Petroleum reservoir simulation using super-element method with local detalization of the solution
A.B. Mazo*, K.A. Potashev**
Kazan Federal University, Kazan, 420008 Russia
E-mail: * ABMazo1956@gmail.com, ** KPotashev@mail.ru
Received June 21, 2017
Abstract
In this paper, we propose a two-stage method for petroleum reservoir simulation. The method uses two models with different degrees of detailization to describe the hydrodynamic processes of different space-time scales. At the first stage, the global dynamics of the energy state of the deposit and reserves has been modeled (characteristic scale of such changes is km/year). The two-phase flow equations in the model of global dynamics operate with smooth averaged pressure and saturation fields, and they are solved numerically on a large computational grid of super-elements with a characteristic cell size of 200–500 m. The tensor coefficients of the super-element model have been calculated using special procedures of upscaling of absolute and relative phase permeabilities. At the second stage, a local refinement of the super-element model has been constructed for calculating small-scale processes (with a scale of m / day), which take place, for example, during various geological and technical measures aimed at increasing the oil recovery of a reservoir. Then we solve the two-phase flow problem in the selected area of the measure exposure on a detailed three-dimensional grid, which resolves the geological structure of the reservoir, and with a time step sufficient for describing fast-flowing processes. The initial and boundary conditions of the local problem have been formulated on the basis of the super-element solution. To demonstrate the proposed approach, we have provided an example of the two-stage modeling of the development of a layered reservoir with a local refinement of the model during the isolation of a water-saturated high-permeability interlayer. We have shown a good compliance between the locally refined solution of the super-element model.
Keywords: super-element method, numerical simulation, petroleum reservoir, local refinement, reservoir treatments simulation, two-phased flow, downscaling
Acknowledgments. The study was supported by the Russian Foundation for Basic Research and the Government of the Republic of Tatarstan (projects nos. 15-41-02698 and 15-41-02699).
Figure Captions
Fig. 1. The dynamics of well water-intake capacity at the instant change of bottom-hole pressure: ? – from the solution of equations (5), (6), solid line – from the solution of equations (5), (7).
Fig. 2. The bottom-hole pressure dynamics in the well at the instant change of water-intake capacity: ? – from the solution of equations (5), (6), solid line – from the solution of the equations (5), (7).
Fig. 3. The projection of on the horizontal plane of the model oil deposit with isolation along the oil-bearing contour.
Fig. 4. Super-element grid coverage of the oil-deposit area, the area of model refinement (on the left) and its coverage by the detail grid (on the right).
Fig. 5. The distribution of saturation in the section of the oil bed as of January 2006 based on the model of the whole oil deposit (on the left), based on the local SEM refinement (on the right).
Fig. 6. The calculated open flow of well no. 52: 1 – accurate solution, 2 – SEM, 3 – SEM with refinement.
References
1. Durlofsky L.J. Coarse scale models of two phase flow in heterogeneous reservoirs: Volume averaged equations and their relationship to existing upscaling techniques. Comput. Geosci., 1998, vol. 2, no. 2, pp. 73–92. doi: 10.1023/A:1011593901771.
2. Beliaev A.Yu. Averaging in Problems of the Theory of Fluid Flow in Porous Media. Moscow, Nauka, 2004. 200 p. (In Russian)
3. Mazo A.B., Potashev K.A. Upscaling of absolute phase permeabilities for superelement modeling of petroleum reservoir. Mat. Model., 2017, vol. 29, no. 6, pp. 89–102. (In Russian)
4. Mazo A.B., Potashev K.A. Upscaling relative phase permeability for superelement modeling of petroleum reservoir engineering. Math. Model. Comput. Simul., 2017, vol. 9, no. 5, pp. 570–579. doi: 10.1134/S207004821705009X.
5. Aarnes J.E., Kippe V., Lie K.-A. Mixed multiscale finite elements and streamline methods for reservoir simulation of large geomodels. Adv. Water Res., 2005, vol. 28, no. 3, pp. 257–271. doi: 10.1016/j.advwatres.2004.10.007.
6. Efendiev Y., Ginting V., Hou T., and Ewing R. Accurate multiscale finite element methods for two-phase flow simulations. J. Comput. Phys., 2006, vol. 220, no. 1, pp. 155–174. doi: 10.1016/j.jcp.2006.05.015.
7. Jenny P., Lee S., and Tchelepi H. Adaptive fully implicit multiscale finite-volume methods for multi-phase flow and transport in heterogeneous porous media. J. Comput. Phys., 2006, vol. 217, no. 2, pp. 627–641. doi: 10.1016/j.jcp.2006.01.028.
8. Pergament A.Kh., Semiletov V.A., Tomin P.Yu. On some multiscale algorithms for sector modeling in multiphase flow in porous media. Math. Models Comput. Simul., 2011, vol. 3, no. 3, pp. 365–374. doi: 10.1134/S2070048211030100.
9. Mazo A.B., Bulygin D.V. Superelements. New approach to oil reservoir simulation. Neft'. Gaz. Novatsii, 2011, vol. 11, pp. 6–8. (In Russian)
10. Mazo A.B., Potashev K.A., Kalinin E.I., Bulygin D.V. Oil reservoir simulation with the superelement method. Mat. Model., 2013, vol. 25, no. 8, pp. 51–64. (In Russian)
11. Bulygin D.V., Mazo A.B., Potashev K.A., Kalinin E.I. Geological and technical aspects of superelement model of oil reservoirs. Georesursy, 2013, vol. 53, no. 3, pp. 31–35. (In Russian)
12. Mazo A., Potashev K., Kalinin E. Petroleum reservoir simulation using super element method. Proc. Earth Planet. Sci., 2015, vol. 15, pp. 482–487. doi: 10.1016/j.proeps.2015.08.053.
13. Mazo A.B., Potashev K.A. Upscaling of absolute permeability for a super-element model of petroleum reservoir. IOP Conf. Ser.: Mater. Sci. Eng., 2016, vol. 158, no. 1, pp. 1–6.
14. Potashev K.A., Abdrashitova L.R. Account for the heterogeneity of waterflooding in the well drainage area for large-scale modeling of oil reservoir. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2017, vol. 159, no. 1, pp. 116–129. (In Russian)
15. Barenblatt G.I., Entov V.M., Ryzhik V.M. Theory of Fluid Flow Through Natural Rocks. London, Kluwer Acad. Publ., 1990. 396 p.
16. Potashev K.A., Mazo A.B., Ramazanov R.G., Bulygin D.V. Analysis and design of a section of an oil reservoir using a fixed stream tube model. Neft'. Gaz. Novatsii, 2016, vol. 187, no. 4, pp. 32–40. (In Russian)
17. Kozeny J. Ueber kapillare Leitung des Wassers im Boden. Sitzungsber. Akad. Wiss., 1927, B. 136, T. 2a, pp. 271–306. (In German)
18. Stiles W.E. Use of permeability distribution in waterflood calculations. J. Pet. Technol., 1949, vol. 1, no. 1, pp. 9–13. doi: 10.2118/949009-G.
19. Dykstra H., Parsons R. Secondary Oil Recovery of Oil in the United States. The Prediction of Oil Recovery by Waterflooding. Washington, DC, API, 1950, pp. 160–174. (In Russian)
20. Bulygin V.Ya. Hydromechanics of the Oil Reservoir. Moscow, Nedra, 1974. 232 p. (In Russian)
For citation: Mazo A.B., Potashev K.A. Petroleum reservoir simulation using super-element mMethod with local detalization of the solution. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2017, vol. 159, no. 3, pp. 327–339.5. (In Russian)
The content is available under the license Creative Commons Attribution 4.0 License.
