Fingering Instability Research Articles

Influences of Dead‐End Pores in Porous Media on Viscous Fingering Instabilities and Cleanup of NAPLs in Miscible Displacements

AbstractImproving the understanding of mechanisms involved in a low miscible displacement efficiency is significant for a wide spectrum of applications in subsurface from environment such as groundwater remediation and sequestration to energy extraction such as enhanced oil recovery and geothermal recovery. Two key limiting factors to the efficiency are viscous fingering (VF) instability and dead‐end pores in porous media. Previous research on VF simply assumes all pores are well connected and fluids can be mobilized by convection. However, fluids trapped in dead‐end pores, such as nonaqueous phase liquids (NAPLs) in groundwater remediation, are inaccessible to convection, resulting in even less efficient displacements. Instead of the classic convection‐diffusion/dispersion equation, in this work, we use a fundamentally different capacitance model to incorporate the mass transfer between two pore types in miscible displacements. The hybrid pseudospectral and high‐order finite difference methods are employed to solve the governing equations in a fixed reference frame for simulating the flow dynamics. We found that the viscous fingering instabilities in well‐connected pore network led to an unstable, nonuniform distribution of trapped NAPLs in dead‐end pores network. Different with a traditional view that NAPLs are easily cleaned up, our study indicates the existence of dead‐end pores results in an inefficient cleanup of NAPLs in swept area because of the slow mass transfer of trapped NAPLs between two pore types. A simple model is developed to accurately predict the NAPL concentration behind finger trailing front, which has not been previously examined. Six flow regimes, four of which are new, are then identified.

Thermocapillary effect on the dynamics of falling self-rewetting fluid films down a heated vertical cylinder

Thermocapillary effect on the dynamics of falling self-rewetting fluid films down a heated vertical cylinder

Viscous normal stresses and fingertip tripling in radial Hele-Shaw flows.

Viscous fingering in radial Hele-Shaw cells is markedly characterized by the occurrence of fingertip splitting, where growing fingered structures bifurcate at their tips, via a tip-doubling process. A much less studied pattern-forming phenomenon, which is also detected in experiments, is the development of fingertip tripling, where a finger divides into three. We investigate the problem theoretically, and employ a third-order perturbative mode-coupling scheme seeking to detect the onset of tip-tripling instabilities. Contrary to most existing theoretical studies of the viscous fingering instability, our theoretical description accounts for the effects of viscous normal stresses at the fluid-fluid interface. We show that accounting for such stresses allows one to capture the emergence of tip-tripling events at weakly nonlinear stages of the flow. Sensitivity of fingertip-tripling events to changes in the capillary number and in the viscosity contrast is also examined.

Inertia Effects in the Dynamics of Viscous Fingering of Miscible Fluids in Porous Media: Circular Hele-Shaw Cell Configuration

We present a numerical study of viscous fingering occurring during the displacement of a high viscosity fluid by low viscosity fluid in a circular Hele-Shaw cell. This study assumes that the fluids are miscible and considers the effects of inertial forces on fingering morphology, mixing, and displacement efficiency. This study shows that inertia has stabilizing effects on the fingering instability and improves the displacement efficiency at a high log-mobility-viscosity ratio between displacing and displaced fluids. Under certain conditions, inertia slightly reduces the finger-split phenomenon and the mixing between the two fluids.

An Observation of Nanoparticle‐Assisted Fingering Instability

AbstractFingering instabilities generally occur in an interface when a lower viscous or higher dense fluid is pushed through another miscible or immiscible fluid of higher viscosity or lower density. Similar fingering phenomena are also seen when hot saline water overlies cold freshwater or two solutes with different diffusivities are employed on two different sides of the interface. In the present study, we report an observation of nanoparticle‐assisted fingering that occurs when a less dense nanofluid lies over a denser liquid in an isothermal condition. Further, we show that the proposed mechanism (i.e., double mass‐diffusive convection) for this kind of fingering works only if nanoparticles are present in the suspending medium but not the microparticles. In the case of microparticles, sedimentation leads to the Rayleigh‐Taylor type of instability.

Viscous banding instabilities: non-porous viscous fingering

We demonstrate a novel instability found within unconfined viscous bands/rims, or free-surface flows involving a longitudinal viscosity contrast. Such instabilities may be described as viscous banding instabilities, non-porous viscous fingering instabilities or unconfined viscous fingering instabilities of free-surface flows involving the intrusion of a less viscous fluid into a band of more viscous fluid. A consequence of this work is that viscous fingering instabilities, widely known to occur in porous media following the seminal work of Saffman & Taylor (Proc. R. Soc. Lond. A, vol. 245, 1958, pp. 312–329), also occur in non-porous environments. Although the mechanism of the viscous banding instability is characteristically different from that of the Saffman–Taylor instability, there are important similarities between the two. The main similarity is that a viscosity contrast leads to instability. A distinguishing feature is that confinement, such as the rigid walls of a Hele-Shaw cell, is not necessary for viscous banding instabilities to occur. More precisely, Saffman–Taylor instabilities are driven by a jump in dynamic pressure gradient, whereas viscous banding instabilities, or non-porous viscous fingering instabilities, are driven by a jump in hydrostatic pressure gradient, directly related to a slope discontinuity across the intrusion front. We examine the onset of instability within viscous bands down an inclined plane, determine conditions under which viscous banding instabilities occur and map out a range of behaviours in parameter space in terms of two dimensionless parameters: the viscosity ratio and the volume of fluid ahead of the intrusion front.

Spreading or contraction of viscous drops between plates: single, multiple or annular drops

The behaviour of a viscous drop squeezed between two horizontal planes (a squeezed Hele-Shaw cell) is treated by both theory and experiment. When the squeezing force $F$ is constant and surface tension is neglected, the theory predicts ultimate growth of the radius $a\sim t^{1/8}$ with time $t$. This theory is first reviewed and found to be in excellent agreement with experiment. Surface tension at the drop boundary reduces the interior pressure, and this effect is included in the analysis, although it is negligibly small in the squeezing experiments. An initially elliptic drop tends to become circular as $t$ increases. More generally, the circular evolution is found to be stable under small perturbations. If, on the other hand, the force is reversed ($F<0$), so that the plates are drawn apart (the ‘contraction’, or ‘lifting plate’, problem), the boundary of the drop is subject to a fingering instability on a scale determined by surface tension. The effect of a trapped air bubble at the centre of the drop is then considered. The annular evolution of the drop under constant squeezing is still found to follow a ‘one-eighth’ power law, but this is unstable, the instability originating at the boundary of the air bubble, i.e. the inner boundary of the annulus. The air bubble is realised experimentally in two ways: first by simply starting with the drop in the form of an annulus, as nearly circular as possible; and second by forcing four initially separate drops to expand and merge, a process that involves the resolution of ‘contact singularities’ by surface tension. If the plates are drawn apart, the evolution is still subject to the fingering instability driven from the outer boundary of the annulus. This instability is realised experimentally by levering the plates apart at one corner: fingering develops at the outer boundary and spreads rapidly to the interior as the levering is slowly increased. At a later stage, before ultimate rupture of the film and complete separation of the plates, fingering spreads also from the boundary of any interior trapped air bubble, and small cavitation bubbles appear in the very low-pressure region, far from the point of leverage. This exotic behaviour is discussed in the light of the foregoing theoretical analysis.

Role of density gradients on miscible Rayleigh–Taylor fingers in porous media

We investigate the effect of density gradients on miscible Rayleigh–Taylor fingers in homogeneous porous media using two families of concentration-dependent density profiles: (a) monotonic and (b) nonmonotonic. The first family consists of linear, quadratic, and cubic functions of the solute concentration, while the latter is described as a quadratic function of the solute concentration such that the density maximum (minimum) appears in time as diffusion relaxes the concentration gradient. With the help of these simple models, we are able to address one of the most puzzling questions about the fingering instabilities with nonmonotonic density profiles. Using linear stability analysis and nonlinear simulations, we show that density gradients play a pivotal role in controlling instability.

کمیسازی وقوع ناپایداری ویسکوز فینگرینگ جریان سیال در محیط متخلخل

In this paper, nonlinear simulation of viscous fingering instability of miscible displacement involving nanofluid is investigated. Using vorticity and stream functions and the spectral method governing equations are obtained. Due to the fractality of fluid-fluid interface in instability phenomena, by using box counting method, its fractal dimension is calculated in different parameters such as deposition rate, mobility ratio and diffusion rates. The results show that increasing the deposition rate reduces the complexity of finger patterns and the diffusion rate of nanofluid has no effect on complexity of finger patterns while increasing the diffusion rate of displaced fluid has significant effect on patterns and makes it more complicated. The fractal analysis also shows that the effect of mobility ratio depends on the deposition rate. By considering deposition rate, although the mobility ratio has no effect on fractal dimension and effective time is constant and equal to 275, start time of instability is delayed by 25 units. It can be concluded that fractal analysis of viscous fingering phenomena can be considered as one of the instability characteristics.

Controlling fingering instabilities in Hele-Shaw flows in the presence of wetting film effects.

In this paper, the interfacial motion between two immiscible viscous fluids in the confined geometry of a Hele-Shaw cell is studied. We consider the influence of a thin wetting film trailing behind the displaced fluid, which dynamically affects the pressure drop at the fluid-fluid interface by introducing a nonlinear dependence on the interfacial velocity. In this framework, two cases of interest are analyzed: The injection-driven flow (expanding evolution), and the lifting plate flow (shrinking evolution). In particular, we investigate the possibility of controlling the development of fingering instabilities in these two different Hele-Shaw setups when wetting effects are taken into account. By employing linear stability theory, we find the proper time-dependent injection rate Q(t) and the time-dependent lifting speed b[over ̇](t) required to control the number of emerging fingers during the expanding and shrinking evolution, respectively. Our results indicate that the consideration of wetting leads to an increase in the magnitude of Q(t) [and b[over ̇](t)] in comparison to the nonwetting strategy. Moreover, a spectrally accurate boundary integral approach is utilized to examine the validity and effectiveness of the controlling protocols at the fully nonlinear regime of the dynamics and confirms that the proposed injection and lifting schemes are feasible strategies to prescribe the morphologies of the resulting patterns in the presence of the wetting film.

A Case Study of the Thermohaline Fine-scale Structure of the Sub-Arctic Front

ABSTRACT This paper analyzes the characteristics of the thermohaline fine-scale structure of the sub-Arctic front (SAF) in the Kuroshio–Oyashio confluence region using temperature, salinity, and current data obtained from surveys in June 2016. Thermohaline intrusion is found to be prevalent in the sea area of the SAF, occurring predominantly in frontal zones with depths less than 200 m and in North Pacific intermediate water (NPIW) layers with depths greater than 350 m. The intrusion in the SAF zones with depths less than 200 m has a higher intensity but smaller dimensions in the NPIW. According to our analysis, the thermohaline intrusion in the SAF sea zone is mainly attributable to the joint effect of double diffusion and flow instability. At depths of 0–350 m, the thermohaline intrusion is vulnerable to the influences of salt fingering, diffusive convection, and symmetric instability. At depths deeper than 350 m, diffusive convection is significant. In addition, shear instability might be a contributor. Considering that the effect of turbulent mixing is primarily governed by symmetric instability in the SAF sea areas, the flow instability characterized by a dominant contribution from symmetric instabilities might exert a more significant influence on thermohaline intrusion. The more distinct thermohaline staircases at layers deeper than 350 m, which are mainly produced by diffusive convection, have larger vertical scales than the staircases in other layers.

Lattice Boltzmann study of miscible viscous fingering for binary and ternary mixtures

The viscous fingering or Saffman-Taylor instability occurs when a less viscous fluid penetrates a more viscous fluid in a porous medium. Fingering patterns refer to the deformation of the interface between the invading and displaced fluids. For mixtures composed of three miscible species, the dynamic of each component varies greatly. In that case, multicomponent diffusion effects are of primary importance, and reverse diffusion could trigger the instability from a configuration that is known to be stable for binary mixtures.

The Effect of Compressibility and Outer Boundaries on Incipient Viscous Fingering

SummaryThe knowledge of the effects of instability and heterogeneity on displacements, primarily enhanced oil recovery, and carbon dioxide storage are well known, although they remain difficult to predict. The usual recourse to modeling these effects is through numerical simulation. Simulation remains the gold standard for prediction; however, its results lack generality, being case-specific. There are also several analytic models for displacements that are usually more informative than simulation results. However, these methods apply to steady-state, incompressible flow.Carbon dioxide injection for storage uses compressible fluids and, in the absence of producers, will not approach steady-state flow (Wu et al. 2017). Consequently, it is unlikely that storage will be in reservoirs of open boundaries (steady-state flow). Flow of compressible fluid necessitates the use of closed or partially sealed boundaries, a factor that is consistent with compressible flow.This work deals with the conditions that cause the onset of incipient viscous fingering or Saffman-Taylor (ST) instability. The actual growth and propagation of fingers, a subject of much recent literature, is not discussed here. The original ST formalism of M > 1 for gravity-free flow is highly restrictive: it is for linear flow of nonmixing incompressible fluids in steady-state flow. In this work, we relax the incompressible flow restriction and thereby broaden the ST criterion to media that have sealing and/or partially sealing outer boundaries. We use the nonlinear partial differential equation for linear flow and developed analytic solutions for a tracer flow analog and also for a two-fluid compressible flow. The analysis is restricted to stabilized flow and to constant compressibility fluids, but we are not restricted to small compressibility fluids.There is no transition (mixing) zone between displacing and displaced fluids; the displacement is piston-like. The absence of a transition zone means that the results apply to both miscible and immiscible displacements, absent dispersion, or local capillary pressure. The assumption of a sharp interface is to focus on the combined effect of mobility ratio and compressibility. We use the product of the fluid compressibility and pressure drop (cfΔP) to differentiate the compressibility groups (Dake 1978; Dranchuk and Quon 1967), where ΔP is defined as the pressure drop within the specific fluid region.The results will be based on proposed analytical solutions compared to numerical simulation. The proposed formulation is less restrictive than the original ST formalism of M > 1 and allows evaluation of viscous fingering initiation or ST stability criterion in the presence of different boundary conditions (open vs. closed boundaries) with compressible fluids under the stated assumptions, which is the scope of this work.The key contribution here is the effect of external boundaries, which consequently makes necessary the use of compressible fluids. Absent compressibility, the necessary condition for the growth of a viscous finger is simply the mobility ratio, M > 1. It is the objective of this work to study how the ST criterion is affected by the presence of sealing and partially sealing outer boundaries with the consequent inclusion of compressible flows as in carbon dioxide storage and enhanced oil recovery by gas injection.The results show that adding compressibility always makes displacements more unstable for stabilized background flow, even for a favorable mobility ratio (M < 1) at extremely large compressibility (e.g., cf > 5×10−3 1/psi). For a sealed external boundary (no production or leakage), displacements will become more stable as a front approaches an external boundary for all mobility ratios (M) investigated.

A novel decoupled second-order time marching scheme for the two-phase incompressible Navier–Stokes/Darcy coupled nonlocal Allen–Cahn model

A novel decoupled second-order time marching scheme for the two-phase incompressible Navier–Stokes/Darcy coupled nonlocal Allen–Cahn model

Thin-film evolution and fingering instability of self-rewetting films flowing down an inclined plane

This paper examines the evolution patterns and essential mechanisms of flow instability of a self-rewetting fluid (SRF) coating on an inclined plane. Considering that the self-rewetting liquid has an anomalous surface tension with temperature change, some interesting phenomena will be found and should be explained. Using the thin-film model, the evolution equation of the air–liquid interface is derived, and the thickness of the liquid film is determined by a fourth-order partial differential equation. Taking T0 (temperature corresponding to the minimum of surface tension) as a cutoff point, two representative cases of the nonlinear flow are comprehensively discussed. One is the case of Ti > T0, and the other is Ti < T0 (interfacial temperature Ti). Based on traveling wave solutions, linear stability analysis (LSA) of the small perturbation applied to the initial condition is given, and the results of LSA are confirmed and explained by the numerical simulations. Results show that the inclined angle and the Weber number always stabilize the free surface, while the Marangoni effect and the Biot number play different roles for the two cases. As Ti − T0 varies from a negative value to a positive value, the Marangoni effect switches to the reverse Marangoni effect. With Ti − T0 < 0, the Marangoni effect enhances the fingering instability, while the Marangoni effect makes the flow more stable if Ti − T0 > 0. The Biot number Bi = 1 corresponds to the most unstable state for Ti < T0 and to the most stable state for Ti > T0.

Lattice Boltzmann Solver for Multiphase Flows: Application to High Weber and Reynolds Numbers.

The lattice Boltzmann method, now widely used for a variety of applications, has also been extended to model multiphase flows through different formulations. While already applied to many different configurations in low Weber and Reynolds number regimes, applications to higher Weber/Reynolds numbers or larger density/viscosity ratios are still the topic of active research. In this study, through a combination of a decoupled phase-field formulation—the conservative Allen–Cahn equation—and a cumulant-based collision operator for a low-Mach pressure-based flow solver, we present an algorithm that can be used for higher Reynolds/Weber numbers. The algorithm was validated through a variety of test cases, starting with the Rayleigh–Taylor instability in both 2D and 3D, followed by the impact of a droplet on a liquid sheet. In all simulations, the solver correctly captured the flow dynamics andmatched reference results very well. As the final test case, the solver was used to model droplet splashing on a thin liquid sheet in 3D with a density ratio of 1000 and kinematic viscosity ratio of 15, matching the water/air system at We = 8000 and Re = 1000. Results showed that the solver correctly captured the fingering instabilities at the crown rim and their subsequent breakup, in agreement with experimental and numerical observations reported in the literature.

MATHEMATICAL SOLUTION OF FINGERING PHENOMENON IN VERTICAL DOWNWARD DIRECTION THROUGH HETEROGENEOUS POROUS MEDIUM

Present study explores the Fingering (Instability) phenomenon's mathematical model that ensues during the process of secondary oil recovery where two not miscible fluids (water and oil) flow within a heterogeneous porous medium as water is injected vertically downwards. Variational iteration method with proper initial and boundary conditions is being used to determine approximate analytic solution for governing nonlinear second order partial differential equation. Whereas MATLAB is applied to acquire the solution's numerical findings and graphical representations.

Experimental analysis of fluid displacement and viscous fingering instability in fractured porous medium: effect of injection rate

Viscous fingering instability has been studied empirically by miscible flow displacement. Despite numerous experimental studies, there is little information regarding the fractured porous medium. The tests have been conducted in a rectangular porous medium, and the fluid was driven in the longitudinal direction. The generated fingering patterns are demonstrated and discussed based on injection rate and fracture orientation. Three porous models with 0°, 45° and 90° fractures were used in order to study the effect of fracture orientation. Quantitative parameters of fingering instability, such as sweep efficiency, headway position, and breakthrough time, have been measured by image processing techniques using MATLAB. It is shown that by increasing the injection rate, the size of the finger patterns slightly increased. Moreover, the effects of fractures on displacement behaviors are shown and explained. According to the results, zero degree fracture altered the rate of sweep efficiency and tip position the more significantly than other fractures. Furthermore, it was observed that by changing the direction of fractures from parallel to perpendicular to the flow direction, the breakthrough time increased consequently.

Linear Stability Analysis and CFD Simulation of Thermal Viscous Fingering Instability in Anisotropic Porous Media

AbstractWater or steam injection in oil fields is a common method for enhanced oil recovery in petroleum engineering. The thermoviscous fingering instability is one of the main problems of a comple...

Fingering instability in spreading epithelial monolayers: roles of cell polarisation, substrate friction and contractile stresses.

Collective cell migration plays a crucial role in many developmental processes that underlie morphogenesis, wound healing, or cancer progression. In such coordinated behaviours, cells are organised in coherent structures and actively migrate to serve different biological purposes. In some contexts, namely during epithelial wound healing, it is well known that a migrating free-edge monolayer develops finger-like instabilities, yet the onset is still under debate. Here, by means of theory and numerical simulations, we shed light on the main mechanisms driving the instability process, analysing the linear and nonlinear dynamics of a continuum compressible polar fluid. In particular, we assess the role of cell polarisation, substrate friction, and contractile stresses. Linear theory shows that it is crucial to analyse the perturbation transient dynamics, since we unravel a plethora of crossovers between different exponential growth rates during the linear regime. Numerical simulations suggest that cell-substrate friction could be the mechanism responsible for the formation of complex finger-like structures at the edge, since it triggers secondary fingering instabilities and tip-splitting phenomena. Finally, we obtain a critical contractile stress that depends on cell-substrate friction and the initial-to-nematic length ratio, characterising an active wetting-dewetting transition. In the dewetting scenario, the monolayer retracts and becomes stable without developing finger-like structures.