Modeling of contaminant transport resulting from dissolution of nonaqueous phase liquid pools in saturated porous media

A mathematical model for transient contaminant transport resulting from the dissolution of a single component nonaqueous phase liquid (NAPL) pool in two-dimensional, saturated, homogeneous porous media was developed. An analytical solution was derived for a semi-infinite medium under local equilibrium conditions accounting for solvent decay. The solution was obtained by taking Laplace transforms to the equations with respect to time and Fourier transforms with respect to the longitudinal spatial coordinate. The analytical solution is given in terms of a single integral which is easily determined by numerical integration techniques. The model is applicable to both denser and lighter than water NAPL pools. The model successfully simulated responses of a 1,1,2-trichloroethane (TCA) pool at the bottom of a two-dimensional porous medium under controlled laboratory conditions.Notation a,a ₁ defined in (45a) and (45b), respectively - b defined in (45c) - b vector of true model parameters (n×1) - vector of estimated model parameters (n×1) - c liquid phase solute concentration (solute mass/liquid volume), M/L³ - c _s aqueous saturation concentration (solubility), M/L³ - C dimensionless liquid phase solute concentration, equal toc/c _s - molecular diffusion coefficient, L²/t - ^e effective molecular diffusion coefficient, equal to / ^*, L²/t - D _x longitudinal hydrodynamic dispersion coefficient, L²/t - D _z hydrodynamic dispersion coefficient in the vertical direction, L²/t - e random vector with zero mean (m×1) - erf[x] error function, equal to (2/ ^1/2) - f vector of fitting errors or residuals (m×1) - Fourier operator - _-1 Fourier inverse operator - g vector of model simulated data (m×1) - k mass transfer coefficient, L/t - average mass transfer coefficient, L/t - K _d partition or distribution coefficient (liquid volume/solids mass), L³/M - pool length, L - _o distance between the pool and the origin of the specified Cartesian coordinate system, L - Laplace operator - _-1 Laplace inverse operator - m number of observations - M Laplace/Fourier function defined in (38) - n number of model parameters - N Laplace/Fourier function defined in (39) - p defined in (46) - Pe _x Péclet number, equal toU _x /D _x - Pe _z Péclet number, equal toU _x /D _z - q defined in (47) - R retardation factor - s Laplace transform variable - S objective function - Sh local Sherwood number, equal tok/ _e - Sh _o overall Sherwood number, equal to l/ _e - t time,t - T dimensionless time, equal toU _x t/ - u dummy integration variable - u vector of independent variables - U _x average interstitial velocity, L/t - x spatial coordinate in the longitudinal direction, L - X dimensionless longitudinal length, equal to (x–)/ - y vector of observed data (m×1) - z spatial coordinate in the vertical direction, L - Z dimensionless vertical length, equal toz/ - Fourier transform variable - defined in (37) - defined in (50) - porosity (liquid volume/aquifer volume), L³/L³ - defined in (52a) and (52b), respectively - decay coefficient, t^–1 - dimensionless decay coefficient, equal to /U _x - bulk density of the solid matrix (solids mass/aquifer volume), M/L³ - dummy integration variable - ^* tortuosity     

Fluid-solid coupling mathematical model of contaminant transport in unsaturated zone and its asymptotical solution

IntroductionThetransportofcontaminantsinunsaturatedzonehascausedmuchattention .Inearly1960s,contaminationproblemsofsoilandgroundwaterhadbeenstudiedathomeandabroad[1].Andinrecentyears ,thetransformationandtransportationofcontaminantshavebeendeeplystudiedinthefieldsofhydrogeology ,petroleumengineering ,environmentalengineeringandsoon[2 ,3].Somecontaminanttransportmodelshavebeenpresentedsofar.Forexample ,Paker[4 ]etal.presentedaconstitutivemodelgoverningparametersofwater,gasandcontaminantswhenth…     

Peristaltic transport and heat transfer of a MHD Newtonian fluid with variable viscosity

The influence of temperature‐dependent viscosity and magnetic field on the peristaltic flow of an incompressible, viscous Newtonian fluid is investigated. The governing equations are derived under the assumptions of long wavelength approximation. A regular perturbation expansion method is used to obtain the analytical solutions for the velocity and temperature fields. The expressions for the pressure rise, friction force and the relation between the flow rate and pressure gradient are obtain. In addition to analytical solutions, numerical results are also computed and compared with the analytical results with good agreement. The results are plotted for different values of variable viscosity parameter β, Hartmann number M, and amplitude ratio ?. It is found that the pressure rise decreases as the viscosity parameter β increases and it increases as the Hartmann number M increases. Finally, the maximum pressure rise (σ=0) increases as M increases and β decreases. Copyright © 2009 John Wiley & Sons, Ltd.     

送风方式对飞机座舱内污染物传播影响研究

采用数值模拟方法研究了波音737-200座舱模型内的气流场以及污染物传播过程,并通过实验数据加以验证;分析了天花板加侧壁送风(混合送风)、侧壁送风以及天花板送风三种送风方式下座舱内的速度场、涡量场、污染物浓度场和空气龄,并且解释了舱内流场结构对污染物传播过程的影响。研究结果表明:不同送风方式下飞机座舱内污染物的传播过程差异明显,这是由流场结构特征的差异造成的,特别是在小尺度范围内,涡结构十分复杂,严重影响污染物的传播规律;在聚集过程中,污染物更容易在漩涡范围内聚集,而变形主导的气流运动会阻碍污染物向更大范围传播;在排除过程中,天花板送风形式下污染物的排除效率最快,混合送风次之,侧壁送风最慢。这对于防控新冠肺炎是有参考价值的。     

Transport of Neutrally Buoyant and Dense Variably Sized Colloids in a Two-Dimensional Fracture with Anisotropic Aperture

The transport of monodisperse as well as polydisperse colloid suspensions in a two-dimensional, water saturated fracture with spatially variable and anisotropic aperture is investigated with a particle tracking model. Both neutrally buoyant and dense colloid suspensions are considered. Although flow and transport in fractured subsurface formations have been studied extensively by numerous investigators, the transport of dense, polydisperse colloid suspensions in a fracture with spatially variable and anisotropic aperture has not been previously explored. Simulated snapshots and breakthrough curves of ensemble averages of several realizations of a log-normally distributed aperture field show that polydisperse colloids exhibit greater spreading than monodisperse colloids, and dense colloids show greater retardation than neutrally buoyant colloids. Moreover, it is demonstrated that aperture anisotropy oriented along the flow direction substantially increases colloid spreading; whereas, aperture anisotropy oriented transverse to the flow direction retards colloid movement.     

Simulation of proppant transport at intersection of hydraulic fracture and natural fracture of wellbores using CFD-DEM

Proppants transport is an advanced technique to improve the hydraulic fracture phenomenon, in order to promote the versatility of gas/oil reservoirs. A numerical simulation of proppants transport at both hydraulic fracture (HF) and natural fracture (NF) intersection is performed to provide a better understanding of key factors which cause, or contribute to proppants transport in HF–NF intersection. Computational fluid dynamics (CFD) in association with discrete element method (DEM) is used to model the complex interactions between proppant particles, host fluid medium and fractured walls. The effect of non-spherical geometry of particles is considered in this model, using the multi-sphere method. All interaction forces between fluid flow and particles are considered in the computational model. Moreover, the interactions of particle–particle and particle–wall are taken into account via Hertz–Mindlin model. The results of the CFD-DEM simulations are compared to the experimental data. It is found that the CFD-DEM simulation is capable of predicting proppant transport and deposition quality at intersections which are in agreement with experimental data. The results indicate that the HF–NF intersection type, fluid velocity and NF aperture affect the quality of blockage occurrence, presenting a new index, called the blockage coefficient which indicates the severity of the blockage.     

Role of Molecular Diffusion in Contaminant Migration and Recovery in an Alluvial Aquifer System

Highly-resolved simulations and flow and transport in an alluvial system at the Lawrence Livermore National Laboratory (LLNL) site explore the role of diffusion in the migration and recovery of a conservative solute. Heterogeneity is resolved to the hydrofacies scale with a discretization of 10.0, 5.0 and 0.5m in the strike, dip and vertical directions of the alluvial-fan system. Transport simulations rely on recently developed random-walk techniques that accurately account for local dispersion processes at interfaces between materials with contrasting hydraulic and transport properties. Solute migration and recovery by pump and treat are shown to be highly sensitive to magnitude of effective diffusion coefficient. Further, transport appears significantly more sensitive to the diffusion coefficient than to local-scale dispersion processes represented by a dispersivity coefficient. Predicted hold back of solute mass near source locations during ambient migration and pump-and-treat remediation is consistent with observations at LLNL, and reminiscent of observations at the MADE site of Columbus Air Force Base, Mississippi. Results confirm the important role of diffusion in low-conductivity materials and, consequently, its impact on efficacy of pump-and-treat and other remedial technologies. In a typical alluvial system on a decadal time scale this process is, in part, fundamentally nonreversible because the average thickness of low-K hydrofacies is considerably greater than the mean-square length of penetration of the solute plume.     

Modeling Microbial Transport and Biodegradation in a Dual-porosity System

A mathematical model describing microbial transport and growth in a heterogeneous aquifer domain, composed of overlapping subdomains of high-permeability and low-permeability materials, is developed. Each material is conceptually visualized as a continuum which occupies the entire considered spatial aquifer domain. Based on the assumption that advection in the low-permeability domain is negligible, the mathematical model is solved by using a publically available reactive transport code. The importance of modeling microbial transport and growth in such a dual-porosity system is demonstrated through a hypothetical case study.     

Modeling fluid and heat transport in the reactive, porous bed of downdraft (biomass) gasifier

A fluid flow and heat transfer model has been developed for the reactive, porous bed of the biomass gasifier to simulate pressure drop, temperature profile in the bed and flow rates. The conservation equations, momentum equation and energy equation are used to describe fluid and heat transport in porous gasifier bed. The model accounted for drag at wall, and the effect of radial as well as axial variation in bed porosity to predict pressure drop in bed. Heat transfer has been modeled using effective thermal conductivity approach. Model predictions are validated against the experiments, while effective thermal conductivity values are tested qualitatively using models available in literature. Parametric analysis has been carried out to investigate the effect of various parameters on bed temperature profile and pressure drop through the gasifier. The temperature profile is found to be very sensitive to gas flow rate, and heat generation in oxidation zone, while high bed temperature, gas flow rate and the reduction in feedstock particle size are found to cause a marked increase in pressure drop through the gasifier. The temperatures of the down stream zones are more sensitive to any change in heat generation in the bed as compared to upstream zone. Author recommends that the size of preheating zone may be extended up to pyrolysis zone in order to enhance preheating of input air, while thermal insulation should not be less than 15 cm.     

Refinement of the plastic-zone boundary in the vicinity of a crack tip for the quasiviscous and viscous types of fracture

A refined solution of the elastoplastic problem of an insulated mode I crack in a thin plate of reasonably large dimensions is obtained. Estimates of the plastic zone in the vicinity of the crack tip are given for quasiviscous and viscous types of fracture.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 1, pp. 126–132, January–February, 2005     

Unsteady hydromagnetic flow of a reactive variable viscosity third‐grade fluid in a channel with convective cooling

This paper examines the combined effects of a transverse magnetic field and variable viscosity on unsteady flow of a reactive third‐grade electrically conducting fluid and heat transfer in a channel with convective cooling at the surface. It is assumed that the fluid has small electrical conductivity and the electromagnetic force produced is very small. The coupled nonlinear partial differential equations governing the problem are derived and solved numerically using a semi‐implicit finite‐difference scheme. Both numerical and graphical results are presented and physical aspects of the problem are discussed with respect to various parameters embedded in the system. It is in general noted that those parameters that increase/decrase one flow quantity (velocity or temperature) also lead to the increase/decrease respectively of the other quantity. Copyright © 2011 John Wiley & Sons, Ltd.     

Thermodynamic driving force in ferroelectric crystals with a rank-2 laminated domain pattern, and a study of enhanced electrostriction

The thermodynamic driving force for domain growth in a rank-2 laminated ferroelectric crystal is derived in this article, and we used it, together with a homogenization theory, to study the issue of enhanced electrostrictive actuation recently reported by Burcsu et al. [2004. Large electrostrictive actuation of barium titanate single crystals. J. Mech. Phys. Solids 52, 823-846]. We derived this force from the reduction of Gibbs free energy with respect to the increase of domain concentration. It is shown that both the free energy and the thermodynamic force consist of three parts: the first arises from the difference in M₀ and M₁, the linear electromechanical compliances of the parent and product domains, respectively, at a given level of applied stress and electric field, the second stems from the electromechanical work associated with the change of spontaneous strain and spontaneous polarization during domain switch, and the third from the internal energy due to the distribution of polarizations strain and electric polarization inside the crystal. We prove that the first term is substantially lower than the second one, and the third one is identically zero with compatible domain pattern. The second one is, however, not exactly equal to the commonly written sum of the products of stress with strain, and electric field with polarization during switch, unless both domains have identical moduli in the common global axes. We also show that, with compatible domain patterns and when M₁=M₀, this driving force is identical to Eshelby's driving force acting on a flat interface due to the jump of energy-momentum tensor. Applications of the theory to a BaTiO₃ crystal subjected to a fixed axial compression and decreasing electric field from the [0 0 1] state reveal that the crystal undergoes a three-stage switching process: (i) the 0→90° switch to form a rank-1 laminate, (ii) the 0→180° switch inside the 0° domain to form a laminate I with a concurrent 90°→−90° switch inside the 90° domain to form laminate II, creating a rank-2-laminated domain pattern, and (iii) finally the 90→180° switch. It is the exchange of stability between the 0, 90°, and 180° domains under compression and electric field that is the origin of the enhanced actuation. We illustrate these intrinsic features by showing the evolution of these domains, and demonstrate how the reported large actuation strain can be attained with a rank-2 laminate.