矿用多匝小回线源瞬变电磁场数值模拟与分布规律研究
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摘要
矿井瞬变电磁法勘探是预测巷道顶、底板和掘进工作面附近突水构造的主要物探方法之一,与地面瞬变电磁法勘探相比,该方法具有分辨率高、体积效应小、灵活和高效等优点;但其基础理论的发展却相对滞后,使之成为矿井瞬变电磁法勘探技术中亟待解决的问题。本文通过数值模拟的方法对全空间介质中的巷道影响以及典型地质-地球物理模型的瞬变电磁响应进行了研究。
推导了全空间多匝小回线源产生的瞬变磁场解析式,分析了瞬变电磁一次场、二次场和总场暂态过程,并对全空间磁偶极源中线性关断电流和半正弦关断电流关断效应的计算方法进行了研究。结果表明:磁偶极子场在r=0处存在奇异点,致使中心回线装置无法使用。多匝小回线源二次场的衰减规律与磁偶极源的二次场相似,但其磁场值在“早期”约为磁偶源场值的10 3倍。关断时间的长短影响着一次场感应电动势的幅值和暂态过程的时间范围,关断时间增长使感应时间增加,电动势幅值降低,总场的衰减速度加快。线性关断电流和半正弦关断电流对瞬变电磁场的影响都主要表现在“早期”,关断时间越长,关断效应对瞬变电磁场的影响越向瞬变“晚期”延伸,半正弦关断电流的影响相对更大。
采用时域有限差分法推导了3D瞬变电磁场差分方程,证明了3维Euler法和DuFort-Frankel法的稳定性,得出了差分计算中初始时间步长和网格步长的关系。结果表明:三维DuFort-Frankel差分算法是无条件稳定的,瞬变电磁场的扩散速度与采样时间间隔和电阻率差异有关。与全空间多匝小回线源解析解作比较,采用非均匀网格技术并选取相应的时间步长计算三维瞬变电磁场差分方程具有较好的精度,在1.05ms时刻,奇异点以外各节点处的磁场值与解析解的最大误差不超过12%。
以廖氏吸收边界条件为基础,推导了修正的廖氏吸收边界条件,并提出了差分计算中巷道边界条件问题。结果表明:新的吸收边界条件能较好的反映瞬变电磁场的扩散规律,对低频电磁波具有更好的吸收效果,当选取α=1、γ=0.98、N=4和初始时间步长在50μs之内时,修正的廖氏吸收边界条件的最大误差不超过0.01。巷道边界条件适用于扩散方程的差分计算并具有很好的稳定性。
定义了巷道影响因子的概念,研究了巷道影响与装置形式、巷道几何大小、围岩导电性、装置位置以及关断时间的关系。结果表明:巷道空间的存在使瞬变电磁场感应电动势值减小;关断时间越长巷道对瞬变电磁场的影响越严重;与分离回线相比,重叠回线的观测结果受巷道影响较大,两种装置形式观测的瞬变晚期场受巷道影响程度近似相同;巷道围岩导电性越强,巷道几何尺寸越大,巷道及其边界对瞬变电磁场的影响越大;重叠回线装置在巷道底板、顶板和侧帮中心点上探测时受到巷道影响程度相同,而当装置位于巷道堵头中心点时,瞬变电磁场受巷道影响最大。
采用了3维切片和2维切片结合的成图方法,研究了典型地质-地球物理模型的瞬变电磁场分布规律和视电阻率异常响应特征,并用数值模拟的方法验证了物理模拟实验的效果。结果表明:全空间瞬变电磁响应的强度和持续时间与介质的电阻率差异、尺寸、形状、至测点的距离等因素相关;层状介质响应与各层间的电阻率差异、层厚相关;断层的断点对瞬变电磁场响应影响最大;对同一地质模型,采用数值模拟和物理模拟实验的方法得出的结论相同。
Mine transient electromagnetic method (MTEM) is one of mainly geophysical methods, which are used to predict water inrush structures nearby the roof, bottom and driving face in roadway. Compared with ground transient electromagnetic method, this technique has many advantages: high resolution, small volume effect, flexibility and high efficiency. However, relatively slow development of its fundamental theory makes it become an immediate problem to be solved. These theoretical problems, roadway effect, transient response for the typical geologic geophysical model in whole-space are studied using numerical modeling method in this paper.
Analysis solution of whole-space multi-turn coils was derived; transient processes of primary, secondary and total field were analyzed; and calculation method of turn-off effects for linear current and half sine current in whole-space magnetic dipole source was also studied. The results indicate that central loop cannot be used because of singular point r=0 in magnetic dipole field. Secondary field of multi-turn coils has a similar attenuation regular with magnetic dipole source, but its magnetic value is 10 3 times than magnetic dipole value in“early time”. Amplitude value of primary field induced electromotive force and time range of transient process are influenced by turn-off time; increase of inductance time, decrease of electromotive force amplitude value, and acceleration of total field attenuation are caused by the longer turn-off time.“Early time”transient field is disturbed heavily by linear and half sine turn-off current; influence to transient field affected by the turn-off effect extends to“later time”for longer turn-off time; and compared with linear turn-off current, the half sine current has more heavy influence on transient field than linear current.
Time domain finite-difference method was used to derive three-dimensional difference equation for transient electromagnetic field; and stability of 3D Euler method and Dufort-Frankel method was proved, and relationship between initial time and grid length in numerical calculation was obtained in this proof process. The results show that, 3D Dufort-Frankel method is unconditional stable; the diffusion velocity of transient field is related to sample interval and resistivity contrast. Numerical method of 3D transient difference equation, using non-uniform grid method and setting relevant time step length parameter, has a high accuracy by comparing numerical solution with theoretical solution of multi-turn coils; actually, the maximum error between numerical and analyzed solution in computational domain beyond the singular point is less than 12% at the moment 1.05ms.
Based on Liao absorbing boundary condition, modified Liao absorbing boundary condition was derived; problem of roadway boundary condition was presented for the first time. The results indicate that, using the new boundary condition can preferably reflect the diffusion regular of transient field, and has a better absorbing effect on low frequency electromagnetic wave. The maximum error of new Liao absorbing boundary condition is less than 0.01, when choosing the parameterα=1,γ=0.98 and N=4, and the initial time step length less than 50μs. Roadway boundary condition has a good applicability and stability in difference calculation of diffusion equation.
Roadway factors were defined; and relationship between roadway effect and equipments, geometry size of roadway, host rock conductivity, equipment position and turn-off time was studied by using roadway factors. The results show that, inductive electromotive force of transient electromagnetic field is much lower because of roadway effect; roadway effect increases with turn-off time. Compared with separated loop, data measured by coincident loop is influenced heavily by roadway space, but transient“later time”field of the two equipments is approximately equal. The bigger geometry size of roadway and the higher conductivity of host rock are, the heavier transient field affected by roadway effect and its boundary is. Roadway effect is same when coincident loop is located at the center of roadway roof, floor and side; and it is heaviest when coincident loop is located at the center of roadway plug.
The method of plotting combining 3D slice with 2D slice was used; the diffusion regular and anomalous characteristics of apparent resistivity for typical geologic-geophysical models in whole-space were studied; the effect of physical model experiment was verified by using numerical method. The results indicate that, the strength and time of transient response is related to several factors in the low-resistivity layer: resistivity, geometric size, shape and the distance from roadway. Transient response of layered medium is determined by resistivity contrast and layer thickness. The influence to transient response at break points in fault is stronger. Numerical method has similar conclusions with physical experiment for the same geologic model.
推导了全空间多匝小回线源产生的瞬变磁场解析式,分析了瞬变电磁一次场、二次场和总场暂态过程,并对全空间磁偶极源中线性关断电流和半正弦关断电流关断效应的计算方法进行了研究。结果表明:磁偶极子场在r=0处存在奇异点,致使中心回线装置无法使用。多匝小回线源二次场的衰减规律与磁偶极源的二次场相似,但其磁场值在“早期”约为磁偶源场值的10 3倍。关断时间的长短影响着一次场感应电动势的幅值和暂态过程的时间范围,关断时间增长使感应时间增加,电动势幅值降低,总场的衰减速度加快。线性关断电流和半正弦关断电流对瞬变电磁场的影响都主要表现在“早期”,关断时间越长,关断效应对瞬变电磁场的影响越向瞬变“晚期”延伸,半正弦关断电流的影响相对更大。
采用时域有限差分法推导了3D瞬变电磁场差分方程,证明了3维Euler法和DuFort-Frankel法的稳定性,得出了差分计算中初始时间步长和网格步长的关系。结果表明:三维DuFort-Frankel差分算法是无条件稳定的,瞬变电磁场的扩散速度与采样时间间隔和电阻率差异有关。与全空间多匝小回线源解析解作比较,采用非均匀网格技术并选取相应的时间步长计算三维瞬变电磁场差分方程具有较好的精度,在1.05ms时刻,奇异点以外各节点处的磁场值与解析解的最大误差不超过12%。
以廖氏吸收边界条件为基础,推导了修正的廖氏吸收边界条件,并提出了差分计算中巷道边界条件问题。结果表明:新的吸收边界条件能较好的反映瞬变电磁场的扩散规律,对低频电磁波具有更好的吸收效果,当选取α=1、γ=0.98、N=4和初始时间步长在50μs之内时,修正的廖氏吸收边界条件的最大误差不超过0.01。巷道边界条件适用于扩散方程的差分计算并具有很好的稳定性。
定义了巷道影响因子的概念,研究了巷道影响与装置形式、巷道几何大小、围岩导电性、装置位置以及关断时间的关系。结果表明:巷道空间的存在使瞬变电磁场感应电动势值减小;关断时间越长巷道对瞬变电磁场的影响越严重;与分离回线相比,重叠回线的观测结果受巷道影响较大,两种装置形式观测的瞬变晚期场受巷道影响程度近似相同;巷道围岩导电性越强,巷道几何尺寸越大,巷道及其边界对瞬变电磁场的影响越大;重叠回线装置在巷道底板、顶板和侧帮中心点上探测时受到巷道影响程度相同,而当装置位于巷道堵头中心点时,瞬变电磁场受巷道影响最大。
采用了3维切片和2维切片结合的成图方法,研究了典型地质-地球物理模型的瞬变电磁场分布规律和视电阻率异常响应特征,并用数值模拟的方法验证了物理模拟实验的效果。结果表明:全空间瞬变电磁响应的强度和持续时间与介质的电阻率差异、尺寸、形状、至测点的距离等因素相关;层状介质响应与各层间的电阻率差异、层厚相关;断层的断点对瞬变电磁场响应影响最大;对同一地质模型,采用数值模拟和物理模拟实验的方法得出的结论相同。
Mine transient electromagnetic method (MTEM) is one of mainly geophysical methods, which are used to predict water inrush structures nearby the roof, bottom and driving face in roadway. Compared with ground transient electromagnetic method, this technique has many advantages: high resolution, small volume effect, flexibility and high efficiency. However, relatively slow development of its fundamental theory makes it become an immediate problem to be solved. These theoretical problems, roadway effect, transient response for the typical geologic geophysical model in whole-space are studied using numerical modeling method in this paper.
Analysis solution of whole-space multi-turn coils was derived; transient processes of primary, secondary and total field were analyzed; and calculation method of turn-off effects for linear current and half sine current in whole-space magnetic dipole source was also studied. The results indicate that central loop cannot be used because of singular point r=0 in magnetic dipole field. Secondary field of multi-turn coils has a similar attenuation regular with magnetic dipole source, but its magnetic value is 10 3 times than magnetic dipole value in“early time”. Amplitude value of primary field induced electromotive force and time range of transient process are influenced by turn-off time; increase of inductance time, decrease of electromotive force amplitude value, and acceleration of total field attenuation are caused by the longer turn-off time.“Early time”transient field is disturbed heavily by linear and half sine turn-off current; influence to transient field affected by the turn-off effect extends to“later time”for longer turn-off time; and compared with linear turn-off current, the half sine current has more heavy influence on transient field than linear current.
Time domain finite-difference method was used to derive three-dimensional difference equation for transient electromagnetic field; and stability of 3D Euler method and Dufort-Frankel method was proved, and relationship between initial time and grid length in numerical calculation was obtained in this proof process. The results show that, 3D Dufort-Frankel method is unconditional stable; the diffusion velocity of transient field is related to sample interval and resistivity contrast. Numerical method of 3D transient difference equation, using non-uniform grid method and setting relevant time step length parameter, has a high accuracy by comparing numerical solution with theoretical solution of multi-turn coils; actually, the maximum error between numerical and analyzed solution in computational domain beyond the singular point is less than 12% at the moment 1.05ms.
Based on Liao absorbing boundary condition, modified Liao absorbing boundary condition was derived; problem of roadway boundary condition was presented for the first time. The results indicate that, using the new boundary condition can preferably reflect the diffusion regular of transient field, and has a better absorbing effect on low frequency electromagnetic wave. The maximum error of new Liao absorbing boundary condition is less than 0.01, when choosing the parameterα=1,γ=0.98 and N=4, and the initial time step length less than 50μs. Roadway boundary condition has a good applicability and stability in difference calculation of diffusion equation.
Roadway factors were defined; and relationship between roadway effect and equipments, geometry size of roadway, host rock conductivity, equipment position and turn-off time was studied by using roadway factors. The results show that, inductive electromotive force of transient electromagnetic field is much lower because of roadway effect; roadway effect increases with turn-off time. Compared with separated loop, data measured by coincident loop is influenced heavily by roadway space, but transient“later time”field of the two equipments is approximately equal. The bigger geometry size of roadway and the higher conductivity of host rock are, the heavier transient field affected by roadway effect and its boundary is. Roadway effect is same when coincident loop is located at the center of roadway roof, floor and side; and it is heaviest when coincident loop is located at the center of roadway plug.
The method of plotting combining 3D slice with 2D slice was used; the diffusion regular and anomalous characteristics of apparent resistivity for typical geologic-geophysical models in whole-space were studied; the effect of physical model experiment was verified by using numerical method. The results indicate that, the strength and time of transient response is related to several factors in the low-resistivity layer: resistivity, geometric size, shape and the distance from roadway. Transient response of layered medium is determined by resistivity contrast and layer thickness. The influence to transient response at break points in fault is stronger. Numerical method has similar conclusions with physical experiment for the same geologic model.
引文
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