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Bayesian methods for the identification of model parameters for water transport in porous media

Paola Stolfi, Elia Onofri, Gabriella Bretti

Abstract

The structure of the nonlinear inverse problem arising from capillarity-driven imbibition in porous media is investigated, considering a degenerate parabolic PDE with compactly supported diffusivity and boundary-driven fluxes as the governing forward model. The inverse problem -- inferring hydraulic model parameters from sparse integral absorption measurements -- is inherently ill-posed: the nonlinear forward operator induces anisotropic parameter sensitivity and structured correlations that render the calibration landscape non-convex and partially unidentifiable. To characterise this structure rigorously, Approximate Bayesian Computation with Sequential Monte Carlo (ABC-SMC) is adopted as a likelihood-free inferential framework, bypassing the analytical intractability of the likelihood while providing full posterior distributions over the parameter space. Two physically motivated parameterisations of the diffusivity function are analysed -- the Natalini-Nitsch (NN) and the BkP formulations. It is shown that the posterior geometry obtained via ABC-SMC encodes, in directly readable form, the sensitivity structure of the nonlinear forward operator: the principal component decomposition of the posterior covariance provides a natural hierarchy of parameter sensitivity, with low-variance eigendirections identifying the parameter combinations to which the forward map is most responsive. This geometric decomposition constitutes a principled and computationally efficient alternative to classical sensitivity analysis, arising as a byproduct of the calibration procedure. These findings are established through both synthetic experiments, confirming accurate parameter recovery, and real laboratory imbibition data from materials of cultural heritage relevance.

Bayesian methods for the identification of model parameters for water transport in porous media

Abstract

The structure of the nonlinear inverse problem arising from capillarity-driven imbibition in porous media is investigated, considering a degenerate parabolic PDE with compactly supported diffusivity and boundary-driven fluxes as the governing forward model. The inverse problem -- inferring hydraulic model parameters from sparse integral absorption measurements -- is inherently ill-posed: the nonlinear forward operator induces anisotropic parameter sensitivity and structured correlations that render the calibration landscape non-convex and partially unidentifiable. To characterise this structure rigorously, Approximate Bayesian Computation with Sequential Monte Carlo (ABC-SMC) is adopted as a likelihood-free inferential framework, bypassing the analytical intractability of the likelihood while providing full posterior distributions over the parameter space. Two physically motivated parameterisations of the diffusivity function are analysed -- the Natalini-Nitsch (NN) and the BkP formulations. It is shown that the posterior geometry obtained via ABC-SMC encodes, in directly readable form, the sensitivity structure of the nonlinear forward operator: the principal component decomposition of the posterior covariance provides a natural hierarchy of parameter sensitivity, with low-variance eigendirections identifying the parameter combinations to which the forward map is most responsive. This geometric decomposition constitutes a principled and computationally efficient alternative to classical sensitivity analysis, arising as a byproduct of the calibration procedure. These findings are established through both synthetic experiments, confirming accurate parameter recovery, and real laboratory imbibition data from materials of cultural heritage relevance.

Paper Structure

This paper contains 15 sections, 36 equations, 18 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Materials and methods
  3. Approximate Bayesian Inference
  4. Models for water absorption
  5. The Natalini-Nitsch (NN) formulation of the absorption function $B'$
  6. The BkP formulation of absorption function $B'$
  7. The experimental setting
  8. The calibration procedure
  9. Academic application
  10. Real data application
  11. Calibration of model parameters for common Brick
  12. Calibration of model parameters for Ajarte
  13. Discussions
  14. Conclusions and future perspectives
  15. Additional Plots

Figures (18)

  • Figure 1: Schematics of boundary conditions \ref{['bc1']}-\ref{['bc2']} reproducing imbibition test made in laboratory on a specimen of porous material.
  • Figure 2: The panels show the performance of the algorithm on the academic application using NN and BkP models.
  • Figure 3: Posterior distributions of the parameters (on-diagonal) and corresponding correlations (off-diagonal) achieved on the academic application for the NN model.
  • Figure 4: Simulated data according NN model (left) and BkP model (right) are plotted against the reference target data. Both panels show high agreement with the reference, suggesting that ABC successfully captures the behaviour of rise phenomenon.
  • Figure 5: Calibrated absorption functions $B'$ for (left) common brick and (right) Ajarte, according to the data provided in Tables \ref{['tab:2']} and \ref{['tab:3']} respectively.
  • ...and 13 more figures