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Calculation of the flow rate between wells in the flow model of an oil reservoir using streamlines

*K.A. Potashev, R.R. Akhunov, A.B. Mazo *

Original article

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To analyze the waterflooding system of an oil reservoir and predict the effectiveness of geological and technical measures, information is required on the distribution of injection rate between the reacting production wells and the reservoir boundary. The most reliable methods for calculating these characteristics are methods based on hydrodynamic modeling of flow. Modern commercial software implement algorithms for these purposes based on the construction and analysis of streamlines. At the same time, there are no reliable estimates of the accuracy of these algorithms and recommendations for choosing the optimal parameters in the available literature.

In this paper, we propose an algorithm for calculating the proportions of the distribution of the total well flow rate between the surrounding wells and the reservoir boundary using streamlines. Streamlines are constructed on the basis of a finite element solution to the flow problem averaged over the formation thickness and determine the boundaries of the streamtubes connecting the corresponding wells. The flow rate through the flow tubes is calculated by numerically integrating the Darcy velocity field of the indicated two-dimensional problem. The algorithm was tested on idealized examples of waterflooding elements of typical well placement schemes, when the exact distribution of the proportions of fluid injected into the formation is known, and on the example of comparison with the solution of the problem of simulating the injection of a tracer into the reservoir. Recommendations for the selection of starting points for tracing streamlines are presented, which allow achieving a minimum level of error in determining the mutual influence of wells in a wide range of the computational grid resolution of the flow model.

A more general application of the described method without significant changes is to equip the high resolution flow model along fixed stream tubes with their rate characteristics.

oil reservoir, well interaction, two-dimensional flow problem, streamtubes, streamlines, numerical simulation

- Albertoni A. Lake, L.W. (2003). Inferring interwell connectivity only from well-rate fluctuations in waterfloods. SPE Reserv. Eval. Eng, 6, pp. 6–16. https://doi.org/10.2118/83381-PA
- Batycky, R.P. (1997). A Three-Dimensional Two-Phase Field Scale Streamline Simulator. Ph.D. Thesis, Department of Petroleum Engineering, School of Earth Science, Stanford University, Stanford, California, USA.
- Buzinov S.N., Umrihin I.D. (1984). Research of oil and gas wells and reservoirs. Moscow: Nedra, 269 p. (In Russ.)
- Chernorubashkin A.I., Makeev G.A., Gavrilenko G.A. (1985). Application of indicator methods to control the development of oil fields. Overview. Moscow: VNIIOENG, 37 p. (In Russ.)
- Holanda R.F., Gildin E., Jensen J.L., Lake L.W., Kabir C.S. (2018). A State-of-the-Art Literature Review on Capacitance Resistance Models for Reservoir Characterization and Performance Forecasting. Energies, 11, 3368, pp. 1–46. https://doi.org/10.3390/en11123368
- Loula A.F.D, Guerreiro J.N.C., Ribeiro F.L.B, Landau L. (1995). Tracer injection simulations by finite element methods. SPE 27047.
- Mazo A.B., Potashev K.A. (2020). Superelements. Petroleum reservoir simulation. Moscow: Infra-M, 220 p. (In Russ.)
- Mazo A.B., Potashev K.A., Baushin V.V., Bulygin D.V. (2017). Numerical Simulation of Oil Reservoir Polymer Flooding by the Model of Fixed Stream Tube. Georesursy = Georesources, 19(1), pp. 15–20. http://doi.org/10.18599/grs.19.1.3
- Mazo A.B., Potashev K.A. (2020). Numerical modeling of local impact on the oil reservoir using fixed flowtubes for typical waterflooding schemes. Georesursy = Georesources, 22(4), pp. 70–78. (In Russ.) https://doi.org/10.18599/grs.2020.4.70-78
- Muskat M., Wyckoff R.D. (1934). A Theoretical Analysis of Waterflooding Networks. Trans., AIME, 107, pp. 62–77. https://doi.org/10.2118/934062-G
- Muskat M., Wyckoff R.D. (1937). The flow of homogeneous fluids through porous media. New York, London, McGraw-Hill Book Co, XIX, 763 p.
- Panin D.A., Potashev K.A. (2020). Modification of FEM-grids near wells in two-dimensional filtration problems. Proc. Conf.: Actual problems of continuum mechanics – 2020. Kazan: pp. 339–344. (In Russ.)
- Pollock D.W. (1988). Semianalytical Computation of Pathlines for Finite-Difference Models. Groundwater, 26(6), pp. 743–750. https://doi.org/10.1111/j.1745-6584.1988.tb00425.x
- Potashev K.A., Ahunov R.R. (2020). Assessment of heterogeneity of reservoir fluid inflow to the cross-sectional contour of a vertical well. Uchenye zapiski Kazanskogo universiteta. Series Physics and Mathematics, 162(2), pp. 180–192. (In Russ.)
- Potashev K.A., Mazo A.B. (2021). Mathematical Modeling of Oil Reservoir Waterflooding Using Fixed Streamtube at Various Values of Viscosity Ratio. Lobachevskii Journal of Mathematics, 42 (8), pp. 2023–2029. https://doi.org/10.1134/S1995080221080254
- Potashev K.A., Mazo A.B., Ramazanov R.G., Bulygin D.V. (2016). Analysis and design of a section of an oil reservoir using a fixed stream tube model. Neft. Gaz. Novacii, 187(4), pp. 32–40. (In Russ.)
- Sauley V.I., Hozyainov M.S., Trenchikov Ju.I. (2004). Comprehensive study of the hydrodynamic relationship between injection and production wells by indicator and geophysical methods. Karotazhnik, 10–11(123–124). (In Russ.)
- Sedov L.I. (1976). Continuum mechanics. Moscow: Nauka, 536 p. (In Russ.)
- Shahvali M., Mallison B., Wei K., Gross H. (2011). An Alternative to Streamlines for Flow Diagnostics on Structured and Unstructured Grids. SPE 146446, pp. 1–16. https://doi.org/10.2118/146446-MS
- Shatsky A.V., Kolesov V.V., Shatsky D.A., Mitrofanov A.D, Bodryagin A.V., Nikitin A.Ju. (2005). Simulation testing and new possibilities of the method of tracer studies. Moscow: VNIIOENG, 8, pp. 50–58. (In Russ.)
- Sokolovsky Je.V., Chizhov S.I., Trenchikov Ju.I. et al. (1989). . Methodological guidance on the technology of indicator studies and the interpretation of their results for the regulation and control of the process of waterflooding of oil deposits. RD 39-014-7428-235-89. Grozny: SevKavNIPIneft. (In Russ.)
- Spirina E.A., Potashev K.A., Mazo A.B. (2019). Evaluation of the reliability of the averaging over the reservoir thickness for the model with a fixed streamtube. Conf. Series: J. of Physics, 1158 042024, pp. 1–6.
- Stepanov S. V., Sokolov S. V., Ruchkin A. A., Stepanov A. V., Knjazev A. V., Korytov A. V. (2018). Problems of assessing the mutual influence of production and injection wells based on mathematical modeling. Tyumen State University Herald. Physical and Mathematical Modeling. Oil, Gas, Energy, pp. 146–164. (In Russ.) https://doi.org/10.21684/2411-7978-2018-4-3-146-164
- Willhite G.P. (1986). Waterflooding. SPE Textbook Series. Richardson, TX, 331 p. https://doi.org/10.1088/1742-6596/1158/4/042024
- Zemel B. (1996). Tracers in the Oil Field. Developments in Petroleum Science, 43. Amsterdam: Elsevier Science.
- Zheltov Ju.P. (1986). Oil field Development. Moscow: Nedra, 332 p. (In Russ.)

Konstantin A. Potashev – DSc (Physics and Mathe-matics), Associate Professor, Head of the Department of Aerohydromechanics, N.I. Lobachevsky Institute of Mathematics and Mechanics, Kazan Federal University, Kazan, Russian Federation

35, Kremlevskaya st., Kazan, 420008, Russian Federation

Rustam Rashid ugli Akhunov – PhD student, Department of Aerohydromechanics, N.I. Lobachevsky Institute of Mathematics and Mechanics, Kazan Federal University, Kazan, Russian Federation

35, Kremlevskaya st., Kazan, 420008, Russian Federation

Aleksandr B. Mazo – DSc (Physics and Mathematics), Professor, Department of Aerohydromechanics, N.I. Lobachevsky Institute of Mathematics and Mechanics, Kazan Federal University, Kazan, Russian Federation

35, Kremlevskaya st., Kazan, 420008, Russian Federation

#### For citation:

Potashev K.A., Akhunov R.R., Mazo A.B. (2022). Calculation of the flow rate between wells in the flow model of an oil reservoir using streamlines. Georesursy = Georesources, 24(1), pp. 27–35. DOI: https://doi.org/10.18599/grs.2022.1.3