With your background in mathematics, you would appreciate this...
High Risk Structures, such as the Oroville Dam & complex, are required to be "Risk managed" such that ANY risk is kept As Low as Reasonably Practical. The higher the risk to life, property, et al. the greater the ALARP factor must be (lower ALARP coefficient).
How this is translated into engineering is providing demonstrate-able proof that each critical component has a KNOWN operational safety margin with an additional factor of safety (FOS) on top of this.
What was so alarming in the initial press statements by DWR - during the Feb crisis - of "sometimes you get a flat tire" and "you run out of oil (engine)".. This reveals a cultural pattern in thinking whereby it is accepted that sometimes you have to "fix it when it breaks". (Prior thread posts reveal DSOD documents where this was actually stated regarding "drummy patches" on the spillway (i.e. "we'll fix it when it breaks")).
This clearly goes against FERC's and other Federal Government Dam Safety Risk Management Guidelines mandate of Dam operation of RIMS (Real-Time, Interactive Risk Management)*.
DWR has zero working piezometers inside the dam - they cannot determine the internal phreatic level (risk to a potential collapse) inside the earthen dam. Piezometers are deemed critical in immediate assessment of dams during and after an earthquake.
DWR built the original spillway on highly erodible material (clay) (and other design defects). The spillway blew up from Stagnation Pressure & Hydraulic forces at 18% of its rated design. Yet their own Final Geology Report clearly revealed these wide swaths of clay - but were ignored for evidence of cost reasons (schedule and cost impact if excavated to the original engineer's requirements to competent rock).
DWR fought having to cover the Emergency Spillway with an Apron - they said that the destructive erosion is within the parameters of the definition of an "Emergency Spillway". DWR also reasoned that the use of the Emergency Spillway would have been extremely rare. Even when asked directly (via 2014 FERC Part 12D analyses and FOR's filing to FERC on the dangers of the ES spillway erosion), DWR engineers stated that the bedrock was competent on the hillside. Yet it failed at 3% of its design rating.
DWR was told not to operate the River Valve without the Baffle ring by UC Davis scientists/engineers - DWR ignored this and operated it anyway, leading to near death of one individual and severe injuries to multiple others from the vacuum effects created due to the baffle ring issue. DWR hid this information from the press for years until OSHA fines and reports exposed what really happened.
DWR has known for nearly two decades about the "end of life" and "cracking" and failures of the anchor tendons. Yet they continue to "calculate", "fudge numbers", and "calculate" scenarios where they can operate the Radial Gates even with losses of more anchor tendons. The gates have exhibited chronic alignment, leaking, and jamming issues.
DWR has been ignoring, but painting with orange paint, a now 16+ foot long crack in a 5 foot thick pier that holds secure one of the trunnion pins to Gate 8. Water has been seeping through this large crack (leaving mineral deposits), likely causing decades of moisture exposure to the trans-secting reinforcement steel (rebar). This rebar is critical to maintain the max 0.017" torsional movement to keep Gate 8 (and Gate 7 by anchor block association) from jamming the gate. Yet DWR just keeps adding orange paint when the crack grows.
There is more evidence that goes on and on in this environment of the opposite of ALARP & RIMS. That is why the "Patch and Pray" moniker evolved. DWR has managed the failure of two spillways (Main and Emergency). What's next?
*Safety Risk Management Guidelines require continuing formal data gathering and analyses that allow future hazard developments to be properly detected, analyzed, and corrected - mitigated when necessary ('Real-Time', Interactive Risk Management).
Long rant coming. Sorry ‘bout that..
Yes, this is a situation that might be modeled mathematically. Plenty of people know how to do this. I’ve even done a little bit of that myself.
It would most likely involve many variables, and thousands of simulations run with separate probability distributions and cross correlations on each of the variables. You would most likely want to make multiple runs using different underlying assumptions as a way of testing the sensitivity of the results to the various estimates used. The final result would be a sensitivity analysis and an outcome probability distribution.
Caveat: The outcome would certainly look “scientific,” but the “garbage in, garbage out” rule would still apply. Anyone who knows how to do this could skew the result if they were so inclined. Spotting what they did would require knowledge of the underlying engineering issues plus knowledge of the model specifics.
I hope someone is actually doing this type of analysis, and the data they use is unbiased. If they are, there seem to be enough things that might go wrong, and a high enough probability associated with each, that the combined risk profile an analysis like this would generate might not look so good... The public would never see it.
HOWEVER...
You don’t need advanced math and a computer to know that the more things that MIGHT go wrong, the higher the odds are that at least one of them WILL go wrong.
You don’t need advanced math and a computer to figure out that if you don’t fix something it is likely to get worse.
You don’t need advanced math and a computer to know that if a dam is leaking enough to create a large green spot you should dam sure figure out why.
The giant underlying exogenous variable is the weather. If this winter is as wet as last winter, then the spillway will again be needed, and the gates/tendons/pillars will again be put to the test. Absent repairs, sooner or later they will fail that test.
Even if the spillway is not used this winter, the green spot will be an issue whenever the lake rises to a level high enough for more water to leak out. This does NOT seem to be a self-sealing leak..