One-dimensional channel network modelling and simulation of flow conditions during the 2008 ice breakup in the Mackenzie Delta, Canada
Introduction
Networks of open channels are common occurrences in complex natural river systems such as braided rivers and river deltas. River channel junctions are key components to river networks and have an impact on how flows are distributed within a network. Junctions are located where two or more channels intersect resulting in channels coming together (converging junctions) or a channel splitting (diverging junctions). The hydrodynamics at channel junctions are complex and are a function of the junction geometry and flow conditions (Ghostine et al., 2009). Over the past six decades, considerable efforts have gone into the development of mathematical junction models. For subcritical flows, which are the most common in natural river systems, the models range from theoretical and partly empirical one-dimensional (1D) approaches (e.g. Garcia-Navarro and Savirón, 1992; Gurram et al., 1997; Hsu et al., 1998; Shabayek, 2002; Ghostine et al., 2013) to three-dimensional (3D) approaches based on the Reynolds-averaged Navier-Stokes equations (e.g. Huang et al., 2002; Ramamurthy et al., 2007; Pandey et al., 2020).
For northern river systems, unsteady flow models with both multi-channel hydraulics and river ice modelling capabilities can be important predictive tools since ice jams (freeze-up and breakup) have a tendency to occur at river confluences (Ettema et al., 1999), near islands (Turcotte and Morse, 2013), and within river deltas (Beltaos et al., 2012). Both one-dimensional and two-dimensional (2D) models with river ice modelling capabilities exist, commercially and within the public domain. 2D models, like CRISSP2D and River2D, provide a better representation of the variability of river ice in complex natural channel or multi-channel systems. Though, they are not typically applied to long reaches or channel networks because costly field data and lengthy computational requirements render them operationally impractical. As a result, it is often more practical to model real-world rivers affected by ice using a 1D model, particularly when modelling large river networks. However, not many 1D models with both multi-channel hydraulics and river ice modelling capabilities exist. Those with junction modelling capabilities and a comprehensive set of river ice modelling capabilities (i.e. water cooling and supercooling, border ice formation, anchor ice formation and release, surface ice evolution and transport, ice cover progression and retreat, and under-cover transport of frazil) are even fewer. The most common approach for modelling junctions in network models is to use conservation of mass and energy at the junction. Because energy losses and differences in velocities heads are difficult to evaluate, the energy equation is typically approximated by equal water levels across the junctions (Shabayek et al., 2002).
HEC-RAS (Hydrologic Engineering Center's River Analysis System) is capable of modelling river networks with stream junctions. For unsteady flow modelling in HEC-RAS, the default option is to force equal water levels across the junction, also called the equality model. A newer option allows for balancing energy across the junction to compute water surfaces at junctions but no documentation is available on how the energy loss at the junction is calculated (Brunner, 2020). Presently HEC-RAS does not have comprehensive ice modelling capabilities and is limited to modelling stationary ice covers and ice jams. CRISSP1D, a fully comprehensive 1D river ice model (Shen, 2005), uses a four-point implicit method to model river networks (Potok and Quinn, 1979). The method assumes equal water levels at junctions. A previous version of River1D, the University of Alberta's comprehensive river ice process model, (Andrishak and Hicks, 2011) also employed the equality model at junctions in order to study the ice effects on flow distributions within the Athabasca Delta. The Danish Hydraulic Institute's Mike11 hydraulic software uses the principles of mass and momentum conservation to predict the depth ratio across a junction (DHI, 2017) but it is not known whether Mike-Ice (Ice Generation and Accumulation add-in module) has been adapted for use with channel networks. Timalsina et al. (2013) applied Mike-Ice to a single reach of the Orkla River in Norway. Yuan et al. (2020) found that for a simple bifurcation Mike11 could not correctly simulate rapidly changing inflows.
The limitations of the equality model have been demonstrated by numerous investigators. Garcia-Navarro and Savirón (1992) found this model is only valid for low Froude numbers. Kesserwani et al. (2008) found that for converging junctions the equality model only agreed well with the compared experimental data when the Froude number immediately downstream of the junction (Fd) remained less than 0.35. They recommended that at higher values of Fd to use a model based on the momentum conservation principle or to handle the junction using a 2D approach.
This paper presents new developments to River1D's channel network modelling capabilities, based on the 1D momentum conservation approach to model channel junctions developed by Shabayek et al. (2002). The adapted approach eliminates the equal water level assumption, and instead takes into account the significant physical effects at channel junctions (such as gravity and flow separation forces, and channel resistance). These effects can be critical to dynamic unsteady flow applications such as ice jam formation/release and severe storm surge events. The adapted approach is also equipped with the ability to dynamically change junction configurations (i.e. diverging to converging or vice versa) as the result of flow reversals.
The new momentum based approach to model junctions is assessed using a series of steady and unsteady tests using a 2D model, the University of Alberta's River2D, for comparison. The model is then applied to the Mackenzie Delta to simulate flow conditions during the 2008 breakup. The work presented in this paper is a step towards the realization of River1D as a comprehensive public-domain river ice process model capable of simulating dynamic ice processes in complex natural river network systems.
Section snippets
Model description
The proposed approach to simulating channel networks is built on the University of Alberta's public-domain 1D model, River1D, which solves the Saint-Venant equations using the characteristic-dissipative-Galerkin (CDG) finite element scheme (Hicks and Steffler, 1990, Hicks and Steffler, 1992). The model was originally developed as a hydrodynamic model for open channel flow in rectangular channels of varying width. Blackburn and She (2019) reformulated the model to accommodate natural channel
Model validation
A series of tests were performed to assess the model's junction capabilities. In Shabayek et al. (2002), the proposed models for converging and diverging junctions were validated but the validation was performed for lab scale problems only. Additionally, Shabayek's models were developed and tested solely for steady flow conditions and were never designed to handle flow reversals. For these reasons, the model was setup to test a variety of junction configurations, which were also simulated using
Model application
River1D was applied to a network of channels in the Mackenzie River Delta (the Delta). The network model, developed for the upper Delta with consideration for the most hydraulically significant channels and junctions, was calibrated and validated using three open water events. The calibrated model was subsequently used to simulate flow conditions during the 2008 breakup of the Delta.
Conclusions and recommendations
This paper presents new network capabilities incorporated into the University of Alberta's one dimensional hydrodynamic model, River1D. The approach used to simulate junctions in the model takes into account the significant physical effects at channel junctions (such as gravity and flow separation forces, and channel resistance) rather than using the simpler assumption of equating water levels across the junction. The model has also been equipped to automatically handle changes to junction
Author statement
The roles and contributions of the authors are listed below:
Julia Blackburn: Methodology, Model development, Model simulation, Formal analysis, Investigation, Writing original draft.
Yuntong She: Conceptualization, Writing – review and editing, Supervision.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgement
This research has been funded through the Beaufort Regional Environmental Assessment program and an NSERC Discovery Grant. Their support is gratefully acknowledged. The authors would like to thank Jennifer Nafziger for assistance with processing the GIS data used in this study. The authors would also like to thank Dr. Spyros Beltaos for providing the water level data collected during the 2008 breakup.
List of symbols
- A
- cross sectional area to the water surface
- AA
- cross sectional area to the water surface at junction node A
- AA(Element B)
- portion of AA contributing to the flow in Element B
- AA(Element C)
- portion of AA contributing to the flow in Element C
- Aan
- cross sectional area of anchor ice
- Ai
- cross sectional area of surface ice
- Aw
- cross sectional area of water under and through the ice
- D
- water depth
- DCM, DTU, DTD, DBU, DBD
- water depths at model sections in parallel channels tests
- DLD, DMD, DMU
- water depths at model sections
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