# A New Method of Estimating Groundwater Evapotranspiration at Sub-Daily Scale Using Water Table Fluctuations

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## Abstract

**:**

_{G}) are important for understanding ecosystem functionality and managing regional water resources. Over the past several decades, various methods have been proposed to estimate groundwater evapotranspiration based on water table fluctuations. However, the majority of methods cannot resolve sub-daily variations in ET

_{G}. In this study, we proposed a new hydraulic theory-based ET

_{G}estimation method at a sub-daily time scale. To evaluate its performance, we employed a variety of measurements (i.e., water table levels, latent heat flux and soil water contents) at a riparian forest (T. ramosissima) in Northwest China from 25 July to 10 October in 2017. The results indicated that the proposed method can successfully estimate ET

_{G}at both sub-daily (R

^{2}= 0.75) and daily (R

^{2}= 0.88) time scales, but the variations in the specific yield under different water table conditions should be carefully taken into account. In addition, we investigated the seasonal variations in water uptake source of the riparian plant, and found that it had strong plasticity in water usage during the study period. That is, it consumed approximately equal amounts of soil water and groundwater when soil moisture was available, and tended to consume more groundwater for survival as the soil moisture was depleted. To verify the seasonal patterns of the water uptake of the riparian forest, systematic isotope-based studies are needed in future study.

## 1. Introduction

_{G}) is critical for understanding ecosystem functionality and managing regional water resources [4]. Up to now, various methods have been proposed to quantify ET

_{G}based on diurnal water table fluctuations [5,6,7,8,9,10]. Due to their cost-effectiveness and simplicity, these ET

_{G}estimation methods have been applied across diverse ecosystems [11,12,13,14,15].

_{G}estimation methods using water table fluctuations. First, the majority of methods assume that groundwater recovery rate is constant during a complete one-day period, and can only calculate ET

_{G}at a daily (or longer) time scale [5,6,9]. As first noted by Troxell [16] and evidenced by numerical simulations [7], the recovery rate is not constant over time, and must be estimated as a function of time. Until now, relatively little attention has been paid to estimate the time-dependent groundwater recovery rate with the exception of that of Loheide [7] and Gribovszki et al. [8]. Second, the key parameter in the water table fluctuation methods, known as specific yield, is difficult to estimate, because neither in situ nor laboratory measurements capture its variability [4]. Previous studies have clearly documented that it not only depends upon the soil hydraulic properties and the water table depth, but is different under different water table conditions (i.e., declining, stable or rising) [17,18,19,20,21]. Unfortunately, systematic investigations of the variations in specific yield under different water table conditions are critically limited in present studies [11]. Finally, identifying the water source of plants is important for understanding the ecohydrological processes of riparian ecosystems [22,23]. Isotopic techniques have been widely used in previous studies [24,25,26,27]. However, sampling the isotopes of various water sources (i.e., soil water, groundwater and plant tissues) is technically challenging, destructive, expensive and time-consuming, which may hamper the assessment of the temporal variations in plant water source during growing seasons [28]. Thus, it is attractive to identify plant water sources by comprehensively using water table fluctuations and flux measurements [12].

_{G}at both sub-daily and daily scales by taking the time dependency of recovery rate into account. Based on 78-day continuous field observations including water table fluctuations, soil moisture and latent heat fluxes over a riparian ecosystem in Northwest China in 2017, the specific objectives of the present study were to: (1) develop a new hydraulic theory-based ET

_{G}estimation method at both sub-daily and daily scales; (2) investigate the variations in specific yield under different water table conditions, and evaluate its impacts on estimations of ET

_{G}; and (3) identify the plant water uptake source during the whole study period.

## 2. Site Descriptions and Data Collection

^{2}[30].

## 3. Methods

#### 3.1. Water Balance in the Riparian Zone

^{−1}) mainly comes from two water sources at our site: groundwater and soil water in the root layer. Therefore, total ET over the riparian zone at a daily scale could be estimated by using the water balance method:

_{G}and ET

_{S}(all in units of (L T

^{−1})) the daily ET from groundwater and soil water, respectively. ET

_{S}can be calculated from the changes in soil water content in the root zone:

_{l}lL] the thickness of the lth soil layer (l = 1, 2, …, n); θ

_{l}(i) and θ

_{l}(i−1) (all in L

^{3}L

^{−3}) are the soil water content in the lth soil layer for the current day i and previous day i−1, respectively. Inspired by previous studies i.e., [5,6,7,8,35], we here proposed a new hydraulic theory-based ET

_{G}estimation method using diurnal water table fluctuations.

#### 3.2. Hydraulic Theory-Based ET_{G} Estimation Method

^{−1}], and the groundwater evapotranspiration rate, ET

_{g}(t) [L T

^{−1}]:

_{y}(–) is the specific yield and WT [L] is the water table depth. Here, the lateral flow is defined as positive toward the riparian zone and the water table depth is defined as zero at the land surface and becomes increasingly negative as the water table becomes deeper. Using Darcy’s law and the Dupuit approximation, the groundwater flow rate toward the riparian zone can be formulated as [36]:

_{s}[L T

^{−1}] is the saturated hydraulic conductivity of the aquifer system; L [L] is the distance from the background to the riparian zone; H

_{0}[L] and h [L] is the groundwater elevation in the background and the riparian zone, respectively (Figure 2); and a

_{1}, a

_{2}and a

_{3}are hydrological coefficients (see details in Supporting Information), which are generally time-consuming and costly to measure directly [10]. We noticed that these coefficients can be derived from the water table records during times of zero ET

_{g}. Combining Equations (3) and (4) by setting ET

_{g}= 0, we arrive at:

_{g}at time t can then be calculated as:

_{G}:

_{G}[L T

^{−1}] is daily ET from groundwater (Figure 3b). Thus, we can estimate groundwater evapotranspiration at both daily [ET

_{G}; L T

^{−1}] and sub-daily [ET

_{g}; L T

^{−1}] scales.

#### 3.3. Determination of Specific Yield

_{y}) is exceedingly difficult to estimate, and is highly variable in different water table conditions [4,5,11]. According to [12], we used the information available from the water balance method to estimate S

_{y}. In brief, the value of S

_{y}was determined by minimizing the residual sum of squares (RSS) of Equation (1):

_{obs}(i) is EC-measured evapotranspiration on the ith day (i = 1, 2, …, T); ET

_{S}(i) + ET

_{G}(i; β) represents estimated evapotranspiration from soil water and groundwater on the ith day using the water balance method (Equation (1)); and β is a coefficient that equals S

_{y}when RSS is minimized. In practice, we used a Monte Carlo (MC) algorithm to find the optimal value of S

_{y}. The pseudo-code of the algorithm is given below:

- Step 1:
- Calculating daily ET
_{S}from the soil moisture observations using Equation (2); - Step 2:
- Selecting a random value of β from a prior interval of specific yield for silt loam soil (i.e., from 0 to 0.2), and calculating ET
_{G}using Equation (7) combined with selected β and diurnal water table records; - Step 3:
- Calculating RSS using Equation (8);
- Step 4:
- Repeating the Steps 2 and 3 for 10,000 times, and selecting the value of β with minimum RSS as the optimal estimate of S
_{y}. The best 500 values of β with minimum RSSs were also used to calculate the standard deviation of S_{y}.

_{y}, we divided the entire observations into three different periods, and the optimal value of S

_{y}for each period was determined by using the MC algorithm. The three periods respectively represented a declining, constant and rising water table condition. Hereafter, we called this calibration procedure the ‘dataset-by-dataset calibration’. In addition, we applied the MC algorithm on entire observations to obtain an optimal S

_{y}value for the whole study period, called the ‘whole dataset calibration’. In this way, we can investigate the variations of S

_{y}under different water table conditions, and its impact on the performances of the proposed method throughout the study period.

#### 3.4. Evaluating the Hydraulic Theory-Based ET_{G} Estimation Method

_{y}were estimated, we used five statistical measures to evaluate the performance of the proposed method. These statistical measures were the coefficient of determination (R

^{2}), slope, y-intercept, bias, root-mean square error (RMSE), and relative error (RE). The calculation formulas can be found in [37]. Among them, R

^{2}ranges between 0 and 1, with higher values indicating a good simulation result. Slope and y-intercept indicate how well the scatter plot between observed and estimated ET fits the 1:1 line. The values of bias, RMSE and RE illustrate the difference between observed and estimated ET, with lower values indicating a better model fit.

## 4. Results

#### 4.1. Environmental Variables

_{n}; W m

^{−2}), air temperature (T

_{a}; °C), soil water content (θ; m

^{3}m

^{−3}), and leaf area index (LAI; m

^{2}m

^{−2}) are illustrated in Figure 4. During the study period (form 25 July to 10 October), the daily mean R

_{n}varied from 33.1 to 208.7 W m

^{−2}with an average value of 124.8 W m

^{−2}. The variation of mean daily T

_{a}has a similar trend to R

_{n}, varying from 5.2 to 29.1 °C with an average value of around 20.1 °C (Figure 4a). The leaf area index (LAI; m

^{2}m

^{−2}) showed a declining trend during the whole study period (Figure 4b). Interestingly, soil moisture (θ; m

^{3}m

^{−3}) only in the middle layers (80–160 cm) was observed to decrease gradually during the study period, and θ in other layers (0–80 cm and 160–200 cm) varied slightly (Figure 4c). The seasonal variation in water table can be divided into three periods (Figure 4d). From 25 July to 18 August, a generally declining trend of water level was observed. After that, the water table remained at a relatively constant level. In late September, the water table began to rise due to ceased plant water uptake.

#### 4.2. Determination of Specific Yield

_{y}) estimated using the minimum RSS method are shown in Figure 5. For the dataset-by-dataset calibration procedure, it was observed that S

_{y}exhibited significant variations under different water table conditions. That is, the maximum value (0.0359) was found during the water table declining period (25 July–18 August), followed by that (0.0206) during the water table stable period (19 August–22 September), and the minimum value (0.0137) occurred during the water table rising period (23 September–10 October). For the whole dataset calibration procedure, the value of S

_{y}was estimated to be 0.0251. This value was similar to that for the water table stable period (0.0206). This may be partly explained by the fact that the dataset during the water table stable period had a comparatively larger number of observations (i.e., 34 out of 77), and subsequently gained more weight in calculating the residual sum of squares.

#### 4.3. Evaluating the Performances of the ET_{G} Estimation Method

_{y}as described above, we can obtain groundwater evapotranspiration at both a half-hourly (ET

_{g}) and daily (ET

_{G}) time step using Equations (6) and (7), respectively. To evaluate the performances of our proposed method, we compared the estimated daily ET using the water balance method with EC-measured values (Figure 6). The results indicated that the estimated daily ET using S

_{y}calibrated by different datasets agreed well with EC-measured values. The points in the plots of measured-versus-estimated daily ET fell tightly along the 1:1 line (slope = 0.91, intercept of 0.31 mm day

^{−1}and a correlation coefficient of 0.88), and the estimated daily ET fluctuated tightly with the measured values (bias= 0.06, RMSE = 0.85 and RE = 0.20) (Figure 6a,c). On the contrary, less satisfactory results were obtained by using S

_{y}calibrated by the whole dataset, with relatively low values of slope (0.66) and R

^{2}(0.76) and high RMSE (1.21) and RE (0.28) ((Figure 6b). In general, the estimated daily ET

_{G}using S

_{y}calibrated by the whole dataset was slightly underestimated and overestimated during the water table declining period and rising period (Figure 6c), respectively. This seems to be due to the significant differences in estimates of S

_{y}between the dataset-by-dataset and multi-dataset procedures during these two periods.

_{g}at a half-hourly time step was expected to be less than 1 (Figure 7). Similar to the results observed at a daily time step, the estimated ET

_{g}using S

_{y}calibrated by the dataset-by-dataset procedure showed better correlations with measured ET than that using S

_{y}calibrated by the whole dataset (Figure 7a,b). In addition, it was observed that the diurnal pattern of estimated ET

_{g}and measured ET are very similar under different water conditions, despite that the magnitudes of ET

_{g}were different for the two S

_{y}calibration procedures (Figure 7c). Thus, it seemed that the method proposed here can successfully estimate groundwater evapotranspiration at different time steps, but the variations in S

_{y}should be carefully taken into account.

#### 4.4. Water Use Pattern

_{S}and ET

_{G}estimated using S

_{y}calibrated by different datasets) in Figure 8. The results indicated that during the whole study period the cumulative ET

_{S}and ET

_{G}were 121 mm and 202 mm (Figure 8b), respectively. The ratio of ET that comes from groundwater (ET

_{G}/ET) and soil water (ET

_{S}/ET) was 0.62 and 0.38, respectively. Thus, groundwater was the main source for plant water consumption during the whole study period. Interestingly, we observed that the riparian plant had strong plasticity in water uptake during the whole study period. When soil moisture is available (i.e., from 25 July to 12 August), the plant consumed approximately equal amounts of soil water and groundwater resources (Figure 8a), and the ratios of ET

_{G}/ET and ET

_{S}/ET were 0.54 and 0.46 (Figure 8b), respectively. As the soil moisture was depleted (i.e., after 13 August), the plant tended to consume more groundwater for survival (Figure 8a), and the ratios of ET

_{G}/ET (>0.64) were larger than that of ET

_{S}/ET (<0.36) (Figure 8b).

## 5. Discussion

_{y}) is of crucial importance for estimating ET

_{G}based on water table fluctuations because its error is translated directly to the final estimates [5,10,38]. In this study, the values of S

_{y}estimated using the minimum RSS method fell within the ranges of the literature reports for silt loam soil. For example, Johnson [39] reported that S

_{y}for silt loam varied from 0.01 to 0.2. Singhal and Gupta [40] documented that the values of S

_{y}for rocky silt loam ranged from 0.02 to 0.05. In addition, we also calculated the S

_{y}values using the method proposed by Loheide et al. [21], which ranged from 0.0297 to 0.0353 with a mean of 0.0339 during the whole study period (see Supporting Information Figure S1). Thus, we thought that the minimum RSS method was suitable to obtain proper S

_{y}values for ET

_{G}estimates in our site. Interestingly, we found that S

_{y}varied significantly under different water table conditions, and its value for a declining water table condition was about three times that of a rising water table condition. This can be explained by the fact that encapsulated air in the aquifer, which can be as high as 20% of soil porosity [41], reduces the value of S

_{y}during the water table rising periods. Therefore, it is important to take the variations in S

_{y}under different water table conditions into account for long-term groundwater evapotranspiration estimations.

^{−1}] to estimate the groundwater recharge rate q(t). For each day, and maximum and minimum q(t) were obtained by selecting the largest positive rate of dWT/dt and the mean of dWT/dt between midnight and 6 a.m., respectively; then it is assumed that q(t) behaves linearly between two consecutive estimations. Loheide [7] proposed to estimate q(t) as a function of the detrended water table. The function, set up using data between midnight to 6 a.m. of two subsequent days, is assumed to be approximately linear during each day. Overall, the performance of our method was very similar to the Gribovszki method (Figure 9). However, we found that during the nighttime period (i.e., from 9 p.m. of the previous night to 6 a.m. in the morning) the estimated q(t) by the Gribovszki method was generally higher than the rate of dWT/dt (Figure 9a). Thus, unreasonable positive ET

_{g}was obtained by this method during the nighttime (Figure 9b). In addition, the maximum q(t) estimated by our method was generally higher and occurred earlier than that estimated by the Gribovszki method. Thus, the values of ET

_{g}estimated by our method were slightly higher than that of the Gribovszki method in the afternoon. On the contrary, the method proposed by Loheide [7] performed relatively unsatisfactorily. This may be mainly because the assumptions of the Loheide method (i.e., the head at the recovery source is constant or follows the general trend of the water table) may not be met at our site. Fahle and Dietrich [10] have systematically compared the performances of six widely used groundwater evapotranspiration estimation methods, and also found that the Loheide method performed considerably worse than other methods.

_{G}/ET by our method agreed with the results using the stable isotopic methods. For example, Wu et al. [43] documented that the ratios of ET

_{G}/ET of T. ramosissima at the southern edge of the Gurbantonggut desert were more than 0.43 in summer and autumn. Zhao et al. [45] reported that the ratio of plant water uptake that from groundwater of T. ramosissima in the lower reaches of Heihe River Basin was 0.49 in wet seasons and increased to 0.90 in dry seasons. To assess the seasonal patterns of water uptake of the riparian plant, a systematic collection of stable isotopes of different water sources (i.e., soil water, groundwater and xylem water) is needed in the future studies.

_{y}. In this study, the profile of soil moisture was measured only at one site, and it may be difficult to capture the spatial variability in soil moisture. Thus, there may be considerable uncertainty in the soil moisture portion of the water balance equation. To overcome this problem, a soil moisture monitoring network is planned to be set up at our site. Second, the periods from midnight to 8 a.m. in the morning and from 7 p.m. to midnight on the day of interest were selected for the recovery analysis at our site. However, this is a somewhat subjective choice of time period. To apply this method at other sites, a slightly different time period may be better for calibrating the parameters in Equation (5). Finally, if hydraulic redistribution is a significant process, plant water uptake from groundwater during the night is not zero. Thus, the fundamental assumption of diurnal water table fluctuation based methods will be violated [46].

## 6. Conclusions

_{y}). Our study revealed that the S

_{y}values determined by the minimum residual sum of squares method were consistent with those reported in the literature for similar soil, and varied considerably under different water table conditions. For long-term groundwater evapotranspiration estimations, the variations of S

_{y}under different water table conditions (i.e., falling, stable and rising) should be properly taken into account. In the future, stable isotope-based studies are needed to evaluate the accuracy of the method in estimating groundwater evapotranspiration rates.

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Conceptual diagram showing groundwater flow toward a riparian zone where it discharges via root uptake by phreato-phytic vegetation. ETs: ET from soil water [L T

^{−}

^{1}]; ET

_{G}: ET from groundwater [L T

^{−}

^{1}]; WT

_{0}and WT: water table depth in the background and the riparian zone [L], respectively; h

_{0}and h: the groundwater elevation in the background and the riparian zone [L], respectively; D: the total depth of the aquifer [L]; L: the distance from the background to the riparian zone [L]; and q: the groundwater flow rate toward the riparian zone [L T

^{−}

^{1}].

**Figure 3.**Illustration of the method proposed here for estimation of groundwater evapotranspiration based on water table fluctuation. (

**a**) The rate of change in the water table dWT/dt as it changes with water table depth; the data show a quadratic relationship for the period of zero groundwater evapotranspiration; (

**b**) diurnal variations in measured water table (WT) level (red line), calculated time-rate of change in measured water table level dWT/dt (blue line), and the groundwater recovery rate q(t) (dash black line) estimates at the study site for a 3-day sample period (10–12 August 2017). The day under investigation is highlighted in gray. Note that the method proposed here operates at the sub-daily time scale and has to be aggregated to obtain daily values ET

_{G}(dark gray).

**Figure 4.**Seasonal variation in daily mean values of (

**a**) R

_{n}(W m

^{−}

^{2}) (black line) and T

_{a}(°C) (red line), (

**b**) LAI (m

^{2}m

^{−}

^{2}); (

**c**) θ (m

^{3}m

^{−}

^{3}) at 2, 20, 80, 120, 160 and 200 cm depth, and (

**d**) water table depth (m) during the study period in the Sidaoqiao station.

**Figure 5.**Specific yield (S

_{y}) calculated by the water balance approach based on different datasets. The standard deviation was calculated using the best 500 values of Sy with minimum RSSs.

**Figure 6.**Comparisons between measured and estimated daily ET based on different Sy calibration procedures: (

**a**) dataset-by-dataset procedure; (

**b**) whole dataset procedure; and (

**c**) seasonal variations of the measured and estimated daily ET based on different Sy calibration procedures.

**Figure 7.**Comparisons between measured ET and estimated ETg at the sub-daily time scale based on different Sy calibration procedures: (

**a**) dataset-by-dataset procedure; (

**b**) whole dataset procedure; and (

**c**) diurnal variations of the measured ET and estimated ETg based on different Sy calibration procedures.

**Figure 8.**Seasonal variations in (

**a**) measured and estimated daily ET, as well as the different components of ET (i.e., ETS and ETG); (

**b**) cumulative ET and its different components (left axis) and the ratios of ETG/ET and ETs/ET at 10 day intervals (right axis) during the whole study period.

**Figure 9.**Illustrations of the performances of different sub-daily groundwater evapotranspiration estimation methods for a 3-day sample period (1–3 September 2017). (

**a**) diurnal variations in measure water table level, calculated time-rate of change in measured water table level, and the groundwater recovery rate q(t) estimated by the 3 different methods; (

**b**) diurnal variations of the measured ET and estimated ETg by the 3 different methods.

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## Share and Cite

**MDPI and ACS Style**

Su, Y.; Feng, Q.; Zhu, G.; Wang, Y.; Zhang, Q.
A New Method of Estimating Groundwater Evapotranspiration at Sub-Daily Scale Using Water Table Fluctuations. *Water* **2022**, *14*, 876.
https://doi.org/10.3390/w14060876

**AMA Style**

Su Y, Feng Q, Zhu G, Wang Y, Zhang Q.
A New Method of Estimating Groundwater Evapotranspiration at Sub-Daily Scale Using Water Table Fluctuations. *Water*. 2022; 14(6):876.
https://doi.org/10.3390/w14060876

**Chicago/Turabian Style**

Su, Yonghong, Qi Feng, Gaofeng Zhu, Yunquan Wang, and Qi Zhang.
2022. "A New Method of Estimating Groundwater Evapotranspiration at Sub-Daily Scale Using Water Table Fluctuations" *Water* 14, no. 6: 876.
https://doi.org/10.3390/w14060876