基于奇异值分解法的核磁测井数据反演方法研究
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摘要
核磁共振测井不仅可以评估岩层渗透率，还能获得岩石的总孔隙度，油、气、水的饱和度、扩散系数，以及油的粘度。对于复杂岩性，它常常是少数几种有效的方法之一。因此核磁测井是近十多年来发展最快的测井方法，现在已经成为石油和天然气勘探综合地球物理测井方法的一个重要组成部分。
核磁测井数据反演方法是核磁测井理论的核心内容。核磁测井数据反演是通过对核磁测井观测到的自旋回波串进行反演计算，得到储集层孔隙中流体的横向弛豫时间等核磁参数。利用核磁参数可以计算岩石的孔隙度、渗透率、油、气、水的饱和度等。自旋回波串是多个单指数衰减信号叠加在一起的多指数衰减弛豫信号。用自旋回波串反演孔隙流体核磁参数的数学模型是第一类弗雷德霍姆积分方程。积分方程的数值解法是将它离散化为线性方程组，再采用奇异值分解法、正则化方法等算法求解。已有的算法在实际应用中存在对信噪比要求高、计算速度慢、谱线不连续等问题。另一方面，由于测井仪器性能及井下环境的限制，核磁测井信号的信噪比很低。因此有必要对已有算法做改进，得到能够适应低信噪比的快速反演算法。
本文通过分析最佳奇异值保留个数与信噪比的关系，提出一种改进的奇异值分解算法——线性截断算法。该算法可以快速有效地实现T2谱反演，而且该算法比传统的奇异值分解算法更加稳定，即在信噪比变化很大情况下解的变化相对不大。线性截断算法可以适用于低信噪比（SNR>10）的T2谱反演，在信噪比很低时仍能较好地保持弛豫谱分布的真实性。进一步分析奇异值分解法与正则化方法的关系，提出了一个正则化因子的经验公式。本文提出的经验因子在低信噪比时更接近各种形态T2谱最佳正则因子的平均取值。另外，考察了初值对联合迭代重建反演算法的影响。建立了以线性截断算法结果为初值的联合迭代重建反演算法。通过与国外软件结果对比，验证了该算法的有效性。研究了布点方式、布点数对反演结果的影响。构造了对数布30点、50点的改进奇异值分解算法。实例表明，在低信噪比情况下采用对数布30点、50点的改进奇异值分解算法对大孔隙具有更好的分辨率，更能反映地层真实孔隙结构。
讨论了利用核磁共振数据进行流体识别的几种方法。介绍了T1、T2搜索、利用流体T1差异进行识别、利用流体T2差异进行识别以及时间域分析的具体算法。
研制了一套核磁共振反演解释系统软件。该系统可以完成基于本文所述几种反演算法的T2谱反演及反演结果的解释处理，可以为油田的勘探、开发提供可靠的地质数据。
NMR logging not only can evaluate permeability of stratum, but also can gettotal porosity of rock, saturation of oil, gas or water, coefficient of diffusion, viscosityof oil. It is usually the one of the several effective methods for complex lithology. Sothe NMR logging is the fastest growing logging method in recent ten years. It has nowbecome an important part of the comprehensive geophysical logging methods forpetroleum and natural gas exploration.
NMR inversion method is the core content of NMR logging theory. NMR datainversion gets transverse relaxation time and other magnetic parameters of pore fluidin stratum by inversion of the spin echo trains from NMR logging. Porosity,permeability and saturation of oil, gas or water can be computed by these magneticparameters. A train of echoes is a multiple exponential relaxation signal which iscomposed of many single exponential relaxation signals. The mathematical model ofthe magnetic parameters inversion by spin echo trains is a Fredholm’s integralequation of the first kind. The numerical solution of the integral equation alwaysreduces it to a system of linear algebraic equations, and then we can use singularvalue decomposition or regularization method to solve those linear algebraicequations. In practical application the existing algorithms have many problems, suchas the requirement of signal to noise is too high, calculation speed is slow orspectrums are not continuous. On the other hand, as a result of logging instrumentperformance and underground environmental constraints, the signal to noise ratio of NMR logging is very low. So it is necessary to improve the existing algorithm for afast inversion algorithm which is able to adapt low signal to noise ratio.
An improved SVD algorithm named linear cutoff algorithm is proposed byanalysis of the relationship between the best number of singular value retained andSNR. It can make the inversion of T2spectrum quickly and effectively, and it isstabler than the traditional SVD algorithm: the difference of the T2spectrumsinversed by linear cutoff algorithm with different signal to noise ratio is not veryapparent. The linear cutoff algorithm can adapt to low SNR (SNR>10) T2spectruminversion. That means the T2spectrum inversed by linear cutoff algorithm can be realeven in low SNR. A formula of regularization factor is proposed by analyzing therelationship between the improved SVD and regularization method. The empiricalregularization factor proposed in this paper is closer to the average value of the bestfactors in low SNR. In addition, the influence of initial value to simultaneous iterativereconstruction inversion algorithm is investigated. Construct a simultaneous iterativereconstruction inversion algorithm, which initial value is the result of linear cutoffalgorithm. We compare the improved algorithm with foreign software results to verifythe effectiveness of the algorithm. Analyze the effects of methods of pre-assignedrelaxation bins and the number of pre-assigned relaxation bins on inversions.Construct the improved SVD algorithm with30or50logarithmic spaced timeconstants. Practical examples show that the improved algorithm with logarithmic30or50logarithmic spaced time constants have better identification capacity to largepore under the condition of low SNR. It can give the true relaxation distributions forthe formation pore structure better.
Several methods for fluids typing using nuclear magnetic resonance data arediscussed. The concrete implementation process of T1, T2search, fluid typing basedon difference of T1, fluid typing based on difference of T2and TDA are introduced.
Software for NMR data inversion and iterpretation is developed. This softwarecan do the inversion of T2spectrum and the iterpretation of the result of inversion. Itcan provide reliable geological data for oilfield exploration and development.
核磁测井数据反演方法是核磁测井理论的核心内容。核磁测井数据反演是通过对核磁测井观测到的自旋回波串进行反演计算，得到储集层孔隙中流体的横向弛豫时间等核磁参数。利用核磁参数可以计算岩石的孔隙度、渗透率、油、气、水的饱和度等。自旋回波串是多个单指数衰减信号叠加在一起的多指数衰减弛豫信号。用自旋回波串反演孔隙流体核磁参数的数学模型是第一类弗雷德霍姆积分方程。积分方程的数值解法是将它离散化为线性方程组，再采用奇异值分解法、正则化方法等算法求解。已有的算法在实际应用中存在对信噪比要求高、计算速度慢、谱线不连续等问题。另一方面，由于测井仪器性能及井下环境的限制，核磁测井信号的信噪比很低。因此有必要对已有算法做改进，得到能够适应低信噪比的快速反演算法。
本文通过分析最佳奇异值保留个数与信噪比的关系，提出一种改进的奇异值分解算法——线性截断算法。该算法可以快速有效地实现T2谱反演，而且该算法比传统的奇异值分解算法更加稳定，即在信噪比变化很大情况下解的变化相对不大。线性截断算法可以适用于低信噪比（SNR>10）的T2谱反演，在信噪比很低时仍能较好地保持弛豫谱分布的真实性。进一步分析奇异值分解法与正则化方法的关系，提出了一个正则化因子的经验公式。本文提出的经验因子在低信噪比时更接近各种形态T2谱最佳正则因子的平均取值。另外，考察了初值对联合迭代重建反演算法的影响。建立了以线性截断算法结果为初值的联合迭代重建反演算法。通过与国外软件结果对比，验证了该算法的有效性。研究了布点方式、布点数对反演结果的影响。构造了对数布30点、50点的改进奇异值分解算法。实例表明，在低信噪比情况下采用对数布30点、50点的改进奇异值分解算法对大孔隙具有更好的分辨率，更能反映地层真实孔隙结构。
讨论了利用核磁共振数据进行流体识别的几种方法。介绍了T1、T2搜索、利用流体T1差异进行识别、利用流体T2差异进行识别以及时间域分析的具体算法。
研制了一套核磁共振反演解释系统软件。该系统可以完成基于本文所述几种反演算法的T2谱反演及反演结果的解释处理，可以为油田的勘探、开发提供可靠的地质数据。
NMR logging not only can evaluate permeability of stratum, but also can gettotal porosity of rock, saturation of oil, gas or water, coefficient of diffusion, viscosityof oil. It is usually the one of the several effective methods for complex lithology. Sothe NMR logging is the fastest growing logging method in recent ten years. It has nowbecome an important part of the comprehensive geophysical logging methods forpetroleum and natural gas exploration.
NMR inversion method is the core content of NMR logging theory. NMR datainversion gets transverse relaxation time and other magnetic parameters of pore fluidin stratum by inversion of the spin echo trains from NMR logging. Porosity,permeability and saturation of oil, gas or water can be computed by these magneticparameters. A train of echoes is a multiple exponential relaxation signal which iscomposed of many single exponential relaxation signals. The mathematical model ofthe magnetic parameters inversion by spin echo trains is a Fredholm’s integralequation of the first kind. The numerical solution of the integral equation alwaysreduces it to a system of linear algebraic equations, and then we can use singularvalue decomposition or regularization method to solve those linear algebraicequations. In practical application the existing algorithms have many problems, suchas the requirement of signal to noise is too high, calculation speed is slow orspectrums are not continuous. On the other hand, as a result of logging instrumentperformance and underground environmental constraints, the signal to noise ratio of NMR logging is very low. So it is necessary to improve the existing algorithm for afast inversion algorithm which is able to adapt low signal to noise ratio.
An improved SVD algorithm named linear cutoff algorithm is proposed byanalysis of the relationship between the best number of singular value retained andSNR. It can make the inversion of T2spectrum quickly and effectively, and it isstabler than the traditional SVD algorithm: the difference of the T2spectrumsinversed by linear cutoff algorithm with different signal to noise ratio is not veryapparent. The linear cutoff algorithm can adapt to low SNR (SNR>10) T2spectruminversion. That means the T2spectrum inversed by linear cutoff algorithm can be realeven in low SNR. A formula of regularization factor is proposed by analyzing therelationship between the improved SVD and regularization method. The empiricalregularization factor proposed in this paper is closer to the average value of the bestfactors in low SNR. In addition, the influence of initial value to simultaneous iterativereconstruction inversion algorithm is investigated. Construct a simultaneous iterativereconstruction inversion algorithm, which initial value is the result of linear cutoffalgorithm. We compare the improved algorithm with foreign software results to verifythe effectiveness of the algorithm. Analyze the effects of methods of pre-assignedrelaxation bins and the number of pre-assigned relaxation bins on inversions.Construct the improved SVD algorithm with30or50logarithmic spaced timeconstants. Practical examples show that the improved algorithm with logarithmic30or50logarithmic spaced time constants have better identification capacity to largepore under the condition of low SNR. It can give the true relaxation distributions forthe formation pore structure better.
Several methods for fluids typing using nuclear magnetic resonance data arediscussed. The concrete implementation process of T1, T2search, fluid typing basedon difference of T1, fluid typing based on difference of T2and TDA are introduced.
Software for NMR data inversion and iterpretation is developed. This softwarecan do the inversion of T2spectrum and the iterpretation of the result of inversion. Itcan provide reliable geological data for oilfield exploration and development.
引文
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