高分辨率阵列侧向测井的数学模型及有限元快速正演

Mathematical model and fast finite element modeling of high resolution array lateral logging

对包含井眼、侵入带、围岩和目的层的轴对称地层模型,推导了无穷远截断边界上的Robin边界条件,建立了高分辨率阵列侧向测井的等值面边值问题模型.Robin边界条件较Dirichlet边界条件更加精确,可大大缩小求解区域而不影响计算精度.考虑到微分方程和边界条件为线性的,利用叠加原理简化了原微分方程边值问题的计算,克服了事先屏蔽电极上电流的不确定性.采用基于地址矩阵的稀疏存贮模式,大大减小了内存需求,且地址矩阵物理意义明确,方便迭代法调用求解有限元方程.引入预条件共轭梯度(PCG)法求解有限元计算形成的大型线性方程组,提高了测井响应的计算速度.利用本文方法定量考察了地层厚度、井径、侵入带等因素对阵列侧向测井响应的影响,为后续阵列侧向测井反演的研究奠定了基础,对实际测井工程具有一定的指导意义.

This paper presents a Robin boundary condition on truncated boundary （instead of infinity）,and establishes an equivalued surface boundary value problem model of array lateral logging in the axisymmetric formation composing of borehole,invaded zone,surrounding rock and target zone.Robin boundary condition is more accurate than the Dirichlet boundary condition,thus the computational domain can be greatly reduced without affecting the simulation accuracy.Taking into account the linearities of differential equations and boundary conditions,we use the principle of superposition to simplify the calculation of the original boundary value problem,and overcome the difficulty of the prior uncertainty of the current on shielded electrode.Sparse storage mode based on the address matrix is proposed,which significantly reduces the memory requirements.At the same time,the physical meaning of the address matrix is clear,so it is convenient to solve finite element （FE） equation by using iteration methods.A preconditioned conjugate gradient （PCG） method is introduced to solve large sparse systems of linear equations derived from FE modeling,which greatly increases the computing speed of the logging responses.The effects of various factors on array lateral logging response,including bed thickness,bore-hole diameter,invasion zone and so on,are quantitatively investigated by the proposed method,which lays the foundation for later inversion of array lateral logging and provides some guiding significance for practical logging engineering.