Analytical solution to the problem of injection or reduction of the formation pressure in the reservoir with a fracture

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Abstract

The problem of injection of Newtonian fluid at a constant flow rate through an injection well into an initially undisturbed infinite reservoir with an erosive vertical main fracture of constant width is considered. Using the Laplace transform method, analytical solutions are obtained for the pressure fields in the fracture and reservoir, the flow velocity in the fracture, as well as the equations for fluid trajectories in the reservoir and in the main fracture are derived. The solutions obtained are also applicable to the problem of fluid withdrawal into a production well intersected by a vertical main fracture. Nonstationary two-dimensional pressure fields in the reservoir, as well as the pressure and velocity fields in the fracture, are constructed.

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About the authors

A. M. Il’yasov

OOO "RKh-BashNIPIneft"

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

V. N. Kireev

Ufa University of Science and Technology

Email: kireev@anrb.ru
Russian Federation, Ufa

References

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2. Fig. 1. The scheme of the problem statement.

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3. Fig. 2. The contour of integration.

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4. Fig. 3. Pressure change in the crack at time points (a–b) – t = 3, 6 and 12 h for different injection rates Q0. Crack width w = 10-4 m. Reservoir permeability k = 1 mD: 1-3 – Q0 = 10, 50, 100 m3/day.

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5. Fig. 4. Change in the flow velocity along the crack length at time points (a–b) – t = 3, 6 and 12 h for different injection rates Q0. Crack width w = 10-4 m. Reservoir permeability k = 1 mD: 1-3 – Q0 = 10, 50, 100 m3/day.

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6. Fig. 5. Pressure propagation in the formation at moments (I–III) – t = 3, 6 and 12 h. Crack width w = 10-4 m. Reservoir permeability k = 1 mD: (a–b) – Q0 = 10, 50, 100 m3/day.

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7. Fig. 6. Pressure propagation in the formation at moments (I–III) – t = 3, 6 and 12 h. Crack width w = 10-4 m. Injection rate Q0 = 50 m3/day: (a–b) – k = 1, 10, 100 mD.

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8. Fig. 7. Pressure propagation in the reservoir at moments (I–III) – t = 3, 6 and 12 h. Reservoir permeability k = 1 mD. Injection rate Q0 = 50 m3/day: (a–b) – w = 5 × 10-5, 10-4, 1.5 × 10-4 M.

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