Volume-surface Integral Equation And Its Application In Numerical Simulation Of Electrical Well-logging

In geophysical well logging, the proportion of highly-deviated well logging and horizontal well longing is getting larger and larger. The tool responses in these borehole environments are different from that in vertical well. Thus, the traditional methods of numerical simulation and data analysis based on vertical well are no longer applicable. Therefore, to explain and analyse the tool responses in these borehole enbironments, it is necessary to carry on research on the forward numerical simulation of electrical well-logging, which is significant for the development of new insruments and engineering practice.Due to the above reasons, this paper presents the study on the volume-surface integral equations(VSIE) and its application in numerical simulation of electrical well-logging. The tradtitonal volume integral equations(VIE) based on SWG basis function requires conformal meshes in geometric discretization and processing of surface electric charges While the nonconformal VIE discussed in this paper is based on piecewise constant basis function, which makes it free from surface electric charges and allow a more flexible geometric discretization of nonconformal meshes. But a hypersingular integral involving in the gradient-gradient operator applied on the Green’s function must be handled. A high-order singularity extraction technique is used to accurately evaluate the integral. The main contents of this paper are as follws:1. The process of solving the surface integral equation(SIE), VIE and VSIE with method of momnets(MOM) is deduced in detail; the surface and volume integrals which are singular or near-singular are evaluated by a high-order singularity extraction technique. The VIE and SIE are based on piecewise constant basis function and the RWG basis function, respectively. The VSIE is formulated by the combination of these two integral eqtqtions, and it’s used to solve the scattering problem of composite objects of metal and dielectric materials exactly.2. The nonconformal VIE solving by point-matching method(PMVIE) is formulated, which makes the calculation of imedence matrix elements faster without the loss of precision. And by combining with the SIE solving by Galerkin method, the VSIE solving by the hybrid of two match methods is established, and its precision is verified by numerical examples.3. An analytical method basd on spherical functions expansion is used to calculate the exact field of loop-coil.4. The different responses of horizontal coils and tilt coils in anisotropic formations are compared. Numerical simulation of the responses of coil array with different tilt angles in isotropic formations are presented, revealing a complicated changes of amplitude ration and phase difference. Besides, the 3-D well logging problems such as tilted formation, wellbore deviation are aslo considered.5. In geometric modeling of borehole and steel mandral, a vertical truncation scheme is used, and the meshs on the borehole boundary surface are nonconformal. To use the same set of meshs for different measuring points, the positions of meshes in the borehole change with the measuring point.