A comprehensive Darcy-type law for viscoplastic fluids: I. Framework

Emad Chaparian

REVIEW 3 major objections 5 minor 48 references

A comprehensive Darcy-type law for viscoplastic fluids: I. Framework

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A Darcy-type law for viscoplastic fluids is proposed and validated: the pressure drop across a porous medium is the sum of a Newtonian-limit drag term and a geometry-only yield term, with no interaction term.

desk verdict Useful engineering correlation for Bingham flow in 2D square-obstacle porous media; the additivity ansatz is unproven but the DNS validation is broad enough to make it credible. read the letter →

arxiv 2506.06184 v3 pith:76HXFIFO submitted 2025-06-06 physics.flu-dyn cond-mat.softmath-phmath.MP

Emad Chaparian This is my paper

classification physics.flu-dyncond-mat.softmath-phmath.MP MSC 76S0576A05 PACS 47.56.+r47.50.-d

keywords Darcy'slawyield-stressfluidsBinghamfluidporousmediapermeabilityyieldlimitcriticalnumberpore-scalesimulation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proposes a single Darcy-type law for yield-stress fluids (fluids that only flow once the applied stress exceeds a yield threshold, here modelled with the Bingham constitutive equation) flowing through porous media, covering the whole range from near-Newtonian flow at tiny Bingham numbers — the ratio of yield stress to characteristic viscous stress — to the yield/plastic limit where flow just starts. The law combines two limit models: a Newtonian-limit drag term fitted from series-based permeability formulas to pore-scale simulations, and a geometry-only yield-limit term carried over from the author's earlier work on the onset of flow. The proposed expression gives the non-dimensional pressure gradient as a function of solid fraction $\phi$ and Bingham number $B$, without requiring a separately tabulated Newtonian permeability for each medium. Pore-scale simulations with an augmented-Lagrangian/adaptive-mesh method are reported to confirm the formula across $B$ from about $10^{-2}$ to $10^{4}$ and $\phi$ from $0.1$ to $0.55$. If correct, the law turns a pore-scale yield-stress flow problem into a one-line macroscopic closure depending only on porosity and Bingham number.

What carries the argument

The load-bearing objects are the two dissipation integrals entering the energy balance, Eq. (19): the viscous dissipation $a(u,u) = \int_{\Omega\setminus\bar{X}} \dot{\gamma}:\dot{\gamma} \, dA$ and the plastic dissipation $j(u) = \int_{\Omega\setminus\bar{X}} \|\dot{\gamma}\| \, dA$, which together fix the pressure gradient $\Delta P/L$ for a prescribed flow rate. The argument's central identity is the additivity ansatz that $\Delta P/L = \lim_{B\to 0} f(\phi,B) + \lim_{B\to\infty} f(\phi,B)$, i.e., that the two limiting regimes exhaust the pressure drop at every $B$ with no interaction term. The yield-limit branch is carried by the geometrical critical-yield-number scale $Y_c = B/(\Delta P/L) \to (1/\pi)(1-\phi)/\phi$, while the Newtonian branch is carried by the series-based drag formula for arrays of cylinders, whose $1/(-\log(R/L) + b + c(R/L) + d(R/L)^2)$ form becomes the $\phi$-dependent denominator of Eq. (30).

What would settle it

Run the same pore-scale simulations at intermediate Bingham numbers and compute the viscous and plastic dissipation ratios $a(u,u)/\int u\, dA$ and $j(u)/\int u\, dA$; if either ratio drifts by more than the statistical error across $B \in [10^{-2}, 10^{4}]$, the additivity assumption underlying Eq. (30) fails. A simpler check is to measure $\Delta P/L$ in a quasi-2D Hele-Shaw cell filled with a yield-stress fluid at fixed flow rate and compare the outcome with Eq. (30) at the same $\phi$ and $B$.

Watch

Extended reading notes

Core claim

The central claim is that the pressure drop of a Bingham fluid through a random two-dimensional porous medium is, to the accuracy of the reported simulations, simply the sum of its Newtonian and yield-limit pressure drops at every Bingham number. Specifically, Eq. (30) states $$\frac{\$\Delta$ P}{L} = \frac{a}{-\log \phi + b + c\phi + d\$phi^{{2}}$} + \frac{\pi\phi}{1-\phi}B,$$ where the first term is the $B\to 0$ viscous limit (with coefficients $a=0.4091$, $b=-2.1954$, $c=4.1141$, $d=-2.1874$ fitted to ensemble-averaged simulations for random square obstacles) and the second term is the $B\to\infty$ yield limit, equivalent to the critical yield number $Y_c = \frac{1}{\pi}\frac{1-\phi}{\phi}$ from the author's previous percolation study. The paper reports that this formula agrees with pore-scale DNS across $B \in [10^{-2}, 10^{4}]$ and $\phi \in [0.1, 0.55]$, with 20 random realizations per solid fraction.

Load-bearing premise

The law assumes that at every intermediate flow strength the pressure drop is exactly the Newtonian-limit pressure drop plus the yield-limit pressure drop, with no extra term due to the flow pattern reorganizing as the yield stress becomes more important.

Editorial extensions

If this is right

For a fixed random 2D geometry of monodisperse square obstacles, the pressure gradient required to sustain a given flow rate can be computed from $\phi$ and $B$ alone, without resolving the pore-scale velocity field.
The effective permeability of a Bingham fluid decreases monotonically with $B$, and the $B\to\infty$ asymptote is fixed purely by geometry, $\pi\phi/(1-\phi)$, independent of the plastic viscosity.
Because the yield-limit term does not involve viscosity, the same combination strategy extends to Herschel–Bulkley fluids by replacing only the Newtonian-limit term with a shear-thinning drag law.
The law provides a macroscale closure that interpolates between the onset-of-flow regime and the near-Newtonian regime, filling the gap where neither limit alone is accurate.
For periodic arrangements of obstacles, the same functional form holds with different fitted coefficients, and the square and triangular array fits bracket the random-media result.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A testable consequence of the additivity ansatz is that the ratios $a(u,u)/\int u\, dA$ and $j(u)/\int u\, dA$ in the energy balance are independent of $B$; checking their drift at intermediate $B$ would either sharpen or refute the model.
The logarithmic denominator of the Newtonian term reflects the two-dimensional Stokes paradox structure, so in three dimensions the same additivity construction would need a different permeability model (e.g., Blake–Carman–Kozeny type) and a three-dimensional yield-limit scale, which the paper notes is still missing.
Since the law is an ensemble-averaged closure, individual realizations scatter around it; for engineering safety factors the upper and lower bound fits for periodic arrays give the natural uncertainty envelope.
The same 'two limits added' construction could in principle be applied to other two-mechanism rheologies, such as Carreau-yield-stress fluids, with only the viscous-limit term modified.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this is a practical correlation, not a derived law. The headline equation (30) combines a fitted Newtonian term with the yield-limit term from your earlier work, and the paper validates it against a decent amount of DNS: 20 realizations per φ, φ from 0.1 to 0.55, B from 1e-2 to 1e4. The agreement in Fig. 5 looks good, though no quantitative error metric is given.

What I think is actually new is the four-parameter fit for the Newtonian permeability of random square obstacles (Eq. 29), and the demonstration that adding the yield-limit term (Eq. 25) gives a correlation that holds across the whole Bingham range for those geometries. The comparison with Castañeda's upper bound is also useful, and it's to the author's credit that his model beats the bound for these realizations.

The soft spots are the ones the stress-test note flags. The energy balance (Eq. 19) gives an exact identity ΔP/L = a(u,u)/I + B j(u)/I. The proposed law (30) is equivalent to assuming each ratio keeps its limiting value for all B. That is not derived, and the paper only validates the sum. It is possible the two ratios compensate. Given how wide the B range is, that would be a surprising coincidence, but the paper should at least report the individual ratios or provide some argument. This is moderate, not fatal, because the correlation is presented as an engineering fit, and it does work over the tested range.

Two smaller quibbles. First, the "comprehensive" in the title overstates the scope: this is 2D, mono-disperse square obstacles only. The author acknowledges this in the conclusion, but it should be in the abstract. Second, the Newtonian coefficients are fitted to the same B=0 data used to display the Newtonian limit; that is standard practice, but it means the only out-of-sample test is the finite-B data, which tests only the sum.

Bottom line: this paper is for people who need a quick engineering estimate for yield-stress flow through a 2D random square-packed medium. Not a fundamental result, but honest, well-documented, and the DNS work is solid. I would send it to review, asking the author to add error metrics, discuss the additivity issue, and tone down the "comprehensive" claim. I'd probably cite it if I worked in this niche.

Referee Report

3 major / 5 minor

Summary. The paper proposes a Darcy-type law for Bingham fluids in two-dimensional porous media, Eq. (30), which is the sum of a Newtonian-limit term (29) and a yield-limit term (28). The Newtonian term is fitted to the author's own DNS data for randomized square obstacles at B=0, and the yield term is taken from the author's prior work [12], Eq. (25). The combined law is then compared with augmented-Lagrangian DNS over φ ∈ [0.1, 0.55] and B ∈ [10^-2, 10^4]. The central claim is that this simple linear combination is valid across the entire range of Bingham numbers. The paper also compares the yield-limit prediction (25) with Castañeda's upper bound (26) for a set of individual realizations.

Significance. If Eq. (30) is indeed valid across the full parameter range, it would provide a practically valuable closure for viscoplastic flow in porous media, replacing earlier models that require prior knowledge of the Newtonian permeability or undetermined prefactors. The computational study is careful in its use of randomized geometries and 20 realizations per solid fraction, and the intermediate-B comparison is genuinely out-of-sample because all fitted constants are determined at B=0. However, the central additivity assumption is not derived, and the quantitative validation is incomplete, so the claimed comprehensive validity is not yet firmly established.

major comments (3)

[Section V, Eq. (30), and Eq. (19)] The proposed law is a linear superposition of the Newtonian and yield limits. From the exact energy balance (19), ΔP/L = a(u,u)/I + B j(u)/I with I = ∫u dA, so Eq. (30) is equivalent to assuming that a(u,u)/I is equal to its B→0 value and that j(u)/I is equal to its B→∞ value for every B. The paper does not derive this assumption and does not test the two dissipation ratios separately; Fig. 5 compares only the total pressure drop with the sum. If the two ratios vary with B while their variations compensate, the law could fit the tested data without exhibiting the claimed limit-combination mechanism. Please compute A(B)=a(u,u)/I and J(B)=j(u)/I from the DNS and report them as functions of B; if they are constant within numerical error, the additivity is confirmed, and if they are not, the law is an interpolation rather than a derived limit combination.
[Section V, Fig. 4 and Appendix C] The coefficients a, b, c, d in Eq. (29) are fitted to the ensemble-average DNS at B=0 using exactly the four solid-volume fractions that appear in the subsequent validation (φ = 0.1, 0.3, 0.5, 0.55). With four free parameters and four data points, the reported R² = 1 is a tautology, not evidence for the functional form. The Newtonian limit is therefore only an interpolation of the same data used to exhibit that limit. To support Eq. (29) as a genuine Darcy law, the authors should either use more solid fractions, impose the asymptotic dilute-limit coefficient from the series solution (14), or compare with independent data from regular arrays with the same fit; in all cases, a measure of predictive error beyond in-sample R² is needed.
[Section V, Fig. 5(b–e)] No quantitative error metric is reported for the comparison between Eq. (30) and the DNS at finite B. The text states 'Great agreement' but does not give the relative error, the maximum deviation, or the deviation relative to the plotted uncertainty bars. Since the central claim is that the law holds 'across the entire range of Bingham numbers', the validation should include, for each φ and B, the mean and maximum relative error of ΔP/L over the 20 realizations. Without such metrics, the intermediate-B agreement remains anecdotal and cannot be compared against alternative models such as Castañeda's bound or the linearized BCK expression (B5).

minor comments (5)

[Eq. (16) description] A typo: 'Bignham model' should be 'Bingham model'.
[Section II C and Eq. (29)] The connection between the drag expansion (14) with O((R/L)^2) and the porosity expansion (29) with O(φ^3) is not explained; note that φ ~ (R/L)^2 in two dimensions, so the orders are consistent only if one accounts for the square root. Please state this explicitly.
[Section IV, Eq. (25)] Equation (25) is presented as a universal scale, but its ingredients, the scalings (24), are inherited from prior work [12]. The present paper re-validates the final expression against 15 realizations in Fig. 3, but it does not re-derive the statistical scaling. This distinction should be stated so that the reader understands which parts of the yield-limit model are newly supported here.
[Fig. 4 caption] The inset showing the square and triangular arrangements is not described in the caption or the text; please label the inset or point the reader to a prior figure.
[Section V, after Eq. (20)] The definition of κ in Eq. (20) involves L_inl/L, but the domain description states L=50 and does not define L_inl separately; please clarify the relationship between the inlet length and the domain length, and whether L_inl equals the side length of the square domain.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the finite-B validation is out-of-sample; the B=0 coefficients are fitted and disclosed, and the yield-limit term is independently re-validated in Fig. 3.

full rationale

The central proposed law, Eq. (30), is a linear combination of a Newtonian-limit term and a yield-limit term. The Newtonian-limit coefficients (a,b,c,d) are explicitly fitted to DNS at B=0 in Fig. 4 and reported in Table I; the paper does not present the B=0 agreement as an out-of-sample prediction, so this is disclosed fitting rather than a fitted input masquerading as a prediction. The yield-limit term, lim_{B->infty} f = pi phi/(1-phi) B, comes from the author's prior work [12], but the present paper independently compares Eq. (25) against new DNS in Fig. 3 and against an upper bound from Castaneda, giving in-paper supporting evidence; thus the self-citation is not load-bearing in a circular way. The finite-B predictions shown in Fig. 5 use coefficients fixed at B=0 and a yield-limit term fixed by the independently checked Yc model, so the intermediate-B DNS data are genuinely out-of-sample. The main weakness is that Eq. (30) assumes, without derivation, that the energy-balance contributions a(u,u)/I and B j(u)/I can be superposed as their B=0 and B->infty limits; this is an untested additivity ansatz that could be flagged as a modeling or correctness risk, but it is not a circularity because the finite-B output is not built into the inputs by construction. No circular reduction, self-definitional fit, or renamed known result is exhibited in the text.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central law depends on four empirically fitted Newtonian coefficients and an empirically fitted yield-limit prefactor inherited from the author's prior work. The additivity ansatz is an ad hoc assumption. No new physical entities are introduced.

free parameters (5)

a (Newtonian fit coefficient) = 0.4091
Coefficient in Eq. (29) fitted to DNS ensemble average at B=0 (Table I).
b (Newtonian fit coefficient) = -2.1954
Coefficient in Eq. (29) fitted to DNS ensemble average at B=0 (Table I).
c (Newtonian fit coefficient) = 4.1141
Coefficient in Eq. (29) fitted to DNS ensemble average at B=0 (Table I).
d (Newtonian fit coefficient) = -2.1874
Coefficient in Eq. (29) fitted to DNS ensemble average at B=0 (Table I).
Yield-limit scaling exponents = 1 and 1
The scalings ⟨hch⟩∼(1−φ) and ⟨Lch/L⟩∼φ, fitted to ~500 simulations in [12], determine the prefactor 1/π in Eq. (25).

assumptions (5)

domain assumption Stokes flow, inertial terms neglected in Eq. (15)
The governing equations (15) omit inertia, valid for slow flow in porous media, but limits applicability.
domain assumption Bingham constitutive model with von Mises yield criterion
Used in Eq. (16); other rheologies (Herschel-Bulkley, elastoviscoplastic) are deferred.
ad hoc to paper Additivity of Newtonian and yield-limit pressure drops
Eq. (30) assumes ΔP/L(B) is exactly the sum of the B=0 and B→∞ limits, without derivation or interaction terms.
domain assumption Ensemble averaging over randomized obstacle realizations
The law is meant for statistically averaged media; individual realizations may deviate (see Sec. IV, realization no.5 error).
domain assumption 2D square monodisperse obstacles
All DNS use square obstacles with edge √π; the fitted coefficients likely change with shape and polydispersity.

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Pith. "Pith review of A comprehensive Darcy-type law for viscoplastic fluids: I. Framework." pith.science (2026). https://pith.science/paper/76HXFIFO

@misc{pith2026250606184,
  author       = {Pith},
  title        = {Pith review of: A comprehensive Darcy-type law for viscoplastic fluids: I. Framework},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/76HXFIFO}},
  note         = {Machine review of arXiv:2506.06184}
}

read the original abstract

A comprehensive Darcy-type law for viscoplastic fluids is proposed. Different regimes of yield-stress fluid flow in porous media can be categorised based on the Bingham number (i.e. the ratio of the yield stress to the characteristic viscous stress). In a recent study (Chaparian, J. Fluid Mech., vol. 980, A14, 2024), we addressed the yield/plastic limit (infinitely large Bingham number), namely, the onset of flow when the applied pressure gradient is just sufficient to overcome the yield stress resistance and initiate the flow. A purely geometrical universal scale was derived for the non-dimensional critical pressure gradient, which was thoroughly validated against computational data. In the present work, we investigate the Newtonian limit (infinitely large pressure difference compared to the yield stress of the fluid - ultra low Bingham number) both theoretically and computationally. We then propose a Darcy-type law applicable across the entire range of Bingham numbers by combining the mathematical models of the yield/plastic and Newtonian limits. Exhaustive computational data generated in this study (using augmented Lagrangian method coupled with anisotropic adaptive mesh at the pore scale) confirm the validity of the theoretical proposed law.

Figures

Figures reproduced from arXiv: 2506.06184 by the authors.

**Figure 2.** FIG. 2. Contours of velocity magnitude for two sample realisations: (a,c) [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** FIG. 3. Critical yield number ( [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4. Non-dimensional pressure gradient at [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5: FIG. 5. Proposed Darcy-type law: (a) surface plot of Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

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Pith tools

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