Evaluation of three lattice Boltzmann models for multiphase flows in porous media

A free energy (FE) model, the Shan–Chen (S–C) model, and the Rothman and Keller (R–K) model are studied numerically to evaluate their performance in modeling two-dimensional (2D) immiscible two-phase flow in porous media on the pore scale. The FE model is proved to satisfy the Galilean invariance through a numerical test and the mass conservation of each component in the simulations is exact. Two-phase layered flow in a channel with different viscosity ratios was simulated. Comparing with analytical solutions, we see that the FE model and the R–K model can give very accurate results for flows with large viscosity ratios. In terms of accuracy and stability, the FE model and the R–K model are much better than the S–C model. Co-current and countercurrent two-phase flows in complex homogeneous media were simulated and the relative permeabilities were obtained. Again, it is found that the FE model is as good as the R–K model in terms of accuracy and efficiency. The FE model is shown to be a good tool for the study of two-phase flows with high viscosity ratios in porous media.     

Multirange multi-relaxation time Shan–Chen model with extended equilibrium

The work presents simulations with the multirange Shan–Chen model developed by Sbragaglia et al. (2007) [18], which improved the Shan–Chen model for the proper surface tension term. Also, by introducing the matrix collision operator and extended equilibrium density distribution function, the density ratio is increased from 100 to 160. The Multi-Relaxation Time (MRT) method attracted the attention of researchers due to several advantages, such as better stability, simulations with Prandtl number different from unity, and possibilities to improve the accuracy of the scheme compared with BGK Single Time Relaxation model. Our recent results have shown that the combination of MRT methods with multiphase flow models can improve the achievable gas–liquid density ratio.     

Applicability of the three-parameter Kozeny–Carman generalized equation to the description of viscous fingering in simulations of waterflood in heterogeneous porous media

This article discusses the applicability of the three-parameter Kozeny–Carman generalized equation to trigger immiscible viscous fingers and describe it in fractal heterogeneous porous media, during numerical simulations of waterflood operations in oil reservoirs. For that purpose, for the first time this equation was incorporated into a model that describes immiscible flows of incompressible two-phase fluids in porous media. Results were generated from intensive simulations, and viscous fingers were visualized graphically for three different well patterns, typical of oil fields: Line-Drive, Five-Spot and Inverted Five-Spot. Such results suggest that this generalization of the Kozeny–Carman equation can be used in numerical simulations of oil recovery processes susceptible to hydrodynamic instability phenomena.     

Application of level-set approach to moving interfaces and free surface problems in flow through porous media

The level-set technique introduced by Osher and Sethian has been widely used in Computational Fluid Dynamics problems involving two or more immiscible fluids with sharp interfaces. This paper presents an application of the level-set to unconfined seepage problems where a free surface or a sharp interface exists. It is based on considering the movement of both fluids, introducing a smooth indicator function with a level-set describing the interface position.The proposed method can be applied to other flow in porous media problems involving immiscible fluids, such as oil recovery applications.     

TOUGHREACT—A simulation program for non-isothermal multiphase reactive geochemical transport in variably saturated geologic media: Applications to geothermal injectivity and CO2 geological sequestration

TOUGHREACT is a numerical simulation program for chemically reactive non-isothermal flows of multiphase fluids in porous and fractured media. The program was written in Fortran 77 and developed by introducing reactive geochemistry into the multiphase fluid and heat flow simulator TOUGH2. A variety of subsurface thermo-physical–chemical processes are considered under a wide range of conditions of pressure, temperature, water saturation, ionic strength, and pH and Eh. Interactions between mineral assemblages and fluids can occur under local equilibrium or kinetic rates. The gas phase can be chemically active. Precipitation and dissolution reactions can change formation porosity and permeability. The program can be applied to many geologic systems and environmental problems, including geothermal systems, diagenetic, and weathering processes, subsurface waste disposal, acid mine drainage remediation, contaminant transport, and groundwater quality. Here we present two examples to illustrate applicability of the program. The first example deals with injectivity effects of mineral scaling in a fractured geothermal reservoir. A major concern in the development of hot dry rock and hot fractured rock reservoirs is achieving and maintaining adequate injectivity, while avoiding the development of preferential short-circuiting flow paths. Rock–fluid interactions and associated mineral dissolution and precipitation effects could have a major impact on the long-term performance of these reservoirs. We used recent European studies as a starting point to explore chemically induced effects of fluid circulation in the geothermal systems. We examine ways in which the chemical composition of reinjected waters can be modified to improve reservoir performance by maintaining or even enhancing injectivity. The second TOUGHREACT application example is related to CO₂ geologic sequestration in a saline aquifer. We performed numerical simulations for a commonly encountered Gulf Coast sediment under CO₂ injection conditions in order to analyze the impact of CO₂ immobilization through carbonate precipitation. Using the data presented in this paper, the CO₂ mineral-trapping capability after 10,000 years can reach 60 kg/m³ of sandstone by secondary carbonate mineral precipitation such as siderite, ankerite, and dawsonite. Most of the simulated mineral alteration pattern is consistent with the field observations of natural CO₂ reservoirs.     

Realistic simulation of mixing fluids

Recently, simulation of mixing fluids, for which wide applications can be found in multimedia, computer games, special effects, virtual reality, etc., is attracting more and more attention. Most previous methods focus separately on binary immiscible mixing fluids or binary miscible mixing fluids. Until now, little attention has been paid to realistic simulation of multiple mixing fluids. In this paper, based on the solution principles in physics, we present a unified framework for realistic simulation of liquid–liquid mixing with different solubility, which is called LLSPH. In our method, the mixing process of miscible fluids is modeled by a heat-conduction-based Smooth Particle Hydrodynamics method. A special self-diffusion coefficient is designed to simulate the interactions between miscible fluids. For immiscible fluids, marching-cube-based method is adopted to trace the interfaces between different types of fluids efficiently. Then, an optimized spatial hashing method is adopted for simulation of boundary-free mixing fluids such as the marine oil spill. Finally, various realistic scenes of mixing fluids are rendered using our method.     

Hydrodynamics in Porous Media: A Finite Volume Lattice Boltzmann Study

Fluid flow through porous media is of great importance for many natural systems, such as transport of groundwater flow, pollution transport and mineral processing. In this paper, we propose and validate a novel finite volume formulation of the lattice Boltzmann method for porous flows, based on the Brinkman–Forchheimer equation. The porous media effect is incorporated as a force term in the lattice Boltzmann equation, which is numerically solved through a cell-centered finite volume scheme. Correction factors are introduced to improve the numerical stability. The method is tested against fully porous Poiseuille, Couette and lid-driven cavity flows. Upon comparing the results with well-documented data available in literature, a satisfactory agreement is observed. The method is then applied to simulate the flow in partially porous channels, in order to verify its potential application to fractured porous conduits, and assess the influence of the main porous media parameters, such as Darcy number, porosity and porous media thickness.     

A Hybrid Approach to Multiple Fluid Simulation using Volume Fractions

This paper presents a hybrid approach to multiple fluid simulation that can handle miscible and immiscible fluids, simultaneously. We combine distance functions and volume fractions to capture not only the discontinuous interface between immiscible fluids but also the smooth transition between miscible fluids. Our approach consists of four steps: velocity field computation, volume fraction advection, miscible fluid diffusion, and visualization. By providing a combining scheme between volume fractions and level set functions, we are able to take advantages of both representation schemes of fluids. From the system point of view, our work is the first approach to Eulerian grid‐based multiple fluid simulation including both miscible and immiscible fluids. From the technical point of view, our approach addresses the issues arising from variable density and viscosity together with material diffusion. We show that the effectiveness of our approach to handle multiple miscible and immiscible fluids through experiments.     

Simulation of incompressible two-phase flows with large density differences employing lattice Boltzmann and level set methods

A hybrid lattice Boltzmann and level set method (LBLSM) for two-phase immiscible fluids with large density differences is proposed. The lattice Boltzmann method is used for calculating the velocities, the interface is captured by the level set function and the surface tension force is replaced by an equivalent force field. The method can be applied to simulate two-phase fluid flows with the density ratio up to 1000. In case of zero or known pressure gradient the method is completely explicit. In order to validate the method, several examples are solved and the results are in agreement with analytical or experimental results.     

TOUGHREACT Version 2.0: A simulator for subsurface reactive transport under non-isothermal multiphase flow conditions

TOUGHREACT is a numerical simulation program for chemically reactive non-isothermal flows of multiphase fluids in porous and fractured media, and was developed by introducing reactive chemistry into the multiphase fluid and heat flow simulator TOUGH2 V2. The first version of TOUGHREACT was released to the public through the U.S. Department of Energy's Energy Science and Technology Software Center (ESTSC) in August 2004. It is among the most frequently requested of ESTSC's codes. The code has been widely used for studies in CO₂ geological sequestration, nuclear waste isolation, geothermal energy development, environmental remediation, and increasingly for petroleum applications. Over the past several years, many new capabilities have been developed, which were incorporated into Version 2 of TOUGHREACT. Major additions and improvements in Version 2 are discussed here, and two application examples are presented: (1) long-term fate of injected CO₂ in a storage reservoir and (2) biogeochemical cycling of metals in mining-impacted lake sediments.     

Degenerate Two-Phase Incompressible Flow IV: Local Refinement and Domain Decomposition

This is the fourth paper of a series in which we analyze mathematical properties and develop numerical methods for a degenerate elliptic-parabolic partial differential system which describes the flow of two incompressible, immiscible fluids in porous media. In this paper we describe a finite element approximation for this system on locally refined grids. This adaptive approximation is based on a mixed finite element method for the elliptic pressure equation and a Galerkin finite element method for the degenerate parabolic saturation equation. Both discrete stability and sharp a priori error estimates are established for this approximation. Iterative techniques of domain decomposition type for solving it are discussed, and numerical results are presented.     

SPH simulation of selective withdrawal from microcavity

The selective withdrawal of weakly compressible fluids is investigated by smoothed particle hydrodynamics (SPH) with a revised model of surface tension. In our model problem, fluid is withdrawn from a two-dimensional microcavity through a narrow outlet above the interface of two immiscible fluids. The outflow boundary is implemented by a particular zone of fluid particles with prescribed velocity, together with the introduction of artificial boundary particles. Based on the average number density of fluid particles, the effective contribution of boundary particles is corrected for the compressible context. It is found that there exists a critical withdrawal rate for each initial interface height, beyond which the lower phase becomes entrained in a thin spout along with the upper phase. Besides, the Froude number with redefinition for this kind of multiphase flow could serve as a criterion of flow behavior. Furthermore, larger surface tension, smaller dynamical viscosity and density of the upper phase all lead to longer threshold time of formation of the spout state, and thus are favorable to the withdrawal of upper phase both in terms of higher efficiency and larger quantity.     

Effect of the skimming layer on electro-osmotic��Poiseuille flows of viscoelastic fluids

An analytical solution is derived for the micro-channel flow of viscoelastic fluids by combined electro-osmosis and pressure gradient forcing. The viscoelastic fluid is described by the Phan-Thien–Tanner model with due account for the near-wall layer depleted of macromolecules. This skimming layer is wider than the electric double layer (EDL) and leads to an enhanced flow rate relative to that of the corresponding uniform concentration flow case. The derived solution allows a detailed investigation of the flow characteristics due to the combined effects of fluid rheology, forcing strengths ratio, skimming layer thickness and relative rheology of the two fluids. In particular, when the EDL is much thinner than the skimming layer and simultaneously the viscosity of the Newtonian fluid inside this layer is much lower than that of the fluid outside, the flow is dominated by the characteristics of the Newtonian fluid. Outside these conditions, proper account of the various fluid layers and their properties must be considered for an accurate prediction of the flow characteristics. The analytical solution remains valid for the flow driven by a pressure gradient and its streaming potential, which is determined in the appendix.     

应用于冷却系统的多孔介质电渗泵研究

对多孔介质电渗泵性能进行研究,分析电渗泵的流率和压力,为多孔介质电渗泵制作提供参考.依据电渗流控制方程,利用Debye-Huckel近似得到单根圆形管电渗流的近似解析模型,采用MEMS数值仿真软件CoventorWare对电渗泵模型进行求解分析,得到电渗泵流场的分布.基于电渗流的近似解析模型,采用截断高斯分布对多孔介质...     

Lattice Boltzmann simulation of viscous fingering phenomenon of immiscible fluids displacement in a channel

In this paper, the viscous fingering phenomenon of two immiscible fluids in a channel is studied by applying the lattice Boltzmann method (LBM). The fundamental physical mechanisms of a finger formation or the interface evolution between immiscible fluids are described in terms of the relative importance of viscous forces, surface tension, and gravity, which are quantifiable via the dimensionless quantities, namely, capillary number, Bond number and viscosity ratio between displaced fluid and displacing fluid. In addition, the effect of wettability on flow behaviour of fluids is investigated for the cases with and without consideration of gravity, respectively. The numerical results provide a good understanding of the mechanisms of viscous fingering phenomenon from a mesoscopic point of view and confirm that the LBM can be viewed as a promising tool for investigating fluid behaviour and other immiscible displacement problems.     

An evaluation of lattice Boltzmann schemes for porous medium flow simulation

We quantitatively evaluate the capability and accuracy of the lattice Boltzmann equation (LBE) for modeling flow through porous media. In particular, we conduct a comparative study of the LBE models with the multiple-relaxation-time (MRT) and the Bhatnagar–Gross–Krook (BGK) single-relaxation-time (SRT) collision operators. We also investigate several fluid–solid boundary conditions including: (1) the standard bounce-back (SBB) scheme, (2) the linearly interpolated bounce-back (LIBB) scheme, (3) the quadratically interpolated bounce-back (QIBB) scheme, and (4) the multi-reflection (MR) scheme. Three-dimensional flow through two porous media—a body-centered cubic (BCC) array of spheres and a random-sized sphere-pack—are examined in this study. For flow past a BCC array of spheres, we validate the linear LBE model by comparing its results with the nonlinear LBE model. We investigate systematically the viscosity-dependence of the computed permeability, the discretization error, and effects due to the choice of relaxation parameters with the MRT and BGK schemes. Our results show unequivocally that the MRT–LBE model is superior to the BGK–LBE model, and interpolation significantly improves the accuracy of the fluid–solid boundary conditions.     

Comparison of two mathematical models for solving the dam break problem using the FEM method

This paper describes and compares two mathematical models for solving the dam break problem using the FEM. The first method solves the Navier–Stokes equations considering two immiscible and incompressible fluids, assigning the density and viscosity according to an indicator function whose zero level set determines the interface position. The second method is based on solving the Shallow Water equations using the well known Two-Step Taylor–Galerkin method. In this case, the problem becomes one of a drying and wetting one. This problem is simply handled by assigning null depth and velocity of the calculated dry nodes when performing the integrals over the elements in the mesh.The comparison of the results from the two methods for a number of test cases reveals the shortcomings of the Shallow Water approach. However, when considering the computational effort required by each method, the Shallow Water approach should be considered as a major candidate for computations involving large domains, leaving the Navier–Stokes approach for fine calculations where knowledge of the 3D structure of the flow is required.     

Towards a continuous microfluidic rheometer

In a previous paper we presented a way to measure the rheological properties of complex fluids on a microfluidic chip (Guillot et al., Langmuir 22:6438, 2006). The principle of our method is to use parallel flows between two immiscible fluids as a pressure sensor. In fact, in a such flow, both fluids flow side by side and the size occupied by each fluid stream depends only on both flow rates and on both viscosities. We use this property to measure the viscosity of one fluid knowing the viscosity of the other one, both flow rates and the relative size of both streams in a cross-section. We showed that using a less viscous fluid as a reference fluid allows to define a mean shear rate with a low standard deviation in the other fluid. This method allows us to measure the flow curve of a fluid with less than 250 μL of fluid. In this paper we implement this principle in a fully automated set up which controls the flow rate, analyzes the picture and calculates the mean shear rate and the viscosity of the studied fluid. We present results obtained for Newtonian fluids and complex fluids using this set up and we compare our data with cone and plate rheometer measurements. By adding a mixing stage in the fluidic network we show how this set up can be used to characterize in a continuous way the evolution of the rheological properties as a function of the formulation composition. We illustrate this by measuring the rheological curve of four formulations of polyethylene oxide solution with only 1.3 mL of concentrated polyethylene oxide solution. This method could be very useful in screening processes where the viscosity range and the behavior of the fluid to an applied stress must be evaluated.