Mathematical modeling of the two-phase fluid flow in inhomogeneous fractured porous media using the double porosity model and finite element method

V.I. Vasilyev*, M.V. Vasilyeva**, A.V. Grigorev***, G.A. Prokopiev****

 M.K. Ammosov North-Eastern Federal University, Yakutsk, 677000 Russia

E-mail: *vasvasil@mail.ru, **vasilyevadotmdotv@gmail.com, ***re5itsme@gmail.com, ****reilroot@gmail.com

Received December 18, 2017

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Abstract

Numerical simulation of the two-phase fluid flow in a fractured porous media using the double porosity model with a highly inhomogeneous permeability coefficient has been studied. A system of equations has been presented for the case of two-phase filtration without capillary and gravitational effects, which is a connected system of equations for pressure and saturation in a porous medium that contains a system of cracks. Different variants of specifying the flow functions between the porous medium and cracks have been considered. The numerical implementation for velocity and pressure approximation is based on the finite element method. To discretize the saturation equation, the classical Galerkin method with counter-flow approximation has been used. The results of numerical calculations for the model problem using various interflow functions have been presented.

Keywords: two-phase filtration, inhomogeneous media, fractured-porous media, double porosity model, interflow functions, finite element method, numerical stabilization

Acknowledgments. The study was supported by the Russian Foundation for Basic Research (project no. 17-01-00732) (V.I. Vasilyev: theory, problem statement), Megagrant of the Government of the Russian Federation (grant no. 14.Y26.31.0013) (M.V. Vasilyeva: numerical algorithm; G.A. Prokopiev: nonlinear case implementation), and by the Russian Science Foundation (project no. 17-71-10106) (A.V. Grigorev: linear case implementation, experiments).

Figure Captions

Fig. 1. Porous medium and fracture network.

Fig. 2. Absolute permeability of fractures, k^f on the logarithmic scale.

Fig. 3. Pressure distribution using the generalized interflow for Variant 1 (σ₁ = 20). On the left – fractures; on the right – porous matrix.

Fig. 4. Saturation at different points of time t_m = 0.001, 0.01, and 0.05 (from top to bottom) using the generalized interflow for Variant 1 (σ₁ = 20). On the left – fractures; on the right – porous matrix.

Fig. 5. Pressure distribution using the generalized interflow for Variant 2 (σ₂ = 100). On the left – fractures; on the right – porous matrix.

Fig. 6. Saturation at different points of time t_m =  0.001, 0.01, and 0.05 (from top to bottom) using the generalized interflow for Variant 2 (σ₂ = 100). On the left – fractures; on the right – porous matrix.

Fig. 7. Saturation at different points of time t_m = 0.0001, 0.0015, and 0.02 (from top to bottom) using the generalized interflow for the nonlinear case. On the left – fractures; on the right – porous matrix.

Fig. 8. The integral characteristic for saturation at the right-hand boundary Γ₂ for different variants of the interflow functions for Variant 1 (σ₁ = 20). On the left – fractures; on the right – porous matrix.

Fig. 9. The integral characteristic for saturation at the right-hand boundary Γ₂ for different variants of the interflow functions for Variant 2 (σ₂ = 100). On the left – fractures; on the right – porous matrix.

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For citation: Vasilyev  V.I.,  Vasilyeva M.V., Grigorev A.V.,  Prokopiev G.A. Mathematical modeling of the two-phase fluid flow in inhomogeneous fractured porous media using the double porosity model and finite element method. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2018, vol. 160, no. 1, pp. 165–182. (In Russian)

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