激发极化测井的三维有限差分模拟

三维柱坐标系下,使用有限差分法模拟激发极化测井,由稳定电流场构造的差分方程,其系数矩阵非对称正定,直接使用共轭梯度法求解各节点电位,计算结果不收敛,这就需要对系数矩阵进行修改。针对这一问题,本文推导出对系数矩阵作预对称的预对称子D和不完全LU分解的L矩阵和U矩阵的构造,采用预对称结合预条件共轭梯度法实现差分模拟程序。通过对比实例模型计算结果与解析式计算结果,验证了差分模拟程序的可行性、有效性。实例结果表明,预对称结合预条件共轭梯度法,不仅使得计算结果收敛,而且大大提高了差分程序的运行速度。

3D Definite Difference Simulation of Induced Polarization Logs

The coefficient matrix of difference equation derived by current field is non-symmetric positive when using definite difference method to simulate induced polarization logs under 3Dcylindrical coordinate system.And the results do not converge when solving potential of each node by using conjugate gradient method directly,so it is necessary to modify coefficient matrix.In order to solve the problem mentioned previously,the structure of matrix Dused in pre-symmetry algorithm and the structure of matrix Land Ucontained in incomplete LUdecomposition algorithm to amend coefficient matrix are derived in the paper.Therefore the finite difference simulation program is achieved by combining pre-symmetry with precondition conjugate gradient method,and its＇ feasibility and effectiveness are verified by establishing an instance model and comparing with the results calculated by analytical expression.It is showed that using pre-symmetry algorithm and preconditioned conjugate gradient method not only makes the results converge and meet the accuracy requirements,but also improves the speed of simulation program.