Analysis of the effect of unsteady reservoir flow and sand excavation on the channel conditions of the Yangtze River (Yibin-Chongqing)

Abstract

This paper systematically describes the current research methods for analyzing unsteady flows in an open channel. In particular, it focuses on elaborating a discrete method to solve the one-dimensional Saint–Venant equations for unsteady flow in an open channel, compares the advantages and disadvantages of different discrete methods, and proposes a new implicit iterative mathematical model applicable to the calculation of the multiwavelength unsteady flow in the Yangtze River from Yibin to Chongqing. This paper also simulates three typical unsteady flow processes and analyzes the effect of sand excavation on the water level in the upper reaches of the Yangtze River from Yibin to Chongqing considering flood, normal-water, and low-water conditions. The spreading characteristics of the daily adjusted unsteady flow were obtained in the Shuifu-Chongqing river reach. Under the conditions of the daily minimum unsteady discharge flow rate, the designed water level of the reach from Yibin to Chongqing increases by 0.19 m compared to the natural water level. With the increase in concentrated sand excavation in the upper reaches of the Yangtze River, a large number of shingle beaches have disappeared, and the riverbed has been drastically lowered. Affected by the excessive and unordered sand excavation in the river channel, the water level of the upper reaches of the Yangtze River from Chongqing to Yibin has decreased by 0.5–2.0 m.

Article highlights

• This work improves the Preissmann implicit difference method to effectively simulate the propagation characteristics of unsteady flow in long reaches.

• The daily adjustment of unsteady flow in the upper reaches increases the river flow during the low-water period so that the designed water level under the action of this daily unsteady flow adjustment increases by approximately 0.19 m compared to that under normal conditions.

• A large number of point bars have been damaged in the wake of sand excavation in long reaches,resulting in a water level decline of approximately 0.5–2.0 m.

1 Introduction

In studies on unsteady flow, there were few notable breakthroughs until the Saint–Venant [1] equations were established. Saint–Venant conducted research to derive the Saint–Venant partial differential equation for unsteady flow. Massau [2] proposed the use of the method of characteristics to solve the equation for unsteady flow in an open channel. Richardson [3] used partial differential equations to analyze unsteady flow, which is considered the first application of numerical simulation in hydraulics. Pereira [4] studied the fourth-order Runge–Kutta scheme, which has good stability. Pappenberger [5] conducted a stability limit test on the Preissmann four-point eccentric implicit scheme based on the Hydraulic Engineering Center’s River Analysis System (HEC-RAS). Their results show that at the limit value, the Preissmann four-point eccentric implicit scheme is still very stable. Freitag [6] employed the improved Thomas chasing method to solve the Preissmann four-point eccentric implicit scheme and achieved good results. With a limited number of research methods, research on this topic has been slow over the years, and few major breakthroughs have been made.

Sneider et al. [7] established an unsteady flow level prediction model for dam discharge to the lower reach. Theiling et al. [8] studied the effect of dam adjustment in the upper reach of the Mississippi River on the flow conditions of the waterway. Sokolov [9] studied the celerity of long waves in channels with complex cross sections. Toro [10] studied how the unsteady flow discharge of the Lagdo Hydropower Station causes a decrease in incoming flow in the lower reach during the low-water period. French [11] combined the Manning–Chezy formula (they assumed that it is applicable) with the Saint–Venant equations, deducing that the unsteady flow rate–water level relationship forms a loop-rating curve. Kundzewicz et al. [12] demonstrated wave attenuation under unsteady flow. Tu and Graf [13] believed that the increasing time of unsteady flow is approximately one-third of the cycle when the unsteady flow propagates for one wavelength. Many studies have shown that wave attenuation and deformation occur during the spreading of unsteady flow Lamberti and Pilati [14]. The wave amplitude decreases as the spreading distance increases. Through mathematical analysis, Kundzewicz and Dooge [15] provided an expression for the attenuated amplitude of unsteady flow waves.

The Yangtze River (Yibin-Chongqing) is 356 km in length. Research results have shown that human activities impact the variation in water and sediment conditions in the upper reaches of the Yangtze River, (Marwan et al. [16], Matos et al. [17], Cheng et al. [18]). Wang et al. [19] showed that the average annual sediment in the upper reaches of the Yangtze River has decreased compared with that before the completion of the reservoir group. In recent years, a large amount of sand and gravel has been extracted in the Yibin to Chognqing reach of the Yangtze River. In addition, the negative effects of sand and gravel mining on river channels have been studied (**ao et al. [20], **ao et al. [21]). Lu et al. [22] analyzed the impact of sand mining on the sedimentation of the Three Gorges Reservoir. Wang et al. [23] investigated the relation between sand mining and runoff-sediment discharge and concluded that the sediment transport intensity was lower in the sand-mined river reach. The previous studies all focused on local river reaches, whereas results concerning long reaches of the upper Yangtze River have rarely been reported.

The **angjiaba Hydropower Station is the last-stage power station in the **sha River hydroelectric cascade development. Ma [24] and Liu [25] showed that the unsteady flow caused by daily adjustments in the dam release changes the natural flow conditions of the river downstream. Currently, studies on the unsteady flow of an open channel in long reaches are still based on the mathematical model of water flow. For the study of unsteady flow downstream of **angjiaba, which is limited by the test conditions, research within 100 km downstream of **angjiaba was selected, but this selection cannot fully indicate the spreading features of the unsteady flow. Zhang [26] showed that sand mining has damaged the channel conditions to different degrees. Until now, few studies have been conducted on the unsteady flow and sand excavation in the Yangtze River (Yibin-Chongqing) given new boundary conditions.

The current study has eight sections. Section 2 presents an overview of the studied river reach. Section 3 presents a review of the numerical model and solution method. Sections 4 and 5 present the propagation characteristics of unsteady flow and the impact of unsteady flow on the water level of the river section. Section 6 presents the gravel excavation pit characteristics in the upstream reach of the Yangtze River based on field measurements and the impact of sand mining on the water level of the channel during different water level periods. Finally, Sect. 7 describes and discusses the results. In this paper, the one-dimensional unsteady flow mathematical model is improved. Studies on the daily changes in the unsteady flow in the reaches downstream of the dam and the characteristics of the water level drop in these reaches after sand excavation can provide technical support for the implementation of channel improvement projects in the upper reaches of the Yangtze River.

2 Overview of the studied river reach

The **sha River is 356 km in length from Shuifu (upstream channel 1076 km) to Jiang** of Chongqing (720 km). At present, the river reach from Shuifu to Yibin is approximately 31 km long. As a Class IV channel, it has dimensions of 2 m × 40 m × 300 m (waterway depth × waterway width × bending radius). Fleets of 220 kW + 2 × 300 t can navigate this reach year round. For the Yibin-Jiang** channel, the technical level reaches Class III, and the maintenance dimensions are 2.9 m × 50 m × 560 m. During the low-water period, this reach can accommodate 1,000-ton ships and their fleets. Figure 1 shows the area of research in the upstream reach of the Yangtze River. The mileage of the main site is shown in Fig. 2.

Fig. 1

Location of the Yangtze River (**angjiaba hydropower station to the Fuling reach) in China

Fig. 2

Schematic diagram of main station waterway mileages

3 Review of the numerical model and solution method

3.1 Comparison of the Saint–Venant equation solution method

Currently, due to calculation accuracy, the instantaneous flow regime method and the microamplitude wave theory typically used to solve the Saint–Venant equations are no longer used. According to the complicated calculation requirements, the method of characteristics needs to classify and solve a variety of flow regimes and requires very complete data. Thus, this method is seldom used. The discrete finite difference method is intuitive and simple and thus has been widely used. Anderson [27] presented a systematic summary of the finite difference method and divided this method into the explicit difference method and the implicit difference method.

The unsteady flows of the Yibin-Chongqing river reach exhibit great changes and a long spreading period. We discuss whether the calculation method and stability of the explicit and implicit difference methods in the existing finite difference method of the Saint–Venant equations are applicable to these flows.

3.1.1 Explicit difference method

3.1.1.1 Calculation error

The unknown $f_{i}^{j + 1}$ is not considered in the partial derivative of the distance calculation. The influence of $f_{i + 1}^{j + 1}$ can cause calculation errors that continue to increase with increasing distance and time step. Clearly, this method cannot be applied to the 500-km-long river reach from Yibin to Chongqing.

3.1.1.2 Scheme stability problem

All explicit schemes exhibit stability problems. The solutions remain stable only when the time interval $\Delta x$ and distance interval $\Delta x$ meet certain conditions.

For example, the stable scheme conditions of the diffusion scheme need to satisfy the relationship shown in Eq. (1):

$$\Delta t \le \frac{\Delta x}{{\left| {\nu \pm \sqrt {g\frac{A}{B}} } \right|}}$$

(1)

where $\nu$ is the flow velocity, A is the section area, and B is the section width.

Daily adjustments in the unsteady flow occur in the river reach from Yibin to Chongqing. For the conditions of incoming flow, since the time interval $\Delta$ t = 3600 s cannot be changed, there are limitations in selecting the section area $\Delta$ x. For a stable diffusion scheme, when $\Delta$ t = 3600 s and if $\left| {\Delta \left. {\nu \pm \sqrt {{\text{g}}\frac{\rm A}{{\text{B}}}} } \right|} \right.$ = 1 m/s, $\Delta$ x $\ge$ 3600 m. For most of the river reaches with dangerous beaches, the length of the channel that obstructs navigation is far shorter than 3.6 km. To study these segments, $\Delta x$ should be less than 300 m, which causes serious instability in the explicit scheme.

3.1.2 Implicit difference method

We discuss the applicability of the Preissmann implicit difference method as an example.

• (1) The Preissmann implicit difference method calculates the conditions where water flow changes only slowly. The unsteady flow of the Yibin-Chongqing river reach changes drastically during the low-water period. In addition, the Preissmann implicit difference method always linearizes the data, which is not suitable for the conditions of this river reach. For example, for $Q_{j}^{n}$,

  $$\left( {Q_{j}^{n} + \Delta Q_{j} } \right)^{2} \approx \left( {Q_{j}^{n} } \right)^{2} + 2Q_{j}^{n} \Delta Q_{j}$$

  (2)

  As shown in Eq. (2), the Preissmann implicit difference method ignores $\Delta Q_{j}^{2}$ in linearization. In practice, the daily adjusted unsteady flow of the Yibin-Chongqing river reach during the low-water period generally changes at ± 1000 m³/s, and the change is sometimes greater than 1500 m³/s, accounting for over 50% of the total flow. In addition, according to the characteristics of the flow discharge of the power station, such a large change is usually completed within an hour, i.e., within a unit time interval $\Delta t$. During the low-water period, the flow Q often fluctuates between 2500 and 4000 m³/s. It is evident that the linearization in the Preissmann implicit difference method is not applicable to hydraulic conditions in which there are dramatic changes.

• (2) The Preissmann implicit difference method is very unfavorable for program debugging. To simplify the method, the continuity and energy equations are simplified as shown in Eqs. (3) and (4).

  $$A_{1j} \Delta Q_{j} + B_{1j} \Delta {\text{z}}_{j} + C_{1j} \Delta Q_{j + 1} + D_{1j} \Delta z_{j + 1} = E_{1j}$$

  (3)
  $$A_{2j} \Delta Q_{j} + B_{2j} \Delta {\text{z}}_{j} + C_{2j} \Delta Q_{j + 1} + D_{2j} \Delta z_{j + 1} = E_{2j}$$

  (4)

  The equations show that both $\Delta Q_{j}$ and $\Delta {\text{z}}_{j}$ are known fixed conditions and that $A_{1j}$, $B_{1j}$, $C_{1j}$, and $D_{1j}$ have no obvious physical meanings. When calculation errors occur, it is not convenient to observe the changes in coefficients caused by various parameters, so debugging is almost impossible.

In conclusion, traditional difference schemes cannot completely meet the calculation conditions. The explicit scheme has inherent errors and may amplify errors due to instability; although the implicit scheme can unconditionally reach stability and calculate over a long duration, the dramatic changes in the unsteady flow of the Yibin-Chongqing river reach cause the Preissmann implicit difference method using the chasing method to generate large errors.

3.2 Proposed implicit iterative method

The Saint–Venant equations are transformed as shown in Eqs. (5) and (6):

$$\frac{\partial A}{{\partial t}} + \frac{\partial Q}{{\partial x}}{ = }0$$

(5)

$$R = gA\left( \frac{Q}{K} \right)^{2} + \frac{\partial Q}{{\partial t}} - \frac{2Q}{A}\frac{\partial A}{{\partial t}} - \left( \frac{Q}{A} \right)^{2} \frac{\partial A}{{\partial x}} + gA\frac{\partial z}{{\partial x}}$$

(6)

When R = 0, the solution meets the conditions of the Saint–Venant equations. In particular, $\partial x$ is the cross-sectional distance, $\partial t$ is the time step, Q is the flow rate, A is the cross-sectional area, h is the water depth, g is the acceleration of gravity, z is the water level, and K is the flow modulus.

This method refers to the four-point grid of the Preissmann implicit scheme and uses the weight coefficient $\theta = 1$, making the scheme fully implicit. The Preissmann format grid layout is shown in Fig. 3.

Fig. 3

Preissmann format grid layout

The continuous difference method form is shown in Eq. (7):

$$\frac{{\Delta A_{j + 1}^{n} + \Delta A_{j}^{n} }}{2\Delta t} + \frac{{Q_{j + 1}^{n + 1} - Q_{j}^{n + 1} }}{\Delta x} = 0$$

(7)

Equation (8) shows the kinematic difference method form:

$$\begin{aligned} R = & \frac{g}{64}(A_{j}^{n} + A_{j}^{n + 1} + A_{j + 1}^{n} + A_{j + 1}^{n + 1} )\left( {\left( \frac{Q}{K} \right)_{j}^{n} + \left( \frac{Q}{K} \right)_{j}^{n + 1} + \left( \frac{Q}{K} \right)_{j + 1}^{n} + \left( \frac{Q}{K} \right)_{j + 1}^{n + 1} } \right)^{2} \\ & + \frac{{\Delta Q_{j + 1}^{n} + \Delta Q_{j}^{n} }}{2\Delta t} - \frac{1}{2}\left( {\left( \frac{Q}{A} \right)_{j}^{n} + \left( \frac{Q}{A} \right)_{j}^{n + 1} + \left( \frac{Q}{A} \right)_{j + 1}^{n} + \left( \frac{Q}{A} \right)_{j + 1}^{n + 1} } \right)\frac{{\Delta A_{j + 1}^{n} + \Delta A_{j}^{n} }}{2\Delta t} \\ & - \frac{1}{16}\left( {\left( \frac{Q}{A} \right)_{j}^{n} + \left( \frac{Q}{A} \right)_{j}^{n + 1} + \left( \frac{Q}{A} \right)_{j + 1}^{n} + \left( \frac{Q}{A} \right)_{j + 1}^{n + 1} } \right)^{2} \frac{{A_{j + 1}^{n + 1} - A_{j}^{n + 1} }}{\Delta x} \\ & + \frac{g}{4}(A_{j}^{n} + A_{j}^{n + 1} + A_{j + 1}^{n} + A_{j + 1}^{n + 1} )\frac{{z_{j + 1}^{n + 1} - z_{j}^{n + 1} }}{\Delta x} \\ \end{aligned}$$

(8)

In these equations, all $f_{j + 1}^{n}$,$f_{j}^{n + 1}$,$f_{j}^{n}$,$\Delta t$ and $\Delta x$ are known. The unknowns include the area $A_{j + 1}^{n + 1}$ of section j + 1 at moment n + 1, the flow modulus $K_{j + 1}^{n + 1}$, the water level $z_{j + 1}^{n + 1}$, and the flow rate $Q_{j + 1}^{n + 1}$.

When the elevation of the section is known, as long as the water level z of a certain section is known, the area A and the flow modulus K of the section can be determined.

With the assignment of values to the unknown water level $z_{j + 1}^{n + 1}$, the value of the unknown flow rate $Q_{j + 1}^{n + 1}$ can be obtained through continuity Eq. (1), and the corresponding area $A_{j + 1}^{n + 1}$ and flow modulus $K_{j + 1}^{n + 1}$ can be determined according to the terrain data. Then, we substitute these parameters into the kinematic equation (Eq. (8)) and set limit values n(n → 0) for R. When R < n, it is deemed to satisfy the kinematic equation (Eq. (8)), and the next step can begin; if R < n is not satisfied, the unknown water level $z_{j + 1}^{n + 1}$ should be reassigned.

Since there are multivariate unknowns in the kinematic equation of the Preissmann scheme and more than two solutions have been obtained, it is necessary to solve these problems through linearization. In the implicit iterative method mentioned herein, the unknown water level $z_{j + 1}^{n + 1}$ is replaced with $z_{j + 1}^{n + 1} { = }z_{j + 1}^{n} + \Delta z_{j + 1}^{{}}$, in which $z_{j + 1}^{n}$ is known and $\Delta z_{j + 1}^{{}}$ is unknown, representing the water level amplitude of section j + 1. When the condition R < n is satisfied but $\Delta z_{j + 1}^{{}}$ takes an incorrect value, a limit value Rz can be set based on experience to recorrect $\Delta z_{j + 1}^{{}}$, and the limit value Rz is selected such that $1 < \left| {R_{z} } \right| < 3$. In fact, the simulation reveals that the error value $\Delta z_{j + 1}^{{}}$ in the multivariate solution is generally greater than 5 m. This restriction condition can prevent errors caused by linearization and ensure a correct calculation result.

3.3 Model accuracy verification

In Sect. 3.1, the advantages and disadvantages of the difference method for understanding the Saint–Venant equations are discussed in detail. The implicit iterative method proposed in this paper has the following advantages over the existing difference method:

• (1) Compared to the explicit difference method, the implicit difference method is characterized by few errors, stability, and few restrictions. The implicit difference method can be used to calculate the multiwavelength unsteady flow in the Yibin-Chongqing river reach.

• (2) Instead of using the implicit difference chasing method, this paper employs an iterative calculation method to eliminate the errors resulting from the linearization of the chasing method. Moreover, owing to the calculation features, the iterative method can better debug the program, and the effect of unsteady flow propagation on various indicators can be observed.

3.3.1 Model accuracy verification (flume test)

Hu [28] studied the characteristics of unsteady flow propagation in a flume. Here, the test was conducted in a water tank 28 m in length, 0.56 m in width, and 0.7 m in height (Fig. 4). Eight ultrasonic water gauge probes were arranged in the water tank to synchronously measure the water level in real time so that the mode could be verified in three aspects: range of variation, outlet flow, and deformation features.

Fig. 4

Schematic diagram of the 28-m variable-slope flume

3.3.1.1 Flow change verification

The model is used to calculate the relative amplitude change and outlet amplitude of the outlet flow. Then, the calculated value is compared with the measured value. The results are shown in Fig. 5a and b. As shown in the figure, this model can clearly present the changes in the outlet flow amplitude.

Fig. 5

Comparison of the calculated and measured outlet amplitude values

3.3.1.2 Outlet flow verification

The working conditions for the verification are as follows: The inlet flow varies between 15 and 40 L, the gradient of the water tank is J = 0.3%, and the waveform with a is a sine wave with a change period of 20 s. The water-taking level with a steady waveform is located at section No. 6, which is 19.05 m away from the edge of the water tank. The moment when the wave trough appears is regarded as 0 s, i.e., the starting point. The verification and comparison in section No. 6 are shown in Fig. 6. The average, maximum, and minimum flow rates under the verification working conditions are as follows: $\overline{Q}{ = }27.5{L/s}$, $Q_{\max } = 40 {L/s,}Q_{\min } = 15 {L/s}$.

Fig. 6

Test in section No. 6: calculation comparison

As shown in Fig. 6, the calculated wave peak and trough in section No. 6 have very small errors compared to the measured values. The maximum error of 0.00278 m (a percentage error of 4%) appears at the wave trough where the period is 100 s.

As shown in Fig. 7, the maximum error in the flow at the wave peak is 0.769 L/s, and the maximum percentage error is 2.5%. The maximum calculated error in the flow at the wave trough is 2.306 L/s, and the maximum percentage error is 13.3%.

Fig. 7

Outlet flow test: measured vs. calculated values

3.3.1.3 Verification of deformation features

The working conditions for the verification are as follows: The inlet flow changes between 15 and 40 L; the changing period is 10 s; the waveform is a sine wave; the following sections are used for verification: No. 2 (6.235 m from the edge of the water tank), No. 4 (12.64 m from the edge of the water tank), and No. 6 (19.05 m from the edge of the water tank). The working conditions for the calculation are as follows: The inlet flow changes between 15 and 40 L; the changing period is 10 s; and the waveform is a sine wave. The glass roughness factor is 0.0095, the time interval Δt is 0.2 s, and the gap between sections is 0.06 m.

Figure 8 shows that the calculated results increase quickly and sharply and then decrease gradually and gently. If the wave trough is point 0, the test vs. calculated wave peaks appear at the following times: 4 s vs. 4 s for section No. 2, 3.5 s vs. 3.6 s for section No. 4, and 3.2 s vs. 3.4 s for section No. 6. The maximum error is 2%.

Fig. 8

Comparison of the water level at section Nos. 4–7: calculated vs. measured waveform

3.3.2 Yibin-Chongqing river reach data verification

The working conditions for verification (1) are as follows: The data at Zhutuo from 0:00 on January 5, 2016, to 23:00 on January 10, 2016, are selected and result in a total of 144 h; the water level change in the initial section is 196.8–198.5 m, and the maximum change is approximately 1.7 m. The calculation time interval Δt is one hour. The initial section flow vs. the water level and the verification comparison results are shown in Fig. 9.

Fig. 9

Verification of the calculated flow rate and water level in Zhutuo

The working conditions for verification (2) are as follows: The daily average flow and water level data measured from the Shimen (781 km) and Shoupayan (899 km) sites from September 4–14, 2017, are selected and result in a total of 10 days. The calculation time interval Δt is 24 h. For the verification comparison result, see Figs. 10 and 11.

Fig. 10

Verification of the calculated flow rate and water level in Shoubayan

Fig. 11

Verification of the calculated flow rate and water level in Shimen

According to Figs. 9 through 11, the water levels, waveforms, and phases in the unsteady flow verification sites coincide. Table 1 shows the mean water level error, minimum water level error, maximum water level error, percentage error, and mean square of errors in the verification sites under the working conditions for verification (1) and (2) (all errors are absolute values.)

Table 1 Verification of the water level in Zhutuo

Table 1 shows that the iterative method has a higher accuracy than the chasing method in terms of the mean water level value errors and in terms of the maximum value of errors at all verification sites.

3.3.3 Application of the model for rapid flow

For this example, the following conditions were used: A rectangular sectional channel with an overall length L = 100 m, width B = 10 m, flow rate Q = 20 m³/s, and Manning coefficient n = 0.03. The boundary conditions were as follows: both upstream and downstream inflows were rapid flows [29]. The calculation parameters were as follows: Δx = 10 m, Δt = 10 s, the total calculation time was 1000 s, and the time weight factor was θ = 1 [30] (Fig. 12).

Fig. 12

Rapid flow example water surface: comparison between the calculated results

In summary, the implicit iterative model proposed herein is more accurate in the process of simulating unsteady flow in long reaches and is thus appropriate for use in the unsteady propagation and calculation in long reaches.

3.3.4 Daily regulated unsteady flow verification

The river reach selected for calculation starts at the **angjiaba Hydropower Station along the **sha River and the Gaochang Hydrologic Station along the Minjiang River and extends to the mainstream of the Yangtze River in Fuling, Chongqing. Using the measured terrain data, we divide all zones into sections. The section spaces were 200–500 m, and the total number of sections was 1180.

• (1) The incoming flow conditions were as follows: the outflow rate of the **angjiaba Hydropower Station along the **sha River was 1380–1540 m³/s, and the daily water level changed by approximately 0.15 m. The flow rate at the Gaochang Hydrologic Station along the Minjiang River was 800–1640 m³/s, and the daily water level amplitude was 0.99 m, which indicates relatively evident daily adjustment.

• (2) The incoming flow conditions were as follows: the outflow rate of the **angjiaba Hydropower Station along the **sha River was 1400–1930 m³/s, the daily flow rate changed by approximately 520 m³/s, and the daily water level changed by approximately 0.84 m, which shows a clear daily adjustment. The flow rate at the Gaochang Hydrologic Station along the Minjiang River was 765–1590 m³/s, the daily adjusted flow rate was approximately 400–705 m³/s, and the daily water level amplitude was 0.49–0.91 m, which indicates relatively evident daily adjustment.

• (3) The incoming flow conditions were as follows: the outflow rate of the **angjiaba Hydropower Station along the **sha River was 2900–5860 m³/s, the daily flow rate changed by approximately 860–2000 m³/s, and the daily water level changed by approximately 0.73–2.25 m, which shows a clear daily adjustment. The flow rate at the Gaochang Hydrologic Station along the Minjiang River was 1700–3180 m³/s, the daily adjusted flow rate was approximately 200–800 m³/s, and the daily water level amplitude was 0.2–1.05 m, which indicates relatively evident daily adjustment.

By verifying the unsteady flow spreading process in the 3 typical periods mentioned above (2 low-water periods and 1 medium flood period), as shown in Fig. 13, we know that the changing process of the simulated water level is essentially consistent with the changes in the measured water level at each station under the three operating conditions. Except for some deviation in the calculated value from the measured value at a few test points, the errors are mainly within ± 10 cm, which meets the calculation requirements.

Fig. 13

Unsteady flow verification diagrams

4 Calculation result of the daily adjusted unsteady flow spreading in the Shuifu-Chongqing river reach

4.1 Calculation conditions

For the **sha River, values were taken according to the typical operating conditions of the daily adjusted unsteady flow and the unsteady flow control conditions at the **angjiaba Hydropower Station. The outflow process is shown in Fig. 14. The incoming flow of the Minjiang River took values based on the typical operating conditions of unsteady flow at the Gaochang Hydrologic Station. The downstream outlet section was Fuling, Chongqing. To ensure that the reach from Chongqing to Jiang** is not affected by backwater from the Three Gorges Reservoir, when the maximum daily and hourly water level amplitudes were determined, the water level of the model’s outlet was considered to be 150 m, i.e., the storage level of the Three Gorges Reservoir.

Fig. 14

Typical daily adjusted unsteady flow conditions

4.2 Daily decrease in the amplitude and flow rate

After the hydropower station along the Jiansha River and Minjiang River began operation, the daily amplitude and maximum hourly amplitude of the daily adjusted unsteady flow in the Shuifu-Chongqing river reach are calculated and are detailed below and are shown in Fig. 15.

• (1) For **angjiaba, under the control conditions, the daily amplitude and maximum hourly amplitude of the daily adjusted unsteady flow were 2.92 m and 0.90 m, respectively; those in Yibin were 2.39 m and 0.41 m, respectively; those in Lizhuang were 1.66 m and 0.27 m, respectively; those in Luzhou were 1.29 m and 0.17 m, respectively; those in Hejiang were 0.86 m and 0.14 m, respectively; those in Zhutuo were 0.90 m and 0.12 m, respectively; and those in Jiulongtan were 0.17 m and 0.03 m, respectively.

• (2) For **angjiaba, under the daily adjustment conditions, the daily amplitude and maximum hourly amplitude of the daily adjusted unsteady flow were 2.50 m and 0.91 m, respectively; those in Yibin were 2.15 m and 0.41 m, respectively; those in Lizhuang were 1.53 m and 0.23 m, respectively; those in Luzhou were 1.03 m and 0.14 m, respectively; those in Hejiang were 0.78 m and 0.11 m, respectively; those in Zhutuo were 0.62 m and 0.10 m, respectively; and those in Jiulongtan were 0.11 m and 0.02 m, respectively.

Fig. 15

Maximum daily and hourly changes in the water level

As shown in Fig. 16, under typical operating conditions with a large average flow rate, the flow rate amplitude of the unsteady flow decreased slowly, and under typical operating conditions with a small average flow rate, the flow rate amplitude of the unsteady flow decreased quickly. The decreases in the flow rate amplitude of the unsteady flow under the three typical operating conditions are as follows: Luzhou, low water 54.8%, normal water 46.1%, and flood 42.0%; Zhutuo, low water 76.8%, normal water 67.1%, and flood 61.7%; Jiang**, low water 85.3%, normal water 76.6%, and flood 71.2%; and Chongqing, low water 90.5%, normal water 82.6%, and flood 77.5%.

Fig. 16

Changes in the flow along the Yibin-Fuling reach at a certain time for the three working conditions

4.3 Spreading of unsteady flow in the Yibin-Chongqing river reach

The spreading speed of unsteady flow is associated with the overall flow rate, and the flow velocity mainly depends on the flow rate. This section calculates the spreading process of unsteady flow for the three typical flow rates (normal-water, low-water, and flood periods) to evaluate the unsteady flow spread along this river reach.

Figure 3 shows the changes in the flow rate along the Yibin-Fuling river reach at a certain moment under the three operating conditions. The flow rate in Yibin reaches a peak at this time. For convenience of comparison, a corresponding fixed value (4000 m³/s for normal water and 8000 m³/s for flood) is subtracted from the flow rates along the path for the typical operating conditions during the normal-water and flood periods.

We know that under low-water conditions, there are approximately 3.5 unsteady flow wavelengths in the Yibin-Fuling reach, and the unsteady flow for low-water conditions thus takes approximately 3–4 days to spread to Fuling. According to the stations along the reach, it takes approximately 21 h for the unsteady flow to spread from **angjiaba to Luzhou, approximately 40 h for the unsteady flow to spread from **angjiaba to Zhutuo, and approximately 65 h for the unsteady flow to spread from **angjiaba to Chongqing.

Under normal-water conditions, there are approximately 2.5 unsteady flow wavelengths in the Yibin-Fuling river reach, which indicates that the unsteady flow under low-water conditions takes approximately 2.5 days to spread to Fuling. According to the stations along the reach, it takes approximately 15 h for the unsteady flow to spread from **angjiaba to Luzhou, approximately 30 h for the unsteady flow to spread from **angjiaba to Zhutuo, and approximately 46 h for the unsteady flow to spread from **angjiaba to Chongqing.

Under flood conditions, there are approximately 2.1 unsteady flow wavelengths in the Yibin-Fuling river reach, indicating that the unsteady flow under flood conditions takes approximately 2.0 days to spread to Fuling. According to the stations along the reach, it takes approximately 12 h for the unsteady flow to spread from **angjiaba to Luzhou, approximately 23 h for the unsteady flow to spread from **angjiaba to Zhutuo, and approximately 40 h for the unsteady flow to spread from **angjiaba to Chongqing.

The spreading speed is mainly related to the overall unsteady flow velocity. Therefore, studies on the spreading time are primarily conducted by identifying the relationship between the spreading time and the overall flow rate.

The fitting formula is shown in Eq. (9):

$$T = {\text{a}}Q^{ - b}$$

(9)

where T is the arrival time and Q is the flow rate in Yibin. For ${\text{a}} = \alpha \times {\text{s}} \times \sqrt {\frac{{BC\sqrt {R{\text{i}}} }}{{\text{g}}}}$, the unit is m^1.5·s^0.5. In particular, s is the distance from a dangerous beach or station to Yibin, B is the width of the river, C is the Chezy coefficient, R is the hydraulic radius, I is the gradient, g is the acceleration of gravity, and α is the constant term. The value of b is generally approximately 0.4.

For the decrease in the flow rate amplitude, the ratio of the flow rate amplitude at a dangerous beach to that in Yibin is used to measure the decrease in the flow rate amplitude at the designated dangerous beach.

The fitting formula is shown in Eq. (10):

$$\Delta Q_{{\text{n}}} = {\text{a}}\Delta Q^{{\text{b}}}_{{{\text{yibin}}}}$$

(10)

where $\Delta Q_{{\text{n}}}$ is the flow rate amplitude at the dangerous beach or station, $\Delta Q_{{{\text{yibin}}}}$ is the flow rate amplitude in Yibin, a is a proportion, and b denotes an exponential term.

For the water level amplitude, the relationship between the flow rate amplitude and water level amplitude is identified.

The fitting formula is shown in Eq. (11):

$$\Delta h = {\text{a}}\Delta Q^{b}$$

(11)

where $\Delta h$ is the water level amplitude; $\Delta Q$ is the flow rate amplitude; ${\text{a}} = \frac{\alpha }{{BC\sqrt {R{\text{i}}} }}$, where B is the river width, C is the Chezy coefficient, R is the hydraulic radius, i is the gradient, and α is the constant term; and b is the exponential term.

By summarizing the spreading time, the flow rate amplitude percent decrease, and the flow rate amplitude–water level amplitude relationship at the corresponding dangerous beach and important downstream stations (Luzhou, Hejiang, Zhutuo, Jiang**, and Chongqing) and by fitting the data obtained, we obtain the empirical formulas for the spreading time, flow rate amplitude percent decrease, and the flow rate amplitude–water level amplitude relationship at important stations.

As shown in Fig. 17, at different flow rates, unsteady flow can reach Zhutuo within 18–40 h (corresponding to a flow rate of approximately 11,000–2250 m³/s). The average flow rate amplitude is 34.7% of the flow rate amplitude of Yibin. When the amplitudes are the same, there is essentially no difference in the changes during flood, normal-water, and low-water periods; when the amplitude is 0–750 m³/s, the water level amplitudes during flood, normal-water, and low-water periods are all 0–0.4 m. The changes in unsteady flow relationships are shown in Table 2.

Fig. 17

Unsteady flow propagation

Table 2 Change in unsteady flow relationship

5 Analysis of the incoming flow conditions calculated with the daily adjusted unsteady flow and designed water level

The designed minimum navigable water level (hereafter referred to as the “designed low water level”) is the lowest water level used in designing waterway projects that allows standard ships or fleets to navigate the waterway. This value is the most basic design parameter in waterway projects, such as in channels and wharfs. Its value is directly related to project cost and project operation benefits. The daily adjustment of the **angjiaba Hydropower Station along the **sha River inevitably has some impact on the hydrological conditions of the downstream channel. This section mainly calculates and analyzes the designed water level given the daily adjusted unsteady flow.

5.1 Determination of the operating conditions for calculation

When calculating the designed water level of the Shuifu-Chongqing river reach, the natural conditions and daily adjustment conditions of the **angjiaba Hydropower Station should be considered. Thus, four types of operating conditions are provided to identify the operating conditions of the designed water level.

5.1.1 Operating condition 1

Natural daily average incoming flow (21 years from 1991 to 2011).

Before the **angjiaba Reservoir was built, according to the data from the hydrological station in the 21 years from 1991 to 2011, with a 98% guaranteed rate of flow, we obtained the designed minimum navigable flow at each main station along the **angjiaba-Chongqing river reach. That of the **shan Station was 1245 m³/s (Table 3).

Table 3 Guaranteed flow rate (21 years) at the main stations from Shuifu to Chongqing (p = 98%)

5.1.2 Operating condition 2

Operating conditions for calculating the minimum outflow rate of the **angjiaba Hydropower Station along the **sha River.

According to the scheduling method of the **angjiaba Hydropower Station, the minimum discharge flow rate after the **angjiaba Reservoir was built reaches 1518 m³/s, and the duration is more than 17–20 h. Therefore, this flow rate is selected for the calculation of the designed water level under steady flow mode (Table 4).

Table 4 Calculated flow from **angjiaba to Chongqing

5.1.3 Operating conditions 3 and 4

Operating conditions for calculating the daily adjusted unsteady flow after the reservoirs along the **sha River and the Minjiang River were built.

After the **angjiaba Reservoir was built, the minimum discharge flow rate was 1518 m³/s. Although the duration exceeds 17–20 h, the unsteady flows discharged from the **angjiaba Hydropower Station are flattened during the spreading process. The minimum flow rate increases minimally when the flows reach the Luzhou and Zhutuo stations. Therefore, operating condition 3 uses this flow rate to calculate the designed minimum water level under unsteady flow mode, and operating condition 4 calculates the daily average water level under unsteady flow mode.

5.2 Numerical results

As shown in Fig. 18, for operating condition 2, when the **angjiaba Hydropower Station discharges at the base flow rate (1518 m³/s), the designed water level of the Yibin-Chongqing river reach is approximately the same as that under natural conditions, although there is a slight increase of 0.03–0.07 m. Under operating condition 3, at the minimum unsteady discharge flow rate, the designed water level of the Yibin-Chongqing river reach increases by 0.11–0.25 m, approximately 0.19 m on average. Under operating condition 4, at the daily average unsteady flow rate, the designed water level of the Shuifu-Chongqing river reach is 0.17–0.32 m higher than that under natural conditions, averaging approximately 0.25 m.

Fig. 18

Water level change under various working conditions

6 Analysis of the effect of sand excavation on the water level

6.1 Characteristic parameters of the Chongqing-Yibin sand excavation zone

The results of a comparison of the topographic maps of the Yibin-Chongqing river reach in 2007 and 2016, indicating the changes in the characteristic parameters of the sand excavation zone, are shown in Table 5. The topographic change caused by improper sand excavation is estimated to be approximately 395 million m³; there are approximately 288 sand pits, including 17 with a sand excavation quantity up to 5 million m³ and 108 with a sand excavation quantity up to 1 million m³. The largest sand pit in the Yibin-Fuling, Chongqing, is 42.5 m deep, located in Niu Shiqi (channel length, 1012.8 km). Among the sand pits, approximately 52 have a depth of 20 m, and approximately 236 are 2.5 m-20 m deep. The smallest sand pit is 2.5 m in depth. The sand excavation depth and distribution of the sand excavation volume and area (Yibin-Jiang**) are shown in Fig. 19. For the sand excavation plan of the river reach under research (Erlongkou-**anglutan), see Fig. 20. This figure shows how the sand pits are distributed before and after sand excavation.

Table 5 Estimation of sand mining characteristic parameters along the Chongqing to Yibin river reach

Fig. 19

Sand mining parameters

Fig. 20

Sand mining area plane distribution map (Erlongkou-**anglutan) (Positive values represent sedimentation along the river section, and negative values represent erosion and sand mining.)

6.2 Analysis of the changes in the water level of the Chongqing-Yibin river reach before and after sand excavation

Since the flow rate of the gauging stations along the path changes extensively during the year, the water level varies with the upstream water level scheduling process, river reach, and different flow rates during low-water, normal-water, and flood periods. Therefore, three flow rates are selected for calculation at Zhutuo Station during the low-water period (Q = 2230 m³/s), normal-water period (Q = 10,500 m³/s), and flood period (Q = 18,500 m³/s) based on the landforms of **angjiaba-Fuling in 2007 (before sand excavation) and 2016 (after sand excavation).

The flood surface lines during low-water, normal-water, and flood periods after sand excavation are shown in Fig. 21. At the three flow rates, the changes in the **angjiaba-Fuling water surface line are essentially the same. The upstream water face line has a larger gradient, while the downstream water face line features a smaller gradient.

Fig. 21

The water surface line change for the three working conditions

As shown in Fig. 22, the changes in the water level reduction amplitudes at the three flow rates are similar, and the water level reduction amplitudes at different flow levels are as follows: low-water period > normal-water period > flood period. However, for some river reaches, the water level reduction values in the normal-water period are larger than those in the flood period. This difference is mainly the result of the presence of a large number of sand pits in the point bar where the water cannot flow out during the low-water period, and during normal-water and flood periods when the high water level is high, the water level reduction amplitude increases due to the impact of the sand pits. Furthermore, the upstream side has more river reaches, with a water level reduction value more than 1 m greater than that of the downstream side. Table 3 shows that the overall water level at the three flow rates decreases by 0–2 m. In particular, the decreases at 700–780 km and 900–1050 km are relatively large at 1–2 m. From 552 to 600 km, which is the Three Gorges Reservoir region, there is a small water level reduction value, i.e., 0.2–0.5 m. The results of comparing the water level reduction value and the amount of sand excavation during the sand excavation along the Fuling-**angjiaba reach in 2016 are shown in Fig. 13.

Fig. 22

Correlation of water surface line change and the extracted sand volume

The sand excavation change characteristics along the path are generally similar to the water level reduction amplitude. The amount of sand excavation gradually increases at 550–670 km, decreases at 670–850 km, suddenly rises at 900 km, and then declines at 900–1076 km. Likewise, the water level reduction amplitude increases abruptly at 550–700 km, gradually decreases at 700–860 km, suddenly increases at 860–900 km, and then gradually decreases. The change trend is almost the same as that of sand excavation.

Due to improper sand excavation, the distance between each sand pit and the amount of sand excavation differs extensively. Therefore, the average value of the sand excavation volume and water level reduction amplitude within every 50 km is summarized to analyze the effect of sand excavation on the reduction in the water level. The sand excavation conditions and water level changes along the path are shown in Figs. 23 and 24.

Fig. 23

Sand extraction volume and water level maximum reduction every 50 km

Fig. 24

The relationship between the amount of sand extracted and the water level reduction every 50 km

According to Table 5, the ratio mainly falls within the range of 1–3 m, and the ranking is still as follows: flood < normal water < low water, which indicates that during the low-water period, sand excavation has the greatest impact on the water level. Additionally, the upstream sand excavation has a larger influence on the water level reduction amplitude. In particular, at 826–676 km, 10 million m³ of sand is removed every 50 km, which causes the water level to drop by 0.2–0.3 m during the low-water period. According to the relationship between the amount of sand excavation per 50 km and the change in the water level (Fig. 23), with the increase in sand excavation, the water level reduction amplitude grows. In particular, when the **angjiaba-Zhutuo and Zhutuo-Fuling river reaches have 10 million m³ of sand excavated per 50 km, the water level decreases by 0.6 m and 0.33, respectively, mainly because the river channel becomes increasingly narrow on the upstream side. A narrower river reach results in a smaller flow area at the same flow rate and thus a larger impact of sand excavation on the change in the water level.

7 Results and discussion

The upstream portion of the Yangtze River along the Yibin-Chongqing river reach will undergo a waterway regulation project. The waterway level in this section will be upgraded from Class III to Class II. The daily regulation of unsteady flow and sand mining in this upstream section will have a great impact on the water level of this reach, and changes in the water flow conditions will directly affect the implementation of the remediation project.

During a low-water period, we reviewed the typical water gauges (at Lizhuang, Yujiawan, Shenbeizui, Zhutuo, Zhengjialiang, and Cuntan) in the section of the Yangtze River from Chongqing to Yibin (Figs. 25, 26). According to the principle of an approximately equal gradient in the low-water period, the water level was determined when the design flow rate was Qzhutuo≈2230 m³/s and Qcuntan≈2900 m₃/s. An overview of the results is shown in Table 6.

Fig. 25

Water gauge check and water level conversion

Fig. 26

Results of unsteady flow and sand mining

Table 6 Water level results at the main sites

The model calculation results given the design flow rate show that the average decreases in the water level corresponding to the 20-km sand extraction near the Cuntan and Zhutuo stations are approximately 0.18 m and 0.26 m, respectively. According to the data released by Mu [31], **ao [20] and the Water Resources Committee of the Yangtze River, the water levels at the Cuntan and Zhutuo stations dropped by approximately 0.15 m and 0.30 m, respectively. The propagation characteristics of unsteady flow and the influence of unsteady flow and sand mining on the design water level calculated by the model in this paper can provide technical support for the waterway regulation of this section of the river. The relationship between the water level and discharge at the Zhutuo hydrological station after sand mining is shown in Fig. 27.

Fig. 27

The relationship between the water level and discharge at the Zhutuo hydrological station

In this paper, only unsteady flow and sand mining were calculated and analyzed under the current channel boundary conditions. After the cascade hub is completed and jointly scheduled upstream of the **sha River, the hydrological conditions will change further. The evolution and regulation of this section are a subject that we will focus on in subsequent work.

8 Conclusion

• (1) The new implicit iterative method proposed in this study is much better than the chasing method. Suitable for the calculation of the spreading of multiwavelength and long-distance unsteady flow, this method can be used to calculate the multiwavelength unsteady flow of the Yibin-Chongqing river reach. The verification with measured data shows good results.

• (2) Through fitting analysis of the spreading time, flow rate amplitude percent decrease, and flow rate amplitude–water level amplitude relationship at important stations (Luzhou, Hejiang, Zhutuo, Jiang**, and Chongqing), we obtained empirical formulas for the spreading time, flow rate amplitude percent decrease, and flow rate amplitude–water level amplitude relationship at these important stations.

• (3) The decreases in the flow rate amplitude of the unsteady flow under the three typical operating conditions are as follows: Chongqing, low water 90.5%, normal water 82.6%, and flood 77.5%. Under low-water conditions, there are approximately 3.5 unsteady flow wavelengths in the Yibin-Fuling reach, and it takes approximately 66 h for the unsteady flow to spread from **angjiaba to Chongqing. Under normal-water conditions, there are approximately 2.5 unsteady flow wavelengths in the Yibin-Fuling reach, and it takes approximately 46 h for the unsteady flow to spread from **angjiaba to Chongqing. Under flood conditions, there are approximately 2.1 unsteady flow wavelengths in the Yibin-Fuling reach, and it takes approximately 39 h for the unsteady flow to spread from **angjiaba to Chongqing.

• (4) At the minimum unsteady flow discharge rate, the designed water level of the Yibin-Chongqing river reach increases by 0.11–0.25 m, approximately 0.19 m on average. Under operating condition 4, at the daily average unsteady flow rate, the designed water level of the Shuifu-Chongqing river reach is 0.17–0.32 m higher than that under natural conditions, averaging approximately 0.25 m.

• (5) By comparing the topographic maps of the Yibin-Chongqing river reach in 2007 and 2016, we find that the topographic change caused by improper sand excavation is estimated to be approximately 395 million m³; there are approximately 288 sand pits. Affected by excessive and improper sand excavation in the river channel, the water level from Chongqing to Yibin in the upper reaches of the Yangtze River is generally reduced by 0.5–2.0 m. The water level in this river reach decreases significantly, especially during low-water periods. As a result, part of the river reach experiences a sharp drop in channel depth and an increase in gradient. Ships cannot navigate if the channel department does not strengthen maintenance.