This is the published version. Copyright Wiley [Commercial Publisher

Brookfield, Andrea E.

Jones, J. P.

Park, Y. J.

Sudicky, E. A.

English

KU ScholarWorks

Reply to comment by J.-P. Renaud et al. on ‘‘An assessment of thetracer-based approach to quantifying groundwater contributionsto streamflow’’E. A. Sudicky,1 J. P. Jones,1 A. E. Brookfield,1 and Y.-J. Park1Received 9 August 2006; revised 11 May 2007; accepted 18 May 2007; published 22 September 2007.Citation: Sudicky, E. A., J. P. Jones, A. E. Brookfield, and Y.-J. Park (2007), Reply to comment by J.-P. Renaud et al. on ‘‘Anassessment of the tracer-based approach to quantifying groundwater contributions to streamflow,’’ Water Resour. Res., 43, W09602,doi:10.1029/2006WR005416.[1] We welcome the comments of Renaud et al. [2007]on our recent paper [Jones et al., 2006]. This dialogprovides us the opportunity to clarify some of the pointswe made regarding this important topic.[2] For the sake of simplicity, a two-component form ofthe mass balance equations discussed by Jones et al. [2006]can be written asQt ¼ Qo þ Qgw ð1aÞCtQt ¼ CoQo þ CgwQgw ð1bÞwhere Q [L3/T] is discharge, C [M/L3] is concentration andthe subscripts t, o and gw refer to the total flow as measuredin the stream, the surface runoff component arising directlyfrom the precipitation event, and the pre-event portion fromthe groundwater system, respectively. As was stated byJones et al. [2006], the Q values in (1a) and (1b) are ofteninterpreted by hydrogeologists and hydrologists to representa bulk hydraulic gradient–driven flow such as, for example,Darcian-type subsurface flow. Renaud et al. [2007] statethat, in addition to the hydraulic gradient driving force,dispersive/diffusive processes influence the motion ofsubsurface water, including isotopically tagged watermolecules. We entirely agree with this statement. Indeed,a major point of Jones et al. [2006] was to illustrate theinfluence of mechanical dispersion and molecular diffusionon an interpretation based on (1a) and (1b) that lumps bothadvective (i.e., hydraulic gradient driven) and dispersive/diffusive processes. Because bulk Darcian subsurface flowis driven by hydraulic gradients, we then demonstrated thatthe implied lumping of advective and dispersive/diffusivetransport processes will lead to an overestimation of theactual Darcian, pre-event contribution to streamflow duringa rainfall event.[3] To further elaborate, let us expand the terms on theright-hand side of (1b) as follows:CtQt ¼ZGocoqo  qoDorcoh i no dGoþZGgwcgwqgw  qgwDgwrcgwh i ngw dGgw ð2Þwhere q [L/T] is the specific discharge, c [M/L3] isconcentration, q is porosity [], G [L2] is the surface-subsurface interface area, D [L2/T] is the hydrodynamicdispersion tensor, and n is the unit vector normal to G. Thesubscripts o and gw here denote overland (i.e., event) andgroundwater (i.e., pre-event) contributions, respectively, andwe have included a ‘‘surface’’ porosity qo to account forrivulet-like overland flow. Note that the hydrodynamicdispersion tensors in (2) combine the effects of mechanicaldispersion and molecular diffusion (i.e., the dispersive/diffusive processes), and the values of c and q appearing in(2) are local values that vary in space and time. Let usconsider the second term that appears on the right-hand sideof (2) for illustrative purposes:CgwQgw ¼ZGgwcgwqgw  qgwDgwrcgwh i ngw dGgw ð3Þwhere cgwqgw and qDgwrcgw are the advective anddispersive/diffusive subsurface fluxes, respectively, contri-buting to the total mass flux CgwQgw. Clearly, the pre-eventtotal mass flux is influenced by hydrodynamic dispersion asdriven by concentration gradients that develop at thestream-subsurface interface as well as the magnitude ofqgwDgw. Values of Qgw calculated from (1a) and the lumpedform of (1b) will therefore be inflated from a purely Darcian(hydraulic gradient-driven) flow contribution unless thehydrodynamic dispersion flux is explicitly accounted for.Our inclusion of Figure 2 for the hypothetical cross sectionfor different values of the dispersion parameters wasprecisely intended to demonstrate this point.[4] Now consider a hypothetical situation where thedispersive/diffusive flux is negligible (Dgw = 0), recogniz-1Department of Earth and Environmental Sciences, University ofWaterloo, Waterloo, Ontario, Canada.Copyright 2007 by the American Geophysical Union.0043-1397/07/2006WR005416W09602WATER RESOURCES RESEARCH, VOL. 43, W09602, doi:10.1029/2006WR005416, 20071 of 2ing that at least molecular diffusion will always be present.Equation (3) would reduce toCgwQgw ¼ZGgwcgwqgwh i ngw dGgw ð4ÞIn the absence of hydrodynamic dispersion, the cgw term isconstant and would therefore equal Cgw in (1b) such that itcan be moved outside of the integral. We are then left withQgw ¼ZGgwqgwh i ngw dGgw ð5Þwhich is by definition the total groundwater contribution tostreamflow that is solely driven by hydraulic gradientsacross the stream-subsurface interface. This is why we setthe dispersion/diffusion parameters to zero in order tocompute the actual hydraulically driven contributions tostreamflow in Figures 11 to 15 of Jones et al. [2006], andto contrast these values with those obtained from a‘‘lumped’’ interpretation based on (1a) and (1b) withdispersion included in the model. Our conclusions drawnfrom these figures therefore remain valid. That is, unless thehydrodynamic dispersion flux is explicitly accounted for ina tracer-based approach to hydrograph separation, thecalculated values of Qgw will be inflated. We would alsolike to point out that, in a given time step in InHM, q iscalculated from the solution of the mixed form of Richards’equation before being used in the tracer transport computa-tions. Therefore any alteration of the dispersive transportparameters in the model will not affect the value of q.[5] Regarding our division of the pre-event waters for theBorden rainfall-runoff experiment into separate unsaturatedand saturated components, we would point out that one ofour ancillary goals was is to determine the source zones ofall of the waters contributing to streamflow generation andthat this simply entailed adding an additional unsaturatedzone tracer in the model that is used to tag the movement ofthe vadose zone water that existed prior to the rainfall event.As was shown in Figure 8 of Jones et al. [2006], the pre-event water initially residing below the water table (i.e., thesaturated zone) did not significantly contribute to thestreamflow produced after the onset of the rainfall event,at least on the basis of the hydraulic gradients that devel-oped. Note that dispersive/diffusive processes were includedin the simulation results provided in Figure 8 and that theconcentration gradients in the saturated zone below thechannel are clearly nonnegligible. The primary source ofthe hydraulically driven pre-event contribution to stream-flow came from the unsaturated zone because of thecapillary fringe effect, although this quantity was also smallcompared to the ‘‘lumped’’ pre-event estimate obtainedfrom the direct application of (1a) and (1b) by Jones etal. [2006]. We do concede that these findings were notemphasized in the paper.[6] Renaud et al. [2007] also bring up a point concerningthe use oxygen isotopes. When isotopically tagged watermolecules diffuse (and mechanically disperse) from thesubsurface to the surface, this does indeed represent themovement of these molecules; however, this does notnecessarily represent the bulk movement of water asdescribed by Darcy’s law, and we again point out that themotion of the tagged water molecules comprise bothadvective (i.e., hydraulically driven) and dispersive/diffusive components. This latter quantity will impact thevalues of the Q terms inferred from lumped mass balanceequations such as (1a) and (1b). That is, a large portion of thewater appearing in the stream may appear to be ‘‘old’’ water,but not all of this pre-event water was hydraulically driveninto the stream as many hydrologists and hydrogeologistscommonly assume.[7] Finally, we concur with Renaud et al. [2007] thattracer-based research has significantly contributed to theadvancement of hydrological sciences and will continue todo so in the future. It needs to be strongly emphasized herethat we are not questioning the veracity of tracer data itself.Instead, we simply questioned the manner in which the dataare commonly interpreted. We remain convinced that theuse of lumped mass balance equations such (1a) and (1b)need to be adapted to account for (or at least approximate)dispersive/diffusive processes if they are to produce a directestimate of the hydraulically driven pre-event contribution,and thus resolve the old water paradox. This is a topic offuture work.ReferencesJones, J. P., E. A. Sudicky, A. E. Brookfield, and Y.-J. Park (2006), Anassessment of the tracer-based approach to quantifying groundwater con-tributions to streamflow, Water Resour. Res., 42, W02407, doi:10.1029/2005WR004130.Renaud, J.-P., H. L. Cloke, and M. Weiler (2007), Comment on ‘‘Anassessment of the tracer-based approach to quantifying groundwater con-tributions to streamflow’’ by J. P. Jones et al., Water Resour. Res., 43,W09601, doi:10.1029/2006WR005157.A. E. Brookfield, J. P. Jones, Y.-J. Park, and E. A. Sudicky, Departmentof Earth and Environmental Sciences, University of Waterloo, Waterloo,ON, Canada N2L 3G1. (sudicky@sciborg.uwaterloo.ca)2 of 2W09602 SUDICKY ET AL.: COMMENTARY W09602

Reply to comment by J.-P. Renaud et al. on “An assessment of the tracer-based approach to quantifying groundwater contributions to streamflow”

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Reply to comment by J.-P. Renaud et al. on “An assessment of the tracer-based approach to quantifying groundwater contributions to streamflow”

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