the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Modeling stable and unstable flow in unsaturated porous media for different infiltration rates
Abstract. The gravity-driven flow in unsaturated porous medium is still one of the biggest unsolved problems in multiphase flow. Sometimes a stable flow with an uniform wetting front is observed, but at other times it is unstable with distinct preferential pathways even if the porous material is homogeneous. The formation of an unstable wetting front in a porous medium depends on many factors such as the type of the porous medium, the initial saturation or the applied infiltration rate. As the infiltration rate increases, the wetting front first transitions from stable to unstable for low infiltration rates, and then from unstable to stable for high infiltration rates. We propose a governing equation and its discretized form, the semi-continuum model, to describe this significant non-monotonic transition. We show that the semi-continuum model is able to capture the influx dependence together with the correct finger width and spacing. We also present that the instability of the wetting front is closely related to the saturation overshoot in 1D. Finally, we demonstrate that the flow can be still preferential even when the porous medium is completely wetted.
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RC1: 'Comment on egusphere-2023-2785', Anonymous Referee #1, 25 Jan 2024
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The manuscript deals with a classical 1D analysis of the unsaturated flow stability during a wetting event. The numerical treatment of the problem at stake, although quite standard and therefore with no new insight, appears to be correct (I went through it, and I didn’t find any error). Instead, the English usage requires a very solid and sounding proofreading. Besides this (marginal) aspect, the main issue which I see with the manus is of methodological nature. In particular, my skepticism is two-fold. First, accounting for the gravity solely (thus neglecting the impact of retention) may be unrealistic especially if one is interested (as it is usually happens in the applications) on the “onset” of the stability vsinstability”. Second, the authors have carried out a long and intensive analysis of the flow rates which make (or not) stable the flow, but what and where is the stability analysis? The very new and innovative insight could have been a Touring analysis of the stable/unstable flow patterns in order to highlight which ones are those parameters (and perhaps the infiltration rate is the most important one) that regulate such a stability. Instead, the manuscript, as it is, is nothing more a numerical analyses (followed by an experimental benchmark), quite similar to many others, already existing in the literature.
Citation: https://doi.org/10.5194/egusphere-2023-2785-RC1 -
AC1: 'Reply on RC1', Jakub Kmec, 29 Jan 2024
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The response to reviewer RC1 was uploaded in the form of a supplement.
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AC1: 'Reply on RC1', Jakub Kmec, 29 Jan 2024
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RC2: 'Comment on egusphere-2023-2785', Anonymous Referee #2, 01 Feb 2024
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The paper introduces a hysteretic model that can generate unstable infiltration fronts and the resulting fingering patterns. The model is tested by reproducing highly idealized experiments in two-dimensional sand tanks, with very limited additional analysis.
The model and its equations are not terribly well explained, which creates some confusion. The model test on experimental data is convincing, but there is no follow up. There is no indication if the model is of any use to apply to natural conditions or to address real world problems. This gives the impression of only half a paper: the presentation of a model and its validation, but no meaningful application.
Major comments (also at the top of the attached annotated manuscript):
The model is not new, and the simulations cover a very limited scope, merely attempting to reproduce laboratory experiments under conditions that have no relevance for the field. This strongly constricts the overall contribution of the paper.
Although the authors discuss the literature at length throughout the paper, they seem to have missed many relevant papers. In the detailed comments I provide numerous references that may be useful to turn this paper into material that deserves publication.
The English needs to be improved a lot. I had a hard time understanding important sections of the paper. In combination with the inadequate explanation of the equations and their variables, it is not really possible to grasp the theoretical elements of the paper.
The model as such is interesting, and could be part of an interesting paper if it is better explained. To make the paper complete, the model would have to be used to carry out simulations that have some relevance for the real world, not the world of 2D tanks filled with air-dry artificial porous media.
The Results are tedious to read.
Too much literature is discussed in the Results section. It slows down the pace and increases the wordiness of the section. The literature review belongs in the Introduction.
You only appear to pursue to reproduce experiments in 2D sand tanks under highly idealized conditions that have no relevance for field scale problems related to preferential flow.What is the point of developing a model that cannot be applied to address problems in natural and agricultural soils, but is entirely focused on experiments with artificial porous media that were carried out to reveal the physical mechanism driving wetting front instability, so that the science could move on and make this knowledge operational for field problems?
You do not convince the reader why your model matters. Confirming experimental results only serves to show that you model works, so you can apply it to problems beyond experimental reach. Instead, you made it the main point of the paper.
Figure out what the real contribution of the paper is, pick the most interesting results, and discuss these intelligently, with an eye to theoretical advances as well as relevance for the real world. Rigourously cut away the obvious stuff. This will require a whole suite of new model runs, new figures, and a more lively discussion.
You present a state-of-the-art hysteretic model, but use it with outdated soil hydraulic functions. Preferential flow under natural conditions will manifest itself during a season with many cycles of rain and evapotranspiration. This is a wonderful playground for a model that can handle hysteresis, but you have not considered even one cylce. This is one of many missed opportunities. See my comments on the Results Section for more thoughts on this.
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AC2: 'Reply on RC2', Jakub Kmec, 29 Feb 2024
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Dear reviewer, we sincerely appreciate your thorough review that can significantly improve the quality of the manuscript. The response was uploaded in the form of a supplement.
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RC3: 'Reply on AC2', Anonymous Referee #2, 06 Mar 2024
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In the response to one of my comments, the authors reply:
‘It has been mathematically proved that the Richards’ equation is unconditionally stable [29] under monotone boundary conditions. The result holds regardless of any particular form of the hydraulic conductivity or the non-decreasing and smooth retention curve, including any type of hysteresis. Therefore, in principle, the Richards’ equation cannot admit finger-like solutions in this case. This is not surprising given its parabolic nature. However, if the saturation overshoot is created “manually”, the Richards’ equation will maintain the overshoot, and hysteresis indeed stabilizes a finger in this case [11, 5]. One of the possibility is to create the overshoot using a time-dependent Dirichlet condition [27] or by defining a bottleneck (zero flux) using a water entry pressure [26]. In our opinion, this is somewhat artificial way to create the overshoot, as the model should ideally be able to generate the overshoot without the need for such ad-hoc threshold directly incorporated into the model. Forming the saturation overshoot should be an output of the model.’
I disagree with the characterization of a water-entry value as an ‘ad-hoc threshold’. I argue instead that the water-entry value is a necessary physical attribute of the soil: not requiring a water-entry value is equivalent to setting the matric potential at which water can enter a dry soil at -∞ (arbitrary units of length, if the matric potential is expressed as energy per unit weight). When the water-entry value is -∞, the Laplace-Young Law stipulates that the following equality holds:
r1-1 + r2-1 = ∞
where r1 and r2 are the principal radii of curvature of the air-water interface at the pore where water enters the soil at infinite matric potential. This equality can only hold if at least one of these radii equals zero. Such a pore can obviously not exist, and if it could, it would be unable to conduct water.
Citation: https://doi.org/10.5194/egusphere-2023-2785-RC3 -
AC3: 'Reply on RC3', Jakub Kmec, 11 Mar 2024
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Dear reviewer, thank you for your response. Our response is again uploaded in the form of a supplement.
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RC4: 'Reply on AC3', Anonymous Referee #2, 26 Apr 2024
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Dear authors,
Apologies for the late reply - field work and a conference interfered.
Just a short response to exppress my appreciation for the insightful response. I find myself in the strange situation that I have learned more from out discussion than from the paper that sparked it. I hope you will find a way to infuse the paper with the thoughts you presented in response to my comments.
Sincerely yours.
Citation: https://doi.org/10.5194/egusphere-2023-2785-RC4
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RC4: 'Reply on AC3', Anonymous Referee #2, 26 Apr 2024
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AC3: 'Reply on RC3', Jakub Kmec, 11 Mar 2024
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RC3: 'Reply on AC2', Anonymous Referee #2, 06 Mar 2024
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AC2: 'Reply on RC2', Jakub Kmec, 29 Feb 2024
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