Microstructural Inference Framework for Saturated Soils Based on the Maximum Entropy Principle: Analytical Formulation, Numerical Implementation, and Validation

José Luis Abrego Salazar

The unit weight γ and the permeability k define an underdetermined inverse problem: infinitely many pore-size distributions p(s) are compatible with the same pair of observations. Empirical correlations resolve this ambiguity by imposing rigid structural forms —valid only for a limited range of materials— that do not hold across the broad spectrum of natural soils.
This work formulates an inference framework based on Jaynes’s Maximum Entropy Principle that selects, among all distributions compatible with γ and k, the one of maximum uncertainty, thereby avoiding unjustified assumptions. The exponential-canonical analytical solution is derived, existence and uniqueness are demonstrated through strict convexity, and a hybrid numerical engine verified by unit tests is implemented.
Validation over 2300 simulations spanning 23 reference soils reveals:
An invariant scaling law Var[s]=a⋅E[s]^b with b=2.0029±0.0227 (R^2=0.831), robust to structural variations.
A systematic deviation from Poiseuille’s law: an empirical slope of E[s] vs. k of 0.186, against the theoretical value of 0.5, quantifying the effective tortuosity of the real medium through a correction factor T≈50.
The framework offers a non-destructive, low-cost alternative to destructive techniques such as MIP —with which a direct comparison is conceptually infeasible, since they operate in different state spaces— and provides an uncertainty filter to guide the use of more sophisticated tests. Its limitations —structural unimodality and the indistinguishability of microstructures with identical γ, k— are unavoidable consequences of the moment problem and are stated explicitly.