Impacts of local dispersion and first-order decay on solute transport in randomly heterogeneous porous media

Stochastic subsurface transport theories either disregard local dispersion or take it to be constant. We offer an alternative Eulerian-Lagrangian formalism to account for both local dispersion and first-order mass removal (due to radioactive decay or biodegradation). It rests on a decomposition of the velocityv into a field-scale componentv , which is defined on the scale of measurement support, and a zero mean sub-field-scale componentv _s , which fluctuates randomly on scales smaller than. Without loss of generality, we work formally with unconditional statistics ofv _s and conditional statistics ofv . We then require that, within this (or other selected) working framework,v _s andv be mutually uncorrelated. This holds whenever the correlation scale ofv is large in comparison to that ofv _s . The formalism leads to an integro-differential equation for the conditional mean total concentration c which includes two dispersion terms, one field-scale and one sub-field-scale. It also leads to explicit expressions for conditional second moments of concentration cc. We solve the former, and evaluate the latter, for mildly fluctuatingv by means of an analytical-numerical method developed earlier by Zhang and Neuman. We present results in two-dimensional flow fields of unconditional (prior) mean uniformv . These show that the relative effect of local dispersion on first and second moments of concentration dies out locally as the corresponding dispersion tensor tends to zero. The effect also diminishes with time and source size. Our results thus do not support claims in the literature that local dispersion must always be accounted for, no matter how small it is. First-order decay reduces dispersion. This effect increases with time. However, these concentration moments c and cc of total concentrationc, which are associated with the scale below, cannot be used to estimate the field-scale concentrationc directly. To do so, a spatial average over the field measurement scale is needed. Nevertheless, our numerical results show that differences between the ensemble moments ofc and those ofc are negligible, especially for nonpoint sources, because the ensemble moments ofc are already smooth enough.     

Methods of Quantum Field Theory in the Physics of Subsurface Solute Transport

The stochastic theory of subsurface solute transport has received stimulus recently from modeling techniques originating in quantum field theory (QFT), resulting in new calculations of the solute macrodispersion tensor that derive from the solving Dyson equation with a subsequent renormalization group analysis. In this paper, we offer a critical evaluation of these techniques as they relate specifically to the derivation of a field-scale advection–dispersion equation. An approximate Dyson equation satisfied by the ensemble-average solute concentration for tracer movement in a heterogeneous porous medium is derived and shown to be equivalent to a truncated cumulant expansion of the standard stochastic partial differential equation which describes the same phenomenon. The full Dyson equation formalism, although exact, is of no importance to the derivation of an improved field-scale advection–dispersion equation. Similarly, renormalization group analysis of the macrodispersion tensor has not yet provided results that go beyond what is available currently from the cumulant expansion approach.     

On the dispersion of soluble matter in a channel flow of non-Newtonian fluid

Summary An analysis of the dispersion of a solute in Rivlin-Ericksen third-order fluid in a parallel plate channel is carried out. It is seen that the solute which is dispersed relative to a plane moving with the mean speed of the flow has its effective Taylor diffusion coefficient which decreases with increasing the non-Newtonian parameter With 1 table     

On the equations of non-linear vibrations

The purpose of this short paper is to show that equations of non-linear vibrations whose linear parts involve two coefficient matrices can be simplified so that the linear portions become uncoupled. Equivalence transformations are utilized in the simplification, which can substantially streamline any subsequent analysis.     

Dispersion of inert solutes in spatially periodic,two-dimensional model porous media

Taylor dispersion of a passive solute within a fluid flowing through a porous medium is characterized by an effective or Darcy scale, transversely isotropic dispersitivity , which depends upon the geometrical microstructure, mean fluid velocity, and physicochemical properties of the system. The longitudinal, and lateral, dispersivity components for two-dimensional, spatially periodic arrays of circular cylinders are here calculated by finite element techniques. The effects of bed voidage, packing arrangement, and microscale Péclet and Reynolds numbers upon these dispersivities are systematically investigated.The longitudinal dispersivity component is found to increase with the microscale Péclet number at a rate less than Pe². This accords with previous calculations by Eidsath et al. (1983), although the latter calculations were found to yield significantly lower longitudinal dispersivities than those obtained with the present numerical scheme. With increasing Péclet number, a Pe² dependence is, however, approached asymptotically, particularly for square cylindrical arrays - owing to the creation of a linear streamline zone between cylinders.     

Process splitting of the boundary conditions for the advection–dispersion equation

Rational strategies are considered for the specification of the intermediate boundary condition at an inflow boundary where process splitting (fractional steps) is adopted in solving the advection–dispersion equation. Three lowest-order methods are initially considered and evaluation is based on comparisons with an analytical solution. For flow and dispersion parameter ranges typical of rivers and estuaries, the given boundary condition for the complete advection–dispersion equation at the end of the complete time step provides a satisfactory estimate of the intermediate boundary value. This was further confirmed by the development and evaluation of two higher-order methods. These required non-centred discrete approximations for spatial derivatives, which offset any special advantages from the higher truncation error order.     

Pollution dispersion analysis using the puff model with numerical flow field data

The computations of the flowfield and pollutant dispersion over a flat plate and the Russian hills of various slopes are described. The Gaussian plume and the puff model have been used to calculate concentration of pollutant. The Reynolds-averaged unsteady incompressible Navier–Stokes equation with low Reynolds k– model has been used to calculate the flowfield. The flow data of a flat plate and the Russian hills from Navier–Stokes equation solutions has been used as the input data for the puff model. The computational results of flowfield agree well with experimental results of both a flat plate and Russian hills. The concentration prediction by the Gaussian plume model and the Gaussian puff model also agrees fairly well with experiments.     

Maximum entropy moment system of the semiconductor Boltzmann equation using Kane’s dispersion relation

It is known that the maximum entropy moment systems of the gas-dynamical Boltzmann equation suffer from severe disadvantages which are related to the non-solvability of an underlying maximum entropy moment problem unless restrictions on the choice of the macroscopic variables are made. In this article, we show that no such difficulties appear in the semiconductor case if Kane’s dispersion relation is used for the energy band of electrons.PACS 73.50-h; 73.61.-r     

On the Influence of Pore-Scale Dispersion in Nonergodic Transport in Heterogeneous Formations

Flow of an inert solute in an heterogeneous aquifer is usually considered as dominated by large-scale advection. As a consequence, the pore-scale dispersion, i.e. the pore scale mechanism acting at scales lower than that characteristic of the heterogeneous field, is usually neglected in the computation of global quantities like the solute plume spatial moments. Here the effect of pore-scale dispersion is taken into account in order to find its influence on the longitudinal asymptotic dispersivity D₁₁we examine both the two-dimensional and the three-dimensional flow cases. In the calculations, we consider the finite size of the solute initial plume, i.e. we analyze both the ergodic and the nonergodic cases. With Pe the Péclat number, defined as Pe=U/D, where U, , D are the mean fluid velocity, the heterogeneity characteristic length and the pore-scale dispersion coefficient respectively, we show that the infinite Péclat approximation is in most cases quite adequate, at least in the range of Péclat number usually encountered in practice (Pe > 10²). A noteworthy exception is when the formation log-conductivity field is highly anisotropic. In this case, pore-scale may have a significant impact on D₁₁, especially when the solute plume initial dimensions are not much larger than the heterogeneities' lengthscale. In all cases, D₁₁ appears to be more sensitive to the pore-scale dispersive mechanisms under nonergodic conditions, i.e. for plume initial size less than about 10 log-conductivity integral scales.     

Non-symmetric CG-like schemes and the finite element solution of the advection–dispersion equation

Seven leading iterative methods for non-symmetric linear systems (GMRES, BCG, QMR, CGS, Bi-CGSTAB, TFQMR and CGNR) are compared in the specific context of solving the advection–dispersion equation by a classic approach: The space derivatives are approximated by linear finite elements while an implicit scheme is used to integrate the time derivatives. Convergence formulas that predict the behaviour of the iterative methods as a function of the discretization parameters are developed and validated by experiments. It is shown that all methods converge nicely when the coefficent matrix of the linear system is close to normal and the finite element approximation of the advection–dispersion equation yields accurate results.     

On the dispersion of a solute in Poiseuille flow of a third-grade fluid in a channel

Gill and Sankarasubramanian's analysis of the dispersion of Newtonian fluids in laminar flow between two parallel walls are extended to the flow of non-Newtonian viscoelastic fluid (known as third-grade fluid) using a generalized dispersion model which is valid for all times after the solute injection. The exact expression is obtained for longitudinal convective coefficient K₁(Γ), which shows the effect of the added viscosity coefficient Γ on the convective coefficient. It is seen that the value of the K₁(Γ) for Γ≠0 is always smaller than the corresponding value for a Newtonian fluid. Also, the effect of the added viscosity coefficient on the K₂(t,Γ) (which is a measure of the longitudinal dispersion coefficient of the solute) is explored numerically. Finally, the axial distribution of the average concentration C_m of the solute over the channel cross-section is determined at a fixed instant after the solute injection for several values of the added viscosity coefficient.     

高强度液晶高分子纺丝纤维取向机制研究

在取向运动输运方程基础上,研究液晶高分子纺丝拉伸流动取向特征、定常取向运动和非定常运动．应用拉格朗日方法研究了轴对称纺丝拉伸流动稳定性．得出液晶高分子纺丝拉伸中分子取向有较高的稳定性,其取向运动具有高抗解取向性,较易形成高取向度,这是制成高强度和高模量纤维、薄膜和模塑制品的基础．     

On problems in the weakly nonlinear theory of hydrodynamic stability and its improvement

There are three main problems in the weakly nonlinear theory of hydrodynamic stability: (1)The radius of convergence with respect to the perturbation parameter is too small and there is no concrete estimation for it. (2)The solution has a special structure, thus in general, it can not satisfy the initial condition posed by many practical problems. (3)When the linear part of its solution does not correspond to a neutral case, there are more than one way in determining the Landau constants, and practically no one knows which is the best way. In this paper, problems(1) and (2)are solved theoretically, and ways for its improvement have been proposed. By comparing the theoretical results with those obtained by numerical simulations, problem(3)has also been clarified. Project supported by the National Natural Science Foundation of China     

On the energy equation for a spatially developing instability wave in the theory of hydrodynamic stability

In this article, it is shown that the energy equation for a spatially developing disturbance used in all the literatures dealing with the problem of hydrodynamic stability suffers from a small, but crucial error.     

On the Uniqueness Theorem for Advective–Diffusive Transport in Porous Media: A Canonical Proof

This paper presents a proof of the uniqueness theorem for the initial boundary value problem governing advective–diffusive transport of a chemical in a fluid-saturated non-deformable isotropic, homogeneous porous medium. The advective Darcy flow in the porous medium results from the gradient of a hydraulic potential, which is derived from a well-posed problem in potential theory. The paper discusses the relevant set of consistent boundary conditions applicable to the potential inducing the advective flow and to the concentration field, which ensures uniqueness of the solution.     

On the field method in non-holonomic mechanics

This paper deals with the generalization of the field method to non-holonomic systems whose motion is subject to either non-linear constraints or those of a higher order, while their motion is modeled by the generalized Lagrange equations of the second kind. Two examples are given to illustrate the theory.The project supported by the Ministry of Science, Technologies and Development, Republic of Serbia (1874)     

On the spatial-temporal averaging method for modeling transport in porous media

The spatial-temporal averaging procedure is considered with a nonhomogeneous distribution of elementary domains in the spatial-temporal space and the probabilistic interpretation of the ST-averaging is also given. Several averaging theorems and corollaries about the averages of spatial and temporal derivatives are presented and rigorously proved which allow elementary domain to vary in space and time. The macroscopic transport equation in the most general condition and the simplified macroscopic equation under the special form of distributions are developed which may be reduced to the classical macroscopic transport equation as the spatial-temporal average degenerates into the volume average.