Auto-weighted Bayesian Physics-Informed Neural Networks and robust estimations for multitask inverse problems in pore-scale imaging of dissolution

TL;DR

This work develops a robust data-assimilation framework that couples dynamical 4D micro-CT imaging with a dimensionless, physics-based dissolution model to address inverse problems in pore-scale reactive flows. By embedding the PDE regularization within Bayesian Physics-Informed Neural Networks and employing Adaptively Weighted Hamiltonian Monte Carlo, the authors achieve reliable uncertainty quantification for both micro-porosity evolution and kinetic parameters, notably the inverse Damköhler numbers and diffusion scaling. The method is validated on synthetic 1D+Time and 2D+Time calcite dissolution problems, producing meaningful posterior distributions for inverse parameters and credible porosity fields, with explicit handling of imaging uncertainties via the Reactive Area of Interest. This framework advances reliable pore-scale predictions under imaging limitations and paves the way for applying multitask BPINNs to real 3D dissolution problems and broader geochemical scenarios, potentially informing CO2 storage risk assessments.

Abstract

In this article, we present a novel data assimilation strategy in pore-scale imaging and demonstrate that this makes it possible to robustly address reactive inverse problems incorporating Uncertainty Quantification (UQ). Pore-scale modeling of reactive flow offers a valuable opportunity to investigate the evolution of macro-scale properties subject to dynamic processes. Yet, they suffer from imaging limitations arising from the associated X-ray microtomography (X-ray microCT) process, which induces discrepancies in the properties estimates. Assessment of the kinetic parameters also raises challenges, as reactive coefficients are critical parameters that can cover a wide range of values. We account for these two issues and ensure reliable calibration of pore-scale modeling, based on dynamical microCT images, by integrating uncertainty quantification in the workflow. The present method is based on a multitasking formulation of reactive inverse problems combining data-driven and physics-informed techniques in calcite dissolution. This allows quantifying morphological uncertainties on the porosity field and estimating reactive parameter ranges through prescribed PDE models with a latent concentration field and dynamical microCT. The data assimilation strategy relies on sequential reinforcement incorporating successively additional PDE constraints. We guarantee robust and unbiased uncertainty quantification by straightforward adaptive weighting of Bayesian Physics-Informed Neural Networks (BPINNs), ensuring reliable micro-porosity changes during geochemical transformations. We demonstrate successful Bayesian Inference in 1D+Time and 2D+Time calcite dissolution based on synthetic microCT images with meaningful posterior distribution on the reactive parameters and dimensionless numbers.

Figures (17)

• Figure 1: From the pore-scale to the reservoir scale: an upscaling principle. Schematic representation of a reservoir scale structure, on the left, with its inherent averaged isotropic macro-properties $\phi$ and $\kappa_0$ computed on a representative elementary volume (REV). Local description of the pore-scale heterogeneities in this REV, on the right, along with its intrinsic micro-scale properties. These are the local micro porosity field $\varepsilon$ (bounded by the least physically possible porosity $\varepsilon_0>0$) and the microscale permeability $K_\varepsilon$, based on the Kozeny-Carman relationship from equation \ref{['eq:KC']}.
• Figure 2: Sequential graph of the potential energy reinforcement: our data assimilation strategy incorporates additional physics-based constraints arising from the PDE model \ref{['eq:inverse_chem_adim_Vf']} through successive sampling steps. The notations are defined in Sect. \ref{['subsec:seq_reinforce']}.
• Figure 3: Initial $\mu$CT images defining the porous sample geometries: synthetic cases with tortuosity indices a) $\beta = 1$ and b) $\beta = 0.5$. The $\mu$CT measurements are normalized, corrupted with noise, and provide the dataset $\mathrm{Im}$ before the dissolution process. The calcite core regions are identified by the double-headed arrows, and correspond to the maximum intensity in the greyscale tomographic scans displayed below.
• Figure 4: Neural Network configuration choice for the surrogate model on the micro-porosity field: a) Computational cost of sampling step 1 with respect to the neural network architectures both in terms of the number of hidden layers and neurons per layer. b) Bayesian Model Average (BMA) error between the surrogate model $\varepsilon_\Theta$ and groundtruth $\varepsilon$, computed as in equation \ref{['BMA_eps']}, for different neural network architectures.
• Figure 5: Uncertainty Quantification on 1D+Time reactive inverse problem with data assimilation: Bayesian Model Average (BMA) predictions on the micro-porosity field $\varepsilon_\Theta$ with their local uncertainties --- given by the standard deviation on the posterior distribution of the predictions --- and mean squared errors (MSE). The top row corresponds to the initial geometry from Fig \ref{['fig:Initial_geo_1D']}a with tortuosity index $\beta = 1$. The bottom row is related to the initial porous sample from Fig \ref{['fig:Initial_geo_1D']}b with $\beta = 0.5$.