声波测井三维交错网格有限差分模拟的漂移失稳及改进方案

交错网格的三维有限差分方法是当前模拟井眼条件下弹性波场传播和声波测井响应的最有效方法之一。在其典型算法中，交错网格的离散方案能够自动地满足流固边界条件。实际情况有时并非如此。在流固边界上流体一侧，这种离散方案给出的差分方程有时并不成立。当计算低频声场时，计算步数少，所以由此引起的累积误差比较小；当模拟高频声场时，计算步数大，这种累积误差变得不可忽略。在实际应用中表现为低频段能给出比较好的模拟结果，当模拟高频声场时存在漂移失稳问题。数值实验显示漂移失稳主要发生在流固边界上。为了消除漂移失稳，提出了一种简单易行的解决方法，计算实例表明该方法能够很好地消除漂移失稳．

On Shifting Instability of 3D Finite Difference Algorithm with Staggered Grid in Acoustic Logging and Its Improvements

The 3D finite difference algorithm with staggered grid is one of the powerful numerical tools to simulate elastic wave propagation in a fluid-filled borehole and responses of the acoustic logging tools. We found that good stimulating results of acoustic logs can be achieved when this algorithm is applied in low frequency range. In high frequency range, however, it often runs into a stability problem. Further analysis indicates that so-called shifting instability easily arises from mismatch of difference equations at fluid-solid boundary on the borehole wall. Numerical experiment shows it should not be taken as granted that the dispersing scheme of staggered grid can automatically meet the requirement of fluid-solid boundary conditions. Actually, we found the difference equations are no longer valid for the fluid side of the fluid-solid boundary. When calculating low frequency elastic wave field, the cumulative errors due to the inconsistence are very small as only few iterations are required. In high frequency band, a substantially large number of iterations are necessary and therefore the cumulative errors could be very large. We subsequently developed a simple formulation to resolve the inconsistence. Numerical example is presented to demonstrate the validity of our method.