Transient hydraulics always characterizes the circulating flow during managed pressure drilling. Therefore, the application of the Godunov scheme to oil-well drilling hydraulics is presented. The numerical model developed describes the treatment process of the initial and boundary conditions from the well geometry and true operational conditions. The well-known finite-volume method and Riemann problem are utilized for building the set of discrete equations. The account of Godunov's simulation describes the profiles of transient pressure and transient flow rate along the well. For attending the oil-field engineering concerns, the drilling parameters discussed are as follows: choke pressure, pumping pressure, bottom-hole pressure, and circulating flow rate. After the comparison between computed and well data, the results show a small difference of less than 7 and 1 % for pumping and bottom-hole pressures, respectively. The main engineering contribution of this work is the solution and application of the first-order Godunov scheme to analyze the transient hydraulics during actual oil-well drilling and also the analysis and interpretation of the pressure wave behavior traveling along the well. The Godunov scheme has high-potential engineering applications for modeling the transient drilling hydraulics, i.e., controlled flow, underbalanced drilling, and foam cementing, as well.