Nonlinear stability analysis of miscible displacements

A simulator for three-dimensional horizontal miscible displacements in porous media is developed. Using this simulator, we examine the initiation and development of instabilities, viscous fingers and gravity tongues.With the only perturbations to the system being truncation and round-off errors, a density ratio (the ratio of the density of the displacing fluid to that of the displaced fluid) different from one is responsible for the initiation of the instabilities, and an unfavorable mobility ratio (the ratio of the viscosity of the displaced fluid to that of the displacing fluid) is responsible for the growth of the instabilities.     

Relationship between spall characteristics and the dimension of fractal fracture structures

The effect of the fractality of a fracture surface and spall contour on the characteristics [fracture time (strength) and spall strength] of the loaded material is studied. It is shown that an increase in the fractal dimensions of the spall contour leads to an increase in the material strength parameter in the tensile wave and spall strength, whereas an increase in the fractal dimension of the fracture surface leads to a decrease in the spall strength. As an example, the spall strength is calculated taking into account the fractality of the fracture surface for Sp. 28 steel.     

无量纲混相驱对流-弥散方程的数值求解

给出了混相驱对流-弥散方程的无量纲形式;采用高阶精度差分格式对其进行数值求解并应用于几个具体算例中。本文数值结果和算例给出的结果对比,充分验证了本文数值模型的可靠性和准确性。该数值模型可以用来研究诸如土壤或地下水污染物输运以及空气污染等领域里的传质问题．     

Variable density and viscosity,miscible displacements in capillary tubes

The displacement of a more viscous fluid by a miscible, less viscous one of lower density in a horizontal capillary tube is studied by means of Stokes flow simulations. Both axisymmetric and three-dimensional simulations are conducted at Péclet numbers up to 10⁴, in order to resolve discrepancies between earlier simulations by [C.Y. Chen, E. Meiburg, Miscible displacements in capillary tubes. Part 2. Numerical simulations, J. Fluid Mech. 326 (1996) 57] and corresponding experiments of [P. Petitjeans, T. Maxworthy, Miscible displacements in capillary tubes. Part 1. Experiments, J. Fluid Mech. 326 (1996) 37] and [J. Kuang, T. Maxworthy, P. Petitjeans, Miscible displacements between silicone oils in capillary tubes, Eur. J. Mech. B Fluids 22 (2003) 271–277]. An initial set of simulations addresses the influence of different viscosity–concentration relations on the quasisteady finger tip velocity. The results indicate that steeper relations generally result in a higher tip velocity. However, the effect is too small to explain the above discrepancies. Further simulations show that a concentration-dependent diffusion coefficient results in a slight reduction of the tip velocity at moderate Pe, but again the effect is too small to fully account for the observed differences.Three-dimensional simulations that include gravitational forces yield a much more significant effect. Consistent with the experiments of [P. Petitjeans, T. Maxworthy, Miscible displacements in capillary tubes. Part 1. Experiments, J. Fluid Mech. 326 (1996) 37], at moderate Pe the tip slows down as the gravity parameter increases, an effect that becomes more pronounced as Pe decreases. However, the three-dimensional simulations do not produce the longitudinal splitting phenomenon observed by [P. Petitjeans, T. Maxworthy, Miscible displacements in capillary tubes. Part 1. Experiments, J. Fluid Mech. 326 (1996) 37]. In order to check for the existence of gravitational instabilities that might cause such a splitting, additional two-dimensional simulations are conducted in cross-sections of the tube. A comparison of these two-dimensional results with corresponding three-dimensional simulations demonstrates that for a wide range of parameters the evolution of the trailing finger sections is governed by a two-dimensional balance between gravitational and viscous forces. However, a gravitational instability along the lines suggested by [P. Petitjeans, T. Maxworthy, Miscible displacements in capillary tubes. Part 1. Experiments, J. Fluid Mech. 326 (1996) 37] was not observed. On the other hand, for some parameter combinations the evolution of a 'dimple' is observed on the lower side of the finger, and close to its tip. This dimple may signal the evolution of a splitting phenomenon after long times, which are beyond the reach of the current simulations. Taken together, the two- and three-dimensional simulations suggest that the splitting phenomenon observed by [P. Petitjeans, T. Maxworthy, Miscible displacements in capillary tubes. Part 1. Experiments, J. Fluid Mech. 326 (1996) 37] likely is caused by the gravity-induced modification of the flow around the tip of the finger, rather than by a gravitational instability per se.     

A computerized technique for measuringin-situ concentrations during miscible displacements in porous media

A technique for measurement of thein-situ concentration in an unconsolidated porous medium has been developed. The method involves measurement of electrical conductivityin-situ, under dynamic conditions, for flow involving brine of differing concentrations, at selected locations along the porous medium and relating it to the brine strength. Data acquisition and analysis is carried out using a Hewlett — Packard micro-computer and its interface. A user-friendly software was designed and developed for the system. The measurement technique was evaluated by studying the effect of brine concentration, brine flow rate, and by conducting miscible displacements experiment. The experimentally measured dispersion coefficients for the porous medium agreed closely with the value predicted by the correlation available in the literature.     

Stability of vertical miscible displacements with developing density and viscosity gradients

Viscous fingering and gravity tonguing are the consequences of an unstable miscible displacement. Chang and Slattery (1986) performed a linear stability analysis for a miscible displacement considering only the effect of viscosity. Here the effect of gravity is included as well for either a step change or a graduated change in concentration at the injection face during a downward, vertical displacement. If both the mobility ratio and the density ratio are favorable (the viscosity of the displacing fluid is greater than the viscosity of the displaced fluid and, for a downward vertical displacement, the density of the displacing fluid is less than the density of the displaced fluid), the displacement will be stable. If either the mobility ratio or the density ratio is unfavorable, instabilities can form at the injection boundary as the result of infinitesimal perturbations. But if the concentration is changed sufficiently slowly with time at the entrance to the system, the displacement can be stabilized, even if both the mobility ratio and the density ratio are unfavorable. A displacement is more likely to be stable as the aspect ratio (ratio of thickness to width, which is assumed to be less than one) is increased. Commonly the laboratory tests supporting a field trial use nearly the same fluids, porous media, and displacement rates as the field trial they are intended to support. For the laboratory test, the aspect ratio may be the order of one; for the field trial, it may be two orders of magnitude smaller. This means that a laboratory test could indicate that a displacement was stable, while an unstable displacement may be observed in the field.     

Velocity fields and streamline patterns of miscible displacements in cylindrical tubes

Displacements of a viscous fluid by a miscible fluid of a lesser viscosity and density in cylindrical tubes were investigated experimentally. Details of velocity and Stokes streamline fields in vertical tubes were measured using a DPIV (digital particle image velocimetry) technique. In a reference frame moving with the fingertip, the streamline patterns around the fingertip obtained from the present measurements confirm the hypothesis of Taylor (1961) for the external patterns, and that of Petitjeans and Maxworthy (1996) for the internal patterns. As discussed in these papers, the dependent variable, m, a measure of the volume of viscous fluid left on the tube wall after the passage of the displacing finger, is a parameter that determines the flow pattern. When m>0.5 there is one stagnation point at the tip of the finger; when m<0.5 there are two stagnation points on the centerline, one at the tip and the other inside the fingertip, and a stagnation ring on the finger surface with a toroidal recirculation in the fingertip between the two stagnation points. The finger profile is obtained from the zero streamline of the streamline pattern.An erratum to this article can be found at     

爆破地震信号的分形盒维数值分析

在岩石场地进行了单段深孔爆破试验,获得了具有该场地特征的爆破地震波传播规律。从理论上推导了爆破地震波振动强度衰减系数K和与分形盒维数D以及盒维数计算中-lg(k) lgNk的双对数拟合直线方程参数b的关系。采用了适用于爆破地震波曲线双尺度特征的矩形盒模型计算了爆破地震波的盒维数D。数据分析表明:在同一场地条件下,爆破地震波的盒维数比较稳定,且爆破地震波振动强度的场地衰减指数与D为两倍的关系;药量和距离对盒维数拟合直线方程参数b的影响明显,且其关系与场地衰减指数对爆破地震波峰值强度的作用相近。通过数据分析,获得了参数b与爆破地震波振动峰值A的关系式:b=0.689lgA+3.0669,其相关系数为0.93。     

On the modelling of miscible displacements in porous media with stagnant fluid

Equations of miscible fluids displacement in porous media presenting a capacitance effect, i.e., porous media with a mobile fraction and a stagnant fraction, are derived by means of a volume or a surface averaging technique in the case of high Peclet numbers. The models thus obtained are constituted by two coupled equations. The first is a convective-dispersive equation related to the transfer in mobile fraction; the second is a first-order rate expression describing mass transfer between the mobile and immobile regions. These derivations justify the equations which can be obtained by means of an heuristic approach and specify their conditions of validity.These models are compared to the models in which the second equation is a diffusion equation; the latter are shown to be erroneous.     

Onset of viscous fingering for miscible liquid-liquid displacements in porous media

The displacement of one fluid by another miscible fluid in porous media is an important phenomenon that occurs in petroleum engineering, in groundwater movement, and in the chemical industry. This paper presents a recently developed stability criterion which applies to the most general miscible displacement. Under special conditions, different expressions for the onset of fingering given in the literature can be obtained from the universally applicable criterion. In particular, it is shown that the commonly used equation to predict the stable velocity ignores the effects of dispersion on viscous fingering.Nomenclature C Solvent concentration - Unperturbed solvent concentration - D _L Longitudinal dispersion coefficient [m²/s] - D _T Transverse dispersion coefficient [m²/s] - g Gravitational acceleration [m/s²] - I _sr Instability number - k Permeability [m²] - K Ratio of transverse to longitudinal dispersion coefficient - L Length of the porous medium [m] - L _x Width of the porous medium [m] - L _y Height of the porous medium [m] - M Mobility ratio - V Superficial velocity [m/s] - V _c Critical velocity [m/s] - V _s Velocity at the onset of instability [m/s] - µ Viscosity [Pa/s] - Unperturbed viscosity [Pa/s] - µ ₀,µ _s Viscosities of oil and solvent, respectively [Pa/s] - Density [kg/m³] - ₀, _s Densities of oil and solvent, respectively [kg/m³] - Porosity - Dimensionless length     

An experimental study of non-isothermal miscible displacements in a Hele-Shaw cell

Non-isothermal miscible displacements in a radial Hele-Shaw cell were experimentally investigated using a scheme in which room temperature liquids of relatively high viscosity were displaced by high-temperature (80 °C), less-viscous liquids. Fundamental characteristics have been presented regarding how the effect of a non-isothermal field on miscible displacement patterns varies in terms of factors such as the viscosity ratio of the more- and less-viscous liquids at 20 °C, M₂₀, the rate of an increase in the pattern’s area, R, and the gap width of the cell, b. The concept of area density was used to quantitatively evaluate the effect of the non-isothermal fields on the patterns. We have found that the effect of the non-isothermal field on the patterns does not monotonically vary with M₂₀ and b. In contrast, it increases with R in the present experimental condition. The experimental results can be explained by introducing an assumption in which heat is transferred mainly to the plates of the cell, in other words, the temperature of the more-viscous liquid remains constant, whereas that of the less-viscous liquid spatiotemporally decreases and the viscosity of it increases along with the temperature decrease. Visualization of non-isothermal field in the cell has been done by means of a thermo sheet and the results support the assumption mentioned above.     

Capillary fingering: Percolation and fractal dimension

We present experimental and theoretical results concerning immiscible displacements (drainage) in 2-dimensional permeable media. When capillary forces are predominant, the injected fluid presents very thin fingers and the Representative Elementary Volume concept cannot be used for describing the partial saturations. The purpose of this paper is to show how this classical concept can be replaced by a statistical approach based on fractal geometry.Communication presented at the International Symposium on the Stochastic Approach to Subsurface Flow, Montvillargenne 4–7 June 1985.     

Explanation of scaling phenomenon based on fractal fragmentation

A significant ‘size effect' is observed in tensile strength of solid particles, such as ice, rock, ceramics and concrete: the tensile strength is not independent of the fragment size, but decreases with increasing size. The Weibull statistical theory was universally used to calculate the size effect observed in solid particles. Recent developments in fractal theory suggest that fractals may provide a more realistic representation of solid particles. In this paper, the scaling phenomenon of ice mechanics is explained using the fractal model for ice particle fragmentation. The Weibull statistics is modified using the fractal crushing of ice, and is compared with the conventional one. Goodness-of-fit statistics show that the modified Weibull statistics fits the experimental data of ice much better than the conventional one. The modified Weibull statistics has only one parameter, the fractal dimension of the fragment size distribution, which has a general value of 2.50 for the ice fragmentations.     

Evaluation of fractal dimension of soft terrain surface

By using a laser profiler, the roughness of sowed and plowed surfaces was obtained. Through evaluation of the precision of fractal dimensions based on the Weierstrass–Mandelbort (W–M) function, we found the rescaled (R/S) analysis method and the structure spectral method were not suitable for the calculation of the fractal dimension on a soft terrain surface. Therefore, the fractal dimension, non-scale range and correlation coefficient for each kind of terrain were analyzed using the following fractal computational methods – (i) variate-difference method, (ii) power spectral density method, and (iii) root mean square method. The results showed that: fractal dimension of plowed terrain was large with small fluctuations, while its internal structure was complex. The power spectral density method was not robust enough to compute the fractal dimension of a soft terrain surface. The fractal dimension computed using the root mean square method was found to be more accurate for the soft terrain surface. Moreover, the correlation coefficient of linear regression when using the root mean square method was good and the range of non-scale variation was small.     

An extended theory to predict the onset of viscous instabilities for miscible displacements in porous media

Many enhanced oil recovery schemes involve the displacement of oil by a miscible fluid. Whether a displacement is stable or unstable has a profound effect on how efficiently a solvent displaces oil within a reservoir. That is, if viscous fingers are present, the displacement efficiency and, hence, the economic return of the recovery scheme is seriously impaired bacause of macroscopic bypassing of the oil. As a consequence, it is of interest to be able to predict the boundary which separates stable displacements from those which are unstable.This paper presents a dimensionless scaling group for predicting the onset of hydrodynamic instability of a miscible displacement in porous media. An existing linear perturbation analysis was extended in order to obtain the scaling group. The new scaling group differs from those obtained in previous studies because it takes into account a variable unperturbed concentration profile, both transverse dimensions of the porous medium, and both the longitudinal and the transverse dispersion coefficient.It has been shown that stability criteria derived in the literature are special cases of the general condition given here. Therefore, the stability criterion obtained in this study should be used for a displacement conducted under arbitrary conditions. The stability criterion is verified by comparing it with miscible displacement experiments carried out in a Hele-Shaw cell. Moreover, a comparison of the theory with some porous medium experiments from the literature also supports the validity of the theory.Nomenclature c solvent concentration - C _g fractional glycerine volume - D molecular diffusion coefficient, cm²/s - D _L longitudinal dispersion coefficient, cm²/s - D _T transverse dispersion coefficient, cm²/s - g gravitational acceleration, cm/s² - h distance between the plates, cm - I _sr dimensionless scaling group - k permeability, cm² - L _x width of the porous medium, cm - L _y height of the porous medium, cm - t time, s - u velocity in thex direction, cm/s - v velocity in they direction, cm/s - V displacement velocity, cm/s - w velocity in thez direction, cm/s - z length of the graded viscosity bank, cm - eigenvalue in thex direction - eigenvalue in they direction - wave number - viscosity, poise - density, g/cc - time constant, s^-1 - porosity     

Evaluation of fractal dimension for mixing and combustion by the schlieren method

The evaluation of fractal dimension values from schlieren flow images has been investigated. It was found for passive mixing heated jet flows that a value of 2.31, close to that obtained by tomographic imaging, was obtained. For turbulent diffusion flames a value of 2.40 was obtained, and this increased slightly with axial movement and with acoustic excitation of the flame. Broadly counteracting effects due to the transverse differentiation and integration along the imaging beam inherent in the schlieren method was found to arise.     

利用分形相关法测量股骨头剖开面的面内位移

由于股骨头内松质骨结构的复杂性,对股骨头内骨小梁的应力和变形的分析与测试十分困难。在已经开展的这一方面的研究工作中主要是进行松质骨试样的宏观力学性能测试。随着计算机和图像技术的发展,图像及图像相关方法开始被用来进行松质骨的生物力学研究,这是非接触的测量方法,可以用来测量松质骨的表面位移场。本文利用图像相关方法在宏观尺度下测量了股骨头受外力作用时冠状面的面内的位移分布,作为进一步探索的开端。尽管松质骨表面的凹凸不平性对测量精度有影响,但是,还是可以分析出面内位移的分布特点。