The boundary element method for 3‐D dc resistivity modeling in layered earth

Geophysics ◽  
2002 ◽  
Vol 67 (2) ◽  
pp. 610-617 ◽  
Author(s):  
Qinzhong Ma
Keyword(s):  
Integral Equation ◽  
Boundary Element Method ◽  
Boundary Element ◽  
Half Space ◽  
Dc Resistivity ◽  
Model Calculations ◽  
Layered Earth ◽  
Inhomogeneous Bodies ◽  
Resistivity Modeling ◽  
Element Method

The integral equation for dc resistivity modeling of 3‐D inhomogeneous bodies buried in a layered earth is derived by using Green's theorem. The main features of this method are (1) the layers above and below the 3‐D object can be included, (2) multiple subsurface inhomogeneous bodies can be embedded in the different layers, and (3) the boundary element method (BEM) is used to solve the integral equation using triangular surface elements. Linear variation of the electrical properties is assumed within each element. The potential on the ground surface is obtained by solving the linear equation system with Gaussian elimination. Model calculations demonstrate that the results obtained by this method compare well with the analytical solution of a sphere in a uniform half‐space and the asymptotic behavior for the solution of a buried body beneath a surficial layer as the layer resistivity approaches that of the half‐space. A comparison of responses over elongate 3‐D bodies with responses over 2‐D bodies of identical cross‐section also shows satisfactory agreement.

A boundary element method for 2-D dc resistivity modeling with a point current source

Geophysics ◽  
1998 ◽  
Vol 63 (2) ◽  
pp. 399-404 ◽  
Author(s):  
Shi‐Zhe Xu ◽  
Shengkai Zhao ◽  
Yi Ni
Keyword(s):  
Boundary Element Method ◽  
Boundary Element ◽  
Relative Error ◽  
Current Source ◽  
Normal Derivative ◽  
Relative Difference ◽  
Dc Resistivity ◽  
Maximum Relative Error ◽  
Resistivity Modeling ◽  
Element Method

A new boundary element method (BEM) is presented for 2-D dc resistivity modeling with a point source. When compared with previously published techniques, the new method has two main features: (1) The normal derivative of potential has been eliminated from the integral equation. The formulation of the present method is simpler and requires less memory and time than the previous published methods. (2) Multiple subsurface inhomogeneous bodies can be modeled. For a simple testing model, the maximum relative error of reciprocity test is 0.24% and the average relative error is 0.05%. For the same model, the maximum relative difference between the BEM solution and the finite‐element method solution is 1.14% and the average relative difference is 0.73%. For a field geoelectrical profile, the responses of the constructed model agree with the observed data quite well.

Removing Non-Uniqueness in Symmetric Galerkin Boundary Element Method for Elastostatic Neumann Problems and its Application to Half-Space Problems

2020 ◽  
Vol 36 (6) ◽  
pp. 749-761
Author(s):  
Y. -Y. Ko
Keyword(s):  
Boundary Element Method ◽  
Rigid Body ◽  
Boundary Element ◽  
Integral Equations ◽  
Half Space ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Neumann Problems ◽  
Galerkin Boundary Element Method ◽  
Element Method

ABSTRACTWhen the Symmetric Galerkin boundary element method (SGBEM) based on full-space elastostatic fundamental solutions is used to solve Neumann problems, the displacement solution cannot be uniquely determined because of the inevitable rigid-body-motion terms involved. Several methods that have been used to remove the non-uniqueness, including additional point support, eigen decomposition, regularization of a singular system and modified boundary integral equations, were introduced to amend SGBEM, and were verified to eliminate the rigid body motions in the solutions of full-space exterior Neumann problems. Because half-space problems are common in geotechnical engineering practice and they are usually Neumann problems, typical half-space problems were also analyzed using the amended SGBEM with a truncated free surface mesh. However, various levels of errors showed for all the methods of removing non-uniqueness investigated. Among them, the modified boundary integral equations based on the Fredholm’s theory is relatively preferable for its accurate results inside and near the loaded area, especially where the deformation varies significantly.

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