Wave Propagation In Porous Media Research Articles

Modelling of 2-D seismic wave propagation in heterogeneous porous media: a frequency-domain finite-element method formulated by variational principles

SUMMARY We propose a frequency-domain finite-element (FDFE) method to model the 2-D P–SV waves propagating in porous media. This specific finite-element method (FEM) is based on the framework of variational principles, which differs from previously widely used FEMs that rely on the weak formulations of the governing equations. By applying the calculus of variations, we establish the equivalence between solving the stress–strain relations, equations of motion and boundary conditions that govern the propagation of P–SV waves, and determining the extremum or stationarity of a properly defined functional. The structured rectangular element is utilized to partition the entire computational region. We validate the FDFE method by conducting a comparison with an analytically-based method for models of a horizontal planar contact of two poroelastic half-spaces, and a poroelastic half-space with a free surface. The excellent agreements between the analytically-based solutions and the FDFE solutions indicate the effectiveness of the FDFE method in modelling the poroelastic waves. Modelling results manifest that both propagative and diffusive natures of the Biot slow P wave can be effectively modelled. The proposed FDFE method simulates wavefields in the frequency domain, allowing for easy incorporation of frequency-dependent parameters and enabling parallel computational capabilities at each frequency point (sample). We further employ the developed FDFE method to model two simplified poroelastic reservoirs, one with gas-saturated sandstone and the other with oil-saturated sandstone. The results suggest that changing the fluid phase of the sandstone reservoir from gas to oil can substantially impact the recorded solid and relative fluid–solid displacements. The modelling suggests that the proposed FDFE algorithm is a useful tool for studying the propagation of poroelastic waves.

Transient high-frequency spherical wave propagation in porous medium using fractional calculus technique

AbstractTransient high-frequency spherical wave propagation in the porous medium is studied using the Biot-JKD theory. The porous media is considered to be a composed of deformable solid skeleton and viscous incompressible fluid inside the pores. In order to treat the fractional proportionality of the dynamic tortuosity to the frequency (or equivalently, to time) due to the viscous interaction between solid and fluid phases, the fractional calculus theory along with the Laplace and Fourier transforms are used to solve the coupled governing partial differential equations of the scaler and vector potential functions obtained from the Helmholtz’s decomposition in the Laplace domain. Both the longitudinal and transverse waves, additionally the reflection and transmission kernels are determined in detail at the Laplace domain. For the Laplace-to-time inversion transform, Durbin’s numerical formula is used and the independence of the results from the involved tuning and accuracy parameters is checked. The effects of the porosity, dynamic tortuosity, characteristics length, etc. on the reflected pressure and stress are investigated. The general pattern of the results is similar to our previous knowledge of wave propagation. Further works and experiments may be conducted in future works for various applications.

High-Order Boussinesq Equations for Water Wave Propagation in Porous Media

To accurately capture wave dynamics in porous media, the higher-order Boussinesq-type equations for wave propagation in deep water are derived in this paper. Starting with the Laplace equations combined with the linear and nonlinear resistance force of the dynamic conditions on the free surface, the governing equations were formulated using various independent velocity variables, such as the depth-averaged velocity and the velocity at the still water level and at an arbitrary vertical position in the water column. The derived equations were then improved, and theoretical analyses were carried out to investigate the linear performances with respect to phase celerity and damping rate. It is shown that Boussinesq-type models with Padé [4, 4] dispersion can be applied in deep water. A numerical implementation for one-dimensional equations expressed with free surface elevation and depth-averaged velocity is presented. Solitary wave propagation in porous media was simulated, and the computed results were found to be generally in good agreement with the measurements.

Deep-learning-based surrogate model for forward and inverse problems of wave propagation in porous media saturated with two fluids

Deep-learning-based surrogate model for forward and inverse problems of wave propagation in porous media saturated with two fluids

Acoustic wave propagation in a porous medium saturated with a Kelvin–Voigt non-Newtonian fluid

SUMMARYWave propagation in anelastic rocks is a relevant scientific topic in basic research with applications in exploration geophysics. The classical Biot theory laid the foundation for wave propagation in porous media composed of a solid frame and a saturating fluid, whose constitutive relations are linear. However, reservoir rocks may have a high-viscosity fluid, which exhibits a non-Newtonian (nN) behaviour. We develop a poroelasticity theory, where the fluid stress-strain relation is described with a Kelvin–Voigt mechanical model, thus incorporating viscoelasticity. First, we obtain the differential equations from first principles by defining the strain and kinetic energies and the dissipation function. We perform a plane-wave analysis to obtain the wave velocity and attenuation. The validity of the theory is demonstrated with three examples, namely, considering a porous solid saturated with a nN pore fluid, a nN fluid containing solid inclusions and a pure nN fluid. The analysis shows that the fluid may cause a negative velocity dispersion of the fast P(S)-wave velocities, that is velocity decreases with frequency. In acoustics, velocity increases with frequency (anomalous dispersion in optics). Furthermore, the fluid viscoelasticity has not a relevant effect on the wave responses observed in conventional field and laboratory tests. A comparison with previous theories supports the validity of the theory, which is useful to analyse wave propagation in a porous medium saturated with a fluid of high viscosity.

Biot-consistent framework for wave propagation with macroscopic fluid and thermal effects

SUMMARY In principle, wave propagation in porous media can simultaneously trigger macroscopic fluid flow and thermal flow, which can be described by Biot's poroelasticity and Lord–Shulman thermoelasticity, respectively. The physical processes of those effects are significantly different, but phenomenologically, they can lead to identical wave attenuation and dispersion and are hard to be distinguished. By using Biot's virtual concept entropy flow, the Biot-consistent General Linear Solid (the GLS) framework and matrix notation, a rigorous and convenient tool is provided to reveal the similarities and disparities between poroelasticity and thermoelasticity. By using the same framework, a Biot-consistent thermo-poroelastic model is proposed to consider macroscopic effects of fluid and thermal flows simultaneously in an elegant way. These similarities allow us to directly translate many of the available results in poroelasticity to thermoelasticity and vice versa by a simple change of notation. The disparities indicate a fundamental difference in physical mechanisms. Plane-wave analysis shows that the primary P-wave modes of thermoelasticity and poroelasticity are all GSLS-equivalent (Generalized Standard Linear Solid) and can be identical if the model parameters are selected properly. However, the corresponding slow-wave modes have significantly different phase velocity dispersion although the attenuation spectra of which are identical. Such a surprising result can be explained by the GSLS non-equivalence of the slow-wave modes and the fundamentally different mechanisms. As expected, the thermo-poroelastic model predicts four wave modes, which are the fast- and slow-P, temperature (T wave) and S waves. Two attenuation peaks due to, respectively, the thermal- and fluid-flow effects are predicted for the fast-P wave. The slow-P wave mode due to fluid flow is influenced by the thermal effects, but the T wave seems unaffected by the fluid flow. The thermo-poroelastic model is then applied to laboratory observations at 200–106 Hz for the brine-saturated tight sandstone under 35 MPa effective pressure. The unified model provides a convenient framework for studying geothermal exploration, thermal-enhanced oil recovery and other applications involving temperature variations within the porous rock.

Theoretically derived pore geometry in carbonates using the extended biot theory

Carbonates typically display a large scatter in velocity-porosity cross plots that is caused by the difference in stiffness that the highly variable pore geometry produces in carbonate rocks at a given porosity. In recent years, digital image analysis (DIA) made it possible to capture objective, quantifiable parameters of the pore geometry to explain the scatter in carbonate velocity-porosity cross plots. The geometrical parameters most influential for acoustic velocity are Perimeter over Area (measuring the complexity of the pore space) and Dominant Size (measuring pore sizes as equivalent diameter). Most theoretical models of acoustic wave propagation in porous media, however, do not incorporate those geometrical characteristics of the natural pore spaces? In contrast, the frame flexibility and coupling factors (fk and γk) embedded in the Extended Biot Theory capture the geometrical characteristics using poroelastic data. Here, these theoretically derived parameters are tested and validated with experimental data. In 95 limestone samples, Extended Biot Theory parameters fk and γk can be derived reliably from measured acoustic data. Both parameters correlate well to Perimeter over Area (r = 0.702, p < 0.0001), and Dominant Size (r = 0.792, p < 0.0001) derived using digital image analysis of thin section photographs. These correlations confirm the existence and the strength of the relationship between theoretical parameters from the Extended Biot Theory and quantitative pore geometry parameters. Thus, this study illustrates that: (1) estimating porosity from acoustic data can be substantially improved by incorporating information on quantitative pore space geometry; and (2) in cases where good estimates of porosity are available (e.g. from well-log data or seismic with extensive well control) quantitative pore geometric characteristics can be estimated directly from acoustic data. These quantitative pore geometry characteristics can be used to estimate permeability indirectly from acoustic data. For example, permeability estimates based on a Kozeny-Carman type function incorporating geometrical information derived from the topological Extended Biot Theory parameters fk and γk, yield a statistically significant, high Pearson correlation coefficient of r = 0.713 (p < 0.0001).

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.

Electromagnetic field excitation during the scattering of an acoustic wave on an inhomogeneity in a poroelastic medium

Elastic wave propagation in porous media containing a movable fluid is frequently accompanied by electromagnetic field generation. Such field can be produced, for instance, by piezoelectric effect or have electrokinetic origin. In the present work, we consider the case of electrolyte-filled pores; thus, the generated electromagnetic field is associated with the electrokinetic effects. In this case, the magnitude of the electric current generated by the elastic wave is proportional to the magnitude of the relative displacement in the fluid and the solid phases. The presence of inhomogeneities in a porous medium can result in appearance of additional fluid flow with respect to the solid phase; thus, an additional electric field is generated. In our work, the magnitude of the electromagnetic field generated during the scattering of an elastic wave on a spherical porous inclusion in a porous matrix is calculated. The inclusion can differ from the matrix by the properties of the fluid in the pores or the properties of the elastic skeleton, such as porosity or permeability. The solution of the dynamic equations of electrokinetics is obtained by using the hypothesis that the inclusion size is much larger than the characteristic size of the pores. We have obtained the dependencies of the amplitude of the generated electromagnetic field on the inclusion parameters and the incident wave frequency. It was shown that the analysis of the additional electromagnetic field generated during the scattering of the elastic wave can provide some useful information about the structure of porous media.

Fast difference scheme for a tempered fractional Burgers equation in porous media

In this paper we present a fast difference scheme for a tempered fractional Burgers equation. The model arises in describing the wave propagation in porous media with the power law and exponential attenuation. To avoid solving the discrete algebraic system by iteration, we present a linearized difference approximation for the nonlinear term. To reduce the numerical computational cost, we propose a fast algorithm for tempered fractional derivative. The fast algorithm is based on the summation of exponentials technique which is developed in Jiang et al. (2017). The theoretical analysis shows that our fast difference scheme is unconditionally stable and convergence. Numerical results demonstrate a competitive performance of the proposed fast difference scheme in comparison to the direct difference scheme.

Biot-Rayleigh-Squirt Theory of Wave Propagation in Porous Media

Biot-Rayleigh-Squirt Theory of Wave Propagation in Porous Media

Virtual elements for sound propagation in complex poroelastic media

We develop a novel Virtual Element Method (VEM) to resolve the mixed Biot displacement-pressure formulation governing wave propagation in porous media. Within this setting, the weak form of the governing equations is discretized using implicitly defined canonical basis functions and the resulting integral forms are computed using appropriate polynomial projections. The projection operator accounting for the solid, fluid, and coupling phases of the problem are presented. Different boundary, interface and excitation conditions are accounted for. The convergence behaviour, accuracy, and efficiency of the method is examined through a set of illustrative examples. A node insertion strategy is proposed to resolve non-conforming interfaces that naturally arise in multilayered systems. Finally the power of the VEM is exploited to examine the acoustic response of composite materials with periodic and non-periodic inclusions of complex geometries.

Evaluation of dynamic behaviour of porous media including micro-cracks by ordinary state-based peridynamics

Reliable evaluation of mechanical response in a porous solid might be challenging without any simplified assumptions. Peridynamics (PD) perform very well on a medium including pores owing to its definition, which is valid for entire domain regardless of any existed discontinuities. Accordingly, porosity is defined by randomly removing the PD interactions between the material points. As wave propagation in a solid body can be regarded as an indication of the material properties, wave propagation in porous media under an impact loading is studied first and average wave speeds are compared with the available reference results. A good agreement between the present and the reference results is achieved. Then, micro-cracks are introduced into porous media to investigate their influence on the elastic wave propagation. The micro-cracks are considered in both random and regular patterns by varying the number of cracks and their orientation. As the porosity ratio increases, it is observed that wave propagation speed drops considerably as expected. As for the cases with micro-cracks, the average wave speeds are not influenced significantly in random micro-crack configurations, while regular micro-cracks play a noticeable role in absorbing wave propagation depending on their orientation as well as the number of crack arrays in y-direction.

Wave Equations of Porous Media Saturated With Two Immiscible Fluids Based on the Volume Averaging Method

Simulating and predicting wave propagation in porous media saturated with two fluids is an important issue in geophysical exploration studies. In this work, wave propagation in porous media with specified structures saturated with two immiscible fluids was studied, and the main objective was to establish a wave equation system with a relatively simple structure. The wave equations derived by Tuncay and Corapcioglu were analyzed first. It was found that the coefficient matrix of the equations tends to be singular due to the inclusion of a small parameter that characterizes the effect of capillary stiffening. Therefore, the previously established model consisting of three governing equations may be unstable under natural conditions. An improved model based on Tuncay and Corapcioglu’s work was proposed to ensure the nonsingularity of the coefficient matrix. By introducing an assumption in which one fluid was completely wrapped by the other, the governing equation of the wrapped fluid was degenerated. In this way, the coefficient matrix of wave equations became nonsingular. The dispersion and attenuation prediction resulting from the new model was compared with that of the original model. Numerical examples show that although the improved model consists of only two governing equations, it can obtain a result similar to that of the original model for the case of a porous medium containing gas and water, which simplifies the complexity of the calculations. However, in a porous medium with oil and water, the predictions of dispersion and attenuation produced by the original model obviously deviate from the normal trend. In contrast, the results of the improved model exhibit the correct trend with a smooth curve. This phenomenon shows the stability of the improved model and it could be used to describe wave propagation dispersions and attenuations of porous media containing two immiscible fluids in practical cases.

Heat generation by ultrasound wave propagation in porous media with low permeability: Theoretical framework and coupled numerical modeling

Abstract Ultrasound wave propagation through a porous medium results in a temperature increase due to mechanical dissipation. This temperature increase has different applications in medical science, food industry, and engineering. A fully coupled dynamic thermo-hydro-mechanical (THM) model is presented for numerical modelling of this phenomenon in deforming porous media. The temperature increase due to wave propagation is taken into account in the energy conservation equation by using a viscous drag force and considering solid-fluid interaction. The solid and fluid displacements ( U → s and U → w ) and the temperature ( T ) are taken as primary unknowns in the model known in literature as uUT model. The finite element method is used for the discrete approximation of the governing partial differential equations. Based on the numerical examples carried out in this study, it is shown that the scale of the simulated sample may have a very pronounced effect on the results, and that results from small scale laboratory samples should not be extrapolated to the actual field condition. Furthermore, the results of ultrasound application in porous media indicate a significant increase in temperature in the vicinity of the source boundary.

High Temperature Effect on Absorption Coefficient of M-MPPs and Sandwich Structures Coupled with MPPs

This paper addresses the effect of high temperature on absorption performance of sandwich material coupled with microperforated panels (MPPs) in multiple configurations using a finite element model (FEM) over a frequency range from 10 to 3000 Hz. The structure is backed with a rigid wall which can either be Aluminium or Al-Alloy used in aeronautic or automobile. The wave propagation in porous media is addressed using Johnson Champoux Allard model (JCA). The FEM model developed using COMSOL Multiphysics software makes it possible to predict the acoustic absorption coefficient in multilayer microperforated panels (M-MPPs) and sandwich structure. It is shown that, when structures made by MPPs or sandwich materials are submitted to high temperature, the absorption performance of the structure is strongly modified in terms of amplitude and width of the bandgap. For application in sever environment (noise reduction in engines aircrafts), Temperature is one of the parameters that will most influence the absorption performance of the structure. However, for application in the temperature domain smaller than 50?C (automotive applications for example), the effect of temperature is not significant on absorption performance of the structure.

Simulation of the seismic wave propagation in porous media described by three elastic parameters

Simulation of the seismic wave propagation in porous media described by three elastic parameters

Acoustical modeling and Bayesian inference for rigid porous media in the low-mid frequency regime.

In this article, a modeling extension for the description of wave propagation in porous media at low-mid frequencies is introduced. To better characterize the viscous and inertial interactions between the fluid and the structure in this regime, two additional terms described by two parameters α 1 and α 2 are taken into account in the representation of the dynamic tortuosity in a Laurent-series on frequency. The model limitations are discussed. A sensitivity analysis is performed, showing that the influence of α 1 and α 2 on the acoustic response of porous media is significant. A general Bayesian inference is then conducted to infer, simultaneously, the posterior probability densities of the model parameters. The proposed method is based on the measurement of waves transmitted by a slab of rigid porous material, using a temporal model for the direct and inverse transmission problem. Bayesian inference results obtained on three different porous materials are presented, which suggests that the two additional parameters are accessible and help reduce systematic errors in the identification of other parameters: porosity, static viscous permeability, static viscous tortuosity, static thermal permeability, and static thermal tortuosity.

Influence of micro-structural parameters on apparent absorption coefficient in porous structures mimicking cortical bone.

Ultrasound wave propagation in porous media is associated with energy loss and attenuation. Attenuation is caused by both scattering and absorption, and is influenced by the microstructure as well as the operating frequency. In the present simulation study, we calculate the attenuation coefficient in porous structures mimicking cortical bone both in presence and absence of absorption to isolate the effects of scattering and absorption on the total attenuation. A parameter called apparent absorption is defined as the difference between the total attenuation and the attenuation exclusively due to scattering . and are estimated in porous structures with varying pore diameters and pore densities at 5MHz and 8MHz. Results show that both scattering and absorption contribute to the total attenuation. They also illustrate that, although absorption only occurs in the solid matrix, the apparent absorption is a function of porosity, presumably due to the presence of multiple scattering. For large values of , an increase in pore size or density does not lead to increase in and only results in an increase of the total attenuation as a result of increase in . On the other hand, in low/intermediate scattering regimes , an increase in either pore size or pore density results in increase in while remains constant.

An overview of the sediment acoustic models and their experimental validations

The Biot theory of acoustic wave propagation in porous media was first adapted to marine sediments by Stoll (1969). To describe the response of a slightly inelastic skeletal frame over a wide frequency range, Stoll introduced the notion of constant complex bulk and shear moduli with small imaginary parts. Stoll’s model predicts f1 frequency-dependence of attenuation at low frequencies and nearly f2 and f1/2 frequency-dependence at frequencies where maximum velocity dispersion occurs. Although the theory well reproduces the velocity and attenuation measurements in marine sediments, the assumption of constant complex moduli slightly violates the causality (Turgut, 1990). The more recent Grain-Shearing (GS) and Viscous Grain-Shearing (VGS) models (Buckingham, 2000 and 2007) are causal and the VGS model also predicts a frequency-dependence of attenuation like that of the Stoll model. With the selection of proper parameter values, both models predict compressional and shear-wave dispersions that are in agreement with those of previous in-situ and laboratory measurements. In addition, the Stoll model predicts the existence of slow compressional waves that have been observed in synthetic porous media (Plona, 1980) but not in natural marine sediments. Several reflection and in-sediment transmission experiments are discussed to facilitate the detection of slow compressional waves in marine sediments. [Work supported by ONR.]