Upscaling,relaxation and reversibility of dispersive flow in stratified porous media

Dispersive tracer released in a unidirectional velocity field belonging to a stratified porous of finite height describes a transition, called relaxation, from a convective dominated behaviour for short times to Fickian behaviour for asymptotic long times. The temporal relaxation state of the tracer is controlled by the transverse mixing term. In most practical applications, the orders of the time and length scales of the relaxation mechanism are such that in an upscaled model of a stratified medium the dispersive flux is in a pre-asymptotic state. Explicit modelling of the relaxation of the dispersive flux in the pre-asymptotic region is required to improve the accuracy. This paper derives a pre-asymptotic one-dimensional upscaled model for the transverse averaged tracer concentration. The model generalises Taylor dispersion (Proc. R. Soc. London 219, 186–203 (1953)) and extends the method of Camacho (Phys. Rev. E 47(2), 1049–1053 (1993a); Phys. Rev. E 48 (1993b)) to dispersion tensors that may vary as function of the transverse direction. In the averaging step, the governing two-dimensional equation is first spectrally decomposed in terms of the eigenfunctions of the transverse mixing term. Next, the resulting modal relaxation equations are combined into an effective relaxation equation for the extended dispersive Taylor flux. Contrary to the one-dimensional Fickian approach, the upscaled model approximates the multi-scale relaxation behaviour as a single scale relaxation process and accounts for the partial reversibility of convective dispersion upon reversal of the flow direction. The upscaled model is evaluated against the original two-dimensional model by means of moment analysis. The longitudinal tracer variance predicted by our model is quantitatively correct in the short and long time limits and is qualitatively correct for intermediate times.     

Analysis of the Time Behaviour of a Diffusive Transport in a Stratified Medium

The paper presents a study of the diffusive transport of passive solute plumes in a two-dimensional non-homogeneous depth stratified flow domain. All the properties of the process are expressed by depth dependent deterministic functions. The method of moments, combined with the method of Green functions are chosen to determine the relevant characteristics of the flow (mass, center of mass, variance, etc.) used to describe the behaviour of the transient motion. General relationships for the n-order concentration moments are proved. Further, it is derived that the transient motion defined by time-dependent parameters tends asymptotically at large time to a stable regime whose characteristics are determined. Consequently, under certain hypotheses, an equivalence between the mean original process and a Fickian diffusive transport at large time may be established. The time required by the process to reach its asymptotic behaviour is also calculated.     

Prediction of interwell tracer flow behaviour in heterogeneous reservoirs

A new numerical approach has been developed for predicting the interwell tracer flow behaviour in heterogeneous porous media typical of water and oil reservoirs. This approach uses a mixed finite-element method with triangular elements to predict pressure and velocity fields, and a novel random walk model to simulate the tracer transport through the reservoir, and to perdict the concentration response at the production well. The mixed finite-element method solves the pressure and velocity simultaneously imposing suitable boundary conditions for both pressure and velocity, which allows the solution of the tracer velocity to be more accurate and to conserve mass more precisely than a standard finite-element method. The random walk model can reflect the tracer flow behaviour directly by tracking the movement of particles representing the tracer input volume.The technique has been validated by comparing the predicted results with analytical solutions of tracer concentration response for a homogeneous five-spot pattern, and with published experimentally observed tracer fronts at breakthrough for homogeneous and three heterogeneous cases of five-spot pattern. Good agreement has been achieved for all cases. The model presented in this paper is general, and can therefore be applied to drive patterns other than the five-spot pattern, and for different types of heterogeneities; it can also include effects such as longitudinal and transverse dispersion and adsorption.     

Non-Fickian Transport in Transparent Replicas of Rough-Walled Rock Fractures

We present an experimental investigation and modeling analysis of tracer transport in two transparent fracture replicas. The original fractures used in this work are a Vosges sandstone sample with nominal dimensions approximately 26 cm long and 15 cm wide, and a granite sample with nominal dimensions approximately 33 cm long and 15.5 cm wide. The aperture map and physical characteristics of the fractures reveal that the aperture map of the granite fracture has a higher spatial variability than the Vosges sandstone one. A conservative methylene blue aqueous solution was injected uniformly along the fracture inlets, and exited through free outlet boundaries. A series of images was recorded at known time intervals during each experiment. Breakthrough curves were subsequently determined at the fracture outlets and at different distances, using an image processing based on the attenuation law of Beer–Lambert. These curves were then interpreted using a stratified medium model that incorporates a permeability distribution to account for the fracture heterogeneity, and a continuous time random walk (CTRW) model, as well as the classical advection–dispersion equation (ADE). The stratified model provides generally satisfactory matches to the data, while the CTRW model captures the full evolution of the long tailing displayed by the breakthrough curves. The transport behavior is found to be non-Fickian, so that the ADE is not applicable. In both stratified and CTRW models, parameter values related to the aperture field spatial variability indicate that the granite fracture is more heterogeneous than the Vosges sandstone fracture.     

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.     

连续小波变换离散化的爆炸振动特征分析

应用连续小波变换的离散化关系,针对一个改进的L-P（littlewood-paley）小波基函数,给出了一种实现频率完全分割的时频特征分析方法,并对爆炸振动时频特征进行了研究。80 kg TNT地面爆炸时地面垂向振动速度的时间能量密度分布情况表明,在质点振动峰值速度到达时刻爆炸振动的频率范围比较宽,而其他时刻的振动频率相对较为集中,时频能量分布的峰值正好对应于爆炸振动速度的峰值到达时间。基于小波变换的爆炸振动频谱特征与Fourier变换的结果具有良好的一致性。此外,还给出了利用小波变换结果建立爆炸振动随机演变理论模型的基本方法。     

Advanced statistics to improve the physical interpretation of atomization processes

This paper reports an analysis of the physics of atomization processes using advanced statistical tools. Namely, finite mixtures of probability density functions, which best fitting is found using a Bayesian approach based on a Markov chain Monte Carlo (MCMC) algorithm. This approach takes into account eventual multimodality and heterogeneities in drop size distributions. Therefore, it provides information about the complete probability density function of multimodal drop size distributions and allows the identification of subgroups in the heterogeneous data. This allows improving the physical interpretation of atomization processes. Moreover, it also overcomes the limitations induced by analyzing the spray droplets characteristics through moments alone, particularly, the hindering of different natures of droplet formation. Finally, the method is applied to physically interpret a case-study based on multijet atomization processes.     

Sorbing and Non-Sorbing Solute Migration in Rough Fractures with a Multi-Species LGA Model: Dispersion Dependence on Retardation and Roughness

A multi-species Lattice-Gas cellular automaton model was applied to the study of the migration of sorbing and non-sorbing tracers in 2D smooth fractures and in a series of increasing roughness fractures. A tenfold increase of the dispersion of the non-sorbing tracer was calculated in the highest roughness case compared to the smooth fracture. Up to a threefold increase of the dispersion of the sorbing tracers was calculated compared to the non-sorbing tracer. These enhanced dispersions were found to be of a Fickian form and were interpreted in terms of the classical Taylor–Aris dispersion. The effects of roughness and retardation over the increase of dispersion were identified and quantified through a semi-empirical relation. These effects were found additive and independent.     

基于首次穿越时间概率密度的时变可靠性分析

基于时变可靠性性能函数首次穿越时间的概率密度F-PTPD(first-passage time probability density)模型,提出了一种求解机械产品全寿命周期可靠性累计概率密度函数的方法(简称F-PTPD方法),为产品在全寿命周期内可靠性分析和设计提供了工具。首先,采用稀疏网络随机配置方法进行时变可靠性性能函数均值的估计,选取性能函数均值为零的第一个时刻点作为首次穿越点;其次,基于均值的首次穿越点将时变可靠性性能函数进行二阶泰勒展开,利用二次函数的性质求解性能函数首次穿越时间关于随机输入变量的函数;再次,针对首次穿越点函数,采用稀疏网络随机配置方法进行首次穿越时间的四阶原点矩估计;最后,基于四阶原点矩利用最大熵概率密度函数估计方法,推导出首次穿越点的概率分布,获得产品寿命周期内时变可靠性的累计概率密度函数。本文方法可获得产品整个寿命周期失效概率的变化趋势,极大地提高了评估效率,对复杂产品的可靠性评估设计有一定的工程指导意义。     

A Solute Transport in Stratified Media

Two-dimensional and steady solute transport in a stratified porous formation is analysed under assumption that the effect of pore-scale dispersion is negligible. The longitudinal dispersion produced as a result of the vertical variation of hydraulic conductivity is analysed by averaging the variability of a solute flux concentration and conductivity. The evolution of the solute flux concentration is expressed with respect to the correlated variable, that is the travel (arrival) time at a fixed location and the averaging procedure is constructed to satisfy the boundary condition where the inlet concentration is a known function of time. In such a statement, a velocity-averaged solute flux concentration is described by a conventional dispersion model (CDM) with a dispersion coefficient which is a function of the arrival time. It is demonstrated that such CDM satisfies the assumption that hydraulic conductivity of the layers is gamma distributed with the parameter of distribution which is chosen to represent a reasonable value of the field scale solute dispersion. The overall behaviour of the model is illustrated by several examples of two-dimensional mass transport.     

Laplace-Transform Based Inversion Method for Fractional Dispersion

Partial differential equations with memory are challenging models for mass transport in porous media where fluid and tracer may be stored by the solid matrix, and then released. Moreover, integral transforms (generalizing time moments) of solutions to such models are linked to the corresponding transport parameters. Inverting that link provides a method to determine model parameters on the basis of solutions. It is checked using numerically generated profiles before passing to experimental data.     

Connectivity in Vuggy Carbonates, New Experimental Methods and Applications

The length and spatial distribution of the touching-vugs channels affect the degree of permeability variations and is the main contributor to heterogeneity in vuggy carbonates. Hence, this article focuses on vug connectivity characterization and its impact on fluid flow. A whole core sample was scanned by X-ray computed tomography (CT). Image segmentation was used to obtain a binarized three-dimensional (3D) model of the vuggy pore space. Analysis of the binarized 3D model is used to calculate the correlation function and correlation length for the vuggy pore space. Connectivity analysis of the binarized 3D model shows that 79% of the vugs connected network spanning along the sample. The remaining 21% vug porosity exists in a large number of isolated vugs. The correlation length for the connected vug network is found to be larger than for vugs in general. NMR T ₂ measurements at increasing capillary pressure is tested on the vuggy material and used to investigate the amount of connected- and isolated vugs. The results verify the large fraction of connected vugs. Application of NMR T ₂ measurements in combination with capillary pressure experiments can also reveal matrix properties that play an important role in recovery processes. The transition between non-Fickian and Fickian regimes for tracer/solute transport is studied by laboratory experiments performed at various sample lengths, from cm to m scale. For the largest sample measured in our experiments show that effluent concentration curve conform to the CDE solution, suggesting that the Fickian regime has been established.     

Impact of the volume of rooms on shock wave propagation within a multi-chamber system

The behavior of a shock wave generated by a hemispherical gaseous charge and propagating within a confined multi-chamber system is analyzed through the evolution of some of the shock parameters (maximum overpressure and positive impulse). The influence of a variation in the volume of the rooms on the pressure history inside the building is also studied. Several small-scale experiments have been carried out using an adjustable model representative of a pyrotechnic workshop. The experimental results show that the pressure histories are very complex. Yet, using a global approach, we were able to link the evolution of the arrival time of the shock wave within the building with the reference obtained in the free field. New parameters were developed to best fit the experimental maximal overpressure in the cells and in the corridor leading to two predictive laws used to estimate the maximal overpressure in the model.     

Distribution-enhanced homogenization framework and model for heterogeneous elasto-plastic problems

Multi-scale computational models offer tractable means to simulate sufficiently large spatial domains comprised of heterogeneous materials by resolving material behavior at different scales and communicating across these scales. Within the framework of computational multi-scale analyses, hierarchical models enable unidirectional transfer of information from lower to higher scales, usually in the form of effective material properties. Determining explicit forms for the macroscale constitutive relations for complex microstructures and nonlinear processes generally requires numerical homogenization of the microscopic response. Conventional low-order homogenization uses results of simulations of representative microstructural domains to construct appropriate expressions for effective macroscale constitutive parameters written as a function of the microstructural characterization. This paper proposes an alternative novel approach, introduced as the distribution-enhanced homogenization framework or DEHF, in which the macroscale constitutive relations are formulated in a series expansion based on the microscale constitutive relations and moments of arbitrary order of the microscale field variables. The framework does not make any a priori assumption on the macroscale constitutive behavior being represented by a homogeneous effective medium theory. Instead, the evolution of macroscale variables is governed by the moments of microscale distributions of evolving field variables. This approach demonstrates excellent accuracy in representing the microscale fields through their distributions. An approximate characterization of the microscale heterogeneity is accounted for explicitly in the macroscale constitutive behavior. Increasing the order of this approximation results in increased fidelity of the macroscale approximation of the microscale constitutive behavior. By including higher-order moments of the microscale fields in the macroscale problem, micromechanical analyses do not require boundary conditions to ensure satisfaction of the original form of Hill's lemma. A few examples are presented in this paper, in which the macroscale DEHF model is shown to capture the microscale response of the material without re-parametrization of the microscale constitutive relations.     

Development of Gas-Particle Euler-Euler LES Approach: A Priori Analysis of Particle Sub-Grid Models in Homogeneous Isotropic Turbulence

A new large eddy simulation (LES) approach for particle-laden turbulent flows in the framework of the Eulerian formalism for inertial particle statistical modelling is developed. Local instantaneous Eulerian equations for the particle cloud are first written using the mesoscopic Eulerian formalism (MEF) proposed by Février et al. (J Fluid Mech 533:1–46, 2005), which accounts for the contribution of an uncorrelated velocity component for inertial particles with relaxation time larger than the Kolmogorov time scale. Second, particle LES equations are obtained by volume filtering the mesoscopic Eulerian ones. In such an approach, the particulate flow at larger scales than the filter width is recovered while sub-grid effects need to be modelled. Particle eddy-viscosity, scale similarity and mixed sub-grid stress (SGS) models derived from fluid compressible turbulence SGS models are presented. Evaluation of such models is performed using three sets of particle Lagrangian results computed from discrete particle simulation (DPS) coupled with fluid direct numerical simulation (DNS) of homogeneous isotropic decaying turbulence. The two phase flow regime corresponds to the dilute one where two-way coupling and inter-particle collisions are not considered. The different particle Stokes number (based on Kolmogorov time scale) are initially equal to 1, 2.2 and 5.1. The mesoscopic field properties are analysed in detail by considering the particle velocity probability function (PDF), correlated velocity power spectra and random uncorrelated velocity moments. The mesoscopic fields measured from DPS+DNS are then filtered to obtain large scale fields. A priori evaluation of particle sub-grid stress models gives comparable agreement than for fluid compressible turbulence models. It has been found that the standard Smagorinsky eddy-viscosity model exhibits the smaller correlation coefficients, the scale similarity model shows very good correlation coefficient but strongly underestimates the sub-grid dissipation and the mixed model is on the whole superior to pure eddy-viscosity model.     

Correspondence Between One- and Two-Equation Models for Solute Transport in Two-Region Heterogeneous Porous Media

In this work, we study the transient behavior of homogenized models for solute transport in two-region porous media. We focus on the following three models: (1) a time non-local, two-equation model (2eq-nlt). This model does not rely on time constraints and, therefore, is particularly useful in the short-time regime, when the timescale of interest (t) is smaller than the characteristic time (?? ₁) for the relaxation of the effective macroscale parameters (i.e., when t?????? ₁); (2) a time local, two-equation model (2eq). This model can be adopted when (t) is significantly larger than (?? ₁) (i.e., when ${t\gg\tau_{1}}$ ); and (3) a one-equation, time-asymptotic formulation (1eq ^??). This model can be adopted when (t) is significantly larger than the timescale (?? ₂) associated with exchange processes between the two regions (i.e., when ${t\gg\tau_{2}}$ ). In order to obtain insight into this transient behavior, we combine a theoretical approach based on the analysis of spatial moments with numerical and analytical results in several simple cases. The main result of this paper is to show that there is only a weak asymptotic convergence of the solution of (2eq) towards the solution of (1eq ^??) in terms of standardized moments but, interestingly, not in terms of centered moments. The physical interpretation of this result is that deviations from the Fickian situation persist in the limit of long times but that the spreading of the solute is eventually dominating these higher order effects.     

An Approach to Transport in Heterogeneous Porous Media Using the Truncated Temporal Moment Equations: Theory and Numerical Validation

In the last decade, the characterization of transport in porous media has benefited largely from numerical advances in applied mathematics and from the increasing power of computers. However, the resolution of a transport problem often remains cumbersome, mostly because of the time-dependence of the equations and the numerical stability constraints imposed by their discretization. To avoid these difficulties, another approach is proposed based on the calculation of the temporal moments of a curve of concentration versus time. The transformation into the Laplace domain of the transport equations makes it possible to develop partial derivative equations for the calculation of complete moments or truncated moments between two finite times, and for any point of a bounded domain. The temporal moment equations are stationary equations, independent of time, and with weaker constraints on their stability and diffusion errors compared to the classical advection–dispersion equation, even with simple discrete numerical schemes. Following the complete theoretical development of these equations, they are compared firstly with analytical solutions for simple cases of transport and secondly with a well-performing transport model for advective–dispersive transport in a heterogeneous medium with rate-limited mass transfer between the free water and an immobile phase. Temporal moment equations have a common parametrization with transport equations in terms of their parameters and their spatial distribution on a grid of discretization. Therefore, they can be used to replace the transport equations and thus accelerate the achievement of studies in which a large number of simulations must be carried out, such as the inverse problem conditioned with transport data or for forecasting pollution hazards.     

Analysis of high-order velocity moments in a strained channel flow

In the current study, model expressions for fifth-order velocity moments obtained from the truncated Gram-Charlier series expansions model for a turbulent flow field probability density function are validated using data from direct numerical simulation (DNS) of a planar turbulent flow in a strained channel. Simplicity of the model expressions, the lack of unknown coefficients, and their applicability to non-Gaussian turbulent flows make this approach attractive to use for closing turbulent models based on the Reynolds-averaged Navier-Stokes equations. The study confirms validity of the model expressions. It also shows that the imposed flow deformation improves an agreement between the model and DNS profiles for the fifth-order moments in the flow buffer zone including when the flow reverses its direction. The study reveals sensitivity of particularly odd velocity moments to the grid resolution. A new length scale is proposed as a criterion for the grid generation near the wall and in the other flow areas dominated by high mean velocity gradients when higher-order statistics have to be collected from DNS.     

A fast non‐Fickian particle‐tracking diffusion simulator and the effect of shear on the pollutant diffusion process

This paper presents a fast method for the generation of non‐Fickian particle paths within a particle‐tracking pollutant diffusion model based on a Fourier spectral representation of fractional Brownian motion (fBm), a generalization of ordinary Brownian motion. Correlated diffusive components in a particle‐tracking algorithm are modelled using fBm increments that have long‐range correlations over numerous spatial and/or temporal scales; hence producing non‐Fickian diffusion. A fast algorithm to generate fBm and its increment by using its power spectral density S(f) in a fast Fourier transform algorithm is given. A general equation for the scaling of fBm within a velocity flow field with simple linear shear is presented. An initial numerical study of the nature of fBm shear dispersion has been conducted by incorporating fBm increments into a non‐Fickian particle‐tracking algorithm. It is shown that the effect of simple (i.e. linear) shear on the diffusion process is to produce enhanced diffusive phenomena with the longitudinal spreading of the plume scaling with exponent ∼1+H, where H is the Hurst exponent used to describe fBm. Finally, a more complex shear zone at the entrance of a coastal bay model is investigated using both a traditional particle‐tracking method and the fBm‐based method. Copyright © 2000 John Wiley & Sons, Ltd.