非均质各向异性地层中方位随钻电磁测井响应三维有限体积法数值模拟算法

方位随钻电磁测井是一种能够实时探测地层边界、实现地质导向与井眼成像的新型测井技术. 本文根据方位随钻电磁测井仪器的典型线圈系结构, 首先引入柱坐标系下非均质完全各向异性地层中电流源并矢Green函数, 并利用电磁场叠加原理给出倾斜发射线圈激发的电场以及倾斜接收线圈上感应电动势的计算公式; 然后应用电流源电场并矢Green函数的混合势克服非均质地层中电磁数值模拟的低感应数问题, 通过ρ 和z方向上Lebedev网格设法降低网格节点个数, 并且利用标准化算法确定柱坐标系下非均质单元上的等效电导率. 在此基础上, 用三维有限体积法建立柱坐标系电场混合势的离散方法, 得到一个交错网格上电场矢势和标势大型代数方程, 并用不完全LU分解以及稳定双共轭梯度法确定数值解. 最后, 通过数据模拟结果对算法的有效性进行检验, 并考察钻铤、线圈倾斜角度以及地层各向异性等参数对仪器响应的影响. 数值结果表明: 在柱坐标系下用三维有限体积法的数值模拟算法处理非均质各向异性层中方位随钻电磁测井响应可以得到很好的结果. 钻铤、电导率各向异性、层边界均对方位随钻电磁波测井响应产生较大的影响; 在电阻率较大的地层, 幅度比和相位差响应越小; 发射线圈和接收线圈同时倾斜时, 幅度比和相位差响应受地层的影响更灵敏.

Three-dimensional finite volume simulation of the response of azimuth electromagnetic wave resistivity while drilling in inhomogeneous anisotropic formation

The azimuth electromagnetic wave resistivity while drilling is a new type of well logging technique. It can real-time detect the formation boundary, realize geosteering and borehole imaging in order to keep the tool always drilling in the some meaning reservoir. For effectively optimizing tool parameters, proper explanation and evaluation of the data obtained by azimuth electromagnetic wave resistivity while drilling, the efficient numerical simulation algorithm is required. In this paper, we use the finite volume algorithm in the cylindrical coordinate to establish the corresponding numerical method so that we can effectively simulate the response of the tool in various complex environments and investigate the influences of the change in formation and tool parameters on the tool response. Therefore, according to the typical coil architecture of the instrument of azimuth electromagnetic wave resistivity while drilling, we first introduce the electrical and magnetic dyadic Green's functions in inhomogeneous anisotropic formation by the electrical current source in the cylindrical coordinate. Through superposition principle, we derive the integral formula to compute the electric field intensity excited by tilted transmitter coils and the induction electrical potential on tilted receiving coils both mounded on the drill collar. Then, we use the coupled electrical potentials of the dyadic Green's functions to overcome the low induction number problem during modeling the electrical fields in inhomogeneous anisotropic formation. Furthermore, we use Lebedev grid in both ρ and z directions to reduce the number of grid nodes, and the standard method to compute the equivalent conductivity in heterogeneous units for enhancing the discrete precision. On the basis, by the three-dimensional finite volume method, we discrete the equations about the coupled electrical potentials in the cylindrical coordinates and obtain the large sparse algebraic equation sets about the coupled electrical potentials field on the Lebedev grid. A combination of incomplete LU decomposition with the bi-conjugate gradient stabilization is used to solve the numerical solution. Finally, we validate the algorithm by comparing the numerical results obtained by two different methods, study the effects of the drill collar, anisotropy, the tilted angles of both coil, and borehole on the instrument response in inhomogeneous anisotropic formation. The numerical results show that the tool response obtained by the three-dimensional finite volume algorithm in the cylindrical coordinate system in anisotropic formation accord with that those obtained by other algorithms. The drill collar, inhomogeneous anisotropic n the formation will lead to both the smaller amplitude ratio and the smaller phase difference. In addition, when the coils of both transmitting and receiving coils are tilted, the amplitude ratio and phase difference of the tool are more sensitive to the change in formation parameter.