Finite Difference Modeling Of DC Electrical Field Including Undulate Topography

Resistivity method is an effective means of orebody location, but along with the increase of target cognition and complicated degree of target itself, resistivity method subject to a certain degree of difficulties and challenges, especially complex surface problems. Along with the development of production practice, and shallow mineral outcrops are relatively easy to explore and development of mineral resources is gradually exhausted. Finding the subject of deep concealed deposits has been paid more and more attention, and most of China's mineral resources locate in the mountainous terrain and varied surface features became the insurmountable difficulties electric exploration, In many major national construction projects, such as the location of dam construction, the pre-exploration of the laying railway, tunnel construction, are also facing the problem of undulate terrain. Undulate topography has a serious impact to the resistivity, as interference to the normal form of apparent resistivity, resulting in lower interpretation accuracy and difficulty to explain or even give wrong conclusions. To solve these problems, researching the electric field distribution and the apparent resistivity under the undulate terrain and topography is very meaningful. This will improve the interpretation of resistivity and the precision of exploration.Early using of physical modeling and theoretical analysis made some interesting results and understanding, but it restricts by the high cost of physical simulation and issues of scale matching, and theoretical analysis applies only to a few simple models. Thus, after widely using digital computer numerical simulation becomes the main method for irregular surface problems.So far, numerical methods mainly have integral equation method, the boun dary element method and finite element method, the finite volume method and finite difference method. Integral equation method and the boundary element method are non-grid method, moreover the finite element method and the finite volume method and the finite difference belong to the grid method. The non-grid method has the feature of strong adaptability to bend boundary and efficient computation, but relatively grid method it appears a poor adaptability to model. The grid method of finite element method and finite volume method also have strong adaptability to bend boundary, while the traditional finite difference handles bend boundary more difficultly. Although the ability in dealing with bend boundary on the finite element method and finite volume method are better than traditional finite difference method, but the finite element method and the finite volume method are complex through the complicated mesh dissection to more precise approximation for bend boundary to achieve its strong bend boundary processing power. Well known complicated mesh dissection is time-consuming, and generated grid quality has a direct effect on numerical calculation.To deal with irregular surface, and avoiding the complex mesh generation of finite element and finite volume method, we introduce transfer method, non-equidistant finite difference method and the ghost Method to made DC field numerical simulation on undulate topography under the Cartesian grid from the research areas of seismic exploration, fluid dynamics and numerical heat transfer, in order to improve the adaptability to the boundary curve of finite difference method.Transfer method is the means that the irregular interior point which near the surface approximate the surface boundary point. We implement 2 dimension 2.5 dimension and 3 dimension numerical modeling by transfer method in this thesis. Meanwhile, the article focuses on the following issues.①The difference scheme on the irregular interior point.②Computation the cosine angle of the direction from source to boundary and the normal direction of surface boundary.③The selection of background field on the irregular interior point.④Power position problem.⑤Undulate topography smooth processing.⑥The selecting of most optimal wave number and wave number values.⑦Accuracy analysis.Non-equidistant finite difference method under the Cartesian mesh dissection uses the non-equidistant difference scheme on the irregular interior point, and directly implements the surface boundary condition at the surface. Such processing avoids that the irregular interior point approximate surface boundary point. Non-equidistant finite difference method have the following three mainly problems involved.①Implement the surface boundary condition.②Valve value to the spacing of non-equidistant difference has impact to the apparent resistivity.③Accuracy analysis.Ghost method uses extrapolating method to extrapolate the potential of ghost point up the surface, so that the irregular interior point can use isometric difference scheme within ghost point. The most striking characteristic of ghost method is the virtual ghost point introduced, making irregular interior point and regular interior point adopt uniform difference scheme. In the ghost method we mainly elaborate the following questions.①Extrapolation the potential of ghost point.②The accuracy of visual resistivity is influenced by the order of polynomial.In addition to topography issues, the article also discusses the computational efficiency of solving linear equations. In the finite difference, the main time overhead focus on the computing time for solving linear equations. Thereby increasing the efficiency of solving linear equations relates to the practicality of the finite difference method. This paper uses multi-grid method which is an ideal and efficient algorithm. In this paper, multi-grid method and the SOR iterative method are compared in terms of computational efficiency and accuracy under the lever surface model and hollow model. The results show that the efficiency of multi-grid method is double than SOR iterative method, meanwhile the accuracy is also better than the SOR. It can be said that in the future the multi-grid is an important method in numerical simulation of DC method for solving linear equations.Through the above three finite difference Numerical simulations of electric field under undulate topography on Cartesian grid, we can have the following understanding.①The numerical simulation of electric field based on Cartesian mesh to solve the surface problem is feasible. Transfer method, non-equidistant finite difference method and the ghost point method can respectively handle the undulate surface problem, and the accuracy can meet the computing requirement.②The finite difference numerical simulation of electric field based on Cartesian mesh avoids the complex mesh generation of finite element and finite volume method. Cartesian grid simplifies the meshing process and saves storage space to grid node locations.③The valve value of non-equidistant grid spacing affects the smoothness of apparent resistivity, so the valve should not get too large value.④Finite difference method based on Cartesian grid can improve the accuracy on the boundary curve. Both non-equidistant finite difference and ghost point method can improve the accuracy of discrete format at undulate surface, thereby enhancing the accuracy of whole potential field.