# The UFD Decay Process Model¶

The Used Fuel Disposition Decay Process Model performs radionuclide isotope decay, ingrowth, and phase partitioning, for the simulation of a nuclear waste repository. It has been developed under the Generic Disposal Systems Analysis (GDSA) work package, which is under the Spent Fuel and Waste Disposition Program of the U.S. Department of Energy (DOE) Office of Nuclear Energy, as part of the Spent Fuel and Waste Science and Technology (SFWST) Campaign. Development of the UFD Decay Process Model is ongoing, and lead by Paul Mariner, Glenn Hammond, and Jennifer Frederick, at Sandia National Laboratories.

## General Algorithmic Design¶

The UFD Decay process model is called each time step of the simulation. Before the simulation begins, the process model initializes the sorbed amount of each isotope in equilibrium with the user-specified aqueous concentration and the material-specific, elemental Kd value, according to,

$C^{sorb}_i = C^{aq}_i Kd_e$

where $$C^{aq}_i$$ is the user-specified aqueous isotope concentration with units of [mol/kg-water], $$Kd_e$$ is the material-specific, elemental Kd value with units of [kg-water/m3-bulk], and $$C^{sorb}_i$$ is the sorbed concentration of isotope $$i$$ with units of [mol/m3-bulk].

At each time step, the total mass of each isotope [mol] is summed up according to,

$M^{total}_i = M^{aq}_i + M^{sorb}_i + M^{ppt}_i$

where $$M^{aq}_i$$ is the mass of isotope $$i$$ in the aqueous phase, $$M^{sorb}_i$$ is the mass of isotope $$i$$ in the sorbed phase, and $$M^{ppt}_i$$ is the mass of isotope $$i$$ in the precipitated phase. The total mass, $$M^{total}_i$$, is then allowed to decay according to the Bateman Equations. These equations are solved according to a 3-generation analytical solution derived for multiple parents and grandparents with non-zero initial daughter concentrations and solved explicitly in time, or a fully implicit solution for any number of generations can be used instead by including the keyword IMPLICIT_SOLUTION within the UFD_DECAY block.

Once the isotopes have gone through the decay and ingrowth calculation, the total mass of each isotope is partitioned back into aqueous, sorbed, and precipitated phases. First, mole fractions are calculated to determine the fraction that each isotope contributes to total element mass,

$X_i = \frac {M^{total}_i} {M^{total}_e}$

where $$X_i$$ is the mass fraction for isotope $$i$$, $$M^{total}_i$$ is the total mass [mol] for each isotope $$i$$, and $$M^{total}_e$$ is the total mass [mol] of the corresponding element $$e$$.

Based on the total element mass and elemetal Kd value, the element aqueous concentration is calculated according to,

$C^{aq}_e = \frac {M^{total}_e} {1000 \left({1+Kd_e/(\rho \phi S_{l})}\right) V \phi S_{l} }$

where $$C^{aq}_e$$ is the aqueous concentration [mol/L] of each element $$e$$, $$M^{total}_e$$ is the total mass [mol] of the element $$e$$, $$\rho$$ is the water density [kg/m3], $$\phi$$ is the material porosity, $$S_l$$ is the pore water saturation, $$V$$ is the grid cell volume, and $$Kd_e$$ is the material-specific, elemental Kd value with units of [kg-water/m3-bulk]. If the aqueous concentration of the element exceeds the elemental solubility limit, then the aqueous elemental concentration is set equal to the elemental solubility limit. The remaining element mass is then partitioned between sorbed and precipitated phases, according to

$C^{sorb}_e = C^{aq}_e Kd_e$

where $$C^{aq}_e$$ is the aqueous elemental concentration with units of [mol/kg-water], $$Kd_e$$ is the material-specific, elemental Kd value with units of [kg-water/m3-bulk], and $$C^{sorb}_e$$ is the sorbed concentration of element $$e$$ with units of [mol/m3-bulk]. If the aqueous element concentration was set to the solubility limit, then the precipitated phase is calculated according to

\begin{align}\begin{aligned}M^{ppt}_e =& M^{total}_e - M^{aq}_e - M^{sorb}_e\\C^{ppt}_e =& \frac {M^{ppt}_e V^{mnrl}_e} {V}\end{aligned}\end{align}

where $$M^{ppt}_e$$ is the precipitated mass [mol] of element $$e$$, $$M^{total}_e$$ is the total mass [mol] of the element $$e$$, $$M^{aq}_e$$ is the aqueous mass [mol] of element $$e$$, $$M^{sorb}_e$$ is the total sorbed mass [mol] of element $$e$$, $$V^{mnrl}_e$$ is the molar volume [m3/mol] of the precipitated element, and $$V$$ is the grid cell volume [m3].

The isotope concentrations are calculated from the partitioned elemental concentrations by multiplying by the isotope mole fractions,

$C_i = X_i C_e$

## 3-Generation Explicit Solve¶

The default routine for solving the decay and ingrowth equations is a 3-generation analytical solution derived for multiple parents and grandparents with non-zero initial daughter concentrations. This approach is documented in Section 3.2.3 of Mariner et al. (2016), SAND2016-9610R.

Mariner, P.E., E.R. Stein, J.M. Frederick, S.D. Sevougian, G.E. Hammond, and D.G. Fascitelli (2016), Advances in Geologic Disposal System Modeling and Application to Crystalline Rock, FCRD-UFD-2016-000440, SAND2016-9610R, Sandia National Laboratories, Albuquerque, NM.

## Implicit Solve¶

The user can specify the keyword IMPLICIT_SOLUTION to solve for decay and ingrowth using an implicit, direct solve of the Bateman equations for any number of generations. The governing equation for isotope decay and ingrowth is,

$\frac {d C_i(t)} {d t} = -\lambda_i C_i(t) + \lambda_p S C_p(t)$

which describes the change in isotope concentration over time ($$\frac {d C_i(t)} {d t}$$) due to its own decay (if any) ($$-\lambda_i C_i(t)$$) plus ingrowth (if any) from the isotope’s parents ($$\lambda_p S C_p(t)$$), where $$\lambda$$ is the decay rate constant [1/sec] and $$S$$ is a stoichiometry coefficient. The equation is discretized and rewritten in terms of a residual equation as follows,

$f\left({c^{k+1,p}}\right) = \frac {c^{k+1,p} - c^{k}} {\Delta t} - R\left({c^{k+1,p}}\right)$

where $$f\left({c^{k+1,p}}\right)$$ is the residual, $$c^{k+1,p}$$ is the solution for concentration at the $$k+1$$ time step and the $$p^{th}$$ iterate, $$\frac {c^{k+1,p} - c^{k}} {\Delta t}$$ is the discretized accumulation term (e.g., the left hand side of the governing equation above), and $$R\left({c^{k+1,p}}\right)$$ is the source or sink term (e.g., the right hand side of the governing equation above).

A Jacobian matrix is formed according to,

$J_{ij} = \frac {\partial f_i(c^{k+1,p})} {\partial c_j^{k+1,p}}$

which is a matrix of all the partial derivatives of the solution with respect to each unknown variable. Using Newton’s method, which solves the following system,

$J \delta c^p = -f(c^{k+1,p})$

the concentration can be updated according to,

$c^{k+1,p+1} = c^{k+1,p} + \delta c^p$

Note: The governing equation is reformuated in terms of isotopes and the isotopes’ daughter(s) in the source code, rather than the isotopes and isotopes’ parent(s) formulation shown here.