Three-dimensional analytical solutions to the advection-dispersion equation in arbitrary Cartesian coordinates

1993 ◽  
Vol 29 (9) ◽  
pp. 3215-3225 ◽  
Author(s):  
T. R. Ellsworth ◽  
G. L. Butters
Keyword(s):  
Dispersion Equation ◽  
Analytical Solutions ◽  
Three Dimensional ◽  
Cartesian Coordinates ◽  
Advection Dispersion

scholarly journals Exact analytical solutions for contaminant transport in rivers 1. The equilibrium advection-dispersion equation

2013 ◽  
Vol 61 (2) ◽  
pp. 146-160 ◽  
Author(s):  
Martinus Th. van Genuchten ◽  
Feike J. Leij ◽  
Todd H. Skaggs ◽  
Nobuo Toride ◽  
Scott A. Bradford ◽  
...  
Keyword(s):  
Surface Water ◽  
Dispersion Equation ◽  
Contaminant Transport ◽  
Analytical Solutions ◽  
Decay Chain ◽  
Transport Processes ◽  
Dispersive Transport ◽  
Chain Reactions ◽  
Surface Water Hydrology ◽  
Advection Dispersion

Abstract Analytical solutions of the advection-dispersion equation and related models are indispensable for predicting or analyzing contaminant transport processes in streams and rivers, as well as in other surface water bodies. Many useful analytical solutions originated in disciplines other than surface-water hydrology, are scattered across the literature, and not always well known. In this two-part series we provide a discussion of the advection-dispersion equation and related models for predicting concentration distributions as a function of time and distance, and compile in one place a large number of analytical solutions. In the current part 1 we present a series of one- and multi-dimensional solutions of the standard equilibrium advection-dispersion equation with and without terms accounting for zero-order production and first-order decay. The solutions may prove useful for simplified analyses of contaminant transport in surface water, and for mathematical verification of more comprehensive numerical transport models. Part 2 provides solutions for advective- dispersive transport with mass exchange into dead zones, diffusion in hyporheic zones, and consecutive decay chain reactions.

scholarly journals Exact Analytical Solutions for Contaminant Transport in Rivers

2013 ◽  
Vol 61 (3) ◽  
pp. 250-259 ◽  
Author(s):  
Martinus Th. van Genuchten ◽  
Feike J. Leij ◽  
Todd H. Skaggs ◽  
Nobuo Toride ◽  
Scott A. Bradford ◽  
...  
Keyword(s):  
Dispersion Equation ◽  
Contaminant Transport ◽  
Analytical Solutions ◽  
Hyporheic Zone ◽  
Decay Chain ◽  
Transport Processes ◽  
Dispersive Transport ◽  
Chain Reactions ◽  
First Order ◽  
Advection Dispersion

Abstract Contaminant transport processes in streams, rivers, and other surface water bodies can be analyzed or predicted using the advection-dispersion equation and related transport models. In part 1 of this two-part series we presented a large number of one- and multi-dimensional analytical solutions of the standard equilibrium advection-dispersion equation (ADE) with and without terms accounting for zero-order production and first-order decay. The solutions are extended in the current part 2 to advective-dispersive transport with simultaneous first-order mass exchange between the stream or river and zones with dead water (transient storage models), and to problems involving longitudinal advectivedispersive transport with simultaneous diffusion in fluvial sediments or near-stream subsurface regions comprising a hyporheic zone. Part 2 also provides solutions for one-dimensional advective-dispersive transport of contaminants subject to consecutive decay chain reactions.

scholarly journals Generalized Analytical Solutions of The Advection-Dispersion Equation with Variable Flow and Transport Coefficients

2021 ◽  
Vol 13 (14) ◽  
pp. 7796
Author(s):  
Abhishek Sanskrityayn ◽  
Heejun Suk ◽  
Jui-Sheng Chen ◽  
Eungyu Park
Keyword(s):  
Dispersion Equation ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Transient Flow ◽  
Dispersion Coefficient ◽  
Transport Coefficients ◽  
Variable Coefficients ◽  
Dispersion Coefficients ◽  
Transform Method ◽  
Advection Dispersion

Demand has increased for analytical solutions to determine the velocities and dispersion coefficients that describe solute transport with spatial, temporal, or spatiotemporal variations encountered in the field. However, few analytical solutions have considered spatially, temporally, or spatiotemporally dependent dispersion coefficients and velocities. The proposed solutions consider eight cases of dispersion coefficients and velocities: both spatially dependent, both spatiotemporally dependent, both temporally dependent, spatiotemporally dependent dispersion coefficient with spatially dependent velocity, temporally dependent dispersion coefficient with constant velocity, both constant, spatially dependent dispersion coefficient with spatiotemporally dependent velocity, and constant dispersion coefficient with temporally dependent velocity. The spatial dependence is linear, while the temporal dependence may be exponential, asymptotical, or sinusoidal. An advection–dispersion equation with these variable coefficients was reduced to a non-homogeneous diffusion equation using the pertinent coordinate transform method. Then, solutions were obtained in an infinite medium using Green’s function. The proposed analytical solutions were validated against existing analytical solutions or against numerical solutions when analytical solutions were unavailable. In this study, we showed that the proposed analytical solutions could be applied for various spatiotemporal patterns of both velocity and the dispersion coefficient, shedding light on feasibility of the proposed solution under highly transient flow in heterogeneous porous medium.

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