Numerical solution of differential and integral equations
The stationary electric field as well as the transient process induced by the vertical current line with finite dimensions can be studied when using the different technologies for ground-borehole electrical survey. The vertical current line is most often placed inside the borehole with the casing pipe. The characteristics of the electromagnetic field which are thereafter interpreted in order to locate the objects being searched are registered on the day surface (i.e. on the EarthТs surface).
Analytical and semi-analytical methods used to date for calculating non-stationary field induced by the vertical current line are mainly applied to the simple model of the medium and make it impossible to take account of axis symmetrical and especially three dimensional objects as well as the subareas with a very high conductivity (the cased borehole).
In this report two methods of finite element modeling of the transient process induced by the vertical current line in the axis symmetrical medium are considered. They make it possible to take into consideration not only the axis symmetrical geological objects but the characteristics of casing pipes as well. The first method is based on the scalar task for phi-component of the magnetic field intensity and the second one is based on the vector task for vector-potential of the magnetic field. The solution of the scalar task is done with the use of nodal FEM (with scalar basis functions) and as for the solution of the vector task, FEM with edge elements is used.
The conducted research showed that in the case of calculating the fields for geoelectrical models without casing pipes, the scalar method has significant advantages (approximately 10 times) in consumption of computing time when obtaining the solution with the required accuracy. In order to calculate the fields for geoelectrical models with casing pipes the vector method is used which allows the solution of the task with the required accuracy almost on the same meshes as in the case without casing pipes. To obtain the same accuracy the scalar method requires much more elaborated meshes and calculation time is two orders more.
The approach based on the division of the desired electromagnetic field into axis symmetrical and three dimensional components has been used for solving of the three dimensional task of the transient process induced by the vertical current line.
For calculation of the three dimensional components two numerical procedures are required. One of them is the procedure of calculating non-stationary three dimensional abnormal field (i.e. the field of three dimensional objects influence) through the values of the normal (axis symmetrical) field. This procedure was developed and applied to the solution of the task including the source in the form of the transmitter loop. However, for calculating the field which is induced by the vertical current line the second procedure of calculating the vector-potential of the stationary magnetic field is required. This procedure was developed on the basis of finite element approximation of the corresponding vector task including the sources defined with three dimensional non-uniformity of investigated medium resistivity.
The methods developed have been used for software design which makes it possible to solve the tasks of the transient process in complex media for sources in the form of the vertical current line.
Note. Abstracts are published in author's edition
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