A thermodynamically consistent, two time-scale theory for multiphase flow in porous media
A two time-scale, thermodynamically consistent model for multiphase flow in a swelling porous medium is developed using three spatial scale hybrid mixture theory. The mesoscale of the medium consists of particles and two bulk phases, for example liquid and vapor. The particles are a combination of vicinal liquid and solid which may swell or shrink as a result of interaction with the other bulk phases; an example of such a medium is clay platelets and water. The theory defines connected and unconnected bulk phases of liquid and vapor phases at the mesoscale to create a dual porosity model at the macroscale. The incorporation of the unconnected and connected phases is useful for modeling unsaturated swelling systems; however, the vapor phase could also he replaced with a liquid that is immiscible with the first. The macroscale solid phase volume fraction is refined in this work from previous hybrid mixture theoretic approaches and fully utilized in the field equations and the constitutive theory. ^ The macroscale equations for each of the phases are presented with bulk regions separated into connected and unconnected regions. A constitutive theory is derived through the use of the entropy inequality for the mixture. Generalized Darcy's laws and the final set of field equations for the system are presented. Comparison with previous hybrid mixture theoretic results for the parallel flow model on similar systems is presented. Such a parallel flow model only considers the unconnected bulk regions and is therefore appropriate for drainage while the current system can be useful for both imbibition and drainage. ^
John H. Cushman, Purdue University.