Long wavelength inner-resonance cut-off frequencies in elastic composite materials

We revisit an ancient paper (Auriault and Bonnet, 1985) which points out the existence of cut-off frequencies for long acoustic wavelength in high-contrast elastic composite materials, i.e. when the wavelength is large with respect to the characteristic heterogeneity length. The separation of scales enables the use of the method of multiple scale expansions for periodic structures, a powerful upscaling technique from the heterogeneity scale to the wavelength scale. However, the results remain valid for non-periodic composite materials which show a Representative Elementary Volume (REV). The paper extends the previous investigations to three-component composite materials made of hard inclusions, coated with a soft material, both of arbitrary geometry, and embedded in a connected stiff material. The equivalent macroscopic models are rigorously established as well as their domains of validity. Provided that the stiffness contrast within the soft and the connected stiff materials is of the order of the squared separation of scales parameter, it is demonstrated (i) that the propagation of long wave may coincide with the resonance frequencies of the hard inclusions/soft material system and (ii) that the macroscopic model presents a series of cut-off frequencies given by an eigenvalue problem for the resonating domain in the cell. These results are illustrated in the case of stratified composites and the possible microstructures of heterogeneous media in which the inner dynamics phenomena may occur are discussed.     

基于多尺度页岩巴西劈裂试验的岩石强度尺度效应根源研究

为探究页岩抗拉强度尺度效应规律及岩石强度尺度效应根源,采集鄂尔多斯盆地延长组长7段陆相页岩野外露头,制作成直径为50 mm,高度分别为15.32 mm,20.39 mm,25.46 mm和30.28 mm四种尺度的岩样并进行巴西劈裂试验,在分析岩样抗拉强度尺度效应规律的基础上,对岩石强度尺度效应根源提出了新的认识。结果表明:页岩最大载荷和破裂面条数随岩样尺度增大而增大,抗拉强度随岩样尺度增大而减小,破裂时间与岩样尺度相关性较差;非均质性和端部摩擦效应是目前解释岩石强度尺度效应根源的主要观点,页岩巴西劈裂试验结果中抗拉强度存在良好的尺度效应现象,这表明岩石强度尺度效应并不是完全根源于试件与压头之间的端部摩擦作用;岩石强度尺度效应由岩石非均质性和端部摩擦效应共同作用而成,对于小尺度岩样,端部摩擦效应占据主导地位,对于大尺度岩样,非均质性占据主导地位,不同类型的岩石,其岩样尺度的“小”与“大”不同。对岩石强度尺度效应规律和根源的研究有助于提高岩体强度预测的准确性,并为揭示岩石破坏机理提供一定借鉴。     

Grain boundary response of aluminum bicrystal under micro scale laser shock peening

Micro scale laser shock peening (μLSP) is a process in which compressive residual stresses are induced in a material surface to improve fatigue life and wear resistance under cyclic loading. Since the diameter of the laser spot used during the process is the same order of magnitude as grain size, the effects of anisotropy and heterogeneity have to be explicitly taken into account in any model of the process. In this study experimental and numerical studies have been performed in order to investigate the response of an aluminum bicrystal under laser shock peening. The grain boundary is shocked to investigate heterogeneity, and single crystals are shocked to study the effect of anisotropy in the absence of heterogeneity. The orientations of the crystals in the bicrystal as well as the reference single crystals have been chosen such that an approximate plane strain condition is achieved. A finite element model which accounts for the anisotropy, heterogeneity and inertia has also been developed based on single crystal micromechanics. Simulation results are compared with experimental findings. The potential benefit of μLSP as a surface treatment for improvement of fatigue life is also discussed.     

Cotransport of Graphene Oxide Nanoparticles and Kaolinite Colloids in Porous Media

Optimising production from heterogeneous and anisotropic reservoirs challenges the modern hydrocarbon industry because such reservoirs exhibit extreme inter-well variability making them very hard to model. Reasonable reservoir models can be obtained using modern geostatistical techniques, but all of them rely on significant variability in the reservoir only occurring at a scale at or larger than the inter-well spacing. In this paper we take a different, generic approach. We have developed a method for constructing realistic synthetic heterogeneous and anisotropic reservoirs which can be made to represent the reservoir under test. The main physical properties of these synthetic reservoirs are distributed fractally. The models are fully controlled and reproducible and can be extended to model multiple facies reservoir types. This paper shows how the models can be constructed and how they have been tested. Reservoir simulation results of a number of generated 3-D heterogeneous and anisotropic models show that heterogeneity, in terms of only the geometric distribution of reservoir properties, has a little effect on oil production from high and moderate quality reservoirs. However, if the effect of heterogeneity on capillary pressure is taken into account, the effect becomes striking, where varying the heterogeneity of reservoirs properties can lead to a 70 % change in the predicted oil production rate and a significant early shift of water breakthrough time. Hence, it is the heterogeneity consequences that are really substantial if not taken into account. These are very significant uncertainties for a hydrocarbon company if the heterogeneity of their reservoir is not well defined.     

Dispersion Measurements in Highly Heterogeneous Laboratory Scale Porous Media

Laboratory experiments and numerical simulations investigate conservative contaminant transport in a heterogeneous porous medium. The laboratory experiments were performed in cylindrical columns 1m long and 3.5cm inside diameter filled with spherical glass beads. Concentration breakthrough curves are measured at a scale much finer than the size of the heterogeneity. Numerical simulations are based on a random walk in a known constant velocity field. The heterogeneity is a distinct, discontinuous change in the local permeability field. Fluid flow is miscible, flowing in a saturated porous medium. Previous work has shown this to be a very poorly understood phenomenon. The measurements reported here help to better understand how dispersion evolves through and past a heterogeneity.     

Stochastic filtering of multiphase transport phenomena

Whether or not a multiphase mixture is to be considered heterogeneous is a function of scale and hence also a function of the measurement process. Heterogeneities manifested through the measurement process and sampling frequency (extrinsic heterogeneity) are discussed in relationship to the intrinsic heterogeneity associated with the true physical environment. The question of whether or not a multiphase environment may be scaled is partially a function of the scale of the extrinsic heterogeneity of interest. Thus, it is also a function of the instrumental apparatus, which implies that the instrument must also be scaled. We review restrictive scaling constraints in frequency space that an instrument must satisfy for proper scaling of an experiment. Cushman (1984 Water Resour. Res. and 1985 Acta Appl. Math) developed a theory that incorporates the scale and measurement process directly into the multiphase transport equations and which in turn allows for correlation of field properties over scales of motion. We discuss a generalization of this theory which accounts for integral representations of distributional stochastic processes as a function of the scale of the instrument used in the measurement of the field variables. This allows us to solve the stochastic transport equations in an operational setting, i.e., in terms of measurable quantities. Throughout the article examples are presented to illustrate the concepts.     

Single- and Dual-domain Models of Solute Transport in Alluvial Sediments: the Effects of Heterogeneity Structure and Spatial Scale

Fine-scale heterogeneity of alluvial aquifers controls solute transport in groundwater at the scales relevant for practical applications: the architecture of sedimentary structures might create preferential flow paths (PFPs) or hydraulic barriers, which affect the breakthrough curves (BTCs). Objective of this paper was the assessment of the relevance of single- and dual-domain models for different heterogeneity patterns and scale lengths in alluvial sediments. Three case studies have been analysed with a classical single-domain model (SDM) and with three dual-domain models (DDMs): a dual-porosity model (DPorM) and two dual-permeability models (DPerM), which differ for the presence or the absence of solute exchange between the two domains. The first case study includes numerical tracer tests in metre-scale blocks of alluvial sediments; the second is a laboratory experiment of tracer injection in a decimetre-scale column of homogeneous sand; the third is a field tracer test performed at hectometre scale at the Cape Cod site. The relevance of the solute exchange in the DDMs is analysed with the characteristic advection and exchange times and with the Péclet and Damköhler numbers. The SDM is satisfactory for alluvial sediments with unstructured heterogeneity. The uncoupled DPerM is shown to be a better approach than the DPorM in sediments with PFPs; in this case, the coupled DPerM does not improve significantly the results of the uncoupled DPerM. A minor difference between the results of the three DDMs is observed for sediments in which the non-Fickian behaviour is not clearly determined by the presence of PFPs.     

Inertia Effects in High-Rate Flow Through Heterogeneous Porous Media

The paper deals with the effects of large scale permeability–heterogeneity on flows at high velocities through porous media. The media is made of a large number of homogeneous blocks where the flow is assumed to be governed by the Forchheimer equation with a constant inertial coefficient. By assuming the validity of the Forchheimer equation at the large scale, an effective inertial coefficient is deduced from numerical simulations. Different media are investigated: serial-layers, parallel-layers and correlated media. The numerical results show that: (i) for the serial-layers, the effective inertial coefficient is independent of the Reynolds number and decreases when the variance and the mean permeability ratio increases; (ii) for the parallel-layers and the correlated media, the effective inertial coefficient is function of the Reynolds number and increases when the variance and the mean permeability ratio increases. Theoretical relationships are proposed for the inertial coefficient as function of the Reynolds number and the characteristics of the media.     

Size effect on mechanical behavior of random fiber networks

Bonded random fiber networks are heterogeneous on multiple scales. This leads to a pronounced size effect on their mechanical behavior. In this study we quantify the size effect and determine the minimum model size required to eliminate the size effect for given set of system parameters. These include the network density, the fiber length and the fiber bending and axial stiffness. The results may guide the definition of models and the selection of the size of representative volume elements in sequential multiscale models of fiber networks. To underline the origins of the size effect, we characterize the network heterogeneity by analyzing the geometry of the network (density distribution), the strain field and the strain energy distribution. The dependence of the heterogeneity on the scale of observation and system parameters is discussed.     

Laboratory observation of nonlocal dispersion

This work presents the results of a one-dimensional experimental investigation of contaminant transport in heterogeneous porous media. The usual transport equations fail to adequately predict dispersion in such systems, and new theories to account for the distinctions have not yet been examined experimentally. We use a one-dimensional porous media which is heterogeneous on the scale of observation to determine if the phenomena predicted by the new theories are observable.The experimental media are constructed from distinct layers of spherical glass beads packed into cylindrical columns of Lucite. Flow was in the direction perpendicular to the layers. Dispersion was measured by recording the concentration of a chloride tracer as a function of time and position. The scale of measurement was finer than the scale of the heterogeneity. The results show that the mixing between miscible fluids was affected by transitions in the system parameters, before the transitions were encountered by the mixing zone. This newly observed phenomenon has been interpreted as a nonlocal effect, and it begins to verify the new predictive theories.     

Analysis of Displacement Variances in Stochastic Nonuniform Flows by Means of a First Order Analytical Model and Comparison with Monte-Carlo Simulations

Flow, particle displacement and particle arrival time statistics in 2D nonuniform flows, without microdispersion, are studied both theoretically, in the framework of stochastic modelling, and numerically by means of Monte-Carlo simulations.Turning, radial convergent and dipolar flow fields are considered. These three types of flow are numerically investigated in heterogeneous media with different levels of heterogeneity. Monte-Carlo simulations show (1) that the scale separation hypothesis, frequently used in fluid mechanics, is justified for one-point flow statistics; and (2) how displacement variances, and consequently the dispersivities defined as their spatial derivatives, depend on the type and the amplitude of flow nonuniformity: in none of the investigated cases does the assumption of scale separation hold for displacement, except in the turning flow when the spatial scale associated to the nonuniformity is much greater than the correlation scale of transmissivity.The theoretical approach of displacement and arrival time statistics relies on the analysis of particle trajectory. Displacement variances expressions are derived by the perturbation method for each type of flow and for different approximation orders. The proposed expressions of displacement variances are, on the whole, in good agreement with the numerical results. On the other hand, the uniform flow approximation – commonly used for the interpretation of tracer experiments – chosen such as to satisfy the mean arrival time to the pumping well, gives the best prediction of the breakthrough curves.     

Experimental investigation of characteristic length scale in periodic heterogeneous porous media

An experimental investigation of scale-dependent dispersion in periodic heterogeneous porous media was conducted. Models with two-, three- and four-layer periodic heterogeneities were constructed to investigate the effect of heterogeneity size on the scale-dependence of dispersion. Longitudinal dispersion coefficients were determined as a function of column length by measuring the breakthrough of a continuous injection of potassium chloride tracer solution. Chloride ion concentration was monitored by recording the millivolt potential of silver/silver chloride electrodes placed at intervals along the length of the column. In all three models, dispersion appeared to be scale dependent up to a distance of approximately 20–30 times the size of the repeated heterogeneity group (hydraulic unit). Because all three models suggested a similar dependence, it was concluded that a medium with periodic heterogeneity may likely be characterized by the scale of its hydraulic unit.     

Intergranular and intragranular behavior of polycrystalline aggregates.Part 2: Results

The global, intergranular and intragranular responses of a polycrystalline aggregate are investigated. It is shown that the heterogeneity of stress and strain dramatically increases from the global to the local level. Plastic deformation structures develop on a scale larger than the grain. Several types of boundary conditions are applied to polycrystalline aggregates, in order to estimate the importance of the surface effect. The results obtained are presented as contour plots on the cube, and a detailed study is performed to relate the variation obtained with the orientation of the grain and the position in the aggregate.     

应变局部化分析的嵌入强间断多尺度有限元法

固体材料的应变局部化行为是导致结构破坏失效的重要因素之一,开展相关数值模拟分析对于结构安全性评估具有重要意义.然而由于材料的非均质和多尺度特性,采用传统数值方法进行求解时通常需要从最小特征尺度离散求解的结构,这将大幅度增加计算规模和成本.针对这一问题,本文提出了一种基于嵌入强间断模型的多尺度有限元方法.该方法从粗细两个尺度离散求解模型,首先在细尺度单元上引入嵌入强间断模型来描述单元间断特性,所附加的跳跃位移自由度则通过凝聚技术进行消除,从而保持细尺度单元刚度阵维度不变.其次,提出了一种增强多节点粗单元技术,其可根据局部化带与粗单元边界相交情况自适应动态地增加粗节点,新构造的增强数值基函数可以捕捉细尺度间断特性,完成物理信息从细单元到粗单元的准确传递以及宏观响应的快速分析;再次,在细尺度解的计算中,将细尺度解分解为降尺度解与单胞局部摄动解,从而消除弹塑性分析时单胞内部的不平衡力.最后,通过两个典型算例分析,并与完全采用细单元的嵌入有限元结果进行对比,验证了所提出算法的正确性与有效性.     

Probabilistic multiscale models and measurements of self-heating under multiaxial high cycle fatigue

Different approaches have been proposed to link high cycle fatigue properties to thermal measurements under cyclic loadings, usually referred to as “self-heating tests.” This paper focuses on two models whose parameters are tuned by resorting to self-heating tests and then used to predict high cycle fatigue properties. The first model is based upon a yield surface approach to account for stress multiaxiality at a microscopic scale, whereas the second one relies on a probabilistic modelling of microplasticity at the scale of slip-planes.Both model identifications are cost effective, relying mainly on quickly obtained temperature data in self-heating tests. They both describe the influence of the stress heterogeneity, the volume effect and the hydrostatic stress on fatigue limits. The thermal effects and mean fatigue limit predictions are in good agreement with experimental results for in and out-of phase tension-torsion loadings. In the case of fatigue under non-proportional loading paths, the mean fatigue limit prediction error of the critical shear stress approach is three times less than with the yield surface approach.     

An Experimental Method for One Dimensional Permeability Characterization of Heterogeneous Porous Media at the Core Scale

Many reservoir simulator inputs are derived from laboratory experiments. Special core analysis techniques generally assume that core samples are homogeneous. This assumption does not hold for porous media with significant heterogeneities. This paper presents a new method to characterize core scale permeability heterogeneity. The method is validated by both numerical and experimental results. The leading idea consists in injecting a high viscosity miscible fluid into a core sample saturated with a low viscosity fluid. In such conditions, the fluid displacement is expected to be piston-like. We investigate the evolution of the pressure drop as a function of time. A continuous permeability profile is estimated along flow direction from the pressure drop assuming that the core sample is a stack of infinitely thin cross sections perpendicular to flow direction. Thus, we determine a permeability value for each cross section. Numerical and laboratory experiments are carried out to validate the method. Flow simulations are performed for numerical models representing core samples to estimate the pressure drop. The selected models are sequences of plugs with constant permeabilities. In addition, laboratory displacements are conducted for both low permeability and high permeability core samples. To investigate whether there is dispersion inside the porous medium, CT scan measurements are performed during fluid displacement: the location of the front is observed at successive time intervals. The results validate the methodology developed in this paper as long as heterogeneity is one dimensional.     

Thermodynamic Automaton Simulations of Fluid Flow and Diffusion in Porous Media

A thermodynamic automaton model of fluid flow in porous media is presented. The model is a nonrelativistic version of a Lorentz invariant lattice gas model constructed by Udey et al. (1998). In the previous model it was shown that the energy momentum tensor and the relativistic Boltzman equation can be rigorously derived from the collision and propagation rules. In the present paper we demonstrate that this nonrelativistic model can be used to accurately simulate well known results involving single phase flow and diffusion in porous media. The simulation results show that (1) one-phase flow simulations in porous media are consistent with Darcy's law; (2) the apparent diffusion coefficient decreases with a decrease in permeability; (3) small scale heterogeneity does not affect diffusion significantly in the cases considered.     

Upscaling Forchheimer law

We investigate the high velocity flow in heterogeneous porous media. The model is obtained by upscaling the flow at the heterogeneity scale where the Forchheimer law is assumed to be valid. We use the method of multiple scale expansions, which gives rigorously the macroscopic behaviour without any prerequisite on the form of the macroscopic equations. We show that Forchheimer law does not generally survive upscaling. The macroscopic flow law is strongly non-linear and anisotropic. A 2-point Padé approximation of the flow law in the form of a Forchheimer law is given. However, this approximation is generally poor. These results are illustrated in two particular cases: a layered composite porous media and a composite constituted by a square array of circular porous inclusions embedded in a porous matrix. We show that non-linearities are sources of anisotropy.     

Invasion Percolation on Correlated and Elongated Lattices: Implications for the Interpretation of Residual Saturations in Rock Cores

The invasion percolation model is used to investigate the effect of correlated heterogeneity on capillary dominated displacements in porous media. The breakthrough and residual saturations are shown to be strongly influenced by the correlations. Correlated heterogeneity leads to lower residual saturations than those observed in random systems and the scatter commonly observed in laboratory core measurements of the residual saturations can be attributed to the presence of such heterogeneity at the pore scale. Invasion percolation computations on elongated lattices, those with a geometry of L ^d–1 × nL where n denotes the aspect ratio, show that residual saturations for systems with correlated heterogeneity exhibit a strong dependence on aspect ratio. This effect is not considered by experimentalists who advocate the use of long (high aspect ratio) cores in order to overcome end-effects in experiments on shorter cores. A new scaling law is proposed for the residual saturations in elongated systems with correlated heterogeneity, and is confirmed by numerical simulations.