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1.
Analytical solutions of one-dimensional advection-diffusion equation with variable coefficients in a finite domain 总被引：1，自引：0，他引：1
Analytical solutions are obtained for one-dimensional advection-diffusion equation with variable coefficients in a longitudinal
finite initially solute free domain, for two dispersion problems. In the first one, temporally dependent solute dispersion
along uniform flow in homogeneous domain is studied. In the second problem the velocity is considered spatially dependent
due to the inhomogeneity of the domain and the dispersion is considered proportional to the square of the velocity. The velocity
is linearly interpolated to represent small increase in it along the finite domain. This analytical solution is compared with
the numerical solution in case the dispersion is proportional to the same linearly interpolated velocity. The input condition
is considered continuous of uniform and of increasing nature both. The analytical solutions are obtained by using Laplace
transformation technique. In that process new independent space and time variables have been introduced. The effects of the
dependency of dispersion with time and the inhomogeneity of the domain on the solute transport are studied separately with
the help of graphs. 相似文献
2.
R. R. Yadav Dilip Kumar Jaiswal Hareesh Kumar Yadav Gulrana 《Environmental Earth Sciences》2012,65(3):849-859
A three-dimensional model for non-reactive solute transport in physically homogeneous subsurface porous media is presented.
The model involves solution of the advection-dispersion equation, which additionally considered temporally dependent dispersion.
The model also account for a uniform flow field, first-order decay which is inversely proportional to the dispersion coefficient
and retardation factor. Porous media with semi-infinite domain is considered. Initially, the space domain is not solute free.
Analytical solutions are obtained for uniform and varying pulse-type input source conditions. The governing solute transport
equation is solved analytically by employing Laplace transformation technique (LTT). The solutions are illustrated and the
behavior of solute transport may be observed for different values of retardation factor, for which simpler models that account
for solute adsorption through a retardation factor may yield a misleading assessment of solute transport in ‘‘hydrologically
sensitive’’ subsurface environments. 相似文献
3.
This paper presents three-dimensional finite element simulations to evaluate diffusion and dispersion tensors in periodic
porous media in the presence of an advective velocity field. These tensors are evaluated in the framework of the double-scale
expansion technique. Two problems, a Newtonian flow and a vector-valued advection–diffusion equation, have to be sequentially
solved at the pore scale. Finite element techniques to approximate these problems are proposed and analyzed. Numerical results
in three-dimensional networks of spheres are presented to quantitatively assess the impact of the pore morphology and of the
advection velocity on the diffusion and dispersion tensors. 相似文献
4.
Leng L. Lim Winston L. Sweatman Robert McKibbin Charles B. Connor 《Mathematical Geosciences》2008,40(2):147-157
Numerous tephra dispersion and sedimentation models rely on some abstraction of the volcanic plume to simplify forecasts of
tephra accumulation as a function of the distance from the volcano. Here we present solutions to the commonly used advection–dispersion
equation using a variety of source shapes: a point, horizontal and vertical lines, and a circular disk. These may be related
to some volcanic plume structure, such as a strong plume (vertical line), umbrella cloud (circular disk), or co-ignimbrite
plume (horizontal line), or can be used to build a more complex plume structure such as a series of circular disks to represent
a buoyant weak plume. Basing parameters upon eruption data, we find that depositions for the horizontal source shapes are
very similar but differ from the vertical line source deposition. We also compare the deposition from a series of stacked
circular disk sources of increasing radius above the volcanic vent with that from a vertical line source. 相似文献
5.
A barrier system based on the hydraulic trap design concept for a landfill was proposed. To study the field scenario in which
a clay liner is underlain by a granular layer functioning as a secondary leachate drain layer, a laboratory advection–diffusion
test was performed to investigate factors controlling the transport of contaminants in a two-layer soil system. The soils
used for this study were Ariake clay and, the underlying layer, Shirasu soil from the Kyushu region of Japan. Potassium (K+) was selected as the target chemical species with an initial concentration of 905 mg L−1. The effective diffusion coefficients (D
e) of K+ for Ariake clay and Shirasu soil were back-calculated using an available computer program, Pollute
V 6.3. Values of D
e derived from this experiment are consistent with previously published ones. The Ariake clay has lower D
e than the Shirasu soil. The hypothesis that mechanical dispersion can be considered negligible is reasonable based on both
the observation that the predicted values well fit the experimental data and the analyses of two dimensionless parameters.
Parametric analyses show that transport of K+ through soils is controlled by advection–diffusion rather than diffusion only, whereas at low Darcy velocity (i.e., ≤10−9 m s−1), transport of K+ will be controlled by diffusion. Applications of the test results and parametric analysis results in practical situations
were reviewed. 相似文献
6.
The transport of chemically reactive solutes (e.g. surfactants, CO2 or dissolved minerals) is of fundamental importance to a wide range of applications in oil and gas reservoirs such as enhanced
oil recovery and mineral scale formation. In this work, we investigate exponential time integrators, in conjunction with an
upwind weighted finite volume discretisation in space, for the efficient and accurate simulation of advection–dispersion processes
including non-linear chemical reactions in highly heterogeneous 3D oil reservoirs. We model sub-grid fluctuations in transport
velocities and uncertainty in the reaction term by writing the advection–dispersion–reaction equation as a stochastic partial
differential equation with multiplicative noise. The exponential integrators are based on the variation of constants solution
and solve the linear system exactly. While this is at the expense of computing the exponential of the stiff matrix representing
the finite volume discretisation, the use of real Léja point or the Krylov subspace technique to approximate the exponential
makes these methods competitive compared to standard finite difference-based time integrators. For the deterministic system,
we investigate two exponential time integrators, the second-order accurate exponential Euler midpoint (EEM) scheme and exponential
time differencing of order one (ETD1). All our numerical examples demonstrate that our methods can compete in terms of efficiency
and accuracy compared with standard first-order semi-implicit time integrators when solving (stochastic) partial differential
equations that model mixing and chemical reactions in 3D heterogeneous porous media. Our results suggest that exponential
time integrators such as the ETD1 and EEM schemes could be applied to typical 3D reservoir models comprising tens to hundreds
of thousands unknowns. 相似文献
7.
Some analytical solutions of one-dimensional advection–diffusion equation (ADE) with variable dispersion coefficient and velocity are obtained using Green’s function method (GFM). The variability attributes to the heterogeneity of hydro-geological media like river bed or aquifer in more general ways than that in the previous works. Dispersion coefficient is considered temporally dependent, while velocity is considered spatially and temporally dependent. The spatial dependence is considered to be linear and temporal dependence is considered to be of linear, exponential and asymptotic. The spatio-temporal dependence of velocity is considered in three ways. Results of previous works are also derived validating the results of the present work. To use GFM, a moving coordinate transformation is developed through which this ADE is reduced into a form, whose analytical solution is already known. Analytical solutions are obtained for the pollutant’s mass dispersion from an instantaneous point source as well as from a continuous point source in a heterogeneous medium. The effect of such dependence on the mass transport is explained through the illustrations of the analytical solutions. 相似文献
8.
In this paper, we continue our analysis of upwind‐mixed methods for advection–diffusion equations, which have been developed
and analyzed by the first author over the past several years. In previous work, our analysis has been limited to low order
approximating spaces, positive definite diffusion coefficients and Dirichlet boundary conditions. In this paper, we extend
our results to higher order approximating spaces, possibly zero diffusion, and more physically realistic boundary conditions.
Moreover, unlike previous papers, we avoid the use of Gronwall's Inequality, which can result in extremely large constants
in the stability and error bounds. Numerical results are presented for constant, linear and quadratic approximating spaces.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
9.
This paper presents a new numerical tool to model the two-dimensional contaminant transport through saturated porous media
using a meshfree method, called radial point interpolation method (RPIM) with polynomial reproduction. In RPIM, an approximate
solution is constructed entirely in terms of a set of nodes and no characterisation of the interrelationship of the nodes
is needed. The advection–dispersion equation with sorption is considered to illustrate the applicability of the RPIM. The
Galerkin weak form of the governing equation is formulated using 2D meshfree shape functions constructed using thin plate
spline radial basis functions. MATLAB code is developed to obtain the numerical solution. Three numerical examples are presented
and the results are compared with those obtained from the finite element method and analytical solutions. In order to test
the practical applicability and performance of the RPIM, two case studies of contaminant transport through landfill liners
are presented. A good agreement is obtained between the results of the RPIM and the field investigation data. 相似文献
10.
K. Muthukrishnavellaisamy G. C. Mishra M. L. Kansal N. C. Ghosh 《Environmental Geology》2009,57(4):899-908
For predicting and forecasting fate of non-conservative pollutants downstream from source using advection–dispersion–decay
equation (ADDE), estimation of three parameters; mean flow velocity (U), longitudinal dispersion co-efficient (D
L) and decay rate co-efficient (λ), is required a priori. In this three parameters model, estimation of D
L holds difficulties and draws interest towards it. The empirical formulae use the field and experimental data of channel and
flow characteristics to estimate D
L. In this paper, an innovative approach has been proposed towards the estimation of D
L using regime channel concept. Having known discharge of flow and silt factor of the riverbed material, hydraulic parameters
of the channel can be determined theoretically, which in turn can be used to estimate D
L appropriately making use of a suitable empirical formula. 相似文献
11.
《Geochimica et cosmochimica acta》1987,51(5):1219-1226
The problem of diffusion of a geochemical component in a natural environment is investigated from the standpoint of mixture theory. The approach here differs from previous diffusion studies in that both the conservation of mass and momentum for the component is considered. This approach avoids parameterizing the diffusive flux in the mass equation by Fick's law. It is shown that when the momentum equation is included with the mass equation, the linear approximation for the space-time distribution of a solute in a binary system is the telegraph equation, well known from electrodynamics. This contrasts with the diffusion equation, which relies on introducing the Fick's law assumption into the conservation of mass equation for the solute. Solutions for both the diffusion and telegraph equation models are obtained and compared for the case of migration of a minor component into the seabed when the sediment-water interface concentration is a prescribed function of time. Although the stationary, steady state solutions of the telegraph and diffusion equations are identical, the former has a transient solution in which fluctuations propagate at finite speed. The Fickian assumption, in contrast, requires an infinite speed of propagation. 相似文献
12.
Long records of monthly salinity observations along the axis of Chesapeake Bay, Delaware Bay, and Long Island Sound are used
to test a simple advection–dispersion model of the salt distribution in linearly tapered estuaries developed in a previous
paper. We subdivide each estuary into three to five segments, each with linear taper allowing a distributed input of fresh
water, and evaluate the dispersion in each segment. While Delaware Bay has weak dispersion and a classical sigmoidal salinity
structure, Long Island Sound and Chesapeake Bay are more dispersive and have relatively small gradients in the central stretches.
Long Island Sound is distinguished by having a net volume and salt flux out of its low-salinity end resulting in a smaller
range of salinity and increasing axial gradients at its head rather than the usual asymptotic approach to zero salinity. Estimates
of residence times based on model transport coefficients show that Long Island Sound has the most rapid response to fresh-water
flux variations. It also has the largest amplitude cycle in river discharge fluctuation. In combination, these cause the large
seasonal variation in the salinity structure relative to interannual variability in Long Island Sound as compared with Chesapeake
Bay and Delaware Bay. 相似文献
13.
依据分形理论和方法,探索溶质在多孔介质中的有效扩散系数的替代预测方法。在多孔介质溶质扩散的弯曲毛细管束模型的基础上,以分形维数作为介质的基本几何特性参数,建立了多孔介质中溶质扩散的分形毛细管束模型,推导出了溶质有效扩散系数与介质孔隙度之间的幂定律关系式,幂指数是介质孔隙分维和表面分维的函数,反映了介质孔隙体积的层次分布与孔隙通道曲折程度对扩散的影响。对粘性土的分形维数测定数据和有效扩散系数试验测定数据的分析表明,利用该关系式预测多孔介质中溶质的有效扩散系数是较为准确可靠的。 相似文献
14.
Ralf Dohmen Hans-Werner Becker Sumit Chakraborty 《Physics and Chemistry of Minerals》2007,34(6):389-407
We have determined Fe–Mg diffusion coefficients in olivines from different sources (Nanga Parbat, Pakistan and San Carlos,
Arizona, USA) at atmospheric pressure as a function of composition, oxygen fugacity (10−5–10−12 Pa) and temperature (700–1200°C) using thin films produced by pulsed laser deposition and RBS to analyze the concentration
profiles. We have characterized the nano-scale structure and composition of the thin films annealed at various conditions
and shown that the nature of the film (e.g. crystallinity, wetting behavior) depends strongly on the annealing conditions.
If these variations are not taken into account in the form of boundary conditions for modeling the diffusion profiles, artifacts
would result in the diffusion data. The diffusion coefficients obtained from 75 experiments reveal that (i) between fO2 of 10−5 and 10−10 Pa, diffusion along all three principal crystallographic directions in olivine, [100], [010] and [001], are described by
a constant activation energy of ∼200 kJ/mol, precluding any temperature dependence of diffusion anisotropy and change of mechanism
of diffusion at temperatures between 950 and 1200°C, (ii) diffusion coefficients increase with oxygen fugacity at fO2 > 10−10 Pa, with an fO2 exponent that lies between 1/4 and 1/7, and (iii) at fO2 below 10−10 Pa, and consequently at temperatures below ∼900°C, diffusion becomes weakly dependent/independent of fO2, indicating a change of diffusion mechanism. Activation energy of diffusion at these conditions is slightly higher, ∼220 kJ/mol. The data, including the change of mechanism, are analyzed in terms of point defect chemistry in Part II of
this work to derive an equation that allows calculation of diffusivities in olivine over its entire field of stability. Availability
of directly measured data at temperatures down to 700°C imply that for the first time diffusion coefficients can be interpolated, rather than extrapolated, for modeling most natural systems. 相似文献
15.
This paper presents an approach conducive to an evaluation of the probability density function (pdf) of spatio-temporal distributions
of concentrations of reactive solutes (and associated reaction rates) evolving in a randomly heterogeneous aquifer. Most existing
approaches to solute transport in heterogeneous media focus on providing expressions for space–time moments of concentrations.
In general, only low order moments (unconditional or conditional mean and covariance) are computed. In some cases, this allows
for obtaining a confidence interval associated with predictions of local concentrations. Common applications, such as risk
assessment and vulnerability practices, require the assessment of extreme (low or high) concentration values. We start from
the well-known approach of deconstructing the reactive transport problem into the analysis of a conservative transport process
followed by speciation to (a) provide a partial differential equation (PDE) for the (conditional) pdf of conservative aqueous
species, and (b) derive expressions for the pdf of reactive species and the associated reaction rate. When transport at the
local scale is described by an Advection Dispersion Equation (ADE), the equation satisfied by the pdf of conservative species
is non-local in space and time. It is similar to an ADE and includes an additional source term. The latter involves the contribution
of dilution effects that counteract dispersive fluxes. In general, the PDE we provide must be solved numerically, in a Monte
Carlo framework. In some cases, an approximation can be obtained through suitable localization of the governing equation.
We illustrate the methodology to depict key features of transport in randomly stratified media in the absence of transverse
dispersion effects. In this case, all the pdfs can be explicitly obtained, and their evolution with space and time is discussed. 相似文献
16.
17.
分数微分对流-弥散方程(FADE)是模拟溶质迁移问题的新理论,但应用FADE来模拟溶质迁移时能否克服弥散的尺度效应尚待验证。利用长土柱实验资料结合FADE的解析解拟合推求FADE的弥散系数,并分析其与尺度之间的相关关系。研究结果表明,FADE的弥散系数具有随尺度增大而增大的现象,且均质土柱中FADE的弥散系数尺度效应小于非均质土柱中弥散系数尺度效应。在均质土柱中,弥散系数与尺度之间成指数相关关系,在非均质土柱中,弥散系数与尺度之间成幂相关关系。考虑了弥散系数分别与迁移时间和迁移距离呈线性递增两种相关关系,进而分别构建了3种考虑弥散尺度效应的FADE模型,并提出了求解的差分方法。利用上述3种考虑弥散尺度效应的FADE来模拟和预测不同空间位置处的溶质迁移过程。结果表明,对均质土柱中的溶质迁移可得到较好的模拟结果;对于非均质土柱,其模拟结果与实测结果仍然存在一定的差异。 相似文献
18.
This paper is concerned with numerical methods for the modeling of flow and transport of contaminant in porous media. The
numerical methods feature the mixed finite element method over triangles as a solver to the Darcy flow equation and a conservative
finite volume scheme for the concentration equation. The convective term is approximated with a Godunov scheme over the dual
finite volume mesh, whereas the diffusion–dispersion term is discretized by piecewise linear conforming triangular finite
elements. It is shown that the scheme satisfies a discrete maximum principle. Numerical examples demonstrate the effectiveness
of the methodology for a coupled system that includes an elliptic equation and a diffusion–convection–reaction equation arising
when modeling flow and transport in heterogeneous porous media. The proposed scheme is robust, conservative, efficient, and
stable, as confirmed by numerical simulations.
相似文献
19.
Laura Donatella Campisi 《Computational Geosciences》2017,21(3):519-531
A method using multilayer perceptrons for analysing diffusion profiles and sketching the temperature history of geological samples is explored. Users of this method can intuitively test and compare results thinking in terms of analytical solutions of the diffusion equation whilst the bulk of the work is made computationally. Being neither completely analytical nor numerical, the method is a hybrid and represents an ideal man-machine interaction. The approach presented in this paper should be preferred when the retrieval of the diffusion coefficients from concentration profiles using dimensionless parameters is not possible and/or there is more than one unknown parameter in the analytical solution of the diffusion equation. Its versatility is a key factor for extending the potential of Dodson’s formulation. The case of a species produced by a radiogenic source and diffusing in a cooling system is therefore discussed. Both the classical change of variable for diffusion coefficients depending on time and an alternative approach decomposing the overall effect of diffusion into a sum of effects due to smaller events could be used to tackle this problem. As multilayer perceptrons can approximate any function, none of the assumptions originally stated by Dodson are necessary. 相似文献