Hydraulic resistance of river ice jams *

Lin Fan , Ze-yu Mao Hung Tao Shen

1. Department of Hydraulic Engineering, Tsinghua University, Beijing 100084, China

2. Development Research Center of the Ministry of Water Resources, Beijing 100036, China

3. Department of Civil and Environmental Engineering, Clarkson University, Potsdam, NY 13676, USA

Abstract: The hydraulic resistance of the river ice jams consists of the resistances due to the seepage flow through the jam and the shear stress on the undersurface of the jam. Existing empirical formulations consider only the the undersurface resistance of the jam,and come up with relations between the jam resistance and the jam thickness with very slight theoretical basis. Based on the analysis of the seepage flow resistance and the flow resistance of the undersurface of the jam, it is shown that the resistance due to the seepage flow is a dominating part of the jam resistance, except for the portion of the jam where the thickness is very small. This analysis also shows that the total jam resistance can be approximated by a linear function of the jam thickness or the ratio of the jam thickness to the flow depth under the jam.

Key words: River ice, hydraulic resistance, ice jam, friction factor, Manning's coefficient

Introduction

The river ice jams can become an extensive blockage of the flow in the channel. The surface ice jams are usually accompanied by a rapid water level rise due to the blockage effect related with the thickness and the hydraulic resistance of the surface ice floe accumulations. Pariset and Hausser[1]developed the classical ice jam theory by considering the static force balance of the surface ice floe accumulations on the water surface, which was then refined by others[2-4]. Two numerical methods were developed for calculating the ice jam profile in river channels: the ICEJAM model developed by Flato and Gerard (1986) and the RIVJAM model developed by Beltaos[4-5]. In both methods, the ice jam thickness equation together with the steady gradually varied flow equations are used. In field applications, due to the difficulty in obtaining the jam thickness profile,the calibration of the ice jam model parameters often relies on the observed water levels. However, since the water surface profile is affected by both the jam thickness and the flow resistance, it is possible to better reproduce the observed water surface profile even if the predicted jam thickness profile is not good.Based on a laboratory flume study, Healy and Hicks(1999) showed that the observed water surface profile can be reproduced with different combinations of the jam thickness profile and the Manning's coefficient of the jam in simulations using the ICEJAM model. This points to the need of a better understanding of the flow resistance of the ice jams. This paper will review and analyze existing formulations, and develop a clear understanding of the flow resistance of the ice jams.

1. Empirical relations based on ice jam roughness

Based on the field data, Nezhikhovskiy related the Manning's roughness coefficientin to the thickness of the ice covers formed by accumulations of ice floes, dense slush, and loose slush. Following empirical formulas were obtained based on Nezhikhovskiy's data and used in HEC-RAS[6]:

For breakup jams

For freeze up jams

Fig. 1 (Color online) Comparison of model outputs with observed data for three different jam roughness formulations

Table 1 The root-mean-squared error (RMSE (m)) and root-mean-squared relative error (RMSRE) of simulated jam thick-

where H is the depth of the flow under the jam, tjis the jam thickness. Equations (1a)-(1c) are in English units[6], both the flow depth and the jam thickness are in ft.

Beltaos[7]used a composite friction factor, f0, to represent the flow resistance due to the ice jams.Based on the field data, Beltaos came up with a formula to calculate the composite friction factor for the channel flow with an ice jam

Shen et al.[8]assumed that the Manning's coefficient of the jam varies linearly with the jam thickness between a minimum value for a single layer juxtaposed cover and a maximum value for a large jam thickness in a dynamic ice jam model.Application results of the model agree well with the field data[9-12].

The above empirical relationships were used in different studies, but without clear theoretical explanations. Since a portion of the water discharge flows through the jam, there is a significant head loss due to the energy dissipation of the seepage flow, especially at the toe of the jam. This aspect was overlooked in previous studies and should be considered. This study will examine the validity of the existing formulas for the ice jam resistance and improve them by including the seepage flow effect.

2. Effect of jam resistance on jam profile

In this section the effect of the flow resistance due to the ice jam on the jam profile will be discussed using the Thames River ice jam data on January 1986[4]. The water surface and the ice jam profiles are calculated using the method of the ICEJAM. In the ICEJAM model, the coupled flow and jam equations are used for the jam thickness and the water level.This approach was adopted in the HEC-RAS[6,13]. The water level downstream the toe, the length of the jam,and the ice thickness at the head of the jam are required as the input. The simulated results are compared with the observed data using different formulations of the jam roughness. The observed leading edge of the ice jam is 42 km downstream the toe. The data is only available up to 39.19 km.Therefore, the simulations are carried out with the jam up to 39.19 km with a 0.5 m leading edge ice thickness, and a bed Manning's coefficient =0.025[4]. Figure 1 shows the simulated water surface and the jam profiles for three different jam roughness formulations, where X is the distance from the downstream, Z is the elevation:

Case 1: Constant jam roughness

In this case, the ice jam Manning's coefficient=0.060as given in the HEC-RAS Example 14[6]is used. The simulated result does not match with the observed data.

Case 2: Nezhikovsky's formula for jam roughness

In this case, Eqs. (1a), (1b) are used for the ice jam roughness. The simulation result compares reasonably well with the data except for the jam toe.

Case 3: Variable roughness along the jam

The jam Manning's coefficient along the channel is calibrated based on the observed jam and water surface profiles. In this case, the comparison agrees best with the observed data, including the jam toe.

Table 1 summarizes the root-mean-square error and the root-mean-square relative error (RMSE and RMSRE) of the simulated results in these three cases.Figure 2 summarizes the jam Manning's coefficient used in all three cases, which shows that the jam resistance coefficient should vary with the thickness in order to accurately describe the jam and water surface profiles. The result in case 3 shows that the Manning's coefficient can be assumed to vary linearly between a minimum and maximum values as proposed by Shen et al.[8].

Fig. 2 The variation of the ice roughness with ice jam thickness in all three cases

3. Energy loss in the ice jam reach

In an ice jammed channel, the total energy loss comprises of those due to the bed shear stress, the shear stress on the undersurface of the jam, and the seepage flow through the jam. The total friction slope,, between two cross sections can be expressed as

where Sfb, Sf1iand Sfi2are the friction slopes corresponding to the bed resistance, the resistance due to the undersurface roughness of the jam, and the resistance due to the seepage flow through the jam,respectively, ρ is the water density,A is the average flow area under the jam between two cross sections,bP ,bτ are the bed wetted perimeter and the shear stress, respectively,i1P ,1iτ are the ice cover wetted perimeter and the shear stress,respectively, andtD is the seepage drag on the ice particles in the jam.

3.1 Energy loss due to the seepage flow

The friction slope Sfi2for a jam element of unit length with a submerged cross section area Aj,containing N ice particles, can be expressed as

where CDis the drag coefficient, v is the seepage velocity, equal to q/ n, q is the apparent velocity,αis the cross-section area of a particle normal to the flow, α is a shape factor, dsis a measure of the particle size, 6/Msand Msis the specific area per unit volume of the solid. Typically,15],is the ice block thickness. The drag coefficient can be expressed as[14]

where λ is a factor representing the effect of the neighboring particles, and C1≈1.0, is a constant varying slightly for different media[14].

The apparent velocity q can be related to the hydraulic gradient as[14]

where a, b are the shape factors,,fν is the kinematic viscosity. Since the flow through the ice jams is fully turbulent, the Reynolds number is very large, in the order of 104[15], while (1 )/a -n b is in the order of 100, λ is in the order of 10[14], the first term in Eqs. (6), (7) can be neglected. Eq. (7) can be written as

Combining Eqs. (4)-( 8), the friction slope Sfi2can be expressed as

For wide river channels, Eq. (9) can be simplified further as

3.2 Energy loss due to the undersurface roughness of the jam

The bed shear stress can be described as

Substituting Eq. (11) into the first term of Eq. (3),we have

wherebA is the flow area associated with the river bed,bn is the bed Manning's coefficient, andcn is the composite Manning's coefficient.

Hence, the friction slope contributed by the shear stress on the bottom surface of the jam can be expressed as

4. Case studies

Three cases are used to analyze the energy loss of the ice jams. These cases are:

Case 1: Jam in a uniform channel

An idealized ice jam formed at the transition area from a steep reach to a flat reach in a rectangular uniform channel is simulated[6]. The simulated jam with a downstream boundary water level of 18 m and an inflow discharge of 3 000 m3/s is shown in Fig. 3.

Fig. 3 Model result for a jam in a rectangular uniform channel

Case 2: Thames River ice jam

The 1986 Thames River ice jam[4], which is simulated in Section 2. The simulated result in Case 3 shown in Figs. 1, 2 will be used.

Case 3: Matapedia River ice jam

An ice jam was observed in the Matapedia River in April, 1986. The simulated result of Beltaos[16]using the RIVJAM model as shown in Fig. 4 will be used.

Fig. 4 Matapedia river ice jam simulated by RIVJAM[16]

4.1 Energy loss

With the water depth and the velocity of each cross section calculated from the ice jam model results,the total energy loss along the jam is determined.Using the total energy loss value from Eqs. (10), (13),with C1=1.0, λ=3π, the energy loss caused by the seepage flow in the ice jams and the shear stresses along the jam undersurface can be obtained. Beltaos[15]gives an average value offrom the field data calibration, and the value of the shape factor β can be estimated as around 0.8.

4.1.1 Uniform channel jam

The flow discharge in this channel is approximately 3 000 m3/s. The seepage coefficient, μ , is taken as 1.0 m/s, and the thickness of the ice block,it ,in the jam is 0.5 m. It is assumed that the porosity, n,is 0.4, the dimensionless coefficient, γ , is 1.67, and the value of the shape factor of the particle β is taken as 0.78. The ratio of the friction slope of the seepage flow in the ice jam, Sfi2, to the total friction slope, Sf, is given in Fig. 5, which shows clearly that the ratio increases with the ratio of the thickness of the jam to the flow depth up to a maximum value of 56%.

Fig. 5 Variation of friction slopes along the jam in the uniform channel

Fig. 6 Variation of friction slopes along the jam in Thames River

4.1.2 Thames River jam

The flow discharge is approximately 290 m3/s,while the seepage coefficient, μ , is 0.6 m/s, and the thickness of the ice block,it , in the jam is 0.2 m[4].The value of the shape factor of the particles, β , is about 0.8. The ratio of the friction slope of the seepage flow in the ice jam to the total friction slope is shown in Fig. 6, which shows clearly that the ratio increases with the jam thickness up to the water depth ratio. The maximum value of Sfi2is about 38% of the total energy loss.

4.1.3 Matapedia river jam

In this case, the flow discharge is approximately 140 m3/s, =1.5 m/sμ , =1.2 md , and the porosity=0.4sn[16]. The value of the shape factor β is about 0.75. Figure 7 shows that the ratio between the friction slope of seepage flow and the total friction slope increases with the jam thickness up to the water depth ratio to a maximum of about 66%.All three cases discussed above show that the energy loss due to the seepage flow through the jam is a dominating part of the total energy loss, especially in the jam toe region.

Fig. 7 Variation of friction slopes along the jam in Matapedia River

4.2 Relationship between and /H

To explain why the resistance coefficient of the ice jam increases with its thickness, the relationship between the ice jam friction factorand /H is examined in this section.

4.2.1 Analysis based on energy slope

The friction factor of the jam can be calculated from the shear stress on the undersurface of the jam,, and the seepage drag,tD . The shear stressi1τ can be calculated from the energy slope Sfi1and the flow depth under the jam controlled by1iτ . Using the shear stress on the undersurface of the ice coveri1τ in conjunction with the drag force on the seepage flow,tD , the ice jam friction factor in Eq. (3) can be expressed as where V is the velocity under the jam, B is the channel width, and.α , we have

Using Eqs. (4)-(7) and

Figure 8 shows the relationship between fiand/H in all three cases.

Fig. 8 Ice friction factor fi versus t j/H

4.2.2 Analysis based on surface roughness of the jam

The friction factor can be related to the roughness height based on the Williamson formula[17]

where f is the friction factor, ksis the surface roughness height, defined as the diameter of the sand grains for the sand bed, R is the hydraulic radius,approximately equal to the flow depth in a natural stream under open water conditions. In the present study, we replace kswith dsto evaluate the friction factor due to the undersurface roughness of the jam, fi1. Combined with the drag force in the seepage flow, the total ice jam friction factor can be expressed as

whereiR is the hydraulic radius of the ice-affected portion of the flow.

Figure 9 shows the relationship between fiand/H calculated from Eq. (17) with the surface roughness method together with those calculated from Eq. (15) with the energy slope method, in all three cases. The results from these two methods are consistent. These results show that the jam friction factors vary linearly with /H, but with a minimum value corresponding to the single layer ice floe thickness. This lower limit is not shown in the Matapedia River jam case since the data for the portion of the jam head is not available.

Fig. 9 Variations of fi with t j/H

4.2.3 Relative contributions of fi1, fi2

Figures 10, 11 show the friction factor fi1associated with the shear stress on the undersurface of the jam and the friction factor fi2associated with the seepage flow through the jam. These figures show that fi1remains nearly constant for a jam, while fi2increases with the jam thickness. Near the head of the jam, the ice jam is thin and the seepage flow is insignificant, the ice friction factor is mainly affected by the shear stress on the undersurface of the jam. As the jam thickness increases, fi2becomes the dominating part of the total ice friction factor. The energy loss due to the undersurface roughness of the jam is relatively small as compared to that due to the seepage flow.

Fig. 10 The friction factor associated with ice shear stresses versus H on Thames River and Matapedia River

Fig. 11 The friction factor associated with seepage flow versus H on Thames River and Matapedia River

4.3 Validation of the Parameter κ

In the energy slope method, empirical values of the parameter κ are used. These values are shown by dash lines in Fig. 12. In this section these κ values are validated using the surface roughness method, Eq. (17), with the calculated fivalues.Figure 12 shows the comparison of the κ values calculated, which shows that the values of κ used in the analysis are reasonable.

4.4 Relationship between and j

Using the value of ficalculated in Fig. 9, the Manning's coefficient of the jam, ni, can be obtained from the relationship. Figures 13, 14 show the jam Manning's niversus tj,. These figures show that the jam Manning's coefficients vary linearly with the jam thickness and the ratio, but with a minimum value corresponding to the roughness of the juxtapose ice cover from a single layer ice floe accumulation. For numerical model applications, it is more convenient to use the relationship between niand tj.

Fig. 12 Validation of the parameter κ

Fig. 13 Manning versusfor uniform channel, Thames River and Matapedia River jams

5. Conclusion

Thehydraulicresistanceisanimportantpara-meter affecting the flow condition associated with an ice jam as well as the jam thickness profile[18-21]. The empirical formulas for the flow resistance are often used in the ice jam models. In these formulas, the jam resistance coefficient varies with the jam thickness due to the undersurface roughness of the ice jam and they were used with little theoretical explanation. The resistance due to the seepage flow through the jam was overlooked. Through a detailed analysis of the seepage flow resistance and the resistance due to the undersurface roughness of jams, this study indicates that the seepage flow resistance increases with the jam thickness and the flow resistance due to the undersurface roughness of the jam remains relatively constant. Moreover, the relative contribution of the resistance due to the undersurface roughness decreases in comparison with the seepage flow resistance when the jam thickness increases. The seepage flow resistance becomes a dominating part of the jam resistance except for a portion of the jam near its head,where the jam thickness is small with negligible seepage flow. The analysis also shows that the total jam resistance in terms of the friction factor or the Manning's coefficient can be approximated by a linear function of the jam thickness or the ratio of the jam thickness to the flow depth under the jam, but with a minimum value corresponding to the juxtaposed ice accumulation.

Fig. 14 Manning versus /H for uniform channel,Thames River and Matapedia River

Acknowledgement

This study was conducted during the first author's visits at Nanyang Technological University and Clarkson University under the support of Tsinghua University and Clarkson University.