Skip to main content
SpringerLink
Log in

Analytical solution of two-dimensional advection–dispersion equation with spatio-temporal coefficients for point sources in an infinite medium using Green’s function method

  • Original Article
  • Published:
Environmental Fluid Mechanics Aims and scope Submit manuscript

Abstract

In the present study analytical solutions of a two-dimensional advection–dispersion equation (ADE) with spatially and temporally dependent longitudinal and lateral components of the dispersion coefficient and velocity are obtained using Green’s Function Method (GFM). These solutions describe solute transport in infinite horizontal groundwater flow, assimilating the spatio-temporal dependence of transport properties, dependence of dispersion coefficient on velocity, and the particulate heterogeneity of the aquifer. The solution is obtained in the general form of temporal dependence and the source term, from which solutions for instantaneous and continuous point sources are derived. The spatial dependence of groundwater velocity is considered non-homogeneous linear, whereas the dispersion coefficient is considered proportional to the square of spatial dependence of velocity. An asymptotically increasing temporal function is considered to illustrate the proposed solutions. The solutions are validated with the existing solutions derived from the proposed solutions in three special cases. The effect of spatially/temporally dependent heterogeneity on the solute transport is also demonstrated. To use the GFM, the ADE with spatio-temporally dependent coefficients is reduced to a dispersion equation with constant coefficients in terms of new position variables introduced through properly developed coordinate transformation equations. Also, a new time variable is introduced through a known transformation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

Abbreviations

a 1 :

Spatial dependence parameter in longitudinal direction, of dimension (L−1)

b 1 :

Spatial dependence parameter in lateral direction, of dimension (L−1)

a 2, b 2 :

Constants

c :

Solute concentration at (x, y), of dimension (ML−3)

C :

Solute concentration in new domain at (X, Y), of dimension (ML−3)

C i :

Initial concentration, of dimension (ML−3)

C 0 :

Reference concentration, of dimension (ML−3)

\( C^{*} \) :

Another concentration variable, of dimension (ML−3)

\( D_{x0} \) :

Longitudinal dispersion coefficient in homogeneous medium, of dimension (L2T−1)

\( D_{y0} \) :

Lateral dispersion coefficient in homogeneous medium, of dimension (L2T−1)

\( k \) :

Asymptotically varying parameter, of dimension (T)

\( m \) :

Temporal dependence parameter, of dimension (T−1)

\( M \) :

Injected pollutant mass, of dimension (ML−2)

\( q \) :

Source term in the old domain, of dimension (ML−3T−1)

\( q_{1} \) :

Transform form of the source term in new domain \( (X,Y,t) \), of dimension (ML−3T−1)

\( Q \) :

Transform form of the source term in new domain \( (X,Y,T) \), of dimension (ML−3T−1)

\( t \) :

Time variable, of dimension (T)

\( T \) :

New time variable, of dimension (T)

\( u_{0} \) :

Longitudinal velocity in homogeneous medium, of dimension (LT−1)

\( v_{0} \) :

Lateral velocity in homogeneous medium, of dimension (LT−1)

\( x,y \) :

Position variables in longitudinal and lateral direction, respectively, of dimension (L)

\( x_{0} ,y_{0} \) :

Injected source location in the domain, of dimension (L)

\( X,Y \) :

New position variables in longitudinal and lateral direction, respectively, of dimension (L)

\( \mu \) :

Decay term of dimension (T−1)

\( \delta ( \cdot ) \) :

Dirac delta function

\( t^{{\prime }} \) :

Dummy variable

\( t_{0} \) :

Dummy variable

\( \zeta \) :

Dummy variable

\( \xi_{1} \) :

Dummy variable

\( \xi_{2} \) :

Dummy variable

\( \chi_{1} \) :

Dummy variable

\( \chi_{2} \) :

Dummy variable

References

  1. Aral M, Liao B (1996) Analytical solutions for two-dimensional transport equations with time dependent dispersion coefficients. J Hydrol Eng 1(1):20–32

    Article  Google Scholar 

  2. Batu V (1993) A generalized two-dimensional analytical solute transport model in bounded media for flux-type finite multiple sources. Water Resour Res 29:2881–2892

    Article  Google Scholar 

  3. Bear J (1979) Hydraulics of groundwater. McGraw-Hill, New York

    Google Scholar 

  4. Chen K, Zhan H, Zhou R (2016) Subsurface solute transport with one-, two-, and three-dimensional arbitrary shape sources. J Contam Hydrol 190:44–57

    Article  Google Scholar 

  5. Crank J (1975) Mathematics of diffusion. Oxford University Press, UK

    Google Scholar 

  6. de Josselin G, de Jong G (1958) Longitudinal and transverse diffusion in granular deposits. Trans Am Geophys Union 39:67–74

    Article  Google Scholar 

  7. Domenico PA (1987) An analytical model for multidimensional transport of a decaying contaminant species. J Hydrol 91:49–58

    Article  Google Scholar 

  8. Domenico P, Robbins G (1984) A new method of contaminant plume analysis. Ground Water 23(4):476–485

    Article  Google Scholar 

  9. Fisher HB, List EJ, Koh RY, Imberger J, Brooks NH (1979) Mixing in inland and coastal waters. Academic Press, Cambridge

    Google Scholar 

  10. Garabedian SP, LeBlanch DR, Gelhar LW, Celia MA (1991) Large-scale natural-gradient tracer test in sand and gravel Cape Code Massachusettes, 2, Analysis of spatial moments for a non-reactive tracer. Water Resour Res 27(5):911–924

    Article  Google Scholar 

  11. Haberman R (1987) Elementry Applied Partial Differential Equations. Prentice-Hall, Englewood Cliffs

    Google Scholar 

  12. Hunt B (1998) Contaminant source solutions with scale-dependent dispersivities. J Hydrol Eng 3(4):268–275

    Article  Google Scholar 

  13. Leij FJ, Priesack E, Schaap MG (2000) Solute transport modeled with Green’s functions with application to persistent solute sources. J Contam Hydrol 41:155–173

    Article  Google Scholar 

  14. Park E, Zhan H (2001) Analytical solutions of contaminant transport from finite one-, two-, and three-dimensional sources in a finite-thickness aquifer. J Contam Hydrol 53:41–61

    Article  Google Scholar 

  15. Rehfeldt KR, Gelhar LW (1992) Stochastic analysis of dispersion in unsteady flow in heterogeneous aquifers. Water Resour Res 28(8):2085–2099

    Article  Google Scholar 

  16. Rumer RR (1962) Longitudinal dispersion in steady and unsteady flow. J Hydraul Div 88(HY4):147–172

    Google Scholar 

  17. Sagar B (1982) Dispersion in three dimensions: approximate analytical solutions. ASCE J Hydraul Div 108(HY1):47–62

    Google Scholar 

  18. Sanskrityayn A, Suk H, Kumar N (2017) Analytical solutions for solute transport in groundwater and riverine flow using Green’s function method and pertinent coordinate transformation method. J Hydrol 547:517–533

    Article  Google Scholar 

  19. Scheidegger AE (1957) The physics of flow through porous media. University of Toronto Press, Toronto

    Google Scholar 

  20. Serrano SE (1992) The form of the dispersion equation under recharge and variable velocity, and its analytical solution. Water Resour Res 28:1801–1808

    Article  Google Scholar 

  21. Serrano SE (1996) Hydrologic theory of dispersion in heterogeneous aquifers. J Hydrol Eng 1:144–151

    Article  Google Scholar 

  22. Singh MK, Singh P, Singh VP (2010) Analytical solution for two-dimensional solute transport in finite aquifer with time-dependent source concentration. J Eng Mech 136(10):1309–1315

    Article  Google Scholar 

  23. Stefanovic DL, Stefan HG (1997) Two-dimensional water quality model for unsteady advection diffusion of non-conservative substances. University of Minnesota St. Anthony Falls Laboratory Engineering Environmental and Geophysical Fluid Dynamics Project Report No. 412

  24. Sudicky EA (1986) A natural gradient experiment on solute transport in a sand aquifer: spatial variability of hydraulic conductivity and its role in the dispersion process. Water Resour Res 22(13):2069–2082

    Article  Google Scholar 

  25. Sykes JF, Pahwa SB, Lantz RB, Ward DS (1982) Numerical simulation of flow and contaminant migration at an extensively monitored landfill. Water Resour Res 18(6):1687–1704

    Article  Google Scholar 

  26. Wexler E (1992) Analytical solutions for a one-, two-, and three-dimensional transport in groundwater systems with uniform flow. In: Techniques of water resources investigations of the United States Geological Survey chap. B7, Book 3, Applications of hydraulics, 53–55

  27. Yadav SK, Kumar A, Kumar N (2012) Horizontal solute transport from a pulse type source along temporally and spatially dependent flow: analytical solution. J Hydrol 412–413:193–199

    Article  Google Scholar 

  28. Yeh GT (1981) AT123D: Analytical transient one-, two-, and three-dimensional simulation of waste transport in the aquifer system. Envir Sci Div 1439 Report ORNL-5602, Oak Ridge, Tennessee, USA

  29. Zoppou C, Knight JH (1999) Analytical solution of a spatially variable coefficient advection–diffusion equation in up to three dimensions. Appl Math Model 23:667–685

    Article  Google Scholar 

Download references

Acknowledgements

The first and third authors acknowledge their gratitude to University Grants Commission, Government of India for financial and academic assistance in the form of Senior Research Fellowship.

Author information

Authors and Affiliations

  1. Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi, 221005, India

    Abhishek Sanskrityayn, Vinod Kumar Bharati & Naveen Kumar

  2. Department of Biological Engineering and Agricultural Engineering and Zachry Department of Civil Engineering, Texas A &M University, 321 Scoates Hall, 2117 TAMU, College Station, TX, 77843-2117, USA

    Vijay P. Singh

Authors
  1. Abhishek Sanskrityayn
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vijay P. Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Vinod Kumar Bharati
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Naveen Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vijay P. Singh.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sanskrityayn, A., Singh, V.P., Bharati, V.K. et al. Analytical solution of two-dimensional advection–dispersion equation with spatio-temporal coefficients for point sources in an infinite medium using Green’s function method. Environ Fluid Mech 18, 739–757 (2018). https://doi.org/10.1007/s10652-018-9578-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10652-018-9578-8

Keywords

Search

Navigation