DOI: https://doi.org/10.15587/2312-8372.2019.148386

### Numerical investigation of the problem of nonlinear three-phase filtration

#### Abstract

The object of research is the numerical simulation of the filtration process of oil, gas and water on adaptive grids taking into account some properties of liquids during their joint flow. To obtain an adequate description of the processes, it is necessary simultaneously take into account the effect of most of these factors on filtration. Mathematically, this leads to solving the systems of nonlinear partial differential equations, the complexity of which does not allow them to be studied deeply enough by analytical methods. Experimental studies of these processes are associated with lengthy and expensive laboratory and field experiments.

One of the most problematic places in the theory of multiphase filtration is that the spatial variable steps should be reduced in areas of abrupt changes not only in the water saturation gradient, but also in the gas saturation gradient. This is because due to the very low viscosity, the free gas under the action of the gradient pressure overtakes the rest of the components. This is because due to the very low viscosity, the free gas under the action of the gradient pressure overtakes the rest of the components of the mixture, such as water and oil.

An algorithm for constructing adaptive grids used, which can be adapted to the properties of the solution. The methods of computational mathematics, including the difference-iterative method in moving grid are used.

The numerical experiments are conducted to assess the impact of the proposed method on the displacement and the size of the oils shaft. The comparative analysis of the results is obtained.

Due to this, it is shown that when the oil shaft approaches to the production well, only gas escapes from the reservoir, and as the oil viscosity decreases, the time that the oil shaft approaches to the production well decreases. It is also shown that with an increase in oil viscosity, the length and growth of the oil shaft decrease, and the decrease in comparison with that occurs at a higher rate. It is also shown that with an increase of oil viscosity, the length lv and growth of the oil shaft hv decrease, and the decrease of lv in comparison with hv occurs at a higher rate. And with an increase of the filtration rate and difference of pressure, the geometric dimensions and the «increment» of the oil shaft increase abruptly.

#### Keywords

capillary forces; three-phase filtration; adaptive grid; Thomas algorithm; viscoplastic fluid

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#### References

Mirzadzhanzade, A. Kh., Kovalev, A. G., Zaytsev, Yu. V. (1972). Osobennosti ekspluatatsii mestorozhdeniy anomal'nykh neftey. Moscow: Nedra, 200.

Pirmamedov, V. G. (1975). Ob odnom raznostno-iteratsionnom metode v podvizhnykh setkakh resheniya nekotorykh nelineynykh zadach teorii fil'tratsii i teploprovodnosti. Dep v VINITI, 2027–75.

Kasumov, S. Yu., Suleymanov, S. G., Niftiev, Ya. M. (1996). O primenenii adaptivnoy setki k odnoy radial'noy zadachi trekhfaznoy fil'tratsii. Izvestiya AN Azerbaydzhanskoy SSR, Seriya fiziko-tekhnicheskikh i matematicheskikh nauk, 139 (1-2).

Peery, J. H., Herron, E. H. (1969). Three-Phase Reservoir Simulation. Journal of Petroleum Technology, 21 (2), 211–220. doi: http://doi.org/10.2118/2015-pa

Sonier, F., Ombret, O. (1973). A Numerical Model of Multiphase Flow Around a Well. Society of Petroleum Engineers Journal, 13 (6), 311–320. doi: http://doi.org/10.2118/3627-pa

Klevchenya, A. A., Taranchuk, V. B. (1981). Chislennoe modelirovanie protsessa neustoychivogo vytesneniya nen'yutonovskoy nefti. Dinamika mnogofaznykh sred, 193–198.

Shalimov, B. V. (1975). Chislennoe modelirovanie odnomernoy trekhfaznoy fil'tratsii. Izvestiya Akademii nauk SSSR, MZHG, 26, 59–66.

Douglas, J. Jr., Peaceman, D. W., Rachford, H. H. (1959). A method for calculating multidimensional immiscible displacement. Trans. SPE of AIME, 216, 297–308.

Pascal, H. (1984). Dynamics of moving interface in porous media for power law fluids with yield stress. International Journal of Engineering Science, 22 (5), 577–590. doi: http://doi.org/10.1016/0020-7225(84)90059-4

Elnaggar, H., Karadi, G., Krizek, R. J. (1971). Effect of non-darcian behavior on the characteristics of transient flow. Journal of Hydrology, 13, 127–138. doi: http://doi.org/10.1016/0022-1694(71)90210-1

Shutler, N. D. (1969). Numerical, Three-Phase Simulation of the Linear Steamflood Process. Society of Petroleum Engineers Journal, 9 (2), 232–246. doi: http://doi.org/10.2118/2233-pa

Lapin, V. (1979). Ob issledovanii nekotorykh nelineynykh zadach teorii fil'tratsii. Zhurnal vychislitel'noy matematiki i matematicheskoy fiziki, 19 (3), 689–700.

Abou-Kassem, J. H., Farouq-Ali, S. M. (2006). Petroleum Reservoir Simulations. Gulf Publishing Company, 445.

Samarskiy, A. A. (1984). Teoriya raznostnykh skhem. Moscow: Nauka, 656.

Killough, J. E. (1976). Reservoir Simulation With History-Dependent Saturation Functions. Society of Petroleum Engineers Journal, 16 (1), 37–48. doi: http://doi.org/10.2118/5106-pa

#### GOST Style Citations

Mirzadzhanzade A. Kh., Kovalev A. G., Zaytsev Yu. V. Osobennosti ekspluatatsii mestorozhdeniy anomal'nykh neftey. Moscow: Nedra, 1972. 200 p.

Pirmamedov V. G. Ob odnom raznostno-iteratsionnom metode v podvizhnykh setkakh resheniya nekotorykh nelineynykh zadach teorii fil'tratsii i teploprovodnosti // Dep v VINITI. 1975. Issue 2027–75.

Kasumov S. Yu., Suleymanov S. G., Niftiev Ya. M. O primenenii adaptivnoy setki k odnoy radial'noy zadachi trekhfaznoy fil'tratsii // Izvestiya AN Azerbaydzhanskoy SSR, Seriya fiziko-tekhnicheskikh i matematicheskikh nauk. 1996. Vol. 139, Issue 1-2.

Peery J. H., Herron E. H. Three-Phase Reservoir Simulation // Journal of Petroleum Technology. 1969. Vol. 21, Issue 2. P. 211–220. doi: http://doi.org/10.2118/2015-pa

Sonier F., Ombret O. A Numerical Model of Multiphase Flow Around a Well // Society of Petroleum Engineers Journal. 1973. Vol. 13, Issue 6. P. 311–320. doi: http://doi.org/10.2118/3627-pa

Klevchenya A. A., Taranchuk V. B. Chislennoe modelirovanie protsessa neustoychivogo vytesneniya nen'yutonovskoy nefti // Dinamika mnogofaznykh sred. 1981. P. 193–198.

Shalimov B. V. Chislennoe modelirovanie odnomernoy trekhfaznoy fil'tratsii // Izvestiya Akademii nauk SSSR, MZHG. 1975. Issue 26. P. 59–66.

Douglas J. Jr., Peaceman D. W., Rachford H. H. A method for calculating multidimensional immiscible displacement // Trans. SPE of AIME. 1959. Vol. 216. P. 297–308.

Pascal H. Dynamics of moving interface in porous media for power law fluids with yield stress // International Journal of Engineering Science. 1984. Vol. 22, Issue 5. P. 577–590. doi: http://doi.org/10.1016/0020-7225(84)90059-4

Elnaggar H., Karadi G., Krizek R. J. Effect of non-darcian behavior on the characteristics of transient flow // Journal of Hydrology. 1971. Vol. 13. P. 127–138. doi: http://doi.org/10.1016/0022-1694(71)90210-1

Shutler N. D. Numerical, Three-Phase Simulation of the Linear Steamflood Process // Society of Petroleum Engineers Journal. 1969. Vol. 9, Issue 2. P. 232–246. doi: http://doi.org/10.2118/2233-pa

Lapin V. Ob issledovanii nekotorykh nelineynykh zadach teorii fil'tratsii // Zhurnal vychislitel'noy matematiki i matematicheskoy fiziki. 1979. Vol. 19, Issue 3. P. 689–700.

Abou-Kassem J. H., Farouq-Ali S. M. Petroleum Reservoir Simulations. Gulf Publishing Company, 2006. 445 p.

Samarskiy A. A. Teoriya raznostnykh skhem. Moscow: Nauka, 1984. 656 p.

Killough J. E. Reservoir Simulation With History-Dependent Saturation Functions // Society of Petroleum Engineers Journal. 1976. Vol. 16, Issue 1. P. 37–48. doi: http://doi.org/10.2118/5106-pa