the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 15 Jun 2026
Beyond hydraulic diffusion: reaction front propagation and timescales in metamorphic (de)volatilization processes
Abstract. Metamorphic (de)volatilization reactions play fundamental roles in many geodynamic processes and are frequently associated with propagating reaction fronts, yet the mechanisms controlling front propagation remain poorly constrained. Here, we derive new analytical solutions for fluid pressure-driven (de)volatilization reaction fronts in porous rocks. The solutions predict that reaction-zone width increases with the square root of time, a scaling that arises from mass conservation across the moving reaction front rather than from classical hydraulic pore-pressure diffusion. Front propagation is governed by an effective diffusivity that depends on both hydraulic and chemical parameters, including reaction-induced density changes. Unlike conventional hydraulic diffusivity, which neglects reaction-induced density variations and generally predicts much faster propagation, the effective diffusivity captures the retardation of reaction fronts caused by density changes, with larger reaction-induced density changes producing slower propagation. In addition to a single-front formulation, we derive an analytical solution for two simultaneously propagating, coupled reaction fronts. The fronts are dynamically linked through mass conservation, preventing independent propagation and causing the leading front to advance faster than the trailing front. Both the single- and two-front solutions closely match numerical simulations. Application of the two-front model to published gypsum dehydration experiments shows that accounting for pore-water to pore-vapor transitions yields more realistic permeability estimates than single-front models. Application to natural metamorphic reactions predicts permeabilities between 10-18 and 10-24 m2, consistent with independent experimental, geophysical, and geological estimates. By analogy, the analytical framework can also be extended to chemically controlled reaction fronts governed solely by chemical diffusion. The analytical solutions provide a quantitative framework for estimating the timescales of natural and experimental (de)volatilization processes involving propagating reaction fronts.
- Preprint
(1790 KB) - Metadata XML
- BibTeX
- EndNote
Status: final response (author comments only)
-
RC1: 'Comment on egusphere-2026-3107', Ali Karrech, 21 Jun 2026
-
AC2: 'Reply on RC1', Liudmila Khakimova, 07 Oct 2026
Dear Prof. Ali Karrech,
We appreciate your careful review and encouraging assessment of our work.
Your comments helped us clarify the model assumptions, the pressure-gradient and permeability notation, and the material properties in the coupled two-front solution. We have also corrected typographical errors.
Please find our detailed point-by-point response in the attached PDF.
On behalf of all authors,
Liudmila Khakimova
-
AC2: 'Reply on RC1', Liudmila Khakimova, 07 Oct 2026
-
AC1: 'Comment on egusphere-2026-3107', Lyudmila Khakimova, 18 Aug 2026
Dear Dr. Karrech,
Thank you for your careful reading of our manuscript, your constructive comments, and your positive assessment of the work.
We have addressed all the points raised in your review. In particular, we have clarified the assumptions and limitations of the analytical model, revised the notation for the pressure gradient and permeability, explained the role of the material properties in the coupled two-front solution, and carefully proofread the manuscript.
Please find attached the revised manuscript and a detailed point-by-point response describing all changes.
We believe that these revisions have improved the clarity and precision of the manuscript, and we thank you again for helping us strengthen the work.
On behalf of all authors,
Liudmila Khakimova
-
RC2: 'Comment on egusphere-2026-3107', Nicolas Riel, 07 Sep 2026
The manuscript entitled “Beyond hydraulic diffusion: reaction front propagation and timescales in metamorphic (de)volatilization processes” by Khakimova and co-authors provide a new derived analytical solution from mass conservation at a moving reaction front.
The manuscript is very well written and the figures of good quality. The single and two-front results are validated against two independent numerical models (own model and Schmalholz et al., 2024). Moreover, the distinction between hydraulic diffusivity and the new reaction-controlled effective is a very interesting result .
The applications (permeability estimates for natural reactions and the gypsum experiment via the two-front water-to-vapor solution) are also well chosen.
I have only one main point requiring clarification:
For the numerical application, the reaction is represented throughout the manuscript using a look-up table relating total density, ρ_total, to fluid pressure, P (Fig. 2a/2e, Fig. 3a/3e).
The manuscript reads that such a relation "can be determined for specific reactions from equilibrium phase calculations using open-access software such as PerpleX ... or Thermolab"
However, I don’t see the description of how the look-up-table for the numerical results was obtained.
- Were these ρ_total-P (or Δρ_total, ΔP) values taken from independent Perple_X/Thermolab calculations run for this study, or from the literature (e.g., Schmalholtz et al., 2024)?
- If from phase-equilibrium calculations, what P-T conditions (or P-T path) and bulk composition were assumed for each reaction?
- What equation of state was used for the fluid/vapor densities, particularly for the water-to-vapor transition in section 4.4?
I think that adding these details is important (even briefly, e.g. as a table in an appendix or the data availability statement) and would help the readers reproduce the reported permeability and timescale estimates which would strengthen the already strong quantitative applications in Section 4.
Nicolas Riel
Minor comment:
L144 -> Perple_X
Citation: https://doi.org/10.5194/egusphere-2026-3107-RC2 -
AC3: 'Reply on RC2', Liudmila Khakimova, 07 Oct 2026
Dear Dr. Nicolas Riel,
Thank you for your constructive comments and positive assessment of our manuscript.
We have clarified the source of the thermodynamic input values and the fluid properties used in the gypsum application. Please find our detailed response in the attached PDF.
Thank you again for your helpful suggestions.
On behalf of all authors,
Liudmila Khakimova
Viewed
| HTML | XML | Total | BibTeX | EndNote | |
|---|---|---|---|---|---|
| 194 | 87 | 32 | 313 | 30 | 31 |
- HTML: 194
- PDF: 87
- XML: 32
- Total: 313
- BibTeX: 30
- EndNote: 31
Viewed (geographical distribution)
| Country | # | Views | % |
|---|
| Total: | 0 |
| HTML: | 0 |
| PDF: | 0 |
| XML: | 0 |
- 1
The manuscript addresses the important problem of reaction front propagation that is, according to the authors, driven essentially by the conservation of mass in the quasi-static regime. The authors derive the governing equations of a single reaction front at a first stage, then they propose two reaction fronts that are coupled. Their derivations lead to a characteristic square-root-of-time scaling of the front propagation that is governed by the conservation of mass across the moving boundary rather than classical non-reactive hydraulic pore-pressure diffusion. Hence, the authors define "effective diffusivities" that incorporate both hydraulic variables (permeability, viscosity, pressure difference) and chemical parameters (variation of density within the front). This result is interesting because it explains why the reaction progress is orders of magnitudes slower than expected based on the intuitive pure hydraulic process. Comparisons with the data of Fusseis (2012) and Schmalholz (2020) seem to support the proposed model.
There are some minor changes that I would like to suggest to improve the readability of the paper:
1- The assumptions and limitations of the proposed model need to be stated clearly. For example, the model is only valid if the matrix is rigid. This assumption makes the model useful in the absence of mechanical loading only. This model may also fail if the front profile changes shape (the case where v_fron is dependent on space) or the reaction presents wave fingers.
2- In equation 9, the gradient is is unusual it should be Delta P/Delta x, because in the end, x-front and x-behind do move.
3- In Darcy law, why is k dependent on z_behind and not on z_front or z_average?
4- In many situations, the coupled two front seem to require that density, permeability and viscosity are all different. (see equations A19, A20 and A21 etc), but this is not required. In fact it sufficient that one of them changes. For example in the case of gypsum, the density and viscosity of water remain the same at room temperature but permeability may change significantly.
5- Please check the text for typos. For example, in page 16, "generate a reaction front with a witdth of 1 m...". Change this to width.
Overall, the manuscript is of good quality in my opinion, and I recommend it for publication.