Bayesian Estimation of Adsorption and Desorption Parameters for Pore Scale Transport

Vasiliy V Grigoriev,Petr N Vabishchevich

doi:10.3390/math9161974

Abstract

Stochastic parameter estimation and inversion have become increasingly popular in recent years. Nowadays, it is computationally reasonable and regular to solve complex inverse problems within the Bayesian framework. Applications of Bayesian inferences for inverse problems require investigation of the posterior distribution, which usually has a complex landscape and is highly dimensional. In these cases, Markov chain Monte Carlo methods (MCMC) are often used. This paper discusses a Bayesian approach for identifying adsorption and desorption rates in combination with a pore-scale reactive flow. Markov chain Monte Carlo sampling is used to estimate adsorption and desorption rates. The reactive transport in porous media is governed by incompressible Stokes equations, coupled with convection–diffusion equation for species’ transport. Adsorption and desorption are accounted via Robin boundary conditions. The Henry isotherm is considered for describing the reaction terms. The measured concentration at the outlet boundary is provided as additional information for the identification procedure. Metropolis–Hastings and Adaptive Metropolis algorithms are implemented. Credible intervals have been plotted from sampled posterior distributions for both algorithms. The impact of the noise in the measurements and influence of several measurements for Bayesian identification procedure is studied. Sample analysis using the autocorrelation function and acceptance rate is performed to estimate mixing of the Markov chain. As result, we conclude that MCMC sampling algorithm within the Bayesian framework is good enough to determine an admissible set of parameters via credible intervals.

Highlights

Reactive transport in porous media is important for many industrial and environmental applications such as water treatment, soil contamination and remediation, catalytic filters, etc
The reactive transport is governed by incompressible Stokes equations in pore-scale geometry, coupled with a convection
Credible intervals have been plotted from sampled posterior distributions for both algorithms

Summary

Introduction

Reactive transport in porous media is important for many industrial and environmental applications such as water treatment, soil contamination and remediation, catalytic filters, etc. The weak point in the computational modeling of reactive transport is the lack of data on the rate of adsorption and desorption at the pore scale (or, more generally, the parameters of heterogeneous reactions). In the case of surface (heterogeneous) reactions at the pore scale, the species’ transport is coupled to surface reaction via Robin boundary conditions. When the reaction rates are not known, their identification belongs to the class of inverse boundary value problems [3,4,5,6]. Different algorithms can be applied for solving parameter identification problems, see, e.g., [7,8]. Any approach that gives only one best value for the parameter vector does not take into account the fact that our measurements contain measurement noise. Deterministic inversion methods do not account for the uncertainty in parameter estimates

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Conclusion