Analysis of an interior penalty discontinuous Galerkin scheme for two phase flow in porous media with dynamic capillarity effects

We present an interior penalty discontinuous Galerkin scheme for two-phase flow with dynamic capillary pressure effects. The mass-conservation laws are approximated directly, without the introduction of a global pressure. We prove existence and convergence of the scheme and obtain error-estimates for sufficiently smooth data

A class of degenerate pseudo-parabolic equations : existence, uniqueness of weak solutions, and error estimates for the Euler-implicit discretization

In this paper, we investigate a class of degenerate pseudo-parabolic equations. Such equations model two-phase flow in porous media where dynamic effects are included in the capillary pressure. The existence and uniqueness of a weak solution are proved, and error estimates for an Euler implicit time discretization are obtained

Regularization schemes for degenerate Richards equations and outflow conditions

We analyze regularization schemes for the Richards equation and a time discrete numerical approximation. The original equations can be doubly degenerate, therefore they may exhibit fast and slow diffusion. Additionally, we treat outflow conditions that model an interface separating the porous medium from a free flow domain. In both situations we provide a regularization with a non-degenerate equation and standard boundary conditions, and discuss the convergence rates of the approximations

Analysis and upscaling of a reactive transport model in fractured porous media involving nonlinear a transmission condition

We consider a reactive transport model in a fractured porous medium. The particularity appears in the conditions imposed at the interface separating the block and the fracture, which involves a nonlinear transmission condition. Assuming that the fracture has thickness e, we analyze the resulting problem and prove the convergence towards a reduced model in the limit e ¿ 0. The resulting is a model defined on an interface (the reduced fracture) and acting as a boundary condition for the equations defined in the block. Using both formal and rigorous arguments, we obtain the reduced models for different flow regimes, expressed through a moderate, or a high Péclet number. Keywords: Fractured porous media; Upscaling; Reactive transport; Nonlinear transmission condition

Analysis and upscaling of a reactive transport model in fractured porous media involving nonlinear a transmission condition

We consider a reactive transport model in a fractured porous medium. The particularity appears in the conditions imposed at the interface separating the block and the fracture, which involves a nonlinear transmission condition. Assuming that the fracture has thickness e, we analyze the resulting problem and prove the convergence towards a reduced model in the limit e ¿ 0. The resulting is a model defined on an interface (the reduced fracture) and acting as a boundary condition for the equations defined in the block. Using both formal and rigorous arguments, we obtain the reduced models for different flow regimes, expressed through a moderate, or a high Péclet number. Keywords: Fractured porous media; Upscaling; Reactive transport; Nonlinear transmission condition

Convergence analysis of mixed numerical schemes for reactive in a porous medium

This paper deals with the numerical analysis of an upscaled model describing the reactive flow in a porous medium. The solutes are transported by advection and diffusion and undergo precipitation and dissolution. The reaction term and, in particular, the dissolution term has a particular, multi-valued character, which leads to stiff dissolution fronts. We consider the Euler implicit method for the temporal discretization and the mixed finite element for the discretization in time. More precisely, we use the lowest order Raviart-Thomas elements. As an intermediate step we consider also a semi-discrete mixed variational formulation (continuous in space). We analyse the numerical schemes and prove the convergence to the continuous formulation. Apart from the proof for the convergence, this also yields an existence proof for the solution of the model in mixed variational formulation. Numerical experiments are performed to study the convergence behavior