Analytical Solution for Multi-Species Contaminant Transport Subject to Sequential First-Order Decay Reactions in Finite Media

Transport equations governing the movement of multiple solutes undergoing sequential first-order decay reactions have relevance in analyzing a variety of subsurface contaminant transport problems. In this study, a one-dimensional analytical solution for multi-species transport is obtained for finite porous media and constant boundary conditions. The solution permits different retardation factors for the various species. The solution procedure involves a classic algebraic substitution that transforms the advection-dispersion partial differential equation for each species into an equation that is purely diffusive. The new system of partial differential equations is solved analytically using the Classic Integral Transform Technique (CITT). Results for a classic test case involving a three-species nitrification chain are shown to agree with previously reported literature values. Because the new solution was obtained for a finite domain, it should be especially useful for testing numerical solution procedures.     

An Analytical Solution for Contaminant Transport in nonuniform Flow

We obtain an analytical solution for two-dimensional steady state mass transport in a trapezoidal embankment in a spatially varying velocity field through its replacement with a hydrologically equivalent rectangular embankment. Application of the Dupuit approximation and conform transformation allow for computation of the concentration field in the resulting rectangle in the complex potential plane. The latter allows deriving expression for the mass flow rate of contaminants, which is analogous to the Dupuit--Forchheimer discharge formula for volumetric water flow rate. Numerical simulation of advection- dispersion in the actual domain compares favorably with these analytical results, and provides limits of the ratio between transverse and longitudinal dispersivities within which the Dupuit approximation is applicable to mass transport problems.     

Approximate Analytical Solution of the Nonlinear Diffusion Equation for Arbitrary Boundary Conditions

A general approximation for the solution of the one-dimensional nonlinear diffusion equation is presented. It applies to arbitrary soil properties and boundary conditions. The approximation becomes more accurate when the soil-water diffusivity approaches a delta function, yet the result is still very accurate for constant diffusivity suggesting that the present formulation is a reliable one. Three examples are given where the method is applied, for a constant water content at the surface, when a saturated zone exists and for a time-dependent surface flux.     

Contaminant Transport in Fractured Media with Sources in the Porous Domain

We study contaminant flow with sources in a fractured porous mediumconsisting of a single fracture bounded by a porous matrix. In the fracturewe assume convection, decay, surface adsorption to the interface, and lossto the porous matrix; in the porous matrix we include diffusion, decay,adsorption, and contaminant sources. The model leads to a nonhomogeneous,linear parabolic equation in a quarter-space with a differential equationfor an oblique boundary condition. Ultimately, we study the problemu _t = u _yy – u + f(x,y,t),x,y>0, t>0, u _t = –u _x + u _y – u on y = 0; u(0,0,t) =u₀(t), t>0,with zero initial data. Using Laplace transforms we obtain the Green'sfunction for the problem, and we determine how contaminant sources in theporous media are propagated in time.     

各种边界条件平行四边形功能梯度板的动力特性解析解

本文基于斜坐标系,建立起平行四边形功能梯度板的基本微分方程及变分方程,用梁函数组合法对平行四边形及菱形功能梯度板进行动力特性分析,提出了适用于每边任取简支、固定、自由边界之一(包括36种边界)平行四边形功能梯度板固有频率与振型的解析解;在简化情况下,给出了各种边界条件平行四边形功能梯度板各阶固有频率解的统一表达式。     

Staggered Finite Volume Modeling of Transport Phenomena in Porous Materials with Convective Boundary Conditions

The aim of this article is to present a one-dimensional finite volume modeling of mass transport phenomena in partially saturated open porous media. The finite volume model is based on a staggered solution strategy which permits to solve the equations sequentially. This appropriate partitioning reduces thus the size of the system to be solved at each time step. Therefore, two iterative algorithms are proposed to solve the discretized problem. Moreover, an original treatment of the convective boundary condition, based on a suitable linearization to derive the algebraic form of this condition, is also presented. The computational model is then specifically used for studying the drying process of a cementitious material. For this purpose, the two proposed algorithms are compared in terms of computational efficiency. Thereafter, the convective boundary condition is analyzed with regard to the drying kinetics. Finally, a one-dimensional drying experimental test is simulated.     

On the Influence of Boundary Conditions when Determining Transport Coefficients from Finite Samples of Porous Media: Assessment for Tomographic Images of Real Materials

Transport in Porous Media - The increasing access to 3d digital images of porous media provides an ideal avenue for the determination of their transport properties, by solving the governing...     

Three-Dimensional Analytical Models for Virus Transport in Saturated Porous Media

Analytical models for virus transport in saturated, homogeneous porous media are developed. The models account for three-dimensional dispersion in a uniform flow field, and first-order inactivation of suspended and deposited viruses with different inactivation rate coefficients. Virus deposition onto solid particles is described by two different processes: nonequilibrium adsorption which is applicable to viruses behaving as solutes; and colloid filtration which is applicable to viruses behaving as colloids. The governing virus transport equations are solved analytically by employing Laplace/Fourier transform techniques. Instantaneous and continuous/periodic virus loadings from a point source are examined.     

The Role of Biofilm Growth in Bacteria-Facilitated Contaminant Transport in Porous Media

Groundwater contaminants adhered to colloid surfaces may migrate to greater distances than predicted by using the conventional advective-dispersive transport equation. Introduction of exogenous bacteria in a bioremediation operation or mobilization of indigenous bacteria in groundwater aquifers can enhance the transport of contaminants in groundwater by reducing the retardation effects. Because of their colloidal size and favorable surface conditions, bacteria can be efficient contaminant carriers. In cases where contaminants have low mobility because of their high partition with aquifer solids, facilitated contaminant transport by mobile bacteria can create high contaminant fluxes. In this paper, we developed a methodology to describe the bacteria-facilitated contaminant transport in a subsurface environment using the biofilm theory. The model is based on mass balance equations for bacteria and contaminant. The contaminant is utilized as a substrate for bacterial growth. Bacteria are attached to solid surfaces as a biofilm. We investigate the role of the contaminant adsorption on both biofilm and mobile bacteria on groundwater contaminant transport. Also, the effect of bacterial injection on the contaminant transport is evaluated in the presence of indigenous bacteria in porous media. The model was solved numerically and validated by experimental data reported in the literature. Sensitivity analyses were conducted to deduce the effect of critical model parameters. Results show that biofilm grows rapidly near the top of the column where the bacteria and contaminant are injected, and is detached by increasing fluid shear stress and re-attach downstream. The adsorption of contaminant on bacterial surfaces reduces contaminant mobility remarkably in the presence of a biofilm. The contaminant concentration decreases significantly along the biofilm when contaminant partition into bacteria. Bacterial injection and migration in subsurface environments can be important in bioremediation operations regardless of the presence of indigenous bacteria.     

Effective Flux Boundary Conditions for Upscaling Porous Media Equations

We introduce a new algorithm for setting pressure boundary conditions in subgrid simulations of porous media flow. The algorithm approximates the flux in the boundary cell as the flux through a homogeneous inclusion in a homogeneous background, where the permeability of the inclusion is given by the cell permeability and the permeability of the background is given by the ambient effective permeability. With this approximation, the flux in the boundary cell scales with the cell permeability when that permeability is small, and saturates at a constant value when that permeability is large. The flux conditions provide Neumann boundary conditions for the subgrid pressure. We call these boundary conditions effective flux boundary conditions (EFBCs). We give solutions for the flux through ellipsoidal inclusions in two and three dimensions, assuming symmetric tensor permeabilities whose principal axes align with the axes of the ellipse. We then discuss the considerations involved in applying these equations to scale up problems in geological porous media. The key complications are heterogeneity, fluctuations at all length scales, and boundary conditions at finite scales.     

Analytical Solution for Constant-Rate Pumping Test in Fissured Porous Media with Double-Porosity Behaviour

An analytical solution for constant-rate pumping test in fissured porous media is developed using the dual porosity approach with pseudo-steady state exchange between fissures and matrix. It is shown how the solution relates to other pumping well functions. For analyses of pumping tests, type curves are presented that enable to estimate aquifer transmissivity, storage coefficients of the fissures and matrix and conductance for fluid exchange between the fissures and matrix. The analytical solution is also linked to a non-linear least squares optimisation algorithm to automatically fit simulated and observed drawdown and estimate hydraulic properties of the aquifer. The usefulness of the method is demonstrated by a practical application using data described in literature from a pumping test conducted in a confined limestone karst aquifer. Results obtained for the aquifer parameters estimates including 95% confidence intervals demonstrate the practical usefulness of the obtained solution.     

Comparison of Kinetic and Hybrid-Equilibrium Models Simulating Colloid-Facilitated Contaminant Transport in Porous Media

The presence of colloidal particles in groundwater can enhance contaminant transport by reducing retardation effects and carrying them to distances further than predicted by a conventional advective/dispersive equation with normal retardation values. When colloids exist in porous media and affect contaminant migration, the system can best be simulated as a three-phase medium. Mechanisms of mass transfer from one phase to another by colloids and contaminants can be kinetic or equilibrium-based, depending on the sorption–desorption reaction rate between the aqueous and solid phases. When the rate of sorption between the water phase and the solid phase(s) is not much greater than the rate of change in contaminant concentration in the water phase, kinetic sorption models may better describe the phenomenon. In some cases of modeling one or more mass transfer processes, a useful simplification may be to introduce the local equilibrium assumption. In this study, the local equilibrium assumption for sorption processes on colloidal surfaces (hybrid equilibrium model) was compared with kinetic-based models. Sensitivity analyses were conducted to deduce the effect of major parameters on contaminant transport. The results obtained from the hybrid equilibrium model in predicting the transport of colloid-facilitated groundwater contaminant are very similar to those of the kinetic model, when the point of interest is not at contaminant and colloid source vicinities and the time of interest is sufficiently long for imposed sources.     

Necessary Conditions at the Boundary for Minimizers in Incompressible Finite Elasticity

New necessary conditions for energy-minimizing states of an incompressible, elastic body are found. These must hold at boundary points at which dead-load traction data is prescribed.     

THE ANALYTICAL SOLUTION FOR SEDIMENT REACTION AND DIFFUSION EQUATION WITH GENERALIZED INITIAL-BOUNDARY CONDITIONS

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Solution of Dynamic Problems for Thin-Walled Structural Elements with Nonuniform Dynamic Boundary Conditions

The characteristics of the stress–strain state of thin-walled structural elements are determined in the case where dynamic boundary loads or displacements described by pulse functions are specified. A general scheme for realization of the method of natural-mode expansion is stated as applied to differential equations with unknown functions of one spatial coordinate and time. Theoretical relations for rods, plates, and shells are given. The potential of the approach developed is illustrated by solving specific problems     

四边任意支承条件下弹性矩形薄板弯曲问题的解析解

利用辛几何法推导出了四边为任意支承条件下矩形薄板弯曲的解析解。在分析过程中首先把矩形薄板弯曲问题表示成Hamilton正则方程，然后利用辛几何方法对全状态相变量进行分离变量，求出其本征值后，再按本征函数展开的方法求出四边为任意支承条件下矩形薄板弯曲的解析解。由于在求解过程中并不需要人为的事先选取挠度函数，而是从弹性矩形薄板弯曲的基本方程出发，直接利用数学的方法求出问题的解析解，使得这类问题的求解更加理论化和合理化。文中的最后还给出了计算实例来验证本文方法的正确性。