Determine Ultrasonic Logging Tool Trajectory, Casing Geometry, and Mud Velocity in Cased Hole From Measured Waveforms

Advanced ultrasonic logging facing significant challenges in non-idealized downhole environments, where tool eccentering, irregular casing geometry, and uncertain borehole mud velocity jointly degrade measurement accuracy. In the current research, there are only a few methods for inverting the logging tool trajectory and casing geometry. However, those existing methods either oversimplify the inverse problem or require impractical sensor configurations. This paper proposes a novel iterative optimization framework that synergizes Dijkstra algorithm-based pathfinding and Boyce-Clark radius shape index regularization to jointly invert tool trajectory, casing geometry, and borehole mud velocity. We initially assume the tool is centered in the borehole, positioned at the origin of the coordinate system. The distance between tool center and casing’s inner surface is the same. Leveraging the characteristics of the two measurements, we calculate the initial velocity using the least squares method. Then, we calculate the distance between tool-housing and the casing’s inner surface for both measurements and determine the initial position for the casing’s inner surface based on the tool trajectory and these distances. We use the Dijkstra algorithm to update the tool trajectory in relation to the casing’s inner surface and the distances. Finally, we iteratively update the casing’s inner surface and the tool trajectory until there are little changes in the tool trajectory. During the iterative optimization process, the ultrasonic wave velocity of borehole fluid is crucial because the known exact velocity makes the inversion of the tool trajectory and casing’s geometric parameters straightforward. However, as the exact velocity is often unknown, we incorporate a velocity inversion constraint with the circumference of the casing during the iterative process. We introduce the Boyce-Clark radius shape index to further correct the trajectory and the borehole mud velocity. By dynamically coupling these parameters, our method eliminates the need for prior fluid velocity knowledge—a key limitation of prior work. Validation on synthetic and field datasets demonstrates sub-millimeter accuracy in casing reconstruction (mean square error: 1.34×10⁻⁴) and fluid velocity estimation with <0.5% relative error. This advancement significantly enhances the reliability of ultrasonic logging in complex wells, providing a foundation for automated well integrity assessment.