Non-isothermal filtration problem: Two-temperature computational model

The article proposes a new computational model of non-isothermal filtration of a two-phase incompressible fluid. From the point of view of applications, we are talking about the displacement of oil by water when hot water enters from an injection well. The specificity of the proposed model is its two-temperature formulation. The two-temperature formulation is understood as thermal heterogeneity, in which at each point of the domain under consideration the temperatures of the two-phase fluid and porous blocks are determined, and the thermal interaction between the two continua is indicated. The mathematical model is presented in a mixed formulation in the form of a system of first-order equations in terms of four scalar functions (liquid pressure, water saturation, liquid and porous medium temperatures) and three vector functions (total liquid velocity and heat conductive fluxes of the liquid and porous medium). The spatial approximation is based on a combination of mixed FEM and centered FVM. The time approximation consists of using an explicit-implicit scheme with upwinding. In particular, IMPES-type method is used for the filtration equations, and the energy equations explicitly consider convective transfer with the choice of time step according to the CFL condition. For heat exchange between the fluid and the porous medium, both explicit and implicit approximations are used. It is shown that the stability condition of the explicit scheme is significantly weaker than the CFL conditions for convective flows in the mass and energy conservation laws at an accuracy coinciding with the accuracy of the implicit scheme. Also, the two-temperature model made it possible to study the role of heat conductive transfer in a liquid.