Delineation of recharge rate from a hybrid water table fluctuation method



[1] The concept of the hybrid water table fluctuation (WTF) method for recharge rate estimation was revisited. To estimate the recharge rate, a physically based WTF equation was established. The concept of transient fillable porosity was proposed and computed with unsaturated hydraulics models. The developed model is tested by applying to the water table fluctuation data from Hongcheon, Korea. In the applications, the recharge and fillable porosity estimates were found to be most sensitive to nonlinearity in the unsaturated water content profile and permeability. Also, the water table level drift, which does not originate from precipitation, serves as a major source of estimation error.

1. Introduction

[2] Delineation of the recharge rate from water table fluctuation (WTF) has been accepted since the 1920s as one of the prospecting methods due to the simplicity of its theoretical background and implementation [Rasmussen and Andreasen, 1959; Healy and Cook, 2002]. With the continuous measurements of groundwater level becoming less expensive, the WTF method for estimating groundwater recharge is gaining more popularity. In the WTF method, groundwater recharge (R) is simply estimated by the amount of water table buildup (Δh) during a precipitation event multiplied by the specific yield (Sy) (i.e., R = SyΔh). However, two misleading points arise in the conventional WTF method. First, the component of lateral flow during a recharge event is ignored. By ignoring the component, the delineated recharge rate can underestimate the actual recharge rate, and the error can accelerate for high permeable aquifers or diffusion dominant recharge processes [Cuthbert, 2010].

[3] Second, the linkage constant of Sy has obscurity. By the WTF method, the same amount of uncertainty in the delineation of Sy is transferred to R [Crosbie et al., 2005], even when Δh is precisely measured. Sophocleous [1991] argued that the usage of a temporally invariant Sy for estimating R in the WTF method can introduce significant errors. Sophocleous also claimed the inappropriateness of Sy in the WTF method, which reflects the drainage characteristic of the unsaturated zone, whereas the water table rise due to recharge is an imbibition process. Alternatively, Sophocleous proposed fillable porosity, which is the volume of water necessary for a unit water table rise per unit area [Kayane, 1983], as a linkage parameter between R and Δh. As a result, Sophocleous proposed the modified WTF method, which is a combination of an unsaturated zone soil water balance and the conventional WTF method. Although the methodology to measure fillable porosity was roughly introduced by Sophocleous [1991], the concept is not well understood or applied in the estimations of recharge rate; instead, Sy is conventionally utilized in the actual implementations of WTF methods [Healy and Cook, 2002].

[4] The concept of varying Sy in the WTF method is not new. Crosbie et al. [2005] have employed depth-variant Sy for the delineation of recharge rate. In this study, an equilibrium water content profile was assumed as a reference and the static Sy was estimated as a function of depth to the water table. By this method, however, the excessive moisture from a preceding precipitation event, resisting against gravitational drainage, cannot be fully reflected and the recharge rate might be overly estimated, especially in humid areas. Therefore, it is somewhat clear that the delineation of recharge rate from the transient Sy or fillable porosity is desirable, yet still not visited as far as the author is aware.

[5] In the present study, the concept of the hybrid-WTF method introduced by Sophocleous [1991] will be revisited with a focus on the definite concept of fillable porosity and an improved WTF model. For this purpose, a physically based equation of the WTF method will be proposed and the concept of transient fillable porosity will be systematically addressed. The developed model will be demonstrated through a case study by delineating the recharge rate and mean fillable porosity with verification through an independent process.

2. Improved WTF Model

[6] Park and Parker [2008] proposed that the rate of water table change is determined by both the recharge rate and net groundwater flux as,

display math

where h is the discharge head [L] representing the groundwater storage within an aquifer region of area A [L2], t is time [T], IO is the net groundwater flow rate [L3 T−1], φ is fillable porosity [-], and R is the recharge rate [LT−1]. Based on equation (1) and an associated analogy [Park and Parker, 2008], together they provide a piecewise solution by assuming a constant recharge rate within a given time interval (Δt) as,

display math

where h(i) and h(i+1) are the discharge head [L] at the ith and i + 1th time steps, respectively, and k is the rate coefficient [T−1] representing the water table decay rate with time through lateral flow [Park and Parker, 2008], which is also directly related to aquifer hydraulic conductivity of the surrounding aquifers [Park et al., 2011]. In equation (2), the first term on the right-hand side is related to the water table recession and the second term is a convolution between recharge rate and aquifer system.

[7] From equaion (2), a piecewise recharge rate during a time interval, Δt, can be delineated as,

display math

by substituting h for the observed head time series. In order to delineate the recharge rate based on equation (3), the experimental determination of two coefficients (k and φ) are required a priori. The rate coefficient, k, can be calibrated while fitting a simulated hydrograph to an observed hydrograph. The procedure and the results are well introduced by Park and Parker [2008]. With equation (3), both of the components of recharge to the water table rise and recharge to the lateral flow increment can be accounted for. In this study, external interferences such as groundwater abstractions, evapotranspiration, ambient pressure, and Lisse effect are not rigorously considered. The external interferences over water table fluctuations are extensively reviewed by Healy and Cook [2002] and the time series method for their exclusions is well introduced by Crosbie et al. [2005], which may be jointly applied with equation (3) for the improved estimations.

3. Transient Fillable Porosity

[8] In equation (3), the most problematic coefficient is φ, which has rather transient characteristics. In the present study, to address the temporally varying recharge rate, the concept of transient φ will be introduced. The φ is given by V/Δh, where V is the required volume of water for the Δh [L] water table rise per unit area [Kayane, 1983; Sophocleous, 1991; Park and Parker, 2008]. In fact, V is a nonlinear function of Δh and unsaturated hydraulics. Figure 1 shows the concept of transient φ. In the figure, the solid line at t is the equilibrium water content profile due to a long period of time without precipitation, and the dashed line at t is a profile, which has not yet reached the equilibrium state. In terms of residual moisture content at a reference elevation, the solid line reaches its irreducible condition, θr, while the dashed line is at interim water content, θtr, because the moisture from the preceding precipitation in the unsaturated zone has not yet drained or stabilized. If there is another precipitation event and the change of the water table is Δh, the engaged water volume corresponding to t and t curves are II and I + II, respectively. The discrepancies in the water volumes originate from the temporal change of the water content profile. From this reasoning, it can be inferred that a recharge rate needs to be decided based both on the water table fluctuation of a saturated zone and the transient water content profile of an unsaturated zone. The temporal change of unstable soil moisture can be described by the kinematic wave model [Charbeneau, 2000; Nachabe, 2002] as,

display math

where the effective saturation Θ [–] is equal to (θθr)/(θsθr), θ is the volumetric water content of porous media [–], θs is saturated water content equivalent to effective porosity [–], θr is the irreducible or residual water content [–], K is the unsaturated hydraulic conductivity [LT−1] given as a function of θ, and z is the distance [L] along the vertical direction toward the ground surface, of which the origin is at the water table. By applying the van Genuchten-Mualem unsaturated hydraulic conductivity model [van Genuchten, 1980] to equation (4), the rate of effective saturation change with time, ρ [T−1], becomes

display math

where m = 1 1/n, Γ = 1 Θ1/m = (αz)n/{1 + (αz)n}, and α, m, and n are the van Genuchten model parameters. Further detail on the derivation of equation (5) is summarized in Appendix A. Therefore, Θ(z, t) is given by,

display math

where Θ0 is the initial effective saturation. Immediately after a water table rises because of recharge, the pores right above the water table are close to saturation. The remaining excessive moisture in this zone is drained by gravitational force with a diffusive pattern and this diffusive component of recharge does not involve a water table rise, except for the case when the aquifer hydraulic conductivity is considerably low. In this study, the diffusive recharge flux is not counted as the recharge rate. Thus, the estimation of recharge rate by the developed methodology could be slightly biased. As previously mentioned, the water content profile before the cessation of drainage (Figure 1) generally deviates from its steady state curve. The excessive soil moisture at this stage is drained relatively quickly by gravitational force until the capillary pressure of the pore reaches the entry threshold, which may be represented by the inflection height of the water content profile. In the present study, it is assumed that the pattern of the transient curve may be approximated by manipulating the transient effective residual water content (θtr). θtr can be delineated from equation (6) with equation (5) by substituting z with a distance from the water table to a reference point which may be set as a height above maximum water table fluctuation. Also, Θ0 for this case is assumed to be that of the inflection height, z*, of the moisture retention curve (i.e., z* = 1/α), and equal 1/2m.

Figure 1.

Schematic diagram of transient fillable porosity.

[9] As an exercise of the derived equations, θtr at 1 m above water table is computed with equations (5) and (6) and the unsaturated hydraulic parameters of sand (θs = 0.43; θr = 0.045; α = 14.5 m−1; n = 2.68; Ks = 7.128 m d−1) [Carsel and Parrish, 1988]. In the computation, it is assumed that the starting value is equal to the entry effective saturation of Θ0 = 0.6476. In 1 d, the effective saturation of sandy loam drops to Θ(z = 1 m, t = 1 d) = 0.2413, which is 37% of its starting value. Based on these observations, one can set θtr for sand in 1 d since a recharge event is θtr = Θtθs + (1 – Θt)θr = 0.2413θs + (1 – 0.2413)θr = 0.1379. From the computed θtr, the fillable porosity corresponds to Δh = 0.1 m water table rise can be computed by the definition, the volume of water for unit rise of water table per unit area (Figure 1) as,

display math

and the value is 0.0661. This value is only 15% of effective porosity of sand.

4. Delineation of Recharge Rate Based on Transient Fillable Porosity

[10] To delineate the recharge rate, equation (3) is used where φ is transiently delineated by the method proposed in section 3 (i.e., equations (5)–(7)). Figure 2 shows the procedure of delineating the recharge rate from the proposed model where the length of preceding dry period [T], Ddry, is counted for each recharge event to compute θtr based on equation (6). In the counting of Ddry, it is considered that the water table does not respond to precipitation under a certain precipitation rate and the threshold precipitation rate (Pmin) is defined to determine the averaged recharge rate and fillable porosity for a given duration of time. With the effective residual saturation and the size of the rise in the water table, Δh for Pi larger than Pmin, φ for each recharge event is determined by equation (7), and finally the recharge rate is delineated from equation (3).

Figure 2.

Proposed procedure for delineating the recharge rate and fillable porosity of this study.

[11] As validation of the estimated recharge rate, equation (2) is employed and the simulated water table is compared to the observed. In the validation process, the mean recharge-precipitation ratio (f) is used, which is given by f = ΣRP, where ΣR and ΣP are the cumulative annual recharge and precipitation rates, respectively. Although the concept of a recharge-precipitation ratio does not exactly explain the actual recharge from precipitation [Cuthbert, 2010], the ratio multiplied by precipitation still reasonably represents the averaged recharge rate for shallow aquifers with slight under- and overestimations during the wet and dry period, respectively, in the prediction stage [Park and Parker, 2008].

5. Application of the Developed Hybrid-WTF Model

[12] To validate the proposed model, we apply the model for the delineations of recharge rate and fillable porosity to the Hongcheon area in South Korea [Park and Parker, 2008]. The area has a temperate climate with an annual mean precipitation of 1508 mm. Daily precipitation data of the study area during the years 2001–2004 were obtained from the Korea Meteorological Administration (Korea Meteorological Administration Annual Climatological Reports, 2001–2004) and groundwater level monitoring data were available from the National Groundwater Information Management and Service Center of Korea (Korea National Groundwater Information Management and Service Center Annual Groundwater Monitoring Reports, 2001–2004), where the stations are about 740 m apart. The average depth to groundwater during the study period was ∼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 ∼38 m d−1 to a depth of ∼12.4 m (116.2 m mean sea level [msl] elevation) and semi-impervious Gneiss bedrock underlying the aquifer. The unsaturated zone mainly consists of sandy loam material with mostly fine to medium sand (>50%) with some silt (<40%) and clay (<20%) [Park and Parker, 2008], the van Genuchten model parameters which are assumed to be equal to those of sandy loam (θs = 0.41; θr = 0.065; α = 7.5 m−1; n = 1.89; Ks = 1.061 m d−1) [Carsel and Parrish, 1988]. Based on the proposed WTF model by Park and Parker [2008], the calibrated k-value is –0.15 d−1 (equation (2)). From the groundwater hydrograph and daily precipitation, the threshold precipitation rate is delineated to be 0.018 m. By applying the procedure proposed (Figure 2) and the unsaturated zone material of sandy loam, the recharge rate and fillable porosity for each year are computed (Table 1).

Table 1. Estimated Recharge Rate and Fillable Porosity During the Years 2001–2004
Precipitation rate (m)1.1361.3661.9741.557
Recharge rate (m)0.4440.3420.8830.661
R/P (-)0.3910.2510.4470.424
Fillable porosity (-)0.0530.0590.0550.052

[13] In Table 1, the maximum cumulative precipitation rate of 1.974 m occurs in 2003 while the minimum of 1.136 m occurs in 2001. The highest recharge rate of 0.883 m occurs in 2003, which corresponds to the highest precipitation of the years studied. On the contrary, in 2002, the recharge rate is lowest (0.342 m) where the precipitation of the year is not the lowest. In terms of recharge-precipitation ratio, 2003 shows the largest ratio of 44.7%, whereas the lowest ratio is 25.1% in 2002. This recharge-precipitation ratio is typically lower than that of the other years, in which in 2001 it is 39.1% and in 2004 it is 42.4%. The fillable porosity is highest (5.9%) for 2002 and lowest (5.2%) for 2004. The average value of the sandy loam fillable porosity in the Hongcheon area is 5.5%, which is almost similar to the average fillable porosity estimated from sandy media in the Great Bend Prairie of Kansas as noted in the work of Sophocleous [1991], which was 5.45%.

[14] To validate the estimated recharge rate, the observed water table level is compared to the simulated level built using equation (2) with the delineated recharge-precipitation ratio and fillable porosity shown in Table 1. Figure 3 shows the observed versus the simulated WTF for the years 2001–2004. The root-mean-square (RMS) error normalized by the difference between the maximum and the minimum water table level for the years 2001–2004 are 4.93%, 8.84%, 5.34%, and 4.96%, respectively. By applying the averages of annual recharge-precipitation ratios, there are systematic underestimations in the simulated water table level based on equation (2) during the high recharge events of 2001–2004 (Figure 3). In most years, both the observed and the simulated fluctuation show an acceptable match, except for 2002. In 2002 (Figure 3b), the simulated curve significantly underestimates the actual curve, which is related to the underestimated recharge rate of 2002. In the recharge-precipitation delineations, 2002 is estimated to be ∼60% of the other years (Table 1). This underestimation may be attributed to measurement error or external drift in the observed WTF time series. In Figure 3b, there exists a clear water table drift from 240 to 330 Julian days, which is not shown for the other years. This observation suggests that the major source of error in the delineation of recharge rate can be an erroneously measured water table level or an external origin of water table drift such as groundwater pumping/injection, significant level changes in adjacent surface water bodies, etc. Therefore, if the water table level includes fluctuations that do not originate from the recharge out of precipitation, the estimated rate may include significant error which can be either over- or underestimations of the actual recharge rate.

Figure 3.

Observed versus simulated WTF for the years: (a) 2001, (b) 2002, (c) 2003, and (d) 2004.

[15] To observe the apportioned importance of the unsaturated model parameters, a global sensitivity approach based on Monte Carlo computations and a variance-based method are applied to the above demonstrated case for the year 2001. In the analysis, a total of 2000 unsaturated hydraulic parameter vectors with four components (i.e., θr, α, n, and Ks) of sandy loam are developed by the correlated random generator proposed by Carsel and Parrish [1988], and 1000 Monte Carlo simulations are performed based on the method proposed by Saltelli et al. [2010]. From the analysis, it is found that n has the largest apportioned importance (80.9%) to the recharge rate out of all unsaturated hydraulic parameters, and Ks is the second largest (18.2%). On the other hand, θr (0.87%) and α (0.03%) are found to be insignificant. The fillable porosity also has similar influence factors of 75.6% for n, 22.6% for Ks, 1.24% for θr, and 0.56% for α. These observations indicate that the nonlinearity in the water content profile and the permeability of the unsaturated zone are the most important factors influencing the estimated recharge rate and the fillable porosity in a given soil type of sandy loam. For a different soil type, however, the apportioned importance of the unsaturated hydraulic parameters could be different.

6. Summary and Conclusion

[16] An analytical form of hybrid-WTF model was newly developed considering the transient characteristics of fillable porosity. To develop it, a physically based water table fluctuation model [Park and Parker, 2008] was rearranged to construct a piecewise expression of the recharge rate, which can address both the recharge components of the water table rise and the lateral flow increment. For the development of transient fillable porosity, the kinematic wave [Charbeneau, 2000] and the van Genuchten-Mualem models [van Genuchten, 1980] were employed to derive the temporally varying effective residual saturation, which was used to delineate fillable porosity by its definition. The procedure for delineating the groundwater recharge rate based on the developed model was also developed.

[17] The developed model was applied to the water table fluctuation data for the years 2001–2004 from Hongcheon, South Korea. Based on the developed model and the procedure, the average recharge rate was ∼37.8% of the annual precipitation rate and the mean fillable porosity was 5.5%. In the application, it was found that the water table level drift, which does not originate from precipitation, serves as a major source of estimation error. Also, for the given soil type of sandy loam, the nonlinearity of the unsaturated water content profile and the permeability showed the dominant influences to the estimations of recharge rate and fillable porosity.

[18] In the present study, no external interferences such as pumping or injection, atmospheric or Lisse effect, or significant level changes of adjacent surface water bodies other than uniformly distributed precipitation are considered. Thus, the estimated recharge rates may significantly under- or overestimate the actual ones. The estimated recharge rate is only based on discrete recharge pulses and may significantly underestimate the actual recharge rate where the diffusive recharge flux is dominant. Also, applications of the developed model to an area where the unsaturated hydraulic properties are not well characterized may create additional uncertainties in the estimations.

Appendix A:: Derivation of Equation (5)

[19] van Genuchten-Mualem expressions of effective saturation and unsaturated hydraulic conductivity are, respectively, given as [van Genuchten, 1980],

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[20] From equation (A1),

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and from equation (A2),

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where Γ = 1 – Θ1/m = (αz)n/{1 + (αz)n}. Therefore, equation (5) with equations (A3) and (A4) gives,

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[21] This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0009900).