Mathematical Modelling of Electric and Magnetic Fields of Main Pipelines Cathodic Protection in Electrically Anisotropic Media

A mathematical model is constructed and computational experiments are performed to study the effect of anisotropy of the specific electrical conductivity of the soil on the distribution of electric and magnetic fields generated by cathodic electrochemical corrosion protection stations of the underground main pipeline (MP). The variation of electric and magnetic fields depending on the azimuth angle of rotation of the specific electrical conductivity tensor of the soil containing the pipeline is analysed.


Introduction
Cathodic electrochemical protection of underground main pipelines slows down the rate of corrosion of the pipe metal by shifting the electrical protective potential at the pipe/soil boundary to a predetermined interval at which the oxidative processes in metal are reduced.
In a cathodic protection system that includes anode beds and pipe, the distribution of electrical properties of the host soil plays a significant role in the dispersion of electric and magnetic fields.
Geological processes in the earth crust, such as sedimentation, shifts, and raising, lead to horizontal, inclined, or vertical stratifications, rock fractures, and, as a result, anisotropy of their physical properties [1]. Electrical anisotropy is observed in rocky areas of mountainous zones of pipelines.
In this paper, we study the electric and magnetic fields [2] that occur in electrochemical cathodic protection systems of main pipelines in anisotropic media by specific electrical conductivity.

Mathematical model
Let there be given a space divided by a plane boundary ^` oX oY oZ , respectively. The specific electrical conductivity of the air 0 σ is assumed to be zero. In the special case, when const , we get a homogeneous isotropic, constant in specific electrical conductivity half-space. Let the half-space 1 Ω contain a pipeline t V centre line geometry is described by a parametric curve . Parameter values s equal to 0 and t L correspond to the marks through which the electric current does not flow along the pipe metal. We'll assume that the radius t R , specific electrical conductivity σ tm of the pipe metal and the cross sectional area of the pipe metal are known.
If the pipe central line  in the matrix Ψ α ( ) correspond to the positive direction of rotation for the right coordinate system. The azimuth rotation of the coordinate system entails an equivalent change in the electrical conductivity tensor. Thus, it is possible to extend the set of considered anisotropic media described by the diagonal tensors to anisotropic media the specific electrical conductivity tensor of which is given by a matrix U P . The mathematical model describing the distribution of the DC field potential in the system has the form:  For solution the problem (1) -(10) the method of fictitious sources was used [3]. In [4], this method is applied to the case of flat-parallel horizontally layered vertically inhomogeneous media. The values of the DC potential at any point in the half-space 1 Ω are given by the formula [4]:  x y z Q ) has the form: The electric current density at any point in the ground anisotropic half-space is found from (11) by the formula We'll assume that the electrically anisotropic halfspace containing the pipeline is homogeneous, isotropic, and constant in its magnetic properties. Then the magnetic induction vector where > @ , is the vector product, 7 0 μ 4π 10 (H/m)is the magnetic constant, and μ is the relative magnetic permeability.

Results of computational experiments
The software implementation of the calculation method described above is made in the C++ programming language.
It is assumed that the physical properties of a pipe 15 m long can be considered constant. Numerical calculations are performed for the case when there are no extended anode beds, and current is injected into the system only from point anodes.
In the numerical experiments, the half-space is anisotropic, with a specific electrical conductivity tensor In experiments as "defective" pipeline segment the segment # 800 with the centre at x=11992.5 m is taken. Contact resistance insulation with "defective" segment is assumed to be 6000.0 Ω*m 2 -40% of the transition resistance isolation with the other segments of pipe.
The dependence of the protective potential along the pipeline on the angle of rotation of the specific electrical conductivity tensor is shown on Fig. 2.    Calculations were made for the azimuth angle of rotation of the soil anisotropy tensor of 0 degrees (the angle is indicated by the lower index). The picture insets show more detailed surface fragments nearby the pipeline "defective" segment. As for the other azimuth angles of rotation the graphs of the distribution of module of magnetic induction vector and the x-gradient of the module of magnetic induction vector are not much distinct visually, the difference between them will show graphs of the modulus of difference of the modules of the magnetic induction vector and the the modulus of difference of gradient on the x-modules vector of magnetic induction.
With this specific electrical conductivity tensor, the maximum value of these functions is obtained for the rotation angles of 0 and 90 degrees. Figures 5 and 6 demonstrate the difference (modulus of difference) of fields in anisotropic media whose anisotropy tensors are azimuthally rotated by 0 and 90 degrees. The difference reaches 9 nT and 0.11 nT/m, respectively.

Conclusions
The computer simulation of electric and magnetic fields demonstrates the influence of soil electrical conductivity anisotropy when calculating the fields of pipeline cathodic protection whose pipelines are located in electrically anisotropic media.
In general, the study allows us to conclude that electric (including protective potential) and magnetic fields differ significantly in areas close to the drainage points of cathode stations, and in the "defective" segment, which entails the necessity to take into account the structure of the medium -its electrical anisotropywhen calculating the parameters of main pipelines, if there are prerequisites for soil layering/fracturing.