Numerical solution to binary quasi-linear partial differential equations for drilling surge pressures based on the characteristic-lines method

Accurately predicting surge pressures caused by tripping drill string or running casing string is fundamental to implementing Managed Pressure Drilling in areas with narrow safety mud density margin. In this work, the region of interest is divided into 3 coupled regions, i.e., inside moving pipe, annulus, and that below the moving pipe. Mass conservation equations and momentum conservation equations depicting the 3 regions are developed considering mud compressibility, pipe elasticity and wellbore wall elasticity. Applying the characteristic-line theory for partial differential equations, a series of discretized equations have been developed for the 3 regions, including internal nodes within uniform size pipe conduit, coupling nodes between two adjacent different size conduits, surface and bottom-hole nodes, and the conjunction node of the 3 regions. Based on the discretized equations, a computational flowchart is programmed to get numerical transient solutions. Both the transient surge pressure variation with time at the given interested point and the surge pressure variation with depth along the whole well axis can be depicted using this method. Two case studies show that computational values are in good consistent with the measurements with a relative error less than 10%.