双轴各向异性介质中回线源瞬变电磁三维拟态有限体积正演算法

Research on the 3D mimetic finite volume method for loop-source TEM response in biaxial anisotropic formation

采用模拟离散的有限体积法实现了双轴各向异性地层回线源瞬变电磁三维正演.首先引入内积定义,采用自然边界条件,将瞬变电磁法的控制方程转化为弱形式表示.将计算区域划分为一系列的控制体积单元,采用交错网格对控制方程进行模拟有限体积空间离散,包括旋度算子离散和空间内积离散.基于斯托克斯定理的旋度积分定义公式实现旋度算子离散.中点平均实现电导率双轴各向异性的空间内积离散,从而得到离散化的控制方程.时间步迭代采用无条件稳定的欧拉后向差分格式.并通过均匀全空间中稳定电流回线源的磁场解析表达式得到回线源初始时刻的电磁场分布.为了同时保证计算精度和效率,本文采用分段等间隔的时间步迭代,利用直接法求解器PARDISO实现其快速求解.最后通过对比层状模型和各向异性半空间模型的正演计算结果,验证了本文算法的计算精度和计算效率;计算三维双轴各向异性模型的正演响应可知,水平方向电导率变化对电磁响应产生显著影响,而垂直方向的电导率变化对电磁响应几乎没有影响.产生这一现象的主要原因是回线源产生的感应电流主要是水平方向的,因此响应主要受到水平方向电导率的影响,垂直方向的电导率影响很小.

In this paper, a 3D forward modeling of loop-source transient electromagnetic （TEM） response in biaxial anisotropic formation is proposed by using the mimetic finite volume method （MFVM）. Firstly, the definition of inner product is introduced, and the governing equation of TEM method is transformed into a weak form under a natural boundary condition. The computational domain is divided into a series of control volume units. The staggered grid is used to simulate the finite volume space discretization of the governing equations, including the curl operator discretization and the spatial inner product discretization. Discretization of the curl operator is based on Stokes＇ Theorem. Discretization of inner product in the biaxial anisotropic formation is based on midpoint average. The backward Euler time stepping method, which is unconditionally stable, is chosen to discretize the governing equation in the time domain. Electromagnetic field distribution at the initial time is obtained by solving the magnetic field of the stable current loop source in the uniform space. When solving the governing equations in the time domain, the MFVM adopts double time step size for each m constant time steps （here m is an input parameter） and uses the direct method solver PARDISO to ensure the accuracy and efficiency where the coefficient matrix just needs to decompose only once for each m time steps. Finally, the computational accuracy and computational efficiency of the proposed algorithm are verified by comparing the results of the forward modeling with a layered model and the anisotropic half-space model. The forward response of the 3D biaxial anisotropy model shows that the horizontal conductivity has a significant impact on the electromagnetic response, while the vertical direction of the conductivity change has little effect. The main reason is that the induced current generated by the loop source is mainly horizontal, so the response is primarily affected by the conductivity in the horizontal direction, while the conductivity in the vertical direction has little effect.