A systematic strategy for strengthening the reliable prediction of crushed salt constitutive models

Friedenberg, Larissa; Kupper, Johannes; Lerche, Svetlana; Liu, Wenting; Rahmig, Michael; the MEASURES project team,

doi:10.5194/sand-5-1-2026

Articles | Volume 5

https://doi.org/10.5194/sand-5-1-2026

© Author(s) 2026. This work is distributed under
the Creative Commons Attribution 4.0 License.

https://doi.org/10.5194/sand-5-1-2026

© Author(s) 2026. This work is distributed under
the Creative Commons Attribution 4.0 License.

Articles | Volume 5

Research article

|

12 Aug 2026

Research article |

| 12 Aug 2026

A systematic strategy for strengthening the reliable prediction of crushed salt constitutive models

Larissa Friedenberg, Johannes Kupper, Svetlana Lerche, Wenting Liu, Michael Rahmig, and the MEASURES project team

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Final revised paper (published on 12 Aug 2026)
Preprint (discussion started on 09 Apr 2026)

Interactive discussion

Status: closed

RC1:
'Comment on sand-2026-6', Thomas Nagel, 19 Apr 2026

This is a nice summary article of aspects of the extensive and structured work done in the KOMPASS and MEASURES projects that is, from my point of view, worth publishing. The research done in these projects is highly relevant to our field. Even though specific results are not discussed in detail as the article rather gives a synopsis and discussion/outlook of the projects themselves, I have some minor comments that could further improve the article. These are some small wording aspects on the one hand and some requests for discussing those results a little more that are selected for presentation in the article.

- In the glossary, I would like to see an aspect added to the definition of the definition of "constitutive model". The description of a specific physical system is usually not achieved by constitutive models alone, but still includes the (more general) balance equations. I'd therefore suggest to rephrase slightly while maintaining the authors' direction: "Conceptual/mathematical description used as a closure relationship for a specific physical system".

- Figure 2: I know this is not the authors intention, but one could interpret the linear triangles on the bottom such that uncertainties will be reduced to zero and we'll end up with a fully robust and accurate model if only we have enough follow-up work. There are diminishing returns here, and residual uncertainty cannot be avoided in geotechnics for a number of reasons. This could easily be indicated in the figure (curved triangles or otherwise).

- L123/4: Can you say something regarding the in-situ applications, ongoing or planned? Are there works that can be referenced?

- L141 is somewhat related: How can the improvement of the constitutive models be demonstrated on the virtual demonstrator, if there's no real-world equivalent at the moment? Say a few words on how this goes beyond just a demonstration of observed differences between different constitutive models and what arguments are used to show the improvement.

- L135: "uncertainties are avoided by using different formulation options [...] and using different methods". Clarify how that avoids uncertainties. It can help capture model-/method-related uncertainties, thus giving you a clearer picture of the scatter when with only one model you'd be flying blind. Is that what is meant here or is it something else?

- L148: Lab data is not used to verify the models according to the definition used in the glossary (which presents a definition to which I agree). Perhaps clarify or take out.

- L149: "benchmarking supports consistency checks between experiments and simulations". In what sense? You already use the terms confirmation and validation for comparison of experiments and simulations. If you use a specific meaning of benchmarking here, you might want to add it to the glossary. From section 4 it seems to me you use it here as a means of getting a handle on model uncertainty, but I'm not sure.

- Fig. 5: Very nice figure, particularly the bottom part. I guess your sigma_v is deviatoric stress? In that case use sigma_d as in text (appears 5 times in the figure, if I didn't miss anything).

- Fig. 6: Volumetric behaviour seems to scatter less between models than deviatoric behaviour (here: axial), which might be interesting to comment on if you have an insight into this. Are these all best-fit results for the models?

- Fig. 7: The creep model for the rock salt is identical for all groups, allowing the comparison of the effect of the crushed salt model. Volumetric behaviour was well reproduced in Fig. 6. Here, quite some scatter remains. In the text, it is mentioned, that calibration reduced the scatter. This is true only in a porosity range below 10%, while in the rest of the porosity domain scatter even increased, with one model standing out. The tests covered 3 to ~17%. The maximum spread between the models occurs in the right graph at roughly those 17% with an order of magnitude spread in stress. Here, the discussion on page 11 should be a bit more detailed and nuanced, rather than simply speaking of improvements, as this is part of the motivation for your follow-up work.

- The reference list consists mainly of reports. List some journal papers from peer-reviewed publication on the topic of crushed salt behaviour as well.

- Personally, I'd refrain from using informal formulations like "family" (better: team or project members) and "the topic is huge" (better: broad, extensive). There are good reasons for scientific articles being formulated on the sober side, even if it can feel boring.

Looking forward to seeing the next set of results of MEASURES.

Citation: https://doi.org/10.5194/sand-2026-6-RC1
- AC1: 'Reply on RC1', Larissa Friedenberg, 05 May 2026
  
  We are sincerely thankful for the review and the comments that helped to improve our paper. In the attached document, we outline how we address the comments in the manuscript.
  
  Citation: https://doi.org/10.5194/sand-2026-6-AC1
RC2:
'Comment on sand-2026-6', Kristopher Kuhlman, 19 Apr 2026

1) In the glossary: The definition of "Constitutive model" could be refined. Currently it is very broad "conceptual/mathematical description of a specific physical system." A more revealing definition (Fish, 2014 -- Practical Multiscaling) might be: "A constitutive equation demonstrates a relation between two physical quantities that is specific to a material or substance and does not follow directly from physical laws"
I think this definition, along with the allied definition of "Modeling" would make your point a bit clearer. Modeling is solving the governing equations based on conservation of energy, mass, etc.
Constitutive modeling can often be performed without the governing equations (e.g., thermal conductivity is fit to a specific function of temperature for a specific material), but usually the constitutive models are needed to fully solve the governing equations and perform "modeling" (i.e., they are needed to close the system of equations to make it solvable). I think this is consistent with what you are presenting in the glossary and in Figure 1, but I think it could be stated a bit more clearly. Often our governing equations are posed to be linear, but non-linearities come into the models through the constitutive laws (not always the case, but often). The governing equation for thermal-mechanical behavior of salt doesn't explicitly involve moisture (if it did you would have to solve for the mass balance of water), but the constitutive behavior may include the effects of moisture.
2) Figure 1: I think in light of this discussion above, the word "modeling" on the black arrow could maybe be changed for clarity. Modeling is an overused word, and I think here it is a bit confusing. It could be "fitting" or "comparing"? Often "modeling" means evaluating the governing equations, which includes the constitutive relationships. Modeling can also mean other things, and I think that is what is going on here. A different word could be used here to make the figure clearer and more consistent with the rest of the paper.

Citation: https://doi.org/10.5194/sand-2026-6-RC2
- AC2: 'Reply on RC2', Larissa Friedenberg, 05 May 2026
  
  Thank you very much for taking the time to review our paper. Your comments helped us to increase the clarity and scientific sharpness of our paper.
  Please find our replys below:
  1) We address this comment together with the first comment of RC1, that goes in the same direction. We changed the definition to: “Conceptual/mathematical description of the physical behavior of a particular material or substance.”
  
  2) Thanks for your comment. We agree that the word modeling is often used with different meanings. We think that our understanding of modeling becomes clear due to the combination of the figure and the glossary.
  Thanks again!
  
  Citation: https://doi.org/10.5194/sand-2026-6-AC2

Peer review completion

AR – Author's response | RR – Referee report | ED – Editor decision | EF – Editorial file upload

AR by Larissa Friedenberg on behalf of the Authors (08 Jul 2026) Author's response Author's tracked changes

EF by Svenja Lange (09 Jul 2026) Manuscript

ED: Publish as is (15 Jul 2026) by Carlo Dietl

AR by Larissa Friedenberg on behalf of the Authors (20 Jul 2026) Manuscript

Short summary

We studied how crushed salt changes over time so that it can safely seal a future waste repository. By combining new laboratory tests with improved computer models, we created a clearer picture of how the material compacts and when it becomes tight. Our results reduce uncertainty in long‑term predictions and support safer planning for storing radioactive waste underground.