HomeMy WebLinkAboutBehavior of Ice Covers Subject to Large Daily Flow and Level Fluctuations 1980I'
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, _ , ry uF ALASKA
··;A STREET
Canadian
Electrical
Association
,,._.E, ALASKA 99!01
CONTRACT 000 G 138
BEHAVIOR OF ICE COVERS
SUBJECT TO LARGE DAILY FLO~l
AND LEVEL FLUCTUATIONS
ACRES CONSULTING SERVICES LIMITED
.//
REPO'R.T FOR THE
CANADIAN ELECTRICAL ASSOCIATION
Research & Develop~ent
Suite 580, One Westmount Square
Montreal, Quebec
H3Z 2P9
CONTP~CT 000 G 138
BEHAVIOR OF ICE COVERS
SUBJECT TO LARGE DAILY FLOW
AND LEVEL FLUCTUATIONS
ACRES CONSULTING SERVICES LIMITED
PREPARED BY
.ACRES CONSULTIN\. SE~VICES LIMITED
5259 Dorchester Road
Niagara Falls, Ontario
L2E 6Wl
DECE~BER 1980
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L E G A L N 0 T I C E
THIS REPORT WAS PREPARED BY ACRES CONSULTING SERVICES
LIMITED AS AN ACCOUNT OF WORK SPONSORED BY THE CANADIAN
ELECTRICAL ASSOCIATION, (CEA) . NEITHER CEA, MEMBERS OF CEA,
NOR ACRES CONSULTING SERVICES LIMITED, NOR ANY PERSON ACTING
ON BEHALF OF EITHER:
(A) MAKES ANY WARRANTY OR REPRESENTATION, EXPRESS OR
IMPLIED, WITH RESPECT TO THE ACCURACY, COMPLETENESS,
OR USEFULNESS OF THE INFORMATION CONTAINED IN THIS
REPORT, OR THAT THE USE OF ANY INFORMATION, APPARATUS,
METHOD, OR PROCESS DISCLOSED IN THIS REPORT MAY NOT
INFRINGE PRIVATELY Ot-1NED RIGHTS.: OR
(B) ASSUMES ANY LIABILITIES WITH RESPECT TO THE USE OF,
OR DAMAGES RESULTING FROM THE USE OF, ANY INFORt\~TION,
APPl.RATUS, METHOD OR PROCESS DISCLOSED IN THIS REPORT.
TFE VIE~·'S AND OPINIONS OF AUTHORS EXPRESSED HEREIN DO NOT
NECESSARILY STATE OR REFLECT THOSE OF THF: CANADIAN F.LECTPICAL
ASSOCIATION OR OF ANY OF ITS MEMBE~S.
ABSTRACT
A research study was conducted to determine the limits to
water level variation to maintain stability of solid ice
covers in reservoirs, lakes and rivers.
Based on a literature survey of relevant European, Russian
and North American publications, it was concluded that there
is no established engineering capability in this area.
Although the serious consequences of solid ice cover breakup
are appreciated, the available documentation of solid ice
cover breakup events is qualitative. Thus, the available
data are inadequate to develop necessary assumptions for
detailed analysis of this phenomenon.
Existing theories for ice cover stability under steady flow
conditions have been evaluated for their application to
variable water level conditions. A preliminary analysis of
the development of stable hinges connecting the solid ice
cover to the shoreline is presented. Recommendations are
made for laboratory and field studies to gather appropriate
data to calibrate these analyses, and to observe the pheno-
menon in order to make assumptions necessary for further
analytical refinements.
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ACKNOWLEDGEMENT
The authors would like to acknowledge the valuable contribu-
tions made by the Canadian Electrical Association Liaison
Engineers, Mr. L. B. Davies, British Columbia Hydro and
Power Authority; Mr. F. Fonseca, Hydro-Quebec; and
Mr. W. R. Jenkner, Ontario Hydro. Their assistance throughout
the study and careful review of the draft report are greatly
appreciated.
The authors would also like to thank the personnel from the
utilities in Canada and the United States who provided
information in response to our survey questionnaires.
Finally, the support of the Generation and Utilization
Subcommittee of the Canadian Electrical Association is
acknowledged.
TABLE OF CONTENTS
LIST OF TABLES
LIST OF FIGURES
LIST OF SYMBOLS
EXECUTIVE SU1'4 ~4 ARV
Page
1 -INTRODUCTION ---------------------------------------1
2 -GENERALIZED APPROACH TO ICE STUDIES ----------------2
3 -FORMATION -----------------------------~------------9
10
10
16
18
21
3.1 -
3 .1.1
3 .1. 2
3.1. 3
3 .1. 4
3.2 -
3.2.1
3.2.2
3.2.3
3.2.4
3.2.5
3.2.6
3.3
Thermal Processes at Formation ---------------
-Heat Balance on Open Water --------------~--
-Frazil Ice Generation ----------------------
-Border Ice Growth --------------------------
-Channel Closures ---------------------------
Mechanical Stability of an
Ice Cover at Formation -==------------------23
-Leading Edge Stability ---------------------23
-Internal Stability at Formation ------------· 26
River Classification by
Governing Ice Process --------------------
-Application to River Closure ---------------
-Accumulation of Slush and Frazil -----------
-Backwater with Ice Cover -------------------
-Application to Variable Flow
and Water Level --------------------------
34
35
37
4S
49
4 -MIDWINTER ------------------------------------------SO
4.1 -Midwinter Thermal Processes ------------------50
4.1.1 -Extent of Open Water-----------------------SO
4.1.2 -Solid Ice Cover Growth---------------------51
4.2 -Mechanical Stability of a
Solid Ice Cover ----------------------------59
4.2.1 -Solid Covers -General Stability -----------59
4.2.2 -Solid Covers -Shoreline Hinging -----------60
5 -BREAKUP --------------------------------------------B2
5.1 -Thermal Processes at Breakup -----------------82
5.2 -Mechanical Stability at Breakup --------------84
5.2.1 -Ice Jamming---------------------~-----------84
6 -INTERACTION BETt~EN ICE AND STRUCTURES -------------86
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6.1 -o~:~!a~~~~t=~-:~-~~===-~=~==-----------------88 L
6. 2 -Damage at Shorelines ----------------------··--91
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7 -CALIBRATION OF SOLID ICE COVER
STABILITY PROCESSES ---------------------------96
7.1 -Observations of Solid Ice Cover
Breakup and Hinging ---------------------96
7.2 -Calibration Data for Hinqing --------------101
8 -CONCLUSIONS AND RECOMMENDATIONS -----------------107
8.1 -Conclusions ------------------~------------107
8.2 -Recommendations ---------------------------110
LIST OF REFERENCES
BIBLIOGRAPHY
APPENDIX A -ANALYSIS OF HINGE STABILITY
WITH WATER LEVEL VARIATION
APPENDIX B -SUMMARY OF INTERVIEWS
LIST OF TABLES
Number
3.1
3.2
3.3
4.1
Title
Insolation Coefficient Values, C ,
from M~chel (.1971) (units conver~ed
to J/m day)
River Classification by Ice Cover Process
Values of Ice Cover Roughness, n., with
Under-Ice Accumulations (from r-til::hel I 1971)
Summary of Solid Ice Cover Stability
for Variable Water Level
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LIST OF FIGURES
Number
3.1
4.1
4.2
4.3
4.4
4.5
4.6
4.7
7.1
7.2
7.3
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Title
Ice Cover Stability Diagram
Definition Sketch of Hinging Mechanism
Stability of an Individual Hinge Link
Sketch of Two-Link Hinge Configurations
for Rising Water Level
Sketch of Two-Link Hinge Configurations
for Falling Water Level
Two-Link Hinge Configurations for Rising
Water Level
Two-Link Hinge Configurations for Falling
Water Level
Formation of Shear Lines and Hinges
Field Measurements of Cover Movement
Field Measurements of Cover Movement
Field Measurements of Cover Movement
Stability of a Single Hinge Link
Definition Sketch for Two-Link Stability
Two-Link Stability -Case 3
LIST OF SYMBOLS
A
Al
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a
B
B.
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2 = cross-sectional flow area (rn )
= ice albedo (average value 0.25)
= supercooled open-water generation_area {m 2 )
= an exponent in the Reynold's analogy
= an empirical coefficient
= channel (top) width (m)
= total border ice growth (m)
b = an empirical coefficient
c
c
D
E
= Chezy roughness coefficient for water passage,
assumed equal for riverbed and underside of ice
cover
= specific heat of water (J/kg°C)
= insolation coefficient (Table 3.1~ which can be
set to zero for overnight heat loss calculation
(J/m2 day) ·
= an experimental coefficient
= accumulated degree-days of freezing (°C-days)
= trashrack bar diameter Cm)
= characteristic dimension of ice fragments taken as
floe thickness or slush blanket thickness (m)
= the elastic modulus of ice (Pa)
e = relative humidity of air
Fr cr = critical Froude number for ice cover progression
= densimetric Froude number at incipient under-ice
block transport
Fr' = densimetric Froude number
f = friction coefficient
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Darcy-Weisb~ck friction factor
wind drag on the upper surface of the cover (Pa)
(positive if acting in the downstream direction)
acceleration due to gravity (m/s 2 )
mean upstream open-water flow depth (m)
summer open-water flow depth (m)
convective heat transfer coefficient from flow t•
ice cover (Jjm2 s °C)
water equivalent of snow precipitation (m/day)
thermal equivalent = 1 N-m/J
roughness height of ice (m)
coefficient of heat transfer from flow 2 to atmosphere
representative of site conditions (J/rn day °C)
thermal conductivity of air (J/m2 day °C)
conduction coefficient (J/m s °C)
thermal conductivity of the fluid (J/m s °C)
thermal conductivity of water (J/m day °C)
latent heat of fusion of ice (J/kg)
progression rate of a packed ice cover (rn/day)
i = hinge link length (rn)
i, i
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=
=
=
a characteristic length (m)
river location along flow axis (m)
length of ice block (rn)
ti, tb = upper (ice) and lower (bed) flow layer depths
m
m
N
= an adhesive parameter
= fraction of hours of clear sky during daylight
hours
N.
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n
n.
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= number of boundaries
= surface area concentration of slush pans (m 2 tm 2 )
= ne t/kf = Nusselt Number
= an experimental coefficient
= Manning's n-value under the ice cover
nb = Manning's n-value of the channel bed
n c
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composite roughness with ice cover
power per unit area (W/m 2 )
Cp~/kf = Prandtl Number (13.6 for water at 0°C)
river discharge (m 3/s)
external artificial addition of heat (J/m2 day}
volume rate of ice production (m 3 /day)
surface ice transport rate (m 3 /s)
total under-ice transport (m 3/s)
rate of total heat transfer to atmosphere per unit
surface area (J/m2 day)
sum of net heat transferred (J/m2 )
river discharge per unit width (m 3;s m)
rate of heat transfer through a solid cover (J/m2 s)
3 2 rate of ice production per unit area (m /s m )
heat from internal friction (J/m2 s)
3 groundwater discharge per unit river length (m /s m)
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terrestrial heat flow (J/m2 s)
ice discharge per unit width under cover weighed under
water with apoarent density 0.08 (N/s rn)
2 heat available from cooling flow (J/m s)
V ip/~ = Reynolds Number u
hydraulic radius of the ice-covered flow (m)
hydraulic radius of upper flow layer (m)
slope of the hydraulic grade line ~qual to slope
of energy gradient and bed slope for uniform flow)
spacing between trashrack bars (m)
total solar radiation
solar radiation absorbed per unit ice ·volume
critical absorbed solar radiation for complete
loss of strength
air temperature (°C)
freezing temperature of water (0°C)
initial water temperature at location t 0 {°C)
upper ice surface temperature (0°C)
water temperature at location t (°C)
c~~ulative degree-days of thawing
equilibrium flow temperature difference from
freezing (°C)
air temperature difference from water freezing
temperature (°C)
= groundwater temperature difference from flow (°C)
t
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x'
z
a
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= equilibrium ice cover thickness (m)
= thickness of fragmented uniform ice cover (m)
= ice cover thickness .(m)
= thickness of slush accumulation (m)
= thickness of thermal ice grown in water voids (m)
= wind speed (m/s)
= wind velocity lS.m above the water surface (m/s)
= mean flow velocity under open water conditions
upstream of the ice cover (m/s)
= water velocity past bars (m/s)
= mean critical upstream velocity of flow (m/s)
= surface flow velocity (mis)
= mean velocity.under the ice cover (m/s)
= height of wind velocity, U, above ice surface (m)
= length of open water reach (m)
= distance downstream from start of supercooling (m)
= distance travelled by slush pan (m)
= water level variation
= horizontal factor for slope of riverbank corres-
ponding to unit vertical factor
= thermal coefficient for Stefan Equation of thermal
ice growth
= maximum hinge link angle limited by geometry
= the limiting angle of the hinge link from hori-
zontal for joint stability
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specific weight of ice (N/rn 3 )
specific weight of water (N/rn 3 )
porosity of slush accumulation (rn 3;m3 )
absolute viscosity (N s/rn2 )
coefficient related to ice cover friction and
internal resistance of the cover
a frictional or contact coefficient reflecting
the strength of the fractured and rotating hinge
joint
= the friction coefficient at the higher hinge joint
= the friction coefficient at the lower hinge joint
=
=
=
=
=
coefficient in Michel's ice jam stability analysis
typically equal to 0.26
coefficient determined by thermal growth relative
to slush pan thickness
specific mass of water (kg/m3 )
. 3
specific mass of ice (kg/m )
air density (kg/m3 )
= bending strength of the ice cover (Pa)
= ice cover bending strength after exposure to solar
radiation (Pa)
= ice cover strength at freezing/thawing temperature
without exposure to solar radiation
= cohesive strength of fragmented icc pack (Pa)
= ice shear strength
= ice shear strength at ooc without thawing (normally
taken as 500 to 700 kPa)
= stress relaxation functions for the ice cover
failure at the shoreline
EXECUTIVE SUMMARY
This report presents the results of the first phase of a
research study of the stability of solid ice covers subjected
to vnrying water levels and discharges, conducted for the
Canadian Electrical Association under Research Contract 000-
G-138.
An outline of data collection and field study requirements
for evaluation of thermal and mechanical ice processes which
govern the ice regime is presented in the Generalized Approach
to Ice Studies. Because of the number of processes involved
and the complex interrelationships between processes, there is
no unique methodology for ice studies. Thus, a thorough
understanding of all ice processes is necessary in order to
icentify critical processes governing the ice regime.
Application of existing criteria for mechanical stability of
fragmented ice covers and thermal regime calculations,
developed for steady flow conditions, to variable flows is
described. It is possible to analyze and predict the response
of an ice cover to water level and flow velocity variation
prior to consolidation. Guidelines to limit flow variation
can be established from these criteria.
This study involved a literature review and a series of inter-
views with hydroelectric power utilities to establish the
state-of-the-art for the specific ice engineering problem of
solid ice cover stability in rivers, reservoirs and lakes.
The breakup of solid ice covers generally results in severe
ice jams, flooding and obstruction to flow. Although the pro-
blem of solid ice cover stability is widely recognized, a
coordinated effort to establish analytical criteria has not
been made. Solutions are devised independently at each site
based on trial-by-error operation or experience. No detailed
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observations or measurements have been collected which could
be used to guide and calibrate the analysis of solid ice
cover stability.
A preliminary analysis of the stability of a hinged contact
between the solid ice cover and shoreline was determined from
the basic concept proposed by Fonseca (1979). A number of
strips of ice, formed at the initial failure of the ice cover,
act as linking members in an inverted arch which connects the
solid central section of the ice cover to the shoreline. With
this support, shearing forces of flow and wind on the ice
cover are transferred to the riverbanks.
Criteria for critical water level variation for hinge stabi-
lity are based on a simplified analysis of the mechanical and
geometrical stability of hinge links throughout a water level
cycle.
The lack of quantitative data on solid ice cover stability
makes the simplified approach necessary at this time. A com-
prehensive field and laboratory program is recommended to
calibrate the criteria developed, observe hinge evolution and
guide further refinements to the analysis.
Development of analytical techniques to define the interaction
of ice covers with shorelines or structures is hampered by the
complexity and large number of parameters involved. Research
has progressed in response to specific technological require-
ments and the criteria developed rely heavily on empirical
coefficients to account for necessary analytical simplifications.
Thus, generalized criteria do not exist at present, and designs
cannot be prepared for many cases of ice/structure, or shoreline,
interaction. Recommendations for research on ice action under
varying flow velocity and water level are made.
l -INTRODUCTION
The winter regime of a river, lake or reservoir depends on
a complex interaction.of meteorological and hydrodynamic
conditions which determine the governing ice processes
throughout the water body. Flow velocity is the most
important parameter. Because of the difference in the nature
of the ice regime during identifiable stages in the life
cycle of an ice cover, three periods have been defined for
further discussion and presentation of equations governing
ice processes in each period. The three periods under
consideration are formation, midwinter and breakup.
Ice processes are further complicated when the flow velocity
and water levels vary sharply. In response to these changes,
the ice regime will reflect the variation of hydrodynamic
conditions. The mechanical ice regime will conform to the
most severe conditions encountered, while the thermal regime
will reflect the cumulative effect of hydrodynamic and
meteorological conditions.
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. 2
2 -GENERALIZED APPROACH
•ro ICE STUDIES
Standard procedures and criteria for evaluating the effects
of flow and level changes on ice covers do not exist.
This is a reflection of the state-of-the-art of ice engineering
and of the unique nature of each water resource project with
respect to its geographic and topographic settings, its purpose
and operating characteristics. It also reflects the complex
dependency of the ice regime on channel geometry and on the
prevailing hydrologic and climatic regimes.
Nevertheless, a broad program of study can be outlined with
modifications to suit the particular problems as
appropriate. Details of methodologies and criteria are
determined when both tile scope of study an0ecific questions
for an individual -project are defined and specific information
on basic data availability is at hand. In general, appropriate
methodologies can be adopted from the available arsenal of
hydrologic, hydraulic, climatological, heat transfer, system
simulation and ice mechanics techniques.
Subsequent sections of this report outline a number of the
fundamental heat transfer and ice mechanics techniques which
have proven useful in previous applications. They also
indicate the state-of-the-art in lesser developed areas of
ice engineering,those areas where considerable ingenuity and r versatility are required to find solutions for the specific
l _ _, problems in hand.
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A general program of study can-·be outlined under four major
activitiesr. ~These-~re-described-as follows.
(a) Ice Surveys
A field program oriented to the collection of ice
study data is conducted at reconnaissance level to
3
-obtain first hand knowledge of channel morphological
and hydraulic characteristics
-identify ice cover processes occurring ~each by reach
~ ,. .. .. ~·~ ~~~-... .: .
in ice cover development, including,channel closures,
ice front locations and velocities, ice generating
reach lengths and open water reaches, border ice
growth and sprin~ breakup sequence
-identify sections or reaches which will be of sp~cific
interest, e.g., locations vulnerable to flood damage
with changes in flow/ice cover/water level regimes, or
sections in which major changes in ice processes may
be anticipated with changes in the flow regime.
Fundamental to successful execution of this level of
survey is an observer with complete knowledge of the
mechanics of ice cover development under different hydro-
~ogic, hydraulic and climatic regimes.
Heat budget and mechanical stability analysis require
detailed surveys to collect
-water level/discharge data at key locations for open
water and ice cover conditions
-river channel cross sections and profiles through key
reaches
-water surface/ice cover profiles through key reaches,
with and without ice cover
ice and snow cover thickness/composition at key locations
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-water temperature profiles throuqh key reaches on
selected days combined with air temperature and
open-water velocity for heat balance calculations,
using therrnographs at key locations
-under ice velocities in key reaches or sections.
This level of survey clearly is more costly than the
reconnaissance survey. However, much of the foregoing
information (e.g., water level/discharge) is often
collected at key locations by various agencies for
other reasons and is usually adequate for studies of
a preliminary nature.
The insights gained from reconnaissance surveys are
particularly valuable in planning effective detailed
surveys to collect data for more detailed studies.
The nature of the specific questions being addressed
will dictate, of course, what items of data in the
foregoing list must be included in a survey program.
(b) Definition of Flow Regime
The single most important variable determining river
stages and the governing ice processes in various river
reaches is discharge. It is thus imperative to define
the flow regime that will prevail at a point of ,interes~
.. !A14 • , :.
prior to undertaking any ice mechanics analysis·.~· 'Y This
definition may be as simple as estimating maximum and/or
minimum values that might reasona~!;_!...., be expected (e.g.,
spring;.!-) breakup} or as complex ,.::;::ti~c~n~~~lc:>~ ... ~~er"/'-'·• "i..; , . i_ ...
hourly variation,..s of, discha~ge;'frc·m, a w'-ter resource )""I~ 79 _...,,,att ~ ....&. •'-"''9'~ ... -_f ~h-<.'f::...i,_-f\_ '/t.r'...-. / ..L-'>-·to·-" '-:'
'developmen~--(e.g. hycfroelectric power plant). Clearly,
proper accounting may have to be made of the natural hydro-
logic regime, its modification by existing and proposed
reservoirs and/or diversions, and even the influence of
externalities such as daily water supply_and electrical._
5
power d;mand patterns .. Routing of water releases from .-
,..~,,. (. ~ ·t. • ~.: ~ .. ;&,. .•• ,. ;(.• ~ ·l~-J .-." -
(storage-diversion points to downstream points.of
interest to account for intermediate channel storage
and friction effects on the discharge pattern under
both open water and ice cover conditions may also be
required.
(c) Definition of
Therma 1 Regime
Reservoirs providing considerable regulation of winter
flows have the secondary effect of modifying the thermal
regime in downstream channels. In the extreme, these
thermal effects will affect winter levels by completely
eliminating an ice .cover. In most cases, ho\-rever, the
:. ct_ !,.!·" .. &."\.
rate and timing o~ .. development is all that is affected
by alteration of the thermal regimes. Nevertheless, it
is sometimes desirable to identify and quantify these
changes as a basis for determining the potential effects
of proposed projects on such thinqs as winter ice roads,
the aquatic environment and incidence of ice~~enera:ted
fog.
Heat balance calculations must properly take into
account ~ocal hydraulic, hydrologic and climatic
conditions, e.g., whether the reach is a natural
channel or a reservoir, open water or ice covered, and
whether or not its thermal regime is influenced by
upstream storages. The calculations may be simply
limited to extreme values for design purposes. Conversely,
the nature of the problem may require a day-by-day
analysis~ The more detailed calculations would permit
. .ti,...Adefinition of the length of open-water reaches day-by-
-;.,..;-',.;:.~; ..:;;..-J·day and, in conjunction with ice mechanics analysis, the
tt,..,..:.~~·v··· length and location of both ice ~enerating and solidly
covered reaches.
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6
(d) Ice Mechanics Analys~s
With a clear understanding of the hydrologic and thermal
reqimes in hand, various ice mechanics analyses can be
undertaken. These analyses lead primarily to stage/
discharge relationships at the key point(s} of interest,
as determined by the prevailing hydrology, the hydraulics
of the channel and the governing ice processes. The
stage/discharge relationships, of course, define the
ranges and extreme water levels under ice conditions for
the range of hydrologic and thermal regimes anticipated,
permitting identification of potentially undesirable
effects and consequent remedial measures that may be
required to make a proposed project feasible.
Analysis will primarily co~prise leading edge (Froude
number) analysis and ~nternal stability analyses in
combination with backwater calculations for key reaches
or sections to determine which process governs water
levels for the range of hydrologic conditions. Sometimes
quite simple "indexes" will clearly show which process
dominates, while in other cases more detailed calcula-
tions may be required throughout a considerable length of
river. The level of detail will also be determined to
some extent by both the intent of the study {e.g. pre-
feasibiltiy or detailed design) and the availability of
basic data. At present, various digital computer programs
whichincorporate many of the fundamental techniques out-
lined in subsequent sections are in use to facilitate
such detailed analyses.
An~ent Acres study of the effect on the ice re9i~
at the Town of Peace River of the proposed Dunvegan power
.. :..-,::; .... ~
development on the Peace Rive~illustrates the generalized
approach to ice studies. The work proqrarn included
reconnaissance surveys of ice cover formation and evolution,
collection of field survey data and review of
existin~ reports on ice condition~ and thermal and -:::::-
wechanical icc regime analysis.
7
The ice reconnaissance surveys were conducted on the
Peace River between the Town of Peace River and the
Dunvegan dam site, 105 kmupstream of the town. Obser-
vation of ice cover formation indicated that leading
i
edge stability was the governing ice process in the
vicinity of Peace River. As the ice cover advanced
through the town, a critical section was reached.
Further progression could not occur until the Froude
number at that section was reduced by staging caused by
deposition of ice under the ice cover.
Subsequent analyses of leading edge and internal sta-
bility confirmed this conclusion. Available river
cross sections and meteorological data were used for
; input to a mathematical model which simulates the
,i _generation, upstream progression and subsequent thick-
,l ~" / . /
~/' •/ .' · ening of an ice cover./ The water levels at various
} P'r/\/ locations in the Town,.-of Peace River were determined
-~ /l -. ?.f! v· asing the model fo.r the appropriate range of operating ,··.~ '.;.: \~~f1 ~~~."discharges at the Dunvegan development. ).-Leading edge
I . . .-~I _.. \"' .
\.·· . . ·!' ... :"'rf stability at formation continued to be the governing ~' .·~ ~~· .//·,. "''"'-~ )~ .-.•· .t H l.. ice process at these higher', winter discharges. Thus, ,. ,.· r. I tc .. ~~ .. ~ L I the critical limit to discharge at formation corres-
r. .r ; .. ~ ... .)
' ~""... r . ;' ~·
ponding to maximum tolerable water levels at flood
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prone areas in the Town of Peace River was detet.:~~n7~.; . ..-"':""'-·~ _-;) L
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;::; balance calculations were. performe~/tc/''·~+~·~~s:·'_,~t~~~,;'-~::~ . .t~·~L:~. [
/ ~,, .. ·#·'--.·~~.-~,·~ I ..,.,.~4,. ...... • ~···.
L ...... ·f:~·-•change j;' n t.hermal regime. Elue-to-reservoir 'in-f-lows and ~ ~ ~ .. ~I' ............ ~ ( .;1J. .1~/ .. J further,.flow regulation~·Although the thermal effects
. r/ of the 1 reservoir on ice' generation and retention of ,. /
__ \ \J _i:~_;rom upstream r~.aches delayed the timing and
~ ;..-:'') !J:;.lf;V' f:;v<..r< ~"A-.......,.. .. .( A. .J.. ~ .... ~U-?f ';f. . 1. 1
_ __/ _/, .J • ' I .L' ,!;e/_ ~ ..... , ....... : t··U -..vr•o~o Y~ ....... ...,._ ..... ~ .... r_..-... ' , ,
.. ._. ... .... F ~ ...
• <( Q f. I c•tc t•«+4
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8
reduced the rate of cover advance throug~ the 1'm"n of
?eace River, the mechanical ice processes were not chan9ed.
. ~ ._.,: .... . .
The Dunvegan stud~.·concJ.usions were--based on a compre-
.. •./?:~ .•..• .,.:.--/
_hensiv~....canalysis o~ thermal and mechanical ice regimes.
The dominance of leading edge stability at the critical
river section was identified in the course of the
reconnaissance survey. The ability to interprete
survey observations of ice processes on local and
global scales is essential.
3 -FORMATION
·w:i.th the onset of subfreezing temperatures, ice cover
formation starts in low-velocity reaches and shoreline
9
areas where the velocity is less than 0.15 m/s in a manner
similar to that on lakes and ponds. Propagation of a
crystalline structure of ice begins at the water surface
and continues as heat is transferred from the water through
the ice to the cold air above.
If the velocity is between 0.15 and 0.3 m/s, surface turbulence
may prevent crystalline propagation of sheet ice. In that
case, the water becomes supercooled and frazil spicules
form. The individual spicules coalesce at the surface to
form "slush pans". Channel closure is achieved by a combination
of rapid border ice growth and, either accumulation of slush
pans at an existing ice cover, ice boom, bridge piers or
other obstructions to surface ice flo~ or bridging of slush
pans at contracting sections, river bends or reaches of
decelerating flow where the ice surface concentration, ice
cohesion (freezing of slush pans) and flow depth are favorable.
In river reaches where the velocity is greater than 0.30 m/s
and the Froude number is less than a critical value, the
cover will progress upstream by juxtaposition of ice against
the upstream edge of ice cover closures formed initially in
quieter river reaches. Border ice growth can assist in
closure of these reaches.
If the velocity is greater than 0.30 m/s and the Froude
number is greater than the critical value for upstream pro-
gression of the ice cover, incoming slush ice will be drawn
under the upstream edge of the cover and deposited down-
stream. The increased resistance to flow caused by thermal
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10
growth of the cover, deposited slush ice and thickening of
the cover to resist increasing internal stresses with the
advance of the ice front results in a steeper backwater
slope. At a lower Froude number resulting from lower
velocity and deeper flow depth, upstream progression of the
ice cover is again possible.
At very high velocities and Froude numbers much higher than
the critical value for ice cover progression, closure of the
section may not be accomplished by winter's end if the open
~ater area to gen~rate frazil ice is limited. Thermal border
ice growth and "buttering" of the border ice with slush ice
will be the governing ice processes.
3.1 -Thermal Processes
at Formation
An understanding of the thermal regime in a river and the
ability to calculate the cooling rate of the flow and rate
of frazil ice generation are necessary to predict the ice
regime in a river. Determination of the mechanical stability
of unconsolidated ice accumulations in the river may be
irrelevant if sustained supercooling required to generate
frazil ice is not achieved.
3.1.1-Heat Balance on
Open Water
Initially the cooling of water to below the freezing
point must be considered. Computation of the cooling
rate from an open-water surface can be based on
empirical relationships or a summation of the various
components of heat loss or heat gain in a heat budget
computation.
Empirical equations take the form
Q* = k <·r -T ) w a
where Q* = rate of heat transfer per unit area
(J/m2 day)
11
k = coefficient of heat transfer representa-
tive of site conditions (J/m2 day °C)
T = water temperature (°C) w
Ta = air temperature (°C}.
( 3 .1)
Although the coefficient k must account for all
aspects of heat transfer for the river reach under
consideration such as latitude and local wind speeds
and exposure, excellent results have been obtained
applying this simplified technique. Typical values
based on field observations are presented by
McLachlan ( 1926}, \iilliams (1959) and Marcotte (1975} •
McLachlan found that for all river reaches and lakes
along the St. Lawrence from Kingston to Montreal, a k
6 2 value of 1.9 x 10 J/m day °C is applicable. Marcotte
discusses the variation of the overall transfer
coefficient with meteorological parameters of wind
speed and air temperature.
With a heat budget approach, the major components of
heat gain or loss, convection, evaporation, radiation
and heat exchange from precipitation, are summed to
give the net heat transfer rate. Discussion and
formulation of heat budget calculations are presented
in Newbury (1966), Carstens (1970), Raphael (1962) and
Michel (1971) • Those of Raphael and Michel are parti-
cularly comprehensive and give a thorough understanding
of the subject. The basic equations developed by
Michel provide a computation method suitable for
engineering applications and are presented here.
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TABI~R 3. l
[ INSOLATION COEFFICIENT VALUES,
C5 , FROl1 MICHEL ~1971) (units
converted to J/rn day)
[
Latitude
[ 40°N 50°N 60°N 70°N
Sept. 1 39,730 31,070 22,400 14,190
[ Sept. 15 36,090 27,340 18,300 10,470
Oct. 1 31,820 22,520 13,980 6,660
Oct. 15 26,800 18,210 9,800 3,680
[ Nov. 1 22,520 13,610 6,150 1,260
Nov. 15 19,300 10,010 3,730 170
[ Dec. 1 15,570 7,750 2,300 ::Q
Dec. 15 14,740 6,870 1,840 ::Q
Jan. 1 14,860 6,950 1,800 ::Q
[ Jan. 15 16,370 8,250 2,510 ::0
Feb. 1 21,480 11,100 4,440 630
[ Feb. 15 23,150 15,490 7,410 2,300
Mar. 1 29,060 20,010 11,260 4,520
!'-tar. 15 34,960 24,740 15,700 8,120
[ Apr. 1 39,730 30,480 21,350 12,310
Apr. 15 42,660 34,160 25,080 16,200
[ May 1 44,760 37,300 28,600 19,550
May 15 46,010 39,270 31,280 22,480
b
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13
The heat loss on a daily basis is given by
Q* =-5.87 u15 [32-(0.66 + 0.33 e ~) {32 + 1.8 Ta)] +
640 Cs (0.25 + 0.75 m) -3.44 (33-1.8 T) -130m
N a N
-9,000 i + 20,260 q s +loo q tJ.T + Q s 9 9 . ex
B
with E;. = l if T a>-18 and E;. = 0 if Ta ~-18
where Q* = rate of total heat transfer per unit
2 surface area (J/ro day)
U15 = wind velocity 15 m above the water
surface (m/s)
e = relative humidity of air
T = air temperature 15 ft above the water a
surface (°C)
(3.2)
Cs = insolation coefficient (Table 3.1) which can
be set to zero for overniqht heat loss
m
N
q
s
h.T g
calculation (J/m2 day) -
= fraction of hours of clear sky during day-
light hours
= water equivalent of snow precipitation
(m/day)
= river discharge per unit width (m 3 /s m)
= slope of flow energy gradient
= groundwater di:,charge per unit river
3 length (m /s m)
= groundwater temperature difference from
water (OC}
= river width (m)
= external artificial addition of heat
(J/m2 day) •
The data required for this formula can be readily
obtained from climatic normals published for different
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111
regions in Canada.
From the determined rate of heat transfer, the tempera-
ture change along a river reach can be calculated from
1
86,400 pw C p
Q* dt
Cl
where T w = water temperature at location t (oC)
To = water temperature at location to ( oc,
t, to = river location along flow axis (m
ow = specific mass of water (kq/m 3 )
cP = specific heat of water (J/kg oc) •
If the rate of heat transfer and river width do not
vary greatly along the reach this becomes
T - T =1.157 x 10-a Q* (.t-.to) w 0
q
For a given temperature change, the length of river
reach along which cooling must take place can be
determined from
q dT
'It I
which for uniform channel width and constant heat
transfer can be simplified to
t -.t
0
= 8. 64 x 10 1 q (Tw -To)
Q*
( 3. 3a)
(3.3b)
(3.4a)
(3.4b)
15
Dingman, Weeks and Yen (1968) present a similar formu-
lation based on <.1 heat budget approach to predict the
length of ice-free river reach downstream from a thermal
inflow site. If thermal pollution input, or a deep-
water reservoir source is considered, the length of the
equilibrium open water reach downstream is given by
X -8.64 10 3 Pw c q J:f dT = X E w
where
k . Q*
0
X = length of open water reach (m)
Tf = freezing temperature of water ( oc)
To = initial water temperature ( oc)
k = coefficient of heat transfer representative
of site conditions {J/m2 day °C)
( 3. 5)
and other variables with units as previously defined.
If the rate of heat loss, Q*, is approximated by
equation 3.5 becomes
x = -8.64 x 10 4 pw cp q
k
ln
(3.1)
(3.5a)
for constant air temperature and coefficient of heat
transfer. Rearranging equation 3.5a and taking antilogs
gives
T - T = T -r f a o a e
4 -(k/8.64 X 10 Pw Cp q) X
Substituting the first two terms of an exponential
series for the right hand side of equation 3.5b gives
or
(3.5b)
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X = -8.64 X 10 4 p C q w p
k lT - T l
f OJ T -T -o a
If Q* in equation 3.4b is taken as the average heat
transfer along the reach, between the initial section
and upstream limit to the cover, given by
Q* = k [-T~o-=;-T~f--Tal
equation 3.4b becomes
~ -2. = 8. 6 4 X 10 J p C q 0 w p
k
( 3. 5d)
{ 3. 4c)
The difference in these equations is due to the exclusion
of Q* from integration in Michel's approach.
3.1.2 -Frazil Ice Generation
Frazil ice production occurs in turbulent water when
the disturbance of the flow surface prevents formation
of a thermal sheet of ice. As pointed out by Tsang
(1976) , supercooling of water below the freezing point
is necessary in order that the release of latent heat
does not immediately heat the water system to above
freezing temperature. Michel (1971) states that maxi-
mum supercooling observed in nature is 0.06°C with a
normal range of O.Ol°C to 0.03°C.
Ice formed in this manner is part of a continuously
evolving process. Small frazil discoids are formed
uniformly throughout the turbulent flow. The invisible
discoids grow into needles or spicules which rise to
the surface of the flow,coalescing into frazil slush
17
clusters. Frazil is only active in the discoid state
during which it adheres readily to other ice particles
or thermally conductive objects. Tsang and t.Uchel give
clear descriptions of frazil evolution. Carstens (1970)
discusses the effects of rate of cooling and turbulence
on laboratory generation of frazil ice and presents
results of field observation programs in Norway.
Carstens concluded that the degree of supercooling is
a function of the rate of heat loss from the free water
. surface, and of the intensity of flow turbulence trans-
porting heat to the water surface. Field experience
indicated an upper limit to surface velocity of 0.6 m/s
for ice covers to form due to flow turbulence.
Anchor ice has been comparatively less studied than
frazil ice. It has been shown that frazil and anchor
ice have similar origins in supercooled turbulent flows,
anchor ice forming at the same time as frazil runs, but
it remains to determine the factors controlling the pre-
domenance of the two ice formations under a given super-
cooling condition.
After the water becomes supercooled, continued heat loss
results in a change of state or frazil production. The
rate of frazil production is given by Michel as
Q* A = o/w pi~
where Q. =volume rate of ice production (m 3 /day) l.
Q* ~heat transfer rate to atmosphere (J/m2 day)
Ao;w== supercooled open water generation area (m 2 )
pi -=specific mass of ice (kg/m3 )
Lf = latent heat of fusion of ice (J/kg} •
( 3. 6)
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The open water area must be adjusted for the effe~t on
width of the border ice growth variation along the
reach and, as noted by Newbury, reduction of ice genera-
tion area due to insulation of the water surface by
accumulating pans of slush ice travelling downstream
and causing the cover to progress upstream in the reach.
3.1.3 -Border Ice Growth
In river reaches where a thermal cover cannot grow
uniformly across the entire channel width, the first
ice to form is shore, or border, ice in areas of low
velocity. This ice grows rapidly from the banks and
islands toward the center of the channels in a solid
sheet.
As the edge of the sheet extends into the more turbu-
lent part of the flow, its progress decreases markedly,
but still continues to advance. The growth of ice
does not require water temperature to be below freezing.
The rate of advance depends on the relative rates of
heat exchange between the ice sheet edge and the tur-
bulent water and between the ic~ sheet artd the atmos-
phere.
The quantity of heat given to the ice boundary by the
warm water is given by
y = 0
where kw = thermal conductivity of water (J/m day °C)
[:;w] y = 0
= water temperature gradient at the
ice boundary along the top water
surface layer (°C/m).
(y -direction)
(3.7a)
19
The heat taken out by convection at the ice boundary
can be similarily expressed as
Q* = k a a
= 0
where k = thermal conductivity of air (J/m day °C) a
(3.7b)
z = 0
= air temperature gradient at the ice
boundary normal to the ice sheet (°C/m).
(z -direction)
Border ice growth occurs if Q*a is greater than Q*w·
Because of the values of k and ~ , the 'air tempera-a w
ture gradient must be 25 times greater than the water
temperature gradient for ice growth. This can be
satisfied for winter wind velocities and air temperature
differences below freezing compared with water velocities
and water temperature differences above freezing {Michel 1971).
Difficulty in obtaining practical measurements of these
temperature gradients which vary appreciably with time
makes it necessary to resort to empirical relationships
to determine border ice growth.
Newbury (1966) developed an empirical border ice growth
~quation based on observations at several river sections
on the Nelson River incorporating an assumption for the
"tractive force" nature of the problem. Newbury
observed that border ice growth may be interrupted by
abrasion of passing slush ice or solid floes in the
open channel. Conversely, border ice growth may be
accelerated by "buttering" of slush ice onto the border
ice edge in layers. The cohesive, or adhesive, strength
of the slush ice and rate of heat exchange determines
the thickness of slush layers adding to the border ice
width.
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'fhe rate of border ice growth across the width for the
Nelson River is given by Newbury as
-5 B. = 2.68 X 10 ~
where B.
~ = total
Nb = number
1:Q* = sum of
border ice growth (m)
of boundaries
net heat transferred
m = an adhesive parameter.
The adhesive parameter was determined as
m = a (_1_0 ___ 7_6_A_S-.) b
where a and b are empirical coefficients
A = cross-sectional flow area Cm 2 )
S = water surface slope (m/m) .
(J/m2)
For the Nelson River a= 5 and b = 2.7 and
( 3. 8)
(3.8a)
-4
Bi = 1.342 x 10 -Nb 1:Q* ().Oc)
(10.76 AS)2 •7
Rearranging,
1::Q* (10.76 AS)2 •7 B. = ~
-4 1. 342 X 10 Nb
In the absence of field data for a given river, this
can be compared with experienced rates of Q* to deter-
mine the time required for closure of a given width
of river reach by border ice growth.
(3.9)
21
In spite of the number of parameters influencing border
ice growth, each of which can vary with time, a simple
observation program to relate cumulative degree-days of
freezing to border ice growth width gives a satisfactory
indication of border ice growth based only on readily
available temperature data. This is due to the relatively
slow thermal response of the border ice cover to daily
fluctuations in hydrologic or meteorological conditions.
3.1.4 -Channel Closures ----------------
In high velocity river reaches where channel closure
cannot be achieved by formation of a thermal cover,
supercooling will occur and frazil ice will be generated.
In reaches where the surface coverage of moving slush
reaches 100 percent (Ni = 1.0}, the blanket of slush
floes may be compressed at channel constrictions
forming a continuous ice-bridge between shore-fast ice
boundaries. Whereas 100 percent surface coverage is
necessary for such bridging, it is not sufficient. For
example, Michel (1971) quotes Dutch experience on two
rivers for which the required conditions were
-surface
-surface
-average
concentration Ni ~1.0 for
velocity V <0.5 m/s for s
air temperature <-9°C.
5 to 8 hours
5 to 8 hours
The coverage, N . , can only "exceed" 1. 0 by compaction
l.
of the slush to form a more dense blanket or by
thickening of the blanket to increase surface discharge
when the product of width and surface velocity is
reduced. The rate of surface ice transport is given
by the continuity equation
{3.10)
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~-
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where Qs = surface ice transport rate (m 3 /s}
N. 1 = surface ice concentration (tenths)
vs = surface velocity (m/s)
ts = slush blanket thickness (m)
B = channel width (m)
3 3 c = porosity of slush accumulation (m /m ).
Thus, for a decelerating flow or converging section
without a corresponding increase in surface velocity
(i~e. simultaneous narrowing and deepening) and 100
percent surface coverage, the slush blanket will
thicken or compress (Ni >1.0).
Velocities lower than 0.5 m/s correspond to a critical
limit to surface shear on the underside of the cover,
and therefore a lower requirement for resisting forces
in the ice to arrest and hold the blanket.
The duration of 5 to 8 hours and average temperatures .
less than -9°C represents an opportunity to develop
sufficient cohesion in the upper slush layer
to provide an average cohesive strength throughout
the entire depth of the blanket which is capable of
resisting the drag and body forces acting to maintain
slush transport.
All of the foregoing elements are represented in the
stability criterion formulated by Pariset.et al, (1961)
which can be applied to slush blanket stability if the
following differences are considered.
-The channel width becomes the width of mpbile slush
blanket between border ice growth rather than the
width between river banks.
There is no downstream barrier for the moving slush
blanket to "pack" against to develop internal buoyant
resistance by thickening of the blanket.
23
Pariset's stability criterion is developed in the
following section on Mechanical Stability at Forma-
tion, and the application to channel closure is
discussed.
3.2 -Mechanical Stability of
an Ice Cover at Formation
3.2.1. -Leading Edge Stability
Upstream progression of incoming ice occurs at an
obstruction to the flow or when the conveyance at
the channel is reduced below the surface inflow of
mobile ice. This can occur at a variety of man-made
or natural obstacles such as bridges, dams or ice
booms, and bends, channel constrictions or thermal
ice covers in lakes or low-velocity river reaches.
When the moving slush blanket reaches the downstream
end of a frazil generating reach, it can progress up-
stream with all ice forming a stable leading edge if
conditions are favorable. If conditions for stable
ice accumulation are not fulfilled, the cover can
still progress with difficulty if the inflow of
slush ice exceeds the rate of transport of ice under
the cover from the leading edge. The rate of advance
of leading edge of the cover in the latter case will
depend on the net rate of ice inflow to the leading
edge of the ice cover.
Initial studies by McLachlan (1926) and a later study
by Cartier (1959) sought to fix a limiting velocity
criterion for which a cover could advance. A wide
range of limiting velocities was obtained for different
ice and river characteristics indicating that further
refinement was required. Kivisild proposed a limiting
Froude number for upstream progression of a packed
ice cover defined as
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Fr = V = 0.08 cr cr
where Vcr= mean flow velocity (m/s)
g
H
= acceleration of gravity (m/s 2 }
= mean flow depth (m)
(3.11)
This criterion has proved to be successful for a wide
variety of hydraulic, ice and meteorological conditions.
Subsequent studies by Pariset, et al (1961), and Michel
(1965) served to confirm the simple criterion of Kivisild.
The criteria of Pariset, et al, and Michel differ only in
the inclusiou of a porosity term in Michel's derivation.
Frazil slush and pans can progress upstream at a
determined thickness as the ice underturns and piles
up under the cover. The limiting Froude number for
incipientinstability of the leading edge of the thick-
ened cover (submersion of the cover and transport of
ice under the cover) is defined as
Frcr = ~ = (p -p')
p
{1 -E) t (1 -!) (3.12)
H H
where
fiH
Vcr=
g =
H =
p =
p' =
t =
£ =
critical upstream mean velocity of flow (m/s)
acceleration of gravity (m/s 2 )
upstream flow depth (m)
water density (kg/m3 ')
3 ice density (kg/m )
equilibrium ice cover thickness (m}
porosity of ~ccumulated ice cover i.e.
ratio of volume of voids filled with
water to the total volume of the ice
accumulation
25
At a relative ice cover thickness to flow depth ratio
t/fl, of 1/3, the restriction to flow under the cover
and upsetting forces on the icc become critical. The
formulation can no longer apply and further thickeninct
of the ice cover leading edge is not possible (Pariset
and Hausser, 1961).
For the observed li.niting ratio of cover thickness
to flow depth of 1/3 and water and ice densities of
1,000 and 920 kg/m3 , respectively, this reduces to
Fr = .154 j 1 -E ( 3 .12a) cr
The limiting case of a solid floe {£ = 0) is 0.154
which has been observed in laboratory studies. The
more typical case encountered in field conditions
is a porosity of 0.72 for a packed cover of frazil
and small floes, hence the typical critical Froude
number of 0.08 reported by Kivisild.
Numerous studies have been conducted to determine the
stability of individual ice blocks [Ashton (1974),
Larsen (1975), Chee and Haggag (1978), Uzuner and
Kennedy (1972)]. From this, the stability of the
leading edge of the cover is inferred. However,
Michel (1971) points out that individual block
stability is not generally a limiting condition as
incoming frazil and slush and pans may underturn and
pack at the leading edge until it is able to progress
upstream at a determined thickness. The minimum
thickness of the leading edge of the cover considering
forces actina on the leading edge can be estimated
from the following derivation of Pariset, et al,
[
[
[
[
[
[
[
[
c
[
[
[
[
[
[
[
[
I'
L
[
[
[
[
L
v 2
u
where Vu = flow velocity under the edge of the ice
cover (m/s)
~rom continuity, this becomes,
," . .;;.+_.:/ __ _
....
2g
r. J ~ ~ . / r,.~l., , t -/It_
where V = flow velocity upstream of the cover (m/s).
An iterative solution i~ required for covers which
are relatively thick compared to flow depth.
The various formulas for individual block stability
are not practical or applicable for the stability of.
(3.13)
(3.13a)
a packed cover comprised of an indeterminable distri-
bution of frazil pan and ice block fragment sizes.
Field experience has demonstrated that, for engineering
purposes, the limiting Froude number of 0.08 and the
minimum thickness for cover progression define leading
edge conditions.
3.2.2 -Internal Stability
at Formation
As the ice cover progresses upstream, the internal
stress increases as the hydrodynamic shearing force
of the flow under the cover, the shearing stress of
wind on the cover and weight of ice along the slope
of the icejwater interface are added to the hydro-
dynamic thrust on the leading edge of the cover. The
vLr1f~
I~
27
increasing forces on the cover as it lengthens must
be balanced by the internal resistance of the cover
created by the buoyancy of the accumulated ice and
the resistance of the banks. Bank reaction is com-
prised of an ice-over-ice frictional term related to
the internal stress transferred to the banks and a
cohesive {freezing) term. If the internal stress
increases beyond the internal resistance of the cover,
the cover will shove or thicken until the resistance
increases sufficiently.
Relationships to equate forces acting on an ice cover
and resisting forces of the cover have been derived
by Pariset, et al (1961), Michel {196~, and Uzuner
and Kennedy (1976). These derivations vary in the
treatment of the resist:ing forces leading to different
forms of coefficients for internal friction and cohesion.
Experience in field application and verification of
coefficient values favour the formulation of Pariset,
et al. The
{[~~ BV 2
C2H2
where B =
.V =
• J 1 .• c =
'-H =
T =
t =
l.1 =
general relationship is given by r 3 + lJ E. ~ t f R \1-~)
(1 -E. ) + w p H
p p H pgH 2 (1 + iL_!) p H/
channel top width (m)
mean flow velocity under open water
conditions (m/s)
(3.14)
Chezy roughness coefficienL, assumed equal fo~
river bed and underside or ice cover {m 1 1 2;s)
o~en-water flow depth (m)
cohesive strength of ice (Pa)
ice cover thickness (m)
coefficient related to ice-over-ice friction
and internal res~stance of the cover.
[
[
[
[
[
[
[
L
[
[
[
[
[
[ 28
[
[
[
[
[
[
[
E
[
E
[
b
.--' I L ..
[
[
L
[
gravitational acceleration 2 g = (m/s )
n' = density of ice (kg/m3 )
n = density of water (kg/m 3 )
fw = wind drag on the upper surface of the cover (Pa)
(positive if acting in the downstream direction)
Pariset, et al, reported values of Manning's n-value
for the composite roughness on the St. Lawrence of 0.06
to 0.05 at formation and 0.04 to 0.03 at later stages
of a stable cover. The Chezy roughness parameter can
be related to Manning's n-value by
c = 1 l1i 1/6
n.
~
where ~ = the hydraulic radius of the ice-covered
flow (m) •
It should be noted that the open-water velocity and
{3.15)
flow depth are those that would exist at the same hydro-
static depth as the flow with an ice cover. Thus, the
open-water depth must consider staging due to the
presence of the ice cover downstream. The ratio of
winter open-water flow depth to summer open-water flow
depth obtained from Manning's equation with constant
discharge equals
H - = 1.32
Ho
(3.16)
where Ho = summer open-water flow depth (m)
nc = composite roughness with ice cover
(see Section 3. 2. 6)
nb = channel bed roughness.
29
For thin covers ( t< ··11 ) with ice roughness near that
0
of the river bed, a 30 percent increase in stage can
be anticipated. If staging and flooding are not
important, ice cover stability can be determined
conservatively from open-water hydraulic parameter
values at the appropriate discharge.
The wind drag can be estimated from the Von Karman
equation by
4! = u2 Pa I {5.75 log (W/K.) + 8.5)2
~w 1
where u = wind speed (m/s)
Pa = air density (kg/m3 )
w = height of wind velocity, U, above ice
surface (m)
K. = roughness height of ice (m) •
1
For a wind speed height of 10 m, ice roughness height
of 0.4 m and air density of 1.28 kg/m3 recommended by
Michel, this reduces to
Solution of the stability equation for the stable ice
thickness, ti, yields
(3.17)
(3.19a)
2T
-~
pgll
-p') [ DV2
-P-C2112
(1 + p • t i ) f lll-
--prr---~-( 1 -fl' ti )3 PCJII2f (3.13)
P II ....:
2 II fl' ( l _ ;r· p .
(I' \ -· " : I
An iterative solution to the above equation with
assumed values of ti is required. At this stage of
complexity, a computer solution of the equation is
recommended. Manual approximations are available,
however, as will be outlined in a following paragraph.
[
[
[
[
[
[
[
[
[
[
[
[
[
[
l '
L_,
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
l
[
Solution of the stability equation has been plotted
in Figure 3.1. For a given value of cohesion, the
area under the corresponding curve, defined by the
pa'""ameter 2-r/pgll, represents all stable combinations
of relative ice thickness, t/H, and the internal
stability ratio, Bv 2 ;c2 H2 • For cover progression
at formation , the limit to the relative thickness
at the leading edge is t/H<l/3. (At a greater thick-
ness, basic assumptions for leading edge stability
become invalid.) As the cover progresses up-
stream, a minimum thickness at the lower, or left-
hand t/H, limit at the appropriate 2t/pgH curve is
required for internal stability corresponding to
the ratio av 2;c2 n2 . Thus, the ice cover thickness
for progression of the leading edge can be less than
or greater than the eventual internal stability
. 2 2 2 thickness, indicated by the values of BV /C ll and
t/H, falling above or below the appropriate 2T/pgH
curve, respectively.
Pariset, Hausser and Gagnon (l966) were unable to
separate the product 1t. They reported values of tt
from 1,100· to 1,300 N/m and a value of lJ equal to 1.28.
Lavender (1973)* was able to reanalyse the data of
Pariset, Hausser and Gagnon (1966) and separate T and ,
·,'
t. This method showed that T could vary frorr 0 to 4,070.
t· :~·itt , .... , . .. i'\.. r~ ·_, '·;.·~ ·'-')/l·,:· .. : .. ,·~,
Pa. A value of 3,110 Pa ·is recommended: 'for typical· ~ce
cover formation conditions. The range of l.J for the
data ranged from 1.39 to 1.60. Thus, the value reported
by Pariset et al is conservative. A minimum design
value of 1.39 is recommended for design, and is used in
this report in Figure 3.1.
*~npublished notes of Acres Employee
Development Committee Seminar, January 1973
.., .
...-..... ..,
I
0
X
~
N I":. om
N
(.)
N>t m Nr.
(.)
a: w
t-w
~
<I a:
~
(!) z
0
<I
0
_J
a:
U.l > 0
(.)
I.&J
(.)
50or---~----,---------~------------------------------~ 2 T
pq li
l --3.001·---·+
! t '
.
·---!--·-----··~~-~
10
0.5
0.1
0.050~----~--~----~----L---~----~~--~--~----~----J
0.2 0.4 0.6 0.8 1.0
COVER THICKNESS I HYDROSTATIC DEPTH-t/H
CANADIAN ELECTRICAL ASSOCIATION
STUDY OF ICE COVERS SUB.JECT TO VARYING WATER LEVEL
....... ICE COVER·· Sf..ABILITY DIAGRAM
FIG. :!.1
[iJ
[
[
[
[
[
[
[
[
[
[
[
L
[
[ 32
I L
[.
[
[
r L
[
[
[
[
[
[
L
r
L
[
[
r _;,
[
For a given ice cover cohesion, the maximum value of
the ice cover loading parameter, sv2 ;c2 H2 , occurs at·
a relative thickness, t/H, from 0.2 to 0.4. Above
this relative thickness, the increased stress imposed
on the cover exceeds the increase in stability due to
increased cover thickness and the upper limit to
2 1 2 2 . h . . I BV C H decreases w1t 1ncreas1ng t H.
If the sv 2 ;c2 H2 value is higher than the maximum point
on the 2T/pgH curve, cover stability.cannot be achieved.
Thickening of the cover by internal collapse, and trans-
port and deposition of ice downstream causes staging
which reduces the value of Bv2 ;c2 H2 sufficiently for
cover stability.
Deposition of ice transported under the cover from
upstream can cause instability if the relative ice
thickness increases beyond the upper, or right~hand,
t/H limit to the 2T/pgH curve. In this case as well,
an increase in depth resulting from staging under the
thicker cover serves to reduce stresses in the cover.
tf some reasonable assumptions are adopted, the solu-
tion for internal stability is greatly simplified.
For design purposes, it is generally appropriate and
conservative to assume a cohesionless cover, i.e.,
T = 0. Also, it is generally desired to design the
channel so that the ice cover is relatively thin, i.e.,
t<<H, and the equation can then be simplified to
~ [1 ~ £:][!].2
+ fw B p p H ---pg
Variable wind direction makes it difficult to account
for the effect of wind drag because fw can contribute
(3.14a)
.13
to instability or Stability of the cover. For that
reason, wind stress is normally ianored for ice cover
stability in rivers. However, for the desian of ice
boom installations in wide river reaches and deter-
mination of river closure by slush bridgin9, wind
stress-can play a majo:.: role and should be considered.
With the precedinq assumptions (T = 0, t<<H, fw = 0),
the stable ice cover thickness, ti, is given by
ti , 3.12 ~ (3.18a)
.t for the substituted values
\/)~1i_ . p' /p = 0.92
1J (t.·}\ ;. " 1..1 = 1. 39 (estimated minir.mrn value obtained by
o· ( Lavender from Pariset's field data).
Pariset, et al, (1961), showed that rivers could be
classified as "wide" or "narrow" depending on the
governing stability criterion for stable ice cover
thickness. If leading edge stability requires a
greater thickness than internal stability, the river
is "narrow". Conversely, if internal stability
requi~es a greater thickness of cover than does
leading edge stability, the river is "wide". To
determine the transition between wide and narrow
rivers, leading edge stability ice thickness is equated
to the approximate solution ~o internal stability ice
thick~ess. Thus, for a wide river
2g
which yields
<38.94
< t.
l.
(3.19)
[
[
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[
[
r
[
[
[
r u
r~
L
[
[ 34
I
[,
[
[
[
[
r l~
[
I L...;;:
[
[
[
[
I '
I L.
[
[
l
r ,
L
Thus, a river is
"wide" if
"narrow"
v2c2
-B--
if v 2 c 2
-B-
< 24
> 24
This ratio is a useful indicator of ice conditions in
the reach. If an ice cover can progress through the
reach, further calculation for internal stability
and ice cover thickness is not necessary if the river
is narrow.
3.2.3 -River Classification
by Governing Ice Process
(3.19a)
(3.19b)
The combinations of leading edge and internal
stability criteria yield the following classifications
of river reaches. The leading edge criterion of
Michel (1971) has been reformulated by substitution of
the Chezy flow relationship as
Fr cr = c j ~ s = 0 .154 j1 ...;. £
I gH
where S = slope of the hydraulic grade line (equal to
bed slope for uniform flow).
For a wide channel
and
cs 1 1 2 = o. 682 /{1 ... E)
(3.12b)
(3.12c)
35
If the greater value of porosity recommended by Michel
of 0.72 is used, this becomes
cs 1 /2 = 0.361
The ice cover can progress upstream without staging
if the Froude number is less than critical and the
product, cs 1 12 , is less than 0.361. This yields the
following Table 3.2 when combined with the internal
stability criterion in equations 3.19a and 3.19b.
3.2.4 -Application to
River Closure
As discussed in Section 3.1.4, closure conditions
{3.12d)
for a river reach covered with a moving slush blanket
can be predicted using the Pariset stability equation.
Because no internal resistance can be developed in the
moving ice, the internal stress must reduce in the
downstream direction within the cover until it becomes
zero. At that point, cohesion at the river boundaries
(border ice or river banks) can support the cover.
The modified formula for this condition is
p' t \ 3 B) (1 -BV 2 ( 2t t fw
s.
s + p H ) (3.14b)
C2H2 = H2 H2 (1 pI. t \ pg pg s\ + p H )
where ts =thickness of the moving slush blanket (m).
The combination of factors which will then contribute
to bridging are
(a) a reduction in discharge to reduce drag and body
forces in the blanket
(b) a wind blowing upstream to help balance the
downstream forces
[
[
['
[
[
[
r
[
[
[
[
[
L
[
[
TABLE 3. 2
F-'-
t',_. J
RIVER CLASSIFICATIO~
BY ICE COVER PROCESS
Leading Edge
Stability
(open water)
Frcr<0.154 [1 -e
cs 112 <0.682 ji -e
(stable without
staging)
Fr >0.154;~.-€-cr
cs 1 12 >0.682/l -e
(stability requires
staging)
~
l. )
Internal Stability
I~
CLASS A -cover forms easily
by juxtaposition without
staging and without
subsequent shoving and
packing.
CLASS C -cover can. advance
only w~th staging but has
sufficient thickness to
maintain internal stability
{wide)
CLASS B -cover advances
easily by juxtaposition
and subsequently shoves
and packs to maintain
internal equilibrium
CLASS D -cover advances
w1th difficulty only after
staging and subsequently
shoves and packs to main-
tain internal equilibrium
37
(c) a reduction in blanket width, B, by border ice
growth
(d) increased heat transfer from the blanket to
increase cohesion, caused by low temperature
and high wind
(e) increased slush blanket thickness, ts' at a
surface contraction (N. >1.0).
~
3.2.5 -Accumulation of
Slush and Frazil
Frazil ice generation depends on the rate of heat
loss over a supercooled open-water area, as discussed
in Section 3.1.2. The rate of ice production per
unit area in an open-water reach is given by
where q. ~ = rate of ice production per unit area
(m 3 /s m2 )
Q* = rate of heat transfer from open-water
surface (J/m2 s)
p. = specific weight of ice (kg/m3 )
~
Lf =latent heat of fusion of ice (J/kg).
(3.20)
As stated previously, the determined surface area for
generation of frazil must consider the presence of
border ice growth and slush pans. If the ice is
assumed to coalesce into pans at the surface having
a typical thickness, ts' the surface concentration of
ice, Ni' will increase from zero at the upstream end
of th.e supercooled reach to a limiting value of 1. 0,
at which point frazil ice generation will effectively
cease.
[
[
[
[
[
[
[
[
f ~
I L:J
[
c
[
[
I"
I
l -
I
I ~
I l __ .,
[
[
[
[
[
r
l~
[
[
[
L:
The surface ice concentration at any point x downstream
in the reach is given by*
Ni = 1 -exp ~ < of*!~ ~.l~s xJ
where N. l.
Q*
X
p •
l.
Lf
e:
ts
v s
The total
= surface area concentration of slush pans
(m2 /m2)
=
=
=
=
=
=
=
rate of surface heat transfer from open
water (J/m2 s °C)
distance downstream from start of super-
coolinq (m)
specific weight of ice (kg/m3 )
latent heat of fusion of ice (J/kg)
porosity of slush accumulation (m3 /m3)
thickness of slush accumulation (m)
surface flow velocity (m/s) .
rate of frazil ice production along the
reach is given by the integral along the river reach
(3.21)
of open-water area multiplied by the rate of open-water
ice production per unit area. The integral for total
frazil ice production in the river reach upstream of
any point along the reachgives
which is, of course, equal to the rate of surface ice
flow at that point in the river reach.
*Complete derivation for Dunvegan Power
Project to be published. Report in preparation.
39
The frazil ice produced alonq the reach in this manner
coalesces and rises to the surface to form slush pans.
These pans occupy a fraction of the total surface area
equal to, N., at a thickness, t , and porosity, e:. As
1 s
the pans move downstream, they thicken through thermal
growth nucleated by the frazil crystals. Employing the
algorithm of Michel and Berenger (1975)for ice growth
in broken ship tracks where part of the surface is
covered with ice, the thickness of ice grown in the
water filled voids is given by
E
1 -e:
f t + l s ~ '...!:..__ r;;;;; -t l } L294 ;-v; s
where 6tt = thickness of thermal ice grown in water
voids (m)
If
If
a
294
6tt
= thermal coefficient for Stefan Equation
of thermal ice growth (see Section 4.1.2)
6T = air temperature difference from water
freezing temperature (°C)
x' = distance travelled by slush pan (m)
~ = coefficient determined by thermal growth
relative to slush pan thickness.
N<t ~ = 1 then ... s ' s
= e: a. /ATV:• (1 -e:) 294
ap. ~ = e: then 294 Vs >ts'
N 6tt = e:2 a. + e: t •
(1 -e:) 294 s
s
(3.23)
(3.23a)
( 3. 23b)
[
[
r L
[
[
[
[
[
[
[
[
[
L
[
I L~
L~
I
l~
[
[
[
[
[
[
[
[
[
l
40
The total volume of additional thermal.ice produced
along the reach to a point x is the sum or integral
of thermal ice production in the open-water voids of
slush pans travelling downstream, or thickness times
area. It is assumed that, from the time frazil ice
is generated at some point in the reach until it
reaches the point x, thermal growth will take place
in the open-water voids. The time to coalesce at the
surface has been ignored. This integral is given by*
Qt = Q* B £ J: yi Lf {1 -£) t X s
-ex (1 - e ) 6tt dx (3.24)
where c = Q*
t v s yi Lf (1 -E) s
As there is no definite integral solution, a computer
solution of a series expansion is required.
The total volume of ice produced along an open-water
reach is given by the summation of frazil ice production
in open water and subsequent thermal ice growth, given by
= Q. + Qt ~ . {3.25)
If the moving blanket of slush ice and thermal growth
reaches a river closure where the Froude number is less
than that critical for progression, the ice may pack
against the closure in part or in total. The rate of
upstream progression will be determined by the net
rate of ice inflow to the leading edge of the ice cover.
The volume of ice underturning and carried away from
the leading edge is assumed to be equal to the under-ice
transport capacity of the flow. This transport of ice
*Complete derivat~on for Dunvegan Power Project
to De published. Report in preparation.
41
away from the leading edge was treated as an inverted
sediment transport problem by Pariset and Hausser
(1961). For the hydraulic radius equal to half the
mean depth, the adapted Meyer-Peter formula is
3,281 v 2
u
c2
2/3 = 12. 3 d
5
+ 0. 84 qu (3.26)
where Vu = mean velocity under the ice cover (m/s)
C = Chezy roughness coefficient for water
passage Cm 1 /2 /s)
ds = characteristic dimension of ice fragments
taken as floe thickness or slush blanket
thickness (m)
qu = ice discharge per unit width under cover
weighed under water with apparent density
0.08 (N/s m).
Rearranging and substituting ts for d,
= (3,920 v 2 -14.7 t }3/2
u s N/s m.
c2
The total transport is given by
= (3,920 v 2 -14.7 t ,31 2
u s B
where Qu =
B =
p =
p' =
g =
~ (p -p')g
p
total under-ice transport (m 3 /s)
channel width (m)
water density (kg/m3 )
ice density {kg/m3 )
acceleration of gravity (m/s 2 ).
(3.26a)
(3.27)
[
[
r
L
[
[
[
[
[
[
[
[
[
[
[
[
[
I ·12
I I
I -
[
L -
[
I'
I
L
[
r~
L
[
[
[
,
l~
[
[
r ,
I
l ~
[
l~
l
r-
I
l_o
The cover will progress upstream at the thickness
re~uired, given by
2g (p -p')
p
(3.13)
Individual floes thicker than tm will come to rest
against the leading edge of tne cover. Thinner floes
and slush in excess of the carrying capacity of the
flow will pack at the leading edge at a porosity, c',
between the porosity of the slush blanket and the
porosity of an accumulation of solid floes, depending
on the amount of thermal ice present and the hydraulic
thrust on the leading edge. So, while individually
the floes are too thin to be stable, the leading edge
will progress. (The porosity of a packed accumulation
of ice fragments and slush reaches a minimum value of
0.4 according to Michel (1978).]
The rate of ice cover progression is given by
6L = 0 s -0 u (3.28)
B tm (1 -e:')
The time required for ice cover progression through a
river reach can be determined from the rate of cover
progression accounting for decreasing ice inflow, Qs'
as the ice production area reduces with upstream
cover progression.
The under-ice transport rate calculated by the Meyer-
Peter method is noted to be no more accurate for ice
transport than it is for sediment transport. The
calculation depends on the characteristic dimension
used to define the distribution of ice fragment sizes.
43
Due to the lack of field data on this size distribution,
more simplified criteria may be adopted for frazil ice
deposition.
In that case, all ice is assumed to pack at the leading
edge at Froude numbers less than critical for cover
progression. Conversely, at higher Froude number, all
ice is carried under the cover at the leading edge~'
Where conditions are favourable, the ice is deposited
under the ice cover increasing cover thickness and
roughness. In some cases where very thick accumulations
occur, a "hanging dam" is formed. A hanging dam may
occur without a great head loss which distinguishes
this phenomenon from an ice jam caused by unstable
collapse of the cover and high head loss.
The simplest criterion for deposition of ice under an
ice cover was a maximum velocitY limit proposed by
Newbury (1966). Based on observations on the Nelson River
in Manitoba, frazil will be deposited at under-ice veloci-
ties less than 0.8 m/s. At velocities above 1.5 m/s,
erosion of unconsolidated slush will occur.
Tesaker (1975) studied frazil ice accumulation under a solid
ice cover for two winters at a Norwegian power plant.
He found that frazil ice accumulated until the flow
velocity reached 0.4 to 0.6 m/s. This corresponded to
a limiting Froude number of the flow under the ice of
0.08 to 0.14 which compares with limiting Froude numbers
for progression of an ice cover. At the suggestion of
Lavender (1975), a densimetric Froude number defined as
I
Fr = v
jgH(p-p'}
p
(3.29}
I
L
r L~
[
[
[
[
[
1-
[
[
[
[
[
l
L
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I L~
[
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[
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6 . ~'~
I L
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L
114
was examined. A densimetric Froude number accounts for
the bouyancy of the particles which is important for
stability. For p'/p of solid ice equal to 0.92, the
densimetric Froude numbers are 0.28 to 0.49 corresponding
to Froude numbers of 0.08 to 0.14, respectively. If a
porous slush pan is considered (e: = 0.75) and p'/p equals
0.98, the densimetric Froude number equals 1.0 (corres-
ponding to a Froude number of 0.14), a well known
critical Froude value in other applications. To arrive
at a useful and general stability criterion, more
knowledge on the density of slush accumulations is
required.
As discussed earlier, the Meyer-Peter approach has
serious limitations relat~d to the determination of a
suitable length parameter for the transported particles.
While this length selection is difficult for slush ice
transport, the method is even less appropriate for
solid ice pans with surface dimensions much different
from pan thickness. Uzuner (1975) analyzed the stability
of solid ice pans under an ice cover and the rate of
transport. The analysis paralleled the stability of
prismatic blocks resting on a river bed. The stability
criterion for block transport obtained from moment
equilibrium is given by
l-~ [ I ; I ' R. ! • 1.596 -_ L277 • b _ ( _ o') ::e 1 t -'/3 ' - l l ...... -H ' , J. :; li s.s+~e -(1-~)~ jte ·
-~ J
(3.30)
45
where Frit = densimetric Froude number at incipient
block transport
v
pI
= flow velocity upstream of the ice cover (m/s)
= ice density (kg/m 3 )
= water density (kg/m 3 )
acceleration due to gravity (m/s 2 )
= equilibrium thickness of fragmented uniform
ice cover ( m)
= depth of approach flow (m)
=length of ice block (m).
A lack of field data to substantiate this criterion
limits its application.
At present, accumulation of sluch ice and floes should
be determined from the simple limiting velocity criteria
until field data have been collected to verify the
analytical criteria for slush and block transport.
3.2.6 -Backwater
with Ice Cover
The effect of an ice cover on the slope of the energy
gradient is two-fold. In a wide river, the hydraulic
radius is equal to half the mean flow depth under the
cover rather than the full flow depth. Secondly, the
composite roughness of the flow boundary is often
rougher than the natural channel bed roughness when a
packed cover of slush and floes is formed. An additional
factor in determining stage is the hydrostatic water
level within the ice cover.
Numerous researchers have attempted to derive a composite
roughness Manning's n-value or Darcy-Weisbach friction
factor to account for the additional roughness of the
ice-covered channel. An excellent summary of this research
is provided by Uzuner (1975) who concluded that the
[
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r
L
L
[
[
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[
r
L
['
_,
[
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{ ' I
L
r~
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L
L
[
[
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f"
I L
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46
derivation of Larsen (1973) was the most complete and
rigorous. In a similar summary by Pratte (1976), that
conclusion was also reached. However, Pratte noted
that the simpler derivation of Torok-Sabaneev (1948) was
more appropriate for engineering use. The composite
roughness of the cross section is given by
+
2
where n. =Manning's n-value under the ice cover
l.
nb = Manning's n-value of the channel bed.
For the ranges of roughness encountered, the formula-
tion of Larsen simplifies to this equation. Further-
more, insufficient field data exist to support the
assumptions of the various researchers. In application,
the errors introduced by selection of single bed and
ice roughness values outweigh the errors in any analy-
tical method. Pratte discusses application of composite
roughness in channels with variable bed characteristics.
Without detailed field measurements of cross section bed
profile and ice thickness variation, a simple approach
is necessary and adequate.
The most complete compilation of field data on Manning's
n-values is reported in Nezhikhovskiy (1964). For
smooth ice covers with no loose ice accumulated under-
neath, the values were found to be consistently
0.010 < n .. < 0.012 (at beginning of freeze-up) J.
0.008 < n .. < 0.010 (middle of winter) . l.
47
The tendency for covers to become smoother as winter
progresses is due to the greater heat transfer caused
by turbulence around projections under the cover com-
bined with faster growth rate in thinner sections.
Carey (1966, 1976), Ashton and Kennedy (1972) and
Larsen (1973) investigated the formation of ripples
on the underside of smooth ice covers. As heat
transfer changes near the end of winter, the smooth
ice cover may erode creating a ripple pattern. The
roughness increases to 0.014 in some cases.
For ice covers with loose ice accumulations under-
neath, roughness values were classified by type and
depth of accumulation as follows
frazil slush
-compact frazil slush with small ice floes
-ice floes.
Following formation of accumulations at freeze~up,
n-values were as given in Table 3.3. Very rapid
decreases in this roughness have been noted, although
a gradual decrease in roughness through the winter is
normal. On the Dniester River, the n-value dropped
from between 0.07 to 0.08 to a final value of 0.008
to 0.0012 in 15 to 25 days. The decrease in rough-
ness varies dramatically from year to year even on
one river, depending mainly on meteorological condi-
tions but also on ice material in the accumulation.
As expected, solid floes decrease in roughness more
quickly than loose frazil slush.
[
[
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1~,
--~
L:
[
[
[
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-L
[
r~
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[
l~
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r~
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48
TABLE 3.3
VALUES OF ICE COVER ROUGHNESS,
WITH UNDER-ICE ACCUfoiULATIONS
(from Michel, 1971)
n.,
1
Initial Ice
Accumulation
Thickness
(m)
0.1
0.3
0.5
0.7
1.0
1.5
2.0
3.0
5.0
Type of Accumulation
Compact Slush
Frazil Slush With Ice Fragments
0.01
0.01
0.02
0.03
0.04
0.04
0.05
0.06
0.013
0.02
0.03
0.04
0.06
0.07
0.08
O.G9
Ice Floes
0.015
0.04
0.05
0.06
0.07
0.08
0.09
0.10
49
Michel (1971} notes that sprinq jams formed from
solid floes had lower values of n. than for freeze-
l.
up accumulations of solid floes. Therefore, a value
of n. for compact frazil slush and small ice floes
1
is recommended for spring jams by Michel.
Beltaos (1978a, 1978b) investigated spring jams at
breakup and concluded that the freeze-up values
reported by Nezhikhovskiy represented extreme values
of roughness. The calculation of composite n-valne
with the Torok-Sabaneev equation may not be applicable
for extreme roughness, the detailed fluid mechanics
of which is not known at present.
The determination of composite roughness for an ice-
covered channel is limited by a lack of field data.
The roughness values listed provide some guidance,
but an engineer requires a great deal of experience
to determine backwater effects under ice-cover
conditions, even when detailed open-water levels and
cross section data are available.
3.3 -Application to Variable
Flow and Water Level
The preceeding formulas have been developed for simple steady,
uniform flow conditions. For mechanical stability, the un-
frozen cover can react to variable water levels and hydraulic
forces. Thus, stability calculations should be made at
extreme discharges and water levels. For thermal calculations,
the effect of discharge variation is cumulative, and average
values should be used.
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50
4-MIDWINTER
The characteristics of the ice cover formed in a river reach
are determined by hydrodynamic and meteorological conditions
at freezeup. In high velocity reaches, a thick, packed cover
is required to develop sufficient internal strength to resist
the stresses imposed by forces acting on the cover.
The strength of the cover increases greatly after the cover
begins to consolidate and then a much thinner cover is required
for stability. In rivers where winter flow is typically low
and relatively constant, the ice regime is dominated by thermal
processes.
A floating, fragmented ice cover can respond to changes in
discharge, attaining the required thickness and level for
stability. A solid cover has greater strength but does not
have the same degree of flexibility to adapt to changes in
water level. Large and rapid water level variations caused
by hydroelectric peaking or spring runoff can break up the
solid cover. High discharge and the sudden reduction in
strength from that of a solid cover to that of a fragmented
cover can result in the most severe ice jams. Variation of
river discharge must be regulated tc ensure that support of
'the cover at the river banks is maintained and rupture of
the solid cover is avoided.
4.1-Midwinter Thermal Processes
4.1.1. -Extent of Open Water
Heat balance on open water and frazil ice generation,
discussed in Section 3.1, determine the time for cover
formation and extent of open water. The cover progresses
51
upstream covering the supercooled section of the river
reach and frazil generation ceases. The leading edge
position migrates upstream and downstream with changing
meteorological conditions following the position of
freezing water temperature. Donchenko (1975) gives
examples of rivers in Russia where the position of the
leading edge varies by as much as 60 km throughout one
winter season as a result of variations in meteorological
conditions and flow regulation. Progression of the
cover can be calculated from the formulas in Sections
3.1.1 and 3.1.2. Recession of the leading edge can be
calculated from the formulas in following sections.
4.1.2 -Solid Ice Cover
Growth
The growth of an ice cover depends upon the net rate
of heat trans· fer from the cover. Heat is taken away
from the cover to the atmosphere and normally heat is
transfered to the cover from the flow. Many factors
influence this heat transfer, as will be discussed
later.
It has been found in practice, however, that ice cover
growth can be reasonably represented by a simple
relationship originally derived for lakes and reservoirs
by Stefan and simplified by others (see Michel 1971).
The formulation is
{4.1)
where ti = thickness of solid ice {m)
a = a form of local heat trnnsfer coefficient
D =accumulated degree~days of freezing (°C-days).
r-
[
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L
The coefficient a is obtained from experience to
represent the time average values of the various
physical and thermal properties of the ice and water,
as well as the highly variable and complex heat trans-
fer between the ice and the atmosphere. Heat transfer
at the ice/water interface is also considered in the
selection of a. Values of the coefficient, derived
from field experience, are
windy lakes with no snow a = 0.8
average lake with snow a = 0.5 to 0.7
average river with snow a = 0.4 to 0.5
sheltered river with rapid flow (l = 0.2 to 0.4.
In spite of the extreme simplification of this technique,
the method has been used successfully. However, this
relationship predicts g=owth of the ice cover as long as
the degree-days of freezing continue to accumulate and
heat is extracted from the underlying body of water.
This implies that ice cover growth is only limited by
duration of winter when, in fact, heat inflow from the
under surface may limit total ice growth. This applies
particularly to recession of the leading edge of the
cover when melting of the undersurface begins.
A complete analysis of heat transfer·at the ice/water
interface can be performed t~ determine ice growth.
The starting assumption is a constant temperature of ooc
at the under surface of the ice and continuous m(~lting
or freezing depending on the direction of net heat
transport. The constant temperature of 0°C facilitates
heat transfer ~calculations by making transfer
processes on either side independent of each other.
Heat transfer from the lower ice surface to the atmos-
phere occurs in two stages. Heat is first conducted to
or from the lower ice surface as described by
= k c
t:-
l.
. where qc = rate of heat transfer (J/m2 s)
k = conduction coefficient (,J/°C m s) c
t. = ice cover thickness {m)
l.
S3
( 4. 2)
Tf = freezing temperature of water (0°C)
Ts = upper ice surface temperature (°C).
Transfer of heat from the upper surface to the atmosphere
occurs by convection and radiation dependent on such
factors as wind speed, air-ice temperature difference
and ice properties. Meteorological conditions have
the strongest influence on heat transf-r.
In general, it is possible to assume that the upper
surface is at air temperature and calculate heat trans-
fer from conduction alone. Baines (1961) notes that
this practice is equivalent to assuming an overcast sky
condition and a strong wind, whereas the average winter
condition is one of moderate sunshine and light wind
which tends to produce thinner covers.
(..._ ... ·, -e.~~f "!" , 1 /J'l 1 .. __ ., I
The validity of the assumption is supported, however, by
data collected on the Nelson River as reported by
Ne\llbury (1966). /Newbury • s anal.ysi:s of. t-he data showed
independence of ice growth from wind conditions and a
transfer coefficient very near the generally
value for conduction through ice (i.e., 2.1R
°C per meter of thickness}. Thermal inertia
no doubt figures significantly in making the
accepted
4 2 x 10 J/m s
t!n the cover
'f \i • approx1ma-
tion reasonable, particularly when using average tem-
peratures over a period of one week or more.
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54
[
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I
Heat transfer from the flow in a river to the ice cover
is a function of the thermal and flow characteristics
of the river. The heat transfer is comprised of three
components; terrestrial heat flow, qt, internal fric-
tion, q f, and cooling of the r i vc r flow, tJw. 'l'hc to tal
heat flow to the cover, qi, is the sum of these components.
( 4. 3)
Very little data is available for the terrestrial heat
flow. Initially, the return flow of heat stored in
the river bed during summer inflow is high. McLachlan
(1926) provides estimates of 17.2-19.7 J/m2 s in
the St. Lawrence River early in winter.
just prior to break up, the heat inflow
In late winter,
is typically
2.09 -2.51 J/m2 s.
rates of 11.72 J/m2 s
Nybrant (1945) estimated correspondin~
pond studies in sweden.
in October to zero in April for
2 A rough average of 4.61 J/m s
hased on Nybrant's work is recommended for an initial
estimate of ice cover thermal equilibrium. Thus,
2 q = 4.61 J/m s t
( 4. 3a)
The heat from internal friction can be calculated from
the head loss
qf = QyS (4.3b)
BJ
where qf =heat from internal friction (J'/m2 s}
Q = river discharge (m 3 /s)
3 y = specific weight of water (N/m }
S = slope of energy gradient equal to slope of
hydraulic grade line in uniform flow
B = river width (m}
J =thermal equivalent =1.00 N-m/J.
Heat from cooling of river flow is given by
a = -Q c dTW '"W -p p
B dx
(4.3c)
55
where qw = heat available from cooling flow (J/m 2 s)
cp = specific heat of water (1 J/kg oc)
p = specific mass of water 3 (kg/m )
T = water temperature ( oc) w
x =distance downstream along the river (m).
With temperature measurements taken under ice-covered
conditions, the cooling of river flow and the rate of
change of ice cover thickness can be calculated from
dti = qc -q.
~ ( 4. 4) dt Lf yi
where qc = rate of heat transfer by conduction through
the ice cover (J/m2 s) .
qi = rate of.heat transfer from the flow to the
ice cover (J/m 2 s}
If water temperature measurements or estimates are
available, the rate of ice cover growth can be calcu-
lated. Integration ~f the differential equation yields
the change in ice cover thickness
6t.=cq -q.> t ~ c 1 (4.4a)
Lf yi
where qc -qi = time average net heat inflow to the
ice cover determined from average flow
and air and water temperatures
t = time for change in ice thickness ( s).
In a long river reach where equilibrium heat flow to
and from the water is achieved, water temperature will
be determined by total heat flow expressed as
where he = convective heat transfer coefficient
(J/m2 s °C)
= equilibrium temperature difference from
freezing ( o C) .
( 4. 5)
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(~
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~ 56
I L,
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.-~
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Convective heat transfer from the flow to the smooth
under surface of the cover can be described by the
Reynold's Analogy (Baines 1961, Ashton 1973, Cowley
and Lavender 1975)
Nu
a 1/3 = f Re Pr
2
where the Nusselt number, Nu = he.t /kf
the Reynolds number, Re = v ~P/~ u
(4.6)
the Prandtl number, Pr = cp~/kf = 13.6 for water
at 0°C
a = an exponent eaual to O.B or 1.0
and f = friction coefficient.
For the dimensionless ratios
,_ . h .<: .<:.<:. • , T I 2 s OC\, u =convective 1eat tranSJ.er coeJ.J..lCJ.ent ,v,m e
c = specific heat of water (J/kg oc)
p
p = specific mass of fluid (kg/m 3 )
kf = thermal conductivity of the fluid (J/m s oc)
.t = characteristic length (m)
absolute viscosity (kg 2
lJ = s/m )
vu = average velocity under the ice cover (m/s).
The Nusselt number provides a measure of the relative
importance of heat transfer by turbulent mixing and
simple conduction. The Reynolds number characterizes
flow turbulence. The Prandtl number repres'ents thermal
properties of the fluid.
For a two-layer flow, the layer depths are proportional
to the shear coefficients of the respective layers.
57
For a wide channel, the hydraulic radius of each
layer is approximately equal to the layer depth. In
the Reynolds Analogy the friction factor is equal to
one-fourth of the Darcy-Weisbach friction factor which
can be related to the Manning coefficient by
f fD 19.6 2 = = n.
l.
4 R. l/3
(4.6a)
l.
where n.
l. = Manning's n-value for the ice cover
R.
l. = hydraulic radius of upper flow layer (m) .
In natural river channels where flow is always turbu-
lent, the Manning equation is applicable. Controversy
in applying the Reynolds Analogy is related to the
appropriate exponent for the Reynolds number which
depends on the treatment of shea.r:.,stress. For example,
---.?~...V Baines, Cowley and Lavender favour-the use of the
exponent "a" equal to 1.0 while Ashton and Michel argue
that an exponent of 0.8 is appropriate for fully rou9h
turbulent flow.
A practical limitation to the heat transfer determina-
tion is the determination of the appropriate n-value
and hydraulic radius for the unknown flow distribution.
Much field and laboratory work is required before the
proper form can be determined. It is sufficient at
this point to solve the Reynolds Analogy in the form
N u
2 = 9.8 ni
R. 1/3
l.
Re Prl/J
~4. 6b)
The layer depths of the upper (Jl.i) or lower (tb}
layers are proportional to the friction factor or the
square of the corresponding n-value. It follows that
the ratio of the layer thickness can be written as
[
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58
The total flow depth under the cover is given by
H = 9.i + R.b
Thus,
An estimate of convective heat transfer coefficient
can be obtained from
he
for
The
can
dt. 1
dt
= 1. 73 X 10 6 [1 + I 1 2]1/3 .nb,
\n.} . ~-
ll/P = 1.78 X 10-6 2 m /s
kf = 0.566 J /m s oc
Pr = 13.6 for water at 0°C.
instantaneous local growth
then be determined from
V n. u ~
2
Hl/3
rate of the ice cover
If this term is negative, melting of the cover occurs
( 4. 7)
( 4. 7 a)
(4.7b)
( 4. 8)
( 4. 9)
and water temperature falls. As ti becomes very small,
the melting rate becomes infinite and the cover recedes
downstream. The melting also transfers heat from the flow
which brings the water temperature down to the freezing
point. A new equilibrium may be established downstream.
4.2 -Mechanical Stability
of a Solid Ice Cover
5~
Research on the mechanical stability of ice covers has been
devoted to the stability of fragmented covers at formation
and breakup. Hydroelectric power plant operation is usually
modified to conform to the stability criteria required for
stable cover formation. After the ice cover has solidified,
normal operation is resumed.
In some cases, rapidly varying velocity and water lev::::l
changes associated with peaking operations continually
cause breakup of the cover. As there are no general stability
criteria available to guide winter peaking operation, this
procedure must be determined by trial-and-error or suspended
for the duration of the \.,.inter.
The criteria presented here have yet to be verified by field
observations. They will not be suitable for design or
operational guidelines until field and laboratory programs
have been conducted to establish coefficient values.
4.2.1 -Solid Covers -
General Stability
Michel (1975) reported the results of experiments
conducted on the stability of a solid ice cover. The
study involved laboratory tests using a poured w~~
layer floating on the water surface to simulate a
solid ice layer subjected to hydraulic forces of flowing
water under the cover. The experimental tests yielded
two criteria; a limiting Froude number of 0.275 for the
stability of the leading edge of the cover and a limit-
ing Froude number for failure of the entire cover by a
process of "oscillatory bending". This Froude number
is defined below.
[
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I L
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[ Gt)
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(Jb t.
l.
I -[ g (p -p ) 821 (4.10)
Fr = v = 0.113 e
/gH
where v = flow velocity under open water conditions
acceleration due to gravity 2 g = (m/s )
H = open water flow depth {m)
bending strength of 2
ob = the ice cover (N/m )
ti = ice cover thickness (m)
p = water density (kg/m3 )
p = ice density (kg/m3 )
B = channel top width (m) •
The results were obtained in a small-scale model of a
prismatic channel. The range of parameters tested was
limited, particularly with respect to channel geometry.
The effect of water level variation causing failure of
the cover at the channel banks was not considered.
Further testing and field verification are required
before the criteria can be used with confidence.
Only qualitative-descriptions of solid ice cover dis-
ruption at hydroelectric peaking generating stations
col'1_d be found so verification of these criteria with
field observations could not be made.
4.2.2 -Solid Covers -
Shoreline Hinging
A hinging failure mechanism of the ice cover contact
at the river banks caused by large variations in stage
related to hydroelectric power plant peaking operations
was identified in a field study by Fonseca (1979).
(m/s)
61
The mechanism is characterized by a minimum number
of three failure lines in the ice cover parallel to
the river banks. Repeated failures and reformation
of the ice cover at varying water levels produce a
cover width dimension greater than the horizontal
width of the river water surface at maximum water
level. The central portion of the cover remains
intact. With the rise and fall of the ice cover
following changes in water level, the fragments or
strips of ice parallel to the river banks form an
inverted arch which supports the central portion
of the ice cover, illustrated in Figure 4.1. The
"hinges" between the fragments, the solid ice cover
and immobile ice in contact with the river bank were
observed to break or crack with each cycle in the
water level.
Kartha (1977) noted the formation of hinge cracks
on the Winnipeg River. A series of tests were con-
ducted to investigate ice cover stability under
peaking operation conditions. Measurements of water
and ice cover levels, and observations of cover
stability were made for three peaking discharge
ranges at seven river cross sections. Unfortunately,
detailed observations of the hinging mechanism were
not given.
Hinging is mentioned specifically in a report by
oonchenko (1975). It is not clear if the same pheno-
menon is considered for this definition of shoreline
contact.
[
[
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c
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1:
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[
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c .J
----·. _ ... ~·"'-
I-Oit----1
"' <I
·,·, '.' ,"·,'
( J ,J
v
CANADIAN ELECTRICAL ASSOCIATION
STUDY OF ICE COVERS SUBJECT TO VARYING WATER LEVEL
DEFINITION SKETCH OF HINGING MECHANISM
I '.
F' IG 4.1
liJ
63
Initial Cover Failure
After the formation of a solid cover at a fixed
water level, the inital failure occurs at the
shoreline. The rise or fall in water level creates
P distributed load in the cover due to the differ-
ence between ice weight and buoyancy of the partially
submerged cover. Bendinq failure at the shoreline
occurs if the water level exceeds a maximum value.
After the initial failure at the shoreline, a
second failure occurs at a point offshore. This
failure sequence has been observed by Shaitan (1967)
and Sinotin and Sokolov (1968) .
The initial peaking operation following cover form-
ation will determine the failure lengths for hinge
link formation. Subsequent failures will occur at
the initial failure points where the cover is weak
{Michel 1970).
Various formulas based on elastic and plastic
analysis have been presented for critical water
level variation at cover failure and vertical loads
on structures caused by water level variation
(Michel 1970, Shaitan 19671 Donchenko 1975). However,
attempts to analyze this phenomenon have been
hampered by a lack of knowledge of the basic creep
properties of ice (Michel 1970). The failure of the
ice cover is highly dependent on the rate of loading,
meteorological conditions, and frequency and range
of water level variations. Shoreline configuration
and ice adhesion to the shoreline play a significant
role in the failure as well. The strength of
adhesion can be as strong as the shear strength of
ice itself for contact with concrete, steel or
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[
[
[
[
64
timber structures. For natural or armoured
riverbanks, the adhesion will be reduced. The
strength of the contact depends on the depth of
freezing of the riverbank (Rossinskij and Nisar-
Muhamedova, 1977).
The generalized formulas do not account for the
highly variable conditions encountered in nature.
In fact, with simplifying.assumptions, the form of
the equations derived for plastic and elastic
failure is the same so the difference in the appli-
cation of the formulas is the choice of elastic or
plastic ice strength for the loading conditions.
The !?roblem is further complicated by the lack of
appropriate field data. As Tsanq (1975) notes, there
are no techniques at present to estimate bending and·
compression strength in the field from readily avail-
able meteorological and hydroloaical data. These
formulas will be of limited use until standardized
testing methods are adopted such as those suggested
by the Workinq Group of the IAHR Section on Ice
Probl~ms (1980). Calculations do support observa-
tions of initial failure for water level variation
les~ than the ice thickness.
Fina 11 y, there
failure of the
link lengths.
are no criteria for the second
ice cover which determines the hinge
These lenqths are i~portant for the
overall stability of the solid ice cover.
The enqineering capability to predict the initial
cover failure is inadequate at present. A system-
atic program of field data collection and observa-
65
tions is requin!d to qui de the analysis of the
phenomenon. However, for the determination of
overall ice cover stability, a rigorous theoretical
solution is not required because
-the initial failures of the cover are only impor-
tant for determination of hinge link len~ths
-it will be impractical to maintain solid ice
cover contact with the shoreline when failure
occurs for water level variation less than ice
thickness.
Therefore, the recommended goal at this ti~ is
to establish a simple empirical correlation between
hinge link length and failure water level variation
which depends on ice thickness and basic meteoro-
logical conditions (such as antecedent degree-days
of freezing/thawing as an ice strength index).
This relationship can be calibrated in the field
with simple data requirements.
Hinge Stability
On the basis of preliminary analyses of the hinqing
mechanism, failure of the hinge throuqh buckling of
the individual fragments or crushing of the ice was
rejected because of the narrow 9araMeter ranqes for
conditions critical to stability. Subsequent
analysis of the hinge develooment utilizing a solid
mechanics approach has yielded the followin~ cri-
teria for stability of hinge sections. Details of
the analyses are given in Appendix A~
[
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I
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[
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~ 66
I
l ~
I
I"
[
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[
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[
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L
r L
[
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L
[
The stability of the hinge depends on the frictic)fJ.:.li
or shearing resistance of the hinge joints to
chanqing buoyant force resolution on the hinge mem-
bers. For a statically determinant solution, a
frictional resistance approach was adopted. If
only one hinge link is free to move, as shown in
Figure 4.2, the limiting water level variation for
stability is given by
= sin a max
where amax is the maximum angle from horizontal
for the hinge link which is in turn given by
max
where ~=a frictional or contact coefficient
reflecting the strength of the fractured
and rotating hinge joint.
The hinge link length at formation is required for
substitution into the above relationship.
The stability of the one-link hinge reveals the
basic stability relationship at the hinge joint.
However, from a practical standpoint, one-link
hinge stability cannot be achieved without elonga-
tion of the center section of the cover or the
( 4. 11)
(..: . lla.)
hinge link. Two or more links are required to
maintain a continuous connection between the shore-
line and free-floating center section of the ice
cover as the water level fluctuates. Various confia-
urations are shown in figures 4.3 and 4.4.
NORMAL
FORCE
r---1
l .J
f
BUOYANT
FORCE
CANADIAN ELECTRICAL ASSOCIATION
STUDY OF ICE COVERS SUBJECT TO VARYING WATER LEVEL
STABILITY OF AN INDIVIDUAL HINGE LINK
FIG. 4.2
•
..
0
E
>-
<]
I~
-0
CANADIAN ELECTRICAL ASSOCIATION
STUDY OF ICE COVERS SUBJECT TO VARYING WATER LEVEL
SKETCH OF TWO-Ll NK HINGE CONFIGURATIONS FOR Rl SING WATER LEVEL
FIG 4.3
[ij]
,----,
l .. J
0
0
0
CANADIAN ELECTRICAL ASSOCIATION
STUDY OF ICE COVERS SUBJECT TO VARYING WATER LEVEL
SKETCH OF TWO-LINK HINGE CONFIGURATIONS FOR FALL lNG WATER LEVEL
FIG 4.4
[iJ
[
I l~
[
[
[
[
[
l
[
[
L
70
Rising and falling water level as they relate to
hinge stability are defined in terms of the arbi-
trary starting position for hinge analysis shown
in figures 4.3 and 4.4. In this position, one
hinge link lies on the channel bank and the other
link is horizontal. A rising water level causes
the central section of the ice cover to move above
the horizontal link level. The horizontal link
follows the cover movement and the second link
rotates off the channel bank.
If two hinge links are free to move, the limiting
criteria become more complex. Based on the basic
stability of a one-link hinge and geometric
relationships for bank slope, hinge link lengths
and link angles, the limiting water level varia-
tion for hinge stability for a rising water level
is given by
!!.y -c + J ~ 2 - 2 c rl -rl r2 :r
where
= sin a max
= 1 -cos a + (z + sin a max max - z cos amax>
f z2 + 1
z is the horizontal component of bank slope for a
unit vertical rise.
( 4. J..:.!)
71
For a fallinq water level
where
1
= 1 -cos a + (z -sin a - z cos a ) max max max
/z 2 + 1
Hinge stability depends on the length of hinge
link for conditions at the initial failure of the
ice cover.
The criteria for hinge stability are summarized
(4.1))
in Table 4.1. In order that the criteria for risins
and falling water level could be compared, plots
were prepared for the geometrical relationship
between ~y/i, n and z.
Hinge link stability can be expressed in terms of
water level variation and ice cover thickness by
substitution of the appropriate hinge link length
for rising or falling water level at formation.
[
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rr---: -.,' _j t j
TABLE 4.1
SUMMARY OF SOLID ICE COVER STABILITY
FOR VARIABLE WATER LEVEL
Hinge Stability
One-Link
Hinge
~=
Rising Water Level
sin a max
Falling Water Level
sin a max
Two-Link
Hinoe
!c 2 -r1
where cr1 = sin amax + 1 fz2 + 1
where
Cr 2 = 1 -cos a max + _c_z_+_s_i~n==a=m=a=x===-z_c_o_s_a...;;m.;.;.;a;.;..x;.;,._)
/ z2 + 1
cf1 + R-2Cf2 -
c£1 = sin amax 1 /z2 + 1
cf 2 = 1 -cos amax + (z -sin amax - z cos amax)
J z 2 + 1
73
For rising water level, maximum water level varia-
tion is governed by both the frictional stability
of the joint and the physical limitation of hinge
elongation. As the water level increases, it can
be seen in figures 4.3 and 4.5 that a, the angle
from horizontal of the hinge link, increases to
a maximum value aa and then decreases until the
point of hinge elongation occurs. For each value
of z there is a corresponding maximum value of aa
that can be achieved, given by
1 sin eta = /z 2 + 1
This corresponds to
hence,
sin aa + 1
1/z 2 + 1
The angular limit line is shown in Figure 4.5.
The physical significance is illustrated in
Figure 4.3 also.
(1.14)
(11.15)
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L
3.0
2.5
~~~
2.0
z
0
1-:g;
0::
~
..J w
> w 1.5 ..J
0:: w
~ ~
w >
1-
4
..J w
0:: 1.0
0.5
/
~ f' // I/
-r=o
II /; /
/
I
ANGULAR LIMIT b,y I
-: SlnQ + ==-,f. 0 ..Ji'E+I
I
10 20 30 40 50 60 70 80 90 100
HINGE ANGLE a
FIG.4.5
CANADIAN ELECTRICAL ASSOCIATION
STUDY OF ICE COVERS SUBJECT TO VARYING WATER LEVEL
TWO-LINK HINGE CONFIGURATIONS FOR RISING WATER LEVEL [i]
75
Another possible limit corresponds to the point
at which the ice cover is level with the hinge
contact at the river bank. If the contact is
poor and the water level rises above the contact
point, the hinge may detach and the entire cover
would become free floating. Thus, the limit to
water level variation equals y 0 and
t.y = y = t 0 r-
1
The line of shoreline contact is shown on Figure
4.5.
It should be noted that the outer hinge link is
considered to he critical for stability because
of the steeper angles attained when both links
are subjected to buoyant uplift forces.
If a , the friction angle for hinge joint stab-max
ility, is less than a , the geometrical angular a
limit, frictional joint stability will govern
hinge stability and the hinge will fail when a
corresponding to t.y/~ and z reaches a • If a max max
is greater than a and the hinge does not fail . a
when a is attained, an increase in water level a
(4.16)
up to the elongation limit is possi~le if the hinge
does not detach at the shoreline.
The shaded area in Figure 4.5 represents the
geometrical configurations for rising \vater level
stability of a two-link hinge for the range of
limiting hinge angles a and bank slopes z max
[
[
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[
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[ 76
[
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E
[
l
L
L
[
possible. The dashed lines in Figure 4.5 represent
geometric solutions to the stability equation which
would only occur after a drop in water level from
the elonqation limit.
The maximum frictional contact angle amax must be
determined from field and laboratory experiments.
With this angle, the limiting rising water level
variation can be determined by following the
appropriate slope curve z to amax· If amax is
greater than the geometric angular limit aa' the
limiting value of ~y/i is at the elongation limit.
For a falling water level, hinge stability is more
restricted, as seen in Figure 4.6. As in the case
of rising water level, the shaded area represents
possible hinge configurations for various bank
slopes and hinge link angles. The elongation limit
is reached before the maximum angular limit is
reached. Unless limited by joint friction, the
elongation limit governs hinge stability. The
dashed lines represent hinge configurations that
could only be achieved by a rise in water level
from the elongation limit, as illustrated in Figure
4.4. The maximum angle which can be formed occurs
in this region of the graph when
sin a = l
a /-z. + 1 z.
(4.17}
At that point,
~ = sin a -1 = 0 a ----R. lzTo (4.18)
~~~
...J w
> w
...J
0::: w
1-
~
w >
1-
<t
...J w
0:::
1.0.----------------
ELONGATION
0.5
0.4
0.3
0.2
0.1
0 10 20
" \
\
\
\
\
\ ANGULAR LIMIT
{:j.y = sina0 -r:ir.-, = o v .Z-+ I
30 40 50 60 70 80 90
HINGE ANGLE a
CANADIAN ELECTRICAL ASSOCIATION
STUDY OF ICE CQVERS SUBJECT TO VARYING WATER LEVEL
FIG.4.6
TWO-LINK HINGE CONFIGURATIONS FOR FALLING WATER LEVEL [i]
"'~ ~ .. ,, .. ·-h' •
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b
78
With the limiting hinge frictional anqle and bank
slope, the maximum falling water level variation
can be determined from Figure 4.6.
The combined limits to rising and falling water
level determine the total limit to water level
variation under peaking operation. With a , a max
limiting hinge angle, determined from field and
laboratory observations, and z, the bank slope
measured in the field, the relative operating
range for peaking operation can be established
from the sum of the rising and falling water level
limits. The absolute range in water level depends
on the length of the hinge links resulting from
ice conditions at cover formation and operating
procedure immediately after cover formation.
Hinge link lengths can be measured in the field or
calculated for the rise or fall in water level
following cover formation. As discussed previously,
a rising water level will create longer hinge links
and permit a larger peaking range after hinge forma-
tion.
The analysis of falling water level below the
starting position may be irrelevant if the hinge
joints have no tensile strength and the links
separate before the elongation limit is reached.
In this case, the lower limit to hinge stability
is the starting position with one link lying on
the channel bank and the second link horizontal.
In the static case of a reservoir with no flow
velocity.. the failure of hinge links is less
important. At high water levels, hinging may
support the cover when subjected to wind stress
preventing piling. At low levels, the hinge links
79
lie on the reservoir banks. In some cases, the
ice freezes to the banks and a new cover freezes
above, (Sinotin and Sokolov 1968, Sokolov 1970),
but for the most part the ice refloats and assumes
its original position in the ice cover at higher
water levels (LeVan and Fonseca 1977, LeVan 1977,
Littlefield interview).
Nith a flow velocity, hinging contributes to
cover stability by transferring forces on the ice
cover to the river banks. If the hinges do not
support the cover, the stresses on the cover
accumulate in the downstream direction and ice
jams can result. Donchenko (1978b) notes that
water level variation with peaking greater than
three to four times the ice cover thickness results
in jamming. Pentland {1973} attributes ice jams on
the South Saskatchewan River to variations in flow
velocity and water level during peaking. Smelyakova
(1970a} cites cases where failure of the cover
occurred with varying water level at discharges
that normally would not cause ice jams. Thus, the
loss of ice cover support from the banks can cause
ice jamming. Finally, Sinotin and Sokolov (1968)
describe the jamming of the ice cover in the con-
necting canal at the Kiev pumped storage power
station where flow velocity approaches 3 m/s, while
the ice cover in the reservoir remains stable for
the same water level variation.
The formulation of Pariset and Hausser (1961) can
be used to determine cover stability between shore-
line hinges. The width of the river between hinges
is used for the channel width, B. An appropriate
value of cohesion, T, must be determined for contact
[
[
r -~
--~
[
[
[
[
[
l
·'
[
[
[
[
[
L
,-
L
[
[
L
[
" I
!
l "
"
l ~
r
l~
[
[
[
[
[
[
C
E
[
c
r L
[
[
[
[
80
at the hinge. If the hinges fail due to excessive
water level variation, the accumulated forces on
the ice cover must be resisted by crushing strength
of the cover where it contacts the shoreline,
structures or intact cover sections.
Another form of hinging which may be considered
a boundary condition of this analysis is the
phenomenon described by Schulte (Calgary Power
interview) • Vertical shear lines in the ice
cover caused by shoving at formation provide
shoreline contact. The upper limit to hinge
stability is the water level which floats the
cover from between the confines of the grounded
ice shear walls. At that point, forces on the ice
cover are no longer transferred to the river banks
through the grounded ice. This type of hinging
may be typical of covers which shove at formation
while the arching hinge formation may be typical
of covers which form in narrow channels where
leading edge stability governs. The formation
sequence of shear walls is shown in Figure 4.7.
For this initial investigation of hinge stability,
the limiting range in water levels has been
related to an arbitrary starting position with the
offshore link horizontal and the nearshore link
lying parallel to the river bank. The relation-
ship of the onshore link position to a normal or
maximum operating water level must be determined.
The formation of the hinge after the initial cover
failure and the frictional contact of hinge joints
must be investigated. The type of hinge formed
must be investigated in terms of flow regulation
and the regime or river classification.
1"'"' c-:r.-tA BY .tuxlA~'OS.tliO•,. v r-.:..z~.. SL.U~"
AI.~ 1(:[ PAPjS 'AT ICE FAQI,T
STAGE I DEVELOPMENT
STAGE 2 DEVELOPMENT
STAGE 3 DEVELOPMENT
riiACTu~<ts 1 H""t~ 1 otvfLOI' ou"'~'
rALLJ••~ STAGE
LOW FLOW STAGE
HIGH FLOW STAGE
r-.. l
LEADING EDGE STABILITY OETER~INES COYER THICKNESS
INITIAllY AS IC£ FRONT ADVANCES THROUGH SECTION
COVER THICKNESS INCRt~SES SUBSEQUENT TO PASSAGE OF
ICE FRONT AS STRESSES INCREASE IN COVER AT SECTION
DUE 10 BODY AND DRAG FORCES AClJ~G ON COYER UPSTREAM
OF SECTION. COVER MAY AlSO THICKEN BY DEPOSITION If
ADDITIONAl ENERGY LOSS REQUIRED TO CREATE SUFFICIENT
STAGE fOR STABLE ICE FRONT AT SOME UPSTREAM SECTION.
SHEAR LINES FORI AS COYER TELESCOPES TO INTERNAL
EQUILIBRIUM THICKNESS.
fLOW UNDER COYER OR SlUSH/FRAZil DEPOSITS SMOOTH
UNDER SURFACE Of ICE COVER, liTH CONSEQUENT DROP IN
STAG£ FOR A GIVEN FLOW.
LARGE DECREASES IN FLOW RESULT IN SAGGING ANO
FRACTURING OF COVER. HINGE ~ECHANISM ~AY FORM.
DEPENDING ON HOI RIGIDlY COVER HAS fROZEN BEFORE
LARGE DROP IN WATER lEVEl. COYER Will FlOAl
BACK UP liTH HOI INCRE~SE. EITHER tiTH HlhGE
M£CHANISM OR AS SHO•N FOR HIGH HOW STAGE.
O{PENDINC ON RIGIDITY OF COVER.
LARGE INCREASES IN FLOW 'Ill flOAT COVER COVER Will
NOT fAIL PROVIDED ITS INITIAl ltHERNAl EQUiliBRIUM
THICKNESS WAS DETERMINED AT HIGH FLOW AND FLOW DOES
NOT REACH STAGE JHICH FLOATS COVER OUT OF CONFINES CANADIAN ELECTRICAL ASSOCIATION
STUDY OF ICE COVERS SUBJECT TO VARYING WATER LEVEL
OF GROUNDED ICE SHEAR WAllS
FORMATION OF SHEAR LINES AND HINGES
FIG. 4.7
•
82
5 -BREAKUP
The breakup process on rivers occurs rapidly and often with
such violence that documentation and measurement are impossible.
The process is further complicated by the number of meteoro-
·logical and hydrodynamic parameters involved. An excellent
description of breakup is provided by Deslauriers (1966).
7he simultaneous weakening of the ice cover and sharp increase
in river discharge resulting from warm weather result in a
sudden release of a large volume of ice at high discharge.
Early breakup when the ice is strong and has no cohesive
strength creates the most severe jams.
5.1 -Thermal Processes
at Breakup
Thermal processes differ from midwinterbecause conduction
reverses and heat is supplied to cover from the atmosphere.
The effect of thermal processes on ice_cover strength is more
important than the reduction in cover thicknes~. After the
cover breaks up, the ice is assumed to have no cohesion and
the stability of the ice is dominated by hydrodynamic condi-
tions. For that reason, prediction of breakup is based only
on statistical records of meteorological conditions (Barbridge
and Lauder 1957) and ice jams are predicted on the basis of
typical spring hydrodynamic conditions. No general attempt has
been made to formulate the dominant meteorological and hy~io
dynamtc parameters that initiate oreakup.
Bu.zin and Kozitskiy (1975) presented a technique to predict
the strength of a river cover during spring thaw. The
relationship took the following form
Bl
( 5 .1)
.where cr = ice cover bendin~ strength after exposure
to solar radiation
cr
0
= ice cover strength at freezing/thawing
temperature without exposure to solar
radiation
sr = solar radiation absorbed per unit ice volume
S 0 = critical absorbed solar radiation for
complete loss of strength.
The recommended value of cr 0 was given as 5.5 x 10 4 N/m2
(80 lb/in.2 ). The critical value of solar heat absorption was
given as 184 J/m3 (4,944 Btu/ft 3 }. The absorbed solar radiation
is given by
where st =
Al =
t. = ~
c,n=
= e
total
-~+-n v'-• ~
solar
ice albedo
radiation
(average value
ice thickness (m)
experimental coefficients.
0. 25)
The values of the coefficients were found to range between
(5.la)
c = 0.541, n = 0.6 and c = 0.243, n = 0.6 for extreme ranges
of ice properties.
With the prediction of ice strength from the above method,
the cover stability criteria of Michel (1975) in Section 4.2.1
can be used to predict the breakup of the ice cover for fore-
cast meteorological and hydrodynamic conditions. Without solar
radiation data, the spring time ice bending strength recommended
by Michel (1970) and fo't'ecast hydrodynamic conditions can be
used to predict breakup from oscillatory l:.·ending.
[
[
r L,
[
[
[
[
[
[
[
[
[
[
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I'
l~
r
L
[
r-::
L
[
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l
84
Similarly, the formulation of Pariset and Hausser (1961) can
be used to forecast breakup. As the cover shear strength
deteriorates rapidly during the spring thaw, a critical
point will be reached and the cover will shear away between
the banks. The e~rical Russian formulation reported in
Tsang (1975) gives the thawing effect on ice shear strength,
'i' as
-r.-. "• ~ h-p.IIJ~) "t..
(5.2)
where T
0
= shear strength at 0°C without thawing,
normally taken asf-~~~ ~o 70_!1 kN/m2 (t..t. k f't,.)
Th =cumulative degree-days of thawing (°C days).
Prediction of spring breakup can be made with forecasted
cover strength based on meteorological conditions and hydro-
logy. On regulated rivers, discharge can be controlled
during critical periods to prevent or prolong breakup, there-
by avoiding serious ice jams.
5.2 -Mechanical Stability
at Breakup
5.2.1 -Ice Jamming
After the cover has broken and the ice drive begins,
prediction of jamming at obstacles to ice passage can
be made using the formula of Pariset and Hausser with
cohesion set to zero. This becomes
( 5. 3}
At maximum value of t/H equal to 1/3, p'/p = 0.92, and
lJ = 1. 39
as indicated by the zero-cohesion curve on Figure 3.1.
The criterion developed by Michel (1965) for stability of
ice jams at breakup is
H 4/-;
lJM .ro-~3.18
where H = flow depth (m)
B = channel width (m)
Q = discharge (m 3/s)
PM = coefficient typically equal to 0.26.
These criteria correspond for a Chezy roughness value C
equal to 27 m1 1 2 ;s. Stability of ice accumulations at
breakup can be determined with either criterion.
85
As in the case of formation, extreme water levels and flow
velocities should be considered for mechanical stability
and time-averaged values for thermal effects.
( 5. 3a)
(5.4)
[
[
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6 -INTERACTION BETWEEN
ICE AND STRUCTURES
Interaction between ice covers and structures has been widely
reported for static and dynamic horizontal loads on structures
and for uplift forces on structures for vertical ice cover
movements. These analyses depend on the geometry of the
structure, ice conditions and the nature of the ice/structure
interaction. It is beyond the scope of this study to summarize
all work on the many complex types of ice cover/structure
interactions. Comprehensive summaries are presented in Michel
{1978) and Korzhavin (1962).
Numerous studies of ice piling on sloping structures have
been conducted as well. Croasdale et al (1978), Bruun and
Johanesson (1971), Tsanq (1974) and Allen (1970), to mention
a few, have described this phenomenon. Readers are referred
to an excellent summary of sea ice observations and analysis
of ice piling by Kovacs and Sodhi (1980) for details.
In the above mentioned studies, the total force developed on
the structure and stability of the entire structure have been
formulated. The total force is related to the ultimate
strength of the ice cover for the appropriate ice failure
mechanism (bending, compression, shear) or to the kinetic
energy of individual ice floes or ice fields. In general,
forces imposed by a frozen, static ice sheet are limited by
the ultimate strength of the ·ice while forces imposed by
moving ice are determined from impulse-momentum or work-energy
relationships (Starosolsky, 1970). For example, compressive
force from thermal expansion is given as a product of ultimate
stress in crushing multiplied by ice thickness times the
length of structure-ice contact. The force of impact by an
ice floe is determined by the kinetic energy of the floe,
and the work done in deformation of the structure and ice
floe as the floe is crushed and stopped.
Iyer (1978) presents a swrunary of existing ice codes in
effect in North America, Scandinavia and the Soviet Union.
Design practices for static and dynamic loads, and the
various methods to account for the effect of ice and structure
geometry to arrive at the appropriaLa ice pressures are com-
pared.
Typical static design loads in North American practice are
expressed as a uniform, distributed load at the water line;
15,000 -22,500 N/m for rigid structures and 7,500 -15,000
N/m for flexible members where creep of the ice and deflection
of the structure relieve the load (Acres, 1970).
Dynamic loads are more difficult to determine because of the
many complex variables involved. Normally, a crushing strength
of 275 Pa is used for dynamic ice pressure and the type of
failure mechanism and contact area are considered in the
application of work=energy principles to determine the total
load.
These methods are considered to be too conservative but
cannot be improved until the actual mechanisms are known.
Structural elnments which cannot be designed practically by
this method to resist dynamic loads are operated to avoid ice
impact entirely. Thus, sluice gates are operated fully open
or fully closea.
These general design practices are found to be inadequate
when applied to specific or unique cases, as illustrated by
the failure of stoplogs in the Montreal Ice Control Structure
{Acres, 1970). The stoplogs were designed to withstand a load
of 7,500 N/m. However, impact of a large ice floe punctured
the steel stoplog and then caused it to buckle. Clearly, the
present design practices must be expanded to include a greater
variety of loading and failure conditions.
[
.~
1
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L_c,
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[
88
Whereas the design practice for determining total loads on
structures is too general and, therefore, inadequate, there
is nothing published to formulate or describe explicitly, the
effect of localized ice action on structure surfaces, pro-
tective rip-rap or natural riverbanks.
6.1 -Damage Related to
Water Level Variation
The theoretical determination of vertical forces on struc-
tures by solic ice covers is difficult due to the time depen-
dency of ice properties {strength, elastic/plastic behavior,
etc). The result is a lack of knowledge on the complex
behavior of a floating ice sheet (Michel, 1978). Many
assumptions have been made to develop formulation for two
analyses--deformation of an infinite plate on an elastic
foundation and brittle failure, or plastic failure at the
ultimate strength of the ice sheet. The formulation for
failure of an infinite plate on an elastic foundation, developed
for ice forces on a straight vertical wall or an isolated pile,
are reported in Michel (1978). The maximum forces and moments
on the structure are equal to the "loading" which causes the
failure of the ice cover. The initial failure is the critical
loading on the structure. Subsequent failures occur at
reduced loads at the weakened initial failure location and are
more difficult to analyze.
Shaitan (1967) observed that the central portion of a solid
cover will follow the smallest changes in water level. The
portion of the ice cover attached to the shore will deform
and acts as a cantilever beam. The analysis of Shaitan calcu-
lates the reactions and moments of the beam transmitted to the
shoreline. The reactions are expressed in terms of design
values of ice stresses which are in turn related to yield
stresses of the ice, allowing for time reduction of stresses.
89
No indication is given, however, of the method of determin-
ation of the appropriate values for ice viscosity and
elastic modulus. Shaitan does note that the initial failure
is critical and that subsequent failures occur when the ice
bears on the initial failure crack. This conclusion reached
by both Shaitan and Michel supports the fundamental assump-
tion of successive cover failures for the formation of hinge
links in Section 4.2.2 of this report. Considering the
difficulty in determining appropriate ice properties for
basic analysis (elastic modulus, ice strength) and the
effect of shoreline contact, the simplified analyses
·presented in Michel (19701. seem most appropriate to estimate
the moment and vertical reaction of the ice cover on the shore.
Michel (1978) notes that the vertical reaction of the ice
cover may be limited by the strength of adhesion of the ice
to the structure surface. There is little data available
on this ice strength which, in the limiting case, may be as
high as the shear strength of the ice itself. This approach
is satisfactory for concrete or steel structure facings but
is inadequate for rip-rap or natural river banks where the
degree of freezing of the banks determines the shoreline
resistance (Rossinskij and Nisar-Mahamedova, 1977). There
is no method to determine the resistance of natural or
placed material to ice action.
Korzhavin (1962) notes that considerable damage to river and
marine protection can be caused by ice action. Individual
stones can be worked from stone masonry by freeze/thaw
cycles, rubble armour lifted from protective layers, and
piles lifted and carried away by floating ice. The damage
caused by impact, rocking and lifting of armour units by ice
can be much greater than the hydrodynamic action on rubble-
mound structures. More damage is caused by spring ice move-
ments (impact) than by midwinter lee action (thermal expansion,
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thrust, lifting), It may be more appropriate to alter ice
conditions than to design a structure to resist the ice forces.
Ice dusting has been used to reduce albedo of the ice surface.
The result is a thinner, weaker cover. Mechanical means such
as blasting, ice cutting and ice breaking with ships have been
used as well. Sommerville and Barnes (1966) describe the use
of an air bubber to prevent further damage, attributed to
rising and falling water level, to the 15-cm thick concrete
reservoir lining of a Winnipeg reservoir. Pekhovich et al
(1970) recommend coating structures with hydrophobic materials
to prevent ice buildup. In the Soviet Union, it is common
practice to coat sea walls and other marine structures with
tar (Tsang, 1975). This has a number of favourable character-
istics. The ice cannot freeze to the tar eliminating vertical
forces and moments, heat absorption by the tar melts the ice
away from the structure reducing horizontal forces due to
thermal expansion, and the layer of tar reduces the freeze-
thaw action by warming and partially sealing the face of
masonry structures.
To prevent ice buildup on isolated structures or mechanical
equipment such as stoplog guides, spillway gates or trash-
racks, electrical heating can be used. Heating must begin
before ice forms on the structure. It is practically
impossible to get rid of the ice once there is a strong
deposit (Michel, 1971). The power required to prevent
freezing is given by
p = 2 • 3 4 ( v b db) 0 • 52
~
where p = power per unit area (W/m2 )
Vb= water velocity past bars (m/s)
db= bar diameter (m)
Sb= spacing between bars (m)
( 6. 1)
9l
6.2 -Damage at Shorelines
Numerous authors discuss the tremendous destructive action
on shorelines due to impact and abrasion of moving ice. The
great mass of moving ice accumulations results in damage even
at low velocities (Tsang, 1975).
Russell, quoted in Bolsenga (1968), relates the observations
of ice damage made in early Arctic geological expeditions. In
shallow areas, the river bed and bank topography is dictated
by ice action which thoroughly removes all vegetation and
creates tremendous furrows in banks and shoals. Bed material
is transported over great distances, huge uoulders pushed up
on the riverbanks, and trees scarred above high water level.
No material ex~ept bedrock can resist the main thrust of ice.
Numerous other authors reported in Bolsenga offer similar
descriptions of great damage and dominance of ice on the river
morphology. Korzhavin (1962) presents many examples of shore-
line and structure damage as well. Details include cutting of
piles by ice abrasion and scour of concrete structures to
depths up to 8 em.
A distinction should be made between the effect of ice on
river morphology and isolated events of ice damage. Newbury
(1968} describes the indirect and direct effects on river
morphology.
Ice affects river morphology indirectly by changing the
hydraulic regime to accommodate ice cover stability require-
ments. A two-layer flow distribution is formed under the
ice cover. If a thin ice cover forms, the change in shear
stress at the river bed is small. If the leading edge Froude
number controls winter flow depth, the flow velocity may be
reduced by staging below summer values and sediment deposition
will occur. Conversely, if a thick cover forms by shoving
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92
or ice deposition, the velocity under the cover will rise and
channel bed erosion may occur.
The direct effects of ice are more obvious. Bed or riverbank
material is lifted or scoured by moving ice. Bed material
carried by moving ice is deposited when the ice melts. The
movement of bed material depends on local ice movements, not
on the bed material. On the Nelson River, movement .of
boulders ranging in size from 0.15 to 4.93 tonnes was moni-
tored. The movement of these boulders ranged from 1 m to
more than 180 m, taking place in shallow bed areas and river-
banks, particularly on islands and in rapids sections where
major shoving and ice jams took place during formation and
breakup.
Evidence of ice action on riverbanks is indicated by the
vegetation trim-line at the maximum level of ice action.
Below this level, a shallow, concave-up groove along the
riverbank is formed by the ice.
Another common feature of ice-covered rivers is side channels,
or terraces, formed by flow bypassing ice jams or staging due
to downstream jams.
Smith (1979) adds that ice processes dominate the river
morphology by enlarging the river cross section. Ice scour
from the ice drive at breakup and ice jams effectively
increase channel cross-sectional area at bankfull flow by as
much as three times. Considering increased velocity, the
cross sections can carry an average of 4.7 times more dis-
charge for the 24 rivers studied in Alberta than for compar-
able rivers in warmer climates. Thus, the recurrence
interval for bankfull flow is increased from 1.6 to 16.7
years.
93
For an established hydrology, ice processes dominate river
morphology. Subsequent ice damage occurs when extreme
combinations of hydrologic and meteorologic conditions
cause correspondingly severe ice jams and movement at break-
up or formation. lf the natural hydrology is altered by
flow diversion, flow regulation or river basin development,
new "damage" may occur as the river morphology adapts to
the new hydrology.
The domination of ice over riverbank characteristics can be
seen at Churchill Falls. With the hydroelectric power develop-
ment, the regulated winter discharge of 1,416 m3/s is much
higher than the natural winter flow of 280 -425 m3 /s. As a
result of the increased winter discharge 1 much thicker ice
covers form at higher water levels. The scouring action of the
ice can be seen as high as 5 m above the normal open-water
elevations. Trees on riverbanks and low islands that were
previously above the ice action have been flattened
ice.
by the
In the analysis of ice piling, the kinetic energy of ice
floes is resisted by the frictional force on the sloping banks
(Croasdale et al, 1978). The total force of the pileup is
expressed as a pressure, analogous to the pressure of a
granular soil. Michel notes that the initial impact of the
ice is generally a greater force than the pressure of the ice
pile. The most significant effect of the ice is to increase
the effective cross-sectional area of the structure in front
of the advancing ice pack.
1~galakov et al (1974) observed that the frequency and limits
to ice pileup on smooth and rough slopes are essentially the
same. The forces on rough slopes, however, are greater. If
the slope roughness is defined in terms of rock armour dia-
meter to ice thickness, then the force of the ice piling on
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94
the slope on a smooth slope (DR/ti = 0} is 39 kN/m, and on
rougher slopes, 49 kN/m (D /t. = 1) and 59 kN/m (D /t = 2) R 1 R i .
The frictional resistance of the slope is assumed on the
basi~ of limited relevant field data for ice/rubble friction
coefficient. This is used to determine the limit of ice
rideup on the sloping structure face. Damage to the face of
the structure is not considered but is generally assumed to
be minor. Michel (1978) states that rock, sandbag or con-
crete block bank protection have poor resistance to scour
when exposed to ice action. Less damage to shorelines
occurs than would be expected, however, because naturally
occurring shorefast ice and ice pileups absorb energy and
protect slopes from scour (Tsang, 1975 and Bruun and
Johanesson, 1971).
Foulds (1974) describes ice piling on the Great Lakes near
several water intakes. At Douglas Point on Lake Huron, ice
mounds reached 13.7 m above water level. On Lake Erie near
Buffalo, ice piled to a height of 9.1 m above water level in
8 m of water.
Kovacs (1980) reviewed reports of ice piling relevant to
arctic or antarctic beaches. Numerous observations of ice
piling in excess of 10 m high were made. Kovacs distinguishes
between ice rideup of sheets of ice and pileups of broken ice.
Pileup seldom occurs more than 10 m inland from the sea but
rideup frequently extends SO m or more inland, regardless of
ice thickness.
At breakup as well, ice jams wash out between vertical shear
lines in the ice cover leaving heavier ice accumulations on
the riverbanks. This ice, grounded or frozen to the river-
banks, protects the banks from scouring action of the moving
ice at breakup. Thus, shorefast ice shculd be promoted.
There are many case studies that attest to the tremendous
forces and damage caused by icc (Bolsenga 1968, Kovacs, 1980).
The Lcports, hOwever, are site-specific, qualitative accounts
which cannot be used to formulate criteria for ice action on
structures because of the number of parameters involved and
the lack of detailed measurements. Because design criteria
are inadequate and the ability of structures to withstand
and direct ice forces requires prohibitively massive struc-
tures, the design approach taken is to eliminate ice action
all together. Protective coatings, mechanical action and
air bubblers are used to eliminate forces from solid covers.
Large hydraulic works are designed to reduce ice impact by
creating low-velocity areas for ice retention or by using
diversion canals, ice booms and ice sluices to direct the
ice away from vulnerable structures.
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96
7 -CALIBRATION OF SOLID ICE
COVER STABILITY PROCESSES
Michel (1971) states that the most severe jams occur when the
ice cover breaks up prematurely. Ice jams formed at an early
breakup are exceptionally strong because the ice cover has not
weakened and a large increase in water level is required to
create sufficient pressure to wash out the jam. This was a
major factor in. the 1968 ice jams on the St. John River
(Atkinson, 1973).
The violence and rapid progression of spring breakup are
responsible for the damage and for the lack of quantitative
data. Data have been collected for calculation of leading
edge stability and internal stability for spring ice jams
but data are not available at the present on ice cover
properties and hydraulic conditions during the sequence of
events leading up to the breakup of the solid cover. It is
not possible, therefore, to calibrate the analysis of the
initial failure of the ice cover at the shoreline or the
stability of hinge formation on the basis of the qualitative
descriptions of solid ice cover breakup given in the few
references located.
1..1 -Observations of Solid Ice
Cover Breakuo and Hinging
The following observations demonstrate the effect of fluctua-
tions in flow on the stability of solid ice covers and support
the assumptions made in hinging analysis.
The analysis of Shaitan (1967), described in section 5.1, gives
an expression for the critical water level variation causing
failure of a solid ice cover frozen to the river banks. The
influence of temperature and rate of loading on ice strengths
97
and stress relax~tion is not clearly rlefined.
Shaitan states that the initial failure is most critical for
structure stability. Subsequent failures occur when the ice
cover bears on the initial failure plane. The formation of
several hinge links can be explained in this way.
There are several reports in the literature that attribute
premature breakup of a solid ice cover to discharge variation
accompanied by water level fluctuation (Townsend 1971,
Gotlib and Razorenov 1970). Smelyakova (1970a) notes more
specifically that ice jams occur at velocities lower than
normally critical for ice jamming when accompanied by fluc-
tuating water level. Many cases are reported where increases
in discharge approaching 100 percent of the normal flow rate
cause disruption of the cover. The zone of influence for
peaking extends downstream of the hydroelectric power plants
over 60 km. No mention of ice problems in forebay reservoirs
or upstream river channels is made. The cases reported are
extreme in terms of discharge variation and the disruption of
the ice cover. Insufficient hydraulic data are presented to
permit analysis of these events or calibration of the concep-
tual model.
Donchenko (1978a, 1978b) states that a water level variation
greater than three or four times the ice thickness will break
the bond between the river banks and the ice cover, and result
in ice jams. The cover initially fails in longshore cracks.
As the ice cover lifts with increasin~ stage, transverse cracks
form and large pans are broken from the leading edge which
then raft and pack. The solution to this instability is to
restrict peaking until a thick cover forms in the area influ-
enced by peaking.
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98
The criterion given for critical rise in water level for
failure of the ice cover contact with the shoreline is
!J.yi = c
where crb = bending strength of the ice (Pa)
t. = ice thickness (m) ~
E =elastic modulus of ice (Pa},
{ i. l)
This criterion has the same form as the relationships presented
by Michel (1970). The coefficient, C, has a value of 34.6 for
"partial connection" to thn shoreline and 17. 3 for "so·-called
hinged attachement". The functions of relaxtion, x and 1/J, are
determined by graphical means, which are not presented in
Donchenko's report.but are referenced to the original Russian
report b~, Kachanov (]947).
The definitions of partial attachment and hinged attachment
are not sufficiently clear. Undoubtedly, the hinged attach-
ment does not correspond to hinging as defined in this report.
It may represent a pinned joint in the beam analysis and
partial attachment may represent a beam with a fixed end.
Flow variation for peaking operation in early winter causes
breakup and ice jams on the South Saskatchewan River (Pentland,
1973}. The solution to this problem is a steady, high discharge
until a strong, solid cover forms to provide maximum conveyance.
Minor flow variation is permitted for the rest of the winter.
No clearly established criteria have been developed.
99
In 1969, a variation of 28 m3;s from the regulated discharge
of 283 m3;s caused breakup of a thin cover shortly after freeze-
up. However, a fluctuation of 113 m3;s in December 1972 caused
only minor jamming. Further studies of the relationship between
ice cover breakup and flow variation were recommended.
On the Winnipeg River, a series of three peaking tests were
performed to evaluate the response of the ice cover to water
level and flow fluctuations {Kartha 1977). The daily peaking
ranges tested were 710 to 1,130 m3 /s, 565 to 1,360 m3;s and
480 to 1,615 m3 /s. Previous peaking operation with a range of
710 m3 /s did not cause ice cover instability so no observations
of ice cover behaviour were made.
At the smallest peaking range tested {710 to 1,130 m3/s); the
ice cover rose and fell with the water level. No cracks or
openings in the cover were observed. The maximum level varia-
tion observed was 0.61 m with an average variation of 0.20 m.
At the middle peaking range. (565 to 1,360 m3/s), hinge cracks
0.15 m wide formed near the shoreline but no breakup of the
cover was observed. No flow over the ice was reported. The
maximum ice cover level variation was 0.86 m.
For the final test (480 to 1,615 m3 /s), the hinge cracks
opened up to a width of 0.76 m. The ice cover broke up at
critical cross sections. Major cover disruption occurred
where the maximum level variation was 1.19 m. Thus, it was
concluded that the critical peaking maximum discharge and flow
variation on the Winnipeg River falls between the second and
third test values.
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100
The interviews with various hydroelectric power utilities
failied to provide quantitative data for calibration of the
conceptual model. The observations made by the various utili-
ties confirm the disruption of the ice cover and formation of
shoreline hinging at peaking stations. The general solution
to cover breakup is a strict curtailment of peaking operation.
The criteria for winter plant operation are established on
the basis of experience. These criteria may be overly conser-
vative and, due to their site specific nature, are of little
help in eliminating ice cover problems at other existing or
proposed developments.
A number of promising field sites for data collection were
identified and are listed in Appendix B. Bearspaw Dam on
the Box River upstream of Calgary specifically mentions pro-
blems with fragmentation and sagging of solid covers (Mr. P.
Dole) • Shulte describes the formation of vertical shear walls
on Alberta rivers between which the solid cover rises and falls
with varying water level. He recommended that covers should be
formed at highest discharge anticipated during the winter
season and that the cover should be "exercised" to maintain a
stable response to fluctuating water levels.
Very few observations of ice problems in reservoirs were
reported. Sokolov (1970) and Sommerville and Burns (1968)
report some damage to concrete reservoir linings and structures
as a result of water level variation. These localized problems
can be solved by weakening the contact with the ice cover with
hydrophobic coatings, air bubblers, heat application or mech-
101
anical action. Similarily, no serious ice problems in reser-
voirs were reported at the Niagara pumped storage plants (Yorke
and Curnmrnons interviews) , the Hollingsworth reservoir (Little-
field interview) or the Northfield Mountain and Blenheim-
Gilboa pumped storage reservoirs.(LeVan 1975). Ice problems
with fluctuating water levels would appear to be a problem only
in the presence of a flow velocity.
Determination of ice cover strength is necessary to apply the
criteria developed in this study. Michel (1978) states that
ice can experience a five-fold decrease in strength in a
single day at breakup. The relationships developed for
compressive strength and ice temperature under laboratory
conditions cannot be applied to field situations until a
remote method of average ice temperature determination is
available. There are no empirical relationships for bending
or compressive strength and degree-days of thawing at present
although such a relationship exists for shear strength
(Tsang, 1975) •
7. 2 -Calibration Data for Hinging
Field data on ice cover hinging are taken from the report
by Fonseca (1979) and a subsequent interview. Field obser-
vations of ice cover levels were taken at two locations on
the Gatineau River downstream from the Paugan Falls generating
station.
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102
At the location identified as KM 37.4, ice cover elevations
were taken at three points across the width of the river on
January 29 and January 30, 1979. These elevations are
shown in Figures 6.1 and 6.2 with water level readings taken
at KM 38. Partial river cross sections are shown as well.
The position of the ice cover on the shoreline at KM 37.4 on
both days was the same at low morning water level. The change
in ice cover levels offshore and onshore on each day reflected
the different water level ranges.
At the location identified as KM 42.7, near the Farrelton
Bridge, horizontal and vertical measurements were taken at
eight locations on the ice cover (see Figure 6.3). In this
case, water level differences,but not absolute water levels,
can be compared with ice cover elevations.
From these figures, the following conclusions can be drawn.
As the water level rises, the central section of the ice
cover follows with an equivalent change in elevation.
-Successive hinge links float free from the banks after the
ice cover offshore rises and the hinge link becomes par-
tially submerged.
These initial data and the photographs in Fonseca's report
identify the hinging process. Some of the as5umptions in
the hinging analysis such as submersion of the links and ro-
tation of links with rising water level have been verified.
There are some questions that cannot be answered about these
field data such as the apparent shortening of the hinge link
101
between points 5 and 6 in Figure 6.3. There is a need for
more photographs and descriptions of the the hinge link and
joint movements before complete calibration of the analysis
of hinging is possible. More field data are required with
wider water level variations to test the extreme limits to
hinge stability.
Observations made by Schulte of ice cover fluctuation between
vertical shear line boundaries illustrate the complexity
and range of hinging action. Any field program is site
specific and, therefore, is limited by the natural range of
parameters at that location. Thus, a laboratory program to
test the complete range of all parameters is required as well
as selective field ~roarams to relate laboratorv results to
prototype field conditions.
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1
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WATER
ICE COVER LEVELS AT km 37.4 (GAT INEAU RIVE:R) JANUARY 29.1979 LEVEL km 38
LEFT BANK
DATUM OF
SHORELINE
PROFILE
APPROXIMATED .
10 20
_1600
2 TIME (HR l
-1700
-1600
= 31700
1640 -1500 1620
1600
IS40
1!$20
ISOO -1400
1440
1420
1400
1340 -1300
1320
1300
1240 -1200
1220
1200
1140 -1100
1120 -900
-1000
1100 IHR)
1040
i100
1020
ICE THICKNESS-30 em
30 40 50 60 70 80 90 100
HORIZONTAL LOCATION (m)
CANADIAN ELECTRICAL ASSOCit\TION
STUDY OF ICE COVERS SUBJECT TO VARYING WATER LEVEL
FIELD MEASUREMENTS OF COVER MOVEMENT
FIG. 7.1
r---"1 l )
9
-E
z
2
.... 8 ~
I.LI
.J
I.LI
.,.
'
ICE COVER LEVELS AT km 37.4 (GATINEAU RIVER} JANUARY 30 1 1979
LEFT BANK
2
DATUM OF
SHORELINE
PROFILE
APPROXIMATED
ICE
0 10 20 30
WATER
LEVEL krn 38
TIME -1700
(HR) -1600
3 -1500
~ -1700 -1400 -== ~ 1600 :§ :~~g
1400 -1300 1:!140
1320
1:!100
1240
-1200
1220
-1200 -== -1100 = 1140 -1000,900 = 1120 ( HR)
1100
1040
1020
9 00
920
THICKNESS-30 em
40 50 60 70 80 90 100
HORIZONTAL LOCATION (m)
CANADIAN ELECTRICAL ASSOCIATION
STUDY OF ICE COVERS SUBJECT TO VARYING WATER LEVEL
FIELD MEASUREMENTS OF COVER MOVEMENT
,-------,
I ' ' J
....----, ' J
FIG 7.2
100
99 -E
z
0 ....
~
LaJ
..J
LaJ
98
. J j
ICE COVER LEVELS AT FARRELTON BRIDGE {GATINE.\U RIVER)-km 42.7 MARCH 7,1979
DATUM OF
SHORELINE
PROFILE
APPROXIMATED
ICE THICKNESS-30 em
5 10
5
TIME
(HR)
7 6
1730
1630
1~50
1510
14$0
1430
1410
13$0
1330
1310
1250
1230
1210
11~0
1130
1110
1050
15 20
WATER
LEVEL AT
FARREL TON
(ARBITRARY
DATUM)
1730
-1650
1550
-1510
1450
-1430
-1410
1350
1330
-1310
-1250
-1230
-1210
-1150
-1130
-1110
-1050
-1030
-1010
-950
-930
(HRl
25
HORIZONTAL LOCATION (m)
CANADIAN ELECTRICAL ASSOCIATION
STUDY OF ICE COVERS SUBJECT TO VARYING WATER LEVEL
. FIELD MEASUREMENTS OF COVER MOVEME1\IT
FIG 73
liJ
8 -CONCLUSIONS AND
RECOMMENDATIONS
8.1 -Conclusions
107
Ice processes may be categorized into those occurring during
initial formation of a solid cover, the continuing competition
of forces in maintaining a dynamic equilibrium of an existing
ice regime with ambient hydrolo~ical and meteorological condi-
tions, and finally those which govern the breakup and melting
of the cover. Characteristic processes pertain to each
channel reach, lake or reservoir depending on the geometry,
climate and the flow management steps which are imposed.
An ice regime is affected by flow velocity and water level
fluctuations during each stage of cover growth, change or
decay. Such effects must be carefully controlled in the
operation of \>rater resource developments in order to avoid
untoward results which could impair the output or benefits of
the project. Similarly, during the design of future develop-
ments, consideration of ice management techniques is an impor-
tant part of maximizing the utility of winter operations.
Based on research performed by many investigators, most of
the ice processes normally encountered can be described by
mathematical formulations and recognized limits. ~orne are
the result of detailed theoretical analyses, while others are
empirical rules derived initially from field observation.
Some are more thoroughly calibrated and verified than others.
The foregoing sections of this report have outlined those
relationships which are most valuable for definition of ice
processes related to typical Canadian utility winter operations.
Taken together, the various formulae and limits form a
conceptual model of ice cover behavior which can serve as a
tool to be applied to both operational and design problems.
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Much research has gone into describing the sequence of initial
ice cover formation, growth and consolidation. Both dynamic
channel processes, and lake effects are well covered
in available literature. The mathematical formulations
currently in use as design techniques adequately predict
prototype behavior in most cases. The effects of varying flow
velocity and/or water level can be assessed by application of
proven relationships for specific circumstances. Appropriate
formulae and guidelines are presented in Sections 3, 4 and 5
of this report.
Midwinter processes during which an ice regime is ~radually
modified by ambient climatic or hydraulic events are also
reasonably well described insofar as the changes in extent
of surface coverage and thickness of a cover are concerned.
Empirical observations have also indicated that a solid
cover is much less sensitive to changes in flow velocity
than a cover during initial formation. Less clear, however,
is the effect of rapid water level changes on an already-
established ice cover. Those relationships pertaining to
midwinter ice processes are covered in Section 4 of this
report.
Because of their dynamic nature, breakup processes are less
well defined than other ice regime stages. Both statistical
and empirical approaches have been outlined in Section 5 of
this report.
Solid ice covers may be susceptable to
when subjected to varying water levels
cover contact with the channel banks.
premature breakup
which destroy the ice
In light of the flood-
ing, power loss and structural damage caused by sudden and
severe ice jams typical of this type of breakup, it is both
surprising and unfortunate that so little quantitative or
qualitative data exist to guide the analysis of solid ice
cover breakup or ice hinge formation and stability. The pre-
109
liminary analysis presented herein (see Section 4.2.2 of this
report) is formulated from a basic hinging concept to develop
simple criteria which can be applied with only basic data
requirements. This analysis is a~propriate for the present
rudimentary understanding of the ice mechanics involved, and
the present level of ability to determine field values of the
parameters needed. Comprehensive laboratory and field studies
are required to confirm the assumptions made in this analysis
and guide its subsequent calibration.
The .annual damages to shoreline structures, \'later resource
developments and the like are a reluctantly accepted part of
life in northern latitudes.
Although cases of severe damage to natural river banks and
man-made structures are reported in the literature, techniques
to forecast ice loadings and assess the probable extent of
damages are unsatisfactory at present. Descriptions of typical
ice effects on structures and shoreline are contained in
Section 6 of this report.
Due to the complexity of ice/structure interaction, it is
likely that research in this area of ice engineering will
continue to be very problem-oriented in scope, addressing
only ad hoc technological requirements. This has been
the case for bridge piers and for artificial islands subject
to drifting Arctic ice pack. In recent years, increases in
hydroelectric peaking operations have created a need to study
the problem of solid ice cover stability in river peaking and
pumped storage hydroelectric plants. Data regarding ice/
structure interaction has, to date, been either site-specific
or only qualitative in nature. Neither approach is of
assistance to the design of more satisfactory shoreline
appurtenances. In the course of future studies, however,
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110
damage to structures, channel linings and reservoir slopes
caused by ice cover movement must also be addressed.
8.2 -Recommendations
On the basis of this literature review and analytical study,
the following recommendations are proposed for further inves-
tigation of solid ice cover stability.
A critical water level variation derived from theoretical
considerations can be calculated from geometrical and mechanical
stability analysis of hinqe link joints to maintain ice cover/
channel bank contact.
Several simplifying assumptions were made in the derivation
and many features of the ice cover contact at the shoreline
could not be expressed mathematically because of the complexity
of the "phenomenon and lack of field observations. A compre-
hensive laboratory and field program is required to calibrate
the analytical criteria developed. The laboratory program
should preceed the field testing. A full range of parameters
can be tested under controlled conditions in order that ex-
treme values of parameters and the various cover failure
mechanisms can be evaluated. Of particular interest is the
evolution of hinge links from initial ice cover failure to
the final hinge configuration, and the influence of ice cover
formation processes and channel geometry on hinge character-
istics. In the course of the laboratory and field tests, the
application of various steady-flow formulae for fragmented
ice cover stability to variable water level and flow velocity
conditions should be investigated. Assumptions made in the
analysis should be verified or new assumptions made based on
test observations and consideration of analytical limitations.
111
The criteria calibrated in laboratory tests should then be
verified in field tests to check possible scale effects in
model testing. Field tests, however, are limited by cost,schedule
and their site-specific nature, as well as the unacceptable
danger of extreme test ran~es. Hence, the need for laboratory
work.
Application of the Pariset and Hausser stability equation for
fragmented ice cover stability to stability of a solid cover
between hinge links should be investigated. The appropriate
values of frictional and cohesive contact across the hinge
links for transfer of cover stresses to the riverbanks must be
determined.
The initial ice cover failure is critical for ice loading on
structures. Ice contact with concrete structures can be deter-
mined from ice adhesion which is limited to ice shear strength.
The ice contact with ar~oured and natural slopes is more complex.
A large number of tests is required under static and flow con-
ditions to account for variations in parameters such as slope
angle, slope covering and underlayer material, and meteorological
conditions affecting ice cover properties and slope freezing.
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ia~ique6 dea Fo4ma~iona d'Embaelea de Glace ~uA le Vnei6~4e,
Poaaibili~l d'un Abaiaaemen~ dea Niveaux aux Embaelea e~
Calcul de. la P4obabili~e de6 Abaiaa emen..ta. Ice Jams and
Frazil in Rivers of the USSR and Measures to Combat Them.
Proc. of the Conferences on Hydrotechnical Coordination.
Vol 56. 1970. Translation from Russian provided by
F. Fonseca, Service Hydraulique, Hydro-Quebec.
Larsen, P. and L. Billfalk. Ice P4oblem6 in Swedi6h Hyd4o
Powe4 Ope4a~ion. IAHR Symposium on Ice Problems, Part 2.
p 235. Lulea, Sweden. 1978.
Llamas, J. Penet4ation o6 Tee Blocka in the Stilling Ba6in
o6 a Vam. IAHR Symposium on Ice Problems. p 123. Lenin-
grad, USSR. 1972.
Maroussenko, Y. I. Action dea Glacea 6u4 le~ 8e4gea de~
Rivie4e6. J. Priroda (Nature). No. 12. 1956. p 91-93.
Translated from Russian by Hydro-Quebec.
Matousek, v. Utilization o6 a Wate4 Re6e4voi4 to Cont4ol
Winte4 Phenomen.a on a Rive4. IAHR Symposium on Ice Pro-
blems. p 196. Leningrad, USSR. 1972.
Michel, B. Th4u6t Exe4ted by an Unconaolidated Ice Coue4
on a Boom. NRC Technical Memorandum T~192. p 163. 1968.
Michel, B. lee Cove46 in Rive46. NRC Technical Memorandum
TM92. Appendix IV(C}. p 231 •. 1968.
Neill, c. B4idge Pivr.a and Simila4 16ala.ted St4uctu4e6.
NRC Technical Memorandum TM92. Appendix III(A). p 191.
1968.
Nevel, D. E. Li6ting Fo4eea Exe4ted by Ice on St4uctu4e6.
NRC Technical Memorandum TM92. p 155. 1968.
Proskuryakov, B. v., v. P.
A Method 6o4 Calculating
Jam& due to Hummocking.
Part 2. Separate Paper.
Berdennikov and S. D. Vinnikov.
St4e66e6 in and Thiekne66 o6 Ice
IAHR Symposium on Ice Problems,
Lulea, Sweden. 1978.
Rozsnyoi, P. and I. Pados. Regulation o6 the Vevelopment o6
1ce-Ba44ie46 in the Reach o6 the Tiaza Rive4 above the
Ba44age o6 Tiazalok 6o4 a Secu4e Winte4 Ope4ation o6 the
Ba44age. IAHR Symposium on Ice Problems. p 279. Hanover,
N.H., USA. 1975 •.
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1:?0
Shlygin, I. A. S~udiea on the E66ee~ o6 lee CoveA on Tidal
Wave PAopaga~ion in the Wate4 A4ea o6 a Tidal PoweA Plant.
INIR Symposium on lee Problems. p 117. Leningrad, USSR.
1972.
Sigurdsson, G. The 8u46ell PAoject: A Caae Study o6 Syatem
Ve6ign 6o4 lee Condition4. IAHR Symposium on Ice Problems.
Paper 4.0. Reykjavik, Iceland. 1970.
Simonsen, C.P.S. and R. W. Carson. lee P4oee44ea Vu4ing
ConatAuetion o6 Limeatone Gene4ating Station. 3rd canadian
Hydrotechnical Conference. p 375. Quebec, Canada. 1977.
Sinotin, v. I. and v. N. Karnovich. On Fo4eea4ting Maximum
Wate4 Stagea Vue to Ice Jama 6oA ~he Caae o6 the Vnieate4
Rive4. IAHR Symposium on Ice Problems. Paper 3.0.
Reykjavik, Iceland. 1970.
Sinotin, v. I. and z. A. Guenkin. E~ude du Plongemen~ dea
Gla~on4 4oua un Obataele. IAHR Symposium on Ice Problems.
Paper 3.7. Reykjavik, Iceland. 1970.
Sinotin, v. I. Speei6ic Featu4e4 o6 lee Jam Fo4mation at
the End o6 the 8ackwate4 Cu4ve. Some Quantitative Regu-
larities. IAHR Symposium on Ice Problems. p 91. Lenin-
grad, USSR. 1972.
Slissky, P. M. Speei6ic Featu4ea o6 Ice Floe 8ehaviou4 in
the Vicinity o6 Ice Retaining S:t4uc:tu4ea. IAHR Symposium
on Ice Problems. p 113. Leningrad, USSR. 1972.
Sokolov, I. N. Augmentation de la Puia4anee de la Cent4ale
Hyd4oelee:tAique de Go4ki Vu4an:t La PiAiode Hive4nal a la
Sui:te d'un AceAoiaament de La Longeu~ de La Polynia dana
La 8ie6 1n6eAieu4. Proceedings of the Conferences on Hydro-
technical Coordination. Vol 42. 1968. Translation from
Russian provided by F. Fonseca, Service Hyoraulique, Hydro-
Quebec.
Sokolov, I. N. FoAmation de Glace A:t:tachle aux PaAoi6 dana
Lea Canaux de t'Ou4at Septent4ionat et Meau4ea PAae:tique4
pouA PAeveni4 Cet:te FoAmation. Fighting Difficulties Due
to Ice in Operation of Hydraulic Installations. Proceedings
of the Conferences on Hydrotechnical Coordination. Vol 81.
1973. Translation from Russian provided by F. Fonseca,
Service Hydraulique, Hydro-Quebec.
Sokolov, I. N. Paaaage du F4azil pa4 un Tunnel de De4iva-
tion de la Cent4al Hyd4oetectAique de ToktogouL. Proceed-
ings of the Conferences on Hydrotechnical Coordination.
Vol 42. 1968. Translation from Russian provided by
F. Fonseca, Service Hydraulique, Hydro-Quebec.
Triquet, c. P~oce~~u~ de Fo~ma~~on de6 Accumula~~on~ de
F~az~l en Amon~ d'un Ba~~age Re~e~vo~~. IAHR Symposium
on Ice Problems. p 103. Leningrad, USSR. 1972.
121
Votruba, L. Accumula~~on e~ Cou~be de Remou~ ~ou6 le Champ
de Glace 6u~ ta R~v~e~e Vta~a. IAHR Symposium on Ice
Problems. Paper 3.5. Reykjavik, Iceland. 1970.
Williams, G. P. F~eeze-up and B~eakup on F~e6h-wa~e~ Lake6.
NRC Technical Memorandum TM92. Appendix IV(A). p 203.
1968.
Zsilak, E. Some New Reta~~on~hip~ o6 ~he Jammed Ice Mo~~on.
IAHR Symposium on Ice Problems. p 243. Hanover, N.H.,
USA. 1975.
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APPENDIX A
ANALYSIS OF HINGE STABILITY
WITH WATER LEVEL VARIATION
A.l-STABILITY OF A
SINGLE-LINK
Detailed observations of hinge joint behaviour are not avail-
able so the mechanisms of joint contact and failure are not
known. The fractured ends of hinge links may assume some
combination of an interlocking shear pin connection and an
ice-over-ice frictional contact. Stability of a single-link
is based on the resistance of hinge joints to buoyant forces
on the submerged hinge links.
The force diagram shown in Figure 4.2 shows that the system
is statically indeterminant with pinned joint connections and
cannot be solved unless a simplified assumption of frictional
resistance is assumed. However, a solution still cannot be
obtained if the frictional coefficient at each joint is the
same. By assuming that the reaction at the higher joint has
a higher frictional resistance than the lower joint, the
force diagram becomes as shown in Figure A-1. For stability,
the summations of forces and moments are zero and the
following equality holds
where amax = the limiting angle of the hinge link
from horizontal
~hl = the fr~ction coefficient at the higher
hinge joint
~h 2 = the friction coefficient at the lower
hinge joint.
(ALl)
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A-2
One limiting case is when 11h 2 equals 11hl anc:l amax equals zero.
'fhe other limit is when uh 2 equals zero and a equals · max
tan-l Cuh 1 /2). The stability of the statically indeterminant
link can only be determined by laboratory and field observa-
tions of the maximum stable link angle. As it is difficult
to separate the individual joints, the limiting case when uh 2
equals zero and a equals tan-l Cuh 1 /2) was selected for max
further consideration.
A.2 -STABILITY OF A
TNO-LINK IIINGF.
A-l
'rhe hinge mechanism will be stable if the frictional resis-
tance of the joints is not exceeded through the water level
cycle. As shown for the case of a single-link hinge, the
joint will be stable if the critical link angle is not
exceeded. The geometrical relationship between water level
change, hinge link length and bank slope was examined starting
at the position shown in Figure A-2. The hinge link lengths
were assumed to be equal, a reasonable assumption if all the·
links were formed within a short time period. The three
cases shown in Figure A-2 were identified as
- a falling water level
- a rising water level up to the level of the river bank
contact
- a rising water level above the level of the river bank
contact.
For the third case, shown in Figure A-3, the following
relationships hold
.2. + z
xl
x2
; 2
I Z. + 1
y
= x3 -
= f:::.y -
= .2.
.2. = y 1 + cosa .2. = constant
2
x2
sin a .2.
x3 p:--;-1
(A2 . 1)
(A2. 2)
(A2. 3)
(A2 • 4)
(A2 .5)
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1\-4
Substituting A2.2 through A2.5 into A2.1
1 + z
j;=z""2-==-+-=l
(A2. 6)
Expanding terms, rearranging and dividing by ~. this becomes
(A2. 7)
!2Y._ + 2 (1 -co sa + ( z
R.
+ sin« -z cosa~ = 0
jz 1 + 1
Therefore, for a rising water level as in Case 3,
= crl + !c 2 - 2 c • rl r2 !
( A2. 8}
where crl = sin a + 1 ----'"L. Jz + 1
cr2 = 1 -cosa + (z + sino. - z cos a)
+ 1
The same solut,ion was obtained for the analysis of Case 2.
For Case 1, a falling water level, the solution obtained was
where
= sin a - 1
fz 2 + 1
= 1 -cos (X + ( z -sin « - z cos a) /z 2 + 1
The solutions to these equations represent geometrically
( A2. 9)
R
I
fLhz • 0
a = tan"
1 (~h)
CANADIAN ELECTRICAL ASSOCIATION
STUDY OF ICE COVERS SUBJECT TO VARYING WATER LEVEL
STABILITY OF A SINGLE HINGE LINK
FIG. AI
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A-'l
possible configurations. The solutions are plotted in
figures 4.5 and 4.6. Interpretation of the curves and their
limitations are discussed in the text of the report.
A-S
possible configurations. The solutions are plotted in
figures 4.5 and 4.6. Interpretation of the curves and their
limitations are discussed in the text of the report.
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RISING WATER LEVEL
ICE COVER ABOVE SHORELINE
CONTACT
RISING WATER LEVEL
ICE COVER BELOW SHORELINE
CONTACT
FALLING WATER LEVEL
r-"" l.
CASE 0
CASE 0
I
I
::---J
CANADIAN ELECTRICAL ASSOCIATION
STUDY OF ICE COVERS SUBJECT TO VARYING WATER LEVEL
DEFINITION SKETCH FOR TWO-LINK STABILITY
. l
FIG.A2
• L ___ _,
,..____,
l,
.-----.,
' J r-l
..
<J I
'--------'-'---'----i!------Jr Y,
l..---------i ------------4.-------r: L --~ ff+l
CANADIAN ELECTRICAL ASSOCIATION
STUDY OF ICE COVERS SUBJECT TO VARYING WATER LEVEL
TWO-LINK STABILITY-CASE 3
r---1 l '
FIG. A3
ill
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APPENDIX B
SUMMARY OF INTERVIEWS
APPENDIX B
B.l -sm~4ARY OF IN'l'ERVIF.WS
Interviews with hydroelectric power utilities were conducted
to identify locations where field data could be obtained to
calibrate the conceptual model. The utilities were questioned
regarding the extent and type of ice problems encountered,
solutions to these problems, field data or documentation
available at present, and specific observations or comments
related to variation in discharge and water level. A list of
Interview Contacts follows this section.
The response of the utilities was favorable and a willingness
to supply available data or to assist in field studies was
expressed, although funding for field studi3s could not be
promised. The observations made by the various utilities
confirm the occurrence of ice cover failure under water level
variation and formation of some forms of shoreline hinging.
However, there is very little documentation available and
that is largely qualitative. Solutions to most severe ice
problems experienced by the utilities have been developed
on the basis of operational experience. Critical limits to
cover stability are now rarely exceeded so assumed stability
limits will be hard to verify.
A specific, follow-up questionnaire on hinge formation was
sent to the utilities where suitable field sites were
identified.
Hinging has been observed in some form on the Peace River in
British columbia, on various rivers in Alberta,on the
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13-2
Winnipeg River in Manitoba, and th(~ Gatincau River in Quebec.
Breakup of solid ice covers has been observed on the Peace
River in British Columbia, on the Box River in Alberta, on
the South Saskatchewan River in Saskatchewan, on the Sturgeon,
Mississagi and Abitibi rivers in Ontario, and on the St. John
River in New Brunswick. The following sites have been identi-
fied as field sites for data collection on the basis of these
interviews.
British Columbia -Peace River downstream of G.M. Shrum
generating station.
Alberta -Box River upstream of Calgary at Bearspaw Dam.
Saskatchewan -South Saskatchewan River at Coteau Creek/
Gardiner Dam power plant.
Manitoba -Winnipeg River from Slave Falls to Lac du Bonnet.
Ontario -Sturgeon River near Sturgeon Falls at the Crystal
Falls generating station.
-Mississagi River north of 'I'hcssalon at Aubrey
Falls generating station.
-Abitibi River near Fraserville between Abitibi
Canyon and Otter Rapids generating stations.
-Niagara River pumped storage generating station.
Quebec -Gatineau River between Paugan and Chelsea.
New Brunswick -St. John River at IJ!actaquac, Beechwood and
Grand Falls generating stations.
Nova Scotia -Wreck Cove generating station.
B-3
ln response tn CJUPSt ions on pumpt'd storaqP •!POI·r-ilt i nq
stations, no reports of damage or serious icc problems were
reported. This is undoubtedly because the ice covers, which
are not subjected to shear stress on the underside, simply
fail at the shoreline and move freely with \v<iter level fluc-
tuations.
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B-4
b.2 -INTERVIEW CONTACTS
The followinq utility personnel were interviewed for this
study.
Mr. W. B. Bancroft, Manager, System Operations, Nova Scotia
Power Corporation
Mr. K. Barrett, Newfoundland Light and Power
Mr. E. Bradford, .Assistant Reaional Operator, Northeastern
Region, Ontario Hydro
Mr. R. W. Carson, Acres Consulting Services, Winni~eg,
Manitoba
Mr. R. Cotanen, Northwestern Region, Ontario Hydro
Mr. Cumm0ns, Lewiston Operations, Niagara Mowhawk Power
Corporation
Mr. L. B. Davies, British Columbia Hydro and Pov1er Authority
Mr. J. Dobson, Saskatchewan Power Corporation
Mr. P. Dole, Calgary Power
Mr. F. Fonseca, Hydraulic Works Division, Hydraulic Servi(:·~,
Hydro-Quebec
Mr. D. M. Foulds, \vater Planning and Management BranGh,
Environment Canada -formerly with Ontario Hydro
Mr. B. Kartha, British Columbia Hydro and Power Authority
Mr. S. T. Lavender, Acres Consultinq Services Limited,
Niagara Falls
Mr. J. Littlefield, Utilities Division, Great Lakes Power
Corporation Limited
Mr. J. Long, Director of Operations, Newfoundland and
Labrador Power
Mr. J. Long, General Manager, Northern Canada Power Commission
Mr. T. Schulte, Calgary Power
Mr. T. Wigle, River Control Engineer, Ontario Hydro
Mr. J. Woodcraft, Eastern Region, Ontario Hydro
Mr. R. Yorke, Niagara River Control, Ontario Hydro