Stability of finite difference numerical simulations of acoustic logging-while-drilling with different perfectly matched layer schemes

In acoustic logging-while-drilling （ALWD） finite difference in time domain （FDTD） simulations, large drill collar occupies, most of the fluid-filled borehole and divides the borehole fluid into two thin fluid columns （radius -27 mm）. Fine grids and large computational models are required to model the thin fluid region between the tool and the formation. As a result, small time step and more iterations are needed, which increases the cumulative numerical error. Furthermore, due to high impedance contrast between the drill collar and fluid in the borehole （the difference is 〉30 times）, the stability and efficiency of the perfectly matched layer （PML） scheme is critical to simulate complicated wave modes accurately. In this paper, we compared four different PML implementations in a staggered grid finite difference in time domain （FDTD） in the ALWD simulation, including field-splitting PML （SPML）, multiaxial PML（M- PML）, non-splitting PML （NPML）, and complex frequency-shifted PML （CFS-PML）. The comparison indicated that NPML and CFS-PML can absorb the guided wave reflection from the computational boundaries more efficiently than SPML and M-PML. For large simulation time, SPML, M-PML, and NPML are numerically unstable. However, the stability of M-PML can be improved further to some extent. Based on the analysis, we proposed that the CFS-PML method is used in FDTD to eliminate the numerical instability and to improve the efficiency of absorption in the PML layers for LWD modeling. The optimal values of CFS-PML parameters in the LWD simulation were investigated based on thousands of 3D simulations. For typical LWD cases, the best maximum value of the quadratic damping profile was obtained using one do. The optimal parameter space for the maximum value of the linear frequency-shifted factor （a0） and the scaling factor （β0） depended on the thickness of the PML layer. For typical formations, if the PML thickness is 10 grid points, the global error can be reduced to 〈1% using the optimal PML parameters, and the error will decrease as the PML thickness increases.

Modeling and verification of comprehensive errors of real-time wear-depth detecting for spherical plain bearing tester

Because of various error factors,the detecting errors in the real-time experimental data of the wear depth affect the accuracy of the detecting data.The self-made spherical plain bearing tester was studied,and its testing principle of the wear depth of the spherical plain bearing was introduced.Meanwhile,the error factors affecting the wear-depth detecting precision were analyzed.Then,the comprehensive error model of the wear-depth detecting system of the spherical plain bearing was built by the multi-body system theory(MBS).In addition,the thermal deformation of the wear-depth detecting system caused by varying the environmental temperature was detected.Finally,according to the above experimental parameters,the thermal errors of the related parts of the comprehensive error model were calculated by FEM.The results show that the difference between the simulation value and the experimental value is less than 0.005 mm,and the two values are close.The correctness of the comprehensive error model is verified under the thermal error experimental conditions.