Showing papers on "Viscous fingering published in 1988"

Numerical models and experiments on immiscible displacements in porous media

Abstract: Immiscible displacements in porous media with both capillary and viscous effects can be characterized by two dimensionless numbers, the capillary number C, which is the ratio of viscous forces to capillary forces, and the ratio M of the two viscosities. For certain values of these numbers, either viscous or capillary forces dominate and displacement takes one of the basic forms: (a) viscous fingering, (b) capillary fingering or (c) stable displacement. We present a study in the simple case of injection of a non-wetting fluid into a two-dimensional porous medium made of interconnected capillaries. The first part of this paper presents the results of network simulators (100 × 100 and 25 × 25 pores) based on the physical rules of the displacement at the pore scale. The second part describes a series of experiments performed in transparent etched networks. Both the computer simulations and the experiments cover a range of several decades in C and M. They clearly show the existence of the three basic domains (capillary fingering, viscous fingering and stable displacement) within which the patterns remain unchanged. The domains of validity of the three different basic mechanisms are mapped onto the plane with axes C and M, and this mapping represents the ‘phase-diagram’ for drainage. In the final section we present three statistical models (percolation, diffusion-limited aggregation (DLA) and anti-DLA) which can be used for describing the three ‘basic’ domains of the phase-diagram.

Simulation of nonlinear viscous fingering in miscible displacement

Abstract: The nonlinear behavior of viscous fingering in miscible displacements is studied. A Fourier spectral method is used as the basic scheme for numerical simulation. In its simplest formulation, the problem can be reduced to two algebraic equations for flow quantities and a first‐order ordinary differential equation in time for the concentration. There are two parameters, the Peclet number (Pe) and mobility ratio (M), that determine the stability characteristics. The result shows that at short times, both the growth rate and the wavelength of fingers are in good agreement with predictions from our previous linear stability theory. However, as the time goes on, the nonlinear behavior of fingers becomes important. There are always a few dominant fingers that spread and shield the growth of other fingers. The spreading and shielding effects are caused by a spanwise secondary instability, and are aided by the transverse dispersion. It is shown that once a finger becomes large enough, the concentration gradient of its front becomes steep as a result of stretching caused by the cross‐flow, in turn causing the tip of the finger to become unstable and split. The splitting phenomenon in miscible displacement is studied by the authors for the first time. A study of the averaged one‐dimensional axial concentration profile is also presented, which indicates that the mixing length grows linearly in time, and that effective one‐dimensional models cannot describe the nonlinear fingering.

Random fluctuations and pattern growth : experiments and models

Abstract: * Contents *.- Course 1: Random Fluctuations and Transport.- Some Themes and Common Tools.- Structure, Elasticity and Thermal Properties of Silica Networks.- Anomalous Diffusion and Fractons in Disordered Structures.- Fractons in Real Fractals.- Anomalous Transport in Disordered Structures: Effect of Additional Disorder.- Information Exponents for Transport in Regular Lattices and Fractals.- Anomalous Transport in Random Linear Structures.- Course 2: Random Growth Patterns: Aggregation.- Morphological Transitions in Pattern Growth Phenomena.- Visualization and Characterization of Microparticle Growth Patterns.- Origin of Fractal Roughness in Synthetic and Natural Materials.- Electrodeposition.- Course 3: Random Fluctuations in Fluid Systems.- Flow Patterns in Porous Media.- Viscous Fingering in a Circular Geometry.- Construction of a Radial Hele-Shaw Cell.- Growth and Viscous Fingers on Percolating Porous Media.- Structure of Miscible and Immiscible Displacement Fronts in Porous Media.- Dynamics of Invasion Percolation.- Course 4: Random Fluctuations in Liquid Crystals.- Directional Solidification of Liquid Crystals.- Interfacial Instabilities of Condensed Phase Domains in Lipid Monolayers.- Pattern Formation of Molecules Adsorbing on Lipid Monolayers.- Course 5: Random Fluctuations in "Solid" Materials.- to Dense Branching Morphology.- The Material Factors Leading to Dense Branching Morphology in Al:Ge Thin Films.- Theoretical Aspects of Polycrystalline Pattern Growth in Al/Ge Films.- Pattern Formation in Dendritic Solidification.- Relaxation of Excitations in Porous Solids.- Soap Bubbles-A Simple Model System for Solids.- Course 6: Fracture of Disordered Solids.- to Modern Ideas on Fracture Patterns.- Rupture in Random Media.- Fracture Experiments on Monolayers of Microspheres.- Simple Models for Colloidal Aggregation, Dielectric Breakdown and Mechanical Breakdown Patterns.- Dielectric Breakdown Patterns with a Growth Probability Threshold.- Course 7: Fluctuation Phenomena in Membranes and Random Surfaces.- The Statistical Mechanics of Crumpled Membranes.- Fluctuations in Fluid and Hexatic Membranes.- Unbinding Transition and Mutual Adhesion in General of DGDG Membranes.- Scaling Properties of Interfaces and Membranes.- Surfactants in Solution: An Experimental Tool to Study Fluctuating Surfaces.- Course 8: Convection, Turbulence and Multifractals.- to Convection.- Onset of Convection.- Waves and Plumes in Thermal Convection.- An Introduction to Multifractal Distribution Functions.- Multifractals in Convection and Aggregation.- Multifractal Analysis of Sedimentary Rocks.- Phase Transition on DLA.- Course 9: Random Fluctuations and Complex Systems.- Disordered Patterns in Deterministic Growth.- 1/f Versus 1/f? Noise.- A Simple Model of Molecular Evolution.- Scale Invariant Spacial and Temporal Fluctuations in Complex Systems.- The Upper Critical Dimension and ?-Expansion for Self-Organized Critical Phenomena.- Self-Organized Criticality and the Origin of Fractal Growth.- Finite Lifetime Effects in Models of Epidemics.- to Droplet Growth Processes: Simulations, Theory and Experiments.- List of Participants.

Theory of fractal growth

Abstract: In the past few years a great deal of activity has been devoted to the study of fractal structures [3] in relation to physical phenomena [4,5]. The prototype fractal growth model is based on a combination of the Laplace equation and a stochastic field. The first model of this class to be formulated was Diffusion Limited Aggregation (DLA) [6]. A few years later the more general Dielectric Breakdown Model (DBM) [7] was introduced. This model used the relation between the random walk and potential theory and made clear that growth could also occur “from inside”. In addition to their intrinsic theoretical interest, these models are now believed to capture the essential features necessary to describe pattern formation in seemingly different phenomena like electrochemical deposition, deudritic growth, dielectric breakdown, viscous fingering in fluids, fracture propagation and others [4,5].

Fundamentals of gas reservoir engineering

Abstract: 1. Introduction. Natural gas. Gas reservoir engineering. Objective and organization. Units and symbols. 2. Reservoir properties. Introduction. Rock types. Porosity. Viscous flow resistance. Inertial flow resistance. Rock compressibility. Capillary pressure. Relative permeability. 3. Gas properties. Introduction. Composition. Phase behaviour. Real-gas law. Z-factor. Compressibility. Condensate/gas ratio. Formation-volume factor. Viscosity. 4. Phase behaviour. Introduction. K-value method. Equation-of-state method. Laboratory experiments. Multistage separation. 5. Recoverable reserves. Introduction. Bulk volume. Pore volume. Hydrocarbon pore volume. Gas and condensate initially-in-place. Recoverable reserves. Uncertainty. 6. Material balance. Introduction. Wet-gas reservoirs. Gas-condensate reservoirs. Non-volumetric depletion. Aquifer influx. 7. Single-phase gas flow. Introduction. Steady-state Darcy flow. Steady-state radial flow. Non-Darcy flow. Transient flow. Linear flow - constant terminal rate. Linear flow - constant terminal pressure. Radial flow - Constant terminal rate. Non-radial flow. 8. Gaswell testing. Introduction. Backpressure equations. Flow-after-flow tests. Isochronal and modified isochronal tests. Transient well-pressure equations. Drawdown tests. Buildup tests. Multiple-rate transient tests. Example of multiple-rate transient test analysis. 9. Wellbore flow mechanics. Introduction. Single-phase flow equations. Pressure distribution in shut-in wells. Rate-dependent pressure losses. Pressure distribution in producing wells. Multi-phase flow. Minimum unloading rate. 10. Water coning. Introduction. Dupuit critical production rate. Schols critical production rate. Cone breakthrough. Water/gas ratio. 11. Natural depletion. Introduction. Development chronology. Reservoir performance. Well-inflow performance. Tubing-flow performance. Well deliverability. Depletion simulator. 12. Gas injection. Introduction. Injection-well performance. Microscopic mixing. Viscous fingering. Gravity overlay. Stratification. Well Pattern. Pattern-flood model. Appendices. Units and conversion factors. Physical and mathematical constants. Physical properties natural-gas components. Author index. Subject index.

Two-phase flow in porous media†‡

Abstract: Two-phase flow in porous media depends on many factors, such as displacement vs steady two-phase flow, saturation, wettability conditions, wetting fluid vs non-wetting fluid is displacing, the capillary number, interfacial tension, viscosity ratio, pressure gradient, uniformly wetted vs mixed-wet pore surface, uniform vs distributed pore throats, small vs large pores, well-connected pores vs pores connected by small throats, etc. These parameters determine how the two fluids are distributed in the pores, e.g. whether they flow in seperate channels or side-by-side in the same channels, either with both fluids being continous or only one fluid being continous and the other discontinuous. In displacement, the capillary number and the viscosity ratio determine whether the displacement front is sharp, or if there is either capillary or viscous fingering.

Detailed Validation of an Empirical Model for Viscous Fingering With Gravity Effects

Abstract: This paper extends to two-dimensional (2D) flows the derivation and validation of an empirical model for viscous fingering previously developed. Fine-scale numerical simulations are used to provide basic data for validating the approximations, and these fingering results are also checked against a range of experiments. The flow rate dependence of gravity segregation in vertical section experiments conducted by van der Poel is examined, where the broadly acceptable agreement of the empirical model is limited by some identified additional features.

Nonlinear unstable viscous fingers in Hele--Shaw flows. I. Experiments

Abstract: Post‐instability viscous fingering in rectilinear flow in a Hele–Shaw cell has been studied experimentally. Of particular interest was the characterization of the range of length scales associated with tip splitting, over a reasonably wide range of parameters. A digital imaging system was used to record the patterns as a function of time, which allowed properties such as the tip velocity, finger width, perimeter, and area to be studied as functions of time and capillary number. The tip velocity was observed to be approximately constant regardless of the occurrence of splitting events, and the average finger width decreased as the degree of supercriticality increased. Quantitative measures of the fact that there is a limit to the complexity of viscous fingers are provided, and that over the range of parameters studied, no evidence for fractal fingering exists. A discussion of the dynamics of tip splitting explains why this is so.

Propagation of Polymer Slugs Through Adsorbent Porous Media

Abstract: The effects of adsorption on the propagation of polymer slugs through oil reservoirs determine the efficiency of polymer flooding for increasing oil recovery. However, up to now the adsorption laws introduced in polymer flood simulation models were oversimplified due to a lack of knowledge. The adsorption properties relevant for polymer propagation, such as instantaneous adsorption, reorganization in adsorbed layer, exchanges of macromolecules between free solution and adsorbed layer, desorption before thermodynamic equilibrium, are analyzed on the basis of both a very careful experimental work and recent polymer adsorption theories taking into account the effects of polymer polydispersity. The influence of surface and adsorbed layer steric exclusion chromatography effects which are in competition with adsorption-desorption chromatography are discussed. The analysis of the respective influence of macromolecular diffusion and hydrodynamic convection shows that this later mechanism governs the overall polymer dispersion. Moreover a determining effect of viscous fingering on the spreading of trailing edge concentration profile is observed. In addition, a theoretical approach of hydrodynamic retention mechanism is proposed. As a practical application, a new methodology to avoid kinetic effect artefacts on the measurement of instantaneous, reversible and irreversible adsorption is described.

Scaling structure of viscous fingering

Abstract: When a fluid in a porous medium attempts to displace one which is more viscous, an initially flat boundary between them is unstable and fingers of the inviscid liquid penetrate the other. We model the medium numerically using a lattice of capillary tubes of random radii. Previous studies by one of the authors (P.R.K.) have already shown that the displacement is compact, but that the boundary between the two fluids is fractal. In this paper we study the distribution of velocities normal to the interface. We find that the distribution is consistent with the hypothesis that, for a system of size L, ${L}^{f(\ensuremath{\alpha})}$ sites have velocities scaling as ${L}^{\mathrm{\ensuremath{-}}\ensuremath{\alpha}}$. The scaling function f(\ensuremath{\alpha}) is measured and its variation with the viscosity ratio and randomness of the medium is found.

Monte Carlo simulations of radial displacement of oil from a wetted porous medium: fractals, viscous fingering and invasion percolation

Abstract: A numerical random-walk method for simulating pore-size radial displacement of oil from wetting porous media under varying conditions of pore-size distribution, wettability, mobility ratio and capillary numbers is described The algorithm involves Monte Carlo decision making, random walks and percolation theory The likelihood of having a walker start from a peripheral site, or from the origin, is determined by the viscosity ratio, M Sticking probabilities, however, depend on the interfacial tension, gamma , and the solid-liquid contact angle, theta Three limiting behaviours are identified in terms of viscosity ratio and capillary number: viscous fingering, plug flow and invasion percolation Numerical experiments are performed for M=13 ( gamma =18 mN m-1, theta =50 degrees ), and for M=76*10-5 ( gamma =66 mN m-1, theta =70 degrees ) at flow rates spanning four decades on a porous network of pores and sites having a log-normal size distribution Typical runs last about 5-10 min Preliminary evidence of partially dendritic growth at high capillary number is discussed Agreement with previously reported experiments is excellent

Fluid capacity distributions of random porous media

Abstract: As a quantitative measure of the microstructure in a statistically homogeneous porous material, we introduce the notion of thefluid capacity at a specified length scale λ. In two dimensions, fluid capacity is the void space per unit area for a square of side λ and in three dimensions it is the void space per unit volume for a cube of side λ. The most random distribution of fluid capacity, for a prescribed mean fluid capacity, corresponds to an exponential distribution. The distribution of fluid capacity is important during unstable fluid displacements in porous media where viscous fingering occurs. For a material with an exponential fluid capacity distribution, an unstable displacement process can be simulated by simple stochastic algorithms related to diffusion-limited aggregation. We measure the two-dimensional fluid capacity distributions of published cross-section photomicrographs of sandstone, salt, and packed beds of glass beads, for various length scales A. The form of the distribution depends upon the magnitude of the length scale λ. For the sandstone and salt packs, appropriate length scales are found on which the fluid capacity has, to a good approximation, an exponential distribution. An exponential distribution appears to be inappropriate for the packed bed of glass beads on all length scales.

Stability of vertical miscible displacements with developing density and viscosity gradients

Abstract: 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.

Simulation of viscous fingering during miscible displacement in nonuniform porous media

Abstract: Numerical simulation is used to study the effect of different factors on unstable miscible displacement and to clarify the mechanisms of finger growth and interaction. The two-dimensional equations of miscible displacement in a rectangular slab are dedimensionalized and the factors that affect their solution are combined into dimensionless parameters. These parameters are the viscosity ratio, the aspect ratio (ratio of longitudinal to transverse dimension), Peclet numbers for molecular, longitudinal and transverse dispersion and the gravity number. To study the effect of the structure of the porous medium, simulations are performed on different random permeability fields, generated by a statistical method, so that they have a given coefficient of permeability variation and a given correlation length. The concentration equation is solved by an implicit finite element modified method of characteristics, which performs backward characteristic tracking. A mixed finite element method is used for the solution of the pressure

Morphological phase transitions in viscous fingering patterns in the liquid crystal 8CB

Abstract: We present the morphological pressure-temperature phase diagram for viscous fingering patterns observed in the isotropic, nematic and smectic. A phases of the liquid crystal 8CB. In addition to the dense branching structure, two distinct dendritic regimes were observed in the nematic and smectic phases. The dependence of characteristic finger width on pressure was studied, and the effects of surface and magnetic field alignment were considered Presentation d'un diagramme de phases P-T pour la morphologie des figures de digitation visqueuses observees dans les phases isotrope, nematique et smectique A de 8CB. En plus de la structure branchee dense, observation de deux regimes dendritiques distincts dans les phases nematique et smectique. Etude de la largeur caracteristique des doigts en fonction de la pression. Etude des effets d'alignement par une surface ou un champ magnetique

Rate Dependence of Unstable Waterfloods

Abstract: Viscous fingering is part of the flow mechanisms that are operative in waterflooding and in a wide range of EOR methods. This work presents a laboratory and computer model analysis of the viscous-fingering dynamics that develop when a less mobile fluid is immiscibly displaced by a more mobile fluid in a permeable medium. Physical experiments of horizontal fluid displacements conducted in as rectangular bead pack show that amplitude/frequency character of the wave-like fingers that form depends on flow rate and mobility ratio. The nature of these wave-like features is shown for two viscosity ratios and several displacement rates. Spatial frequency domain analyses of finger shapes at fixed time intervals were conducted on digitized records of the laboratory experiments. The results of these analyses indicate that fluctuations comprising the fingered zone have a wave number corresponding to a maximum growth rate. The analyses suggest that the amplitude of flow fluctuations can be characterized by a root mean square (RMS) growth rate that is linear in time. Finite-difference solutions of a set of two-dimensional (2D) flow-in-porous-media equations were made to exhibit similar frontal instability. The linear growth rate of the mixing zone suggests that fractional flow relationships can provide an adequate andmore » practical method of representing space-averaged two-phase flow variations for many reservoir engineering applications.« less

Two-phase flow in Hele-Shaw cells: numberical studies of sweep efficiency in a five-spot pattern

Abstract: The unsteady Hele-Shaw problem is a model nonlinear system that, for a certain parameter ranger, exhibits the phenomenon known as viscous fingering. While not directly applicable to multiphase porous-media flow, it does prove to be an adequate mathematical model for unstable dieplacement in laboratory parallel-plate devices. We seek here to determine, by use of an accurate boundary-integral frount-tracking scheme, the extent to which the simplified system captures the canonical nonlinear behavior of displacement flows and, in particular, to ascertain the role of noise in such systems. We choose to study a particular pattern of injection and production “wells.” The pattern chosen is the isolated “five-spot,” that is a single source surrounded by four symmetrically-placed sinks in an infinite two-dimensional “reservoir.” In cases where the “pusher” fluid has negligible viscosity, sweep efficiency is calculated for a range of values of the single dimensionless parameter τ, an inverse capillary number. As this parameter is reduced, corresponding to increased flow rate or reduced interfacial tension, this efficiency decreases continuously. For small values of τ, these stable displacements change abruptly to a regime characterized by unstable competing fingers and a significant reduction in sweep efficiency. A simple stability argument appears to correctly predict the noise level required to transit from the stable to the competing-finger regimes. Published compilations of experimental results for sweep efficiency as a function of viscosity ratio showed an unexplained divergence when the pusher fluid is less viscous. Our simulations produce a similar divergence when, for a given viscosity ratio, the parameter τ is varied.

Viscous Fingering in Porous Media

Abstract: The problem of viscous fingering in porous media is of central importance in oil recovery It is also an interesting problem in hydrodynamics and in the physics of porous media It has recently been shown that viscous fingering in porous media is fractal (Maloy et al, 1985a, b; Chen and Wilkinson, 1985) We begin with an introduction to the viscous fingering problem in a two-dimensional geometry, the Hele-Shaw cell, and present some of the relevant experimental results We then present experimental results on fingering in porous media and discuss in particular the very recent evidence that the fingering is fractal

Growth by Gradients: Fractal Growth and Pattern Formation in a Laplacian Field

Abstract: The formation of a wide variety of patterns is controlled by the strength of the gradient of a field at the interface between the structure and the outside. For example, the rate of growth in aggregation processes [1, 2] depends on the concentration of diffusing particles, in solidification [3] it depends on the temperature gradient, in dielectric breakdown [4] it is proportional to some power of the electric field, in electrodeposition [5, 6] it depends on the gradient of the electrical voltage, and in viscous fingering [7, 8] it is the pressure gradient. It has long been speculated [9] that concentration gradients of nutrients, light and energy resources are the factors controlling the development of a variety of biological patterns as well, even though much less is known about pattern formation in biological systems [9].

Viscous Fingering in a Circular Geometry

Abstract: The instability giving rise to Saffman-Taylor viscous fingering occurs at the interface between two fluids moving between narrowly spaced solid plates (i.e., a Hele Shaw cell). The interface is unstable when it is the less viscous fluid which forces the more viscous fluid to recede. The flow of each fluid is dominated by the viscous dissipation on the plates and the mean velocity in the cell’s plane is given by the Darcy law $$V = - \frac{{{{b}^{2}}}}{{12\mu }} abla p,$$ (1) where b is the plates spacing and μ the viscosity of the fluid. The pressure p, because of the incompressibility of the fluid, obeys a Laplace law $$ \Delta p = 0. $$ (2) Surface tension has a stabilizing influence on the interface. It is taken into account by adding a boundary condition for the pressure jump at the interface $$ \delta p = T\kappa . $$ (3) where T is the surface tension and κ the local two dimensional curvature.

The Effect of Dispersion on Fingering in Miscible Displacements

Abstract: The term “viscous fingering” generally refers to the instability associated with the displacement of a more viscous fluid by a less viscous one within a porous medium. As we will see, the mechanism of the instability is associated with the forces acting on a front as a result of a viscosity contrast near that front. However, the forces causing or mitigating the instability may also be due to density differences acting together with gravity, so perhaps the more general term, “fingering” is more appropriate.

Disordered Patterns in Deterministic Growth

Abstract: Methods developed for the investigation of dynamical systems are used to characterize the degree of randomness of spatially disordered chains of atoms and growing unstable interfaces. Our calculations indicate that the experimentally observed geometry of viscous fingering patterns can not be described in terms of low-dimensional chaos.

A renormalisation group approach to the viscous-finger fractal at finite viscosity ratio

Abstract: A real-space renormalisation group method is applied to viscous fingering to analyse the fractal nature at a finite viscosity ratio. The renormalisation group transformation is obtained for the permeability of the surface layer which plays the role of coupling constant in the renormalisation group of the phase transition. A stable fixed point is found as a function of the viscosity ratio. The growth probabilities on the perimeter bonds of the injected fluid are evaluated at the fixed point. The multifractal scaling properties are found to be described by the generalised dimensions and the alpha -f spectra. It is shown that at a finite viscosity ratio the displaced area is compact with a surface fractal dimension between 1 and the DLA result with increasing viscosity ratio. This result is consistent with that of King's numerical simulation (1987).

Multifractal Viscous Fingering and Non-Newtonian Growth

Abstract: The first part of this paper is in collaboration with H.E.Stanley and G. Daccord. We have calculated for experimental viscous fingers the hierarchy of fractal exponents which describe the growth of the two fluid interface. Our analysis is based on a “coastline” method and is applied to the growth of Newtonian viscous fingers. In the second part we show how present statistical models for the viscous finger instability can be extended to enable the study of non-Newtonian viscous fingers. We present results on the change of finger thickness as a function of the shear-thinning index m.

Recent Results of Experiments with Saffman-Taylor Flow

Abstract: In this paper we summarize our recent work on the viscous fingering probleml,2. In its initially planar form, Saffman-Taylor flow3, viscous fingering represents the simplest of pattern formation problems. Thus the observation of its details provides a valuable opportunity for direct testing of the computer calculations which are now just becoming feasible for pattern formation4, Despite its extreme simplicity, Saffman- Taylor flow has much in common with the Mullins-Sekerka instability5 which gives rise to dendritic growth in alloys. In this paper we first discuss the formal similarity between the dispersion relations for the Saffman-Taylor and Mullins-Sekerka instabilities, then we set out some of our recent results on the Saffman-Taylor problem1.