Hybrid Mixed Methods Applied to Miscible Displacements with Adverse Mobility Ratio

We propose stable and locally conservative hybrid mixed finite element methods to approximate the Darcy system and convection-diffusion problem, presented in a mixed form, to solve miscible displacements considering convective flows with adverse mobility ratio. The stability of the proposed formulations is achieved due to the choice of non-conforming Raviart-Thomas spaces combined to upwind scheme for the convection-dominated regimes, where the continuity conditions, between the elements, are weakly enforced by the introduction of Lagrange multipliers. Thus, the primal variables of both systems can be condensed in the element level leading a positive-definite global problem involving only the degrees of freedom associated with the multipliers. This approach, compared to the classical conforming Raviart-Thomas, present a reduction of the computational cost because, in both problems, the Lagrange multiplier is associated with a scalar field. In this context, a staggered algorithm is employed to decouple the Darcy problem from the convection-diffusion mixed system. However, both formulations are solved at the same time step, and the time discretization adopted for the convection-diffusion problem is the implicit backward Euler method. Numerical results show optimal convergence rates for all variables and the capacity to capture the formation and the propagation of the viscous fingering, as can be seen in the comparisons of the simulations of the Hele-Shaw cell with experimental results of the literature.