Stabilizing effect of diffusion in enhanced oil recovery and three-layer Hele-Shaw flows with viscosity gradient

In the presence of diffusion, stability of three-layer Hele-Shaw flows which models enhanced oil recovery processes by polymer flooding is studied for the case of variable viscosity in the middle layer. This leads to the coupling of the momentum equation and the species advection-diffusion equation the hydrodynamic stability study of which is presented in this paper. Linear stability analysis of a potentially unstable three-layer rectilinear Hele-Shaw flow is used to examine the effects of species diffusion on the stability of the flow. Using a weak formulation of the disturbance equations, upper bounds on the growth rate of individual disturbances and on the maximal growth rate over all possible disturbances are found. Analytically, it is shown that a short-wave disturbance if unstable can be stabilized by mild diffusion of species, where as an unstable long-wave disturbance can always be stabilized by strong diffusion of species. Thus, an otherwise unstable three-layer Hele-Shaw flow can be completely stabilized by a large enough diffusion, i.e., by increasing enough the magnitude of the species diffusion coefficient. The magnitude of this diffusion coefficient required to completely stabilize the flow will depend on the magnitude of interfacial viscosity jumps and the viscosity gradient of the basic viscous profile of the middle layer.     

Miscible displacements in Hele–Shaw cells: Nonmonotonic viscosity profiles

The influence of nonmonotonic viscosity–concentration relationships on viscous fingering of neutrally buoyant, miscible fluids in a Hele–Shaw cell has been investigated. In a first step, quasisteady base states are obtained by means of nonlinear Stokes simulations. The properties of these base states are analyzed as a function of the Péclet number, the viscosity ratio, and the profile parameters. Subsequently, the stability of these base states is investigated by means of a linear stability analysis. Overall, the nonmonotonicity of the viscosity–concentration relationship is seen to have a much smaller influence on Hele–Shaw displacements than on corresponding Darcy flows. The reason for this difference lies in the nature of the respective base states. For Darcy flows, the base state is characterized by constant velocity and a diffusively decaying concentration (and hence viscosity) profile. This base viscosity profile is strongly affected by the nonmonotonicity. On the other hand, for Hele–Shaw displacements the quasisteady base states are convectively dominated and characterized by sharp fronts, so that their shape depends only weakly on the details of the viscosity–concentration relationship. Hence, for Hele–Shaw displacements both the eigenfunctions and the associated growth rates are quite similar for monotonic and nonmonotonic profiles, in contrast to the findings by [O. Manickam, G.M. Homsy, Stability of miscible displacements in porous media with nonmonotonic viscosity profiles, Phys. Fluids A 5 (1993) 1356–1367] for Darcy flows.     

On the stability of finite difference method for solving three-dimensional unsteady gasdynamic equations with artificial viscosity terms

The artificial viscosity method for three—dimensional unsteady gas flow is developed. The stability of finite difference scheme in this case is investigated. The necessary and sufficient conditions for the stability are obtained; these conditions formally agree with the two-dimensional result in Rusanov's paper.     

Stability of the Couette flow of ideal rigid-plastic bodies

The generalized Rayleigh stability problem is studied for the plane flows of ideal rigid-plastic bodies. The stationary scattering theory is used for the Couette flow to describe the structure of continuous and point spectra and to construct an expansion in eigenfunctions and in generalized eigenfunctions. Some integral estimates are proposed for the domain containing the spectrum of the problem to prove the stability of this flow.     

Three-Phase Fractional Flow Analysis for Foam-Assisted Non-aqueous Phase Liquid (NAPL) Remediation

Among numerous foam applications in a wide range of disciplines, foam flow in porous media has been spotlighted for improved/enhanced oil recovery processes and shallow subsurface in situ NAPL (non-aqueous phase liquid) remediation, where foams can reduce the mobility of gas phase by increasing effective gas viscosity and improve sweep efficiency by mitigating subsurface heterogeneity. This study investigates how foams interact with and displace oleic contaminants in remediation treatments by using MoC (Method of Characteristics)-based three-phase fractional flow theory. Six different scenarios are considered such as different levels of foam strength (i.e., gas mobility reduction factors), different initial conditions (i.e., initially oil/water or oil/water/gas present), foam stability affected by water saturation $({S}_{\mathrm{w}})$ and oil saturation $({S}_{\mathrm{o}})$ , and uniform versus non-uniform initial saturations. The process is analyzed by using ternary diagrams, fractional flow curves, effluent histories, saturation profiles, time–distance diagrams, and pressure and recovery histories. The results show that the three-phase fractional flow analysis presented in this study is robust enough to analyze foam–oil displacements in various conditions, as validated by an in-house numerical simulator built in this study. The use of numerical simulation seems crucial when the foam process becomes very complicated and faces multiple possible solutions.     

Exact solutions for fractional diffusion equation in a bounded domain with different boundary conditions

We give an analytical treatment of a time fractional diffusion equation with Caputo time-fractional derivative in a bounded domain with different boundary conditions. We use the Fourier method of separation of variables and Laplace transform method. The solution is obtained in terms of the Mittag-Leffler-type functions and complete set of eigenfunctions of the Sturm–Liouville problem. Such problems can be used in the context of anomalous diffusion in complex media, as well as for modeling voltammetric experiment in limiting diffusion space.     

Energy analysis of development of kinematic perturbations in weakly inhomogeneous viscous fluids

The deformation stability relative to small perturbations is analyzed for weakly inhomogeneous viscous media on the assumption that both the main flow and perturbation field are three-dimensional. To test the damping or growth of initial perturbations, sufficient estimates based on the use of variational inequalities in different function spaces (energy estimates) are obtained. The choice of function space determines the measures of the parameter deviations, which may be different for the initial and current parameters. The unperturbed process chosen is a fairly arbitrary unsteady flow of homogeneous incompressible viscous fluid in a three-dimensional region of Eulerian space. At the initial instant, not only the kinematics of the motion but also the density and viscosity of the fluid are disturbed and the medium is therefore called weakly inhomogeneous. On the basis of the integral relation methods developed in recent years, sufficient integral estimates are obtained for lack of perturbation growth in the mean-square sense (in theL ₂ space measure). The rate of growth or damping of the kinematic perturbations depends linearly on the initial variations of the kinematics, density and viscosity. Illustrations of the general result are given. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 56–67, March–April, 2000. The work was supported by the Russian Foundation for Basic Research (projects No. 99-01-00125 and No. 99-01-00250) and by the Federal Special “Integration” Program (project No. 426).     

The Proper Orthogonal Decomposition, wavelets and modal approaches to the dynamics of coherent structures

We present brief précis of three related investigations. Fuller accounts can be found elsewhere. The investigations bear on the identification and prediction of coherent structures in turbulent shear flows. A second unifying thread is the Proper Orthogonal Decomposition (POD), or Karhunen-Loève expansion, which appears in all three investigations described. The first investigation demonstrates a close connection between the coherent structures obtained using linear stochastic estimation, and those obtained from the POD. Linear stochastic estimation is often used for the identification of coherent structures. The second investigation explores the use (in homogeneous directions) of wavelets instead of Fourier modes, in the construction of dynamical models; the particular problem considered here is the Kuramoto-Sivashinsky equation. The POD eigenfunctions, of course, reduce to Fourier modes in homogeneous situations, and either can be shown to converge optimally fast; we address the question of how rapidly (by comparison) a wavelet representation converges, and how the wavelet-wavelet interactions can be handled to construct a simple model. The third investigation deals with the prediction of POD eigenfunctions in a turbulent shear flow. We show that energy-method stability theory, combined with an anisotropic eddy viscosity, and erosion of the mean velocity profile by the growing eigenfunctions, produces eigenfunctions very close to those of the POD, and the same eigenvalue spectrum at low wavenumbers.Prepared for presentation at International Union of Theoretical and Applied Mechanics Symposium Eddy Structure Identification in Free Turbulent Shear Flows, Poitiers, France, 12–14 October 1992. Supported in part by: the U.S. Air Force Office of Scientific Research, The U.S. Office of Naval Research (Mechanics Branch and Physical Oceanography Program), and the U.S. National Science Foundation (program in Physical Oceanography).     

Electroviscous potential flow in nonlinear analysis of capillary instability

We study the nonlinear stability of electrohydrodynamic of a cylindrical interface separating two conducting fluids of circular cross section in the absence of gravity using electroviscous potential flow analysis. The analysis leads to an explicit nonlinear dispersion relation in which the effects of surface tension, viscosity and electricity on the normal stress are not neglected, but the effect of shear stresses is neglected. Formulas for the growth rates and neutral stability curve are given in general. In the nonlinear theory, it is shown that the evolution of the amplitude is governed by a Ginzburg–Landau equation. When the viscosities are neglected, the cubic nonlinear Schrödinger equation is obtained. Further, it is shown that, near the marginal state, a nonlinear diffusion equation is obtained in the presence of viscosities. The various stability criteria are discussed both analytically and numerically and stability diagrams are obtained. It is also shown that, the viscosity has effect on the nonlinear stability criterion of the system, contrary to previous belief.     

Diffusion Mass Transfer in Miscible Oil Recovery: Visual Experiments and Simulation

Co-and counter-current type transfers due to diffusion and -free- convection caused by the buoyant forces between fracture and matrix were studied experimentally using 2-D glass-bead models. Mineral oil and kerosene were used as the displaced phase. The model saturated with oil was exposed to solvent phase (pentane) under static conditions (no flow in fracture) to mimic matrix-fracture interaction during gas or liquid solvent injection in naturally fractured reservoirs. Displacement fronts and patterns were analyzed and quantified using fractal techniques to obtain correlations between the fractal properties and displacement type. Displacements resulted in a mixture of bulk diffusion and -free- convection mainly depending on the interaction type (co- or counter-current), oil type, and displacement direction (horizontal and vertical). Conditions yielding different types of displacement patterns were identified. Finally, a stochastic model that was inspired from invasion percolation and diffusion limited aggregation algorithms was developed for the horizontal displacement cases. The experimental observations were matched to the displacement patterns obtained through the stochastic modeling.     

Interface stabilization in two-layer channel flow by surface heating or cooling

The linear stability of plane Poiseuille flow of two immiscible Newtonian liquids in a differentially heated channel is considered. The equations of motion and energy are fully coupled via temperature-dependent fluid-viscosity coefficients. A long-wave asymptotic formulation of the stability problem is presented together with numerical results for disturbances of arbitrary wavelength. Two combinations of immiscible liquids are analyzed: silicone/water and oil/water (water at the bottom layer in both cases). It is shown that an imposed wall temperature difference can be stabilizing or destabilizing depending on the disturbance wavenumber and layer thickness ratio. Interfacial tension has a stabilizing effect on the interface. Stabilizing influence of interfacial tension is observed at intermediate and large wavenumbers. Most importantly, for certain ranges of the controlling dimensionless parameters, stable interfaces at all disturbance wavelengths can be attained.     

粘性可压混合层时间稳定性对称紧致差分求解

基于可压扰动方程组的一阶改型 ,将高精度对称紧致格式引入边值法数值线性稳定性分析。对所获非线性离散特征值问题给出了一个通用形式二阶迭代局部算法 ,实现了时间模式和空间模式的统一求解 ,并将扰动特征值及其特征函数同时得到。据此分析了可压平面自由混合层时间稳定性 ,涉及二维 /三维扰动波、粘性 /无粘扰动波、第一 /第二模态、特征函数、伪特征值谱等。研究表明 ,压缩性效应和粘性效应对最不稳定扰动波数和增长率呈相似的减抑作用 ;在 Mc=1附近 ,从高波数段开始 ,粘性效应可强化二维不稳定扰动波由第一模态向第二模态的过渡     

Fluid transfer between tubes in interacting capillary bundle models

The interacting capillary bundle model proposed by Dong et al. [Dong, M., Dullien, F.A.L., Zhou, J.: Trans. Porous Media 31, 213–237 (1998); Dong, M., Dullien, F.A.L., Dai, L., Li, D.: Trans. Porous Media 59, 1–18 (2005); Dong, M., Dullien, F.A.L., Dai, L., Li, D.: Trans. Porous Media 63, 289–304 (2006)] has simulated correctly various aspects of immiscible displacement in porous media, such as oil production histories at different viscosity ratios, the effects of water injection rate and of the oil–water viscosity ratio on the shape of the displacement front and the independence of relative permeabilities of the viscosity ratio. In the interacting capillary bundle model pressure equilibrium was assumed at any distance x measured along the bundle. Interaction between the capillaries also results in transfer of fluids across the capillaries. In the first part of this paper the process of fluid transfer between two capillaries is analysed and an algebraic expression for this flow is derived. Consistency with the assumption of pressure equilibration requires that all transfer must take place at the positions of the oil/water menisci in the tubes without any pressure drop. It is shown that fluid transfer between the tubes has no effect on the predictions obtained with the model. In the second part of the paper the interacting tube bundle model is made more realistic by assuming fluid transfer between the tubes all along the single phase flow regions across a uniform resistance, resulting in pressure differences throughout the single phase regions between the fluids present in the different tubes. The results of numerical simulations obtained with this improved interacting capillary bundle model show only small differences in the positions of the displacement front as compared with the predictions of the idealized model.     

A von Neumann–Smagorinsky turbulent transport model for stratified shear flows

A simple subgrid turbulent diffusion model based on an analogy to the von Neumann–Richtmyer artificial viscosity is explored for use in modelling mixing in turbulent stratified shear flow. The model may be more generally applicable to multicomponent turbulent hydrodynamics and to subgrid turbulent transport of momentum, composition and energy. As in the case of the von Neumann artificial viscosity and many subgrid-scale models for large-eddy simulation, the turbulent diffusivity explicitly depends on the grid size and is not based on a quantitative model of the unresolved turbulence. In order to address the issue that it is often not known a priori when and where a flow will become turbulent, the turbulent diffusivity is set to zero when the flow is expected to be stable on the basis of a Richardson/Rayleigh–Taylor stability criterion, in analogy to setting the von Neumann artificial viscosity to zero in expanding flows. One-dimensional predictions of this model applied to a simple shear flow configuration are compared to those obtained using a K–ε model. The density and velocity profiles predicted by both models are shown to be very similar.     

Stability of resonant interfacial waves in the presence of uniform magnetic field

The aim of this work is to study the instability of interacting waves between two immiscible magnetic liquids. The effects of gravitation and a uniform normal magnetic field are taken into account. The method of multiple scales is used to determine the stability criteria of the considered problem. The various stability criteria are discussed both analytically and graphically. According to the numerical examples, we have remarked that the increase of the ratio of the permeability of the liquids appears to be the destabilizing effect of the magnetic field. The short waves below the critical wavenumbers are stable whereas a number of long waves are unstable. The viscosity effect on the stability criteria is a dual-role one, depending on the strength of the applied magnetic field.     

Finite element and sensitivity analysis of thermally induced flow instabilities

This paper presents a finite element algorithm for the simulation of thermo‐hydrodynamic instabilities causing manufacturing defects in injection molding of plastic and metal powder. Mold‐filling parameters determine the flow pattern during filling, which in turn influences the quality of the final part. Insufficiently, well‐controlled operating conditions may generate inhomogeneities, empty spaces or unusable parts. An understanding of the flow behavior will enable manufacturers to reduce or even eliminate defects and improve their competitiveness. This work presents a rigorous study using numerical simulation and sensitivity analysis. The problem is modeled by the Navier–Stokes equations, the energy equation and a generalized Newtonian viscosity model. The solution algorithm is applied to a simple flow in a symmetrical gate geometry. This problem exhibits both symmetrical and non‐symmetrical solutions depending on the values taken by flow parameters. Under particular combinations of operating conditions, the flow was stable and symmetric, while some other combinations leading to large thermally induced viscosity gradients produce unstable and asymmetric flow. Based on the numerical results, a stability chart of the flow was established, identifying the boundaries between regions of stable and unstable flow in terms of the Graetz number (ratio of thermal conduction time to the convection time scale) and B, a dimensionless ratio indicating the sensitivity of viscosity to temperature changes. Sensitivities with respect to flow parameters are then computed using the continuous sensitivity equations method. We demonstrate that sensitivities are able to detect the transition between the stable and unstable flow regimes and correctly indicate how parameters should change in order to increase the stability of the flow. Copyright © 2009 John Wiley & Sons, Ltd.     

Fast numerical evaluation of flow fields with vortex cells

A vortex cell (in this paper) is an aerodynamically shaped cavity in the surface of a body, for example a wing, designed specially to trap the separated vortex within it, thus preventing large-scale unsteady vortex shedding from the wing. Vortex stabilisation can be achieved either by the special geometry, as has already been done experimentally, or by a system of active control. In realistic conditions the boundary and mixing layers in the vortex cell are always turbulent. In the present study a model for calculating the flow in a vortex cell was obtained by replacing the laminar viscosity with the turbulent viscosity in the known high-Reynolds-number asymptotic theory of steady laminar flows in vortex cells. The model was implemented numerically and was shown to be faster than solving the Reynolds-averaged Navier–Stokes equations. An experimental facility with a vortex cell was built and experiments performed. Comparisons of the experimental results with the predictions of the model are reasonably satisfactory. The results also indicate that at least for flows in near-circular vortex cells it is sufficient to have accurate turbulence models only in thin viscous layers, while outside the viscosity should only be small enough to make the flow effectively inviscid.     

Stress gradient-induced migration effects in the Taylor–Couette flow of a dilute polymer solution

We present an investigation of the phenomenon of stress-induced polymer migration for dilute polymer solutions in the Taylor–Couette device, consisting of two infinitely long, concentric cylinders rotating at constant angular velocities. The underlying physical model is represented by the dilute limit of a two-fluid Hamiltonian system involving two components: one (the polymer) is viscoelastic and obeys the Oldroyd-B constitutive equation, and the other (the solvent) is viscous Newtonian. The two components are considered to be in thermal, but not mechanical equilibrium, interacting with each other through an isotropic drag coefficient tensor. This allows for stress-induced diffusion of polymer chains. The governing equations consist of the continuity and the momentum equations for the bulk velocity, the constitutive model for the polymer chain conformation tensor and the diffusion equation for the polymer concentration. The diffusion equation contains an extra source term, which is proportional to gradients in the polymer stress, so that polymer concentration gradients can develop even in the absence of externally imposed fluxes in the presence of stress inhomogeneities. The solution to the steady-state purely azimuthal flow is obtained first using a spectral collocation method and an adaptive mesh formulation to track the steep changes of the concentration in the flow domain. The calculations show the development of strong polymer migration towards the inner cylinder with increasing Deborah number (De) in agreement with experimental observations. The migration is enhanced for increasing values of the gap thickness resulting in concentration changes by several orders of magnitude in the area between the inner and outer cylinder walls. The extent of the migration also depends strongly on the ratio of the solvent to the polymer viscosity. In addition to a strongly inhomogeneous polymer concentration, significant deviations from the homogenous flow are also observed in the velocity profile. Next, results are reported from a linear stability analysis around the steady-state solution against axisymmetric disturbances corresponding to various wavenumbers in the axial direction. The calculations show that the steady-state solution remains stable up to moderate values of the Deborah number, explaining why some of the predicted stress-induced migration effects should be experimentally observable. The role of the Peclet number (Pe) on the stability of the system is elucidated.     

On Nonlinear Instability and Stability for Stratified Shear Flow

An example of stratified shear flow is presented in which an explicit construction is given for unstable eigenvalues with smooth eigenfunctions for the Taylor--Goldstein equation. It is proved for any stratified, plane parallel shear flow that the unstable spectrum of the linear operator is purely discrete. A general theorem is then invoked to prove that the specific example is nonlinearly unstable. A sufficient condition for nonlinear stability for stratified shear flow is discussed.     

Dynamic analysis for axially moving viscoelastic panels

In this study, stability and dynamic behaviour of axially moving viscoelastic panels are investigated with the help of the classical modal analysis. We use the flat panel theory combined with the Kelvin–Voigt viscoelastic constitutive model, and we include the material derivative in the viscoelastic relations. Complex eigenvalues for the moving viscoelastic panel are studied with respect to the panel velocity, and the corresponding eigenfunctions are found using central finite differences. The governing equation for the transverse displacement of the panel is of fifth order in space, and thus five boundary conditions are set for the problem. The fifth condition is derived and set at the in-flow end for clamped–clamped and clamped-simply supported panels. The numerical results suggest that the moving viscoelastic panel undergoes divergence instability for low values of viscosity. They also show that the critical panel velocity increases when viscosity is increased and that the viscoelastic panel does not experience instability with a sufficiently high viscosity coefficient. For the cases with low viscosity, the modes and velocities corresponding to divergence instability are found numerically. We also report that the value of bending rigidity (bending stiffness) affects the distance between the divergence velocity and the flutter velocity: the higher the bending rigidity, the larger the distance.