Numerical simulation of single-phase fluid flow in fractured porous media

M. V. Vasilyeva ; V. I. Vasilyev ; A. A. Krasnikov ; D. Ya. Nikiforov

Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 159 (2017) no. 1, pp. 100-115

Résumé

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.

Détail

Keywords: mathematical simulation, single-phase fluid flow, fractured porous media, nonhomogeneous coefficients, discrete model of fractures, method of finite differences, method of finite elements.
Mots-clés : filtration

Bibliographie

[1] Aziz K., Settari A., Petroleum Reservoir Simulation, Appl. Sci. Publ. Ltd., London, 1979, 476 pp.

[2] Bear J., Dynamics of Fluids in Porous Media, Elsevier, N.Y., 1972, 764 pp. | Zbl

[3] Chen Z., Huan G., Ma Y., Computational Methods for Multiphase Flows in Porous Media Siam, Southern Methodist Univ., Dallas, Texas, 2006, 569 pp. | MR

[4] Vasilyeva M. V., Vasilyev V. I., Timofeeva T. S., “Numerical solution of the convective and diffusive transport problems in a heterogenous porous medium using finite element method”, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 158, no. 2, 2016, 243–261 (In Russian)

[5] Talonov A., Vasilyeva M., “On numerical homogenization of shale gas transport”, J. Comput. Appl. Math., 301 (2016), 44–52 | DOI | MR | Zbl

[6] Vabishchevich P., Vasil'eva M., “Iterative solution of the pressure problem for the multiphase filtration”, Math. Modelling and Analysis, 17:4 (2012), 532–548 | DOI | MR | Zbl

[7] Zaslavskii M. Yu., Tomin P. Yu., On modeling of multiphase flows in fractured media with application to history matching problem, Preprint no. 45, IMP im. M. V. Keldysh, Moscow, 2010, 20 pp. (In Russian)

[8] Tomin P. Yu., Mathematical simulation of filtration processes in fractured reservoirs, Cand. Phys.-Math. Diss., Moscow, 2011, 147 pp. (In Russian)

[9] Gong B., Karimi-Fard M., Durlofsky L. J., “Upscaling discrete fracture characterizations to dual-porosity, dual-permeability models for efficient simulation of flow with strong gravitational effects”, Soc. Pet. Eng. J., 13:1 (2008), 58–67

[10] Karami-Fard M., Gong B., Durlofsky L. J., “Generation of coarse-scale continuum flow models from detailed fracture characterizations”, Water Resour. Res., 42:10 (2006), Art. W10423, 13 pp.

[11] Karami-Fard M., Durlofsky L. J., Aziz K., “An efficient discrete fracture model applicable for general purpose reservoir simulators”, Soc. Pet. Eng. J., 9:02 (2004), 227–236

[12] Barenblatt G. I., Zheltov I. P., Kochina I. N., “Basic concept in the theory of seepage of homogeneous liquids in fissured rocks (strata)”, J. Appl. Math. Mech., 24:5 (1960), 1286–1303 | DOI | Zbl

[13] Arbogast T., Douglas J. (Jr.), Hornung U., “Derivation of the double porosity model of single phase flow via homogenization theory”, SIAM J. Math. Anal., 21:4 (1990), 823–836 | DOI | MR | Zbl

[14] Kazemi H., Merrill L. S. (Jr.), Porterfield K. L., Zeman P. R., “Numerical simulation of water-oil flow in naturally fractured reservoirs”, Soc. Pet. Eng. J., 16:6 (1976), 317–326 | DOI

[15] Warren J. E., Root P. J., “The behavior of naturally fractured reservoirs”, Soc. Pet. Eng. J., 3:3 (1963), 245–255 | DOI

[16] Vabishchevich P. N., Grigoriev A. V., “Numerical simulation of a fluid flow in anisotropic fractured porous media”, Sib. Zh. Vychisl. Mat., 19:1 (2016), 61–74 (In Russian) | MR

[17] Lee S. H., Jensen C. L., Lough M. F., “Efficient finite-difference model for flow in a reservoir with multiple length-scale fractures”, Soc. Pet. Eng. J., 5:3 (2000), 268–275

[18] Li L., Lee S. H., “Efficient field-scale simulation of black oil in a naturally fractured reservoir through discrete fracture networks and homogenized media”, SPE Reservoir Evaluation Engineering, 11:4 (2008), 750–758 | DOI

[19] Karimi-Fard M., Firoozabadi A., “Numerical simulation of water injection in 2d fractured media using discrete-fracture model”, SPE Annual Technical Conference and Exhibition, 2001, SPE Paper 71615

[20] Kim J.-G., Deo M. D., “Finite element, discrete-fracture model for multiphase flow in porous media”, AIChE J., 46:6 (2000), 1120–1130 | DOI | MR

[21] Efendiev Y., Lee S., Li G., Yao J., Zhang N., “Hierarchical multiscale modeling for flows in fractured media using generalized multiscale finite element method”, Int. J. Geomath., 6:2 (2015), 141–162 | DOI | MR | Zbl

[22] Akkutlu I. Y., Efendiev Y., Vasilyeva M., “Multiscale model reduction for shale gas transport in fractured media”, Comput. Geosci., 20:5 (2016), 953–973 | DOI | MR | Zbl

[23] Samarskii A. A., Theory of Difference Schemes, Nauka, Moscow, 1989, 616 pp. (In Russian) | MR

[24] Samarskii A. A., Nikolaev E. S., Methods for Solving Grid Equations, Nauka, Moscow, 1978, 589 pp. (In Russian) | MR

[25] Brenner S., Scott R., The Mathematical Theory of Finite Element Methods, Springer-Verlag, N.Y., 2007, XVIII, 400 pp. | MR