the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Modeling liquid transport in the Earth’s mantle as twophase flow: Effect of an enforced positive porosity on liquid flow and mass conservation
Changyeol Lee
Nestor G. Cerpa
Dongwoo Han
Ikuko Wada
Abstract. Fluid and melt transport in the solid mantle can be modeled as a twophase flow in which the liquid flow is resisted by the compaction of the viscously deforming solid mantle. Given the wide impact of the liquid transport on the geodynamical and geochemical evolution of the Earth, the socalled “compaction equations” are more and more incorporated in geodynamical modeling studies. The implementation of these equations requires a regularization technique to handle the porosity singularity in dry mantle. Moreover, it is also common to enforce a positive porosity (liquid fraction) to avoid unphysical negative values of porosity. However, the effects of this “capped” porosity on the liquid transport and mass conservation have not been quantitatively evaluated. Here, we investigate these effects using a series of 1 and 2dimensional numerical models using the commercial finite element package COMSOL Multiphysics®. The results of benchmarking experiments against a semianalytical solution for 1 and 2D solitary waves illustrate the successful implementation of the compaction equations. We show that the solutions are accurate when the element size is smaller than half of the compaction length. Furthermore, in timeevolving experiments where the solid is stationary (immobile), we show that the mass balance errors are similarly low for both capped and uncapped experiments (i.e., allowing negative porosity). When Couette flow, convective flow, or subduction corner flow of the solid mantle is assumed, the capped porosity leads to overestimations of the mass of liquid in the model domain and mass flux of liquid across the model boundaries, resulting in intrinsic errors in mass conservation even if high mesh resolution is used. Despite the errors in mass balance, however, the general trends of porosity evolution in the capped experiments are similar to those in the uncapped experiments. Hence, the use of the regularization of the compaction equations with the enforced positive porosity is reasonable for modeling fluid and melt transport in a deforming mantle.
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Changyeol Lee et al.
Status: final response (author comments only)

RC1: 'Comment on egusphere20231719', Anonymous Referee #1, 22 Aug 2023
The paper discusses an interesting and important topic related to the issue of negative porosity appearing in numerical simulations.
Some issues that should be addressed follow.
1. Abstract, line 17: "The implementation of these equations requires ..." Perhaps it is best to soften the statement, since there are "positivity preserving" numerical methods used in other fields. Such methods may be adaptable to mantle dynamics.
2. Line 115: The singular behavior is due to (10) and (11), not just (10).
3. Lines 122124: I assume that you mean that the computed phi is replaced by max(0,phi). Please define precisely what it means to "cap" the porosity, so there is no confusion.
4. Line 151: "equals" should be "equal".
5. Line 157: This is the first statement that the porosity is greater than 1. It also occurs in the numerical results. Since the porosity can not exceed 1 by definition, perhaps some explanation is needed here.
6. Example 4.1: The explanation of results seems to miss the obvious. Figure (e) implies that the boundary fluxes agree, so (d) is due solely to capping. The amount of capping is significant.
7. Examples 4.24: It is difficult to compare the two numerical results in these problems. Figure (g) implies a large capping effect, while (b) shows a large error in the uncapped result (there is a lot od negative porosity). Thus it is diffucult to determine the correct behavior. Can you run a refined example to determine te correct behavior of the systems?
8. Line 440: The phrase "increasing mesh resolution does not significantly reduce the error" seems incorrect, and the statement is unsubstantiated. Please run some examples to either show whether this is the case.
9. The conclusion that it is reasonable to cap the porosity (appearing several places, such as the abstract and conclusions) seems a bit strong. Only the overall system behavior agrees between the two numerical methods in the tested scenarios. But the behavior also differs substantially in important ways.
Citation: https://doi.org/10.5194/egusphere20231719RC1 
RC2: 'Comment on egusphere20231719', Samuel Butler, 30 Aug 2023
A review of “Modeling liquid transport in the Earth’s mantle as twophase flow: Effect of an enforced positive porosity on liquid flow and mass conservation”
In this paper, the authors solve the compaction equations in the small porosity limit in a number of scenarios, including solitary waves, in a fixed solid background, in a Couette flow solid background, in a convecting solid background and in a corner flow. For the solitary waves, the authors compare their solutions to analytical solutions while for the other cases, the authors monitor the quality of their solutions by considering the global conservation of liquid. The authors compare solutions in which the porosity is “capped” or forced to remain between 0 and 1 and “uncapped” in which case it sometimes becomes negative. They show that conservation of liquid is generally better obeyed for the uncapped solutions. However, they conclude that the general behaviour of the capped and uncapped solutions is similar and that capped solutions are generally reasonable.
The paper explores an interesting range of models and is generally well written. I have made a number of comments on the annotated manuscript that I would like for the authors to act on.
I have the following additional comments:
 The main purpose of the paper is to compare the behaviour of “capped” and “uncapped” solutions. However, the authors do not describe in the paper how they “cap” the solutions. If the porosity solution becomes negative during time stepping, do they simply set it to 0 in the model? Also, how is the capping implemented in Comsol? Both of these questions need to be addressed.
Additionally, I would think that the error would depend on how the capping is carried out. Butler (2017) introduced an analytical transformation that caused the porosity to always fall between 0 and 1. I think that authors should compare this type of “capping” and perhaps others in terms of the liquid volume conservation errors.
 The authors only consider conservation of liquid volume as an error measure. However, they could also consider the global balances represented by the other governing equations of their system. For instance, integrating equation 2 over the domain gives a mechanical energy balance equation i.e. the work done on the system by forces exerted on the boundaries is balanced by the work done by gravity. Including these additional balances would significantly increase the level of interest in this paper.
Reference:
Shearinduced porosity bands in a compacting porous medium with damage rheology, S.L. Butler, Physics of the Earth and Planetary Interiors, 2017, 264, 717

CC1: 'Comment on egusphere20231719', Chenyu Tian, 31 Aug 2023
In this paper, the authors did a great job discussing imposing capped porosity in numerical simulation to avoid unphysical negative situations.I have some comments as follows.
1. The authors indicated using direct fullycoupled PARDISO solver. Can you explain more on the parallelism and the order of accuracy of time integration scheme? Can you include cpu time used with different mesh sizes?
2. The presentation of figure 2 is not good. It is hard to differentiate lines with similar blues especially (5,3,1) and (7,3,1) lines. The font used in figure 2 is too small.
3. Figure 8 shows the relative volumebalance error of both capped and uncapped experiments, which indicates that capped porosity produced very large relative error up to 1e3 in some cases. Such large relative error jeopardize the claim made in this paper.
4. The authors claim Figure 8 a) shows a significant decrease in the relative volumebalance error. I would suggest run several more smaller mesh size to determine whether the relative error will go negative or not.
5. In this paper, the authors run experiment on four different mesh sizes. I would suggest run more mesh sizes and obtain more data points. The authors can state in the paper if the limitation of computational resources prevent them from using more data points.
6. The authors used a userdefined small porosity for the capped simulation. I would suggest the authors explore different definitions and scaling of the userdefined small porosity. A different definition or scaling of the small porosity might help with decreasing the relative volume balance error.
Citation: https://doi.org/10.5194/egusphere20231719CC1
Changyeol Lee et al.
Changyeol Lee et al.
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