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本文阐述了用点源二维有限差分法计算电测深曲线的原理、数学模型及计算实例。此法采用不等间隔的网格,将置于柱坐标系中的半无限介质进行离散化。由于原点放在点源的位置上,仅须在四分之一空间对电场作二维有限差分数值模拟。为了不漏掉那些厚度小于网格间距的“薄层”对电场分布的影响,利用各电性层的纵向电导和横向电阻等所谓Dar Zarrouk参数计算差分方程中的系数,从而取得“差分网格疏而不漏”的效果。用逐次超松弛(SOR)法求解差分方程组,每迭代五次优选一次松弛系数,直至相继两次系数值的变化小于10~(-2)。计算结果表明,有限差分法较之数字滤波法要占用较长的计算时间,它不仅适用于对称四极梯度装置形式,而且也能解决各向异性介质的电阻率数值的模拟问题。
In this paper, the principle, mathematical model and calculation example of calculating electric sounding curve with point source two-dimensional finite difference method are expounded. This method uses a grid of unequal intervals to discretize semi-infinite media placed in a cylindrical coordinate system. Since the origin is placed at the point source, only two-dimensional finite difference numerical simulations of the electric field must be made in a quarter space. In order not to miss the influence of the “thin layer” whose thickness is less than the grid spacing on the electric field distribution, the so-called Dar Zarrouk parameters such as the longitudinal conductivity and the transverse resistance of each electric layer are used to calculate the coefficients in the difference equation so as to obtain the “ Sparse but not leaked ”effect. Difference equations are solved by successive over relaxation (SOR) method. The relaxation coefficient is optimized five times per iteration until the change of the coefficient value is less than 10 ~ (-2) two consecutive times. The calculation results show that the finite difference method takes longer time than the digital filter method. It not only can be used in the form of symmetrical quadrupole gradient device, but also can solve the problem of numerical simulation of resistivity in anisotropic media.