(untitled) (European Journal of Pure and Applied Mathematics, 2025)
In this article, we present a numerical investigation of two-phase flow involving liquid and solid particles (cuttings) during drilling operations in wellbores. The dynamics of this complex system are simulated using a high-resolution second-order Total Variation Diminishing (TVD) Runge-Kutta Discontinuous Galerkin (RKDG) method, enhanced with a Weighted Essentially Non-Oscillatory (WENO) limiter. Unlike traditional slope-limiter techniques, the integration of WENO methodology with the RKDG scheme significantly improves the accuracy and stability of the numerical solution by effectively capturing sharp interfaces and discontinuities without spurious oscillations. A two-phase drift-flux model is employed to account for the relative motion between the liquid and solid phases, enabling realistic simulation of the transient and dynamic behavior of cuttings transport under various well conditions. Through a series of test cases, we analyze the influence of varying rates of penetration (ROP) on critical parameters such as bottom hole pressure (BHP), cuttings concentration, and pressure evolution over time. The results demonstrate that the proposed scheme offers enhanced accuracy and computational efficiency compared to conventional approaches.