Métodos de elementos finitos híbridos estáveis e estabilizados para escoamentos miscíveis em meios porosos heterogêneos

Abstract

Miscible displacement in porous media is modeled mathematically by coupling the Darcy problem with transport. These equations are strongly linked, especially in cases with adverse mobility ratio. In addition to the nonlinearity and the strong coupling between the equations, there are still numerical difficulties related to the predominance of convective effects, heterogeneity of the porous media and the compatibility between approximation spaces. In this context, we propose the study of stable and stabilized hybrid finite element methods capable of overcoming the typical difficulties of this problem and, in some cases, ensuring local mass conservation. The so-called stable methods are characterized by the use of Raviart-Thomas approximation spaces, while the stabilized methods incorporate residual terms into the formulation. In this way, we developed mixed hybrid methods for the Darcy problem and for the transport problem, aiming to generate formulations for the transport equation compatible with the formulations for the Darcy problem, where the stable methods for Darcy are combined with the stable ones and the same goes for stabilized methods. Therefore, to maintain this correspondence between the formulations, the choice for Lagrange multipliers, in both approaches, is of the same nature, associated with the trace of the scalar variable. The stable and stabilized methods for the transport problem are combined with an upwind scheme, used to soften the effects of the predominantly convective regime, and discretized in time by a second-order approach based on the Crank-Nicolson method. In this context, the stable and stabilized methods employing continuous and discontinuous multipliers are tested and validated, through convergence studies, independently in order for the Darcy and transport problems and in a coupled way for the Darcy-transport problem. In addition, scenarios are simulated in homogeneous and heterogeneous environments assuming unitary and adverse mobility ratios where the proposed methods demonstrate an ability to capture the viscous fingers generation phenomena and fractal patterns generated by this type of approach, in addition to presenting a reduction in the oscillatory effects from the dominant convection that is inherent to these applications.