finite element magnetic field response of an exponential conductivity ground … · 2015-03-27 ·...

Applied Mathematical Sciences, Vol. 9, 2015, no. 52, 2579 - 2594HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/ams.2015.52116

Finite Element Magnetic Field Response of

an Exponential Conductivity Ground Profile

Priyanuch Tunnurak, Nairat Kanyamee and Suabsagun Yooyuanyong

Department of Mathematics, Faculty of ScienceSilpakorn University and Centre of Excellence in Mathematics, Thailand

Copyright c© 2015 Priyanuch Tunnurak, Nairat Kanyamee and Suabsagun Yooyuanyong.

This is an open access article distributed under the Creative Commons Attribution License,

which permits unrestricted use, distribution, and reproduction in any medium, provided the

original work is properly cited.

Abstract

In this paper, mathematical model of finite element method for themagnetic field of an exponential conductivity ground profile is presentedand computed to find the magnetic field at various locations by assumingthat the Earth structure having exponential conductivity profile. Thereis a source providing a DC voltage and a receiver on the ground surface.Numerical technique the finite element method (FEM) is introducedby using the Galerkin’s method of Weighted Residuals to find approx-imate solutions of partial differential equation. Matlab programing isconducted to calculate and plot graphs of magnetic field at various lo-cations. The results perform very well to the intensity of magnetic fieldfor cross sections of ground structure.

Mathematics Subject Classification: 86A25

Keywords: Finite element, Magnetic, Galerkin

1 Introduction

Currently, human studied the Earth structure widely in order to utilizethe natural resources embedded beneath the Earth for developing the agricul-tural sector and industrial sectors in their countries. They use knowledge ofgeophysics which is a branch of science concerned with the Earth survey. The

2580 Priyanuch Tunnurak et al.

survey uses mathematics, physics and the physical properties of the Earth suchas the resistivity, conductivity, electric potential, magnetic field and electricfield to search for the natural resources.

We create a mathematical model by using magnetometric resistivity methodto find the value of magnetic field beneath the Earth surface. In 2003, Chenand Oldenburg[7] assume that the Earth structure consists of horizontallystratified layers having constant conductivity at certain depths except thelast layer where the conductivity having the same varying through the restof the layer. They derived the magnetic field directly by solving a bound-ary value problem of a horizontally stratified layered Earth with homogeneouslayers. However, in the real situation there are cases where the subsurfaceconductivities vary exponentially, linearly or binomially with depth. Thereexists a considerable amount of research about mathematical modeling whichassumes that the Earth structure consists of horizontally stratified multilayerwith one or more layers having exponentially, linearly or binomially varyingconductivities at certain depths except the last layer where the conductivityhaving the same varying through the rest of the layer. Stoyer and Wait[28]studied the problem of computing apparent resistivity for a structure with ahomogeneous overburden overlying a medium whose resistivity varies exponen-tially with depth. Banerjee et al.[1] gave expressions for apparent resistivityof a multilayered Earth with a layer having exponentially varying conductiv-ity. Kim and Lee[14] derived a new resistivity kernel function for calculatingapparent resistivity of a multilayered Earth with layers having exponentiallyvarying conductivities. Siew and Yooyuanyong[29] studied the electromagneticresponse of a thin disk beneath an inhomogeneous conductive overburden andexpressions for the electric fields in the overburden. Ketchanwit[15] studied theEarth surface layers using time-domain electromagnetic field by constructingthree mathematical models having exponentially varying and constant vary-ing conductivities. Sripunya[30] derived solutions of the steady state magneticfield due to a DC current source in a layered Earth with some layer havingexponentially or binomially or linearly varying conductivity.

In this paper, mathematical model is presented by using numerical tech-niques for finding approximate solutions. The finite element method (FEM)is used to find the numerical solutions of the magnetic field under the Earthsurface. We assume that the Earth structure contains only one layer havingexponential conductivity (σ(z) = σ0 + (σ1−σ0)e−bz). This method is differentfrom the Hankel transform approach which is difficult to solve for some complexproblems such as all the research mentioned above. There are a few researchusing FEM by applying the Galerkin’s method of Weighted Residuals to findthe solution of the magnetic field. For instance, Lee[16] presented a numericalmethod of computing the electromagnetic response of two-dimensional Earthmodels to an oscillating magnetic dipole. Velimsky and Martince[36] intro-

Finite element magnetic field response 2581

duced a time-domain method to solve the problem of geomagnetic inductionin a heterogeneous Earth excited by variations of the ionospheric and mag-netospheric currents with arbitrary spatiotemporal characteristics. Mitsuhataand Uchida[20] presented a finite element algorithm for computing magneticfield response for 3D conductivity structures. Therefore we are interested ap-proximate techniques in finding the magnetic field beneath the Earth by usingthe Galerkin’s method of Weighted Residuals.

2 Mathematical Formulations

In this section, we use finite element method (FEM) for constructing ap-proximate solutions of our problems. Assuming that the Earth structure con-tains only one layer having exponential conductivity (σ(z) = σ0+(σ1−σ0)e−bz)and there are a source providing a DC voltage and a receiver on the groundsurface which picks up the signal from r = 10 m to r = 190 m as shown inFigure 2.1.

Figure 2.1: Geometric model of the Earth structure.

We define z as the depth of an object from the Earth surface (meter), r asthe distance between source and receiver of magnetic field on the Earth surface(meter) and σ(z) as the conductivity of the medium which is a function of z(S/m), where σ0, σ1 and b are positive constants.

From Maxwell’s equations , the relationship between the electric and mag-netic fields[29,30,31,37] written in cylindrical coordinates (r, φ, z) is as follows.

∇× ~E = ~0 (2.1)

and∇× ~H = σ ~E, (2.2)

where ~E is the electric field vector, ~H is the magnetic field vector, σ is the


conductivity of the medium and ∇ is the gradient operator in cylindrical co-ordinates (r, φ, z) [17,27] defined by

∇ =∂

∂rer +

1

r

∂

∂φeφ +

∂

∂zez,

where er is the unit vector in radial direction (r), eφ is the unit vector in thedirection of φ, ez is the unit vector in the direction of z.

Substituting equation (2.2) into (2.1) , we obtain

∇× 1

σ(∇× ~H) = ~0. (2.3)

Substitute equation ∇× 1σ(∇× ~H) in cylindrical coordinates (r, φ, z) [17,27]

into (2.3) , we obtain

1

r

[∂

∂φ

(1

r

(1

σ

∂(rHφ)

∂r− 1

σ

∂Hr

∂φ

))− ∂

∂z

(1

σ

∂Hr

∂z− 1

σ

∂Hz

∂r

)]er

+

[∂

∂z

(1

r

(1

σ

∂Hz

∂φ− 1

σ

∂(rHφ)

∂z

))− ∂

∂r

(1

r

(1

σ

∂(rHφ)

∂r− 1

σ

∂Hr

∂φ

))]eφ

+1

r

[∂

∂r

(1

σ

∂Hr

∂z− 1

σ

∂Hz

∂r

)− ∂

∂φ

(1

r

(1

σ

∂Hz

∂φ− 1

σ

∂(rHφ)

∂z

))]ez = ~0, (2.4)

where Hr, Hφ and Hz are the components of ~H in er, eφ and ez directions,respectively. Since the magnetic field is axisymmetric, it depends only on rand z and not on the azimuth φ and from electromagnetic theory, we knowthe magnetic field has only the azimuthal component, i.e. ~H = Hφ(r, z)eφ[19].Simplifying equation (2.4) yields

1

σ

∂2H

∂z2+∂H

∂z

∂

∂z

(1

σ

)+

1

σ

[1

r

∂2(rH)

∂r2+

∂

∂r

(1

r

)∂(rH)

∂r

]= 0.

We denote conductivity σ as a function of depth z only, i.e. σ = σ(z), andwe now have

∂2H

∂z2+ σ

∂H

∂z

∂

∂z

(1

σ

)+∂2H

∂r2+

1

r

∂H

∂r− 1

r2H = 0. (2.5)

Since the Laplace equation and the problem is axisymmetric, our problembecomes

∆H + σ∂H

∂z

∂

∂z

(1

σ

)− 1

r2H = 0. (2.6)

The next step, we use finite element method to establish a numerical solu-tion of our problem. We apply the Galerkin’s Method of Weighted Residuals


to equation (2.6).We transform equation (2.6) into weak formulation to find H ∈ H1. Let

V = v ∈ H1 : v is a continuous function on Ω, ∂v∂r

and ∂v∂z

are piecewisecontinuous on Ω and v = 0 on ∂Ω .The weak formulation of equation (2.6) is denoted by

(∆H, v) +

(σ∂H

∂z

∂

∂z

( 1

σ

), v

)− (

1

r2H, v) = 0 , v ∈ V

or ∫Ω

∆HvdΩ +

∫Ω

σ∂H

∂z

∂

∂z

( 1

σ

)vdΩ−

∫Ω

1

r2HvdΩ = 0. (2.7)

By Green’s identity[32] and v = 0 on ∂Ω, we obtain

−∫

Ω

∇H · ∇vdΩ +

∫Ω

σ∂H

∂z

∂

∂z

( 1

σ

)vdΩ−

∫Ω

1

r2HvdΩ = 0. (2.8)

Using cylindrical co-ordinates (r, φ, z) [21] and since the problem is ax-isymmetric and H has only the azimuthal component in cylindrical coordinate,equation (2.8) becomes

−∫

Ω

r∇H · ∇vdrdz +

∫Ω

rσ∂H

∂z

∂

∂z

( 1

σ

)vdrdz −

∫Ω

1

rHvdrdz = 0, (2.9)

where Ω is the 2D cross-section of domain Ω (φ is fixed), i.e. Ω = (r, z), 10 ≤r ≤ 190, 0 ≤ z ≤ 180.

Next we consider the two dimensional domain of equation (2.9). By divid-ing the domain into rectangular elements, we discretize r into 9 subintervalsequally, discretize z into 9 subintervals equally and (ri, zj) is a node of Ω onthe non overlapping rectangles such that the horizontal and vertical edges ofthese rectangles are parallel to the r and z coordinate axes,respectively, i.e.

ri = 10 + 20i , i = 0, . . . , 9,

zj = 20j , j = 0, . . . , 9.

Since the form of equation (2.9) suggests that the finite elements can havean arbitrary shape and position in space computing integrals over their ele-ment domains is a bit tricky. To overcome this difficulty, one uses a projec-tion method which maps the coordinates of a well known reference elementto the coordinates of an arbitrary element in space by a mapping the valuesrange from -1 to +1, and the reference coordinates are as (ξ1, η1) = (−1,−1),(ξ2, η2) = (1,−1) , (ξ3, η3) = (1, 1) , (ξ4, η4) = (−1, 1) that represent in Figure2.2(b).


(a) A rectangularelements.

(b) Thereferenceelement.

Figure 2.2: The coordinate transformation (r, z) in terms of the local coordi-nates (ξ, η)

Figure 2.3: The nodes Hi100i=1 of the elements.

For simplicity and to avoid any confusion, we use H(Xi), i = 1, 2, . . . , 100for Hi, i = 1, 2, . . . , 100. In other words, we define nodes Xi, i = 1, 2, . . . , 100for (ri, zj), i, j = 0, 1, . . . , 9 as shown in Figure 2.3.For each i = 1, 2, . . . , 100 define ϕj as basis function such that

ϕj(Xi) =

1 , i = j0 , i 6= j

.

A function v ∈ V can be written in the form of linear combination of basisfunction ϕi

v(X) =100∑i=1

αiϕi(X).

We obtain v(Xj) = αj by choosing appropriate values for αj. Equation (2.9)becomes

−(r∇H,∇ϕi) +

(rσ∂H

∂z

∂

∂z

( 1

σ

), ϕi

)− (

1

rH, ϕi) = 0.


Substituing (σ(z) = σ0 + (σ1 − σ0)e−bz) where σ0, σ1 and b are positive con-stants, we obtain

−(r∇H,∇ϕi) +

(br

((σ1 − σ0)e−bz

σ0 + (σ1 − σ0)e−bz

)∂H

∂z, ϕi

)− (

1

rH, ϕi) = 0,

for i = 1, 2, . . . , 100.Next, we consider the solution in the form of linear combination of basis

function ϕj

H(X) =100∑j=1

Hjϕj(X),

when Hj is the unknown constants to be found.Then equation (2.9) can be written in the form of linear combination as

follows

100∑j=1

Hj

[−∫

Ω

r(∇ϕj·∇ϕi)drdz+

∫Ω

br

((σ1 − σ0)e−bz

σ0 + (σ1 − σ0)e−bz

)∂ϕj∂z

ϕidrdz−∫

Ω

1

rϕjϕidrdz

]= 0,

(2.10)for each i = 1, 2, . . . , 100.

We need a transformation from an original element to a reference elementas shown in Figure 2.2

r = rk +h

2(1 + ξ), dr =

h

2dξ,

z = zk +h

2(1 + η), dz =

h

2dη. (2.11)

where k = 0, 1, . . . , 8.The basis functions can be written in the form of ξ and η as follows

N1(ξ, η) =1

4(1− ξ)(1− η),

N2(ξ, η) =1

4(1 + ξ)(1− η),

N3(ξ, η) =1

4(1 + ξ)(1 + η),

N4(ξ, η) =1

4(1− ξ)(1 + η). (2.12)

We transform r, z to ξ, η by using the transformation equation (2.11) to-


gether with basis functions equation (2.12), so equation (2.10) becomes

100∑j=1

Hj

[−∫ 1

−1

∫ 1

−1

(rk +

h

2(1 + ξ)

)[ ˆ∂ϕj∂ξ

ˆ∂ϕi∂ξ

+ˆ∂ϕj∂η

ˆ∂ϕi∂η

]dξdη

+bh

2

∫ 1

−1

∫ 1

−1

(rk +

h

2(1 + ξ)

)( (σ1 − σ0)e−b(zk+h

2(1+η)

)σ0 + (σ1 − σ0)e−b

(zk+h

2(1+η)

)) ˆ∂ϕj∂η

ϕidξdη

− h2

4

∫ 1

−1

∫ 1

−1

ϕjϕi(rk + h

2(1 + ξ)

)dξdη] = 0, (2.13)

for each i = 1, 2, . . . , 100.The value of equation (2.13) can be written in the form of matrix as follows(

−A+B − C)

ui =(a + d + c

),

where (−A + B − C), the stiffness matrix is an 100 × 100 matrix, ui and(a + d + c) are 100-vectors for all i = 1, 2, . . . , 100.

3 Numerical Results

Since the Galerkin’s Method of Weighted Residuals was applied to equation(2.5), we obtained the values of magnetic field at various positions of theearth’s structure with one layer having exponentially decreasing conductivityσ = σ0 + (σ1 − σ0)e−bz when σ0 < σ1. There is a source providing a DCvoltage and a receiver on the ground surface which picks up the signal fromr = 10 m to r = 190 m. We discrete the depth into 9 subintervals equallyof the size h = 20 m, i.e. we consider z = 0, 20, . . . , 180 m. We use constantσ1 = 1.5 S/m, σ0 = 0.5 S/m and b = 0.01, 0.05, 0.1, 0.2 and 0.3 m−1. Thenumerical solutions of the magnetic field at each node is calculated by usingMatlab program.

Graphs of the relationship between magnetic field and distance of receiver fromsource at various depths.

(a) (b) (c)


(d) (e)

Figure 3.1: The relationship between magnetic field and distance of receiverfrom source when z is fixed for (a) b = 0.01 m−1 (b) b = 0.05 m−1 (c) b =0.1 m−1 (d) b = 0.2 m−1 (e) b = 0.3 m−1

From Figure 3.1(a) to (e), when b = 0.01, 0.05, 0.1, 0.2 and 0.3 m−1, respec-tively, we can see that the value of magnetic field decreases exponentially as rincreases and it decreases as z increases as well. The value of magnetic field ishighest when b = 0.01 m−1 and it decreases when b = 0.05 m−1 and b = 0.1m−1, respectively and increases slowly when b = 0.2 m−1 and b = 0.3 m−1,respectively. These results are caused by the conductive of the ground and thevertically location of its.

(a) (b)

(c) (d)

Figure 3.2: The relationship between magnetic field and distance of receiverfrom source when b varies from 0.01, 0.05, 0.1, 0.2 and 0.3 m−1 and z is fixed.(a) z=20 m (b) z=60 m (c) z=100 m (d) z=140 m


Figure 3.2(a) to (d) represents the value of magnetic field which are plottedagainst r when b varies and z is fixed at 20, 60, 100 and 140 m, respectively.We can see that the value of magnetic field where b = 0.01, 0.05, 0.1, 0.2 and0.3 m−1 decreases exponentially as r increases and it has similar values whenz increases because the value of magnetic field decreases to zero and has valuenear zero when z increases as b varies. From Figure 3.2(b), (c) and (d), thethree curves representing the value of magnetic field when b = 0.1, 0.2 and 0.3m−1have a different behavior from the others. The value of magnetic field isgreater than that when b = 0.05 m−1, as we can see the curves cross those twolines for b = 0.05 m−1. These results take place according to the conductiveof ground and the spacing distance between source and receiver.

Contour graphs of the relationship between magnetic field and distance ofreceiver from source at various depth are shown as in Figure 3.3.

(a) (b) (c)

(d) (e)

Figure 3.3: Contour graphs of magnetic field at different distances of receiverfrom source and different depths when (a) b = 0.01 m−1 (b) b = 0.05 m−1

(c) b = 0.1 m−1 (d) b = 0.2 m−1 (e) b = 0.3 m−1

From Figure 3.3(a) to (e), when b = 0.01, 0.05, 0.1, 0.2 and 0.3 m−1, thered color shows the area when the value of magnetic field is high and the bluecolor shows the area when the value of magnetic field is low. The value ofmagnetic field decreases from b = 0.01 to b = 0.1 m−1 and it increases slowlyfrom b = 0.1 to b = 0.3 m−1, as we can see in Figure 3.3(a) to (e).

Turning to the case of increasing conductivity σ = σ0 + (σ1−σ0)e−bz whenσ0 > σ1, σ1 = 0.5 and σ0 = 1.5 (S/m), the numerical solutions of the magneticfield at each node is calculated by using Matlab program as shown in Figure.


Graphs of the relationship between magnetic field and distance of receiver fromsource at various depths.

(a) (b) (c)

(d) (e)

Figure 3.4: The relationship between magnetic field and distance of receiverfrom source at various depths when (a) b = 0.01 m−1 (b) b = 0.05 m−1 (c) b= 0.1 m−1 (d) b = 0.2 m−1 (e) b = 0.3 m−1

From Figure 3.4(a) to (e), when b = 0.01, 0.05, 0.1, 0.2 and 0.3 m−1, respec-tively, we can see that the value of magnetic field decreases exponentially as rincreases and it decreases as z increases as well. The value of magnetic fieldincreases when b increases and it decreases slowly from b = 0.2 m−1 to b = 0.3m−1. This results is caused by the conductive of the ground and the verticallylocation of its which is as same as the case of decreasing conductivity profile.

(a) (b) (c) (d)

Figure 3.5: The relationship between magnetic field and distance of receiverfrom source when b varies from 0.01, 0.05, 0.1, 0.2 and 0.3 m−1 and z is fixed.(a) z=20 m (b) z=60 m (c) z=100 m (d) z=140 m


Figure 3.5(a) to (d)represents the values of magnetic field which are plottedagainst r when b varies and z is fixed at 20, 60, 100 and 140 m, respectively.We can see that the value of magnetic field where b = 0.01, 0.05, 0.1, 0.2 and0.3 m−1 decreases exponentially as r increases and it has similar values whenz increases because the value of magnetic field decreases to zero and has valuenear zero when z increases as b varies. From Figure 3.5(b), (c) and (d), thethree curves representing the values of magnetic field which are plotted againstr when b = 0.1, 0.2 and b = 0.3 m−1 have a different behavior from the others.The value of magnetic field is smaller than that when b = 0.2 m−1, as we cansee the curve cross the line for b = 0.2 m−1. These results occur according tothe conductive of ground and the spacing distance between source and receiver.

Contour graphs of the relationship between magnetic field and distance ofreceiver from source at various depths.

(a) (b) (c)

(d) (e)

Figure 3.6: Contour graphs of magnetic field at different distances of receiverfrom source and different depths when (a) b = 0.01 m−1 (b) b = 0.05 m−1

(c) b = 0.1 m−1 (d) b = 0.2 m−1 (e) b = 0.3 m−1

From Figure 3.6(a) to (e), when b = 0.01, 0.05, 0.1, 0.2 and 0.3 m−1, thered color shows the area when the value of magnetic field is high and the bluecolor shows the area when the value of magnetic field is low. The value ofmagnetic field increases when b increases and it decreases slowly from b = 0.2m−1 to b = 0.3 m−1, as we can see in Figure 3.6(a) to (e).


4 Conclusions

A mathematical model of the magnetic fields is conducted by using partialdifferential equations. Numerical solutions of partial differential equations arecomputed by using finite element method (FEM) to find the value of the mag-netic field at various locations of the ground. In our models, the ground have anexponentially decreasing and increasing conductivity profiles. The Galerkin’sMethod of Weighted Residuals is applied in finite element method. There isa source providing a DC voltage and a receiver on the ground surface whichpicks up the signal from r = 10 m to r = 190 m. We begin our study withthe relationship between magnetic and electric fields. The vector equationsfor computing magnetic and electric field is very difficult. Thus, in most re-search, we change the vector equations into scalar equations by considering thecomponents of the vector. We finally arrive with partial differential equationdescribing the system and it is surprisingly that our equation is independentfrom the electric current. The numerical results is calculated and plotted byusing MATLAB R2008b program to show the behavior of magnetic field atdifferent depths and distances from source point. In our research, the behav-ior of magnetic field decreases to zero when the depth of soil increases. As wellas the case of increasing the space between source-receiver, the magnetic fielddecreases to zero too. The value of b is an important role for the conductionof ground and effect to the magnetic field quantities as well. The comparisionof the quantities of magnetic field for the case of σ as an increasing function ishigher than the case of σ as a decreasing function according to the advantage ofthe DC source that better reflex at very high depth than on the ground surface.

Acknowledgements. The authors would like to thank the Departmentof Mathematics, Faculty of Science, Silpakorn University and Centre of Excel-lence in Mathematics for continuous financial and equipments support.

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Received: February 23, 2015; Published: March 27, 2015

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