GEOLOGINEN TUTKIMUSLAITOS
GEOLOGICAL SURVEY OF FINLAND
TUTKIMUSRAPORTTI N:o 32
REPORT OF INVESTIGATION No. 32
Lauri Eskola
The solution of the eleetrie field
intensity and potential of a point
eurrent souree in a two-dimensional
environment with anisotropie
eonduetivity
Part I: Theory
Espoo 1978
GEOLOGINEN TUTKIMUSLAITOS
Tutkimusraportti n: 0 32
Lauri Eskola
THE SOLUTION OF THE ELECTRIC FIELD INTENSITY AND POTENTIAL OF A
POINT CURRENT SOURCE IN A TWO-DIMENSIONAL ENVIRONMENT WITH ANISOTROPIC CONDUCTIVITY
PART I: THEORY
Espoo 1978
2
Eskola, L. 1978. The solution of the eleetric field intensity and potential of a point
eurrent souree in a two-dimensional environment with anisotropie eonduetivity. Part
I: Theory. Geological Survey of Finland, Report of Investigation No. 32. 18 pages,
2 figures.
A method is given for solving the de eleetrodynamie field problem of a threedimensional point current souree in a two-dimensional struetural model. Parallel to the
strike the field is represented by a Fourier series. On the plane perpendicular to the
strike the problem is solved by means of the method of subseetions where the first
degree behaviour of field over the subseetions is assumed. The effeet of anisotropy is
also eonsidered.
The applieations of the method and the eonclusions will be published in a later
number of this series.
ISBN 951 - 690- 087 - 9
VTIOFFSETPAINO 19 78
617 /6
3
CONTENTS
Introduction .. ............. ... . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The integral equation for the field intensity . . . . . . . . . . . . . . . . . . . . . . . . . . .
The solving of the integral equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Primary potential and primary field intensity ... . . . . . . . . . . . . . . . . . . . . . .
The Green's functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Page
5
6
8
12
14
18
5
INTRODUCTION
Numerical solutions of the electric potential problem frequently entail resolving
a set of algebraic equations. To solve the field problem of a three-dimensional model, Eskola (1975) introduced the method of subareas, whereby the surface integral equation
of the field intensity reduces into a set of simultaneous algebraic equations. For real
conductivities the number of equations in the set is equal to the number of subareas;
for complex conductivities it is twice that number. Thus, when the size of the threedimensional model is large , the set of equations rapidly exceeds the equation-solving
memory capacity of a computer.
Geological structures, however, often persist in one horizontal direction to such
an extent that they can be assumed to be two-dimensional. This greatly eases the numeric
task of the solution. To solve the electric field problem of a two-dimensional model, Eskola and Hongisto (1978) have proposed a method of line subsections in which the crossseetion of the boundaries is represented by line segments parallel to the coordinate axes
that run perpendicular to the geological strike. The line integral equation for the field
intensity reduces into a set of linear algebraic equations. The method requires that also
the transmitter electrodes be two-dimensional, i.e. the line electrodes parallel to the strike.
Eskola et al. also suggests procedures that allow the galvanic effects of a two-dimensional
structure measured in the field of the point current electrodes to be transformed into the
effects that would be obtained for the same structure by applying the line current electrodes. In certain circumstances, however, transformation is difficult to realize, for example,
when the effective zero level of the field intensity remains outside the measuring range.
The transformations cannot be applied at all for electrode configurations in which the
receiving electrodes do not move, in relation to the transmitter electrodes, perpendicular
to the measuring direction (e.g. the dipole-dipole array).
This paper gives a method for solving the fjeld problem of a two-dimensional structure with anisotropie conductivity excited by a three-dimensional point current source.
The boundaries of the model are represented by arbitrarily oriented line segments in
the plane perpendicular to the strike. On this plane the problem is solved by means
of the method of subsections. Parallel to the strike the field is represented by a
Fourier series. A set of algebraic equations is now obtained separately for each harmonie of the field . The total field is obtained by superposing the solutions of the
sets of equations. The method includes the solution of the two-dimensional problem
with a line eurrent electrode. This special case is given by the harmonie zero of the
field. Snyder (1976) has presented a somewhat similar method for solving the field
Snyder (1976) has presented a somewhat similar method for solving the field
problem of a two-dimensional structural model. The methods differ in that Snyder
uses the field intensity behaviour of degree zero over each subsection and the Fourier
integral transform theory in the direction of the strike , whereas , in this paper, the
first degree behaviour of the field over the subsections and the Fourier series expansion in the direction of the strike is used. The effeet of anisotropy is also considered.
The applications of the method and the conc\usions will be published in a
later number of this series .
6
THE INTEGRAL EQUATION OF THE FIELD INTENSITY
For the electric field intensity the following equation holds (Stratton 1941,
pp. 186 ... 188) :
(1)
where a is the surface charge_density and w the volume charge density , S the boundaries
of the space, V the volume, R the field point and R o the source point. Gk denotes the
Green's function for the k-component of the field intensity. In a homogeneous linear
medium
-
w = Eo div E = 0,
and hence, the last integral vanishes from (I). Let the coordinates x, y, z be chosen so
that z is positive downwards and the surface of the ground at plane z = O. The field
of the surface charge on plane z = 0 can be substituted by the field of the electrical
image of the charge distribution for the half space z > O. Thus, (I) develops into
(Ja)
Now,
~
is the half-space Green's function for the k-component of the field intensity :
~ = (k - ~) / IR - Ri + (k - k~) / IR - R~13,
Using the following three boundary conditions for steady flow of current
(Stratton 1941 , p. 483):
(2)
~t
-
EH = 0,
7
and Ohm's law
T=
gE,
a can be defined by means of the field intensity. The indices sand t denote the normal
and the tangential components of the field vectors, respectively. is positive towards the
medium 2. Let us make the following assumptions : the principal axes of anisotropy of
conductivity are ä, band c; b is parallel to the y-axis and adenotes the angle between
In a two-dimensional structural model, when the y-axis (directhe positive x-axis and
tion of the geological strike) is one of the principal axes of anisotropy, the effect of
anisotropy on the surface charge density can be treated by projecting the conductivity
tensor to the plane y = O. Let us consider the normal component js of current density
at a point of the boundary between media land 2 (see Fig. I) . Let ß denote the angle
between the positive x-axis and the tangent of the boundary. js can now be expressed
separately for media land 2 by means of the components in the directions of the
principal axes ä and C of anisotropy as follows:
s
a.
js = - ja sin(ß - a) + je cos(ß - a).
(3)
The field intensity required to produce js has both the normal and the tangential
component:
Es =
I;
- ~ sin(ß -
a)
= ~ cos(ß - a)
+ Ee
+ Ee
cos(ß - a)
(4)
sin(ß - a).
Applying the scalar Ohm's law in the directions of the principal axes
ä and C
x
C
2
z
Fig. 1. The principal axes of anisotropy.
8
we get from (3) and (4):
(5)
~s = ~ .
~)
sin2 (ß -
~t = (&: -
~) .
+ &: . eos2 (ß
sin(ß -
~)
Now, from (2) and (5) we get for
-
. eos(ß -
~)
~) .
0 :
(6)
THE SOLVING OF THE INTEGRAL EQUATION
Together, (la) and (6) form an integral equation of the seeond kind , whieh is
solved numerieally. Let the geologieal model be two-dimensional and the y-axis parallel
to the strike. Further, let the point eurrent eleetrode (or eleetrodes) be loeated at
y = O. The boundaries of the model on plane y = 0 are represented by polygons whose
sides are divided into line subseetions Pi-1Pi whieh are so small that the field ean be
taken as linearly varying over eaeh of them. Parallel to the y-axis, the field intensity is
represented by the Fourier series expansion. By virtue of the above assumptions, the
field intensity is symmetrie in relation to plane y = 0 and ean therefore be expressed
as a eosine series between -L / 2~y~L / 2 (Stuart 1961)
E(x, y, z) =
N
n
~ E
n=O
(x , z) eos(21Tny / L).
The optimal values of N and L depend on the aeeuraey required and the geometry of
the model. When the method of subseetions and the Fourier series expansion are applied,
the term in (1 a) referring to the seeondary field beeomes
~ec(x, y, z) = n~oE; sec (x, z)
M
L(2 Pi
En sec (X , Z) =.2: J
k
n
f QE
eos(21Tny / L) ,
where
(p) . eos(21Tnyo /L) .
,=1 ·L(2 Pi- l
~x ,
0, Z!p , Yo) dp dyo .
(7)
9
x
y
z
Fig. 2. The geometry of the model.
The p is a point of subsection Pi- 1Pi. The summing over the index i refers to the
summing over the subsections i (= Pi-1Pi). In the integral, the subindex referring to
the medium I has been dropped from the field intensity . The subsections are numbered
in such a way that the subsection with Pi-l and Pi as terminal points is the ith in sequence. We are looking in the direction of increasing i when the positive normal of the
boundary (medium 2) remains on our right hand side (see Fig. 2) . In principle , the
model may contain any number of one- or two-dimensional bodies.
If the linear behaviour of the field between the terminal points of the subsections
is applied, E(p) can be defined as
E(p)
= (I -
w(i)) . E(Pi_ l) + w(i) . E(p),
Denoting
Vk( 1') = Vk( Pi- l Pi ) = fPi w(i)' Gk(i) dp ,
where
10
where i in
fact that
~ (i)
denotes that the source point lies on subseetion i, and applying the
(7) takes the form
(8)
For a closed boundary, Po = PM and PM+l = Pl' G~ may be called the half-space
Green's function for the nth harmonie of the fjeld intensity. An approximation
equation can now be written for (I a)
~(x ,
y, z) = Eok(x, y , z)
+
N
~
IR)
Q En(p) . cos(21Tny / L) .
(9)
Similarly, the potential is expressed by
Q En (p) . cos(21Tny / L) ,
(10)
where Gn may be called the half-space Green's function for the nth harmonie of the
potential.
To solve (9), a set of simultaneous algebraic equations are formed for each
harmonie n. The calculation points of the fjeld are the terminal points of the subsections.
As the corner points of the boundary polygon are singular points of the charge density,
they cannot be used as terminal points of subsections. Thus, for calculating the fjeld
intensity in the vicinity of a corner point this must be treated as a single bent subseetion
consisting of two parts one on each side of the corner point. Let q be a corner point
on subsection Pi- lI>. . The effect of q on the Green's function can be treated by substituting
and
for Uk and Vk as follows :
u:
v:.
11
It must be noted also that , as a consequence of difference in orientation,
<4
and q
take different values along the different parts Pi-lq and qPi of the source subsection
Pi-I qPi ' The singularity of the Green 's function at the source point is eliminated by
su bsti tu ting
R'S
=
- a12E0
for the field intensity due to the charge density at the point and by excluding it from
calculations. When the left hand side of (9) and the primary field are also represented
as Fourier series, the following equation holds at every point Pi :
j~-l, G~ePiIPj- lPjI\+I) . (QsE~ePj) + qI;ePj))
i,i+1
j~- I , G~ePiIPj_lPjPj+l)
eQsE~(p) + qEtep)) +
i+l
[IR,. .G:
(p; 11\_, Pj!\H) .
(o.E; (1\) + o.F,"(Pj))
(11 )
12
In (11)
(12)
er
The G~ and
are Green's funetions that are obtained by excluding small surroundings
of point Pi from the integration ranges in funetions G~ and Gt. The ßi is the angle between the positive x-axis and the subseetion i:
When the environment is isotropie the terms involving the produet OtI; vanish and
the number of unknowns equals the number of subseetions . For anisotropie environment , the number of unknowns equals twiee the number of subseetions.
After (11) has been solved for eaeh value of n, the field intensity ean be
ealculated at any point in the half-spaee z > by means of (9), and the potential
by means of (10). These proeedures have to be earried out separately for eaeh plane
y = eonstant eontaining eurrent eleetrodes . The resulting field or potential ean be obtained by the prineiple of superposition .
°
PRIMARY POTENTIAL AND PRIMARY FIELD INTENSITY
Let the eurrent eleetrode 1 be loeated at point ( Xl ' 0 , ~) of the system of
eoordinatesx, y , z, and let this point eoineide with the origin of the system of prineipal
axes a, b, e of anisotropy. The primary potential 0 0 due to the eurrent strength \ emitted by eleetrode 1 ean be written (Bhattaeharya and Patra 1968, p. 14)
Pk = l /~ , k = a, b, e.
The eomponents of the eurrent density j are
Using the relation
1( = J T . ds , where
s
13
S is a surface enc10sing the current source, we obtain
Transforming into the system of coordinates x, y, z by
a
=
(x -
~)
. cos a + (z -
• sin a
~)
b=y
c = - (x -
~)
. sin a
+ (z
~)
-
. cos a ,
where the a-axis makes an angle a with the positive x-axis, and taking into consideration
the effect of the ground surface, yields
(13)
P
= P(z"
a)
=T .
(x - ~i
2L . (x -
+
Pb •
~)(z
I +K.
(z - ~f
+
- ~)
P' = P( - ~ , - a)
T = Pa • cos 2 a
+
2
Pe • sin a
K = Pa • sin2 a + Pe • co'; a
L = (Pe - Pa) • sin a . cos a .
The primary field is as folIows :
(14 )
Fk
= Fk (~,
a) , F~
= Fk ( - ~,
(T . (x - ~)
Fx(z" a)
=
Fz(z" a)
= (K .
(z - ~)
+
-a)
L . (z - z,)) /p3f2
+L•
(x - ~)) /p3f2.
14
If there are several eurrent eleetrodes on plane y = 0, the resultant potential and field
intensity ean be ealeulated by the prineiple of superposition.
The harmonies of the primary fjeld intensity are obtained as follows:
~k(X , z) = O ll)
E~k (x , z) = (2/L)
f(2 Eok(x,
y, z) dy for n=O,
(15)
-L(2
r Eok(x, y, z) . eos(21Tny/L) dy
-L(2
( ISa)
for nfO .
(15) and (ISa) are ea1culated by numerieal integration . Alternatively, (15) mayaIso be
expressed as an elementary funetion by direet integration .
THE GREEN'S FUNCTIONS
The Green's funetion for the nth harmonie of the fjeld intensity is defined by
(8). Let the equation of the line that coincides with the souree subseetion be (when
not horizontal)
~ =
aZu
+
b.
Consequently, the equation of the line with whieh the image of the souree subseetion
coincides beeomes
x~ = -az~
+
b.
By applying these notations, the following equations are obtained for funetions
Gase ~-l =f Zi (the su bseetion is not horizontal):
II (i) = f
Pi
~ dp
=
= ~(Zu ,
a),
u~
= ~(z~,
and \k '
(16)
Pi-l
~
II
-a)
(7)
(18)
15
Vk(i)
=f
p.
1
Pi-l
wGk dp
=
(19)
v'k -- -
('
Vk ZO'
-
~- l' -a)
(21 )
t
= 17;
- 7;- 11 / (7; -Zi_l)
a = (~ - ~-1) /(7; - 7;-1)
b = (~-17; - ~7;-1) /(7; -7;-1)
+
z) /(a2
(2A
+
7;-1) - b
h= - (X - b)(A
+
L;....1) - aA(~_1
A = - (a(x - b)
c
e
= X
+a .
= 2A + 1,-1 +
m = - (Z
+
+
I)
+
A)
Z
A)(L;....1
+
A).
Ca se 7;-1 = 7; (the subseetion is horizontal):
(16a)
(17a)
16
Uz(Zo) = - d . (x - ~)(z - zo) /(y~
=
Vk(i)
Vx(zo)
Xi
I (Vk
xo=Xi_1
= Ix,
+
-
-
(z - zof) • IR - Rol
+ V;)
(l8a)
(l9a)
- x,-II-I • (ln(x -
+
Xo
-
-
IR - ~I)
+
(20a)
(21 a)
Correspondingly, the Green's funetion for the nth harmonie of the potential
is defined:
2
f (UU + J) - Lf2
Gn(i) =
V(i
+
1)
+
V(i)) • cos(21Tnyo/ L) dyo '
(22)
where
U(i) = U(Pi- IP) = fPi G dp
Pi- I
p.
V(i)
G
=
= V(Pi_ lPi) =
-
-
l / IR- Rol
J w(i)'
Pi-I
G dp
--
+ l/IR -
R~I .
Case Zi_ 1 1= Zi (the subseetion is not horizontal):
(23)
(24)
17
V(i)
=
v
I
+
zi_1
v'
(25)
z~= ~i
(26)
v' =-v(Z~, - Zi_ l' - a)
Case Zi_ 1 = Zi (the subseetion is horizontal):
U(i) =
u = ü"tz o) = -
V(i) =
x 1·
I
-
-
d . ln(x - Xo + IR - ~I)
(v
+ v')
(24a)
(25a)
Xo~i_ 1
-
-
v = v(zo) = lXi - Xi_ I I- I . (IR - Rol - (x - Xi_I)
ln(x - Xo
+
-
-
IR - Rol))
(26a)
The integrals (8) and (22) are ealculated by numerieal integration. The harmonies
zero ~ and 0' mayaiso be expressed as elementary functions by direet integration .
When LJ1 is substituted for 11 the harmonie zero represents a solution of a two-dimensional problem with the transmitter eleetrode as a long line eleetrode . The J1 is the
eurrent strength emitted by the line electrode I per unit length.
The prineiple of solution presented in this paper ean also be applied to problems
involving Green's funetions of more eomplieated spaees, i.e., of a layered half-spaee.
Manuscript received February 6, 1978.
18
REFERENCES
Bhattacharya , P.K. & Patra, H.P., 1968 . Direct current geoelectric sounding. Elsevier,
Amsterdam , 135 p .
Eskola , L. , 1975 . A method for caJculating galvanic effects in an inhomogeneous environment. Geol. Surv. Finland , Rep. Invest . 9 , 24 p.
Eskola , L. & Hongisto, H ., 1978. Kaksidimensionaalisen mallin sähköisen kenttäprobleeman ratkaiseminen osajanojen menetelmällä . Geol. Surv. Finland , Rep.
Invest. 31 , in press.
Snyder, D.D ., 1976. A method for modeling the resistivity and IP response of twodimensional bodies . Geophysics 41 , 997 ... 1015.
Stratton , J .A., 1941. Electromagnetic theory. McGraw-Hill, New York , 6 15 p .
Stuart , R.D. , 196 I . An introduction to Fourier analysis. Methuen & Co LTD and
Science Paperbacks, 128 p .
ISBN 951-690-087-9