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.     

Upscaling,relaxation and reversibility of dispersive flow in stratified porous media

Dispersive tracer released in a unidirectional velocity field belonging to a stratified porous of finite height describes a transition, called relaxation, from a convective dominated behaviour for short times to Fickian behaviour for asymptotic long times. The temporal relaxation state of the tracer is controlled by the transverse mixing term. In most practical applications, the orders of the time and length scales of the relaxation mechanism are such that in an upscaled model of a stratified medium the dispersive flux is in a pre-asymptotic state. Explicit modelling of the relaxation of the dispersive flux in the pre-asymptotic region is required to improve the accuracy. This paper derives a pre-asymptotic one-dimensional upscaled model for the transverse averaged tracer concentration. The model generalises Taylor dispersion (Proc. R. Soc. London 219, 186–203 (1953)) and extends the method of Camacho (Phys. Rev. E 47(2), 1049–1053 (1993a); Phys. Rev. E 48 (1993b)) to dispersion tensors that may vary as function of the transverse direction. In the averaging step, the governing two-dimensional equation is first spectrally decomposed in terms of the eigenfunctions of the transverse mixing term. Next, the resulting modal relaxation equations are combined into an effective relaxation equation for the extended dispersive Taylor flux. Contrary to the one-dimensional Fickian approach, the upscaled model approximates the multi-scale relaxation behaviour as a single scale relaxation process and accounts for the partial reversibility of convective dispersion upon reversal of the flow direction. The upscaled model is evaluated against the original two-dimensional model by means of moment analysis. The longitudinal tracer variance predicted by our model is quantitatively correct in the short and long time limits and is qualitatively correct for intermediate times.     

Analysis of the Time Behaviour of a Diffusive Transport in a Stratified Medium

The paper presents a study of the diffusive transport of passive solute plumes in a two-dimensional non-homogeneous depth stratified flow domain. All the properties of the process are expressed by depth dependent deterministic functions. The method of moments, combined with the method of Green functions are chosen to determine the relevant characteristics of the flow (mass, center of mass, variance, etc.) used to describe the behaviour of the transient motion. General relationships for the n-order concentration moments are proved. Further, it is derived that the transient motion defined by time-dependent parameters tends asymptotically at large time to a stable regime whose characteristics are determined. Consequently, under certain hypotheses, an equivalence between the mean original process and a Fickian diffusive transport at large time may be established. The time required by the process to reach its asymptotic behaviour is also calculated.     

Method of homogenization applied to dispersion in porous media

The theory of homogenization which is a rigorous method of averaging by multiple scale expansions, is applied here to the transport of a solute in a porous medium. The main assumption is that the matrix has a periodic pore structure on the local scale. Starting from the pores with the Navier-Stokes equations for the fluid motion and the usual convective-diffusion equation for the solute, we give an alternative derivation of the three-dimensional macroscale dispersion tensor for solute concentration. The original result was first found by Brenner by extending Brownian motion theory. The method of homogenization is an expedient approach based on conventional continuum equations and the technique of multiple-scale expansions, and can be extended to more complex media involving three or more contrasting scales with periodicity in every but the largest scale.     

Toward consistent formulation of Reynolds stress and scalar flux closures

Langevin stochastic differential equations provide a consistent basis for Reynolds stress, scalar transport and p.d.f. models. However, the stochastic equations must be capable of representing existing closures, like the General Linear Model, or the Rotta and Monin return to isotropy formulations. A consistent approach to derive both Reynolds stress and scalar flux transport equations, starting from a stochastic differential equation for velocity fluctuations, is presented here. A set of algebraic relations for the dispersion tensor is derived for homogeneous shear flow and for the log-layer.     

Equivalent Diffusion in a Thin Film Flow with a Non-One-Dimensional Velocity Field and Anisotropic Diffusion Tensor

For describing the mass transfer processes in channels, Taylor's dispersion theory is widely used. This theory makes it possible, with asymptotic rigor, to replace the complete diffusion (heat conduction) equation with a convective term that depends on the coordinate transverse to the flow by an effective diffusion (dispersion) equation with constant coefficients, averaged over the channel cross-section. In numerous subsequent studies, Taylor's theory was generalized to include more complex situations, and novel algorithms for constructing the dispersion equations were proposed. For thin film flows a theory similar to Taylor's leads to a matrix of dispersion coefficients.In this study, Taylor's theory is extended to film flows with a non-one-dimensional velocity field and anisotropic diffusion tensor. These characteristics also depend to a considerable extent on the spatial coordinates and time. The dispersion equations obtained can be simplified in regions in which the effective diffusion coefficient tensor changes sharply.     

Contaminant contraction in two-dimensional oscillatory flows

If the vertically-mixing time is comparable with that of period oscillatory current,the contaminant contraction may occur.The coefficient of shear dispersion is negative(singularity).According to the two-dimensional delay-diffusion equation derived by theauthor:where u(t),v(t)are vertically-averaged velocities,the equations for X(t),Y(t),central displacement,dispersion tensor,had been derived.αD_(ij)/τis positivewhenτis small.If theτis large,the memory functions may be negative.Also theexpressions for D_(ij)and X,Y had been obtained.     

A Fickian diffusion model for the spreading of liquid plumes infiltrating in heterogeneous media

Infiltration of water and non-aqueous phase liquids (NAPLs) in the vadose zone gives rise to complex two- and three-phase immiscible displacement processes. Physical and numerical experiments have shown that ever-present small-scale heterogeneities will cause a lateral broadening of the descending liquid plumes. This behavior of liquid plumes infiltrating in the vadose zone may be similar to the familiar transversal dispersion of solute plumes in single-phase flow. Noting this analogy we introduce a mathematical model for ‘phase dispersion’ in multiphase flow as a Fickian diffusion process. It is shown that the driving force for phase dispersion is the gradient of relative permeability, and that addition of a phase-dispersive term to the governing equations for multiphase flow is equivalent to an effective capillary pressure which is proportional to the logarithm of the relative permeability of the infiltrating liquid phase. The relationship between heterogeneity-induced phase dispersion and capillary and numerical dispersion effects is established. High-resolution numerical simulation experiments in heterogeneous media show that plume spreading tends to be diffusive, supporting the proposed convection-dispersion model. Finite difference discretization of the phase-dispersive flux is discussed, and an illustrative application to NAPL infiltration from a localized source is presented. It is found that a small amount of phase dispersion can completely alter the behavior of an infiltrating NAPL plume, and that neglect of phase-dispersive processes may lead to unrealistic predictions of NAPL behavior in the vadose zone.     

A Solute Transport in Stratified Media

Two-dimensional and steady solute transport in a stratified porous formation is analysed under assumption that the effect of pore-scale dispersion is negligible. The longitudinal dispersion produced as a result of the vertical variation of hydraulic conductivity is analysed by averaging the variability of a solute flux concentration and conductivity. The evolution of the solute flux concentration is expressed with respect to the correlated variable, that is the travel (arrival) time at a fixed location and the averaging procedure is constructed to satisfy the boundary condition where the inlet concentration is a known function of time. In such a statement, a velocity-averaged solute flux concentration is described by a conventional dispersion model (CDM) with a dispersion coefficient which is a function of the arrival time. It is demonstrated that such CDM satisfies the assumption that hydraulic conductivity of the layers is gamma distributed with the parameter of distribution which is chosen to represent a reasonable value of the field scale solute dispersion. The overall behaviour of the model is illustrated by several examples of two-dimensional mass transport.     

Reactive Solute Transport in a Nonstationary,Fractured Porous Medium: A Dual-Porosity Approach

A numerical method of moment is developed for solute flux through a nonstationary, fractured porous medium. Solute flux is described as a space-time process where time refers to the solute flux breakthrough and space refers to the transverse displacement distribution at a control plane. A first-order mass diffusion model is applied to describe interregional mass diffusion between fracture (advection) and matrix (nonadvection) regions. The chemical is under linear equilibrium sorption in both fracture and matrix regions. Hydraulic conductivity in the fracture region is assumed to be a spatial random variable. In this study, the general framework of Zhang et al.(2000) is adopted for solute flux in a nonstationary flow field. A time retention function related to physical and chemical sorption in the dual-porosity medium is developed and coupled with solute advection along random trajectories. The mean and variance of total solute flux are expressed in terms of the probability density function of the parcel travel time and transverse displacement. The influences of various factors on solute transport are investigated. These factors include the interregional mass diffusion rate between fracture and matrix regions, chemical sorption coefficients in both regions, water contents in both regions, and location of the solute source. In comparison with solute transport in a one-region medium, breakthrough curves of the mean and variance of the total solute flux in a two-region medium have lower peaks and longer tails. As compared with the classical stochastic studies on solute transport in fractured media, the numerical method of moment provides an approach for applying the stochastic method to study solute transport in more complicated fractured media.     

Thermal Dispersion–Radiation Effects on Non-Darcy Natural Convection in a Fluid Saturated Porous Medium

The effects of thermal dispersion and thermal radiation on the non-Darcy natural convection over a vertical flat plate in a fluid saturated porous medium are studied. Forchheimer extension is considered in the flow equations. The coefficient of thermal diffusivity has been assumed to be the sum of molecular diffusivity and the dispersion thermal diffusivity due to mechanical dispersion. Rosseland approximation is used to describe the radiative heat flux in the energy equation. Similarity solution for the transformed governing equations is obtained. Numerical results for the details of the velocity and temperature profiles which are shown on graphs have been presented. The combined effect of thermal dispersion and thermal radiation, for the two cases Darcy and non-Darcy porous medium, on the heat transfer rate which are entered in tables is discussed.     

A theory of macrodispersion for the scale-up problem

Dispersion is the result, observable on large length scales, of events which are random on small length scales. When the length scale on which the randomness operates is not small, relative to the observations, then classical dispersion theory fails. The scale up problem refers to situations in which randomness occurs on all length scales, and for which classical dispersion theory necessarily fails. The purpose of this article is to present non-Fickian, theories of dispersion, which do not assume a scale separation between the randomness and the observed consequences, and which do not assume a single length scale.Porous media flow properties are heterogeneous on all length scales. The geological variation on length scales below the observational length scale can be regarded as unknown and unknowable, and thus as a random variable.We develop a systematic theory relating scaling behavior of the geological heterogeneity to the scaling behavior of the fluid dispersivity. Three qualitatively distinct regimes (Fickian, non-Fickian and nonrenormalizable) are found. The theory gives consistent answers within several distinct analytic approximations, and with numerical simulation of the equations of porous media flow.Comparison to field data is made. The use of Kriging to generate constrained ensembles for conditional simulation is discussed.     

Mixed convection effects on heat and mass transfer in a non Newtonian fluid with chemical reaction over a vertical plate

This paper studies mixed convection, double dispersion and chemical reaction effects on heat and mass transfer in a non-Darcy non-Newtonian fluid over a vertical surface in a porous medium under the constant temperature and concentration. The governing boundary layer equations, namely, momentum, energy and concentration, are converted to ordinary differential equations by introducing similarity variables and then are solved numerically by means of fourth-order Runge-Kutta method coupled with double-shooting technique. The velocity, temperature concentration, heat and mass transfer profiles are presented graphically for various values of the parameters, and the influence of viscosity index n, thermal and solute dispersion, chemical reaction parameter χ are observed.     

Upscaling of Nonwetting Phase Residual Transport in Porous Media: A Network Approach

Based on the volume averaging method, a macroscopic model is developed for the upscaling of NAPL transport in a porous medium idealised by a network model. Under the assumption of local mass non-equilibrium, a macroscopic equation involving a dispersion tensor, additional convective terms and a linear form for the interfacial mass flux is obtained. The resolution of the two local closure problems obtained allow the determination of the local properties without adjustable parmeters. These problems are solved in a semi-analytical, semi-numerical manner on the network. The originality of this work is the association of the upscaling by volume averaging method with the network approach. The local properties, including the dispersion tensor and the mass exchange coefficient, can therefore be calculated over a large number of pore-bodies and pore-throats in a computationaly tractable manner, thus leading to more significant results. Results are presented for 3D, spatially periodic models of porous media.     

A Numerical Method of Moments for Solute Transport in Nonstationary Flow Fields

A Lagrangian perturbation approach has been applied to develop the method of moments for predicting mean and variance of solute flux through a three-dimensional nonstationary flow field. The flow nonstationarity may stem from medium nonstationarity, finite domain boundaries, and/or fluid pumping and injecting. The solute flux is described as a space–time process where time refers to the solute flux breakthrough and space refers to the transverse displacement distribution at the control plane. The analytically derived moment equations for solute transport in a nonstationary flow field are too complicated to solve analytically, a numerical finite difference method is implemented to obtain the solutions. This approach combines the stochastic model with the flexibility of the numerical method to boundary and initial conditions. This method is also compared with the numerical Monte Carlo method. The calculation results indicate the two methods match very well when the variance of log-conductivity is small, but the method of moment is more efficient in computation.     

Similarities between the quasi‐bubble and the generalized wave continuity equation solutions to the shallow water equations

Two common strategies for solving the shallow water equations in the finite element community are the generalized wave continuity equation (GWCE) reformulation and the quasi‐bubble velocity approximation. The GWCE approach has been widely analysed in the literature. In this work, the quasi‐bubble equations are analysed and comparisons are made between the quasi‐bubble approximation of the primitive form of the shallow water equations and a linear finite element approximation of the GWCE reformulation of the shallow water equations. The discrete condensed quasi‐bubble continuity equation is shown to be identical to a discrete wave equation for a specific GWCE weighting parameter value. The discrete momentum equations are slightly different due to the bubble function. In addition, the dispersion relationships are shown to be almost identical and numerical experiments confirm that the two schemes compute almost identical results. Analysis of the quasi‐bubble formulation suggests a relationship that may guide selection of the optimal GWCE weighting parameter. Copyright © 2004 John Wiley & Sons, Ltd.     

Delay Model for a Cycling Transport Through Porous Medium

A solute transport through a porous medium is examined provided that the fluid leaving the porous sample returns back in a continuous way. The porous medium is thus included into a closed hydrodynamic circuit. This cycling process is suggested as an experimental tool to determine porous medium parameters describing transport. In the present paper the mathematical theory of this method is developed. For the advective type of transport with solute retention and degradation in porous medium, the system of transport equations in a closed circuit is transformed to a delay differential equation. The exact analytical solution to this equation is obtained. The solute concentration manifests both the oscillatory and monotonous behaviors depending on system parameters. The number of oscillation splashes is shown to be always finite. The maximum/minimum points are determined as solutions of a polynomial equation whose degree depends on the unknown solution itself. The cyclic methods to determine porous medium parameters as porosity and retention rate are developed.     

饱和砂土中泥浆渗透的变形-渗流-扩散耦合计算模型

传统的泥浆渗透计算中没有考虑土体变形和浆液流速的影响.根据泥浆颗粒的质量守恒定律推导了耦合流速的浓度扩散方程,并通过在浓度方程中引入沉积系数进一步计算得到沉积颗粒的质量;同时,以沉积量作为耦合项对毕奥固结方程中的水量连续方程进行了修正,在此基础上建立了变形-渗流-扩散耦合的控制方程及其变分原理. 采用有限单元法求解基本方程,运用了时间增量法与直接迭代法,并利用一维试验验证计算方法的可靠性,并与赫齐格的经典模型的计算结果进行了比较,结果表明,本文建立的模型的计算结果可以较好地预测各组试验中颗粒的沉积规律,且吻合程度优于仅考虑颗粒对流和扩散的传统计算方法. 最后,将泥浆在槽壁中的渗透简化为二维问题并进行了计算,计算结果与工程认识相符合,泥浆的沉积填充效应随深度的增加而增大,施工时需要严格控制浅层作业段的机械垂直度;成槽机的下斗抓挖时机可以根据地层填充的致密程度进行计算,对现场施工具有一定的指导意义.      

A Boussinesq model with alleviated nonlinearity and dispersion

The classical Boussinesq equation is a weakly nonlinear and weakly dispersive equation, which has been widely applied to simulate wave propagation in off-coast shallow waters. A new form of the Boussinesq model for an uneven bottoms is derived in this paper. In the new model, nonlinearity is reduced without increasing the order of the highest derivative in the differential equations. Dispersion relationship of the model is improved to the order of Pade （2,2） by adjusting a parameter in the model based on the long wave approximation. Analysis of the linear dispersion, linear shoaling and nonlinearity of the present model shows that the performances in terms of nonlinearity, dispersion and shoaling of this model are improved. Numerical results obtained with the present model are in agreement with experimental data.