Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-16T01:20:07.923Z Has data issue: false hasContentIssue false
  • English
  • Français

Flow and residence time in a two-dimensional aquifer recharged by rainfall

Published online by Cambridge University Press:  23 April 2021

V. Jules Open the ORCID record for V. Jules [Opens in a new window] ,
E. Lajeunesse Open the ORCID record for E. Lajeunesse [Opens in a new window] ,
O. Devauchelle Open the ORCID record for O. Devauchelle [Opens in a new window] ,
A. Guérin ,
C. Jaupart  and
P.-Y. Lagrée
V. Jules
Affiliation:
Université de Paris, Institut de physique du globe de Paris, CNRS, F-75005Paris, France
E. Lajeunesse*
Affiliation:
Université de Paris, Institut de physique du globe de Paris, CNRS, F-75005Paris, France
O. Devauchelle
Affiliation:
Université de Paris, Institut de physique du globe de Paris, CNRS, F-75005Paris, France
A. Guérin
Affiliation:
MSC, Université Paris Diderot, CNRS-UMR 705775013Paris, France
C. Jaupart
Affiliation:
Université de Paris, Institut de physique du globe de Paris, CNRS, F-75005Paris, France
P.-Y. Lagrée
Affiliation:
Sorbonne Université, CNRS - UMR 7190, Institut Jean Le Rond d'Alembert, F-75005Paris, France
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate the flow of water in a two-dimensional laboratory aquifer recharged by artificial rainfall. As rainwater infiltrates, it forms a body of groundwater which can exit the aquifer only through one of its sides. The outlet, located high above the aquifer bottom, drives the flow upwards. Noting that the water table barely departs from the horizontal, we linearize the boundary condition at the free surface, and solve the flow equations in steady state. We find an approximate expression for the velocity potential, which accounts for the shape of the streamlines, and for the propagation of dye through the aquifer. Based on this theory, we calculate the travel time of water through the experiment, and find that its distribution decays exponentially, with a characteristic time that depends on the shape of the aquifer. We find that the hydrodynamic dispersion that occurs at the pore scale does not matter much for this distribution, which depends essentially on the geometry of the groundwater flow.

JFM classification

Low-Reynolds-number flows: Porous media Mass Transport: Dispersion Mixing: Mixing
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 917 , 25 June 2021 , A13
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Andreotti, B., Forterre, Y. & Pouliquen, O. 2013 Granular Media: Between Fluid and Solid. Cambridge University Press.CrossRefGoogle Scholar
Boussinesq, J. 1903 Sur un mode simple d’écoulement des nappes d'eau d'infiltration à lit horizontal, avec rebord vertical tout autour lorsqu'une partie de ce rebord est enlevée depuis la surface jusqu'au fond. CR Acad. Sci. 137 (5), 11.Google Scholar
Bresciani, E., Davy, P. & de Dreuzy, J.-R. 2014 Is the dupuit assumption suitable for predicting the groundwater seepage area in hillslopes? Water Resour. Res. 50 (3), 2394–2406.CrossRefGoogle Scholar
Brutsaert, W. & Nieber, J.L. 1977 Regionalized drought flow hydrographs from a mature glaciated plateau. Water Resour. Res. 13 (3), 637–643.CrossRefGoogle Scholar
Cardenas, M.B. 2007 Potential contribution of topography-driven regional groundwater flow to fractal stream chemistry: residence time distribution analysis of tóth flow. Geophys. Res. Lett. 34 (5), L05403.CrossRefGoogle Scholar
Cartwright, I., Cendón, D., Currell, M. & Meredith, K. 2017 A review of radioactive isotopes and other residence time tracers in understanding groundwater recharge: possibilities, challenges, and limitations. J. Hydrol. 555, 797–811.CrossRefGoogle Scholar
Charlaix, E., Hulin, J.P. & Plona, T.J. 1987 Experimental study of tracer dispersion in sintered glass porous materials of variable compaction. Phys. fluids 30 (6), 1690–1698.CrossRefGoogle Scholar
Dagan, G. 1964 Second order linearized theory of free-surface flow in porous media. La Houille Blanche 1964 (8), 901–910.CrossRefGoogle Scholar
Darcy, H. 1856 Les fontaines publiques de la ville de Dijon. Dalmont.Google Scholar
Davis, S.N. & Bentley, H.W. 1982 Dating Groundwater: A Short Review. ACS Publications.CrossRefGoogle Scholar
De Marsily, G. 1986 Quantitative hydrogeology. Paris School of Mines, Fontainebleau.Google Scholar
Dentz, M., Le Borgne, T., Englert, A. & Bijeljic, B. 2011 Mixing, spreading and reaction in heterogeneous media: a brief review. J. Contam. Hydrol. 120, 1–17.CrossRefGoogle ScholarPubMed
Devauchelle, O., Petroff, A.P., Seybold, H.F. & Rothman, D.H. 2012 Ramification of stream networks. Proc. Natl Acad. Sci. USA 109 (51), 20832–20836.CrossRefGoogle ScholarPubMed
Dupuit, J. 1863 Études théoriques et pratiques sur le mouvement des eaux dans les canaux découverts et à travers les terrains perméables. Dunod.Google Scholar
Farlow, S.J. 1993 Partial Differential Equations for Scientists and Engineers. Courier Corporation.Google Scholar
Gelhar, L.W., Welty, C. & Rehfeldt, K.R. 1992 A critical review of data on field-scale dispersion in aquifers. Water Resour. Res. 28 (7), 1955–1974.CrossRefGoogle Scholar
Gleeson, T., Befus, K.M., Jasechko, S., Luijendijk, E. & Cardenas, M.B. 2016 The global volume and distribution of modern groundwater. Nat. Geosci. 9 (2), 161–167.CrossRefGoogle Scholar
Godsey, S.E., et al. 2010 Generality of fractal 1/f scaling in catchment tracer time series, and its implications for catchment travel time distributions. Hydrol. Process. 24 (12), 1660–1671.CrossRefGoogle Scholar
Guérin, A. 2015 Dynamics of groundwater flow in an unconfined aquifer. Theses, Université Paris Diderot (Paris 7), Sorbonne Paris Cité.Google Scholar
Guérin, A., Devauchelle, O. & Lajeunesse, E. 2014 Response of a laboratory aquifer to rainfall. J. Fluid Mech. 759, R1.CrossRefGoogle Scholar
Guérin, A., Devauchelle, O., Robert, V., Kitou, T., Dessert, C., Quiquerez, A., Allemand, P. & Lajeunesse, É. 2019 Stream-discharge surges generated by groundwater flow. Geophys. Res. Lett. 46 (13), 7447–7455.CrossRefGoogle Scholar
Guyon, E., Hulin, J.P. & Petit, L. 2001 Hydrodynamique Physique. EDP Sciences/CNRS Editions.Google Scholar
Haitjema, H.M. & Mitchell-Bruker, S. 2005 Are water tables a subdued replica of the topography? Groundwater 43 (6), 781–786.Google ScholarPubMed
Hamming, R.W. 1973 Numerical methods for scientists and engineers, 2nd edition. McGraw Hill.Google Scholar
Haria, A.H. & Shand, P. 2004 Evidence for deep sub-surface flow routing in forested upland wales: implications for contaminant transport and stream flow generation. Hydrol. Earth Syst. Sci. Discuss. 8 (3), 334–344.CrossRefGoogle Scholar
Hecht, F. 2012 New development in freefem++. J. Numer. Math. 20 (3–4), 251–265.CrossRefGoogle Scholar
Hillel, D. 2003 Introduction to Environmental Soil Physics. Elsevier.Google Scholar
Ingebritsen, S.E. & Sanford, W.E. 1999 Groundwater in Geologic Processes. Cambridge University Press.Google Scholar
Kirchner, J.W., Feng, X. & Neal, C. 2000 Fractal stream chemistry and its implications for contaminant transport in catchments. Nature 403 (6769), 524–527.CrossRefGoogle ScholarPubMed
Le Borgne, T., de Dreuzy, J.-R., Davy, P. & Bour, O. 2007 Characterization of the velocity field organization in heterogeneous media by conditional correlation. Water Resour. Res. 43 (2), W02419.CrossRefGoogle Scholar
Lehr, J.H. 1963 Groundwater: flow toward an effluent stream. Science 140 (3573), 1318–1320.CrossRefGoogle ScholarPubMed
Lobkovsky, A.E., Jensen, B., Kudrolli, A. & Rothman, D.H. 2004 Threshold phenomena in erosion driven by subsurface flow. J. Geophys. Res. 109 (F4), F04010.Google Scholar
Maher, K. 2010 The dependence of chemical weathering rates on fluid residence time. Earth Planet. Sci. Lett. 294 (1–2), 101–110.CrossRefGoogle Scholar
Nauman, E.B. & Buffham, B.A. 1983 Mixing in Continuous Flow Systems. John Wiley and Sons.Google Scholar
Petroff, A.P., Devauchelle, O., Abrams, D.M., Lobkovsky, A.E., Kudrolli, A. & Rothman, D.H. 2011 Geometry of valley growth. J. Fluid Mech. 673 (6), 245–254.CrossRefGoogle Scholar
Polubarinova-Kochina, P.Y. 1962 Theory of Ground Water Movement. Princeton University Press.Google Scholar
Powers, W.L., Kirkham, D. & Snowden, G. 1967 Orthonormal function tables and the seepage of steady rain through soil bedding. J. Geophys. Res. 72 (24), 6225–6237.CrossRefGoogle Scholar
Read, W.W. 1993 Series solutions for Laplace's equation with nonhomogeneous mixed boundary conditions and irregular boundaries. Math. Comput. Model. 17 (12), 9–19.CrossRefGoogle Scholar
Rempe, D.M. & Dietrich, W.E. 2014 A bottom-up control on fresh-bedrock topography under landscapes. Proc. Natl Acad. Sci. USA 111 (18), 6576–6581.CrossRefGoogle ScholarPubMed
Rubinstein, L.I. 2000 The Stefan Problem, vol. 8. American Mathematical Society.Google Scholar
Saffman, P.G. 1959 A theory of dispersion in a porous medium. J. Fluid Mech. 6 (3), 321–349.CrossRefGoogle Scholar
Souzy, M., Lhuissier, H., Méheust, Y., Le Borgne, T. & Metzger, B. 2020 Velocity distributions, dispersion and stretching in three-dimensional porous media. J. Fluid Mech. 891, A16.CrossRefGoogle Scholar
Toth, J. 1963 A theoretical analysis of groundwater flow in small drainage basins. J. Geophys. Res. 68 (16), 4795–4812.CrossRefGoogle Scholar
Troch, P.A., et al. 2013 The importance of hydraulic groundwater theory in catchment hydrology: the legacy of Wilfried Brutsaert and Jean-Yves Parlange. Water Resour. Res. 49 (9), 5099–5116.CrossRefGoogle Scholar
Van de Giesen, N.C., Parlange, J.-Y. & Steenhuis, T.S. 1994 Transient flow to open drains: comparison of linearized solutions with and without the dupuit assumption. Water Resour. Res. 30 (11), 3033–3039.CrossRefGoogle Scholar
Varni, M. & Carrera, J. 1998 Simulation of groundwater age distributions. Water Resour. Res. 34 (12), 3271–3281.CrossRefGoogle Scholar
Zwietering, T.N. 1959 The degree of mixing in continuous flow systems. Chem. Engng Sci. 11 (1), 1–15.CrossRefGoogle Scholar

Jules et al. supplementary movie

See pdf file for movie caption

Download Jules et al. supplementary movie(Video)
Video 9 MB
Supplementary material: PDF

Jules et al. supplementary material

Caption for movie file
Download Jules et al. supplementary material(PDF)
PDF 192.4 KB