Capillary Gravity Research Articles

Capillary hysteresis and gravity segregation in two phase flow through porous media

We study the gravity driven flow of two fluid phases in a one dimensional homogeneous porous column when history dependence of the pressure difference between the phases (capillary pressure) is taken into account. In the hyperbolic limit, solutions of such systems satisfy the Buckley-Leverett equation with a non-monotone flux function. However, solutions for the hysteretic case do not converge to the classical solutions in the hyperbolic limit in a wide range of situations. In particular, with Riemann data as initial condition, stationary shocks become possible in addition to classical components such as shocks, rarefaction waves and constant states. We derive an admissibility criterion for the stationary shocks and outline all admissible shocks. Depending on the capillary pressure functions, flux function and the Riemann data, two cases are identified a priori for which the solution consists of a stationary shock. In the first case, the shock remains at the point where the initial condition is discontinuous. In the second case, the solution is frozen in time in at least one semi-infinite half. The predictions are verified using numerical results.

Gravity effects on oil–water two-phase displacement in homogeneous porous media

Gravity plays an important role in enhanced oil recovery and groundwater hydrology. A two-dimensional visual homogeneous micromodel was used in this study to describe the role of gravity in displacement processes. A theoretical analysis is proposed for three flow modes, i.e., vertical-upward, vertical-downward, and horizontal displacements, in which water and decane are used for the displacing and the displaced phases, respectively. A relatively compact displacement front was obtained at high flow rates in the three displacement modes, and the front gradually became unstable with a decrease in the flow rate. Compared with horizontal displacement, in vertical-upward displacements, gravity can hinder the evenness of the flow and aggravate the front finger formations at the inlet. This process forces the heavier displacing phase to expand horizontally at the midpoint and weakens the front's fingers. In the vertical-downward displacement process, two states occurred at the same low flow rate: stable flow and unstable flow. Unstable flows occurred more frequently with a decrease in the flow rate. To better understand the role of gravity in displacement, we proposed a theoretical prediction model for the flow state transition of the three displacement modes by combining the capillary force, viscous force, and gravity based on pore-filling events. Finally, to predict the final recovery factor for various displacement modes, four dimensionless formulations were produced using the capillary number, the gravity number, the bond number, and the viscosity ratio.

Experiments in Surface Gravity–Capillary Wave Turbulence

The last decade has seen a significant increase in the number of studies devoted to wave turbulence. Many deal with water waves, as modeling of ocean waves has historically motivated the development of weak turbulence theory, which addresses the dynamics of a random ensemble of weakly nonlinear waves in interaction. Recent advances in experiments have shown that this theoretical picture is too idealized to capture experimental observations. While gravity dominates much of the oceanic spectrum, waves observed in the laboratory are in fact gravity–capillary waves, due to the restricted size of wave basins. This richer physics induces many interleaved physical effects far beyond the theoretical framework, notably in the vicinity of the gravity–capillary crossover. These include dissipation, finite–system size effects, and finite nonlinearity effects. Simultaneous space-and-time-resolved techniques, now available, open the way for a much more advanced analysis of these effects.

Gravity–capillary multi-component wave patterns generated by a single-frequency wave-maker oscillation and subsequent resonances

Cross-waves are standing waves with crests perpendicular to a wave-maker; they are subharmonic waves excited by parametric instability. The modulational and chaotic behaviours of nonlinear cross-waves have been studied widely since the 1970s. Most of the previous work has focused on gravity waves where surface tension can be neglected. In this work we study cross-waves that are highly dependent on surface tension as well as gravity. By oscillating a planar wave-maker either vertically or horizontally with frequencies of 25 Hz through 40 Hz at one end of a rectangular basin, two-dimensional multi-component surface patterns are realized. Using the free-surface synthetic Schlieren technique to measure the surface elevations, multi-dimensional Fourier transforms are utilized to track the evolutionary spectrum of the water surface in both the temporal and spatial domains. Wavelet transforms are implemented to show the development of the various frequency components. Three-wave resonances with and without first subharmonics are observed for small nonlinearity. Three-dimensional oblique propagating cross-waves are generated at higher nonlinearity; unlike most previous cross-wave experiments, this staggered pattern propagates far downstream. Experimental evidence shows that two oblique propagating waves form a two-dimensional short-crested pattern, and that the lateral component of the waves develops into parametric sloshing modes corresponding to the width of the tank. Two regimes of nonlinear wave patterns, resonant triads and oblique propagating cross-waves, are delineated.

Gravity–capillary flows over obstacles for the fifth-order forced Korteweg–de Vries equation

The aim of this work is to investigate gravity–capillary waves resonantly excited by two topographic obstacles in a shallow water channel. By considering the weakly non-linear regime the forced fifth-order Korteweg–de Vries equation arises as a model for the free-surface displacement. The water surface is initially taken at rest over a uniform flow and the initial value problem for this equation is computed numerically using a pseudospectral method. We study nearly-resonant flows with intermediate capillary effects. Details of the wave interactions are analysed for obstacles with different sizes. Our numerical results indicate that the flow is not necessarily governed by the larger obstacle.

Numerical bifurcation and stability for the capillary–gravity Whitham equation

We adopt a robust numerical continuation scheme to examine the global bifurcation of periodic travelling waves of the capillary–gravity Whitham equation, which combines the dispersion in the linear theory of capillary–gravity waves and a shallow water nonlinearity. We employ a highly accurate numerical method for space discretization and time stepping, to address orbital stability and instability for a rich variety of the solutions. Our findings can help classify capillary–gravity waves and understand their long-term dynamics.

Large amplitude capillary-gravity solitary waves in a stratified fluid sandwiched between two deep homogeneous layers

The study addresses the propagation of plane capillary gravity solitary waves of permanent form in a three layer formulation. The intermediate fluid is assumed to be stratified, while the upper and lower ones are homogeneous and infinitely deep. One or both interfaces separating these layers are subject to capillarity. The research can be applied to the case of two deep fluids when one of these fluids is stratified near the interface. The latter formulation is relevant to studies of capillary gravity waves in the transitional area between sea water and liquid carbon dioxide in the deep ocean. This has become an issue of importance for the secure storage of carbon dioxide, which is an environmental/technological problem in modern days. Therefore, we address a capillary-gravity wave motion beyond the well-examined cases of a free surface or two fluid flows. It is shown that in the considered formulation, capillary-gravity solitary waves of finite amplitude obey an integro-differential equation. This equation contains both Korteweg-de Vries (KdV) and Benjamin-Ono (BO) dispersion laws and a specific nonlinearity, which depends on the properties of the stratified layer. Capillary (KdV-type) dispersion dominates if the thickness of the stratified layer is d≪d∗. When d≫d∗, the gravitational (BO-type) dispersion determines the flow. The value d∗ depends on the mode number, gravitational acceleration, and capillarity effects. Analytical solutions for the amplitude function and the streamline patterns are presented.

Mathematical modeling of gravity and buoyancy effect on low interfacial tension spontaneous imbibition in tight oil reservoirs

AbstractConsidering capillary force, viscous force, and gravity and buoyancy effects due to the oil–water density difference during the imbibition process, a new mathematical model for core‐scale spontaneous imbibition considering gravity and buoyancy is established. The imbibition rates with and without gravity and buoyancy in high‐, low‐, and ultralow‐interfacial tension (IFT) solutions are compared. The study shows that oil buoyancy can accelerate the imbibition rate, and with increasing core permeability and decreasing IFT, the differences in imbibition rate with and without gravity and buoyancy become more significant. At a high IFT of dozens of mN/m, the gravity and buoyancy effects are nonsignificant and can be ignored, but at a low IFT such as at 0.355 mN/m and a core permeability above 1 mD, gravity and buoyancy must be considered in spontaneous imbibition simulations. In low‐IFT solutions, the critical inverse Bond number when gravity and buoyancy must be considered is approximately 1 × 10−2.

A new computational technique for the analytic treatment of time-fractional Emden–Fowler equations

This paper presents the study of fractional Emden–Fowler (FEF) equations by utilizinga new adequate procedure, specifically the q-homotopy analysis transform method (q-HATM). The EF equation has got greater significance in both physical and mathematical investigation of capillary and nonlinear dispersive gravity waves. The projected technique is tested by considering four illustrations of the time-fractional EF equations. The q-HATM furnish ℏ, known as an auxiliary parameter, by the support of ℏ we can modulate the various stages of convergence of the series solution. Additionally, to certify the resolution and accurateness of the proposed method we fitted the suitable numerical simulations. The redeem results guarantee that the proposed process is more convincing and scrutinizes the extremely nonlinear issues emerging in the field of science and engineering.

Angled Crested Like Water Waves with Surface Tension: Wellposedness of the Problem

We consider the capillary–gravity water wave equation in two dimensions. We assume that the fluid is inviscid, incompressible, irrotational and the air density is zero. We construct an energy functional and prove a local wellposedness result without assuming the Taylor sign condition. When the surface tension $$\sigma $$ is zero, the energy reduces to a lower order version of the energy obtained by Kinsey and Wu (Camb J Math 6(2):93–181, 2018) and allows angled crest interfaces. For positive surface tension, the energy does not allow angled crest interfaces but admits initial data with large curvature of the order of $$\sigma ^{-\frac{1}{3}+ \epsilon } $$ for any $$\epsilon >0$$ .

Analytical Investigation into Effects of Capillary Force on Condensate Film Flowing over Horizontal Semicircular Tube in Porous Medium

A theoretical investigation is performed into the problem of laminar filmwise condensation flow over a horizontal semicircular tube embedded in a porous medium and subject to capillary forces. The effects of the capillary force and gravity force on the condensation heat transfer performance are analyzed using an energy balance approach method. For analytical convenience, several dimensionless parameters are introduced, including the Jakob number Ja, Rayleigh number Ra, and capillary force parameter Boc. The resulting dimensionless governing equation is solved using the numerical shooting method to determine the effect of capillary forces on the condensate thickness. A capillary suction velocity can be obtained mathematically in the calculation process and indicates whether the gravity force is greater than the capillary force. It is shown that if the capillary force is greater than the condensate gravity force, the liquid condensate will be sucked into the two-phase zone. Under this condition, the condensate film thickness reduces and the heat transfer performance is correspondingly improved. Without considering the capillary force effects, the mean Nusselt number is also obtained in the present study as N u ¯ | V 2 ∗ = 0 = 2 R a D a / J a 1 / 2 ∫ 0 π 1 + cos θ 1 / 2 d θ .

ANALYTICAL SOLUTIONS OF COUPLED NONLINEAR SCHRODINGER EQUATIONS FOR TWO NON LINEARLY INTERACTING STOKES WAVE TRAINS OVER INFINITE DEPTH

An attempt to find the exact analytical solutions of the two coupled nonlinear Schrodinger equations of 3rd order occurring from the oblique interaction of two capillary gravity wave trains in the case of crossing sea states in deep water is the main premise of the present paper. The solutions obtained here are due to the nonlinear interaction of two Stokes wave trains in one spatial dimension. Graphs have been plotted to investigate the influence of capillarity on the amplitudes of such wave trains. From 3D figures it has been observed that the capillarity has diminishing influence on the amplitudes of the either wave packet.

Wave Patterns of Gravity–Capillary Waves from Moving Localized Sources

We study wave patterns of gravity–capillary waves from moving localized sources within the classic setup of the problem of ship wakes. The focus is on the co-existence of two wave systems with opposite signatures of group velocity relative to the localized source. It leads to the problem of choice of signs for phase functions of the gravity (“slow”) and capillary (“fast”) branches of the dispersion relation: the question generally ignored when constructing phase patterns of the solutions. We detail characteristic angles of the wake patterns: (i) angle of demarcation of gravity and capillary waves—“the phase Mach” cone, (ii) angle of the minimal group velocity of gravity–capillary waves—“the group Mach” cone, (iii, iv) angles of cusps of isophases that appear after a threshold current speed. The outer cusp cone is naturally associated with the classic cone of Kelvin for pure gravity waves. The inner one results from the effect of capillarity and tends to the “group Mach” pattern at high speeds of current. Amplitudes of the wave patterns are estimated within the recently proposed approach of reference functions for the problem of propagation of packets of linear dispersive waves. The effect of shape is discussed for elliptic reference sources.

Numerical Simulation of a Horizontal Well With Multi-Stage Oval Hydraulic Fractures in Tight Oil Reservoir Based on an Embedded Discrete Fracture Model

Tight oil is a kind of unconventional oil and gas resource with great development potential. Due to the unconventional characteristics of low porosity and low permeability in tight oil reservoirs, single wells generally have no natural productivity, and industrial development is usually conducted in combination with horizontal wells and hydraulic fracturing techniques. To capture the flow behavior affected by fractures with complex geometry and interaction, we adopted embedded discrete fracture models (EDFMs) to simulate the development of fractured reservoirs. Compared with the traditional discrete fracture models (DFMs), the embedded discrete fracture models (EDFMs) can not only accurately represent the fracture geometry but also do not generate a large number of refine grids around fractures and intersections of fractures, which shows the high computational efficiency. To be more consistent with the real characteristic of the reservoir and reflect the advantage of EDFMs on modeling complex fractures, in this work, the hydraulic fractures are set as oval shape, and we adopted 3-dimensional oil–gas two-phase model considering capillary forces and gravity effects. We developed an EDFM simulator, which is verified by using the fine grid method (FGM). Finally, we simulated and studied the development of tight oil without and with random natural fractures (NFs). In our simulation, the pressure varies widely from the beginning to the end of the development. In real situation, tight oil reservoirs have high initial pressure and adopt step-down bottom hole pressure development strategy where the bottom hole pressure of the last stage is below the bubble point pressure and the free gas appears in the reservoir. Modeling studies indicate that the geometry of fracture has a great influence on the pressure and saturation profiles in the area near the fractures, and dissolved gas flooding contributes to the development of tight oil, and NFs can significantly improve production, while the effect of the stress sensitivity coefficient of NFs on production is more significant in the later stage of production with lower reservoir pressure.

Time-dependent wave motion with undulated bottom

In the present manuscript, the time-dependent capillary gravity wave motion in the presence of a current and an undulated permeable bottom is analyzed. The spectral method is used to simulate the time-dependent surface elevation. Also, the Laplace–Fourier transform method is used to obtain the integral form of the surface elevation, and the asymptotic form of the associated highly oscillatory integral is derived using the method of stationary phase. The reflection and transmission coefficients due to the small bottom undulation are obtained using the perturbation method and the Fourier transform method and also alternatively using Green’s function technique and Green’s identity. The nature of wave energy propagation obtained from plane capillary gravity wave motion is verified through time domain simulation and using the spectral method. It is found that, in the case of co-propagating waves, the wave energy propagates faster and also the surface profiles in terms of wave packets move faster for larger values of the Froude number. Also, the maximum value of the reflection and transmission coefficients decreases due to increasing values of the Froude number. For the sinusoidal bottom topography, the Bragg resonance occurs if the ratio of the wave numbers of the wave and the rippled bed is one by two.

An efficient computational technique for time‐fractionalKaup‐Kupershmidtequation

AbstractIn this article, an efficient novel technique, namely theq‐homotopy analysis transform method (q‐HATM) is applied to find the solution for the time‐fractional Kaup‐Kupershmidt (KK) equation. The KK equation plays a vital role while studying the capillary gravity waves and nonlinear dispersive waves. To check the effectiveness and pertinency of the projected method, we consider three distinct cases of the fractional nonlinear KK equation. Theq‐HATM provides the auxiliary parameterℏ, called convergence control parameter, with the help of that we can manipulate and adjust the area of convergence of the series solution. Moreover, to authenticate the accuracy and reliability of the considered technique the numerical simulations have been presented. The retrieved results ensure that the projected scheme is effortless to carry out and analyze the highly nonlinear problems arising in science and technology.

Enhanced Recovery From Naturally Fractured Gas Reservoirs With Seismic Vibrations

Abstract Water level rising in fracture networks of a naturally fractured gas reservoir is extremely challenging and can significantly decrease the ultimate recovery due to reservoir heterogeneity. Although capillary drainage and gravity force can enhance the displacement of gas recovery from matrix to fracture, these forces may not be so effective in mobilizing a large amount of trapped gas through the matrix. So called, the use of seismic wave can be suggested as a low cost and environmentally friendly enhanced method compared with the other conventional enhanced methods. This article is aimed to examine the ability of seismic vibration in generating an efficient driving force for moving the remaining gas into the fracture which, to the best of the author’s knowledge, has not been reported so far. To this end, an in-house numerical simulator has been developed to investigate this enhanced recovery method and also to evaluate the effect of wave characteristics as well as rock properties on the ultimate recovery. The governing equations are solved numerically using finite difference approach and the accuracy of these equations was compared with a commercial simulator for verification. The results are very encouraging and show substantial gas recovery enhancement by applying seismic waves. Our investigation also shows that this stimulation method is more efficient at lower frequencies and also in higher permeable matrix and fractures.

Tidal Currents Detected in Kraken Mare Straits from Cassini VIMS Sun Glitter Observations

Abstract We present Cassini Visual and Infrared Mapping Spectrometer (VIMS) observations of sun glitter—wave-induced reflections from a liquid surface offset from a specular point—on Kraken Mare. Sun glitter reveals rough sea surfaces around Kraken Mare, namely the coasts and narrow straits. The sun glitter observations indicate wave activity driven by the winds and tidal currents in Kraken Mare during northern summer. T104 Cassini VIMS observations show three sun glitter features in Bayta Fretum indicative of variegated wave fields. We cannot uniquely determine one source for the coastal Bayta waves, but we lean toward the interpretation of surface winds, because tidal currents should be too weak to generate capillary–gravity waves in Bayta Fretum. T105 and T110 observations reveal wave fields in the straits of Seldon Fretum, Lulworth Sinus, and Tunu Sinus that likely originate from the constriction of tidal currents. Coastlines of Bermoothes and Hufaidh Insulae adjoin rough sea surfaces, suggesting a complex interplay of wind-roughened seas and localized tidal currents. Bermoothes and Hufaidh Insulae may share characteristics of either the Torres Strait off Australia or the Åland region of Finland, summarized as an island-dense strait with shallow bathymetry that hosts complex surface circulation patterns. Hufaidh Insulae could host seafloor bedforms formed by tidal currents with an abundant sediment supply, similar to the Torres Strait. The coastlines of Hufaidh and Bermoothes Insulae likely host ria or flooded coastal inlets, suggesting that the Insulae may be local peaks of primordial crust isolated by an episode of sea-level rise or tectonic uplift.

Lagrangian motion of fluid particles in gravity–capillary standing waves

A third-order analytical solution for the gravity–capillary standing wave is derived in Lagrangian coordinates through the Lindstedt–Poincare perturbation method. By numerical computation, the dynamical properties of nonlinear standing waves with surface tension in finite water depth, including particle trajectory and surface profile are investigated. We find that the presence of surface tension leads to a change of the crest form. Moreover, we also find that the particle trajectories near the surface oscillate back and forth along the arcs which will change from concave to convex as the inverse Bond number increases. There is no mass transport of the particles in a wave period.

Resonances for water waves over flows with piecewise constant vorticity

We investigate three and four-wave resonances of capillary–gravity water waves arising as the free surface of water flows exhibiting piecewise constant vorticity. More precisely, our type of flow has a jump in the vorticity distribution that separates a rotational layer at the top (commonly generated by wind-shear) from another rotational layer adjacent to the sea-bed, region that accommodates strong sheared currents. Instrumental in deriving our findings is the dispersion relation for such flows having a jump in the distribution of the vorticity. In general, for rotational flows of non-constant vorticity, the dispersion relation is very intricate. However, we show that a disentanglement occurs in the case of capillary–gravity water waves for which the ratio “thickness of the near-surface vortical layer/the wavelength of the surface wave” is sufficiently large. More precisely, we find explicitly two solutions, λ0 and λ1 that represent the (relative) surface wave speed. We then confirm analytically that λ1 gives rise to three-wave resonances for capillary–gravity water waves with wavelengths not exceeding 2 cm. However, for significant wavelength range, we establish that λ1 does not lead to four-wave resonances. In contrast to the previous conclusion λ0 does not bring about three-wave resonances, but is able to generate four-wave resonances.