Keywords

1 Introduction

Reservoirs have been deployed as long-term investments to ensure water and energy security. However, the high transient flow triggered by the discharge regulation of reservoirs alters the hydrodynamic conditions of the natural river, which also exerts an impact on the navigation safety of vessels in the waterway to some extent (Liu et al. 2012). To satisfy the requirements of water supply, power generation, shipping and flood control, the discharge regulation of reservoirs generally involves a combination of regulation schemes (Tefs et al. 2021); the differences in hydraulic parameters such as river flow variation rate and flow amplitude during the regulation of reservoirs throughout the year result in periodic rising and falling water along the river.

In recent years, most of the studies on wave propagation triggered by the flow regulation of reservoirs have been studied for hydraulic elements such as wave period and hydraulic gradient, with prototype observations, physical model tests, and numerical simulation techniques being the general technical tools employed to investigate the hydrodynamic phenomenon. Although prototype observation techniques intuitively reflect the real-world hydraulic variation pattern of the river, the hydrodynamic field coupled with external complicated environmental factors (e.g., ship-generated waves) in the large scale plane makes it costly and difficult to obtain accurate data; on the other hand, although the physical model addresses the issue of a larger planar scale, the scale effect between the physical model and prototype as well as the accuracy control of experimental apparatus greatly restrict the improvement of the agreement between the model test and prototype measuring results. In light of this, to overcome the unfavourable factors such as long observation periods and high costs, numerical modelling techniques have become an accepted technical aid for the shipping community to investigate navigable flow conditions. Nowadays, the famous shallow water equations have been widely used in the field of river hydrodynamics, and many methodologies upon the solutions of shallow water equations have been investigated (Guinot 2003; Yoon and Kang 2004). Simultaneously, the hydrodynamic processes have been reported in engineering applications using commercial software (Nguyen and Zheng 2012; Muñoz et al. 2021). However, for a multi-branched channel, in recent years, several literature has reported the relationship between sediment distribution and flow rate (Das et al. 2022; Du et al. 2016) rather than the navigational flow conditions in the approach channel.

Previous research shows that gravity wave flows driven by reservoir scheduling have a greater impact on safe navigation in the river reaches (Bravo and Jain 1991; Shang et al. 2017). Due to the limitations of the sampling frequency of the prototype data, there are many research results involving steady flow, while the unsteady flow is mainly focused on the analysis of the flow field and wave characteristics (Wan et al. 2020). When the wave generated by the joint operation of the hub evolves into a lock approach connecting the lock head with the upstream and downstream navigation channels, due to the small cross-section coefficient of the channel and the significant shallow water effect (PIANC 1997; Vantorre 2003; Kazerooni and Seif 2013), gravity long-wave and ship hull coupling may affect the comfort of vessels and additional safety risks for navigating vessels. More importantly, considering the operation of the ship lock and ship lift, the safe docking of vessels at the lower lock head of the ship lift and the constraint of the reverse hydraulic head of the mitre gate further hinder the enhancement of the navigation guarantee rate (Xu et al. 2020). Therefore, in the context of the development of inland navigation, previous studies on the navigable flow conditions of the approach channel have been conducted (Zhao et al. 2012; Ioan et al. 2016; Xie et al. 2016). However, these studies have mainly focused on single or double approach channels under single-stage reservoir operations, and the driving factor of water-level fluctuations has not been highlighted, nor have the results of the available analyses been in-depth. With the reconstruction or expansion of inland navigation structures (Duviella et al. 2018; Li et al. 2021; Yan et al. 2019), the arrangement of multi-lane navigation structures may emerge with multiple channels coupled, and their hydrodynamic processes threaten the safety of ship navigation.

With the increasing demand for hydropower industry and inland waterway transport freight, water-level fluctuations in multiple approach channels under the joint operation of the two reservoirs have become a hot topic, and its specificity is manifested in two aspects: (i) the scale, number of branches and layout type of approach channels; and (ii) the flow regulation mode under the joint operation of the two reservoirs. In light of this, to investigate the evolution of the waves in bifurcating approach channels driven by the joint regulation of the two reservoirs, and considering the flow regulation of the dual reservoirs and the typical layout of the bifurcating approach channels, this study attempts to explore the influence of the discharge regulation mode of the two reservoirs on water-level fluctuations in bifurcating approach channels based on a two-dimensional (2-D) hydrodynamic model and to reveal the intrinsic link between the hourly variation of water level in the approach channel and flow regulation mode of the two reservoirs.

2 General Description of the Study Area

The TGD and the GZD located along the Yangtze River are nearly 38.0 km apart, and the Three Gorges and the Gezhouba reservoirs constitute a typical dual-reservoir. The river section is mostly presented as U- or V-shaped in cross-section. To be specific, the river reach from the TGD site to Letianxi is a wide valley section around 9.6 km in length, which the river reach from Letianxi to Nanjinguan Station is 26.49 km with several bends such as Letianxi, Liantuo, Shipai, and Nanjinguan. For reservoir functioning, the Gezhouba reservoir serves as a counter-regulation reservoir for the TGD with a limited regulating capacity of merely 0.86 billion m3, and the water level in front of the GZD varies within the range of 63.0–66.5 m.

In the waterway, the typical bifurcating approach channels consist of a ship lift approach, an existing lock approach, and a newly-planned lock approach. The ship lift approach channel merges into an existing lock approach 1,100 m downstream of the lock head and enters the main channel of the Yangtze River. The bottom width of the lock approach is around 180–200 m, and the elevation of the riverbed is 56.5 m. Several sampling sites in bifurcating approach channels are shown in Fig. 1.

Fig. 1.
figure 1

Sampling locations of the study area.

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3 Methodology

3.1 Hydrodynamic Model

In modelling the waterway flow, due to the horizontal scale being much larger than the vertical scale, the variation in water depth and velocity along the vertical direction is much smaller than that along the horizontal direction. Therefore, the three-dimensional Navier-Stokes equations can be averaged vertically to derive a set of depth-averaged two-dimensional equations, resulting in the following well-known two-dimensional St. Venant equations, including the continuity equation and momentum equation (without considering the effect of the Coriolis force and wind stress):

$$\frac{\partial h}{{\partial t}} + \frac{\partial hu}{{\partial x}} + \frac{\partial hv}{{\partial y}} = 0$$
(1)
$$\frac{\partial uh}{{\partial t}} + \frac{\partial uuh}{{\partial x}} + \frac{\partial uvh}{{\partial y}} = \frac{\partial }{\partial x}(\gamma_t h\frac{\partial u}{{\partial x}}) + \frac{\partial }{\partial y}(\gamma_t h\frac{\partial u}{{\partial x}}) - gh\frac{\partial z}{{\partial x}} - g\sqrt {u^2 + v^2 } /C^2$$
(2)
$$\frac{\partial vh}{{\partial t}} + \frac{\partial uvh}{{\partial x}} + \frac{\partial vvh}{{\partial y}} = \frac{\partial }{\partial x}(\gamma_t h\frac{\partial v}{{\partial x}}) + \frac{\partial }{\partial y}(\gamma_t h\frac{\partial v}{{\partial y}}) - gh\frac{\partial z}{{\partial y}} - g\sqrt {u^2 + v^2 } /C^2$$
(3)

Where t is time; x and y are horizontal Cartesian coordinates; h is water depth; u and v are depth-averaged velocity components in x and y directions, respectively; g is gravitational acceleration; z is bed elevation; C is the Chezy coefficient; γt is the horizontal eddy viscosity coefficient.

The spatial discretization of the equations is performed using a cell-centred finite volume method, and the second-order spatial accuracy is achieved via employing a linear gradient reconstruction technique. The time integration of the governing equations is performed using the second-order Runge-Kutta method. This study focuses merely on water levels without considering turbulence effects and the code used is programmed based on the finite volume method. In addition, the integrated hydraulic roughness of the model is determined by the water level of the river at steady-state conditions.

3.2 Mesh Generation and Boundary Conditions

The computation domain of the river reach between the TGD and the GZD is gridded using triangular mesh, with an encrypted grid number of 51,717 in the approach channel and a grid scale of 5–10 m; and a grid scale of 20–50 m in the river channel. The number of nodes and cells is 90,643 and 174,970 in the model, respectively. The layout of the grid within the approach channel is shown in Fig. 2.

Fig. 2.
figure 2

Mesh generation in bifurcating approach channels.

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Boundary conditions adopted for the model are set as follows: (i) the outflow boundary of the Three Gorges reservoir is set as the flow boundary; and (ii) the outflow boundary of the Gezhouba reservoir can be set as the flow boundary and water level boundary according to the specific scenarios investigated in this case study. The remaining boundaries are set as the land boundaries; normal fluxes were forced to zero for all variables along the closed boundaries.

3.3 Performance Evaluation Indices

To check and compare the performance of modelling approaches, two performance evaluation indices, including root mean square error (RMSE) and determination coefficient (R2), are selected to assess the adaptability and feasibility of the simulation results. The expressions of these performance evaluation indices are listed as follows:

$$RMSE = \sqrt {\frac{1}{N}\sum_{i = 1}^N {(M_i - S_i )^2 } }$$
(4)
$$R^2 = {{\left[ {\sum_{i = 1}^N {(M_i - \overline{M})(S_i - \overline{S})} } \right]^2 } / {\sum_{i = 1}^N {(M_i - M)^2 \sum_{i = 1}^N {(S_i - \overline{S})^2 } } }}$$
(5)

Where Mi and Si are the measured values and calculated ones, respectively; while \(\overline{M}\) and \(\overline{S}\) are the mean of the measured and simulated values, respectively; N represents the number of observations.

4 Results and Discussion

4.1 Model Validation

To assess the reliability and stability of the established hydrodynamic model, an operational scenario is assumed as follows: the discharge flow of the upstream reservoir is set to 6,000 m3/s and the outflow variation rate of the downstream reservoir is set to 2,000 m3/s/15 min. As shown in Fig. 3, two indices, R2 and RMSE, are utilized to evaluate the simulated and measured relative water depths corresponding to the initial water depths at the lower lock head of the ship lift and the entrance of the common approach channel.

Fig. 3.
figure 3

Numerical simulations versus physical model tests. (a) The lower lock head of the ship lift; and (b) The entrance of the common approach channel.

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A good correlation between the measured values and calculated ones can be found in Fig. 3. The simulated relative depth at the lower lock head of the ship lift is in accordance with that of the measured results from the physical model test, and the RMSE can reach 0.018. Overall, the established numerical model in this study can simulate the wave propagation in the approach channel.

4.2 Uncertainty Analysis of the Outflow Boundary of Reservoirs

Generally, the upstream reservoir is set as the flow boundary to satisfy the flow regulation of the dual reservoirs, whereas the boundary conditions of the downstream reservoir subjected to counter-regulation functioning can be divided into three types: constant water level, constant flow, and the outflow variation of the downstream reservoir is in accordance with that of the upstream reservoir. Therefore, assuming the following scenarios: the discharge flow of the Three Gorges reservoir is set to 6,000 m3/s and the variation rate is set to 2,000 m3/s/15 min; the outflow boundary at the GZD is set at the water surface elevation of 63.0 m, a constant flow of 6,000 m3/s, and the same flow variation rate as that of the Three Gorges reservoir. Figure 4 shows the amplitude of water-level fluctuations in bifurcating approach channels.

Fig. 4.
figure 4

The amplitude of water-level fluctuations under different boundary conditions

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The retrograde wave generated by the constant discharge flow of the downstream reservoir enables the water level fluctuating amplitude at the lower lock head of the ship lift to rise continuously, nevertheless, the amplitude at the lower lock head of the ship lift under the constant water level boundary oscillates periodically. Furthermore, the wave evolves from the entrance to the channel, with a small variation of water level at the entrance and a greater fluctuating amplitude at the lock head. In addition, the water level variation at the lower lock head of the ship lift under the constant water level and flow boundary increases by around 25% from the entrance of the ship lift approach channel to the lower lock head, whereas the consistency of the flow variation process of the two reservoirs increases water level fluctuating amplitude at the lower lock head of the ship lift by 42.5%, which highlights the amplification effect of the water level variation at the lower lock head of the ship lift. Overall, under the flow variation rate of 2,000 m3/s/15 min, the maximum hourly variation of the water level at the lower lock head of the ship lift under three types of boundary conditions exceeds 0.5 m (MOT 2018), therefore, the safe docking of the vessels at the lower lock head of the ship lift fails to meet the engineering application requirements.

4.3 Synchronous Variation of Flow Processes at the Two Reservoirs

The flow conditions in the river reach between the TGD and the GZD should meet the requirements for safe navigation of vessels as well as the maximum hourly variation of the water level at the vicinity of the lower lock head of the ship lift. Considering the influence of flow variation and variation rate on the amplitude of water-level fluctuations in bifurcating approach channels, under the scenario that the outflow process of the dual reservoirs is consistent, the initial outflows from the upstream and downstream reservoirs are set to 6,000 m3/s, and the various rates of the flow increment are set to 2,000 m3/s/5 min, 2,000 m3/s/15 min, and 2,000 m3/s/30 min, respectively. The water level fluctuating amplitudes are shown in Fig. 5.

Fig. 5.
figure 5

The maximum hourly amplitude of water-level fluctuations at various flow variation rates and the minimum water depth in bifurcating approach channels. (a) The maximum hourly amplitude of water-level fluctuations, and (b) Minimum water depth in the approach channel.

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As shown in Fig. 5, the water level variation in the approach channel increases with the growth of the flow variation rate. The water level variation at the lower lock head of the ship lift under the flow variation rate of 2,000 m3/s/5 min is 2.25 times that under the flow variation rate of 2,000 m3/s/30 min, and thus flow variability has a significant effect on the water-level fluctuation at the lower lock head of the ship lift. Notably, the amplification effect of wave amplitude is more obvious from the entrance of the ship lift approach channel to the lower lock head, especially when the flow variation rate is 2,000 m3/s/5 min, and the water level variation at the lower lock head of the ship lift under this scenario increases by 1.06 times. Furthermore, the minimum water depth corresponding to the trough in the approach channel meets the requirement of ship draft. In general, in the case of the flow variation rate of 2,000 m3/s/30 min, the amplitude of water level variation at the lower lock head of the ship lift is within the threshold value.

4.4 Time-Delay Effect of Flow Regulation

The flow regulation process of the two reservoirs has both synchronous and lagging characteristics, and in exceptional regulation situations there may be a lag in the flow regulation time of the downstream reservoirs affecting the water-level fluctuations in bifurcating approach channels. To clarify this issue, this subsection takes the river reach between the TGD and the GZD as an example and assumes that there is a hysteresis in regulation time for the two reservoirs. The base flow from the upstream reservoir is set to 6,000 m3/s, the initial water level of the downstream reservoir is set to 63.0 m, and the outflow variation rates of the reservoirs vary by 2,000 m3/s within 15 min. The propulsive wave reaches the GZD site for around 31 min. For this reason, 0, 31, 40, and 45 min following the occurrence of a flow pulse from the upstream reservoir are selected as the lag periods for the analysis, the amplitudes of water-level fluctuations at the entrance of the approach channel and the lower lock head of the ship lift at the flow variation rate of 2,000 m3/s/15 min are shown in Fig. 6.

Fig. 6.
figure 6

Water level fluctuating amplitude considering flow regulation lag periods.

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The linearized increment of pulse flow triggers the nonlinear process of water level variation in the river channel and approach channel. When the flow variation rates of the reservoirs increase by 2,000 m3/s/15 min synchronously, as presented in Fig. 6, the positive wave triggered in the upstream reservoir interacts with the negative falling wave triggered in the downstream reservoir, which leads to an increase in wave amplitude in the approach channel. As the flow regulation time of the downstream reservoir is extended, the initial base flow from the upstream reservoir increases from 6,000 to 8,000 m3/s, and the water level fluctuating amplitude decreases due to the increment of base flow. Hence, the lag of flow regulation could turn the scenario of the dual reservoirs regulation into a single-step regulation mode, and the water level fluctuating amplitude in the approach channel can be reduced to some extent.

4.5 The Constant Flow Operation Mode of Reservoirs

4.5.1 Three Gorges Reservoir with Constant Outflow

To explore the effect of flow amplitude on the water-level fluctuations in the approach channel, let the base flow from the Three Gorges reservoir be 6,000 m3/s and kept constant, the initial water level of the waterway is set to 64.0 m, and the outflow of Gezhouba reservoir is increased by 1,000 m3/s, 2,000 m3/s, and 3,000 m3/s respectively within 15 min, then the maximum hourly variation of water level in the approach channel under different flow amplitudes is shown in Fig. 7(a). Additionally, considering the effect of the retrograde wave triggered by the flow variation rate of the downstream reservoir outflow on the water-level fluctuations in the approach channel, the variation rates of the flow regulation of Gezhouba reservoir are set to 1,000 m3/s/5 min, 1,000 m3/s/15 min, and 1,000 m3/s/25 min, respectively, then the maximum hourly variation of water level at typical monitoring points in the approach channel under various flow variation rates are shown in Fig. 7(b).

Fig. 7.
figure 7

Maximum hourly variation of water level at various flow amplitude and variation rates. (a) Flow amplitude, and (b) Flow variation rate.

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The differences in flow amplitude and flow variation rate during the regulation of the downstream reservoir lead to the retrograde falling waves propagating upstream and evolving into the bifurcated approach channel, and the flow pulse magnitude and variation rate directly affect the amplitudes of water-level fluctuations in the approach channel. As shown in Fig. 7(a), the flow amplitude and maximum hourly water level fluctuating amplitude show a linear correlation, in particular, for every additional 1,000 m3/s, the water level variation at the lower lock head of the ship lift increases by 0.23–0.25 m; however, the water level fluctuating amplitude at various flow variation rates under the flow amplitude of 1,000 m3/s satisfies the requirements for safe docking at the lower lock head of the ship lift (Fig. 7(b)).

4.5.2 Gezhouba Reservoir with Constant Outflow

Under the scenario of a constant discharge flow of 6,000 m3/s from the Gezhouba reservoir, the following scenarios are assumed to analyze the amplitude of water-level fluctuations in a multi-bifurcated approach channel: (i) the discharge flow from the Three Gorges reservoir increases by 1,000 m3/s, 2,000 m3/s, and 3,000 m3/s within 15 min, respectively; and (ii) the flow variation rates are 2000 m3/s/5 min, 2,000 m3/s/15 min, and 2,000 m3/s/25 min, respectively. The maximum hourly variation of the water level at typical sampling sites in the approach channel under these scenarios is shown in Fig. 8.

Fig. 8.
figure 8

Maximum hourly variation of water level at various flow amplitudes and variation rates. (a) Flow amplitude, and (b) Flow variation rate.

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When the discharge flow of the GZD remains constant, the operation of the Three Gorges reservoir results in a prograde rising wave in the river and approach channel. Compared with the water level variations in other stations in the approach channel, it is found that flow transient regulation leads to a significant amplification effect of the water level fluctuating amplitude at the lower lock head of the ship lift. In particular, when the flow variation rate exceeds 2,000 m3/s/15 min, the lower lock head of the ship lift fails to meet the safety docking requirements of the ship-bearing compartment. Additionally, in comparison with the constant flow conditions of the TGD, the retrograde wave generated by the flow regulation of the Gezhouba reservoir has a larger wave energy loss due to the boundaries reflection by the bends such as Nanjinguan, Shipai, Liantuo, and Letianxi. With the same flow variation rate, the rising wave amplitude at the lower lock head of the ship lift triggered by the outflow of the upstream reservoir is larger than the falling wave amplitude caused by the operation of the downstream reservoir.

5 Conclusions

In this study, a hydrodynamic model is applied to investigate the influence of unsteady flow induced by the flow regulation of the two reservoirs on the water-level fluctuations in bifurcating approach channels, and combined with the typical operating conditions in the course of the regulation of reservoirs, flow amplitude and variation rate suitable for the safe docking of vessels at the lower lock head of the ship lift is explored, which may provide quantitative control for shipping safety. The following conclusions can be drawn from this study.

  1. (1)

    In the navigational scheduling system including the ship lift approach and the multi-lane lock approach, wave energy transport in bifurcating approach channels is driven by various boundary conditions, which leads to the most significant wave amplitude at the lower lock head of the ship lift. In particular, the special narrow hydraulic section of the ship lift approach channel causes the wave height to be amplified at the lock head during the evolution of gravity waves.

  2. (2)

    The difference in the scheduling mode of the dual reservoirs triggers high transient flow, which breaks the volumes of water balance between the two dams, and the flow amplitude and variation rate have a great impact on the water-level fluctuations in bifurcating approach channels.

  3. (3)

    When the variation rate of the reservoir outflow is 2,000 m3/s/15 min, the maximum hourly variation of the water level at the lower lock head of the ship lift exceeds the critical threshold value (0.5 m). The actual operation of the project should be avoided. However, the maximum hourly variation of water level at the lower lock head of the ship lift can be reduced provided that the simultaneous regulation of the dual-stage reservoir is generalized to interval scheduling.

Finally, it should be noted that the effect of flow variation rate and the flow amplitude on the water-level fluctuations in the approach channel is merely discussed from the perspective of the typical flow regulation modes, whereas the propagation processes of the gravity wave in the approach channel under more complex conditions are further enhanced in the subsequent study.