# Conservative difference scheme for filtering problems in fractured media

• Vasilyeva Maria V., vasilyevadotmdotv@gmail.com M. K. Ammosov North-Eastern Federal University, Institute of Mathematics and Informatics, 42 Kulakovsky Street, Yakutsk 677000, Russia
• Vasil’ev Vasily I., vasvasil@mail.ru M. K. Ammosov North-Eastern Federal University, Institute of Mathematics and Informatics, 42 Kulakovsky Street, Yakutsk 677000, Russia
• Tyrylgin Aleksei A., koc9tk@mail.ru M. K. Ammosov North-Eastern Federal University, Institute of Mathematics and Informatics, 42 Kulakovsky Street, Yakutsk 677000, Russia
Keywords: fractured porous medium, multicontinuum models, dual porosity model, mixed dimensional formulation, numerical simulation, filtration problem, embedded fracture model, convervative difference scheme

### Abstract

We consider filtration problems in the fractured media that are necessary when modeling the processes of extracting hydrocarbons from unconventional reservoirs, geothermal fields development, underground disposal of radioactive waste in aquifers, etc. Fracture networks in such oil pools can exist on different scales and differ in the nature of their occurrence. We discuss a mathematical model of ﬂuid filtration in fractured porous media described by coupled equations of mixed dimension with assigning of a special ﬂow function. The problem approximation is constructed through the finite difference method on structured grids using the embedded fracture model, which makes possible creating grids for the porous medium matrix independently of the fracture network grid. The construction of a conservative difference scheme is given for the matrix of porous medium with the use of an integro-interpolation method and generalized for coupled equations describing mathematical models of multicontinuum with hierarchical representation of fracture networks. The results of the numerical implementation of the two-dimensional model problem are presented.

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How to Cite
Vasilyeva, M., Vasil’ev, V. and Tyrylgin, A. (&nbsp;) “Conservative difference scheme for filtering problems in fractured media”, Mathematical notes of NEFU, 25(4), pp. 84-101. doi: https://doi.org/10.25587/SVFU.2018.100.20556.
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Mathematical Modeling