Impact of Transient Flow on Subsurface Solute Transport with Exponentially Time-Dependent Flow Velocity
Publication: Journal of Hydrologic Engineering
Volume 23, Issue 7
Abstract
The groundwater flow velocity might be temporally variable instead of being a constant as most analytical solutions of solute transport in the subsurface commonly assume. This study investigates the impact of transient flow on solute transport in the subsurface with time-dependent groundwater flow velocity. This study is based on the analysis of breakthrough and leaching processes of solute transport in a one-dimensional (1D) setting. As an example, the flow velocity is assumed to follow an exponential function of time and eventually approaches its steady-state value. Analytical solutions of such models are obtained using the Laplace transform assuming a homogeneous media and Fickian type of dispersion, and the impacts of different parameters of the temporally exponential function of the groundwater flow velocity on solute transport are thoroughly analyzed. The results indicate that a larger power index in the temporally and exponentially decreasing velocity equation results in a faster solute transport process. A sensitivity analysis of parameters shows that the solute transport is most sensitive to the initial flow velocity for the case with exponentially increasing velocity, whereas it is most sensitive to the final steady-state velocity for the case with exponentially decreasing velocity. The general conclusion is that groundwater flow transiency usually has significant impacts on the solute transport process and should not be overlooked.
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Acknowledgments
This research was partially supported by the National Science Foundation of China (Grant Nos. 41372253, 41772259, and 41521001). The authors sincerely thank two anonymous reviewers for their critical and constructive comments which helped substantially improve the quality of the paper.
References
Basha, H. A., and F. S. El-Habel. 1993. “Analytical solution of the one-dimensional time-dependent transport equation.” Water Resour. Res. 29 (9): 3209–3214. https://doi.org/10.1029/93WR01038.
Benson, D. A. 1998. “The fractional advection-dispersion equation: Development and application.” Ph.D. dissertation, Univ. of Nevada.
Benson, D. A., and M. M. Meerschaert. 2009. “A simple and efficient random walk solution of multi-rate mobile/immobile mass transport equations.” Adv. Water Resour. 32 (4): 532–539. https://doi.org/10.1016/j.advwatres.2009.01.002.
Berkowitz, B., A. Cortis, M. Dentz, and H. Scher. 2006. “Modeling non-Fickian transport in geological formations as a continuous time random walk.” Rev. Geophys. 44 (2): 177–186. https://doi.org/10.1029/2005RG000178.
Berkowitz, B., J. Klafter, R. Metzler, and H. Scher. 2002. “Physical pictures of transport in heterogeneous media: Advection-dispersion, random-walk, and fractional derivative formulations.” Water Resour. Res. 38 (10): 9-1–9-12. https://doi.org/10.1029/2001WR001030.
Berkowitz, B., and H. Scher. 1995. “On characterization of anomalous dispersion in porous.” Water Resour. Res. 31 (6): 1461–1466. https://doi.org/10.1029/95WR00483.
Biggar, J. W., and D. R. Nielsen. 1976. “Spatial variability of the leaching characteristics of a field soil.” Water Resour. Res. 12 (1): 78–84. https://doi.org/10.1029/WR012i001p00078.
Bronswijk, J. J. B., W. Hamminga, and K. Oostindie. 1995. “Field-scale solute transport in a heavy clay soil.” Water Resour. Res. 31 (3): 517–526. https://doi.org/10.1029/94WR02534.
Brown, D. L., and W. D. Silvey. 1977. Artificial recharge to a freshwater-sensitive brackish-water sand aquifer, Norfolk, Virginia. Vol. 939 of US Geological Survey professional paper. Washington, DC: US Government Printing Office.
Cortis, A., Y. Chen, H. Scher, and B. Berkowitz. 2004. “Quantitative characterization of pore-scale disorder effects on transport in ‘homogeneous’ granular media.” Phys. Rev. E 70 (4): 041108. https://doi.org/10.1103/PhysRevE.70.041108.
Dagan, G. 1984. “Solute transport in heterogeneous porous formations.” J. Fluid Mech. 145: 151–177. https://doi.org/10.1017/S0022112084002858.
Feehley, C. E., C. Zheng, and F. J. Molz. 2000. “A dual-domain mass transfer approach for modeling solute transport in heterogeneous aquifers: Application to the Macrodispersion Experiment (MADE) site.” Water Resour. Res. 36 (9): 2501–2515. https://doi.org/10.1029/2000WR900148.
Fox, P. J. 2007. “Coupled large strain consolidation and solute transport. I: Model development.” J. Geotech. Geoenviron. Eng. 133 (1): 3–15. https://doi.org/10.1061/(ASCE)1090-0241(2007)133:1(3).
Gao, G., H. B. Zhan, S. Feng, B. Fu, and G. Huang. 2012. “A mobile-immobile model with an asymptotic scale-dependent dispersion function.” J. Hydrol. 424–425 (10): 172–183. https://doi.org/10.1016/j.jhydrol.2011.12.041.
Gao, G., H. B. Zhan, S. Feng, G. Huang, and X. Mao. 2009. “Comparison of alternative models for simulating anomalous solute transport in a large heterogeneous soil column.” J. Hydrol. 377 (3–4): 391–404. https://doi.org/10.1016/j.jhydrol.2009.08.036.
Gelhar, L. W., A. Mantoglou, C. Welty, and K. R. Rehfeldt. 1985. A review of field scale physical solute transport processes in saturated and unsaturated porous media. Palo Alto, CA: Electric Power Research Institute.
Haggerty, R., and S. M. Gorelick. 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.
Huang, Y. C., and H. D. Yeh. 2007. “The use of sensitivity analysis in on-line aquifer parameter estimation.” J. Hydrol. 335 (3–4): 406–418. https://doi.org/10.1016/j.jhydrol.2006.12.007.
Jaiswal, D. K., A. Kumar, and R. R. Yadav. 2011. “Analytical solution to the one-dimensional advection-diffusion equation with temporally dependent coefficients.” Water Resour. Prot. 3 (1): 76–84. https://doi.org/10.4236/jwarp.2011.31009.
Keely, J. F., and C. F. Tsang. 1983. “Velocity plots and capture zones of pumping centers for ground-water investigations.” Ground Water 21 (6): 701–714. https://doi.org/10.1111/j.1745-6584.1983.tb01941.x.
Kumar, A., D. K. Jaiswal, and N. Kumar. 2010. “Analytical solutions to one-dimensional advection-diffusion equation with variable coefficients in semi-infinite media.” J. Hydrol. 380 (3–4): 330–337. https://doi.org/10.1016/j.jhydrol.2009.11.008.
Kumar, A., D. K. Jaiswal, and N. Kumar. 2012. “One-dimensional solute dispersion along unsteady flow through a heterogeneous medium, dispersion being proportional to the square of velocity.” Hydrolog. Sci. J. 57 (6): 1223–1230. https://doi.org/10.1080/02626667.2012.695871.
Kumar, N., and M. Kumar. 1998. “Solute dispersion along unsteady groundwater flow in a semi-infinite aquifer.” Hydrol. Earth Syst. Sci. 2 (1): 93–100. https://doi.org/10.5194/hess-2-93-1998.
Kumar, S. G., M. Sekhar, and D. Misra. 2008. “Time-dependent dispersivity of linearly sorbing solutes in a single fracture with matrix diffusion.” J. Hydrol. Eng. 13 (4): 250–257. https://doi.org/10.1061/(ASCE)1084-0699(2008)13:4(250).
Lautz, L. K. 2008. “Estimating groundwater evapotranspiration rates using diurnal water-table fluctuations in a semi-arid riparian zone.” J. Hydrol. 16 (3): 483–497. https://doi.org/10.1007/s10040-007-0239-0.
Lee, J. G., P. J. Fox, and J. J. Lenhart. 2009. “Investigation of consolidation-induced solute transport. I: Effect of consolidation on transport parameters.” J. Geotech. Geoenviron. Eng. 135 (9): 1228–1238. https://doi.org/10.1061/(ASCE)GT.1943-5606.0000047.
Li, L. P., H. Y. Zhou, and J. J. Gómez-Hernández. 2011. “A comparative study of three-dimensional hydraulic conductivity upscaling at the macro-dispersion experiment (MADE) site, Columbus Air Force Base, Mississippi (USA).” J. Hydrol. 404 (3–4): 278–293. https://doi.org/10.1016/j.jhydrol.2011.05.001.
Liu, D., A. P. Jivkov, L. Wang, G. Si, and J. Yu. 2017. “Non-Fickian dispersive transport of strontium in laboratory-scale columns: Modelling and evaluation.” J. Hydrol. 549: 1–11. https://doi.org/10.1016/j.jhydrol.2017.03.053.
Loheide, S. P., J. J. Butler, and S. M. Gorelick. 2005. “Estimation of groundwater consumption by phreatophytes using diurnal water table fluctuations: A saturated-unsaturated flow assessment.” Water Resour. Res. 41 (7): 372–380. https://doi.org/10.1029/2005WR003942.
Markwalder, T. M., P. Grolimund, R. W. Seiler, F. Roth, and R. Aaslid. 1984. “Dependency of blood flow velocity in the middle cerebral artery on end-tidal carbon dioxide partial pressure: A transcranial ultrasound Doppler study.” J. Cereb. Blood Flow Metab. 4 (3): 368–372. https://doi.org/10.1038/jcbfm.1984.54.
Ogata, A., and R. B. Banks. 1961. A solution of the differential equation of longitudinal dispersion in porous media. Reston, VA: US Geological Survey.
Padilla, I. Y., T. C. J. Yeh, and M. H. Conklin. 1999. “The effect of water content on solute transport in unsaturated porous media.” Water Resour. Res. 35 (11): 3303–3313. https://doi.org/10.1029/1999WR900171.
Sanskrityayn, A., H. Suk, and N. Kumar. 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. https://doi.org/10.1016/j.jhydrol.2017.02.014.
Singh, M. K., S. Ahamad, and V. P. Singh. 2014. “One-dimensional uniform and time varying solute dispersion along transient groundwater flow in a semi-infinite aquifer.” Acta Geophys. 62 (4): 872–892. https://doi.org/10.2478/s11600-014-0208-7.
Singh, M. K., N. K. Mahato, and N. Kumar. 2015. “Pollutant’s horizontal dispersion along and against sinusoidally varying velocity from a pulse type point source.” Acta Geophys. 63 (1): 214–231. https://doi.org/10.2478/s11600-014-0244-3.
Singh, M. K., N. K. Mahato, and P. Kumar. 2011. “Comparative study of analytical solutions for time-dependent solute transport along unsteady groundwater flow in semi-infinite aquifer.” Int. J. Geosci. 2 (4): 457–467. https://doi.org/10.4236/ijg.2011.24048.
Singh, M. K., N. K. Mahato, and P. Singh. 2008. “Longitudinal dispersion with time-dependent source concentration in semi-infinite aquifer.” J. Earth Syst. Sci. 117 (6): 945–949. https://doi.org/10.1007/s12040-008-0079-x.
Singh, M. K., V. P. Singh, P. Singh, and D. Shukla. 2009. “Analytical solution for conservative solute transport in one-dimensional homogeneous porous formations with time-dependent velocity.” J. Eng. Mech. 135 (9): 1015–1021. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000018.
Sudicky, E. A. 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. https://doi.org/10.1029/WR022i013p02069.
Suk, H. 2016. “Generalized semi-analytical solutions to multispecies transport equation coupled with sequential first-order reaction network with spatially or temporally variable transport and decay coefficients.” Adv. Water Resour. 94: 412–423. https://doi.org/10.1016/j.advwatres.2016.06.004.
Thorbjarnarson, K. W., and D. M. Mackay. 1994. “A forced-gradient experiment on solute transport in the Borden aquifer. 2: Transport and dispersion of the conservative tracer.” Water Resour. Res. 30 (2): 385–399. https://doi.org/10.1029/93WR02652.
van Genuchten, M. T., and P. J. Wierenga. 1976. “Mass transfer studies in sorbing porous media. I: Analytical solutions.” Soil Sci. Soc. Am. J. 40 (4): 473–480. https://doi.org/10.2136/sssaj1976.03615995004000040011x.
Yadav, S. K., A. Kumar, and N. Kumar. 2012. “Horizontal solute transport from a pulse type source along temporally and spatially dependent flow: Analytical solution.” J. Hydrol. 412 (1): 193–199. https://doi.org/10.1016/j.jhydrol.2011.02.024.
Yadava, R. R., R. R. Vinda, and N. Kumar. 1990. “One-dimensional dispersion in unsteady flow in an adsorbing porous medium: An analytical solution.” Hydrol. Process. 4 (2): 189–196. https://doi.org/10.1002/hyp.3360040208.
Zaheer, M., Z. Wen, H. B. Zhan, X. L. Chen, and M. G. Jin. 2017. “An experimental study on solute transport in one-dimensional clay soil columns.” Geofluids 2017: 17. https://doi.org/10.1155/2017/6390607.
Zamani, K., and F. A. Bombardelli. 2014. “Analytical solutions of nonlinear and variable-parameter transport equations for verification of numerical solvers.” Environ. Fluid Mech. 14 (4): 711–742. https://doi.org/10.1007/s10652-013-9325-0.
Zhan, H. B., and S. W. Wheatcraft. 1996. “Macrodispersivity tensor for nonreactive solute transport in isotropic and anisotropic fractal porous media: Analytical solutions.” Water Resour. Res. 32 (12): 3461–3474. https://doi.org/10.1029/95WR02282.
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Journal of Hydrologic Engineering
Volume 23 • Issue 7 • July 2018
Copyright
©2018 American Society of Civil Engineers.
History
Received: Jul 16, 2017
Accepted: Feb 1, 2018
Published online: May 15, 2018
Published in print: Jul 1, 2018
Discussion open until: Oct 15, 2018
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