Variational asymptotic micromechanics modeling of heterogeneous porous materials

This paper presents a numerical technique to predict the effective elastic properties of heterogeneous fluid-filled porous media where the heterogeneity may result from dissimilar solid and fluid phase properties or due to mismatch in porous microstructure. The technique is based on the variational asymptotic method of homogenization where finite element method is employed for discretization. Biot’s theory of poroelasticity is used to describe porous media where both solid and fluid phase motions (u ? U formulation) are considered with associated strain measures. The method estimates the poroelastic constitutive law in single analysis which makes it very efficient compared to other finite element based homogenization techniques. The method is also general enough to compute all 28 elements of an anisotropic constitutive matrix. Other than estimating the effective properties the micro-stress/strain distribution is also obtained at no additional cost.The method is successfully applied for homogenization of porous media, fluid-filled cavity and finally for effective property estimation of bone lamella. In absence of any other direct method of porous media homogenization, the present technique is compared with classical homogenization methods with fluid approximated as solid of very high Poisson’s ratio. The suitability of this approximation and various other alternatives are also discussed. It is shown that the present homogenization method can be an efficient tool for bone property estimation where fluid-filled porous hierarchical micro-/nanostructure must be respected at all steps.     

Double porosity in fluid-saturated elastic media: deriving effective parameters by hierarchical homogenization of static problem

We propose a model of complex poroelastic media with periodic or locally periodic structures observed at microscopic and mesoscopic scales. Using a two-level homogenization procedure, we derive a model coherent with the Biot continuum, describing effective properties of such a hierarchically structured poroelastic medium. The effective material coefficients can be computed using characteristic responses of the micro- and mesostructures which are solutions of local problems imposed in representative volume elements describing the poroelastic medium at the two levels of heterogeneity. In the paper, we discus various combinations of the interface between the micro- and mesoscopic porosities, influence of the fluid compressibility, or solid incompressibility. Gradient of porosity is accounted for when dealing with locally periodic structures. Derived formulae for computing the poroelastic material coefficients characterize not only the steady-state responses with static fluid, but are relevant also for quasistatic problems. The model is applicable in geology, or in tissue biomechanics, in particular for modeling canalicular-lacunar porosity of bone which can be characterized at several levels.     

Pore-Scale Simulation of Shear Thinning Fluid Flow Using Lattice Boltzmann Method

The present work attempts to identify the roles of flow and geometric variables on the scaling factor which is a necessary parameter for modeling the apparent viscosity of non-Newtonian fluid in porous media. While idealizing the porous media microstructure as arrays of circular and square cylinders, the present study uses multi-relaxation time lattice Boltzmann method to conduct pore-scale simulation of shear thinning non-Newtonian fluid flow. Variation in the size and inclusion ratio of the solid cylinders generates wide range of porous media with varying porosity and permeability. The present study also used stochastic reconstruction technique to generate realistic, random porous microstructures. For each case, pore-scale fluid flow simulation enables the calculation of equivalent viscosity based on the computed shear rate within the pores. It is observed that the scaling factor has strong dependence on porosity, permeability, tortuosity and the percolation threshold, while approaching the maximum value at the percolation threshold porosity. The present investigation quantifies and proposes meaningful correlations between the scaling factor and the macroscopic properties of the porous media.     

Homogenized modeling for vascularized poroelastic materials

A new mathematical model for the macroscopic behavior of a material composed of a poroelastic solid embedding a Newtonian fluid network phase (also referred to as vascularized poroelastic material), with fluid transport between them, is derived via asymptotic homogenization. The typical distance between the vessels/channels (microscale) is much smaller than the average size of a whole domain (macroscale). The homogeneous and isotropic Biot’s equation (in the quasi-static case and in absence of volume forces) for the poroelastic phase and the Stokes’ problem for the fluid network are coupled through a fluid-structure interaction problem which accounts for fluid transport between the two phases; the latter is driven by the pressure difference between the two compartments. The averaging process results in a new system of partial differential equations that formally reads as a double poroelastic, globally mass conserving, model, together with a new constitutive relationship for the whole material which encodes the role of both pore and fluid network pressures. The mathematical model describes the mutual interplay among fluid filling the pores, flow in the network, transport between compartments, and linear elastic deformation of the (potentially compressible) elastic matrix comprising the poroelastic phase. Assuming periodicity at the microscale level, the model is computationally feasible, as it holds on the macroscale only (where the microstructure is smoothed out), and encodes geometrical information on the microvessels in its coefficients, which are to be computed solving classical periodic cell problems. Recently developed double porosity models are recovered when deformations of the elastic matrix are neglected. The new model is relevant to a wide range of applications, such as fluid in porous, fractured rocks, blood transport in vascularized, deformable tumors, and interactions across different hierarchical levels of porosity in the bone.     

Effect of porosity on the properties of strain localization in porous media under undrained conditions

Within the general framework of mixture theory and by introducing the fictitious “fluid phase” as a mixture of a liquid and a gas, the conditions for localization of deformation into a shear band in the incremental response of partially saturated and fully saturated elastic–plastic porous media under undrained conditions are derived. The effect of porosity is included in the derivation. The explicit analytical expressions of the direction of shear band initiation and the corresponding hardening modulus of the porous media for the plane strain case are deduced, and a parametric analysis is made of the influence of the porosity on the properties of strain localization based on Mohr–Coulomb yield criterion. It is found that the dependence of the shear banding properties of partially saturated porous media on the porosity is related to the stress states and Poisson's ratio. However, the properties of the strain localization for the fully saturated porous media are almost independent of Poisson's ratio. Finally, on the basis of Mohr–Coulomb yield criterion, some solutions of the shear banding orientation for water-saturated granular materials are obtained, which are proved to be in good agreement with the experimental results reported by other researchers.     

Generalized Macroscopic Models for Fluid Flow in Deformable Porous Media I: Theories

This study developed generalized mathematical models to describe the motion of fluids in porous media, and applied these models to harmonic excitation applications. The problem of fluid flow in small channels of a periodic elastic solid matrix was studied at the pore scale, and the homogenization technique was applied to predict the macroscopic behavior of reservoirs. Based on the homogenization study, five separate characteristic macroscopic models were identified according to the relation between a length scale parameter and a property contrast number. These five models can be used to interpret the corresponding responses of a saturated porous medium. The relation to existing theories, such as Darcy's law, the Telegrapher's equation and Biot's theory, was investigated. The numerical results and applications are presented in Part II of the study.     

多孔材料中声波的传播与演化

采用两相多孔介质的拉格朗日模型来描述一种理论流体充填的多孔弹性固体材料,其中孔隙度的变化满足一个附加的平衡方程。     

The PELskin project—part III: a homogenized model of flows over and through dense poroelastic media

In the context of the PELskin project, a homogenized model to study flows over and inside poroelastic media has been developed. It allows to simulate the fluid–structure interaction between a fluid and an extremely dense poroelastic medium, without limitations on physical and geometrical parameters such as the density of the elastic material, the porosity or the number of periodic microstructures which constitute the medium. The model is applied to the case of the flow in a channel driven by an oscillating pressure gradient, with half the channel covered by a carpet of flexible, densely packed fibers, connected to each other to allow for the propagation of the deformation field.     

A computational framework for fluid–porous structure interaction with large structural deformation

We study the effect of poroelasticity on fluid–structure interaction. More precisely, we analyze the role of fluid flow through a deformable porous matrix in the energy dissipation behavior of a poroelastic structure. For this purpose, we develop and use a nonlinear poroelastic computational model and apply it to the fluid–structure interaction simulations. We discretize the problem by means of the finite element method for the spatial approximation and using finite differences in time. The numerical discretization leads to a system of non-linear equations that are solved by Newton’s method. We adopt a moving mesh algorithm, based on the Arbitrary Lagrangian–Eulerian method to handle large deformations of the structure. To reduce the computational cost, the coupled problem of free fluid, porous media flow and solid mechanics is split among its components and solved using a partitioned approach. Numerical results show that the flow through the porous matrix is responsible for generating a hysteresis loop in the stress versus displacement diagrams of the poroelastic structure. The sensitivity of this effect with respect to the parameters of the problem is also analyzed.

     

海洋地震工程流固耦合问题统一计算框架

海底地震动的模拟以及海洋工程结构的地震反应分析中,涉及到海水、饱和海床、弹性基岩、结构之间的相互耦合.传统的方法分别采用声波方程描述理想流体、Biot方程描述饱和海床、弹性波方程描述基岩和结构,分别考虑相互之间的耦合,十分不便.本文基于理想流体、固体分别为饱和多孔介质的特殊情形(孔隙率分别为1和0),由饱和多孔介质的Biot方程可退化得到理想流体的声波方程和固体的弹性波方程.然后,以饱和多孔介质方程为基础,经集中质量有限元离散,考虑不同孔隙率的饱和多孔介质之间耦合的一般情形,建立了该耦合情形的求解方法.进一步论证了该一般情形的耦合计算方法可分别退化到流体与固体、流体与饱和多孔介质、固体与饱和多孔介质之间的耦合计算,从而将流体、固体、饱和多孔介质间的耦合问题纳入到统一计算框架,并编制了相应的三维并行分析程序.以P-SV波垂直入射时,半无限层状海水-饱和海床、海水-弹性基岩、海水-饱和海床-弹性基岩三种情形的动力分析为例,采用统一计算框架结合透射边界条件进行求解,并与传递矩阵方法得到的解进行对比,验证了该统一计算框架的有效性以及并行计算的可行性.      

Homogenization of acoustic waves in strongly heterogeneous porous structures

We consider acoustic waves in fluid-saturated periodic media with dual porosity. At the mesoscopic level, the fluid motion is governed by the Darcy flow model extended by inertia terms and by the mass conservation equation. In this study, assuming the porous skeleton is rigid, the aim is to distinguish the effects of the strong heterogeneity in the permeability coefficients. Using the asymptotic homogenization method we derive macroscopic equations and obtain the dispersion relationship for harmonic waves. The double porosity gives rise to an extra homogenized coefficient of dynamic compressibility which is not obtained in the upscaled single porosity model. Both the single and double porosity models are compared using an example illustrating wave propagation in layered media.     

液饱和多孔介质中三维应力波的传播

为了深入研究液饱和多孔介质中应力波的传播，提出了三维两相细观计算模型．基于此模型。应用Galerkin余量法并计及流-固耦合界面的耦合效应，利用直接耦合的技术，开发了三维流-固混合显式动力有限元计算程序．在此基础上对冲击载荷作用下液饱和多孔介质中三维应力波的传播现象进行了数值模拟，并详细讨论了孔隙率，孔隙形状等因素对应力波传播主导波形的影响．     

On Determining the Percolation Reynolds Number and the Characteristic Linear Dimension for Ideal and Fictitious Porous Media

Various versions of representations of the percolation Reynolds number for porous media with isotropic and anisotropic flow properties are considered. The formulas are derived and the variants are analyzed with reference to model porous media with a periodic microstructure formed by systems of capillaries and packings consisting of spheres of constant diameter (ideal and fictitious porous media, respectively). A generalization of the Kozeny formula is given for determining the capillary diameter in an ideal porous medium equivalent to a fictitious medium with respect to permeability and porosity and it is shown that the capillary diameter is nonuniquely determined. Relations for recalculating values of the Reynolds number determined by means of formulas proposed earlier are given and it is shown that taking the microstructure of porous media into account, as proposed in [1, 2], makes it possible to explain the large scatter of the numerical values of the Reynolds number in processing the experimental data.     

海洋地震工程流固耦合问题统一计算框架——————不规则界面情形

海底地震动场及海洋声场的模拟中,需要考虑复杂海床介质及海底地形的影响,涉及到海水、饱和海床、弹性基岩之间的相互耦合.传统的方法分别采用声波方程描述理想流体、Biot方程描述饱和海床、弹性波方程描述基岩,分别进行空间离散和界面耦合, 十分不便.本文基于理想流体、固体分别为饱和多孔介质的特殊情形(孔隙率分别为1和0),由饱和多孔介质的Biot方程可退化得到理想流体的声波方程和固体的弹性波方程.然后, 以饱和多孔介质方程为基础, 经集中质量有限元离散,严格考虑不同孔隙率的饱和多孔介质在不规则界面的耦合条件,通过求解法向和切向界面力的途径,建立了不同孔隙率的饱和多孔介质耦合情形的求解方法,将流体、固体、饱和多孔介质间的耦合问题纳入到统一计算框架,并编制了相应的三维并行分析程序.考虑海水--弹性基岩、海水--饱和海床--弹性基岩体系中凹陷地形情形,采用本文提出的统一计算框架, 结合透射边界条件,分析了P波入射时的动力反应, 并通过结果是否满足界面条件,验证了该统一计算框架的有效性以及并行计算的可行性.      

Acoustical characterization of fluid-saturated porous media with local heterogeneities: Theory and application

A linear dynamic model of fully saturated porous media with local (either microscopic or mesoscopic) heterogeneities is developed within the context of Biot’s theory of poroelasticity. Viscoporoelastic behavior associated with local fluid flow is characterized by the notion of the dynamic compatibility condition on the interface between the solid and the fluid. Complex, frequency-dependent material parameters characterizing the viscoporoelasticity are derived. The complex properties can be obtained through determining the quasi-static poroelastic parameters, the properties of individual constituents, and the relaxation time of the dynamic compatibility condition on the interface. Relationships among various quasi-static poroelastic parameters are developed. It is shown that local fluid flow mechanism is significant only in the porous media with local heterogeneities. The relaxation time of the compatibility condition on the interface depends upon the details of local structure of porous media that control local fluid pressure diffusion. The new model is used to describe the velocity dispersion and attenuation in fully saturated porous media. The proposed model provides a theoretical framework to simulate the acoustical behavior of fully saturated porous media over a wide range of frequencies without making any explicit assumption about the structure of local heterogeneities.     

基于迭代法的非线性弹性均质化研究

微观结构对复合材料的宏观力学性能具有至关重要的影响, 通过合理设计复合材料微观结构可以得到期望的宏观性能. 均质化方法作为一种有效的设计方法, 它从微观结构的角度出发, 利用均匀化的概念, 实现了对复合材料宏观力学性能的预测和设计. 而当考虑非线性因素, 均质化的实现就非常困难. 本文利用双渐近展开方法, 将位移按照宏观位移和微观位移展开, 推导了非线性弹性均质化方程. 通过直接迭代法, 对非线性弹性均质化方程进行了求解, 并给出了具体的迭代方法和实现步骤. 本文基于迭代步骤和非线性弹性均质化方程编写MATLAB 程序, 对3种典型本构关系的周期性多孔材料平面问题进行了计算, 对比细致模型的应变能、最大位移和等效泊松比, 对程序及迭代方法的准确性进行了验证. 之后对一种三元橡胶基复合材料进行多尺度均质化, 将其分为芯丝尺度和层间尺度. 用线弹性的均质化方法得到了芯丝尺度的等效弹性参数, 并将其作为层间尺度的材料参数. 在层间尺度应用非线性弹性均质化方法对结构进行计算, 得到材料的宏观等效性能, 并以实验结果为基准进行评价.      

不可压饱和多孔弹性简支梁的动力响应

在杆件弯曲小变形的假定下,考虑杆件的侧向变形因素,根据多孔介质理论,本文首先建立了不可压饱和多孔弹性梁弯曲变形时动力响应的控制方程。其次,基于所建立的控制微分方程,利用变量分离法,研究了两端可渗透的饱和多孔弹性简支梁在梁中间集中载荷作用下的动力响应,得到了不同物性参数下简支梁动态弯曲时挠度和孔隙流体压力等效力偶等随时间的响应曲线。研究发现由于孔隙流体和固相骨架的相互作用,不可压饱和多孔弹性梁挠度的动力响应具有粘性特征,同时,随着时间的增加,饱和多孔弹性梁的挠度、弯矩等最终趋于经典弹性梁的静挠度、弯矩,此时,孔隙流体压力为零,梁的固相骨架承担所有的外载荷。     

简支饱和多孔弹性梁的非线性弯曲

基于饱和多孔介质理论和弹性梁的大挠度弯曲假设,在多孔弹性梁轴线不可伸长,孔隙流体仅沿轴向方向扩散的限制下,建立了微观不可压饱和多孔弹性梁大挠度拟静态响应的一维非线性数学模型.在此基础上,利用Galerkin截断法,分析了两端可渗透的简支多孔弹性梁在突加横向均布载荷作用下的非线性弯曲,给出了梁弯曲时挠度、弯矩以及孔隙流体压力等效力偶随时间的响应曲线.数值结果表明:当载荷较小时,大挠度非线性与小挠度线性理论的结果相差很小,而当载荷较大时,非线性大挠度理论的结果小于相应线性小挠度理论的结果,并且这种差异随着载荷的增大而增大.同时,在载荷突加于梁上时,多孔弹性梁骨架起初不变形,孔隙流体压力等效力偶由零突增为非零,其值与外载荷保持平衡.随着时间的增加,梁的挠度增加,等效力偶逐渐减小为零,最终多孔梁骨架承担全部的外载荷.