Carina Bringedal

Abstract

We consider a model for the flow of two immiscible fluids in a two-dimensional thin strip of varying width. This represents an idealization of a pore in a porous medium. The interface separating the fluids forms a freely moving interface in contact with the wall and is driven by the fluid flow and surface tension. The contact-line model incorporates Navier-slip boundary conditions and a dynamic and possibly hysteretic contact angle law. We assume a scale separation between the typical width and the length of the thin strip. Based on asymptotic expansions, we derive effective models for the two-phase flow. These models form a system of differential algebraic equations for the interface position and the total flux. The result is Darcy-type equations for the flow, combined with a capillary pressure–saturation relationship involving dynamic effects. Finally, we provide some numerical examples to show the effect of a varying wall width, of the viscosity ratio, of the slip boundary condition as well as of having a dynamic contact angle law.

Abstract

Drops on a free-flow/porous-medium-flow interface have a strong influence on the exchange of mass, momentum and energy between the two macroscopic flow regimes. Modeling droplet-related pore-scale processes in a macro-scale context is challenging due to the scale gap, but might be rewarding due to relatively low computational costs. We develop a three-domain approach to model drop formation, growth, detachment and film flow in a lower-dimensional interface domain. A simple upscaling technique allows to compute the drop-covered interface area fraction which affects the coupling fluxes. In a first scenario, only drop formation, growth and detachment are taken into account. Then, spreading and merging due to lateral fluxes are considered as well. The simulation results show that the impact of these droplet-related processes can be captured. However, extensions are necessary to represent the influence on the free flow more precisely.

Abstract

Mineral precipitation and dissolution processes in a porous medium can alter the structure of the medium at the scale of pores. Such changes make numerical simulations a challenging task as the geometry of the pores changes in time in an apriori unknown manner. To deal with such aspects, we here adopt a two-scale phase-field model, and propose a robust scheme for the numerical approximation of the solution. The scheme takes into account both the scale separation in the model, as well as the non-linear character of the model. After proving the convergence of the scheme, an adaptive two-scale strategy is incorporated, which improves the efficiency of the simulations. Numerical tests are presented, showing the efficiency and accuracy of the scheme in the presence of anisotropies and heterogeneities.

Abstract

We propose an efficient numerical strategy for solving non-linear parabolic problems defined in a heterogeneous porous medium. This scheme is based on the classical homogenization theory and uses a locally mass-conservative formulation at different scales. In addition, we discuss some properties of the proposed non-linear solvers and use an error indicator to perform a local mesh refinement. The main idea is to compute the effective parameters in such a way that the computational complexity is reduced but preserving the accuracy. We illustrate the behavior of the homogenization scheme and of the non-linear solvers by performing two numerical tests. We consider both a quasi-periodic example and a problem involving strong heterogeneities in a non-periodic medium.

Abstract

The modelling and simulation of the unsaturated flow or the flow of two immiscible fluid phases in a porous medium is challenging as this flow takes place through the pores of the medium, which form a highly complex domain. Next to the complexity of the domain, a major challenge is to account for the interface separating the fluids, or the unsaturated fluid from the inert filling part, as the location of this interface is not known a-priori. The evolution of this interface depends on the flow of both fluids and of the surface tension. Moreover, the surface tension may depend on the concentration of a surfactant dissolved in one fluid phase. In this work, such aspects are taken into account, and effective, Darcy-scale models are derived based on the known physics at the pore scale. In this sense a thin strip is used as the representation of a single pore in the porous medium. The Darcy-scale models are derived for various regimes, accounting for different pore-scale processes. Numerical examples show that the upscaled models are a good approximation of the transversal average of the solution to the pore-scale models, as the ratio of the width and the length of the pore approaches zero.

Abstract

Bringedal, CarinaWe present a phase-field model for single-phase flow and reactive transport where ions take part in mineral precipitation/dissolution reactions. The evolving interface between fluid and mineral is approximated by a diffuse interface, which is modeled using an Allen--Cahn equation. As the original Allen--Cahn equation is not conservative, we apply a reformulation ensuring conservation of the phase-field variable and address the sharp-interface limit of the reformulated model. This model is implemented using a finite volume scheme and the discrete conservation of the reformulated Allen--Cahn equation is shown. Numerical examples show how the discrete phase-field variable is conserved up to the chemical reaction.