# Contaminant Transport from a Deep Geological Repository: Lumped Parameters Derived from a 3D Hydrogeological Model

^{*}

## Abstract

**:**

## 1. Introduction

#### Some Basic Notions

## 2. Used Data and Models

^{93}Mo, with a half-life of 4000 a. Linear sorption was considered in the 3D model and the LPM (Kd concept), the K

_{d}values were 0.0005 m

^{3}/kg for Se and 0.01 m

^{3}/kg for Cs. The source concentration was set (as fictitious) to 0.001 kg/m

^{3}, the same for all tracers, constant in time—to determine the transport pathway parameters, not for a real temporal evaluation in the safety assessment. Elements I, Se, and Cs were selected as inactive tracers, primarily to demonstrate the effects of their different sorption coefficients. The

^{93}Mo was selected as an active non-sorbing tracer because its half-life is of a similar order of magnitude as the transport time. Nuclides with higher half-lives may appear stable on this time scale, and nuclides with shorter half-lives do not reach the interface with the biosphere at all.

## 3. LPM Parameter Calculation

#### 3.1. Concept of the Procedure

**Inflow**(i.e., groundwater flow from the storage area to the geosphere).**Length**(i.e., the distance traveled by groundwater from the storage area to the geosphere/biosphere interfaces).**Flow time**(i.e., the time taken for groundwater to travel the distance from the storage area to the interface with the biosphere).**Dilution**(the outflow/inflow ratio).- ○
- Alternatively,
**outflow**can be considered instead (the flow rate of contaminated groundwater from the geosphere to the biosphere).

**Volume**of the transport path.- Data on the
**porosity**of the transport path (minimum, average, maximum). **Flow areas**:- ○
**Outflow area of the source**, i.e., an area where the contaminant outflow from the storage space into the “geosphere” is present—geosphere inflow area.- ○
**Total contaminated outflow area**(from the geosphere to the biosphere).

**Breakthrough curves**of selected tracers, i.e., the time courses of concentrations at the effluent into the biosphere, which are responses to unit jumps of concentrations at the source at time zero. These curves provide the data needed to determine the flow time.- The same list of data for individual river basins (parts of the model).

#### 3.2. Lumped Parameters Calculation

^{−8}is used). For the outflow area, the tolerance was based on the ratio of the flux through the assigned “contaminated” area to the total flux out of the model. A value of 0.99999 was used for this case. The example of the tracer-accessed volume visualization is presented in Figure 3. We chose a contaminant outflow fraction of 0.99999 (99.999%); this choice was to limit the effect of the excessive dilution of the total flux. This number was a matter for the evaluator/evaluation team performing the safety analysis. It was possible to choose, e.g., 0.5 (50%), or to exclude flows with such low concentrations that the substance would not be present at all; that is, those where the concentration was lower than would correspond to the presence of at least one atom/molecule.

- “Geosphere” inflow area ${S}_{G}^{in}$. Considering the repository volume is a part of the model domain, the interface repository/geosphere is supposed to be defined in the model input. The area is composed of the element sides between the repository subdomain and the geosphere subdomain with water flux in the respective direction. The procedure could be affected by the backflow described in Section 2. A compensation factor of the ratio between the total tracer outflow from the model (asymptotic in time) and the total tracer outflow from the repository (including the backflow), is used for correction.
- Water inflow into the “geosphere” ${Q}_{G}^{in}$ —calculated as a sum of fluxes through the respective element sides, considering an eventual correction for the backflow.
- Length of the transport pathway $L$. It is calculated by backtracking the tracer in the mesh. Starting from each outflow contaminated element, a neighbor element with the highest tracer flux is considered upstream in the path, and the distance of the gravity center is added to the length. Then, the average of all paths is made and weighted by fluxes in the respective outflow elements.
- Contaminated outflow area ${S}_{B}^{out}$—the sum of element sides based on the above threshold choice.
- Outflow rate of contaminated water ${Q}_{B}^{out}$—the sum of the fluxes through the “contaminated outflow area”.
- Volume $V$, porosity $\eta $ of the “path”—the sum of element volumes except the backflow cases, minimum ${\eta}_{min}$ and maximum ${\eta}_{max}$ values along the path.
- The dilution factor is postprocessed as the ratio of the outflow to the inflow values above.

#### 3.3. Processing of Dirac Pulse Model Results

_{i}is the tracer retardation factor of tracer i, [-], t is the time step index, t′

_{t,i}is the transformed time in step t, for time series tracer i, [T], t

_{t}is the time in step t, [T]

^{2}[T

^{2}] is calculated as

^{2}·T

^{−1}], T is the mean flow time, [T], L is the total length of the transport pathway, [L], v is the mean flow velocity, [L·T

^{−1}].

## 4. Example Site Ordering

- Inflow, i.e., the flow of groundwater from the repository space into the “geosphere”.
- Dilution of the outflow element with the maximum tracer concentration.
- Dilution of the outflow element with the maximum tracer flux.
- Mean dilution on the outflow elements in the “contaminated outflow area” (0.99999 shares of the total tracer flux).
- Flow time to the outflow element with the maximum concentration.
- Flow time to the outflow element with the maximum tracer flux.
- Mean flow of the outflow elements in the “contaminated outflow area”.
- Flow length to the outflow element with maximum tracer concentration.
- Flow length to the outflow element with the maximum flux of the tracer.
- Minimum flow length over all elements with a tracer flux, i.e., included in the “contaminated outflow area”.
- Mean flow length over the outflow elements in the “contaminated outflow area”.

## 5. Comparison of LPM and the 3D Model

#### 5.1. Evaluation and Optimization Procedure

_{3D}

_{,j}is the tracer j outflow according to the 3D model [M·T

^{−1}] fl

_{LP,j}is the tracer j outflow according to LPM [M·T

^{−1}], j is the tracer index, t is the time [T], T is the simulation time [T]. The normalization (the denominator in the formula) helps to obtain comparable results of the model fit for different evaluated sites or model geometries.

- The 3D model (Flow123d results as the example).
- LPM with the parameters calculated by the integral method (Section 3).
- LPM with optimized parameters based on the defined criterion (GoldSim processed).

#### 5.2. Example of One Sample Site Processing

^{2}] of the “Pipe” component is calculated by Equation (6) from other evaluated data

^{3}·T

^{−1}] (i.e., outflow ${Q}_{B}^{out}$ introduced above), T is the travel time [T], and η is the porosity of the filling [-]. It means that the derived generic transport path parameters overdetermine the pipe component parameters—the evaluated areas ${S}_{G}^{in}$ and ${S}_{B}^{out}$ were not used at all to keep the model consistent with the travel time according to (6). Additional parameters not necessary for the LPM calculation are included in Document S3.

^{2}(that is about 1000 times larger than the model area). This is due to the fact that the entire “Pipe” component comprises the flow rate equal to the “outflow” (flow of groundwater from the geosphere to the biosphere), which is orders of magnitude larger than a realistic inflow. To comply with the transport time, Equation (6) leads to a large cross-section. This problem is resolved in Section 5.4 and Section 5.5.

#### 5.3. Results of the Single Pipe Model for All Sites

#### 5.4. Results of Two Cascade Pipes Variant

#### 5.5. Results of Three Cascade Pipes Variant

#### 5.6. Results of Parallel Pipe Structures

## 6. Conclusions and Outlook

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- GoldSim Technology Group LLC. GoldSim Contaminant Transport Module, User's Guide; GoldSim Technology Group LLC: Seattle, WA, USA, 2018; Volume Contaminant Transport Module Version 7.1. [Google Scholar]
- Chopra, M.; Nair, R.N.; Sunny, F.; Sharma, D.N. Migration of radionuclides from a high-level radioactive waste repository in deep geological formations. Environ. Earth Sci.
**2015**, 73, 1757–1768. [Google Scholar] [CrossRef] - Becker, D.A.; Cormenzana, J.L.; Delos, A.; Duro, L.; Grupa, J.; Hart, J.; Landa, J.; Mariovoet, J.; Orzechovski, J.; Schröder, T.J.; et al. Safety Indicators and Performance Indicators, deliverable (D-N°:3.4.2), PAMINA—Performance Assessment Methodologies in Aplication to Guide the Development of the Safety Case; European Commission: Brussel, Belgium, 2009. [Google Scholar]
- SKB. Radionuclide Transport and Dose Calculations for the Safety Assessment SR-PSU, Revised ed.; SKB TR-14-09; Svensk Kärnbränslehantering AB: Stockholm, Sweden, 2015. [Google Scholar]
- Geier, J.E.; Lindgren, G.A.; Tsang, C.-F. Simplified Representative Models for Long-Term Flow and Advective Transport in Fractured Crystalline Bedrock. Hydrogeol. J.
**2019**, 27, 595–614. [Google Scholar] [CrossRef] - Rechard, R.P.; Wilson, M.L.; Sevougianc, S.D. Progression of performance assessment modeling for the Yucca Mountain disposal system for spent nuclear fuel and high-level radioactive waste. Reliab. Eng. Syst. Saf.
**2014**, 122, 96–123. [Google Scholar] [CrossRef] - Vopálka, D.; Lukin, D.; Vokál, A. Modelling of processes occurring in deep geological repository—development of new modules in the GoldSim environment. Czechoslov. J. Phys.
**2006**, 56, D623–D628. [Google Scholar] - Huff, K.D. Rapid methods for radionuclide contaminant transport in nuclear fuel cycle simulation. Adv. Eng. Softw.
**2017**, 114, 268–281. [Google Scholar] [CrossRef] - Gylling, B.; Romero, L.; Moreno, L.; Neretnieks, I. Transport from the canister to the biosphere: Using an integrated near- and far-field model. MRS Online Proc. Libr.
**1996**, 465, 1037. [Google Scholar] [CrossRef] - Říha, J.; Královcová, J. Generation of Transport Paths in Fractured Porous Media. Acta Polytech.
**2017**, 57, 348–354. [Google Scholar] [CrossRef] - Jeong, J.; Lee, Y.-M.; Kim, J.-W.; Cho, D.-K.; Ko, N.Y.; Baik, M.H. Progress of the Long-Term Safety Assessment of a Reference Disposal System for High Level Wastes in Korea. Prog. Nucl. Energy
**2016**, 90, 37–45. [Google Scholar] [CrossRef] - Robinson, B.A.; Chu, S. A residence-time-based transport approach for the groundwater pathway in performance assessment models. Comp. Geosci.
**2013**, 52, 155–163. [Google Scholar] [CrossRef] - Hagedorn, B.; Clarke, N.; Ruane, M.; Faulkner, K. Assessing aquifer vulnerability from lumped parameter modeling of modern water proportions in groundwater mixtures: Application to California’s South Coast Range. Sci. Total Environ.
**2018**, 624, 1550–1560. [Google Scholar] [CrossRef] [PubMed] - Delsman, J.R.; de Louw, P.G.B.; de Lange, W.J.; Oude Essink, G.H.P. Fast calculation of groundwater exfiltration salinity in a lowland catchment using a lumped celerity/velocity approach. Environ. Model. Softw.
**2017**, 96, 323–334. [Google Scholar] [CrossRef] - Murakami, H.; Ahn, J. Development of compartment models with Markov-chain processes for radionuclide transport in repository region. Ann. Nucl. Energy
**2011**, 38, 511–519. [Google Scholar] [CrossRef] - Caldini, F.; De Sanctis, J.; Girotti, T.; Zio, E.; Luce, A.; Taglioni, A. Monte Carlo estimation of radionuclide release at a repository scale. Ann. Nucl. Energy
**2010**, 37, 861–866. [Google Scholar] [CrossRef] - Březina, J.; Stebel, J.; Exner, P.; Flanderka, D. Flow123d. 2021. Available online: http://flow123d.github.com (accessed on 30 June 2022).
- Říha, J.; Uhlík, J.; Grecká, M.; Baier, J.; Černý, M.; Gvoždík, L.; Havlová, V.; Královcová, J.; Maryška, J.; Milický, M.; et al. Transport Models; Final Report; TZ 324/2018/ENG; Radioactive Waste Repository Authority: Prague, Czechia, 2018; p. 105. [Google Scholar]
- Milický, M.; Uhlík, J.; Gvoždík, L.; Polák, M.; Černý, M.; Baier, J.; Jankovec, J.; Královcová, J.; Grecká, M.; Rukavičková, L. Final Detailed 3D Groundwater Flow Models of Potential Deep Repository Sites; Final Report; TZ 323/2018/ENG; Radioactive Waste Repository Authority: Prague, Czechia, 2018; p. 196. [Google Scholar]
- Vokál, A.; Havlová, V.; Hercík, M.; Landa, J.; Lukin, D.; Vejsada, J. Update of the Reference Project of a Deep Geological Repository in a Hypothetical Locality, Stage III Initial Safety Report Study, C. Documentation Part C.2 Long-Term Safety Evaluation of Deep Geological Repository; ÚJV Řež, a.s.: Husinec, Czech Republic, 2010; p. 133. [Google Scholar]
- Landa, J. Modelling the Impact of the Radionuclides Migration Parameters from the Repository into the Biosphere. Ph.D. Thesis, Czech Technical University in Prague(CTU), Prague, Czechia, 2012. [Google Scholar]

**Figure 1.**Scheme of a conceptual model of the deep geological repository for radioactive waste, highlighting the granite massif (i.e., the “geosphere” compartment) as the barrier of interest and the meaning of the LPM representation.

**Figure 2.**Possible flow cases through source elements. The source elements are filled in brown–red. Green arrows indicate the inflow of clean (uncontaminated) water, red arrows indicate the outflow of contaminated water from the source. Hereinafter, three possible types of backflow are indicated here: (1) due to “source jagging” (black arrow), not actually the case in this work if the geometry is defined appropriately, (2) due to the division of the storage space into more discontinuous parts (purple arrows), and (3) due to the actual backflow caused by curved groundwater flow lines (blue arrows).

**Figure 3.**Concentration field of the continual tracer injection from the repository volume with a threshold value to detect the transport pathways. In this figure, the total transport path; that is, all the elements that are affected by the advection of the tracer from the source. The source itself has a dark red color; above it, the arrow indicates the main inflow of groundwater, and other arrows indicate the flow from the source elements further into the “geosphere”. This example shows the backflow in which the edge parts of the storage space are affected.

**Figure 4.**Workflow of the developed method, with two final steps: the lower is the intended use of the method and is presented in Section 4 as an example; the upper is only considered as a part of this paper for justification and was done once.

**Figure 5.**(

**a**) Comparison of the values of the “total outflow of contaminants” for the tested sites, (

**b**) contribution (ratio) of the non-sorbing radioactive tracer to the total value of the indicator.

**Figure 6.**Comparison of tracer fluxes at site l1 from the Flow123d model (blue curves), from one “Pipe” component set from the postprocessing data (green curves), and the optimized “Pipe” component (red curves).

**Figure 7.**Tracer travel times for test sites for the non-sorbing tracer, the average of stable tracers (“stable”) and for optimized parameters (“opt”); there are also arithmetic (“aver.”) and geometric (“geom.”) averages of all sites.

**Figure 8.**Comparison of outflows of tracers at site l3 according to Flow123d (blue curves), according to the three cascade “Pipe” components set from postprocessing data of the non-sorbing tracer (green curves), and according to the optimized cascade of the three “Pipe” components (red curves).

**Figure 9.**Arrangement of the GoldSim model of the three parallel paths, formed by pairs of cascade “Pipes”.

**Figure 10.**Comparison of tracer outflows at site l1 according to Flow123d (blue curves), according to the three pairs of cascade “Pipe” components set from postprocessing data of the non-sorbing tracer (green curves), and according to the optimized group of three pairs of cascade “Pipe” components (red curves).

**Table 1.**Example of two variants (weight choice explained in the text) of the site evaluation, resulting in the order of suitability (selected indicators shown due to limited space: 1,2,4,5,7,8,10,11).

Id | Inflow [m^{3}/a] | Dilution [-] | Transport Time [a] | Pathway Length [m] | Evaluation | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Max Conc. Elm. | Mean of All Elms. | Max Conc. Elm. | Mean of All Elms. | Max Conc. Elm. | Min. Length Elm. | Mean of All Elms. | Variant 1 | Variant 2 | ||||

(1) | (2) | (4) | (5) | (7) | (8) | (10) | (11) | Sum Indic. | Order | Sum Indic. | Order | |

l1 | 792 | 545 | 4205 | 11,589 | 25,478 | 1194 | 995 | 2597 | 37 | 3 | 63 | 1 |

l2 | 2116 | 15 | 531 | 2059 | 4725 | 1670 | 728 | 2674 | 69 | 6 | 164 | 8 |

l3 | 1637 | 589 | 4695 | 15,785 | 16,646 | 4647 | 1012 | 4414 | 25 | 1 | 65 | 2 |

l4 | 1648 | 69 | 2742 | 14,217 | 17,406 | 4323 | 650 | 3609 | 51 | 5 | 112 | 5 |

l5 | 1143 | 153 | 4162 | 49,095 | 20,071 | 7142 | 762 | 3813 | 29 | 2 | 68 | 3 |

l6 | 2085 | 44 | 1538 | 1105 | 13,096 | 855 | 663 | 2220 | 80 | 8 | 154 | 7 |

l7 | 4599 | 5 | 202 | 4087 | 4960 | 1604 | 723 | 1734 | 84 | 9 | 201 | 9 |

l8 | 2369 | 245 | 1416 | 5844 | 18,495 | 2490 | 1013 | 3971 | 46 | 4 | 142 | 6 |

l9 | 729 | 69 | 914 | 11,438 | 14,788 | 740 | 491 | 1381 | 74 | 7 | 111 | 4 |

**Table 2.**Parameters obtained by the postprocessing and by the optimization used to set one “Pipe” component. The last row represents the evaluation of fit between the LPM and the 3D model.

LPM Parameter | Units | Postprocessing | Optimized |
---|---|---|---|

Flow length | m | 2597 | |

Cross section | km^{2} | 3724 | 5516 |

Longitudinal dispersivity | m | 4707 | 5161 |

Porosity of the filling medium | % | 0.8774 | |

Inflow | m^{3}/a | 791.9 | |

Outflow | m^{3}/a | 3,329,904 | |

Dilution | - | 4205 | |

Transport time | a | 25,478 | 37,741 |

Value of rel. opt. criterion (5) | % | 0.2 | 0.007 |

**Table 3.**Longitudinal dispersivities (m) for test sites postprocessed from the non-sorbing tracer, from the average of the stable tracers (“stable”) and the optimized values. There are also (arithmetic) and geometric averages over all sites.

Site | l1 | l2 | l3 | l4 | l5 | l6 | l7 | l8 | l9 | Aver. | Geom. Aver. |
---|---|---|---|---|---|---|---|---|---|---|---|

Non-sorb. | 4.707 | 6.079 | 12.597 | 6.902 | 7.015 | 20.273 | 6.955 | 7.904 | 6.223 | 8.739 | 7.909 |

Stable | 5.488 | 7.065 | 13.391 | 8.788 | 8.764 | 20.862 | 9.150 | 9.719 | 7.162 | 10.043 | 9.322 |

Optimized | 5.162 | 5.374 | 7.288 | 12.012 | 18.973 | 25.219 | 4.311 | 5.174 | 3.573 | 9.676 | 7.677 |

**Table 4.**Values of the relative criteria (%) for test sites for models derived from the non-sorbing tracer data, from the average of the stable tracers (“stable”), and for the optimized model. There are also (arithmetic) and geometric averages over all sites.

Site | l1 | l2 | l3 | l4 | l5 | l6 | l7 | l8 | l9 | Aver. | Geom. Aver. |
---|---|---|---|---|---|---|---|---|---|---|---|

Non-sorb. | 0.208 | 0.095 | 0.488 | 0.038 | 0.128 | 0.074 | 0.25 | 0.57 | 0.555 | 0.27 | 0.19 |

Stable | 0.086 | 0.042 | 0.332 | 0.026 | 0.148 | 0.035 | 0.241 | 0.425 | 0.477 | 0.2 | 0.13 |

Optimized | 0.0068 | 0.0025 | 0.0032 | 0.0137 | 0.0372 | 0.0066 | 0.0007 | 0.0022 | 0.0015 | 0.008 | 0.004 |

**Table 5.**Postprocessed and optimized parameters of both the pipes in the cascade configuration and their total values (sum or average, respectively), for site l1. The variant “min–max” for the porosity and the optimization option with two independent dispersivities are shown.

Parameter | Units | First Pipe | Second Pipe | Total | |||
---|---|---|---|---|---|---|---|

Postproc. | Optimized | Postproc. | Optimized | Postproc. | Optimized | ||

Flow length | m | 2129 | 1987 | 468.0 | 609.3 | 2597 | |

Cross section | m^{2} | 2,390,830 | 514,868 | Not defined | |||

Longitudinal dispersivity | m | 4707 | 4123 | 4707 | 3650 | 4707 | 4012 |

Porosity of the filling medium | % | 0.3964 | 0.6051 | 5.0 | 0.6045 | 0.8774 | |

Inflow | m^{3}/a | 791.9 | 791.9 | 791.9 | |||

Outflow | m^{3}/a | 791.9 | 3,329,904 | 3,329,904 | |||

Dilution | - | 1 | 4205 | 4205 | |||

Transport time | a | 25,474 | 36,304 | 3.62 | 4.71 | 25,478 | 36,309 |

Opt. criterion | kg^{2}/s | 0.1969 | 0.02919 | ||||

Rel. criterion | % | 0.35 | 0.051 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Landa, J.; Hokr, M. Contaminant Transport from a Deep Geological Repository: Lumped Parameters Derived from a 3D Hydrogeological Model. *Energies* **2022**, *15*, 6602.
https://doi.org/10.3390/en15186602

**AMA Style**

Landa J, Hokr M. Contaminant Transport from a Deep Geological Repository: Lumped Parameters Derived from a 3D Hydrogeological Model. *Energies*. 2022; 15(18):6602.
https://doi.org/10.3390/en15186602

**Chicago/Turabian Style**

Landa, Jiří, and Milan Hokr. 2022. "Contaminant Transport from a Deep Geological Repository: Lumped Parameters Derived from a 3D Hydrogeological Model" *Energies* 15, no. 18: 6602.
https://doi.org/10.3390/en15186602