裂隙挤喷流对孔隙介质排水体积模量的影响

实际岩石比如沉积形成的岩石往往是裂隙和孔隙并存的孔隙介质. 由于扁状的裂隙与近似球形或圆管形的孔隙具有不同的可压缩性,当孔隙介质受压时,液体会从易压缩的裂隙中挤出流入不易压缩的孔隙中,这种挤喷流会引起弹性模量的频散和能量的耗散. 着重研究了裂隙挤喷流和液体可压缩性对孔道变形的影响,推导出了动载荷作用下排水体积模量的表达式. 与挤喷流相关的裂隙附加柔度会引起排水体积模量随频率变化,使得孔隙介质呈现黏弹性. 频率越高,模量的实部越大,岩石抵抗变形的能力越强. 而模量的虚部体现了挤喷流对能量的耗散. 裂隙密度主要决定模量频散的幅度以及能量耗散的强度,且裂隙密度越大,模量频散幅度越大,能量耗散也越强. 裂隙的纵横比主要决定模量频散速率最快或能量耗散最强时对应的特征频率. 若孔隙介质中不含有裂隙,即裂隙密度是0时,排水体积模量退化为Biot理论中的排水体积模量.     

Determination of elastic moduli of composite medium containing bimaterial matrix and non-uniform inclusion concentrations

Reservoir porous rocks usually consist of more than two types of matrix materials,forming a randomly heterogeneous material.The determination of the bulk modulus of such a medium is critical to the elastic wave dispersion and attenuation.The elastic moduli for a simple matrix-inclusion model are theoretically analyzed.Most of the efforts assume a uniform inclusion concentration throughout the whole single-material matrix.However,the assumption is too strict in real-world rocks.A model is developed to estimate the moduli of a heterogeneous bimaterial skeleton,i.e.,the host matrix and the patchy matrix.The elastic moduli,density,and permeability of the patchy matrix differ from those of the surrounding host matrix material.Both the matrices contain dispersed particle inclusions with different concentrations.By setting the elastic constant and density of the particles to be zero,a double-porosity medium is obtained.The bulk moduli for the whole system are derived with a multi-level effective modulus method based on Hashin's work.The proposed model improves the elastic modulus calculation of reservoir rocks,and is used to predict the kerogen content based on the wave velocity measured in laboratory.The results show pretty good consistency between the inversed total organic carbon and the measured total organic carbon for two sets of rock samples.     

Prediction of elastic modulus and Poisson’s ratio for unsaturated concrete

Many concrete structures are located in water environment, but the underwater concrete is usually unsaturated even though it has been soaked in the water for a very long time. Some experiments have proven that the mechanical properties of concrete are affected by the saturation degree of fluid and aspect ratio of pores. Several publications discussed the saturated concrete qualitatively, but few gave quantitative analysis especially for the unsaturated concrete. In terms of the microstructure of unsaturated concrete, equivalent medium and inclusion-based theory of composite materials, a model is proposed to explain the changes happened in the wet concrete and to predict the elastic parameters (including elastic modulus and Poisson’s ratio) of unsaturated concrete. The viscosity of water in pores, micro-cracks and the further hydration of cement are taken into account in this paper by means of the definition of saturation concept according to the effect of pore water on the modulus of concrete. In this model, both stiff effect of water and soft effect of cracks on the concrete are introduced to describe the bulk modulus, at the same time the effect of shear rate on the shear modulus is considered. The comparison between the theoretical models and experimental results in the extreme state indicates that the model proposed in this paper is valid to predict the elastic properties of unsaturated concrete.     

The elastic moduli estimation of the solid-water mixture

Conceptually, the undrained elastic constants estimated by the poroelasticity theory should be identical to the effective moduli of the two-phase composite of a porous material saturated with pore water. Here we show numerically that the undrained elastic constants determined by an effective moduli estimate are almost identical with those calculated by poroelasticity theory, and if pore shapes are not exactly known and the porosity is around 50%, estimating the elastic constant as the average value of its Voigt and Reuss bounds is reasonably accurate. This is the situation in bone and dentin, the materials that are our primary intended application. This result will hold for situations in which the totally enclosed water phase is constrained to small deformations by virtue of its confinement. Importantly, in this work we assume that water is an isotropic elastic solid with a shear modulus that is 10^?4 times the bulk modulus of the water. Note that it is compressible, but almost incompressible with a Poisson’s ratio of 0.4999.     

Shear wave dispersion and attenuation in periodic systems of alternating solid and viscous fluid layers

The attenuation and dispersion of elastic waves in fluid-saturated rocks due to the viscosity of the pore fluid is investigated using an idealized exactly solvable example of a system of alternating solid and viscous fluid layers. Waves in periodic layered systems at low frequencies are studied using an asymptotic analysis of Rytov’s exact dispersion equations. Since the wavelength of shear waves in fluids (viscous skin depth) is much smaller than the wavelength of shear or compressional waves in solids, the presence of viscous fluid layers necessitates the inclusion of higher terms in the long-wavelength asymptotic expansion. This expansion allows for the derivation of explicit analytical expressions for the attenuation and dispersion of shear waves, with the directions of propagation and of particle motion being in the bedding plane. The attenuation (dispersion) is controlled by the parameter which represents the ratio of Biot’s characteristic frequency to the viscoelastic characteristic frequency. If Biot’s characteristic frequency is small compared with the viscoelastic characteristic frequency, the solution is identical to that derived from an anisotropic version of the Frenkel–Biot theory of poroelasticity. In the opposite case when Biot’s characteristic frequency is greater than the viscoelastic characteristic frequency, the attenuation/dispersion is dominated by the classical viscoelastic absorption due to the shear stiffening effect of the viscous fluid layers. The product of these two characteristic frequencies is equal to the squared resonant frequency of the layered system, times a dimensionless proportionality constant of the order 1. This explains why the visco-elastic and poroelastic mechanisms are usually treated separately in the context of macroscopic (effective medium) theories, as these theories imply that frequency is small compared to the resonant (scattering) frequency of individual pores.     

Effective elastic properties of porous materials: Homogenization schemes vs experimental data

In this paper, we focus on the prediction of elastic moduli of isotropic porous materials made of a solid matrix having a Poisson's ratio vm of 0.2. We derive simple analytical formulae for these effective moduli based on well-known Mean-Field Eshelby-based Homogenization schemes. For each scheme, we find that the normalized bulk, shear and Young's moduli are given by the same form depending only on the porosity p. The various predictions are then confronted with experimental results for the Young's modulus of expanded polystyrene (EPS) concrete. The latter can be seen as an idealized porous material since it is made of a bulk cement matrix, with Poisson's ratio 0.2, containing spherical mono dispersed EPS beads. The Differential method predictions are found to give a very good agreement with experimental results. Thus, we conclude that when vm=0.2, the normalized effective bulk, shear and Young's modulus of isotropic porous materials can be well predicted by the simple form (1 − p)² for a large range of porosity p ranging between 0 and 0.56.     

Squirt-Flow in Fluid-Saturated Porous Media: Propagation of Rayleigh Waves

Propagation of attenuated waves is studied in a squirt-flow model of porous solid permeated by two different pore regimes saturated with same viscous fluid. Presence of soft compliant microcracks embedded in the grains of stiff porous rock defines the double-porosity formation. Microcracks and pores respond differently to the compressional effect of a propagating wave, which induces the squirt-flow from microcracks to pores. Elastodynamics of constituent particles in porous aggregate is represented through a single-porosity formulation, which involves the frequency-dependent complex moduli. This formulation is deduced as a special case of double-porosity formation allowing the wave-induced flow of pore-fluid. This squirt-flow model of porous solid supports the attenuated propagation of two compressional waves and one shear wave. Superposition of these body waves, subject to stress-free surface, defines the propagation of Rayleigh wave. This wave is governed by a complex irrational dispersion equation, which is solved numerically after rationalising into an algebraic equation. For existence of Rayleigh wave, a complex solution of the dispersion equation should represent a leaky wave, which decays for propagation along any direction in the semi-infinite medium. A numerical example is solved to analyse the effects of squirt-flow on phase velocity, attenuation and polarisation of the Rayleigh waves, for different combinations of parameters. Numerical results suggest the existence of an additional (second) Rayleigh wave in the squirt-flow model of dissipative porous solids.     

The evaluation of the influence of various parameters on the velocity of elastic wave propagation in a rock medium

A full waveform recording in a borehole during acoustic logging makes it possible to determine the elastic parameters of a medium under in-situ conditions.The velocity of elastic wave propagation in rocks and elastic moduli are influenced by factors connected with its macrostructure and microstructure, as well as with rock overburden and porous pressure and temperature.The results of the calculations of the relationships between the elastic and reservoir parameters of sedimentary rocks are presented in this paper. The theoretical Kuster and Toksöz model has been applied.The influence of the porosity, the pore space coefficient, and the saturation of different media of porous rocks on elastic moduli and on compressional and shear wave propagation have been considered in this model. The complex composition of the skeleton and the influence of clay material in the porous rock are taken into account.     

含液多孔介质力学问题的边界元方法

提出了一种含液多孔介质力学问题的边界元求解方法.首先将问题分解为一系列含单孔流体夹杂的子问题,然后针对每个子问题建立了流体孔体积变化率与流体压力之问的函数关系,进一步采用边界元方法建立了以各流体孔压力为基本未知量的线性代数方程组,最后根据所求出的各流体孔的压力计算含液多孔介质内各点的位移、变形和应力.为了说明方法的有效...     

P-wave velocity prediction in porous medium with liquid-pocket patchy saturation

It becomes increasingly clear that non-uniform distribution of immiscible fluids in porous rock is particularly relevant to seismic wave dispersion. White proposed a patchy saturation model in 1975, in which spherical gas pockets were located at the center of a liquid saturated cube. For an extremely light and compressible inner gas, the physical properties can be approximated by a vacuum with White's model. The model successfully analyzes the dispersion phenomena of a P-wave velocity in gas-watersaturated rocks. In the case of liquid pocket saturation, e.g., an oil-pocket surrounded by a water saturated host matrix, the light fluid-pocket assumption is doubtful, and few works have been reported in White's framework. In this work, Poisson's ratio, the bulk modulus, and the effective density of a dual-liquid saturated medium are formulated for the heterogeneous porous rocks containing liquid-pockets. The analysis of the difference between the newly derived bulk modulus and that of White's model shows that the effects of liquid-pocket saturation do not disappear unless the porosity approaches zero. The inner pocket fluid can no longer be ignored. The improvements of the P-wave velocity predictions are illustrated with two examples taken from experiments, i.e., the P-wave velocity in the sandstone saturated by oil and brine and the P-wave velocity for heavy oils and stones at different temperatures.     

Spring-network model of red blood cell: From membrane mechanics to validation

Red blood cell membrane is highly elastic and proper modeling of this elasticity is essential for biomedical applications that involve computational experiments with blood flow. Inseparable and often some of the most difficult parts of modeling process are verification and validation. In this work, we present a revised model, which uses a spring network to represent the cell membrane immersed in a fluid and has been successfully used in blood flow simulations. We demonstrate the validation steps by first deriving the theoretical relations between the bulk properties of elastic membranes—shear modulus and area compressibility modulus—and parameters of the model that enter the nonlinear stretching and local area conservation computational moduli. We verify the theoretically derived relations using computer simulations of deformable triangular mesh. We calibrate the model by performing a computational version of the optical tweezers experiment. And finally, we validate the modeled cell behavior by investigating the cell rotation frequency when it is subjected to shear flow and cell deformation in narrow channels. The supplementary material contains an extensive dataset that can be used for setting different elastic properties for each cell in simulations of dense suspensions, while still conforming to the biological data. This work contains a complete model development process: From modelling of basic mechanical concepts (the spring network) and advanced biomechanical concepts (such as elasticity of the membrane), through calibration process towards the final stage of model validation.     

Modeling and prediction of bulk properties of open-cell carbon foam

The emerging ultralightweight material, carbon foam, was modeled with three-dimensional microstructures to develop a basic understanding in correlating microstructural configuration with bulk performance of open-cell foam materials. Because of the randomness and complexity of the microstructure of the carbon foam, representative cell ligaments were first characterized in detail at the microstructural level. The salient microstructural characteristics (or properties) were then correlated with the bulk properties through the present model. In order to implement the varying anisotropic nature of material properties in the foam ligaments, we made an attempt to use a finite element method to implement such variation along the ligaments as well as at a nodal point where the ligaments meet. The model was expected to provide a basis for establishing a process-property relationship and optimizing foam properties.The present model yielded a fairly reasonable prediction of the effective bulk properties of the foams. We observed that the effective elastic properties of the foams were dominated by the bending mode associated with shear deformation. The effective Young's modulus of the foam was strongly influenced by the ligament moduli, but was not influenced by the ligament Poisson's ratio. The effective Poisson's ratio of the foam was practically independent of the ligament Young's modulus, but dependent on the ligament Poisson's ratio. The effective Young's modulus of the carbon foam was dependent more on the transverse Young's modulus and the shear moduli of the foam ligaments, but less significantly on the ligament longitudinal Young's modulus. A parametric study indicated that the effective Young's modulus was significantly improved by increasing the solid modulus in the middle of the foam ligaments, but nearly invariant with that at the nodal point where the ligaments meet. Therefore, appropriate processing schemes toward improving the transverse and shear properties of the foam ligaments in the middle section of the ligaments rather than at the nodal points are highly desirable for enhancing the bulk moduli of the carbon foam.     

Linear rheology of compressible soft nanocomposites

The influences of interfacial tension and compressibility to the linear viscoelastic properties of nanocomposite and nanoporous materials are considered theoretically. The effective bulk and shear moduli of the systems are calculated within the generalized composite sphere model which takes into account the effect of interfacial tension. It is found that frequency dependence of the effective dynamic shear and bulk moduli of nanocomposites with the compressible elastic matrix and viscous inclusions may be represented in terms of the Zener model comprising of the viscoelastic Kelvin element in series with the elastic spring. The relations of the Zener model parameters with the material characteristics are revealed. The physical interpretation of the frequency behavior of the dynamic shear and bulk moduli against the interfacial tension, component compressibility, viscosity, and inclusion volume fraction is discussed. Victor G. Oshmyan deceased.     

The total creep of viscoelastic composites under hydrostatic or antiplane loading

The problem of bounding the total creep (or total stress relaxation) of a composite made of two linear viscoelastic materials and subjected to a constant hydrostatic or antiplane loading is considered. It is done by coupling the immediate and the relaxed responses of the composite, which are pure elastic. The coupled bounds provide the possible range of the total deformation at infinite time as a function of the initial deformation of the composite. For antiplane shear existing bounds for coupled two-dimensional conductivity yield the required coupled bounds, and these are attained by doubly coated cylinder assemblages. The translation method is used to couple the effective bulk moduli of a viscoelastic composite at zero and infinite time. A number of microgeometries are found to attain the bulk modulus bounds. It is shown that the Hashin's composite sphere assemblage does not necessarily correspond to the maximum or minimum overall creep, although it necessarily attains the bounds for effective bulk moduli. For instance, there are cases when the doubly coated sphere microstructure or some special polycrystal arrangements attain the bounds on the total creep.     

Thin interphase/imperfect interface in elasticity with application to coated fiber composites

The imperfect interface conditions which are equivalent to the effect of a thin elastic interphase are derived by a Taylor expansion method in terms of interface displacement and traction jumps. Plane and cylindrical interfaces are analyzed as special cases. The effective elastic moduli of a unidirectional coated fiber composite are obtained on the basis of the derived imperfect interface conditions. High accuracy of the method is demonstrated by comparison of solutions of several problems in terms of the imperfect interface conditions or explicit presence of interphase as a third phase. The problems considered are transverse shear of a coated infinite fiber in infinite matrix and effective transverse bulk and shear moduli and effective axial shear modulus of a coated fiber composite. Unlike previous elastic imperfect interface conditions in the literature, the present ones are valid for the entire range of interphase stiffness, from very small to very large.     

Theoretical Analyses of the Effects of Solute Dispersion on Chemical-Dissolution Front Instability in Fluid-Saturated Porous Media

This article deals with the theoretical aspects of chemical-dissolution front instability problems in two-dimensional fluid-saturated porous media including solute dispersion effects. Since the solute equilibrium concentration is much smaller than the molar density of the dissolvable mineral in a mineral dissolution system, a limit case, in which the ratio of the solute equilibrium concentration (in the pore fluid) to the molar density of the dissolvable mineral (in the solid matrix of the porous medium) approaches zero, is considered in the theoretical analysis. Under this assumption, the critical condition under which a planar chemical-dissolution front becomes unstable has been mathematically derived when solute dispersion effects are considered. The present theoretical results clearly demonstrated that: (1) the propagation speed of a planar chemical-dissolution front in the case of considering solute dispersion effects is the same as that when solute dispersion effects are neglected. This indicates that solute dispersion does not affect the propagation speed of the planar chemical-dissolution front in a fluid-saturated porous medium. (2) The consideration of solute dispersion can cause a significant increase in the critical Zhao number, which is used to judge whether or not a planar chemical-dissolution front may become unstable in the fluid-saturated porous medium. This means that the consideration of solute dispersion can stabilize a planar chemical-dissolution front, because an increase in the critical Zhao number reduces the likelihood of the planar chemical-dissolution front instability in a fluid-saturated porous medium. In addition, the present results can be used as benchmark solutions for verifying numerical methods employed to simulate detailed morphological evolution processes of chemical dissolution fronts in two-dimensional fluid-saturated porous media.     

Moving singularities in elasto-diffusive solids with applications to fracture propagation

Solutions are derived for steady-state motion of a singularity class, which includes point sources and dislocations, through a medium in which the elastic stress-field can evolve with time due to the diffusion of an internal second-phase species, such as a pore-fluids and lattice impurity concentrations in a crystalline solid, or the transfer of heat. The technique is to integrate the known influence functions for a stationary singularity. Attention is focused on the most tractable aspect, namely the stress field on the trajectory of motion: this suffices for simulation of growing shear and tensile fractures (e.g. in a porous fluid-saturated solid). Continuous densities of fluid sources and point discontinuities (dislocations) are suitably distributed (as determined by solving the resulting singular integral equations) to satisfy solid stress and fluid pressure or flow conditions on the fracture surfaces. Alternative methods for finding the complete dislocation influence function are discussed and comparisons with existing source solutions are made. Substantial stabilization effects are found in fracture propagation.     

Stress-Dependent Porosity and Permeability of Porous Rocks Represented by a Mechanistic Elastic Cylindrical Pore-Shell Model

The stress dependency of the porosity and permeability of porous rocks is described theoretically by representing the preferential flow paths in heterogeneous porous rocks by a bundle of tortuous cylindrical elastic tubes. A Lamé-type equation is applied to relate the radial displacement of the internal wall of the cylindrical elastic tubes and the porosity to the variation of the pore fluid pressure. The variation of the permeability of porous rocks by effective stress is determined by incorporating the radial displacement of the internal wall of the cylindrical elastic tubes into the Kozeny–Carman relationship. The fully analytical solutions of the mechanistic elastic pore-shell model developed by combining the Lamé and Kozeny–Carman equations are shown to lead to very accurate correlations of the stress dependency of both the porosity and the permeability of porous rocks.

     

Effect of an inhomogeneous interphase zone on the bulk modulus and conductivity of a particulate composite

A model is presented of a particulate composite containing spherical inclusions, each of which are surrounded by a localized region in which the elastic moduli vary smoothly with radius. This region may represent an interphase zone in a composite, or the transition zone around an aggregate particle in concrete, for example. An exact solution is derived for the displacements and stresses around a single inclusion in an infinite matrix, subjected to a far-field hydrostatic compression, and is then used to derive an approximate expression for the effective bulk modulus of a material containing a random dispersion of these inclusions. The analogous conductivity (thermal, electrical, etc.) problem is then discussed, and it is shown that the expression for the normalized effective conductivity corresponds exactly to that for the normalized effective bulk modulus, if the Poisson ratios of both phases are set to zero.     

On effective moduli of inhomogeneous media characterized by several small parameters

Composites and porous media of elongated structure, as well as materials with pores or inclusions having the shape of parallelepipeds or channels of rectangular cross-section, are considered under certain conditions on the inclusion-to-matrix modulus ratio and the volume fraction of inclusions (pores). The effective moduli are calculated by the method of mathematical homogenization theory. Numerical results on the dependence of the effective moduli on the prolateness of the structure, the shape of the inclusions (pores), the inclusion-to-matrix modulus ratio, and the volume fraction of inclusions (pores) are given. The effective moduli computed according to the algorithm of mathematical homogenization theory are compared with those given by the explicit approximate formulas earlier developed by the authors.