Application of the creeping flow restoration method to an analogue model

Schuh-Senlis, Melchior; Caumon, Guillaume; Cupillard, Paul

doi:https://doi.org/10.5194/egusphere-2023-2039

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https://doi.org/10.5194/egusphere-2023-2039

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28 Sep 2023

| 28 Sep 2023

Application of the creeping flow restoration method to an analogue model

Melchior Schuh-Senlis, Guillaume Caumon, and Paul Cupillard

Abstract. Structural restoration is commonly used to assess the deformation of geological structures and to reconstruct past basin geometries. For this, most methods use numerical simulations to compute the deformation of geological models from a chosen deformation mechanism for each geological layer, and conditions applied on the boundaries depending on geological knowledge. For example, geomechanical restoration classically uses elastic motion, considers faults as frictionless contact surfaces, and imposes boundary conditions such as interface flattening to estimate the paleo-deformation. To bring more physical behavior and better handle large deformations, we use a reverse time Stokes-based method with negative time step advection. In order to test the method on complex models including various rheology and faults, we apply it to an analogue experiment model. We first show that reasonable restored geometries can be obtained using classical kinematic boundary conditions. We then show that it is possible to relax the imposed kinematic conditions and replace them by more physical boundary conditions. These conditions, however, imply a larger impact of the material properties on the restoration results. Finally, we show that relaxing the boundary conditions and using the previous imposed conditions as choice criteria allows both the assessment of the value of the effective material properties, and the improvement of the restoration results.

Received: 05 Sep 2023 – Discussion started: 28 Sep 2023

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Melchior Schuh-Senlis, Guillaume Caumon, and Paul Cupillard

Status: final response (author comments only)

RC1:
'Comment on egusphere-2023-2039', Peter Lovely, 26 Oct 2023
“Application of the creeping flow restoration method to an analogue model,” by Melchior Schuh-Senlis, et al. leverages a laboratory scale structural analogue model, which has been imaged through its evolution by a time-series of X-ray tomography images, to validate a novel approach to structural restoration. The restoration method assumes that deformation may be explained by a viscous rheology modeled using Stokes flow equations, and applies reverse time-stepping to reverse deformation of the structure(s). Is see this line of research as a critical next step in the development and demonstration of Stokes flow restoration methods.

However, I have significant concerns about the results that are presented. Most importantly, the analysis of restoration results and comparison with time-series imaging of the analogue model focuses almost entirely (with the exception of Figure 6) on the first restoration step, corresponding to the final stage (18 minutes?) of the forward model. This is only a small amount of the total deformation of the model. The misfit between restoration and forward model at this step is difficult to discern (this is a good thing) but I am sure it becomes greater as the restoration goes further back in time. The authors do not explain their rationale for focusing only on this final step of the forward model, and it leaves me concerned that an incomplete picture of the study is portrayed in the manuscript.

Additionally, the pressure (Neumann) boundary condition applied to the right hand side, which is suggested as an enhancement to the model relative to kinematic boundary condition, is not sufficiently justified, from my perspective. In viscous flow models, deviatoric stress is related to viscosity and strain rate. How does Poisson’s ratio help to calculate an optimal horizontal traction in this case, and, thus, what makes this boundary condition more physical than the previously applied kinematic condition?

Perhaps it is beyond the scope of this study, but the final results left me feeling somewhat unsatisfied, with no benchmark how the Stokes flow restoration compares to other restoration methods. I believe that Chauvin, et al. (2018) benchmarked an elastic restoration using the same numerical analog model. Some comparison of the results could enhance the impact of the manuscript without a lot more work.

Finally, there are many opportunities to improve the organization of the manuscript, as well as clarity and precision of the writing. I have attempted to identify and assist with examples in the comments below and in the attached, annotated version of the manuscript.

For these reasons, I recommend major revisions before accepting this manuscript for publication.

Specific comments:
The abstract is difficult to follow. My perspective is that too much focus is placed on general background related to restoration, and too little focus on the background of stokes flow-based restoration. The last few sentences, focused on the contribution of this work, are vague and worded in such a manner the results and implications are difficult to understand.
The introduction provides a relatively detailed, and helpful, overview of the history of restoration and various methods. However,
The detail is imbalanced. Mechanical restoration methods receive quite a lot of focus, whereas kinematic restoration methods are described only very briefly

Validation of geologic interpretation is an important application of restoration methods, but is never addressed

The direction that the manuscript is going to take could be made clear earlier in the introduction. The review of restoration methods is good, but left me wondering until many paragraphs into the introduction where the authors are going. Perhaps the introduction could be shortened and a separate section or subsection added to review kinematic and geomechanical restoration methods.

The paragraph introducing analogue modeling methods comes out of nowhere. The purpose of the analogue model for this study needs to be introduced first.

The final paragraphs in which the direction of this manuscript is laid out might be explained more clearly (see annotations)

Section 2.1 (Creeping Flow Restoration) is a good overview of the method. Though published previously, it is sufficiently important to reiterate in this manuscript. A few minor suggestions include:
Describe somewhere the viscosity model you use. I believe you assume a linear viscous fluid with constant viscosity, but this should be clarified

A brief introduction to the section on what is stokes flow and why it is used here would be helpful.

Section 2.2 (The FAIStokes code) is summarized significantly and references Schuh-Senlis et al (2020) appropriately.
Because the numerical method is not the focus of this publication, I wonder if this section could be abbreviated further. Is Figure 2 (flowchart showing numerical implementation) necessary?

How is the free surface is implemented and (how) is the material above it is addressed in the numerical scheme.

In Section 3 (Presentation of the Analogue model), I have made quite a few suggestions and posed several questions to the authors. To summarize,
The model could be explained better. Specifically, what is the right-hand-side boundary of the model? How are the layers deposited through time? How do the material properties of the silicon, sand, and pyrex scale in both space and time to geologic materials and structures? See annotations for additional detail.

The description of X-ray tomography and how it is used may not be clear to an unfamiliar reader. Need to specify that it is an imaging method.

The header title “Available data” is vague and unclear. I would suggest merging this section with the analogue model description in the prior paragraph and relabeling the section something like “Analogue Model Description.”

Section 3.2 (Creation of the Numerical Model): Please clarify what is meant by “numerical model.” This section refers specifically to the geometry to be restored and its discretization on the mesh. Stokes flow and the implementation in the FAIStokes code is another numerical model not addressed here.

In section 3.2, please address the mesh resolution and particle density per cell, in addition to the swarm.

The final paragraph addressing the use of X-ray tomography images to determine the height of the topography after the deposition of each layer seems out of place. Boundary conditions are not otherwise addressed in this section, and the time-series of tomography images has not been adequately introduced.

Section 4 (Boundary condition analysis)
It might help to put this section in the context of how boundary conditions have been dealt with in previous applications of Stokes flow restoration, especially the 2020 paper by the same authors. I tried to read this manuscript in the context of that prior work and was left wondering because I think that this study begins with a different approach (this may be fine, but I found it confusing).

I have made several suggestions to clarify wording in the annotated manuscript

Section 4.1 (Restoration using kinematic boundary conditions)
Please specify if gravity is applied in this model

The paragraph beginning line 223 seems primarily relevant to the numerical implementation, and I’m concerned it will be a diversion to the average geologist reading the manuscript (it took me some effort to understand). Perhaps it could be moved to the numerical implementation section, or explained in a more intuitive manner?

Figure 6 is a nice time series of the restoration model, but only (b) is compared to the tomography reference image. Why? Also, why is this the only time-series of restoration results? I would like to see a similar time series and comparison to tomography images for the preferred model described in section 5 also. The absence of this rigorous comparison leaves me feeling like the analysis presented in this manuscript is incomplete.

I have made several suggestions to clarify the captions of figures 7 and 8

In figure 8, is d_reference(x) always positive? Or does it represent an absolute value? I cannot see in figure

Section 4.2 (Upgrading the kinematic conditions to natural boundaries)
At the start of section 4.2.1, please explain more completely the restoration model to be performed. In particular, what is the boundary condition applied to the top of the model in this case? I had to read on some way for this to be clear.

It is unclear to me how the pressure boundary condition applied to the right side of the model relates to the forward analogue model. In viscous deformation, deviatoric stress relates to strain rate. Why is Poisson’s ratio used here? What makes this boundary condition more physically realistic than the kinematic boundary condition applied previously?

Building on this, it is unclear to me (and not addressed sufficiently in the paragraph beginning line 273) what might be the implications for model precision.

In figure 11, it would be helpful to include the topography before restoration as an additional reference, beside the expected flat topography.

The legend of figure 11 is unclear, because (a) and (b) refer to the right hand side boundary condition, but this is not specified. Reading the text, it says that the top surface is treated as free in fig. 11.

The caption of Table 6 could be shortened by referencing Table 5 (similar to what is done with Figures 8 & 10

The last sentence of this section makes a broad statement that is, so far, unjustified. This should be presented as a hypothesis or it should be specified that this will be demonstrated in the next section.

Section 5 (Model parameters analysis)
The section header I believe refers specifically to material properties… please say so.

In line 305 a statement is made regarding the ratio of viscosity between overburden and silicone. Please explain this better. Where does this ratio come from?

I like figure 13! It’s a good visual display of the DOE study.

In line 325, I do not follow what are the “computational instabilities” shown in Section 4.2.2.

In the sequence of figures 14-16, the authors focus on results at the final time step of each simulation. However, as a geologist, I am most interested in the optimal time step (i.e. the results when each curve in Fig. 14 is minimized). Could the same analysis be completed on the optimal time step (i.e. show it in Fig. 15) or could an explanation be provided why the final time step is better?

Line 345 presents a hypothesis which has thus far not been demonstrated as fact. Please present it as a hypothesis or with the caveat that this will be demonstrated in the following section.

At the start of section 5.2, please specify (in addition to the parameters) which experiment number (from prior section) is selected as optimal.

Line 353 says a range of multipliers from 0.75 to 3 are used, but Table 7 shows multipliers ranging from 0.75 to 7. Please reconcile.

Does figure 17 need to be its own figure? It’s just adding one curve to figure 14.

Table 8 includes the exact same information as Tables 5 & 6, plus a new row. Please consider if tables 5 & 6 need to be included as stand-alone tables.

In figure 18, the difference between restored geometry and reference target (image) is difficult to discern. As discussed previously, please add restoration steps with more deformation or address why only the first restoration step (in which deviation is presumably smallest) is chosen for comparison.

Section 6: discussion
Table 8, row 3, column 1: the use of “Kinematic restoration” confuses kinematic restoration methods with kinematic boundary conditions in a mechanical (stokes flow) restoration. Please clarify.

Line 377: I believe this references silicone viscosity, not actually salt.

Please clarify the caption for Table 8 as requested in annotated manuscript.

Line 379: please specify what you mean by times scales ranging from seconds to hours. The experiment was 4 hours long.

The paragraph beginning line 384 overlooks the fact that the last case (optimized material properties) produced the best restoration results. This is key to conclusions and should be highlighted here.

Paragraph beginning line 390 seems a bit of an aside to the point of the manuscript. It’s relevant, but maybe could be moved later in the discussion.

Line 401: The sentence beginning “It poses…” touches on questions around the magnitude of traction applied to the right boundary. I would like to see this addressed more thoroughly (in relation to comment on section 4.2)

The numerical improvement from the first case to the last is relatively small (see total of first two rows in Table 8). I would like to see more discussion of if it’s worth the (large) additional effort of to optimize the model.

Throughout the manuscript, wording is frequently awkward and/or imprecise. I have tried to identify and assist with the more significant examples in the attached, annotated version of the manuscript.
Citation: https://doi.org/10.5194/egusphere-2023-2039-RC1
- AC3: 'Reply on RC1', Melchior Schuh-Senlis, 14 Jan 2024
  
  Thank you for the review and all the comments on the manuscript. I will address all the specific comments in a revised version of the manuscript. I was not able to download the annotated version of the manuscript (404 Error ...), but if you could send it to me using the e-mail address present in the preview page, I will gladly address all the issues you pointed out there as well. In the following, I will also try to answer some of your questions.
  Indeed, the abstract, introduction and organization of the document have to be improved. I realized, reading your review, that the presented method was raised too much as an improvement to previous geomechanical restoration methods, without presenting results that are significantly better to sustain this assertion. This misleads from what I consider the main point of the article, which is the material properties analysis, made possible by choosing more natural boundary conditions (which do not enforce the velocity on the model). I will revise the manuscript to put forward this point rather than a comparison which is not really done.
  The focus was made on the first restoration step, corresponding to the final stage of deformation, to reduce computation time for the simulations. This part of the simulation shows all the deformation of the top layer of the model, and finishes with a flat top surface, which was considered enough for testing purposes. It allowed, rather than testing the ability of the method to restore the model as well as other methods, to study the impact of the boundary conditions and the material properties on the restoration results.
  On the right hand side of the model, the velocity is not known, and can only be estimated from the deformation of the model. This neglects the vertical variations of velocity, which have a large impact on the overall model deformation. Having kinematic conditions on all boundaries also puts an over-parameterization on the simulation system. The presented Neumann conditions, while simplifying the physics by using Poisson's ratio, both remove this over-parameterization, and are based on known properties inside the model. Moreover, using a traction on the right hand side of the model removes the enforcement of the velocity at this boundary, which was necessary for the material properties analysis.
  Kind Regards,
  Melchior Schuh-Senlis
  
  Citation: https://doi.org/10.5194/egusphere-2023-2039-AC3
RC2:
'Comment on egusphere-2023-2039', Anonymous Referee #2, 28 Nov 2023

Review of the manuscript entitled “Application of the creeping flow restoration method to an analogue model” by Schuh-Senlis et al. submitted to EGU Solid Earth
Schuh-Senlis et al. present a meticulous study aimed at determining the optimal method for restoring an analogue experiment that represents a geological event. This is achieved through the application of an inversion scheme utilising Stokes' equations, with velocities inverted to facilitate backward stepping in time (restoration).
The study draws attention to the impact of velocity and pressure boundary conditions on the success rate of inverting a numerical model. The numerical model, derived from the initial conditions of a late stage analogue experiment, demonstrates that while pressure boundary conditions, despite their greater realism, do not yield significantly superior results, they underscore the importance of material properties, such as viscosities along individual faults.
As a reader, the expectation was to find a robust solution to the problem. Towards the end, however, there is a sense of frustration at the need to delve into material properties individually through a parameterisation study. Nonetheless, this meticulous approach is considered positive, as it necessitates a detailed exploration to establish a constitutive relationship capable of inverting a broad spectrum of geological events. Yet, achieving such a relationship appears incomplete, demanding further studies that may involve more sophisticated tools, such as machine learning. Conversely, the authors could opt for a simpler model, if the analogue part exists, involving one or two faults with varying properties. Then, the authors would increase the complexity of the problem in subsequent chapters.
Before proceeding with publication, I advise considering the following points:
1. In your inversion model, it is noted that the heat transport equation is not solved. Could you elaborate on the potential impact of this factor on your results?
2. On line 35, clarification is sought regarding why the aforementioned methods are particularly limited in capturing the dynamics of salt basins. Is it related to layering or other characteristics of salt basins?
3. Have you assessed whether the resolution of your model may significantly impact the results?
4. Would you consider exploring periodic boundary conditions on the side walls? It is presumed that faults may restore as the model equilibrates.

Citation: https://doi.org/10.5194/egusphere-2023-2039-RC2
- AC1: 'Reply on RC2', Melchior Schuh-Senlis, 13 Jan 2024
  
  Thank you for the review and the kind remarks. I will try to give some answers to your questions in the following, and improve the linked parts of the manuscript accordingly.
  Indeed, I understand the frustration at the need to delve into material properties individually at the end of the manuscript. I will try to improve this section and the discussion part to highlight potential methods that were discussed to automatize this methodology, such as inversion schemes.
  Indeed, the heat transport equation was not solved in the presented simulations, for several reasons: first, the temperature during the laboratory experiment was considered to be mostly constant and stable, and as such have no impact on the experiment itself. Second, in a more practical view, numerical simulations were already long enough, and simplifications had to be made. In this sense, heat transport was considered as having sufficiently low impact to be neglected compared to other factors such as viscosity variations in the faults.
  The limitations of previous mechanical restoration methods to restore salt basins are simply due to the fact that elastic deformation alone (and the physics behind) cannot be used to compute the creeping flow of salt layers in time. I will try to add more explanations on this in the introduction.
  Indeed, the resolution of the computation grid has an impact on the results, and specifically on the behaviour of the faults, which is why specific care was put to having an adaptively refined grid. The amount of the impact itself was not studied, but could be an interesting perspective for future work.
  Periodic boundary conditions were not considered for the analogue model studied here, given the experiment layout. On other models, however, they could indeed be considered. It could then be interesting to see how they affect the simulations and the effective material properties compared to the boundary conditions presented in the manuscript.
  Kind Regards,
  Melchior Schuh-Senlis
  
  Citation: https://doi.org/10.5194/egusphere-2023-2039-AC1
- AC2: 'Reply on RC2', Melchior Schuh-Senlis, 14 Jan 2024
  
  Thank you for the review and the kind remarks. I will try to give some answers to your questions in the following, and improve the linked parts of the manuscript accordingly.
  Indeed, I understand the frustration at the need to delve into material properties individually at the end of the manuscript. I will try to improve this section and the discussion part to highlight potential methods that were discussed to automatize this methodology, such as inversion schemes.
  Indeed, the heat transport equation was not solved in the presented simulations, for several reasons: first, the temperature during the laboratory experiment was considered to be mostly constant and stable, and as such have no impact on the experiment itself. Second, in a more practical view, numerical simulations were already long enough, and simplifications had to be made. In this sense, heat transport was considered as having sufficiently low impact to be neglected compared to other factors such as viscosity variations in the faults.
  The limitations of previous mechanical restoration methods to restore salt basins are simply due to the fact that elastic deformation alone (and the physics behind) cannot be used to compute the creeping flow of salt layers in time. I will try to add more explanations on this in the introduction.
  Indeed, the resolution of the computation grid has an impact on the results, and specifically on the behaviour of the faults, which is why specific care was put to having an adaptively refined grid. The amount of the impact itself was not studied, but could be an interesting perspective for future work.
  Periodic boundary conditions were not considered for the analogue model studied here, given the experiment layout. On other models, however, they could indeed be considered. It could then be interesting to see how they affect the simulations and the effective material properties compared to the boundary conditions presented in the manuscript.
  Kind Regards,
  Melchior Schuh-Senlis
  
  Citation: https://doi.org/10.5194/egusphere-2023-2039-AC2
EC1:
'Comment on egusphere-2023-2039', Patrice Rey, 20 Dec 2023

This paper is not ready. It needs major revisions before it is considered for publication in Solid Earth. Please carefully consider the reviewers comments, as well as the following.
The paper tackles an interesting subject. However, writing lacks rigor, e.g.:

• Line 70 “The idea is to choose materials that present the same deformations as those observed in geological models,…” You probably mean "similar rheology".

• Line 158: “These point swarms are denser than the material particle swarm and are one dimension lower (i.e. lines in our 2D cases).” One dimension lower? More simply: … the surface of our 2D numerical model is tracked using a set of passive tracers with an initial horizontal spacing of X m.

• Line 202: What do you mean by flattening? Do you mean removing any topography to make the surface horizontal? What does "... tying the fault lines ..." mean?

• Analog or analogue, choose one and be consistent.
The introduction section is largely borrowed from a previous paper from the same authors, please rewrite it.
The analysis focuses on the comparison of the first increment of restoration with the last increment of extension. Yet, X-ray tomography at 2-minute intervals allows for a more thorough comparison between the restoration and the analog model.
In Figure 6, panel h shows that the top surface of the basal viscous layer doesn’t seem to return to its initial horizontal position as restoration proceeds. This suggests some serious limitations of the method. Have you considered that the viscous/viscous interface in the numerical model may not capture the physics of the granular/viscous interface in the analog model?
The purpose of the paper was lost when it became a game of tweaking the viscosities, following a painful trial and error approach, to match the tempo of the analog model. Have you considered to what extent the tempo of a 3D model involving a basal viscous layer can be compared to that of 2D model?
The description of the reference analog model needs more details:

• What are the dimensions of the model in x, y and z?

• Also give the scaling and the dimensions in kilometer.

• Give an idea of velocity of the right end of the model.

• Were the walls lubricated to approach free slip?

• How was sedimentation implemented, continuous or in stage?

• How was the right boundary condition implemented?

• What was the final total amount of extension?
Many claims are made that dynamic boundary conditions are more “natural”, “less unphysical”, or “more realistic”. Fair enough, but can you explain why?
The heat equation is not included. Given the dependence between viscosity and temperature, please justify this choice.
Add the dimension to all parameters in Table 1 and 4.
In the conclusion, it is stated that “… conclusive results can be obtained while changing the consideration of salt layers and faults to a more physical behavior, compared to previous

geomechanical restoration schemes using elastic behavior”. Yet no such a comparison is made in the paper.
In the conclusion, it is stated that “… the creeping flow restoration of this analogue experiment model showed that this restoration scheme can be applied to complex real-case structural models, … ”. I do not think that this is demonstrated in this paper.
Kind regards,

Patrice

Citation: https://doi.org/10.5194/egusphere-2023-2039-EC1
- AC4: 'Reply on EC1', Melchior Schuh-Senlis, 18 Jan 2024
  
  Thank you for the review and the kind remarks. I will improve the manuscript the manuscript with your comments, and answer some of your questions in the following.
  Indeed, the flattening condition removes any topography to make the top surface flat. "Tying the fault lines" relates to applying a deformation condition to the lines representing the contact between the horizon and fault surfaces, in order to remove the fault throw on these horizons. I will make those points clearer in the manuscript.
  I will rewrite the introduction to remove the parts borrowed from the previous manuscript, improve the manuscript organization and make clearer the purpose of the article.
  Indeed, Figure 6 shows that the restoration of the lower layers cannot be achieved properly with these boundary conditions and material properties. This is one of the reasons which lead to the following sections trying to assess their impact on the results, to improve the restoration process. The viscous/viscous interface could indeed be part of the problem, although the difference in viscosity should be able to mimic the granular/viscous interface. What could also be problematic is that the numerical solution and advection of the particles follow the grid discretization and not the interface itself, which may overly simplify the deformation at this interface. I will work on the analysis on these results to better show this limitation.
  What the analysis of the material properties showed, is that the main factor changing the results is the viscosity of the faults. The main difference between the 2D and 3D cases would then be, in my sense, the horizontal variation of the fault throws. This was not yet studied, however.
  The analogue experiment was done outside the scope of the study, so I do not possess all the informations on it, but I will try to add those requested to the manuscript.
  The dynamic conditions are considered more "natural" here in comparison to kinematic conditions because they do not impose a deformation on points where the deformation is not known, and try instead to build on known physical parameters or behaviour.
  There is indeed a dependency between viscosity and temperature, particularly when doing simulations at very large scales (mantle, subduction,...). Here, the temperature during the laboratory experiment was considered to be mostly constant and stable, and as such have no impact on the experiment itself. Moreover, the scaling of the model aims to represent basin models, where the temperature should not have large variations at the studied time scales. To simplify the problem and be able to assess the impact of other parameters, the variation of temperature was then neglected. In future studies, it could still be interesting to see its impact on simulations.
  Indeed, no comparison is made to other geomechanical restoration schemes, as stated in reply to RC1, I will remove these statements. I will also rewrite the conclusion to mirror the organization changes I will do on the manuscript.
  Kind Regards,
  Melchior Schuh-Senlis
  
  Citation: https://doi.org/10.5194/egusphere-2023-2039-AC4

Melchior Schuh-Senlis, Guillaume Caumon, and Paul Cupillard

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Short summary

This paper presents the application of a numerical method for restoring models of the subsurface to a previous state in their deformation history, acting as a numerical time machine for geological structures. The method is applied to a model based on a laboratory experiment. The results show that using more natural conditions in the computation of the deformation allows us to assess the value of some previously unknown physical parameters of the different materials inside the model.


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