Variational inequalities for modeling flow in heterogeneous porous media with entry pressure

Abstract

One of the driving forces in porous media flow is the capillary pressure. In standard models, it is given depending on the saturation. However, recent experiments have shown disagreement between measurements and numerical solutions using such simple models. Hence, we consider in this paper two extensions to standard capillary pressure relationships. Firstly, to correct the nonphysical behavior, we use a recently established saturation-dependent retardation term. Secondly, in the case of heterogeneous porous media, we apply a model with a capillary threshold pressure that controls the penetration process. Mathematically, we rewrite this model as inequality constraint at the interfaces, which allows discontinuities in the saturation and pressure. For the standard model, often finite-volume schemes resulting in a nonlinear system for the saturation are applied. To handle the enhanced model at the interfaces correctly, we apply a mortar discretization method on nonmatching meshes. Introducing the flux as a new variable allows us to solve the inequality constraint efficiently. This method can be applied to both the standard and the enhanced capillary model. As nonlinear solver, we use an active set strategy combined with a Newton method. Several numerical examples demonstrate This work was supported in part by IRTG NUPUS. the efficiency and flexibility of the new algorithm in 2D and 3D and show the influence of the retardation term.Keywords Porous media · Entry pressure · Variational inequality · Mortar · Active set strategy MotivationFlow processes in porous media involving two immiscible fluids need to be understood and predicted when dealing with subsurface hydrosystems or industrial applications. For example, in the unsaturated zone, the spatial distribution of the water and air phase, as well as their fluxes, serves as a basis for modeling transport of contaminants, such as pesticides or heavy metals (e.g., [8]). As examples for industrial applications in two-phase flow, the movement of fluids through a filter or the infiltration of ink into paper (see [24]) can be considered. All these applications have in common that the porous media structure is, in general, highly heterogeneous. The challenges are to combine the complex multiphase flow processes with the heterogeneity distribution of the porous media properties.The physical-mathematical model underlying simulations of two-phase flow on the Darcy scale usually requires a constitutive relationship between (wetting phase) saturation S w and the capillary pressure p c . Traditionally, one assumes that this relationship is determined under quasistatic or steady-state conditions but can also be applied to any transient flow processes fulfilling the Reynolds number criterion. However, recently, some works have questioned this assumption (see, e.g., [25]). The authors were able to improve numerical simulation results by applying a model

Semismooth Newton methods for variational problems with inequality constraints

“…As the first example, we consider an incompressible two-phase flow process in heterogeneous porous media [35]. Assuming that we have no source/sink term and no gravity, the mass balance for the wetting phase (α = w) and the nonwetting phase (α = n) within the body…”

“…Due to the jump in the material parameters, we need additional relations to describe the behavior of the flow at the interface Γ. Here, we restrict ourselves to the (physically more interesting) case that the nonwetting phase is penetrating from the subdomain with higher permeability to the one with higher entry pressure, i.e., from master to slave side; for the numerical treatment of the reverse case, we refer to [35] and Remark 4.1. As derived in [38], the pressure p α is only continuous along Γ if the phase α is present on both sides.…”

“…For the spatial discretization, we apply a node-centered finite volume scheme [39] by introducing a shape-regular triangulation T r h of the subdomain Ω r , r ∈ {s, m}, into simplices or quadrilaterals/hexahedrals and the corresponding dual mesh T * ,r h (see [35,39] for details). We remark that the meshes do not need to be matching at the interface Γ.…”

Error estimates for a mixed finite element discretization of a two-phase porous media flow model with dynamic capillarity

“…Numerical schemes for two phase flow through heterogeneous media are discussed in [35] for cases without an entry pressure. For situations including an entry pressure, variational inequality approaches have been considered in [36]. Further, we refer to [21] for coupling conditions between heterogeneous blocks under the dynamic effect.…”

Two-Phase Flow in Porous Media: Dynamic Capillarity and Heterogeneous Media

“…This situation is similar to the case analyzed in Cuesta and Pop (2009), where the interface is replaced by a discontinuity in the initial conditions. Also, variational inequality approaches have been considered in Helmig et al (2009) for situations including an entry pressure. However, the conditions are simply postulated and no derivation is presented.…”