Analytical Solution for Advection-Dispersion Equation of the Pollutant Concentration using Laplace Transformation

2021 ◽  
Vol 4 (1) ◽  
pp. 33-40
Author(s):  
Keshav Paudel ◽  
Prem Sagar Bhandari ◽  
Jeevan Kafle
Keyword(s):  
Analytical Solution ◽  
Dispersion Equation ◽  
Laplace Transformation ◽  
Dispersion Coefficient ◽  
Oxygen Demand ◽  
Pollutant Concentration ◽  
One Dimension ◽  
Advection Dispersion ◽  
State Case ◽  
Saturated Oxygen

We present simple analytical solution for the unsteady advection-dispersion equation describing the pollutant concentration C(x; t) in one dimension. In this model the water velocity in the x-direction is taken as a linear function of x and dispersion coefficient D as zero. In this paper by taking k = 0, k is the half saturated oxygen demand concentration for pollutant decay, we can apply the Laplace transformation and obtain the solution. The variation of C(x; t) with different times t upto t → ∞ (the steady state case) is taken into account advection-dispersion equation in our study.

scholarly journals Analytical Solution of 1D Advection-Dispersion Equation In Finite Groundwater Reservoir With Spatially Dependent Dispersivity And Two Inputs Sources With Source-Sink Term Impact

2021 ◽  
Author(s):  
Thomas TJOCK-MBAGA ◽  
Patrice Ele Abiama ◽  
Jean Marie Ema'a Ema'a ◽  
Germain Hubert Ben-Bolie
Keyword(s):  
Analytical Solution ◽  
Linear Function ◽  
Dispersion Equation ◽  
Source Term ◽  
Numerical Solutions ◽  
Integral Transform ◽  
Liouville Problem ◽  
Finite Domain ◽  
Advection Dispersion ◽  
Sink Term

Abstract This study derives an analytical solution of a one-dimensional (1D) advection-dispersion equation (ADE) for solute transport with two contaminant sources that takes into account the source term. For a heterogeneous medium, groundwater velocity is considered as a linear function while the dispersion as a nth-power of linear function of space and analytical solutions are obtained for and . The solution in a heterogeneous finite domain with unsteady coefficients is obtained using the Generalized Integral Transform Technique (GITT) with a new regular Sturm-Liouville Problem (SLP). The solutions are validated with the numerical solutions obtained using MATLAB pedpe solver and the existing solution from the proposed solutions. We exanimated the influence of the source term, the heterogeneity parameters and the unsteady coefficient on the solute concentration distribution. The results show that the source term produces a solute build-up while the heterogeneity level decreases the concentration level in the medium. As an illustration, model predictions are used to estimate the time histories of the radiological doses of uranium at different distances from the sources boundary in order to understand the potential radiological impact on the general public.

Analytical Solution of Advection-Dispersion Equation with Spatially Dependent Dispersivity

2017 ◽  
Vol 143 (11) ◽  
pp. 04017126 ◽  
Author(s):  
Vinod Kumar Bharati ◽  
Vijay P. Singh ◽  
Abhishek Sanskrityayn ◽  
Naveen Kumar
Keyword(s):  
Analytical Solution ◽  
Dispersion Equation ◽  
Advection Dispersion

scholarly journals Two-Dimension Hydrodynamic Dispersion Equation with Seepage Velocity and Dispersion Coefficient as Function of Space and Time

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Abdon Atangana ◽  
S. C. Oukouomi Noutchie
Keyword(s):  
Dispersion Equation ◽  
Dispersion Coefficient ◽  
Analytical Techniques ◽  
Hydrodynamic Dispersion ◽  
Seepage Velocity ◽  
Transform Method ◽  
Space And Time ◽  
Advection Dispersion ◽  
Homotopy Decomposition ◽  
Geological Formation

The contamination through the geological formation cannot move and disperse with the same speed and dispersion coefficient, respectively, due to the variability of the geological formation. This paper is therefore first devoted to the description of the hydrodynamic advection dispersion equation with the seepage velocity and dispersion coefficient as function of space and time. Secondly the equation is solved via two analytical techniques: the homotopy decomposition method and the differential transform method. The numerical simulations of the approximated solutions are presented.

ONE-DIMENSIONAL TEMPORALLY DEPENDENT ADVECTION-DISPERSION EQUATION IN POROUS MEDIA: ANALYTICAL SOLUTION

2010 ◽  
Vol 23 (4) ◽  
pp. 521-539 ◽  
Author(s):  
R. R. YADAV ◽  
DILIP KUMAR JAISWAL ◽  
HAREESH KUMAR YADAV ◽  
GUL RANA
Keyword(s):  
Porous Media ◽  
Analytical Solution ◽  
Dispersion Equation ◽  
One Dimensional ◽  
Advection Dispersion

scholarly journals Numerical routing of tracer concentrations in rivers with stagnant zones

2016 ◽  
Vol 17 (3) ◽  
pp. 825-834 ◽  
Author(s):  
Abbas Parsaie ◽  
Amir Hamzeh Haghiabi
Keyword(s):  
Numerical Model ◽  
Analytical Solution ◽  
Dispersion Equation ◽  
Splitting Technique ◽  
Advection Dispersion ◽  
The Uk ◽  
Engineering Modeling ◽  
Pollution Transmission ◽  
Good Agreement ◽  
Numerical Approaches

Modeling pollution transmission in rivers is an important subject in environmental engineering studies. Numerical approaches to modeling pollution transmission in rivers are useful tools for managing the water quality. The advection-dispersion equation is the governing equation in the transport of pollution in rivers. Recently, due to advances in fractional calculus in engineering modeling, the simulation of pollution transmission in rivers has been improved using the fractional derivative approach. In this study, by solving the fractional advection-dispersion equation (FRADE), a numerical model was developed for the simulation of pollution transmission in rivers with stagnant zones. To this purpose, both terms of the FRADE equation (advection and fractional dispersion) were discretized separately and the results were connected using a time-splitting technique. The analytical solution of a modified advection-dispersion equation (MADE) model and observed data from the Severn River in the UK were used to demonstrate the model capabilities. Results indicated that there is a good agreement between observed data, the analytical solution of the MADE model, and the results of the developed numerical model. The developed numerical model can accurately simulate the long-tailed dispersion processes in a natural river.

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.

scholarly journals An Analytical Solution of the Advection Dispersion Equation in a Bounded Domain and Its Application to Laboratory Experiments

2011 ◽  
Vol 2011 ◽  
pp. 1-14 ◽  
Author(s):  
M. Massabó ◽  
R. Cianci ◽  
O. Paladino
Keyword(s):  
Porous Medium ◽  
Analytical Solution ◽  
Dispersion Equation ◽  
Saturated Porous Medium ◽  
Infinite Domain ◽  
Noninvasive Technique ◽  
Advection Dispersion ◽  
Experimental Apparatus ◽  
Plate Geometry ◽  
Mixed Types

We study a uniform flow in a parallel plate geometry to model contaminant transport through a saturated porous medium in a semi-infinite domain in order to simulate an experimental apparatus mainly constituted by a chamber filled with a glass beads bed. The general solution of the advection dispersion equation in a porous medium was obtained by utilizing the Jacobiθ3Function. The analytical solution here presented has been provided when the inlet (Dirac) and the boundary conditions (Dirichelet, Neumann, and mixed types) are fixed. The proposed solution was used to study experimental data acquired by using a noninvasive technique.

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