Non-linear Compaction Function

• Non-linear compaction function is a constitutive relationship that connects stress, pressure, and porosity, capturing transitions from elastic to viscous, plastic, or fractal-dominated regimes.
• It is defined through non-linear equations—such as Athy’s law and Maxwell-type viscoelastic laws—and is essential in modeling sedimentary basins, granular materials, and cosmological phenomena.
• Advanced numerical methods, including implicit finite difference schemes, adaptive finite elements, and fractal interpolation techniques, facilitate accurate simulation of these complex compaction processes.

A non-linear compaction function is a constitutive relationship that connects state variables (such as stress, pressure, porosity, or fractional mass excess) in a non-linear manner, governing the evolution of compaction and densification in highly heterogeneous, multiphysics systems. Non-linear compaction functions appear prominently in the modeling of sedimentary basin evolution, porous media mechanics, granular materials, high-pressure geomaterials, fractal structures, and cosmological perturbations associated with primordial black hole formation. They underpin the transition from reversible elastic responses to irreversible plastic, viscous, or even fractal-dominated deformation regimes and are essential for physically accurate, predictive models of compaction, permeability evolution, and localization behaviors.

1. Fundamental Theory and Model Formulations

Non-linear compaction functions arise naturally when constitutive relationships among key state variables must encode multiphysics behaviors—such as the transition between poroelastic and viscous (pressure-solution) compaction, or the non-linear densification of unconsolidated granular aggregates. A generalized viscoelastic (Maxwell-type) law for sediment compaction, for example, is written as:

$$u_z^s = -\frac{1}{K(\phi)}\, \frac{D p_e}{D t} - \frac{1}{\xi(\phi)}\, p_e$$

where $u^s_z$ is the vertical gradient of solid velocity, $p_e$ is the effective pressure, $K(\phi)$ is a poroelastic modulus (commonly, $K(\phi) = C_s (1 - \phi)^2/\phi$), and $\xi(\phi)$ is a bulk viscosity, typically $\xi(\phi) = \xi_0 (\phi_0/\phi)^n$ for $n \gtrsim 2$ in cemented rocks (Yang, 2010). This compaction law blends time-dependent viscous deformation and instantaneous elastic responses.

In geological applications, the pressure–porosity relationship (Athy's law) is often employed as a non-linear compaction function:

$$p_e(\phi) = \ln \frac{\phi_0}{\phi} - (\phi_0 - \phi)$$

suited for shallow, mechanically dominated compaction; for deeper pressure-solution-dominated regimes, a viscous law is used:

$p_e = -\xi \left(\nabla \cdot {\bf u}^s\right)$

where $\xi$ is the effective viscosity (Yang, 2010, Yang, 2010).

In cosmological settings, the compaction function is formulated as a quadratic, non-linear map between the spatially-averaged density fluctuation and the fractional mass excess, relevant for gravitational collapse criteria:

$$\mathcal{C}(R, x) = g_R(x) \left[1 - \frac{3}{8} g_R(x)\right]$$

with $g_R(x)$ a dimensionless density contrast normalized to the smoothing scale $R$ (Germani et al., 2019, Harada et al., 2023).

2. Non-linear Compaction in Physical Media: Applications and Regimes

Sedimentary Basins and Porous Media

Nonlinear compaction functions capture the transition from near-surface poroelasticity to depth-dependent, viscous, pressure–solution-dominated compaction. In hydrocarbon basins, this transition is modeled using nonlinear $K(\phi)$ and $\xi(\phi)$, and strongly nonlinear permeability $k(\phi) = (\phi/\phi_0)^m$ with $m \sim 8$ for shales. Numerical solutions reveal:

• Near the surface: porosity decays nearly exponentially with depth ($\phi(z) \approx \phi_0\exp[-C_s(h_0-z)]$).
• At depth: viscous effects lead to nearly uniform, low-porosity and overpressuring zones.
• Nonlinear compaction is also coupled to mineral reactions (e.g., smectite–illite transition), whose temperature-dependent kinetics generate sharp reaction windows (Yang, 2010, Yang, 2010).

Granular Aggregates and Unconsolidated Sands

Laboratory triaxial tests on unconsolidated quartz aggregates reveal highly nonlinear stress–strain (compaction) relations and the absence of traditional, sharply defined yield points. The response is characterized by:

$\sigma(\epsilon) = E_e\epsilon + f(\phi(\epsilon))$

where as porosity decreases through progressive grain crushing/rotation, the differential stress envelope ($Q$ vs $P$) hardens continuously (Hangx et al., 2019). Empirical laws for the evolution of compaction with porosity typically take the form:

$Q = Q_0 + A \left[1-\frac{\phi}{\phi_0}\right]^k$

encoding the distributed, non-localized nature of compaction and hardening.

Granular Mixtures: Rigid + Deformable Particles

Simulations using non-smooth contact dynamics and hyper-elastic particle modeling show that the packing fraction $\phi$ increases nonlinearly with stress and approaches an asymptotic maximum $\phi_{max}$, governed by mixture composition and particle friction. The coordination number $Z$ evolves via a power law with porosity:

$$Z - Z_0 = \xi (\phi - \phi_0)^\alpha,\quad \alpha \approx 0.5$$

and the compaction-pressure relationship is

$$P(\phi, \kappa)/E \simeq -\frac{b\phi}{2\pi}[Z_0 + \xi(\phi - \phi_0)^\alpha]\ln\left(\frac{\phi_{max}(\kappa) - \phi}{\phi_{max}(\kappa) - \phi_0}\right)$$

where $E$ is particle Young modulus, $b$ empirical, and $\kappa$ mixture ratio (Cárdenas-Barrantes et al., 2020).

3. Non-linear Statistical and Geometric Compaction in Cosmology

The compaction function is fundamental in primordial black hole (PBH) formation modeling. Defined as the ratio of local mass excess to areal radius,

$$\mathcal{C}(r) = \frac{2 (M(r, t) - M_b(r, t))}{R(r, t)}$$

the compaction function's nonlinear nature (usually quadratic in the linear density contrast) sets the cloud collapse threshold beyond which a region forms a PBH. In practice, the compaction function is evaluated at its maximum, using a prescription $C_{th}$ obtained from numerical simulations (Germani et al., 2019, Yang et al., 29 Aug 2024, Harada et al., 9 Sep 2024).

Notably, the Shibata–Sasaki compaction function has a precise geometric interpretation as the compactness of an extremal surface (in a static spacetime obtained by factoring out cosmic expansion):

$$\mathcal{C}_{SS}(r) = \frac{1}{2}\left[1 - \Psi^{-4}(r)\left(\frac{d}{dr}[\Psi^2(r)\,r]\right)^2\right]$$

which attains its theoretical maximum $1/2$ at bifurcating trapping horizons and photon spheres (Harada et al., 9 Sep 2024). This geometric origin underpins its robustness as a black hole formation criterion.

4. Non-linear Compaction in Fractal and Disordered Systems

Nonlinear compaction functions generalize linear contraction maps to nonlinear (Edelstein) contractions in the context of fractal interpolation functions (FIFs) on PCF (post-critically finite) self-similar sets (Shah et al., 15 May 2025). Functional iteration equations such as

$f^*(l_n(x)) = s_n(f^*(x)) + h_n(x)$

with $s_n(\cdot)$ nonlinear compaction maps, permit attractors with prescribed box dimensions and regularity properties essential for modeling rough, fractal landscapes.

The energy, $\mathcal{E}_{m}(u)$, and normal derivatives at boundary points are used to quantify regularity and local "flux" properties, and box dimensions can be estimated in terms of the contraction strengths $L(s_n)$ of the vertical compaction functions. This formalism is instrumental in signal processing, data compression, and natural texture modeling.

5. Numerical Methods and Solution Strategies

Nonlinear compaction equations are governed by strongly coupled, nonlinear PDEs. Solution strategies include:

• Implicit, two-stage finite difference schemes for basin-scale compaction (Yang, 2010, Yang, 2010)
• Adaptive stabilized finite elements for strongly nonlinear and spatially localized solutions with singular periodic (cnoidal wave) solutions (Cier et al., 2020)
• Asymptotic analysis and traveling-wave solutions for models with highly localized reaction–compaction coupling (Yang, 2010)

For non-linear compaction functions arising in statistical (cosmological) regimes, abundance predictions require joint Gaussian integrals constrained by the nonlinear mapping from curvature perturbations to compaction, and precise evaluation of threshold exceedance probabilities (Germani et al., 2019, Yang et al., 29 Aug 2024).

6. Physical Implications and Observational Consequences

Non-linear compaction functions predict a range of physically and industrially significant phenomena:

• The porosity, overpressure, and mineralogical stratification in sedimentary basins; overpressure zones and low-permeability seals in hydrocarbon exploration (Yang, 2010, Yang, 2010)
• Extended, irreversible, shear-enhanced compaction and elasto-plastic transition in mudstones, with direct impact on wellbore stability and hydraulic fracturing design (Hasbani et al., 2023)
• Nonlinear compaction and morphological transformation in crumpled wires, governed by self-avoiding random walks, with the maximum packing density showing anomalous scaling with filament thickness (Shaebani et al., 2015)
• Step-function/threshold behavior in soot aggregate restructuring during atmospheric and combustion processing, critically affecting aerosol dynamics and light absorption (Corbin et al., 2022)
• Threshold behavior for PBH formation, insensitive to long-wavelength cosmological gauge artifacts but precisely determined via the compaction function, driving accurate constraints from $\mu$-distortion on the CMB (Harada et al., 2023, Yang et al., 29 Aug 2024, Harada et al., 9 Sep 2024)

7. Extensions and Theoretical Developments

Recent research highlights several active directions:

• Extension of non-linear compaction to general quadrature/integration frameworks: Nonlinear quadrature rules, exact for special function classes (e.g., exponentials), provide higher accuracy for integrals of sums of exponentials and other well-structured functions, with potential applications in the Padé–Laplace method (Hippel, 2022).
• Fractal compaction functions utilizing Edelstein contractions suggest new classes of interpolants with tunable smoothness and fractal dimensions (Shah et al., 15 May 2025).
• Ongoing development of robust numerical solvers and regularization techniques for highly nonlinear, potentially singular compaction PDEs (Cier et al., 2020).

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In conclusion, non-linear compaction functions are central to accurately modeling, simulating, and interpreting the evolution of complex, compressive systems across geology, materials science, and cosmology. They encode the transition across physical regimes (elasticity, viscosity, plasticity, self-avoidance, reaction-driven instability, and fractality), enabling robust predictions of both local and global phenomena under nontrivial loading, chemical, and environmental histories. The breadth and richness of their applications continue to drive developments in both theory and computational methodology.