Influence of Pore Size Distribution on the Electrokinetic Coupling Coefficient in Two-Phase Flow Conditions

Round 1

Reviewer 1 Report

This manuscript submitted by Vinogradov, Hill & Jougnot is hybrid method, improving on previous publications, to predict the streaming potential coupling coefficient as a function of water saturation.

I think the manuscript is well written, attacks a long-standing complication in the modeling of electrokinetics in two-phase flow, and does a good comparison between the proposed method and two previous methods.

Since the method is a hybrid, building on two existing methods, it necessarily is quite similar to them and builds incrementally on two methods that use the bundle of capillary tubes concept to predict saturation-dependent coupling coefficient. Previous methods have either considered peaked pore-size distributions or considered periodic radius pores, but this is the first method to consider both. 

I think the manuscript can be accepted after a few minor modifications, including references related to traditional two-phase flow approaches and some very minor fixes to grammar in places.

Larger points:

1. Introduction: There are often two scales that come up in studies like this. A pore or micro scale where everything is known but the physics are more complex, and the macro or porous medium scale where nothing is know but the effective physics are simpler. I think the method chosen depends on what scale your data come from and what scale you hope to make your predictions in. In the discussion of the 4 different main approaches (lines 86-96) the choices depend on what scale you hope to make predictions at. If you want to make pore-scale predictions, then pore-scale modeling (#1 and 2) would make sense. If you wish to make large-scale predictions, then macro-scale methods (#3 and 4) make more sense. I think the author's approach makes sense, and it relates back to a large body of work using bundle-of-capillary-tubes models in hydrology and soil science, which is still practical and useful.
2. I think the authors could include some more relevant references from the soil science and hydrology literature. The approach taken by Jackson (2008), and those who came after him, seems to be developed without much reference to this large body of literature.  This will increase the impact of your work, since it will hopefully tie it back to approaches done by many other people.
3. I find it puzzling in Eq 1 and Figures 3 and 4 (similar to Jackson, 2008), the authors work with the extensive quantity "total number of capillary tubes" which then needs to be normalized and doesn't have much physical meaning. Especially given the comment on line 290: "it is imperative to ensure that either the total number of capillaries is kept constant, or the total volume of capillaries is constant." It would make more sense to work in terms of a probability density function (pdf) or other normalized form of the pore size distribution.  I don't think it will change things much beyond rescaling the axes.
4. The idea of using non-monotonic pore size distributions is obviously quite old, and the authors should cite some more of the soil science literature on this. For example, Brutsaert (1966) shows a pore radius distribution very similar to the one the author's developed (his Fig 4), but he then goes on to fit incomplete gamma and lognormal distributions to it. Pore-size distributions can lead to well-known functions: lognormal model leads to that of Kosugi (modified by Malama & Kuhlman 2015 for finite r_max and r_min), and the beta distribution leads to the van Genuchten model (Haverkamp & Parlange, 1986). 
   1. Brutsaert, W. (1966). Probability laws for pore-size distributions. Soil Science, 101(2), 85-92.
   2. Haverkamp, R. T., & Parlange, J. Y. (1986). Predicting the water-retention curve from particle-size distribution: 1. Sandy soils without organic matter. Soil Science, 142(6), 325-339.
   3. Malama, B., & Kuhlman, K. L. (2015). Unsaturated Hydraulic Conductivity Models Based on Truncated Lognormal Pore‐Size Distributions. Groundwater, 53(3), 498-502.
5. The authors indicate their method doesn't require as much pore-scale characterization data as other approaches, but then they indicate that they develop the method using estimates of pore size and pore throats "obtained from direct pore size distribution measurements" (line 134 and line 222). Line 353 also indicates that micro-CT data and pore-network modeling was used to estimate a and c values. I get what you are saying, but maybe you could indicate that you _can_ use this information if it is available, but it is not absolutely necessary. Pore-scale reconstructions or ball-and-stick models depend more on pore-scale characterization (or at least they would involve assuming a lot more things than you are with your model).
6. Where the periodic pore is introduced (lines 198-220), it would be useful to state what exactly is meant by "more realistic representation" (line 201). I assume you mean the ability to include effects of hysteresis and a non-zero residual liquid saturation. 
7. A key point is sort buried in line 494, which I think should be given a separate bullet in the final conclusions: "We do not present results obtained with constant capillary radii as they are essentially identical to those with the alternating radii."
8. Tortuosity is not a function of saturation in this model, correct? In your case, tortuosity is a medium parameter, but tortuosity typically increases significantly with decrease in saturation (e.g., Millington & Quirk, 1960). It is stated that prediction tortuosity is a critical point (line 273). 

Minor/editorial points

• line 58: "thermodynamic property" might more accurately be "thermodynamic driving force," in the nomenclature of Onsager
• line 70: "interpreted" could be "estimated" since it is a deterministic equation.
• line 71: change to "routinely measured in the subsurface and laboratory."
• line 78: "among scientists" might be "in the literature"
• line 100: word missing in "the ^ produced"
• line 129: "In this study, the authors use two hydrodynamic functions (...) to extract the size distribution of equivalent straight capillaries" maybe "infer" would be better than "extract"?
• line 190: "diminutive" might be "smaller" and "r_c and r_c+dr_c" might be "r_c to r_c+dr_c"
• Equations 24 & 25: I think typically introducing two varieties of a quantity should have two different names (even if you don't use them later). These are both called C_r(S_w).
• Line 445: "probed" could be "used"

Author Response

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Author Response File: Author Response.docx

Reviewer 2 Report

Review Water Manuscript ID: water-1276779 -- Influence of pore size distribution on the electrokinetic coupling coefficient in two phase flow conditions, Vinogradov J, Hill R & Jougnot D

This paper is questionable on at least two major grounds.  First, it is model-on-model tinkering without being grounded in field data.  Second, its conceptual framework is confused at best and just plain wrong at worst.   

To begin at the beginning, the authors base their work on their perceptions of Berea sandstone, a hydrocarbon industry standard rock.  In blindly copying the hydrocarbon industry assessment of sandstone flow properties (and crustal/reservoir rock in general), the authors fall into fatal error: “The Berea sandstone core sample, as discussed in Moore et al. (2004), was modelled as a bundle of tortuous capillary tubes, assuming the system was water wet.”

Based on extensive well-log, well-core and well-flow data worldwide, the flow properties of crustal rock in general and reservoir rock in particular bear no relation to the hydrocarbon industry assumptions the authors adopt.  These assumptions are: (i) porosity and permeability are spatially uncorrelated; (ii) permeability is linked to porosity through the Carman-Kozeny relation κ ~ φ³/(1 – φ)² or some version of this; (iii) large scale formation flow properties are some form of “up-scaled” small scale formation flow properties.  Author surprise notwithstanding, we will see that, scientifically speaking, these industry assumptions (i)-(iii) are all simply/physically wrong.

The author conceptual problems don’t stop there.  The authors are dealing with shallower crustal flow systems than most if not all hydrocarbon reservoirs.  In assuming that Berea sandstone is directly relevant to their shallower formations, the authors do not allow for the possibility that the properties of deeper crustal rock are modified by the reduced stress conditions of shallower formations.  Basing their modelling on a misrepresentation of Berea sandstone flow properties, the authors don’t allow shallow flow systems to speak for themselves.  We see that the authors are locked in a modelling qua modelling world rather than modelling qua field data world. 

There is a good chance if not a near certainty that the authors are navigating in a backwater of model fiddling with no connection to shallow rock formation flow reality.

To come to grips with fluid flow in crustal formations, the authors must heed the following fluid-rock interaction empirics:

• From well-log spectral properties worldwide, we see that rock-fluid interactions are spatially correlated at all scales from mm to km; the key well-log spectral property is spatial fluctuation power-spectra scaling inversely with spatial frequency, S(k) ~ 1/k, across five decades of scale length, 1/km < k < 1/cm; i.e., rock-fluid interactions create pink noise spatial fluctuations; by contrast, the standard industry assumption of a flat or white noise spectral condition S(k) ~ 1/k⁰ = const, which the authors accept, is comprehensively invalidated by observation.
• From well-core spatial correlation systematics worldwide, we see that fluctuations in porosity are closely linked to fluctuations in the logarithm of permeability, δφ ~ δlog(κ); if porosity is spatially correlated at all scales (i.e., porosity spatial fluctuations obey spectral scaling relation), then connectivity between pores is spatially correlated as well; in mathematical terms, if porosity is expressed as a numerical density n, the ability of pores to link together to produce larger scale permeability is proportional to the combinatorial factor n! = n(n-1)(n-2)….. It then follows that spatial fluctuations in pore density create changes in pore connectivity, giving a simple physical interpretation to the empirical relation through Stirling’s formula, log(n!) ~ nlog(n) by which δlog(n!) ~ δn to recover the well-core empiric δφ ~ δlog(κ).
• From well-flow empirics worldwide, we see that spatially correlated properties of crustal rock-fluid interactions lead to lognormally distributed well-flow as given by the integrated poroperm relation κ(x,uy,z) ~ exp(αφ(x,y,z)), where empirical parameter α is seen to obey the condition 3 < αφ < 5 across 2 decades of reservoir and basement rock formation mean porosity, .003 < φ < 0.3.

Crustal rock-fluid interaction empirics (1)-(3) derive from a physical interpretation of grain-scale rock-fluid interaction that has little or nothing to do with industry/author notions of grain-scale microphysics.  The key empirical property to understand is the physical origin of power-law scaling spatial fluctuations S(k) ~ 1/k of empiric (1).  This property can be derived from critical state thermodynamical phase transitions occurring in binary systems such as ferromagnets, water at its critical state pressure/temperature, or binary fluid critical state opalescence.  In such critical state phase transitions, otherwise local short-range spatial fluctuations between binary energy states become long-range spatial fluctuations (in ferromagnets, Angstrom-scale fluctuations scale up to mm-scale magnetic domains; in critical state water and binary fluids, density Angstrom-scale fluctuations grow to 100s of nanometers to scatter light).  In analogy with these phenomena, grain-scale local grain-scale flow/no-flow fluctuations scale upwards to cm-to-km scale spatial fluctuation noise to give the macroscopic poroperm relation κ(x,uy,z) ~ exp(αφ(x,y,z)) seen in lognormal well-flow distributions worldwide for hydrocarbon reservoirs, groundwater aquifers, geothermal systems and fossil-flow-system mineral deposits.

Critical state empirics (1)-(3) arise in sedimentary and igneous rock in the course of rock compaction/burial, with the empirics maintained over geological time by continuous tectonic finite-strain deformation countered by geochemical healing processes.  Crustal rock near the surface is not subject to the high confining stresses below, say, 1km, and is subject to rapid weathering, hence can in principle deteriorate by rock-fluid interactions that undo the deeper rock empirics (1)-(3). 

This possible state of near-surface rock formations is, however, inconsistent with the authors’ use of Berea sandstone as their near-surface “type rock”.  The authors are thus twice at fault: wrong assessment of Berea sandstone, followed by possible failure of Berea sandstone core recovered from depth to reflect long-term shallow/low-stress rock formation fluid-rock interaction empirics.  The authors here engage instead with a decades-long parade of incorrect rock-fluid interaction formulations without acquiring relevant field data by which to gauge the physical properties of the shallow rock formations they purport to study.

For the authors to be relevant to shallow formation rock-fluid interactions, they need to judge to shallow formations in terms of shallow rock that is either (i) like deeper rock, or (ii) like a deteriorated from of deeper rock.  This means that the authors must allow for large-scale spatial fluctuations in porosity and related permeability distributions inherited from the formation’s existence at deeper depths, or establish by field-scale observation that such large-scale spatial fluctuations have been decisively altered by their formation’s shallow low-stress environment.

It is possible that the electrokinetic couple model and its relation to field data acquisition and processing can be usefully applied to real surveys over real formations, but the authors must demonstrate this in terms of physical accurate crustal rock-fluid interaction empirics and appropriate field-scale simulation data.  

A more detailed discussion of crustal rock-fluid interaction empirics is given in Volume 2017 |Article ID 9687325 | https://doi.org/10.1155/2017/9687325

As this paper presently stands, author tinkering with invalid or unvalidated grain-scale rock-fluid interaction properties without reference to field-scale data that responds to primal properties of crustal rock-fluid interactions is not a science contribution. 

Comments for author File: Comments.pdf

Author Response

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Reviewer 3 Report

Thank you