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The method for calculating the productivity of wells under transient behavior with an account of the lift characteristic of the well

UDK: 622.276.5.001
DOI: 10.24887/0028-2448-2019-6-84-89
Key words: well productivity, lift curve, analytical solution
Authors: E.A. Kanin (Moscow Institute of Physics and Technology (National Research University), RF, Moscow), V.A. Krasnov (Rosneft Oil Company, RF, Moscow)

For calculating the productivity of wells working at the reservoirs with simple configuration (cylinder, parallelepiped), quite often the models based on the analytical solutions of the filtration equation are used. In such models at the wellbore (the internal boundary of the reservoir) the following boundary conditions are set: constant bottomhole pressure (the first kind) or constant flow rate (the second kind. However, these boundary conditions do not always adequately reflect the processes occurring at the mutual work of the reservoir and well. For this reason, in this work, the new model for calculating transient productivity of wells is developed. It is based on the third kind boundary condition: the linear dependence between downhole pressure and flow rate which is a linear approximation of the lift curve of the well in the operating range of well’s flow rates. This approach developed by Rosneft’s specialists is applicable for any configuration of wells (vertical, horizontal well, hydraulic fracture) and to homogeneous external boundary conditions. The deduced relations allow obtaining solutions for the downhole pressure and flow rate of the well, provided that the solution of the problem of constant rate production (constant terminal rate solution) for the same configuration of the well and external boundary conditions is known. These relations also allow considering wellbore storage. For problem-solving we apply the method of Laplace transform. Using the developed approach, it is possible to calculate the productivity of wells accounting for the performance of submersible equipment, and, as a result, to estimate potential transient production of the well taking into account the real conditions of its operation.

References

1. Van Everdingen A.F., Hurst W., The application of the Laplace transformation to flow problems in reservoirs, Journal of Petroleum Technology, 1949, V. 1, no. 12, pp. 305–324.

2. Brown K.E. et al., Nodal systems analysis of oil and gas wells, Journal of Petroleum Technology, 1985, V. 37, no. 10, pp. 1751–1763.

3. Brill J., Mukherjee H., Multiphase flow in wells, Richardson, Texas, 1999, 384 p.

4. Khasanov M.M., Krasnov V.A., Musabirov T.R., The solution of the problem of the interaction of the formation with the borehole in the conditions of non-stationary inflow (In Russ.), Nauchno-tekhnicheskiy Vestnik OAO “NK “Rosneft'”, 2007, no. 2, pp. 41-46.

5. Shchelkachev V.N., Osnovy i prilozheniya teorii neustanovivsheysya fil’tratsii (Fundamentals and applications of the theory of unsteady filtration), Moscow: Neft’ i gaz Publ., 1995, 586 p.

6. Ozkan E. et al., Supplement to new solutions for well-test-analysis problems. Part 1, SPE 18615-PA, 1991, https://doi.org/10.2118/18615-PA.

7. Stehfest H., Algorithm 368: Numerical inversion of Laplace transforms [D5], Communications of the ACM, 1970, V. 13, no. 1, pp. 47–49.

8. Chen C.C., Rajagopal, R., A multiply-fractured horizontal well in a rectangular drainage region, SPE 37072-PA, 1997, https://doi.org/10.2118/37072-PA.

9. Khasanov M.M. et al., Express method to estimate target bottomhole pressure in pumping oil well (In Russ.), SPE 171303-MS, 2014, https://doi.org/10.2118/171303-MS.

10. Bedrin V.G. et al., Comparison of ESP technologies for operation at high gas content in pump based on NK Rosneft field tests (In Russ.), SPE 117414-MS, 2008, https://doi.org/10.2118/117414-MS.

11. Beggs D.H. et al., A study of two-phase flow in inclined pipes, Journal of Petroleum Technology, 1973, V. 25, no. 5, pp. 607–617.

12. Khasanov M.M. et al., A simple mechanistic model for void-fraction and pressure-gradient prediction in vertical and inclined gas/liquid flow (In Russ.), SPE 108506-PA, 2009, https://doi.org/10.2118/108506-PA.

13. Ansari A.M. et al., A comprehensive mechanistic model for upward two-phase flow in wellbores (In Russ.), SPE 108506-PA, 1990, https://doi.org/10.2118/108506-PA.

14. Krasnov V. et al., Monitoring and optimization of well performance in Rosneft Oil Company - The experience of the unified model application for multiphase hydraulic calculations (In Russ.), SPE 104359-MS, 2006, https://doi.org/10.2118/104359-MS.

15. Pashali A. et al., Real time optimisation approach for 15 000 ESP wells (In Russ.), SPE 112238-MS, 2008, https://doi.org/10.2118/112238-MS.

For calculating the productivity of wells working at the reservoirs with simple configuration (cylinder, parallelepiped), quite often the models based on the analytical solutions of the filtration equation are used. In such models at the wellbore (the internal boundary of the reservoir) the following boundary conditions are set: constant bottomhole pressure (the first kind) or constant flow rate (the second kind. However, these boundary conditions do not always adequately reflect the processes occurring at the mutual work of the reservoir and well. For this reason, in this work, the new model for calculating transient productivity of wells is developed. It is based on the third kind boundary condition: the linear dependence between downhole pressure and flow rate which is a linear approximation of the lift curve of the well in the operating range of well’s flow rates. This approach developed by Rosneft’s specialists is applicable for any configuration of wells (vertical, horizontal well, hydraulic fracture) and to homogeneous external boundary conditions. The deduced relations allow obtaining solutions for the downhole pressure and flow rate of the well, provided that the solution of the problem of constant rate production (constant terminal rate solution) for the same configuration of the well and external boundary conditions is known. These relations also allow considering wellbore storage. For problem-solving we apply the method of Laplace transform. Using the developed approach, it is possible to calculate the productivity of wells accounting for the performance of submersible equipment, and, as a result, to estimate potential transient production of the well taking into account the real conditions of its operation.

References

1. Van Everdingen A.F., Hurst W., The application of the Laplace transformation to flow problems in reservoirs, Journal of Petroleum Technology, 1949, V. 1, no. 12, pp. 305–324.

2. Brown K.E. et al., Nodal systems analysis of oil and gas wells, Journal of Petroleum Technology, 1985, V. 37, no. 10, pp. 1751–1763.

3. Brill J., Mukherjee H., Multiphase flow in wells, Richardson, Texas, 1999, 384 p.

4. Khasanov M.M., Krasnov V.A., Musabirov T.R., The solution of the problem of the interaction of the formation with the borehole in the conditions of non-stationary inflow (In Russ.), Nauchno-tekhnicheskiy Vestnik OAO “NK “Rosneft'”, 2007, no. 2, pp. 41-46.

5. Shchelkachev V.N., Osnovy i prilozheniya teorii neustanovivsheysya fil’tratsii (Fundamentals and applications of the theory of unsteady filtration), Moscow: Neft’ i gaz Publ., 1995, 586 p.

6. Ozkan E. et al., Supplement to new solutions for well-test-analysis problems. Part 1, SPE 18615-PA, 1991, https://doi.org/10.2118/18615-PA.

7. Stehfest H., Algorithm 368: Numerical inversion of Laplace transforms [D5], Communications of the ACM, 1970, V. 13, no. 1, pp. 47–49.

8. Chen C.C., Rajagopal, R., A multiply-fractured horizontal well in a rectangular drainage region, SPE 37072-PA, 1997, https://doi.org/10.2118/37072-PA.

9. Khasanov M.M. et al., Express method to estimate target bottomhole pressure in pumping oil well (In Russ.), SPE 171303-MS, 2014, https://doi.org/10.2118/171303-MS.

10. Bedrin V.G. et al., Comparison of ESP technologies for operation at high gas content in pump based on NK Rosneft field tests (In Russ.), SPE 117414-MS, 2008, https://doi.org/10.2118/117414-MS.

11. Beggs D.H. et al., A study of two-phase flow in inclined pipes, Journal of Petroleum Technology, 1973, V. 25, no. 5, pp. 607–617.

12. Khasanov M.M. et al., A simple mechanistic model for void-fraction and pressure-gradient prediction in vertical and inclined gas/liquid flow (In Russ.), SPE 108506-PA, 2009, https://doi.org/10.2118/108506-PA.

13. Ansari A.M. et al., A comprehensive mechanistic model for upward two-phase flow in wellbores (In Russ.), SPE 108506-PA, 1990, https://doi.org/10.2118/108506-PA.

14. Krasnov V. et al., Monitoring and optimization of well performance in Rosneft Oil Company - The experience of the unified model application for multiphase hydraulic calculations (In Russ.), SPE 104359-MS, 2006, https://doi.org/10.2118/104359-MS.

15. Pashali A. et al., Real time optimisation approach for 15 000 ESP wells (In Russ.), SPE 112238-MS, 2008, https://doi.org/10.2118/112238-MS.


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