Technologies of non-stationary reservoir flooding are a settled oil recovery method of oil production and reservoir pressure maintenance under the development of most hydrocarbon deposits in the Russian Federation. First of all, this is due to the feasibility, accessibility and low price of water sources for waterflooding. However, the water injection of creates a deferred problem – the inevitable, often water breakthrough, provoked by a sudden and irreversible change in water saturation. The problem of optimization and effective management of the process of non-stationary flooding remains an urgent task. The two-phase flowtheory of Buckley and Leverett does not take into account the loss of stability of the displacement front, provoking a step change and the triplicity of the water saturation value. Therefore, at one time, a mathematically simplified approach was proposed - a repeatedly differentiable approximation to exclude a "jump" in water saturation. Such a simplified solution results in negative consequences well-known from the practice of flooding, recognized by experts as "viscous instability of the displacement front", "finger-shaped displacement front", "dagger flooding of well products", "premature water breakthrough in producing wells", "fractal geometry of displacement front movement". The core of the problem is an attempt to predict the beginning of the loss of stability of the oil displacement front and to prevent its negative consequences on the water flooding in difficult conditions of interaction of hydrothermodynamic, capillary, molecular, inertial and gravitational forces.

In this study, the methods of catastrophe theory are used as a new approach for the analysis of nonlinear polynomial dynamical systems. For this purpose, a mathematical model of growth is selected and by solving the inverse problem, the initial coefficients of the system of differential equations of a two-phase flow are defines. A unified control parameter has been identified, which is used as a discriminant criterion of oil and water growth models for monitoring, control and optimizing the process of water flooding.

References

1. Kreyg F.F., Razrabotka neftyanykh mestorozhdeniy pri zavodnenii (Applied waterflood field development), Moscow: Nedra Publ., 1974, 191 p.

2. Dake L.P., The practice of reservoir engineering, Elsevier Science, 2001, 570 p.

3. Aziz Kh., Settari A., Petroleum reservoir simulation, Applied Science Publishers, 1979, 476 p.

4. Rose W., Rose D.M., “Revisiting” the enduring Buckley – Leverett ideas, Journal of Petroleum Science and Engineering, 2004, V. 45, pp. 263–290, DOI:10.1016/j.petrol.2004.08.001

5. Abbasi J., Ghaedi M., Riazi M., A new numerical approach for investigation of the effects of dynamic capillary pressure in imbibition process, Journal of Petroleum Science and Engineering, 2018, V. 162, pp. 44–54, DOI:10.1016/j.petrol.2017.12.035

6. Yonggang D., Ting L., Mingqiang W. et al., Leverett analysis for transient two-phase flow in fractal porous medium, CMES, 2015, V. 109–110, no. 6, pp. 481–504.

7. Charnyy I.A., Podzemnaya gidrogazodinamika (Underground hydraulic gas dynamics), Moscow – Leningrad: Gostoptekhizdat Publ., 1963, 396 p.

8. Barenblatt G.I., Entov V.M., Ryzhik V.M., Dvizhenie zhidkostey i gazov v prirodnykh plastakh (Movement of liquids and gases in natural reservoirs), Moscow: Nedra Publ., 1982, 211 p.

9. Nigmatullin R.I., Dinamika mnogofaznykh sred (The dynamics of multiphase media), Part 2, Moscow: Nauka Publ., 1987, 360 p.

10. Shakhverdiev A.Kh., Sistemnaya optimizatsiya protsessa razrabotki neftyanykh mestorozhdeniy (System optimization of oil field development process), Moscow: Nedra Publ., 2004, 452 p.

11. Mirzadzhanzade A.Kh., Shakhverdiev A.Kh., Dinamicheskie protsessy v neftegazodobyche: sistemnyy analiz, diagnoz, prognoz (Dynamic processes in the oil and gas production: systems analysis, diagnosis, prognosis), Moscow: Nauka Publ., 1997, 254 p.

12. Mandrik I.E., Panakhov G.M., Shakhverdiev A.Kh., Nauchno-metodicheskie i tekhnologicheskie osnovy optimizatsii protsessa povysheniya nefteotdachi plastov (Scientific and methodological and technological basis for EOR optimization), Moscow: Neftyanoe khozyaystvo Publ., 2010, 288 p.

13. Shakhverdiev A.Kh., Once again about oil recovery factor (In Russ.), Neftyanoe khozyaystvo = Oil Industry, 2014, no. 1, pp. 44–48.

14. Shakhverdiev A.Kh., System optimization of non-stationary floods for the purpose of increasing oil recovery (In Russ.), Neftyanoe khozyaystvo = Oil Industry, 2019, no. 1, pp. 44–49, DOI:10.24887/0028-2448-2019-1-44-49

15. Shakhverdiev A.Kh., Shestopalov Yu.V., Mandrik I.E., Aref'ev S.V., Alternative concept of monitoring and optimization water flooding of oil reservoirs in the conditions of instability of the displacement front (In Russ.),Neftyanoe khozyaystvo = Oil Industry, 2019, no. 12, pp. 118–123, DOI:10.24887/0028-2448-2019-12-118-123

16. Shakhverdiev A.Kh., Some conceptual aspects of systematic optimization of oil field development (In Russ.), Neftyanoe khozyaystvo = Oil Industry, 2017, no. 2, pp. 58–63, DOI: 10.24887/0028-2448-2017-2-58-63

17. Shakhverdiev A.Kh., And Shestopalov Yu.V., Qualitative analysis of quadratic polynomial dynamical systems associated with the modeling and monitoring of oil fields, Lobachevskii journal of mathematics, 2019, V. 40, no. 10, pp. 1691–1706.

18. Shakhverdiev A.Kh., Shestopalov Yu.V., Kachestvennyy analiz dinamicheskoy sistemy podderzhaniya plastovogo davleniya s tsel'yu povysheniya nefteotdachi zalezhey (Qualitative analysis of a dynamic system for maintaining reservoir pressure in order to increase oil recovery), Proceedings of 14 International Conference “Novye idei v naukakh o Zemle” (New ideas in earth sciences), Moscow, 2-5 April 2019. – https://www.mgri.ru/science/scientific-practical-conference/2019-doc/tom%205.pdf

19. Thompson J. M. T, Instabilities and catastrophes in science and engineering, John Wiley & Sons, 1982, 226 p.

20. Arnol'd V.I., Teoriya katastrof (Catastrophe theory), Moscow: Nauka Publ., 1990, 128 p.

21. Nicolis G., Prigogine I., Self-organization in nonequilibrium systems: From dissipative structures to order through fluctuations, John Wiley & Sons, 1977, 512 p.

22. Gaiko V.A., On global bifurcations and Hilbert’s sixteenth problem, Nonlinear Phenomena in Complex Systems 3, 2000, no. 1, pp. 11–27.

Technologies of non-stationary reservoir flooding are a settled oil recovery method of oil production and reservoir pressure maintenance under the development of most hydrocarbon deposits in the Russian Federation. First of all, this is due to the feasibility, accessibility and low price of water sources for waterflooding. However, the water injection of creates a deferred problem – the inevitable, often water breakthrough, provoked by a sudden and irreversible change in water saturation. The problem of optimization and effective management of the process of non-stationary flooding remains an urgent task. The two-phase flowtheory of Buckley and Leverett does not take into account the loss of stability of the displacement front, provoking a step change and the triplicity of the water saturation value. Therefore, at one time, a mathematically simplified approach was proposed - a repeatedly differentiable approximation to exclude a "jump" in water saturation. Such a simplified solution results in negative consequences well-known from the practice of flooding, recognized by experts as "viscous instability of the displacement front", "finger-shaped displacement front", "dagger flooding of well products", "premature water breakthrough in producing wells", "fractal geometry of displacement front movement". The core of the problem is an attempt to predict the beginning of the loss of stability of the oil displacement front and to prevent its negative consequences on the water flooding in difficult conditions of interaction of hydrothermodynamic, capillary, molecular, inertial and gravitational forces.

In this study, the methods of catastrophe theory are used as a new approach for the analysis of nonlinear polynomial dynamical systems. For this purpose, a mathematical model of growth is selected and by solving the inverse problem, the initial coefficients of the system of differential equations of a two-phase flow are defines. A unified control parameter has been identified, which is used as a discriminant criterion of oil and water growth models for monitoring, control and optimizing the process of water flooding.

References

1. Kreyg F.F., Razrabotka neftyanykh mestorozhdeniy pri zavodnenii (Applied waterflood field development), Moscow: Nedra Publ., 1974, 191 p.

2. Dake L.P., The practice of reservoir engineering, Elsevier Science, 2001, 570 p.

3. Aziz Kh., Settari A., Petroleum reservoir simulation, Applied Science Publishers, 1979, 476 p.

4. Rose W., Rose D.M., “Revisiting” the enduring Buckley – Leverett ideas, Journal of Petroleum Science and Engineering, 2004, V. 45, pp. 263–290, DOI:10.1016/j.petrol.2004.08.001

5. Abbasi J., Ghaedi M., Riazi M., A new numerical approach for investigation of the effects of dynamic capillary pressure in imbibition process, Journal of Petroleum Science and Engineering, 2018, V. 162, pp. 44–54, DOI:10.1016/j.petrol.2017.12.035

6. Yonggang D., Ting L., Mingqiang W. et al., Leverett analysis for transient two-phase flow in fractal porous medium, CMES, 2015, V. 109–110, no. 6, pp. 481–504.

7. Charnyy I.A., Podzemnaya gidrogazodinamika (Underground hydraulic gas dynamics), Moscow – Leningrad: Gostoptekhizdat Publ., 1963, 396 p.

8. Barenblatt G.I., Entov V.M., Ryzhik V.M., Dvizhenie zhidkostey i gazov v prirodnykh plastakh (Movement of liquids and gases in natural reservoirs), Moscow: Nedra Publ., 1982, 211 p.

9. Nigmatullin R.I., Dinamika mnogofaznykh sred (The dynamics of multiphase media), Part 2, Moscow: Nauka Publ., 1987, 360 p.

10. Shakhverdiev A.Kh., Sistemnaya optimizatsiya protsessa razrabotki neftyanykh mestorozhdeniy (System optimization of oil field development process), Moscow: Nedra Publ., 2004, 452 p.

11. Mirzadzhanzade A.Kh., Shakhverdiev A.Kh., Dinamicheskie protsessy v neftegazodobyche: sistemnyy analiz, diagnoz, prognoz (Dynamic processes in the oil and gas production: systems analysis, diagnosis, prognosis), Moscow: Nauka Publ., 1997, 254 p.

12. Mandrik I.E., Panakhov G.M., Shakhverdiev A.Kh., Nauchno-metodicheskie i tekhnologicheskie osnovy optimizatsii protsessa povysheniya nefteotdachi plastov (Scientific and methodological and technological basis for EOR optimization), Moscow: Neftyanoe khozyaystvo Publ., 2010, 288 p.

13. Shakhverdiev A.Kh., Once again about oil recovery factor (In Russ.), Neftyanoe khozyaystvo = Oil Industry, 2014, no. 1, pp. 44–48.

14. Shakhverdiev A.Kh., System optimization of non-stationary floods for the purpose of increasing oil recovery (In Russ.), Neftyanoe khozyaystvo = Oil Industry, 2019, no. 1, pp. 44–49, DOI:10.24887/0028-2448-2019-1-44-49

15. Shakhverdiev A.Kh., Shestopalov Yu.V., Mandrik I.E., Aref'ev S.V., Alternative concept of monitoring and optimization water flooding of oil reservoirs in the conditions of instability of the displacement front (In Russ.),Neftyanoe khozyaystvo = Oil Industry, 2019, no. 12, pp. 118–123, DOI:10.24887/0028-2448-2019-12-118-123

16. Shakhverdiev A.Kh., Some conceptual aspects of systematic optimization of oil field development (In Russ.), Neftyanoe khozyaystvo = Oil Industry, 2017, no. 2, pp. 58–63, DOI: 10.24887/0028-2448-2017-2-58-63

17. Shakhverdiev A.Kh., And Shestopalov Yu.V., Qualitative analysis of quadratic polynomial dynamical systems associated with the modeling and monitoring of oil fields, Lobachevskii journal of mathematics, 2019, V. 40, no. 10, pp. 1691–1706.

18. Shakhverdiev A.Kh., Shestopalov Yu.V., Kachestvennyy analiz dinamicheskoy sistemy podderzhaniya plastovogo davleniya s tsel'yu povysheniya nefteotdachi zalezhey (Qualitative analysis of a dynamic system for maintaining reservoir pressure in order to increase oil recovery), Proceedings of 14 International Conference “Novye idei v naukakh o Zemle” (New ideas in earth sciences), Moscow, 2-5 April 2019. – https://www.mgri.ru/science/scientific-practical-conference/2019-doc/tom%205.pdf

19. Thompson J. M. T, Instabilities and catastrophes in science and engineering, John Wiley & Sons, 1982, 226 p.

20. Arnol'd V.I., Teoriya katastrof (Catastrophe theory), Moscow: Nauka Publ., 1990, 128 p.

21. Nicolis G., Prigogine I., Self-organization in nonequilibrium systems: From dissipative structures to order through fluctuations, John Wiley & Sons, 1977, 512 p.

22. Gaiko V.A., On global bifurcations and Hilbert’s sixteenth problem, Nonlinear Phenomena in Complex Systems 3, 2000, no. 1, pp. 11–27.