1. Introduction The backwater interaction between mainstem and tributary water levels is a key process governing hydrological regimes and geomorphic evolution in river systems. Existing studies have investigated backwater effects mainly through long-term hydrological observations, statistical analyses, and hydrodynamic numerical modeling. Early studies on large river systems showed that downstream controls at river confluences or river–lake junctions can alter stage–discharge relationships, raise water levels under the same discharge conditions, and extend backwater influence far upstream. For example, Meade et al. analyzed long-term records of water level and discharge in the Amazon River basin and found that mainstem backwater could significantly affect tributary water levels and cause extensive adjustment of the water surface slope. From the perspective of flood wave propagation, Trigg et al. further demonstrated that backwater can strongly control water level responses and flood wave movement in large low-gradient rivers [ 9, 10]. In river–lake systems, Petersen-Øverleir and Reitan reported that gauging stations affected by downstream controls, such as lakes, may exhibit non-unique stage–discharge relationships [ 11]. Hidayat et al. further showed that backwater effects can reduce the reliability of conventional discharge estimation methods [ 12]. More recently, Meng et al. re-examined the water surface slope at the interaction between Poyang Lake and the Yangtze River. They found that lake outflow can substantially reduce the water surface slope of the Yangtze River near the confluence, extend the backwater influence approximately 74 km upstream, and cause a significant increase in water level under unfavorable discharge combinations [ 13]. These studies provide an important basis for identifying backwater extent, water level response, and water surface slope variation. However, previous studies on backwater effects have mainly focused on water level rise, backwater length, stage–discharge relationships, and longitudinal water surface slope variation. Less attention has been paid to how backwater-induced local hydraulic changes affect navigation flow conditions. For tributary estuaries affected by mainstem backwater, navigation risk is not determined solely by water level, but is also closely related to the spatial distribution of navigation depth, local flow velocity, flow direction, transverse velocity gradients, and the position of the dominant flow path. Therefore, explaining backwater effects only from the perspective of water level or water surface slope is insufficient for fully characterizing their influence on navigation safety. Navigation assessment commonly relies on indicators such as navigation depth, flow velocity, channel width, vessel resistance, and safety margins. For mountainous rivers and rapids, Jiang et al. systematically reviewed methods for determining navigable hydraulic indices and indicated that flow velocity, water surface slope, vessel resistance, and upstream navigation capability are important hydraulic factors affecting navigation conditions [ 18]. In addition, Ye et al. showed that inland waterway navigation risk is influenced by multiple factors, including hydrological conditions, channel characteristics, vessel-related factors, and traffic conditions [ 19]. These studies provide an important indicator basis for navigation safety assessment. However, most existing approaches still rely mainly on single indicators or threshold-based judgements, and insufficient attention has been paid to the linkage between spatially distributed hydrodynamic conditions and navigation-risk zoning under mainstem backwater effects. To address these gaps, this study develops a depth-averaged two-dimensional hydrodynamic model for the Min River estuary to investigate the responses of key hydraulic elements, including flow velocity redistribution, water level variation, and hydrodynamic axis migration, under different backwater conditions. This study integrates simulated results with navigation safety constraints, including insufficient navigation depth, excessive local flow velocity, and increased vessel resistance, to delineate navigation risk zones and identify critical estuarine water level thresholds. It extends conventional backwater research beyond the analysis of water level response, backwater extent, and water surface slope variation to navigation-oriented spatial hydrodynamic analysis. This reveals how mainstem backwater modifies navigable depth, velocity distribution, and the hydrodynamic axis. Furthermore, it improves on traditional navigation risk assessments that rely on single thresholds by developing a risk zoning method that combines multiple hydrodynamic factors and vessel resistance constraints. This approach can provide a physical basis and methodological reference for navigation risk assessment in similar confluence reaches affected by mainstem backwater and upstream reservoir regulation. 2. Study Area The estuarine reach of the Min River and Jinsha River (the headwaters of the Yangtze River) is located in Yibin City, Sichuan Province (28°46′30″–28°47′30″ N, 104°35′30″–104°37′30″ E), in the southern Sichuan Basin. This area comprises the confluence of the lower reaches of the Jinsha River and Min River, as well as the initial section of the main Yangtze River channel. The Min River originates in the Min Mountains and flows through northwestern Sichuan before converging with the Jinsha River in Yibin City. The Jinsha River enters the area from the west via Pingshan County and flows eastward, exiting at Shulou Town. Downstream of this confluence, the combined flow is officially designated as the Yangtze River. This reach benefits from convenient transportation and serves as a regional water–land transport node. The topography of the Min River basin is highly complex. Hydrodynamic conditions in the lower estuarine reach are constrained not only by inflow and sediment supply from upstream but also by discharge from the Jinsha River, leading to a pronounced water level backwater effect. Furthermore, this river reach is influenced by reservoir regulation from both mainstem and tributary hydropower systems. Therefore, this study focuses on the lower estuarine reach of the Min River. The study domain encompasses a 34 km section of the Min River upstream of the Hejiangmen estuary, an approximately 2 km section of the Jinsha River upstream of the estuary, and a 3 km section of the main Yangtze River channel downstream of the estuary. The total elevation drop along the studied Min River reach is 15.48 m, with an average slope of 0.5‰. An overview of the study area is shown in Figure 1. 3. Methods and Data 3.1. Methods This study focuses on the backwater effect of the Jinsha River on the lower Min River and its impact on navigable flow conditions. To fully capture the flow characteristics of the Min River estuary under backwater influence, a depth-averaged two-dimensional hydrodynamic model was developed using MIKE 21 HD (MIKE Zero 2014, Version 14.0.0.9034), based on the hydrological, hydraulic, and topographic characteristics of the study reach. The model domain extends approximately 34 km, from the Pingshan Min River Bridge upstream to the Min River estuary in Yibin downstream, and incorporates sections of the Jinsha River (2 km upstream of the estuary) and the main Yangtze River channel (3 km downstream of the estuary). Based on field observations, the model adequately reproduces the backwater effect at the Min River estuary. The locations of the model domain and monitoring stations are shown in Figure 1. 3.1.1. Mathematical Model Governing Equations and Boundary Conditions The two-dimensional hydrodynamic model is based on the depth-averaged conservation of mass and momentum, incorporating source and sink terms, turbulent stresses, and bottom and surface shear effects. The governing equations are as follows: Continuity equation: ∂ h ∂ t + ∂ h u ପ୍ତ ∂ x + ∂ h v ପ୍ତ ∂ y = h S (1) Momentum equations: ∂ h u ପ୍ତ ∂ t + ∂ h u ପ୍ତ ∂ x 2 + ∂ h u v ପ୍ତ ∂ y = f u ପ୍ତ h − g h ∂ Z ∂ x − h ρ 0 ∂ p a ∂ x − g h 2 2 ρ 0 ∂ ρ ∂ x + τ s x ρ 0 − τ b x ρ 0 + ∂ ∂ x h T x x + ∂ ∂ y h T x y + h u s S (2) ∂ h v ପ୍ତ ∂ t + ∂ h v ପ୍ତ ∂ y 2 + ∂ h u v ପ୍ତ ∂ x = − f u ପ୍ତ h − g h ∂ Z ∂ y − h ρ 0 ∂ p a ∂ y − g h 2 2 ρ 0 ∂ ρ ∂ y + τ s y ρ 0 − τ b y ρ 0 + ∂ ∂ x h T x y + ∂ ∂ y h T y y + h v s S (3) where xy are the Cartesian coordinates; t is the time; h is the total water depth (m), defined as h = Z + d; Z is the water level (m); d is the bathymetric depth below the reference datum (m); uv are the velocity components in the xy directions (m/s); S is the source/sink intensity (m/s); f = 2 ωsinφ is the Coriolis parameter; g is the gravitational acceleration (m/s 2); Pa is the atmospheric pressure (Pa); ρ0 is the reference water density; and ρ is the local fluid density. For the incompressible and homogeneous flow assumption adopted in this study, ρ was treated as constant; τsxτsy are the surface wind stresses in the xy directions (N/m 2); τbxτ are the bottom stresses in the xy directions (N/m 2); Txx, Txy, Tyx, and Tyy are the horizontal viscous stresses (N/m 2); and usvs are the velocity components associated with the source or sink terms in the x and y directions, respectively (m/s). The two-dimensional hydrodynamic model adopts the following basic assumptions: (1) The pressure distribution is assumed to be hydrostatic, and vertical acceleration is neglected. This assumption is appropriate for the present river reach, where the horizontal length scale is much larger than the water depth and the flow is predominantly horizontal. (2) The fluid is assumed to be incompressible and homogeneous, and density variation is not considered. The flow motion is governed by the depth-averaged two-dimensional shallow water equations. (3) A zero-equation turbulence model is adopted to represent horizontal turbulent diffusion. This approach is suitable for large-scale depth-averaged simulations of natural river and estuarine reaches. (4) Riverbed roughness is parameterized using Manning’s coefficient. The composite roughness coefficients were calibrated separately under flood flow, normal flow, and low flow conditions, with final values ranging from 0.030 to 0.035. (5) The simulated water levels and velocity distributions were then validated against observed data, and the errors were within the allowable range specified by the relevant standards. The boundary conditions are as follows: (1) Closed boundaries along natural riverbanks are defined as no-slip boundaries, where the flow velocity is set to 0. (2) Inflow discharges at the open boundary are specified using historically measured discharge data from the Gaochang Hydrological Station and the Xiangjiaba Station, with flow direction set perpendicular to the inlet cross-section. (3) The downstream outlet of the model is located near the Yibin station. As the lower reaches of the Min River are continuously affected by the backwater effect of the Jinsha River, downstream boundary conditions are defined using the water level at the Yibin station. As the backwater effect and the unsteady characteristics of the runoff process are not considered at this stage, this water level is determined by the stage–discharge relationship at the estuary corresponding to various discharges from the Gaochang station (Min River) and the Xiangjiaba station (Jinsha River). (4) The initial water level was set slightly lower than the average water level of the upstream and downstream boundaries to facilitate numerical stabilization during model initialization. Grid Partitioning and Sensitivity Analysis The model employs an unstructured triangular mesh. Given the numerous floodplains in the Min River reach, the mesh is refined in the main navigation channel and coarsened in floodplain areas. Local mesh refinement is applied to critical regions, including existing bridge piers, regulation structures, designed dredging areas, planned navigation channels, and the estuarine zone, to more accurately represent topographic variations. This approach aims to accurately simulate flow dynamics in the river reach while reducing the total number of mesh elements to improve computational efficiency. A mesh sensitivity analysis was conducted to verify the appropriateness of the mesh configuration and the accuracy of the numerical simulation results. Three mesh schemes were tested, with minimum mesh areas of 6.01 m 2, 10.26 m 2, and 11.09 m 2, respectively. The computational efficiency of these schemes was evaluated under consistent computational conditions (see Table 1). Using the simulation results of the finest mesh scheme (minimum mesh area 6.01 m 2) as the reference, water level data at three key cross-sections along the study reach were selected to calculate the root-mean-square error (RMSE) of water levels for the coarse and medium mesh schemes relative to the reference scheme. The results show that the coarse mesh yields a water level RMSE of 0.031 m and a maximum water level error of 0.048 m, indicating that mesh resolution has a significant influence on the results. For the medium mesh, the water level RMSE is 0.007 m, which is considerably smaller than its maximum water level error of 0.015 m. From the medium mesh to the fine mesh, further grid refinement leads to negligible improvement in water level simulation accuracy, indicating that the results have essentially converged and stabilized, thereby satisfying the requirements for mesh independence and convergence. Subsequently, by balancing computational accuracy, computational cost, and time efficiency, the final mesh configuration was determined as follows: a grid spacing of 15 m along the flow direction, a spacing of approximately 10–15 m perpendicular to the flow direction, and a local minimum mesh area of 10.26 m 2 at the bridge piers. The computational domain contains a total of 104,075 nodes and 204,936 elements (see Figure 2). Model Uncertainty Analysis Considering that discharge is not constant during actual flood events but varies over time, the roughness coefficient may also change accordingly due to variations in water depth, vegetation submergence, and channel bed resistance. Roughness variation is relatively large under low flow conditions with shallow water, and becomes smaller under flood conditions with greater water depth. Based on this, and the roughness range of ( n = 0.030–0.035) obtained from multiple model calibrations, the upper and lower limits of the roughness coefficient were specified under different discharge conditions. By calculating the water level differences under different discharge levels and their corresponding roughness values, their combined effect on the uncertainty of the simulation results were analyzed. The results are shown in Figure 3. They indicate that a larger roughness coefficient leads to a higher simulated water level, with a maximum increase of approximately 0.07 m, whereas a smaller roughness coefficient results in a lower simulated water level, with a maximum decrease of approximately 0.08 m. Therefore, considering the topographic differences between the floodplain and the main channel, different roughness values within the calibrated range were adopted to reflect model parameter uncertainty and evaluate the robustness of the simulation results. The comparison between observed and calculated water levels shows that the model accuracy satisfies the requirements of the Technical Code for Simulation Test of Water Transport Engineering (JTS/T 231–2021) [ 26]. This analysis demonstrates that, under the combined variation in discharge and roughness, the model results remain within a small fluctuation range, indicating the stability and reliability of the model and providing a hydrodynamic basis for the subsequent identification of critical water level thresholds and navigation risk zoning. Model Parameter Settings The model parameters are set as follows [ 27]: (1) Roughness Coefficient. The river channels in the Min River basin are mostly characterized by compound cross-sections, consisting of a main channel and floodplains. The main channel alternates between wide and narrow sections with varying elevations, while the floodplains are wide and discontinuous. In the calculations, the roughness coefficient was determined through interpolation based on model calibration and validation results, with a value range of 0.030–0.035 for the study area. (2) Dry and wet water depths. The simulation accuracy for floodplain and shoal areas is improved by adopting a wetting–drying dynamic boundary scheme: the dry water depth is set to 0.1 m, the inundation depth to 0.5 m, and the wet water depth to 0.2 m. (3) Horizontal eddy viscosity coefficient. To prevent excessive turbulent diffusion terms from distorting the flow field and to enhance model stability, the Smagorinsky coefficient is set to 0.28. 3.1.2. Model Validation The model was validated using measured water level and flow-velocity data collected from the study reach. Water level observation data were collected under different discharge conditions of the Min River: Q = 1030 m 3/s (low-flow period, 1 April 2024), Q = 3020 m 3/s (normal-flow period, 30 August 2024), and Q = 8420 m 3/s (flood period, July 2025). The measured flow-velocity data were collected on 12 April 2024, corresponding to a discharge of Q = 1510 m 3/s in the Min River. The validation results of the model-predicted water surface profile are shown in Figure 4, while the validation results for flow velocity and direction are presented in Figure 5. The evaluation metrics are shown in Table 2. Through model parameter calibration, the channel roughness was determined to be 0.030–0.035. According to the Technical Code for Simulation Test of Water Transport Engineering (JTS/T 231-2021), the difference between calculated and measured water levels for mountain rivers should be within ±0.1 m. The validation results show that the differences range from −0.01 to 0.09 m, meeting the code requirements. 3.2. Stage–Discharge Relationship at the Estuary Under the Mainstem Backwater Effect The Min River converges with the Jinsha River in Yibin City before joining the Yangtze River. The magnitude of the backwater effect on the Min River is directly related to the water level at Yibin Station and the discharge of the Min River. The Gaochang Hydrological Station, located upstream of the Min River estuary, is approximately 27 km from the downstream estuary (Hejiangmen). The Yibin water level station is located at the estuary (Hejiangmen). On the Jinsha River, Xiangjiaba Station is located 9 km upstream of its confluence with the Min River. Approximately 19 km downstream of the estuary, the Lizhuang water level station is located on the mainstem of the Yangtze River. The locations of these monitoring stations are shown in Figure 1. Using long-term water level and discharge data from these stations, the study characterizes the hydrology and runoff of the river reach and defines the boundary conditions for the numerical simulation. 3.2.1. Water Level and Discharge Processes Figure 7 illustrates the relationships between the water level at Yibin Station and the discharges at Gaochang Station, Xiangjiaba Station, and Lizhuang Station. The water level at the Min River estuary does not follow a simple linear relationship with the upstream Min River inflow, as in conventional rivers, but is primarily driven by discharge from the Jinsha River. This results in elevated Yibin water levels even when the Min River discharge is low, indicating a strong backwater effect of the Jinsha River on the lower Min River. 3.2.2. Analysis of the Relationship Between Estuarine Water Level and Inflow from the Min River The water level variation characteristics in the study area indicate that, under the influence of the mainstem backwater effect, a pronounced multivalued relationship exists between the estuarine water level and upstream inflows from the Min and Jinsha Rivers. To quantify this relationship and systematically analyze the impact of mainstem backwater on the hydrodynamic characteristics of the estuarine reach under varying Min River inflow conditions, a “water level difference parameter” was introduced to establish the stage–discharge relationship at the estuary. Specifically, a series of stage–discharge rating curves ( Figure 8) was developed using the measured discharge from Gaochang Station and the water level difference between Gaochang and Yibin Stations. For a given upstream inflow from the Min River, the corresponding water level at the Hejiangmen estuary can be determined by selecting the appropriate rating curve based on the measured water level difference between Gaochang and Yibin. Furthermore, by integrating the stage–discharge relationship at the Lizhuang Hydrological Station and interpolating based on the measured water level at Yibin Station, the upstream Jinsha River inflow can be derived. This approach effectively decouples the multivalued relationship and provides critical boundary conditions for subsequent analyses of hydrodynamic responses to backwater effects. Δ Z = Z G a o c h a n g − Z Y i b i n (4) where Δ Z is the water level difference parameter (m), and ZGaochangZYibin are the water levels at Gaochang Station and Yibin Station, respectively (m). 3.2.3. Simulation Scenarios To analyze the navigable flow conditions in the river reach and investigate the variation in backwater effects under different inflows and estuarine water levels, interpolation calculations were performed based on the stage–discharge relationship at the estuarine reach ( Figure 8). A total of eight representative discharges were configured for the simulations: Q1 = 900 m 3/s (low-flow period, minimum navigable discharge), Q2 = 1840 m 3/s (medium discharge), Q3 = 2250 m 3/s (regulation flow), Q4 = 5570 m 3/s (flood flow period 1), Q5 = 8650 m 3/s (flood period 2), Q6 = 10,000 m 3/s (flood period 3), Q7 = 12,000 m 3/s (flood period 4), and Q8 = 15,000 m 3/s (flood flow period 5). For each discharge level, ten water levels were specified, sequentially designated as Q- Z1 through Q- Z10, resulting in a total of 80 computational scenarios. The water levels ranged from 258.40 m to 275.38 m. The simulation scenario settings are shown in Table 3. 4. Results and Analysis 4.1. Flow Conditions and Analysis of Backwater Effects 4.1.1. Water Surface Gradient and Backwater Length In estuarine zones, the interaction between mainstem and tributary flows, modulated by inflow conditions and channel topography, generates backwater effects of varying intensity. When tributary outflow is impeded, the longitudinal water surface becomes flatter, reflecting a reduced water surface gradient. Figure 9 illustrates these longitudinal profiles under different combinations of discharge and estuarine water levels, highlighting key indicators of backwater intensity and spatial extent relevant to navigation safety. To quantitatively assess the impact of estuarine backwater effects on the flow regime of the Min River, two scenarios are defined: (1) a natural scenario, representing the state with the lowest estuarine water level for a given discharge, where the water surface gradient approximates the riverbed slope, and (2) a backwater-affected scenario, defined as the state in which the water level along the reach is elevated by 0.1 m or more relative to the natural scenario for the same discharge. The upstream limit of the backwater-affected zone is used to determine the backwater length. According to the simulation and statistical results ( Figure 10), backwater length in the estuarine reach exhibits a strong positive correlation with estuarine water level for a constant Min River discharge. Taking QMin = 900 m 3/s as an example, the reach within 5.43 km upstream of the estuary is identified as the perennial backwater zone. Within this area, the water surface gradient ranges from 0.003‰ to 0.23‰. In the fluctuating backwater zone (5.43–17.33 km upstream), the water surface gradient increases markedly to 0.11–0.79‰, indicating a progressive attenuation of the backwater effect with distance from the estuary. Furthermore, the results indicate that both the extent and intensity of the backwater effect increase stepwise as the Min River discharge increases. At QMin = 5570 m 3/s, the estuarine backwater influence zone begins to extend upstream. The terminus of the perennial backwater zone shifts to 6.52 km upstream, with gradients ranging from 0.05‰ to 0.42‰, remaining relatively low. The fluctuating backwater zone extends to 19.30 km, with gradients increasing to 0.24–0.72‰, indicating continued weakening of the backwater effect upstream. When the Min River discharge reaches 15,000 m 3/s, the backwater effect reaches its maximum extent, and the perennial backwater zone terminus extends to 15.16 km upstream of the estuary, with backwater influence spanning the entire study reach. This indicates that under flood conditions, the imbalance between flow inertia and channel resistance creates an extended backwater zone. Thus, estuarine backwater can influence water surface morphology and flow characteristics along the entire 34 km study reach upstream of the estuary. In summary, the flow conditions in the study reach are governed by the nonlinear coupling between the upstream inflow and estuarine water level. Using nonlinear fitting methods, empirical equations describing the relationship between backwater distance and estuarine water level under different flow conditions were established (see Figure 10). Ninety-five percent confidence intervals ( p 3.5 m/s) expands significantly to encompass the entire river, with maximum velocities exceeding 5.0 m/s. This indicates highly concentrated flow energy and intense turbulence, posing a substantial threat to navigation safety. For the reach downstream of Sipoqi, the maximum longitudinal velocity should be controlled below 3.5 m/s to ensure navigation safety. The minimum estuarine water levels required to satisfy this velocity criterion under various discharge levels are presented in Table 7. The results indicate that as the discharge of the Min River increases from 900 m 3/s to 15,000 m 3/s, the corresponding minimum estuarine water level must be progressively raised from 261.96 m to 272.49 m. Under this regulated water level regime, the maximum longitudinal flow velocity remains stable within the range of 3.23–3.48 m/s. These results indicate that, with proper water level regulation, the flow velocity in this reach can be controlled below 3.5 m/s within the examined discharge–water-level scenarios, thereby providing a hydraulic reference for assessing navigation flow conditions in the studied reach. 4.2.3. Analysis of Upstream Navigation Resistance The capability of a vessel to navigate upstream through rapids is typically characterized by the “self-propelled upstream navigation hydraulic index”. It represents the limiting combination of flow velocity and water surface slope that a vessel can overcome under rated load and power conditions [ 34]. Based on actual vessel operations in the lower Min River and the Class III waterway construction standard, a 1000-ton motor vessel was selected as the representative vessel. The principal dimensions of the vessel are 68 m in overall length, 12.8 m in molded breadth, 2.4 m in design draft, and 800 kW engine power. The hydraulic index for upstream navigation through rapids was calculated for the representative vessel using the calculation method recommended in the Waterway Engineering Handbook. The results are listed in Table 8, and the relevant formulas are as follows: Vessel thrust and navigation resistance can be measured through full-scale ship trials. When full-scale trials are not feasible, they may be estimated using the following equations. Estimation of effective thrust of motor vessels: T 0 = e ୍ଠ 75 H P V S (5) where T0 is the effective thrust of the motor vessel (kgf); Hp is the total engine power (kW), which is 800 kW according to the representative design vessel; e is the effective thrust coefficient, which is related to the propeller shape and diameter, and is taken as 0.38, generally consistent with the experimental value; Vs is the relative velocity between the upstream vessel and the water flow (m/s), V s = V f + U w ; and Vf is the surface flow velocity at the control section of the rapid entrance (m/s). The control section refers to the location along the navigation route where the maximum flow velocity of the rapid occurs within the length of the vessel or fleet. In this study, Vf was set to 3.8 m/s based on the rapid suppression hydraulic index for mountainous rivers and the upstream navigation capacity of the representative design vessel; Uw is the minimum bank-relative speed that the vessel should maintain when navigating upstream through the rapid, which may be taken as 0.2–0.5 m/s. Selecting the moderately conservative value of 0.3 m/s within the recommended range can reasonably balance engineering experience, computational stability, and the actual upstream navigation capability of the representative vessel. Slope resistance: R j = β 1 W J (6) where W is the total displacement of the tow (kgf), calculated based on the vessel breadth, length, and draft; J is the water surface slope at the control section of the rapids under the most turbulent water level conditions, taken as the average slope over the ship or tow length; and β 1 is the correction coefficient accounting for local increases in water surface slope during upstream navigation, which is taken as 1.05 for large ships. Flow resistance: R V 1 = f Ω V 1 1.83 + ξ 1 δ A m V 1 1.7 + 4 F r (7) where f is the friction coefficient, taken as 0.17 for steel hulls; V1 is the corrected relative flow velocity between water and ship (m/s), accounting for shallow water, narrow channels, turbulence, etc.; and V1 = ηVS, where η is the correction coefficient for relative flow velocity, introduced to account for shallow water, narrow channels, and turbulence, and may be taken as 1.15–1.30. The lower-limit value of 1.25 was adopted because of the sufficient water-depth and channel-width margins in the study reach; Ω is the wetted surface area of the ship (m 2), defined as Ω = L( C1T + δB), where L, B, and T denote the ship length (m), beam (m), and draft (m), respectively; C1 is a correction coefficient, taken as 1.8 for motor vessels; δ is the ship block coefficient, taken as 0.7–0.8 for motor vessels; Am is the midship sectional area of the submerged portion of the ship (m 2), defined as Am = βBT, where β is the midship section coefficient, taken as 0.93 for motor vessels; ξ1 is the residual resistance coefficient of a motor vessel; and Fr is the Froude number of the ship, calculated as follows: ξ 1 = 17.7 δ 2.5 L 6 B 3 + 2 , F r = V 1 g L . Navigation resistance: R = R j + σ R V 1 (8) where R is the navigation resistance of the tow (kgf); Rj is the slope resistance (kgf); RV1 is the navigation resistance of the motor vessel (kgf); and σ is the formation coefficient of the tow, taken as 1 based on full-scale ship tests. Five typical rapid shoal reaches in the Min River affected by backwater, including Sipoqi and Tongluowan, were selected for analysis (as shown in Figure 17Figure 18). In these figures, the critical line defines the hydraulic threshold for self-propelled upstream navigation through the shoal reaches. For any data point, if both flow velocity and water surface slope are below the corresponding thresholds, the hydraulic condition indicates relatively low navigation resistance. Otherwise, self-propelled upstream navigation becomes more difficult. Considering the shallow-water constraints during the normal- and low-flow periods, discharges below 5570 m 3/s were selected for analysis. The results indicate that the resistance characteristics vary considerably among different water level drop intervals. (1) At Qiaoban, Pianchuangzi, and Yangjia Shoals, velocity and water surface slope are mainly concentrated in the lower-left region of the critical line, indicating relatively low hydraulic resistance in these reaches. (2) Tongluowan and Sipoqi Shoals are characterized by low water levels and rapid flow. As water level and discharge increase, flow concentration weakens. As a result, flow velocity decreases, local water surface slope is reduced, and hydraulic data points shift from the high-velocity/high-slope region to the low-velocity/low-slope region. Specifically, at a discharge of 900 m 3/s, significant navigation-obstructing conditions occur at Sipoqi Shoal, and the hydraulic conditions satisfy the self-propelled upstream navigation criterion only when the estuarine water level rises to 266.51 m. As discharge increases to 1840 m 3/s, hydraulic data points under all water level conditions are concentrated in the lower-left zone of the critical line, indicating that the hydraulic conditions satisfy the criterion for self-propelled upstream navigation. Tongluowan Shoal shows a similar trend: with increasing water level and discharge, the number of hydraulic data points within the navigation-obstructing zone decreases noticeably. (3) Yangjiaoshi and Wuchashu Shoals show more pronounced navigation obstruction under normal flow discharges. Under low estuarine water levels, the number of hydraulic data points in the navigation-obstructing region increases significantly with rising discharge. However, as the estuarine water level rises, the hydraulic data points tend to cluster in the low-velocity/low-slope region. At a discharge of 1840 m 3/s, the estuarine water level must exceed 263.85 m to satisfy the upstream navigation criterion. When discharge further increases to 2250 m 3/s, these shoals satisfy the criterion only when the estuarine water level exceeds 264.24 m ( Table 9). In summary, the five rapid shoal reaches in the Min River basin, influenced by backwater effects, exhibit distinct flow-resistance characteristics. Shoals such as Qiaoban, Pianchuangzi, and Yangjia Shoals show relatively low hydraulic resistance under the examined scenarios. For low-water rapid reaches such as Tongluowan and Sipoqi, navigation obstruction tends to decrease with increasing water level and discharge. In contrast, Yangjiaoshi and Wuchashu Shoals show more pronounced hydraulic constraints under normal flow conditions and satisfy the upstream navigation criterion only under specific water level and discharge combinations. Therefore, the navigation safety atlas provides a scenario-based reference for identifying potential navigation risk zones and understanding the hydraulic causes of navigation obstruction. Its application is limited to the representative vessel type, simulated flow conditions, and simplified hydrodynamic indicators considered in this study, and should not be interpreted as a direct real-time navigation decision tool. 4.2.4. Optimal Navigation Conditions Figure 19 illustrates the water depth characteristics corresponding to the minimum estuarine control water levels for the reach downstream of Sipoqi Shoal under different discharge conditions. Under each discharge condition, the requirements for flow velocity, navigable depth, and navigation resistance are satisfied. A comparison of the three navigation control conditions shows that the water depth required by the navigable depth criterion is greater than that required by the flow velocity and navigation-resistance criteria across all tested discharges. This indicates that the navigable-depth requirement is the dominant constraint for this shoal reach. Accordingly, the navigation control depths for the three discharge conditions are determined as 2.60 m, 2.65 m, and 3.09 m, respectively. As shown in Table 10, for discharges between 900 m 3/s and 2250 m 3/s, the corresponding estuarine control water levels satisfying navigation requirements are 267.96 m, 263.85 m, and 263.46 m, respectively. For high-discharge conditions ranging from 5570 m 3/s to 15,000 m 3/s, only water depth and flow-velocity criteria are applied to shallow shoal reaches. As discharge increases, the corresponding minimum required estuarine water levels for each discharge class are 262.08 m, 268.57 m, 270.19 m, 271.17 m, and 272.49 m. These water level thresholds can serve as reference values for scenario-based navigation assessment under the examined discharge conditions. 5. Conclusions 5.1. Main Conclusions In this study, the lower Min River, a tributary influenced by backwater from the Jinsha River, was selected as a case study. A depth-averaged two-dimensional hydrodynamic model was developed for the estuarine reach, and the reach’s hydrodynamic response characteristics were analyzed under different estuarine water levels and tributary inflows. On this basis, a navigation risk atlas was constructed using three indicators—water depth, flow velocity, and navigation resistance. The main conclusions are as follows: (1) Mainstem backwater significantly restructures the hydrodynamic distribution in the tributary estuarine reach. By reducing the channel water surface slope and flattening the longitudinal profile, backwater weakens the longitudinal flow-driving force, leading to discharge redistribution and velocity field restructuring, and ultimately promoting an earlier shift in the hydrodynamic axis toward the concave bank. Within the estuarine zone, backwater induces local flow stagnation and non-uniform velocity, generating a high-shear flow field. The backwater length is positively correlated with the estuarine water level and increases stepwise with discharge. Under high-flow conditio
