Parameter unidentifiability in Hoek–Brown calibration under sparse triaxial datasets: a bootstrap-based diagnostic framework

Abstract

Abstract The Hoek–Brown criterion is widely used for rock-mass strength characterization; however, its material constant m i is often calibrated from limited triaxial datasets without formal uncertainty quantification. This study shows that such calibrations can be practically non-identifiable even when residual diagnostics are unremarkable and the optimizer converges consistently. A fifteen-specimen triaxial dataset from a hydrothermally altered igneous rock was analysed using nonlinear calibration, BCa bootstrap resampling (B = 2000), leave-one-out diagnostics, sample-size analysis, a UCS-anchored joint calibration, uncertainty propagation, sensitivity analysis, and variance decomposition. A fifteen-specimen uniaxial vein dataset served as a methodological control, with petrography and X-ray diffraction providing mechanistic interpretation. The triaxial-only calibration yielded m i = 8.25 and σ ci = 90.31 MPa with severe parameter coupling (ρ = − 0.982); leave-one-out removal of single specimens altered m i by up to 890% (CV = 147.7%). Bootstrap analysis showed 48.9% of resampled fits reaching parameter bounds, an interior-only BCa 95% interval for m i of 0.11–36.89, and propagated 95% parameter-uncertainty intervals of 83–106% of the median rock-mass strength at engineering-relevant confinements. Anchoring the calibration with the fifteen uniaxial determinations already in hand, which observe σ ci directly because the criterion reduces to σ 1 = σ ci at σ 3 = 0, eliminated bound-hitting (f = 0%) and shifted m i from 8.25 to 19.25, exposing the trade-off ridge empirically while leaving precision modest (CV ≈ 37%). Petrography and diffraction identified a silicic-to-argillic alteration gradient that provides a plausible independent explanation for the strength dispersion (CV = 37.8%) obscuring the confinement signal; the vein control showed plausible m i assumptions diverging by 36% at σ 3 = 10 MPa in the absence of triaxial data. The bootstrap bound-hitting frequency is proposed as a practical identifiability diagnostic with provisional 10%/30% screening thresholds, embedded in a tiered engineering workflow and transferable to other empirical strength criteria affected by parameter coupling.