Analytical Solution of the Problem on Bi-Linear Flow in a Formation with a Finite Auto-Fracture

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Abstract

The problem of unsteady bilinear flow of a single-phase Newtonian fluid in a formation with a finite auto-fracture connecting an injection and production well is considered. The wells simultaneously begin to operate at constant pressures in an initially undisturbed infinite formation with a vertical main fracture of constant width. Using the Laplace transform method, analytical solutions were obtained for the pressure fields in the fracture and formation, as well as the flow velocity in the fracture. An approximate model is considered that uses a self-similar solution to the problem of filtration of an incompressible fluid in an elastic half-space with constant pressure at the boundary to simulate filtration leaks. It was found that for a number of model parameters a simple analytical solution of the approximate model gives acceptable results.

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About the authors

A. M. Ilyasov

RN-BashNIPIneft, LLC

Author for correspondence.
Email: amilyasov67@gmail.com
Russian Federation, Ufa

V. N. Kireev

Ufa University of Science and Technology

Email: kireevvn@uust.ru
Russian Federation, Ufa

References

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  3. Khabibullin I.L., Khisamov A.A. Unsteady flow through a porous stratum with hydraulic fracture // Fluid Dyn., 2019, vol. 54, no. 5, pp. 594–602.
  4. Khabibullin I.L., Khisamov A.A. Modeling of unsteady fluid filtration in a reservoir with a hydraulic fracture // J. Appl. Mech. Tech. Phys., 2022, vol. 63, pp. 652–660.
  5. Il’yasov A.M., Kireev V.N. Unsteady flow in a reservoir with a main fracture crossing an injection or production well // J. Appl. Mech. Tech. Phys., 2023, vol. 64, no. 5, pp. 840–852.
  6. Il’yasov A.M., Kireev V.N. Analytical solution to the problem of injection or reduction of the formation pressure in the reservoir with a fracture // fluid dyn., 2024, vol. 59, no. 2, pp. 189–201.
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2. Fig. 1. Schematic of the problem statement

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3. Fig. 2. Integration contour

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4. Fig. 3. Pressure variation in the fracture at time points (a-c) - t = 1, 24 and 72 h for different reservoir permeabilities k. The fracture width is w = 10-4 m: 1-3 - k = 1, 10, 100 mD

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5. Fig. 4. Pressure variation in the fracture at the moments of time (a-c) - t = 1, 24 and 72 h for different values of fracture width w. Permeability of the formation is k = 1 mD: 1-3 - w = 5 ∙ 10-5, 10-4, 1.5 ∙ 10-4 m

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6. Fig. 5. Variation of velocity in the fracture at the moments of time (a-c) - t = 1, 24 and 72 h for different permeabilities of the formation k. The fracture width is w = 10-4 m: 1-3 - k = 1, 10, 100 mD

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7. Fig. 6. Variation of velocity in the fracture at the moments of time (a-c) - t = 1, 24 and 72 h for different values of fracture width w. Permeability of the formation is k = 1 mD: 1-3 - w = 5 ∙ 10-5, 10-4, 1.5 ∙ 10-4 m

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8. Fig. 7. Variation of fluid leakage rate into the reservoir along the fracture length at time points (a-c) - t = 1, 24 and 72 h for different reservoir permeabilities k. Fracture width is w = 10-4 m: 1-3 - k = 1, 10, 100 mD

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9. Fig. 8. Pressure propagation in the reservoir at moments (I-III) - t = 1, 24 and 72 h. Fracture width is w = 10-4 m: (a-c) - k = 1, 10, 100 mD

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10. Fig. 9. Pressure propagation in the reservoir at moments (I-III) - t = 1, 24 and 72 h. Permeability of the formation is equal to k = 1 mD: (a-c) - w = 5 ∙ 10-5, 10-4, 1.5 ∙ 10-4 m

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11. Fig. 10. Comparison of velocity in the fracture at different time moments (a-c) - t = 1, 24 and 72 h at different fracture widths: thick lines - velocity calculated by formula (3.18), thin lines - velocity calculated by simplified model, formula (4.6). Formation permeability k = 1 mD: 1-3 - w = 5 ∙ 10-5, 10-4, 1.5 ∙ 10-4 m

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