Rigid Porous Medium Research Articles

Measuring audible acoustic frequencies parameters of rigid porous media via an innovative impedance tube method - Solving the inverse problem

This study introduces an experimental approach to measuring audible acoustic frequency parameters of a rigid porous medium using an impedance tube. Grounded in the equivalent fluid model, a derivative of Biot's theory, our investigation delves into the propagation of waves within porous media, emphasizing the importance of effective density and dynamic compressibility of the saturated fluid. Our primary focus is on resolving the inverse problem by minimizing both experimental and theoretical absorption coefficient expressions within the audible frequency range. Concurrently, we determine four pivotal parameters: viscous and thermal permeability, the inertia factor introduced by Norris, and the thermal tortuosity introduced by Lafarge. The outcomes include a comparative analysis between experimental and simulated absorption coefficients, leveraging optimized parameters, across two distinct polyurethane foam samples.

Sound absorbers from walnut shell waste: measurement and models

Waste walnut shells have been cleaned, dried, chopped, glued, pressed, and molded to form porous cylinders for measurement of normal incidence sound absorption coefficient spectra in an impedance tube. Porosities and flow resistivities have been measured non-acoustically. Field Emission Scanning Electron Microscopy images, used to obtain fragment size for determining shell density, reveal that the fragment surfaces are rough, which may influence acoustical performance. Measured absorption spectra have been compared with predictions of two analytical models for acoustical properties of rigid porous media, which assume microstructures of identical uniform parallel slanted slits (SS) and non-uniform cylindrical pores with a log normal radius distribution (NUPSD) respectively. Measured values of flow resistivity and porosity are used in the models but tortuosity is adjusted to give best agreement with the quarter wavelength resonance in the measured absorption coefficient spectra. Predictions agree best with data for samples made from the smallest shell fragments. These samples have the highest values of tortuosity, flow resistivity and offer good absorption at speech frequencies which suggests that recyclable and sustainable sound absorbers made from walnut shell wastes could be useful for controlling indoor reverberation..

Phase and group speeds of airborne surface waves over porous layers and periodically rough hard surfaces.

Sound fields above porous layers or rough acoustically hard boundaries may include airborne surface waves. The surface wave properties depend on the effective surface admittance. Analytical expressions for surface wave speeds are derived from models for the acoustical properties of rigid porous media. Surface wave effects on measurements of level difference spectra over porous asphalt are investigated and predictions of phase, group speeds, and vertical attenuation of the surface waves over externally reacting hard backed layers corresponding to a porous asphalt are compared. Predictions of surface wave characteristics above an identical vertical slit medium are compared with data obtained over arrays of parallel aluminum strips on an acoustically hard surface. Group speeds of surface waves over lattices, parallel regularly spaced strips, and snow obtained by numerical differentiation of the phase speed spectra corresponding to admittance spectra deduced from complex excess attenuation are found to compare well with those estimated from time domain data. An effective admittance, deduced from a boundary element method simulation of the excess attenuation spectrum over regularly spaced ribs so that the frequency of the peak corresponds with that in the measured spectrum, is used to estimate the group speed of the associated surface wave.

Capillary rise in partially saturated rigid porous media

We explore predictions of two models of one-dimensional capillary rise in rigid and partially saturated porous media. One is an existing one from the literature and the second is a free-boundary model based on Richards’ equation with two moving boundaries of the evolving partially saturated region. Both models involve the specification of saturation-dependent functions for local capillary pressure and permeability and connect to classical models for saturated porous media. Existing capillary-rise experiments show two notable regimes: (i) an early-time regime typically well-described by classical capillary-rise theory in a fully saturated porous media, and (ii) a long-time regime that has anomalous dynamics in which the capillary-rise height may scale with a non-classical power law in time or have more complicated dynamics. We demonstrate that the predictions of both models compare well with experimental capillary-rise data over early- and long-time regimes gathered from three independent studies in the literature. The model predictions also shed light on recent scaling laws that relate the capillary pressure and permeability of the partially saturated media to the capillary-rise height. We use these models to probe computationally observed permeability relationships to capillary-rise height. We demonstrate that a recently proposed permeability scaling for the anomalous capillary-rise regime is indeed realized and is particularly apparent in the lower portion of the partially saturated media. For our free-boundary model we also compute capillary pressure measures and show that these reveal the linear relation between the capillary pressure and capillary-rise height expected for a capillarity–gravity balance in the upper portion of the partially saturated porous media.

Peridynamic modeling of seepage in multiscale fractured rigid porous media

AbstractThe numerical simulation of seepage in fractured porous media holds significant relevance for subsurface energy development. The presence of fractures at various scales profoundly influences the hydraulic properties of porous media during seepage. A peridynamic (PD)‐based frame is proposed for the seepage problem analysis of multiscale fractured porous media, in which micro‐fractures are implicitly represented like the porous media, macro‐fractures are explicitly represented. For fluid flow modeling, the peridynamic bonds are divided into three types: pore seepage bond, fracture seepage bond, and fluid exchange bond. The micro‐permeabilities of different bonds are derived by Darcy's law, Poiseuille's law, and fluid exchange model, respectively. The proposed peridynamic model can naturally simulate the seepage problem of porous rock with multiscale fractures in a unified framework. Then, four cases of pore seepage and fracture seepage are analyzed using the proposed model, fractured porous media seepage with micro‐fractures and macro‐fractures are respectively considered. The predicted results are compared with the available theoretical solutions, the close agreement shows that the wide validity and applicability of the proposed model in multiscale seepage problems.

Wood chip sound absorbers: Measurements and models

Normal incidence absorption coefficient spectra of samples made from glued wood chips have been measured for various mesh sizes, bulk densities, thicknesses, and air gaps. Increasing thickness introduces additional layer resonance peaks and shifts the initial peak towards lower frequencies. The wood chip samples composed of the smallest mesh sizes were found to offer the highest sound absorption, comparable with that of the same thickness of materials made from synthetic fibers. Measured absorption spectra are compared with predictions of four models for the acoustical properties of rigid porous media. These include a model for slanted parallel identical uniform slits (SS), the Johnson-Champoux-Allard (JCA) and Johnson-Champoux-Allard-Lafarge (JCAL) models for arbitrary pore structures, and model for a non-uniform pore size distribution (NUPSD). Porosity and flow resistivity values have been determined non-acoustically. However, the tortuosity and characteristic lengths required for the JCA model have been obtained by fitting the measured absorption spectra. The thermal permeability required for the JCAL model has been deduced indirectly from the fitted tortuosity through a relationship with standard deviation of the pore size distribution due to the NUPSD model. JCAL and JCA models give the best agreement overall, but predictions of the SS and NUPSD models that use only the fitted tortuosity in addition to measured porosity and flow resistivity are found to give comparable agreement with data for many samples. SS and NUPSD predictions are improved by increasing the tortuosity values compared with those obtained by fitting the JCA model. The study should encourage the creation of sustainable sound-absorbing materials from wood chip wastes.

Filter Approximations for Random Vibroacoustics of Rigid Porous Media

Abstract An approximate efficient stochastic dynamics technique is developed for determining response statistics of linear systems with frequency-dependent parameters, which are used for modeling wave propagation through rigid porous media subject to stochastic excitation. This is done in conjunction with a filter approximation of the system frequency response function. The technique exhibits the following advantages compared to alternative solution treatments in the literature. First, relying on an input–output relationship in the frequency domain, the response power spectrum matrix is integrated analytically for determining the stationary response covariance matrix, at zero computational cost. Second, the proposed filter approximation facilitates a state-variable formulation of the governing stochastic differential equations in the time domain. This yields a coupled system of deterministic differential equations to be solved numerically for the response covariance matrix. Thus, the nonstationary (transient) response covariance can be computed in the time domain at a relatively low computational cost. Various numerical examples are considered for demonstrating the accuracy and computational efficiency of the herein developed technique. Comparisons with pertinent Monte Carlo simulation (MCS) data are included as well.

Comparing Different Coupling and Modeling Strategies in Hydromechanical Models for Slope Stability Assessment

The dynamic interaction between subsurface flow and soil mechanics is often simplified in the stability assessment of variably saturated landslide-prone hillslopes. The aim of this study is to analyze the impact of conventional simplifications in coupling and modeling strategies on stability assessment of such hillslopes in response to precipitation using the local factor of safety (LFS) concept. More specifically, it investigates (1) the impact of neglecting poroelasticity, (2) transitioning from full coupling between hydrological and mechanical models to sequential coupling, and (3) reducing the two-phase flow system to a one-phase flow system (Richards’ equation). Two rainfall scenarios, with the same total amount of rainfall but two different relatively high (4 mm h−1) and low (1 mm h−1) intensities are considered. The simulation results of the simplified approaches are compared to a comprehensive, fully coupled poroelastic hydromechanical model with a two-phase flow system. It was found that the most significant difference from the comprehensive model occurs in areas experiencing the most transient changes due to rainfall infiltration in all three simplified models. Among these simplifications, the transformation of the two-phase flow system to a one-phase flow system showed the most pronounced impact on the simulated local factor of safety (LFS), with a maximum increase of +21.5% observed at the end of the high-intensity rainfall event. Conversely, using a rigid soil without poroelasticity or employing a sequential coupling approach with no iteration between hydromechanical parameters has a relatively minor effect on the simulated LFS, resulting in maximum increases of +2.0% and +1.9%, respectively. In summary, all three simplified models yield LFS results that are reasonably consistent with the comprehensive poroelastic fully coupled model with two-phase flow, but simulations are more computationally efficient when utilizing a rigid porous media and one-phase flow based on Richards’ equation.

Modified Picard with multigrid method for two‐phase flow problems in rigid porous media

AbstractTwo‐phase flow problems in porous media can be found in several areas, such as Geomechanics, Hydrogeology, Engineering and Biomedicine, for example. Typically, these processes are mathematically modeled by a highly nonlinear system of coupled partial differential equations. The nonlinearity of the system makes the design and implementation of robust numerical solvers a challenging task. In this work we consider the flow of two immiscible and incompressible fluids within a non‐deformable porous medium. A mixed pressure‐saturation formulation is adopted, allowing the transition from the unsaturated to saturated zones and maintaining numerical mass conservation. A cell‐centered finite volume method and an implicit Euler scheme are considered for the spatial and time discretization of the problem. In this work, we propose a solution method for two‐phase flow problems which is based on the combination of the modified Picard linearization method and a very simple cell‐centered multigrid algorithm that performs efficiently even for heterogeneous random media. This is shown in the numerical experiments, where two test problems are presented to demonstrate the robustness of the proposed solver.

Comparison between Van Genuchten and Brooks-Corey Parameterizations in the Solution of Multiphase Problems in Rigid One-Dimensional Porous Media

Multiphase problems in porous media involve the complex flow of multiple fluid phases within porous structures, covering areas of Engineering, Medicine, Geology, among others. Understanding these phenomena is crucial in natural processes, resource management, and Engineering system design. Due to its great importance, several researchers have contributed by developing different formulations to describe the multiphase problem. Through laboratory experiments, research lines emerged to numerically determine relative permeability and capillary pressure, parameters that were previously obtained only through experiments, giving rise to two distinct models to solve the flow problem: the van Genuchten and Brooks-Corey parameterizations. This work aims to analyze, through numerical simulations, the resolution of multiphase problems in rigid porous media using both parameterization models, in order to compare the efficiency of each method in different settings. Apparently, this study is unique in its focus, uncovering valuable insights into performance and applicability of both parameterization, providing a comprehensive view of their advantages and constraints in several settings. The results of this study have indicated an advantage in employing Brooks-Corey parameterization, particularly in reducing the error between the numerical and analytical solutions. The findings of this study can help, for example, to better understand oil recovery from naturally fractured reservoirs.

A reinterpreted discrete fracture model for Darcy–Forchheimer flow in fractured porous media

We propose a novel hybrid-dimensional model for the Darcy–Forchheimer flow in fractured rigid porous media, with a natural applicability to non-conforming meshes. Motivated by the previous work on the reinterpreted discrete fracture model (RDFM) for Darcy flows, we extend its key idea to non-Darcy flows. Coupling the Darcy’s law in the matrix and the Forchheimer’s law in the fractures through the introduction of the Dirac-δ functions to characterize the fractures, we derive a relationship between the total flow velocity in the porous media and the fluid velocity in the fractures. With this relation, it is natural to model the Darcy–Forchheimer flow in the whole computational domain by one equation. The local discontinuous Galerkin (LDG) method is applied for numerical discretization of the steady-state single-phase flow problem and a time-marching method is adopted to find the solution of the resulting nonlinear system. Besides, we construct a direct solver for the nonlinear equation of the fluid velocity in fracture to save computational cost. As an application, we discuss a simple transport model coupled with the flow equation. Several numerical experiments validate the performance of the model with its effectiveness on non-conforming meshes. We observe that the Darcy–Forchheimer model effectively reduce the flow rates and make the predictions more realistic in case of excessive fracture velocities.

An analytical model for solute transport in a large-strain aquitard affected by delayed drainage

This paper proposed a model for solute transport in a large-strain aquitard coupling effects of diffusion, adsorption, and consolidation processes. The analytical solutions describing the drawdown and solute transport in the aquitard were derived under the condition of variable drawdowns in adjacent confined aquifer. The breakthrough time (BTs) of solute transport in the large-strain aquitard (LD model) was analyzed, and compared with that of the rigid porous medium (ND model) before consolidation, and the equivalent rigid porous medium (NDf model) at end of the consolidation, respectively. Results reveal that greater Darcy flow velocity, stronger solute diffusivity, and weaker adsorption of soil particles result in shorter BTs of solute transport in the large-strain aquitard, and the impacts of specific storage and void ratio on the BTs are influenced by the adsorption of soil particles. Compared with the ND model, the delayed drainage accelerates solute transport with weak adsorption, while it has the opposite effect with strong adsorption. BTs predicted by the LD model is always smaller than that of the NDf model. It is also found that the larger leakage coefficient of the aquitard significantly enlarges the difference of BTs between the NDf and LD models (Δt), and the consolidation factor of the aquitard is positively correlated with the Δt. The ratio of Δt over the BTs of the LD model has a linear positive relationship with the average cumulative water release of the per unit volume aquitard under the fixed drawdown in the adjacent confined aquifer.

Sensitivity of acoustic parameters on the reflected signal from a rigid porous medium in the low-frequency range of ultrasound

Porous materials are two-phase media consisting of pores saturated with a fluid emerging from a solid skeleton. There are several parameters describing the porous medium. These parameters are classified into two categories: the high-frequency parameters and the low-frequency parameters. During the excitation by an acoustic wave, two cases can occur; the vibration of both phases simultaneously in the case where the structure is flexible, this case is well studied by Biot theory. For a rigid structure, it is the movement of the fluid that is taken into account; this last case is treated by the equivalent fluid theory which is a particular case of Biot theory. Previous work [Proc. Mtgs. Acoust. 45, 045004 (2021)] has shown the influence of the parameters involved in the corrected Johnson-Allard model recently introduced by Sadouki [Phys. Fluids 33, (2021)] on the transmitted signal from a rigid-porous material in the low-frequency regime of ultrasound. The objective of this work is to show, once again, the influence of the same parameters on the reflected signal by providing a comparative study between transmitted and reflected modes.

Existence of solutions for a quasilinear problem with fast nonlocal terms

In this paper, it is considered the existence of a solution for a nonlocal equation involving the p-Laplacian operator which is related to several applications such as vibration problems, flow of fluids through a homogeneous isotropic rigid porous medium, nuclear reactor's dynamics and in population dynamics. The approach is based on sub-supersolution arguments and Schauder's fixed point theorem. An important fact is that our result is valid for nonlocal terms which may exhibit a growth higher than p.

A macroscopic model for immiscible two-phase flow in porous media

This work provides the derivation of a closed macroscopic model for immiscible two-phase, incompressible, Newtonian and isothermal creeping steady flow in a rigid and homogeneous porous medium without considering three-phase contact. The mass and momentum upscaled equations are obtained from the pore-scale Stokes equations, adopting a two-domain approach where the two fluid phases are separated by an interface. The average mass equations result from using the classical volume averaging method. A Green's formula and the adjoint Green's function velocity pair problems are used to obtain the pore-scale velocity solutions that are averaged to obtain the upscaled momentum balance equations. The macroscopic model is based on the assumptions of scale separation and the existence of a periodic representative elementary volume allowing a local description as usually postulated for upscaling. The macroscopic momentum equation in each phase includes the generalized Darcy-like dominant and viscous coupling terms and, importantly, an additional compensation term that accounts for surface tension effects to momentum transfer that is, otherwise, incompletely captured by the Darcy terms. This interfacial term, as well as the dominant and viscous coupling permeability tensors, can be predicted from the solutions of two associated closure problems that coincide with those reported in the literature. The relevance of the compensation term and the upscaled model validity are assessed by comparisons with direct numerical simulations in a model two-dimensional periodic structure. Upscaled model predictions are found to be in excellent agreement with direct numerical simulations.

Assessment of simplified momentum equations for free surface flows through rigid porous media

In many applications, free surface flow through rigid porous media has to be modeled. Examples refer to coastal engineering applications as well as geotechnical or biomedical applications. Albeit the frequent applications, slight inconsistencies in the formulation of the governing equations can be found in the literature. The main goal of this paper is to identify these differences and provide a quantitative assessment of different approaches. Following a review of the different formulations, simulation results obtained from three alternative formulations are compared with experimental and numerical data. Results obtained by 2D and 3D test cases indicate that the predictive differences returned by the different formulations remain small for most applications, in particular for small porous Reynolds number ReP < 5000. Thus it seems justified to select a simplified formulation that supports an efficient algorithm and coding structure in a computational fluid dynamics environment. An estimated accuracy depending on the porous Reynolds number or the mean grain diameter is given for the simplified formulation.

Two-equation continuum model of drying appraised by comparison with pore network simulations

A two-equation non-local-equilibrium (NLE) continuum model of isothermal drying is assessed by comparison with pore network simulations considering a rigid capillary porous medium that is fully saturated initially. This continuum model consists of a transport equation for the liquid and of a transport equation for the vapor. The two main variables are the liquid saturation and the vapor partial pressure. The two equations are coupled by a phase-change term and mass transport at the medium surface is modeled by considering the individual boundary conditions for the two continuum model equations. The macroscopic parameters that appear in the NLE continuum model include classical parameters such as the effective liquid and vapor diffusivities, as well as non-classical and new parameters such as the specific interfacial area and the fraction of dry surface pores. These parameters are determined for the porous microstructure corresponding to the cubic network used to perform the pore network simulations. The results obtained by the two-equation NLE continuum model are compared with pore network simulation data. Comparisons reveal that the two-equation NLE continuum model can capture with a reasonable degree of accuracy the NLE effect as well as the phase distributions and drying kinetics of the pore network model drying simulations.

NUMERICAL SIMULATION OF FLOW THROUGH ABSORBING POROUS MEDIA PART 1: RIGID POROUS MEDIA

NUMERICAL SIMULATION OF FLOW THROUGH ABSORBING POROUS MEDIA PART 1: RIGID POROUS MEDIA

Energy analysis and discretization of the time-domain equivalent fluid model for wave propagation in rigid porous media

The equivalent fluid model (EFM) describes the acoustic properties of rigid porous media by defining the intra-pore fluid phase as a fluid with an effective density and an effective compressibility. Their definitions are based on the dynamic tortuosity α and the normalized dynamic compressibility β. These physical quantities are complex-valued functions depending on the frequency, and can be irrational as in the Johnson-Champoux-Allard-Pride-Lafarge (JCAPL) model. Hence, the system of equations derived from the EFM can involve fractional derivatives in the time domain. This paper presents an approach to formulate the EFM equations described by the JCAPL model in the time domain, leading to an augmented system for which a proof of stability is given. From the EFM, a model for numerical simulation is built with α and β approximated using a multipole model. Sufficient stability conditions are then provided for the multipole-based EFM. Lastly, a numerical analysis is carried out in order to illustrate the theoretical results and a simulation of the impedance tube experiment is presented.

Physical parameters influence on the transmitted signal through a rigid porous medium in the low-frequency range of ultrasound

The acoustic characterization of porous materials with rigid structure plays an important role in the prediction of the acoustic behavior of these media. In order to obtain an accurate prediction of its response as a function of frequency, several parametric models have been proposed. However, semi-phenomenological models are the most used to describe the acoustic behavior of porous materials with rigid or flexible structures according to a certain number of macroscopic physical properties related to the internal structure of the material and essential for their valuation. Among these models, the one proposed by Johnson et al. to describes the complex density of an acoustic porous material with an immobile skeleton having arbitrary pore shapes. The objective of this work is to study the effect of these parameters involved in Johnson's model as well as other new parameters, recently introduced by Sadouki [ Phys. Fluids 33, (2021)), on the transmitted signal in the low-frequency regime of ultrasound.