Performance of Gradient and Gradient-Free Optimizers in Transient Hydraulic Tomography

Abstract

Sub-surface characterization in fractured aquifers is challenging due to the co-existence of contrasting materials namely matrix and fractures. Transient hydraulic tomography (THT) is proved to be an efficient and robust technique to estimate hydraulic (K_m, K_f) and storage (S_m, S_f) properties in such complex hydrogeologic settings. However, performance of THT is governed by data quality and optimization technique used in inversion. We assessed the performance of gradient and gradient-free optimizers with THT inversion. Laboratory experiments were performed on a two-dimensional, granite rock (80 cm × 45 cm × 5 cm) with known fracture pattern. Cross-hole pumping experiments were conducted at 10 ports (located on fractures), and time-drawdown responses were monitored at 25 ports (located on matrix and fractures). Pumping ports were ranked based on weighted signal-to-noise ratio (SNR) computed at each observation port. Noise-free, good quality (SNR > 100) datasets were inverted using Levenberg–Marquardt: LM (gradient) and Nelder–Mead: NM (gradient-free) methods. All simulations were performed using a coupled simulation-optimization model. Performance of the two optimizers is evaluated by comparing model predictions with observations made at two validation ports that were not used in simulation. Both LM and NM algorithms have broadly captured the preferential flow paths (fracture network) via K and S tomograms, however LM has outperformed NM during validation ($R_{\mathrm{LM}}^2=0.76,{\text{RMSE}}_{\mathrm{LM}}=1.75\mathrm{cm},R_{\mathrm{NM}}^2=0.73,{\text{RMSE}}_{\mathrm{NM}}=1.77\mathrm{cm}$). Our results conclude that, while method of optimization has a trivial effect on model predictions, exclusion of low quality (SNR ≤ 100) datasets can significantly improve the model performance.