Surrogate-Assisted Bayesian Inference of Fracture Network Parameters from Elastic Waves: A Sensitivity-Guided Approach

Le Zhang, Qinghua Lei, Chuanyin Jiang, Longjun Dong, Thomas Hermans 

We develop a sensitivity-guided, surrogate-assisted Bayesian framework to infer fracture network parameters from elastic waves. Synthetic fracture networks characterized by power-law length exponent \(a\), fracture density \(d\), and percolation parameter \(p\) are constructed. Elastic wave propagation through these fracture networks is then simulated across a range of dimensionless specific stiffness values \(\tilde\kappa\). From 2560 Monte Carlo simulation runs, we extract two wave transport metrics: the inverse quality factor \(Q^{-1}\) and the normalized transmitted energy \(E\). Distance-based generalized sensitivity analysis in the \((Q^{-1},E)\) space reveals stiffness-dependent wave transport regimes (propagation, superdiffusion, normal diffusion, subdiffusion, and localization) and quantifies the contributions of parameters \(a\), \(d\), \(p\) and \(\tilde\kappa\) in each regime. A random forest surrogate for the mapping of \((a,d,p,\tilde\kappa)\mapsto(Q^{-1},E)\) is then embedded in a Metropolis-Hastings scheme to perform Bayesian inversion of fracture network parameters. When the dimensionless stiffness \(\tilde\kappa\) is near 1, both \(a\) and \(d\) are reliably recovered, with posterior probabilities for the true values well above their uniform priors. For large \(\tilde\kappa\) (corresponding to the propagation and superdiffusion regimes), the fracture stiffness itself becomes highly identifiable, and complementary inversions in which \(\tilde\kappa\) is treated as unknown show that it can be robustly recovered from \((Q^{-1},E)\) in these regimes. For small \(\tilde\kappa\) (subdiffusion and localization regimes), multiple scattering prevails and all parameters become more weakly resolved. Our results demonstrate that wave attenuation and energy metrics can be used to jointly invert fracture stiffness and network geometry, provided that the inversion targets wavefield regimes where these parameters are most sensitive.