A mixture theory model for the forced convection flow through an unsaturated wellbore

This work studies the dynamics of the filling up of a rigid cylindrical porous matrix by a Newtonian fluid and the heat transfer associated phenomenon. A mixture theory approach is employed to obtain a preliminary local description for non-isothermal flows through a wellbore. The mixture consists of three overlapping continuous constituents, representing the porous matrix (solid constituent), the Newtonian fluid (liquid constituent) and an inert gas included to account for the compressibility of the mixture as a whole. Assuming the convective flow on radial direction only, a set of four non-linear partial differential equations describes the problem. Its hydrodynamic part—a non-linear hyperbolic system—is approximated by means of a Glimm’s scheme, combined with an operator splitting technique, while an implicit finite difference scheme is used to simulate the thermal part. Some examples illustrate the proposed strategy.