Physics-Informed Hybrid Stochastic-AI Framework Integrating Galle and Woods Mechanics, Monte Carlo Simulation, and Dual-Track AI Decision Support for ROP Optimization, Cutter Wear Mitigation, and NPT Reduction: A Tawakul-1 Well Case Study (Block-8, Sudan)

Abstract Rate of Penetration (ROP) optimization in heterogeneous, structurally complex sedimentary basins remains a significant technical challenge due to non-linear interdependencies among surface operational controls, unquantified downhole uncertainty, and progressive cutter degradation. Classical empirical models (e.g., Galle and Woods) provide essential mechanical rigor but are inherently deterministic and struggle with field telemetry noise, whereas purely data-driven machine learning models lack physical interpretability and dynamic risk awareness. To bridge this gap, this study introduces a physics-informed, hybrid stochastic-AI decision-support framework that seamlessly integrates the depth-based Galle and Woods drillability core, high-throughput Monte Carlo simulation (MCS), local supervised machine learning, and cloud-based Large Language Model (LLM) generative reasoning. The architecture was calibrated, evaluated, and validated using depth-aligned operational telemetry across a 263 m target claystone interval (MD 3167--3430 m) encompassing Polycrystalline Diamond Compact (PDC) bit Runs 8 and 9 of the Tawakul-1 exploratory well in Block-8 (Sinnar State, Blue Nile Basin, Sudan). This selection ensured consistent lithological and drilling fluid baselines to minimize field perturbations. Automated preprocessing algorithms sanitized raw field telemetry by filtering off-bottom operational states and suppressing sensor noise, yielding 131 sanitized records for Run 8 (from 147 initial rows) and 376 records for Run 9 (from 381 initial rows). High-throughput Monte Carlo simulations (13,100 iterations for Run 8; 37,600 iterations for Run 9) transformed static parameters into risk-quantified probabilistic response surfaces, decoding multi-variable non-linear interactions and establishing bit dulling characteristics. A local machine learning regression engine (XGBoost) demonstrated exceptional predictive accuracy (Testing R 2 = 0.9842, RMSE = 0.412 ft/hr, AAPE = 2.15%), outperforming Artificial Neural Network (ANN) and Support Vector Machine (SVM) architectures to define upper-bound mathematical performance potential. To mitigate premature tool failure, stochastic outputs were serialized via a secure HTTPS REST API JSON protocol to the Google Gemini LLM engine. Governed by closed-domain system prompts and low-entropy decoding (T = 0.1), the LLM evaluated point-density clusters and thermal-wear boundaries to establish actionable operational envelopes. For Run 9, Gemini selected a wear-balanced equilibrium at Weight on Bit (W = 24.0 klb) and Rotary Speed (N = 161 RPM). This risk-averse selection achieved a stabilized ROP of 47.63 ft/hr and representing a 28.56% improvement over historical field performance (37.05 ft/hr); while holding the cutter wear derivative to a sustainable threshold (dF/dD = 10.8675), suppressing wear rates by 49.2% relative to unconstrained local machine learning peaks. By preventing entry into critical thermal-wear regimes, the proposed framework effectively delineates distinct ROP and cutter degradation signatures across drilling intervals, directly eliminates major causes of Non-Productive Time (NPT) and Invisible Lost Time (ILT), and provides an uncertainty-aware decision-support architecture for maximizing total Cost per Foot (CPF) efficiency and tool longevity in mature sedimentary basins.