Two-phase Problem Research Articles

Spatial segregation of multiple species: A singular limit approach

The spatial segregation of the populations occurs commonly in ecology. One typical way to understand this phenomenon is to consider strong competition in some species. In this paper, we shall consider multiple-species competition-diffusion models. Under the condition that some interspecies competition rates are large, we show that the segregation phenomenon occurs. Furthermore, we derive some two-phase Stefan-like problems appearing as the singular limit, which may provide some modeling interpretation for free boundary problems studied in the literature.

On a weighted two-phase boundary obstacle problem

On a weighted two-phase boundary obstacle problem

PHYSICS-INFORMED NEURAL NETWORKS BASED ON SEQUENTIAL TRAINING FOR CO2 UTILIZATION AND STORAGE IN SUBSURFACE RESERVOIR

CO<sub>2</sub> utilization and storage (CCUS) simulation in subsurface reservoirs with complex heterogeneous structures necessitates a model that can capture multiphase compositional flow and transport. The governing equations are highly nonlinear due to the complex thermodynamic behavior, which involves the appearance and disappearance of multiple phases. Accurate simulation of these processes necessitates the use of stable numerical methods. While machine learning (ML) approaches have been used to solve a variety of nonlinear computational problems, a new approach based on physics-informed neural networks (PINNs) has been proposed for solving partial differential equations (PDEs). Unlike typical ML algorithms that require a large dataset for training, PINNs can train the network with unlabeled data. The applicability of this method has been explored for multiphase flow and transport in porous media. However, for nonlinear hyperbolic transport equations, the solution degrades significantly. This work proposes sequential training PINNs to simulate two-phase transport in porous media. The main concept is to retrain the neural network to solve the PDE over successive time segments rather than train for the entire time domain simultaneously. We observe that sequential training can capture the solution more accurately concerning the standard training for conventional two-phase problems. Furthermore, we extend the sequential training approach for compositional problems in which nonlinearity is more significant due to the complex phase transition. Our approach was tested on miscible and immiscible test cases and showed higher accuracy than the standard training method.

Local Well-Posedness for the Magnetohydrodynamics in the Different Two Liquids Case

We consider the free boundary problem of MHD in the multi-dimensional case. This problem describes the motion of two incompressible fluids separated by a closed interface under the action of a magnetic field. This problem is overdetermined, and we find an equivalent system of equations which is uniquely solvable locally in time in the Lp-Lq maximal regularity class, where 1<p,q<∞ and 2/p+N/q<1. As a result, the original two-phase problem for the MHD is solvable locally in time.

Second law analysis of MHD convection of a radiating nanofluid within the gap between two inclined concentric pipes

This paper theoretically examined the inherent irreversibility in hydromagnetic mixed convection of a radiating adjustable viscosity nanofluid between two concentric inclined cylindrical pipes. Thermodynamics’ first and second laws are incorporated into the two-phase nanofluid flow model problem to explore the repercussions of thermophoresis, Brownian motion, inclination angle, Joule heating, buoyancy forces, viscous dissipation, thermal radiation and entropy generation rate on the overall flow structure with temperature and nanoparticles concentration distribution. The nonlinear model equations of differential types are obtained and numerically addressed through shooting quadrature in conjunction with the Runge–Kutta–Fehlberg integration scheme. Relevant outcomes are graphically represented and discussed. The findings indicate that a rise in the inclination angle lessens the buoyancy effects and diminishes the entropy generation rate in the annular region of the concentric pipes. Within the annulus, the irreversibility due to heat and mass transfer dominates the entropy generation rate. In contrast, an upsurge in magnetic field intensity decreases the entropy generation rate and the Bejan number.

Fully nonlinear Hamilton–Jacobi equations of degenerate type

We examine Hamilton–Jacobi equations driven by fully nonlinear degenerate elliptic operators in the presence of superlinear Hamiltonians. By exploring the Ishii–Jensen inequality, we prove that viscosity solutions are locally Lipschitz-continuous, with estimates depending on the structural conditions of the problem. We close the paper with an application of our findings to a two-phase free boundary problem.

A sharp interface framework based on the inviscid Godunov-Peshkov-Romenski equations: Simulation of evaporating fluids

A core aspect in the simulation of compressible fluid flow is the solution of Riemann problems. The solution is well understood for single-phase flows and is used to define numerical fluxes for finite volume and discontinuous Galerkin schemes. However, the solution of two-phase Riemann problems with phase transitions is still an active field of research. This is due to the strong coupling of complex thermodynamics and flow dynamics that leads to severe challenges in finding solutions. In this paper, we consider the Riemann problem based on the Godunov-Peshkov-Romenski formulation of the continuum flow equations and propose a novel strategy to model two-phase flows with phase transitions. We take into account heat transfer, which is modeled in this system of equations by hyperbolic terms. Our solver is applied to two-phase shock tubes with an evaporating fluid. It is validated against molecular dynamics simulations for the Lennard-Jones fluid with truncated and shifted potential.

Fractional Stefan Problem Solving by the Alternating Phase Truncation Method

The aim of this paper is the adaptation of the alternating phase truncation (APT) method for solving the two-phase time-fractional Stefan problem. The aim was to determine the approximate temperature distribution in the domain with the moving boundary between the solid and the liquid phase. The adaptation of the APT method is a kind of method that allows us to consider the enthalpy distribution instead of the temperature distribution in the domain. The method consists of reducing the whole considered domain to liquid phase by adding sufficient heat at each point of the solid and then, after solving the heat equation transformed to the enthalpy form in the obtained region, subtracting the heat that has been added. Next the whole domain is reduced to the solid phase by subtracting the sufficient heat from each point of the liquid. The heat equation is solved in the obtained region and, after that, the heat that had been subtracted is added at the proper points. The steps of the APT method were adapted to solve the equations with the fractional derivatives. The paper includes numerical examples illustrating the application of the described method.

Riemann solvers for phase transition in a compressible sharp-interface method

In this paper, we consider Riemann solvers with phase transition effects based on the Euler–Fourier equation system. One exact and two approximate solutions of the two-phase Riemann problem are obtained by modelling the phase transition process via the theory of classical irreversible thermodynamics. Closure is obtained by appropriate Onsager coefficients for evaporation and condensation. We use the proposed Riemann solvers in a sharp-interface level-set ghost fluid method to couple the individual phases with each other. The proposed sharp-interface method is validated against molecular dynamics data of evaporating Lennard–Jones truncated and shifted fluid. We further study the effects of phase transition on a shock-drop interaction with the novel approximate Riemann solvers.

Generalized finite difference method based meshless analysis for coupled two-phase porous flow and geomechanics

This paper applies GFDM to analyze coupled two-phase flow and geomechanics for the first time, to our knowledge, is also the first meshless solver for two-phase fluid-solid coupling problems in porous media. Firstly, the implicit GFDM-based discrete schemes of the hyperbolic two-phase flow equation and elliptic stress equilibrium equation coupled by Biot's poroelastic theory are derived. Then the nonlinear solver based on Newton's method is used to solve the fluid pressure, water saturation, and displacement at each node simultaneously. Two technical details are found critical to the implementation of this method. The one is using different radii of the node influence domain to discretize the flow equation and the stress equilibrium equation respectively, and the other one is the selection criterion of the two radii of the node influence domain, which improves the implementation flexibility of the multi-physics coupling calculation, reduces the dissipation error of the calculation results of the convection-dominated water saturation distribution, and ensure a low computational cost. Several numerical test cases are implemented to analyze the computational efficiency and accuracy, and illustrate that GFDM can achieve good computational performance in case of regular domains, irregular domains, and different point clouds.Overall, this work may provide an important reference for building a GFDM-based general-purpose numerical simulator for multi-physics flow problems in porous media.

Remarks on the Control of Two-Phase Stefan Free-Boundary Problems

This paper concerns the null controllability of the two-phase 1D Stefan problem with distributed controls. This is a free-boundary problem that models solidification or melting processes. In each phase, a parabolic equation, completed with initial and boundary conditions, must be satisfied; the phases are separated by a phase change interface, where an additional free-boundary condition is imposed (the so-called Stefan condition). We assume that two localized sources of heating/cooling controls act on the system (one in each phase). We prove the following local null controllability result: the temperatures can be steered to zero and, simultaneously, the interface can be steered to a prescribed location provided the initial data and the interface position are sufficiently close to the targets. The ingredients of the proofs are a compactness-uniqueness argument (which gives appropriate observability estimates adapted to constraints) and a fixed-point formulation and resolution of the controllability problem (which gives the result for the nonlinear system). We also prove a negative result corresponding to the case where only one control acts on the system and the interface does not collapse to the boundary.

Inverse two-phase nonlinear Stefan and Cauchy-Stefan problems: A phase-wise approach

We develop a novel phase-wise sequential numerical approach based on the method of fundamental solutions (MFS) for inverse two-phase nonlinear Stefan and Cauchy-Stefan problems in one dimension (1D). By treating each phase independently, the inverse two-phase nonlinear Stefan problem splits into two single-phase inverse problems: an inverse nonlinear boundary identification problem and an inverse linear one-phase Stefan problem. Along with the reconstruction of boundary data, the simultaneous reconstruction of the boundary and initial data is also considered. Numerical investigations show the robustness and efficiency of the proposed method in reconstructing the data.

Two-phase inverse Stefan problems solved by heat polynomials method

In this paper, two-phase inverse Stefan problem are investigated using the method of heat polynomials. The approximate solution of the problem is presented in the form of linear combination of heat polynomials. We present numerical results illustrating an application of the heat polynomials method for typical benchmark test examples. Due to ill-posedness of the problem, the regularization will be taken into account. Numerical results are discussed and compared with results obtained by another method.

Higher order imperfect interface models of conductive spherical interphase

This paper presents a study of the Bövik-Benveniste methodology for high order imperfect interface modeling of steady-state conduction problems involving coated spherical particles. Two types of imperfect interface models, that reduce the original three-phase configuration problem to the two-phase configuration problem, are discussed. In one model, the effect of the layer is accounted for via jumps in the field variables across the traces of its boundaries, while in the other via corresponding jumps across the trace of its mid-surface. Explicit expressions for the jumps are provided for both models up to the third order. The obtained higher order jump conditions are incorporated into the unit cell model of spherical particle composite. The multipole expansion method is used to derive the convergent series solutions to the corresponding boundary value problems. Numerical examples are presented to demonstrate that the use of higher order imperfect interface models allows for accurate evaluation of the local fields and overall conductivity of composites reinforced with coated spherical particles.

A new meshless method to solve the two-phase thermo-hydro-mechanical multi-physical field coupling problems in shale reservoirs

The efficient development of shale reservoirs relies on multi-stage hydraulic fracturing technology in horizontal wells involving the difficulty of multi-scale flow for either gas or water phases, which is characterized by the changes in the stress and temperature fields. The original simulation methods are unable to accurately solve the multi-physics field-coupled problems under such complex mechanical conditions or have low stability for two-phase gas-water problems. For this purpose, this study conducted the shale gas production simulation by the meshless finite difference method (GFDM) for the first time to ensure the stability and accuracy of calculations. Bases on the previous multi-sector coupled physical model for shale gas development and proposes a coupled thermo-hydro-mechanical multi-physical field mathematical model considering two-phase gas-water flow. The model is postulated based on different sectors considering various characteristics of gas/water flow, namely desorption attributes, deformation degree and heat transfer characteristics. This method was used to calculate the variation of the multi-physical field and compare the properties of gas-water flow within different reservoir conditions without any non-convergence situation during the calculation. The simulation results indicate that the stress field has a crucial effect on the yield, such that discard of its coupling causes the yield to be overestimated by 12.5%. In contrast, the temperature field has a marginal effect (less than 1%) and can be neglected according to the operating conditions. Furthermore, the stress field significantly characterizes the micro-fracture sector, which is also the major flow-contributor, such that the yield is overestimated by 11% in absence of its coupling, while the effect of stress field in the matrix sector is lower than 1%.

Homogenisation of a two-phase problem with nonlinear dynamic Wentzell-interface condition for connected–disconnected porous media

AbstractWe investigate a reaction–diffusion problem in a two-component porous medium with a nonlinear interface condition between the different components. One component is connected and the other one is disconnected. The ratio between the microscopic pore scale and the size of the whole domain is described by the small parameter $\epsilon$ . On the interface between the components, we consider a dynamic Wentzell-boundary condition, where the normal fluxes from the bulk domains are given by a reaction–diffusion equation for the traces of the bulk solutions, including nonlinear reaction kinetics depending on the solutions on both sides of the interface. Using two-scale techniques, we pass to the limit $\epsilon \to 0$ and derive macroscopic models, where we need homogenisation results for surface diffusion. To cope with the nonlinear terms, we derive strong two-scale convergence results.

Physics-informed neural network simulation of multiphase poroelasticity using stress-split sequential training

Physics-informed neural networks (PINNs) have received significant attention as a unified framework for forward, inverse, and surrogate modeling of problems governed by partial differential equations (PDEs). Training PINNs for forward problems, however, pose significant challenges, mainly because of the complex non-convex and multi-objective loss function. In this work, we present a PINN approach to solving the equations of coupled flow and deformation in porous media for both single-phase and multiphase flow. To this end, we construct the solution space using multi-layer neural networks. Due to the dynamics of the problem, we find that incorporating multiple differential relations into the loss function results in an unstable optimization problem, meaning that sometimes it converges to the trivial null solution, other times it moves very far from the expected solution. We report a dimensionless form of the coupled governing equations that we find most favorable to the optimizer. Additionally, we propose a sequential training approach based on the stress-split algorithms of poromechanics. Notably, we find that sequential training based on stress-split performs well for different problems, while the classical strain-split algorithm shows an unstable behavior similar to what is reported in the context of finite element solvers. We use the approach to solve benchmark problems of poroelasticity, including Mandel’s consolidation problem, Barry–Mercer’s injection-production problem, and a reference two-phase drainage problem. The Python-SciANN codes reproducing the results reported in this manuscript will be made publicly available at https://github.com/sciann/sciann-applications.

A phase field model for mass transport with semi-permeable interfaces

In this paper, a thermaldynamical consistent model for mass transfer across permeable moving interfaces is proposed by using the energy variation method. We consider a restricted diffusion problem where the flux across the interface depends on its conductance and the difference of the concentration on each side. The diffusive interface phase-field framework used here has several advantages over the sharp interface method. First of all, explicit tracking of the interface is no longer necessary. Secondly, interfacial conditions can be incorporated with a variable diffusion coefficient. Finally, topological changes of interfaces can be handed easily. A detailed asymptotic analysis confirms the diffusive interface model converges to the existing sharp interface model as the interface thickness goes to zero. An energy stable numerical scheme is developed to solve this highly nonlinear coupled system.Numerical simulations first illustrate the consistency of theoretical results on the sharp interface limit. Then a convergence study and energy decay test are conducted to ensure the efficiency and stability of the numerical scheme. To illustrate the effectiveness of our phase-field approach, several examples are provided, including a study of a two-phase mass transfer problem where droplets with deformable interfaces are suspended in a moving fluid.

An Augmented Lagrangian Model for Signal Segmentation

In this paper, we provide a new insight to the two-phase signal segmentation problem. We propose an augmented Lagrangian variational model based on Chan–Vese’s original one. Using both energy methods and PDE methods, we show, in the one-dimensional case, that the set of minimizers to the proposed functional contains only binary functions and it coincides with the set of minimizers to Chan–Vese’s one. This fact allows us to obtain two important features of the minimizers as a by-product of our analysis. First of all, for a piecewise constant initial signal, the jump set of any minimizer is a subset of the jump set of the given signal. Second, all of the jump points of the minimizer belong to the same level set of the signal, in a multivalued sense. This last property permits to design a trivial algorithm for computing the minimizers.

Quantitative stability estimates for a two-phase Serrin-type overdetermined problem

In this paper, we deal with an overdetermined problem of Serrin-type with respect to a two-phase elliptic operator in divergence form with piecewise constant coefficients. In particular, we consider the case where the two-phase overdetermined problem is close to the one-phase setting. First, we show quantitative stability estimates for the two-phase problem via a one-phase stability result. Furthermore, we prove non-existence for the corresponding inner problem by the aforementioned two-phase stability result.