the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 16 Jun 2026
Modelling contaminant transport of soil and landfills: A comparison study of three numerical codes
Abstract. The negligibly radioactive materials from a control area can be fed into further material cycles, to an incineration plant or disposed according to the type of clearance after clearance procedure, which is described in the German Radiation Protection Ordinance (Deutscher Bundestag, 2024b). The clearance values of the Radiation Protection Ordinance are based on the 10 µSv concept that limits the additional radiation exposure for the population or worker. In the past, analytical models were used to calculate radionuclide concentrations for groundwater path dose estimation. In this work, numerical groundwater models are used to simulate the distribution of radionuclides and provide spatial information on concentrations, which can be used as input data for dose estimation.
The transport processes of advection, diffusion, dispersion, sorption and decay are implemented in the simulation codes “distributed density-driven flow (d3f++)” (Fein and Schneider, 1999; Fein, 2004), “Dune for Multiphase flow and transport (DuMux )” (Flemisch et al., 2011; Koch et al., 2021) and “Simulation of Processes in Groundwater (SPRING)” (delta h, 2024). To compare the codes and their implemented transport processes, a simple 2D column and a generic 2D landfill body were modelled. The study demonstrated a good agreement between the three computational codes, thereby strengthening trust in numerical modelling for future applications in dose estimation. The differences in the concentration breakthrough curves can be attributed to the differences in the implementation of initial conditions (IC) for saturation, the influence of dispersion in the different codes and the upwind methods, highlighting the sensitivity to these parameters and numerical solvers. Although minor discrepancies emerge in the results, the study demonstrates that the concentrations and the times of maximum concentration are largely comparable.
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Status: final response (author comments only)
- RC1: 'Comment on sand-2026-11', Anonymous Referee #1, 14 Jul 2026
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RC2: 'Comment on sand-2026-11', Anonymous Referee #2, 22 Jul 2026
Benchmarking has become a standard method for quality assurance and confidence building in numerical analysis. The authors intend to achieve this goal in the field of waste disposal using three codes. The authors have made a very thorough attempt to identify the differences in the various physical processes and mathematical implementations. Numerous parameter variations were carried out based on two highly simplified examples. However, the differences are so great that, after reading through it, one loses confidence in the numerical results.
Two Codes, SPRING and d3f++ are based on the Richards’ formulation, a simplified multiphase flow method, supply similar results. The result of DuMux differs most from the other two codes in its temporal development 10% and also in its amplitude 50%.
As code developers, the authors must be able to identify and correct these differences before publication. In fact, it should be possible to simplify the two-phase flow to a Richards’ formulation to obtain the same results.
Specifically, the following comments still need to be taken into account:
- In terms of scientific publications, there is a lack of literature reviews on the state of the art in the field.
The phrase ‘the radionuclide .. groundwater model’ repeated in the manuscript, suggests that the authors are conflating two distinct processes: flow and transport.
Indeed, there are lots of numerical simulation dealing with the radionuclide spread since last thirty years, the statement ‘mostly analytical methods’ seems too abstract. - To verify the code implementation, a comparison between the numerical and analytical solutions are needed.
- In the example, the transport is simulated based on a steady state flow condition, so phase for flow using different formulation should not be the reason for the overall differences. The flow velocity from different codes should be analysed.
- The decay of radionuclides was mentioned in the context of the transport process but not addressed in detail. No calculation example was provided in this context. This needs to be added for the sake of completeness.
- Space distributions were mentioned, but no figure showing these results. It may helpful to find the reason for these differences.
Overall, the current version is not yet ready for publication. Taking the aforementioned comments into account, further work is required, and the manuscript should be rewritten.
Citation: https://doi.org/10.5194/sand-2026-11-RC2 - In terms of scientific publications, there is a lack of literature reviews on the state of the art in the field.
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RC3: 'Comment on sand-2026-11', Anonymous Referee #3, 20 Aug 2026
The authors present a benchmarking study comparing the results of three different codes, which are used for simulating solute transport having radionuclide transport in mind. The authors compare the results of DuMux, d3f++, and SPRING considering advection, being the result of saturated or unsaturated flow simulations, diffusion, dispersion, and sorption. I believe the topic is important for the community of radioactive waste management and fits to the scope of SaND, though the scientific novelty of the specific results the authors present is debatable. It would be nice to highlight more what is new about the code comparison. The sole use of numerical models for solute transport is not new. Numerical simulations of radionuclide transport specifcally are done by the international scientific community since decades with great success. The vast majority of this literature is not mentioned in the manuscript, this should be improved. Also, the results for different codes used in the manuscript significantly deviate from each other. Peak concentrations from the different codes show deviations up to a factor 2-3, such differences are not small in the context of benchmarking. At best this should be fixed by the authors, otherwise it must be extremely carefully shown, where these differences come from. The latter is tried by the authors, but the explanations can not fully justify the mismatch in my opinion, they also should be backed by (more) additional modelling attempts. The description of methods and codes is already very detailed, still some relevant information on numerical setups are missing. In the context of mismatching modelling results, I miss a more detailed explanation on the numerical model setup. This includes differences of the numerical methods for discretization in space, i.e., the finite volume method used in d3f++ vs. vertex-centered finite volumes/box-method in DuMux vs. finite elements as well as time discretiaztion/steps and numerical grids in use. Also, an analysis of global mass balance would help.
Equations for transport are presented in different fashions, this for itself is fine considering the scope of the manuscript. However, they are not fully consistent. Here, some additional work with associated code and model checks should be undertaken.
At the present stage I do not recommend to publish the manuscript. However, it may be reconsidered for publication in SaND, if and only if, the manuscript will be thoroughly revised. To this end, I would wish that my former general comment and the following specific comments are properly addressed.
Additional, more specific major comments:
- Lines 13-18: I do not share the statement that the discrepancies in results are minor in the context of benchmarking codes against each other (Line 16-17). Concentrations at peak arrival time differs up to a factor between 2 and 3. In my opinion, this should be investigated in more detail. This part of the abstract should be rewritten.
- Lines 39-41: As written in my general comment, it is misleading to write that there was not much research carried out on numerical simulations on coupled flow and radionuclide transport. Please extend the literature review and adapt your statements.
- Please check thoroughly that all variables in the equations are properly defined. It is common practice to abbreviate units of physical quantities.
- In the text you define solute transport equations using mass fractions and using concentrations. In principle, both versions are correct / can be correct. However, the authors use the same variable for both, mass fraction and concentration. This is not fine. It leads to implicit re-definitions and confusion. Use an other variable for concentrations, e.g., "c" to differentiate both. First occasion: Equation 6 (mass-fraction based) vs. Line 111 and Equation 10 as well as further transport equations 11, 12, 14, 15.
- In the minor comments, some mistakes in the governing equations are highlighted. Besides of only considering them, I strongly advise to check the simulations codes / the models accordingly.
- Line 148: A tortuosity of 1 in a porous medium is weird. It would imply a straight path, which implies that there are no grains. This is odd. Maybe, you use a common simplification taking the inverse of the water content? Check also again what SPRING does here.
- Although radionuclide transport is simulated neither geochemistry nor radioactive decay is considered. While the first may be deemed acceptable in very specific real-life situations and is fine in the context of a code comparison, I believe that radioactive decay should be considered. Otherwise, the whole analysis is limited to very long-lived nuclides, which should be made explicit, if intended.
- Consider to enlarge the Appendix for including more model runs (Line 199-200)
- Figures 4, 5, 7, 8, 9: For me it looks like there is a considerable difference of overall solute mass in the systems considering DuMux vs. SPRING and d3f++. Please check the global mass balances.
- Redo captions of Figures, especially the x-axis. Do you mean solute concentration? Why is the concentration mass based? Did you divide by water density? All this needs explanations and a more thorough description. I would capitalize the "M" in DuMux.
- Please adapt discussion and conclusions such that your manuscript alterations upon working on the revisions are considered properly.
Minor comments:
- It is common practice to write full tensors in bold to differentiate them from scalars.
- Equation 2: If you use Darcy's law in the pressure-based formulation, I miss the gravity term here. Especially, as you also simulate cases with vertical cross sections. Please also check your models accordingly.
- Equations 6, 8, and 11: This may be considered a bit picky but in principle, there is a water content missing in front of the dispersion tensor. Additionally, there is a water content missing on the left-hand side of Equation 8. However, considering both alterations together one ends up again with Equation 11, which is correct. But consider also my next comment.
- Equation 7: Tortuosity is commonly defined in hydrogeology as the real path length over the unit path. As such one wants to divide by tortuosity here and not multiply. However, considering Line 148, I realize that DuMux defined tortuosity inversely. Mathematically, this is fine, however, it needs to be stated as such to not confuse the transport and specifically the diffusion community.
- Equation 9: This is not an equation. Please make an equation out of it.
- Equation 12: Definition of dispersion tensor is missing. Here the multiplication with the water content is correct, assuming that the dispersion tensor is the one of Scheidegger. However, now tortuosity for scaling the diffusion coefficient for the transport equation of SPRING is missing.
- Equation 14: Again, tortuosity for scaling the diffusion coefficient is missing for the transport equation of d3f++.
- Equation 15: I don't understand the red highlighting. The Equations 11, 12, and 14 have this pre-factor. Please clarify again, why the highlighting. Please also be more explicit with the explanations after Equation 15. Lines 141-142 need a more detailed description.
- Equations 6, 11, 12, 14, 15: In Equation 6 the authors use mass fractions, in all other equations they use concentrations. For both they use the same variable sign. This should be improved.
- Line 189: "constant" instead of "steady"
- Lines 201-202: This is hydraulic conductivity not permeability
- Lines 212-213: Not identical but similar
- Line: 214-216: Please, redo your search considering what is written in this review.
- Line: 238: tone?
- Line: 341-342: I don't think that the agreement is that good.
- Line: 342-343: I don't believe that these are all differences. If it would be only that only the spread of the plumes/distribution of concentration should be different. For me, it looks like that for some models there are also different amounts of substance mass in the system. Please check, if this is the case.
Citation: https://doi.org/10.5194/sand-2026-11-RC3
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The paper compares different numerical codes and their results for two simplified flow and transport problems and tries to keep a balance between numerical issues and the application to a real life problem. Anyway, the explanation of differences between resulting breakthrough curves is not fully consistent yet. It is recommended to clarify these parts, and check the overall structure again. Finally, a native English speaker should proofread.