地球电磁三维数值模拟的多重网格方法及其应用研究
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摘要
地球电磁勘探方法在能源及矿产资源勘探开发、水文工程地质勘察以及地壳上地幔深部研究中均发挥着重要作用。随着地质勘探目标日益复杂,三维精细结构探测成为迫切需求,导致复杂地下结构的地球物理三维数值模拟在计算效率上面临着越来越大的挑战。就目前地球电磁三维数值模拟的有限差分、有限元法而言,大规模非均匀网格、大的电性差异以及复杂的电性结构等因素,使得最后形成的大型线性方程组的系数矩阵条件数很差,严重影响着现有迭代解法的收敛速度(如不完全Cholesky共轭梯度法,即ICCG法)。多重网格(MG)方法为近年来计算数学领域发展的新的高效算法,在数学上被证明是求解线性椭圆型偏微分方程的最优解法,其收敛速度与网格节点数无关。但由于诸多困难,在地球电磁三维数值模拟中的应用尚极少见。
论文首先针对地球物理中常见的Poisson方程及Helmholtz方程进行了几何多重网格法数值模拟,并与目前应用于电磁三维数值模拟中较好的ICCG迭代算法进行比较。结果表明,多重网格算法收敛速度均与网格节点数或网格尺度无关;而ICCG法的收敛速度则随网格节点数增加迅速减慢。几何多重网格法的计算速度比ICCG法要快数倍,证明了多重网格法三维数值模拟的高效性。
然而以上几何多重网格(GMG)法应用于直流电阻率三维数值模拟遇到很大困难。点源直流电阻率法的数值模拟是一个开域的问题,三维正演中的网格高度非均匀,同时大的电性差异(不连续电性界面)、点电源的奇异性等问题都使得几何多重网格法无法获得其固有的高效性。Zhebel(2006)在其数学系博士论文中改进了传统的几何多重网格法,应用基于矩阵的多重网格法求解带电性差异的直流电阻率三维正演问题,但收效甚微。
代数多重网格(AMG)法近10年发展非常快,在计算流体力学领域的应用表明该方法在非均匀网格和物性系数不连续的情况下依旧能保持其固有的收敛性。论文对直流电阻率三维数值模拟的有限差分格式进行求解,针对大规模非均匀网格、大的电性差异等主要问题进行了研究,实现了电阻率三维数值模拟的代数多重网格算法,并与ICCG法进行系统的比较。大量的模型计算结果表明,AMG算法收敛速度与网格节点数、计算区域、网格精细度、电性差异及模型尺寸等因素基本无关,迭代收敛次数均在7至8次左右,其计算时间随网格节点数增加基本呈较低的线性增长;ICCG法收敛曲线则呈振荡型,受上述各种因素的影响,其迭代收敛次数从数百次变化至数千次,计算时间随网格节点数增加呈非线性快速增长。一般大规模网格情况下,单个点源的三维数值模拟,AMG法的计算速度约为ICCG法的3至4倍;对多个点源进行求解时,如21个点电源,AMG方法的计算速度约为ICCG法的9倍,优势将更加明显。
在电阻率三维正演的代数多重网格快速算法基础上,将其应用于巷道超前探测的三维数值模拟研究中。近年来,国内煤矿屡次发生透水矿难,造成巨大的人员伤亡和财产损失,直流电阻率巷道超前探测面临着迫切的生产需求,然而由于可利用的观测空间有限、巷道掘进面前方异常信号弱以及巷道空腔和旁侧异常体的影响等诸多问题,目前仍是国际上的研究难点。本文在大量地下巷道模型的电阻率三维数值模拟基础上,总结分析了巷道掘进面前方异常体真实位置与观测得到的视电阻率曲线极值点位置之间的关系,首次给出了定量估计巷道掘进面前方异常体真实位置的预测模型。文中还分析了异常体尺度对视电阻率异常曲线的影响,据此,可以参照理论预测模型,定性推断前方异常体的大小。同时,对巷道旁侧异常体的影响及定位也进行了细致的研究。研究结果对巷道超前探测的生产实践有重要的理论指导意义。
多相介质的有效电导率研究在测井资料定量解释中非常重要,也是建立壳幔深部电导率模型的基础。本文应用电阻率三维数值模拟代数多重网格快速算法,发展了一种新的多相介质有效电导率反演计算方法,在两相介质电导率差异变化很宽(达108)的情况下,大量模型的有效电导率计算结果与有限元数值模拟ElecFEM3D的计算结果基本一致,表明本文的计算结果是可靠的。
上述巷道超前探测和多相介质有效电导率计算的应用研究中,均涉及大规模三维网格、大电性差异的大量模型的电阻率三维数值模拟,工作量巨大。因此,电阻率三维数值模拟代数多重网格快速算法发挥了重要作用。
论文最后研究了大型复系数矩阵的代数多重网格算法并探索其在大地电磁三维数值模拟中的应用。结果表明,代数多重网格(AMG)算法在解大型复数方程组时可以获得了多重网格算法的最优效率,但应用于大地电磁三维数值模拟还需要对AMG更深层次的数学问题进行研究才有可能获得更好的效果。
Geo-electromagnetic methods are widely used and play an important role in engineering geological survey, mining exploration and detection of the deep structure of earth’s crust and mantle. With the growing complexity of the exploration targets, the requirements for investigation of 3D fine structure become more and more urgent, which greatly challenges the efficiency of 3D geophysical numerical modeling for complex underground structures. However, the convergence rates of standard iteration solvers (such as incomplete cholesky conjugate gradient, abbr. ICCG) for forward modeling will be severely affected by large scale of non-uniform grids, big conductivity contrast, and complex electrical structure and so on, since the condition number of the coefficient matrix becomes worse. Multigrid method is of high numerical efficiency in solving linear equations arisen from boundary value problem of partial differential equation (PDE), which is newly developed by the computational mathematicians in the past decades, but seldom used in 3D geo-electromagnetic modeling so far.
First, the geometric multigrid (GMG) method is used to solve linear equation systems arisen from 3D Poisson equation and 3D Helmholtz equation, and compared to ICCG method that widely used in 3D geo-electromagnetic modeling at present. The convergence rate of GMG shows independency of the number of unknowns, while ICCG is apparently slow down as the number of unknowns growing. Moreover, GMG algorithm is much faster than ICCG algorithm in the same conditions, showing its superior efficiency.
However, the GMG method mentioned above has difficulties when applied to 3D DC resistivity modeling. The modeling of DC resistivity method with point source is an open boundary problem, with highly non-uniform 3D grids. Furthermore, big conductivity contrast (discontinuous electrical interface) and singularity of point source are also inevitable. Because of these problems, the GMG method tends to lose its intrinsic high efficiency. Zhebel(2006) improves the standard GMG method in her mathematical PH.D. thesis, using matrix-dependent multigrid (MDMG) method to solve 3D DC resistivity modeling problem with conductivity contrast, it seems that the convergence performance is not good enough.
Algebraic multigrid (AMG) method is developing very fast in the past ten years, its application in computational fluid mechanics shows it remains its intrinsic high efficiency with non-uniform grids and discontinuous coefficients. In this paper, we developed a fast 3D DC resistivity modeling algorithm using AMG to solve large linear equation system derived from finite difference simulation, and systematically compared our results with those from ICCG. Numerical calculations for a large number of models show the convergence rate of AMG algorithm is independent of the number of unknowns or the grid size, the conductivity contrast, the area of computation and the size of the inhomogeneity. The iteration number for AMG to reach convergence is around 7 to 8 times, and its computational time increases slowly and linearly as the number of unknowns grows. While the convergence curves of ICCG algorithm are oscillating, it requires hundreds or thousands of iterations to reach convergence with the influences mentioned above, and its computational time increases fast and non-linearly as the number of unknowns grows. AMG method is about 2 to 3 times faster than ICCG method for 3D DC resistivity modeling with a single point source, furthermore, AMG is about 8 times faster than ICCG for multi-sources case (21 sources for example), shows much more efficiency. Based on the fast algebraic multigrid solver, 3D DC resistivity modeling is applied to tunnel forward prediction, which is very important to the security of workers in underground mining engineering. Because of limit investigation space, weak anomalous signal in front of the tunnel, and the influences of tunnel cavity itself and the anomalies around the tunnel, the tunnel forward prediction remains a challenging task. In this paper, we carry out 3D DC resistivity modeling for a large number of underground tunnel models and analyze the relationship between the actual position of the anomaly in front of the tunnel and the position of the characteristic point(e.g. minimum point) on apparent resistivity curve. A quantitative prediction model is established, for the first time, to predict the actual position of the anomaly in front the tunnel. The influence of the size of the anomaly on observed apparent resistivity is also investigated, which could be used to qualitatively estimate the size of the anomaly in front of the tunnel. Meanwhile, a detail study on the identification and location of the anomalies around the tunnel is also carried out. The result would be very useful for the practice of tunnel forward prediction.
The effective conductivity of complex multi-phase medium is important to the interpretation of logging data, and also it is the foundation to simulate conductivity model of the deep earth’s crust and mantle which is composed of multi-phase minerals. A new method is developed to calculate the effective conductivity of the multi-phase medium using 3D resistivity modeling and inversion. With a wide range of variation of conductivity contrast (up to 10~8) of two-phase medium, the effective conductivities of two-phase medium calculated by our method are in good agreement with those from finite element modeling program ElecFEM3D, which shows our results is reliable.
Both applications in tunnel forward prediction and effective conductivity of multi-phase medium involve large scale of 3D non-uniform grids, big conductivity contrast and 3D resistivity modeling for a large number of models, requiring huge amount of numerical works. Thus, fast AMG solver for 3D resistivity modeling plays a very important role in both applications.
Finally, algebraic multigrid is applied to solve large linear equations with complex coefficients and consequently 3D magnetotelluric(MT) modeling using AMG is also investigated. Our results show that AMG algorithm obtains its intrinsic high efficiency for solving large linear equations with complex coefficients. However, we need a deep probe into the mathematical problems of AMG to get a better performance for solving 3D MT modeling.
论文首先针对地球物理中常见的Poisson方程及Helmholtz方程进行了几何多重网格法数值模拟,并与目前应用于电磁三维数值模拟中较好的ICCG迭代算法进行比较。结果表明,多重网格算法收敛速度均与网格节点数或网格尺度无关;而ICCG法的收敛速度则随网格节点数增加迅速减慢。几何多重网格法的计算速度比ICCG法要快数倍,证明了多重网格法三维数值模拟的高效性。
然而以上几何多重网格(GMG)法应用于直流电阻率三维数值模拟遇到很大困难。点源直流电阻率法的数值模拟是一个开域的问题,三维正演中的网格高度非均匀,同时大的电性差异(不连续电性界面)、点电源的奇异性等问题都使得几何多重网格法无法获得其固有的高效性。Zhebel(2006)在其数学系博士论文中改进了传统的几何多重网格法,应用基于矩阵的多重网格法求解带电性差异的直流电阻率三维正演问题,但收效甚微。
代数多重网格(AMG)法近10年发展非常快,在计算流体力学领域的应用表明该方法在非均匀网格和物性系数不连续的情况下依旧能保持其固有的收敛性。论文对直流电阻率三维数值模拟的有限差分格式进行求解,针对大规模非均匀网格、大的电性差异等主要问题进行了研究,实现了电阻率三维数值模拟的代数多重网格算法,并与ICCG法进行系统的比较。大量的模型计算结果表明,AMG算法收敛速度与网格节点数、计算区域、网格精细度、电性差异及模型尺寸等因素基本无关,迭代收敛次数均在7至8次左右,其计算时间随网格节点数增加基本呈较低的线性增长;ICCG法收敛曲线则呈振荡型,受上述各种因素的影响,其迭代收敛次数从数百次变化至数千次,计算时间随网格节点数增加呈非线性快速增长。一般大规模网格情况下,单个点源的三维数值模拟,AMG法的计算速度约为ICCG法的3至4倍;对多个点源进行求解时,如21个点电源,AMG方法的计算速度约为ICCG法的9倍,优势将更加明显。
在电阻率三维正演的代数多重网格快速算法基础上,将其应用于巷道超前探测的三维数值模拟研究中。近年来,国内煤矿屡次发生透水矿难,造成巨大的人员伤亡和财产损失,直流电阻率巷道超前探测面临着迫切的生产需求,然而由于可利用的观测空间有限、巷道掘进面前方异常信号弱以及巷道空腔和旁侧异常体的影响等诸多问题,目前仍是国际上的研究难点。本文在大量地下巷道模型的电阻率三维数值模拟基础上,总结分析了巷道掘进面前方异常体真实位置与观测得到的视电阻率曲线极值点位置之间的关系,首次给出了定量估计巷道掘进面前方异常体真实位置的预测模型。文中还分析了异常体尺度对视电阻率异常曲线的影响,据此,可以参照理论预测模型,定性推断前方异常体的大小。同时,对巷道旁侧异常体的影响及定位也进行了细致的研究。研究结果对巷道超前探测的生产实践有重要的理论指导意义。
多相介质的有效电导率研究在测井资料定量解释中非常重要,也是建立壳幔深部电导率模型的基础。本文应用电阻率三维数值模拟代数多重网格快速算法,发展了一种新的多相介质有效电导率反演计算方法,在两相介质电导率差异变化很宽(达108)的情况下,大量模型的有效电导率计算结果与有限元数值模拟ElecFEM3D的计算结果基本一致,表明本文的计算结果是可靠的。
上述巷道超前探测和多相介质有效电导率计算的应用研究中,均涉及大规模三维网格、大电性差异的大量模型的电阻率三维数值模拟,工作量巨大。因此,电阻率三维数值模拟代数多重网格快速算法发挥了重要作用。
论文最后研究了大型复系数矩阵的代数多重网格算法并探索其在大地电磁三维数值模拟中的应用。结果表明,代数多重网格(AMG)算法在解大型复数方程组时可以获得了多重网格算法的最优效率,但应用于大地电磁三维数值模拟还需要对AMG更深层次的数学问题进行研究才有可能获得更好的效果。
Geo-electromagnetic methods are widely used and play an important role in engineering geological survey, mining exploration and detection of the deep structure of earth’s crust and mantle. With the growing complexity of the exploration targets, the requirements for investigation of 3D fine structure become more and more urgent, which greatly challenges the efficiency of 3D geophysical numerical modeling for complex underground structures. However, the convergence rates of standard iteration solvers (such as incomplete cholesky conjugate gradient, abbr. ICCG) for forward modeling will be severely affected by large scale of non-uniform grids, big conductivity contrast, and complex electrical structure and so on, since the condition number of the coefficient matrix becomes worse. Multigrid method is of high numerical efficiency in solving linear equations arisen from boundary value problem of partial differential equation (PDE), which is newly developed by the computational mathematicians in the past decades, but seldom used in 3D geo-electromagnetic modeling so far.
First, the geometric multigrid (GMG) method is used to solve linear equation systems arisen from 3D Poisson equation and 3D Helmholtz equation, and compared to ICCG method that widely used in 3D geo-electromagnetic modeling at present. The convergence rate of GMG shows independency of the number of unknowns, while ICCG is apparently slow down as the number of unknowns growing. Moreover, GMG algorithm is much faster than ICCG algorithm in the same conditions, showing its superior efficiency.
However, the GMG method mentioned above has difficulties when applied to 3D DC resistivity modeling. The modeling of DC resistivity method with point source is an open boundary problem, with highly non-uniform 3D grids. Furthermore, big conductivity contrast (discontinuous electrical interface) and singularity of point source are also inevitable. Because of these problems, the GMG method tends to lose its intrinsic high efficiency. Zhebel(2006) improves the standard GMG method in her mathematical PH.D. thesis, using matrix-dependent multigrid (MDMG) method to solve 3D DC resistivity modeling problem with conductivity contrast, it seems that the convergence performance is not good enough.
Algebraic multigrid (AMG) method is developing very fast in the past ten years, its application in computational fluid mechanics shows it remains its intrinsic high efficiency with non-uniform grids and discontinuous coefficients. In this paper, we developed a fast 3D DC resistivity modeling algorithm using AMG to solve large linear equation system derived from finite difference simulation, and systematically compared our results with those from ICCG. Numerical calculations for a large number of models show the convergence rate of AMG algorithm is independent of the number of unknowns or the grid size, the conductivity contrast, the area of computation and the size of the inhomogeneity. The iteration number for AMG to reach convergence is around 7 to 8 times, and its computational time increases slowly and linearly as the number of unknowns grows. While the convergence curves of ICCG algorithm are oscillating, it requires hundreds or thousands of iterations to reach convergence with the influences mentioned above, and its computational time increases fast and non-linearly as the number of unknowns grows. AMG method is about 2 to 3 times faster than ICCG method for 3D DC resistivity modeling with a single point source, furthermore, AMG is about 8 times faster than ICCG for multi-sources case (21 sources for example), shows much more efficiency. Based on the fast algebraic multigrid solver, 3D DC resistivity modeling is applied to tunnel forward prediction, which is very important to the security of workers in underground mining engineering. Because of limit investigation space, weak anomalous signal in front of the tunnel, and the influences of tunnel cavity itself and the anomalies around the tunnel, the tunnel forward prediction remains a challenging task. In this paper, we carry out 3D DC resistivity modeling for a large number of underground tunnel models and analyze the relationship between the actual position of the anomaly in front of the tunnel and the position of the characteristic point(e.g. minimum point) on apparent resistivity curve. A quantitative prediction model is established, for the first time, to predict the actual position of the anomaly in front the tunnel. The influence of the size of the anomaly on observed apparent resistivity is also investigated, which could be used to qualitatively estimate the size of the anomaly in front of the tunnel. Meanwhile, a detail study on the identification and location of the anomalies around the tunnel is also carried out. The result would be very useful for the practice of tunnel forward prediction.
The effective conductivity of complex multi-phase medium is important to the interpretation of logging data, and also it is the foundation to simulate conductivity model of the deep earth’s crust and mantle which is composed of multi-phase minerals. A new method is developed to calculate the effective conductivity of the multi-phase medium using 3D resistivity modeling and inversion. With a wide range of variation of conductivity contrast (up to 10~8) of two-phase medium, the effective conductivities of two-phase medium calculated by our method are in good agreement with those from finite element modeling program ElecFEM3D, which shows our results is reliable.
Both applications in tunnel forward prediction and effective conductivity of multi-phase medium involve large scale of 3D non-uniform grids, big conductivity contrast and 3D resistivity modeling for a large number of models, requiring huge amount of numerical works. Thus, fast AMG solver for 3D resistivity modeling plays a very important role in both applications.
Finally, algebraic multigrid is applied to solve large linear equations with complex coefficients and consequently 3D magnetotelluric(MT) modeling using AMG is also investigated. Our results show that AMG algorithm obtains its intrinsic high efficiency for solving large linear equations with complex coefficients. However, we need a deep probe into the mathematical problems of AMG to get a better performance for solving 3D MT modeling.
引文
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