Structural Identifiability and Comparative Calibration of Water Retention Curves for Imbibition in Porous Media

TL;DR

This study compares four water retention curve models for porous media imbibition, revealing that only their combined effect can be identified from data and providing a unified, open-source calibration framework.

Contribution

It introduces a unified mathematical reformulation of different WRC models and demonstrates their calibration against the same dataset within an open-source platform.

Findings

Hydraulic conductivity and capillary pressure scale cannot be independently identified.

The effective diffusion function shape governs model discrimination.

First cross-calibration of diverse WRCs on identical data within a unified framework.

Abstract

This paper investigates the structural identifiability and a comparative calibration of four water retention curves (WRCs) within the framework of the Richards equation coupled with Darcy's law for capillary imbibition in porous media. The considered models -- two classical physically-based laws and two abstract parametrisations developed for building stones -- are consistently reformulated by expressing the hydraulic conductivity $K(\Theta)$ and capillary pressure $\psi(\Theta)$ independently, allowing the nonlinear diffusion coefficient $D(\Theta)$ to be reconstructed in a unified structural form. This common representation enables a rigorous mathematical comparison across models with different theoretical foundations. All models are calibrated against the same experimental imbibition dataset using a grid-based optimisation strategy with adaptive refinement. The analysis reveals a structural property of the associated inverse problem: the hydraulic conductivity and the capillary pressure scale enter the governing equation multiplicatively and therefore cannot be independently identified from imbibition data. Only their product acts as an observable diffusion parameter, where model discrimination is primarily governed by the shape of the resulting effective diffusion function. To the best of our knowledge, this is the first study providing a coherent cross-calibration of these WRCs against an identical dataset within a unified computational framework. Our open-source implementation, released within the Stoneverse platform, provides a reproducible baseline for further developments, including probabilistic inversion and learning-based approaches.