In petroleum drilling, carbonate formations characterized by natural fractures can result in troublesome gas–liquid gravity displacement, which refers to the phenomenon that the drilling mud leakage and gas kick are simultaneously triggered. This work focuses on clarifying the mechanism of gas–liquid displacement in vertical fractures during the drilling of carbonate formations and investigating the characteristics of gas–liquid displacement under various conditions. First, the bottom hole pressure allowing for gas–liquid gravity displacement is analyzed, which determines the coexistence condition of leakage and kick in vertical fractures. Then, a theoretical model of gas–liquid displacement flow in a vertical fracture is established. To verify the reliability and accuracy of the model, the results of numerical simulation are compared with those of a visualization experiment. The development process and flow characteristics of gas–liquid displacement in the fracture under different conditions are numerically simulated. The effects of pressure difference, drilling mud property, and fracture geometry on the gas–liquid displacement rate are analyzed. It is found that the drilling mud leakage rate increases with the increase of fracture width, fracture height, and drilling mud density, while it decreases with the increase of pressure difference and fracture length. The gas invasion rate increases with the increase of fracture width, fracture height, and pressure difference, while it decreases with the increase of drilling mud density and fracture length. The equations for leakage rate and gas invasion rate are derived by the response surface method, and the methods for mitigating gas–liquid gravity displacement are discussed. It is expected that the present work provides a better understanding of the gas–liquid gravity displacement in carbonate formations.

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To analyze the risks caused by the uncertainty of formation parameters to bullheading killing, a method for quantitatively evaluating the bullheading killing risks is established. Firstly, considering the influence of gas invasion volume, formation fracture, and killing parameters, a bullheading killing model is established based on a gas-liquid two-phase flow. Then, the uncertainties of formation parameters (formation pressure and permeability) are quantified. Based on the shut-in wellhead information, the range of formation pressure is predicted with the gas column model and multiphase flow model. Considering the influence of formation fracture on the permeability, Monte Carlo random sampling is applied to predict the range of formation permeability. Based on industry standards, a safety pressure value is set up, and the wellbore pressure corresponding to all value combinations of formation parameters under the given killing parameters is obtained by killing model. Moreover, according to the probability and degree that wellbore pressure exceeds the safety value, the risks are rated to quantify the risk of bullheading killing. Under this circumstance, the feasibility and accuracy of this method are validated by practical cases, and it is found by simulation that flow rate can affect the risk of wellhead damage to the greatest extent, and there exists a critical rate. When the flow rate is greater than the critical rate, the increase in flow rate will greatly improve the risk probability. In such case, improving the density of kill fluid can reduce the risk of wellhead damage in a limited way, but it will greatly increase the risks of formation fracture and casing damage. Therefore, for bullheading killing, it is not advisable to employ high-density kill fluid. By this method, the bullheading killing risks can be fully assessed before actual construction, thus providing reference for determining reasonable construction parameters of bullheading killing.

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