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A simple model for water table fluctuations in response to precipitation
Journal of Hydrology (2008) 356, 344– 349
available at www.sciencedirect.com
journal homepage: www.elsevier.com/locate/jhydrol
A simple model for water table fluctuations in response to precipitation Eungyu Park a b
a,*
, J.C. Parker
b
Department of Geology, Kyungpook National University, 1370 Sangyeok-dong, Buk-gu, Daegu, South Korea Civil and Environmental Engineering Department, University of Tennessee, Knoxville, TN 37996-2010, United States
Received 15 October 2007; received in revised form 18 April 2008; accepted 25 April 2008
KEYWORDS Water table fluctuations; Precipitation; Recharge; Discharge potential; Hongcheon Korea
A simple physically-based model is developed for quantifying groundwater fluctuations in response to precipitation time-series. A semi-analytical solution of the governing differential equation is derived for relevant initial conditions and an efficient numerical algorithm is presented. Model performance is assessed by comparing predicted and observed groundwater fluctuations over a multi-year period in response to precipitation data for a site in the Hongcheon area of South Korea. Groundwater fluctuation and precipitation time-series during the year 2001 are used for model calibration. The calibrated model is then used to predict groundwater fluctuations for 2002–2004 from measured daily precipitation data. Prediction variance is only slightly larger than the calibration variance. Results indicate that model parameters are stable over time and that reliable water level fluctuation predictions can be made by the model from actual or projected precipitation data following calibration for a limited time-series. ª 2008 Elsevier B.V. All rights reserved.
Summary
Introduction Measurements of groundwater level fluctuations in response to precipitation events can provide a practical means of estimating temporally and spatially variable groundwater recharge rates (Rasmussen and Andreasen, 1959; Sophocleous, 1991; Healy and Cook, 2002; Rai and Singh, 1995; Bierkens, 1998; Rai and Manglik, 1999; Knotters and Bierkens, 2000; Coulibaly et al., 2001; Rai et al., 2006). Furthermore, * Corresponding author. Tel.: +82 53 950 5356; fax: +82 53 950 5362. E-mail address: [email protected] (E. Park).
groundwater level fluctuations themselves can directly impact the migration potential of free-phase hydrocarbon and dissolved phase contaminant plumes and for evaluating contaminant volatilization rates from groundwater due to water table ‘‘pumping’’ (e.g., Parker et al., 1994; Parker, 2003). Rasmussen and Andreasen (1959) presented a simple model relating water level fluctuations in response to recharge events for the purpose of estimating groundwater recharge from precipitation events. The latter authors assumed that the change of water level over a given time multiplied by the specific yield is equal to the change in storage due to precipitation, extraction/injection, evapotranspiration, etc. Healy and Cook (2002) pointed out that
0022-1694/$ - see front matter ª 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jhydrol.2008.04.022
A simple model for water table fluctuations in response to precipitation the accuracy of this method is limited by its inability to account for discharge. Sophocleous (1991) proposed a practical way of estimating groundwater recharge using a hybrid water-fluctuation method in which estimation errors are mitigated based on a soil water balance. Rai and Singh (1995), Rai and Manglik (1999) and Rai et al. (2006) derived an analytical solution by linearizing the two-dimensional Boussinesq equation to predict time-varying water table height based on groundwater recharge as a function of time. Bierkens (1998) modeled water table fluctuation by solving a stochastic differential equation. Knotters and Bierkens (2000) developed a piecewise analytical approximation based on the empirical soil water balance equation. Coulibaly et al. (2001) showed the applicability of an artificial neural network model in predicting water table fluctuations. The objective of this paper is to present an extension of the Rasmussen and Andreasen model to describe water table fluctuations in response to precipitation time-series that exhibit piecewise constant characteristics in a discrete record. Up to now, several different analytical models have been developed including Hantush (1967), Rao and Sarma (1984), Latnopoulos (1985), Rai and Singh (1995), Rai and Manglik (1999) and Rai et al. (2006). Those analytical solutions require either constant (Rao and Sarma, 1984) or continuous analytical functions (Latnopoulos, 1985; Rai and Singh, 1995) or rather complicated piecewise functions (Rai and Manglik, 1999; Rai et al., 2006). In this study, a practical analytical model is developed to predict water table fluctuations based on discrete record of precipitation such as daily or monthly precipitation data. The relationship of the proposed model to the one-dimensional groundwater flow equation will be shown. We present a semi-analytical solution to the problem and demonstrate the model by calibrating it to a limited time-series of precipitation and water level data and verifying the calibrated model using an independent post-calibration data set.
Model development For a given aquifer region, the rate of change in groundwater storage may be expressed as the net difference between inflow and outflow rates (Fig. 1a). We define discharge potential, v (L3), as the groundwater storage within an aquifer region of area A (Rasmussen and Andreasen, 1959) as v ¼ nAh
ð1Þ
where n is the fillable porosity (L0) defined by Sophocleous (1991) as the volume of water per unit area required for a unit rise of the water table. The concept of fillable porosity is illustrated in Fig. 1b, where area III + I divided by Dh is intrinsic porosity of the aquifer, I + II is infiltration volume per area, I is the amount of recharge required for a unit change in the water table elevation, and I divided by Dh is fillable porosity. As noted by Sophocleous (1991), fillable porosity is always smaller than intrinsic porosity. Non-linear dynamics between unsaturated and saturated zones are ignored by assuming n is independent of h (Bierkens (1998). Discharge head, h, is defined as H Hmin where H is groundwater elevation relative to a reference elevation (e.g., msl) and Hmin is the minimum groundwater level in
345
Figure 1 (a) Schematic drawing of precipitation and recharge. (b) Schematic diagram for fillable porosity. hr is irreducible residual saturation, hs is maximum saturation and Dh is unit increment of discharge head, h.
the modeled area. No groundwater discharge is assumed to occur at or below Hmin. We assume that the modeled aquifer region is relatively small and the hydraulic gradient inside domain is approximately constant. Assuming no groundwater pumping, significant evapotranspiration and water table fluctuation due to entrapped air (Healy and Cook, 2002; Todd and Mays, 2005), a mass balance indicates that the rate of change in discharge potential is dv ¼ I O þ AR dt
ð2Þ
where R is the net specific recharge rate (L T1), I is the groundwater inflow rate (L3 T1) and O is the groundwater outflow rate (L3 T1). We assume net storage change due to groundwater flow to be proportional to discharge potential such that I O ¼ kv
ð3Þ
where k is a rate coefficient (T1). To explain Eq. (3), consider the region of interest D1 and surrounding domains D0 and D2 (Fig. 2) in which the groundwater level increases due to recharge. Water will flow from the upgradient domain D2 to D1 and from D1 to D0. Summing water fluxes, O, through the boundaries of D1, assuming a constant hydraulic conductivity K yields Dh1 h1 þ K Dh2 h2 I O ¼ K ð4Þ Dx Dx
346
E. Park, J.C. Parker Equating fillable porosity and aquifer specific yield, Sy, gives dh d dh ¼K h ð10Þ Sy dt dx dx which is the one-dimensional groundwater flow equation for a water table aquifer. Therefore, the suggested relationship between net storage change due to groundwater flow and discharge potential given by Eq. (4) satisfies the conventional groundwater flow equation subject to the noted assumptions. We further assume that recharge rate can be approximated as a fixed fraction of precipitation as R ¼ aP
ð11Þ
where a is the recharge-precipitation ratio. Substituting (1), (3) and (11) into (2) yields Figure 2 domains.
Schematic
drawing
of
neighboring
hydraulic
where the width perpendicular to y- and z-axes is of unit length. The volume of water stored in D1 (discharge potential) is given by v¼n
h1 þ h2 L ¼ nhL 2
ð5Þ
where L is length of the domain along the direction of is the arithmetic mean hydraulic hydraulic gradient and h head in the domain. If the local groundwater gradient in D1 is the same as the regional gradient, then Eq. (4) can be simplified to I O ¼ K
Dh h1 h2 Dh iL L ¼ K Dx Dx L
ð6Þ
where i = (h1 h2)/L is mean hydraulic gradient of the domain and Dh = Dh1 = Dh2 where we assume that the head change is uniform within the small region. From Eqs. (5) and (6) when no external sources or sinks are imposed we may write I O ¼ kv
ð7Þ
where Ki Dh k¼ nh Dx
ð8Þ
and Eq. (2) reduces to dv ¼IO dt
ð9aÞ
Combining Eqs. (1), (5), (7), (8) and (9a) gives the following equation: dv Ki Dh dh Ki Dh ¼ kv ¼ v () ¼ h dt dt nh Dx nh Dx
ð9bÞ
Taking the finite difference approximation of Eq. (9b) becomes unity when L ! Dx ! 0, we get assuming h=h dh K Dh h1 h2 K h1 Dh h2 Dh K d dh ¼ ¼ lim ¼ lim h Dx!0 n L dt L!Dx n Dx Dx 2 n dx dx Dx!0
ð9cÞ
dh aP ¼ þ kh dt n
ð12Þ
Eq. (12) is similar to the water table fluctuation model of Healy and Cook (2002) except that it includes the discharge term kh, which circumvents the assumption in earlier models that the recharge period is sufficiently short. The solution of (12) in the Laplace domain is obtained by introducing the recharge time-lag in the unsaturated zone (T), tlag, as h0 expðtlag pÞ aP0 expðtlag pÞ h ¼ K1 þ ð13Þ pk nðp kÞ where K1 is the inverse Laplace transform, p is the Laplace transform variable, P 0 is P in the Laplace domain and h0 is h at t = 0. Assuming time-lag is small (tlag 0), inversion of (13) yields (Carslaw and Jaeger, 1959) Z 1 t h ¼ h0 expðktÞ þ aPðt sÞ expðksÞds ð14Þ n 0 where P(t s) is the precipitation rate at time t s. The derived semi-analytical solution is similar to the solution developed by Knotters and Bierkens (2000, Eq. (7)), although the latter derive their model based on an unsaturated zone mass balance, while we approach the problem from the saturated zone flow equations. For a period in which P is constant, (14) may be integrated to obtain h ¼ h0 expðktÞ þ
aPðexpðktÞ 1Þ kn
ð15Þ
Since precipitation data is generally available at discrete intervals, we apply (14) to tabulated time-intervals treating precipitation as a piecewise-constant function, according to the algorithm Step 1. Set hi = h0 for the first time-step (i = 1) Step 2. Compute the groundwater elevation at the end of the time-step from the average precipitation rate Pi for the period as hiþ1 ¼ hi expðkDti Þ þ Hiþ1 ¼ hi þ Hmin
aP i ðexpðkDti Þ 1Þ kn
ð16aÞ ð16bÞ
where Hmin is the lowest observed water level elevation. Step 3. Increment the time-step and repeat Step 2 until the last time-step has been computed.
A simple model for water table fluctuations in response to precipitation
347
From (16) by equating hi+1 to hi, it may be observed that groundwater level will increase if Pi > hikn/a. At lower precipitation rates, water level will decrease. A FORTRAN95 code for the foregoing algorithm was written to calculate groundwater level fluctuations from recharge versus time data or precipitation versus time plus the recharge-precipitation ratio, a and with values for the parameters k, n and Hmin. The FORTRAN code was coupled with the nonlinear regression model PEST (Doherty, 2002) to enable unknown model parameters to be estimated from measurements of water level and recharge or precipitation versus time. The program is available from the first author upon request.
Application to field data To test the proposed model, we investigate the relationship between precipitation and groundwater elevations in the Hongcheon area of South Korea. The area has a temperate climate with average annual precipitation of 1508 mm, of which more than half falls in July and August (Fig. 3). Daily precipitation data for the study area from 2001 through 2004 were obtained from the Korea Meteorological Administration.1 The monitoring well is located about 740 m southeast of the rainfall gauging station. Daily water level measurements were available from the National Groundwater Information Management and Service Center of Korea.2 The average depth to groundwater in the study area is approximately 8.9 m with a seasonal fluctuation range of up to 3.7 m. The water table aquifer consists of fluvial sand and gravel with a hydraulic conductivity of about 38 m d1 to a depth of about 12.4 m (116.2 m msl elevation) Gneiss bedrock underlying the fluvial deposits has a hydraulic conductivity of about 0.019 m d1 that diminishes with depth as fracture size and density decrease. The vadose zone consists of sandy loam material with mostly fine to medium sand (more than 50%) sand with some silt (less than 40%) and clay (less than 20%). Water levels were recorded electronically. Daily water level data were used for this study. Water table depth at the beginning of the study period (January 1, 2001) was 9.15 m (119.5 m msl elevation). Hydraulic interferences caused by municipal water usage are poorly investigated in the area. However, the effects are assumed to be negligible based on the absence of correlation with water level data. The model was calibrated to measured daily water levels for calendar year 2001 with daily precipitation data as input. Values of Hmin = 119.5 m, h0 = 0 m, k = 0.15 d1 and a/n = 8 were determined by nonlinear regression to minimize the root mean square (RMS) deviation between observed and predicted water levels for the calibration period (i.e., 2001). In the current study, a and n in a/n are not separable. Those parameters, however, may be decided uniquely using the proposed hybrid method by Sophocleous (1991). The RMS error for the calibration was 0.15 m and the standard deviation was 0.36 m (Fig. 4). 1
Korea Meteorological Administration Annual Climatological Reports, 2001–2004. 2 Korea National Groundwater Information Management and Service Center Annual Groundwater Monitoring Reports, 2001– 2004.
Figure 3 Average monthly precipitation (mm) in the Hongcheon area from 2001 to 2004.
During the calibration period, overprediction occurred at approximately 10 and 50 d, which correspond to the end of dry periods. Conversely, underprediction occurred with wet antecedent conditions (e.g., between 190 and 220 d). These tendencies likely reflect increases in unsaturated zone capacity with time during dry periods and decreases during wet periods. To validate the model, measured and predicted water levels were compared for the out-of-sample period from 2002 through 2004. Predictions were made using measured daily rainfall data as input using model parameters calibrated to year 2001 data. The results (Fig. 5) exhibit an RMS error of 0.20 m, a standard deviation of 0.44 m and a maximum deviation of 3.79 m. Deviations for the validation period are only slightly greater than those for the calibration data. The results indicate that model parameters are stable over time and that reliable water level fluctuation predictions can be made by the model from actual or projected precipitation data following calibration for a limited time-series.
Summary and discussion A semi-analytical solution for estimating water table fluctuations in response to precipitation was developed based on the equation for one-dimensional flow in unconfined aquifers. To test the model performance, precipitation data and observed water table fluctuations from Hongcheon, South Korea for the year 2001 was used to calibrate model parameters. Water table fluctuations for 2002–2004 were predicted using the calibrated model parameters and precipitation data. A comparison of computed and observed water levels from 2002 to 2004 resulted in an RMS error only slightly greater than the model error during the calibration period indicating the method can be successfully calibrated with limited data and applied over an extended future period. The model derivation involves several assumptions, which place limitations on the model that should be recognized. No external sources or sinks other than uniformly distributed precipitation is considered in the model. Thus, if groundwater pumping or locally variable recharge occurs within the area of interest that results in a significantly spatially variable hydraulic gradient, the model may not effectively predict groundwater fluctuation. Also we assume negligible time-lag from precipitation to water table
348
E. Park, J.C. Parker
Figure 4 Results for calibration period showing: (a) observed and predicted water table fluctuations versus time and (b) observed daily precipitation versus time for the year 2001 at Hongcheon Station, Kangwon, Korea.
Figure 5 Results for verification period showing: (a) observed and predicted water table fluctuations versus time and (b) observed daily precipitation versus time for the period from 2002 through 2005 at Hongcheon Station, Kangwon, Korea.
response. If the water table is deep, the unsaturated hydraulic conductivity is small and/or the time resolution of data used for calibration is small the lag-time may limit model accuracy. The derivation also assumes a uniform minimum water level, Hmin. If the area of interest is large and/ or a high hydraulic gradient occurs, deviations from this assumption may reduce model accuracy. Further studies involving applications of the model and other methods such as the water table fluctuation method and soil moisture balance approach of Sophocleous (1991) would be desirable to
assess the accuracy and applicability of this and other models under a range of conditions.
Acknowledgement This work was supported by the Korea Research Foundation Grant, funded by the Korean Government (MOEHRD) under Contract KRF-2005-015-C00513.
A simple model for water table fluctuations in response to precipitation
References Bierkens, M.F.P., 1998. Modeling water table fluctuations by means of a stochastic differential equation. Water Resour. Res. 34 (10), 2485–2499. Carslaw, H.S., Jaeger, J.C., 1959. Conduction of Heat in Solids, second ed. Clarendon Press, Oxford, UK. Coulibaly, P., Anctil, F., Aravena, R., Bobe ´e, B., 2001. Artificial neural network modeling of water table depth fluctuations. Water Resour. Res. 37 (4), 885–896. Doherty, J., 2002. PEST: model independent parameter estimation. Watermark Numer. Comput.. Hantush, M.S., 1967. Growth and decay of ground water mounds in response to uniform percolation. Water Resour. Res. 3 (1), 227– 234. Healy, R.W., Cook, P.G., 2002. Using groundwater levels to estimate recharge. Hydrogeol. J. 10, 91–109. Knotters, M., Bierkens, M.F.P., 2000. Physical basis of time series models for water table depths. Water Resour. Res. 36 (1), 181–188. Latnopoulos, P., 1985. Analytical solutions for periodic well recharge in rectangular aquifers with third-kind boundary conditions. J. Hydrol. 77 (1985), 293–306. Parker, J.C., 2003. Physical processes affecting natural depletion of volatile chemicals in soil and groundwater. Vadose Zone J. 2, 222–230.
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Parker, J.C., Zhu, J.L., Johnson, T.G., Kremesec, V.J., Hockman, E.L., 1994. Modeling free product migration and recovery at hydrocarbon spill sites. Groundwater 32, 119–128. Rai, S.N., Singh, R.N., 1995. Two-dimensional modelling of water table fluctuation inresponse to localized transient recharge. J. Hydrol. 167, 167–174. Rai, S.N., Manglik, A., 1999. Modelling of water table variation in response to time-varying recharge from multiple basins using the linearised Boussinesq equation. J. Hydrol. 220, 141–148. Rai, S.N., Manglik, A., Singh, V.S., 2006. Water table fluctuation owing to time-varying recharge pumping and leakage. J. Hydrol. 324, 350–358. Rao, N.H., Sarma, P.B.S., 1984. Recharge to aquifers with mixed boundaries. J. Hydol. 74, 43–51. Rasmussen, W.C., Andreasen, G.E., 1959. Hydrologic Budget of the Beaverdam Creek Basin, Maryland. US Geological Survey WaterSupply Paper 1472, 106 pp. Sophocleous, M., 1991. Combining the soil water balance and water level fluctuation method to estimate natural groundwater recharge: practical aspects. J. Hydrol. 124, 229–241. Todd, D.K., Mays, L.W., 2005. Groundwater Hydrology, third ed. John Wiley & Sons Inc., New York.
available at www.sciencedirect.com
journal homepage: www.elsevier.com/locate/jhydrol
A simple model for water table fluctuations in response to precipitation Eungyu Park a b
a,*
, J.C. Parker
b
Department of Geology, Kyungpook National University, 1370 Sangyeok-dong, Buk-gu, Daegu, South Korea Civil and Environmental Engineering Department, University of Tennessee, Knoxville, TN 37996-2010, United States
Received 15 October 2007; received in revised form 18 April 2008; accepted 25 April 2008
KEYWORDS Water table fluctuations; Precipitation; Recharge; Discharge potential; Hongcheon Korea
A simple physically-based model is developed for quantifying groundwater fluctuations in response to precipitation time-series. A semi-analytical solution of the governing differential equation is derived for relevant initial conditions and an efficient numerical algorithm is presented. Model performance is assessed by comparing predicted and observed groundwater fluctuations over a multi-year period in response to precipitation data for a site in the Hongcheon area of South Korea. Groundwater fluctuation and precipitation time-series during the year 2001 are used for model calibration. The calibrated model is then used to predict groundwater fluctuations for 2002–2004 from measured daily precipitation data. Prediction variance is only slightly larger than the calibration variance. Results indicate that model parameters are stable over time and that reliable water level fluctuation predictions can be made by the model from actual or projected precipitation data following calibration for a limited time-series. ª 2008 Elsevier B.V. All rights reserved.
Summary
Introduction Measurements of groundwater level fluctuations in response to precipitation events can provide a practical means of estimating temporally and spatially variable groundwater recharge rates (Rasmussen and Andreasen, 1959; Sophocleous, 1991; Healy and Cook, 2002; Rai and Singh, 1995; Bierkens, 1998; Rai and Manglik, 1999; Knotters and Bierkens, 2000; Coulibaly et al., 2001; Rai et al., 2006). Furthermore, * Corresponding author. Tel.: +82 53 950 5356; fax: +82 53 950 5362. E-mail address: [email protected] (E. Park).
groundwater level fluctuations themselves can directly impact the migration potential of free-phase hydrocarbon and dissolved phase contaminant plumes and for evaluating contaminant volatilization rates from groundwater due to water table ‘‘pumping’’ (e.g., Parker et al., 1994; Parker, 2003). Rasmussen and Andreasen (1959) presented a simple model relating water level fluctuations in response to recharge events for the purpose of estimating groundwater recharge from precipitation events. The latter authors assumed that the change of water level over a given time multiplied by the specific yield is equal to the change in storage due to precipitation, extraction/injection, evapotranspiration, etc. Healy and Cook (2002) pointed out that
0022-1694/$ - see front matter ª 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jhydrol.2008.04.022
A simple model for water table fluctuations in response to precipitation the accuracy of this method is limited by its inability to account for discharge. Sophocleous (1991) proposed a practical way of estimating groundwater recharge using a hybrid water-fluctuation method in which estimation errors are mitigated based on a soil water balance. Rai and Singh (1995), Rai and Manglik (1999) and Rai et al. (2006) derived an analytical solution by linearizing the two-dimensional Boussinesq equation to predict time-varying water table height based on groundwater recharge as a function of time. Bierkens (1998) modeled water table fluctuation by solving a stochastic differential equation. Knotters and Bierkens (2000) developed a piecewise analytical approximation based on the empirical soil water balance equation. Coulibaly et al. (2001) showed the applicability of an artificial neural network model in predicting water table fluctuations. The objective of this paper is to present an extension of the Rasmussen and Andreasen model to describe water table fluctuations in response to precipitation time-series that exhibit piecewise constant characteristics in a discrete record. Up to now, several different analytical models have been developed including Hantush (1967), Rao and Sarma (1984), Latnopoulos (1985), Rai and Singh (1995), Rai and Manglik (1999) and Rai et al. (2006). Those analytical solutions require either constant (Rao and Sarma, 1984) or continuous analytical functions (Latnopoulos, 1985; Rai and Singh, 1995) or rather complicated piecewise functions (Rai and Manglik, 1999; Rai et al., 2006). In this study, a practical analytical model is developed to predict water table fluctuations based on discrete record of precipitation such as daily or monthly precipitation data. The relationship of the proposed model to the one-dimensional groundwater flow equation will be shown. We present a semi-analytical solution to the problem and demonstrate the model by calibrating it to a limited time-series of precipitation and water level data and verifying the calibrated model using an independent post-calibration data set.
Model development For a given aquifer region, the rate of change in groundwater storage may be expressed as the net difference between inflow and outflow rates (Fig. 1a). We define discharge potential, v (L3), as the groundwater storage within an aquifer region of area A (Rasmussen and Andreasen, 1959) as v ¼ nAh
ð1Þ
where n is the fillable porosity (L0) defined by Sophocleous (1991) as the volume of water per unit area required for a unit rise of the water table. The concept of fillable porosity is illustrated in Fig. 1b, where area III + I divided by Dh is intrinsic porosity of the aquifer, I + II is infiltration volume per area, I is the amount of recharge required for a unit change in the water table elevation, and I divided by Dh is fillable porosity. As noted by Sophocleous (1991), fillable porosity is always smaller than intrinsic porosity. Non-linear dynamics between unsaturated and saturated zones are ignored by assuming n is independent of h (Bierkens (1998). Discharge head, h, is defined as H Hmin where H is groundwater elevation relative to a reference elevation (e.g., msl) and Hmin is the minimum groundwater level in
345
Figure 1 (a) Schematic drawing of precipitation and recharge. (b) Schematic diagram for fillable porosity. hr is irreducible residual saturation, hs is maximum saturation and Dh is unit increment of discharge head, h.
the modeled area. No groundwater discharge is assumed to occur at or below Hmin. We assume that the modeled aquifer region is relatively small and the hydraulic gradient inside domain is approximately constant. Assuming no groundwater pumping, significant evapotranspiration and water table fluctuation due to entrapped air (Healy and Cook, 2002; Todd and Mays, 2005), a mass balance indicates that the rate of change in discharge potential is dv ¼ I O þ AR dt
ð2Þ
where R is the net specific recharge rate (L T1), I is the groundwater inflow rate (L3 T1) and O is the groundwater outflow rate (L3 T1). We assume net storage change due to groundwater flow to be proportional to discharge potential such that I O ¼ kv
ð3Þ
where k is a rate coefficient (T1). To explain Eq. (3), consider the region of interest D1 and surrounding domains D0 and D2 (Fig. 2) in which the groundwater level increases due to recharge. Water will flow from the upgradient domain D2 to D1 and from D1 to D0. Summing water fluxes, O, through the boundaries of D1, assuming a constant hydraulic conductivity K yields Dh1 h1 þ K Dh2 h2 I O ¼ K ð4Þ Dx Dx
346
E. Park, J.C. Parker Equating fillable porosity and aquifer specific yield, Sy, gives dh d dh ¼K h ð10Þ Sy dt dx dx which is the one-dimensional groundwater flow equation for a water table aquifer. Therefore, the suggested relationship between net storage change due to groundwater flow and discharge potential given by Eq. (4) satisfies the conventional groundwater flow equation subject to the noted assumptions. We further assume that recharge rate can be approximated as a fixed fraction of precipitation as R ¼ aP
ð11Þ
where a is the recharge-precipitation ratio. Substituting (1), (3) and (11) into (2) yields Figure 2 domains.
Schematic
drawing
of
neighboring
hydraulic
where the width perpendicular to y- and z-axes is of unit length. The volume of water stored in D1 (discharge potential) is given by v¼n
h1 þ h2 L ¼ nhL 2
ð5Þ
where L is length of the domain along the direction of is the arithmetic mean hydraulic hydraulic gradient and h head in the domain. If the local groundwater gradient in D1 is the same as the regional gradient, then Eq. (4) can be simplified to I O ¼ K
Dh h1 h2 Dh iL L ¼ K Dx Dx L
ð6Þ
where i = (h1 h2)/L is mean hydraulic gradient of the domain and Dh = Dh1 = Dh2 where we assume that the head change is uniform within the small region. From Eqs. (5) and (6) when no external sources or sinks are imposed we may write I O ¼ kv
ð7Þ
where Ki Dh k¼ nh Dx
ð8Þ
and Eq. (2) reduces to dv ¼IO dt
ð9aÞ
Combining Eqs. (1), (5), (7), (8) and (9a) gives the following equation: dv Ki Dh dh Ki Dh ¼ kv ¼ v () ¼ h dt dt nh Dx nh Dx
ð9bÞ
Taking the finite difference approximation of Eq. (9b) becomes unity when L ! Dx ! 0, we get assuming h=h dh K Dh h1 h2 K h1 Dh h2 Dh K d dh ¼ ¼ lim ¼ lim h Dx!0 n L dt L!Dx n Dx Dx 2 n dx dx Dx!0
ð9cÞ
dh aP ¼ þ kh dt n
ð12Þ
Eq. (12) is similar to the water table fluctuation model of Healy and Cook (2002) except that it includes the discharge term kh, which circumvents the assumption in earlier models that the recharge period is sufficiently short. The solution of (12) in the Laplace domain is obtained by introducing the recharge time-lag in the unsaturated zone (T), tlag, as h0 expðtlag pÞ aP0 expðtlag pÞ h ¼ K1 þ ð13Þ pk nðp kÞ where K1 is the inverse Laplace transform, p is the Laplace transform variable, P 0 is P in the Laplace domain and h0 is h at t = 0. Assuming time-lag is small (tlag 0), inversion of (13) yields (Carslaw and Jaeger, 1959) Z 1 t h ¼ h0 expðktÞ þ aPðt sÞ expðksÞds ð14Þ n 0 where P(t s) is the precipitation rate at time t s. The derived semi-analytical solution is similar to the solution developed by Knotters and Bierkens (2000, Eq. (7)), although the latter derive their model based on an unsaturated zone mass balance, while we approach the problem from the saturated zone flow equations. For a period in which P is constant, (14) may be integrated to obtain h ¼ h0 expðktÞ þ
aPðexpðktÞ 1Þ kn
ð15Þ
Since precipitation data is generally available at discrete intervals, we apply (14) to tabulated time-intervals treating precipitation as a piecewise-constant function, according to the algorithm Step 1. Set hi = h0 for the first time-step (i = 1) Step 2. Compute the groundwater elevation at the end of the time-step from the average precipitation rate Pi for the period as hiþ1 ¼ hi expðkDti Þ þ Hiþ1 ¼ hi þ Hmin
aP i ðexpðkDti Þ 1Þ kn
ð16aÞ ð16bÞ
where Hmin is the lowest observed water level elevation. Step 3. Increment the time-step and repeat Step 2 until the last time-step has been computed.
A simple model for water table fluctuations in response to precipitation
347
From (16) by equating hi+1 to hi, it may be observed that groundwater level will increase if Pi > hikn/a. At lower precipitation rates, water level will decrease. A FORTRAN95 code for the foregoing algorithm was written to calculate groundwater level fluctuations from recharge versus time data or precipitation versus time plus the recharge-precipitation ratio, a and with values for the parameters k, n and Hmin. The FORTRAN code was coupled with the nonlinear regression model PEST (Doherty, 2002) to enable unknown model parameters to be estimated from measurements of water level and recharge or precipitation versus time. The program is available from the first author upon request.
Application to field data To test the proposed model, we investigate the relationship between precipitation and groundwater elevations in the Hongcheon area of South Korea. The area has a temperate climate with average annual precipitation of 1508 mm, of which more than half falls in July and August (Fig. 3). Daily precipitation data for the study area from 2001 through 2004 were obtained from the Korea Meteorological Administration.1 The monitoring well is located about 740 m southeast of the rainfall gauging station. Daily water level measurements were available from the National Groundwater Information Management and Service Center of Korea.2 The average depth to groundwater in the study area is approximately 8.9 m with a seasonal fluctuation range of up to 3.7 m. The water table aquifer consists of fluvial sand and gravel with a hydraulic conductivity of about 38 m d1 to a depth of about 12.4 m (116.2 m msl elevation) Gneiss bedrock underlying the fluvial deposits has a hydraulic conductivity of about 0.019 m d1 that diminishes with depth as fracture size and density decrease. The vadose zone consists of sandy loam material with mostly fine to medium sand (more than 50%) sand with some silt (less than 40%) and clay (less than 20%). Water levels were recorded electronically. Daily water level data were used for this study. Water table depth at the beginning of the study period (January 1, 2001) was 9.15 m (119.5 m msl elevation). Hydraulic interferences caused by municipal water usage are poorly investigated in the area. However, the effects are assumed to be negligible based on the absence of correlation with water level data. The model was calibrated to measured daily water levels for calendar year 2001 with daily precipitation data as input. Values of Hmin = 119.5 m, h0 = 0 m, k = 0.15 d1 and a/n = 8 were determined by nonlinear regression to minimize the root mean square (RMS) deviation between observed and predicted water levels for the calibration period (i.e., 2001). In the current study, a and n in a/n are not separable. Those parameters, however, may be decided uniquely using the proposed hybrid method by Sophocleous (1991). The RMS error for the calibration was 0.15 m and the standard deviation was 0.36 m (Fig. 4). 1
Korea Meteorological Administration Annual Climatological Reports, 2001–2004. 2 Korea National Groundwater Information Management and Service Center Annual Groundwater Monitoring Reports, 2001– 2004.
Figure 3 Average monthly precipitation (mm) in the Hongcheon area from 2001 to 2004.
During the calibration period, overprediction occurred at approximately 10 and 50 d, which correspond to the end of dry periods. Conversely, underprediction occurred with wet antecedent conditions (e.g., between 190 and 220 d). These tendencies likely reflect increases in unsaturated zone capacity with time during dry periods and decreases during wet periods. To validate the model, measured and predicted water levels were compared for the out-of-sample period from 2002 through 2004. Predictions were made using measured daily rainfall data as input using model parameters calibrated to year 2001 data. The results (Fig. 5) exhibit an RMS error of 0.20 m, a standard deviation of 0.44 m and a maximum deviation of 3.79 m. Deviations for the validation period are only slightly greater than those for the calibration data. The results indicate that model parameters are stable over time and that reliable water level fluctuation predictions can be made by the model from actual or projected precipitation data following calibration for a limited time-series.
Summary and discussion A semi-analytical solution for estimating water table fluctuations in response to precipitation was developed based on the equation for one-dimensional flow in unconfined aquifers. To test the model performance, precipitation data and observed water table fluctuations from Hongcheon, South Korea for the year 2001 was used to calibrate model parameters. Water table fluctuations for 2002–2004 were predicted using the calibrated model parameters and precipitation data. A comparison of computed and observed water levels from 2002 to 2004 resulted in an RMS error only slightly greater than the model error during the calibration period indicating the method can be successfully calibrated with limited data and applied over an extended future period. The model derivation involves several assumptions, which place limitations on the model that should be recognized. No external sources or sinks other than uniformly distributed precipitation is considered in the model. Thus, if groundwater pumping or locally variable recharge occurs within the area of interest that results in a significantly spatially variable hydraulic gradient, the model may not effectively predict groundwater fluctuation. Also we assume negligible time-lag from precipitation to water table
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Figure 4 Results for calibration period showing: (a) observed and predicted water table fluctuations versus time and (b) observed daily precipitation versus time for the year 2001 at Hongcheon Station, Kangwon, Korea.
Figure 5 Results for verification period showing: (a) observed and predicted water table fluctuations versus time and (b) observed daily precipitation versus time for the period from 2002 through 2005 at Hongcheon Station, Kangwon, Korea.
response. If the water table is deep, the unsaturated hydraulic conductivity is small and/or the time resolution of data used for calibration is small the lag-time may limit model accuracy. The derivation also assumes a uniform minimum water level, Hmin. If the area of interest is large and/ or a high hydraulic gradient occurs, deviations from this assumption may reduce model accuracy. Further studies involving applications of the model and other methods such as the water table fluctuation method and soil moisture balance approach of Sophocleous (1991) would be desirable to
assess the accuracy and applicability of this and other models under a range of conditions.
Acknowledgement This work was supported by the Korea Research Foundation Grant, funded by the Korean Government (MOEHRD) under Contract KRF-2005-015-C00513.
A simple model for water table fluctuations in response to precipitation
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