Abstract
In the present study, analytical solutions of the advection dispersion equation (ADE) with spatially dependent concave and convex dispersivity are obtained within the fractal and the Euclidean frameworks by using the extended Fourier series method. The dispersion coefficient is considered to be proportional to the nth power of a non-homogeneous quadratic spatial function, where the index n is considered to vary between 0 and 1.5 so that the spatial dependence of dispersivity remains within the limit to describe the heterogeneity in the fractal framework. Real values like \( n = \) 0.5 and 1.5 are considered to delineate heterogeneity of the aquifer in the fractal framework, whereas integral values like n = 1 represent the same in the Euclidean sense. A concave or convex variation is free from demanding a limiting value as in the case of linear variation, hence it is more appropriate in the ambience of many disciplines in which ADE is used. In this study, concentration at the source site remains uniform until the source is present and becomes zero once it is annihilated forever. The analytical solutions, validated through the respective numerical solutions, are obtained in the form of an extended Fourier series with only first five terms. They are convergent to the desired concentration pattern and are stable with the Peclet number. It has been possible because of the formulation of a new Sturm–Liouville problem with advective information. The analytical solutions obtained in this paper are novel.
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Abbreviations
- \( a_{1} \) :
-
heterogeneity parameter \( ({\text{L}}) \)
- \( a_{2} \) :
-
heterogeneity parameter \( ({\text{L}}^{ - 2} ) \)
- \( b_{0} \) :
-
constant
- \( b_{1} \) :
-
non-dimensional heterogeneity parameter \( ( = a_{1} \ell ) \)
- \( b_{2} \) :
-
non-dimensional heterogeneity parameter \( ( = a_{2} \ell^{2} ) \)
- \( c \) :
-
concentration variable \( ({\text{ML}}^{ - 3} ) \)
- \( c_{0} \) :
-
reference concentration parameter \( ({\text{ML}}^{ - 3} ) \)
- \( C \) :
-
non-dimensional concentration variable
- \( C^{*} \) :
-
non-dimensional concentration variable
- \( D_{0} \) :
-
uniform dispersion coefficient in homogeneous medium \( ({\text{L}}^{ 2} / {\text{T}}) \)
- \( D \) :
-
variable dispersion coefficient in heterogeneous medium \( ({\text{L}}^{ 2} / {\text{T}}) \)
- \( \varvec{D} \) :
-
diagonal matrix of order \( m \times p \)
- \( \varvec{I} \) :
-
identity matrix of order \( m \times p \)
- \( \ell \) :
-
length of domain (L)
- N :
-
number of terms in the EFS
- \( n \) :
-
index of the quadratic expression in equation (3)
- \( Pe \) :
-
Peclet number
- \( \varvec{Q} \) :
-
matrix of order \( m \times p \)
- \( \varvec{R} \) :
-
matrix of order \( m \times p \)
- \( \varvec{R}_{1} \) :
-
matrix of eigenvectors of order \( m \times p \)
- \( s \) :
-
Laplace variable
- \( t \) :
-
time variable
- \( t_{0} \) :
-
time when the source has been removed
- \( T \) :
-
non-dimensional time variable
- \( T_{0} \) :
-
non-dimensional source removal time
- \( u \) :
-
variable velocity in heterogeneous medium
- \( u_{0} \) :
-
uniform velocity in homogeneous medium \( ({\text{L/T}}) \)
- \( x \) :
-
position variable
- \( X \) :
-
non-dimensional position variable
- \( \beta_{m} \) :
-
time-dependent coefficients of EFS
- \( \psi_{m} \) :
-
orthogonal eigenfunctions of SLP
- \( \phi_{m} \) :
-
orthonormal eigenfunctions of SLP
- \( \lambda_{m} \) :
-
eigenvalues of SLP
- \( \nu_{m} \) :
-
eigenvalues of \( \varvec{B} \)
- \( \alpha_{m} \) :
-
roots of transcendental equation
References
Adler P M 1985 Transport processes in fractals, II, stokes flow in fractal capillary networks; Int. J. Multiph. Flow 11 241–254.
Basu S K, Kumar N and Srivastava J P 2010 Modeling NPK release from spherically coated fertilizer granules; Simul. Model. Pract. Theory 18 820–835.
Bear J and Bachmat Y 1967 A generalized theory on hydrodynamic dispersion in porous media; Int. Assoc. Sci. Hydrol. Publ. 72, OSTI ID: 6035908.
Bharati V K, Singh V P, Sanskrityayn A and Kumar N 2017 Analytical solution of advection diffusion equation with spatially dependent dispersivity; J. Eng. Mech. 143(11) 1–11.
Bharati V K, Singh V P, Sanskrityayn A and Kumar N 2018 Analytical solutions for solute transport from varying pulse source along porous media flow with spatial dispersivity in fractal & Euclidean framework; Eur. J. Mech. B Fluids 72 410–421.
Cassol M, Wortmann S and Rizza U 2009 Analytic modeling of two-dimensional transient atmospheric pollutant dispersion by double GITT and Laplace transform technique; Env. Model. Softw. 24 144–151.
Castellões F V, Quaresma J N N and Cotta R M 2010 Convective heat transfer enhancement in low Reynolds number flows with wavy walls; Int. J. Heat Mass Transf. 53 2022–2034.
Chen J-S, Ni C F and Liang C P 2008 Analytical power series solution to the two-dimensional advection-dispersion equation with distance-dependent dispersivites; Hydrol. Process. 22(24) 4670–4678.
Chen J, Chen J, Liu C, Liang C and Lin C 2011 Analytical solutions to two-dimensional advection-dispersion in cylindrical coordinates in finite domain subject to first- and third-type inlet boundary conditions; J. Hydrol. 405 522–531.
Cotta R M 1990 Hybrid numerical-analytical approach to nonlinear diffusion problems; Numer. Heat Transfer B Fundam. 127 217–226.
Cotta R M 1993 Integral transforms in computational heat and fluid flow; CRC Press, Boca Raton, USA.
Cotta R M, Ungs M J and Mikhailov M D 2003 Contaminant transport in finite fractured porous medium: Integral transforms and lumped-differential formulations; Ann. Nucl. Energy 30 261–285.
Cotta R M, Cotta B P, Naveira-Cotta C P and Cotta-Pereira G 2010 Hybrid integral transforms analysis of the bio-heat equation with variable properties; Int. J. Thermal Sci. 49 1510–1516.
Cussler E L 1997 Diffusion: Mass transfer in fluid systems; Cambridge University Press, Cambridge, UK, pp. 142–184.
Dale P D, Sherrat J A and Maini P K 1996 A mathematical model for collagen fiber formation during foetal and adult dermal wound healing; Proc. Roy. Soc. London B 263 653–660.
Di Federico V and Neuman S P 1998 Transport in multiscale log conductivity fields with truncated power variograms; Water Resour. Res. 34(5) 963–973.
Gao G, Zhan H, Feng S, Fu B, Ma Y and Huang G 2010 A new mobile-immobile model (MIM) for reactive solute transport with scale-dependent dispersion; Water Resour. Res. 46 W08533, https://doi.org/10.1029/2009wr00870.
Gelhar L W, Welty C and Rehfeldt K R 1992 A critical review of data on field-scale dispersion in aquifers; Water Resour. Res. 28(7) 1955–1974.
Guenther R B and Lee J W 1988 Partial differential equations of mathematics physics and integral equation; Prentice-Hall, Englewood Califfs, NJ.
Guerrero J S P, Pimentel L C G, Skaggs T H and Th van Genuchten M 2009 Analytical solution of the advection-diffusion transport equation using a change-of-variable and integral transform technique; Int. J. Heat Mass Transf. 52 3297–3304.
Guerrero J S P, Pimentel L C G and Skaggs T H 2013 Analytical solution for the advection- dispersion transport equation in layered media; Int. J. Heat Mass Transf. 56 274–282.
Haberman R 1987 Elementary applied partial differential equations; Prentice Hall, New Jersey.
Hiester N K and Vermeulen T 1952 Saturation performance of ion-exchange and adsorption columns; Chem. Eng. Process. 48 505–516.
Houghton P A, Madurawe R U and Hatton T A 1988 Convective and dispersive models for dispersed phase axial mixing, the significance of poly-dispersivity effects in liquid-liquid contactors; Chem. Eng. Sci. 43 617–639.
Huang K, Th Van Genuchten M and Zhang R 1996 Exact solutions for one-dimensional transport with asymptotic scale-dependent dispersion; Appl. Math. Model. 20 298–308.
Hunt B 1998 Contaminant source solutions with scale-dependent dispersivities; J. Hydrol. Eng. 3(4) 268–275.
Hunt B 2002 Scale-dependent dispersion from a pit; J. Hydrol. 7(2) 168–174.
Kinzelbach W P and Ackerer P 1986 Modelisation de la propogation d’ un champ d’ écoulement transitoire; Hydrogeologie 2 197–206.
Kumar A, Jaiswal D K and Kumar N 2009 Analytical solution of one-dimensional advection- diffusion equation with variable coefficients in a finite domain; J. Earth Syst. Sci. 118(5) 539–549.
Kreyszig E 2014 Advanced engineering mathematics; Wiley, New Delhi.
Liu C, Szecsody J E, Jonh M Z and Ball W P 2000 Use of the generalized integral transform method for solving equations of solute transport in porous media; Adv.Water Resour. 23 483–492.
Metheron G and De Marsily G 1980 Is transport in porous always diffusive? A counter example; Water Resour. Res. 16(5) 901–917.
Molz F J, Güven O and Melville J G 1983 An examination of scale-dependent dispersion coefficients; Ground Water 21 715–725.
Najafzadeh M, Ghaemi A and Emamgholizadeh S 2018 Prediction of water quality parameters using evolutionary computing-based formulations; Int. J. Env. Sci. Technol., https://doi.org/10.1007/s13762-018-2049-4.
Neuman S P 1990 Universal scaling of hydraulic conductivities and dispersivities in geologic media; Water Resour. Res. 26(8) 1749–1758.
Neuman S P 2016 Comment on “Is unique scaling of aquifer macrodispersivity supported by field data?” By A Zech, S Attinger, V Cvetkovic, G Dagan, P Dietrich, A Fiori, Y Rubin and G Teutsch; Water Resour. Res. 52 4199–4202.
Pang L and Hunt B 2001 Solutions and verification of a scale-dependent dispersion model; J. Contam. Hydrol. 53(1–2) 21–39.
Pickens J F and Grisak G E 1981 Modeling of scale-dependent dispersion in hydrogeologic systems; Water Resour. Res. 1701–1711.
Qiu Y, Deng B and Kim C N 2011 Analytical solution for spatially dependent solute transport in streams with storage zone; ASCE J. Hydrol. Eng. 16 689–694.
Rehfeldt K R and Gelhar L W 1993 Stochastic analysis of dispersion in unsteady flow in heterogeneous aquifers; Water Resour. Res. 28 2085–2099.
Sauty J P 1980 An analysis of hydrodispersion transfer in aquifers; Water Resour. Res. 16 145–158.
Scheidegger A E 1954 Statistical hydrodynamics in porous media; J. Appl. Phys. 25(8) 994–1001.
Schugart R C, Friedman A, Zhao R and Sen C K 2008 Wound angiogenesis as a function of tissue oxygen tension: A mathematical model; Proc. Natl. Acad. Sci. 105 2628–2633.
Sudicky E A, Cherry J A and Frind E O 1983 Migration of contaminants in groundwater at a landfill: A case study, 4, a natural gradient dispersion test; J. Hydrol. 63 81–108.
Thorpe G R and Whitaker S 1992 Local mass and thermal equilibrium in ventilated grain bulks, part I: The development of heat and mass conservation equations; J. Stored Prod. Res. 28 15–27.
Thorpe G R, Ochoa-Tapia J A and Whitaker S 1991 The diffusion of moisture in food grains, II: Estimation of effective diffusivity; J. Stored Prod. Res. 27 11–30.
Todd A 1989 On the simulation of incompressible miscible displacement in a naturally fractured petroleum reservoir (Fench summary) RAIRO model; Math. Anal. Numer. 23 5–51.
Wadi A S, Dimian M F and Ibrahim F N 2014 Analytical solutions for one-dimensional advection-dispersion equation of the pollutant concentration; J. Earth Syst. Sci. 123(6) 1317–1324.
Wheatcraft S W and Tyler S W 1988 An explanation of scale dependent dispersivity in heterogeneous aquifer using concepts of fractal geometry; Water Resour. Res. 24(4) 566–578.
Yadav S K, Kumar A, Jaiswal D K and Kumar N 2011 One-dimensional unsteady solute transport along unsteady flow through inhomogeneous medium; J. Earth Syst. Sci. 120 205–213.
Yates S R 1990 An analytical solution for one-dimensional transport in heterogeneous porous media; Water Resour. Res. 26 2331–2338.
You K and Zhan H 2013 New solutions for solute transport in a finite column with distance dependent dispersivities and time-dependent solute sources; J. Hydrol. 487 87–97.
Zech A, Attinger S, Cvetkovic V, Dagan G, Dietrich P, Fiori A, Rubin Y and Teutsch G 2015 Is unique scaling of aquifer macrodispersivity supported by field data?; Water Resour. Res. 51 7662–7679.
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The first and third authors acknowledge the University Grants Commission, Government of India for the financial and academic assistance in the form of a senior research fellowship.
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Appendix
Appendix
Here the analytical solutions are provided with seven and eight terms of the EFS in equation (26) using the concave variation of dispersivity for the index n = 1.5 in equation (3). Peclet number is \( {\text{Pe}} = \) 1.4 in the domain of length \( \ell = \) 1.0 km. Other input values are \( u_{0} = \) 0.14 km/yr and \( D_{0} = \) 0.1 km2/yr. The solution with seven terms in the presence of the source is as below:
where \( \alpha_{m} \) are the roots of the transcendental equation associated with (21), and the seven time-dependent coefficients may be obtained from equation (31) as
where \( \upsilon_{1} = 359.9983, \) \( \upsilon_{2} = 267.6400, \) \( \upsilon_{3} = 174.8626, \) \( \upsilon_{4} = 3.8192, \) \( \upsilon_{5} = 21.4275, \) \( \upsilon_{6} = 107.4537, \) \( \upsilon_{7} = 55.7066 \) are the seven eigenvalues of the matrix R in equation (28).
Similarly, the solution with eight terms in equation (26) may be written in the presence of the source as
where the eigenvalues of the matrix R in (28) are obtained as \( \upsilon_{1} = 481.1673, \) \( \upsilon_{2} = 373.6767, \) \( \upsilon_{3} = 260.3778, \) \( \upsilon_{4} = 176.3123, \) \( \upsilon_{5} = 3.8211, \) \( \upsilon_{6} = 21.4225, \) \( \upsilon_{7} = 55.8164, \) \( \upsilon_{8} = 107.1349 \), in terms of which the time-dependent coefficients of (A3) are obtained from equation (31) as
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Bharati, V.K., Singh, V.P., Sanskrityayn, A. et al. Analytical solution for solute transport from a pulse point source along a medium having concave/convex spatial dispersivity within fractal and Euclidean framework. J Earth Syst Sci 128, 203 (2019). https://doi.org/10.1007/s12040-019-1231-5
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DOI: https://doi.org/10.1007/s12040-019-1231-5