Biot Model Research Articles

An augmented fully mixed formulation for the quasistatic Navier–Stokes–Biot model

Abstract We introduce and analyze a partially augmented fully mixed formulation and a mixed finite element method for the coupled problem arising in the interaction between a free fluid and a poroelastic medium. The flows in the free fluid and poroelastic regions are governed by the Navier–Stokes and Biot equations, respectively, and the transmission conditions are given by mass conservation, balance of fluid force, conservation of momentum and the Beavers–Joseph–Saffman condition. We apply dual-mixed formulations in both domains, where the symmetry of the Navier–Stokes and poroelastic stress tensors is imposed in an ultra-weak and weak sense. In turn, since the transmission conditions are essential in the fully mixed formulation, they are imposed weakly by introducing the traces of the structure velocity and the poroelastic medium pressure on the interface as the associated Lagrange multipliers. Furthermore, since the fluid convective term requires the velocity to live in a smaller space than usual, we augment the variational formulation with suitable Galerkin-type terms. Existence and uniqueness of a solution are established for the continuous weak formulation, as well as a semidiscrete continuous-in-time formulation with nonmatching grids, together with the corresponding stability bounds and error analysis with rates of convergence. Several numerical experiments are presented to verify the theoretical results and illustrate the performance of the method for applications to arterial flow and flow through a filter.

Mathematical effects of linear visco-elasticity in quasi-static Biot models

We investigate and clarify the mathematical properties of linear poro-elastic systems in the presence of classical (linear, Kelvin-Voigt) visco-elasticity. In particular, we quantify the time-regularizing and dissipative effects of visco-elasticity in the context of the quasi-static Biot equations. The full, coupled pressure-displacement presentation of the system is utilized, as well as the framework of implicit, degenerate evolution equations, to demonstrate such effects and characterize linear poro-visco-elastic systems. We consider a simple presentation of the dynamics (with convenient boundary conditions, etc.) for clarity in exposition across several relevant parameter ranges. Clear well-posedness results are provided, with associated a priori estimates on the solutions. In addition, precise statements of admissible initial conditions in each scenario are given.

A locking-free and mass conservative H(div) conforming DG method for the Biot's consolidation model

A locking-free and mass conservative H(div) conforming DG method for the Biot's consolidation model

Parameter‐robust mixed element method for poroelasticity with Darcy‐Forchheimer flow

AbstractIn this paper, the Biot's consolidation model is considered, and Darcy‐Forchheimer equation is used to describe the relationship between the fluid velocity and pressure. The model is nonlinear and the unknown variables are the solid displacement, the fluid velocity and pressure. The Bernardi‐Raugel element is used for displacement and the RT mixed element is used for the fluid velocity and pressure. The parameter‐robust BR‐RT0‐P0 method presented here is uniformly stable not only with respect to the Lamé constant , but also with respect to the constrained specific storage coefficient . We demonstrate the well‐posedness of the BR‐RT0‐P0 finite element solutions, and give the error estimates of the finite element approximations using the monotonicity of the nonlinear Forchheimer term. The error estimates are divided into two cases, that is, is positive and is nonnegative. Numerical experiments are presented to verify theoretical analysis. Moreover we focus on the variation of pressure by several poroelasticity problems.

Vibrational damping properties of finite microperforated plates

Microperforated plates (MPP) are traditionally used to advantageously absorb acoustic waves in sound control technologies. However, less is known concerning the additional structural damping they can induce. The latter is activated through exchanges in the viscous and thermal boundary layers near the fluid-structure interface of the microperforations. MPPs therefore offer an alternative to, or can be used together with, viscoelastic materials, commonly implemented to damp vibrations at medium and high frequencies. In this work, the structural damping capabilities of MPP are investigated. To this end, the damping performance of a finite size MPP is explored analytically through an alternative form of the Biot model, classically devoted to porous plates, and considering energy dissipation through viscous friction mechanisms. Analytical results are compared to experimental measurements of structural damping factors on various MPP samples. The model is validated and confirms the damping effect added by the microperforations in the low frequency range. A sensitivity analysis on the perforation rate and perforation diameter provides a condition for which additional damping is maximized.

An iterative decoupled algorithm with unconditional stability for Biot model

This paper is concerned with numerical algorithms for Biot model. By introducing an intermediate variable, the classical 2-field Biot model is written into a 3-field formulation. Based on such a 3-field formulation, we propose a coupled algorithm, some time-extrapolation based decoupled algorithms, and an iterative decoupled algorithm. Our focus is the analysis of the iterative decoupled algorithm. It is shown that the convergence of the iterative decoupled algorithm requires no extra assumptions on physical parameters or stabilization parameters. Numerical experiments are provided to demonstrate the accuracy and efficiency of the proposed method.

A lumped model for long bone behavior based on poroelastic deformation and Darcy flow

The present paper provides a simplified model for compact bone behavior by accounting for bone fluid flow coupled to the elasticity of the porous structure. The lumped model considers the bone material as a layered poroelastic structure and predicts normal pressure versus displacement, i.e, a stress–strain curve. There is a parametric dependency on porosity and permeability but, in addition, on pressure history. Specifically, the pressure impulse (the integral of pressure versus time) plays a key role. This factor is alluded to in several past studies, but not highlighted in a simplified fashion. Based on a global flow balance, bone displacement depends on the fluid flow in a channel according to the classical Darcy model of 1856, and on the rate of change of fluid within the porous solid according to the 1941 classical model of Biot. The present results agree with those of Perrin et al. which, in turn, agree with results of a detailed numerical simulation.

Incorporating the Nearly Constant Q Models Into 3-D Poro-Viscoelastic Anisotropic Wave Modeling

The Earth is often characterized by viscoelastic rocks, porous sediments and anisotropic structures. Poro-elasticity with Biot’s theory is considered fundamental to describe the interaction between the deformation of the elastic porous solid and the flow of fluid in the porous structure. The quality factor ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i> ) in the theory of viscoelasticity relates seismic wave attenuation and dispersion to physical properties of the Earth’s interior, e. g. temperature, stress and composition. However, the constant <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i> wave equation in its time-domain differential form remains difficult to solve when describing the attenuation in an explicitly specified <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i> parameter. Here, we introduce the first-and second-order nearly constant <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i> models capable of describing the attenuation of the solid skeleton, thereby extending the Biot and Biot-squirt (BISQ) models to poro-viscoelastic media. The bulk and shear moduli of the solid frame are represented by the modified relaxation function. By presenting examples with finite-difference time-domain (FDTD) numerical modeling for seismic wavefields in anisotropic, viscoelastic porous media including transversely isotropic media with a vertical symmetry axis (VTI) and orthorhombic meida, we demonstrate that the extended Biot and BISQ models provide good descriptions of the wave propagation in poro-viscoelastic anisotropic media and can thus help better understand the Earth’s interior.

Elastic wave propagation characteristics in unsaturated double-porosity medium under capillary pressure

Rock pores often contain two-phase or multi-phase fluids, so it is important to understand how the wave-induced fluid pressure diffusion affects dispersion and attenuation of elastic waves for resource exploration. To describe the propagation of elastic wave in a double-porosity medium saturated by two-phase fluids, a wave propagation model, including both global and local flow mechanisms and considering the effect of capillary pressure, is derived. The dispersion and attenuation characteristics of three longitudinal waves (P1, P2, P3) and one transverse wave (S wave) are investigated by analyzing a plane wave, and the effects of physical parameters, such as inclusion radius, water saturation, permeability and porosity, on the propagation characteristics of P1 wave are investigated. Theoretical analysis shows that the model derived in this work can be degenerated into the Biot model under specific conditions. According to the numerical simulation results, due to the coupling of global flow and local flow, the P1 wave velocity may decrease below the Gassmann-Wood limit. The physical explanation of this phenomenon is as follows: when considering the effect of capillary pressure, the coupling effect of global flow and local flow will break the basic assumption that rock is undrained. The relationship between physical parameters of porous medium and the dispersion and attenuation characteristics of elastic wave is complicated and nonlinear. Compared with Santos model, elastic modulus predicted by Santos-Rayleigh model is in good agreement with the experimental data in the low frequency band, which proves that this model has good reliability in modeling the velocity field of seismic exploration.

Study on Sound Velocity and Attenuation of Underwater Cobalt-Rich Crust Based on Biot and BISQ Theories

A prediction model of the sound velocity and sound attenuation of underwater cobalt-rich crusts (CRCs) was established to solve the problem that it is difficult to predict the sound velocity in thickness measurements of cobalt-rich crusts. Based on Biot theory and BISQ theory, a simplified Biot and BISQ model was proposed for the prediction of the sound velocity and sound attenuation of CRCs by using the Kozeny–Carman (KC) equation. The models could calculate the sound velocity and attenuation by the porosity and detection frequency. Based on the physical and mechanical properties of CRCs, a similarity model of the sound velocity and sound attenuation of CRCs was made by using the similarity theory to solve the problem that it is difficult to measure the acoustic propagation characteristics of CRCs. The sound velocity and sound attenuation of CRC similarity models with different porosities were measured by an underwater transmission experiment and the results of the simplified model calculation and experimental measurements were compared. The results showed that the simplified Biot model was suitable for the CRC sound velocity prediction and the simplified BISQ model was suitable for the CRC sound attenuation prediction, which had a high prediction accuracy.

Energy flow analysis of poroelastic media

• Heat-conduction type 1D energy models are developed for isotropic poroelastic media. • A notion of equivalent longitudinal bulk modulus and density of poroelastic media is introduced. • Criteria for the evaluation of energy models are provided in terms of the intrinsic parameters of the materials. • Developed models are validated by comparing with Biot's model at high frequencies. Energy flow analysis (EFA) has been used to formulate heat conduction-type energy models, which are effective in predicting high-frequency vibrational responses of structures. In this paper, energy models for poroelastic media are presented within the EFA framework. A one-dimensional case in which two dilatational waves propagate is considered. The derivation procedures rely on the notion of the equivalent longitudinal bulk modulus and density, which are defined based on the elastic and inertial coefficients of the frame and interstitial fluid and the relative wave motion between them. These parameters have roles similar to those of the equivalent bulk modulus and density of the fluid models for rigid- and limp-frame porous material and are used to represent the energy behavior of each propagating mode in terms of volume-averaged quantities. The resulting energy equations describe the propagation of the energy-related quantities of the two waves, and their capabilities are illustrated in cases in which a poroelastic layer is used to fill the cavity between two panels. The predicted results of energy models show a good approximation of the exact energy distributions of each wave, obtained from Biot's displacement formulation at high frequencies.

A mixed virtual element method for Biot's consolidation model.

In this paper, we shall present a weak virtual element method for the standard three field poroelasticity problem on polytopal meshes. The flux velocity and pressure are approximated by the low order virtual element and the piecewise constant, while the elastic displacement is discretized by the virtual element with some tangential polynomials on element boundaries. A fully discrete scheme is then given by choosing the backward Euler for the time discretization. With some assumptions on the exact solutions, we prove that the convergence order is of order 1 with respect to the meshsize and the time step, and the hidden constants are independent of the parameters of the problems. Some numerical experiments are given to verify the results.

Numerical modelling of the poto-poto by coupling of finite elements and boundary elements for acoustic characterization

Sustainable development encourages the construction of buildings based from local raw materials because they are ecological, renewable, available, low energy consumption, less costly, low pollution, and therefore without risk to human health. The poto-poto is a widespread material, made of soil material and bamboo, two natural materials widely used in construction throughout the world and particularly in Central African countries, due to their affordable cost or the low technical requirements for their implementation. This material (poto-poto) although widely used, remains very insufficiently explored as well from the point of view of its mechanical characteristics as from the normative point of view necessary for its modernization. This is why, given the extent and large scale of its use, we considered it important to explore it by establishing its physical characteristics. In this paper, we have chosen to continue the characterization of poto-poto already started in the previous paper by also addressing its acoustic characteristics.Indeed, the effects of climate change are manifested by abnormal variations in temperature against seasons. Hence the urgency to master the physical properties in order to make the most of it both in terms of acoustic implementation and/or thermal insulation as regards its use as a green and modern construction material. In this work, we have, on the basis of the Biot model, carried out a 2D numerical modelling of a poto-poto wall by coupling finite elements method (FEM) and boundary elements method (BEM). The MATLAB software was used to obtain the sound absorption curves of our model. We then made several specimens of our eco-material (3 cm, 4 cm and 5 cm) dosed with 2%, 4% and 6% of bamboo fibers, on which Kundt tube measurements were performed to determine the experimental sound absorption curves. The comparison of these curves with those of the model showed acceptable standard deviations and above all allowed us to conclude that the poto-poto presents an acoustic absorption in the range of frequencies 500–1250 Hz and can thus be effectively used for the acoustic implementation of the walls and certain rooms of the buildings.

Viscoelastic free vibration analysis of in-plane functionally graded orthotropic plates integrated with piezoelectric sensors: Time-dependent 3D analytical solutions

This paper proposes an accurate three-dimensional framework for elastic and viscoelastic free vibration investigation of in-plane functionally graded (IPFG) orthotropic rectangular plates integrated with piezoelectric sensory layers. The developed analytical framework is capable of considering layer-wise unidirectional linear functional gradation in both stiffness and density of the orthotropic composite layers. 3D piezoelasticity-based governing equations of motion are formulated in mixed form by employing Hamilton’s principle, and further solved analytically for Levy-type support conditions using the power-series-based extended Kantorovich method (EKM) jointly with Fourier series. The displacements, stresses, and electrical variables (electric field and electric potential) are solved as the primary variables that ensure the point-wise interlayer continuity and electro-mechanical support conditions. The viscoelastic property of the orthotropic interlayer is defined by employing Biot model, which is similar to the standard linear viscoelastic model. The correctness and efficacy of the present mathematical model are established by comparing the present numerical results with published literature and 3D finite element results, obtained by utilizing user material subroutine in the commercial FE software ABAQUS. An extensive numerical study is performed for various configurations and thickness ratios to investigate the influences of in-plane gradation, viscoelasticity and their coupled effects on the free-vibration response of hybrid laminated plates. It is found that in-plane gradation of stiffness and density remarkably alters the flexural frequencies and corresponding mode shapes of the hybrid intelligent rectangular plates. The flexural frequencies and stresses in the plate can be modified by selecting suitable grading indexes. Another interesting observation is that the in-plane gradation shows a considerably less effect on the electrical response of piezoelectric layers, which can play a vital role in the design of sensors and actuators for dynamic applications. Further, the numerical study demonstrates a potential time-dependent structural behaviour based on the present viscoelastic modelling. The consideration of viscoelasticity could be crucial for analysing the mechanical behaviour of a wide range of polymer composites more realistically and for prospective temporal programming in smart structural systems by exploiting the viscoelastic effect. Although the present analytical solution has been proposed for the free-vibration investigation of smart in-plane functionally graded (IPFG) viscoelastic plates, it can also be utilized directly to analyze the symmetric and asymmetric laminated piezoelectric smart plates with constant properties.

A mixed element analysis of the Biot's model with Darcy–Forchheimer flow

AbstractIn this article, we consider the three‐field Biot's consolidation model, in which the flow's velocity is described by Darcy–Forchheimer law. The model is nonlinear and the unknown variables are the displacement of the solid phase, fluid flux, and pore pressure. The linear finite element is proposed to approximate the solid displacement and the RT mixed finite element is proposed to approximate the fluid flux and pore pressure. The error estimates is given for semidiscrete scheme and fully discrete scheme. When analyzing the fully discrete scheme, no time step restriction is needed. Numerical tests for this method confirm the accuracy of the P1‐RT0‐P0 method when the constraint storage coefficient is positive and is null.

A decoupled stabilized finite element method for the time–dependent Navier–Stokes/Biot problem

In this paper, we propose a decoupled stabilized finite element method for the time–dependent Navier–Stokes/Biot problem by using the lowest equal‐order finite elements. The coupling problem is divided into two subproblems which can be solved in parallel: One is the Navier–Stokes model by treating the nonlinear term explicitly, and the other is the Biot model. In the numerical scheme, we use the implicit backward Euler method in time, while treat the coupling terms explicitly. The stability analysis and error estimates are established for the proposed fully discrete scheme. Numerical results are provided to justify the theory.

Prediction of frequency and spatially dependent attenuation of guided waves propagating in mounted and unmounted A380 parts made up of anisotropic viscoelastic composite laminates

Monitoring damage in composite structures using guided wave-based techniques is particularly effective due to their excellent ability to propagate over relatively long distance and hence to cover a large area with few testing time and equipment. The industrialization of this method is highly tributary of the number and placement of the active elements. Yet, the optimal sensorization of a structure relies on the decrease in amplitude of guided waves over propagation distance. A reliable prediction of attenuation of guided waves is still a challenge especially for anisotropic viscoelastic composite materials which exhibit complex changes of attenuation with propagation direction and thus a spatial dependency of attenuation. In this paper, the damped global matrix method (dGMM), having stable and efficient merits, is developed to predict the frequency and spatially dependent attenuation of waves propagating in anisotropic composite materials. dGMM integrates three damping models (Hysteretic, Kelvin-Voigt, and Biot models) into the conventional undamped GMM to consider viscoelasticity of composite laminates. The proposed dGMM is first theoretically validated by numerical comparison with the semi-analytical finite element method. In addition, two industrial case studies, parts of an A380 nacelle at scale one, are employed to experimentally validate the proposed attenuation prediction method. The first one is a fan cowl structure and the second one is an inner fixed structure, both either unmounted or mounted on an actual instrumented A380 plane. This makes the validation extremely valuable for both the scientific and industrial communities. The proposed attenuation prediction method thus paves the way to optimally deploy sensor network for structural health monitoring of anisotropic viscoelastic composite structures.

Applicability of Time Domain Transform Methods for Frequency Dependent Dynamic Stiffness

Some types of dynamic stiffness, such as the dynamic ground stiffness used in soil-structure interaction analyses and the viscoelastic body used in vibration control systems, have strong frequency dependency. To perform seismic response analysis considering this frequency dependence and the nonlinearity of the model, the dynamic stiffness in the frequency domain must be transformed into the time domain, and a time-history nonlinear response analysis is required. Therefore, many studies on these time-domain transforms have been conducted. One of the present authors has already studied and proposed transform methods for this purpose, and some of their results were used to design new types of damping models. In the present study, the outline and characteristics of the proposed methods (A to C) for this transform are described first. Next, typical problems with strong frequency dependency (i.e., the dynamic soil stiffness, Maxwell element, viscoelastic body, Biot model, and causal hysteretic damping) were transformed into the time domain using these transform methods. The applicability of the transform methods was examined. Subsequently, the characteristics of each problem in the frequency domain and the characteristics of the obtained impulse response in the time domain were analyzed. Finally, it was confirmed that the proposed methods were applicable to all studied problems. These studies are important to understand the physical meaning of these problems, which have strong frequency dependency.

Wave equations for porous media described by the Biot model

Single-mode equivalent space-time representations of the acoustic wave propagating in a Biot poroelastic medium have previously been found only for asymptotic cases: In the low frequency regime, where the viscous skin depth is greater than the characteristic pore size, the time domain equivalent is represented with integer order temporal and spatial loss terms, whereas in the high frequency regime, it is represented with fractional order temporal and spatial loss terms. In the current work, a time domain representation in terms of a partial differential equation is proposed for all three wave solutions of the Biot model across all frequencies, and it is derived from the material response function of the Biot poroelastic medium with suitable approximations for the compressional modes and the dynamic permeability. The dynamic permeability in the time domain is represented by a fractional pseudo-differential operator. Optimal correction factors are introduced into the wave equation to compensate for the discrepancies in the compressional wave dispersion and attenuation. Additionally, the method for incorporating the squirt flow mechanism into the wave equation via the Extended Biot poroviscoelastic model is described. The proposed wave equation has a physical basis and satisfies the passivity criterion.

Generalized Effective Biot Theory and Seismic Wave Propagation in Anisotropic, Poroviscoelastic Media

AbstractThe interior of the Earth is quite complex due to the actual geometrical structure and the presence of complex rheological materials, including viscoelastic rocks, porous sediments and the presence of anisotropy. Seismic wavefield forward modeling in such media forms the basis of most wave equation‐based methods for investigating the structure of the Earth and processing and imaging of seismic data, e.g., seismic full waveform inversion. Numerical modeling using Biot's equations that describe the physics of poroelasticity provides a useful framework to investigate wave attenuation and dispersion. Poroelasticity describes the interaction between the deformation of an elastic porous solid and the fluid flow in the porous structure. We present a time‐domain finite‐difference method for seismic wavefield modeling, taking into account both attenuation and anisotropy of poroviscoelastic Earth structure. The mathematical formulations and modeling methods are based on (a) the linear Biot's poroelastic theory and double porosity models in anisotropic media and (b) a relaxation function that uses the generalized standard linear solid model with anisotropic τ parameter for magnitude of attenuation and nearly constant Q model. Using a modified relaxation function that is suggested to describe the anisotropy attenuation, we develop a generalized anisotropic Biot model in the anisotropic, viscoelastic media. In addition to the anisotropic poroelasticity based on Biot theory, we implement the generalized anisotropic Biot model with complex moduli for an effective Biot's model in which the attenuative anisotropic viscoelastic model with the generalized standard linear solid model is used to approximate the attenuation factor function. The method generalizes the linear poroviscoelastic model based on effective Biot theory for seismic wave modeling to the attenuative, anisotropic case. In the anisotropic, poroviscoelastic model, we represent the bulk modulus and shear modulus of the solid frame by the modified relaxation function. We present time‐domain finite‐difference modeling for seismic wavefields in anisotropic, viscoelastic porous media including transversely isotropic media with a vertical, tilted or horizontal symmetry axis (VTI, TTI, and HTI). We also consider the extension of the two new models to nearly constant Q viscoelastic anisotropy.