Skip to main content

Determination of groundwater flow regimes based on the spatial non-local distribution of hydraulic gradient: Model and validation


The groundwater flow in natural aquifers can change from the Darcy flow to the non-Darcian flow due to a variety of causes, such as the increase of the Reynolds number in the highly permeable media or the decrease of the hydraulic gradient below a threshold in the low-permeability media, while the representative flow regime cannot be reliably determined using the traditional criteria. To address this challenge, this paper proposes a new term called the equivalent hydraulic gradient (EHG) by generalizing the differential form of the Darcy’s law using a spatial integral of the upstream hydraulic head. The nonlocal spatial variation of the hydraulic head difference between upstream and downstream zones is assumed to be the potential cause of the transition of the groundwater flow regimes. This assumption is analogous to the common assumption used for quantifying the anomalous pollutant transport in the geological media. Applications of this idea show that the EHG concept could distinguish three main flow regimes, namely the Super-Darcy flow, the Darcy flow, and the Sub-Darcy flow, although the Super-Darcy flow regime is rarely observed in the laboratory column flow experiments. Results of this study therefore shed lights on the interpretation of the fundamental dynamics of the groundwater moving in various heterogeneous aquifers, and may lead to the rebuilding of the hydrodynamics of the surface water, the groundwater, and the soil.

This is a preview of subscription content, access via your institution.


  1. de Graaf I. E. M., Gleeson T., van Beek L. P. H. R. et al. Environmental flow limits to global groundwater pumping [J]. Nature, 2019, 574(7776): 90–94.

    Article  Google Scholar 

  2. Bear J. Dynamics of fluids in porous media [M]. New York, USA: Elsevier, 1972.

    MATH  Google Scholar 

  3. Soni J. P., Islam N., Basak P. An experimental evaluation of non-Darcian flow in porous media [J]. Journal of Hydrology, 1978, 38(3–4): 231–241.

    Article  Google Scholar 

  4. Zimmerman R. W., Al-Yaarubi A., Pain C. C. et al. Nonlinear regimes of fluid flow in rock fractures [J]. International Journal of Rock Mechanics and Mining Sciences, 2004, 41(3): 164–169.

    Google Scholar 

  5. Cardenas M. B., Slottke D. T., Ketcham R. A. et al. Effects of inertia and directionality on flow and transport in a rough asymmetric fracture [J]. Journal of Geophysical Research, 2009, 114(B6): B06204.

    Article  Google Scholar 

  6. Quinn P. M., Cherry J. A., Parker B. L. Relationship between the critical Reynolds number and aperture for flow through single fractures: Evidence from published laboratory studies [J]. Journal of Hydrology, 2019, 581: 124384.

    Article  Google Scholar 

  7. Hölting B., Coldewey W. G. Hydrogeology [M]. Berlin, Germany: Springer, 2019.

    Book  Google Scholar 

  8. Darcy H. Les fontaines publiques de la ville de Dijon [M]. Paris, France: Victor Dalmont, 1856.

    Google Scholar 

  9. Hubbert M. K. Darcy’s law and the field equations of the flow of underground fluids [J]. International Association of Scientific Hydrology, 1957, 2(1): 23–59.

    Article  Google Scholar 

  10. Jr. Fetter C. W. Applied hydrogeology [M]. Fourth edition, London, UK: Pearson, 2014.

    Google Scholar 

  11. Reynolds O. An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels [J]. Proceedings of the Royal Society of London, 1883, 35(224–226): 84–99.

    MATH  Google Scholar 

  12. Chaudhary K., Cardenas M. B., Den W. et al. Pore geometry effects on intrapore viscous to inertial flows and on effective hydraulic parameters [J]. Water Resources Research, 2013, 49(2): 1149–1162.

    Article  Google Scholar 

  13. Dejam M., Hassanzadeh, H., Chen Z. Pre-Darcy flow in porous media [J]. Water Resources Research, 2017, 53(10): 8187–8210.

    Article  Google Scholar 

  14. Luo Q., Yang Y., Qian J. et al. Spring protection and sustainable management of groundwater resources in a spring field [J]. Journal of Hydrology, 2020, 582: 124498.

    Article  Google Scholar 

  15. Qian J., Zhan H., Chen Z. et al. Experimental study of solute transport under non-Darcian flow in a single fracture [J]. Journal of Hydrology, 2011, 399(3–4): 246–254.

    Article  Google Scholar 

  16. Sedghi-Asl M., Ansari I. Adoption of extended Dupuit-Fichheimer assumptions to non-Darcy flow problems [J]. Transport in Porous Media, 2016, 113(3): 457–469.

    MathSciNet  Article  Google Scholar 

  17. Siddiqui F., Soliman M. Y., House W. et al. Pre-darcy flow revisited under experimental investigation [J]. Journal of Analytical Science and Technology, 2016, 7: 2.

    Article  Google Scholar 

  18. Zhou J. Q., Li C., Wang L. et al. Effect of slippery boundary on solute transport in rough walled rock fractures under different flow regimes [J]. Journal of Hydrology, 2021, 598: 126456.

    Article  Google Scholar 

  19. Zeng Z., Grigg R. A criterion for non-Darcy flow in porous media [J]. Transport In Porous Media, 2006, 63(1): 57–69.

    Article  Google Scholar 

  20. Javadi M., Sharifzadeh M., Shahriar K. et al. Critical Reynolds number for nonlinear flow through rough-walled fractures: The role of shear processes [J]. Water Resources Research, 2014, 50(2): 1789–1804.

    Article  Google Scholar 

  21. Zhou J. Q., Hu S. H., Fang S. et al. Nonlinear flow behavior at low Reynolds numbers through rough-walled fractures subjected to normal compressive loading [J]. International Journal of Rock Mechanics and Mining Sciences, 2015, 80: 202–218.

    Article  Google Scholar 

  22. Wheatcraft S. W., Meerschaert M. M. Fractional conservation of mass [J]. Advances in Water Resources, 2008, 31(10): 1377–1381.

    Article  Google Scholar 

  23. Moutsopoulos K. N., Papaspyros I. N. E., Tsihrintzis V. A. Experimental investigation of inertial flow processes in porous media [J]. Journal of Hydrology, 2009, 374(3–4): 242–254.

    Article  Google Scholar 

  24. Edelen D. G. B. Nonlocal field theories (Eringen A. C. Continuum physics) [M]. New York, USA: Academic Press, 1976, 75–204.

    Google Scholar 

  25. Zhang Y., Benson D. A., Reeves D. M. Time and space nonlocalities underlying fractional derivative models: Distinction and literature review of applications [J]. Advances in Water Resources, 2009, 32(4): 561–581.

    Article  Google Scholar 

  26. Mueller E. V., Gallagher M. R., Skowronski N. et al. Approaches to modeling bed drag in pine forest litter for wildland fire applications [J]. Transport In Porous Media, 2021, 138: 637–660.

    Article  Google Scholar 

  27. Qian J. Z., Chen Z., Zhan H. B. et al. Solute transport in a filled single fracture under non-Darcian flow [J]. International Journal of Rock Mechanics and Mining Sciences, 2011, 48(1): 132–140.

    Article  Google Scholar 

  28. Zhou J. Q., Chen Y., Wang L. et al. Universal relationship between viscous and inertial permeability of geologic porous media [J]. Journal of Geophysical Research, 2019, 46(3): 1441–1448.

    Google Scholar 

  29. Sivanesapillai R., Steeb H., Hartmaier A. Transition of effective hydraulic properties from low to high Reynolds number flow in porous media [J]. Geophysical Research Letters, 2014, 41(14): 4920–4928.

    Article  Google Scholar 

Download references


This work was supported by the public welfare geological survey program of Anhui Province (Grant No. 2015-g-26), the Key Research and Development Program of Anhui Province (Grant No. 201904a07020071).

Author information

Authors and Affiliations


Corresponding author

Correspondence to Jia-zhong Qian.

Additional information

Project supported by the National Natural Science Foundation of China (Grants Nos. 41831289, 41877191 and 42072276).

Biography: Xiu-xuan Wang (1993-), Male, Ph. D.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Wang, Xx., Qian, Jz., Ma, L. et al. Determination of groundwater flow regimes based on the spatial non-local distribution of hydraulic gradient: Model and validation. J Hydrodyn 34, 299–307 (2022).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

Key words

  • Hydraulic head distribution
  • spatially non-local effect
  • flow regime
  • Forchheimer number
  • Reynolds number