178
12 Consequences
situation. It is indeed hard to believe that the maximum is 46.94 − 45.14 = 1.8
distance to the mean when the standard deviation is 1.63, leading to 1.1 sigma bounds.
That situation can’t even be modelled with a J-shaped beta empirical distribution.
The graph shown in Fig. 12.2 has other blatant anomalies regarding its skewness.
Given the strong non-linearity of τ y and μ m , working with “arbitrarily selected
average normal and high” C v values from Figs. 12.1 and 12.2 is hazardous. Indeed
this could lead to under- or over-estimates of flooding in the dam break analyses.
Since there is a chance that a probabilistic analysis would lead to more favourable,
reasonable parameters, it is worthwhile to devote some extra effort to it.
As both τ y and μ m are function of a single variable C v (see Eqs. 12.1 and 12.2)
we can get estimates for both variables using two point estimates of C v using point
estimates methods as shown in Table 12.4. Below are the P+, P− point estimates
(see Rosenblueth 1975 for details) for the graph and values of Fig. 12.1:
P− = 45.14 − 1.63 = 43.51
P+ = 45.14 + 1.63 = 46.77
The interesting result brought by this example is that assuming “normal tailings”
at the average value of C v = 45% leads to underestimate the average value of τ y
and μ m by at least 4%. Furthermore, given the large COV due to the non-linearity of
the τ y and μ m functions, using the average value does not appear to be a reasonable
selection.
Instead, considering the two variables and building four point estimates, would
make it possible to perform a point estimate of the flood model and determine probabilistically the hazard zones.
One more important note. The rheology of a particular material can be highly
type-dependent, and chemical and mineralogical properties can influence the tailings
rheology. Thus a fluid’s parameters can vary from one mine to another due to various
reasons, including difference in ore and processing methods. Finally, the tailings
Table 12.4 Estimates for τ y and μ m using two point estimates of C v using the point estimates
methods (Rosenblueth 1975)
C v %
%
Evaluated
standard
deviation
Value for
average
Variation
%
C.O.V.%
Average
45.14
C v +
46.77
0.4677
Standard
deviation
1.63
C v −
43.51
0.4351
τ+
16.10
τ average 12.63
12.15
3.99
τ−
9.17
τ var
12.00
3.46
27.43
μ+
5.52
μ average 4.27
4.09
4.54
μ−
3.03
μ var
1.55
1.25
29.15
12 Consequences
situation. It is indeed hard to believe that the maximum is 46.94 − 45.14 = 1.8
distance to the mean when the standard deviation is 1.63, leading to 1.1 sigma bounds.
That situation can’t even be modelled with a J-shaped beta empirical distribution.
The graph shown in Fig. 12.2 has other blatant anomalies regarding its skewness.
Given the strong non-linearity of τ y and μ m , working with “arbitrarily selected
average normal and high” C v values from Figs. 12.1 and 12.2 is hazardous. Indeed
this could lead to under- or over-estimates of flooding in the dam break analyses.
Since there is a chance that a probabilistic analysis would lead to more favourable,
reasonable parameters, it is worthwhile to devote some extra effort to it.
As both τ y and μ m are function of a single variable C v (see Eqs. 12.1 and 12.2)
we can get estimates for both variables using two point estimates of C v using point
estimates methods as shown in Table 12.4. Below are the P+, P− point estimates
(see Rosenblueth 1975 for details) for the graph and values of Fig. 12.1:
P− = 45.14 − 1.63 = 43.51
P+ = 45.14 + 1.63 = 46.77
The interesting result brought by this example is that assuming “normal tailings”
at the average value of C v = 45% leads to underestimate the average value of τ y
and μ m by at least 4%. Furthermore, given the large COV due to the non-linearity of
the τ y and μ m functions, using the average value does not appear to be a reasonable
selection.
Instead, considering the two variables and building four point estimates, would
make it possible to perform a point estimate of the flood model and determine probabilistically the hazard zones.
One more important note. The rheology of a particular material can be highly
type-dependent, and chemical and mineralogical properties can influence the tailings
rheology. Thus a fluid’s parameters can vary from one mine to another due to various
reasons, including difference in ore and processing methods. Finally, the tailings
Table 12.4 Estimates for τ y and μ m using two point estimates of C v using the point estimates
methods (Rosenblueth 1975)
C v %
%
Evaluated
standard
deviation
Value for
average
Variation
%
C.O.V.%
Average
45.14
C v +
46.77
0.4677
Standard
deviation
1.63
C v −
43.51
0.4351
τ+
16.10
τ average 12.63
12.15
3.99
τ−
9.17
τ var
12.00
3.46
27.43
μ+
5.52
μ average 4.27
4.09
4.54
μ−
3.03
μ var
1.55
1.25
29.15