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

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 $K(Θ)$ and capillary pressure $ψ(Θ)$ independently, allowing the nonlinear diffusion coefficient $D(Θ)$ 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.

Figures (12)

• Figure 1: Roman pillars made of bricks in the archaelogical park of Ostia Antica.
• Figure 2: Experiment from bracciale used to to calibrate the models. \ref{['fig:benchmark-a']} Depiction of the experimental setup. \ref{['fig:benchmark-b']} Imbibition curves obtained from the common brick. The three series of reference data (grey crosses) are averaged to create a single reference curve (dashed line).
• Figure 3: Finite-difference stencil for the nonlinear diffusion equation. At time level $t^k$, nodal values $\Theta_j^k$ are stored at grid points, while interface diffusivities $D_{j\pm\frac{1}{2}}^k$ and numerical fluxes $F_{j\pm\frac{1}{2}}^k$ are evaluated at cell interfaces. The update of $\Theta_j^{k+1}$ results from the symmetric flux difference across the control volume surrounding node $j$.
• Figure 4: Optimization results for BC model (IT = 2).
• Figure 5: Optimization results for vG model (IT = 1). It is important to note that $\alpha$ is reported instead of $c = 1/\alpha$ and $n$ is reported in place of $\lambda = n-1$.

Theorems & Definitions (7)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Remark 6
• Remark 7