Practical Application of Telluric Modelling for Pipelines

Ken LAX1, David H. BOTELER2, Charalambos A. CHARALAMBOUS3, Risto PIRJOLA4

1 Corroconsult UK Ltd, Telford, UK, ken.lax@corroconsult.com

2 Natural Resources Canada, Ottawa, Canada, david.boteler@canada.ca

3 Dept. of Electrical and Computer Engineering, U. of Cyprus, Nicosia, Cyprus,

cchara@ucy.ac.cy 4 Natural Resources Canada, Ottawa, Canada, ristojuhani.pirjola@canada.ca

Abstract Telluric currents in pipelines are driven by the electric fields produced by geomagnetic disturbances and are a growing concern for maintaining proper cathodic protection potentials for pipelines. Considerable theoretical work has been done to develop modelling methods that allow calculation of the pipeline potentials produced by specified electric fields. This has resulted in an integrated model, utilising transmission line theory and network analysis that can be combined with calculated electric fields to show the telluric potential variations that will occur during actual geomagnetic disturbances.

To illustrate the use of this integrated model two example calculations are presented. The first example shows the modelling of a single straight pipeline, while the second shows modelling of a pipeline network with typical features such as bends, changes in pipeline dimensions, junctions with branch lines and insulating flanges. Numerical values for each step of the process are provided so that the examples can be used as test cases for telluric modelling software. The second model is the same as used in the draft ISO standard 21857 and the explanations provided here go into more detail than is possible in the standard and are intended to provide additional insights that will allow practising engineers to make informed use of the telluric modelling techniques.

Keywords: Telluric currents, pipeline modelling, nodal admittance matrix method

Introduction

Telluric currents in pipelines have been observed in many parts of the world [1]. They are driven by naturally occurring geo-electric fields generated by variations in the Earth’s magnetic field or by tidal seawater movements through the Earth’s magnetic field [2,3]. The telluric currents give rise to variations in pipe-to-soil potentials that interfere with the cathodic protection of pipelines [4]. The use of higher resistance coatings on modern pipelines has not stopped the flow of telluric currents, rather it has increased the pipeline voltages associated with the potential drop caused by the telluric currents flowing on or off a pipeline. The development of methods for modelling telluric effects [5, 6, 7] has shown that the telluric potentials are predictable and are greater at specific locations that can be determined from the characteristics of a pipeline. There is good agreement between modelled potentials and telluric observations [8]: the observations validating the modelling, and the modelling providing a theoretical framework for understanding the observations.

To make telluric modelling more accessible for use by pipeline engineers we show that the standard modelling approach can be considerably simplified by representing a pipeline as a series of short sections. Two worked examples are then presented. The first being a single straight pipeline. The second being a simple pipeline network containing typical features such as bends, insulating flanges and branch lines found in a typical pipeline. This is the same pipeline model being used in the draft ISO standard 21857 and the information presented here is designed to go beyond the information in the standard and provide practical details to enable a practicing engineer to replicate the calculations.

Pipeline Modelling Geomagnetic induction in a pipeline can be modelled by considering the pipeline as a transmission line with series impedance Z and parallel admittance Y, as shown in Figure 1.

Figure 1. Schematic of electromagnetic induction in a pipeline and the equivalent circuit for a short pipeline section. The series impedance, Z, is given by the resistance per unit length along the pipeline. This can be calculated from the steel resistivity ρ and the cross-sectional area of the pipeline steel

Ω km-1 (1)

Where ro and ri are the outer radius and inner radius of the pipe respectively. The parallel admittance, Y, is determined from the coating conductance, C, and the surface area per unit distance of the pipeline.

S km-1 (2)

These parameters can be used give the propagation constant, γ, and the characteristic impedance, Zo, that describe the electrical response of the pipeline.

ZY (3a)

0

ZZ

Y (3b)

The geo-electric field parallel to each pipeline section, E//, is represented by voltage sources distributed along each transmission line section. For an electric field, E, the component parallel to a pipeline section is given by

�// = � ���� (4)

where θ is the angle between the pipeline and the electric field, E.

2 2( )

o i

Zr r

2 oY C r

Using Distributed-Source Transmission Line (DSTL) theory, the telluric voltages and currents within a pipeline section of length, L, between nodes ‘i’ and ‘k’ are given by [5]

�(�) = ������ − ��

��� − ����� ���(���) + �

����� − ��

��� − ����� ����

(5)

�(�) = −1

���

����� − ��

��� − ����� ���(���) +

1

���

����� − ��

��� − ����� ���� +

�//

���

(6)

where Vi and Vk are the voltages at nodes ‘i’ and ‘k’ at the ends of the pipeline section. To construct a model for a complete pipeline it is helpful to represent each pipeline section by an equivalent-pi circuit [5] comprising a series admittance, YE, in parallel with a current source, IE, and with admittance to ground, Y/2, at each end of the section as shown in Figure 2.

Figure 2. Equivalent-pi circuit for a pipeline section The equivalent-pi circuit components are given by

0

1

sinhEY

Z L (7)

(8)

�� = �//

� (9)

0

1cosh 1

2 sinh

YL

Z L

The equivalent-pi circuits can be combined to form a nodal admittance network that can be used to calculate the telluric potentials in any general pipeline network. Consider a pipeline network with n nodes connected by straight pipeline sections where the pipeline characteristics and induced electric field are uniform within each pipeline section. The equivalent-pi circuit for each pipeline section can be used to construct a nodal admittance network as shown in Figure 3 where jik is the equivalent current source between nodes i and k, yik is the admittance of the line between nodes i and k, and yi is the admittance to ground from node i. The latter is the sum of the admittances to ground of the equivalent-pi circuits for all of the branches going to that node plus the admittance of any ground bed added at that node.

Figure 3. Nodal admittance network for modelling geomagnetic induction in a pipeline system. Applying Kirchhoff’s current law for each node leads to a set of equations that can be written in matrix form:

J = Y V (10)

where [J] is the current source vector in which each term is the sum of the current sources, IE, directed into a particular node.

�� = � ���

�

���

� ≠ �

(11)

[Y] is the admittance matrix in which the diagonal elements are the sum of the admittances of all paths connected to node i, and the off-diagonal elements are the negatives of the admittances between nodes i and k, i.e.

1

N

ii i kik

Y y y

k ≠ i (12a)

ki kiY y (12b)

The voltages of the nodes are then found by taking the inverse of the admittance matrix and multiplying by the nodal current sources:

-1V = Y J (13)

These voltages at the "nodes" in the network obtained from equation (13) represent the values

for iV and kV at the ends of each pipeline section. Substituting these into equations (5) and (6)

then gives the telluric voltage ( )V x and current ( )I x in each pipeline section.

The above equations using the transmission line approach can be used as they are, but do not have any obvious connection to the pipeline parameters, Y and Z. Charalambous [9,10] developed a more intuitive method using a resistive network approach. This splits a pipeline into multiple short segments with each segment represented by a longitudinal resistance and a resistance to Earth. Then, depending on the energization principles, Kirchhoff’s Current Law is applied to find the voltages at each node. Inspired by this approach we re-examine the transmission line equations (5) and (6) and the equivalent-pi circuit equations (7), (8) and (9) and derive new equations for pipeline sections that are “electrically short”. The term “electrically short” means that the length, L, of the pipeline section is much less than the adjustment distance, 1/γ (i.e. the inverse of the propagation constant, γ). For pipelines with a high resistance coating, the adjustment distance can be of the order of 100 km so a pipeline section 10 km long or less would be considered “electrically short”. Modelling Equations for Electrically-Short Sections For a pipeline section that is electrically short, the expression for the pipeline voltage and for the components of the equivalent-pi circuit can be significantly simplified. Starting with the general transmission line equation (5) and rearranging by multiplying by the exponential terms and collecting terms in Vk and Vi gives

�(�) = ��(��������)� �����(���)����(���)�

�������� (14)

Expanding the exponential terms in equation (14) using the Taylor series expansion

�� = 1 + � + ��

�!+

��

�! .. gives

��� − ���� = 2�� +(��)�

�+ ⋯ (15)

and equivalent expressions for the exponential terms involving �� and �(� − �). For a short pipeline section such that the length, L, is much less than the adjustment distance 1/γ then ��� ≪ 1 and we can drop terms in (��)� and above. The distances � and (� − �) are both less than or equal to L so the same terms can be dropped in their expansions. Under these conditions the equation for V reduces to:

�(�) = �� ��� � �� ��(���)

��� (16)

Cancelling common terms gives

�(�) = ���

� + ��

���

� (17)

Which corresponds to simple linear interpolation between the end values Vi and Vk.

The equivalent-pi circuit components for a pipeline section are given by equations (7), (8) and (9). For a pipeline section length, L, that is much less than the adjustment distance 1/γ, then γL << 1. Under these conditions, we can use a Taylor series expansion of sinh and cosh and drop the higher terms, leading to the approximations

sinh �� ≈ �� cosh �� − 1 ≈ (��)�

� (18)

Also noting that �� � = � and �

��= �, the parameters for the equivalent-pi circuit are now

given by:

�� = �

�� (19)

�

� =

��

� (20)

And the current source is still

�� = �//

� (21)

Thus the equivalent-pi circuit for an electrically-short pipeline section is as shown in Fig. 4.

Figure 4. Equivalent-pi circuit for an electrically-short section

Equivalent-pi circuits for each pipeline section can be combined to form a lumped-component model of the pipeline network. The current sources, IE, from the equivalent-pi sections become the current sources, jik, in the nodal network

��� = �� (22) The series admittance, YE, of the equivalent-pi sections becomes the series admittance, yik, of the corresponding section in the nodal network

��� = �� (23)

The admittances to ground at the nodes, Yi, are the sum of the equivalent-pi admittance to ground Y'/2 of each section connected to that node:

�� = ∑��

� (24)

To illustrate this, consider a pipeline comprising 4 sections with the equivalent-pi circuits A, B, C and D shown in Figure 5a. The current source and series admittances values from the equivalent-pi circuits become the values for the corresponding components in the lumped component model. The admittances to ground from the equivalent-pi circuits are combined, according to equations (24), to give the admittance to ground for each node. At the ends of the sections there is only one Y'/2 admittance to ground so this becomes the nodal admittance value:

�� =��

�

� �� =

���

� �� =

���

� (25)

The junction between sections A and B becomes node 2. Thus the admittance to ground for node 2 is the sum of the Y'/2 values from sections A and B:

�� = ��

�

� +

���

� (26)

Further along, after section B, the pipeline branches into sections C and D. Thus at node 3 the admittance to ground is calculated by summing the Y'/2 values from all these sections:

�� = ��

�

� +

���

� +

���

� (27)

This gives the nodal admittance network shown in Figure 5b

Figure 5. Example pipeline represented as a) Equivalent-pi sections, b) a nodal admittance network

Practical Procedure for Modelling Telluric Effects on Pipelines The methods described above can be broken down into a step-by-step procedure as shown in Table 1. Steps 1-5 involve using the pipeline parameters to set up the pipeline model. These parameters include the pipe dimensions, the resistivity of the steel and the coating conductance. Information about the coating would be obtained from manufacturer’s specification sheets, however practical conductance values need to take account of holidays in the coating. Standard reference works [11,12] contain information about coating conductance values that can occur. The next step is to specify the telluric electric field amplitude and direction to use as inputs to the model. This choice will typically be made based on a knowledge of the telluric activity levels in the area. Calculations can be made using magnetic observatory data and an Earth model for the area of the pipeline [13]. Magnetic observatory data from around the world is available through the Intermagnet web site: www.intermagnet.org. In Canada this has been used to provide an online telluric calculation service for pipeline operations [14]. For telluric hazard assessment, convenient electric field values to use are 1 V/km northward and 1 V/km eastward. Later, if the impact of a larger (or smaller) electric field is required, the modelling results for 1 V/km can simply be scaled up (or down). Telluric potentials for electric fields in other directions can be obtained by an appropriate combination of the northward and eastward results. Steps 6-8 describe using the model with the chosen input to calculate the pipeline voltages. The final step is to plot the results. Note that just plotting the nodal voltages with straight lines between points will give the required in-between voltages that correspond to linear interpolation. To repeat the calculations for different electric field inputs, e.g. northward and eastward electric field, just requires repeating steps 6 and 7 to determine the column matrix [J]. Once the matrix inversion of the admittance matrix [Y] has been done it does not need to be repeated and [Y]-1 can be multiplied by any new [J] to give new results. To illustrate the use of the step-by-step procedure we present two example calculations. Appendix A contains the calculations for a single straight pipeline. Appendix B contains the calculations for a more complicated pipeline network. The method described above can be applied to any size of pipeline network. The examples shown deal with small pipeline lengths or simple pipeline networks just so that the number of terms in the equations are small enough that tabulated values can be provided. This enables the above results to be used as a test case for software designed for modelling telluric effects on pipelines. The length of sections into which to subdivide a pipeline network is a decision that has to be made. A lumped component circuit representation is correct as long as each pipeline section is “electrically short”. For example, if the length of a section is less than a quarter of the adjustment distance then the error in dropping higher order terms in the series expansions of the exponential terms is only 1 %. The adjustment distances for modern pipelines are typically in the range from 70 km to 130 km, depending on the dimensions and type of coating, so a section length of 10 km or less can usually be considered electrically short.

Table 1. Step-by-step procedure for modelling telluric effects on pipelines Step 1.

Gather the necessary pipeline dimensions and characteristics. This will include: i) Pipeline outside diameter and wall thickness ii) Steel resistivity (usually assumed to be 0,18 10-6 ohm-m) iii) Coating conductance iv) Pipeline network information (i.e. directions and lengths of parts of the

pipeline, positions of bends, ground beds, insulating flanges and branches).

Step 2. Use the pipe dimensions, steel resistivity and coating conductance in equations (1) and (2) to calculate the series impedance, Z, and parallel admittance, Y, of each pipeline segment. Step 3. Split the pipeline into electrically short sections (e.g. 1km or 10-km lengths) and use the Z and Y values from step 2 in equations (19) and (20) to calculate the equivalent-pi circuit for each section. [The calculation of IE is deferred till later.] Step 4. Construct the lumped element model of the pipeline, using the equivalent-pi YE values for the series admittances and combining the equivalent-pi Y’/2 values to form the admittance from each node to ground (equations 22, 23, 24). Step 5. Construct the network admittance matrix [Y] for the lumped element model of the pipeline. Diagonal elements of the matrix are the sum of the admittances of all paths connected to each node (equation 11). Off-diagonal elements are the negative of the admittance between nodes i and k, (equation 12). Step 6 For the chosen electric field amplitude and direction, calculate the electric field component parallel to each pipeline section (equation 4). Use these E// values, with Z, in equation (21) to calculate the equivalent-pi current source IE for each section. Step 7 Sum the current sources IE connected to each node (equation 11) to give the nodal current sources Ji. Put all these current source values into a column matrix [J]. Step 8. Do the matrix inversion of the admittance matrix [Y] and multiply by column matrix [J] (equation 13). The calculations can be done in a dedicated software package or spreadsheet program like Excel. The result is a column matrix [V] that has the voltages for the nodes in the network. Step 9. Plot the results. The matrix calculation gives the nodal voltages. These need to be paired with the distances of the nodes along the pipeline. The voltages between nodes are given by simple linear interpolation between the nodal voltages (equation 17).

Conclusions Methods are available that enable modelling of the telluric potentials produced on pipelines. A simplified method is shown here which splits a pipeline into electrically short sections between nodes. Each section is represented by an equivalent-pi section with parameters derived from the series impedance, parallel admittance and length of each section. The use of electrically short sections allows the equivalent-pi components to be much simpler than in previous derivations. The equivalent-pi components are then combined to produce a lumped element model of the pipeline for which we construct a network admittance matrix [Y] for the pipeline. The geo-electric fields that drive telluric currents are also used to calculate a current source column matrix [J]. The telluric voltages on the pipeline are then obtained by taking the matrix inversion of [Y] and multiplying by the current source column matrix [J] to give the nodal voltages. The pipeline potential between nodes, for the electrically short sections, is simply obtained by linear interpolation between the nodal voltages. References

1. Boteler, D.H. and Trichtchenko, L., Telluric Influence on Pipelines, Chapter 21, Oil and Gas Pipelines: Integrity and Safety Handbook, edited by R. Winston Revie, John Wiley & Sons, Inc., Hoboken, p 275-288, 2015. 2. Gummow, R., Boteler, D.H., and Trichtchenko, L., Telluric and ocean current effects on buried pipelines and their cathodic protection systems, Report for Pipeline Research Council International, No. L51909, 2002. 3. D.H. Boteler, Telluric Currents and their Effect on Cathodic Protection of Pipelines, Paper No 04050, Proc. CORROSION/2004, NACE, Houston, March 2004. 4. Rix, B.C. and Boteler, D.H. Telluric current considerations in the CP design for the Maritimes and Northeast Pipeline, Paper 01317, Proc., CORROSION 2001, NACE, Houston, March 11-16, 2001 5. Boteler, D.H., Distributed-source transmission line theory for electromagnetic induction studies, Proc. 1997 Zurich EMC Symposium, Feb. 18-20, URSI supplement, 401-408, ETH, Zurich, 1997. 6. Boteler, D.H., Assessing pipeline vulnerability to telluric currents, Paper No 07686, Proc. CORROSION/2007, NACE, Houston, March 2007. 7. Boteler, D.H., A New Versatile Model of Geomagnetic Induction of Telluric Currents in Pipelines, Geophysical Journal International; doi: 10.1093/gji/ggs113, 2013. 8. Boteler, D.H. and Seager, W.H., Telluric currents: A meeting of theory and observation, Corrosion, 54, 751-755, 1998. 9. C. A. Charalambous, I. Cotton, P. Aylott, ‘’A simulation tool to predict the impact of soil topologies on coupling between a light rail system and buried third party infrastructure’’ , Vehicular Technology, IEEE Transactions on, Vol. 57, No. 3, pages: 1404-1416, May 2008. 10. C. A. Charalambous, "Comprehensive Modeling to Allow Informed Calculation of DC Traction Systems’ Stray Current Levels," in IEEE Transactions on Vehicular Technology, vol. 66, no. 11, pp. 9667-9677, Nov. 2017. doi: 10.1109/TVT.2017.2748988 11. NACE Standard TM0102-2002, Item No. 21241, Measurement of Protective Coating Electrical Conductance on Underground Pipelines, NACE, Houston, 2002. 12. Peabody’s Control of Pipeline Corrosion, ed. R.L. Bianchetti, NACE, Houston, 2nd Edition, 2001. 13. Trichtchenko, L. and Boteler, D.H., Modeling of geomagnetic induction in pipelines, Annales Geophysicae, 20, 1063-1072, 2002. 14. Trichtchenko, L., D.H. Boteler, and P. Fernberg, Space Weather Services for Pipeline Operation, Proceedings, ASTRO 2008, Montreal, April 29-May 1, 2008.

Appendix A. Modelling Telluric Effects on a Single Pipeline Consider an 8-km long 30 inch pipeline with 15.6 mm wall thickness as shown in Figure A.1

Figure A.1. a) 8-km long pipeline subject to an Electric Field, E b) pipeline split into 1-km long sections To examine the telluric voltages produced by an electric field at an angle to the pipeline we

will need to use the equations shown in Table A1. In this example the electric field is at an

angle of 60° to the pipeline.

The first steps are to collect the pipeline dimensions and parameters and use equations (1) and

(2) to calculate the series impedance, Z, and parallel admittance, Y. These are then used in

equation (3) to calculate the propagation constant, γ, and the adjustment distance, 1/γ ( Tables

A.2 and A.3).

Then split the pipeline into sections that are electrically short, i.e. much shorter than the

adjustment distance. The adjustment distance is 130 km (Table A.3), so any lengths of 10 km

or less would be satisfactory (the sections do not need to be of equal length). For this example,

we choose to split the pipeline into 8 sections, each of length L = 1km. For each section we

calculate the equivalent-pi circuit components. In this example, the pipeline is uniform so each

equivalent-pi circuit is the same and has the parameters shown in Table A.4.

Table A.1. Equations for telluric modelling of the pipeline. Step 2 Calculate the pipeline series impedance (eq

1)

Ω km-1

Step 2 Calculate the pipeline parallel admittance (eq 2)

S km-1

Step 3 Calculate the equivalent-pi series admittance (eq 19)

�� = �

�� S

Step 3 Calculate the equivalent-pi end admittances to ground (eq 20)

�

� =

��

� S

Step 4 Calculate nodal admittances to ground (eq 24)

�� = ����

2

�

���

� ≠ �

Step 5 Construct the admittance matrix [Y] – diagonal elements (eq 12a) 1

N

ii i kik

Y y y

k ≠ i

Step 5 Construct the admittance matrix [Y] – off-diagonal elements (eq 12b)

ki kiY y

Step 6 Calculate electric field parallel to each pipeline section (eq 4)

�// = � ����

Step 6 Calculate equivalent-pi current source (eq 9)

�� = �//

�

Step 7 Sum the current sources into each node (eq 11) �� = � ���

�

���

� ≠ �

Step 8 Calculate the matrix inversion and multiply by the current column matrix (eq 13)

-1V = Y J

Table A.2: Pipeline Dimensions

Outside diameter

(inch)

Outside diameter

(mm)

Wall thickness

(mm)

Outside radius

(m)

Inside radius

(m)

Type A 30 762 15,6 0,381 0,3654

Table A.3: Pipeline Electrical Characteristics Steel

resistivity,

ρ (Ω.m)

Coating

conductance,

C (S.m-2)

Series

Impedance,

Z (Ω.km-1)

Parallel

Admittance,

Y (S.km-1)

Propagation

Constant,

γ (km-1)

Adjustment

Distance,

1/γ (km)

Type A 0,18 10-6 5 10-6 0,004921 0,01197 0,007675 130,3

2 2( )

o i

Zr r

2 oY C r

Table A.4: Equivalent-Pi Circuit and Lumped Components

YE

(S)

Y’/2

(S)

Y’

(S)

Type A 203,21 0,005985 0,01197

Step 4 is to combine the equivalent-pi circuits for each section into a nodal network to give the lumped component network shown in Figure A.2. with the component values in Table A.4.

Fig. A.2.. Nodal Admittance Model of an 8-section pipeline

This lumped component model corresponds to a circuit model that is derived by simply

distributing the series impedance and parallel admittance of the pipeline according to the length

of the sections. The admittances to ground are split into 7 ‘YL’ values at the mid-section nodes

and the eighth ‘YL’ value divided into two ‘YL/2’ values: one each at nodes 1 and 9.

Step 5 is to use the admittances for the 8-section pipeline in Figure A.2 in equations (12a) and (12b) to construct the admittance matrix [Y]. Diagonal elements in the admittance matrix, Yii (e.g. Y11, Y22, Y33, etc) are the sum of all admittances connected to each node i (equation 12a). For the pipeline in Figure A.2 these are ��� = 2�� + �� for nodes 2 to 8 and ��� = �� + ��/2 for nodes 1 and 9. Off-diagonal elements in the admittance matrix, Yij (e.g. Y12, Y13, Y14, etc) represent the connections between pairs of nodes i and j. For example Y12 represents the connection between nodes 1 and 2. For node pairs that are connected, the matrix element value is -YE (equation 12b). However, there are many node pairs for which there is no connection (for example, node 1 is not directly connected to node 3) and the Yik values in these cases are zero. The end result is the admittance matrix shown in Table A.5. [Note: some of the values have been rounded off for display in the table.]

Table A.5: Admittance Matrix [Y] for the Pipeline Model in Figure A.2

1 2 3 4 5 6 7 8 9

1 203,216 -203,21 0 0 0 0 0 0 0

2 -203,21 406,432 -203,21 0 0 0 0 0 0

3 0 -203,21 406,432 -203,21 0 0 0 0 0

4 0 0 -203,21 406,432 -203,21 0 0 0 0

5 0 0 0 -203,21 406,432 -203,21 0 0 0

6 0 0 0 0 -203,21 406,432 -203,21 0 0

7 0 0 0 0 0 -203,21 406,432 -203,21 0

8 0 0 0 0 0 0 -203,21 406,432 -203,21

9 0 0 0 0 0 0 0 -203,21 203,216

Step 6 involves calculating the component of the electric field that is parallel to each pipeline section. In this example, all the pipeline sections are in the same direction, at an angle of 60° to the electric field direction, so the parallel electric field for each section is:

�// = � cos(60) (A.1)

For an electric field amplitude of 2 V/km, and noting that cos(60)=0.5, this gives E// = 1V/km.

For this value of E// the equivalent current source given by equation (21), IE is equal to 203.21

A.

In step 7 these current source values are summed to give the total current source into each node

(equation 11). For nodes 2 to 8, there is the same current source directed into the node on one

side as directed out of the node on the other side so the algebraic sum of the current sources at

these nodes is zero. At node 1, there is current source IE directed out of the node (so current

in = -IE). At node 9, there is a current source IE directed into the node. This gives the column

matrix shown in Table A.6.

Step 8 involves taking the matrix inversion of [Y] (Table 5) to give [Y]-1 and doing the

matrix multiplication (equation 13) of this with [J] (Table 6) to give the nodal voltages shown

in Table A.7. The voltages between nodes are obtained by simple linear interpolation as given

in equation (17). The results are shown in Figure A.3.Table A.6: Column Matrix [J]

Node Ji

1 -203,21

2 0

3 0

4 0

5 0

6 0

7 0

8 0

9 203,21

Table A.7: Nodal Voltages [V]

Node Vi

1 -4,0

2 -3,0

3 -2,0

4 -1,0

5 0,0

6 1,0

7 2,0

8 3,0

9 4,0

Figure A.3. Voltage produced in an 8-section pipeline by a geo-electric field of 2 V/km at angle 60° to the pipeline as shown in Figure A.1.

Appendix B. Modelling Telluric Effects on a Pipeline Network To further illustrate how the network modelling can be used, consider the pipeline network shown in Figure B.1. This pipeline contains features such as a change in pipeline dimensions, a bend, a junction with a branch line and insulating flanges that would occur on a real pipeline and influence the telluric potentials that are produced. The network is constructed from three different types of pipe, labelled A, B, and C in Figure B.1, with dimensions as in Table B.1 and steel resistivity and coating conductance obtained from manufacturer’s specifications and shown in Table B.2.

Figure B.1. Example pipeline Layout

Table B.1: Pipeline Dimensions

Outside

diameter (inch)

Outside

diameter (mm)

Wall thickness

(mm)

Outside radius

(m)

Inside radius

(m)

Type A 30 762 15,6 0,381 0,3654

Type B 20 508 12,5 0,254 0,2415

Type C 15 381 10,0 0,1905 0,1805

Table B.2: Pipeline Electrical Characteristics

Steel

resistivity,

ρ (Ω.m)

Coating

conductance,

C (S.m-2)

Series

Impedance,

Z (Ω.km-1)

Parallel

Admittance,

Y (S.km-1)

Propagation

Constant,

γ (km-1)

Adjustment

Distance,

1/γ (km)

Type A 0,18 10-6 5 10-6 0,004921 0,01197 0,007675 130,3

Type B 0,18 10-6 5 10-6 0,009251 0,00798 0,008592 116,4

Type C 0,18 10-6 5 10-6 0,015444 0,00598 0,00961 104,1

The first steps in pipeline modelling are to use the pipeline dimensions and parameters in equations (1) and (2) to calculate the series impedance, Z, and parallel admittance, Y. These are then used in equation (3) to calculate the propagation constant, γ, and the adjustment distance, 1/γ. The values for the pipeline types in Figure B.1 are shown in Table B.2.

Step 3 is to split the pipeline into sections and determine the equivalent-pi circuit for each section. The pipeline can be split into any length of sections as long as they are electrically short. Table B.2 shows that the adjustment distances for the pipeline type are all greater than 100 km, so a length of 10 km or less would be considered as meeting the “electrically short” requirement. In practice, it may be easier to adopt a short length such as 1 km that can be expected to always meet this requirement. Then each part of the pipeline network shown in Figure B.1 would be comprised of multiple sections. Using 1-km lengths for the pipeline in Figure B.1 would result in a lumped element network with 72 nodes resulting in a 72 x 72 admittance matrix which is inverted to give the nodal voltages.

To illustrate the modelling process we split the pipeline in Figure B.1 into sections that are 10-km long or less. This produces a 9-node lumped element network (Figure B.2) resulting in a 9 x 9 admittance matrix which is small enough to show in the paper. The pipeline type and length of each section are shown in Table B.3. For each section the series impedance, Z, parallel admittance, Y, and section length, L, are used in equations (19) and (20) to calculate the equivalent-pi circuit components YE and Y’/2 shown in Table B.3.

Table B.3: Section Type and Length and Equivalent-Pi Circuit Components

Section Pipeline

Type

Length

(km)

YE Y’/2

1-2 B 10 10,81 0,0399

2-3 A 10 20,32 0,05985

3-4 A 10 20,32 0,05985

4-5 A 10 20,32 0,05985

5-6 A 10 20,32 0,05985

6-7 A 9 22,58 0,053865

7-8 A 8 25,40 0,04788

7-9 C 5 12,95 0,01495

Step 4 involves using the equivalent-pi circuit components to form the lumped component network shown in Figure B.2. The admittance between nodes is simply the series admittance, YE, from the corresponding equivalent-pi circuit. The admittance to ground from each node is the sum of the Y’/2 values from all the equivalent-pi circuits connected to that node (equation 24). The inserts in Figure B.2 show two examples of this. Insert (a) gives an example of the components from adjacent sections that combine to form the admittance to ground at node 2. Insert (b) shows that when there is a junction with a branch line, the admittance to ground at

the junction node is simply the sum of the three equivalent-pi admittances to ground from the main line sections and the branch line section. The resulting value for the admittance to ground at each node are shown in Table B.4.

Figure B.2. Pipeline of Figure B.1 split into 10-km or shorter sections

and the corresponding lumped element model. The inserts (a) and (b) shown the components combined to form

the admittance to ground at nodes 2 and 7 respectively.

Table B.4: Nodal Admittances, Yi in the Lumped Component Model

Node Sections

involved

Yi

1 1-2 0,0399

2 1-2, 2-3 0,09975

3 2-3, 3-4 0,1197

4 3-4, 4-5 0,1197

5 4-5. 5-6 0,1197

6 5-6, 6-7 0,113715

7 6-7, 7-8, 7-9 0,116695

8 7-8 0,04788

9 7-9 0,01495

Step 5: the admittance values obtained are now used in equations (12a) and (12b) to form the elements of the network admittance matrix. Diagonal elements, Yii are the sum of the admittances of all paths connected to node i,. In Table B.5, this is addition of the Yi and Yki values to give the Yii values. Off-diagonal elements, Yki are the negatives of the admittances between nodes i and k.

Table B.5. Admittance Matrix for the Pipeline Network in Figure B.2.

1 2 3 4 5 6 7 8 9

1 10,83672 -10,79685 0 0 0 0 0 0 0

2 -10,79685 31,19894 -20,30240 0 0 0 0 0 0

3 0 -20,30240 40,72443 -20,30240 0 0 0 0 0

4 0 0 -20,30240 40,72443 -20,30240 0 0 0 0

5 0 0 0 -20,30240 40,72443 -20,30240 0 0 0

6 0 0 0 0 -20,30240 42,97849 -22,56243 0 0

7 0 0 0 0 0 -22,56243 61,01141 -25,38697 -12,94536

8 0 0 0 0 0 0 -25,38697 25,43483 0

9 0 0 0 0 0 0 -12,94536 0 12,96032

Step 6 involves calculating the electric field parallel to each pipeline section. To assess the

possible telluric impact on the pipeline it is necessary to consider geo-electric fields that can

occur in any direction. The simplest way to do this is to perform the modelling calculations

for a northward electric field of 1 V/km and an eastward electric field of 1 V/km (Figure B.3).

The telluric potentials for any amplitude and direction of electric field can then be found by a

simple scaling and addition of the results for these northward and eastward electric fields.

Figure B.3. The pipeline network exposed to electric field 1 V/km with direction

a) Northward

b) Eastward

In each case, northward or eastward electric field, the procedure is to calculate the component of the electric field parallel to each pipeline section. This simply involves multiplying the electric field amplitude by the cosine of the angle between the electric field and the pipeline section. However, care must be taken in choosing the angle that is used. In Figure B.3 we are numbering the nodes from left to right, so the positive direction of the pipeline is from 1 to 2, 2 to 3, etc. Thus the angle between the northward electric field and the pipeline direction is 120° as shown in Figure B.3a (not 60°). Cos(120) = -0.5 so using this angle automatically gives the correct sign for the values of E// and IE that are calculated.

For a northward electric field of 1 V/km the electric field parallel to sections 1-2, 2-3, and 3-4 is E// = 1.0 * cos(120), while for sections 4-5, 5-6, 6-7 and 7-8 E// = 0 because the electric field is perpendicular to these sections (cos(90)=0.0); section 7-9 is parallel to the electric field, so for this section E// = 1,0.

For an eastward electric field of 1 V/km the electric field parallel to sections 1-2, 2-3 and 3-4 is E// = 1.0 * cos (30), while for sections 4-5, 5-6, 6-7 and 7-8 E// = 1,0 because this electric field is parallel to these sections, but the field is now perpendicular to section 7-9 so this has E// = 0 .

The parallel electric field is then used in equation 9 to calculate the equivalent current source, IE. (Note that different sections have different values for the series impedance, Z: see Table B.2.)

Table B.6 Calculation for Northward E Field

Section Angle to E field

E// IE

1-2 120 -0,5 -54,048

2-3 120 -0,5 -101,605

3-4 120 -0,5 -101,605

4-5 90 0,0 0,0

5-6 90 0,0 0,0

6-7 90 0,0 0,0

7-8 90 0,0 0,0

7-9 0 1,0 64.75

Table B.7: Calculation for Eastward E Field

Section Angle to E field

E// IE

1-2 30 0,866 93.612

2-3 30 0,866 175,98

3-4 30 0,866 175,98

4-5 0 1,0 203,21

5-6 0 1,0 203,21

6-7 0 1,0 203,21

7-8 0 1,0 203,21

7-9 90 0,0 0,0

For each node the equivalent current sources directed into each node are then combined (equation 11) to give the nodal current sources [J]. Care must be taken to use the correct sign for each component. The current sources IE from Tables B.6 or B.7 represent the current sources directed to the higher node number. For example, the current source in section 2-3 is directed towards node 3. Thus in summing the contributions going into node 3 the current source I23 should be used with the value exactly as it is in Tables B.6 or B.7. For node 3 we also want the contribution going into the node from section 3-4. A current going into node 3 would be directed from node 4 to node 3 and labelled I43. However, the values in Tables B.6 and B.7 are for current sources directed from node 3 to node 4 and labelled I34. This difference in direction means that I43 = - I34. Thus the sign of the IE value from Tables B.6 and B.7 will have to be changed if it represents a current source directed away from a node instead of towards a node.

For a northward electric field of 1 V/km, the resulting values in the nodal current source column matrix [J] are shown in Table B.8. Similarly, for an eastward electric field of 1 V/km, the resulting values for nodal current source column matrix [J] are shown in Table B.9.

Step 8: Having constructing the admittance matrix [Y] and the Current Source Column Matrix [J], the next step is to do the matrix inversion of [Y] and multiply this by the current source column matrix [J] to give the nodal voltages [V] (equation 13). The nodal voltages produced by a northward electric field are shown in Table B.10. The nodal voltages for an eastward electric field are shown in Table B.11. The voltages between the nodes are simply obtained, as in Appendix A, by linear interpolation between the nodal voltages (equation 17). This gives the results shown in Figures B.4 and B.5.

Table B.8: Column Matrix [J] for a Northward E Field = 1 V/km

Node Ji

1 54,05

2 47,56

3 0

4 -101,61

5 0

6 0

7 -64,75

8 0

9 64,75

Table B.9: Column Matrix [J] for an Eastward E Field = 1V/km

Node Ji

1 -93,62

2 -82,38

3 0

4 -27,23

5 0

6 0

7 0

8 203,22

9 0

Table B.10: Nodal Voltages [V] for a Northward E Field = 1 V/km

Node Vi

1 11,97

2 7,00

3 2,06

4 -2,88

5 -2,83

6 -2,79

7 -2,77

8 -2,76

9 2,23

Table B.11: Nodal Voltages [V] for an Eastward E Field = 1 V/km

Node Vi

1 -31,91

2 -23,36

3 -14,87

4 -6,46

5 3,24

6 12,97

7 21,79

8 29,74

9 21,76

a)

b)

Figure B.4. Pipeline Voltages produced by a Northward Electric field of 1 V/km

a) Main Pipeline b) Branch Pipeline

Voltage t

o E

art

h (

V)

Voltage t

o E

art

h (

V)

a)

b)

Figure B.5. Pipeline Voltages produced by an Eastward Electric field of 1 V/km

a) Main Pipeline b) Branch Pipeline

Vo

ltag

e to

Ea

rth

(V

)V

oltage t

o E

art

h (

V)