Modified Saint-Venant equations for flow simulation in tidal rivers

Xiao-qin ZHANG*, Wei-min BAO,

1. College of Hydrology and Water Resources, Hohai University, Nanjing 210098, P. R. China

2. State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, Nanjing 210098, P. R. China

Modified Saint-Venant equations for flow simulation in tidal rivers

Xiao-qin ZHANG*1, Wei-min BAO1,2

1. College of Hydrology and Water Resources, Hohai University, Nanjing 210098, P. R. China

2. State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, Nanjing 210098, P. R. China

Flow in tidal rivers periodically propagates upstream or downstream under tidal influence. Hydrodynamic models based on the Saint-Venant equations (the SVN model) are extensively used to model tidal rivers. A force-corrected term expressed as the combination of flow velocity and the change rate of the tidal level was developed to represent tidal effects in the SVN model. A momentum equation incorporating with the corrected term was derived based on Newton’s second law. By combing the modified momentum equation with the continuity equation, an improved SVN model for tidal rivers (the ISVN model) was constructed. The simulation of a tidal reach of the Qiantang River shows that the ISVN model performs better than the SVN model. It indicates that the corrected force derived for tidal effects is reasonable; the ISVN model provides an appropriate enhancement of the SVN model for flow simulation of tidal rivers.

tidal river; tidal effect; dynamic water pressure; Saint-Venant equations; corrected force; flow simulation

1 Introduction

Hydrological processes in tidal rivers are nonlinear and complicated as the water stage constantly changes due to the interaction between upstream floods and downstream tides. Consequently, flood routing in tidal rivers usually poses a complex problem. Compared with hydrological physical conceptual models based on empirical storage-flow relationships (Franchini and Lamberti 1994; Qu et al. 2009) and mathematical black-box models based on data analysis (Wu et al. 2008; Hsu et al. 2010), hydrodynamic models based on the Saint-Venant equations (the SVN model) can describe the dynamic mechanism of unsteady flow well, and they are widely used for tidal rivers (Lamberti and Pilati 1996; Su et al. 2001). The mechanics analysis of these SVN models is restricted to the gravity, friction, hydrostatic water pressure, and inertial force. They can perform well for most rivers, but their performance may prove unsatisfactory when applied to rivers with significant tidal effects.

Tide and upstream flow are the two major external forcing mechanisms controlling flow routing in tidal rivers. The Saint-Venant equations do not explicitly consider the influence of tides other than by using the downstream boundary condition in its applications, the gravity, friction, and hydrostatic water pressure being considered the main driving forces involved. A general point of view is that the tidal effect is included in the SVN models as a form of a pressure gradient force at the downstream boundary introduced by the periodic tidal level variation in the estuary. Tsai (2005) pointed out that there are two physical mechanisms that account for the downstream backwater effect: the propagation of the secondary wave moving from lower reaches to upper reaches and the instantaneous transmission of the pressure gradient term. Theoretically the flow controlled by gravity moves downstream, while tidal river flow can move upstream in flood scenarios due to the tidal level variation at the downstream boundary. Under the influence of tides and the water balance imposed by the continuity equation, the Saint-Venant equations can describe bidirectional flows. Generally, the direction of the water pressure gradient is upstream during the flood tide and downstream during the ebb tide. The larger the rate the tide rises or ebbs, the larger the water pressure is, which may make one think that the water pressure can account for tidal effects. However, under the condition that the influence of incoming flood excesses that of downstream tide during the flood tide, the direction of water pressure is downstream, which can not reflect the tidal effect with the upstream direction (Bao et al. 2010). The hydrostatic water pressure is employed in the Saint-Venant equations. Tides propagating upstream imposed by tidal momentum can produce significant dynamic water pressure that hydrostatic water pressure cannot account for, especially in some scenarios such as tidal bores (Mazumder and Bose 1995) and the coincidence of astronomical tides and storm surges. Hence, it is desirable to develop a simple technique with physical concepts accommodating tidal effects.

The interaction between tides and upstream flow is complicated (Sobey 2001; Horrevoets et al. 2004; Garel et al. 2009). The transition of propagation direction upstream or downstream may be changed with the variability controlled not only by the relative magnitude of upstream inflow and tidal effect, but also by channel geometry, as well as initial and boundary conditions (Phillips and Slattery 2007; Pan et al. 2007). The tidal effect in tidal channels is not directly forced by the astronomical forces due to the sun and moon but arises instead as a side effect of fluctuation in the deep ocean, propagating through shallower coastal waters as an isolated wave or as a combination or train of waves. Although the backwater effects are extensively studied (Tsai 2005; Munier et al. 2008), few studies particularly focus on the tidal effect on flood routing. Since Longuet-Higgins and Stewart (1964) first defined the radiation stress as the excess momentum due to the presence of waves, it has been fairly widely used to reflect the effects of waves on water (Zheng and Yan 2001; Newell et al. 2005). Extending the radiation stress concept to tidal rivers, the tidal effect takes the form of horizontal stress action on river flow, which provides a good way to consider the tidal effect as a corrected force in terms oftidal energy. It therefore seems worthwhile to develop a corrected force in the SVN model to better represent flow movement in tidal rivers.

In this study, a force-corrected term was established to account for comprehensive tidal effects. Furthermore, an improved SVN model with the corrected force (the ISVN model) was constructed to improve flow simulation in tidal rivers, which hopefully extends our understanding of the dynamics of flood wave propagation in tidal rivers and provides a significant enhancement of the corresponding SVN model. In the case study the improved model was used for stage simulation in the tidal channel of the Qiantang River.

2 Study area and data

The Qiantang River is located in Zhejiang Province in China (Fig. 1). It discharges into the Hangzhou Bay, and then into the East China Sea. It is well known for its spectacular tidal bore and has become a famous scenic location, attracting thousands of tourists annually. The astronomical tide in the Qiantang River Estuary is of the irregular semi-diurnal kind. The Qiantang River Basin has a subtropical monsoon climate, being warm and humid with abundant precipitation, a sufficient amount of sunshine, and four distinct seasons. In summer and autumn, typhoons frequently occur. The basin has a large population and is extensively developed. The coincidence of the upstream flow and downstream storm tide may result in heavy damage.

Fig. 1 Location of study area

Hangzhou Bay possesses a unique funnel shape about 100 km wide at its mouth, and narrows down upstream to 20 km wide at Ganpu Hydrological Station, 89 km away from the mouth. In the rapid narrowing channel, significant tidal energy accumulates (Friedrichs and Aubrey 1994) with the increase of tidal range by up to 75% at Ganpu (Pan et al. 2007). During the spring tide, the tidal effect can extend upstream to Fuchun Power Station. Analysis of tide propagation in the Qiantang River shows that the water quantity flowing upstream or downstream due to the tide wave propagating upstream or downstream (WQ) is closely relatedto the change of water level at the downstream boundary,(referred asHT), wherezis the water level andtis time. Fig. 2 displays the relationship betweenWQandHTat Luchaogang Hydrological Station using the collected data from Nov. 1 to Nov. 30 in 2009 along the Qiantang River. From Fig. 2, the correlation coefficientRis 0.871, indicating a good relationship betweenWQandHT.

Fig. 2 Relationship betweenWQandHTat Luchaogang Hydrological Station

The reach from Tonglu Hydrological Station to Fuyang Hydrological Station was selected as the study reach. Tonglu Hydrological Station is the upstream boundary dominated by incoming flow and Fuyang Hydrological Station is the downstream boundary affected by tides. The cross-sections along the river are of a compound nature with various main channels and floodplains. The channel width ranges from 400 m to 700 m near Tonglu Hydrological Station, and from 500 m to 900 m near Fuyang Hydrological Station. Fifty-nine cross-sections from Tonglu to Fuyang hydrological stations were surveyed (Fig. 1), including Tonglu, Zhaixi, and Fuyang hydrological stations. The length is about 15.6 km from Tonglu to Zhaixi hydrological stations and about 28.6 km from Zhaixi to Fuyang hydrological stations.

3 Method

3.1 Force-corrected term for tidal effect

From a physical perspective, tide propagation is dominated by tidal energy; and its driving force is essentially created by the change of tidal momentum. The accumulation of tidal energy is related to the gravitational field of the moon and, to a lesser extent, the sun as well as the geometry of the estuary. In an estuary of specific geometry, the change in tidal energy can be determined from the change rate of the tidal level. Generally, the change rate of the tidal level reflects tidal energy and tidal velocity.

In a tidal channel, for an increase Δhin tidal level occurring at the downstream boundary over the incremental time period of Δtin a flood tide, we assume that the corresponding tide wave moves upstream a distance Δxlongitudinally, due to Δh, as sketched in Fig. 3. Obviously, the value of Δxis small when Δhis small, and vice versa. As an approximation, the increased Δhduring the time period Δtis assumed to be directly proportional to the propagation distance Δx, as shown in Fig. 3. Mathematically, the linearrelationship between Δhand Δxcan be described as

whereDis defined as the velocity coefficient of tide wave propagation.

Fig. 3 Micro-unit analysis diagram

With this assumption the relationship according to the water balance can then be expressed as

wherevis the flow velocity,bis the channel width, and ΔAis the change of cross-sectional area caused by the rising and falling of the tide.

In a rectangular chann el Eq. (2) can be rewritten as

Combining Eqs. (1) and (3) and letting Δtapproach zero in the limit, Eq. (4) can be obtain ed:

It indicates that the tide wave velocity is directly proportional to the change rate of the water level at the downstream boundary.

Because the tidal effect is closely related to the change in the accumulation of its kinetic energy, the form of the fo rce accounting for tidal effect,Fc, can be assumed to be

whereDkis the transfer coefficient relating to the tidal kinetic energy and its driving force, andρis the water density.

On the basis of Eq. (4) and denotingDkDasDm, the corrected force is rewritten as

3.2 Improved SVN model (ISVN model)

Accounting for one-dimensional flow in a prismatic channel, the component of the force due to gravity,FG, the friction resistance force,FJ, and that due to hydrostatic water pressure,FP, along the flow direction can be expressed respectively as

whereS0is the riverbed gradient,gis the gravitational acceleration,Ris the hydraulic radius, andcis the Chezy coefficient.

In accordance with Newton’s second law, the momentum equation incorporating the force-corrected term is given by

Substituting Eqs. (6) through (9) into Eq. (10), and then dividing by the water mass,ρΔAdx, the modified momentum equation is expressed as

The hydrodynamic model based on the ISVN model is

WhenDm=0, the ISVN model is equal to the conventional SVN model. The classical four-point implicit difference Preissmann schem e was used for model solution.

4 Results and discussion

4.1 Model application

Data from 13 flood events were collected for the simulation of stages at Zhaixi Hydrological Station using the stages at Tonglu and Fuyang hydrological stations. These flood events were split into two independent groups: namely nine flood events (those occurring from 1986 through 1989 and in 2006) for model calibration and four flood events (those occurring from 1987 through 1989 and in 2006) for model verification. The roughness value used for the ISVN model refers to that estimated by the SVN model. It may influence the model performance, but allows the performance of the ISVN and SVN models to be compared under the same conditions.

In a tidal river, generally, the longer the distance from a specific cross-section to the downstream boundary is, the smaller the tidal effect at that cross-section is. The value ofDmmay vary with the distance. For simplicity, the expressioncan be used to reflect the variation inDmalong the river, wherelis the total length from the upstream boundary to the downstream boundary,kis the length from the specific cross-section to the downstream boundary, andDxis a parameter that needs to be estimated, which is considered a constant value in this study.

The relative peak stage error (ERPS), time-at-peak error (ET), root mean square error(ERMS), Nash-Sutcliffe coefficient (EC) (Nash and Sutcliffe 1970), and the decreased proportion of the number of stage errors in excess of 30 cm (PD) with the ISVN model relative to that with the SVN model were calculated to evaluate the model performance. WhenERPS＜0, it indicates that the simulated stage is larger than the observed stage, and vice versa; whenET＜0, it indicates that the calculated peak stage occurs after the observed stage, and vice versa.PDis expressed as

whereNSis the number of the stage errors in excess of 30 cm using the SVN model andNIis the number when the ISVN model is used.

4.2 Results

The ISVN and SVN models were applied to the tidal reach of the Qiantang River from Tonglu to Fuyang hydrological stations for stage simulation. The calibration and verification results are respectively listed in Tables 1 and 2. Selected flood hydrographs are displayed in Fig. 4 for calibration and Fig. 5 for verification. As noted earlier, the value ofDxwas estimated to be 23 in the calibration process.

Table 1 Calibration results at Zhaixi Hydrological Station

Table 2 Verification results at Zhaixi Hydrological Station

Fig. 4 Stage hydrograph of flood 870620 calculated using SVN and ISVN models at Zhaixi Hydrological Station

Fig. 5 Stage hydrograph of flood 890720 calculated using SVN and ISVN models at Zhaixi Hydrological Station

As shown in Table 1, the ISVN model performs better than the SVN model. Compared with the SVN model, the average value of the relative peak stage error (ERPS) of the ISVN model decreases from ?5.6% to ?3.3%, and the averageERMSvalue from 0.437 m to 0.307 m, while the averageECvalue increases from 0.818 to 0.890. The average value ofPDis 42.1% and the maximum value is 64.9% for flood 870620; this indicates that the ISVN model can reduce the stage errors effectively. The peak stage time errors (ET) of the ISVN and SVN models are seen to be not more than 2 h. From the verification results in Table 2, the ISVN model also produces better results than the SVN model, with theERPSvalue for flood 870722 decreasing from ?9.4% to ?1.6%, theERMSvalue for flood 060517 decreasing from 0.244 m to 0.176 m, and theECvalue for flood 880820increasing from 0.599 to 0.696. This indicates that the calibrated value ofDmis reasonable, and the ISVN model with theDxvalue of 23 can adequately account for tidal effects in the study reach of the Qiantang River.

As shown in Fig. 4(a), there exist systematic errors between the calculated and observed stages for flood 870620 when the SVN model is used, whereas, as shown in Fig. 4(b), the systematic errors nearly disappear with the ISVN model. Fig. 5 shows that the stagescalculated using the ISVN model can capture the stage fluctuation better than the SVN model. From the hydrographs simulated using the ISVN model with those using the SVN model, it can be seen that the degree of agreement between the observed and calculated stages of the ISVN model is better than that of the SVN model.

Fig. 6 shows the comparison of observed and calculated stages with the SVN and ISVN models at Zhaixi Hydrological Station, which demonstrates that the simulations using the ISVN model are better matched to the observed values than those using the SVN model, both in the model calibration and verification processes, as the values of the correlation coefficientRare seen to be 0.948 and 0.931 respectively for calibration and verification with the SVN model whereas they are 0.974 and 0.947 respectively for the calibration and verification with the ISVN model. Moreover, it can be seen that the ISVN model has better capability to simulate high water stages than the SVN model.

Fig. 6 Comparison of observed and calculated stages with SVN and ISVN models at Zhaixi Hydrological Station

From the analysis above, it is concluded that the ISVN model can obtain better accuracy than the SVN model in flow simulation of tidal rivers. This also indicates that the expression of the corrected force proposed in this study is reasonable and can effectively reflect tidal effects.

4.3 Discussion

These simulation results also show that the ISVN model provides a large improvement onERPSfor flood 880616, onERMSandPDfor flood 870620, and onECfor flood 060508. One of the reasons is the relative effect of upstream flow and downstream tides. Generally, the larger the tide effect is, the better the ISVN model performs. To issue an accurate warning with sufficient lead time for tidal rivers, some specific problems should be considered, as along with the improvement of the accuracy of upstream runoff and downstream tide forecasting.

TheDmvalue was estimated to vary with the distance from the downstream boundary, i.e.,, using a constant calibrated value ofDx, in order to obtain a simple model difference solution. There are three possible ways to analyze the change in simulations caused by a change in theDmvalue. One way is that flood events could be classified into small floods dominated by tidal effects and large floods controlled by upstream flow withDmbeing separately estimated for small floods and large floods. In real-time flood forecasting for tidal rivers, we could choose theDmvalue according to the magnitude of upstream flow. Another way would be to empirically establish the relationship betweenDmand the flow depth. The third way would be to use other solution methods to solve the ISVN model by incorporating theDmexpression. It is recommended that such methods should be attempted in future studies.

In the case study, the roughness values for the ISVN model were those estimated using the SVN model, and the values ofDmwere optimized in the calibration process. It is known that the roughness is not a constant but can vary with flow. Additionally, some interactions between roughness andDmmay exist. More parameter analysis is needed in further studies.

The estimation of initial conditions was simplified in our case. The initialhvalues between Tonglu and Zhaixi hydrological stations were calculated using the linear interpolation of the initial observed stages at Tonglu and Zhaixi hydrological stations, while those between and Fuyang hydrological stations were linearly interpolated using the initial values at these two cross-sections. The initialvvalues along the river were simply estimated using Manning’s equation with the estimated initialh, which may result in large errors at the beginning of simulations, such as flood 890720 in Fig. 5(b). A better way of representing initial conditions is critical for improving the accuracy at the beginning of numerical simulations. Other methods should be explored.

The momentum equation with the force-corrected term includes some assumptions, such as the linear relationship between Δhand Δxin Eq. (1), and the general form of the force-corrected term in Eq. (5). It is recommended that these assumptions be studied in detail and verified with more simulations of other tidal rivers.

5 Conclusions

The tidal effect has significant impacts on flow routing in tidal rivers, resulting in highhydrodynamic water pressure, which is not commonly considered in current hydrodynamic models based on the SVN model. In this paper, a corrected force derived from physical analysis was introduced into the SVN model to account for the comprehensive effects of tide. It consists of the combination of the flow velocity and change in tidal level. The results in the tidal reach of the Qiantang River demonstrate that the modified model with the force-corrected term (the ISVN model) performs better than the SVN model. The approach of the force-corrected term is seen to provide an effective way to account for unconsidered force due to tidal effect.

With a view to further application, more extensive further research should be devoted to the expression of the force-corrected term and parameter estimation, as well as the estimation of initial conditions. To systematically verify the capability of the ISVN model for predicting water level in tidal rivers, more accurate data are required to produce numerical simulations. We hope that the ISVN model can provide a good platform to study the dynamic mechanism of tidal river flow and to develop practical tools for simulating the transport of land-derived material such as sediment in tidal rivers.

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This work was supported by the National Key Technologies R&D Program of China for the Eleventh Five-Year Plan Period (Grant No. 2008BAB29B08-02) and the Program for the Ministry of Education and State Administration of Foreign Experts Affairs of China (Grant No. B08408).

*Corresponding author (e-mail:zxqin403@163.com)

Received Feb. 18, 2011; accepted Sep. 6, 2011