Modeling of fluid inflow towards multistage hydraulic fractures of infinite permeability using stream tubes

K.A. Potashev ∗ , A.B. Mazo ∗∗ , M.V. Mukhina ∗∗∗ , A.A. Uraimov ∗∗∗∗ , D.V. Maklakov ∗∗∗∗∗ , M.R. Khamidullin ∗∗∗∗∗∗

Kazan Federal University, Kazan, 420008 Russia

E-mail: ∗kpotashev@mail.ru, ∗∗abmazo1956@gmail.com, ∗∗∗mvvoler@gmail.com, 

∗∗∗∗aauraimov@gmail.com, ∗∗∗∗∗dmaklak@kpfu.ru, ∗∗∗∗∗∗marseille.ksu@gmail.com

Received January 28, 2022


ORIGINAL ARTICLE

Full text PDF

DOI: 10.26907/2541-7746.2022.1.101-121

For citation: Potashev K.A., Mazo A.B., Mukhina M.V., Uraimov A.A., Maklakov D.V., Khamidullin M.R. Modeling of fluid inflow towards multistage hydraulic fractures of infinite permeability using stream tubes. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2022, vol. 164, no. 1, pp. 101–121. doi: 10.26907/2541-7746.2022.1.101-121. (In Russian)


Abstract

This article is devoted to the problem of a fundamental reduction of the machine time for numerical solution of non-steady state problems of the flow in the vicinity of multistage hydraulic fractures along horizontal wells in a petroleum reservoir. This issue arises when it is necessary to solve inverse problems associated either with the identification of fracture parameters based on the results of their hydrodynamic studies or with their optimization to obtain specified production indicators. As a way to reduce computational costs, we previously proposed replacing the spatial problem of flow in the reservoir with a set of one-dimensional problems along the stream tubes. In this case, the problems for pressure in each fracture are solved taking into account the distributed inflow of reservoir fluid from the stream tubes adjacent to the fracture edges. Decomposition of the spatial problem into a set of one-dimensional problems along the stream tubes reduces the required machine time for numerical simulation of the non-steady state flow by orders of magnitude. The object of this research is the functions of the relative width along the stream tubes distribution and their lengths, which are necessary to calculate the local inflow to the fractures and are the key parameters of the model that determine its accuracy.

The parameterization of the functions of length and distribution of the relative width along the stream tubes adjacent to the edges of vertical multistage hydraulic fractures is performed. The well between the fractures is assumed to be non-perforated. The case of fractures of infinite permeability in a homogeneous reservoir is considered, when the problem is reduced to a two-dimensional formulation in a horizontal plane. The results are also applicable without any changes to a stratified heterogeneous formation. Analytical expressions are obtained for the listed properties of stream tubes with a difference between the inner and outer edges of the fractures. For this purpose, analytical solutions of the corresponding model problems are used. An algorithm for the parametrization of stream tubes for the case of fractures differing in length is proposed. The proposed simplification of the spatial model is tested, and the range of values of the initial parameters of the system, which allows for an acceptable level of error of the simplified model for evaluative calculations, is shown.

Keywords: oil reservoir, multistage hydraulic fracturing, single-phase filtration, stream tubes, analytical solution, numerical simulation

Acknowledgments. This study was supported by the Kazan Federal University Strategic Academic Leadership Program (“PRIORITY-2030”).

References

  1. Kanevskaya R.D. Matematicheskoe modelirovanie razrabotki mestorozhdenii nefti i gaza s primeneniem gidravlicheskogo razryva plasta [Mathematical Modeling of Oil and Gas Field Development Using Hydraulic Fracturing]. Moscow, OOO “Nedra-Biznestsentr”, 1999. 212 p. (In Russian)
  2. Mazo A.B., Hamidullin M.R. Explicit-Implicit Algorithms for accelerating the calculation of two-phase inflow to a horizontal well with multistage hydraulic fracturing. Vychisl. Metody Program., 2017, vol. 18, no. 3, pp. 204–213. (In Russian)
  3. Mazo A.B., Sattarov R.I., Khamidullin M.R., Potashev K.A., Trifonof T.V. Interpretation of indicator studies of multistage fracturing. SPE Russ. Petroleum Technology Conf., Moscow, 16–18 Oct., 2017, 2017, art. SPE-187762-MS, pp. 1–23. doi: 10.2118/187762-MS.
  4. Mazo A.B., Potashev K.A. Superelementy. Modelirovanie razrabotki neftyanykh mestorozhdenii [Superelements. Oil Field Development Modeling: A Monograph]. Moscow, INFRA-M, 2020. 220 p. (In Russian)
  5. Mukherjee H., Economides M.J. A parametric comparison of horizontal and vertical well performance. SPE Form. Eval., 1991, vol. 6, no. 2, pp. 209–216. doi: 10.2118/18303-PA.
  6. Guo G., Evans R.D. Inflow performance of a horizontal well intersecting natural fractures. SPE Production Operations Symp., Oklahoma City, Oklahoma, Mar. 1993, 1993, pp. 851– 865. doi: 10.2118/25501-MS.
  7. Prats M. Effect of vertical fractures on reservoir behavior–incompressible fluid case. SPE J., 1961, vol. 1, no. 2, pp. 105–118. doi: 10.2118/1575-G.
  8. Raghavan R., Joshi S.D. Productivity of multiple drainholes or fractured horizontal wells. SPE Form. Eval., 1993, vol. 8, no. 1, pp. 11–16. doi: 10.2118/21263-PA.
  9. Hu C., Lu J., He X. Productivity formulae of an infinite-conductivity hydraulically fractured well producing at constant wellbore pressure based on numerical solutions of a weakly singular integral equation of the first kind. Math. Probl. Eng., 2012, vol. 2012, art. 428596, pp. 1–18. doi: 10.1155/2012/428596.
  10. Muskat M. Physical Principles of Oil Production. Boston, I.H.R.D.C., 1981. 922 p.
  11. Kanevskaya R.D., Kats R.M. Exact solutions of problems of fluid inflow into a well with a vertical hydrofracture and their use in numerical models of flow through porous media. Fluid Dyn., 1996, vol. 31, pp. 854–864. doi: 10.1007/BF02030104.
  12. Giger F.M. Horizontal wells production techniques in heterogeneous reservoirs. Proc. SPE Middle East Oil Technical Conf. Exhib., Bahrain, Mar. 1985, 1985, pp. 239–246. doi: 10.2118/13710-MS.
  13. Barenblatt G.I., Entov V.M., Ryzhik V.M. Dvizhenie zhidkostei i gazov v prirodnykh plastakh [Movement of Liquids and Gases in Natural Reservoirs]. Moscow, Nedra, 1984. 211 p. (In Russian)
  14. Loitsyanskii L.G. Mekhanika zhidkosti i gaza [Mechanics of Liquids and Gases]. Moscow-Leningrad, Gostekhizdat, 1950. 676 p. (In Russian)
  15. Abramovits M., Stigan I. Spravochnik po spetsial'nym funktsiyam [A Handbook of Special Functions]. Moscow, Nauka, 1979. 832 p. (In Russian)
  16. Whittaker E.T., Watson G.N. Kurs sovremennogo analiza [A Course of Modern Analysis]. Pt. 2: The transcendental functions. Moscow, Gos. Izd. Fiz.-Mat. Lit., 1963. 501 p. (In Russian)
  17. Mazo A.B., Potashev K.A., Khamidullin M.R. Mathematical model of fluid filtration to a multistage hydraulically fractured horizontal well. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2015, vol. 157, no. 4, pp. 133–148. (In Russian)

The content is available under the license Creative Commons Attribution 4.0 License.