the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 01 Apr 2026
Integrating Muli-Step-Flux-Method for full range soil hydraulic characterisation: From saturation to oven-dryness
Abstract. Soil hydraulic properties (SHP), defined by the water retention curve (WRC) and the hydraulic conductivity curve (HCC), are crucial to describe water storage and flow in soils. Several methods have been developed and combined to measure these two fundamental curves across the full range from saturation to oven dryness. However, for the HCC, there is still a data gap between approximately -1 to -100 hPa, which is expected to be affected by soil structure. We present an experimental workflow in which the multi-step-flux (MSF) method is integrated into the well-established combination of methods for measuring SHP, namely the falling head method, the simplified evaporation method, and the dew point method, specially designed to be applied to the same sample. The MSF is an adaptive, direct measurement of the HCC based on applying series of steady-state water flows to a soil sample characterised by unit hydraulic gradient. Once equilibrium is achieved, the sample is characterised at each step by a constant pressure head, constant water content and constant unsaturated hydraulic conductivity which is equal to the applied flux. We tested the method for three different soil columns: a repacked sand with a very well-defined air-entry pressure and two undisturbed structured silt loams. For the sand, the MSF results coincide with the saturated hydraulic conductivity value, measured with the falling head method. For the undisturbed loam samples, the structural effect on the HCC is clearly visible. Integrating the MSF into the common lab-workflow to characterise SHP will help future studies to investigate soil structure effect on SHP.
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Status: final response (author comments only)
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RC1: 'Comment on egusphere-2026-1405', Hans-Jörg Vogel, 13 Apr 2026
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AC1: 'Reply on RC1', Mathilde B. S. Nielsen, 15 Jul 2026
Dear Editors and dear Reviewers,
We would like to thank the reviewers for their constructive comments and valuable suggestions, which helped us to improve the manuscript considerably. Hans-Jörg Vogel and reviewer 2 requested additional information on the benefits of using a Mariotte bottle as a water reservoir, clarification of the hydraulic conductivity modelling for the Loam 2 sample, and further discussion of the applicability of the MSF method for dual-permeability approaches along with a discussion about limitations and perspectives. In addition, important questions were raised regarding the equilibrium conditions of the MSF method. Jos C. van Dam mainly provided suggestions of how to improve the display of Figure 2 and to unify units of pressure heads.
We have addressed and implemented these points in the revised manuscript and provided detailed responses to the individual comments below.
Furthermore, to better capture the air-entry region of the sand hydraulic conductivity curve, we performed an additional MSF experiment covering the range from saturation to a pressure head of −69 cm. These additional measurements strengthen the evaluation of the method and have been incorporated into the revised Figure 2.
To address a concern of Reviewer 2 about the uncertainty in the zero-pressure head gradient arising from fluctuations of the pressure head beneath the sample around the target pressure inside the sample, we now provide the raw pressure regulation data of the MSF from Loam 1 in the supplementary material, along with the corresponding median and mean values for both the tensiometer and the pressure sensor.
Overall, we carefully considered all comments and revised the manuscript accordingly. Lines given below refer to the new version of the manuscript with tracked changes. Text that are bold is reviewer original comments.
Thank you very much.
On behalf of all co-authors
Mathilde Nielsen
Referee 1, Hans-Jörg Vogel:
In this technical note the authors demonstrate how different methods covering different ranges in water potential can be combined to get a consistent description of soil hydraulic properties. The key point is how the results of the Multi-Step-Flux methods to measure hydraulic conductivity at high water potentials coincides with the more traditional methods. The results look promising, which is good news. This is true for the three materials used in this study were only drainage was considered. The paper is well structured and largely well written. I have one general remark and some technical details that should be considered prior to publication.
General remark:
The focus is more on the technical aspects and less on the potential of practical applications. Nonetheless, I miss at least a short discussion of the limitations of the applicability of these lab-measured SHPs for modeling water dynamics in natural soil (e.g. using Richards equation). These limitations come from well known features such as hysteresis and non-equilibrium effects (only shortly mentioned in the introduction) but also due to the boundary conditions (i.e. natural atmospheric boundary condition vs well define flux rates and water potentials adjusted in the lab). The MSF method is an excellent tool to study these important features which could be mentioned as well. I am a little worried that this paper sends the message that with MSF we now filled a critical gap and that we can now carry on with our traditional beloved concepts - it is not like that!Thank you very much for your comments. We have tried to address all your comments and have added in a Perspective and limitations part to the manuscript to address the issue rest in your general remark.
Technical details:L30 right reference? falling head method is much older.
Correct. We have changed the reference to (Klute and Dirksen, 1986; Darcy, 1856). (Line 30)
L37ff What about tension disc infiltrometers that covering at least a part of this gap?Thank you for this comment. We agree that disc infiltrometers are commonly used to determine near-saturated hydraulic conductivity and can cover part of the pressure head range usually between -1 and -10 cm. In the revised manuscript, we have therefore expanded the introduction (Line 40–45) and perspectivation and limitation (Line 306–308) of disc infiltrometer methods and their role in bridging the near-saturated HCC data gap. Furthermore, we now discuss the work of Sarkar et al. ( 2019), who proposed a modified disc-infiltrometer-based approach implemented on a KSAT apparatus to obtain hydraulic conductivity data in the near-saturated range. We also briefly discuss the advantages and limitations of disc infiltrometer methods compared with the MSF approach, including issues related to hydraulic contact between the porous disc and the soil surface and the non-uniform flow field resulting from localized infiltration (Line 40–45).
L78: How deep was the sample taken in this subsoil?
The sample was taken at a depth of 30-35 cm. We provide this information in the new version of the manuscript (Line 74).
L87 Conductivity was only measured for decreasing water contents (i.e. decreasing flux rates?)
Yes, this was already specified in lines 136–137. Please note that the well-established protocol combining KSAT, HYPROP, and WP4C characterizes the primary drainage curve, from assuming full saturation to oven dryness. We now highlight this in line 30 in new version. To integrate the MSF as a comprehensive method into the procedure and to reduce the effects of hysteresis and air entrapment, we focused on the drainage curve as well.
L89 Split this sentence in two?
Thanks for that suggestion.
It has been implemented: “To assess the reproducibility of the repacked sand, three replicate samples were used for KSAT and HYPROP. For the MSF measurements, two additional replicates were used, as the prolonged flow experiment may induce a settlement of repacked samples.” (Line 87–89)
L105-112: You could mention that this MSF method was introduced by Weller et al (2011)
We have added the reference to the materials and methods section of the new version of the manuscript. (Line 107).
L109: hydraulic heat?
Thanks for noticing. It should have been “hydraulic head”. We corrected the typo. (Line 109)
L122: Why do you need a Mariotte bottle in addition to a peristaltic pump?
The Mariotte bottle serves as a water reservoir and ensures a constant pressure head at the inlet of the peristaltic pump, independent of the water level in the reservoir. As the reservoir empties during the experiment, the Mariotte bottle maintains a constant hydraulic head and thereby helps to minimize unwanted variations in the applied flow rate. The peristaltic pump controls the target flux, while the Mariotte bottle provides stable boundary conditions for the water supply.
We have clarified this explanation in the revised manuscript: “The flux rate was controlled by a peristaltic pump that transported water from a reservoir to a drip irrigation head mounted on top of the sample. To ensure a constant water flux even as the water level in the reservoir decreased, the pressure head in the reservoir was maintained at +3 cm using a Mariotte bottle.” (Line 132–134)
L124: Please provide units for the delay values, I suppose they are in cm
It is in cm. It has now been added to the article: “. For pressure heads between 0 to -5 cm, the delta values were set to 1.5–2 cm; for pressure heads between -5 and -10 cm, it was set to 2–3 cm; and for pressure heads below -10 cm, it was set to 2–4 cm.” (Line 126–127)
L132: Could you justify the duration the flux steps of „at least one hour“ based on data (for example tensiometer readings during this hour)? When looking at Weller et.al. (2011) the water potential was still far from equilibrium after one hour. I think perfect equilibrium cannot be reached but may be some range of uncertainty for the water potential in K(h) could be given?
We agree that equilibrium conditions may take too long to be reached and that we are still in the equilibration phase when defining the tension. However, the time needed to reach close to equilibrium conditions is dependent on sample dimensions, the applied flux, soil structure and soil texture. As shown by Weller et al. (2011) (Fig. 4) and as observed in our measurements, biggest changes occurred within the first 30 min after changing the flux rate. Since our samples are only 5 cm in height a vertical adjustment of the matrix potential can be expected much faster than for the Weller et al. (2011) experiments (10 cm in height). In the new version of the manuscript, we specify that the stable condition is when the weight of the sample (water content) only changes is less than 2 g for 20 min, and the pressure head changes to less than 0.8 cm. (Line 138–140)
Furthermore, we would like to highlight that the two main other methods (falling head, simplified evaporation) are performed under dynamic conditions. To quantify the effect of dynamic/equilibrium state on the derived water retention curve, we performed an additional test (experiment not presented in the manuscript). For the same repacked sand, we combined pressure heads for the water retention curve derived by MSF and HYPROP under dynamic/equilibrium state. For the MSF, we compared the pressure head derived from the maximum overshoot after changing the flux with the pressure head equilibrated after 14 hours. For the HYPROP, we compared pressure heads before and after covering the soil surface and stop evaporation for 14 hours (no change in water content but change in pressure head). In both cases we could see that under dynamic conditions the pressure head was more negative as under equilibrium conditions (see Figure S1 in Supplement to reply) and that the WRC at dynamic conditions of the HYPROP and MSF are closer to each other.
L174: S-sharp?
Thank you for identifying the typo. We rephrased the paragraph according to the requests by reviewer 2. In the new manuscript the shape of the curve is described by: “Where K(h)total is the full model for loam 2, K(h)mac describes the macropore region by using a van Genuchten effective saturation function for hydraulic conductivity (van Genuchten, 1980):”(Line 182–183)
L208ff: The MSF flux rates are all in the range above the air entry point while it should be easy to adjust lower fluxes in MSF (which is an important strength of the method). Why did you not do this?
We agree. In order to show the strength of the method we performed an extra MSF measurement of the repacked sand which included a pressure head lower than the air-entry value (see new version of the manuscript, Fig. 2 and Fig. S2 Supplement to reply).
In Fig. 2 I see only two data points while from what is written in line 199 there should be four. Could you explain?
The first three sand MSF data points are very close to each other (at pF 0.54, 0.57, 0.59). We changed the dot size and included an outline for better visibility in the final version. (Figure 2 new manuscript)
L216: Again, tension infiltrometers are an established method in this range close to saturation. However, this is rather a field measurement but it would be interesting how it compares (in another paper).
We agree. It would indeed be interesting to compare the MSF with other methods to better understand and quantify the differences between field and lab observations.
We included the idea in the limitations and perspective section: “One way to address this limitation would be to use the MSF method to investigate hydraulic nonequilibrium and equilibrium processes (Vogel et al., 2023) and to compare the method with other approaches operating in a similar measurement range, such as disc infiltrometers (Sarkar et al., 2019) and multistep outflow experiments (Vereecken et al., 1997; Weller et al., 2011). Like most laboratory methods which are used to measure SHP, the described integration of the MSF into a standard workflow characterizes a defined volume of soil under controlled conditions and focus on the drainage curve of an initially saturated sample. Because natural soils exhibit considerable spatial variability in hydraulic properties, highly dynamic climatic conditions, and are rarely completely saturated, discrepancies between laboratory and field measurements are expected (Basile et al., 2003; Kumar et al., 2010; Vogel, 2019). Future studies should therefore compare MSF measurements with field-scale methods such as tension disc infiltrometers, mini disc infiltrometers, and lysimeters.” (Line 310–317).
L222ff: A note on the presentation in Fig.2: It is a bit difficult to assess curve shapes and “small step drops” when the symbols are plotted so large that such details are barely visible (but the figure probably looks much more convincing this way).
We have revised the figure. The repacked sand now includes the additional measurements, the data points have been made smaller, and the points have been outlined with a dark line for better contrast. (Figure 2 new manuscript)
L263: I have difficulties to follow this statement. Why is a sharp drop in conductivity near saturation physically unrealistic? That is exactly what we would expect (and what is frequently observed) if a relatively dense material is permeated by macropores, such as earthworm burrows (as expected for Loam 2). And besides, that is exactly what was measured! The physical problem that Ippisch rightly pointed out, is that the conductivity curve cannot be derived from the WRC using Mualem’s concept when the van Genuchten parameter n is significantly less than 2 - but this is a different story.
Thank you very much for your comment. But this is exactly what is happening in our sample. The Ksat is 1000 cm/day, but at a pF of 0, we have a conductivity two orders of magnitude lower, and the model predicts a smooth transition from Ksat to the first MSF point, which indicates that unrealistically large pores are draining. For example, for pF: -2, we can calculate an equivalent diameter of 30 cm, and at pF: -1, a diameter of 3 cm. At the same time, the water retention curve shows full saturation, even at pF: 0. This is a well-known limitation of the Mualem van Genuchten model, particularly when the value of n is much less than 2. The fitted n value of 1.03 also confirms this unrealistic behavior.
We have now added a short explanation in the revised manuscript: „This behaviour implies that unrealistically large pores, with equivalent diameters of several centimetres, would need to drain within a very narrow suction range near saturation, which is physically implausible, given that the measured retention curve is extremely close to saturation for the same pF range. “(Line 281–284)
L280: I am wondering if the well known dual porosity models are not suitable to address exactly this problem? It is obvious that additional MSF measurement are highly valuable, but do we really need new models to cover this phenomenon?
Thank you for this valuable comment. The other reviewer had similar concerns therefor the replays are identical. We would like to clarify a distinction that was not sufficiently clear in the original conclusion. There are two types of models relevant here: (1) SHP models (e.g., unimodal van Genuchten type models, bimodal van Genuchten-type models), which aim to describe the water retention and hydraulic conductivity curves continuously from saturation to oven dryness, and (2) flow models such as dual-porosity and dual-permeability models, which simulate water flow by explicitly accounting for distinct pore domains. Based on our results, the current bimodal SHP models do not adequately describe our near-saturation data, suggesting that alternative model concepts may be needed for this first type. For the second type of models, we agree that dual-porosity and dual-permeability models are well suited to this purpose, and we propose that the near-saturation data provided by the MSF can be used to parameterize these models, enabling potentially more data-based water flow simulations.
We have revised the conclusion to clarify this distinction: “In contrast, for process-based flow models such as dual-porosity and dual-permeability models, the near-saturation data provided by the MSF can serve as a valuable basis for parameterization, supporting more accurate characterization of structural pore domains and potentially improved water flow simulations.” (Line 334–336)
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AC1: 'Reply on RC1', Mathilde B. S. Nielsen, 15 Jul 2026
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RC2: 'Comment on egusphere-2026-1405', Anonymous Referee #2, 29 May 2026
Review of ”Integrating multi-step-flux-method for full range soil hydraulic characterisation: From saturation to oven dryness” by Nielsen et al 2026 Soil
This manuscript presents a method for measuring the water retention curve and the hydraulic conductivity function from the wet- to the dry end. The included examples show that the inclusion of multi-step flux experiments where a unit hydraulic gradient is applied for each step provide data with additional information. The manuscript is well organised and generally well written.
I have only minor comments that should be addressed before the manuscript is considered for publication.
Minor comments
L = Language can be improved.
L30. Use different symbols for “minus” and “range/dash”.
L38. Please clarify in what type of experimental setup the humidity sensors can be used.
L46. Change “to be” to “of”.
L46. “Integrating” into what?
L50. L. Consider changing “or” to e.g. “i.e”.
L52. Typo “gab”.
L54. All data points are affected by soil structure if you consider soil structure to be the spatial arrangement of solids and pores. Please rephrase.
L67. Unclear what “structure effects” means.
L72-73. Check language.
L76. Please, give soil texture data and organic carbon content for this silt loam soil.
L80. Change “μ” to “μm”.
L96. Remove “on”.
L109. “heat”?
L113. “tubes”
L120. Unclear to me what the pump was doing. Did it only supply water for the Marriot bottle?
L126. Consider changing “under pressure” to “suction” and add “pressure” at the end of the sentence.
L127. Earlier you used hPa for pressure potential. Be consistent. Change “low” to “high”?
L128. Units for delta? If it is cm the delta values seem very large.
L133. The term hydraulic “equilibrium conditions” usually imply no flow. Here you have a zero gradient in pressure potential. Please rephrase.
L134. Please give numbers for “relatively stable”.
L137. Change second half of sentence to e.g. “…we also calculated the water content for each step:”
L148. L.
L149-150. L.
L150-151. Please be more specific here.
L155-158. This text should not be under 2.1.4.
L169. What does “relative conductivity” mean. Is this term commonly used?
L173. What is a “S-sharp”?
L174. What is the origin or reasoning behind equation 10? As I understand it, it does not account for the large increase in conductivity with macropore size (i.e. within the macropore range).
L194. Unclear to me where these parameter values came from.
L206-207. Unclear to me what this means. The pressure potential ranges covered by the two methods did not overlap.
L241. You can conclude that the samples had different soil structure. From a statistical perspective you cannot say anything about possible effects of management.
L264. Consider removing “to represent the behaviour mathematically”.
L265. Remove “Only”. There could be other approaches that you did not test that would work well.
L268-269. A better way to phrase this would be that a small volume of water in the largest pores resulted in a large increase in hydraulic conductivity.
L269-271. Yes, the MSF data are useful for getting the correct shape of the curve. However, these phenomena are well known and the basis for dual-permeability models that have been around since early 1990s. The sentence could be rephrased to better reflect this point and not exaggerate the novelty of this study.
L280 Please be more specific than “during modelling”.
L280. Unclear to me what is meant by structurally complex soils.
L280-281. Possibly, but dual-permeability models are models built to handle these processes. Are they not good enough?
L344. Capital letters?
Figures
Figure 1. You could indicate (e.g. with an arrow pointing down) that the KSAT setup used falling head.
Citation: https://doi.org/10.5194/egusphere-2026-1405-RC2 -
AC2: 'Reply on RC2', Mathilde B. S. Nielsen, 15 Jul 2026
Dear Editors and dear Reviewers,
We would like to thank the reviewers for their constructive comments and valuable suggestions, which helped us to improve the manuscript considerably. Hans-Jörg Vogle and reviewer 2 requested additional information on the benefits of using a Mariotte bottle as a water reservoir, clarification of the hydraulic conductivity modelling for the Loam 2 sample, and further discussion of the applicability of the MSF method for dual-permeability approaches along with a discussion about limitations and perspectives. In addition, important questions were raised regarding the equilibrium conditions of the MSF method. Jos C. van Dam mainly provided suggestions of how to improve the display of Figure 2 and to unify units of pressure heads.
We have addressed and implemented these points in the revised manuscript and provided detailed responses to the individual comments below.
Furthermore, to better capture the air-entry region of the sand hydraulic conductivity curve, we performed an additional MSF experiment covering the range from saturation to a pressure head of −69 cm. These additional measurements strengthen the evaluation of the method and have been incorporated into the revised Figure 2.
To address a concern of Reviewer 2 about the uncertainty in the zero-pressure head gradient arising from fluctuations of the pressure head beneath the sample around the target pressure inside the sample, we now provide the raw pressure regulation data of the MSF from Loam 1 in the supplementary material, along with the corresponding median and mean values for both the tensiometer and the pressure sensor.
Overall, we carefully considered all comments and revised the manuscript accordingly. Lines given below refer to the new version of the manuscript with tracked changes.
Text in bold is the reviewer's original comment.
Thank you very much.
On behalf of all co-authors
Mathilde Nielsen
Referee 2, Anonymous:
Review of ”Integrating multi-step-flux-method for full range soil hydraulic characterisation: From saturation to oven dryness” by Nielsen et al 2026 Soil
This manuscript presents a method for measuring the water retention curve and the hydraulic conductivity function from the wet- to the dry end. The included examples show that the inclusion of multi-step flux experiments where a unit hydraulic gradient is applied for each step provide data with additional information. The manuscript is well organised and generally well written.
I have only minor comments that should be addressed before the manuscript is considered for publication.
Thank you very much for your comments. We have tried to address all your comments below.
Minor comments
L = Language can be improved.
L30. Use different symbols for “minus” and “range/dash”.
Thank you for bringing that to our attention. It has changed throughout the text.
L38. Please clarify in what type of experimental setup the humidity sensors can be used.
The experiment was designed as a simplified evaporation experiment, comparable to the classic HYPROP measurements. In addition to the tensiometers, two relative humidity sensors were used to extend the hydraulic conductivity measurements to potential > pF 3.
It has be reformulated in the new version of the manuscript: “The latter has most recently been addressed by using relative humidity sensors in the simplified evaporation method, which extend the HCC curve to pF 5.5 for sand and pF 6 for loam (Bosse et al., 2025).” (Line 39).
L46. Change “to be” to “of”.
Corrected. However, the entry of the sentence has changed (Line 44).
L46. “Integrating” into what?
Corrected. The word “integrating” was misleading and have been changes to “using” (Line 47–48).
L50. L. Consider changing “or” to e.g. “i.e”.
The sentence has been reformulated:” The MSF operates by applying a constant water flux via drip irrigation while regulating the hydraulic head to maintain a zero-pressure head gradient. This is achieved by continuously adjusting the pressure head below the sample to match the pressure head measured near the top of the sample (Weller et al., 2011). ” (Line 48–50)
L52. Typo “gab”.
Corrected. (Line 52)
L54. All data points are affected by soil structure if you consider soil structure to be the spatial arrangement of solids and pores. Please rephrase.
We agree that the original version was misleading. Therefore, we rephrased this sentence. In the new version, we specify our intention: ”Including the MSF as a standard measurement not only closes the data gap from saturation to field capacity, but it also provides the possibility to study structure driven macro-pore flow in detail (Jarvis et al., 2016; Bonetti et al., 2021; Leuther et al., 2019).” (Line 52–53)
L67. Unclear what “structure effects” means.
We rephrased the sentence to “We further investigated how well the measurement protocol is suited for studying the effect of soil macro-pore networks on SHP.” (Line 63–64)
L72-73. Check language.
We rephrased the sentence to “To further evaluate the workflow and the significance of the MSF measurements, both single-porosity and bimodal van Genuchten–PDI models (Peters, 2013; Iden and Durner, 2014) were used to describe the WRC and HCC. Model parameters were estimated by fitting the models to the experimental data both with and without inclusion of the MSF measurements.” (Line 68–70)
L76. Please, give soil texture data and organic carbon content for this silt loam soil.
The soil texture of the undisturbed soil samples was a silty loam28% clay, 57% silt, 14% sand. The organic carbon content was 9.5 mg g-1.
It has been added to the manuscript: ”Two undisturbed soil cores (volume of 250 cm3 and a surface area of 50 cm2) of the same texture (a silty loam (clay 28%, silt 57%, sand 14%, organic carbon 9.5 mg g-1 (Sponagel, 2005)) were taken from a conventional ploughed topsoil (5–10 cm) with a bulk density of 1.50 g cm-3 (Loam 1) and from a subsoil (30–35cm) under a disk harrow tillage system with a bulk density of 1.61 g cm-3 (Loam 2) by gently hammering a steel cylinder into the soil” (Line 73–74)
L80. Change “μ” to “μm”.
Corrected. (Line 77–78)
L96. Remove “on”.
Corrected. (Line 96)
L109. “heat”?
Corrected. (Line 109)
L113. “tubes”
Corrected. (Line 112)
L120. Unclear to me what the pump was doing. Did it only supply water for the Marriot bottle?
The Mariotte bottle serves as a water reservoir to maintain constant pressure head of +3 cm in the tubing system, even as the water level in the bottle drops. The peristaltic pump conveys the water through the tubing to the irrigation head. The pump's rotational speed thus regulates the water flow applied to the sample.
We have rephrased that in the manuscript: “The flux rate was controlled by a peristaltic pump that transported water from a reservoir to a drip irrigation head mounted on top of the sample. To ensure a constant water flux even as the water level in the reservoir decreased, the pressure head in the reservoir was maintained at +3 cm using a Mariotte bottle.” (Line 133–135)
L126. Consider changing “under pressure” to “suction” and add “pressure” at the end of the sentence.
Thank you. We rephrased the sentence accordingly: “This adjustment was achieved by two regulating valves that alternately connected the chamber either to a container with negative pressure of around -60 cm or to atmospheric pressure.” (Line 122–124)
L127. Earlier you used hPa for pressure potential. Be consistent. Change “low” to “high”?
Thank you for pointing this out. In the revised manuscript, pressure head is consistently expressed in centimetres [cm]. To improve clarity, qualitative descriptions have been replaced with explicit numerical values: “For pressure heads between 0 to -5 cm, the delta values were set to 1.5–2 cm; for pressure heads between -5 and -10 cm, it was set to 2–3 cm; and for pressure heads below -10 cm, it was set to 2–4 cm.” (Line 126–127).
L128. Units for delta? If it is cm the delta values seem very large.
The value is expressed in centimetres. The pressure head in the compartment below the sample (the adjusted pressure head) is continuously fluctuating around the target (inside the sample). Since the regulation against air and negative pressure occurs quickly and with the same deviation (positive and negative) from the target value, the average pressure in the chamber is close to the actual pressure inside sample. As an example, we now provide the raw data of the Loam 1 regulation as well as the median and the average for both tensiometer and pressure sensor in the supplementary material for the the revised manuscript (Fig. S1 and Table S1).
We further rephrased the sentence in the manuscript: “Consequently, the adjusted pressure head oscillated around the target value. However, because these fluctuations occurred rapidly, the mean pressure head closely approximated the actual pressure head. An example of the Loam 1 pressure regulation as well as the median and the average in pressure heads for both tensiometer and pressure sensor are provided in the supplementary material (Fig. S1 and Table S1).” (Line 127–131)
Table S1: The three MSF steps for Loam 1. The last 10 minutes of each flux, and the corresponding median and mean measurements of the tensiometer and the pressure sensor in the chamber beneath the sample.
Flux
[Log10(cm/d)]
Median tensiometer [cm]
Standard error [cm]
Median pressure sensor [cm]
Standard error [cm]
Mean tensiometer [cm]
Standard error [cm]
Mean pressure sensor [cm]
Standard error [cm]
1.43
-6.85
0.08
-6.45
0.29
-6.93
0.06
-6.76
0.23
1.08
-8.31
0.08
-9.25
0.53
-8.31
0.07
-8.65
0.42
-0.4
-11.52
0.05
-12.05
0.55
-11.52
0.04
-11.77
0.44
L133. The term hydraulic “equilibrium conditions” usually imply no flow. Here you have a zero gradient in pressure potential. Please rephrase.
We have rephrased:” Each step was maintained for at least one hour to ensure that near steady-state flow conditions were reached, defined by changes in pressure head and sample mass of <0.8 cm and <2 g, respectively, over a period of at least 20 minutes.” (Line 138–140)
L134. Please give numbers for “relatively stable”.
We now define relatively stable in the manuscript as weight changes < 2 g for 20 min and pressure <0.8 cm for 20 min. (Line 138–140)
L137. Change second half of sentence to e.g. “…we also calculated the water content for each step:”
We rephrased the sentence: “For each step, we calculated the water content by:” (Line 146)
L148. L.
We rephrased the whole paragraph 2.1.3 Simplified evaporation method, HYPROP:
“After re-saturation, the WRC (pF -1–3.8) and HCC (pF around 1.8–3.8) were determined using the HYPROP device. Two tensiometers were installed at depths of 1.25 cm and 3.75 cm, and the entire setup was placed on an electronic balance connected to a computer. Water was allowed to evaporate from the soil sample until the air-entry value of both tensiometers was reached. WRC data points were calculated at 10 min intervals by relating the average pressure head measured by the two tensiometers to the corresponding water content, which was determined from the measured weight loss. The pressure head gradient between the tensiometers enabled calculation of the HCC using a modified form of Darcy’s law (Peters and Durner, 2008):
K_i (h_i )=-q_i/((∆h_i)/∆z+1) (7)
Where hi is the average pressure head of the two tensimeters at time step i, Ki is the hydraulic conductivity at hi, qi is the water flux at time step i, Δhi is the pressure head difference between the tensiometers at time step i, and Δz is the vertical distance between them.” (Line 153–163)
L149-150. L.
We rephrased the whole paragraph 2.1.3 Simplified evaporation method, HYPROP:
“After re-saturation, the WRC (pF -1–3.8) and HCC (pF around 1.8–3.8) were determined using the HYPROP device. Two tensiometers were installed at depths of 1.25 cm and 3.75 cm, and the entire setup was placed on an electronic balance connected to a computer. Water was allowed to evaporate from the soil sample until the air-entry value of both tensiometers was reached. WRC data points were calculated at 10 min intervals by relating the average pressure head measured by the two tensiometers to the corresponding water content, which was determined from the measured weight loss. The pressure head gradient between the tensiometers enabled calculation of the HCC using a modified form of Darcy’s law (Peters and Durner, 2008):
K_i (h_i )=-q_i/((∆h_i)/∆z+1) (7)
Where hi is the average pressure head of the two tensimeters at time step i, Ki is the hydraulic conductivity at hi, qi is the water flux at time step i, Δhi is the pressure head difference between the tensiometers at time step i, and Δz is the vertical distance between them.” (Line 153–163)
L150-151. Please be more specific here.
We rephrased the whole paragraph 2.1.3 Simplified evaporation method, HYPROP and added the equation describing how the conductivity is calculated:
K_i (h_i )=-q_i/((∆h_i)/∆z+1) (7)
Where hi is the average pressure head of the two tensimeters at time step i, Ki is the hydraulic conductivity at hi, qi is the water flux at time step i, Δhi is the pressure head difference between the tensiometers at time step i, and Δz is the vertical distance between them.” (Line 160–163)
L155-158. This text should not be under 2.1.4.
We respectfully disagree, as this paragraph specifically describes the determination of the dry end of the water retention curve using subsamples analysed via the dew point method. For this reason, it should remain in this section.
Nevertheless, to improve clarity, the paragraph has been rephrased: “Subsamples (~3 g) were collected from the top, middle, and bottom of each soil column after the HYPROP experiment to cover a pressure head range of pF 4.5–6.5. The moist mass was determined using a high-precision balance, and the corresponding pressure head was measured with a WP4C device. Subsequently, all subsamples and the soil column were oven-dried at 105°C for at least 24h to determine gravimetric water content, total dry mass, and bulk density. Gravimetric water content was then converted to volumetric water content using the bulk density.” (Line 164 –169)
L169. What does “relative conductivity” mean. Is this term commonly used?
is a function that varies between 0 and 1 and the hydraulic conductivity function is calculated by scaling the relative conductivity by the saturated hydraulic conductivity. In the PDI model, we can define 2 relative functions, one for the capillary part and one for the non-capillary (film dominated) part. We believe that these terms are well established in the soil physics community.
It has been rephrased in the new manuscript: “where Ksat is the saturated hydraulic conductivity [cm day-1], is the conductivity for the capillary part and is the conductivity for the adsorptive (film) part, and is a weighting factor.” (Line 178-180)
L173. What is a “S-sharp”?
Thank you for identifying the typo. We rephrased the paragraph.
In the new manuscript the shape of the curve is described by: “Where K(h)total is the full model for loam 2, K(h)mac describes the macropore region by using a van Genuchten effective saturation function for hydraulic conductivity (van Genuchten, 1980):”(Line 184–185)
L174. What is the origin or reasoning behind equation 10? As I understand it, it does not account for the large increase in conductivity with macropore size (i.e. within the macropore range).
This is an additional component that we added to the hydraulic conductivity function to describe the large drop between the saturated hydraulic conductivity and the first MSF data point. Current bimodal models cannot describe this drop (for example bimodal van Genuchten), because they predict an equal drop in the water retention curve, which is not supported by our data. In principle, this additional component of the conductivity in the wet range should be related to the macropore/large-pore distribution. To account for this, we use the usual effective saturation function, with a well-defined alpha value and a high n value. In principle, these parameters can be estimated by X-ray µCT. Again, this is a workaround to show how the ideal model for K(h) should look.
It has been rephrased in the new manuscript: “Since the VG-PDI could not describe the WRC and the HCC for Loam 2, we made a slight modification of the model HCC. We add a macropore conductivity component to the VG-PDI HCC model by using a van Genuchten model for hydraulic conductivity (van Genuchten, 1980). The modification model for loam 2 HCC:
〖K(h)〗_total=〖K(h)〗_mac+〖K(h)〗_matrix (10)
Where K(h)total is the full model for loam 2, K(h)mac describes the macropore region by using a van Genuchten effective saturation function for hydraulic conductivity (van Genuchten, 1980):
K(h)_mac=〖K_(sat_mac )*(1+〖(α_mac*|h|)〗^(n_mac ))〗^(m_mac ) (11)” (Line 181–185)
L194. Unclear to me where these parameter values came from.
Please refer to our reply to your previous comment. The parameters were set by us as a work around to describe this 2 orders of magnitude drop observed in our data.
L206-207. Unclear to me what this means. The pressure potential ranges covered by the two methods did not overlap.
HYPROP and WP4C methods do not overlap, which we would not expect, as they have different measuring ranges. What we mean is that they extend each other.
We have tried to clarify that in the text by rephrasing: “The water contents measured with WP4C ranged from 1.95 to 0.03 vol.%. The WP4C data formed a smooth continuation of the HYPROP data, demonstrating good consistency between the two methods (Fig 2). Similar consistency between the evaporation method and the dew point method was reported by Schelle et al. (2013). ” (Line 223–125)
L241. You can conclude that the samples had different soil structure. From a statistical perspective you cannot say anything about possible effects of management.
This is correct, it was also not our intention to discuss in this paper management effects based on two samples.
We have rephrased: “Therefore, one possible explanation is that Loam 1 contains more medium-sized pores, compared to Loam 2.” (Line 258).
L264. Consider removing “to represent the behaviour mathematically”.
Removed.
L265. Remove “Only”. There could be other approaches that you did not test that would work well.
Removed.
L268-269. A better way to phrase this would be that a small volume of water in the largest pores resulted in a large increase in hydraulic conductivity.
Yes, thank you for that suggestion.
Corrected in new version of the manuscript: ”This simple extension of the van Genuchten-PDI model is not meant to be the solution for modelling these kinds of data, but it highlights the need for a new modelling approach to represent this behaviour in SHP models, because a small volume of water in the large pores results in a large increase in the drives the sharp decline in the hydraulic conductivity curve.” (Line 288–290)
L269-271. Yes, the MSF data are useful for getting the correct shape of the curve. However, these phenomena are well known and the basis for dual-permeability models that have been around since early 1990s. The sentence could be rephrased to better reflect this point and not exaggerate the novelty of this study.
Thank you we have now rephrased this sentence to: “Additionally, the MSF workflow can provide high-resolution data near saturation that are well suited as input for existing modelling approaches, such as dual-permeability models (Gerke and van Genuchten, 1993; Roulier and Jarvis, 2003; Holbak et al., 2021), which explicitly account for structural effects on conductivity and water flow.“ (Line 291–293)
L280 Please be more specific than “during modelling”.
This paragraph has been rephrased:” Furthermore, for soils in which saturated hydraulic conductivity is substantially higher than the hydraulic conductivity measured by the MSF, current bimodal SHP models may not adequately describe the data, and alternative model concepts may be required to represent SHPs mathematically from saturation to oven dryness.” (Line 331–334)
L280. Unclear to me what is meant by structurally complex soils.
This paragraph has been rephrased:” Furthermore, for soils in which saturated hydraulic conductivity is substantially higher than the hydraulic conductivity measured by the MSF, current bimodal SHP models may not adequately describe the data, and alternative model concepts may be required to represent SHPs mathematically from saturation to oven dryness.” (Line 331–334)
L280-281. Possibly, but dual-permeability models are models built to handle these processes. Are they not good enough?
Thank you for this valuable comment. The other reviewer had similar concerns therefor the replays are identical. We would like to clarify a distinction that was not sufficiently clear in the original conclusion. There are two types of models relevant here: (1) SHP models (e.g., unimodal van Genuchten type models, bimodal van Genuchten-type models), which aim to describe the water retention and hydraulic conductivity curves continuously from saturation to oven dryness, and (2) flow models such as dual-porosity and dual-permeability models, which simulate water flow by explicitly accounting for distinct pore domains. Based on our results, the current bimodal SHP models do not adequately describe our near-saturation data, suggesting that alternative model concepts may be needed for this first type. For the second type of models, we agree that dual-porosity and dual-permeability models are well suited to this purpose, and we propose that the near-saturation data provided by the MSF can be used to parameterize these models, enabling more accurate water flow simulations.
We have revised the conclusion to clarify this distinction: “In contrast, for process-based flow models such as dual-porosity and dual-permeability models, the near-saturation data provided by the MSF can serve as a valuable basis for parameterization, supporting more accurate characterization of structural pore domains and improved water flow simulations.” (Line 334–336)
L344. Capital letters?
Thanks. Changes in the manuscript. (Line 421)
Figures
Figure 1. You could indicate (e.g. with an arrow pointing down) that the KSAT setup used falling head.
Splendid idea! We also altered the figure to be more accreted in the pF values and a “water movement arrow” for the MSF (see new version of the manuscript, Fig. 1).
Citation: https://doi.org/10.5194/egusphere-2026-1405-AC2
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AC2: 'Reply on RC2', Mathilde B. S. Nielsen, 15 Jul 2026
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RC3: 'Comment on egusphere-2026-1405', Jos C. van Dam, 15 Jun 2026
Accurate soil hydraulic functions are indispensable for reliable modeling of soil water infiltration and redistribution. Especially near saturation and in structured soils, the hydraulic conductivity functions may show strong non-linearities, which may have a significant impact on the simulation of soil water flow. The Multi-Step-Flux (MSF) method covers the critical range near saturation.
The authors applied MSF to 3 soils, which are quite different in air entry value and soil structure. The results are promising. The MSF method seems very useful to collect relevant soil hydraulic data in combination with other laboratory measurement techniques.
The paper is well organized and reads very well.
In addition to the two already submitted reviews, I have just three remarks:
Title: Improve spelling Multi-Step-Flux-Method
L4: In the paper, you use consequently cm as unit for pressure head. Therefore, instead of hPa, use cm also for the pressure head range with the data gap.
Figure 2: The log scale, large symbols, and small size of the panels make it hard to see differences between the points and lines. I suggest placing the legend at the top, which increases the size of the panels, and decreasing the size of the symbols.
Citation: https://doi.org/10.5194/egusphere-2026-1405-RC3 -
AC3: 'Reply on RC3', Mathilde B. S. Nielsen, 15 Jul 2026
Dear Editors and dear Reviewers,
We would like to thank the reviewers for their constructive comments and valuable suggestions, which helped us to improve the manuscript considerably. Hans-Jörg Vogel and 2 requested additional information on the benefits of using a Mariotte bottle as a water reservoir, clarification of the hydraulic conductivity modelling for the Loam 2 sample, and further discussion of the applicability of the MSF method for dual-permeability approaches along with a discussion about limitations and perspectives. In addition, important questions were raised regarding the equilibrium conditions of the MSF method. Jos C. van Dam mainly provided suggestions of how to improve the display of Figure 2 and to unify units of pressure heads.
We have addressed and implemented these points in the revised manuscript and provided detailed responses to the individual comments below.
Furthermore, to better capture the air-entry region of the sand hydraulic conductivity curve, we performed an additional MSF experiment covering the range from saturation to a pressure head of −69 cm. These additional measurements strengthen the evaluation of the method and have been incorporated into the revised Figure 2.
To address a concern of Reviewer 2 about the uncertainty in the zero-pressure head gradient arising from fluctuations of the pressure head beneath the sample around the target pressure inside the sample, we now provide the raw pressure regulation data of the MSF from Loam 1 in the supplementary material, along with the corresponding median and mean values for both the tensiometer and the pressure sensor.
Overall, we carefully considered all comments and revised the manuscript accordingly. Lines given below refer to the new version of the manuscript with tracked changes. Text in bold is the reviewer's original comment.
Thank you very much.
On behalf of all co-authors
Mathilde Nielsen
Referee 3, Jos C. van Dam
Accurate soil hydraulic functions are indispensable for reliable modeling of soil water infiltration and redistribution. Especially near saturation and in structured soils, the hydraulic conductivity functions may show strong non-linearities, which may have a significant impact on the simulation of soil water flow. The Multi-Step-Flux (MSF) method covers the critical range near saturation.
The authors applied MSF to 3 soils, which are quite different in air entry value and soil structure. The results are promising. The MSF method seems very useful to collect relevant soil hydraulic data in combination with other laboratory measurement techniques.
The paper is well organized and reads very well.
Thank you very much for your comments. We have tried to address all your comments below.
In addition to the two already submitted reviews, I have just three remarks:
Title: Improve spelling Multi-Step-Flux-Method
Thank you for bringing it to our attention. This will be corrected in final version.
L4: In the paper, you use consequently cm as unit for pressure head. Therefore, instead of hPa, use cm also for the pressure head range with the data gap.
In the new version of the manuscript we are consistent with using cm instate of hPa.
Figure 2: The log scale, large symbols, and small size of the panels make it hard to see differences between the points and lines. I suggest placing the legend at the top, which increases the size of the panels, and decreasing the size of the symbols.
Thank you for this suggestion. In the new version the figurer will have smaller points outlined with a dark line, slimmer lines for the models and the legend have moved to the bottom (See new version of the manuscript, Fig. 2).
Citation: https://doi.org/10.5194/egusphere-2026-1405-AC3
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AC3: 'Reply on RC3', Mathilde B. S. Nielsen, 15 Jul 2026
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- 1
In this technical note the authors demonstrate how different methods covering different ranges in water potential can be combined to get a consistent description of soil hydraulic properties. The key point is how the results of the Multi-Step-Flux methods to measure hydraulic conductivity at high water potentials coincides with the more traditional methods. The results look promising, which is good news. This is true for the three materials used in this study were only drainage was considered. The paper is well structured and largely well written. I have one general remark and some technical details that should be considered prior to publication.
General remark:
The focus is more on the technical aspects and less on the potential of practical applications. Nonetheless, I miss at least a short discussion of the limitations of the applicability of these lab-measured SHPs for modeling water dynamics in natural soil (e.g. using Richards equation). These limitations come from well known features such as hysteresis and non-equilibrium effects (only shortly mentioned in the introduction) but also due to the boundary conditions (i.e. natural atmospheric boundary condition vs well define flux rates and water potentials adjusted in the lab). The MSF method is an excellent tool to study these important features which could be mentioned as well. I am a little worried that this paper sends the message that with MSF we now filled a critical gap and that we can now carry on with our traditional beloved concepts - it is not like that!
Technical details:
L30 right reference? falling head method is much older.
L37ff What about tension disc infiltrometers that covering at least a part of this gap?
L78: How deep was the sample taken in this subsoil?
L87 Conductivity was only measured for decreasing water contents (i.e. decreasing flux rates?)
L89 Split this sentence in two?
L105-112: You could mention that this MSF method was introduced by Weller et al (2011)
L109: hydraulic heat?
L122: Why do you need a Mariotte bottle in addition to a peristaltic pump?
L124: Please provide units for the delay values, I suppose they are in cm
L132: Could you justify the duration the flux steps of „at least one hour“ based on data (for example tensiometer readings during this hour)? When looking at Weller et.al. (2011) the water potential was still far from equilibrium after one hour. I think perfect equilibrium cannot be reached but may be some range of uncertainty for the water potential in K(h) could be given?
L174: S-sharp?
L208ff: The MSF flux rates are all in the range above the air entry point while it should be easy to adjust lower fluxes in MSF (which is an important strength of the method). Why did you not do this?
In Fig. 2 I see only two data points while from what is written in line 199 there should be four. Could you explain?
L216: Again, tension infiltrometers are an established method in this range close to saturation. However, this is rather a field measurement but it would be interesting how it compares (in another paper).
L222ff: A note on the presentation in Fig.2: It is a bit difficult to assess curve shapes and “small step drops” when the symbols are plotted so large that such details are barely visible (but the figure probably looks much more convincing this way).
L263: I have difficulties to follow this statement. Why is a sharp drop in conductivity near saturation physically unrealistic? That is exactly what we would expect (and what is frequently observed) if a relatively dense material is permeated by macropores, such as earthworm burrows (as expected for Loam 2). And besides, that is exactly what was measured! The physical problem that Ippisch rightly pointed out, is that the conductivity curve cannot be derived from the WRC using Mualem’s concept when the van Genuchten parameter n is significantly less than 2 - but this is a different story.
L280: I am wondering if the well known dual porosity models are not suitable to address exactly this problem? It is obvious that additional MSF measurement are highly valuable, but do we really need new models to cover this phenomenon?