Waves In Porous Media Research Articles

Wettability effect on wave propagation in saturated porous medium.

Micro-fluid mechanics studies have revealed that fluid slip on the boundary of a flow channel is a quite common phenomenon. In the context of a fluid-saturated porous medium, this implies that the fluid slippage increases with the increase of the hydrophobicity, which is the non-wetting degree. Previous studies find that wettability of the pore surface is strongly related to the slippage, which is characterized by slip length. To accurately predict acoustical properties of a fluid-saturated porous medium for different wettability conditions, the slippage of the wave-induced flow has to be taken into account. This paper introduces the slip length as a proxy for wettability into the calculation of the viscous correction factor, dynamic permeability, and dynamic tortuosity of the Biot theory for elastic waves in a porous medium. It demonstrates that, under different wettability conditions, elastic waves in a saturated porous medium have different phase velocity and attenuation. Specifically, it finds that increasing hydrophobicity yields a higher phase velocity and attenuation peak in a high-frequency range.

Elastic waves in porous media saturated with non-wetting fluid

Elastic waves have widely been used as a non-destructive probing method in oilfield exploration and development, and the most well-known applications are in seismic exploration and borehole sonic logging. For waves in porous media, it is popular to use the Biot theory, which incorporates the wave-induced global flow, accounting for the frictional attenuation. The Biot theory assumes that the fluid is wetting to the solid matrix. However, the fluid is not always wetting the rock in real reservoirs. It was previously revealed that a non-wetting fluid parcel tends to slip on the solid wall pore boundary where the intermolecular potential between the fluid and solid wall is weaker than in wetting fluid conditions. This particular slippage feature means that the coupling relationship between the fluid and solid frame and frictional dissipation is likely to be very different between non-wetting and wetting fluid situations. We characterise this wave-induced slippage using an apparent viscosity for the non-wetting fluid within the thin viscous boundary layer. This apparent viscosity is smaller than the viscosity of the bulk fluid. We demonstrate that the slip correction affects the dynamic permeability and dynamic tortuosity and results in slippage/wettability dependent phase velocities and attenuation of the fully fluid-saturated rock.

Parametric Space for Elastic Waves in Porous Media with Complex Rheology

We evaluate the influence of mechanical parameters on the decay rate of elastic waves in fluid-saturated media with bubbles. The parameters describe the complex rheological scheme adopted as a model of the grainy porous medium. Areas of validity of the model in the space of mechanical parameters are determined.

Subsonic detonation waves in porous media

The existence of subsonic detonation wave (processes occurring at high speeds) for a system that is related to the combustion model for highly hydraulically resistant porous media is studied. The porous media combustion model consists of a system of coupled partial differential equations. We apply some topological arguments to prove the existence of heteroclinic orbits for the system which governs ordinary differential equations.

Compression waves in porous media containing gas hydrate

Wave processes in a porous medium containing water, gas and gas hydrate are numerically investigated. The two-velocity model of porous medium is used. The nonlinearity of the oscillations of the bubbles is taken into account in the equation of gas state and the equation for bubble radius. The transmission of compression of step waves from the water into hydrate-containing porous medium is investigated. Beyond the line of phase equilibrium the association of gas hydrate is possible. The influence of the medium parameters and the intensity of the incident wave on the evolution of waves in this porous medium is studied. When entering the porous medium, the initial pulse is divided into fast (deformation) and slow (filtration) waves. In a slow wave damped oscillations associated with pulsations of bubbles are observed. The increase in the initial equilibrium pressure in the medium leads to an increase in the velocities of the deformation and filtration waves. With the initial volume content of the gas in the pore space unchanged, an increase in the initial hydrate content leads to a small increase in the velocities and amplitudes of fast and slow waves. At constant initial volume content of hydrate, the increase of gas content leads to a noticeable decrease of slow wave speed and decrease of amplitudes of both fast and slow waves. It was found that the accounting of the processes of formation of gas hydrate in a porous medium at the considered time intervals does not have a noticeable effect on the character of wave propagation.

Shock waves in porous media

The experimental data [1, 2] on the effect of the parameters of plane incident shock waves and the characteristics of a porous medium on the attenuation and reflection of waves from a rigid obstacle are examined. The propagation of waves with a pressure jumps of up to ΔP/P0 = 20 (P0 = 0.1 MPa) in polyurethane foam blocks with a density of 20 kg/m3 and 35 kg/m3 and a length of up to 0.8 m, tightly adjacent to the walls and the end face of the driven section of the shock tube is studied. At lengths of the porous block greater than the extent of the shock wave, the porous block effectively attenuated the shock wave. For a wave of limited duration with a triangular pressure profile, a stronger attenuation of the wave in the polyurethane foam block was observed in comparison with that for an extended wave pulse. The reflection pressures of the waves in polyurethane foam with pressure jumps of ΔP/P0 ≥ 12 exceed the reflection pressures of shock waves in air. The degree of attenuation or amplification of a wave depends on the material density and the ratio of the lengths of the foamed polyurethane block and the pressure pulse.

Extended Boussinesq equations for waves in porous media

We develop an extended Boussinesq model to predict the propagation of waves in porous media. The inertial and drag resistances are taken into account in the developed model. When these resistances are neglected, the developed equations are reduced to the extended Boussinesq equations of Madsen and Sørensen (1992) [Coastal Eng. 18, 183–204] for non-porous media The model can be applied for non-porous media as well as porous media because it does not need any matching condition at the interface between the porous and non-porous media. The developed model is verified by comparing numerical solutions to experimental data with several cases of solitary waves or sinusoidal waves interacting with a porous breakwater in one- and two-dimensional domains.

Velocity and attenuation of elastic wave in a developed layer with the initial inner percolation in the pores

An improved model based on Biot poroelastic theory is presented incorporating the initial seepage flow in the matrix. The proposed model quantifies wave propagation in the oil field development process as VSP, transient well tests, and seismic production technology. Porosity variation, fluid–solid compressibility, and pseudo-threshold pressure gradient in low permeability reservoir are simultaneously considered, resulting in a modified form of continuity and motion equations. Integrating the equations and decomposition for the fluid–solid displacements helps to derive the analytical wave vectors of the fast P, slow P, and S waves in porous media. The sensitivity of affecting factors, such as petrophysical parameter, fluid property, and vibration frequency, on the wave velocity and attenuation is subsequently evaluated. Furthermore, the Biot model can be considered a special situation when the initial flow rate in this model tends to zero. The increase in initial percolating rate, expressed by the ratio of initial flow rate and solid scalar potential, causes an obvious decrease in the fast P wave velocity and an increase in its quality factor. Propagation of the slow P wave and the S wave is scarcely influenced. Low vibration frequency and low permeability also contribute to a large difference between the wave propagation parameters of the improved and Biot models. Cognition of the proportion in the inertia and viscosity effects is helpful in analyzing the complicated multiple mechanisms in wave propagation in the actual developed layer.

Rayleigh wave at the surface of a general anisotropic poroelastic medium: derivation of real secular equation

A secular equation governs the propagation of Rayleigh wave at the surface of an anisotropic poroelastic medium. In the case of anisotropy with symmetry, this equation is obtained as a real irrational equation. But, in the absence of anisotropic symmetries, this secular equation is obtained as a complex irrational equation. True surface waves in non-dissipative materials decay only with depth. That means, propagation of Rayleigh wave in anisotropic poroelastic solid should be represented by a real phase velocity. In this study, the determinantal system leading to the complex secular equation is manipulated to obtain a transformed equation. Even for arbitrary (triclinic) anisotropy, this transformed equation remains real for the propagation of true surface waves. Such a real secular equation is obtained with the option of boundary pores being opened or sealed. A numerical example is solved to study the existence and propagation of Rayleigh waves in porous media for the top three (i.e. triclinic, monoclinic and orthorhombic) anisotropies.

Propagation of elastic waves in saturated porous medium containing a small amount of bubbly fluid

It is very important to understand the acoustical properties of porous medium. To study the relationship between acoustical and other physical properties of porous medium will help us to use acoustical tools for determining the physical properties of porous medium. Many researchers have paid much attention to the properties of acoustic wave propagation in the gassy marine sediments based on the Biot model which is popularly used to predict the dispersion and attenuation of sound in saturated porous medium. The patchy model which contains gas inside the spherical water predicts that the existence of gas just has little effect on the propagation of acoustic wave in porous medium when the gas content is very small. However, the presence of a small number of bubbles in a fluid saturated sediment will lead to different acoustic responses. As is well known, the bubble vibration theory proposed by Keller and Miksis shows that a small number of bubbles existing in the liquid will have a great influence on sound velocity and attenuation. Therefore, in order to study the effect of a small amount of gas existing in fluid saturated porous medium on the property of acoustic wave propagation, we investigate a bubbly liquid saturated porous medium and consider the case of the bubbles vibrating linearly under the action of sound waves. First, we derive the continuity equation of the seepage according to the mass conservation of the pore fluid and the relationship between porosity differentiation and pore fluid pressure differentiation. Then, the bubble linear vibration theory given by Commander is used to deal with the time derivative of gas volume fraction in the continuity equation of the seepage, The bubble linear vibration theory gives the relationship between instantaneous bubble radius and background pressure of the medium. Through this relationship, we obtain the equation of time derivative of gas volume fraction and time derivative of pore fluid pressure. Then, we combine the obtained equation with the continuity equation of seepage, and obtain the modified continuity equation of seepage whose form is similar to that of Biot model. Finally, the modified Biot's equations for fluid saturated porous medium containing a small amount of bubbly fluid is obtained. As is well known, an effective density fluid model for acoustic propagation in sediments, derived from Biot theory, just can predict the acoustic properties of the fast compressional waves. However, the present model can predict the acoustic properties of fast, slow compressional waves and shear waves propagating in sediments. Through numerically calculating the dispersion, attenuation, amplitude ratios of pore fluid displacement to solid displacement for fast and slow compressional waves, it is found that the existence of a small number of bubbles has an influence on the acoustic properties of both the fast compressional waves and the slow compressional waves, especially the velocity of the fast compressional wave. In addition, the low-frequency speed approximation formula for the fast compressional wave is also presented. The approximate formula directly indicates the relationship between the velocity of fast compressional wave and the parameters of porous medium such as the gas volume fraction and the bubble radius. This study shows that the influence of a small number of bubbles in fluid saturated on acoustic wave propagation is noticeable. The modified Biot model presented in this paper provides one model to study the properties of acoustic waves in fluid saturated porous medium with a small number of bubbles.

Waves in porous media saturated with bubbly liquid

Wave processes in a porous medium saturated with bubbly liquid, taking into account the nonlinearity of the oscillations of the bubbles, are numerically investigated. The nonlinearity is accounted in the equation of gas state and the equation of Rayleigh - Lamb. Numerical study in the framework of nonlinear model is performed for the first time. The dispersion relations for linear waves in a porous medium containing bubbly liquid, as well as for the main acoustic mode in a cylindrical cavity in such a medium, are obtained. Frequency dependences of velocity and attenuation of harmonic waves in a cylindrical waveguide are calculated. The influence properties of the bubbly liquid saturated porous medium on the wave velocity and attenuation and the penetration depth of perturbation into the surrounding porous medium are analyzed. The passage of the step wave from fluid into a porous medium saturated with bubbly liquid is investigated. The influence of porous medium parameters and intensity of the incident wave on the wave evolution in bubbly liquid saturated porous medium is studied. The calculation results show good qualitative agreement with the experimental data of other authors.

Nonlinear wave propagation in porous materials based on the Biot theory.

Nonlinearity must be considered with some porous granular media because of the large deformation under seismic waves. In this study, the propagation of nonlinear waves in porous media is studied based on the Biot theory and the governing equations are obtained by the Lagrangian formulation. Three new nonlinear parameters are introduced to consider the coupled nonlinearity between the solid and fluid components in porous media. It is shown that an additional nonlinear wave with a double frequency is generated by the coupling effect of linear fast and slow waves. When only a shear wave is applied at the source, no double-frequency nonlinear wave is predicted and three nonlinear longitudinal waves are generated. On the basis of the practical case studies, the effect of strong nonlinearity is computed under the influence of a one-dimensional single longitudinal wave source and a single shear wave source.

The structure of steady shock waves in porous metals

The paper aims at developing an understanding of steady shock wave propagation in a ductile metallic material containing voids. Porosity is assumed to be less than 0.3 and voids are not connected (foams are not considered). As the shock wave is traveling in the porous medium, the voids are facing a rapid collapse. During this dynamic compaction process, material particles are subjected to very high acceleration in the vicinity of voids, thus generating acceleration forces at the microscale that influence the overall response of the porous material. Analyzing how stationary shocks are influenced by these micro-inertia effects is the main goal of this work. The focus is essentially on the shock structure, ignoring oscillatory motion of pores prevailing at the tail of the shock wave. Following the constitutive framework developed by Molinari and Ravichandran (2004) for the analysis of steady shock waves in dense metals, an analytical approach of steady state propagation of plastic shocks in porous metals is proposed. The initial void size appears as a characteristic internal length that scales the overall dynamic response, thereby contributing to the structuring of the shock front. This key feature is not captured by standard damage models where the porosity stands for the single damage parameter with no contribution of the void size. The results obtained in this work provide a new insight in the fundamental understanding of shock waves in porous media. In particular, a new scaling law relating the shock width to the initial void radius is obtained when micro-inertia effects are significant.

Shallow water waves in porous medium for coast protection

This paper extends the Hirota-Satsuma equation in continuum mechanics to its fractional partner in fractal porous media in shallow water for absorbing wave energy and preventing tsunami. Its derivation is briefly introduced using the fractional momentum law and He?s fractional derivative. The fractional complex transform is adopted to elucidate its basic solution properties, and a modification of the exp-function method is used to solve the equation. The paper concludes that the kinetic energy of the travelling wave tends to be vanished when the value of the fractional order is less than one.

Modeling attenuation and dispersion of acoustic waves in porous media containing immiscible non viscous fluids

Este artículo reporta resultados de la propagación de ondas P en medios porosos, simulada solucionando en diferencias finitas las ecuaciones generalizadas de Biot. En modelos saturados se observó que cuando avanza la onda, la amplitud máxima del espectro se desplaza hacia frecuencias menores, y que esta amplitud máxima y su frecuencia están directamente relacionadas. Además, que el factor de calidad disminuye con la porosidad y la saturación. Por ende, la atenuación aumenta con la porosidad, la saturación y la frecuencia pero tiende asintóticamente a un valor constante. Se observó que la fase de la onda cambia linealmente con la frecuencia a una rata de cambio que aumenta linealmente con el tiempo de viaje. Esta rata aumenta con la saturación pero disminuye ligeramente con la porosidad. Este trabajo ignora la divergencia esférica y la retro dispersión, concentrándose en la atenuación intrínseca causada por la fricción, en particular entre líquido y partículas sólidas.

Propagation of nonlinear thermoelastic waves in porous media within the theory of heat conduction with memory: physical derivation and exact solutions

In this paper, we study a mathematical model of nonlinear thermoelastic wave propagation in fluid‐saturated porous media, considering memory effect in the heat propagation. In particular, we derive the governing equations in one dimension by using the Gurtin–Pipkin theory of heat flux history model and specializing the relaxation function in such a way to obtain a fractional Erdélyi–Kober integral. In this way, we obtain a nonlinear model in the framework of time‐fractional thermoelasticity, and we find an explicit analytical solution by means of the invariant subspace method. A second memory effect that can play a significant role in this class of models is parametrized by a generalized time‐fractional Darcy law. We study the equations obtained also in this case and find an explicit traveling wave type solution. Copyright © 2016 John Wiley & Sons, Ltd.

Effect of irregularity on torsional surface waves in an initially stressed anisotropic porous layer sandwiched between homogeneous and non-homogeneous half-space

The present paper is concerned with the propagation of torsional surface waves in an initially stressed anisotropic porous layer sandwiched between homogeneous and non-homogeneous half-space. We assume the quadratic inhomogeneity in rigidity and density in the lower half-space and irregularity is taken in the form of rectangle at the interface separating the layer from the lower half-space. The dispersion equation for torsional waves has been obtained in a closed form. Velocity equation is also obtained in the absence of irregularity. The study reveals that the presence of irregularity, initial stress, porosity, inhomogeneity and anisotropy factor in the dispersion equation approves the significant effect of these parameters in the propagation of torsional waves in porous medium. It has also been observed that for a uniform media, the velocity equation reduces to the classical result of Love wave.

A semi-analytical solution for the transient response of one-dimensional unsaturated single-layer poroviscoelastic media

This paper develops a semi-analytical solution for the transient response of an unsaturated single-layer poroviscoelastic medium with two immiscible fluids by using the Laplace transformation and the state-space method. Using the elastic–viscoelastic correspondence principle, we first introduce the Kelvin–Voigt model into Zienkiewicz’s unsaturated poroelastic model. The vibrational response for unsaturated porous material can be obtained by combining these two models and assuming that the wetting and non-wetting fluids are compressible, the solid skeleton and solid particles are viscoelastic, and the inertial and mechanical couplings are taken into account. The Laplace transformation and state-space method are used to solve the basic equations with the associated initial and boundary conditions, and the analytical solution in the Laplace domain is developed. To evaluate the responses in the time domain, Durbin’s numerical inverse Laplace transform method is used to obtain the semi-analytical solution. There are three compressional waves in porous media with two immiscible fluids. Moreover, to observe the three compressional waves clearly, we assume the two immiscible fluids are water and oil. Finally, several examples are provided to show the validity of the semi-analytical solution and to assess the influences of the viscosity coefficients and dynamic permeability coefficients on the behavior of the three compressional waves.

Propagation of Love-Type Wave in Porous Medium over an Orthotropic Semi-Infinite Medium with Rectangular Irregularity

Propagation of Love-type wave in an initially stressed porous medium over a semi-infinite orthotropic medium with the irregular interface has been studied. The method of separation of variables has been adopted to get the dispersion relation of Love-type wave. The irregularity is assumed to be rectangular at the interface of the layer and half-space. Finally, the dispersion relation of Love wave has been obtained in classical form. The presence of porosity, irregularity, and initial stress in the dispersion equation approves the significant effect of these parameters in the propagation of Love-type waves in porous medium bounded below by an orthotropic half-space. The scientific effect of porosity, irregularity, and initial stress in the phase velocity of the Love-type wave propagation has been studied and shown graphically.

Upscaling diffusion waves in porous media

The aim of this work is to derive the effective-medium equations and to estimate the related effective diffusivities for diffusion waves in porous media. Effective diffusivities are estimated within the framework of the volume averaging method, where they are obtained from the solution of the associated closure problems in 2D and 3D periodic unit cells. The results showed that the transport of diffusion waves are governed by the diffusion and co-diffusion mechanisms of harmonic waves. In addition, numerical results showed that the effective diffusivities increase with frequency, while the effective co-diffusivities display a resonance-like behavior. Our results also indicate that geometry plays a more significant effect over the predictions of the co-diffusion coefficient at moderate frequencies and it mainly influences the predictions of the direct diffusivity at low frequencies (i.e.,ω∗≪1).