Formation of the region of phase transformations in the case of the conversion of CH$_4$ hydrate into CO$_2$ hydrate in a porous medium

We study the problem with an unknown moving boundary of the conversion of $\mathrm{CH}_4$ hydrate into $\mathrm{CO}_2$ hydrate in a porous medium. We assume that in the initial state, methane hydrate coexists with water and free methane in the thermodynamic equilibrium state. Calculations show that the assumptions of the existence of the front conversion mode and of the constancy of saturations before the front in the mathematical model lead to the methane hydrate supercooling. We propose a generalized mathematical model that takes into account phase transitions in an extended region before the front. We find a self-similar solution of the problem in the linear approximation. Our made calculations show that the carbon dioxide injection with the conversion of methane hydrate into carbon dioxide hydrate is accompanied by the formation of methane hydrate before the front. We show that an amount of formed methane hydrate before the front increases with increasing injection pressure and permeability. We find that the hydrate formation in the mixture region increases the conversion front velocity.