Effective equations for energy transport in petroleum reservoirs

Thermal enhanced oil recovery methods such as hot water flooding, steam drive, cyclic steam injection, steam-assisted gravity drainage, in-situ combustion, and electrical heating involve multiphase and multicomponent thermal flow with phase change and thermal displacement which improves the oil mobility inside the porous rock. Mostly, these methods are modeled by applying mass, momentum, and energy balances at the macroscale assuming that fluids and rock grains get instantaneously the same temperature, namely thermal equilibrium. In this paper, the macroscale energy equations are theoretically derived via the volume averaging method, considering: (1) local thermal non-equilibrium with and without phase change (four-equations models), (2) pseudo-local thermal non-equilibrium (two-equations model), and (3) local thermal equilibrium (one-equation model). The starting point for the derivation of the equations is the governing energy equations at the microscale for a four-phase system (water, oil, gas, and rock). The different effective equations are obtained as well as the closure problems allowing the numerical computation of the associated effective coefficients, for instance, the effective thermal conductivity tensor contained in the one-equation model was numerically calculated and compared against available experimental data, getting reasonable accuracy. The aforementioned models will allow studying the effect of considering the local thermal non-equilibrium behavior on thermal oil recovery methods.