An experimental study on scale dependency of fractional dispersion coefficient

Abstract

Six series of solute transport experiments were conducted with an objective to compare scale decency of fractional dispersion coefficient (Df) in space fractional advection–dispersion equation (s-FADE) with that of dispersion coefficient (D) in advection–dispersion equation (ADE) at three flow rates. The experiments were performed in a 2.5 m × 0.1 m × 0.6 m (length × width × height) sandbox filled with a homogeneous or heterogeneous soil. The parameters of the s-FADE and ADE were estimated using an inverse method. The results indicated that the scale dependency of Df was significantly less than that of D in the heterogeneous soil. The maximum variations of Df and D for the heterogeneous soil were observed at the flow rate of 200 cm3 min–1, in which the ratios of the maximum Df and D to the minimum ones were 6.009 and 39.039, respectively. However, in the homogeneous soil, the scale dependency of Df was relatively similar to that of D. In this soil, the ratios of the maximum Df and D to the minimum ones were 3.109 and 5.041, respectively, which were observed at the flow rate of 548 cm3 min–1. Contrary to Df and D, the v values of the s-FADE and ADE for both soils varied in relatively similar ranges at the flow rates studied. Also, the fractional differentiation order (α) of the s-FADE described properly the heterogeneity degrees of the soils used. In a nutshell, the s-FADE is an efficient solute transport model to explain the solute transport process in natural porous media with different heterogeneity degrees.

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Abbreviations

ADE:

Advection–dispersion equation

s-FADE:

Space fractional advection-dispersion equation

C :

Concentration

\( \overline{C} \) :

Average of measured concentrations

v :

Average pore-water velocity

D :

Dispersion coefficient

x :

Spatial coordinate (distance)

t :

Temporal coordinate (time)

L :

Size of spatial domain

OF :

Objective function

n :

Positive integer number

N :

Total number of measurement points

RMSE :

Root mean square error

r 2 :

Determination coefficient

meas :

Measured

calc :

Calculated

f :

Fractional

α:

Order of fractional differentiation (heterogeneity degree)

ξ :

Auxiliary variable

τ :

Integration variable

Γ(⋅) :

Gamma function

References

  1. Benson DA, Wheatcraft SW, Meerschaert MM (2000a) Application of a fractional advection-dispersion equation. Water Resour Res 36(6):1403–1412. https://doi.org/10.1029/2000WR900031

    Article  Google Scholar 

  2. Benson DA, Wheatcraft SW, Meerschaert MM (2000b) The fractional-order governing equation of Lévy motion. Water Resour Res 36(6):1413–1423. https://doi.org/10.1029/2000WR900032

    Article  Google Scholar 

  3. Benson DA, Schumer R, Meerschaert MM (2001) Fractional dispersion, Lévy motion, and the MADE tracer tests. Transp Porous Med 42:211–240. https://doi.org/10.1023/A:1006733002131

    Article  Google Scholar 

  4. Berkowitz B, Scher H (1995) On characterization of anomalous dispersion in porous and fractured media. Water Resour Res 31(6):1461–1466. https://doi.org/10.1029/95WR00483

    Article  Google Scholar 

  5. Berkowitz B, Cortis A, Dentz M, Scher H (2006) Modeling non-Fickian transport in geological formations as a continuous time random walk. Rev Geophys 44:RG2003. https://doi.org/10.1029/2005RG000178

    Article  Google Scholar 

  6. Chen J-S, Ni C-F, Liang C-P, Chiang C-C (2008) Analytical power series solution for contaminant transport with hyperbolic asymptotic distance dependent dispersivity. J Hydrol 362:142–149. https://doi.org/10.1016/j.jhydrol.2008.08.020

  7. Clarke DD, Meerschaert MM, Wheatcraft SW (2005) Fractal travel time estimates for dispersive contaminants. Groundwater 43(2):401–407. https://doi.org/10.1111/j.1745-6584.2005.0025.x

    Article  Google Scholar 

  8. Cushman JH (1987) Development of stochastic partial differential equations for subsurface hydrology. Stoch Hydrol Hydraul Risk Assess 1(4):241–262. https://doi.org/10.1007/BF01543097

  9. Gao G, Zhan H, Feng SH, Huang G, Mao X (2009) Comparison of alternative models for simulating anomalous solute transport in a large heterogeneous soil column. J Hydrol 377:391–404. https://doi.org/10.1016/j.jhydrol.2009.08.036

    Article  Google Scholar 

  10. Gelhar LW, Welty C, Rehfeldt KR (1992) A critical review of data on fieldscale dispersion in aquifer. Water Resour Res 28(7):1955–1974. https://doi.org/10.1029/92WR00607

    Article  Google Scholar 

  11. Ghavanloo E, Rafii-Tabar H, Fazelzadeh SA (2019) Essential concepts from nonlocal elasticity theory. In: Computational continuum mechanics of nanoscopic structures. Springer Tracts in Mechanical Engineering. Springer, Cham, pp 241–260

    Google Scholar 

  12. Guerrero JSP, Skaggs TH (2010) Analytical solution for one-dimensional advection–dispersion transport equation with distance-dependent coefficients. J Hydrol 390:57–65. https://doi.org/10.1016/j.jhydrol.2010.06.030

    Article  Google Scholar 

  13. Haggerty R, Gorelick SM (1995) Multiple-rate mass transfer for modeling diffusion and surface reactions in media with pore-scale heterogeneity. Water Resour Res 31(10):2383–2400. https://doi.org/10.1029/95WR10583

    Article  Google Scholar 

  14. Hokmabadi NN, Sarfarazi V, Moshrefifar MR (2016) Investigation of separation non-persistent faults in fracture mechanism of rock bridge. Civil Eng J 2(7):348–357

    Article  Google Scholar 

  15. Huang K, Toride N, Van Genuchten MT (1995) Experimental investigation of solute transport in large, homogeneous and heterogeneous, saturated soil columns. Transp Porous Med 18:283–302. https://doi.org/10.1007/BF00616936

    Article  Google Scholar 

  16. Huang G, Huang Q, Zhan H (2006) Evidence of one-dimensional scale-dependent fractional advection–dispersion. J Contam Hydrol 85(1):53–71. https://doi.org/10.1016/j.jconhyd.2005.12.007

    Article  Google Scholar 

  17. Huang Q, Huang G, Zhan H (2008) A finite element solution for the fractional advection–dispersion equation. Adv Water Resour 31:1578–1589. https://doi.org/10.1016/j.advwatres.2008.07.002

    Article  Google Scholar 

  18. Kelly JF, Meerschaert MM (2019) The fractional advection-dispersion equation for contaminant transport. In: Tarasov VE (ed) Application in physics, part B, 1st edn. De Gruyter, Berlin, Boston, pp 129–150. https://doi.org/10.1515/9783110571721-006

    Google Scholar 

  19. Khafagy MM, Abd-Elmegeed MA, Hassan AE (2020) Simulation of reactive transport in fractured geologic media using random-walk particle tracking method. Arab J Geosci. 13. https://doi.org/10.1007/s12517-019-4952-5

  20. Khan N, Gaurav D, Kandl T (2013) Performance evaluation of Levenberg-Marquardt technique in error reduction for diabetes condition classification. Procedia Comput Sci 18(2629):2637–2637. https://doi.org/10.1016/j.procs.2013.05.455

    Article  Google Scholar 

  21. Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. Elsevier, Amsterdam

    Google Scholar 

  22. Lee J, Rolle M, Kitanidis PK (2018) Longitudinal dispersion coefficients for numerical modeling of groundwater solute transport in heterogeneous formations. J Contam Hydrol 212:41–54. https://doi.org/10.1016/j.jconhyd.2017.09.004

    Article  Google Scholar 

  23. Lu S, Molz FJ (2002) Possible problems of scale dependency in applications of the three-dimensional fractional advection-dispersion equation to natural porous media. Water Resour Res. 38:4-1–4-7. https://doi.org/10.1029/2001WR000624

    Article  Google Scholar 

  24. Martinez FSJ, Pachepsky YA, Rawls WJ (2010) Modelling solute transport in soil columns using advective–dispersive equations with fractional spatial derivatives. Adv Eng Softw 41:4–8. https://doi.org/10.1016/j.advengsoft.2008.12.015

    Article  Google Scholar 

  25. Mehdinejadiani B (2017) Estimating the solute transport parameters of the spatial fractional advection-dispersion equation using Bees Algorithm. J Contam Hydrol 203:51–61. https://doi.org/10.1016/j.jconhyd.2017.06.004

    Article  Google Scholar 

  26. Mehdinejadiani B, Naseri AA, Jafari H, Ghanbarzadeh A, Baleanu D (2013) A mathematical model for simulation of a water table profile between two parallel subsurface drains using fractional derivatives. Comput Math Appl 66:785–794. https://doi.org/10.1016/j.camwa.2013.01.002

    Article  Google Scholar 

  27. Moradi G, Mehdinejadiani B (2018) Modeling solute transport in homogeneous and heterogeneous porous media using spatial fractional advection-dispersion equation. Soil Water Res 13:18–28. https://doi.org/10.17221/245/2016-SWR

    Article  Google Scholar 

  28. Pachepsky Y, Benson D, Rawls W (2000) Simulating scale-dependent solute transport in soils with the fractional advective–dispersive equation. Soil Sci Soc Am J 64(4):1234–1243. https://doi.org/10.2136/sssaj2000.6441234x

    Article  Google Scholar 

  29. Pang L, Hunt B (2001) Solutions and verification of a scale-dependent dispersion model. J Contam Hydrol 53:21–39. https://doi.org/10.1016/S0169-7722(01)00134-6

    Article  Google Scholar 

  30. Pickens JF, Grisak GE (1981) Modeling of scale-dependent dispersion in hydrogeologic systems. Water Resour Res 17(4):1191–1211. https://doi.org/10.1029/WR017i006p01701

    Article  Google Scholar 

  31. Saleem HA, Subyani AM, Elfeki A (2019) Solute transport model for groundwater contamination in Wadi Bani Malik, Jeddah. Saudi Arabia. Arab J Geosci. 12. https://doi.org/10.1007/s12517-019-4319-y

  32. Sanskrityayn A, Kumar N (2016) Analytical solution of advection–diffusion equation in heterogeneous infinite medium using Green’s function method. J Earth Syst Sci 125(8):1713–1723. https://doi.org/10.1007/s12040-016-0756-0

    Article  Google Scholar 

  33. Schumer R, Benson DA, Meerschaert MM, Wheatcraft SW (2001) Eulerian derivation of the fractional advection–dispersion equation. J Contam. Hydrol 48(1):69–88. https://doi.org/10.1016/S0169-7722(00)00170-4

    Article  Google Scholar 

  34. Schumer R, Meerschaert MM, Baemuer B (2009) Fractional advection-dispersion equations for modeling transport at Earth surface. J Geophys Res 114:F00A07. https://doi.org/10.1029/2008JF001246

    Article  Google Scholar 

  35. Sharma PK, Agarwal P, Mehdinejadiani B (2020) Study on non-Fickian behavior for solute transport through porous media. ISH J Hydraul Eng.:1–9. https://doi.org/10.1080/09715010.2020.1727783

  36. Singh MK, Das P (2015) Scale dependent solute dispersion with linear isotherm in heterogeneous medium. J Hydrol 520:289–299. https://doi.org/10.1016/j.jhydrol.2014.11.061

    Article  Google Scholar 

  37. Wałowski G (2018) Experimental assessment of porous material anisotropy and its effect on gas permeability. Civil Eng J 4(4):906–915. https://doi.org/10.28991/cej-0309143

    Article  Google Scholar 

  38. Xiong Y, Huang G, Huang Q (2006) Modeling solute transport in one-dimensional homogeneous and heterogeneous soil columns with continuous time random walk. J Contam Hydrol 86(3):163–175. https://doi.org/10.1016/j.jconhyd.2006.03.001

    Article  Google Scholar 

  39. Yates SR (1992) An analytical solution for one-dimensional transport in porous media with an exponential dispersion function. Water Resour Res 52:2149–2154. https://doi.org/10.1029/92WR01006

    Article  Google Scholar 

  40. Zhang Y, Benson DA, Reeves DM (2009) Time and space nonlocality underlying fractional-derivative models: distinction and literature review of filed applications. Adv Water Resour 32:561–581. https://doi.org/10.1016/j.advwatres.2009.01.008

    Article  Google Scholar 

  41. Zhang Y, Meerschaert MM, Neupauer RM (2016) Backward fractional advection dispersion model for contaminant source prediction. Water Resour Res 52:2462–2473. https://doi.org/10.1002/2015WR018515

    Article  Google Scholar 

  42. Zhou L, Selim HM (2003) Application of the fractional advection-dispersion equation in porous media. Soil Sci Soc Am J 67:1079–1084. https://doi.org/10.2136/sssaj2003.1079

    Article  Google Scholar 

  43. Zhou R, Zhan H, Chen K, Peng X (2018) Transport in a fully coupled asymmetric stratified system: comparison of scale dependent and independent dispersion schemes. J Hydrol X 1:100001. https://doi.org/10.1016/j.hydroa.2018.10.001

    Article  Google Scholar 

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Acknowledgments

The authors would like to express thanks for financial support from University of Kurdistan (UOK). The authors also greatly appreciate anonymous reviewers for their valuable comments and suggestions to improve the manuscript.

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Correspondence to Behrouz Mehdinejadiani.

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Responsible Editor: Broder J. Merkel

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Moradi, G., Mehdinejadiani, B. An experimental study on scale dependency of fractional dispersion coefficient. Arab J Geosci 13, 409 (2020). https://doi.org/10.1007/s12517-020-05438-z

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Keywords

  • Fractional dispersion coefficient
  • Fractional order
  • Non-Fickian transport
  • Non-locality
  • Scale dependency