The effect of Maxwellian fluid on wave propagation in porous media

Fansheng Xiong; Weitao Sun

REVIEW 3 major objections 5 minor 2 references

The effect of Maxwellian fluid on wave propagation in porous media

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read By inserting a fractional-derivative Maxwell constitutive law into Biot's poroelastic equations, this paper claims that viscoelastic pore fluids create multiple attenuation peaks and velocity oscillations that a Newtonian fluid cannot…

desk verdict The fractional-Maxwell extension of Biot's dissipation function is a real derivation, but the sandstone wave predictions are unreproducible as written because the pore radius is never reconciled with Table 1's permeability. read the letter →

arxiv 1908.10495 v2 pith:5AMGZKOG submitted 2019-08-27 physics.geo-ph physics.flu-dyn

Weitao Sun , Fansheng Xiong This is my paper · ORCID

classification physics.geo-phphysics.flu-dyn

keywords wavepropagationporoelasticmediafractionalderivativeMaxwellmodelviscoelasticfluidseismicattenuationdynamicpermeabilityDeborahnumberBiottheory

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 attempts to show that the standard assumption of a Newtonian pore fluid is too restrictive for rocks saturated with viscoelastic fluids, and that replacing it with a fractional derivative Maxwell model changes the predicted wave behavior in a way experiments support. The authors derive a frequency-dependent frictional dissipation function $F(\chi)$ from the fractional Maxwell constitutive relation and insert it into Biot's poroelastic wave equations, yielding analytical dispersion and attenuation expressions for P-, slow P-, and S-waves. The central result is that a Maxwellian saturant produces pulsatile velocity enhancement and multiple attenuation peaks, while a Newtonian saturant produces a single smooth peak; the fractional derivative orders control the transition from dissipative to elastic fluid-solid coupling. In the Newtonian limit the equations recover Biot's theory, and for a Maxwellian fluid the predicted velocity peaks in an oscillating tube align with laboratory observations. If right, the work matters because seismic and ultrasonic measurements in reservoirs containing heavy oil or polymer solutions would carry a viscoelastic fingerprint, and the frequency location of attenuation peaks would report on fluid rheology rather than just viscosity.

What carries the argument

The load-bearing object is the viscous dissipation function $F(\chi)$ of Eq. (12), a parameter-free correction to the friction term in Biot's equations. It comes from solving oscillating flow of the fractional Maxwell fluid in a circular tube with no-slip walls, and it depends on the dimensionless frequency $\chi=a\sqrt{\omega/\nu}$, the Deborah number $De=\lambda/\lambda_\nu$, and the fractional derivative orders $\alpha$ and $\beta$. Replacing the Newtonian viscous term by $\eta\phi^2 F(\chi)/\kappa$ in the Biot equations makes the rheology of the saturating fluid control the wave field: $F(\chi)$ determines the dispersion and attenuation of the P-, slow P-, and S-waves, sets the frequency-dependent characteristic frequency, and encodes the transition from a dissipative to an elastic coupling regime. It is the single point where the constitutive model enters the wave equations.

What would settle it

Measure P-wave phase velocity and attenuation in a rock of known porosity and permeability saturated with a viscoelastic fluid whose relaxation time and viscosity are measured independently, sweeping frequency from tens of hertz to tens of megahertz. The central claim fails if the attenuation curve shows one smooth Biot-like peak rather than multiple peaks at lower frequencies, or if the measured centerline velocity peaks in an oscillating tube occur at frequencies that do not match the model's predicted locations.

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Extended reading notes

Core claim

The paper's central claim is that a poroelastic medium saturated by a fractional derivative Maxwellian fluid supports elastic waves whose dispersion and attenuation are qualitatively different from Biot's Newtonian prediction, and that this difference is controlled by a frequency-dependent frictional dissipation function derived from the fractional Maxwell constitutive law. In the Newtonian limit the equations reduce to Biot's theory; for a viscoelastic fluid such as CPyCl/NaSal they produce pulsatile velocity enhancement and multiple attenuation peaks at resonance frequencies, with the fractional derivative orders determining whether the fluid-solid coupling is in a dissipative or an elastic regime. The same formulation predicts centerline fluid velocities in an oscillating tube whose peak locations match laboratory measurements for a Maxwellian fluid, which the paper offers as evidence that viscoelastic fluid effects can account for velocity oscillations observed in experiments. If correct, the result means that seismic and ultrasonic wave trains in rocks saturated by heavy oil or polymer solutions carry a measurable viscoelastic fingerprint that a Newtonian model cannot reproduce.

Load-bearing premise

The argument assumes that the friction felt by waves in a real rock is adequately represented by the solution for a single straight cylindrical pore with no-slip walls, under a long-wave approximation, once that friction is inserted into the Biot equations.

Editorial extensions

If this is right

When the pore fluid is viscoelastic, attenuation curves for P- and S-waves develop multiple peaks and the main transition shifts to lower frequencies than Biot's theory predicts for the same viscosity.
Increasing the fractional derivative order $\beta$ smooths the resonance peaks and moves the attenuation peak to intermediate frequencies, so the model spans the range from Newtonian to ideal Maxwell behavior.
For Newtonian pore fluids the new equations reduce to Biot's results, so existing interpretations remain valid where the fluid is truly Newtonian.
The slow P-wave in a Maxwellian-fluid-saturated rock has much lower attenuation at low frequencies, so it should be observable where it is usually lost in Newtonian saturation.
The characteristic frequency below which the poroelastic model is valid becomes frequency dependent and grows exponentially with frequency for the Maxwellian fluid, extending the model's range beyond Biot's high-frequency limit.

Reading between the lines

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

The same frequency-dependent friction function could be used to invert field seismic or sonic-log data for a reservoir fluid's relaxation time and fractional order, since the predicted peak locations are sensitive to those parameters; the paper does not build that inversion.
The single-tube no-slip geometry is likely to understate how much real pore networks and surface roughness smear the resonance peaks, so the sharpest test of the mechanism would compare model predictions with waves measured in rocks of independently characterized pore geometry.
A practical extension, not pursued in the paper, is to test the low-attenuation slow P-wave prediction directly in a laboratory sandstone saturated with a wormlike micelle solution.
If the fractional derivative orders can be tied to the fractal geometry of the pore network, wave dispersion could become a probe of pore structure rather than only fluid rheology; that link is suggested by work the paper cites but not established here.

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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

The paper does something genuinely new: it generalizes Biot's viscous dissipation function to a fractional-derivative Maxwell fluid, deriving F(χ) from the constitutive equation with Bessel-function analysis, and it recovers the Newtonian limit. That part is solid.

The reason I'd be cautious: the numerical section that claims multiple attenuation peaks in sandstone saturated by CPyCl/NaSal (Figs. 9-11) never states which pore radius was used. In Sec. IV.A, a=1.0 mm is used to illustrate F(χ), giving De≈1.1e5. But the sandstone in Table 1 has κ0=1.1e-13 m², and the paper itself gives the Hagen-Poiseuille relation κ0=a²/8, so a≈0.94 μm and De≈1.2e11. Those two choices put you on completely different branches of F(χ): with the small radius, χ stays below 1 up to ~10 GHz, while with a=1 mm, χ is hundreds at 1 GHz. The paper doesn't say which was used. If it was 1 mm, the implied permeability contradicts Table 1 by six orders; if it was the small radius, the resonance peaks plotted may not appear in that frequency band. Either way, Figs. 9-11 are not reproducible as written. This is the load-bearing soft spot.

Second soft spot: the abstract says the predicted fluid velocities are 'consistent with' the lab data, but the actual match required tuning α from 1.0 to 1.03, and the text admits the peak amplitudes are not perfect. That's a fit, not an independent confirmation. Should be disclosed as such.

Third: the claim that the fdMaxwell model is valid at 'extremely high frequency' because the characteristic frequency grows exponentially is overreach. The scattering limit of Biot's theory is a physical grain-size effect; redefining the characteristic frequency doesn't remove it.

The derivation itself is the contribution, and the experimental tube-velocity peak locations are a point in its favor. The paper deserves a serious referee, but as written it needs major revision: state the radius used, rerun the sandstone predictions with a permeability-consistent radius and report sensitivity, and temper the experimental and high-frequency claims. No code or data is provided, which makes the numerical curves even harder to check.

Referee Report

3 major / 5 minor

Summary. The paper derives a viscous dissipation function F(χ) for a fractional-derivative Maxwellian fluid flowing in a circular tube, incorporates it into Biot's poroelastic wave equations, and obtains analytical expressions for P- and S-wave dispersion and attenuation. It shows that the Newtonian limit recovers Biot's classical results, and it applies the model to sandstone saturated with brine and with the viscoelastic solution CPyCl/NaSal. The predicted tube-center fluid velocities are compared with laboratory data from Castrejon-Pita et al. (2003), and the paper argues that viscoelastic fluid behavior produces multiple attenuation peaks and a dissipative-to-elastic transition that are absent for Newtonian saturants.

Significance. If the results hold, this is a useful extension of Biot theory to non-Newtonian saturants: the derivation of F(χ) from the fractional Maxwell constitutive equation is self-contained, and the check that the Newtonian limit reproduces Biot's F(χ) is a genuine strength. The paper also offers a concrete mechanism for velocity oscillations in viscoelastic-fluid-saturated rocks. However, the sandstone wave predictions depend on an unspecified and potentially inconsistent pore radius, and the experimental agreement in Figure 5 is obtained only after tuning the fractional order α to 1.03; these issues currently limit the strength of the central claims.

major comments (3)

[Section IV.A, IV.C, Eq. (12), Table 1] The pore radius used in the sandstone wave predictions in Figures 9-11 is never stated. Section IV.A fixes a=1.0 mm to illustrate F(χ), while Section IV.C uses French Vosgian sandstone with κ=1.1e-13 m² and, in Figure 3, invokes the Hagen-Poiseuille relation κ0=a²/8, which implies a≈0.94 μm. The two radii differ by three orders of magnitude, changing De=νλ/a² from about 1.1e5 to about 1.2e11 and changing χ=a√(ω/ν) by a factor of 10^3 at fixed frequency. With the permeability-consistent radius, χ remains below unity up to about 10 GHz, so the resonant/elastic regime displayed in Figures 2 and 9-11 may not be probed at all. The multiple attenuation peaks in Figures 9-11 may therefore be artifacts of an unstated, permeability-inconsistent radius. Please state the radius used and recompute the sandstone results using a derived from κ0=a²/8, or justify an alternative pore-geometry relation.
[Section IV.B, Figure 5] The claimed consistency with laboratory observations is weakened because the fractional derivative order α is adjusted from 1.0 to 1.03 to make the predicted peaks lower and closer to the experimental data. This is a calibration step rather than an independent prediction, so the abstract's statement that 'the predicted fluid velocities are consistent with the laboratory observations' overstates the evidence. Please report the fitting procedure, the misfit before and after adjustment, and a sensitivity analysis over α and β, and reframe the comparison as a calibration plus qualitative validation rather than a direct prediction.
[Section III, Appendix, Eqs. (13)-(16)] The wave equations insert the friction term ηϕ²F(χ)/κ using the static permeability κ from Table 1, while F(χ) is evaluated with a pore radius a that is not specified for Figures 9-11. If the radius inside F(χ) is not the one implied by κ through κ0=a²/8, the friction term mixes incompatible length scales. This is not just a presentation issue: because De and χ change by orders of magnitude with a, the qualitative behavior of the predicted dispersion and attenuation curves depends directly on this choice. The manuscript must make the value of a explicit for every figure and ensure it is consistent with the stated permeability.

minor comments (5)

[Section IV.C] Typo: 'pock parameters' should be 'rock parameters'.
[Section II, Eq. (4)] The fractional derivative in Eq. (4) is defined as a Riemann-Liouville derivative; please state explicitly whether the constitutive equation uses this definition or the Caputo definition, since initial-condition treatment differs.
[Section IV.A, Figure 2 caption] The text mentions 'α=1 and β=1.5' while the caption says fractional order pairs (1.0,1.0) and (1.0,1.5); please make the notation for the order pair consistent throughout.
[Section IV.B] Figure 5 would be much more informative with error bars or individual experimental points; the current comparison is only qualitative, and the claim 'peak locations are consistent' is not quantified.
[Section III] The statement that the fdMaxwell model 'would be valid even at extremely high frequency' is stronger than the evidence: the model still ignores scattering and other high-frequency effects that limit Biot theory, and the large characteristic frequency alone does not establish validity.

Circularity Check

1 steps flagged · score 4.0 of 10

Partial circularity: the velocity-amplitude 'prediction' is a post-hoc adjustment of the fractional order α to the same experimental data, while the central F(χ)-into-Biot derivation is self-contained.

fitted input called prediction [Section IV.B, Figure 5 (and abstract conclusion)]
"The theoretical predictions in this study predicted the typical viscoelastic resonance behavior as shown in Figure 5 (b). Although the predicted peak amplitudes are not perfect, the peak locations are consistent. The amplitude discrepancy may arise from the local high shear rate (Castrejon-Pita et al. 2003).Adjusting the fractional derivative order from 1.0 to 1.03 shows that the predicted peaks are much lower and closer to the experimental data."

The fractional derivative order α is a free parameter of the model. After seeing the same Castrejon-Pita et al. velocity data, the paper changes α from 1.0 to 1.03 specifically to bring the computed amplitudes 'closer to the experimental data,' then presents the curve as a prediction and the abstract asserts that 'the predicted fluid velocities are consistent with the laboratory observations.' For the amplitude feature, this is a one-parameter fit to the same dataset, not an independent forecast. The peak locations at α=1 are not fitted and remain genuine content, so the circularity is partial.

full rationale

The core derivation is not circular. F(χ) is obtained analytically from the fractional Maxwell constitutive equation and the cylindrical no-slip flow solution (Eqs. 7-12), with no parameters fitted to the wave data, and the Newtonian limit is independently checked against Biot's theory and glycerol data. The wave dispersion/attenuation predictions (Figs. 9-11) are parameter studies using literature rheology for CPyCl/NaSal and a measured sandstone, not fits. There are no load-bearing self-citations: the supporting references (Biot, Tsiklauri and Beresnev, Castrejon-Pita et al.) are external. The only reduction-by-construction I find is the abstract's velocity-consistency claim: the α=1.03 curve is tuned post hoc to the same Castrejon-Pita experiment whose match is then advertised as a prediction; the α=1 peak locations are independent, so this is partial rather than total circularity. The unstated pore radius used for the sandstone wave predictions (a=1.0 mm illustrative vs. a≈0.94 μm implied by Table 1's κ=1.1e-13 m²) is a serious reproducibility/correctness concern, but it is not a circularity because neither choice is defined in terms of the predicted wave quantities.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

No new physical entities (particles, forces, dimensions) are introduced. The 'dynamic permeability' κ(ω) and 'characteristic frequency fc^Maxwell' are derived quantities, not independent entities.

free parameters (3)

Fractional derivative order α for CPyCl/NaSal = 1.03 (tuned from 1.0)
In Section IV.B and Figure 5, α is adjusted to 1.03 to bring predicted peak amplitudes closer to the experimental velocities of Castrejon-Pita et al. (2003).
Fractional derivative order β = 1.0 and 1.5 in examples; not tuned to data
The retardation-order β is varied parametrically to show smoothing of resonance; not fitted to laboratory data.
Relaxation time λ for brine = 10 ns (chosen small)
Assigned a tiny value to simulate the Newtonian limit in the sandstone examples; a limiting parameter rather than a fit.

assumptions (4)

domain assumption Fractional Maxwell constitutive equation τ + λ^α D^ατ = 2η(γ̇ + λ^β D^βγ̇) governs the pore fluid rheology
Invoked as Eq. (3) in Section II; the physical validity for real reservoir fluids is not independently established in the paper.
domain assumption Fluid is incompressible, pore radius is much smaller than wavelength, and flow is laminar with no-slip at the pore wall
Stated in Section II before deriving the Bessel solution; limits application to long-wavelength, low-Reynolds conditions.
domain assumption Biot's poroelastic equations and coefficients Q, R, and tortuosity α∞ apply, with the dynamic permeability replacing the static resistance term
Biot's framework is adopted (Section III) without re-derivation; it imposes isotropy, homogeneity, and linear elasticity.
standard math Riemann-Liouville fractional derivative definition (Eq. 4)
Standard mathematical tool used in Eq. (4); no uniqueness or physical grounding claimed.

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Cite this review

Pith. "Pith review of The effect of Maxwellian fluid on wave propagation in porous media." pith.science (2026). https://pith.science/paper/5AMGZKOG

@misc{pith2026190810495,
  author       = {Pith},
  title        = {Pith review of: The effect of Maxwellian fluid on wave propagation in porous media},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5AMGZKOG}},
  note         = {Machine review of arXiv:1908.10495}
}

read the original abstract

This study investigates the effect of a Maxwellian fluid on the propagation of waves in poroelastic media. Based on a fractional derivative stress-strain relation, a viscous dissipation function is obtained to measure the viscoelastic fluid-solid coupling effect. With the viscous dissipation function, elastic waves propagation is formulated in poroelastic media saturated by a fractional derivative Maxwellian fluid, and the analytical expression of the P- and S-wave dispersion/attenuation is presented. Numerical examples show that the fractional derivative Maxwell strain-stress relation has a significant influence on wave velocities and causes the fluid-solid coupling transition from a dissipative regime to an elastic regime. In addition, the predicted fluid velocities are consistent with the laboratory observations of viscoelastic fluids under an oscillating pressure gradient. The results indicate that a viscous-elastic fluid effect may account for the velocity oscillation observed in laboratory. The method elucidates dynamical differences for viscous and viscoelastic fluid in porous medium, which may be of great importance to unconventional oil/gas exploration industry as well as theoretical researches.

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Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

[1]

Bacri, J.-C., and D. Salin. 1986, Sound velocity of a sandstone with oil and brine at different concentrations. GEOPHYSICAL RESEARCH LETTERS, 13, no. 4,326-328. Bagley, R. L., and P. J. Torvik. 1983, A Theoretical Basis for the Application of Fractional Calculus to Viscoelasticity Journal of Rheology, 27, no. 3,201-210. 23 Balankin, A. S., and B. E. Eliza...

work page 1986
[72]

Fellah, Z

doi: 10.1121/1.1528592. Fellah, Z. E. A., A. Wirgin, M. Fellah, N. Sebaa, C. Depollier, and W. Lauriks. 2005, A time-domain model of transient acoustic wave propagation in double-layered porous media. The Journal of the Acoustical Society of America, 118, no. 2,661-670. doi: 10.1121/1.1953247. Friedrich, C. 1991, Relaxation and retardation functions of th...

work page doi:10.1121/1.1528592 2005

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Bacri, J.-C., and D. Salin. 1986, Sound velocity of a sandstone with oil and brine at different concentrations. GEOPHYSICAL RESEARCH LETTERS, 13, no. 4,326-328. Bagley, R. L., and P. J. Torvik. 1983, A Theoretical Basis for the Application of Fractional Calculus to Viscoelasticity Journal of Rheology, 27, no. 3,201-210. 23 Balankin, A. S., and B. E. Eliza...

work page 1986

[2] [72]

Fellah, Z

doi: 10.1121/1.1528592. Fellah, Z. E. A., A. Wirgin, M. Fellah, N. Sebaa, C. Depollier, and W. Lauriks. 2005, A time-domain model of transient acoustic wave propagation in double-layered porous media. The Journal of the Acoustical Society of America, 118, no. 2,661-670. doi: 10.1121/1.1953247. Friedrich, C. 1991, Relaxation and retardation functions of th...

work page doi:10.1121/1.1528592 2005