Monitoring of Pollutant Diffusion into Clay Liners by Electrical Methods

A non-destructive test was carried out on a liner material—sand bentonite mixture (SB) with a continuous concentration diffusion of NaCl electrolyte. The work reported studied the spacio-temporal variation of the electrical conductivity $\sigma ^{*}_{\mathrm{s}}$ (z, t) in a diffusion soil column with different heights. A relationship between the interstitial pore fluid concentration of SB and the electrical conductivity of the solution has been established by mixing and compacting samples of sand bentonite with NaCl electrolytes at different concentrations. Electrical conductivity of compacted specimens was measured with a two-electrode cell. The conductivity measurements were used to quantify the pore fluid concentration and effective diffusion coefficient of SB liners. It is concluded here that the electrical conductivity of compacted specimens depends mainly on the salt concentration in the pore fluid and it could be used to measure ionic movement through liners during diffusion. The experimental diffusion coefficient reached theoretical diffusion coefficient when sample height is equal to 40 cm.     

On Boundary Permeation Through the Walls of Blocked Cerebral Capillaries: A Non-Linear Model

In this paper, we present a mathematical model of a blockage in a cerebral capillary where the first stage of cerebrospinal fluid formation takes place. The cerebral capillary is modelled as a slightly bent cylindrical tube, and the plasma in the tube as a Newtonian fluid. The principles and methods of the “effective area” for studying the penetration of fluid into permeable walls were used to investigate the filtrate’s momentum flux. The non-linear Stokes constitutive equation for incompressible fluid is supplemented by a permeable boundary condition, to prove the existence of a unique weak solution, which depends on the viscosity and the nature of the curvature of the specialized capillary.     

Numerical Simulation of Drying of a Saturated Deformable Porous Media by the Lattice Boltzmann Method

A numerical study is reported here to investigate the drying of saturated deformable porous rectangular plate based on the Darcy–Brinkman extended model. All walls of the plate are maintained to a convective heat flux as well as the top and bottom faces are also subjected to a mass flux. The model for the energy transport is based on the local thermodynamic equilibrium between the fluid and the solid phases. The lattice Boltzmann method is used for solving the governing differential equations system. A comprehensive analysis of the influence of the Poisson’s coefficient, the Young’s modulus and the permeability on macroscopic fields is investigated throughout this work.     

Numerical and experimental study of heat transfer in a tall vertical closed cavity

In this work the numerical and experimental results of heat transfer in a vertical tall closed cavity are presented. The cavity has an aspect ratio of 20, one of the vertical walls receive a constant and uniform heat flux, while the opposite wall is kept at a constant temperature. The remaining walls are assumed adiabatic. The cavity is full of air. The computational fluid dynamics software Fluent 6.3 was used for the simulation and an experimental prototype was built to obtain the heat transfer coefficients. The air temperature and the fluid velocity values are higher when emissivity (ε) is 0.03 (almost pure natural convection). The experimental total heat transfer coefficient increases between 119.9 and 159.9 % when the emissivity of the walls changes from 0.03 to 0.95.     

Homotopy Analysis Method to Walter’s B fluid in a vertical channel with porous wall

In this research the steady three-dimensional flow of a Walter’s B fluid in a vertical channel with porous wall, through which the fluid is injected uniformly into the channel through one side of the channel, is studied analytically using Homotopy Analysis Method (HAM). The channel is assumed to be infinite and uniform. The effects of the elasticity of the fluid on the flow and heat transfer on the walls of the channel are discussed.     

Stress concentration around a hole in a radially inhomogeneous plate

The stress concentration factor around a circular hole in an infinite plate subjected to uniform biaxial tension and pure shear is considered. The plate is made of a functionally graded material where both Young’s modulus and Poisson’s ratio vary in the radial direction. For plane stress conditions, the governing differential equation for the stress function is derived and solved. A general form for the stress concentration factor in case of biaxial tension is presented. Using a Frobenius series solution, the stress concentration factor is calculated for pure shear case. The stress concentration factor for uniaxial tension is then obtained by superposition of these two modes. The effect of nonhomogeneous stiffness and varying Poisson’s ratio upon the stress concentration factors are analyzed. A reasonable approximation in the practical range of Young’s modulus is obtained for the stress concentration factor in pure shear loading.     

On the centrifugal separation of a bulk mixture

The centrifugal separation of a mixture of particles and fluid in an axisymmetric container is examined. The flow consists of three distinct regions—mixture, sediment and purified fluid—with Ekman boundary layers at the interfaces and walls. In the settling process, the mixture and pure fluid acquire retrograde and prograde rotations relative to the tank. This flow pattern, and the shape and locus of the interface which are easily determined, provide another simple means to compare mixture theory and experiment. It is shown that when the Coriolis force is important, the pure fluid layer on the “outwardly” inclined wall is not thin. Moreover the interface between the mixture and the pure fluid is not perpendicular to the centrifugal force. Both features contrast those of the gravitational Boycott effect. As a consequence, there is no obvious enhancement of settling due to geometrical configuration.     

On the propagation of waves through porous solids

This note reexamines Biot's model for the propagation of acoustic waves in a material such as cohensionless sand, infused with a fluid, within the context of mixture theory. Instead of the standard entropy equation that is used in mixture theory, an inequality for the viscous dissipation is employed here due to a conceptual difficulty that one encounters in applying the standard equation to a mixture of sand and a fluid. The wave equations are reformulated by taking the velocity field, instead of the displacement, for the fluid as a primary quantity. By recognizing and thereby exploiting the dependence of the stored energy of the sand on the pore fluid pressure and choosing an appropriate form for the rate of dissipation, a set of governing equations are obtained which are equivalent to those derived by Biot [J. Acoust. Soc. Am. 28(1956) 168, 179; J. Appl. Phys. 33(1962) 1482]. A differential equation for the pore fluid pressure is derived and the effects of drag and virtual mass are dealt with in a unified fashion. The procedure allows us to develop generalizations to Biot's equations in a rational manner.     

Experimental Study on the Pore Structure Fractals and Seepage Characteristics of a Coal Sample Around a Borehole in Coal Seam Water Infusion

In this study, a two-dimensional fully coupled computational model is developed for simulation of proppant settlement in hydro-fractures with the use of the extended finite element framework. The porous domain is governed by the well-known \((\mathbf{u}-p)\) formulation, which consists of the momentum balance equation of the bulk, in conjunction with the momentum balance and continuity equations of the pore fluid. The hydro-fracture inflow is modeled as a 1D flow on the basis of the Darcy law, in which fracture permeability is incorporated by means of the cubic law. Contact constraints are elaborated to eliminate the overlap of fracture edges and the leak-off flow. Proppant settlement is conducted on the basis of Stokes’ law for particle terminal velocity, in which the effects of fracture walls, concentration, viscosity and bridging are incorporated into the model. A fixed-point algorithm is introduced to achieve the optimum combination for the proppant injection. Using the extended finite element method, the strong discontinuity in the displacement field due to crack body, as well as the weak discontinuity in the pressure field due to leakage, is included in the model with the use of the Heaviside and modified level set enrichment functions, respectively. The robustness and versatility of the proposed numerical algorithm in determining the optimum proppant injection is examined through several numerical simulations.     

Radial Viscous Fingering in Case of Poorly Miscible Fluids

The displacement of a viscous fluid from an annular Hele-Shaw cell with a source of finite radius by a less viscous one is investigated. A special case of poorly miscible fluids is considered when corresponding dimensionless criteria—capillary and Peclet numbers—both tend to infinity. Brinkman model which additionally takes into account small viscous forces in a plane of the cell is used to describe the displacement process. Linear analysis shows a stabilizing effect of viscous forces and reveals a geometrical similarity criterion, namely the ratio of the interface’s radius to the gap between the cell’s plates. The displacement patterns, obtained numerically under Brinkman model, are very sensitive to the discovered criterion. The comparison with available experimental data is acceptable.     

单裂缝中携砂液流动规律研究

裂缝中携砂液流动是一种固液两相流,携砂液的运移与支撑剂的铺置是水力压裂裂缝保持导流能力的关键.本文基于FLUENT流体计算软件,采用双流体模型,将颗粒看作拟流体,携砂液按照牛顿流体处理,分析了支撑剂体积分数α_s、阿基米德数Ar、颗粒雷诺数Re以及裂缝入口边界对流动规律的影响.研究结果表明:携砂液在裂缝中的流动过程中,发展成为支撑剂体积分数不同的四个区域,包括砂堤区、颗粒悬浮区、颗粒滚流区和无砂区;支撑剂的沉降程度随着支撑剂体积分数和阿基米德数的增加而增加,而随着雷诺数增加而降低;入口为网眼型时,进入裂缝后过流面积的增加导致流速突降,使得支撑剂更容易在入口处产生堆积,在同一入口流速下,较均匀入口的工况铺砂高度大.     

Hydrodynamics of a submerging drop: Lined structures on the crown surface

Macrophotography and high-speed videofilming are used to investigate the material transfer in a falling drop upon collision with the surface of a fluid at rest. In the experiments the drops of colored water, milk, mineral oil, and seed oil fell in pure or colored water. Emphasis was placed on recording the pattern of the drop material spreading over the surface of the receiving fluid. On the continuous surface of the primary cavity and the crown the drop material is concentrated in the form of thin fibers which form a regular streaky or netlike pattern in which triangular, quadrangular, and pentagonal cells are expressed. The cell rows are ordered in the form of layers on the lateral walls and the bottom of the cavity. The fiber dimensions and the degree of their expressiveness vary in the process of flow evolution. The upper row of structures on the crown surface is formed by vertical fibers.     

Macroscale Properties of Porous Media from a Network Model of Biofilm Processes

We demonstrate how a network model can predict porosity and permeability changes in a porous medium as a result of biofilm buildup in the pore spaces. A biofilm consists of bacteria and extracellular polymeric substances (EPS) bonded together and attached to a surface. In this case, the surface consists of the walls of the porous medium, which we model as a random network of pipes.Our model contains five species. Four of these are bacteria and EPS in both fluid and adsorbed phases. The fifth species is nutrient, which we assume to reside in the fluid phase only. Bacteria and EPS transfer between the adsorbed and fluid phases through adsorption and erosion or sloughing. The adsorbed species influence the effective radii of the pipes in the network, which affect the porosity and permeability.We develop a technique for integrating the coupled system of ordinary and partial differential equations that govern transport of these species in the network. We examine ensemble averages of simulations using different arrays of pipe radii having identical statistics. These averages show how different rate parameters in the biofilm transport processes affect the concentration and permeability profiles.     

Modeling species transport by concentrated brine in aggregated porous media

Basic equations governing the transport of species by concentrated brine flowing through an aggregated porous medium are developed. Some simple examples are solved numerically. The medium is considered to be composed of porous rock aggregates separated by ‘macropores’ through which the brine flows and transport of salt and low-concentration species takes place. The aggregates contain dead-end pores, cracks, and stationary pockets collectively called ‘micropores’. The micropore space does not contribute to the flow, but it serves as a storage for salt and species. Adsorption of fluid species takes place at internal surface of aggregates where it is assumed that a linear equilibrium isotherm describes the process. The effects of high salt concentrations are accounted for in the brine density relation, the viscosity relation, Darcy's and Fick's laws, and the rate of mass transfer between macropores and micropores. Mass balance equations, supplemented by extended forms of Darcy's and Fick's laws, are employed to arrive at two sets of equations. One set consists of seven coupled equations for the salt mass fraction and fluid density in macropores, salt mass fraction in micropores, fluid velocity vector, and the fluid pressure. The other set consists of two coupled equations to be solved for the mass fractions of low-concentration species in micropores and macropores. Based on these equations, a mathematical model called TORISM is developed. Using this model, the potential significance of modifications to Darcy's Law are demonstrated.     

Numerical simulation of mass transfer to micropolar fluid flow past a stenosed artery

A mathematical model of unsteady non‐Newtonian blood flow together with the mass transfer through constricted arteries has been developed. The mass transport refers to the movement of atherogenic molecules, i.e. blood‐borne components, such as low‐density lipoproteins from flowing blood into the arterial walls or vice versa. The flowing blood is represented as the suspension of all erythrocytes assumed to be Eringen's micropolar fluid and the arterial wall is considered to be rigid having cosine‐shaped stenosis in its lumen. The mass transfer to blood is controlled by the convection–diffusion equation. The governing equations of motion accompanied by the appropriate choice of the boundary conditions are solved numerically by Marker and Cell method and the results obtained are checked for numerical stability with the desired degree of accuracy. The quantitative analysis carried out finally includes the respective profiles of the flow‐field and the mass concentration along with their distributions over the entire arterial segment as well. The key factors, such as the wall shear stress and Sherwood number, are also examined for further quantitative insight into the flow and the mass transport phenomena through arterial stenosis. The present results show consistency with several existing results in the literature which substantiate sufficiently to validate the applicability of the model under consideration. Copyright © 2010 John Wiley & Sons, Ltd.     

饱和砂土对地震的动力响应

本文用“工程近似”方法研究了饱和砂土对地震的动力响应。根据饱和砂土对循环荷载动态响应的机理建立了简化模型，给出了描述动态响应的动力方程和描述孔压产生与消散的扩散方程。用Ｎｅｗｍａｒｋ方法与有限差分数值求解这些方程。求得了饱和砂土层对地震的动力响应，包括孔压、剪应力与表面加速度的变化，并对数值结果进行了分析、讨论。     

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.     

Analysis of thermo-mechanical behaviour of a crack using XFEM for Leak-before-Break assessments

The stresses near a crack which has a fluid escaping through it are presented in this paper. The pressure and heat flux, due to the fluid acting on the crack walls, are imposed as boundary conditions in a new finite element tool which has been developed specifically for Leak-before-Break. This special tool uses the extended finite element method to include information about the problem on a sub element level. It is shown to be as accurate as standard finite element models which use very refined meshes, but having the added benefit of being much quicker to implement, and vastly reducing postprocessing. This means that leak rates can be investigated more efficiently. The model is thermo-elastic, and plasticity is accounted for by a correction to the crack opening displacement based on the R6 method. Both crack opening area and peak stresses are shown to decrease when the walls of the crack are hotter than the background plate temperature. The consequences of this for Leak-before-Break assessments are discussed in the paper.     

Low-frequency model of a granular medium saturated with a viscous fluid

A mathematical model of a granular medium saturated with a viscous homogeneous fluid is constructed. The steady-state one-dimensional oscillations of cylindrical granules and an incompressible fluid under the action of a plane sonic wave whose length is significantly greater than the cell dimensions are investigated. The steady-state flow of the medium across the cell cross-section and the mean fluid velocity (Darcy’s law) are determined by means of passages to the limits with respect to the frequency and granule mass. The expressions obtained for the soil permeability coefficient under the action of a gravitational hydraulic head are compared with the representations of other authors.     

Simulation of three-dimensional polymer mould filling processes using a pseudo-concentration method

Mould filling processes, in which a material flow front advances through a mould, are typical examples of moving boundary problems. The moving boundary is accompanied by a moving contact line at the mould walls causing, from a macroscopic modelling viewpoint, a stress singularity. In order to be able to simulate such processes, the moving boundary and moving contact line problem must be overcome. A numerical model for both two- and three-dimensional mould filling simulations has been developed. It employs a pseudo-concentration method in order to avoid elaborate three-dimensional remeshing, and has been implemented in a finite element program. The moving contact line problem has been overcome by employing a Robin boundary condition at the mould walls, which can be turned into a Dirichlet (no-slip) or a Neumann (free-slip) boundary condition depending on the local pseudo-concentration. Simulation results for two-dimensional test cases demonstrate the model's ability to deal with flow phenomena such as fountain flow and flow in bifurcations. The method is by no means limited to two-dimensional flows, as is shown by a pilot simulation for a simple three-dimensional mould. The reverse problem of mould filling is the displacement of a viscous fluid in a tube by a less viscous fluid, which has had considerable attention since the 1960's. Simulation results for this problem are in good agreement with results from the literature. © 1998 John Wiley & Sons, Ltd.