Bayesian Inversion for Anisotropic Hydraulic Phase-Field Fracture

TL;DR

This paper develops a Bayesian inversion framework for anisotropic hydraulic fracture modeling, incorporating pressure, displacement, and phase-field data, and employs advanced algorithms to estimate key parameters under uncertainty.

Contribution

It introduces a novel Bayesian inversion approach tailored for anisotropic hydraulic fractures, including a new crack driving state function and parameter estimation method.

Findings

Successfully estimates fracture parameters from observational data.

Demonstrates robustness of the Bayesian framework in numerical examples.

Extends inversion techniques to anisotropic hydraulic fracture models.

Abstract

In this work, a Bayesian inversion framework for hydraulic phase-field transversely isotropic and orthotropy anisotropic fracture is proposed. Therein, three primary fields are pressure, displacements, and phase-field while direction-dependent responses are enforced (via penalty-like parameters). A new crack driving state function is introduced by avoiding the compressible part of anisotropic energy to be degraded. For the Bayesian inversion, we employ the delayed rejection adaptive Metropolis (DRAM) algorithm to identify the parameters. We adjust the algorithm to estimate parameters according to a hydraulic fracture observation, i.e., the maximum pressure. The focus is on uncertainties arising from different variables, including elasticity modulus, Biot's coefficient, Biot's modulus, dynamic fluid viscosity, and Griffith's energy release rate in the case of the isotropic hydraulic fracture while in the anisotropic setting, we identify additional penalty-like parameters. Several numerical examples are employed to substantiate our algorithmic developments.