9 th -13 th November, 2009 – Curitiba, Brazil
LRC - Lightning Research Center - Federal University of Minas Gerais, Brazil
liviaraggi@yahoo.com.br - visacro@cpdee.ufmg.br
Abstract - This paper presents a study about the relevance of computing a third layer in a horizontal stratified soil. The analyses is based on finding an equivalent two-layer model that satisfactory represents soils originally modelled from measurements as three-layer ones. The two-layer model has a very simple implementation and converges rapidly, when compared with other multilayer models. The investigated equivalence has relevant practical interest as it allows diminishing the computational time required to assess the performance of large grounding-electrode arrangements during the grid design procedure.
1 INTRODUCTION
The design of grounding grids is basically defined to ensure safety conditions during short-circuit events. The grid design considers the proper position of electrodes to ensure a smooth distribution of the electrical potentials developed on the soil surface during the flow of shortcircuit currents, assuming that all the electrodes have the same electrical potential during the flow of low-frequency currents [1].
The fundamental parameters that govern the design are the grounding resistance and the distribution of electric potentials developed on the soil surface.
The first step in any procedure related to such design consists in obtaining a reliable representation for the soil.
Basically this representation is derived from the results of measurements corresponding to the application of the
Frank Wenner Method [2]. According to this traditional method, four short rods are placed at the soil spaced a meters apart and reaching the same depth b . The ratio of the voltage measured between the two internal rods to the current injected into the soil by the two external rods provide a Resistance R
F
K
F
R
F
, whose value is proportional to the soil resistivity by the Frank Wenner constant K
F
), being this constant given by 2 π a . for a >> b .
( ρ =
Though this method was developed to determine the resistivity of a homogeneous semi-infinite medium, it has been applied using different values of space a between rods to identify horizontally stratified soils. This kind of application results in curves similar to the ones presented in Fig 1.
600
500
400
300
200
100
0
3 layers ( ρ 1> ρ 2< ρ 3)
2 layers ( ρ 1> ρ 2)
1 layer
2 4 8 16 32 64 a (m)
Fig. 1 – ρ a xa curves ( between rods, ρ
1
, ρ
ρ a
: resistiviy as a function of the space a
2
, ρ
3
: ground resistivity of layers 1, 2 ,3)
From curves ρ a xa such as the ones above, models of soils are derived, most frequently the homogeneous, two- and three-layer stratified soils.
The experience has shown that only in some cases the soil is adequately represented as a semi-infinite homogeneous medium. In most situations this representation requires a two-layer-soil-stratification model and, less frequently, a three-layer representation is required. In rare cases, to match the experimental ρ a layers are required. xa curves models with four soil
It is prudent to consider that the use of the measurements derived from the application of Frank Wenner Method as input to model earth resistivity has some intrinsic limitations, including uncertainties of measurements that affect significantly the reliability and the accuracy of developed stratified models. Also, as a matter of fact, in the design of grounding grids the behaviour of the parameters of interest, notably the grounding resistance and the potential distribution on the soil surface, expresses the average effect of the whole current distributed to the soil through electrodes, which are not

critically affected by variations of resistivity in the deepest layers of soil.
On the other hand, the increase in the number of layers raises significantly the processing time of the computational procedures used to design real grids.
All the aspects mentioned above denotes a practical interest and the reasonability to achieve a simplified representation of soil that allows a good balance between the accuracy of the calculation in design procedures and their time consumption. The authors believe this balance might be achieved by a two-layer representation for the soil and that, considering the mentioned limitations of the modelling, this representation is more consistent than more detailed representations.
The literature provides a traditional procedure to reduce a three-layer soil representation into two-layer one, based only on averaging the values of second and high order layer, eventually attributing a weight to them related to layer width. In most cases this rough procedure yields significant errors.
This picture motivated the authors to investigate some simplified procedures to produce consistent two-layer representations of soils originally modelled from measurements as three-layer ones. This work reports some results developed to find equivalent models as indicated in Fig. 2.
Fig. 2 – Equivalent soil-stratification models. ( ρ ground resistivity of layers 1, 2 ,3 and 2 layers 1 and 2) eq
; H
1
1
, ρ
, H
2
2
, ρ
3
, ρ
2eq
: length of
:
2 DEVELOPMENTS
2.1 Developed Approach
A heuristic approach was developed to find a simplified algorithm to derive a two-layer stratified soil model from a given three-layer model.
Particularly, in this approach the values of ρ
1
and H
1 the three layers model are preserved and the value of ρ of
2eq that replaces layers 2 and 3 of that model is determined.
Based on their experience the authors proposed simplified expressions to find ρ
2eq
The quality of the proposed solutions was verified by matching the main designparameters of defined grid arrangements found for both representations the original three-layer and the equivalent two-layer model, particularly the grid ground resistance and the distribution of potentials on the soil surface caused by current injected into the soil through electrodes.
The method of the complex images described in reference
[3] was used to calculate the grounding resistance and the potential on the soil surface generated by both models the three- and the two-layer-stratified soils.
2.2 Expression proposed for concentrated electrodes
The evaluations developed in this work shown that for ground systems consisting of short single electrodes, for example a 4m horizontal electrode, the value ρ 0
2eq
from expression (1) provides a consistent estimate for ρ
2eq
, resulting on maximum errors of the design parameters smaller than 3 %.
ρ
0
2 eq
=
( ρ
2 k
⋅
⋅
H
2
(H
1
+
+
ρ
3
H
⋅
2
H
)
1
)
(1)
Where k is given by: k =
( ρ
2
ρ
2
−
−
ρ
1
ρ
1
) + ( ρ
2
+ ρ
2
−
−
ρ
3
ρ
3
)
⋅ 0 , 3 + 1 (2)
2.3 Algorithm proposed for large grids
Long electrodes, including complex arrangements, require extending the modelling process to find ρ
2eq
in order to take into account the grounding-system dimension in addition to the parameters of the original three-layer-soil model.
In this case the value of ρ 0
2eq provided by expression (1) is only an initial value within an interactive procedure to determine the resistivity of the equivalent second layer
ρ
2eq
. This initial value is varied until the smallest estimated error of the grounding-design parameter is found in certain defined conditions, following the procedure described below.
First, the initial value ρ 0
2eq
given by expression (1) is incremented and decremented by N 3%-step variations, to define a group of 2N+1 possible values for ρ i
2eq
:
ρ i
2 eq
= ( 1 + 0 , 03 ⋅ i ) ⋅ ρ 0
2 eq
, − N < i < N (3)

Then a set of p positions are defined corresponding to the points where the potential rise promoted by the current injected into the soil by a punctual current source placed at soil surface will be evaluated and compared considering both conditions the three- and two-layer-soil stratification. The position of the current source q s
is indicated at the soil-air interface at the soil side in Fig. 3 along with the points p
1
to p
P
, where the electric potential is calculated.
The number of points P is defined according to the value of the first layer depth H and of the largest linear dimension of the grid L , Fig.3. Such points are placed at the horizontal plane that contains the deepest point of the electrodes h (supposed to be entirely contained in the first layer) and at least five positions are considered at this plane along the L orientation, spaced 0.25 L , from the source position (points: h,0; h,0.25L; h,0.5L; h,0.75L; h,L ).
The position of the points at the depth h was chosen because the critical errors along the electrodes are found there. Also, decreasing the displacement to values shorter than 0.25 L would increase the accuracy of the process.
3 RESULTS
The application of this heuristic approach to a large number of general electrode arrangements showed quite good results. This section illustrates some of such cases considering two different electrode arrangements buried in both equivalent models, two- and three-stratified soils, considering the injection of a low frequency current. In each case, the ground resistance and the potential distribution over the soil were calculated employing the complex image method and, then, compared. To perform these evaluations, a computational tool developed by the
LRC research group and based on a constant potential approach for electrodes was used in the calculations [6].
3.1 Example 1
The first example consists of a 10x10m grid immersed 0.4 deep in the modelled soils, illustrated by Fig. 4. The parallel conductors are spaced 2m and the electrodes diameter is 0.005m. The diagonal line crossing the grid in
Fig. 4 defines the points along the soil surface where potentials are calculated.
Fig. 3 – Analysed points.
Once these points are defined, the potentials generated by the current source at points p
1
, p
2
, …p
P
, are calculated and compared for the 2N+1 possible values of ρ i
2eq
, considering both conditions the three- and two-layer-soil stratification ( V
2eq
(p j
) and V
3
(p j
) ), using the complex image method.
The quadratic error ε smallest value of ε i
2 i
2 defined by expression (4) below is calculated. The value of ρ i
2eq
corresponding to the
is assumed as the best estimate of ρ
2eq for the equivalent two-layer-soil model.
ε i
2
= j
P
= 1


V
3
( )
V
3
− V
2
( ) eq
( )



2
(4)
This procedure has a very simple implementation and converges quite fast, when compared with other procedures required by multilayer models.
Fig. 4 – Example 1: 10x10m grid.
The values of L and h were respectively 10 m and 0.4m and the points were spaced 0.25L
. Table 1 summarizes some of the results of sensitivity analyses performed to calculate different soil stratification conditions. It indicates the equivalent two-layer-soil solution for each stratification condition.
Table 1: equivalent models
Case
(
ρ
Ω
1 m)
ρ
2
( Ω m)
ρ
3
( Ω m)
H
1
(m)
H
2
(m) (
ρ
Ω
2eq m)

The corresponding ground resistances were calculated.
The error between the values obtained using each model is given by ∆ R and the values are displayed in Table 2, where:
∆ R(%) =
(
R
3
− R
2 eq
)
⋅ 100 (5)
R
3
Table 2: ground resistances
Case R
3
( Ω )
R
(
2eq
Ω )
∆ R
(%)
1 37.92 37.36 1.48
2 66.39 66.76 -0.56
3 13.53 13.9 -2.73
4 11.37 11.23 1.23
5 32.32 32.33 -0.03
6 47.55 47.96 -0.86
7 16.1 16.22 -0.75
8 14.97 14.96 0.07
The errors are reduced, lower than 3% and this is considered a quite good result in terms of grounding applications.
Fig. 5 illustrates a further result generated by the indicated approach, depicting the distribution of the potential rise at the soil surface along the diagonal, considering the injection of a 1kA low-frequency current at the grid in the conditions defined by the first case of
Table 1, (
1000 Ω m, H
1
= 2.5m, H
2
ρ
1
= 500
= 5m, ρ
Ω
2eq m, ρ
2
= 2500
= 1480 Ω m).
Ω m, ρ
3
=
40
38
36
34
32
30
28
26
24
22
20
3 layers
2 layers
0 5
(m)
10 15
Fig. 5 – Example 1: Potential rise
Fig. 6 depicts the ρ a xa curves that would be obtained from measurements by the application of the Wenner Method , for both models. In this figure, ρ a
is the apparent resistivity for the rods spaced a .
1600
1400
1200
1000
800
600
400
3 layers
2 layers
200
0
0 10 20 a (m)
30 40 50
Fig. 6 – Example 1: ρ a xa curve
3.2 Example 2
The second example consists of a 20x20m grid, with eight 2m-long vertical rods distributed along the grid perimeter (Fig 7). The parallel conductors are spaced by
4m and the electrodes diameter is 0.005m. It’s simulated the injection of a 1kA current.
4m
4m
5 500 1000 5
6 500 7500 5
20m
20m
Fig. 7 – Example 2: 20x20m grid plus 8 rods.
Tables 3 and 4 show respectively the equivalent models and the errors (in terms of grounding resistances) found for the same cases considered. Fig. 8 and Fig. 9 describe potential rise at soil surface and ρ the seventh case of Table 3 ( ρ
= 250 Ω m, H
1
= 5m, H
2
1
= 10m, ρ a
2eq xa curves, considering
= 500 Ω m, ρ
2
= 100 Ω m, ρ
3
= 162 Ω m ) .
Case
(
ρ
Ω
1
Table 3: equivalent models m)
ρ
2
( Ω m) ( Ω
ρ
3 m)
H
1 500 1000
1
(m)
H
2
(m) (
ρ
Ω
2eq m)
2 500 7500

Table 4: ground resistances
Case
(
R
Ω
3
)
R
(
2eq
Ω )
∆ R
(%)
1 21.27 20.90 1.74
2
7
47.2 47.52 -0.68
3
6
6.25 6.34 -1.44
4 4.51 4.4 2.44
5 19.07 18.87 1.05
33.8 34.35 -1.63
7.09 7.25 -2.26
8 6.04 5.97 1.16
6
5
4
3
8
7
2
1
0
0
3 layers
2 layers
5 10 15
(m)
20
Fig. 8 – Example 2: Potential rise
25 30
600
500
400
300
200
100
3 layers
2 layers
0
0 20 40 a (m)
60 80 100
Fig. 9 – Example 2: ρ a xa curve
Various electrodes arrangements and soil configurations were simulated. The difference between ground resistance and potential rise on soil surface for both models are much lower than 4%, reaching this limit in the critical cases.
4 CONCLUSION
The main conclusion of this work is that it is possible to develop a consistent two-layer representation of soils that were originally modelled as three-layer stratified soils.
Furthermore, a simplified procedure that can be easily implemented to develop such two-layer equivalent model from the three layer representation is proposed in the paper.
These results are considered of practical interest since they allow diminishing the computational time required to assess the performance of large grounding-electrode arrangements during the grid design procedure.
The next step in this investigation is to determine this equivalent two-layer representation directly from the ρ a xa curves obtained from measurements performed by the application of the Frank Wenner Method.
5 REFERENCES
[1] S. Visacro, Low frequency grounding resistance and lightning protection , Chapt 8 - Lightning Protection book, edited by Vernon Cooray, Elsevier - to be published 2009.
[2] S. Visacro, Grounding and Earthing: Basic Concepts,
Measurements and Instrumentation, Grounding Strategies
(in Portuguese), 2nd ed. São Paulo, Brazil : ArtLiber Edit.,
2002, pp. 1–159.
[3] Y. L. Chow, J. J. Yang, K. D. Srivastava, "Complex Images of a Ground Electrode in Layered Soils", J. of Applied
Physics , vol. 71, n. 15, pp. 569-574, Jan 1992.
[4] P. Min-fang, Y. Dong-jiang, T. Xiang-bo, H. Yi-gang,
"Calculation of a Grounding Grid in Multi-layer Medium",
International Conference on Robotics, Intelligent Systems and Signal Processing , pp.1317-1321, Changsha, China,
Oct. 2003.
[5] G. Kindermann and J. M. Campagnolo, Aterramento
Elétrico , Florianópolis, Author’s Edition, 2002.
[6] S. Visacro, Grounding: Practices and Design (in
Portuguese), book to be published by ArtLiber Edit., São
Paulo, Brazil,.