A Review of Diffuse Interface-Capturing Methods for Compressible Multiphase Flows

A 6M digital twin for modeling and simulation in subsurface reservoirs

One of the fundamental problems in simulating the motion of sharp interfaces between immiscible fluids is a description of the transition that occurs when the interfaces merge and reconnect. It is well known that classical methods involving sharp interfaces fail to describe this type of phenomena. Following some previous work in this area, we suggest a physically motivated regularization of the Euler equations which allows topological transitions to occur smoothly. In this model, the sharp interface is replaced by a narrow transition layer across which the fluids may mix. The model describes a flow of a binary mixture, and the internal structure of the interface is determined by both diffusion and motion. An advantage of our regularization is that it automatically yields a continuous description of surface tension, which can play an important role in topological transitions. An additional scalar field is introduced to describe the concentration of one of the fluid components and the resulting system of equations couples the Euler (or Navier–Stokes) and the Cahn–Hilliard equations. The model takes into account weak non–locality (dispersion) associated with an internal length scale and localized dissipation due to mixing. The non–locality introduces a dimensional surface energy; dissipation is added to handle the loss of regularity of solutions to the sharp interface equations and to provide a mechanism for topological changes. In particular, we study a non–trivial limit when both components are incompressible, the pressure is kinematic but the velocity field is non–solenoidal (quasi–incompressibility). To demonstrate the effects of quasi–incompressibility, we analyse the linear stage of spinodal decomposition in one dimension. We show that when the densities of the fluids are not perfectly matched, the evolution of the concentration field causes fluid motion even if the fluids are inviscid. In the limit of infinitely thin and well–separated interfacial layers, an appropriately scaled quasi–incompressible Euler–Cahn–Hilliard system converges to the classical sharp interface model. In order to investigate the behaviour of the model outside the range of parameters where the sharp interface approximation is sufficient, we consider a simple example of a change of topology and show that the model permits the transition to occur without an associated singularity.

A diffuse interface method for complex three-dimensional free surface flows

A simple diffuse interface approach for compressible flows around moving solids of arbitrary shape based on a reduced Baer–Nunziato model

This paper is devoted to the numerical approximation of a hyperbolic non‐equilibrium multiphase flow model with different velocities on moving meshes. Such goal poses several difficulties. The presence of different flow velocities in conjunction with cell velocities poses difficulties for upwinding fluxes. Another issue is related to the presence of non‐conservative terms. To solve these difficulties, the discrete equations method (J. Comput. Phys. 2003; 186(2):361–396; J. Fluid. Mech. 2003; 495:283–321; J. Comput. Phys. 2004; 196:490–538; J. Comput. Phys. 2005; 205:567–610) is employed and generalized to the context of moving cells. The complementary conservation laws, available for the mixture, are used to determine the velocities of the cells boundaries. With these extensions, an accurate and robust multiphase flow method on moving meshes is obtained and validated over several test problems with exact or experimental solutions. Copyright © 2007 John Wiley & Sons, Ltd.

The multiphase fluid flow in porous media is one of the most fundamental phenomena in various physical processes, such as oil/gas flow in reservoir, subsurface contamination dispersion, chemical separation, etc. Due to its importance, the efficient and accurate solution and prediction of multiphase flow in porous media is highly required in engineering applications and mechanism studies, which has been a research hot spot with increasing interest in recent years. However, the strong nonlinearity implicated in the multiphase flow model has brought great challenges for the computation and analysis. In addition, the permeability in Darcy-type pressure equation is always represented as a high contrast coefficient, which further complicates the simulation difficulty especially for high fidelity prediction and decision-making process. In this study, we attempt to apply one popular global model reduction method, namely Proper Orthogonal Decomposition (POD), to accelerate the simulation of incompressible multiphase flow in porous media. The cornerstone of POD is processing information from a sequence of snapshots (or instantaneous solutions) to identify a set of basis functions to construct a low-dimensional space, the most energetic part of these basis functions is then selected to represent the solution of the original system. In our work, one key point is how to establish the accurate POD reduced order model (ROM) for incompressible mul-tiphase porous flow problems. According to Li et al. [1], the traditional Galerkin projection method (directly project the original governing equation into the low-dimensional space spanned by basis functions) is incapable of introducing the fluid saturation and permeability located on the cell face in POD-ROM for incompressible multiphase flow whether in homogeneous or heterogeneous porous media, thus we adopt the matrix operation method (project the discrete governing equation instead of the original governing equation into the low-dimensional space spanned by basis functions) to build the POD-ROM in this study. Based on the proposed POD-ROM framework, we use the implicit method to solve the (non-) wetting-phase pressure and saturation equations. However, in the implicit solution of the pressure and saturation equations, the standard POD-ROM alone sometimes cannot yield an expected acceleration owing to two main reasons: (1) the nonlinearity between pressure and saturation is sensitive, thus the saturation would change largely even with slight pressure change. And the small error of pressure results would lead to a large deviation of saturation, which in turn exerts negative impacts on the computation of pressure; (2) the complexity for computing a projected nonlinear term in POD-ROM still depends on the dimension of the original full-order system, it is unavoidable in conventional POD-ROM. Therefore, more POD modes are needed to be used in order to achieve the prescribed numerical accuracy, which could worsen the speed up efficiency of proposed POD-ROM. To further reduce the computational complexity of POD-ROM to speed up the implicit solution of pressure and saturation equations, another model reduction method called Discrete Empirical Interpolation Method (DEIM) [2] is applied to treat the projected nonlinear term in the proposed incompressible multiphase porous flow POD-ROM. Different from previous studies, we develop an adaptive DEIM (ADEIM) in this work to achieve a more flexible selection of the interpolation points. Combined the POD-ROM with ADEIM, a POD-ADEIM-ROM for incompressible multiphase flow in porous media is established. The computational efficiency and numerical accuracy of the proposed model is validated through several representative numerical examples. Numerical results indicate that an outstanding acceleration (2~3 orders of magnitude faster) without sacrificing numerical accuracy significantly is obtained from the proposed model when comparing to the traditional solution method that without any acceleration technique. In addition, the effects of POD mode number, the distribution of permeability field, and the mesh density on the overall performances of the proposed model are investigated in detail.

In the second year of the project, the Eulerian-Lagrangian formulation for analyzing three-phase slurry flows in a bubble column is further developed. The approach uses an Eulerian analysis of liquid flows in the bubble column, and makes use of the Lagrangian trajectory analysis for the bubbles and particle motions. An experimental set for studying a two-dimensional bubble column is also developed. The operation of the bubble column is being tested and diagnostic methodology for quantitative measurements is being developed. An Eulerian computational model for the flow condition in the two-dimensional bubble column is also being developed. The liquid and bubble motions are being analyzed and the results are being compared with the experimental setup. Solid-fluid mixture flows in ducts and passages at different angle of orientations were analyzed. The model predictions were compared with the experimental data and good agreement was found. Gravity chute flows of solid-liquid mixtures is also being studied. Further progress was also made in developing a thermodynamically consistent model for multiphase slurry flows with and without chemical reaction in a state of turbulent motion. The balance laws are obtained and the constitutive laws are being developed. Progress was also made in measuring concentration and velocity of particles of different sizes near a wall in a duct flow. The technique of Phase-Doppler anemometry was used in these studies. The general objective of this project is to provide the needed fundamental understanding of three-phase slurry reactors in Fischer-Tropsch (F-T) liquid fuel synthesis. The other main goal is to develop a computational capability for predicting the transport and processing of three-phase coal slurries. The specific objectives are: (1) To develop a thermodynamically consistent rate-dependent anisotropic model for multiphase slurry flows with and without chemical reaction for application to coal liquefaction. Also establish the material parameters of the model. (2) To provide experimental data for phasic fluctuation and mean velocities, as well as the solid volume fraction in the shear flow devices. (3) To develop an accurate computational capability incorporating the new rate-dependent and anisotropic model for analyzing reacting and nonreacting slurry flows, and to solve a number of technologically important problems related to Fischer-Tropsch (F-T) liquid fuel production processes. (4) To verify the validity of the developed model by comparing the predicted results with the performed and the available experimental data under idealized conditions.

Hazardous dense nonaqueous phase liquids (DNAPLs), such as chlorinated solvents, are slightly water soluble and pose a serious threat to soil and groundwater supplies in many portions of the United States. The migration and entrapment of DNAPLs in the subsurface environment is typically believed to be controlled by physical heterogeneities; i.e, layers and lenses of contrasting soil texture. The rationale for this assumption is that capillarity, as determined by the soil texture, is the dominant transport mechanism. Capillarity also depends on interfacial tension and medium wettability. Interfacial tension and medium wettability may be spatially and temporally dependent due to variations in aqueous phase chemistry, contaminant aging, and/or variations in mineralogy and organic matter distributions. Such chemical heterogeneities have largely been ignored to date, even though they are known to have dramatic effects on the hydraulic property relations. Numerical multiphase flow and transport models typically assume that solids are water-wet and that interfacial tension is constant. The primary objective of this research is to investigate the influence of coupled physical and chemical heterogeneities on the migration and entrapment of DNAPLs. This objective will be accomplished through a combination of laboratory and numerical experiments. Laboratory experiments will be conducted to examine: (i) aqueous phase chemistry effects on medium wettability and interfacial tension; and (ii) relative permeability-saturation-capillary pressure relations for chemically heterogeneous systems. An important objective of this research is to modify a two-dimensional multiphase flow and transport model to account for chemically and physically heterogeneous systems. This numerical simulator will be used in conjunction with independently measured parameters to simulate two-dimensional DNAPL infiltration experiments. Comparisons of simulated and laboratory data will provide a means to experimentally validate this model. The validated numerical simulator will subsequently be employed to investigate various innovative remediation schemes such as the use of surfactants and in situ wettability alteration. The accomplishment of the research herein will: (i) lead to a better understanding of the way aqueous phase chemistry changes medium wettability; (ii) validate and/or lead to the development of methods to predict and model wettability on hydraulic property relations; (iii) lead to the development of a multiphase flow simulator that accounts for fractional wettability and concentration dependent interfacial properties; (iv) lead to an improved knowledge of the effects of pore-scale variability on scale-up issues in multiphase systems; (v) provide an understanding of the interaction of chemical and physical heterogeneity on DNAPL flow and entrapment; (vi) provide two-dimensional laboratory data sets to validate multiphase flow models for physically and chemically heterogeneous systems; and (vii) facilitate the development and implementation of innovative remediation strategies.

Numerical modeling of multiphase compressible flows with the presence of shock waves using an interface-sharpening five-equation model

In this paper, we demonstrate the ability to run an adaptation on field data for a steady-state multiphase pipe flow model for calculating the bottomhole pressure of an inclined well using the deviation survey data, wellhead parameters (pressure, phases flowrates and properties), and the temperature distribution along the wellbore. In the underlying model published earlier in different modifications (Baryshnikov et al., 2020; Kanin et al., 2019), we divide the well into segments and perform the hydraulic calculation of the pressure distribution along the wellbore via the marching algorithm in which the set of PVT correlations and the segment model are utilized. The distinguishing feature of the utilized approach is a data-driven segment model required for calculating the pressure drop along the chosen part of the pipe. This model is based on Artificial Neural Networks (ANNs), which firstly trained on the synthetic data and lab measurements, and after that, additionally tuned on the real field data collected on two wells during the flowback period conducted via the free-flow production method. During the optimization of the ANNs on the field data, we imply the quasi-steady-state approximation of the multiphase flow, i.e. the transient flow is considered as a sequence of the stationary states. In work, we also carry out the comparison of the prediction capability of the constructed pipe segment model with several most common multiphase flow correlations and mechanistic models (within the frame of the marching algorithm) and demonstrate its more accurate outcome. As a result, we develop a method for calculating the flowing bottomhole pressure value adaptable on the required field data. The model adapted on field data allows estimating the flowing bottomhole pressure at each time moment (assuming the quasi-steady-state flow) which is particularly important in the absence of the bottomhole pressure gage at the well.

The focus of the underlying research work is on the macroscopic modeling of unstable multiphase fluid flow in deformable porous media, where a lower‐viscous fluid is displaced by a more viscous fluid. This process leads to the formation of channel‐like networks, called viscous fingering. The instability effect is involved in a wide range of different fields in engineering. Some of the most common applications include carbon sequestration to store carbon dioxide (CO2) in underground reservoirs, contaminant transport in geostructures, in industrial processes, such as filtration, catalytic reactions, and in the operation of polymer electrolyte membrane fuel cells with multiphase flow in the gas diffusion layers. In this work, ideal miscible water–glycerin fluids are considered. It is assumed that the interacting fluids in the deformable porous media are incompressible. In addition, a dispersion–diffusion law is applied to capture the fluid–fluid interactions. The presence of a deformable porous material adds additional effects to the problem of the multiphase flow in porous media. The stresses in the porous solid matrix lead to changes in the porosity and influence the flow velocity of the fluids. To couple the deformable porous media with the multiphase flow, a macroscopic approach is used that relies on the theory of porous media. For the porous solid matrix, a linear elastic material model within small strains assumption is applied. The influence of the deformation‐dependent porosity on the instability is studied for 2D simulations, such as a multilayered geometry with different elastic parameters. The presented coupled nonlinear system of differential equations is simulated with the finite element method. Furthermore, a stabilization technique based on the quasi‐compressibility method is used.

Aiming to understand the effect of various parameters such as liquid velocity, surface tension, and wetting phenomena, a Volume‐of‐Fluid (VOF) model was developed to simulate the multiphase flow in high‐pressure trickle‐bed reactor (TBR). As the accuracy of the simulation is largely dependent on mesh density, different mesh sizes were compared for the hydrodynamic validation of the multiphase flow model. Several model solution parameters comprising different time steps, convergence criteria and discretization schemes were examined to establish model parametric independency results. High‐order differencing schemes were found to agree better with the experimental data from the literature given that its formulation includes inherently the minimization of artificial numerical dissipation. The optimum values for the numerical solution parameters were then used to evaluate the hydrodynamic predictions at high‐pressure demonstrating the significant influence of the gas flow rate mainly on liquid holdup rather than on two‐phase pressure drop and exhibiting hysteresis in both hydrodynamic parameters. Afterwards, the VOF model was applied to evaluate successive radial planes of liquid volume fraction at different packed bed cross‐sections. © 2009 American Institute of Chemical Engineers AIChE J, 2009

An algorithm was developed for determining the implications of multiphase flow uncertainties. The specific method employed combines the classical uncertainty analysis of Kline and McClintock with multiple random variable probability theory. The initial results indicate that in addition to uncertainty prediction, the algorithm can also be used to statistically reduce the sharp discontinuities common in multiphase flow modeling, thus leading to improved simulation convergeance. Introduction Flow assurance and risk assessment are major issues in the offshore petroleum industry today. While there is a push for increasingly more complex flow models it is important not to lose the understanding of the accuracy of such models and their sensitivity to input parameters. Error bars for flow assurance simulations can greatly affect field planning, equipment sizing, and risk assessment. At present, uncertainty calculations for multiphase flow conditions are not well understood and are rarely reported. Over short distances, uncertainties in prediction of pressure drop in the design stage can be large without causing significant production problems. However as multiphase transport in pipelines extends into tens of miles, a significant error can occur in predicting capacity. The method described can also be extended to the prediction of such values as corrosion rate, arrival temperature, and wax deposition rate. In developing a method for determining uncertainty, it is desired that re-coding of the model not be necessary, implementation be simple, practical computational constraints be considered (e.g., no Monte Carlo approach). This led to an approach that was discrete and also was able to lessen the effects of sharp discontinuities. Specifically, existing model functions can be evaluated at selected intervals with the results being probability-weighted to arrive at a solution. Statement of Theory and Definintions Error bars represent the confidence interval for a given confidence level. When engineering results are specified, error bars are attached to the data to give the reader an idea of the level of uncertainty associated with the results. A list of values for common confidence intervals is given in Table 1. It is important to note that when the confidence level does not accompany the error bars, it is common to assume 2.15 - 1 odds. The values in Table 1 were generated using Eq. 1 in conjuntion with the Gaussian distribution.

Many plastic products are processed in extruders. These devices exhibit complex flow phenomena, especially when multiple phases are present, as seen in processes such as grafting and foaming. This thesis introduces comprehensive, multiphysics computational fluid dynamics models of multiphase flow in extruders. The couplings between mass transfer, heat transfer, shear and concentration dependent viscosity and chemical reactions are accounted for - together with various multiphase flow models - to model the complex flows present in extruders. Results show that this multiphysics approach is necessary to predict complex phenomena such as e.g., backflow, plasticization and sputtering at the die. Additionally, the final chapter, which discusses life cycle assessment, shows that linear and static models may lead to erroneous conclusions and incorrect policy insights.

The lattice Boltzmann equation (LBE) method is a promising technique for simulating fluid flows and modeling complex physics. Because the LBE model is based on microscopic models and mesoscopic kinetic equations, it offers many advantages for the study of multi-component or multiphase flows. However, there are still challenges encountered when dealing with thermal effects and multiphase flows, particularly at small scales or in varying geometries. In this paper, we discuss some techniques to overcome these challenges. First, we present an overview of the LBE method, and show how it can be extended to model multiple phases and thermal effects. Next, we describe our multi-component and multiphase (MCMP) LBE method for high density ratios. While the original formulation of Shan and Chen’s (SC) model can incorporate some multiphase and component scenarios, the density ratio of the different components is restricted (less than approximately 2.0), which limits the applications. Hence, based on the SC model and improvements in the single-component multiphase (SCMP) flow model reported by Yuan and Schaefer, we have developed a new model that can simulate a MCMP system with a high density ratio. An example of that system is shown. Finally, we have developed a parallel computation LBE method based on the Compute Unified Device Architecture for NVIDIA GPUs. Using this method, we are able to efficiently model a number of phases and length scales, examples of which are presented.