A Revisit of Delayed Yield Phenomenon in Unconfined Aquifer

Transcription

1 A Revisit of Delayed Yield Phenomenon in Unconfined Aquifer Deqiang Mao, Li Wan, Tian Chyi J. Yeh, Cheng Haw Lee, Kuo Chin Hsu, Jet Chau Wen, and Wenxi Lu Water Resour. Res. doi: /2010wr009326,2011 National Central University, Taiwan, ROC. 11/21/2014

2 Unconfined Aquifer Pumping Test Unsaturated zone Saturated zone Variably saturated governing equation h θ h [ Kh (, x) ( h+ z)] = ωss( x) + = ( ωss( x) + Ch (, x) ) t t t Kh ( ) = Kexp( αh) s θ( h) = θ + ( θ θ )exp( βh) r s r 21st El Dia de Agua 2

3 Drawdown (m) 10 z= 1.5m z= 3.0m z= 6.0m z= 7.1m z= 7.5m Hantush z=1.5m z=3.0m S-shaped Drawdown Curve Drawdown (m) Hantush z=1.5m z=3.0m z=6.0m z= 1.5m z= 3.0m z= 6.0m z= 7.1m z= 7.5m 10-3 Hantush z=6.0m a) 10-4 b) Time (min) Time (min) Figure1. Log-log drawdown time curves at five different elevations (z =1.5, 3.0, 6.0, 7.1, 7.5 m) at a) r = 5 m and b) r = 30 m. Solid lines denote the results based on the solution by Hantush at three observation elevations in the saturated zone z = 1.5, z = 3.0, z = 6.0m. In Figure a, the difference between the Hantush solution at z=1.5 m and z=3.0 m is not distinguishable. In Figure b, the solutions are the same at all three elevations.

4 Previous Study Boulton[1954, 1963] use an empirical delay coefficient to the specific yield term to represent the slow water release process. 2 1 S y t α( t γ) 2 0 s s S s s + = +α e d γ r r r T t T t 21st El Dia de Agua 4

5 Based on the concept of instantaneous and complete drainage at the water table, Neuman [1972] use downward hydraulic head gradient below the water table to produce the S-shaped curve. Delayed Water Table response The free-surface boundary condition for the water table s s ξ Kr nr + KZ nz = Sy I nz r z t 2 2 s Kr s s s Kr + + K 2 z = S 2 s r r r z t

6 S-shaped Drawdown Curve 10 0 Drawdown (m) Early time Horizontal flow Estimate Ss Intermediate time Vertical> horizontal flow Late time Horizontal flow Water table falls Estimate Sy Time (min) 21st El Dia de Agua 6

7 Nwankwor et al. [1992] explained the S-shaped timedrawdown behavior as a consequence of changes in vertical hydraulic gradients and expansion of capillary fringe. Narasimhan and Zhu [1993] advocated that unsaturated zone above the water table is important. It must be included in the analysis. Mathias and Butler[2006], Tartakovsky and Neuman [2007] includes the unsaturated flow to improve the analytical solution.

8 Rate Change of Storage h θ h [ Kh (, x) ( h+ z)] = ωs( ) ( ( ) (, )) s x + = ωss x + Chx t t t ( ωs ( x) Ch (, x) ) Ω= + s h t Rate change in storage represents the net flux in a unit volume of the porous media per time! Ss: expansion of water and compaction of porous media. C (h): drainage of water from pores. 21st El Dia de Agua 8

9 Rate Change of Storage t= 0.02min t= 0.07min t= 0.3min t= 1.0min t= 3.0min Elevation (m) Elevation (m) r= 2.5m r= 5.0m r= 10.0m r= 20.0m r= 30.0m Elevation (m) r= 2.5m r= 5.0m r= 10.0m r= 20.0m r= 30.0m a) 1 b) 1 c) Ω (1/min) Ω (1/min) Ω (1/min) Early time Intermediate time Late time Drainage of initial unsaturated zone Elastic effect Drainage of pores 21st El Dia de Agua 9 falling water table

10 1D Example Fully saturated model Exponential model 100 Drawdown (cm) 10 1 z=10.0cm z=40.0cm z=70.0cm 0 Q Time (min) Drawdown curve 21st El Dia de Agua 10

11 1D Example Elevation (cm) WT WT t= 0.01min t= 3.0min t=100.0min Cumulative volume of water (cm 2 ) Compaction of aquifer Initiallly unsaturated zone Lowering of water table Ω (1/min) Rate change of storage Time (min) Cumulative water release 21st El Dia de Agua 11

12 Effect of Initial Unsaturated Zone W/ unsaturated zone W/o unsaturated zone 100 Drawdown (cm) 10 1 z=20.0cm z=50.0cm Q Time (min) Drawdown curve 21st El Dia de Agua 12

13 What is the Cause of S-shaped drawdown-time curve? It is merely the transition of water release mechanisms during vertical flow from 1) expansion of water/compaction of the porous medium to 2) drainage of water from the unsaturated zone above initial water table, and 3) initially saturated pores as the water table falls during the pumping of the aquifer. The widely accepted terms delayed yield and delayed water table response do not elucidate the transition of the two water release mechanisms and these terms may be misleading! 21st El Dia de Agua 13

14 Effect of Parameter Heterogeneity 1st order approximation H H H H(,) x t = H(,) x t + p(,) x t = H(,) x t + f() x + s() x + a() x + ln K ( x) ln S ( x) ln α( x) H H H b( x) + ts( x) + tr( x) ln β( x) ln θ ( x) ln θ ( x) Head covariance Cross correlation s R = J R J + J R J + J R J + J R J + J R J + J R J r T T T T T T pp pf ff pf ps ss ps pa aa pa pb bb pb pt t t pt pt t t pt S S s ss s r rr r ρ ( x, x,) t pf i j Rpf ( xi, x j, t) Jpf ( xi, x j, t) R ff ( xi, x j) = = σ ( x,) t σ σ ( x,) t σ p i f p i f 21st El Dia de Agua 14

15 Effect of Parameter Heterogeneity K S S S α Drawdown(m) Mean drawdown var=1.0 var.=0.1 Drawdown(m) Mean drawdown var=1.0 var.=0.1 Drawdown(m) Mean drawdown var=1.0 var.= Time(min) Time(min) Time(min) β θ S θ r Drawdown(m) Mean drawdown var=1.0 var.=0.1 Drawdown(m) Mean drawdown var=1.0 var.=0.1 Drawdown(m) Mean drawdown var=1.0 var.= Time(min) Time(min) Time(min) 21st El Dia de Agua 15

16 z z Cross Correlation Analysis Between head and Ks ρ hf Obs ρ hf Obs Time=0.08min x Pw Time=10.0min x Pw WT WT cross correlation cross correlation z ρ hf Obs Time=1000.0min x Pw WT cross correlation st El Dia de Agua 16

17 Cross Correlation Analysis Between head and S s Time=0.08min z Obs Pw WT cross correlation x Time=1.0min z Obs Pw WT cross correlation z x Time=10.0min Obs x Pw WT cross correlation 21st El Dia de Agua

18 Cross Correlation Analysis Between head and θ s ρ hθs Time=10min z Obs Pw WT cross correlation x ρ hθs Time=1000min z Obs Pw WT cross correlation x Cross correlation analysis shows an observation location in the aquifer is not equally influenced by heterogeneity everywhere in the aquifer. As a consequence, applications of a model assuming homogeneity to estimation of parameters may require a large number of spatial observations in order to yield representative parameter values. 21st El Dia de Agua 18

19 Conclusion Transition of water release mechanisms and vertical flow explain the S-shaped drawdown curve. Heterogeneous variably saturated governing equation h θ h [ Kh (, x) ( h+ z)] = ωss( x) + = ( ωss( x) + Ch (, x) ) t t t considers the transition of water release mechanisms and heterogeneity would provide a more realistic representation of flow processes in unconfined aquifer. Variability in the saturated zone parameters has greater impacts than that in the vadose zone on the S curve. Hydraulic tomography would be a viable approach for delineating heterogeneity in unconfined aquifers. 21st El Dia de Agua 19

20 FLOW TO A WELL IN A HETEROGENEOUS UNCONFINED AQUIFER: INSIGHTS FROM INTERMEDIATE SCALE SANDBOX EXPERIMENTS Steven J. Berg and Walter A. Illman University of Waterloo Canada

21 Figure 1: Photograph of the sandbox showing all sensor locations ( = pressure transducers; without pressure transducers; = tensiometers; and x = water content probes) and the various layers that were packed. Sensor locations are approximate.

22 Figure 2: Schematic diagram showing the ports that were pumped for the cross-hole pumping tests. Solid black circles represent locations of pressure transducers and solid grey circles represent ports that were not instrumented. Solid squares are pumping locations used for the estimation of homogeneous and heterogeneous K and Ss fields. Dashed squares are pumping locations for the validation of the results from transient hydraulic tomography.

23 Figure 4: a) K and b) Ss tomograms computed using the transient hydraulic tomography algorithm of Zhu and Yeh [2005] SSLE with 8 crosshole pumping tests. Solid squares are pumping locations used for transient hydraulic tomography dashed squares are pumping locations for the validation of the results from transient hydraulic tomography.

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25 Figure 5: a) Simulated versus observed drawdowns from 8 cross-hole pumping tests used for calibration purposes and b) additional tests used for the validation of the K and S s tomograms.

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27 Figure 6: Moisture characteristic curves determined through the hanging column method of: a) F-35; b) F-45; c) F110; d) Sil-co-Sil 53; e) Sil-co-Sil 106; and f) F35 and F45 matched simultaneously. The squares represent F35 and gradient symbols represent F35.

28 Figure 7: Schematic diagram of the sandbox showing an array of sensors ( = pressure transducers; = port without pressure transducers; = tensiometers; and x = water content sensors) utilized to monitor the pumping test in a heterogeneous unconfined aquifer. The box indicates the port at which the unconfined pumping test was performed.

29 Figure 9: Spatial distribution of: a) pressure head and b) volumetric water content in the upper half of the sandbox with time during the unconfined aquifer pumping test at port 3. Symbols indicate the position of various sensors used to monitor the pumping test ( = pressure transducers; = tensiometers; and x = water content sensors)).

30 Case 1. Homogeneous vadose Zone and homogeneous aquifer Figure 10: Simulated (solid line) and observed (dashed line) drawdown from selected: a) pressure transducers; and, b) tensiometers during the unconfined aquifer pumping test (Case 1). A comparison at all ports is available online as Figure S2.

31 Case 2. Homogeneous vadose Zone and heterogeneous aquifer Figure 11: Simulated (solid line) and observed (dashed line) drawdown from selected: a) pressure transducers; and, b) tensiometers during the unconfined aquifer pumping test (Case 2). A comparison at all ports is available online as Figure S3.

32 Case 3. Heterogeneous vadose Zone and aquifer Figure 12: Simulated (solid line) and observed (dashed line) drawdown from selected: a) pressure transducers; and, b) tensiometers during the unconfined aquifer pumping test (Case 3). A comparison at all ports is available online as Figure S4.