A multi-length-scale theory of the anomalous mixing-length growth for tracer flow in heterogeneous porous media

We develop a multi-length-scale (multifractal) theory for the effect of rock heterogeneity on the growth of the mixing layer of the flow of a passive tracer through porous media. The multifractal exponent of the size of the mixing layer is determined analytically from the statistical properties of a random velocity (permeability) field. The anomalous diffusion of the mixing layer can occur both on finite and on asymptotic length scales.     

A hierarchical model for random walks in random media

We show that the random walk generated by a hierarchical Laplacian in ^d has standard diffusive behavior. Moreover, we show that this behavior is stable under a class of random perturbations that resemble an off-diagonal disordered lattice Laplacian. The density of states and its asymptotic behavior around zero energy are computed: singularities appear in one and two dimensions.     

基于连续时间无规行走模型研究反常扩散

基于连续时间无规行走(CTRW)理论，数值研究了布朗粒子的欠扩散、正常扩散和超扩散三种扩散行为.解决了CTRW模型的跳跃步长和等待时间分布函数的可实现化问题，对Metropolis抽样方法进行了改进以适用于周期势.探讨了布朗马达依靠闪烁棘轮和摇摆棘轮整流反常扩散所获得的定向速度，结果显示，闪烁布朗马达定向流极大值出现在超扩散条件下；摇摆布朗马达定向流最大值出现在弹道扩散条件下.     

基于连续时间无规行走模型研究反常扩散

基于连续时间无规行走(CTRW)理论,数值研究了布朗粒子的欠扩散、正常扩散和超扩散三种扩散行为.解决了CTRW模型的跳跃步长和等待时间分布函数的可实现化问题,对Metropolis抽样方法进行了改进以适用于周期势.探讨了布朗马达依靠闪烁棘轮和摇摆棘轮整流反常扩散所获得的定向速度,结果显示,闪烁布朗马达定向流极大值出现在超扩散条件下;摇摆布朗马达定向流最大值出现在弹道扩散条件下. 关键词： 无规行走 反常扩散 Metropolis抽样 棘轮势     

Distribution functions for fluids in random media

A random medium is considered, composed of identifiable interactive sites or obstacles equilibrated at a high temperature and then quenched rapidly to form a rigid structure, statistically homogeneous on all but molecular length scales. The equilibrium statistical mechanics of a fluid contained inside this quenched medium is discussed. Various particle-particle and particle-obstacle correlation functions, which differ from the corresponding functions for a fully equilibrated binary mixture, are defined through an averaging process over the static ensemble of obstacle configurations and application of topological reduction techniques. The Ornstein-Zernike equations also differ from their equilibrium counterparts.     

The hierarchical random field Ising model

We introduce and solve explicitly a hierarchical approximation to the random field Ising model. This approximation is defined in terms of Peierls' contours. It exhibits a spontaneous magnetization ind>2 and illustrates some of the ideas used in the proof of that result for the real RFIM. Ind=2, there is no spontaneous magnetization, but a very slow decay of correlations. However, we argue that this latter property is an artifact of the approximation. For the real RFIM, we expect exponential decay of correlations for any value of the disorder.     

A new class of long-tailed pausing time densities for the CTRW

We present some asymptotic results for the family of pausing time densities having the asymptotic (t) property(t) [t ln¹⁺(t/T)]^–1. In particular, we show that for this class of pausing time densities the mean-squared displacement r ²(t) is asymptotically proportional to ln(t/T), and the asymptotic distribution of the displacement has a negative exponential form.     

A difference-fractal model for the permeability of fibrous porous media

In this Letter, a difference-fractal model for the permeability of viscous flow through fibrous porous media is proposed. Since fractal objects have well-defined geometric properties, and are discrete and discontinuous, we apply the difference approach to developing the fractal model. The model of non-dimensional permeability is expressed as a function of porosity and fractal dimension. To verify the validity of the proposed model, the predicted permeability values are compared with those of experimental measurements. A good agreement between the prediction of the fractal model and the existing experimental data from the literature is found.     

随机多孔介质逾渗模型渗透率的临界标度性质

研究了一类非零键渗透率满足均匀分布的随机多孔介质逾渗模型-数值计算了该模型系统渗透率在临界点处的标度指数-结果表明该指数并不能看作是普适常数,而与均匀分布的参数有关-这意味着即使非零键渗透率值的概率密度函数满足负一阶矩存在条件,系统渗透率在逾渗临界点处的标度指数仍然依赖于分布函数的具体参数,并不是常数-这一数值结果与Sahimi对此问题的结论不同- 关键词： 逾渗 随机多孔介质 标度指数 渗透率     

Numerical model of two-dimensional heterogeneous combustion in porous media under natural convection or forced filtration

A novel mathematical model and original numerical method for investigating the two-dimensional waves of heterogeneous combustion in porous media are proposed and described in detail. The mathematical model is constructed within the framework of the model of interacting interpenetrating continua and includes equations of state, continuity, momentum conservation and energy for solid and gas phases. Combustion, considered in the paper, is due to the exothermic reaction between fuel in the porous solid medium and oxidiser contained in the gas flowing through the porous object. The original numerical method is based on a combination of explicit and implicit finite-difference schemes. A distinctive feature of the proposed model is that the gas velocity at the open boundaries (inlet and outlet) of the porous object is unknown and has to be found from the solution of the problem, i.e. the flow rate of the gas regulates itself. This approach allows processes to be modelled not only under forced filtration, but also under free convection, when there is no forced gas input in porous objects, which is typical for many natural or anthropogenic disasters (burning of peatlands, coal dumps, landfills, grain elevators). Some two-dimensional time-dependent problems of heterogeneous combustion in porous objects have been solved using the proposed numerical method. It is shown that two-dimensional waves of heterogeneous combustion in porous media can propagate in two modes with different characteristics, as in the case of one-dimensional combustion, but the combustion front can move in a complex manner, and gas dynamics within the porous objects can be complicated. When natural convection takes place, self-sustaining combustion waves can go through the all parts of the object regardless of where an ignition zone was located, so the all combustible material in each part of the object is burned out, in contrast to forced filtration.     

Tortuosity for streamlines in porous media

An analysis of tortuosity for streamlines in porous media is presented by coupling the circle and square models. It is assumed that some particles in porous media do not overlap and that fluid in porous media is incompressible. The relationship between tortuosity and porosity is attained with different configurations by using a statistical method. In addition, the tortuosity fractal dimension is expressed as a function of porosity. Those correlations do not include any empirical constant. The percolation threshold and tortuosity fractal dimension threshold of porous media are also presented as: c = 0.32, D T c = 1.07. The predicted correlations of the tortuosity and the porosity agree well with the existing experimental and simulated results.     

Diffusion of oriented particles in porous media

Diffusion of particles in porous media often shows subdiffusive behavior. Here, we analyze the dynamics of particles exhibiting an orientation. The features we focus on are geometrical restrictions and the dynamical consequences of the interactions between the local surrounding structure and the particle orientation. This interaction can lead to particles getting temporarily stuck in parts of the structure. Modeling this interaction by a particular random walk dynamics on fractal structures we find that the random walk dimension is not affected while the diffusion constant shows a variety of interesting and surprising features.     

A fractional equation for anomalous diffusion in a randomly heterogeneous porous medium

A fractional partial differential equation is derived for the spreading of matter in a saturated porous medium starting from precise hypotheses concerning the medium itself, which is a collection of intertwisted tubes with randomly distributed slopes, filled with quiescent fluid. Examining the fundamental solution of the fractional equation indicates that the second moment is not proportional to time, which is the signature of anomalous diffusion. The equation derived preserves non-negativity and also the total mass of matter.     

Asymptotic solutions of the continuous-time random walk model of diffusion

Asymptotic distributions of the Montroll-Weiss equation for the continuous-time random walk are investigated for long times. It is shown that, for a certain subclass of the hopping waiting time distributions belonging to the domain of attraction of stable distributions, these asymptotic distributions are of stable form. This indicates that the realm of applicability of the diffusion equation is limited. The Montroll-Weiss equation is rederived to include the influence of the initial waiting interval and the role of the stable distributions in physical problems is briefly discussed.     

A stochastic mixed finite element heterogeneous multiscale method for flow in porous media

A computational methodology is developed to efficiently perform uncertainty quantification for fluid transport in porous media in the presence of both stochastic permeability and multiple scales. In order to capture the small scale heterogeneity, a new mixed multiscale finite element method is developed within the framework of the heterogeneous multiscale method (HMM) in the spatial domain. This new method ensures both local and global mass conservation. Starting from a specified covariance function, the stochastic log-permeability is discretized in the stochastic space using a truncated Karhunen–Loève expansion with several random variables. Due to the small correlation length of the covariance function, this often results in a high stochastic dimensionality. Therefore, a newly developed adaptive high dimensional stochastic model representation technique (HDMR) is used in the stochastic space. This results in a set of low stochastic dimensional subproblems which are efficiently solved using the adaptive sparse grid collocation method (ASGC). Numerical examples are presented for both deterministic and stochastic permeability to show the accuracy and efficiency of the developed stochastic multiscale method.     

The propagation of ultrasound in porous media

The equations of Ying and Truell, and Waterman and Truell, describing the propagation of ultrasound in two-phase materials are solved numerically for porous solids, and are found to give unphysical results for high porosity. A new self-consistent theory, which can be solved analytically, is presented and is shown to have reasonable behaviour at high porosity.     

An exponential integrator for advection-dominated reactive transport in heterogeneous porous media

We present an exponential time integrator in conjunction with a finite volume discretisation in space for simulating transport by advection and diffusion including chemical reactions in highly heterogeneous porous media representative of geological reservoirs. These numerical integrators are based on the variation of constants solution and solving the linear system exactly. This is at the expense of computing the exponential of the stiff matrix comprising the finite volume discretisation. Using real Léja points or a Krylov subspace technique compared to standard finite difference-based time integrators. We observe for a variety of example applications that numerical solutions with exponential methods are generally more accurate and require less computational cost. They hence comprise an efficient and accurate method for simulating non-linear advection-dominated transport in geological formations.     

Enhanced anomalous diffusion of sputtered atoms in nanosized pores

Plasma sputtering deposition of platinum catalysts in porous anodic aluminum oxide (AAO) templates is shown to generate an anomalous superdiffusion concentration profile. The growth of an overlayer between the hexagonal array of pores is shown to enhance the diffusion into the pores, leading to a diffusion coefficient having superlinear time dependence. The Pt clusters in the pores have a mean size of 10 nm and almost concentrate on the inner pore surface, and are present up to a depth of 6 μm.     

Computational analysis of the roles of biochemical reactions in anomalous diffusion dynamics

Most biochemical processes in cells are usually modeled by reaction–diffusion(RD) equations. In these RD models,the diffusive process is assumed to be Gaussian. However, a growing number of studies have noted that intracellular diffusion is anomalous at some or all times, which may result from a crowded environment and chemical kinetics. This work aims to computationally study the effects of chemical reactions on the diffusive dynamics of RD systems by using both stochastic and deterministic algorithms. Numerical method to estimate the mean-square displacement(MSD) from a deterministic algorithm is also investigated. Our computational results show that anomalous diffusion can be solely due to chemical reactions. The chemical reactions alone can cause anomalous sub-diffusion in the RD system at some or all times.The time-dependent anomalous diffusion exponent is found to depend on many parameters, including chemical reaction rates, reaction orders, and chemical concentrations.