Empirical formulas for groundwater use under wheat growth conditions in the Huaibei Plain of China Open Access

Groundwater is valuable as it can be used for potable water (Panchal et al. 2020), industrial applications (Bachman et al. 2008), and crop water consumption (Gao et al. 2022). The calculation of crop groundwater use is an important research topic in the field of water balance computation (Wang & Hou 2008; Costelloe et al. 2014), which plays a key role in sustainable water resource management (Danielopol et al. 2003; Yang 2008; Costelloe et al. 2014), irrigation, and drainage design (Sepaskhah et al. 2003). Groundwater use is the delivery of phreatic water to the aeration zone (Yang 2008) via evapotranspiration (Yan & Zhou 2002; Wang & Hou 2008), one part of which is used for crop evapotranspiration and another part of which is used for ground evaporation between plants. Groundwater use can be observed using a Mariotte bottle, the groundwater level was set to a constant level (Dietrich et al. 2016), and the groundwater consumption by crops was measured using a flowmeter. Zhu & Wang (2013) obtained crop groundwater use by subtracting the phreatic water evaporation without crops from that with crops, which does not include the ground evaporation between plants. However, in mature crops, the blade shade restricts the ground evaporation between plants as a result of the minus groundwater use (Zhu & Wang 2013). Ground evaporation between plants occurs when the crops utilize groundwater. Therefore, in the present study, groundwater use includes ground evaporation between plants.

Many scholars have studied the groundwater use calculation. Initially, some scholars researched phreatic water evaporation without crops. Aviriyanover (1985) developed empirical formulae involving groundwater depth, water evaporation capability, and phreatic water evaporation. Subsequently, the Ye Shuiting formula (Ye et al. 1982), hyperbola formula (Hu et al. 2004), and Tsinghua formula (Zhidong et al. 1988) appeared in China. Torres & Hanks (1989) and Feddes et al. (1978) constructed a mechanistic numerical simulation formula from the perspective of soil water dynamics. Because the empirical formulae are simple and have convenient calculations, they provide a validation method for numerical simulation, and the empirical formulae provide an important method to study phreatic water evaporation (Luo et al. 2013).

To calculate phreatic water evaporation in crops, some scholars applied the phreatic water formula of bare land to directly calculate groundwater use; however, these formulae faced problems because crop growth has been neglected as one of the vital factors that influence groundwater use (Cheng 1993). Regarding these shortcomings, Mao et al. (1999) and Zhu et al. (2002) added new parameters or changed the parameters of the formulae to calculate groundwater use. Luo et al. (2013) further established the relationship between groundwater use and crop growth duration to reveal the influence of different crop growth periods on groundwater use. However, Luo et al. (2013), Mao et al. (1999), and Zhu et al. (2002) used the same empirical parameter in different crop growth stages in the empirical formulas, which did not consider the influence of different crop growth stages on the groundwater use. Grismer & Gate (1988), Li et al. (1992), and Sepaskhah et al. (2003) developed a linear relation formula for the ratio of groundwater use, crop water demand, and groundwater depth, but did not consider the non-linear effect of the groundwater depth on the groundwater use.

Therefore, to consider the influence of different crop growth stages and the non-linear effect of the groundwater depth on groundwater use, three types of groundwater use expressions were developed by referring to the phreatic evaporation formula without crops. According to the observation data, the groundwater depth, wheat evapotranspiration, and groundwater use points were fitted by the three formulae, and Accelerated Gene Algorithm (AGA) was used to optimize the formula parameters so that the final groundwater formula could be recommended according to their performance.

For the formula method, phreatic evaporation is calculated with atmospheric evaporation capacity and groundwater depth, primarily using the Aviriyanover, Ye Shuiting, and power function formulae (Wang et al. 2009), with the Aviriyanover formula being the most widely used (Zhou et al. 2015). To calculate phreatic evaporation under wheat conditions and understand the influence of wheat transpiration and soil evaporation on phreatic evaporation (Wang & Hou 2008), the atmospheric evaporation capacity in the phreatic evaporation formula is changed into wheat evapotranspiration. Meanwhile, to eliminate dimensional effects, the groundwater depth is replaced by the ratio of actual groundwater depth and the groundwater depth where the wheat groundwater use is 0.

To conveniently calculate the field water balance, the phreatic evaporation is called the groundwater use in this study (Yan & Zhou 2002).

Referring to the formula of phreatic evaporation, the following can be obtained:

Crop constants reveal the relationship parameters between actual crop evapotranspiration and reference crop evapotranspiration, which are affected by soil, climate, crop growth conditions, management patterns, and other factors. Table 1 shows the crop parameters of wheat for all months.

There are three statistical assessment indices for the fitting and validation of results:

The daily crop evapotranspiration was calculated based on the sunshine duration, relative humidity degree, wind speed, maximum temperature, and minimum temperature data of a wheat field from 1991 to 2005. Meanwhile, the groundwater use of wheat for nine groundwater depths, including 0.2, 0.4, 0.6, 0.8, 1.0, 1.5, 2, 3, and 4 m from 1991 to 2005 was also observed. Because crop growth periods are a main factor influencing groundwater use (Wang & Hou 2008; Luo et al. 2013), growth periods should be considered when analyzing the relationships. Table 2 shows the growth periods of wheat.

The following characteristics from Figures 3,⁴⁵⁶–7 were observed:

• (1)

  Groundwater use and wheat evapotranspiration. Groundwater use increased with the increase in wheat evapotranspiration due to the crop evapotranspiration fraction from groundwater. For example, in the heading stage of wheat, when the daily crop evapotranspiration is 0.96 mm, the daily groundwater use of wheat is 1.04 mm; when the daily crop evapotranspiration is 4.42 mm, the daily groundwater use of wheat is 4.38 mm.

• (2)

  Groundwater use and groundwater depth. With the increase in groundwater depth, the slope of the straight fit line gradually decreased, and the correlation coefficients of wheat evapotranspiration and groundwater use gradually decreased. For example, in the jointing stage, the correlation coefficient between groundwater use and crop evapotranspiration is 0.63 when the depth of the groundwater is 0.2 m; the correlation coefficient is 0.48 when the depth of the groundwater is 0.8 m. This means that with increasing groundwater depth, groundwater use and the degree of dependence gradually decreased.

• (3)

  The correlation coefficients of groundwater and wheat evapotranspiration, as observed in Figure 7, do not reach 1, as the soil water content consumed by the wheat evapotranspiration is partly derived from groundwater, surface runoff, and precipitation. Therefore, wheat evapotranspiration was not exclusively from groundwater use, and the correlation coefficient did not reach 1.

• (4)

  Groundwater use and different growth stages of wheat. Figure 7 shows that the correlation coefficient increased from the seedling to jointing stage of crop growth, then changed slightly from the jointing to heading stage of crop growth, and finally decreased at the same groundwater depth at 0.2, 0.4, 0.6, 0.8, and 1.0 m. For example, when the groundwater depth is 0.2 m, the correlation coefficient increased from 0.55 to 0.64 from the seedling to jointing stage of crop growth, then changed slightly from 0.64 to 0.63 from the jointing to heading stage of crop growth, and finally decreased from 0.63 to 0.44 from heading stage to mature stage of crop growth. This illustrates that the groundwater use in the seedling stage and mature stages was low and in the jointing and heading stages was high. Wheat root at the seedling stage is shallow in depth, the leaf area index is small (Luo et al. 2008, 2013), and the evapotranspiration is low, owing to which groundwater use is less. Wheat grows strong during the jointing and heading stage and roots grow gradually, thereby increasing leaf area index and evapotranspiration. Consequently, root water absorption increases, hence decreasing the soil water potential in the unsaturated zone, which in turn increases the gradient of the submerged surface to the unsaturated zone (Luo & Sophocleous 2010; Luo et al. 2013). This accelerates the evaporation rate and the final higher groundwater use, and wheat grows slowly in the mature period with less water, resulting in reduced groundwater use.

To show the relationships among groundwater depth, wheat evapotranspiration, and groundwater use, the three-dimensional scatter diagram is plotted in Figure S1. Figure S1 shows that with the increase in groundwater depth, wheat evapotranspiration and groundwater use decrease gradually. When the groundwater depth of wheat was up to 3 m, the groundwater use was approximately 0. Therefore, for the groundwater depth H_max where the corresponding groundwater use was 0, in formulae (1)–(3), H_max of wheat was 3 m. During the seedling, jointing, and heading stages, the groundwater use of wheat increased gradually and decreased at the mature stage.

The results obtained after AGA optimization and calculation of statistical indices are listed in Table 3 (according to the analysis in the last section, H_max was 3 m). In the seedling stage, the performance of the power function formula for fitting data of wheat was the best with the largest correlation coefficient of 0.696; in the jointing stage, heading stage, and mature stage, the performance of the Aviriyanover formula was the best with the largest correlation coefficient of 0.721, 0.708, and 0.638 respectively.

Table 3

Results and statistical indices of fitting data of wheat with three formulae

Equations .	Parameters .	Seedling stage .	Jointing stage .	Heading stage .	Mature stage .	Average .
Aviriyanover formula	n	5.126	1.925	1.517	2.143	2.678
	RME	0.701	1.044	1.053	1.034	0.958
	RMSPE	1.064	1.979	2.203	2.458	1.926
	R	0.677	0.721	0.708	0.638	0.686
Ye Shuiting formula	a	5.000	2.330	1.886	2.605	2.955
	RME	0.763	1.139	1.169	1.098	1.042
	RMSPE	1.239	2.128	2.421	2.495	2.071
	R	0.658	0.714	0.702	0.632	0.676
Power function formula	a	0.064	0.251	0.292	0.231	0.210
	b	0.922	0.529	0.515	0.522	0.622
	RME	0.647	1.318	1.338	1.320	1.155
	RMSPE	1.053	2.467	2.791	2.734	2.261
	R	0.696	0.667	0.664	0.587	0.653

Equations .	Parameters .	Seedling stage .	Jointing stage .	Heading stage .	Mature stage .	Average .
Aviriyanover formula	n	5.126	1.925	1.517	2.143	2.678
	RME	0.701	1.044	1.053	1.034	0.958
	RMSPE	1.064	1.979	2.203	2.458	1.926
	R	0.677	0.721	0.708	0.638	0.686
Ye Shuiting formula	a	5.000	2.330	1.886	2.605	2.955
	RME	0.763	1.139	1.169	1.098	1.042
	RMSPE	1.239	2.128	2.421	2.495	2.071
	R	0.658	0.714	0.702	0.632	0.676
Power function formula	a	0.064	0.251	0.292	0.231	0.210
	b	0.922	0.529	0.515	0.522	0.622
	RME	0.647	1.318	1.338	1.320	1.155
	RMSPE	1.053	2.467	2.791	2.734	2.261
	R	0.696	0.667	0.664	0.587	0.653

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The results obtained after comparing the indices in the fitting results of all the formulae are listed in Table 4. For Table 4, the average performance of error analysis of the Aviriyanover formula was the best, which indicates that the fitting effect is better than that for the other formulae. Therefore, the improved Aviriyanover formula is recommended to fit scatters of groundwater use and crop evapotranspiration based on the results in Table 4.

The parameters suitable for the local wheat in the formulae can be obtained in Table 3. The data from 2002 to 2004 were used for validation. First, the evapotranspiration and groundwater depth data from 2002 to 2004 was applied to the formulae (1)–(3) to calculate the groundwater use x_i. Meanwhile, the observed value y_i of groundwater use was collected and combined with formulae (6)–(8), the assessment values of the validating results in all formulae were obtained. The computational results are listed in Tables 5 and 6.

Table 5

Validation results of groundwater use of wheat data using the three formulae

Equations .	Parameters .	Seedling stage .	Jointing stage .	Heading stage .	Mature stage .	Average .
Aviriyanover formula	RME	0.822	1.121	1.060	0.373	0.844
	RMSPE	1.422	2.059	1.981	0.525	1.497
	R	0.629	0.598	0.738	0.715	0.670
Ye Shuiting formula	RME	0.973	1.138	1.084	0.416	0.903
	RMSPE	1.680	2.064	2.019	0.571	1.584
	R	0.606	0.596	0.735	0.699	0.659
Power function formula	RME	0.798	1.155	1.135	0.482	0.892
	RMSPE	1.374	2.078	2.144	0.666	1.565
	R	0.650	0.625	0.744	0.660	0.670

Equations .	Parameters .	Seedling stage .	Jointing stage .	Heading stage .	Mature stage .	Average .
Aviriyanover formula	RME	0.822	1.121	1.060	0.373	0.844
	RMSPE	1.422	2.059	1.981	0.525	1.497
	R	0.629	0.598	0.738	0.715	0.670
Ye Shuiting formula	RME	0.973	1.138	1.084	0.416	0.903
	RMSPE	1.680	2.064	2.019	0.571	1.584
	R	0.606	0.596	0.735	0.699	0.659
Power function formula	RME	0.798	1.155	1.135	0.482	0.892
	RMSPE	1.374	2.078	2.144	0.666	1.565
	R	0.650	0.625	0.744	0.660	0.670

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Table 6 shows that the average rank of the errors of the Aviriyanover formula is excellent at 1 with the large average correlation coefficient of 0.670, and formulae with a lower rank of the errors are the power function and Ye Shuiting formulae. Because of the high precision and simplicity of the improved Aviriyanover formula with only one parameter, this formula is selected to further calculate groundwater use.

To compare the results of our work with others, the groundwater use method of Luo et al. (2013) was used to calculate the errors of validation (shown in Table 7). Table 7 shows that the accuracy of the improved Aviriyanover formula is higher than that of Luo et al. (2013) in the indices of RME, RMSPE, and R, because Luo et al. (2013) did not consider the influence of different crop growth stages on the groundwater use.

Based on Figure S1, three-dimensional diagrams of fitting scatters with Aviriyanover formula of formula (1) are drawn.

Compared with a two-dimensional diagram, a three-dimensional diagram includes more productive content, in which all the values can be compared, and the variation trend can be observed from an overall three-dimensional perspective. The fitting surface in Figure 9 shows that, with a decrease in groundwater depth and an increase in wheat evapotranspiration, groundwater use increases gradually. Overall, the fitting surface is a curved surface inclined toward the positive direction of the axis of wheat evapotranspiration. After comparing the fitting surface diagrams of the four wheat growth periods, the average curvature of the surface in the heading stage was the least, and the stages with more curvature followed the order of the jointing, mature, and seedling stages, indicating that compared with the heading stage, groundwater use at the seedling stage decreased more quickly with increasing groundwater depth and decreasing wheat evapotranspiration. This is relevant to the parameter n in the Aviriyanover formula, and the n in the heading stage was the least, followed by the jointing and mature stages with more curvature; the seedling stage had the highest n value. This indicates that the n value is linked to growth characteristics and groundwater use. The greater the n value, the larger the wheat groundwater use from shallow groundwater.

To overcome the problem of the minus value of the calculated groundwater use, considering the influence of wheat evapotranspiration and wheat growth periods on groundwater use, three types of groundwater formulae were developed referencing the formulae of phreatic evaporation without crops: the Aviriyanover, Ye Shuiting, and power function.

Use of groundwater under different groundwater depths considering wheat evapotranspiration from data collected during 1991–2005 was calculated. The results were used to plot two-dimensional wheat evapotranspiration and groundwater use and three-dimensional scatter diagrams of wheat evapotranspiration and groundwater use under different groundwater depths in various growth periods. From the scatterplots, we observed that groundwater use increased gradually with the decreasing groundwater depth and the increasing wheat evapotranspiration, and the correlation coefficients of wheat evapotranspiration and groundwater use were small in the seedling and mature stages whereas the correlation coefficients were large in the jointing and heading stages. This is because wheat grows strong during the jointing and heading stage and root water absorption increases.

The parameters of groundwater use formulae were calibrated using AGA with the help of groundwater use data (collected during 1991–2005) under different groundwater depths and evapotranspiration of wheat. The calibrated formulae were validated with the data of the last two years, and the validating effects were assessed and sorted with average relative errors, root mean square errors, and correlation coefficients. It was found that the fitting and validation errors of the Aviriyanover formula were the least with the correlation coefficients of 0.686 and 0.670, and this formula is recommended to calculate groundwater use.

In the fitting surface of scatters of groundwater depth, wheat evapotranspiration, and groundwater use, it was observed that the surface is a curved surface inclined toward the positive direction of the axis of crop evapotranspiration, the average curvature of the surface in the heading stage was the least, and the stages with less curvature followed the order of the jointing, mature, and seedling stages. This is relevant to the parameter n in the Aviriyanover formula. The greater the value of n in the Aviriyanover formula, the larger the groundwater use to wheat from shallow groundwater.

The recommended Aviriyanover formula can be used in some hydrology models and crop growth models. Many hydrology models do not consider the influence of groundwater use on the water balance equation and water cycle. The Aviriyanover formula can be added to the groundwater module of hydrology models and the calculated groundwater use by the formula should be added to the water balance equation of groundwater. Besides, the Aviriyanover formula can be used in crop growth models and there may be a larger transpiration of crops. The Aviriyanover formula can also be used in irrigation planning to save more water.

Although the Aviriyanover formula is easy to be applied, the precision is not very high. Some other methods, such as neural networks, will be tried to compute the groundwater use. Only wheat were studied in this paper, there are some other crops to be studied in the future, and more formulas can be used in the hydrology models and crop growth models.

This work was financially supported by the National Key R & D Program of China (2018YFC1508306), Anhui Provincial Natural Science Foundation (2208085QE179), the Fundamental Research Funds for the Central Universities (JZ2022HGQB0213, JZ2021HGTA0165), National Natural Science Foundation of China (U2240223, 51409002, 51779067, 51709071), the Key Projects of Natural Science Research in Universities of Anhui Province (KJ2020A0745).

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.