Difference schemes based on the support operator method are considered as applied to fluid dynamics in underground collectors containing gas hydrate deposits. A system of mass and energy balances describing fluid dynamics in a porous medium containing gas hydrate deposits is given. A dissipative hydrate equation is derived that determines the thermodynamic evolution of the parameters of the system. It is shown that the jumps in specific volumes and internal energy occurring in phase transitions play a crucial role in the stability of the evolution of the system in the dissipation thermodynamic module of the system. A family of rotation-invariant difference schemes of the support operator method on unstructured meshes is constructed for numerical computations. The schemes are tested on a series of model problems. Their numerical solutions are presented.

The purpose of this paper is to present a one-dimensional model that simulates the thermo-physical processes for methane hydrate decomposition in porous media. The mathematical model consists of equations for the conservation of energy, gas, and liquid as well as the thermodynamic equilibrium equation for temperature and pressure $\left(P-T\right)$ in the hydrate stability region. The developed model is solved numerically by using the implicit finite difference technique on the grid system, which correctly describes the appearance of phase, latency, and boundary conditions. The Newton–Raphson method was employed to solve a system of nonlinear algebraic equations after defining and preparing the Jacobean matrix. Additionally, the proposed model describes the decomposition of methane hydrate by thermal catalysis of the components that make up the medium through multiple phases in porous media. In addition, the effect of thermodynamic processes during the hydrate decomposition on the pore saturation rate with hydrates a7nd water during different time periods was studied in a one-dimensional model. Finally, in a one-dimensional model over various time intervals, $t=1,\hspace{0.17em}10,\hspace{0.17em}50\hspace{0.17em}\mathrm{s}$, the pressure and temperature distributions during the decomposition of methane hydrates are introduced and investigated. The obtained results include more accurate solutions and are consistent with previous models based on the analysis of simulations and system stability.

The work is devoted to numerical modeling of the processes of heat and mass transfer of a two-phase fluid in the environment of a production well, which is necessary for monitoring the development of fractured-porous reservoirs. This work proposes an efficient approach to constructing a solution to the problem. To solve the problem, a model of the “double medium” type is used, where the pore part of the reservoir is considered as the first medium, and the system of natural fractures is considered as the second medium. For the resulting mathematical model, the difference schemes with time weights are constructed based on the algorithm of splitting by physical processes, which ensure the correctness and consistency of fluxes in the fracture system and the pore reservoir. In the numerical solution, the approximations of differential operators obtained in the framework of the finite difference method are used. For the parallel implementation of the developed numerical approach, the domain decomposition method and the matrix sweep algorithm are chosen. The program implementation is made using the MPI standard. Computational experiments are carried out, the results of which confirm the effectiveness of the developed numerical algorithm and its parallel implementation. In numerical experiments, the distributions of pressure and temperature near an operating production well are obtained, on the basis of which it is possible to adjust the operation of wells in order to increase production.