M.V. Vasilyeva*, V.I. Vasilyev**, A.A. Krasnikov***, D.Ya. Nikiforov****
North-Eastern Federal University, Yakutsk, 677000 Russia
E-mail: *vasilyevadotmdotv@gmail.com, **vasvasil@mail.ru, ***aleksey.a.krasnikov@gmail.com, ****dju92@mail.ru
Received January 31, 2017
Abstract
The models of single-phase fluid filtration in the fractured medium have been considered. Fractures have a significant impact on filtration processes, because they act as highly conductive channels. The mathematical model has been described by a parabolic pressure equation. Two approaches to flow approximation in fractures have been discussed: Approach 1 (by defining the nonhomogeneous coefficient for a cell occupied by the fracture); Approach 2 (by using a discrete model of fractures). Both approaches enable the explicit flow simulation in fractures by means of grid methods. Approximation of the problem has been performed using the method of finite differences and the method of finite elements. Numerical comparison of the two methods based on the model two-dimensional problem has been carried out. The results of simulation for the three-dimensional case have been presented.
Keywords: mathematical simulation, single-phase fluid flow, filtration, fractured porous media, nonhomogeneous coefficients, discrete model of fractures, method of finite differences, method of finite elements
Acknowledgments. The study was supported by the Russian Foundation for Basic Research (project no. 17-01-00732 a) and the megagrant of the Government of the Russian Federation (project no. 14.Y26.31.0013).
Figure Captions
Fig. 1. Fractured porous medium.
Fig. 2. The computational grid for the fractured porous medium with explicit account of the flow in fractures: on the left – Approach 1; on the right – Approach 2.
Fig. 3. The computational grid for the fractured porous medium with explicit account of the flow in fractures: on the left – Approach 1; on the right – Approach 2.
Fig. 4. The results of the numerical solution of the model problem using finite-difference approximation for various time layers: m = 1, 5 and 20: above – Approach 1a; in the center – Approach 2: below – Approach 1b.
Fig. 5. The results of the numerical solution of the model problem using the finite-element approximation for various time layers: m = 1, 5 and 20: above – Approach 1a; below – Approach 2.
Fig. 6. The numerical comparison of Approaches 1 and 2. Error at various moments of time: on the left – method of finite differences; on the left – method of finite elements.
Fig. 7. The computational grid for the two-dimensional discrete model of fractures.
Fig. 8. The distribution of the pressure field for the two-dimensional problem at various moments of time: 1 h, 1 day, and 7 days.
Fig. 9. The computational grid and geometry of fractures for the three-dimensional problem of filtration in the fractured porous medium.
Fig. 10. The distribution of the pressure field at various moments of time in the three-dimensional case: 6 h, 12 h, and 1 day.
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For citation: Vasilyeva M.V., Vasilyev V.I., Krasnikov A.A., Nikiforov D.Ya. Numerical simulation of single-phase fluid flow in fractured porous media. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2017, vol. 159, no. 1, pp. 100–115. (In Russian)
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