井地电位法的异常电位边值问题及其等价的变分问题

Anomalous potential boundary value problem of borehole-to-surface electrical resistivity method and its equivalent variational problem

井地电位法是地下井中供电,并在供电地表观测的一种充电法。该方法在煤气田压裂成像、城市地下空间成像等地下动态导体的监测与成像领域应用广泛。井地电位法一般采用具有较高精度的异常电位有限单元法进行正演数值模拟。本文从电磁场论出发,推导了地下点电流源的总电位和异常电位的边值问题,并针对地下点电流源在无穷远边界上边界条件的解析表达式进行了详细的理论推导和论证;基于变分原理从地下点电流源的边值问题出发,推导、论证并得到了与之等价的变分问题,并结合地下任意线电流源正常电位的解析公式,推导得到地下任意线电流源异常电位的变分问题;基于互易定理验证了对上述结果的正确性。

The borehole-to-surface electrical resistivity method(BSERT)is a mise-à-la-masse method in which power is supplied in underground wells and the potential is observed at the supplied surface.The method is widely used in the field of monitoring and imaging underground dynamic conductors such as gas field fracturing imaging and urban underground space imaging.The BSERT generally uses the finite element method for the anomalous potential which has higher accuracy for the forward numerical simulation.The boundary value problems of the total potential and the abnormal potential of the underground point current source are derived from the electromagnetic field theory,and the analytical expressions of the boundary conditions of the underground point current source on the infinite boundary are deduced and demonstrated in detail.Based on the variational principle,starting from the boundary value problems of the underground point current source,the equivalent variational problem is derived and demonstrated.Combined with the analytical formula of the normal potential of the underground arbitrary line current source,the variational problem of the abnormal potential of the underground arbitrary line current source is derived.The correctness of the aforementioned results is confirmed by satisfying the reciprocity theorem.