基于水平集方法的毛管上升特征研究

毛管上升现象与许多行业密切相关,系统地对此现象进行研究具有重大意义。与传统理论研究方法不同,本文使用N-S方程耦合水平集方法模拟毛管气液上升行为。通过与简化条件的解析解对比,验证了模拟方法的可靠性。此外,详细地研究了毛管振荡现象,并分析了影响毛管振荡行为的主要因素。结果表明,水平集方法能够精确地表征毛管振荡现象,与数值解相比具有更高的精度。毛管长度的增加能够减弱液柱振荡,主要归结于非湿相气体的粘滞力作用;湿相密度和湿相粘度同样对毛管振荡现象影响显著。湿相密度越大,惯性力越大,促进了毛管振荡;而湿相粘度变大,会增大粘滞力作用,因此减弱了毛管振荡现象。毛管振荡是由多种影响因素共同控制的,流体的惯性力是造成毛管振荡的主要原因,而粘滞力是减弱毛管振荡行为的主要因素,使液柱振荡逐渐衰减,并稳定至平衡高度。     

Forced convective film condensation inside vertical tubes

A theoretical study of forced convective film condensation inside vertical tubes is presented. We propose a unified procedure for predicting the pressure gradient and condensation heat transfer coefficient of a vapor flowing turbulently in the core and associated with laminar or turbulent film on the tube wall. The analysis for the vapor flows is performed under the condition that the velocity profiles are locally self-similar. The laminar and turbulent film models equate the gravity, pressure and viscous forces, and consider the effect of interfacial shear. The transition from laminar to turbulent film depends not only on the liquid Reynolds number but also on the interfacial shear stress. In this work we also proposed a new eddy viscosity model which is divided into three regions: the inner region in liquid condensate near the wall; the interface region including both liquid and vapor; and the outer region for the vapor core. Comparisons of the theory with some published experimental data showed good agreement.     

A Non-isothermal Pore Network Drying Model with Gravity Effect

The concept of immiscible displacement as an invasion percolation (IP) process driven by heat and mass transfer is used in a pore network model for convective drying of capillary porous media. The coupling between heat and mass transfer occurs at the liquid–gas interface through temperature-dependent equilibrium vapor pressure and surface tension as well as the phase change enthalpy (in evaporation and condensation). The interfacial effects due to capillary forces and gravity are combined in an invasion potential; viscous forces are neglected. Simulation results show stabilized invasion patterns and finite drying front width by the influence of gravity.     

Stability of a liquid jet into incompressible gases and liquids: Part 2. Effects of the irrotational viscous pressure

In this paper we investigate the effects of an irrotational, viscous pressure on the stability of a liquid jet into gases and liquids. The analysis extends our earlier work (part 1) in which the stability of the viscous jet was studied assuming that the motion and pressure are irrotational and the viscosity enters through the jump in the viscous normal stress in the normal stress balance at the interface. The liquid jet is always unstable; at high Weber numbers the instability is dominated by capillary instability; at low W the instability is dominated by Kelvin–Helmholtz (KH) waves generated by pressures driven by the discontinuous velocity. In the irrotational analysis the viscosity is important but the effects of shear are neglected. In fact a discontinuous velocity is not compatible with the continuity of the tangential components of velocity and shear stress so that KH instability is not properly posed for exact study using the no-slip condition but some of the effects of viscosity can be ascertained using viscous potential flow. The theory is called viscous potential flow (VPF). Here we develop another irrotational theory in which the discontinuities in the irrotational tangential velocity and shear stress are eliminated in the global energy balance by selecting viscous contributions to the irrotational pressure. These pressures generate a hierarchy of potential flows in powers of the viscosity, but only the first one, linear in viscosity, in the irrotational viscous stress, is thought to have physical significance. The tangential velocity and shear stress in an irrotational study cannot be made continuous, but the effects of the discontinuous velocity and stress in the mechanical energy balance can be removed “in the mean.” This theory with the additional viscous pressure is called VCVPF, viscous correction of VPF. VCVPF is VPF with the additional pressures. The theory here cannot be compared with an exact solution, which would not allow the discontinuous velocity and stress. In other problems, like capillary instability, in which VCVPF can be compared with an exact solution, the agreements are uniformly excellent in the wave number when one of the fluids is gas and in good but not uniform, agreement when both fluids are liquids.     

Shock calculation based on second viscosity using local differential quadrature method

Based on the second viscosity, the local differential quadrature (LDQ) method is applied to solve shock tube problems. It is shown that it is necessary to consider the second viscosity to calculate shocks and to simulate shock tubes based on the viscosity model. The roles of the shear viscous stress and the second viscous stress are checked. The results show that the viscosity model combined with the LDQ method can capture the main characteristics of shocks, and this technique is objective and simple.     

Large eddy simulation and experiment of shear breakup in liquid-liquid jet: Formation of ligaments and droplets

Understanding the shear breakup in jet flows and the formation of droplets from ligaments is important to determine the final droplet size distribution (DSD). The initial droplet size, which affects the final DSD, is considered to be generated by the shear breakup. Large eddy simulation (LES) was performed to investigate the shear breakup in liquid-liquid jet flows. The explicit Volume of Fluid (VOF) model with the geometric reconstruction scheme was used to capture the oil-water interface. The estimated oil distribution including wave peaks, ligaments, droplets and water streaks were compared to the experiments with a good agreement. The estimated DSD matched with the measurements favorably well. In the simulation, the formation of droplets with a smooth and curved surface from ligaments or sheet-like structures was obtained. Different mechanisms were observed along with the shear layer including the formation of droplets from ligament through the capillary forces, breakage of a droplet into smaller ones and attachment of a droplet to a ligament. The destructive shear forces and resisting surface tension forces were quantified on stretching and retracting ligaments. The influence of internal viscous force was found to be negligible due to low oil viscosity. The critical capillary number was found to be larger than 5.0 for ligaments breaking with the shear breakup. The capillary number was below unity for retracting ligaments. The coalescence of two equal-sized droplets was obtained in the shear breakup region. The shear stress magnitude at the contact region increased more than two folds. The total surface area decreased nearly 20% after the coalescence.     

A Dynamic Pore Network Model for Oil Displacement by Wettability-Altering Surfactant Solution

A dynamic pore network model, capable of predicting the displacement of oil from a porous medium by a wettability-altering and interfacial tension reducing surfactant solution, is presented. The key ingredients of the model are (1) a dynamic network model for the displacement of oil by aqueous phase taking account of capillary and viscous effects, (2) a simulation of the transport of surfactant through the network by advection and diffusion taking account of adsorption on the solid surface, and (3) the coupling of these two by linking the contact angle and interfacial tension appearing in the dynamic network simulation to the local concentration of surfactant computed in the transport simulation. The coupling is two-way: The flow field used to advect the surfactant concentration is that associated with the displacement of oil by the injected aqueous phase, and the surfactant concentration influences the flow field through its effect on the capillarity parameters. We present results obtained using the model to validate that it reproduces the displacement patterns observed by other authors in two-dimensional networks as capillary number and mobility ratio are varied, and to illustrate the effects of surfactant on displacement patterns. A mechanism is demonstrated whereby in an initially mixed-wet medium, surfactant-induced wettability alteration can lead to stabilization of displacement fronts.     

Pore-Scale Simulations of Gas Displacing Liquid in a Homogeneous Pore Network Using the Lattice Boltzmann Method

A lattice Boltzmann high-density-ratio model, which uses diffuse interface theory to describe the interfacial dynamics and was proposed originally by Lee and Liu (J Comput Phys 229:8045–8063, 2010), is extended to simulate immiscible multiphase flows in porous media. A wetting boundary treatment is proposed for concave and convex corners. The capability and accuracy of this model is first validated by simulations of equilibrium contact angle, injection of a non-wetting gas into two parallel capillary tubes, and dynamic capillary intrusion. The model is then used to simulate gas displacement of liquid in a homogenous two-dimensional pore network consisting of uniformly spaced square obstructions. The influence of capillary number (Ca), viscosity ratio ( $M$ M ), surface wettability, and Bond number (Bo) is studied systematically. In the drainage displacement, we have identified three different regimes, namely stable displacement, capillary fingering, and viscous fingering, all of which are strongly dependent upon the capillary number, viscosity ratio, and Bond number. Gas saturation generally increases with an increase in capillary number at breakthrough, whereas a slight decrease occurs when Ca is increased from $8.66\times 10^{-4}$ 8.66 × 10 - 4 to $4.33\times 10^{-3}$ 4.33 × 10 - 3 , which is associated with the viscous instability at high Ca. Increasing the viscosity ratio can enhance stability during displacement, leading to an increase in gas saturation. In the two-dimensional phase diagram, our results show that the viscous fingering regime occupies a zone markedly different from those obtained in previous numerical and experimental studies. When the surface wettability is taken into account, the residual liquid blob decreases in size with the affinity of the displacing gas to the solid surface. Increasing Bo can increase the gas saturation, and stable displacement is observed for $Bo>1$ B o > 1 because the applied gravity has a stabilizing influence on the drainage process.     

Shear cell rupture of nematic liquid crystal droplets in viscous fluids

We model the hydrodynamics of a shear cell experiment with an immiscible nematic liquid crystal droplet in a viscous fluid using an energetic variational approach and phase-field methods [86]. The model includes the coupled system for the flow field for each phase, a phase-field function for the diffuse interface and the orientational director field of the liquid crystal phase. An efficient numerical scheme is implemented for the two-dimensional evolution of the shear cell experiment for this initial data. The same model reduces to an immiscible viscous droplet in a viscous fluid, which we simulate first to compare with other numerical and experimental behavior. Then we simulate drop deformation by varying capillary number (independent of liquid crystal physics), liquid crystal interfacial anchoring energy and Oseen–Frank distortional elastic energy. We show the number of eventual droplets (one to several) and “beads on a string” behavior are tunable with these three physical parameters. All stable droplets possess signature quadrupolar shear and normal stress distributions. The liquid crystal droplets always possess a global surface defect structure, called a boojum, when tangential surface anchoring is imposed. Boojums [79], [32] consist of degree +1/2 and ?1/2 surface defects within a bipolar global orientational structure.     

Rheology of an emulsion of viscoelastic drops in steady shear

Steady shear rheology of a dilute emulsion with viscoelastic inclusions is numerically investigated using direct numerical simulations. Batchelor's formulation for rheology of a viscous emulsion is extended for a viscoelastic system. Viscoelasticity is modeled using the Oldroyd-B constitutive equation. A front-tracking finite difference code is used to numerically determine the drop shape, and solve for the velocity and stress fields. The effective stress of the viscoelastic emulsion has three different components due to interfacial tension, viscosity difference (not considered here) and the drop phase viscoelasticity. The interfacial contributions – first and second normal stress differences and shear stresses – vary with Capillary number in a manner similar to those of a Newtonian system. However the shear viscosity decreases with viscoelasticity at low Capillary numbers, and increases at high Capillary numbers. The first normal stress difference due to interfacial contribution decreases with increasing drop phase viscoelasticity. The first normal stress difference due to the drop phase viscoelasticity is found to have a complex dependence on Capillary and Deborah numbers, in contrast with the linear mixing rule. Drop phase viscoelasticity does not contribute significantly to effective shear viscosity of the emulsion. The total first normal stress difference shows an increase with drop phase viscoelasticity at high Capillary numbers. However at low Capillary numbers, a non-monotonic behavior is observed. The results are explained by examining the stress field and the drop shape.     

Influence of photodegradation and aging on the viscosity of antimisting polymer solutions

The effect of aging and exposure to light on the viscosity of an antimisting polymer solution was investigated in laminar capillary flows. A particular feature of this solution is the abrupt increase in its viscosity that occurs if a critical shear rate is exceeded. Whereas storage in the dark induced only slight modification of the viscous characteristics of the samples, exposure to light was found to increase the value of the critical shear rate, substantially decrease the maximum post-gelation viscosity and reduce the susceptiblity to high-shear mechanical degradation.     

Is there a relation between the relaxation time measured in CaBER experiments and the first normal stress coefficient?

We investigate a variety of different semidilute polymer solutions in shear and elongational flow. The shear flow is created in the cone-plate-geometry of a commercial rheometer. We use capillary thinning of a filament that is formed by a polymer solution in the Capillary Breakup Extensional Rheometer (CaBER) as an elongational flow. We compare the relaxation time measured in the CaBER with relaxation times based on the first normal stress difference and the zero shear polymer viscosity that we measure in our rheometer. All of these three measurable quantities depend on different fluid parameters—the viscosity of the solvent, the polymer concentration within the solution, and the molecular weight of the polymers—and on the shear rate (in the shear flow measurements). Nevertheless, we find that the first normal stress coefficient depends quadratically on the CaBER relaxation time. Several scaling laws are presented that could help to explain this empirical relation.     

Relative Permeability Analysis of Tube Bundle Models

The analytical solution for calculating two-phase immiscible flow through a bundle of parallel capillary tubes of uniform diametral probability distribution is developed and employed to calculate the relative permeabilities of both phases. Also, expressions for calculating two-phase flow through bundles of serial tubes (tubes in which the diameter varies along the direction of flow) are obtained and utilized to study relative permeability characteristics using a lognormal tube diameter distribution. The effect of viscosity ratio on conventional relative permeability was investigated and it was found to have a significant effect for both the parallel and serial tube models. General agreement was observed between trends of relative permeability ratios found in this work and those from experimental results of Singhal et al. (1976) using porous media consisting of mixtures of Teflon powder and glass beads. It was concluded that neglecting the difference between the average pressure of the non-wetting phase and the average pressure of the wetting phase (the macro-scale capillary pressure) – a necessary assumption underlying the popular analysis methods of Johnson et al. (1959) and Jones and Roszelle (1978) – was responsible for the disparity in the relative permeability curves for various viscosity ratios. The methods therefore do not account for non-local viscous effects when applied to tube bundle models. It was contended that average pressure differences between two immiscible phases can arise from either capillary interfaces (micro-scale capillary pressures) or due to disparate pressure gradients that are maintained for a flow of two fluids of viscosity ratio that is different from unity.     

Relative permeability and capillary pressure functions of porous media as related to the displacement growth pattern

Visualization experiments of the unsteady immiscible displacement of a fluid by another are performed on glass-etched pore networks of well-controlled morphology by varying the fluid system and flow conditions. The measured transient responses of the fluid saturation and pressure drop across the porous medium are introduced into numerical solvers of the macroscopic two-phase flow equations to estimate the non-wetting phase, k_rnw, and wetting phase, k_rw, relative permeability curves and capillary pressure, P_c, curve. The correlation of k_rnw, k_rw, and P_c with the displacement growth pattern is investigated. Except for the capillary number, wettability, and viscosity ratio, the immiscible displacement growth pattern in a porous medium may be governed by the shear-thinning rheology of the injected or displaced fluid, and the porous sample length as compared to the thickness of the frontal region. The imbibition k_rnw increases as the flow pattern changes from compact displacement to viscous fingering or from viscous to capillary fingering. The imbibition k_rw increases as the flow pattern changes from compact displacement or capillary fingering to viscous fingering. As the shear-thinning behaviour of the NWP strengthens and/or the contact angle decreases, then the flow pattern is gradually dominated by irregular interfacial configurations, and the imbibition k_rnw increases. The imbibition P_c is a decreasing function of the capillary number or increasing function of the injected phase viscosity in agreement with the linear thermodynamic theory.     

Shear Rate Determination from Pore-Scale Flow Fields

Aqueous solutions with polymer additives often used to improve the macroscopic sweep efficiency in oil recovery typically exhibit non-Newtonian rheology. In order to predict the Darcy-scale effective viscosity \(\mu _{\mathrm{eff}} \) required for practical applications often, semi-empirical correlations such as the Cannella or Blake–Kozeny correlation are employed. These correlations employ an empirical constant (“C-factor”) that varies over three orders of magnitude with explicit dependency on porosity, permeability, fluid rheology and other parameters. The exact reasons for this dependency are not very well understood. The semi-empirical correlations are derived under the assumption that the porous media can be approximated by a capillary bundle for which exact analytical solutions exist. The effective viscosity \(\mu _{\mathrm{eff}} (v_{\mathrm{Darcy}} )\) as a function of flow velocity is then approximated by a cross-sectional average of the local flow field resulting in a linear relationship between shear rate \(\gamma \) and flow velocity. Only with such a linear relationship, the effective viscosity can be expressed as a function of an average flow rate instead of an average shear rate. The local flow field, however, does in general not exhibit such a linear relationship. Particularly for capillary tubes, the velocity is maximum at the center, while the shear rate is maximum at the tube wall indicating that shear rate and flow velocity are rather anti-correlated. The local flow field for a sphere pack is somewhat more compatible with a linear relationship. However, as hydrodynamic flow simulations (using Newtonian fluids for simplicity) performed directly on pore-scale resolved digital images suggest, flow fields for sandstone rock fall between the two limiting cases of capillary tubes and sphere packs and do in general not exhibit a linear relationship between shear rate and flow velocity. This indicates that some of the shortcomings of the semi-empirical correlations originate from the approximation of the shear rate by a linear relationship with the flow velocity which is not very well compatible with flow fields from direct hydrodynamic calculations. The study also indicates that flow fields in 3D rock are not very well represented by capillary tubes.     

Linear rheology of compressible soft nanocomposites

The influences of interfacial tension and compressibility to the linear viscoelastic properties of nanocomposite and nanoporous materials are considered theoretically. The effective bulk and shear moduli of the systems are calculated within the generalized composite sphere model which takes into account the effect of interfacial tension. It is found that frequency dependence of the effective dynamic shear and bulk moduli of nanocomposites with the compressible elastic matrix and viscous inclusions may be represented in terms of the Zener model comprising of the viscoelastic Kelvin element in series with the elastic spring. The relations of the Zener model parameters with the material characteristics are revealed. The physical interpretation of the frequency behavior of the dynamic shear and bulk moduli against the interfacial tension, component compressibility, viscosity, and inclusion volume fraction is discussed. Victor G. Oshmyan deceased.     

Compositional gravity drainage 1. Equilibrium solutions and controlling Bond numbers for a two-phase,three-component system

The amount of wetting phase that is recovered by gravity drainage is determined by an interplay of gravitational and capillary forces. The relative importance of those forces is often expressed in terms of a Bond number. For compositional gravity drainage, where the initial and displacing fluids are not in chemical equilibrium, there is no single Bond number, as phases that form during a displacement will be associated with a different interfacial tension and density for each equilibrium tie line encountered as the compositions change during flow. We study vertical compositional displacements to determine how the Bond numbers of the initial and displacing fluids control the ultimate recovery. We find analytical solutions to the capillary/gravity equilibrium for a simplified model three-component, two-phase system. The equilibrium phase composition versus distance profiles are different than those predicted from standard viscous dominated displacements. We calculate the recovery as a function of the Bond numbers of the initial and displacing phases, and the degree of diffusion for this simple system. We discuss the important role of molecular diffusion in the ultimate recovery for condensing displacements. Finally, we find that the simple numerical average of the Bond numbers provides a reasonable estimate of an effective Bond number for calculating the retained wetting phase for many compositional gravity drainages.     

Imbibition of Anionic Surfactant Solution into Oil-Wet Capillary Tubes

The displacement of oil by anionic surfactant solutions in oil-wet horizontal capillary tubes is studied. The position of the oil–water interface is recorded with time. The surfactant solution used is a mixture of several different surfactants and co-solvents tailored to produce ultra-low interfacial tension (IFT) for the specific oil used in the study. The surfactant solution results in ultra-low IFT at optimum salinity and room temperature. Several experimental parameters including the capillary tube radius and surfactant solution viscosity are varied to study their effect on the interface speed. Two different models are used to predict the oil–water interface position with time. In the first model, it is assumed that the IFT is constant and ultra-low throughout the experiments. The second model involves change of wettability and IFT by adsorption of surfactant molecules to the oil–water interface and the solid surface. Comparing the predictions to the experimental results, it is observed that the second model provides a better match, especially for smaller capillary tubes. The model is then used to predict the imbibition rate for very small capillary tubes, which have equivalent permeability close to oil reservoirs. The results show that the oil displacement rate is limited by the rate of diffusion of surfactants to the interface.     

Rheological characterization of commercial highly viscous alginate solutions in shear and extensional flows

The rheological properties of sodium alginate in salt-free solutions were studied by steady shear, dynamic oscillatory and extensional measurements. This biopolymer consists of mannuronic and guluronic acid residues that give a polyelectrolyte character. We applied the scaling theories and checked their accordance with polyelectrolyte behaviour for low concentrations with a shift to neutral polymer behaviour at larger concentrations. This nature was supported by the effect of the concentration on the specific viscosity, the relaxation times from steady shear and the longest relaxation times from small amplitude oscillatory shear (SAOS) measurements. To analyze the extensional behaviour of the samples, we conducted a study of dimensionless numbers and time scales where filament thinning driven by viscous, capillary or elastic forces is at play. We conclude that an exponential filament thinning followed by breakup results in the best regimes that describe the experimental data. Besides, the data pointed out that alginate in salt-free concentrated solutions shows strain thinning of the extensional viscosity and chain rigidity, behaviours that cannot be inferred from the shear rheometry.