Prediction of deep rock drilling resistance by integrating fractal theory and logging data: the Kuqa Depression block in the Tarim Basin

Abstract

The evaluation of drilling resistance in deep and ultra-deep structurally complex formations is commonly constrained by limited core data, strong lithological heterogeneity, and spatially variable logging responses. For the deep formations of the Kuqa Depression in the Tarim Basin, this study proposes a logging-constrained empirical–fractal framework for estimating drilling resistance-related parameters. The workflow integrates core-based experimental data, conventional logging curves, empirical regression models, rescaled range analysis, and stochastic midpoint displacement interpolation, thereby extending discrete core measurements to continuous single-well profiles and interwell spatial estimates. Based on 115 sets of rock mechanical and rock drillability experimental data, several key parameters were analyzed, including uniaxial compressive strength, elastic modulus, Poisson’s ratio, indentation hardness, abrasiveness, and rock drillability grade. Empirical relationships between these parameters and depth, density, and acoustic slowness were established. Model selection was based on a combined assessment of R², RMSE, MAE, AIC, and physical interpretability. The results indicate that the relationships of K_d–AC and P_y–AC show relatively good fitting performance and can provide constraints for the continuous estimation of rock drillability grade and indentation hardness. In contrast, density-related models mainly reflect regional empirical trends in rock mechanical parameters. Rescaled range analysis shows that the Hurst exponents of the DEN, AC, and GR curves from wells D1–D3 are all greater than 0.5, indicating positive correlation and persistence in the logging responses, which can provide spatial-continuity constraints for fractal interpolation. The interwell distributions constructed using the stochastic midpoint displacement method suggest that DEN, GR, and AC exhibit clear interwell variability. The estimated distributions of K_d and P_y mainly inherit the spatial variation of AC, whereas UCS and abrasiveness are primarily controlled by depth-dependent empirical trends. The results suggest that the proposed workflow can improve the continuity of drilling resistance evaluation under limited core data and sparse well control. It provides a practical reference for identifying difficult-to-drill intervals and supporting preliminary drilling engineering evaluation in deep structurally complex formations.