Shear Critical Porosity Below Bulk Critical Porosity: Generalizing the Gassmann–Nur Hyperbola

1. Introduction

The critical-porosity model relates the elastic moduli of a porous rock to its porosity through a single structural parameter: the porosity at which the load-bearing frame gives way and the dry moduli vanish (Nur et al., 1998). In the Gassmann–Nur form used for amplitude and pore-pressure analysis, the dry bulk and shear moduli fall linearly to zero at that critical porosity, and Gassmann’s relations map the dry frame to the saturated rock (Higginbotham et al., 2012; Higginbotham, 2020, 2025). A consequence is that the compressional and shear velocities of a mineral’s rocks trace a hyperbola in the V_p–V_s plane (Mavko and Mukerji, 1998), the curve on which much critical-porosity interpretation rests.

The published model carries a tacit assumption: that bulk and shear rigidity vanish at the same porosity, ϕ_s=ϕ_c. As Section 3 makes exact, that single condition is equivalent to holding the dry-frame Poisson’s ratio constant across all porosities. It is a convenient idealization — it lets one critical porosity stand for the whole frame — but it is a strong physical claim about how a frame loses its two rigidities together as it opens toward a suspension.

This paper makes one new claim and relies on the published corpus for everything else. The claim is that the two critical porosities are generally distinct, with the shear critical porosity at or below the bulk one,

ϕ_s ≤ ϕ_c, so that the dry-frame Poisson’s ratio is not constant but rises as porosity approaches the transition. The shear-supporting backbone of a granular frame, which must transmit tangential traction across locked contacts, gives out at a lower porosity than the bulk-supporting backbone, which resists compression even though grains are poorly connected.

The single-critical-porosity content — the hyperbola, the anchor and slope expressions, the critical-porosity determination, and the pore-pressure method — is published and is cited here, not re-derived. What is new is the generalization to ϕ_s ≤ ϕ_c: its physical basis, the dry-frame measurement that establishes it, and its consequences in the high-porosity regime. Section 2 proves that the hyperbola exists for any constant pair of critical porosities, so the equal-porosity condition fixes only where the anchor (the value of the compressional velocity as the shear velocity approaches zero) sits, not whether the curve exists. Section 3 gives the physical basis for ϕ_s ≤ ϕ_c and shows it is exactly the ordering that preserves the hyperbola. Section 4 measures both critical porosities in the dry sandstone frames of Han (1986) and rejects the constant-Poisson condition within the data. Section 5 traces the consequences where they matter — the shallowest sediments and undercompacted, overpressured intervals — and connects them to the published pore-pressure framework. Section 6 identifies the same separation as the dry-rock residual of the published fluid parameter d — an independent corroboration, in dry shale, of a result the rest of the paper draws from quartz sandstone.

2. The Hyperbola Does Not Require A Single Critical Porosity

A porous rock can have empty pores or pores filled with fluid. Additionally the rock matrix is composed of mineral grains. The following subscript notation will be required to designate associated variables.

d → dry rock, the pores of the rock are empty of fluid

w → wet rock, the pores of the rock are filled with fluid

m → the mineral grains

The published form of the Gassmann–Nur model assigns a single critical porosity to a given mineral: the dry bulk and shear moduli are taken to vanish at the same porosity, ϕ_s=ϕ_c, which is the content of the condition K_d/K_m=μ_d/μ_m (Higginbotham et al., 2012; Higginbotham, 2020, 2025; the hyperbola itself is Mavko and Mukerji, 1998). That condition is sufficient for the hyperbolic V_p–V_s relation, but it is not necessary. This section shows that the hyperbola exists for arbitrary constant bulk and shear critical porosities, and that the equal-critical-porosity condition fixes only the value of the anchor, not the existence of the curve.

Throughout, the pore fluid velocity means the compressional velocity through the pore fluid alone, and the suspension velocity the compressional velocity through a suspension of mineral grains in that fluid. These are close in value but distinct quantities.

2.1. Setup

Let K_m, μ_m, ρ_m be the mineral bulk modulus, shear modulus, and density; K_f, ρ_f the pore-fluid bulk modulus and density; ϕ_c the bulk critical porosity and ϕ_s the shear critical porosity, both taken constant after Nur’s model. Define

${\mu }_d=\left(1-\varphi /{\varphi }_s\right){\mu }_m,\varphi \le {\varphi }_s,K_d=\left(1-\varphi /{\varphi }_c\right)K_m,\varphi \le {\varphi }_c\mathrm{and}{\varphi }_c\text{'}={\varphi }_c+{\displaystyle \frac{K_f}{K_m-K_f}}.\left(2.1\right)$The first two equations generalize Nur’s model to two critical porosities. The third, the saturated-bulk critical porosity, is produced by Gassmann’s mapping of the linear dry bulk frame. Restrict attention to

$\varphi \text{<}min\left({\varphi }_s,{\varphi }_c\right)$, where the porous frame is shear supporting; there the piecewise caps in the critical-porosity transformation are inactive, and the Gassmann–Nur moduli can be written as:

$K_w=\left(1-{\displaystyle \frac{\varphi }{{\varphi }_c\text{'}}}\right)K_m,{\mu }_w=\left(1-{\displaystyle \frac{\varphi }{{\varphi }_s}}\right){\mu }_m,\rho ={\rho }_m-\left({\rho }_m-{\rho }_f\right)\varphi .\left(2.2\right)$Constraints: ${\mu }_w$ vanishes for $\varphi \text{>}{\varphi }_s$ ; $K_w$ follows (2.2) for $\varphi \le {\varphi }_c$ and passes to the Reuss suspension modulus beyond ${\varphi }_c$ (see 2.10).

Notice that even when ${\varphi }_s={\varphi }_c$ there is a critical porosity like ${\varphi }_c\text{'}\text{>}{\varphi }_s$ present in (2.2).

The shear and compressional velocities follow from their definitions,

$V_s^2={\displaystyle \frac{{\mu }_w}{\rho }},V_p^2={\displaystyle \frac{K_w+\frac{4}{3}{\mu }_w}{\rho }}.\left(2.3\right)$No assumption relating ϕ_s and ϕ_c has been made.

This is the central claim. The shear modulus vanishes at ϕ_s, whereas the saturated bulk modulus carries its own critical porosity ${\varphi }_c\text{'}$, displaced above the dry value ϕ_c by the fluid term K_f/(K_m − K_f). A separation between a bulk and a shear critical porosity is thus not a new device; Gassmann’s mapping of the linear dry frame already produces one through fluid loading alone. The conventional single critical-porosity model is the special case in which the two are tied because the dry bulk and shear moduli are assumed to vanish together: the porosity is called critical because the shear modulus goes to zero there and the bulk modulus passes over, as Gassmann’s relation then requires (see section 2.2, equation 2.5), to a suspension of mineral grains in the fluid. The one new element of the present model is to relax that coincidence — to let the dry bulk frame remain load-bearing over an interval of porosity beyond the point at which the shear modulus vanishes, ϕ_s ≤ ϕ_c, so that bulk and shear critical porosities differ in the dry frame as well, and not only through the fluid shift that (2.1) already carries.

.2.2. The hyperbola is Preserved

The hyperbola we seek can be written as:

${\displaystyle \frac{V_p^2}{a^2}}-{\displaystyle \frac{V_s^2}{b^2}}=1.\left(2.4\right)$This follows from combining Gassmann’s equations with the two critical porosity model (2.2), a generalization of Nur’s single critical porosity model. Here Gassmann’s equations are reproduced for reference.

$K_w=K_d+{\displaystyle \frac{{\left(1-K_d/K_m\right)}^2}{\frac{\left(1-\varphi -K_d/K_m\right)}{K_m}+\varphi /K_f}},{\mu }_w={\mu }_d.\left(2.5\right)$Because $V_p^2$ and $V_s^2$ carry the same affine (linear-in-ϕ) density denominator, eliminating porosity leaves an exact line for any constant pair—the affine denominator theorem. The following verifies this symbolically and finds explicit constants for $a^2\mathrm{and}{b}^2$ .

Form the ratio $V_p^2/V_s^2$this removes the explicit denominator, solve $V_s^2$ of equation (2.3) for porosity using (2.2), substitute, and simplify. The result is the following.

Define:

$\mathrm{For}{V}_s^2\text{>}0:X={\displaystyle \frac{1}{{\varphi }_c\text{'}}}-{\displaystyle \frac{1}{{\varphi }_s}},Y=1-{\displaystyle \frac{1}{{\varphi }_s}},Z=1-{\displaystyle \frac{1}{{\varphi }_c\text{'}}}.\left(2.6\right)$Then,

$V_p^2{\displaystyle \frac{{\rho }_{m}Y-{\rho }_f}{K_{m}X}}-V_s^2\left({\displaystyle \frac{{\rho }_{m}Z-{\rho }_f}{{\mu }_{m}X}}+{\displaystyle \frac{4}{3}}{\displaystyle \frac{{\rho }_{m}Y-{\rho }_f}{K_{m}X}}\right)=1.\left(2.7\right)$This is (2.4) with the constants written out: its V_p² coefficient is 1/a² and its V_s² coefficient is 1/b², with a, b as given in §2.3 (verified symbolically). The porosity has been eliminated entirely; the two critical porosities survive only through ϕ_s (in Y) and ϕ_c^′ (in Z), and the common factor 1/X . Note the pairing: Y, built from the shear critical porosity ϕ_s, multiplies the K_m (compressional) terms ; Z, built from ϕ_c^′, multiplies the μ_m term. This cross-pairing is correct, not a transposition. This derivation procedure leads to the above equation multiplied by the shear velocity squared, so the above equation is only mathematically forced when the shear velocity is non-zero, but this is where the hyperbola connects $V_p$ to $V_s$ while $V_s$ lives as a non-zero value. The messy case of X → 0 turns out to be interesting and is discussed in Appendix A.

Unrelated to the X → 0 issue of Appendix A, equation (2.7) is forced for $V_s^2\ne 0$ and extends by continuity to the limit $V_s^2\to 0$ ; negative $V_s^2$ , which the algebra permits, is unphysical imaginary shear velocities.

The shear velocity goes to zero when $\varphi ={\varphi }_s$ forcing ${\mu }_w=0.$ In the single critical porosity model the two critical porosities are equal so this would mean $K_d=0$ and examination of (2.5) indicates that $K_w$ becomes the formula for a fluid suspension of the mineral grains. However, in the two critical porosity model ${\varphi }_c$ is above ${\varphi }_s$ so $K_d$ is not forced to zero. Shear has vanished but bulk support persists until $\varphi \text{>}{\varphi }_c$ .

Requiring ϕ_s=ϕ_c as a precondition is unnecessary: that condition enters the problem at a different point, identified next.

2.3. The Anchor and the Slope in Physical Constants

The anchor a is the compressional velocity in the limit V_s → 0. Because μ_w=(1 − ϕ/ϕ_s)μ_m, the shear velocity vanishes at ϕ=ϕ_s, so the anchor is V_p evaluated at the shear critical porosity:

$a^2={\displaystyle \frac{K_a}{{\rho }_a}},K_a=\left(1-{\displaystyle \frac{{\varphi }_s}{{\varphi }_c\text{'}}}\right)K_m=K_w\left({\varphi }_s\right),{\rho }_a=\left(1-{\varphi }_s\right){\rho }_m+{\varphi }_s{\rho }_f.\left(2.8\right)$Both K_a and ρ_a are evaluated at the same porosity ϕ_s — the porosity at which V_s → 0.

The slope of the line, equivalently the hyperbolic parameter b, follows in closed form.

With r ≡ (ρ_m − ρ_f)/ρ_m,

${\displaystyle \frac{a^2}{b^2}}={\displaystyle \frac{4}{3}}+{\displaystyle \frac{K_m}{{\mu }_m}}{\displaystyle \frac{\frac{1}{{\varphi }_c\text{'}}-r}{\frac{1}{{\varphi }_s}-r}}.\left(2.9\right)$Equations (2.8) and (2.9) give a and b for arbitrary ϕ_c, ϕ_s; both reduce to the published single critical porosity expressions when ϕ_s=ϕ_c. Relations (2.4)–(2.5) and (2.7)–(2.9) have been verified symbolically.

2.4. What the Equal-Porosity Condition Actually Forces

Setting ϕ_s=ϕ_c in (2.8) collapses the anchor onto the Reuss suspension modulus evaluated at the bulk critical porosity,

$K_a{|}_{{\varphi }_s={\varphi }_c}=\left(1-{\displaystyle \frac{{\varphi }_c}{{\varphi }_c\text{'}}}\right)K_m={\displaystyle \frac{K_m{K}_f}{{\varphi }_c{K}_m+\left(1-{\varphi }_c\right)K_f}},\left(2.10\right)$so that a becomes the velocity of a fluid suspension of mineral grains at porosity ϕ_c — for water-saturated quartz, the familiar value a few percent above the pore fluid. The condition K_d/K_m=μ_d/μ_m is therefore not a condition for the hyperbola to exist; it is the condition under which the anchor sits at the suspension velocity, or equivalently the condition under which shear rigidity and bulk-frame stiffness vanish at the same porosity. When ϕ_s<ϕ_c the curve is equally exact, but its anchor lies above the suspension velocity, by an amount that grows as the separation ϕ_c − ϕ_s widens.

2.5. Determining the Critical Porosity From The Anchor

The anchor needed for these expressions is itself recoverable from data. Eliminating b between the hyperbola (2.4) and the slope (2.9) gives the anchor from any single measured (V_p, V_s) pair on the curve, using only the mineral constants:

$a^2={\displaystyle \frac{V_p^2-\left(\frac{4}{3}+\frac{K_m}{{\mu }_m}\right)V_s^2}{1-\frac{{\rho }_m}{{\mu }_m}{V}_s^2}}.\left(2.11\right)$This relation is exact for arbitrary ϕ_c, ϕ_s — it does not assume the two critical porosities are equal — so the anchor can be read from the data without first committing to a value of either critical porosity. However Appendix E considerations apply in general.

The anchor in turn constrains the critical porosities. Writing the anchor (2.8) as a²ρ_a=K_a and clearing the saturated-bulk denominator ϕ_c^′ yields a quadratic in 1 − ϕ_s,

(1 − ϕ_s)²+(1 − ϕ_s)C₁ − C₀=0,  (2.12)

with, writing δ ≡ ϕ_c − ϕ_s for the separation,

$C_1={\displaystyle \frac{{\rho }_f}{{\rho }_m-{\rho }_f}}-{\displaystyle \frac{K_m}{K_m-K_f}}-\delta ,\left(2.13\right)$$C_0={\displaystyle \frac{{\rho }_f}{{\rho }_m-{\rho }_f}}{\displaystyle \frac{K_m}{K_m-K_f}}\left(1-{\displaystyle \frac{K_f}{{\rho }_f}}{\displaystyle \frac{1}{a^2}}\right)+\delta {\displaystyle \frac{{\rho }_f}{{\rho }_m-{\rho }_f}}\left(1-{\displaystyle \frac{K_m}{{\rho }_f}}{\displaystyle \frac{1}{a^2}}\right).\left(2.14\right)$The unknown is 1 − ϕ_s: the anchor is V_p as V_s → 0, which is reached at the shear critical porosity, so it is ϕ_s that the quadratic returns, the bulk critical porosity entering only through the separation δ. The two δ-terms are the entire two-porosity content; setting δ=0 collapses (2.13)–(2.14) to the published single critical porosity determination (Higginbotham, 2020, 2025 — equations (13)–(15) ) . If the anchor is additionally taken at the pore-fluid velocity, a²=K_f/ρ_f then C₀=0 and, keeping in mind that ${\varphi }_c={\varphi }_s$ here, the admissible root is the closed form

${\varphi }_c={\displaystyle \frac{{\rho }_f}{{\rho }_m-{\rho }_f}}-{\displaystyle \frac{K_f}{K_m-K_f}},\left(2.15\right)$about 0.59 for quartz saturated with a formation brine (K_f = 2.69 GPa, ρ_f = 1.06 g/cm³), for which the pore-fluid velocity is a = 1590 m/s as used in Appendix B; pure water (K_f = 2.2 GPa, ρ_f = 1.0 g/cm³, a = 1483 m/s) gives 0.54 instead. The closed form and the anchor are fixed by the same two fluid constants and move together, so the fluid must be specified before either is quoted. The closed form also belongs to the degenerate case alone: for δ ≠ 0 the constant term C₀ no longer vanishes at the pore-fluid anchor and the discriminant of (2.12) turns negative, so the quadratic has no real root at all. That is the algebraic form of the statement in §2.4 that a frame with ϕ_s < ϕ_c carries its anchor above the suspension velocity, and hence above the pore-fluid velocity. In either case the value returned lies far above the 0.40 suggested by Nur et al. (1998), indicating that the anchor for the single critical porosity theory is not the pore-fluid velocity. Nevertheless, the hyperbola associated with using 1500 m/s as a fixed anchor, fits a lot of data quite well—see Figure 1. Further discussion of Nur et al. (1998) related to these equations can be found in Appendix B.

As an inversion from a single measured anchor, (2.12) carries two unknowns — ϕ_s and the separation δ — so one anchor fixes a relation between them, not either alone. The δ=0 form is a usable single critical porosity inversion, but applied to a frame whose critical porosities actually differ it omits the linear-in-δ corrections in (2.13)–(2.14) and misreads the rock: a frame with ϕ_c=0.40 and ϕ_s=0.35 produces an anchor near 2110 m/s, which the δ=0 quadratic reads as an apparent critical porosity of about 0.16 — well below both — because it ascribes the elevated anchor entirely to a lower porosity. How steeply the inferred porosity tracks the assumed separation is itself fixed by (2.12): its derivative with respect to δ is about +4.8 at the quartz operating point of Appendix C, so the inferred shear critical porosity inherits some four to five times any error in the assumed δ — the local-slope counterpart of the 0.35 → 0.16 misreading — and it diverges where the two roots of (2.12) coincide, the locus on which a lone anchor cannot be inverted at all. The single critical porosity inversion is therefore not a safe estimator of either critical porosity once the two differ; separating ϕ_s from ϕ_c requires data that fix the dry moduli directly, treated in Section 4.

2.6. Relation to Mavko and Mukerji (1998)

Mavko and Mukerji obtained the same hyperbola by assuming the dry frame’s moduli are proportional — equivalently, that the dry-frame Poisson ratio is constant with porosity, which is exactly ϕ_s=ϕ_c. The result above shows the curve does not depend on that assumption: it is a consequence of the shared affine denominator alone, and it survives ϕ_s ≠ ϕ_c. The distinction is not merely formal. As the following sections argue on physical grounds and then show in dry-frame data, the shear critical porosity is in fact below the bulk one, so the dry-frame Poisson ratio is not constant — and the generalized relation, not its constant-Poisson special case, is the one that describes real rock.

2.7. The hyperbola in Data

The hyperbola is not only a theorem but a measured fact. Figure 1 shows the compressional and shear velocities of sandstones (Castagna et al., 1993, pp. 138–139) against the curve (2.4). With the anchor held at a specific brine pore fluid velocity a = 1500 m/s — the equal-critical-porosity value of §2.4 — and only the single parameter b = 972 m/s fitted (Higginbotham et al., 2011), the relation tracks the data across the full velocity range at a root-mean-square misfit of 191 m/s. That misfit is primarily the heterogeneity of a mixed compilation, not a defect of the form: the single curve fixes one anchor and one slope, whereas the samples span a range of pore fluids, which displace the anchor, and mineralogies, which displace the slope, and either source alone — across the brine velocities and clay contents these rocks actually carry — would produce a scatter of this size. The same single-parameter curve, anchored at the same specific pore fluid velocity, also fits carbonates and a range of crystalline and evaporite rocks for which no fluid-substitution argument applies (Higginbotham et al., 2011; Higginbotham, 2025), consistent with the relation being a structural property of an affine frame rather than an artifact of Gassmann’s fluid mapping (exceptions discussed in Appendix E). By the theorem of §2.2 the curve exists for any constant pair (ϕ_c, ϕ_s); the anchor at 1500 m/s is the equal critical porosity special case defaulting to the pore fluid. A separation ϕ_s < ϕ_c would raise the anchor above the suspension velocity (already higher than the pore fluid velocity) (§2.4) without altering the curve shape. The fit confirms the form, not the separation; the dry-frame data of Section 4 carry the latter.

3. Physical Basis for ϕ_s ≤ ϕ_c

Section 2 showed that the hyperbola exists for any constant pair ϕ_c, ϕ_s and that ϕ_s=ϕ_c fixes the anchor at the grain-in-fluid suspension velocity rather than being required for the curve. The question is then which situation is physically generic: equal critical porosities, or distinct ones. The argument is that equality is the special case needing justification, that the generic frame has the shear critical porosity below the bulk one, and that this regime is exactly the one in which the hyperbola of Section 2 remains exact. The reasoning is offered as physical motivation, not proof: it rests on idealizations — frictionless packings, single Hertzian contacts — that do not map exactly onto cemented sedimentary rock.

3.1. Critical Porosity as a Load-Bearing Threshold

Critical porosity has a direct mechanical reading: it is the porosity at which the load-bearing fraction of the solid vanishes for a given deformation mode. With the dry-frame moduli in critical-porosity form,

$K_d=\left(1-{\displaystyle \frac{\varphi }{{\varphi }_c}}\right)K_m,{\mu }_d=\left(1-{\displaystyle \frac{\varphi }{{\varphi }_s}}\right){\mu }_m,\left(3.1\right)$each modulus falls to zero when the stress-supporting backbone for its mode ceases to span the rock. There is no reason the two modes lose their backbones at the same porosity. A grain that is present but not locked into the contact network contributes nothing to shear rigidity — it cannot transmit tangential traction — yet it still resists hydrostatic compression, because a volume change presses on every grain whether or not it is well connected. A second mechanism operates even where the contact network is intact. A grain contact may carry normal load while sustaining no tangential traction: the grains slide rather than grip, so the frame develops no restoring stress against shear even though it resists compression through the same contact. In the contact mechanics of §3.3 this is the vanishing of the tangential stiffness while the normal stiffness is unaffected.

Shear rigidity therefore depends on connectivity more steeply than bulk rigidity, and the shear-supporting fraction vanishes at a lower porosity than the bulk-supporting fraction: ϕ_s ≤ ϕ_c.

The consequence for the dry-frame Poisson’s ratio is exact. From (3.1),

${\displaystyle \frac{K_d}{{\mu }_d}}={\displaystyle \frac{K_m}{{\mu }_m}}{\displaystyle \frac{1-\varphi /{\varphi }_c}{1-\varphi /{\varphi }_s}},\left(3.2\right)$which is independent of porosity if and only if ϕ_s=ϕ_c. Since the dry-frame Poisson’s ratio depends on the moduli only through x ≡ K_d/μ_d,

${\nu }_d={\displaystyle \frac{3x-2}{2\left(3x+1\right)}}={\displaystyle \frac{3K_d-2{\mu }_d}{2\left(3K_d+{\mu }_d\right)}},\left(3.3\right)$a strictly increasing function of x, constancy of ν_d with porosity is exactly equivalent to ϕ_s=ϕ_c. Notice that if ${\mu }_d=0$ while $K_d$ remains finite, then ${\nu }_d=0.5$ .

Setting the two critical porosities equal is thus the same as asserting that the dry frame holds the same Poisson’s ratio at every porosity, from the mineral point to the transition — a strong and directly testable claim. The load-bearing argument predicts the opposite: ϕ_s<ϕ_c, hence a dry-frame Poisson’s ratio that is not constant but drifts upward as the rock approaches the transition and shear gives out first.

Let $\delta ={\varphi }_c-{\varphi }_s$ as in section 2.5. Then for $\delta \ll {\varphi }_s$$\nu ={\displaystyle \frac{\left(3K_m-2{\mu }_m\right)\left({\varphi }_s-\varphi \right)+\frac{\varphi }{{\varphi }_s}3K_m\delta }{2\left(3K_m+{\mu }_m\right)\left({\varphi }_s-\varphi \right)+\frac{\varphi }{{\varphi }_s}6K_m\delta }}.\left(3.4\right)$Equation (3.4) is the idealized small-δ trend — the dry moduli taken linear and pinned to the mineral values at $\varphi =0$ — for which Poisson’s ratio rises gently across the consolidated range and turns sharply toward 0.5 only as $\varphi \to {\varphi }_s$.

3.2. Real Dry Frames do not Hold Poisson’s Ratio Constant

Measured dry-rock data do not support a porosity-independent dry-frame Poisson’s ratio. The dry and saturated sandstone velocities of Han (1986) give a dry-frame V_p/V_s ratio — and therefore a ν_d — that varies systematically with porosity and clay content rather than holding fixed.

3.3. The Direction of the Inequality

Three independent lines of reasoning place the shear critical porosity at or below the bulk one, and they are descriptions of one fact rather than three separate mechanisms.

The first is the load-bearing argument of §3.1: shear requires a connected, tangentially locked backbone, while bulk stiffness does not, so shear is the first rigidity lost as the frame opens.

The second is grain-scale contact mechanics. In the Hertz–Mindlin description (a granular frame as a random pack of identical elastic spheres) of a granular frame the normal and tangential stiffnesses of a contact are distinct functions of the grain moduli (Mindlin, 1949), so the assembly’s bulk and shear moduli have no reason to track porosity in lockstep. The model further reproduces measured bulk moduli while systematically overpredicting shear moduli — the empirical reduced-shear-factor correction (Bachrach and Avseth, 2008) — indicating that tangential coupling at contacts is the weaker and more variable of the two, and hence the more fragile as the frame loosens.

The third is the loss of rigidity in granular packings. In simulations of frictionless sphere packings the bulk modulus, shear modulus, and pressure all vanish at the jamming (isostatic) point, with the shear-to-bulk ratio vanishing as that point is approached from above (O’Hern et al., 2003; Liu and Nagel, 2010; Makse et al., 2000). This idealized result — for frictionless, isotropically compressed packings, and distinct from cemented rock and from shear-jammed states, where the ratio instead tends to a constant — again makes shear the first casualty of an opening frame.

Two thresholds are at work here, related but distinct. Percolation is geometric — the porosity at which a connected, sample-spanning path of load-bearing contacts first forms; jamming is mechanical — the porosity at which a disordered grain pack first acquires finite rigidity, nonzero bulk and shear moduli that support static stress, coinciding for frictionless spheres with isostaticity, the mean number of contacts per grain reaching twice the spatial dimension (six in three dimensions). Jamming is in this sense rigidity percolation: not merely that contacts span the frame, but that the spanning network can bear load.

This mechanical-percolation reading of critical porosity is not new here. Sayers (2025) frames critical porosity explicitly as a jamming or percolation threshold, building on the sphere-pack results of Makse et al. (2000). The refinement proposed here — that bulk and shear rigidity percolate at slightly different thresholds, ϕ_s ≤ ϕ_c — sits within that established picture rather than departing from it.

3.4. A Single Structural Critical Porosity, with Shear Displaced Below It

The proposal is not to replace one critical porosity with two independent, freely floating ones. It is a single structural critical porosity with the shear threshold displaced slightly below it, a small and constant separation that holds at every porosity, not one that switches on as the transition is approached. The two critical porosities do not coincide away from the transition and diverge only on its approach; they are separated by the same amount throughout the load-bearing range. What becomes pronounced near the transition is the consequence of that fixed separation rather than the separation itself: the shear modulus reaches zero first, so the dry-frame Poisson’s ratio rises across the whole range and turns sharply toward 0.5 near the transition, and the bulk and shear modulus lines, falling at different constant rates, cross within the measured data (Figure 2) — a crossing the equal-porosity case forbids. The hyperbolic form is essentially preserved (§2.2); its details are not. The separation lifts the anchor above the suspension velocity and pulls the slope below its mineral value, so a single critical porosity fit stays serviceable only because the separation is small, carrying a systematically low anchor whose error is largest at the vertex — where the data turn down below the hyperbola at low velocity and, across the crush point, onto a distinct lower-velocity segment that may have to be analyzed on its own (§2.7, Appendix E).

The reading also identifies which intercept of a modulus–porosity line is trustworthy. The modulus-equals-zero intercept — the critical porosity — sits next to the high-porosity data, where the mechanism lives, and is the physical quantity. The zero-porosity intercept — the mineral modulus — is the unreliable one: extrapolating from a partial-backbone regime to ϕ=0 projects the stiffness of a fraction of the solid rather than the single-crystal value. Mineral moduli are accordingly best taken from mineral physics, with the ϕ=0 intercept used only as a consistency check (Nur et al., 1998). In clay-rich rocks the two-segment, crush-point structure of the modulus–porosity trend (Higginbotham, 2025; the crush porosity likely the consolidation-threshold porosity of Vernik and Kachanov, 2010a, 2010b) makes this explicit: below the crush porosity the backbone is complete and the segment extrapolates honestly toward the mineral modulus, while above it a growing fraction of grains sits out of the backbone, so the change in slope across the crush point measures the non-load-bearing fraction.

Returning briefly to Figure 1 of section 2.7, notice in this figure that the low velocity points fall below the line and the high velocity points are above the line. This points to the issue discussed above. Those low velocity points in the figure are above the crush point porosity and have characteristics that differ from the higher velocity consolidated rocks below the crush point. The true hyperbola for those higher velocity points has anchor velocity above the value shown in Figure 1 which fits the high velocity points more accurately. This crush point issue is discussed further in Appendix E.

3.5. The Admissible Regime is the Regime with a Real Anchor

These results combine cleanly with Section 2. By the affine-denominator theorem the hyperbola (2.4) exists for any constant pair over 0 ≤ ϕ < min (ϕ_c, ϕ_s); the ordering of the two critical porosities does not bear on its existence (§2.4) but on its anchor — V_p at the porosity ϕ = ϕ_s where shear is lost. The dry bulk modulus carried there is K_d(ϕ_s) = (1 − ϕ_s/ϕ_c)K_m. When ϕ_s=ϕ_c it vanishes and the anchor is the suspension velocity (2.10); when ϕ_s<ϕ_c the bulk frame still carries stiffness where shear is lost, so the anchor lies above the suspension velocity by a margin that grows with the separation — the closed-form anchor (2.8), $K_d$ surviving beyond ${\mu }_d$. When ϕ_s>ϕ_c that bulk modulus is negative: the vertex would require shear rigidity to outlast bulk rigidity as the frame opens — a frame that has lost every load-bearing contact still resisting shear — for which there is no mechanism. The physically admissible ordering ϕ_s ≤ ϕ_c is therefore exactly the regime in which the vertex is reached with the bulk frame intact, its anchor a real velocity at or above the suspension value.

The generalization to two critical porosities thus costs nothing in the central result and buys the porosity-dependent dry-frame Poisson’s ratio that real rocks display. It also turns the anchor into a measurement: the amount by which a fitted anchor exceeds the grain-in-fluid suspension velocity is an index of how far the shear critical porosity sits below the bulk one. Well-cemented frames, anchoring at the suspension velocity, report ϕ_s ≈ ϕ_c; looser or clay-rich frames, anchoring above it, report ϕ_s<ϕ_c. Section 4 turns to dry-frame measurements, which estimate ϕ_c and ϕ_s directly and so test the inequality without the fluid ambiguity that attends the anchor of a mixed-fluid compilation.

4. Dry-Frame Evidence: The Han (1986) Sandstones

Section 2 and Section 3 leave a specific empirical question. The anchor of a mixed-fluid compilation cannot separate the two critical porosities (§2.5), and the physical argument for ϕ_s<ϕ_c (§3) is motivation, not measurement. The quantity that settles it is the dry frame, where μ_d and K_d are obtained directly and there is no fluid ambiguity. The most suitable existing dataset is Han’s (1986), which tabulates dry and water-saturated ultrasonic velocities, porosity, and clay content for 69 sandstones at 40 MPa confining pressure, including ten clay-free samples (Fontainebleau, St. Peter, and Beaver) spanning porosities 0.055–0.224.

4.1. Dry-Frame Moduli

The dry-frame moduli follow from the tabulated dry velocities and porosity (Appendix F),

${\mu }_d={\rho }_d{V}_{sd}^2,K_d={\rho }_d\left(V_{pd}^2-{\displaystyle \frac{4}{3}}{V}_{sd}^2\right),{\rho }_d=\left(1-\varphi \right){\rho }_g,\left(4.1\right)$with grain density ρ_g=2.65 g/cm³. For the ten clay-free samples, straight-line fits in the critical-porosity form (3.1) give

K_d=(35.3 ± 1.3) − (87.7 ± 8.2)ϕ GPa, μ_d=(40.4 ± 0.9) − (115.5 ± 5.9)ϕ GPa, (4.2)

The bulk-modulus intercept recovers the quartz value within a few percent (35.3 vs 36.6 GPa).

These are ultrasonic measurements, and Gassmann’s relations are derived for the low-frequency limit, in which pore pressure equilibrates across the connected pore space within a wave period. At ultrasonic frequency that condition need not hold, and the pore-fluid bulk modulus then ceases to behave as a well-defined fluid property (Higginbotham, 2025). The present analysis does not invoke it. The critical porosities are read as the zero-modulus intercepts of the dry and saturated modulus–porosity lines — geometric properties of the frame — rather than computed from a fluid modulus, which is the fluid-independent form of the model (Higginbotham, 2025). The determination of ϕ_c and ϕ_s from the dry frame involves no fluid at all and is on that account frequency-robust; the saturated ϕ_c^′ is simply the value the data give at the measurement frequency. Reading the critical porosities directly from slopes is what admits ultrasonic data into the test.

4.2. The Shear Critical Porosity Lies Below the Bulk One

The zero-modulus intercepts of (4.2) give the critical porosities

ϕ_c=0.402 ± 0.025, ϕ_s=0.350 ± 0.011,

the bulk value agreeing with the canonical sandstone figure near 0.40 (Nur et al., 1998). Their separation,

ϕ_c − ϕ_s=+0.052 ± 0.027,

has the sign predicted in §3: shear rigidity extrapolates to zero at lower porosity than bulk rigidity. The separation is positive across every subset tested and is a two-to-three sigma determination (a resampling (bootstrap) estimate — refitting on samples drawn with replacement — gives +0.055 ± 0.019, with the separation positive in essentially every resample). It is also not an artifact of the lowest-porosity frames, whose long reach to the zero-modulus intercepts gives them the most leverage: under leave-one-out no single sample shifts the separation by more than 0.0163, and removing the lowest-porosity frames in turn leaves it between 0.05 and 0.07. Because the zero-modulus intercepts lie beyond the highest measured porosity (0.224), these critical porosities are extrapolations; the stronger statement, below, requires none.

4.3. The Dry-Frame Poisson’s Ratio is Not Constant

By the equivalence of §3.1, ϕ_s=ϕ_c holds if and only if the dry-frame Poisson’s ratio is independent of porosity. That can be tested inside the measured range, with no extrapolation, through the ratio K_d/μ_d — a monotonic proxy for ν_d. For the clean samples it rises with porosity,

${\displaystyle \frac{K_d}{{\mu }_d}}=\left(0.833\pm 0.036\right)+\left(0.95\pm 0.23\right)\varphi ,\left(4.3\right)$a slope 4.1σ from zero, with intercept 0.833 against the quartz value K_m/μ_m=0.813. The corresponding dry-frame Poisson’s ratio climbs from about 0.09 at low porosity to about 0.14 at ϕ ≈ 0.22. A porosity-independent dry Poisson’s ratio — the assumption underlying Krief et al. (1990) and the single critical porosity reading of the Nur model — is rejected by this clean dry-quartz suite, and with it the condition ϕ_s=ϕ_c; the strength and even the sign of this trend, however, are not reproduced in an independent clean suite (§4.5).

See Appendix G for details associated with these fit computations and their associated errors.

4.4. Clay Versus Porosity, and the Bounds of the Test

Extending to all 69 samples with a linear clay term sharpens the result and adds a discriminant. Clay reduces K_d and μ_d by nearly equal amounts (about −27 and −29 GPa per unit clay fraction), so its coefficient on the ratio K_d/μ_d is +0.09 ± 0.11 — consistent with zero — while the porosity coefficient on the ratio is +1.06 ± 0.23 (about 4.7σ). Clay softens the frame in a Poisson-preserving way; porosity softens shear preferentially. This is precisely what the load-bearing argument of §3 requires: pore-filling clay degrades the backbone wholesale, whereas added porosity dismantles the shear-supporting network before the bulk-supporting one. It also matches Han’s own characterization — his poorly cemented samples (notably St. Peter) populate the high-K_d/μ_d cluster, the well-cemented Fontainebleau and Beaver samples sit near the mineral ratio.

Two consistency checks support the analysis. The saturated bulk-modulus line for the clean samples crosses zero at ϕ_c^′=0.460 ± 0.020, a measured fluid shift ϕ_c^′ − ϕ_c=+0.058. This is the ultrasonic value, read directly from the slope. The low-frequency Gassmann prediction K_f/(K_m − K_f), evaluated for deionized water at the measurement conditions (K_f ≈ 2.2 GPa; Batzle and Wang, 1992), is 0.064 with the mineral-physics quartz modulus K_m=36.6 GPa, and 0.066 with the suite’s own dry-frame intercept K_m=35.3 GPa (the clean-ten fit, 4.2); the measured shift falls modestly short of both, the shortfall being the high-frequency departure expected for these permeable clean sandstones — and absorbed, in the fluid-independent form (§4.1), by taking ϕ_c^′ as measured rather than as ϕ_c+K_f/(K_m − K_f). And the mean ratio of saturated to dry shear modulus is 1.04, the small ultrasonic departure from μ_w=μ_d that Han documents.

Three limitations bound the claim. The critical porosities are extrapolations beyond the measured range, so their separation, though consistently positive, is a 2–3σ result; the porosity dependence of the dry-frame Poisson’s ratio, by contrast, is established within the data at better than 4σ and is a porosity effect, not a clay effect. One sample (DETANBUF) shows a dry V_p marginally above its saturated value and is flagged as a probable transcription or measurement artifact; excluding it does not change the fits. All measurements are ultrasonic and at a single effective stress. Within those bounds, the dry frame contradicts the constant-Poisson idealization and places the shear critical porosity below the bulk one in this suite — the separation that the anchor of §2.5 registers but cannot, by itself, resolve, and that an independent clean-quartz suite does not reproduce (§4.5).

4.5. Independent Data and the Difficulty of Confirmation

Testing the ordering against independent data is harder than the within-suite result suggests, and the data presently available do not confirm it. An independent set of dry, clean Fontainebleau frames at the same effective stress (Gomez et al., 2010) does not reproduce the sign: their dry K_d/μ_d trends downward with porosity, at a slope near −1.5, the reverse of Han and — at face value — placing ϕ_s above ϕ_c, the ordering §3.5 identifies as the one without a mechanism. The reversal is not, by itself, decisive: it is only about 1–2σ from zero, it is carried by the low-porosity, microcrack-controlled plugs (where K_d/μ_d scatters from 0.79 to 1.16 and the shear-modulus intercept falls below the quartz value), and with data only to ϕ ≤ 0.25 the critical-porosity extrapolation is essentially unconstrained (a resampled separation of −0.06 ± 1.3). The suite thus neither confirms the ordering nor cleanly rejects constancy; it simply fails to reproduce Han. The same fragility is visible inside Han: the rise is steepest in the poorly cemented St. Peter samples, with the Fontainebleau samples alone rising about half as fast (+0.7 against the clean-suite +0.95).

The datasets that might instead reach toward critical porosity fall short for their own reasons. A mixed-fluid V_p–V_s compilation (Castagna et al., 1993) confirms the hyperbola across four lithologies but cannot resolve the vertex, because brine and fitting-method anchor swings (≈ 250–340 m/s) exceed the ≈ 100 m/s softening signature. Ultrasonic measurements on clean unconsolidated sands (Zimmer, 2003) show dry frames nearly porosity-independent at fixed pressure — the contact-stiffness regime — corroborating the reduced-shear (frictionless-grain) mechanism of §3.3 but supplying no porosity ordering.

The result that survives is therefore the one carried by Han: within that suite the dry-frame Poisson’s ratio departs from constancy at better than 4σ, so the equal-porosity idealization ϕ_s=ϕ_c is a poor description there. But that departure is not reproduced in an independent clean suite, and the two disagree in direction, so the ordering ϕ_s ≤ ϕ_c rests on the Han frames and the physical argument of §3, not on independent confirmation. The clean, well-characterized dry-frame data near critical porosity that would settle it are scarce, and that scarcity is likely structural rather than accidental: as the frame approaches the transition the shear modulus and its load-bearing backbone go marginal together (§3.3), so the shear arrival — slow, weak, and strongly attenuated in friable frames — is the first measurement to degrade, exactly where the test most needs it (Hamilton, 1976; Zimmer, 2003). This is a stated expectation, not a licence: it raises the bar for what counts as a clean test rather than excusing the ordering from needing one, and it does not touch the Gomez reversal, which arose at the low-porosity, crack-controlled end. Until such data exist, the ordering is best carried as a physically grounded conjecture with one supporting dataset rather than an established fact.

5. Consequences in the High-Porosity Regime

The result of Section 2, Section 3 and Section 4 is modest in size — a shear critical porosity a few hundredths below the bulk one — a constant separation rather than one that grows toward the transition. Its consequences, though present at every porosity, become pronounced where porosity approaches critical: in the shallowest sediments and in undercompacted, overpressured intervals. There, and only there, the two-porosity model departs significantly from its single critical porosity special case, and the departures are specific enough to look for. The applications below lean on the published Gassmann–Nur–Aplin framework (Higginbotham, 2020) and treat the decoupling as a perturbation of it.

5.1. A Near-Seafloor Interval of Bulk Stiffness Without Shear

A sediment is deposited near its critical porosity — effectively a grain-in-fluid suspension at the seafloor — and compacts as porosity falls monotonically with depth (Higginbotham, 2020). In the two-porosity picture the two rigidities switch on at different depths. Bulk-frame stiffness appears once porosity drops below ϕ_c; shear rigidity appears only once it drops below the lower ϕ_s. Because ϕ_s<ϕ_c, there is a shallow depth interval,

ϕ_s < ϕ < ϕ_c,  (5.1)

in which the frame carries compressional stiffness but essentially no shear — the rock sits at the vertex of the hyperbola, near the anchor velocity (2.8), with V_s → 0 while V_p stays finite and above the suspension value. This is a falsifiable prediction with a familiar counterpart: the very low shear velocities and anomalously high V_p/V_s of shallow, soft marine sediments. The single critical porosity model places the onset of both rigidities at the same depth and so admits no such interval; the two-porosity model predicts a band of finite thickness, and the published compaction forward model (Higginbotham, 2020, Appendix II) converts a compaction coefficient (Aplin, 1995) into its depth and extent. A measured near-seafloor zone of finite V_p with vanishing V_s, ending at the depth where porosity reaches ϕ_s, would be direct evidence for the separation.

5.2. Pore-Pressure Prediction in Undercompacted Intervals

Velocity-based pore-pressure prediction (Higginbotham, 2020) is the business of overpressured zones, and overpressure is undercompaction: porosity is retained at values higher than the normal-compaction trend, that is, nearer critical. This is exactly the regime in which the shear decoupling is largest (Section 3.4). The pore-pressure formula is built on the density–velocity constants C_ρ, s, and α², which the single critical porosity reading evaluates at ϕ_s=ϕ_c. Those constants are robust to modest decoupling — they shift by a few percent across the plausible separation (Section 2.4) — so the practical effect is small. But it is not zero, and it is systematic: a calibration set in a normally pressured (lower-porosity) interval and applied in an overpressured (higher-porosity) one carries a porosity-dependent bias, because the separation between critical porosities is constant, but its effect grows as porosity nears critical, where the dry-frame Poisson’s ratio is farthest from its mineral value. The contribution of the two-porosity model here is not a new predictor but a statement of where the existing one is least reliable — the high-porosity overpressured intervals it most needs to get right — and of the direction of the error, which a single critical porosity calibration cannot represent.

5.3. Composite Rocks and the Critical-Porosity Mixer

Real sediments are mineral mixtures, and the published framework mixes the component critical porosities harmonically: for quartz fraction f_q and clay fraction f_s, with subscript “r” indicating the composite rock,

${\displaystyle \frac{1}{{\varphi }_{cr}}}={\displaystyle \frac{f_q}{{\varphi }_{cq}}}+{\displaystyle \frac{f_s}{{\varphi }_{cs}}}\left(5.2\right)$(Higginbotham, 2020, eq. B7), the reciprocal critical porosity — the modulus-softening rate — adding linearly by volume. In the two-porosity framework this rule applies separately to the bulk and shear critical porosities, so a composite carries both a mixed ϕ_cr and a mixed ϕ_sr, each by its own (5.2). The structure of Section 2, Section 3 and Section 4 passes through composite mixing unchanged.

This supplies the mechanism for the clay discriminant of Section 4.4. If a clay shares quartz’s ratio of shear to bulk critical porosity, ϕ_sx=kϕ_cx for both components, then the composite preserves it,

${\varphi }_{sr}=k{\varphi }_{cr},\left(5.3\right)$at any clay fraction — so adding clay leaves the dry-frame Poisson’s ratio fixed while softening both moduli. That is precisely the behavior measured in the Han suite, where clay’s coefficient on K_d/μ_d was consistent with zero even as porosity softened shear preferentially. Fitting the mixer (5.2) to all 69 frames (Appendix D) makes this quantitative: clay enters with an effective critical porosity near 0.12, essentially the same for the bulk and shear branches as Poisson-preservation requires, and accounts for roughly a third of the dry-modulus scatter about the clean-quartz frame. It is the deviation Higginbotham (2025, Figure 3) draws explicitly — the blue “quartz with some clay content” curve falling below the clean-quartz curve and tracking the lower edge of the measured velocities. The mixer separates the two effects cleanly: clay mixing preserves the critical-porosity ratio, while porosity growth within a component softens shear preferentially — the signature of shear critical porosity less than bulk critical porosity. The same decomposition cautions against reading a composite’s critical porosities as a single mineral’s — the harmonic blend (5.2), not the component values, sets where the composite frame loses each rigidity.

6. An independent Confrontation: The Dry-Rock D Residual

The Han suite of Section 4 is one confrontation with data; the published corpus already records a second, in different rocks and by a different route, that the two-porosity model resolves. The modified Gassmann–Nur model eliminates the pore-fluid bulk modulus in favor of a single measured quantity d, the gap between two critical porosities read as the difference of two modulus–porosity slopes (Higginbotham, 2025): the saturated bulk modulus supplies one, the rock shear modulus the other. In the notation of this paper the shear-slope value is ϕ_s and the bulk-slope value is the saturated bulk critical porosity ϕ_c^′ of (2.1), so

d=ϕ_c′ − ϕ_s. (6.1)

(The published work writes the shear-slope value as ϕ_c; under the equal-porosity assumption that one porosity serves as both the bulk and the shear critical value, and the present paper’s contribution is precisely to separate them — so the published ϕ_c, being read from the shear modulus, is this paper’s ϕ_s.) For a permeable rock at low frequency d takes the Gassmann value K_f/(K_m − K_f); but the published work records that this prediction fails for dry rocks, where it gives zero yet the measured d does not vanish, so that d must instead be read from a graph (Higginbotham, 2025).

The two-porosity model identifies that residual. Substituting the saturated-bulk critical porosity ϕ_c^′ = ϕ_c + K_f/(K_m − K_f) of (2.1) into (6.1),

$d={\displaystyle \frac{K_f}{K_m-K_f}}+\left({\varphi }_c-{\varphi }_s\right),\left(6.2\right)$so d carries two terms: the fluid shift, which the single critical porosity reading already accounts for, and the bulk–shear separation ϕ_c − ϕ_s, which it does not. In the dry limit K_f → 0 the fluid term vanishes and what remains is d = ϕ_c − ϕ_s: the dry-rock residual that the Gassmann prediction misses is the shear–bulk decoupling of this paper. The single critical porosity model, setting ϕ_s=ϕ_c, has no term for it and so is forced to carry it as an unexplained correction.

The Han clean-quartz frames put a number on the decomposition. Their separation ϕ_c − ϕ_s = 0.052 (§4.2) and their measured saturated fluid shift 0.058 (§4.4) are roughly equal halves of a total d ≈ 0.110. Read through the modified-model relation K_f,eff = K_md/(1 + d) as if it were all fluid, that d implies an effective fluid modulus near 3.6 GPa — well above water’s 2.2–2.5 GPa — the inflation being the decoupling charged to the fluid. The residual appears, independently, in dry shale: the published modulus–porosity extraction for the dry smectite and kaolinite of Mondol et al. (2008) returns a bulk critical porosity above the shear one by about 0.03–0.04 with no fluid present (Higginbotham, 2025, supplementary material) — the same ordering and a comparable magnitude to the Han sandstones, in a mineral system entirely different from quartz.

This is a second line of evidence for ϕ_s ≤ ϕ_c, and its independence is the point. The Han result (Section 4) reads the two critical porosities directly from clean quartz sandstone; the d residual reads them from the failure of a fluid prediction in dry shale. Two minerals, two methods, the same separation and the same sign. The shale determination is the weaker of the two — the bulk-versus-shear ordering in the Mondol frames shifts with the extrapolation method, the clay shear modulus is chemistry-dominated, and the crush-point porosity at which the modulus–porosity trend bends is hard to locate (Higginbotham, 2025) — so it corroborates rather than proves. But it corroborates in rocks that share none of Han’s mineralogy, and it does so by accounting for a discrepancy the single critical porosity model had already been obliged to record. The decoupling is not a device introduced to fit the Han suite; it was latent in the published model as the dry-rock residual of d, waiting for a name.

7. Conclusion

The Gassmann–Nur critical-porosity model does not require a single critical porosity. A theorem (Section 2): because the dry moduli and the density are each affine in porosity and V_p² and V_s² share the affine denominator ρ, the (V_s², V_p²) locus is an exact straight line — equivalently, the V_p–V_s hyperbola — for any constant pair ϕ_c, ϕ_s. The equal-porosity condition ϕ_s=ϕ_c, equivalent to a porosity-independent dry-frame Poisson’s ratio, does not generate the curve; it only places the anchor at the grain-in-fluid suspension velocity.

A conjecture (Section 3): the generic frame has its shear critical porosity at or below its bulk one, ϕ_s ≤ ϕ_c. The shear-supporting backbone fails before the bulk-supporting one as a frame opens — a single statement read three ways, through the load-bearing threshold, through Hertz–Mindlin contact mechanics with its empirical reduced-shear correction, and through the loss of rigidity at jamming — and it is exactly the ordering under which the hyperbola of Section 2 remains exact. This is offered as physical motivation, not proof.

One dataset (Section 4): the dry sandstone frames of Han (1986) place the shear critical porosity below the bulk one, ϕ_c = 0.402 against ϕ_s = 0.350 (extrapolated intercepts), and show — within the measured range, with no extrapolation — the dry-frame Poisson’s ratio rising with porosity at better than four sigma. This rejects the equal-porosity idealization within the Han suite, but it is not independently confirmed: a second clean-quartz suite does not reproduce the sign, and the clean data near critical porosity that would settle the ordering remain scarce (§4.5). The ordering ϕ_s ≤ ϕ_c is thus carried by Han and the physical argument of §3, a grounded conjecture rather than an established fact. Clay softens the frame in a Poisson-preserving way; porosity softens shear preferentially, as the load-bearing picture requires.

A second, independent confrontation (Section 6): the dry-rock residual of the published fluid parameter d — long carried as an unexplained correction, since d ≈ K_f/(K_m − K_f) predicts zero for dry rocks yet the measured value does not vanish — is exactly the bulk–shear separation ϕ_c − ϕ_s. It surfaces in dry shale (Mondol et al., 2008), a mineral system sharing none of Han’s quartz, at the same sign and a comparable magnitude. Method-sensitive, and so corroborating rather than proving, it is nonetheless independent of the Han frames and resolves a discrepancy the single critical porosity model had already been obliged to record.

Three consequences (Section 5): a near-seafloor interval of bulk stiffness without shear, matching the very low shear velocities of shallow marine sediments; a systematic, porosity-dependent bias in single critical porosity pore-pressure prediction in exactly the overpressured intervals it most needs to get right; and a harmonic critical-porosity mixer that, applied separately to bulk and shear, reproduces the measured clay discriminant.

The generalization costs nothing in the central result — the hyperbola shape survives — and buys the porosity-dependent dry-frame Poisson’s ratio that real rock displays. It rests throughout on the published single critical porosity corpus (Higginbotham et al., 2012; Higginbotham, 2020, 2025; Mavko and Mukerji, 1998; Nur et al., 1998), generalizing it at one point: the single critical porosity becomes two, separated by a small constant amount at every porosity, its effects pronounced only near the transition, with shear the first rigidity to go.