Convergence - Divergence of the Finite Volume Method for the Fluid and Heat Flow in Heterogeneous Porous Rocks

Abstract

Fluid velocity has an important impact in the Finite Volume numerical method (FVM) used to model fluid and energy flows in heterogeneous porous media. This formulation is based on the energy flow conservation equations under local thermal non-equilibrium conditions (LTNE). The heat transfer from the solid matrix to the moving non-isothermal fluid inside the pores is simulated for several velocities and global different heat transfer coefficients. During the numerical simulation of coupled heat/mass flow in a heterogeneous porous system it is necessary to average highly variable physical parameters at the boundaries between different domains represented geometrically in the computational mesh of the FVM. This can be effectively achieved using appropriate average techniques at the contact interfaces of each domain. The averaging process should correctly represent the behaviour of the fluid velocity crossing different areas of the reservoir. Many numerical divergence problems arise from the fluid velocity value and the interfacial interactions at the boundaries of different continua. The averages have a decisive influence on the numeric results of the simulation. A very stable, convergent averaging scheme for the FVM is presented herein and compared to new analytical diffusion-convection models. Especial attention is paid to the dynamic process of cold water (50°C) injected into a geothermal reservoir at higher temperature (350°C). The fluid temperature profile is calculated when the fluid migrates at constant speed through a permeable corridor from a fluid injection point to an extraction zone. The physical reason for the divergence of a numerical model, and the exploration of the range of validity of the LTNE hypothesis during the injection of cold water into a hot reservoir are both explained.