American Institute of Mathematical Sciences

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2013, 2013(special): 457-466. doi: 10.3934/proc.2013.2013.457

Analytical approach of one-dimensional solute transport through inhomogeneous semi-infinite porous domain for unsteady flow: Dispersion being proportional to square of velocity

Atul Kumar 1, and  R. R. Yadav 1,

1. 

Department of Mathematics and Astronomy, Lucknow University, Lucknow, 226007, Uttar Pradesh, India, India

Received  August 2012 Revised  December 2012 Published  November 2013

In this study, we present an analytical solution for solute transport in a semi-infinite inhomogeneous porous domain and a time-varying boundary condition. Dispersion is considered directly proportional to the square of velocity whereas the velocity is time and spatially dependent function. It is expressed in degenerate form. Initially the domain is solute free. The input condition is considered pulse type at the origin and flux type at the other end of the domain. Certain new independent variables are introduced through separate transformation to eliminate the variable coefficients of Advection Diffusion Equation (ADE) into constant coefficient. Laplace transform technique (LTT) is used to get the analytical solution of ADE concentration values are illustrated graphically.
Keywords: unsteady parameter., Advection-Diffusion Equation, inhomogeniety, semi-infinite medium, porous domain.
Mathematics Subject Classification: Primary: 37N10, 65L10; Secondary: 44A1.
Citation: Atul Kumar, R. R. Yadav. Analytical approach of one-dimensional solute transport through inhomogeneous semi-infinite porous domain for unsteady flow: Dispersion being proportional to square of velocity. Conference Publications, 2013, 2013 (special) : 457-466. doi: 10.3934/proc.2013.2013.457
References:
[1]

J. S. Chen, C. W. Liu, H. T. Hsu, C. M. Liao, A Laplace transformed power series solution for solute transport in a convergent flow field with scale-dependent dispersion, Water Resources Research, 39(8) (2003), doi:10.1029/2003WR002299.

[2]

J. S. Chen, Two-dimensional power series solution for non-axisymmetrical transport in a radially convergent tracer test with scale-dependent dispersion, Advances in Water Resources, 30(3) (2007), 430-438.

[3]

J. S. Chen, C. F. Ni and C. P. Liang, Analytical power series solutions to the two-dimensional advection-dispersion equation with distance-dependent dispersivitie, Hydrological Processe, 22(24 (2008), 46700-4678.

[4]

J. S. Chen and C. W. Liu, Generalized analytical solution for advection-dispersion equation in finite spatial domain with arbitrary time-dependent inlet boundary condition, Hydrol. Earth Syst. Sci. Discuss., 8 (2011), 4099-4120.

[5]

J. Crank, "The Mathematics of Diffusion," Oxford University Press, London, 2nd ed.

[6]

E. H. Ebach and R. White, Mixing of fluids flowing through beds of packed solids, Journal of American Institute of Chemical Engineering, 4 (1958), 161-164.

[7]

K. S. M. Essa, S. M. Etman and M. Embaby, New analytical solution of the dispersion equation, Atmospheric Research, 84 (2007), 337-344.

[8]

A. Fedi, M. Massabo, O. Paladino and R. Cianci, A New Analytical Solution for the 2D Advection-Dispersion Equation in Semi-Infinite and Laterally Bounded Domain, Applied Mathematical Sciences, 4(75) (2010), 3733-3747.

[9]

M. Th. van Genuchten, Non-equilibrium transport parameters from miscible displacement experiments, U. S. Salinity Laboratory, USDA, A. R. S., Riverside, CA, 119 (1981).

[10]

M. Th. van Genuchten and W. J. Alves, Analytical solutions of the one-dimensional convective-dispersive solute transport equation, Technical Bulletin No 1661, US Department of Agriculture, 1982.

[11]

J. S. P. Guerrero, L. C. G. Pimentel, T. H. Skaggs and M. Th. van Genuchten, Analytical solution of the advection-diffusion transport equation using a change-of-variable and integral transform technique, International Journal of Heat and Mass Transfer , 52 (2009), 3297-3304.

[12]

B. Hunt, Scale-dependent dispersion from a pit, Journal of Hydrologic Engineering, 7(4) (2002), 168-174.

[13]

D. K. Jaiswal, A. Kumar, N. Kumar and R. R. Yadav, Analytical solutions for temporally and spatially dependent solute dispersion of pulse type input concentration in one-dimensional semi-infinite media, Journal of Hydro-environment Research, 2 (1963), 257-263.

[14]

S. A. Kartha and R. Srivastava, Effect of immobile water content on contaminant transport in unsaturated zone, Journal of Hydro-environment Research, 1 (2009), 206-215.

[15]

I. Kocabas and M. R. Islam, Concentration and temperature transients in heterogeneous porous media. Part I: Linear transport, Journal Petroleum Science and Engineering, 26 (2000a), 221-233.

[16]

I. Kocabas and M. R. Islam, Concentration and temperature transients in heterogeneous porous media. Part II: Radial transport, Journal Petroleum Science and Engineering, 26 (2000b), 211-220.

[17]

A. Kumar, D. K. Jaiswal and N. Kumar, Analytical solutions of one-dimensional advection-diffusion equation with variable coefficients in a finite domain, Journal of Earth System Sciemce, 118(5) (2009), 539-549.

[18]

A. Kumar, D. K. Jaiswal and N. Kumar, Analytical solutions to one-dimensional advection- diffusion equation with variable coefficients in semi-infinite media, Journal of Hydrology, 380 (2010), 330-337.

[19]

D. Kuntz and P. Grathwohl, Comparision of steady-state and transient flow conditions on reactive transport of contaminants in the vadose soil zone, Journal of Hydrology, 369 (2009), 225-233.

[20]

F. T. Lindstrom and L. Boersma, Analytical solutions for convective-dispersive transport in confined aquifers with different initial and boundary conditions, Water Resources Research, 25 (1989), 241-256.

[21]

M. A. Marino, Flow against dispersion in non-adsorbing porous media, Journal of Hydrology, 37 (1978), 149-158.

[22]

D. M. Moreira, M. T. Vilhena, D. Buske and T. Tirabassi, The state-of-art of the GILTT method to simulate pollutant dispersion in the atmosphere, Journal of Hydrology, 92 (2009), 1-17.

[23]

A. Ogata, "Theory of dispersion in granular media,'' U. S. Geol. Sur. Prof. Paper 411I, 34, 1970.

[24]

A. Ogata and R. B. Bank, A solution of differential equation of longitudinal dispersion in porous media, U. S. Geol. Surv. Prof. Pap. 411, A1-A7, 1961.

[25]

J. P. Sauty, An analysis of hydrodispersive transfer in aquifer, Water Resource Research, 16(1) (1980), 145-158.

[26]

A. E. Scheidegger, "The Physics of Flow through Porous Media," University of Toronto Press, 1957.

[27]

M. K. Singh, P. Singh and V. P. Singh, Analytical Solution for Solute Transport along and against Time Dependent Source Concentration in Homogeneous Finite Aquifers, Adv. Theor. Appl. Mech., 3(3) (2010), 99-119.

[28]

P. Singh, One Dimensional Solute Transport Originating from a Exponentially Decay Type Point Source along Unsteady Flow through Heterogeneous Medium, Journal of Water Resource and Protection, 3 (2011), 590-597.

[29]

F. D. Smedt, Analytical solution and analysis of solute transport in rivers affected by diffusive transfer in the hyporheic zone, Journal of Hydrology, 339 (2007), 29-38.

[30]

N. Su, G. C. Sander, F. Liu, V. Anh and D. A. Barry, Similarity solutions for solute transport in fractal porous media using a time- and scale-dependent dispersivity, Applied Mathematical Modeling, 29 (2005), 852-870.

[31]

S. Wortmanna, M. T. Vilhenaa, D. M. Moreirab and D. Buske, A new analytical approach to simulate the pollutant dispersion in the PBL, Atmospheric Environment, 39 (2005), 2171-2178.

show all references

References:
[1]

J. S. Chen, C. W. Liu, H. T. Hsu, C. M. Liao, A Laplace transformed power series solution for solute transport in a convergent flow field with scale-dependent dispersion, Water Resources Research, 39(8) (2003), doi:10.1029/2003WR002299.

[2]

J. S. Chen, Two-dimensional power series solution for non-axisymmetrical transport in a radially convergent tracer test with scale-dependent dispersion, Advances in Water Resources, 30(3) (2007), 430-438.

[3]

J. S. Chen, C. F. Ni and C. P. Liang, Analytical power series solutions to the two-dimensional advection-dispersion equation with distance-dependent dispersivitie, Hydrological Processe, 22(24 (2008), 46700-4678.

[4]

J. S. Chen and C. W. Liu, Generalized analytical solution for advection-dispersion equation in finite spatial domain with arbitrary time-dependent inlet boundary condition, Hydrol. Earth Syst. Sci. Discuss., 8 (2011), 4099-4120.

[5]

J. Crank, "The Mathematics of Diffusion," Oxford University Press, London, 2nd ed.

[6]

E. H. Ebach and R. White, Mixing of fluids flowing through beds of packed solids, Journal of American Institute of Chemical Engineering, 4 (1958), 161-164.

[7]

K. S. M. Essa, S. M. Etman and M. Embaby, New analytical solution of the dispersion equation, Atmospheric Research, 84 (2007), 337-344.

[8]

A. Fedi, M. Massabo, O. Paladino and R. Cianci, A New Analytical Solution for the 2D Advection-Dispersion Equation in Semi-Infinite and Laterally Bounded Domain, Applied Mathematical Sciences, 4(75) (2010), 3733-3747.

[9]

M. Th. van Genuchten, Non-equilibrium transport parameters from miscible displacement experiments, U. S. Salinity Laboratory, USDA, A. R. S., Riverside, CA, 119 (1981).

[10]

M. Th. van Genuchten and W. J. Alves, Analytical solutions of the one-dimensional convective-dispersive solute transport equation, Technical Bulletin No 1661, US Department of Agriculture, 1982.

[11]

J. S. P. Guerrero, L. C. G. Pimentel, T. H. Skaggs and M. Th. van Genuchten, Analytical solution of the advection-diffusion transport equation using a change-of-variable and integral transform technique, International Journal of Heat and Mass Transfer , 52 (2009), 3297-3304.

[12]

B. Hunt, Scale-dependent dispersion from a pit, Journal of Hydrologic Engineering, 7(4) (2002), 168-174.

[13]

D. K. Jaiswal, A. Kumar, N. Kumar and R. R. Yadav, Analytical solutions for temporally and spatially dependent solute dispersion of pulse type input concentration in one-dimensional semi-infinite media, Journal of Hydro-environment Research, 2 (1963), 257-263.

[14]

S. A. Kartha and R. Srivastava, Effect of immobile water content on contaminant transport in unsaturated zone, Journal of Hydro-environment Research, 1 (2009), 206-215.

[15]

I. Kocabas and M. R. Islam, Concentration and temperature transients in heterogeneous porous media. Part I: Linear transport, Journal Petroleum Science and Engineering, 26 (2000a), 221-233.

[16]

I. Kocabas and M. R. Islam, Concentration and temperature transients in heterogeneous porous media. Part II: Radial transport, Journal Petroleum Science and Engineering, 26 (2000b), 211-220.

[17]

A. Kumar, D. K. Jaiswal and N. Kumar, Analytical solutions of one-dimensional advection-diffusion equation with variable coefficients in a finite domain, Journal of Earth System Sciemce, 118(5) (2009), 539-549.

[18]

A. Kumar, D. K. Jaiswal and N. Kumar, Analytical solutions to one-dimensional advection- diffusion equation with variable coefficients in semi-infinite media, Journal of Hydrology, 380 (2010), 330-337.

[19]

D. Kuntz and P. Grathwohl, Comparision of steady-state and transient flow conditions on reactive transport of contaminants in the vadose soil zone, Journal of Hydrology, 369 (2009), 225-233.

[20]

F. T. Lindstrom and L. Boersma, Analytical solutions for convective-dispersive transport in confined aquifers with different initial and boundary conditions, Water Resources Research, 25 (1989), 241-256.

[21]

M. A. Marino, Flow against dispersion in non-adsorbing porous media, Journal of Hydrology, 37 (1978), 149-158.

[22]

D. M. Moreira, M. T. Vilhena, D. Buske and T. Tirabassi, The state-of-art of the GILTT method to simulate pollutant dispersion in the atmosphere, Journal of Hydrology, 92 (2009), 1-17.

[23]

A. Ogata, "Theory of dispersion in granular media,'' U. S. Geol. Sur. Prof. Paper 411I, 34, 1970.

[24]

A. Ogata and R. B. Bank, A solution of differential equation of longitudinal dispersion in porous media, U. S. Geol. Surv. Prof. Pap. 411, A1-A7, 1961.

[25]

J. P. Sauty, An analysis of hydrodispersive transfer in aquifer, Water Resource Research, 16(1) (1980), 145-158.

[26]

A. E. Scheidegger, "The Physics of Flow through Porous Media," University of Toronto Press, 1957.

[27]

M. K. Singh, P. Singh and V. P. Singh, Analytical Solution for Solute Transport along and against Time Dependent Source Concentration in Homogeneous Finite Aquifers, Adv. Theor. Appl. Mech., 3(3) (2010), 99-119.

[28]

P. Singh, One Dimensional Solute Transport Originating from a Exponentially Decay Type Point Source along Unsteady Flow through Heterogeneous Medium, Journal of Water Resource and Protection, 3 (2011), 590-597.

[29]

F. D. Smedt, Analytical solution and analysis of solute transport in rivers affected by diffusive transfer in the hyporheic zone, Journal of Hydrology, 339 (2007), 29-38.

[30]

N. Su, G. C. Sander, F. Liu, V. Anh and D. A. Barry, Similarity solutions for solute transport in fractal porous media using a time- and scale-dependent dispersivity, Applied Mathematical Modeling, 29 (2005), 852-870.

[31]

S. Wortmanna, M. T. Vilhenaa, D. M. Moreirab and D. Buske, A new analytical approach to simulate the pollutant dispersion in the PBL, Atmospheric Environment, 39 (2005), 2171-2178.

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