轴对称地层中高分辨率阵列侧向测井信赖域反演法

Trust region inversion algorithm of high-resolution array lateral logging in axisymmetric formation

本文研究轴对称地层中高分辨率阵列侧向测井(HRLA)的多参数信赖域反演方法.首先对前期HRLA的有限元正演方法进行改进,提出基于叠加原理和并行直接稀疏求解器PARDISO的快速正演方案,更适合于反演计算.将HRLA反演问题转化为非线性最小二乘问题,利用信赖域算法求解.为提高反演速度,推导了目标函数对优化参数偏导数的具体计算公式.对典型地层模型,与已有文献中Jacobi预条件共轭梯度法(JCG)计算结果比较,发现PARDIDO比JCG快10倍以上,验证了本文正演程序的正确性与高效性.利用信赖域算法求解了电阻率四参数反演和传统的三参数反演.研究结果表明:并行直接稀疏求解器PARDISO能有效求解此类HRLA正演问题,对6次不同探测深度的测井模拟,所形成的有限元刚度矩阵完全相同,只须进行一次矩阵分解,大大加快了正反演的速度.信赖域算法收敛速度快,且具有全局收敛性.HRLA的信赖域反演结果几乎不依赖于初值的选取,从较差初值出发仍能得到满意的反演结果.另外信赖域算法抗噪能力比较强,即使对测井数据添加信噪比为10dB(甚至5dB)的高斯白噪声,仍能通过反演得到较为准确的地层参数.

The high-resolution laterolog array （HRLA） tool can simultaneously provide six curves of logging responses with different depths of investigation, which give abundant information for the inversion of geological formation parameters. This paper proposes a trust region algorithm to recover several parameters in the axisymmetric formation from HRLA data, including the mud-invasion radius, mud resistivity and original formation resistivity.Firstly, we propose an accurate and efficient method to solve the forward problem in order to obtain the logging responses of HRLA tool in the axisymmetric formation, where the superposition principle is used to simplify the calculation of resistivity logging responses, and the parallel direct solver PARDISO is employed to solve the linear system with multiple right-hand sides resulting from finite element approximation. The decomposition of the stiffness matrix is needed only once when using PARDISO to solve these linear systems, which is cheap in terms of work and fast in time. Secondly, the problem of identifying formation parameters is transformed into a nonlinear least squares problem that can be solved by the trust region algorithm.In order to speed up the optimization process, we derive an explicit formula for the partial derivative of the objective function with respect to each resistivity parameter. Finally, for a typical formation model we present resistivity four-parameter and traditional three-parameter inversion results by using different initial guesses.In order to test the validity and effectiveness of the proposed method, we design a typical formation model with borehole radius rh=0.1016 m, invasion radius ri=0.4016 m, permeable formation thickness H=2 m, borehole mud resistivity Rm=1 Ωm, surrounding rock resistivity Rs=10 Ωm and original formation resistivity Rt=500 Ωm. Comparing the forward modeling result with the result in the existing literature, we find that the maximum relative error between the two results is only 0.01%, which verifies the correctness of the forward algorithm program. It takes less than half a second to perform forward modeling with 24881 grid points using PARDISO on a personal laptop. The computational time of the Jacobi conjugate gradient （JCG） method is more than ten times of that of PARDISO, and PARDISO provides a more accurate solution than the JCG. We use the trust region algorithm to invert formation parameters with four different initial values, one of which is far away from the real model. The inversion results show that the method can converge to the true model quickly even with very poor initial guesses. When using noisy data mixed with Gaussian white noise with signal-to-noise ratio of 10dB or even 5dB, the trust inversion algorithm still gives satisfactory results.Trust region algorithm combined with PARDISO is successfully used to multi-parameter inversion of HRLA data. Numerical results show that the trust region inversion algorithm converges fast, and has a global convergence property. Another advantage of the proposed method is the robustness to noise.