Decoupled, Linear, and Energy Stable Finite Element Method for the Cahn-Hilliard-Navier-Stokes-Darcy Phase Field Model

In this paper, we consider the numerical approximation for a phase field model of the coupled two-phase free flow and two-phase porous media flow. This model consists of Cahn—Hilliard—Navier—Stokes equations in the free flow region and Cahn—Hilliard—Darcy equations in the porous media region that are coupled by seven interface conditions. The coupled system is decoupled based on the interface conditions and the solution values on the interface from the previous time step. A fully discretized scheme with finite elements for the spatial discretization is developed to solve the decoupled system. In order to deal with the difficulties arising from the interface conditions, the decoupled scheme needs to be constructed appropriately for the interface terms, and a modified discrete energy is introduced with an interface component. Furthermore, the scheme is linearized and energy stable. Hence, at each time step one need only solve a linear elliptic system for each of the two decoupled equations. Stability of the model and the proposed method is rigorously proved. Numerical experiments are presented to illustrate the features of the proposed numerical method and verify the theoretical conclusions. © 2018 Society for Industrial and Applied Mathematics

Mixed framework for Darcy-Stokes mixtures

textWe consider the system of equations arising from mantle dynamics introduced by McKenzie (J. Petrology, 1985). In this multi-phase model, the fluid melt velocity obeys Darcy's law while the deformable "solid" matrix is governed by a highly viscous Stokes equation. The system is then coupled through mass conservation and compaction relations. Together these equations form a coupled Darcy-Stokes system on a continuous single-domain mixture of fluid and matrix. The porosity φ, representing the relative volume of fluid melt to the bulk volume, is assumed to be much smaller than one. When coupled with solute transport and thermal evolution in a time-dependent problem, the model transitions dynamically from a non-porous single phase solid to a two-phase porous medium. Such mixture models have an advantage for numerical approximation since the free boundary between the one and two-phase regions need not be determined explicitly. The equations of mantle dynamics apply to a wide range of applications in deep earth physics such as mid-ocean ridges, subduction zones, and hot-spot volcanism, as well as to glacier dynamics and other two-phase flows in porous media. Mid-ocean ridges form when viscous corner flow of the solid mantle focuses fluid toward a central ridge. Melt is believed to migrate upward until it reaches the lithospheric "tent" where it then moves toward the ridge in a high porosity band. Simulation of this physical phenomenon required confidence in numerical methods to handle highly heterogeneous porosity as well as the single-phase to two-phase transition. In this work we present a standard mixed finite element method for the equations of mantle dynamics and investigate its limitations for vanishing porosity. While stable and optimally convergent for porosity bounded away from zero, the stability estimates we obtain suggest, and numerical results show, the method becomes unstable as porosity approaches zero. Moreover, the fluid pressure is no longer a physical variable when the fluid phase disappears and thus is not a good variable for numerical methods. Inspired by the stability estimates of the standard method, we develop a novel stable mixed method with uniqueness and existence of solutions by studying a linear degenerate elliptic sub-problem akin to the Darcy part of the full model: [mathematical equation], where a and b satisfy a(0)=b(0)=0 and are otherwise positive, and the porosity φ ≥ 0 may be zero on a set of positive measure. Using scaled variables and mild assumptions on the regularity of φ, we develop a practical mass-conservative method based on lowest order Raviart-Thomas finite elements. Finally, we adapt the numerical method for the sub-problem to the full system of equations. We show optimal convergence for sufficiently smooth solutions for a compacting column and mid-ocean ridge-like corner flow examples, and investigate accuracy and stability for less regular problemsComputational Science, Engineering, and Mathematic

A fully-coupled discontinuous Galerkin method for two-phase flow in porous media with discontinuous capillary pressure

In this paper we formulate and test numerically a fully-coupled discontinuous Galerkin (DG) method for incompressible two-phase flow with discontinuous capillary pressure. The spatial discretization uses the symmetric interior penalty DG formulation with weighted averages and is based on a wetting-phase potential / capillary potential formulation of the two-phase flow system. After discretizing in time with diagonally implicit Runge-Kutta schemes the resulting systems of nonlinear algebraic equations are solved with Newton's method and the arising systems of linear equations are solved efficiently and in parallel with an algebraic multigrid method. The new scheme is investigated for various test problems from the literature and is also compared to a cell-centered finite volume scheme in terms of accuracy and time to solution. We find that the method is accurate, robust and efficient. In particular no post-processing of the DG velocity field is necessary in contrast to results reported by several authors for decoupled schemes. Moreover, the solver scales well in parallel and three-dimensional problems with up to nearly 100 million degrees of freedom per time step have been computed on 1000 processors

Challenges and progress on the modelling of entropy generation in porous media: a review

Depending upon the ultimate design, the use of porous media in thermal and chemical systems can provide significant operational advantages, including helping to maintain a uniform temperature distribution, increasing the heat transfer rate, controlling reaction rates, and improving heat flux absorption. For this reason, numerous experimental and numerical investigations have been performed on thermal and chemical systems that utilize various types of porous materials. Recently, previous thermal analyses of porous materials embedded in channels or cavities have been re-evaluated using a local thermal non-equilibrium (LTNE) modelling technique. Consequently, the second law analyses of these systems using the LTNE method have been a point of focus in a number of more recent investigations. This has resulted in a series of investigations in various porous systems, and comparisons of the results obtained from traditional local thermal equilibrium (LTE) and the more recent LTNE modelling approach. Moreover, the rapid development and deployment of micro-manufacturing techniques have resulted in an increase in manufacturing flexibility that has made the use of these materials much easier for many micro-thermal and chemical system applications, including emerging energy-related fields such as micro-reactors, micro-combustors, solar thermal collectors and many others. The result is a renewed interest in the thermal performance and the exergetic analysis of these porous thermochemical systems. This current investigation reviews the recent developments of the second law investigations and analyses in thermal and chemical problems in porous media. The effects of various parameters on the entropy generation in these systems are discussed, with particular attention given to the influence of local thermodynamic equilibrium and non-equilibrium upon the second law performance of these systems. This discussion is then followed by a review of the mathematical methods that have been used for simulations. Finally, conclusions and recommendations regarding the unexplored systems and the areas in the greatest need of further investigations are summarized

A Framework for Modeling Subgrid Effects for Two-Phase Flows in Porous Media

In this paper, we study upscaling for two-phase flows in strongly heterogeneous porous media. Upscaling a hyperbolic convection equation is known to be very difficult due to the presence of nonlocal memory effects. Even for a linear hyperbolic equation with a shear velocity field, the upscaled equation involves a nonlocal history dependent diffusion term, which is not amenable to computation. By performing a systematic multiscale analysis, we derive coupled equations for the average and the fluctuations for the two-phase flow. The homogenized equations for the coupled system are obtained by projecting the fluctuations onto a suitable subspace. This projection corresponds exactly to averaging along streamlines of the flow. Convergence of the multiscale analysis is verified numerically. Moreover, we show how to apply this multiscale analysis to upscale two-phase flows in practical applications

Finite element analysis of non-isothermal multiphase porous media in dynamics

This work presents a mathematical and a numerical model for the analysis of the thermo-hydro-mechanical (THM) behavior of multiphase deformable porous materials in dynamics. The fully coupled governing equations are developed within the Hybrid Mixture Theory. To analyze the THM behavior of soil structures in the low frequency domain, e.g. under earthquake excitation, the u-p-T formulation is advocated by neglecting the relative acceleration of the fluids and their convective terms. The standard Bubnov-Galerkin method is applied to the governing equations for the spatial discretization, whereas the generalized Newmark scheme is used for the time discretization. The final non-linear and coupled system of algebraic equations is solved by the Newton method within the monolithic approach. The formulation and the implemented solution procedure are validated through the comparison with other finite element solutions or analytical solutions