Coupling with transport
The implementation of the smart Kd-concept into the transport code d3f++ is depicted in Fig. 1.
Fig. 1: Diagram of the smart Kd-concept implemented in d3f++ including the smart Kd-matrix generated with PHREEQC/UCODE.
As described in Smart Kd-values as multidimensional matrices the Kd‑matrices are calculated by coupling procedure of UCODE with PhreeqC. All parameters impacting radionuclide sorption need to be treated in the transport code. This comprises the ionic strengths, calculated by the density distribution, and the explicit transport of the components Ca, DIC, protons (representative for pH), aluminum and sulfate. The respective Kd‑values for the radionuclides are then depicted from the pre-calculated Kd‑matrices. The points in the multidimensional matrices of smart Kd‑values of course will not exactly correspond to parameter combinations encountered in a specific time/space step of the reactive transport code. Thus, the retrieval of a realistic smart Kd‑value includes algorithms for multidimensional nearest-neighbor searches and coupled averaging, see Fig. 2 for a generic illustration.
Fig. 2: Illustration for a multidimensional matrix of smart Kd‑values and a search for nearest neighbors.
Finally, a post-processing step is performed to equilibrate the components considering calcite (gibbsite) dissolution or precipitation and based on that update the Kd‑values and radionuclide concentrations.
A first implementation into the reactive transport code d³f++ together with an application case served as proof-of-concept. Whereas the site-specific field data are unique for each application, the thermodynamic data required to fully describe aqueous speciation, surface speciation, and precipitation/dissolution processes are standardized. This also applies for parameters of activity coefficient models required at moderate or high ionic strengths.