 24, pp. 1172–1188.
3. Samarskiy A.A., Zhurnal vychislitel'noy matematiki i matematicheskoy
fiziki – Computational Mathematics and Mathematical Physics, 1967, V. 7,
pp. 62–93.
4. Basniev K.S., Dmitriev N.M., Kanevskaya R.D., Maksimov V.M., Podzemnaya
gidromekhanika (Subsurface hydromechanics), Moscow – Izhevsk: Publ. of
Institute of Computer Science, 2006, 488 p.
5. Metodicheskie rekomendatsii po proektirovaniyu razrabotki neftyanykh i
gazoneftyanykh mestorozhdeniy (Guidelines for the design of oil and gas
field development), Appendix to the Order of MNR of Russia no. 61 of
21.03.07.
6. Aziz Kh., Settari A., Petroleum reservoir simulation, London: Applied Science Publishers, 1979, 476 p. 

Key words: scaling of functions, relative permeability, method of regularization, numerical diffusion, reservoir simulation.

The article dwells on the problem of simulating discontinuous solutions in numerical models as exemplified by the oil-water displacement problem in porous medium with neglected capillary pressure (Buckley-Leverett problem). It has been shown that coarse sampling in numerical models enhances the process of numerical diffusion, leading to “blurring” of the displacement front. The work presents a new method to regularize the numerical solution of the water-oil displacement problem in porous medium by scaling the relative permeability functions, which enables reconstruction of the displacement front’s movement using coarse grid with no changes in the numerical scheme. The carried out test calculations have confirmed the applicability of this method.

References
1. Bakhvalov NS., Zhidkov N.P., Kobel'kov G.N., Chislennye metody (Numerical methods), Moscow: BINOM Publ., 2003, 632 p.
2. Petrov I.B., Kholodov A.S., Zhurnal vychislitel'noy matematiki i matematicheskoy fiziki – Computational Mathematics and Mathematical Physics, 1984, V. 24, pp. 1172–1188.
3. Samarskiy A.A., Zhurnal vychislitel'noy matematiki i matematicheskoy
fiziki – Computational Mathematics and Mathematical Physics, 1967, V. 7,
pp. 62–93.
4. Basniev K.S., Dmitriev N.M., Kanevskaya R.D., Maksimov V.M., Podzemnaya
gidromekhanika (Subsurface hydromechanics), Moscow – Izhevsk: Publ. of
Institute of Computer Science, 2006, 488 p.
5. Metodicheskie rekomendatsii po proektirovaniyu razrabotki neftyanykh i
gazoneftyanykh mestorozhdeniy (Guidelines for the design of oil and gas
field development), Appendix to the Order of MNR of Russia no. 61 of
21.03.07.
6. Aziz Kh., Settari A., Petroleum reservoir simulation, London: Applied Science Publishers, 1979, 476 p. 


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