Modelagem matemática e computacional do transporte bifásico de fluidos com gravidade e do transporte e retenção de partículas em meios porosos

Abstract

In this work, we develop mathematical and computational models capable of accurately quantifying the phenomena of fluid transport and particle transport and retention in porous media. For fluid transport, we consider the water-oil immiscible two-phase flow with gravitational effects, described by the mass conservation of the phases together with Darcy’s law. In the scenarios studied, the resulting model is a partial differential equation with a non-linear and non-convex flow function. In its general form, the equation is known in the literature as the Buckley-Leverett equation. Furthermore, we consider particle transport and retention based on the theory of multiple retention mechanisms. In the model, we quantify retention phenomena by filtration and adsorption kinetics and adsorption isotherms. Additionally, we obtain simplified models, in one-dimensional form, of the systems of governing equations. We then developed analytical solutions for the onedimensional models using the method of characteristics and the Lax and Oleinik entropy conditions. An important contribution of this work is the development of novel analytical solutions for pure gravitational segregation scenarios. For computational modeling, we apply the high-order finite volume method central-upwind to solve the two-dimensional transport equations. Moreover, we solve the retention kinetics using the 3rd order RungeKutta method. We then propose several numerical simulations in order to compare the analytical solutions developed with the numerical approximations obtained. It is important to highlight that there is no formal proof in the literature of the convergence of the central-upwind method for the physical solution of equations with non-convex flow functions. In this context, the results show that the method is capable of capturing the developed analytical solutions with accuracy and stability. Finally, we use the analytical and numerical solutions to quantify the empirical parameters of the model by adjusting experimental data available in the literature.