Characteristics and calculation method of gas‑liquid flow rates for coexisting kick and loss in fractured oil and gas reservoirs

Gas kick and lost circulation often coexist in fractured reservoirs due to gas‑liquid gravity displacement within the same pressure system, posing serious well‑control challenges. However, quantitative characterization of the gas‑liquid exchange rate at the wellbore‑fracture interface remains lacking. This study simulates gas‑liquid two‑phase flow between wellbore and fracture under constant‑pressure boundary conditions using the Volume of Fluid (VOF) model, validated against published numerical data with good agreement. We then systematically examine the effects of mud properties (density, viscosity, velocity) and fracture geometry (width, height, pressure) on steady‑state kick and loss rates. Dimensional analysis via the Buckingham π theorem identifies the Pressure‑Capillary ratio ( P/Ca ), Bond number (Bo), and gas‑liquid viscosity ratio as key dimensionless groups. Multivariate nonlinear regression was then employed to establish dimensionless power-law prediction formulas for gas and liquid exchange rates, achieving coefficients of determination (R²) of 0.93 and 0.94, respectively. Over 90% of the predicted points fell within the 85% prediction interval, and the minimum prediction error for the validation set was 4.4%, confirming the engineering reliability of the proposed correlations. The results provide an efficient approach for estimating gas–liquid exchange rates under coexisting kick–loss conditions. Nevertheless, further extension of the model’s applicability to more extreme downhole temperature and pressure conditions is still required, so as to offer more robust theoretical support for real-time well control decisions and managed pressure drilling optimization in fractured formations.