On non-uniqueness of pressures in problems of fluid filtration in fractured-porous media. (English) Zbl 1531.65108
Summary: The paper proposes a new approach to the numerical solution of the singular Neumann problem for a system of elliptic equations, in which, for example, the problem of filtration of a two-phase incompressible fluid in a fractured-porous medium is reduced within the framework of the dual porosity model. The one-dimensionality of the kernel of the corresponding operator is established, after which an extended generalized mixed formulation is considered in which the singularity is eliminated by introducing some additional bilinear form. The extended formulation is approximated by the mixed finite element method, and the properties of the resulting system of linear algebraic equations with a saddle point matrix are established. Moreover, some numerical examples illustrate the theoretical results.
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 76S05 | Flows in porous media; filtration; seepage |
| 76T06 | Liquid-liquid two component flows |
| 76M20 | Finite difference methods applied to problems in fluid mechanics |
| 35J48 | Higher-order elliptic systems |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
| 35Q35 | PDEs in connection with fluid mechanics |
Keywords:
fractured porous media; double porosity; mixed generalized formulation; mixed finite element method; saddle point system; matrix kernelReferences:
| [1] | Barenblatt, G. I.; Zheltov, I. P.; Kochina, I. N., Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks, J. Appl. Math. Mech., 24, 5, 1286-1303 (1960) · Zbl 0104.21702 |
| [2] | Warren, J. E.; Root, P. J., The behavior of naturally fractured reservoirs, Soc. Petrol. Eng. J., 3, 3, 245-255 (1963) |
| [3] | Kazemi, H.; Merrill, L. S.; Porterfield, K. L.; Zeman, P. R., Numerical simulation of water-oil flow in naturally fractured reservoirs, Soc. Petrol. Eng. J., 16, 6, 317-326 (1976) |
| [4] | Gilman, J. R.; Kazemi, H., Improved calculations for viscous and gravity displacement in matrix blocks in dual porosity simulators, Soc. Petrol. Eng. J., 40, 1, 60-70 (1988) |
| [5] | Douglas, J.; Arbogast, T., Dual porosity models for flow in naturally fractured reservoirs, (Cushman, J. H., Dynamics of Fluids in Hierarchical Porous Media (1990), Academic Press: Academic Press London), 177-221 |
| [6] | Ivanov, M. I.; Kremer, I. A.; Laevsky, Y. M., On the streamline upwind scheme of solution to the filtration problem, Sib. Elect. Math. Rep., 16, 757-776 (2019) · Zbl 1422.76126 |
| [7] | Ivanov, M. I.; Kremer, I. A.; Laevsky, Yu. M., Solving the pure Neumann problem by a mixed finite element method, Numer. Anal. Appl., 15, 4, 316-330 (2022) |
| [8] | Krjizhek, M., On approximation of the Neumann problem by the penalty method, Appl. Math., 38, 6, 459-469 (1993) · Zbl 0795.65075 |
| [9] | Bochev, P.; Lehoucq, R. B., On the finite element solution of the pure Neumann problem, SIAM Rev., 47, 1, 50-66 (2005) · Zbl 1084.65111 |
| [10] | Dai, X., Finite element approximation of the pure Neumann problem using the iterative penalty method, Appl. Math. Comput., 186, 1367-1373 (2007) · Zbl 1117.65153 |
| [11] | Ivanov, M. I.; Kremer, I. A.; Urev, M. V., Solving the pure Neumann problem by a finite element method, Numer. Anal. Appl., 12, 4, 433-446 (2019) |
| [12] | Ivanov, M. I.; Kremer, I. A.; Laevsky, Yu. M., A computational model of fluid filtration in fractured porous media, Numer. Anal. Appl., 14, 2, 126-144 (2021) · Zbl 1507.76106 |
| [13] | 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 Univ. Ser. Fiz.-Mat. Nauki, 160, 1, 165-182 (2018), (in Russian, abstract in English) |
| [14] | Brezzi, F.; Fortin, M., Mixed and Hybrid Finite Element Methods (1991), Springer-Verlag: Springer-Verlag New York · Zbl 0788.73002 |
| [15] | Raviart, P.-A.; Thomas, J. M., A mixed finite element method for second order elliptic problems, (Galligani, I.; Magenes, E., Mathematical Aspects of the Finite Element Method. Mathematical Aspects of the Finite Element Method, Lectures Notes in Math, vol. 606 (1977), Springer-Verlag: Springer-Verlag New York) · Zbl 0362.65089 |
| [16] | Girault, V.; Raviart, P.-A., Finite Element Methods for Navier-Stokes Eequations (1986), Springer-Verlag: Springer-Verlag Heidelberg · Zbl 0585.65077 |
| [17] | Boffi, D.; Brezzi, F.; Fortin, M., Mixed Finite Element Methods and Applications (2013), Springer: Springer Berlin, Heidelberg · Zbl 1277.65092 |
| [18] | Ciarlet, P. G., The Finite Element Method for Elliptic Problems (1978), North-Holland: North-Holland Amsterdam · Zbl 0383.65058 |
| [19] | Sheldon, J. W.; Zondek, B.; Cardwell, W. T., One-dimensional, incompressible, noncapillary, two-phase fluid flow in a porous medium, petroleum transactions, AIME, 216, 290-296 (1959) |
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