Self-similar solution of the problem of hydrate formation in snow massifs

A mathematical model of the process of injection of hydrate forming gas (methane) into the snow massif saturated with the same gas is constructed. It is assumed that the initial temperature of the “snow+gas” system is lower than the melting point of ice, and pressure is less than the equilibrium pressure of phase transitions of the “snow+gas+hydrate” system. The value of temperature of the injected gas lies above the equilibrium temperature of phase transitions for the “snow+gas+hydrate” system. It is shown that, depending on the initial condition of the «snow+gas» system and the gas injection intensity, it is possible to distinguish three zones in the filtration field, namely the near-field area saturated with snow and gas, the intermediate area, in which gas, snow and hydrate are in the state of phase equilibrium, and the distant area filled with gas and snow. Accordingly, two front boundaries are introduced: the boundary between the distant and intermediate zones, where the transition of snow into the structure of hydrate begins, and the boundary between the near-field and intermediate zones, on which the hydrate formation process ceases. Self-similar solutions describing the temperature and pressure of fields and the distribution of saturations of snow, hydrate and gas saturation in the massif are constructed. Analytical solutions are obtained for the near-field and distant regions. For the intermediate zone, the system consisting of three ordinary differential equations in self-similarity coordinates is obtained. The numerical realization of the problem was carried out using the Runge-Kutta fourth order method and the shooting method. In the numerical experiment, it has been established that the volume formation zone of the hydrate decreases with increasing temperature of the injected gas. It is also shown that as the permeability of the massif increases the intermediate zone becomes wider, and at the same time the value of saturation of hydrates on the nearest boundary decreases. It has been found that with the growth of initial snow accumulation in the massif the heated zone is narrowed, and the greatest extent of the volume hydrate formation region is observed in the snow massifs having low temperature.