Radionuclide solubility by the Simple Functions

When a canister has failed and water has come in, the concentration of a radionuclide in the water inside is limited by the solubility of the least soluble solid it can form. The Simple Functions of Grivé et al. (SKB TR-10-61) turn a groundwater composition into that limit for 20 elements — Sr, Ra, Zr, Nb, Tc, Ni, Pd, Ag, Sn, Se, Th, Pa, U, Np, Pu, Am, Cm, Sm, Ho and Pb — from a small, fixed speciation scheme per element and a few candidate solids chosen by expert judgement. SKB used them in SR-Site, as an Excel workbook sampled with @Risk (SKBdoc 1282962), and in the PSAR: the equilibrium constants are sampled within their uncertainties and the groundwater is drawn from a table of compositions, which gives a distribution of solubilities per element for the transport calculations. This page is that calculation. Nothing is uploaded: the files you open stay in the browser.

How a solubility is calculated

1. Activity corrections

Every constant is corrected from I = 0 to the ionic strength of the water with the Oelkers & Helgeson form of the Davies equation (TR-10-61, equation 2), for a species of charge z:

log γ_z = −z²·0.5091√I / (1 + 1.5√I) − log(1 + 0.0180153 I) + 0.064 I

log K′ = log K° + Σ_reactants ν log γ − Σ_products ν log γ

Water and O₂(g) count as neutral species in the sums, as they do in the workbooks, and the pH is taken as −log[H⁺], with the H⁺ activity coefficient carried in K′. The correction is written here from each reaction as it stands (the Data tab shows it); the SR-Site workbooks wrote it by hand, and where the two differ the workbook's is used unless you tick every term from the reactions (see below).

2. The water: free ligands and the redox state

The input is the total Ca (Ca + Mg), Cl, Na, SO₄ and Si, the ionic strength and the free HCO₃⁻. Carbonate follows from the bicarbonate, [CO₃²⁻] = [HCO₃⁻] / (K′_HCO3[H⁺]); free sulphate, calcium and sodium are solved together from their mass balances over CaOH⁺, CaCO₃, CaHCO₃⁺, CaSO₄, NaCO₃⁻, NaHCO₃, NaSO₄⁻, HSO₄⁻, FeSO₄ and FeHSO₄⁺; chloride is corrected for FeCl⁺. No carbonate is reduced to methane and no sulphate to sulphide, and calcite is not allowed to precipitate: when the water is oversaturated the Summary says so, as the workbook's warning did.

Version B (SR-Site, the PSAR): the water inside the canister is taken to be in equilibrium with the corrosion products of the cast-iron insert. The redox state is the magnetite/goethite boundary at the pH of the water,

Eh = (0.059/4)·(−4(−logK_mag − 3(−logK_goe)) + 83.1 − 4 pH)·1000 mV,   pe = Eh/59.16,   pO₂ = 10^{−83.1 + 4 pH + 4 pe}

with log K = 5.49 for 3Fe²⁺ + 3H₂O + ½O₂ = Fe₃O₄ + 6H⁺ and 8.75 for Fe²⁺ + 1.5H₂O + ¼O₂ = FeOOH + 2H⁺ (TR-10-61, equations 14–17), and [Fe²⁺] follows from goethite at that pO₂. The constants in the Eh are those at I = 0 and the one in [Fe²⁺] is corrected to the ionic strength, as in the workbooks; the 0.059 is theirs too (RT ln10/F is 0.05916 V, an option).

Version A (TR-10-61, section 3.1): the Eh and the total iron of the water are inputs, pO₂ follows from the Eh, and [Fe²⁺] from the iron mass balance over FeOH⁺, Fe(OH)₃, Fe(OH)₄⁻, FeCO₃, FeHCO₃⁺, FeSO₄, FeHSO₄⁺ and FeCl⁺, solved together with the sulphate balance. The table needs columns for Eh (mV) and [Fe]tot.

3. One element

Each aqueous species j of the element is a complex of n_j atoms of its master species M (UO₂²⁺, Np⁴⁺, SeO₄²⁻, …) with the ligands, so its concentration is K′_j[M]^n_jΠ[L]^ν. Summing by nuclearity gives Eq₁ = 1 + Σ_monomers, Eq₂, Eq₃. Each candidate solid fixes the free master species: for a solid that releases m atoms of M,

P = [M] = (K′_s Π[L]^ν)^1/m,   S = P·Eq₁ + P²·Eq₂ + P³·Eq₃

and the solubility of the element is the lowest S over its candidate solids. The candidate that gives it is the controlling solid. Above 0.01 mol/kg the workbooks labelled the element “not solubility limited” (the concentration would then be set by the inventory, not by a solid); the value itself went into the results all the same, and it does here, with the count of such realisations reported.

Uncertainty

Every constant with an uncertainty ΔlogK is sampled from a normal distribution with mean logK° and standard deviation ΔlogK/2 (so ΔlogK is about a 95 % range), independently of the others, with a Latin hypercube per constant when that box is ticked — as @Risk does. The groundwater of each realisation is drawn from the table: at random with replacement, as SR-Site's RiskIntUniform, or without replacement, or in order, or one water throughout. Unticking sample the equilibrium constants gives the variability from the groundwater alone; choosing one water only gives that from the constants alone. Every constant has its own random stream, keyed on its name, so changing one constant or adding a species does not change the draws of the others, and the same seed gives the same numbers.

One water. The tab gives the solubility of each element at the nominal constants, the solubility of every candidate solid, the aqueous speciation at the controlling solid, and a linearised uncertainty: σ_{log S}² = Σ (∂logS/∂logK_i·σ_i)² with σ_i = ΔlogK_i/2, the derivatives taken by central differences through the whole calculation (the major-ion mass balances included), and the share f_Z of each constant (TR-10-61, equations 5–13). The ± shown is 2σ_{log S}, the same confidence as the ΔlogK. TR-10-61's tables quote a ± 2.3 times smaller: its workbooks take ΔK = K·ΔlogK where the propagation of a logarithmic range needs K·ln10·ΔlogK, and report 0.434·ΔS/S. For the example of its table 3-2 this page gives the report's solubilities and controlling solids exactly, and its ± after division by ln 10, except for Nb, Pa and Pd, whose workbook derivatives have slips of their own.

The thermodynamic data

The dataset is the one of the SR-Site calculations: the reactions and constants of the SR-Site data report (TR-10-52, tables 3-29 to 3-32) with the digits of the workbooks, and the plutonium correction of February 2011 (Pu(CO₃)₃³⁻ has ΔlogK = 1.4; the first workbooks had 10.6, which put the upper tail of the plutonium solubility above 1 mol/kg). The Data tab lists every reaction with its constant, its uncertainty, the activity term and the ligand dependence derived from it, and a check that it balances. Any of them can be edited, species and solids added or removed, and a solid left out of the minimum; Save data writes the dataset as JSON for Open data… to read back. A reaction is written with the master species of the element, the solid (for a solid) and the basis species H⁺, H₂O, O₂(g), CO₃²⁻, SO₄²⁻, Cl⁻, Ca²⁺, Na⁺, Fe²⁺ and H₄SiO₄, charges sign first (Th+4, SO4-2, Fe(OH)4-), and terms separated by “ + ” with spaces.

Where the SR-Site workbooks differ from their own reactions

Writing every term out from the reactions and comparing with the workbooks cell by cell turned up these. They are kept by default, so that the SR-Site and PSAR numbers come out; each has an option.

Ticking the first four changes the SR-Site distributions by less than 0.03 log units for every element but Sm (about 0.09 lower); the fifth changes no distribution, only how Am and Cm go together.

Groundwater tables

One row per water, with a header row naming the columns. The names are recognised loosely: pH, [Ca]tot (m) or Ca, [Cl]tot (m), [Na]tot (m), [SO4-2]tot (m), [Si]tot (m), IS (mol/kg) or I, [HCO3-] (m), and for Version A Eh (mV) and [Fe]tot (m); a column Mg is added to Ca when that box is ticked. CSV, semicolon- or tab-separated text and Excel workbooks are read; from a workbook the sheet called Groundwater is taken if there is one, otherwise the first sheet with such a header, so the parameter workbook of the PSAR can be dropped as it is. Rows with a missing value are left out and counted. Concentrations are mol/kg water; where one is not known the workbooks asked for a very small number rather than zero (10⁻²⁰).

The built-in examples are the reference waters of TR-10-61, table A-1 (from TR-06-09), with Mg apart, the more reducing Eh where two are given, and 10⁻²⁰ for silicon, which the table does not give; and the example water of its table 3-1. The Simple Functions were built for I ≤ 0.2 mol/kg, 25 °C, 6 < pH < 11 and −8 < pe < 14; waters outside are calculated and flagged.

What comes out

What the method leaves out

• The radionuclide does not act back on the water: pH and ligand concentrations are those of the groundwater whatever dissolves. For elements that reach 10⁻³–10⁻² mol/kg (Ni, Sr in some waters) a full equilibrium calculation would let the dissolution shift the pH and use up ligands.
• In Version B, [Fe²⁺] is what goethite allows at the fixed pH and pO₂. It goes as 10^12−2pH roughly: 10⁻² mol/kg at pH 7, 10⁻¹ at pH 6.5. Such a water would precipitate siderite, and its ionic strength is not updated for the iron; iron sulphate complexes then lower the free sulphate (raising the Sr and Ra solubilities), and the Se solids, whose solubility falls with [Fe²⁺], come out lower. The One water tab shows [Fe²⁺] and the Summary its range.
• Temperature is 25 °C throughout; TR-10-61 found up to two log units of difference at 0 and 100 °C.
• Activity corrections are of the Davies type, meant for I ≤ 0.2 mol/kg; TR-10-61 found no significant difference from SIT up to 1.9 mol/kg for U, Th, Se, Np and Ni.
• Only the species that make up at least 10 % of an element in the range the scheme was built for are present, and only the candidate solids chosen by expert judgement can form.

Checking it

The arithmetic reproduces the cached results of the SR-Site workbooks (SKBdoc 1282962) for two groundwaters to 13 significant digits in every element, solid, ligand and activity coefficient; its distributions over the PSAR groundwater set are indistinguishable from the PSAR solubility data by the two-sample Kolmogorov–Smirnov test for all 20 elements; and in Version A it reproduces the example of TR-10-61 (table 3-2) — every solubility, every controlling solid and the ranking of the constants. node resources/tests/simplefunctions/test-model.js runs the checks that need only public data.

References