Evaluating hysteresis mechanisms through pore networks simulations

Ben-Noah, Ilan; Friedman, Shmulik P.; Mualem, Yechezkel

doi:10.5194/egusphere-2026-4185

Preprints

https://doi.org/10.5194/egusphere-2026-4185

Preprints

27 Jul 2026

| 27 Jul 2026

Evaluating hysteresis mechanisms through pore networks simulations

Ilan Ben-Noah, Shmulik P. Friedman, and Yechezkel Mualem

Abstract. Hysteresis in capillary head and relative permeability relationships with saturation degree is an important phenomenon in granular and porous media. Accounting for hysteresis is essential for accurately predicting flow in multiphase systems in hydrological applications such as groundwater management and soil water dynamics. The observed hysteresis loops reflect complex, history-dependent interactions between fluids and pore structures. This work uses three-dimensional (3D) pore network models to systematically investigate how media properties. We analyze the influence of the pore size distribution, correlation, and connectivity on the combined and decoupled mechanisms causing hysteresis: geometric ink bottle effects, non-wetting fluid trapping, and network-dependent effects arising from complex pore accessibility.

By leveraging controlled simulated drainage and imbibition scenarios, namely invasion vs random percolation, bond vs site-governed displacement, and with vs without trapping, we decoupled these mechanisms. The different mechanisms present distinct effects on the hysteretic loops. In particular, trapping primarily affects retention curves at high saturation degrees of the wetting phase (S_w) and dramatically reduces wetting-phase relative permeability (k^W_r). In comparison, the ink-bottle effect, driven by pore geometry, is visible across the entire capillary head (h_c) range. In contrast, network hysteresis effects significantly the shape of the retention curve (S_w(h_c) loops and drives k_r(S_w) hysteresis at low S_w.

Furthermore, the impact of these mechanisms is highly dependent on medium structure. Increasing the spread of the pore size distribution enhances non-wetting phase trapping volume while mitigating ink bottle effects. Correlation between pore bodies' and throats' radii strongly increases the impact of trapping on k^W_r. Conversely, increasing connectivity (i.e., higher coordination number) reduces the trapped fluid fraction and generally mitigates ink bottle and network hysteresis effects in retention evaluation. These results provide necessary mechanistic understanding, supporting the inverse interpretation of hysteretic loops to deduce the underlying topological structure of porous media.

Received: 13 Jul 2026 – Discussion started: 27 Jul 2026

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Ilan Ben-Noah, Shmulik P. Friedman, and Yechezkel Mualem

Status: final response (author comments only)

RC1:
'Comment on egusphere-2026-4185', Allen G. Hunt, 29 Jul 2026

I would like to be clear from the outset. This article represents an important viewpoint for the readers of HESS, which has virtually not been represented in analogous publications, and the few publications that are relevant have not been cited in this forum. Further, I use the word major only to make sure that the revised manuscript gets re-reviewed. I think that the revisions needed may be substantial, but it is mainly due to a lack of discussion of two closely related works that appear to have been overlooked, which does not indicate that there are major defects in the manuscript that require a complete rewriting.
One paper that needs to be cited is: Hunt, A. G., 2004, Continuum percolation theory for water retention and hydraulic conductivity of fractal soils: 2. Extension to non-equilibrium, Advances in Water Resources, 27, 245-257. I have provide this paper as an attachment. In particular, Figure 2 of this paper looks quite similar to Figure 5 and, I think, for the same combination of ink-bottle and connectivity contributions to the hysteresis. However, the basis for Figure 2 of Hunt (2004) is somewhat simpler; instead of addressing all the effects of percolation at once, it only addressed the connection to the infinite cluster, as given by the accessibility function. Thus, there is no air trapping at the wet end of the curve in that paper. Other distinctions are: the 2004 paper addresses a real medium, not a simulation, and it thus does not have the capacity to resolve different effects. On the other hand, it does produce an exceptional agreement with experiment, effectively without adjustable parameters as to the shapes of the curves, only the characteristic pressures - and these were adjusted to make them agree with the experiment at the wet end. There is a lot of detailed material in there that the authors may or may not find useful; I think that the discussion of the time scales involved that might not allow maximum hysteresis (which was modeled using Eq. (8)). The other paper that is worth mentioning is Hunt, A. G., and Gee, G. W., 2003, Wet-end deviations from scaling of the water retention characteristics of fractal porous media, Vadose Zone Journal, 2, 759-765. In Hunt and Gee (2003), it is pointed out that it may be a frequent occurrence (and was so in a set of around 50 Hanford site soils) that the wet end "air-entry pressure" is not the "bubbling pressure" for the same reason that there is hysteresis, namely air can enter beyond just the edges only when all the air-allowable pores actually percolate. By analyzing the data from this perspective, it was possible to show that the critical air fraction for percolation, at least in coarser media with small clay contents, was nearly the same as the water fraction for percolation at the dry end. Thus, the overshoot in drying at the wet end was governed by the same accessibility function with the same critical volume fraction as then inability to wet the medium at the dry end. It is possible that the scaling of invasion percolation and the use of the accessibility function in random percolation give very similar understanding - but it may be that the similarities in results are not indicative of the same understanding. This topic also needs further discussion. Next, the authors use different sized domains to check for possible limitations of finite-sizes. But there is some potential problem here. As pointed out by the inventors of invasion percolation, Wilkinson and Willemsen, whom the authors cite, there are two percolation transitions in any transit from wet to dry in porous media. These transitions imply that the correlation length at these to moisture contents must diverge. Indeed, in a third paper, Steenhuis, T., Hunt, A. G., Parlange, J.-Y., and Ewing, R. P., 2005, Assessment of the application of percolation theory to water-repellent soils, Australian Journal of Soil Research, 43: 357-360, we show that this divergence makes it possible for as incorporation of as little as 5% water repellent particles into a sand will cause the entire medium to be water repellent. Thus, I recommend that the authors consider the use of finite-size scaling in their analyses, which would allow the finite-size effects to be eliminated completely without the necessity of estimating the magnitude of the effect. The last important link between the present manuscript and Hunt (2004) is the latter's use of an analogy to the glass transition in viscous liquids to establish the relevance of non-equilibrium effects, which can be directly evaluated from the pressure saturation curve and the use of critical path analysis to find the hydraulic conductivity as a function of the tension. This equation is Eq. (24). The analogy to the cooling rate in a glass is the rate at which the tension h is increased.
I have one last important comment, but it casts no aspersion on the authors. When I submitted the manuscript to Advances in Water Resources in 2003, it went to Rink van Dijke as a referee, at that time in Scotland. He pointed out that Muhammad Sahimi with his co-author Heiba had already published something similar in 1982. I was unable to find the reference, but luckily, Robert P. (Toby) Ewing knew of it. It was in the petroleum engineering literature. Indeed, those authors had developed the language "allowable" and "accessible" to describe pores with radii that would accept the one type of fluid in question and those that were also connected to the infinite cluster. They applied this formalism to address hysteresis on Bethe lattices, but without producing specific formulas that could be compared with experiment. Given that 22 years later, the porous media community was unaware of their work and now, another 22 years later, it is unaware of what I contributed in 2004, it might be time for the porous media community to come to grips with the fact that there has been a successful application of percolation theory to predict hysteresis in a real system. That. however, does not take anything away from what the authors are doing, just as the work of Sahimi and Heiba did not invalidate my work as new. But it would be to the community's advantage not to have to wait another 22 years. All of these works support Irwin Fatt's argument from the 1950's that it is time to use network models, rather than bundles of capillary tube models. That is now 70 years ago.
I thank the authors for moving forward again on this critical problem and the editor Bo Guo for giving me the opportunity to review it.

Citation: https://doi.org/10.5194/egusphere-2026-4185-RC1
- AC1: 'Reply on RC1', ilan Ben-Noah, 11 Oct 2026
  
  Replies to Reviewer 1 (typeset bold and italic)
  Reviewer 1:
  
  I would like to be clear from the outset. This article represents an important viewpoint for the readers of HESS, which has virtually not been represented in analogous publications, and the few publications that are relevant have not been cited in this forum. Further, I use the word major only to make sure that the revised manuscript gets re-reviewed. I think that the revisions needed may be substantial, but it is mainly due to a lack of discussion of two closely related works that appear to have been overlooked, which does not indicate that there are major defects in the manuscript that require a complete rewriting.
  
  -> We thank the reviewer for the positive evaluation of the work, and thank the reviewer for directing us to these interesting reading materials.
  One paper that needs to be cited is: Hunt, A. G., 2004, Continuum percolation theory for water retention and hydraulic conductivity of fractal soils: 2. Extension to non-equilibrium, Advances in Water Resources, 27, 245-257. I have provide this paper as an attachment.
  
  \res{We added, in line (35), a mentioning of this paper "Hunt (2004) used percolation theory while addressing only the connection to the infinite cluster as given by the accessibility function to analyze non-equilibrium effects on hysteresis in fractal soils, which showed close agreement with experimental data (despite overlooking air trapping). These dynamic effect are beyond the scope of the present study that focuses on capillary equilibrium conditions."}
  In particular, Figure 2 of this paper looks quite similar to Figure 5 and, I think, for the same combination of ink-bottle and connectivity contributions to the hysteresis. However, the basis for Figure 2 of Hunt (2004) is somewhat simpler; instead of addressing all the effects of percolation at once, it only addressed the connection to the infinite cluster, as given by the accessibility function. Thus, there is no air trapping at the wet end of the curve in that paper. Other distinctions are: the 2004 paper addresses a real medium, not a simulation, and it thus does not have the capacity to resolve different effects. On the other hand, it does produce an exceptional agreement with experiment, effectively without adjustable parameters as to the shapes of the curves, only the characteristic pressures - and these were adjusted to make them agree with the experiment at the wet end. There is a lot of detailed material in there that the authors may or may not find useful; I think that the discussion of the time scales involved that might not allow maximum hysteresis (which was modeled using Eq. (8)).
  -> In this context, we added in line (66): "The time scale in which dynamic effects can be ignored, and its dependency on the water content and lattice properties are discussed in Hunt (2004)."
  The other paper that is worth mentioning is Hunt, A. G., and Gee, G. W., 2003, Wet-end deviations from scaling of the water retention characteristics of fractal porous media, Vadose Zone Journal, 2, 759-765. In Hunt and Gee (2003), it is pointed out that it may be a frequent occurrence (and was so in a set of around 50 Hanford site soils) that the wet end "air-entry pressure" is not the "bubbling pressure" for the same reason that there is hysteresis, namely air can enter beyond just the edges only when all the air-allowable pores actually percolate. By analyzing the data from this perspective, it was possible to show that the critical air fraction for percolation, at least in coarser media with small clay contents, was nearly the same as the water fraction for percolation at the dry end. Thus, the overshoot in drying at the wet end was governed by the same accessibility function with the same critical volume fraction as then inability to wet the medium at the dry end. It is possible that the scaling of invasion percolation and the use of the accessibility function in random percolation give very similar understanding - but it may be that the similarities in results are not indicative of the same understanding. This topic also needs further discussion.
  -> We added this interesting observation in line 531: "This conclusion is in accordance with the results of Hunt and Gee (2003) in coarse media, with a probably more uniform pore size distribution (low CV), the saturation degree at percolation was found to be closer to the complementary disconnection saturation degree. The same can be observed here in Fig. 8a when comparing the S_w of ISRT to 1-S_w of DBIN at their percolation point (Δ)."
  Next, the authors use different sized domains to check for possible limitations of finite-sizes. But there is some potential problem here. As pointed out by the inventors of invasion percolation, Wilkinson and Willemsen, whom the authors cite, there are two percolation transitions in any transit from wet to dry in porous media. These transitions imply that the correlation length at these to moisture contents must diverge. Indeed, in a third paper, Steenhuis, T., Hunt, A. G., Parlange, J.-Y., and Ewing, R. P., 2005, Assessment of the application of percolation theory to water-repellent soils, Australian Journal of Soil Research, 43: 357-360, we show that this divergence makes it possible for as incorporation of as little as 5\% water repellent particles into a sand will cause the entire medium to be water repellent. Thus, I recommend that the authors consider the use of finite-size scaling in their analyses, which would allow the finite-size effects to be eliminated completely without the necessity of estimating the magnitude of the effect.
  -> The discussion regarding the Lattice size effects in the limitation section and the analysis in the appendix is meant to only validate that large enough lattices were used for analyzing the hysteresis mechanisms and not explore correlation lengths and fine-size effects. Our analysis shows that the effects of the lattice size on the hysteresis loops are indeed small. This is in accordance with others' results (cited in the text). Therefore, scaling our numerically obtained curves (using mostly empirical relations) will probably not change our conclusions.
  
  The last important link between the present manuscript and Hunt (2004) is the latter's use of an analogy to the glass transition in viscous liquids to establish the relevance of non-equilibrium effects, which can be directly evaluated from the pressure saturation curve and the use of critical path analysis to find the hydraulic conductivity as a function of the tension. This equation is Eq. (24). The analogy to the cooling rate in a glass is the rate at which the tension h is increased.
  -> This link is not obvious, in our manuscript we focus specifically on (limit the scope to) quasi-static conditions. Discussing the (limited) analogy of this case to the non-equilibrium glass transition or overshoot might confuse the readers.
  
  We are now in the preparation of a following study dealing with different dynamic hysteresis mechanism which is more suited to include this interesting discussion about the analogy between different rate limiting hysteretic processes.
  I have one last important comment, but it casts no aspersion on the authors. When I submitted the manuscript to Advances in Water Resources in 2003, it went to Rink van Dijke as a referee, at that time in Scotland. He pointed out that Muhammad Sahimi with his co-author Heiba had already published something similar in 1982. I was unable to find the reference, but luckily, Robert P. (Toby) Ewing knew of it. It was in the petroleum engineering literature. Indeed, those authors had developed the language "allowable" and "accessible" to describe pores with radii that would accept the one type of fluid in question and those that were also connected to the infinite cluster. They applied this formalism to address hysteresis on Bethe lattices, but without producing specific formulas that could be compared with experiment. Given that 22 years later, the porous media community was unaware of their work and now, another 22 years later, it is unaware of what I contributed in 2004, it might be time for the porous media community to come to grips with the fact that there has been a successful application of percolation theory to predict hysteresis in a real system. That. however, does not take anything away from what the authors are doing, just as the work of Sahimi and Heiba did not invalidate my work as new. But it would be to the community's advantage not to have to wait another 22 years. All of these works support Irwin Fatt's argument from the 1950's that it is time to use network models, rather than bundles of capillary tube models. That is now 70 years ago.
  -> Thank you for this insight and historical perspective. We hope that in the (revised) article we cited the main contributions to the topic and that in the current study we have "placed one more small brick in this building.".
  I thank the authors for moving forward again on this critical problem and the editor Bo Guo for giving me the opportunity to review it.
  -> We thank the reviewer for carefully reading our manuscript and for the insightful comments.
  
  Citation: https://doi.org/10.5194/egusphere-2026-4185-AC1
RC2:
'Comment on egusphere-2026-4185', Anonymous Referee #2, 15 Sep 2026

The manuscript is technically interesting and provides insights into the origins of the hysteresis phenomena due to imbibition/drainage in porous media. Technically, the underlying model is sensible and ideal for the scope of the draft.
However, the manuscript is difficult to read at times. I have following proposals to improve the manuscript:
- Add dedicated figures to illustrate the various mechanisms in section 2.2

- The acronyms are sensible, however their constant use in the full text next to a few other abbreviations does impair the readability.

- For section 3, I encourage the authors to present their findings according to the results most interesting to the average reader. Specifically, I propose to dedicate sections for Relative Permeability curves, Retentions curves, etc. and in each section discuss the relevant mechanisms. Effectively vice versa to the current state.

- A dedicated outlook is missing and the impact of the finding is not clear. Please discuss future developments and the relevance of the insights to the field.

Citation: https://doi.org/10.5194/egusphere-2026-4185-RC2
- AC2: 'Reply on RC2', ilan Ben-Noah, 11 Oct 2026
  
  Replies to Reviewer 2 (typeset in bold and italic)
  Reviewer 2:
  The manuscript is technically interesting and provides insights into the origins of the hysteresis phenomena due to imbibition/drainage in porous media. Technically, the underlying model is sensible and ideal for the scope of the draft.
  -> We thank the reviewer for the positive evaluation of the work.
  However, the manuscript is difficult to read at times. I have following proposals to improve the manuscript:
  - Add dedicated figures to illustrate the various mechanisms in section 2.2
  -> We added a schematic figure (Fig. 1 see attached) to explain the three mechanisms of hysteresis we are considering in this research.
  - The acronyms are sensible, however their constant use in the full text next to a few other abbreviations does impair the readability.
  -> We revised the necessity of all abbreviations and removed/changed them when possible (e.g., in lines 248, 360, 472, 476, 480, 484, 548, 569, and 577).
  - For section 3, I encourage the authors to present their findings according to the results most interesting to the average reader.
  -> We thank the reviewer for these constructive comments regarding paper readability.
  
  We acknowledge that the manuscript prioritizes presenting a comprehensive set of insights, which creates a dense structure.
  
  We removed some of the less important insights, for example in lines 568-570, and 660-662 to improve readability.
  Specifically, I propose to dedicate sections for Relative Permeability curves, Retentions curves, etc. and in each section discuss the relevant mechanisms. Effectively vice versa to the current state.
  -> When preparing the original draft, we carefully evaluated organizing the paper by separating the retention and permeability analyses (and even considered splitting the manuscript into two distinct papers). Ultimately, we retained the current unified structure for two primary reasons:
  
  1) Interdependence of Mechanisms: Several media effects on permeability curves are directly explained by their underlying impacts on the retention curves. Presenting both within the same integrated analysis allows readers to see these direct causal links (rather than repetitive reading in separate chapters).
  
  2) Modular Readability: Readers often target specific media properties rather than reading the paper linearly from end to end. Organizing Section 3 systematically by media property allows these readers to quickly isolate all relevant impacts (both retention and permeability) in one place. To support readers looking for curve-specific overviews, we transposed this structure in the Conclusion, summarizing the effects on retention and permeability separately.
  A dedicated outlook is missing and the impact of the finding is not clear. Please discuss future developments and the relevance of the insights to the field.
  -> we added an Outlook and future research subsection at the end of the conclusion section:
  "Building upon this quasi-static baseline, future research should focus on three key developments:
  
  1) Transitioning to Dynamic Regimes: Extending the model beyond equilibrium displacement to incorporate dynamic, rate-dependent phenomena, such as viscous coupling, film flow, and ganglia mobilization at higher capillary numbers ($Ca$).
  
  2) Application to Complex Real-World Topologies: Validating these decoupled mechanisms on 3D micro-CT imaged digital rock and soil cores, incorporating heterogeneous wettability and contact angle hysteresis.
  
  3) Macroscopic Upscaling and Constitutive Modelling: Translating these microscopic insights into improved closed-form constitutive models (e.g., refining classical independent domain or Mualem-type models) to embed topological parameters directly into field-scale continuum simulators."
  
  Citation: https://doi.org/10.5194/egusphere-2026-4185-AC2

Ilan Ben-Noah, Shmulik P. Friedman, and Yechezkel Mualem

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Short summary

Water retention and flow in soils depend on wetting history, a process called hysteresis. We used three-dimensional pore networks simulations to elucidate the distinct role of three basic hysteresis mechanisms: "ink-bottle", "fluid trapping", and "pore accessibility". We discovered that the water content of the system and the material's internal structure dictate which mechanism dominates.


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