Uniquely solvable and energy stable decoupled numerical schemes for the Cahn–Hilliard–Stokes–Darcy system for two-phase flows in karstic geometry

Abstract

We propose and analyze two novel decoupled numerical schemes for solving the Cahn-Hilliard-Stokes-Darcy (CHSD) model for two-phase flows in karstic geometry. In the first numerical scheme, we explore a fractional step method (operator splitting) to decouple the phase-field (Cahn-Hilliard equation) from the velocity field (Stokes-Darcy fluid equations). To further decouple the Stokes-Darcy system, we introduce a first order pressure stabilization term in the Darcy solver in the second numerical scheme so that the Stokes system is decoupled from the Darcy system and hence the CHSD system can be solved in a fully decoupled manner. We show that both decoupled numerical schemes are uniquely solvable, energy stable, and mass conservative. Ample numerical results are presented to demonstrate the accuracy and efficiency of our schemes.Keywords Cahn-Hilliard-Stokes-Darcy system · two phase flow · karstic geometry · interface boundary conditions · diffuse interface modelMany natural and engineering applications involve multiphase flows in karstic geometry, i.e., geometry with both conduit (or vug) and porous media [25]. Such kind of problems are intrinsically difficulty due to the multi-scale multi-physics nature. In [25], the authors utilized Onsager's extremum principle to derive a diffuse interface model, the so-called Cahn-Hilliard-Stokes-Darcy system (CHSD), for two-phase incompressible flows with matched densities in the karstic geometry. Existence and weak-strong uniqueness of weak solutions for the CHSD system have been proved recently in [28]. For complex systems like the CHSD model, efficient and accurate numerical schemes are highly desirable. There are several challenges associated with the system.First, due to the relative slow motion of fluid in porous media, long time simulations are needed in order to capture physically important phenomena. In particular, we would like to have numerical schemes that inherit, with modification if needed, the energy law of the continuous model. Second, the CHSD model, similar to all phase field model with a sharp interface limit, is stiff due to the existence of relatively steep transition regions.This stiffness leads to a severe time-step restriction if one adopts classical explicit time stepping. Third, the CHSD system involves at least three coupled physical processes: the dynamics of the phase field variable (governed by the Cahn-Hilliard equation), the fluid flow in the conduit (governed by the Stokes equation), and the fluid flow in the porous media (governed by the Darcy system). Efficient and accurate numerical schemes for each of the sub-models do exist. Therefore, it would be highly desirable to have numerical schemes that decouple these subsystems while maintaining the long time stability. Such decoupled schemes would reduce the complexity of the computation and allow the possibility of the utilization of legacy codes.In this paper, we introduce and analyze two novel decoupled schemes for the CHSD system. In particular, we show that both schemes are uniquely solvab...

A decoupled numerical method for two-phase flows of different densities and viscosities in superposed fluid and porous layers

“…based on (2.2). We inherit the idea from [14] that we can solve a Cahn-Hilliard equations on the whole domain Ω. This is an alternative to [28], where two Cahn-Hilliard equations are solved on Ω m and Ω c separately.…”

Error estimate of a decoupled numerical scheme for the Cahn-Hilliard-Stokes-Darcy system

“…Define the total energy of the system as follows: (1.16) where F (ϕ) = 1 4 (ϕ 2 −1) 2 . The CHSD system (1.1)-(1.15) obeys a dissipative energy law [5]:…”

“…Well-posedness of a variant of the CHSD model is studied in [13]. A decoupled unconditionally stable numerical algorithm for solving the CHSD system is proposed in [5]. Here we focus on the error analysis of a similar decoupled numerical scheme (cf.…”

“…We will focus on the error analysis of the following unconditionally energy stable scheme that decouples the computation of the Cahn-Hilliard flow from that of fluid equations, i.e., for a totally decoupled scheme; see the related descriptions in [5]. FGiven 0…”

“…To avoid a well-known difficulty associated with the coupling between the Cahn-Hilliard equation and the fluid motion, we make use of the operator-splitting in the numerical scheme, so that these two solvers are decoupled, which in turn would greatly improve the computational efficiency. The unique solvability and the energy stability have been proved in [5]. In this work, we carry out a detailed convergence analysis and error estimate for the fully discrete finite element scheme, so that the optimal rate convergence order is established in the energy norm, i.e.,, in the ∞ (0, T ; H 1 ) ∩ 2 (0, T ; H 2 ) norm for the phase variables, as well as in the ∞ (0, T ; H 1 ) ∩ 2 (0, T ; H 2 ) norm for the velocity variable.…”