Phase Flow Problems Research Articles

Effects of a spherical slip cavity filled with micropolar fluid on a spherical micropolar droplet

In this work, a two-fluid phase flow problem involving an axisymmetrical quasi-steady motion of a spherical micropolar droplet translating at a concentric point in a second non-mixable micropolar fluid within a spherical impermeable cavity with a slip surface is analysed under low Reynolds numbers. The two fluid phases that have a microstructure (micropolar fluid) are the case that is being focused on. The Stokes equations are solved inside and outside the droplet for the velocity fields. In addition, based on the concentric position, general solutions in terms of spherical coordinates are obtained. In this case, tangential couple stress and continuity of microrotation are used. For different cases, the normalised drag forces acting on the droplet are represented via graphs for different values of relative viscosity, droplet-to-cavity radii ratio, and the parameter that connects the tangential couple stress with microrotation. The normalised drag force is found to be a monotonically increasing function of the drop-to-cavity radii ratio. It is found that when the droplet-to-cavity radii ratio approaches zero, there is a very strong interaction between the droplet and the cavity. When comparing a solid sphere to a gas bubble, the normalised drag force is larger. Additionally, the results showed that permitting spin and slip at the cavity’s interior surface improved the wall correction factor influencing the droplet. The present study is important in the fields of natural, industrial, and biomedical processes such as raindrop formation, liquid–liquid extraction, suspension rheology, sedimentation, coagulation, and the motion of blood cells in an artery or vein.

Influence of installation height of a submersible mixer on solid‒liquid two‒phase flow field

With the increasingly severe situation of water pollution control, optimal design of the mixing flow field of submersible mixers and improving the mixing uniformity of activated sludge have become key research issues. At present, the research on the submersible mixer is mostly focused on water as the medium, and the flow field characteristics of solid-liquid two-phase flow, which is closer to the actual scene, still need more systematic research. This paper presented numerical simulations of the solid‒liquid two‒phase flow problem at various installation heights based on the coupled CFD‒DEM method in the Euler‒Lagrange framework. The velocity distribution, dead zone distribution, particles’ velocity development, particles’ mixing degree, and particles’ aggregation of the flow field were compared and analyzed for different installation heights. The results show that the flow field has two flow patterns: single‒ and double‒circulation, due to different installation heights, in which the velocity and turbulent kinetic energy of the flow field of the double‒circulation flow pattern are more uniform. The installation height affects the moment particles enter the impeller and the core jet zone, thus affecting the degree of particle mixing and the mixing time. The adjustment of the installation height also has an impact on particle aggregation. These findings indicate that the installation height significantly affects the flow field characteristics and the particle motion distribution. The coupled CFD‒DEM method can analyze the macroscopic phenomenon of the solid‒liquid two‒phase flow field of the submersible mixer from the scale of microscopic particles, which provides a theoretical approach for the optimal design of the mixing flow field. It can provide better guidance for engineering practice.

Well-posedness of the free boundary problem in incompressible MHD with surface tension

In this paper, we study the two phase flow problem with surface tension in the ideal incompressible magnetohydrodynamics. First, we prove the local well-posedness of the two phase flow problem with surface tension. Second, for the initial data satisfying a Syrovatskij type stability condition, we prove that as surface tension tends to zero, the solution of the two phase flow problem with surface tension converges to the solution of the two phase flow problem without surface tension.

Deep Learning of Two-Phase Flow in Porous Media via Theory-Guided Neural Networks

Summary A theory-guided neural network (TgNN) is proposed as a prediction model for oil/water phase flow in this paper. The model is driven by not only labeled data, but also scientific theories, including governing equations, boundary and initial conditions, and expert knowledge. Two independent neural networks (NNs) are built in the TgNN for oil/water phase flow problems, with one approximating pressure and the other approximating saturation. The two networks are connected by loss functions, which include a data mismatch term, as well as theory-guided terms. The desired parameters in NNs are trained by a certain optimization algorithm to decrease the value of the loss function. The training process uses a two-stage strategy as follows: (1) after one of the two NNs obtains a satisfactory result, parameters in the network with better performance are fixed in calculating the nonlinear terms and (2) the other NN continues to be trained until satisfactory performance is also obtained. The proposed TgNN offers an effective way to solve the coupled nonlinear two-phase flow problem. Numerical results demonstrate that the proposed TgNN achieves better accuracy than the traditional deep neural network (DNN). This is because the governing equation can constrain spatial and temporal derivatives, and other physical constraints (i.e., boundary and initial conditions, expert knowledge) can make the outputs more scientifically consistent. The effect of sparse data (including labeled data and collocation points) is tested, and the results show that more labeled data and collocation points lead to improved long-term prediction performance. However, the TgNN can also be successfully trained in the absence of labeled data by merely adhering to the above-mentioned scientific theories. In addition, several more complicated scenarios are tested, including the existence of data noise, changes in well condition, transfer learning, and the existence of different levels of dynamic capillary pressure. Compared with the traditional DNN, TgNN possesses superior stability with the guidance of theories for the considered complex situations.

Numerical simulation for hydrocarbon production analysis considering Pre-Darcy flow in fractured porous media

Fracture has significant effect on the production of hydrogen–carbon compound reservoirs. Underground fluid always flows through different spaces which involves different flow patterns. In this study, the embedded discrete fracture method (EDFM) to couple the flow in matrix and fracture is introduced. The Darcy flow is considered in fracture while the Pre-Darcy flow is employed in matrix. The model of two phase flow problem is established. The governing equations are coupled through multiple spatial scales. The structured mesh is applied to discrete the whole matrix domain. The fracture grids are embedded to the matrix meshes. The verification shows good accuracy of the fully implicit method for solving the model. Based on the numerical simulation, the results of 1D, 2D flow problems considering Pre-Darcy flow are presented. The characteristic of pressure and saturation is analyzed. Some numerical experiment examples are conducted to demonstrate the capability of the method including multiple wells system, complex fracture problem, and hydraulic fractured horizontal well system. Furthermore, we extend the model to 3D transport problem. The newly developed numerical simulation method improves the limitations of EDFM in Pre-Darcy flow. The newly developed simulation method can be taken as an effective tool for hydrocarbon production analysis when Pre-Darcy flow happens in fractured porous media.

Well-posedness of the two-phase flow problem in incompressible MHD

<p style='text-indent:20px;'>In this paper, we study the two phase flow problem in the ideal incompressible magnetohydrodynamics. We propose a Syrovatskij type stability condition, and prove the local well-posedness of the two phase flow problem with initial data satisfies such condition. This result shows that the magnetic field has a stabilizing effect on Kelvin-Helmholtz instability even the fluids on each side of the free interface have different densities.</p>

An improved multi-scale two phase method for bubbly flows

Multiple spatial scale is an important characteristic of two phase flow phenomena. The micro-scale and macro-scale flow structures are obviously different in flow state and have different effects on the mass, momentum and energy transfer between two phases. Different modeling approaches have been developed for each scale physical phenomenon in traditional numerical simulations. However, it is difficult to simulate two phase flow systems with multi-scale flow structures simultaneously. In order to address this problem, a multi-scale two phase method is developed based on the combination of Volume of Fluid (VOF) interface capture method and Euler–Lagrange particle tracking method. The fundamental assumption of the present method is that there is a clear scale separation between VOF interfaces and bubbles. Therefore, VOF method with artificial compressive algorithm is used to simulate the dynamic evolution of macro-scale air-water interface. While Euler–Lagrange method is used to track the micro-scale bubbles that cannot be captured in grids. Collision, coalescence and breakup of Lagrange bubbles and two-way coupling are fully considered to construct a comprehensive micro-bubble solving procedure. Transformation criteria and the corresponding algorithms between micro-scale and macro-scale flow structures are designed and discussed in detail. In addition, a new curvature-based algorithm for the transformation from VOF interface to Lagrange bubbles is proposed. Simulations of typical two phase flow problems involving multi-scale flow transformation are carried out to test the performance of the multi-scale solver. Results indicate that the multi-scale two phase method performs significantly better than the pure VOF method in capturing micro-scale phenomena. Besides, the curvature-based transformation algorithm proposed in this paper is proved to be more precise and efficient than the previous identify-based one. From the perspective of simulation accuracy and efficiency, the multi-scale two phase method is more promising for the simulation of actual complex two-phase flows.

A coupled volume-of-fluid and level set method in interDyMFoam solver

InterDyMFoam is a solver for calculating immiscible two-phase flow in an open source OpenFOAM software. The solver uses the volume-of-fluid method (VOF) for capturing the fluid-fluid interface. The solver is modified to work with the coupled VOF and level set (LS) method. In the proposed method, the VOF is used to capture the interface since it is a mass conserving and the LS is used to calculate the curvature and physical quantities near the interface. A bubble rising in column liquids is used to compare the solvers. The proposed method gives an accurate calculations for the physical properties in two phase flow problems and handles the topological change of the interface.

Efficient deep learning techniques for multiphase flow simulation in heterogeneous porousc media

We present efficient deep learning techniques for approximating flow and transport equations for both single phase and two-phase flow problems. The proposed methods take advantages of the sparsity structures in the underlying discrete systems and can be served as efficient alternatives to the system solvers at the full order. In particular, for the flow problem, we design a network with convolutional and locally connected layers to perform model reductions. Moreover, we employ a custom loss function to impose local mass conservation constraints. This helps to preserve the physical property of velocity solution which we are interested in learning. For the saturation problem, we propose a residual type of network to approximate the dynamics. Our main contribution here is the design of custom sparsely connected layers which take into account the inherent sparse interaction between the input and output. After training, the approximated feed-forward map can be applied iteratively to predict solutions in the long range. Our trained networks, especially in two-phase flow where the maps are nonlinear, show their great potential in accurately approximating the underlying physical system and improvement in computational efficiency. Some numerical experiments are performed and discussed to demonstrate the performance of our proposed techniques.

An approximate semianalytical method for two-phase flow analysis of liquid-rich shale gas and tight light-oil wells

Two-phase flow of oil and gas is quite common in the reservoir during the depletion of liquid-rich shale gas and tight light-oil. Many analytical models proposed for multistage fractured horizontal wells completed in unconventional reservoirs are restricted to single phase flow problem. In this paper, an approximate semianalytical method is proposed to handle the complexities of phase change, pressure dependent PVT properties, two-phase flow behavior and complex fracture networks. The black oil model is used to model phase change and two-phase flow of oil and gas. The secondary fractures in the stimulated reservoir volume and the reservoir beyond the main fractures are considered in the physical model to make long-term production forecasts. The forecasted producing GOR is used to decrease the equations of the two-phase flow model, so only the flow equations of primary phase are used to develop the mathematic model. Then an approximate method is used to linearize the model and the analytical solution is obtained in Laplace space for constant pressure inner boundary problem. The Duhamel's principle is applied to correct the production rate under variable flowing pressure inner boundary, and a procedure is proposed for history matching of field production data and making forecasts. Finally, the semianalytical model is validated with two simulation examples, the average relative error is shown to be less than 10%. After that, two field examples are used to show its utility.

On Local Fourier Analysis of Multigrid Methods for PDEs with Jumping and Random Coefficients

In this paper, we propose a novel nonstandard local Fourier analysis (LFA) variant for accurately predicting the multigrid convergence of problems with random and jumping coefficients. This LFA method is based on a specific basis of the Fourier space rather than the commonly used Fourier modes. To show the utility of this analysis, we consider, as an example, a simple cell-centered multigrid method for solving a steady-state single phase flow problem in a random porous medium. We successfully demonstrate the predictive capability of the proposed LFA using a number of challenging benchmark problems. The information provided by this analysis could be used to estimate a priori the time needed for solving certain uncertainty quantification problems by means of a multigrid multilevel Monte Carlo method.

Python framework for hp-adaptive discontinuous Galerkin methods for two-phase flow in porous media

• Presentation of a framework for solving two phase flow problems in porous media . • Easy to use python frontend and a state-of-the-art C++ backend based on well established software package DUNE. • hp-adaptive Discontinuous Galerkin Method for two phase flow in porous media. • Comparison of five different solution techniques from fully coupled fully implicit to loosely coupled splitting schemes. In this paper we present a framework for solving two-phase flow problems in porous media. The discretization is based on a Discontinuous Galerkin method and includes local grid adaptivity and local choice of polynomial degree. The method is implemented using the new Python frontend Dune-FemPy to the open source framework Dune. The code used for the simulations is made available as Jupyter notebook and can be used through a Docker container. We present a number of time stepping approaches ranging from a classical IMPES method to a fully coupled implicit scheme. The implementation of the discretization is very flexible allowing to test different formulations of the two-phase flow model and adaptation strategies.

A space–time domain decomposition approach using enhanced velocity mixed finite element method

A space–time domain decomposition approach is presented as a natural extension of the enhanced velocity mixed finite element (EVMFE), introduced by Wheeler et al. in (2002) [26], for spatial domain decomposition. The proposed approach allows for different space–time discretizations on non-overlapping, subdomains by enforcing a mass continuity at non-matching interfaces to preserve local mass conservation inherent to the mixed finite element methods. To this effect, we consider three different model formulations: (1) a linear single phase flow problem, (2) a non-linear slightly compressible flow and tracer transport, and (3) a non-linear slightly compressible, multiphase flow and transport. We also present a numerical solution algorithm for the proposed domain decomposition approach where a monolithic (fully coupled in space and time) system is constructed that does not require subdomain iterations. This space–time EVMFE method accurately resolves advection–diffusion transport features, in a heterogeneous medium, while circumventing non-linear solver convergence issues associated with large time-step sizes for non-linear problems. Numerical results are presented for the aforementioned, three, model formulations to demonstrate the applicability of this approach to a general class of flow and transport problems in porous media.

Adaptive mesh method applied to the thermal hydraulic program of system code with a judging criterion based on the matrix error

In this paper, an adaptive mesh method is described and applied to the one-dimension thermal hydraulic program in a system code for simulating the two-phase flow problems. The adaptive mesh method has been widely used to solve various engineering problems, but it is still rarely applied in existing thermal hydraulic system codes which are of great significance for reactor design and analysis. Compared with the uniform mesh, the mesh size could be adjusted automatically. It is helpful to locate the discontinuity points or deal with the large gradient solutions with this method, especially for some local phenomena that occur in phase transfer or two phase flow problems. Based on it, the computational accuracy and efficiency are improved, when solving the partial difference equations. The applicability of the adaptive mesh method in a system code is a major issue, and the first step is studying the feasibility of the adaptive mesh method in thermal hydraulic programs. In this paper, the finite volume method and the Newton-Raphson algorithm are applied in the code as the numerical method of two-fluid model. Furthermore, the code also includes heat transfer models, friction models and a flow regime map, which are important in simulating hydraulic phenomena. Depending on the algorithm, a judging criterion for determining the location where mesh refinement is needed is proposed. Three typical numerical examples are given to verify the feasibility and effectiveness of the adaptive mesh algorithm in solving the one-dimension two-phase flow problems of local phenomena and show great agreement with theoretical or experiment results.

On the efficiency of the IMPES method for two phase flow problems in porous media

In this work, we show a method to efficiently solve the multiphase flow through a porous media in the near wellbore region. The numerical discretisation is based on the IMplicit Pressure Explicit Saturation (IMPES) approach.While most of the works in the literature rely on Fully Implicit Methods (FIM) to simulate the reservoir, this is not suited for the near wellbore region, where smaller computational and physical times are required, therefore parametric models with history matching are used in this zone. However, parametric models present several uncertainties that can affect the estimation of the pressure drop. An accurate and fast simulation of that region is therefore required.Here, we propose a new and robust implementation of the IMPES scheme, aiming to reduce the large computational cost of the (implicit) pressure system of equations and increase the robustness and reliability of the scheme. The method takes advantage of the short physical time steps between iterations, observed in the near wellbore region, to produce a new pressure field at the cost of a numerical explicit scheme. Two and three dimensional numerical experiments are carried out to show the validity of the proposed discretisation. We show that the cost of the method is reduced in a factor of 7–19 when compared to the classical IMPES.

A dual grid, dual level set based cut cell immersed boundary approach for simulation of multi-phase flow

A numerical methodology based on dual grid dual level set function is presented for simulating 2D multiphase flows through complex geometries on a non-body fitted Cartesian computational grid. A Cut Cell based immersed boundary method (IBM) is used to sharply resolve complex geometries. The cut cells near the immersed boundary are identified using a fictitious level set function. The governing equations are discretized using the finite volume method (FVM) on a m × n staggered grid. The dynamics of fluid-fluid interface is tracked by an other level set function defined on dual grid (2m × 2n). The use of dual grid arrangement for fictitious level set function results in an accurate calculation of diffusion and advection fluxes at the cut cell near the immersed boundary. A module by module validation of the developed numerical methodology is carried out with several test cases. Two single phase flow problems corresponds to Poiseuille flow and flow through a convergent-divergent section while the two multiphase problems selected are the standard Young’s Laplace law test – semicircular fluid rod on a horizontal surface – and the bubble rise phenomenon in an inclined channel. The obtained results show an excellent agreement with those derived analytically or taken from literature. Furthermore, the performance study carried out on the present model shows that the method is between first and second order accurate.

Hydraulic conductivities identification via Ensemble Kalman Filtering with transformed data considering the risk of systematic bias

In subsurface hydrology, Ensemble Kalman Filtering (EnKF) has been coupled with groundwater flow and transport models to solve the inverse problem. Several extensions of the EnKF have been proposed to improve its performance when dealing with non-multi-Gaussian random field models of the hydraulic conductivity. One such variant is the EnKF with transformed data (tEnKF), which uses Gaussian anamorphosis within a conditioning step. Although this transformation has been used in the past to identify hydraulic conductivities, previous studies have ignored the risk of introducing a systematic bias in the spatiotemporal evolution of the hydraulic head field during the forecast steps that the update steps may not correct over time. This paper proposes that in order to evaluate the performance of tEnKFs, applications in synthetically generated random porous media should take into account this risk by incorporating prior knowledge with a multi-Gaussian conductivity correlation structure, and by adopting a reference field with asymmetric correlation structure. As an example of this application, hydraulic conductivities using the tEnKF were identified by solving a one-dimensional, single phase flow problem in a continuous random porous medium. Common concepts in Geostatistics are used to explain the hypothesis underlying both EnKF and tEnKF and to establish a clear link between the tEnKF and the stochastic simulation of conditional random fields.

A parallel mixed–hybrid finite element method for two phase flow problems in porous media using MPI

A parallel mixed–hybrid finite element method for two phase flow problems in porous media using MPI

A circulating gravity wave in a cylindrical tank

AbstractA circular cylindrical vertical tank is partially filled with a liquid (Water) and a gas (air) above it. The top lid of the container rotates around the cylinder axis and induces a flow in the gas and the liquid. Above a critical rotational speed a large amplitude circulating gravity wave forms. The two phase flow problem will be reduced to a free surface single phase flow problem and the critical parameters, when the axis‐symmetric flow becomes unstable with respect to the circulating gravity wave will be determined. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Mixed finite element analysis of lognormal diffusion and multilevel Monte Carlo methods

This work is motivated by the need to develop efficient tools for uncertainty quantification in subsurface flows associated with radioactive waste disposal studies. We consider single phase flow problems in random porous media described by correlated lognormal distributions. We are interested in the error introduced by a finite element discretisation of these problems. In contrast to several recent works on the analysis of standard nodal finite element discretisations, we consider here mass-conservative lowest order Raviart–Thomas mixed finite elements. This is very important since local mass conservation is highly desirable in realistic groundwater flow problems. Due to the limited spatial regularity and the lack of uniform ellipticity and boundedness of the operator the analysis is non-trivial in the presence of lognormal random fields. We establish finite element error bounds for Darcy velocity and pressure, as well as for a more accurate recovered pressure approximation. We then apply the error bounds to prove convergence of the multilevel Monte Carlo algorithm for estimating statistics of these quantities. Moreover, we prove convergence for a class of bounded, linear functionals of the Darcy velocity. An important special case is the approximation of the effective permeability in a 2D flow cell. We perform numerical experiments to confirm the convergence results.