the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Full prediction of unsaturated hydraulic conductivity – comparison of four different capillary bundle models
Andre Peters
Sascha Christian Iden
Wolfgang Durner
Abstract. To model the water, solute and energy transport in porous media, it is essential to have accurate information about the soil hydraulic properties (SHP), i.e. the water retention curve (WRC) and the soil hydraulic conductivity curve (HCC). Having reliable data information to parameterize these models is important, but equally critical is the selection of appropriate SHP models. While various expressions for the WRC are commonly compared, the capillary conductivity model proposed by Mualem (1976a) is widely used but seldom compared to alternatives. The objective of this study was to compare four different capillary bundle models in terms of their ability to accurately predict the HCC without scaling the conductivity function by a measured conductivity value. These expressions include two simpler models proposed by Burdine (1953) and Alexander and Skaggs (1986), which assume a bundle of parallel capillaries with tortuous flow paths, and two more sophisticated models based on statistical cut-and-random-rejoin approaches, namely those proposed by Childs and Collis-George (1950) and the aforementioned model of Mualem (1976a). In order to check whether different parametrizations of the WRC interfere with the suitability of the conductivity models, we utilized four different capillary saturation models in combination with each of the conductivity prediction models, resulting in a total of 16 SHP model schemes. All schemes were calibrated using 12 carefully selected datasets that provided water retention and hydraulic conductivity data over a wide saturation range. Subsequently, the calibrated models were tested and rated by their ability to predict the hydraulic conductivity of 23 independent datasets of soils with varying textures. The statistical cut-and-random-rejoin models, particularly the Mualem (1976a) model, outperformed the simpler capillary bundle models in terms of predictive accuracy. This was independent of the specific WRC model used. Our findings suggest that the widespread use of the Mualem model is justified.
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Andre Peters et al.
Status: final response (author comments only)
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RC1: 'Comment on hess-2023-134', Anonymous Referee #1, 28 Jun 2023
Overall, the paper is good, and the subject matter worth publishing. I provided detailed comments throughout the manuscript to improve clarity.
There are two main concerns I have that I believe should be addressed:
- In the Introduction, the literature overview on soil water retention curves is not up to date. I offer a few references below that might be of help. But there are additional recent papers worth citing in the overview.Li et al. WRR 2023 doi: 10.1029/2022WR033160
Wang et al. WRR 2022. doi: 10.1029/2021WR031297 (has some good references)
Rudiyanto et al. J. Hydrol. 2020. doi: 10.1016/j.jhydrol.2020.125041
Weber et al. WRR 2019 doi: 10.1029/2018WR024584 (some of you were involved)
- Partially as a consequence of this, the authors use outdated expressions for the WRC. For two of those (vGm and FX), recent papers proposed improvements that address the shortcomings that are highly relevant for this paper. References are provided below. This weakens the paper considerably. Section 4.3 looks awkward because of this.
Ippisch et al., Adv. Water Resour. 2006. doi: 10.1016/j.advwatres.2005.12.011
de Rooij, HESS, 2022. doi: 10.5194/hess-26-5849-2022
Wang et al. WRR 2022. doi: 10.1029/2021WR031297.
Some minor points that are not limited to a single place in the paper:
- the color scheme of the graphs involves two very light shades that I could not see terribly well
- the English needs a little work - commas appear in strange places, for instance. But nothing that hampers the readability of the text.
- I do not think 'Table' is abbreviated in HESS, or in any other journal.
- The paper requires familiarity with earlier work by this group. The referencing is adequate, so readers can easily find the earlier papers if needed. I therefore do not consider this a problem.I am in limbo about recommending minor or major revisions. Because the use of outdated WRC models really worries me and I realize that taking care of this will require some effort I am gravitating toward recommending major revisions. But the editor has the final say, of course.
- AC1: 'Reply on RC1', Andre Peters, 14 Jul 2023
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RC2: 'Comment on hess-2023-134', Anonymous Referee #2, 09 Jul 2023
Review of “Full prediction of unsaturated hydraulic conductivity – comparison of four different capillary bundle models” by Peters et al., HESS 2023
The approach advanced here by the authors (and in previous publications) rests on pragmatic arguments for representing the complexity of WRC and HCC which considerable number of ad-hoc approximations and parameters. This is not meant to judge such efforts as “useless” – these effects have their practical value and place as long as authors and readers remember these are simplifications made for practical applications by Peters and Durner (2008 and the string of studies thereafter) and should not be confused with physically-based theoretical frameworks – while the so-called PDI model contains several physically based elements, it is also riddled with empiricism that makes the title read a bit overstated. Moreover, in contrast with the calibrated language of Peters et al. (2021, WRR) statements are now made as if the simplifications made over the years are now “facts”. Various important assertions are made with no basis, such as: "representing the soil hydraulic properties in functional form is mandatory for simulation of water..." this simply not correct – many numerical models have been doing fine with lookup tables and other data-based information. The use of parametric models is a mere computational convenience as these models contain no new information (about SHP) other than what is contained in the data used to estimate parameters.
Many of the authors' pragmatic applications and approximations are useful, however, these need to be constantly stated to avoid confusion – my primary concern with this work is with the hyped representation of what is essentially an empirical model as theoretically-derived and physically-based. The cutoff critical pressures for non-capillary water, the arbitrary interpolation of the WRC to the dry end; the basis for the hypothesis of a universal tortuosity scaling coefficient; even the fact that eq. 22 in Peters et al. 2023 showing proportionality of Ks with alpha^2 has not been tested with hard data... Until “predictions” are made for unknown soils and broadly tested I would refrain from using terms such as "full prediction of unsaturated hydraulic conductivity (function)…”. I don’t want to belittle the hard work of the authors, however, such a proclamation in the title sounds more like a mantra at this stage.
In the following I make a few observations to place this work in context:
- Kv (vapor transport) is not and should never become a component of the HYDRAULIC conductivity function. While capillary and film flow components obey viscous flow (liquid under shear), Kv emerges from and obeys Ficks law of diffusion... I found referencing to Saito et al (2006) insufficient to resolving this fundamental physical discrepancy. Like many others, the reviewer fully understands the dilemma of HCC representation at the dry end, yet I would stay away from mixing two different physical transport processes by using the same function even for the sake of practicality (calling it K_effective is not helping much).
- In general, I found the motivation for the present study to be poorly defined. While the authors have been motivating their refinements of already existing models under the “umbrella” of pragmatism, how is comparing 4 or even10 fundamentally flawed bundle of capillaries (BCC) models is advancing the field? None of these models ever claimed to represent real soils, they do not consider film flow and cannot possibly address issues of liquid topology and connectivity other than via some ad-hoc tortuosity factor. So, whether a certain approach scales then normalizes or vice versa makes no difference in the fundamental fact that these are very poor toy approximations.
- To be specific regarding the motivation, the authors state in the abstract "The objective of this study was to compare four different capillary models in terms of their ability to accurately predict the HCC without scaling the conductivity function by a measured conductivity value". Upon reading this very specific objective it reads arbitrary and contrived (almost as it has been formulated after the study was completed...). Given the nature of these BCC models - how is comparing different BCC models with numerous parameters, thresholds and simplifying assumptions addresses the overall goals of the opening statements? (reliable data, accurate information, and SHP models). I am also puzzled by the notion of "accurately" - measured how? Is this the Akaike information criteria (Peters et al. 2021) is this proximity to a set of incomplete measurements as shown in the supporting materials? The point is that these models have been proposed and are used because they capture salient features of the capillary retention and (with many additional assumptions) also capillary flow processes. Devising another ad hoc set of parameters is not likely to lead to new insights or advance the field. I was able to count at least 20 parameters (from several pressure threshold criteria, to extrapolation schemes for WRC dry end and HCC, to smoothing parameters, and so on). I am thus left confused and wondering if the authors seriously consider these nuanced additions to be of practical use for say, watershed hydrologic models or Earth system modeling? or even for predicting solute transport in their own backyards?
- While there is nothing wrong with the mathematics, nor with various decisions made by the authors such as: saturation of 0.75 is a reasonable threshold for the end of capillarity and the onset of adsorption - the question I am grappling with is how this exercise in mathematical parameterization will advance our science? Wouldn’t any machine learning algorithm outperform all these assumptions if “accurately” is defined as in Peters et al., 2021? I wish the authors would have elaborated more on the innovation, novelty, generalization, etc. (declaring a new tortuosity factor “universal” is not enough – it needs to be shown).
- To properly review this study, I had to look back into a string of publications from the same authors. It is probably unintentional, however, when these studies are read in a sequence, one can notice the gradual “erosion” of qualifying statements that appeared in early work with some of the assumptions are now presented as established “facts” referring to previous publications (I much preferred the calibrated language of Peters et al. 2021; WRR). Perhaps a section on assumptions and approximations can help readers navigate through the physical and empirical elements of the study.
- Even the choice of using the capillary part of the WRC to deduce capillary flow under the HCC lacks depth – how the authors envision different liquid configurations in the pore spaces, liquid topology, continuity etc. – all of which must affect the “Full prediction of unsaturated HCC”
- Other than lip service to the role of film flow in the unsaturated HCC – not much is presented here to support a “Full prediction”. Can one use characteristic “grain diameter” as a substitute for surface area as in Kozeny-Carman or even Lebeu and Konrad – of course, not. There are hundreds of papers in the geophysical literature that show the failure of such approximation in shaly formations (clay soils). Where is the key parameter of soil specific surface area responsible for most of the films in the dry end of the WRC – this is part of my skepticism of the physical basis of the derivations here. Until WRC models embrace this critical (physical) parameter affecting many soil processes, there will always remain a representation gap of adsorption processes. No amount of BCC representation and tortuosity acrobatics is likely to fill in this gap!
I have other specific comments regarding presentation (similar colors of lines with no legends), and how inferences were made (how performance was measured), however, these are minor points. I am debating between recommending a major revision and an outright rejection of the manuscript until the key issues above are resolved – in any case, the decision rests with the editor.
Citation: https://doi.org/10.5194/hess-2023-134-RC2 -
AC2: 'Reply on RC2', Andre Peters, 14 Jul 2023
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RC3: 'Reply on AC2', Anonymous Referee #2, 15 Jul 2023
To keep this exchange short and constructive, I limit my response to making a few additional points for context and perspective:
- It is unlikely that the authors would retract inaccurate assertions made in previous publications such as, that the use of parametric models is “mandatory” (see introduction in Peters et al. 2023), their recurring argument that because certain information has been published previously precludes it from further scrutiny (in this review) is inconsistent. Not only because of the permanently tentative basis of the scientific method, but also considering potential confusion of young students exposed to such assertions without knowledge of the broader picture.
- Considering that each equation and statement you are making in this work is subject to scrutiny, the authors can choose among several options. If vapor transport or film flow are NOT part of the proposed theory (which they shouldn’t as non-capillary components of the representation), the right thing to do is (1) to disclose these limitations upfront (hence, “full prediction” is properly qualified) and/or (2) not repeat these derivations here and treat these as “facts”.
- For context – I have not seen a fundamental explanation for why Kv (vapor transport component of K) should be lumped with the HCC. This perpetual misconception was mentioned again here (A.5) and commented that “Kv as a function of the invoked WRC” (how is this exactly a function of the WRC – via eqs. 12 and 13 in Saito et al. 2006?). If there is a sense of “somewhat disparaging style of the review” it is probably my subconscious response to the shockingly sloppy science that places the burden of proof for such blunder on a numerical study by Saito et al. 2006 (attempting to solve the Richards equation with vapor transport) – I give the authors more credit than reflected in this off-hand response and reference to Saito et al. 2006. Not only that this is categorically not hydraulic conductivity and cannot be linked with coefficients that describe shear flow of liquid in proportion to pressure gradients (or potentials); the underlying gradient of potential energy that drives flow of liquids in porous media is not the proper gradient for vapor transport as discussed (many years ago) by: L. Monteith, G.S. Campbell, 1980, Diffusion of water vapour through integuments—Potential confusion, Journal of Thermal Biology, 5(1) 7-9 “The appropriate potential is the concentration of water vapour in air or the vapour pressure. The free energy of water is not an appropriate potential and Toolson's (1978) analysis for arthropods is therefore incorrect”. I hope the authors would take this into consideration and refrain from lumping these different processes into the HCC.
- Finally, for perspective – (i) the authors’ claim of “full prediction of HCC” – but also argue that “Notably, our title does not contain any statement of “physics based” or similar. For us, "full prediction of HCC" simply means that we predict HCC without conductivity data, and this is the case after we calibrate the model”. In physics, “full prediction” implies generality afforded by physical principles otherwise we correlate, calibrate, estimate etc.; (ii) it would have been appropriate to mention the pioneering study of: Jackson, R.D., 1972. On the calculation of hydraulic conductivity. Soil Sci. Soc. Am. Proc. 36:380-383. (and an earlier study by Jackson et al., 1965, WRR very similar to this one – both have been cited by van Genuchten 1980). Jackson 1972 was published over half a century ago and uses similar line of reasoning and tools as the authors are reporting here (and in Peters et al. 2023) to “fully” predict the unsaturated hydraulic conductivity function. What remains from this early effort are: (1) the motivation of van Genuchten 1980 and others to convert WRC to HCC; and (2) a nice classroom exercise regarding how this might work using the BCC usually accompanied with a stern warning that there is no predictability power here simply because liquid organization (the capillary flow pathways) are NOT representable in this lumping exercise. Fifty years later, the authors are claiming to have solved the problem – yet other than curve fitting (not that different from Jackson et al. 1965, WRR) there is no fundamental physical explanation of how this magical tortuosity factor works and how general is it.
Citation: https://doi.org/10.5194/hess-2023-134-RC3 - AC3: 'Reply on RC3', Andre Peters, 21 Aug 2023
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RC3: 'Reply on AC2', Anonymous Referee #2, 15 Jul 2023
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RC4: 'Comment on hess-2023-134', John R. Nimmo, 18 Jul 2023
Review by John Nimmo.
This paper evaluates the relative merits of four different capillary bundle models, applied within the framework established in the earlier paper P23 (Peters and others, 2023), for predicting unsaturated hydraulic conductivity from retention data. The tests are rigorous and conducted for data from twenty-three widely different soils, and using four different commonly used formulas for representing the soil water retention. The results are satisfyingly definitive in showing that the Mualem model gives the best results. Another contribution of this paper is in exploring and elucidating the function of the saturated tortuosity coefficient introduced in P23.
An important insight revealed in lines 230-241, and noted in line 179 and elsewhere, is that the value of the saturated tortuosity coefficient τs depends on the particular conductivity model it is used with. This is not surprising, though it brings out the fact that unless τs can be computed independent of a K model, it is not universal and not a property of the medium. This feature seems at odds with the hypothesis of a universal value as described by P23. It should be explained and perhaps elaborated in the discussion.
This paper needs added material in the introduction or a separate section that reviews previous tests and comparisons of capillary bundle models (e.g. van Genuchten and Nielsen, 1985; Hoffmann-Riem and others, 1999; Kosugi, 1999). That will help make clear the context of this work and the contribution it adds to the existing literature. Although I hesitate to mention my own work in a manuscript review, a paper of mine (Nimmo and Akstin, 1988) is directly relevant and has some parallels with the present work. In it, we tested four capillary bundle models, three of which are among the four tested in this new manuscript. Our test was done on different samplings of identical soil material, with variations in packing and preparation to produce samples that varied modestly in porosity and hydraulic properties. As in the present work, the model of Mualem (1976) was found to be preferable. The test made a convincing demonstration of the basic utility of capillary bundle models in showing that measured retention curves for different samples, plugged into the capillary bundle models, gave rise to predicted conductivity curves that differed from each other, in direction and in approximate magnitude, in the same way the four sets of measured conductivity data differed. At the time of that study, this result raised my previously dubious regard for the usefulness of capillary bundle models.
Though the work in this manuscript shows little real innovation, it has value in its thorough testing of widely used models and in providing helpful information for anyone considering the hydraulic conductivity-predicting model put forth in P23. It should be published after moderate revision.
Other comments:
28: Reword. Functional form is not mandatory. There are alternatives, like tabulated values, though little used.
33: “Any liquid flow ceases” is too definite a statement. Better to just say vapor flow becomes the dominant transport process.
50: Seems like a misplaced comma.
92: Better to say “particles” than “molecules” because molecules are subject to Brownian motion and do not individually follow a streamline.
99: Subscript sc is reversed in equation.
133: Define θs and θr.
161: Insert “among”—they are among the most commonly used . . .
220: In Fig. 1, curves for CCG and Bur are faint and hard to see—should be thicker. Also colors should be different to show more contrast than between the blue and green shown. Similar effects in Fig. 4.
311: To help the reader, for the left side of Fig. 6, note briefly what is different to give four slightly different retention curves when the same FX model is used for each.
345-387. Appendix A1 is highly duplicative of original publications and the appendix in P23. It should be omitted, except possibly for part A1.3. The material in part A1.3 might be better placed in the main text.
411: There is no Figure 7. Must be Figure A4.
Hoffmann-Riem, H., van Genuchten, M.T., and Flühler, H., 1999, General model of the hydraulic conductivity of unsaturated soils, in van Genuchten, M.T., Leij, F.J., and Wu, L., eds., Proceedings of the international workshop on Characterization and measurement of the hydraulic properties of unsaturated porous media: Riverside, CA, University of California, p. 31-42.
Kosugi, K., 1999, General model for unsaturated hydraulic conductivity for soils with lognormal pore-size distribution: Soil Science Society of America Journal, v. 63, 270-277 p.
Mualem, Y., 1976, A new model for predicting the hydraulic conductivity of unsaturated porous media: Water Resources Research, v. 12, no. 3, 513-522 p.
Nimmo, J.R., and Akstin, K.C., 1988, Hydraulic conductivity of a sandy soil at low water content after compaction by various methods: Soil Science Society of America Journal, v. 52, no. 2, 303-310 p.
Peters, A., Hohenbrink, T.L., Iden, S.C., van Genuchten, M.T., and Durner, W., 2023, Prediction of the absolute hydraulic conductivity function from soil water retention data: Hydrol. Earth Syst. Sci., v. 27, no. 7, doi:10.5194/hess-27-1565-2023, 1565-1582 p.
van Genuchten, M.T., and Nielsen, D.R., 1985, On describing and predicting the hydraulic properties of unsaturated soils: Annales Geophysicae, v. 3, no. 5, 615-628 p.
Citation: https://doi.org/10.5194/hess-2023-134-RC4 - AC4: 'Reply on RC4', Andre Peters, 21 Aug 2023
Andre Peters et al.
Andre Peters et al.
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