Waves In Porous Media Research Articles

On propagation characteristics of ultrasonic guided waves in layered fluid-saturated porous media using spectral method.

Fluid-saturated porous media plays an increasingly important role in emerging fields such as lithium batteries and artificial bones. Accurately solving the governing equations of guided wave is the key to the successful application of ultrasonic guided wave nondestructive testing technology in fluid-saturated porous media. This paper derives the Lamb wave equation in layered fluid-saturated porous materials based on Biot theory and proposes the spectral method suitable for solving complex wave equations. The spectral method reconstructs the fundamental wave equations in the form of a matrix eigenvalue problem using spectral differentiation matrices. It introduces boundary conditions by replacing corresponding rows in the wave equation matrix with stress or displacement in matrix form. For complex differential equations, such as the governing equations of guided waves in porous media, the spectral method has the significant advantages of faster computation speed, less root loss, and easier encoding process. The spectral method is used to calculate the acoustic field characteristics under different boundary conditions and environments of the layer fluid-saturated porous media. Results show that the surface treatment details and environment of fluid-saturated porous materials play an important role in the propagation of guided waves.

Some three- and n-component waves in porous media

Based on experimental results and self-consistent physical theory, a three-component description of nonlinear body waves in porous media is constructed. Applications of this result to two-component fluid flow in dynamic porous media and seismic wave propagation in multiphase porous media are presented. This description is important to petroleum reservoir simulation, groundwater hydrology, and seismic analysis of the earth. The two-body analogue of these results has been shown, in the past, to be inconsistent with the two-body linear models currently used by petroleum engineers and groundwater hydrologists. In seismic theory, simple linear models are generally used and much of the information obtained from the theory presented here is referred to as noise. The processes observed and described here have been patented and applied commercially to oil production and groundwater remediation. It is shown here that even if the correct nonlinear equations are used, three-component wave descriptions of porous media cannot be constructed solely from the equations of motion for the components. This is because of the introduction of the complexity of multiple scales into this nonlinear field theory. Information about the coupling between the components is required to obtain a physical description. It is observed that the fields must be coupled in phase and out of phase, and this result is consistent with the description of three- and n-body gravitational fields in Newtonian gravity and general relativity.

Break-up and mobilization of DNAPL by acoustic excitation: Experimental evidence and pore network modeling

Dense non-aqueous phase liquids (DNAPLs) are long-term groundwater contaminants due to their high toxicity and slight solubility in water. The use of acoustic waves to remobilize trapped ganglia in subsurface porous systems have some advantages over pre-existing solutions including eliminating the bypassing effect and new environmental hazards. Designing an effective acoustically assisted remediation method for such purposes relies on understanding the underlying mechanisms and developing validated models. In this work, pore-scale microfluidic experiments were run to investigate the interplay between break-up and remobilization under sonication at different levels of flow rate and wettability conditions. Based on the experimental observation and pore-scale physical characteristics, a pore network model was developed and verified against the experimental results. Such a model was developed based on a two-dimensional network and scaled up to three-dimensional networks. In the experiments, processing of two-dimensional images showed that acoustic waves can remobilize trapped ganglia. The other observed effect of vibration is to break up blobs and reduce the mean ganglia size. Recovery enhancements were greater in hydrophilic micromodels as compared to hydrophobic system. A strong correlation was found between the remobilization and breakup indicating that the trapped ganglia are breaking up due to acoustic stimulation firstly and then a background viscous force may get them flowing under the new generated fluid distribution. In modeling, the simulation results of residual saturation reasonably matched with experimental observations. The differences between the prediction by the model and the experimental data at verification points is less than 2% for data before and after the acoustic excitation. The transitions from three-dimensional simulations were used to propose a modified capillary number. This study gives a better understanding of the mechanisms behind the effect of acoustic waves in porous media and provides a predictive tool for evaluating enhancement in fluid displacement.

Emission and propagation of sound waves in porous media with active inner heat sources

Wave propagation through porous media with active inner heat sources is investigated in this paper. Through the use of the two-scale asymptotic method of homogenisation, it is found that a macroscopic non-homogeneous wave equation describes the emission and propagation of sound waves in such active porous media. The upscaled model is validated numerically for the cases of single and double porosity media, and shows that the general properties of the effective parameters of the porous media are not altered by the inner heat sources. Instead, the inner heat sources contribute to the non-homogeneous term in the upscaled wave equation. The paper also explores the potential of using active inner heat sources for controlling sound waves incident to the porous media in practical scenarios.

A hybrid FV/FD scheme for a novel conservative form of extended Boussinesq equations for waves in porous media

A hybrid FV/FD scheme for a novel conservative form of extended Boussinesq equations for waves in porous media

Propagation of acoustic wave and analysis of borehole acoustic field in porous medium of heterogeneous viscous fluid

Sound field in fluid-saturated porous medium is closely related to the viscosity of fluid and the heterogeneity of porous medium. In order to improve the physical mechanism of wave propagation in porous medium and expand its application in borehole acoustic field, the shear stress of porous fluid and the heterogeneity of pore structure are considered. The wave theory in porous medium containing viscous fluid is deduced based on the Biot theory. The influence of porous medium parameters on slow shear wave is analyzed, and the dispersion and attenuation of elastic wave caused by shear stress balance in porous fluid under the influence of inhomogeneous pore structure are studied. The analytical solution of borehole acoustic field in porous medium containing viscous fluid is further derived. The phase velocity and attenuation of borehole mode waves in heterogeneous porous medium, and the waveform of borehole full wave are calculated. The influence of pore fluid viscosity on borehole full wave is analyzed. The results show that there are slow shear waves in the porous medium containing viscous fluid. The slow shear wave is characterized by low velocity and large attenuation. In heterogeneous porous medium, the balance process of shear stress related to slow shear wave not only leads to the dispersion and attenuation of fast P-wave, but also affects the propagation characteristics of borehole pseudo Rayleigh wave and Stoneley wave. In addition, the pore fluid viscosity has a great influence on the borehole Stoneley wave. The present work improves the physical mechanism of acoustic wave propagation in porous medium and provides theoretical guidance for the explanation and application of borehole acoustic waves in porous formations.

Enhancing solute transport by pressure-wave driven flow in unsaturated porous media

Solutes and their limited mobilization in heterogeneous porous media may lead to subsurface salinization and contamination evident in various environmental settings. Previous studies showed that flow and transport driven by pressure waves in porous media under saturated conditions significantly enhance solute displacement compared to drag-dominant (Darcy) flow and transport, resulting in more thorough remediation. However, experimental evidence lacks demonstration of such transport under unsaturated conditions. We investigate solute leaching in a flow cell, experimentally modeling solute displacement from an unsaturated, saline, fine-textured soil region to the surrounding coarse-textured soil. The extent of mobilization efficiency is evaluated by comparing solute breakthrough measurements between two leaching treatments, with drag-dominant continuous flux (CF) and pressure-wave pulse flux (PPF). The results indicate that the PPF treatment provided an additional 75% solute mass that leached from the fine-textured region per unit volume of water compared to the CF. The effect is partly due to the impact of the pressure waves on water imbibition into the medium. The PPF induced relatively high pressure gradients at the wetting front compared to percolation during the CF, thus wetting the media in a more spatially uniform pattern. Consequently, the PPF treatment led to higher water content at and behind the front, which increased the continuity of the water phase across the fine-textured and coarse-textured soils and enhanced transport. The study is a proof of concept for pressure-wave driven flow to enhance local wetting and solute transport in unsaturated soil.

Effect of mesoscopic-flow loss on seismic reflections in media with penny-shaped inclusions

Summary We obtain the amplitude and energy reflection coefficients of seismic waves in porous media with penny-shaped inclusions, based on the generalized Biot-Rayleigh model that takes into account the attenuation due to mesoscopic local fluid flow (LFF). We consider two cases, including a contact between two porous media having either different fluids (gas-water contact) or crack density/aspect ratio, as well as a water half-space overlying a porous medium, and study the frequency-dependent reflection-transmission (scattering) coefficients for open- and sealed-pore boundary conditions. Our examples show that the LFF mechanism mainly reduces the reflection coefficients (amplitude and energy) at the gas-water contact and at a water/porous-medium interface for frequencies less than 10 kHz, due to the fact that the velocity in the lower medium decreases. For the latter case, if the fluid is gas, the LFF effect becomes only important at frequencies between 0.0001 and 10 Hz for the open-pore case. This is due to the fact that the acoustic impedance contrast between water and gas is high. At frequencies less than 0.0001 Hz, the interface is equivalent to a water/elastic-medium one, and hence the results are the same as those of the sealed-pore case. Moreover, the crack density and aspect ratio affect the mesoscopic attenuation and relaxation frequency, and therefore the reflection coefficients.

Wave propagation characteristics in porous medium containing a solid in pores

Aiming at the propagation characteristics of acoustic waves in a porous medium containing a solid in pores, the equations of motion and constitutive relation are deducted in the case of two-solid porous media. The frequency dispersion and attenuation characteristics of wave modes are analyzed by a plane wave analysis. In addition, based on the first-order velocity-stress equations, the time-splitting high-order staggered-grid finite-difference algorithm is proposed and constructed for understanding wave propagation mechanisms in such a medium, where the time-splitting method is used to solve the stiffness problem in the first-order velocity-stress equations. The generation mechanisms and energy distributions of different kinds of waves are investigated in detail. In particular, the influences of the friction coefficient between solid grains and pore solid as well as frequency on wave propagation are analyzed. It can be known from the results of plane wave analysis that there are two compression waves (P1 and P2) and two shear waves (S1 and S2) in a porous medium containing a solid in pores. The attenuations of P2 wave and S2 wave are much larger than those of P1 wave and S1 wave. This is due to the friction between the solid grains and the pore solid. The results show that our proposed numerical simulation algorithm can effectively solve the problem of stiffness in the velocity-stress equations, with high accuracy. The excitation mechanisms of the four wave modes are clearly revealed by the simulation results. The P1 wave and S1 wave propagate primarily in the solid grain frame, while P2 wave and S2 wave are concentrated mainly in the pore solid, which are caused by the relative motion between the solid grains and the pore solid. Besides, it should be pointed out that the wave diffusions of the P2 wave and S2 wave are influenced by the friction coefficient between solid grains and pore solid. The existence of friction coefficient between two solids makes P2 wave and S2 wave attenuate to a certain extent at high frequency, but the attenuation is much smaller than that at low frequency. This is the reason why it is difficult to observe the slow waves in practice. However, because the slow waves also carry some energy, it may not be ignored in the studying of the energy attenuation of acoustic waves in porous media.

Velocity and Attenuation of Linear Waves in Porous Media Saturated with Gas and its Hydrate

Velocity and Attenuation of Linear Waves in Porous Media Saturated with Gas and its Hydrate

An efficient ADER-DG local time stepping scheme for 3D HPC simulation of seismic waves in poroelastic media

Many applications from the fields of seismology and geoengineering require simulations of seismic waves in porous media. Biot's theory of poroelasticity describes the coupling between solid and fluid phases and introduces a stiff reactive source term (Darcy's Law) into the elastodynamic wave equations, thereby increasing computational cost of respective numerical solvers and motivating efficient methods utilising High-Performance Computing.We present a novel realisation of the discontinuous Galerkin scheme with Arbitrary High-Order DERivative time stepping (ADER-DG) that copes with stiff source terms. To integrate this source term with a reasonable time step size, we utilise an element-local space-time predictor, which needs to solve medium-sized linear systems – each with 1,000 to 10,000 unknowns – in each element update (i.e., billions of times). We present a novel block-wise back-substitution algorithm for solving these systems efficiently, thus enabling large-scale 3D simulations. In comparison to LU decomposition, we reduce the number of floating-point operations by a factor of up to 25, when using polynomials of degree 6. The block-wise back-substitution is mapped to a sequence of small matrix-matrix multiplications, for which code generators are available to generate highly optimised code.We verify the new solver thoroughly against analytical and semi-analytical reference solutions in problems of increasing complexity. We demonstrate high-order convergence of the scheme for 3D problems. We verify the correct treatment of point sources and boundary conditions, including homogeneous and heterogeneous full space problems as well as problems with traction-free boundary conditions. In addition, we compare against a finite difference solution for a newly defined 3D layer over half-space problem containing an internal material interface and free surface. We find that extremely high accuracy is required to accurately resolve the slow, diffusive P-wave at a or near a free surface, while we also demonstrate that solid particle velocities are not affected by coarser resolutions. We demonstrate that by using a clustered local time stepping scheme, time to solution is reduced by a factor of 6 to 10 compared to global time stepping. We conclude our study with a scaling and performance analysis on the SuperMUC-NG supercomputer, demonstrating our implementation's high computational efficiency and its potential for extreme-scale simulations.

Propagation behavior of Love-type waves in poro-elastic medium using staggered grid finite difference Scheme

The analysis of the propagation behaviour of seismic waves in porous medium requires a mathematical backup. That is pictured here. The present study investigates the propagation behavior of Love-type waves in poro-elastic medium. A staggered grid finite-difference (SFD) scheme in time-space domain formulation for the Biot’s equation of poro-elasticity, is presented and applied to the problem. Complete theoretical seismograms for the horizontal and vertical components of displacement are obtained. The dispersion curves are evaluated considering the material parameters of different models. The stability analysis of the above numerical scheme is deliberated. From different graphs, it is noticed that there is difference in phase velocity for various models. By observing the curves behaviour, we can understand the nature of the composite structure. Our outcomes also endorse the finite difference modeling is an important numerical tool to acquire knowledge about the transmission of seismic waves in porous medium. The agreement is excellent.

Numerical Study of Global ELF Electromagnetic Wave Propagation with Respect to Lithosphere–Atmosphere–Ionosphere Coupling

Before and after earthquakes, abnormal physical and chemical phenomena can be observed by gathering ground-based and satellite data and interpreted by the lithosphere–atmosphere–ionosphere coupling (LAIC) mechanism. In this study, we focused on the mechanism of LAIC electromagnetic radiation and investigated the seismic electromagnetic (EM) wave generated in the lithosphere by earthquakes and its global propagation process from the lithosphere through the atmosphere and into the bottom of ionosphere, in order to analyze the abnormal disturbance of ground-based and space-based observation results. First, analytic formulas of the electrokinetic effect were used to simulate the generation and propagation process of the seismic EM wave in the lithosphere, interpreted as the conversion process of the seismic wave and EM wave in porous media. Second, we constructed a three-dimensional Earth–ionosphere waveguide by applying the finite-difference time-domain (FDTD) algorithm to model the global propagation process of the seismic EM wave into the atmosphere and cavity between the bottom of the ionosphere and the surface of the Earth. By combining the model of the electrokinetic effect in the lithosphere with the numerical model of the Earth–ionosphere waveguide in the atmosphere and ionosphere, we numerically simulated the global transmission process of extremely low-frequency (ELF: 3 Hz–3000 Hz) EM waves which are related to earthquakes. The propagation parameters of coseismic ELF EM waves with different duration times and center frequencies were analyzed and summarized. The simulation results demonstrate that the distribution characteristics of an electric field along longitude, latitude and altitude with time are periodic and the time interval during which an EM wave travels around the whole Earth is approximately 0.155 s when adopting the conductivity of the knee profile. We also compared the observation data with the simulation results and found that the attenuating trends of the ELF electric field are consistent. This proposed ELF EM wave propagation model of lithosphere–atmosphere–ionosphere coupling is very promising for the explanation of abnormal disturbances of ground-based and space-based observation results of ELF EM fields which are associated with earthquakes.

Nonlinear plane waves in saturated porous media with incompressible constituents

We consider the propagation of nonlinear plane waves in porous media within the framework of the Biot–Coussy biphasic mixture theory. The tortuosity effect is included in the model, and both constituents are assumed incompressible (Yeoh-type elastic skeleton, and saturating fluid). In this case, the linear dispersive waves governed by Biot’s theory are either of compression or shear-wave type, and nonlinear waves can be classified in a similar way. In the special case of a neo-Hookean skeleton, we derive the explicit expressions for the characteristic wave speeds, leading to the hyperbolicity condition. The sound speeds for a Yeoh skeleton are estimated using a perturbation approach. Then we arrive at the evolution equation for the amplitude of acceleration waves. In general, it is governed by a Bernoulli equation. With the present constitutive assumptions, we find that longitudinal jump amplitudes follow a nonlinear evolution, while transverse jump amplitudes evolve in an almost linearly degenerate fashion.

Numerical modeling of heterogeneous combustion with phase transitions in porous metal-containing media

In this paper, the heterogeneous combustion of porous metal-containing media is studied taking into account the phase transitions of metal in condensed and gas phases. Such process takes place when applying a novel promising method for the extraction of rare metals from various combustible materials, such as coal and petroleum wastes, by means of filtration combustion. A novel mathematical model, based on the conservation laws, is developed for investigating the physicochemical and gas-dynamic processes during the propagation of combustion waves in porous media with admixtures of metals. Using numerical experiments, the possibilities of metal mass transfer and accumulation of condensed metal in a narrow zone of the porous reactor are shown. The effect of the gas velocity at the reactor inlet and initial metal concentration on combustion of porous metal-containing media is studied. It is revealed that by changing the gas velocity at the reactor inlet, it is possible to control the process of metal transfer and accumulation during the extraction of rare metals from various combustible media. It is shown that the investigated method of rare metals extraction can be very effective for combustible media with a low initial metal concentration.

Theory of shock electrodialysis I: Water dissociation and electrosmotic vortices

Shock electrodialysis (shock ED), an emerging electrokinetic process for water purification, leverages the new physics of deionization shock waves in porous media. In previous work, a simple leaky membrane model with surface conduction can explain the propagation of deionization shocks in a shock ED system, but it cannot quantitatively predict the deionization and conductance (which determines the energy consumption), and it cannot explain the selective removal of ions in experiments. This two-part series of work establishes a more comprehensive model for shock ED, which applies to multicomponent electrolytes and any electrical double layer thickness, captures the phenomena of electroosmosis, diffusioosmosis, and water dissociation, and incorporates more realistic boundary conditions. In this paper, we will present the model details and show that hydronium transport and electroosmotic vortices (at the inlet and outlet) play important roles in determining the deionization and conductance in shock ED. We also find that the results are quantitatively consistent with experimental data in the literature. Finally, the model is used to investigate design strategies for scale up and optimization.

Effect of flow-independent viscosity on the propagation of Rayleigh wave in porous media

The propagation characteristics of Rayleigh wave near the surface of fluid-saturated poroviscoelastic medium are investigated in this paper. The effective stress in porous media is defined by the linear viscoelasticity theory based on Biot model and Kelvin-Voigt model. The dispersion equations of Rayleigh wave are derived analytically and then solved numerically to obtain its propagation speed and attenuation coefficient in the poroviscoelastic solid half-space. The effects of relaxation time and viscous coupling on the dispersion and attenuation of the Rayleigh wave as well as the body waves are discussed. The results indicate that the flow-independent viscosity has important influence on the propagation characteristics of Rayleigh waves, compared with the flow-dependent effect of viscous coupling between the pore fluid and solid skeleton.

Plane longitudinal waves in a fluid-saturated porous medium with a nonlinear relationship between deformations and displacements of the liquid phase

A mathematical model is presented to describe the propagation of a plane longitudinal wave in a fluid saturated porous medium, taking into account the geometric nonlinearity of the liquid component of the medium. The nonlinear relationship between deformations and displacements refines the classical Biot’s theory, within the framework of which a fluid saturated porous medium is considered. Evolutionary equations for the displacements of the skeleton of the medium and fluid in the pores are obtained. It is shown that, if the liquid is confined in the pores, then the wave propagation is described by an equation that generalizes the well-known Burgers equation and has a solution in the form of a stationary shock wave resulting from mutual compensation of the effects of nonlinearity and dissipation. The dependence of the width of the shock wave front on the viscosity of the fluid saturating the pores and the shock wave amplitude is determined. As the viscosity coefficient increases, the wave profile becomes steeper, i.e., the wave front width decreases. With an increase in the wave amplitude, the front width can either increase or decrease, depending on the other parameters of the original system. The limiting cases of the obtained generalized Burgers equation are analyzed with respect to the fluid viscosity parameter. If the liquid flows freely in the pores, then the system of evolutionary equations is reduced to a single equation of a simple wave, i.e., the propagation of a plane longitudinal wave in a porous medium can be described by the well-known equation of nonlinear wave dynamics - the Riemann equation. The equation describes the nonlinear waves, which are characterized by a steepening of the leading edge with subsequent overturning resulting from the growth of nonlinear effects in the absence of compensating factors such as dispersion and dissipation.

Geometric variational approach to the dynamics of porous medium, filled with incompressible fluid

We derive the equations of motion for the dynamics of a porous media filled with an incompressible fluid. We use a variational approach with a Lagrangian written as the sum of terms representing the kinetic and potential energy of the elastic matrix, and the kinetic energy of the fluid, coupled through the constraint of incompressibility. As an illustration of the method, the equations of motion for both the elastic matrix and the fluid are derived in the spatial (Eulerian) frame. Such an approach is of relevance e.g. for biological problems, such as sponges in water, where the elastic porous media is highly flexible and the motion of the fluid has a 'primary' role in the motion of the whole system. We then analyze the linearized equations of motion describing the propagation of waves through the media. In particular, we derive the propagation of S-waves and P-waves in an isotropic media. We also analyze the stability criteria for the wave equations and show that they are equivalent to the physicality conditions of the elastic matrix. Finally, we show that the celebrated Biot's equations for waves in porous media are obtained for certain values of parameters in our models.

SeismicQof inhomogeneous plane waves in porous media

Seismic waves in an attenuative porous medium are generally inhomogeneous waves, which have different directions of propagation and attenuation. The dissipation factors (1/ Q) of inhomogeneous waves are strongly dependent on the degree of wave inhomogeneity and cannot be expressed correctly with the usual 1/ Q expressions valid only for homogeneous waves. We have used the differing definitions of 1/ Q for inhomogeneous waves (i.e., the ratio of the time-averaged dissipated energy density to the time-averaged strain energy density or time-averaged total energy density) and the complex form of the energy balance equations of poroviscoelastic media to derive concise and explicit expressions for the dissipation factors. They are given as simple functions of the material parameters and the wave inhomogeneity parameter for inhomogeneous SV-waves and fast and slow P-waves. The isotropic, poroviscoelastic medium under consideration is upscaled from effective Biot theory for a double-porosity solid, which is the most general theory to describe wave propagation in a reservoir. We find that, if the inhomogeneity parameter is infinite (i.e., the inhomogeneity angle is 90°) for all three Biot waves, then the dissipation factors only depend on the ratio of the imaginary to the real part of the complex shear modulus. Our explicit expressions for the dissipation factors of poroviscoelastic materials also are reduced to obtain their counterparts for viscoelastic media as a special case. The inhomogeneous waves in an example poroviscoelastic material are used to demonstrate that the 1/ Q values of the three Biot waves strongly depend on the inhomogeneity parameters and furthermore the different definitions may cause significant differences of 1/ Q values. We find that the dissipation factor of fast P-waves may decrease with the increasing degree of inhomogeneity, which contradicts previously published results.