### Post-graduate student Department of Computer Tecnologies in Automatics and Control Systems Mathematical modelling of diffusion processes in the shale gas extracting technology realization

Андрій Петрович Олійник, Лідія Омелянівна Штаєр, Оксана Ігорівна Клапоущак

#### Abstract

The article studies the problem of evaluation of the penetration of substances into the environment during implementation of technology of shale gas extraction, as well as the current state of research in this area. A mathematical model of diffusion based on two-dimensional unsteady diffusion equation in Cartesian coordinate system, taking into account a variable diffusion coefficient, initial and boundary conditions that take into account features of the technology of shale gas extraction. A numerical method of the model realization based on the method of variable directions was suggested. The stability of corresponding difference schemes using spectral characteristics of stability was studied.

The method of mathematical formalization of boundary conditions that permit to simulate various intensities and duration of emissions was suggested. A numerical algorithm for solving the problem was developed and its implementation was carried out in the form of complex applications. For the model parameters test calculations were performed that showed correspondence of simulation results with the actual physical picture of the process. The geometric characteristics of the zone of penetration of substances in the test environment were determined. The features of filtration of substances with the variable diffusion coefficient in a region and the various intensity of substances emission were studied. The value of the zone of emission of substances in the studied model area was assessed, the directions for future research were determined

#### Keywords

Shale gas; diffusion; equations of mathematical physics; stability of calculations; sweep method; concentration

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