Two-Phase Newtonian Returning Fluid Model via a System of Variational–Hemivariational Inequalities

A steady-state Oseen problem for incompressible fluid with nonsmooth unilateral and frictional type boundary conditions is studied. The problem serves as a new mathematical model for the gas–liquid two-phase fluid problem in shale gas development. First the physical setting of a two-phase returning fluid model is described. Then, the weak formulation which consists of a coupled system of two elliptic variational–hemivariational inequalities and an elliptic equation is delivered. The main existence theorem is proved by solving three auxiliary problems and applying a Schauder–Tychonoff fixed-point argument. Finally, a partial pressure-recovery result is provided.