Comment on Experimental observations of saltwater up-coning

Transcription

1 1 Comment on Experimental observations of saltwater up-coning H. Zang 1,, D.A. Barry 2 and G.C. Hocking 3 1 Griffit Scool of Engineering, Griffit University, Gold Coast Campus, QLD 4222, Australia. Tel.: ; Fax: ong.zang@griffit.edu.au 2 Laboratoire de tecnologie écologique, Institut d ingénierie de l environnement, Station 2, Ecole polytecnique fédérale de Lausanne (EPFL), 1015 Lausanne, Switzerland. Tel.: ; Fax: andrew.barry@epfl.c 3 Department of Matematics and Statistics, Murdoc University, WA 6150, Australia. Tel.: ; Fax: g.ocking@murdoc.edu.au Submitted to Journal of Hydrology, 13 January, 2011 Autor to wom all correspondence sould be addressed.

2 2 Keywords: Steady witdrawal, Saltwater up-coning, Subcritical witdrawal, Critical witdrawal, Supercritical witdrawal, Aquifer pumping Werner et al. (2009) presented 2D experimental data on time-dependent saltwater upconing using controlled sand-tank experiments in wic freswater overlying a saltwater layer was pumped at a single extraction point, leading to saltwater up-coning. Te experimental setup imposed constant ead boundary conditions for bot fres- and saltwater. Te experimental results were compared to a sarp-interface perturbation-based approximate analytical solution to te governing model (Dagan and Bear, 1968). Tis approximation was derived assuming tat: (i) te fres-salt water interface is sarp, (ii) te interface extends to infinity, were it remains undisturbed, and (iii) it applies to any pumping rate, weter it is subcritical, critical or supercritical. Werner et al. (2009) noted tat few analytical solutions are available for up-coning, and so tey used te approximation of Dagan and Bear (1968) to compare wit teir experimental data altoug te experimental conditions and model assumptions do not coincide exactly. Te accuracy of te analytical approximation of Dagan and Bear (1968) is dependent on te movement of te freswater-saltwater interface relative to te initial eigt, d, of te witdrawal point above te interface. Teir approximation is considered reasonable for interface movement of up to about d/3. Te need to place te boundary condition at infinity was relaxed in a series of analytic and numerical analyses of up-coning and down-coning (Zang and Hocking, 1996; Zang et al., 1997, 1999, 2009). In tese studies, an impermeable boundary was placed symmetrically at a fixed distance, x L, from te pumping well, a situation tat is closer to te experimental setting of Werner et al. (2009) tan te model of Dagan and Bear (1968). At tese boundaries, te interface position is fixed. In brief, Zang and Hocking (1996) provided te analytical

3 3 solution for steady critical and subcritical witdrawal wen te pump is located at te top impermeable boundary of te flow domain; Zang et al. (1997) found te analytic solution for steady critical witdrawal for various pump locations; Zang et al. (1999) solved te timedependent interface response using te boundary element metod; wile Zang et al. (2009) provided te analytical solution for steady supercritical witdrawal from two layered fluids. Table 1 gives te relevant experimental and dimensionless parameters, te latter indicated by a superscripted asterisk. An important parameter is te critical pumping rate, wic is defined as te rate for wic te saltwater up-coning will just reac te extraction point. Supercritical flow rates, i.e., tose greater tan te critical flow rate, always result in saltwater breaktroug into te extracted water. For subcritical flow rates, saltwater never reaces te extraction point. Of course, te sarp interface assumption ignores mixing across te interface but neverteless te computed critical flow rate as obvious practical value. Te critical pumping rate was discussed by Werner et al. (2009). Tey observed for all teir experiments tat according to te definitions of Bear (1979) and Bear and Dagan (1964), initial up-coning plumes were expected to ave a convex sape (near te plume apex) and stable plumes were expected to develop. However, up-coning proceeded until te interface intercepted te well in te experiments of tis study, and terefore te steady-state conditions of criticality tat oters ave reported (e.g. Bower et al., 1999) do not appear to be transferable to te current analysis. Tat is, te experiments of Werner et al. (2009) do not ave convex saltwater up-coning sapes for most of teir experiments. For example, for Experiment 1, teir Fig. 3f sows saltwater breaktroug into te extraction well. Teir Fig. 4f sows te same beaviour for Experiment 2. For bot Experiments 1 and 2, te breaktroug sape was similar, as noted by Werner et al. (2009). Interestingly, teir Fig. 5f sows te saltwater cone is extended, wit a long tail reacing te extraction point, suggesting tat saltwater breaktroug into te extraction well is minimal. Tis is confirmed

4 4 by teir Fig. 9a, wic sows only a small increase in salinity in te pumped water. By contrast, for Experiment 4, teir Fig. 6f sows tat te pumping rate is clearly subcritical in tat te peak of te saltwater mound sows a convex sape. Tis figure and teir Fig. 9b sow, owever, tat some saltwater reaces te pumping well. In Table 1, q cr is te scaled dimensionless critical flow rate, wic was computed for a given impermeable boundary location, and a given pump location, s. Comparison of q and q cr in Table 1 sows tat te pumping rates in Experiments 1 and 2 were bot supercritical, tat for Experiment 3 was also supercritical, but close to critical, wile for Experiment 4 te pumping rate was clearly subcritical. Tese results are all consistent wit te experimental results sown in Figs. 3-6 of Werner et al. (2009), as discussed in te foregoing paragrap. Because Experiment 3 of Werner et al. (2009) is close to te critical pumping rate, it is possible to ceck furter te steady-state analytical solution given by Zang et al. (1997), wic was derived for tis case. For te above given apparatus dimensions and water depts, te critical interface sape for s = 0.43 and x L = 0.61 were computed using Eqs. (3.8) and (3.9) of Zang et al. (1997), giving te interface sape plotted in Fig. 1. A few interface locations in Fig. 5f (rigt side) of Werner et al. (2009) were traced by and and compared wit te calculated interface sape using te model of Zang et al. (1997). Fig. 1 sows close agreement between te experimental data and model predictions, altoug te analytical solution consistently over-predicts te data. It is possible tat tis is due to te difference in boundary conditions in te model and experiment. In te model, te interface is fixed at x L, but te flow above and below te interface comes from. Tis situation is probably a reasonable approximation for an experiment wit fixed ead conditions at x L. In te experiments, Werner et al. (2009) noted tat te conditions at te sides of te experimental apparatus were ead-dependent flux conditions. Suc a condition would provide more

5 5 resistance to flow tan a fixed-ead condition, and is consistent wit te over-prediction evident in te model s predictions. In Fig. 1, te dimensionless crest eigt (i.e., actual crest eigt scaled by a) given by te model is alf widt scaled by a) as c = Te model also predicts te dimensionless alf plume widt (actual w = 0.12 at z = s /2 = At tis same location, te corresponding data for Experiment 3 are c 0.35 ( c 33 cm, te last eigt measurement before breaktroug to te extraction, as read from Fig. 7 of Werner et al., 2009) and 0.07 (w = 7 cm from Table 4 in Werner et al., 2009,). Note tat in Fig. 2. of Werner et al. w (2009), W(t) is defined as te full widt of up-coning. However, a close examination of te up-coning plume in Fig. 5f and te data in Table 4 suggest tat alf-widts are listed in te latter, i.e., w = W/2 rater tan W is given in Table 4. Te model s estimate of w overpredicts te experimental measurement, consistent wit te over-prediction of te interface evident in Fig. 1. Overall, given te uncertainty in te experimental boundary condition noted by Werner et al. (2009), it is evident tat te model predictions are in good agreement wit te experimental data, and are far superior tan te predictions of te perturbation approximation of Dagan and Bear (1968), not surprisingly since te range of application of te latter approximation is limited. Previous studies (Zang and Hocking, 1996; Zang et al., 1997, 1999, 2009) ave sown tat te boundary location as a significant effect on up-coning predictions in twolayer, sarp interface models. Additionally, as noted above, tere is a discrepancy between te boundary conditions used in te analytical solution of Zang et al. (1997) and te experiments reported by Werner et al. (2009). Despite te difference in boundary conditions, te above analysis ad led to a caracterisation of te experiments into subcritical, critical and supercritical witdrawal cases tat accord wit te images given by Werner et al. (2009).

6 6 Furtermore, for te critical witdrawal case, te location of te steady interface and interface caracteristics predicted by te up-coning model of Zang et al. (1997) matc well te experimental data of Werner et al. (2009). Te over-prediction of te model is most likely due to te boundary flux condition of te experiments rater tan a fixed ead condition. Te boundary flux condition implies te presence of a resistance to flow at te boundary. Even so, we conclude from te comparison wit te experimental data tat te matematical conditions imposed in obtaining te solution are a reasonable approximation to Werner et al. (2009) s experimental setting. References Bear, J., Hydraulics of Groundwater. McGraw-Hill Book Company, New York, USA. 569 pp. Bear, J., Dagan, G., Some exact solutions of interface problems by means of te odograp metod. Journal of Geopysical Researc 69, Bower, J.W., Motz, L.H., Durden, D.W., Analytical solution for determining te critical condition of saltwater upconing in a leaky artesian aquifer. Journal of Hydrology 221, Dagan, G., Bear, J., Solving te problem of local interface upconing in a coastal aquifer by te metod of small perturbations. Journal of Hydraulic Researc 6, Werner, A.D., Jakovovic, D., Simmons C.T., Experimental observations of saltwater up-coning. Journal of Hydrology, Zang, H., Hocking, G.C., Witdrawal of layered fluid troug a line sink in porous media. Journal of te Australian Matematics Society, Series B 38,

7 7 Zang, H., Hocking, G.C., Axisymmetric flow in an oil reservoir of finite dept caused by a point sink above an oil-water interface. Journal of Engineering Matematics 32, Zang, H., Hocking, G.C., Barry, D.A., An analytical solution for critical witdrawal of layered fluid troug a line sink in a porous medium. Journal of te Australian Matematics Society, Series B 39, Zang, H., Barry, D.A., Hocking, G.C., Analysis of continuous and pulsed pumping of a preatic aquifer. Advances in Water Resources 22, Zang, H., Hocking, G.C., Seymour, B., Critical and supercritical witdrawals of layered fluid troug a line sink in porous media. Advances in Water Resources 32,

8 8 Table 1 Summary of te experimental parameters (Zang et al., 1997; Werner et al., 2009). Experiment Q (m 3 /s) d a s K (m/s) q s x L q cr Number (m) (m) (kg/m 3 ) Notation: Q is te pumping rate; d is te pump location above te initial interface position; a is te freswater layer tickness; s is te saltwater density; K = kg( s 0 )/μ is te relative ydraulic conductivity; 0 is te freswater density; is te water viscosity; g is te magnitude of gravitational acceleration; q = Q/(πKaB) is te scaled non-dimensional pumping rate; B is te tickness of te sand tank; te boundary (were te interface position is fixed); x L = x L /a is te non-dimensional location of s = d/a is te dimensionless vertical distance between te pumping well and te initial interface location and q cr is te critical pumping rate, i.e., te scaled rate for wic te saltwater will just reac te extraction point.

9 9 Figure Caption Fig. 1. Te modelled and measured interface sape comparison for te critical flow case (Experiment 3) of Werner et al. (2009). Te dimensionless orizontal distance from te pumping location is x = x/a, were x is te actual orizontal distance. Te corresponding vertical distance, measured from te point on te interface directly below te pump is z = z/a, were z is te actual vertical distance.

10 Fig