Buffer erosion and canister corrosion, hole by hole

For a KBS-3 repository the question is: in how many deposition holes does flowing, dilute groundwater erode the bentonite buffer far enough that sulphide in the water reaches the copper canister by advection, and how many canisters does that sulphide corrode through within the assessment time? SKB answered it in SR-Site and again in the PSAR with an Excel workbook, ErosionCorrosionModel_2_0 (SKBdoc 1895159, documented in SKBdoc 1895157), later ported to Python. This page is that calculation, restructured: the arithmetic in one module, the file readers in another, and every parameter carrying the name it has in the workbook and in the Python port. Nothing is uploaded; the files you open stay in the browser.

The answer takes three inputs: a table of every deposition hole from the hydrogeological modelling (flow, aperture, travel time, transport resistance, and which holes are cut by large fractures), a distribution of sulphide concentrations, and the parameter set on the left. What comes back is the table of failure times that feeds the radionuclide transport calculations, the key outputs of the workbook’s Info sheet, the distribution plots of its five plot sheets, and a histogram of the failure times with the canisters and the sulphide values behind them.

What happens to each hole

1. Rejection

Holes that would not be used are rejected by criteria on what could be observed during construction. A hole is rejected if any selected criterion holds: FPC ≠ 0 (a fracture that intersects the full tunnel perimeter also cuts the hole); EFPC ≥ the number of consecutive holes (a fracture cuts this and four more neighbouring holes); FLEN above the fracture-length limit; transmissivity above 10⁻⁶ m²/s and FLEN above 443 m (the T/L criterion); optionally a Darcy flux above a limit, or membership of an inflow-rejection list. Rejected holes take no part in what follows but stay in the total: the corrected outputs are scaled by N_canisters / (N_tot − N_rejected), 6000 canisters by default.

Transmissivity is not in the hydro file; it is back-calculated from the transport aperture δ with Doe’s relation T = (δ/0.5)² (TR-09-20, equation 3-2), or with the cubic law T = (δ/a)³ if you choose it.

2. Flow at the hole

The hydro model gives the Darcy flux U₀ in the fracture crossing the hole. The water velocity in that fracture is

v = U₀·w/δ

with w = 5 m, the height Serco used to turn a fracture flow into a Darcy flux, so that using it again recovers the flow (R-09-20, equation 3-8 and D-7). A DarcyTools table carries the velocity itself and is used as it is. The flow through the hole once the buffer has gone is

q_eb = f_conc·U₀·w·2r_h

and the equivalent flow rate that carries sulphide to the copper (TR-06-33, chapter 6; TR-10-42, appendix) is the smaller of q_eb and 1.13·√(q_eb·D_w·V_zone)/d_buffer, where V_zone = d_buffer·π(r_h² − r_can²)/2 is the volume of the eroded half of the buffer. The two cross at q_lim, 0.0885 m³/yr with the default geometry.

3. Time to advective conditions

The buffer is advective once m_buffadv = 1200 kg has been lost. The loss rate depends on the model chosen:

t_adv = m_buffadv / (rate · f_tdilute)

where f_tdilute is the fraction of the time the water is dilute enough to erode: 0.25 in SR-Site, 0.5 in the PSAR. A hole with no flowing fracture, or a rejected one, never becomes advective (1E+99 in the tables). “Initial advection” sets t_adv = 0 everywhere, the case of no buffer from the start.

4. Corrosion and the failure table

With the buffer advective, sulphide at concentration [HS⁻] corrodes the copper at the rate Q_eq·[HS⁻]·M_Cu/(f_HS·ρ_Cu·A_corr) over the area A_corr = πr_can·h_corr (TR-10-66, 4.3.2–4.3.3), so the time to corrode through d_can = 47 mm is

t_corr = CorrHoleFact / (Q_eq·[HS⁻]),   CorrHoleFact = d_can·A_corr·ρ_Cu / (f_HS·M_Cu)

the copper in the corroded height, in kmol of sulphide. The sulphide distribution is a table of values; for each accepted hole the model finds the smallest value that fails the canister before the assessment time (HSmin), counts how many table values exceed it, and writes one row per such value: t_fail = t_adv + t_corr, with the hole’s F, t_w and q beside it. Holes whose particle did not reach the surface (OKFLAG ≠ 0) are left out of the table but not rejected. The mean number of failed canisters is the number of rows divided by the number of sulphide values, and the corrected figure scales that to the normalising number of canisters over the accepted holes.

5. The distribution plots

Cumulative distributions over the ensemble of holes of: the hydro model’s own Q_eq, with and without the addition for a thermally spalled zone (TR-10-66, equations 4-9 to 4-12); the corrosion rate through an intact buffer at a fixed sulphide concentration, with and without spalling (TR-10-66, equation 4-19a); the buffer loss rate of the chosen model; the corrosion rate through an eroded buffer at the fixed concentration; the hydraulic gradient; the aperture; the velocity; and the time to advection. Each is drawn for all holes and again after rejection, with the reference lines the workbook draws (5 cm of copper in 10⁶ and 10⁵ years; 1200 kg in the assessment time and in the dilute fraction of it).

6. The failure histogram

A hole that fails for m of the n sulphide values has m rows in the failure table, one per value, and each row stands for 1/n of a canister: the chance that the hole meets that value. The histogram adds 1/n for each row whose failure time falls in a bin, scaled by N_canisters / (N_tot − N_rejected) as the corrected mean is, so its bars add up to the mean number of failed canisters. A finer sulphide distribution gives the same canister more rows, not taller bars. Counting rows instead is one of the choices; those bars add up to the length of the table. With several realisations a bar is the mean over them.

The bars can be split by the decade of the sulphide value behind each row, which shows the part of the distribution that the failures at each time come from, or by canister: the eight that contribute most on their own, the rest together. The legend gives each colour’s share of the failures, its part of the bars. A table under the chart sets each decade’s share of the sulphide values beside its share of the failures: a decade holding few of the values can give most of the failures.

The split can also be made by the failures themselves: the sulphide values that fail a canister, highest first, in two to five groups that each give the same share of the failures, the legend naming the range of [HS⁻] each group holds. With five, the cuts are at 20, 40, 60 and 80 % of the failures, each moved to the nearest boundary between two sulphide values, since a group holds whole values; cuts that land on the same boundary are one. Where single values give more than a group’s share there are fewer groups, and the legend and the table give the shares as made.

Or by how probable the sulphide values are: by their percentiles in the table, a value’s percentile being the share of the table below it, so that the highest of n values lies at 100·(n − 1)/n. The table is cut at 50, 90, 99 and 99.9, which resolve the upper tail where the failures come from, at its quintiles or quartiles, or at up to four percentiles of one’s own. The legend gives each group’s percentiles and share of the failures, and under them the range of [HS⁻] the group holds; a table too short to fill a group leaves it empty, and the table under the chart says so.

Below the bars each canister is a curve of its failure time against the sulphide concentration,

t_fail = t_adv + CorrHoleFact / (Q_eq·[HS⁻]),

from the highest value of the table down to the hole’s HSmin, where the curve meets the assessment time. At high concentrations it turns up towards t_adv: the buffer has to be eroded before any sulphide reaches the copper. Beside the curves is the share of the sulphide table at or above each concentration, with a dot for each canister at its HSmin, level with the foot of its curve. The share there is the canister’s chance of failing in time, and those chances add up, over the canisters, to the mean number of failed canisters. The second table under the chart has one row per canister that fails, however many sulphide values fail it.

Hydro files

Four shapes are read, detected from the content rather than the extension:

Every file becomes one realisation. Load ten and you have the ten-realisation semi-correlated case of the PSAR: the summary shows min, mean and max over realisations, the failure table carries the realisation number, and the distributions can be pooled. The page never keeps your files beyond the visit; settings are kept in the browser.

Sulphide

HSForsmark and HSLaxemar are the workbook’s tables to the digit; HSTest is the ten-value table of the code test. The generated distribution replaces the workbook’s one-point HSGeneric: a shifted lognormal by mean, standard deviation and shift (the 2.4 port’s _gen_hs_generic), or a log10-normal by μ and σ (the @Risk workbook’s 10^RiskNormal(−5.79757, 0.63849)), drawn as midpoint quantiles so that the same parameters always give the same table. A custom list is any text with numbers in it.

Checking it

The code documentation comes with a hydro test file built to be checked by hand (TestCaseHydro_2_0, SKBdoc 1895160). It is SKB’s and not part of this page: open your copy here with the SR-Site settings (OldKTH, 25 % dilute, HSTest), and the failure table and the key outputs are the ones SKBdoc 1895157 documents. When the page was built it was also checked against the PSAR base case, where the PSAR defaults give 612 rejected holes, 41 advective positions at one million years, three at 100,000, four failure times, a corrected mean of 0.0827632 and an earliest failure at 189,441.71 years — the numbers cached in the workbook itself — and against the Python port’s failure tables in six configurations. resources/tests/erosion_corrosion/test-model.js exercises the test case and the TR-16-11 rates read off that report’s figures 4-5 and 4-7 (given a local copy of the test file), the closed forms, single holes through every model and option, the readers for every file format, and that the bars of the failure histogram add up to the key outputs however long the sulphide table is.

Two places where the sources differ from each other, and what this page does: the workbook’s VBA Lambert W stops at a residual of 10⁻³ while the Python port uses SciPy’s — this page solves to machine precision, which matters only with “force rRSS” off; and the Python port’s HSForsmark table is rounded to six digits, so its failure times sit about 10⁻⁶ from the workbook’s — this page carries the workbook’s digits. The count “failed canisters at the highest sulphide concentration” follows the workbook (holes with at least one failing value), which the Python port matches with the default filters on.

Parameters

Every setting and parameter, with its default (the PSAR workbook’s), its name in the workbook and its attribute in the Python port. A changed value is marked in the panel and listed on the Summary tab; Save case writes them all to a JSON file that Open case… reads back, and the Excel export puts them on its first sheet.

References