Modeling and Analysis of Rice Root Water Uptake under the Dual Stresses of Drought and Waterlogging

Abstract

The development of an accurate root water-uptake model is pivotal for evaluating crop evapotranspiration; understanding the combined effect of drought and waterlogging stresses; and optimizing water use efficiency, namely, crop yield [kg/ha] per unit of ET [mm]. Existing models often lack quantitative approaches to depicting crop root water uptake in scenarios of concurrent drought and waterlogging moisture stresses. Addressing this as our objective; we modified the Feddes root water-uptake model by revising the soil water potential response threshold and by introducing a novel method to calculate root water-uptake rates under simultaneous drought and waterlogging stresses. Then, we incorporated a water stress lag effect coefficient, $\phi \left(W_s\right)$, that investigated the combined effect of historical drought and waterlogging stress events based on the assumption that the normalized influence weight of each past stress event decreases with an increase in the time interval before simulation as an exponential function of the decay rate. Further, we tested the model parameters and validated the results obtained with the modified model using data from three years (2016–2018) of rice (Oryza sativa, L) trails with pots in Bengbu, China. The modified Feddes model significantly improved precision by 9.6% on average when calculating relative transpiration rates, particularly post-stress recovery, and by 5.8% on average when simulating soil moisture fluctuations during drought periods. The root mean square error of relative transpiration was reduced by 60.8%, and soil water was reduced by 55.1%. By accounting for both the accumulated impact of past moisture stress and current moisture conditions in rice fields, the modified model will be useful in quantifying rice transpiration and rice water use efficiency in drought–waterlogging-prone areas in southern China.

1. Introduction

Global climate change, along with surface conditions and intensive human activities, have led to more frequent and severe shifts between drought and waterlogging events. These climatic extremes present significant challenges to maintaining sustainable food security [1,2,3]. The Intergovernmental Panel on Climate Change (IPCC) report highlights the impact of these phenomena: billions of people have been affected by droughts and floods this century, with economic losses reaching hundreds of billions of dollars for droughts and tens of billions for floods [4]. Moreover, the losses attributed to drought and waterlogging in the context of climate change have escalated markedly compared with those at the end of the last century, underscoring the pressing need for more effective risk prevention and control strategies [5].

Rice (Oryza sativa, L), a major staple crop in southern China, faces severe challenges to its growth and water use efficiency, namely, crop yield [kg/ha] per unit of ET [mm], owing to both drought and waterlogging disasters. These challenges have led to estimated annual rice yield reductions exceeding 7600 tons in China, primarily in the middle and lower reaches of the Yangtze River and the Huang-Huai Plain, significantly impeding local agricultural sustainability [6].

The roots of rice plants, serving as active organs for absorption and synthesis, play a pivotal role in influencing the growth and development of aboveground parts, as well as in facilitating the supply of water, nutrients, and grain yield formation. Under unrestricted natural conditions, climatic factors determine root water absorption, while under field cultivation, soil hydraulic properties dictate root water uptake [7]. Roots absorb water from the soil, which then enters the xylem through root hair cells, is transported to the stem, diffuses to the leaves, and eventually transpires into the atmosphere through the leaf stomata. This process forms the soil–plant–atmosphere continuum (SPAC), with root water uptake serving as the driving force behind this entire cycle [8].

The role of roots is prominent in the context of rice agriculture affected by drought and waterlogging disasters [9,10]. Research indicates that the more developed the crop roots, the greater the capacity for absorbing water and nutrients from the soil, resulting in the increased efficiency of soil nutrients and water use [11]. Well-developed roots possess robust capabilities for water and nutrient absorption and transportation. Conversely, excessively large or weak roots, by either increasing or decreasing individual competitiveness, will become less adaptable [12]. As an integral component of the hydrological cycle, root water uptake regulates processes such as infiltration, rainfall, evaporation, transpiration, and drainage. Quantitative simulations of root water uptake play a role in guiding agricultural water management. Therefore, the accurate calculation of spatiotemporal patterns in plant root water uptake during soil moisture fluctuations is indispensable in managing regional water resources.

Macroscopic models describing root water uptake are commonly formulated as functions involving weighting factors such as root distribution and moisture stress, along with potential crop evapotranspiration. The Feddes model [13] is widely used because of its simplicity, comprehensive consideration of parameters, and accurate simulation of evapotranspiration. This model not only accounts for the influence of moisture stress on root water uptake but also considers the distribution of roots within the soil profile. To adapt to diverse root–soil environments, numerous scholars have proposed root water-uptake models with compensatory mechanisms based on the Feddes model [14,15,16,17,18]. For instance, Lai and Katul [19] and Li et al. [20] suggested that the root water-uptake rate is related to both root distribution and soil moisture conditions across the entire root zone, independent of the ratio of actual transpiration [Ta, cm d⁻¹] and potential transpiration [Tp, cm d⁻¹]. They expressed the water-uptake term as a linear function of Tp and introduced an additional compensatory weighting function to correct the disparities between the local water content and overall root zone moisture. Feddes et al. [21] advocated incorporating more complex and comprehensive physical mechanisms into existing modeling approaches while preserving the simplicity of the macroscopic models. Lier et al. [22] omitted the plant stress index and introduced the function M, linked to the matrix potential, establishing a root water-uptake model with physical mechanisms and implicit compensation.

Moreover, some researchers have decoupled the moisture stress function in compensation, viewing water-uptake compensation as a redistribution process resulting from uneven root–soil interface hydraulic head distribution [23,24,25]. Nevertheless, these models still rely heavily on root density distribution. Albasha et al. [26] argued that root water-uptake compensation should be independent of the plant’s stress state and should be described as a response to heterogeneous soil moisture distribution, aiming to depict root water-uptake behavior through a more detailed characterization of root density distribution dynamics. Carminati et al. [27] explained the delayed effect of root water uptake with the presence of mucilage in roots. Wu et al. [28] reintroduced a moisture stress recovery coefficient to describe root water-uptake compensation behavior during the recovery period of drought stress, positing that root water uptake is primarily related to the plant’s moisture status from the previous day. Berardi and Girardi [29] provided a comprehensive mathematical justification of the work by Wu et al. [28], framing the problem of delayed root water uptake in the concept of ecological memory. Consequently, the root water-uptake compensation coefficient can be simplified as an exponential function of the previous day’s moisture stress level.

In summary, over the past few decades, numerous root water-uptake modeling approaches have been proposed to elucidate the complex processes of root water uptake [30,31,32]. However, these models, constrained by various limitations, have not effectively captured the root water-uptake dynamics of rice under the dual stresses of drought and waterlogging. This study attempts to rectify these shortcomings by revising the soil water potential response threshold and developing a water stress lag effect coefficient, $\phi \left(W_s\right)$, to further improve the Feddes root water-uptake model and, consequently, to simulate plant transpiration dynamics, root water-uptake rates, and soil water movement under abrupt drought–waterlogging alternating conditions. This was achieved by quantifying the dynamic effects of water stresses on rice using data from three years (2016–2018) of pot experiments. This study assumed that the effects of environmental factors on all plants were the same before water stresses were applied, and the rice roots were assumed to have adapted to the previous drought before encountering the subsequent contrasting waterlogging. Our overall objective is to develop practical water management strategies for rice production for regions that experience frequent and unpredictable drought and waterlogging events.

2. Materials and Methods

2.1. Pot Experiments

Pot experiments were carried out at the Xinmaqiao Comprehensive Experiment Station of Irrigation and Drainage, Bengbu, China (117°21′34″ E, 33°08′56″ N), during the rice growing season (May to September) in 2016–2018. The annual average temperature and rainfall in this area are 14.3 °C and 911 mm, respectively. More than 60% of the rainfall is concentrated between July and September and falls in the form of torrential rain (see Supplementary Material Figure S1). Frequent drought is another climate defect in this region. Thus, local crops are often at a high risk of abrupt drought–waterlogging alternation stresses.

The study site was laid out in outdoor pots designed with three replications each year (Figure 1). This outdoor pot experiment was carried out in a multistep circular water pool with an internal diameter of 8 m. It was designed with seven steps of different depths to satisfy the requirements of different submergence depths. The dimensions of the experimental pots were 35 cm in inner diameter and 45 cm in height. Tiny holes of 2.5 mm at the bottom of the pots and the sand filter layer were composed of a water-permeable filter cloth, 3 cm coarse gravel, and 2 cm fine sand from bottom to top, enabling the free seepage of water. The soil in the pots was a typical black sandy loam (sub-clays) containing 8.6 g/kg of organic matter, 630 g/kg of total N, 90 mg/kg of alkali-hydrolyzable N, 16 g/kg of available P₂O₅, and 94 mg/kg of available K₂O. The PH of the soil was 7.5. The bulk density and a field water capacity of 0–40 cm were 1.54 g/cm³ and 0.42 cm³/cm³, respectively. On the south side of the pool was a movable rain shelter (8.0 m long, 8.0 m wide, and 2.5 m high) that protected the drought-stressed pots from rainfall.

The experiments between 2016 and 2018 involved inducing drought stress during the rice tillering stage and subjecting the rice to waterlogging stress during the stages from stem elongation to panicle initiation and heading–flowering. Two control groups were established to investigate the effects of drought and waterlogging conditions. The experimental setup for the rice panicle initiation stage from 2016 to 2018, incorporating rapid shifts between drought and waterlogging under controlled water conditions, is shown in

Table 1. Drought and waterlogging rapid shift water control plan during the rice jointing–booting period for 2016–2018.

Treatment	Drought Phase (Stage I)						Waterlogging Phase (Stage II)
	Drought Severity (%FC)		Duration (Days)	Start–End Dates (Month/Day)			Waterlogging Depth (Plant Height)	Duration (Days)	Start–End Dates (Month/Day)
				2016	2017	2018			2016	2017	2018
Sudden Drought–Waterlogging Alternation Group	RSDW1	50	5	7/25–7/29	7/30–8/3	7/28–8/1	1/1	7	7/30–8/5	8/4–8/10	8/2–8/8
	RSDW2	50	10	7/25–8/3	7/25–8/3	7/23–8/1	1/2	9	8/4–8/12	8/4–8/12	8/2–8/10
	RSDW3	50	15	7/25–8/8	7/20–8/3	7/18–8/1	3/4	5	8/9–8/13	8/4–8/8	8/2–8/6
	RSDW4	60	5	7/25–7/29	7/30–8/3	7/28–8/1	3/4	9	7/30–8/7	8/4–8/12	8/2–8/10
	RSDW5	60	10	7/25–8/3	7/25–8/3	7/23–8/1	1/1	5	8/4–8/8	8/4–8/8	8/2–8/6
	RSDW6	60	15	7/25–8/8	7/20–8/3	7/18–8/1	1/2	7	8/9–8/15	8/4–8/10	8/2–8/8
	RSDW7	70	5	7/25–7/29	7/30–8/3	7/28–8/1	1/2	5	7/30–8/3	8/4–8/8	8/2–8/6
	RSDW8	70	10	7/25–8/3	7/25–8/3	7/23–8/1	3/4	7	8/4–8/10	8/4–8/10	8/2–8/8
	RSDW9	70	15	7/25–8/8	7/20–8/3	7/18–8/1	1/1	9	8/9–8/17	8/4–8/12	8/2–8/10
Single Drought Group	SD1~SD3, SD7~SD9 (2016) SD1~SD9 (2017, 2018)						Normal Water Level (2~3 cm)
Single Waterlogging Group	Normal Water Level (2~3 cm)						SW1~SW3, SW7~SW9 (2016) SW1~SW9 (2017, 2018)
Control Group	Normal Water Level (2~3 cm)

Note: FC represents the field water capacity. RSDW represents the rapid shift between drought and waterlogging treatment groups. SD represents the single drought treatment group SW represents the single waterlogging treatment group. Control represents the normal moisture treatment group.

.

The rice plants used in this study were a hybrid indica rice variety known as Long Liangyou Huazhan, which typically undergoes physiological maturation 123~140 days from germination. The seed breeding commenced in mid-May of each year, followed by transplantation 30 days later, typically around mid-June. Given the local annual climate change situation, the entire water management procedure was divided into three distinct stages: drought (Stage I), waterlogging (Stage II), and recovery to normal water level (Stage III). During the seedling period, the pots were all on the first step of the pool, keeping a 2–3 cm water level above the soil surface. Approximately three weeks after transplantation, parts of the pots were shifted onto the south side of the pool for free draining. The drought-stressed pots were weighed at 7 a.m. and 6 p.m. every day, and water was added according to the requirement of the target soil moisture content of each treatment (Table 1). At the end of Stage I, the drought-stressed pots were moved into the pool. The depth of the designed submergence was adjusted by adding bricks (235 mm × 115 mm × 53 mm) at the bottom of the pots (Figure 1b). At the end of Stage II, the pots were moved onto the first step until the harvest. During the entire period, the water table of the pool was measured by a ruler at 9 a.m. every day. Water was supplemented or drained out as required.

2.2. Sampling and Measurements

During Stage I (drought), the pot soil water changes were measured at 6:00 am and 6:00 pm. We added a certain amount of moisture according to Table 1 requirements. The weighing range of bucket weighing and reuse was 100 kg, with an accuracy of 2 g. The soil gravimetric water content was gravimetrically measured in 0.1 m depth intervals. Then, the volume water content was obtained through the ratio of soil gravimetric water content and soil dry bulk weight (1.54 g/cm³). During Stage II (waterlogging), the flood depth in each pot was measured twice a day (06:00 and 18:00) with a ruler. To maintain a constant water depth of 2–3 cm in the buckets within the normal treatment group, a specific volume of water was added or drained, and the amounts of water or drainage were measured and recorded.

Throughout the moisture stress process, destructive tests were conducted five times: at the onset of drought stress when soil moisture met the control requirements; at the conclusion of drought stress marking the commencement of waterlogging stress; at the cessation of waterlogging stress; and on the 10th and 20th days following the return to normal water levels. The morphophysiological indicators pertaining to plant root growth were measured, with each measurement conducted in three replicates. Throughout the rice growing season, all agronomic practices, except those related to moisture treatment, strictly adhered to the prescribed cultivation methods for shrimp rice fields.

To derive the root distribution parameters, a destructive sampling technique was used, as described by Schuurman et al. [33]. This involved recording the maximum root depth, followed by the selection of roots that exhibited a white or light-yellow color, indicative of their viability [34,35]. The selected roots were carefully extracted using tweezers and stored in bags at a temperature below 4 °C for future use. In accordance with the treatment sequence, root samples were arranged on transparent rectangular trays (dimensions: 250 mm × 200 mm × 20 mm) filled with a shallow water layer of 2 mm to minimize overlapping and crossing. Subsequently, the roots were scanned using a scanner (Epson Perfection V800 Photo Scanner; Seiko Epson Corp, Tokyo, Japan). The scanned images were analyzed using the WinRHIZO root analysis system (Regent Instruments Inc., Quebec City, QC, Canada), which enabled the acquisition of morphological indicators, such as root length, root diameter, root surface area, and root volume, for different soil layers within the various treatment soil columns.

2.3. Methodology

2.3.1. Improved Conceptual Framework for the Feddes Root Water-Uptake Model

The absorption of water by plant roots is a function of the water content surrounding the roots and is also influenced by the crop’s previous exposure to drought, i.e., water stress. Historical moisture stress experienced by the crop introduces a feedback mechanism that affects the current root water uptake. Therefore, the functional expression of the root water-uptake model, which accounts for the delayed effect of soil moisture stress, is expressed as follows, per Feddes et al. [13].

$$S\left(h,z,t\right)=\phi \left(W_s\right)\mathsf{\alpha }\left[\mathrm{h}\left(h,z\right)\right]S_{max}\left(h,z,t\right)={\phi \left(W_s\right)\mathsf{\alpha }\left[\mathrm{h}\left(h,z\right)\right]R}_{lp}{L}_d(h,z)$$

(1)

where $S\left(h,z,t\right)$ represents the actual root water-uptake rate on the $t$ day after transplant at a soil depth of $h$, measured in cm³ cm⁻³ d⁻¹; $S_{max}\left(h,z,t\right)$ denotes the maximum root water-uptake rate in cm³ cm⁻³ d⁻¹; $\phi \left(W_s\right)$ is the function reflecting the lag effect of moisture stress on the crop, where $0\le \phi \le 1$ indicates a stress-induced lag effect, and $\phi >1$ signifies a compensatory lag effect; $W_s$ refers to the cumulative index of historical moisture stress experienced by the crop; $R_{lp}$ is the potential root water-uptake coefficient per unit root length measured in cm³ cm⁻¹ d⁻¹; $\alpha \left[h\right(\mathrm{h},z\left)\right]$ represents the soil water potential response function, which is expressed as a linear function of soil water potential, $h$ (in cm); and $L_d(\mathrm{h},z)$ denotes root length density in cm cm⁻³.

A power function characterizes the lag effect of moisture stress experienced by the crop, with the functional expression of $\phi \left(W_s\right)$ being

$$\phi \left(W_s\right)={(1-W_s)}^{\lambda }$$

(2)

Here, $\lambda$ is a fitting coefficient, with $\lambda \ge 0$.

Theoretically, the influence of past moisture stress events on root water uptake gradually diminishes as the time intervals increase, particularly when considered on a daily scale. In this context, the more recent a historical moisture stress event is in relation to the present, the more pronounced its effect on the current rate of root water uptake. Consequently, the weight assigned to the moisture stress event from the previous day exerts the most substantial influence on current root water uptake [36]. The cumulative effect of historical moisture stress experienced by the crop is provided by

$$W_s={\int }_{t_0}^{t-1}{\omega }_{\tau }K\left(t-\tau \right)d$$

(3)

where $\tau$ represents any historical moment; $t_0$ is the start time of moisture stress; $t$ denotes the current time; $t-\tau$ indicates the proximity of the moment, $\tau$, to the current time, with larger values of $\tau$ signifying closer proximity to the current time; ${\omega }_{\tau }$ is the index of the plant water deficit degree at time $\tau$; and $K(t-\tau )$ is a weak lag kernel function, which varies with $t-\tau$, as illustrated in Figure 2. Equation (3) can be interpreted as follows: the transition of lag, $\tau$, from $t_0$ to $t-1$ effectively traces back from the current time, $t$, to the onset of moisture stress. Considering that moisture stress levels vary at different times, especially under multiple moisture stress conditions such as drought and waterlogging, the integral form of the model encapsulates the cumulative influence of both past and current moisture stresses on the plant.

The ${\omega }_{\tau }$ in Equation (3) is estimated using the measured values, and its functional expression is

$${\omega }_{\tau }=\frac{e_p-e_a}{e_p}$$

(4)

where $e_p$ represents the instantaneous evaporation rate of rice leaves at time $\tau$ under normal moisture conditions, measured in cm d⁻¹.

$e_a$ is the evaporation rate of rice leaves at time $\tau$ under moisture stress conditions, also measured in cm d⁻¹.

The function, $K(t-\tau )$, changes with $t-\tau$, and its expression is

$$K\left(t-\tau \right)=e^{\frac{-\left(t-\tau \right)}{t}}$$

(5)

Given that the soil moisture response threshold for rice differs from that of most xerophytic crops (as shown in Figure 3), we revised the soil water potential response function, $\alpha \left[h\right(z\left)\right]$, in the Feddes model, as detailed in Equation (6):

$$\mathsf{\alpha }\left[\mathrm{h}\left(h,z\right)\right]=\left\{\begin{array}{c}\frac{0 h_1\le h}{\frac{h_1-h}{h_1-h_2} h_2\le h<h_1}\hfill \\ 1 h_3\le h<h_2\hfill \\ \frac{\frac{h-h_4}{h_3-h_4} h_4\le h<h_3}{0 h\le h_4}\hfill \end{array}\right.$$

(6)

In this equation, $h$ represents the current soil water potential, measured with a tensiometer (cm) in the wet range or with soil psychrometers in the dry range. $h_1$ denotes the anaerobic threshold water potential, cm. Since rice possesses well-developed aerenchyma, anaerobic conditions are only likely when the plant is completely submerged; hence, $h_1$ is set to the actual height of the plant. $h_2$ is the critical water potential for oxygen stress, which corresponds to the soil water potential under normal rice growth conditions and is set to the normal water depth for rice, 3 cm for this study. $h_3$ indicates the critical soil water potential for drought stress, −400 cm for this study. $h_4$ represents the water potential at the wilting point of the plant, −15,000 cm for this study.

2.3.2. Validation and Quantification

For the entire root zone layer, the equations are as follows:

$$T_p={\int }_0^{d_r}{S}_{max}(h,z)dz={\int }_0^{d_z}{R}_{lp}{L}_d(h,z)dz=R_{lp}{RL}_{PA}$$

(7)

$$\begin{gathered} T_a={\int }_0^{d_r}\mathsf{\phi }\left(W_s\right){\mathsf{\alpha }\left[\mathrm{h}\left(h,z\right)\right]S}_{max}(h,z)dz={\int }_0^{d_r}\mathsf{\phi }\left(W_s\right)\mathsf{\alpha }\left[\mathrm{h}\left(h,z\right)\right]R_{lp}{L}_d(h,z)dz \\ ={\mathsf{\phi }\left(W_s\right)R}_{lp}{\int }_0^{dr}\mathsf{\alpha }\left[\mathrm{h}\left(h,z\right)\right]L_d(h,z)dz \end{gathered}$$

(8)

In this context, $T_p$ denotes the potential transpiration intensity of rice, measured in cm d⁻¹; $T_a$ represents the actual transpiration intensity, also measured in cm d⁻¹; $d_r$ is the maximum growth depth of the root system (cm); and ${RL}_{PA}$ signifies the total root length per unit area within the soil root zone layer (cm ²). Based on the measured total root length and maximum root depth, Equation (8) can be used to simulate the actual transpiration intensity, $T_a$ (simulated value), under the rapid shift stress conditions of drought and waterlogging for various treatments. These simulated values are then compared with values calculated through water balance during Stage I (drought; see Equations (9)–(11)) or via the Penman–Monteith (PM) model for transpiration intensity, $T_a^{\prime }$ (during other stages, referred to as measured values; see Supplementary Material and Equations (10) and (11)).

The bottoms of the pots were sealed, and thus, the water balance for the rice plants during the drought growth stage can be provided by

$$\mathsf{\Delta }H=P+R_{in}-{ET}_a^{\prime }$$

(9)

In this equation, $\mathsf{\Delta }H$ represents the changes in soil water potential, cm; $P$ denotes the rainfall, cm; $R_{in}$ represents the irrigation, cm; and ${ET\prime }_a$ is the actual evaporation transpiration of rice, mm d⁻¹. The calculation formula is provided in the Supplementary Materials.

${ET\prime }_a$ includes the crop leaf transpiration strength, $T_a^{\prime }$, and the bare soil evaporation, $E_s$, between crops, where $E_s$ can be calculated by the following, as per Ritchie [37].

$$T_a^{\prime }={ET\prime }_a-E_s$$

(10)

$$E_s=\left\{\begin{array}{c}{ET\prime }_a\times \left(1-0.43\times LAI\right) LAI\le 1\\ \\ {ET\prime }_a\times \exp \left(-0.4\times LAI\right)/1.1 LAI>1\end{array}\right.$$

(11)

where $E_s$ is the bare soil evaporation between crops, mm d⁻¹ and mmd⁻¹; $LAI$ is the leaf area index of the rice.

2.3.3. Model Application

The improved Feddes root water-uptake model (as in Equation (1)) can be used not only to calculate the relative transpiration rate, $T_a/T_p$ (in conjunction with Equations (7) and (8)), but also to simulate soil moisture dynamics using the Richards equation of van Genuchten et al. [38] and Vogel et al. [39]. Its basic form is as follows:

$$C\left(h\right)\frac{\partial h}{\partial t}=\frac{\partial }{\partial z}\left[K\left(h\right)\frac{\partial h}{\partial z}\right]-\frac{\partial K\left(h\right)}{\partial z}-S\left(h,z,t\right)$$

(12)

$$h\left(z,0\right)=h_0\left(z\right),0\le z\le z_m$$

(13)

$$-K\left(h\right)\frac{\partial h}{\partial z}+\frac{\partial K\left(h\right)}{\partial z}{\mid }_{z=0}=P\left(t\right)-E\left(t\right),t>0$$

(14)

$$h\left(z_m,t\right)=h_{z_m}\left(t\right),t>0$$

(15)

where $C\left(h\right)$ represents the specific soil water capacity, defined as $\mathrm{C}\left(h\right)=d\theta /dh$, measured in 1/cm. Here, $\theta$ denotes the soil water content on a volume basis, cm³/ cm³; $\mathrm{K}\left(\mathrm{h}\right)$ is the soil hydraulic conductivity, measured in cm d⁻¹; $\mathrm{h}$ denotes the soil pressure head in cm; z is the depth below the ground surface in cm; $h_0\left(z\right)$) indicates the initial soil water potential distribution in cm; $z_m$ is the thickness of the simulated soil layer in cm; $h_{z_m}\left(t\right)$ represents the soil water potential at the lower boundary in cm; $P\left(t\right)$ is the irrigation amount or rainfall during the simulation period, measured in cm d⁻¹; and $P\left(t\right)$ is the evaporation amount at the upper boundary surface, also measured in cm d⁻¹.

The parameters required for input in the model primarily include meteorological data; soil data such as soil moisture content and parameters related to soil moisture characteristic curves [40]; irrigation and drainage data; water levels in flooding pools; root length density; root depth; plant height; leaf area index; and stomatal resistance. Only one parameter, λ, needs to be determined by the measured data.

Subsequently, the revised root water-uptake model was applied to the drought stage in the experiment (RSDW1, RSDW4, and RSDW7) to simulate the dynamics of relative transpiration rate, $T_a/T_p$, via Equations (7) and (8) and soil water content changes in the soil profile via Equations (12)–(15) between three consecutive water stress treatment events. Similarly, a traditional root water-uptake model, Equation (1), was also established with additionally optimized parameters and applied during the experiment for cases not considering $\phi \left(W_s\right)$($\phi =1$). Finally, for verification, the simulated results regarding relative transpiration rate and soil water content were compared with the measured data.

2.3.4. Parameter Calibration

Two-thirds of the above input data collected during the three-year study period were used for the determination of model parameter λ, while the remaining one-third of the data, including measurements from the rapid shifts between the drought and waterlogging groups (RSDW1, RSDW4, and RSDW7)—as well as the corresponding single drought group (SD1, SD4, SD7) and the single waterlogging group (SW1, SW4, SW7)—were reserved for validation purposes.

The parameter identification process was carried out using the nonlinear least squares method [41], which involves minimizing the sum of squared errors between the simulated and measured values. The implicit difference method [42] was used to solve Equations (9)–(12), resulting in a tridiagonal equation system. An iterative approach [43] was used to linearize the equations, and ultimately, the matrix chasing–catching method [44] was used to solve them. The numerical simulation was performed with a time step of one day and a spatial step of 10 cm.

2.3.5. Validation Metrics

The performance of the model was assessed by applying several metrics [45], including the root mean square error (RMSE), mean relative error (MRE), model efficiency (ME), and coefficient of determination (R²). The RMSE was used to gauge the magnitude of the model’s estimation error, and lower values were preferred, indicating more accurate calculations. The MRE computes the relative error, and smaller values are generally considered better. The simulation results are typically deemed good if the MRE is less than 20%. The ME ranges from −1 to 1, where values between 0 and 1 indicate the model’s applicability and effectiveness, while values between −1 and 0 indicate that the model is unsuitable for the given data. The coefficient of determination, R², measures how well the model aligns with the measured data. Higher R² values, closer to 1, signify a better fit between the model and measured data for the relative transpiration rates and the soil moisture fluctuations.

3. Results

3.1. ${ET}_a^{\prime }$

Figure 4, which presents a linear regression comparing the actual and PM-equation-derived daily evaporation transpiration, ${ET}_a^{\prime }$, for rice from 2016 to 2018 reveals a distribution pattern where lower ${ET}_a^{\prime }$ values are predominantly situated above the 1:1 line, while higher ${ET}_a^{\prime }$ values fall below it. This suggests that the ${ET}_a^{\prime }$ values calculated via PM values tend to be lower during favorable weather conditions and higher during overcast or rainy conditions. This discrepancy may arise from measurement errors, particularly the omission of heat dissipation from the container walls under sunny conditions, which led to the erroneous attribution of this water loss to plant transpiration. Conversely, during rainy or overcast weather, atmospheric moisture and surface water from grassland can impact readings. In general, the approximation between calculated and measured values was within the allowable range of error. Therefore, the daily evaporation transpiration data computed using the PM formula proved reliable for use in root water-uptake models for rice under various moisture stress conditions.

Figure 5 illustrates the variation in the daily evaporation transpiration rates of rice during the growth period under different stresses. Under sudden RSDW and SW conditions, a sharp decrease in ${ET}_a^{\prime }$ was observed due to waterlogging stress, with greater reductions corresponding to higher water depths. This trend can be attributed to the depth of submergence, which affects the area of rice leaves exposed above the water surface. Deeper submergence results in a smaller leaf area available for ${ET}_a^{\prime }$. The trend for single drought stress (SD) generally aligns with that of the control group, with only a slight decrease during the drought phase, and the degree of reduction increases with the severity of drought. Upon re-establishing normal water levels, the ${ET}_a^{\prime }$ of the single waterlogging group rapidly recovers, even exceeding normal levels. In contrast, recovery in the RSDW group is less pronounced than in the single waterlogging group. Although rapid recovery is also observed, ${ET}_a^{\prime }$ in the RSDW group remains significantly low, especially in severe RSDW1 (Figure 5a). However, compared with the single drought group, ${ET}_a^{\prime }$ in the RSDW group gradually surpasses that of the single drought group over time, even exceeding that of the control group (RSDW2, Figure 5b). This indicates that later-stage waterlogging stress can help mitigate the suppressive effects of early drought stress on plant evaporation transpiration.

3.2. Simulation of Relative Transpiration Rate

In this section, we assess the modified Feddes root water-uptake model by simulating the relative transpiration rate,$T_a/T_p$, which was tested against various combinations of drought and waterlogging stress conditions (severe drought–waterlogging alternation group, RSDW1; moderate drought–waterlogging alternation group, RSDW4; and mild drought–waterlogging alternation group, RSDW7), along with their corresponding single drought (SD1, SD4, and SD7) and single waterlogging stress groups (SW1, SW4, and SW7), as depicted in Figure 6 and

Table 2. Comparison of the measured and estimated relative transpiration rates with the improved Feddes model and the original Feddes model.

Combination	Treatment	Improved Feddes Model			Original Feddes Model
		RMSE	MRE (%)	R2	RMSE	MRE (%)	R2
Calibration Period
Sudden Drought-Flood	RSDW Group	0.067	21.55	0.93	0.129	30.1	0.67
Single Drought	SD Group	0.045	6.35	0.90	0.074	11.6	0.72
Single Flood	SW Group	0.061	8.29	0.91	0.107	18.37	0.65
Validation Period
Sudden Drought-Waterlogging Alternation	RSDW 1 Group	0.097	24.5	0.95	0.197	48.9	0.76
	RSDW 4 Group	0.099	16.73	0.89	0.135	27.59	0.68
	RSDW 7 Group	0.039	16.17	0.88	0.068	29.22	0.52
Single Drought	SD1 Group	0.046	6.61	0.98	0.085	11.6	0.88
	SD4 Group	0.048	6.55	0.90	0.098	13.29	0.60
	SD7 Group	0.027	6.98	0.91	0.059	14.03	0.63
Single Waterlogging	SW1 Group	0.038	16.05	0.96	0.114	22.1	0.78
	SW4 Group	0.071	8.13	0.91	0.158	17.94	0.58
	SW7 Group	0.047	5.56	0.92	0.083	14.27	0.73

Note: RSDW represents the rapid shift between drought and waterlogging treatment groups, SD represents the single drought treatment group, SW represents the single waterlogging treatment group, and control represents the normal moisture treatment group.

.

The dynamic $T_a/T_p$ simulation results presented in Figure 6 demonstrate that the modified Feddes model exhibits a higher level of agreement between the simulated and measured values compared with the original root water-uptake models, which exhibit certain deviations. Specifically, in groups with more severe drought or waterlogging stress, the original model tends to overestimate (e.g., RSDW1, SD1, and SW1), whereas in less severe cases, underestimation is observed during the rewatering period (e.g., RSDW4, SD4, and SW4). This discrepancy may arise from the original models’ assumption of uniform water uptake capabilities across all root systems, neglecting changes in these capabilities under moisture stress conditions. For instance, after a period of severe drought stress, the water-uptake capacity per unit of root length may differ from that under optimal moisture conditions. Hence, this assumption can lead to deviations from the actual measurements in certain scenarios, such as during the rewatering stage. The deviations are smaller when the historical moisture stress is simpler (e.g., SD1, SD4, and SD7) and become significantly more pronounced with complex historical moisture stress (e.g., RSDW1). The modified Feddes model, incorporating parameters to account for the lagging effect of historical moisture stress on root water-uptake capability, such as the plant water deficit index and a time-delay influence function, effectively compensates for this limitation in the traditional Feddes model.

The trend in relative transpiration rates also reveals that severe (RSDW1) drought and waterlogging stress significantly inhibits root water-uptake capabilities, with recovery post-rewatering being challenging. In contrast, mild (RSDW7) or moderate drought and waterlogging stress only suppresses root water uptake during the stress period, but recovery to normal levels is generally achievable post-rewatering. After rewatering, the relative transpiration rate in the sudden drought–waterlogging alternation group (RSDW) remains suppressed consistently below 1.0, in contrast to the single drought or waterlogging stress groups. Single moderate/light drought (SD4; SD7) or waterlogging (SW4; SW7) stress typically exhibits a compensatory response in root water uptake post-rewatering, whereas no such compensatory effect can be observed after severe drought (SD1) or complete waterlogging (SW1).

Comparing the performance metrics (Table 2), the modified Feddes model demonstrates significant enhancements in RMSE, MRE, and R² compared with the original model, particularly during the rewatering phase post-moisture stress. In the RSDW1 group, during the 52–72-day rewatering phase post-transplantation, the gap between the simulated values of the two models is most pronounced. The modified Feddes model’s relative transpiration rate reduces the discrepancy from 0.24 at the end of the sudden RSDW to 0.13 after 10 days of rewatering and further to 0.06 after 20 days. In contrast, the original model’s discrepancy increases from 0.08 post-alternation to 0.25 after 10 days, gradually decreasing to 0.14 and consistently maintaining a higher level of deviation. During the moisture stress phase, both models exhibit smaller gaps between the simulated and actual values compared with the rewatering phase. This indicates that the improved model, by accounting for the lagging effect of prior moisture stress, significantly enhances simulation accuracy during rewatering. The fluctuation of soil moisture during the stress phase has a more pronounced effect on root water uptake, making moisture stress the dominant factor affecting root uptake during this period.

Compared with single drought or single waterlogging stresses, both models perform less effectively in simulating RSDW groups. Among these, RSDW1 shows the poorest simulation accuracy, followed by RSDW4, with RSDW7 being the most accurate. This suggests that dynamic changes in soil moisture affect the precision of the models, with more severe sudden alterations leading to greater inaccuracies.

3.3. Application of the Modified Model

Figure 7 shows the soil moisture content distribution within the soil column profile (10 cm, 20 cm, 30 cm, and 40 cm) during the drought stress phase, taking the RSDW5 treatment as an example to illustrate the simulation accuracy of the modified model and the traditional model. The profile distribution reveals non-uniform moisture content after drought stress, characterized by a dry upper layer and a more saturated lower region. Notably, the severity of drought stress is directly correlated with the aridity of the topsoil layer. In contrast to empirical measurements, the enhanced Feddes model demonstrates superior fidelity in simulating soil moisture distribution compared with the original model. In instances of moderate to severe drought conditions, the traditional model tends to overestimate the water uptake capacity of the root system, resulting in substantial disparities between the simulated and observed soil moisture content. However, under mild drought conditions, both models yield relatively congruent simulation results.

An analysis of

Table 3. Comparison of soil moisture content calculated by different root absorption models.

Treatment	Improved Feddes Model		Original Feddes Model
	RMSE (cm3 cm−3)	MRE (%)	RMSE (cm3 cm−3)	MRE (%)
Severe Drought	0.009	8.3	0.012	10.0
Moderate Drought	0.008	6.0	0.024	10.9
Mild Drought	0.005	5.5	0.013	8.4

reveals that, in contrast to the original root water-uptake model, the improved Feddes model manifests a reduced MRE, ranging from 5.5% to 8.3%. Moreover, the RMSE exhibits a lower magnitude in the improved Feddes model (ranging from 0.005 to 0.009 cm³ cm⁻³) when juxtaposed with the traditional Feddes model (ranging from 0.011 to 0.024 cm³ cm³). Consequently, it is evident that the enhanced Feddes model, as established in this study, more accurately represents the profile distribution of soil moisture content under drought stress conditions in contrast to the conventional root water-uptake model.

4. Discussions

Current research suggests that while soil moisture levels can swiftly return to optimal ranges following rehydration after drought, the recovery of plant root water uptake and transpiration functions is a relatively gradual process. This observation highlights the disparity between soil moisture effectiveness and the plant’s water status [46,47,48]. Various models have been proposed to elucidate the recovery mechanism of root water uptake, including micro-models based on the water potential gradient at the root–soil interface [49]. However, these models are infrequently employed because of the challenges associated with parameter acquisition. In contrast, macro-root models, such as the Feddes model [50], are more widely utilized because of their reduced input parameters and computational demands. Under conditions of ample water supply, the Feddes model assumes that roots absorb water at their maximum uptake rate and that plants transpire water at potential rates. Nevertheless, this assumption can lead to notable inaccuracies when simulating root water uptake after sudden drought–waterlogging transitions because it disregards the recovery process of plant roots.

Numerous models have introduced parameters to simulate the compensatory mechanisms of root water uptake. Jarvis et al. [14] conceived a macro-uptake compensation model by employing the plant stress index function, ω. Li et al. [51] proposed a weighted stress function linked to soil moisture and root distribution to depict localized uptake compensation. Wu et al. [28] introduced a moisture stress recovery coefficient to quantify the compensatory effect of alternating dry–wet stress on root water uptake. However, these models often possess limitations in scope, involve parameters that are challenging to quantify, or fail to effectively simulate the root water-uptake process under sudden drought–waterlogging stress conditions owing to simplifications. Building on prior research, this research redefined the soil water potential response function and introduced a moisture stress hysteresis influence function, thereby establishing a modified Feddes root water-uptake model tailored to sudden drought–waterlogging alternation conditions.

The moisture stress hysteresis influence index, $\phi \left(W_s\right),$ a key parameter in the improved Feddes model, reflects the capacity of the plant to recover from moisture stress to a certain extent. A higher $\phi$ value indicates the stronger recovery ability of the root system post-moisture stress. Figure 8 presents the calibration process of $\phi$ under different moisture stress conditions and the cumulative moisture stress index, $W_s$. It can be observed that $\phi$ decreases with an increase in $W_s$, implying that more severe historical moisture stress leads to poorer root water-uptake recovery post-rewatering. In both the sudden drought–waterlogging alternation condition (RSDW) and single drought (SD) groups, $W_s$ is almost always greater than zero, with $\phi <1$, indicating that both types of moisture stress inhibit root water-uptake capacity to some extent. In contrast, a significant portion of the single waterlogging (SW) group, especially in 50% and 75% moderate/light waterlogging conditions, shows $W_s$ values less than zero, where $\phi >1$, suggesting a compensatory response in the root water-uptake ability after single waterlogging events. The value of the parameter, λ, reflects the extent of the inhibitory effect of moisture stress on root water uptake. A larger λ results in a smaller $\phi \left(W_s\right)$. The λ values for groups RSDW, SD, and SW are 0.548, 0.447, and 0.503, respectively. This indicates that, under equivalent historical moisture stress, the root water-uptake performance in the groups experiencing a sudden shift from drought to waterlogging is severely inhibited and faces the greatest challenge in recovery. They are followed by the single waterlogging group, with the single drought group being the least affected.

Compared with traditional root water-uptake models, the modified Feddes model notably improves the accuracy of the root water-uptake simulation (Table 2 and Table 3). This improvement primarily stems from the model’s consideration of the cumulative effect of historical moisture stress, which enables it to capture the persistent effects of sudden drought–waterlogging stress. The modified model has certain limitations. Specifically, the introduction of a compensatory coefficient to adjust root water-uptake capability lacks a well-established physical mechanism. The Feddes model assumes that, under optimal soil moisture conditions, the contributions of different root layers to plant transpiration are equal. It treats root systems in all soil layers as equally proficient mini-absorbers, with the rate of water uptake by roots in each layer determined solely by the root density distribution. However, Adiku et al. [52] argued that root water uptake follows the principle of minimal energy consumption, suggesting that the uptake pattern is not solely dependent on root length density distribution but highly correlates with soil moisture content distribution. Further experimental research is necessary to determine which assumption more accurately reflects actual root water uptake. It is essential to recognize that the root water-uptake process is governed not only by physical energy regulation but also by molecular biological mechanisms within the plant. An ideal root water-uptake simulation should seamlessly integrate macroscopic physical mechanisms with microscopic molecular mechanisms, starting with the root structure itself, to establish a model that closely aligns with the adaptive patterns of the root system.

5. Conclusions

In this study, the relative transpiration rate and soil water in root water uptake were investigated and quantified under the dual stresses of drought and waterlogging on the basis of three-year experiments, and a traditional root water-uptake model was improved by redefining the threshold values of the soil water potential response function and by introducing a lag effect to the moisture stress coefficient. In contrast to traditional root water-uptake model, the modified root water-uptake model offered 45.7%, 48.3%, and 45.2% increases in MRE and 41.2%, 50.0%, and 56.1% increases in RMSE when simulating the relative transpiration rate under RSDW, SD, and SW conditions, respectively, and it offered 33.9%, 45.0%, and 16.4% increases in MRE, and 61.5%, 66.7%, and 25.0% increases in RMSE when simulating the soil volumetric water content change under mild drought, moderate drought, and severe drought conditions.

This study showed that the threshold values of the soil water potential can be adjusted according to the water depth of rice in a changing environment, and the lag effect of the moisture stress coefficient could be well described as an exponential function of the general extent of historical water stresses. Although this simplified method might be at the expense of slightly lowering the accuracy for RSDW, it was still reasonable and reliable in evaluating the combined effect on root water uptake under the dual stresses of drought and waterlogging and thus was adopted in this study. In comparison with the traditional root water uptake model, significant improvement was found in estimating the relative transpiration rate and simulating soil water dynamics when the lag effect of water stresses was taken into account, especially during the recovery periods of water stresses. The quantification of root water uptake in a changing environment might be helpful not only to understanding the mechanism of water transport in soil–plant systems but also to plant growth simulation and irrigation scheduling. However, further research is needed to validate and improve this finding in the more complex context of soil, plant, and climate conditions.