ruary 1987 (Daly and Arcone 1989). The wide vari-
Table 5. Representative coefficients of variation
ability in piece size given in this figure is also evi-
(COV) based on data from Harr 1987 and some
dent in qualitative descriptions contained in the
of the ice properties discussed in text.
IJDB of ice piece size provided for breakup jams.
For example, White (1996): "The ice pieces near
COV
Property
Material
(%)
the toe appear to be thermally grown ice, four to
eight inches thick, with occasional thicker pieces.
Porosity (p)
Soil
10
The pieces are neither exceptionally large or small,
Specific gravity (γ)
Soil
2
Internal angle of friction (φ)
and are in the range of two to six feet in diameter,
Soil
10
Internal angle of friction for sand (φ)
with the mode closer to the smaller size." White
Soil
7
Internal angle of friction for gravel(φ)
Soil
12
(1993) and Tuthill (1993) also made observations
Cohesion (c)
Soil
40
of piece size and distribution for two ice jams that
Thickness
Ice
17
formed on the Black River in Coventry, Vermont,
Flexural strength
Ice
20
in 1993. The first jam formed in early January and
Crushing strength
Ice
13
was made of blocks generally 8 inches to 10 inches
Flow velocity
Ice
33
35a
Ice roughness (ni)
Ice
thick, ranging up to 18 inches. This jam froze in
19b
Porosity (frazil ice in laboratory)
Ice
place and was followed by a later jam in March,
32c
Intrinsic permeability of frazil
Ice
for which ice pieces averaged 1 to 3 feet by 0.5 ft
(laboratory)
thick. Lever et al. (1997) reported that ice piece size
69d
Intrinsic permeability of frazil
Ice
near the toe of breakup jams ranged from three to
(field)
100e
Piece size
Ice
four times the ice thickness, similar to estimates
by Wuebben and Stewart (1978).
aCalculated
from data presented in White and Daly (1997).
bCalculated from data presented in Andersson and Daly (1992).
cWhite
(1991).
dWhite
and Lawson (1992).
RELIABILITY CONSIDERATIONS
eCalculated from data presented in Daly and Arcone (1989).
In selecting an appropriate range of values for
use in modeling, it is valuable to have some idea
tory is reasonably reliable. It is clear that additional
of the reliability of the properties described above.
work is required before the intrinsic permeability
The coefficient of variation (COV) can be obtained
of frazil ice in the field reaches an acceptable level
from the distribution of known values:
of reliability. The same is true for the coefficient of
σ(x)
internal strength () and field values of porosity,
COV =
(30)
E(x)
whether for breakup jams or freezeup jams. Based
solely on experience with soils, one would expect
where σ(x) is the standard deviation of the sample
that values of porosity and friction angle for ice
could be developed with some reliability, while it
and E(x) is its expected value, or mean.
might be difficult to achieve the same level of reli-
Generally, sample sizes of the ice properties dis-
ability for ice cohesion. It is likely that further
cussed above are too small to develop this infor-
study could result in reasonably reliable estimates
mation; the exception would probably be ice
of frazil particle size; however, it is unlikely that
roughness. However, knowledge of the COV for
the same can be developed for ice piece size.
various soil properties will provide an indication
of the order of COV for similar ice properties. Harr
(1987) summarizes the COVs for a number of soil
and ice properties and provides references as well
SUMMARY AND CONCLUSIONS
(see Table 5). He notes that a COV less than about
An overview of ice jam hydraulics (juxtaposi-
25% indicates a level of reliability acceptable in
tion, undercover deposition, and the formation of
most engineering applications, while a COV
an ice jam from shoving and internal collapse)
greater than about 40% indicates unreliability.
identified a group of variables important in hy-
Using the information in this table as a rough
draulic and physical processes, namely
guide, we could consider that present knowledge
of ice-influenced velocity, ice roughness, porosity
Ice cover roughness (ni)
of frazil ice deposits in the laboratory, and perhaps
intrinsic permeability of frazil ice in the labora-
Cohesion (Ci)
24
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