DSpace at VNU: Applicating of secondary charge method for solving favourable problem in partially inhomogeneous models

This method is used for calculating electric field of the generating electrodes in a medium with given specific conductivity distribution and electric field of the secondary charge elect

V N U J O U R N A L O F S C IE N C E , M a th e m a tic s - P h ysics, T.xx, N02, 20 04

A P P L I C A T I N G O F S E C O N D A R Y C H A R G E M E T H O D

F O R S O L V I N G F A V O U R A B L E P R O B L E M

I N P A R T I A L L Y I N H O M O G E N E O U S M O D E L S

V u D u e M in h

College o f Science, V N U

method and results of application of this method for several partially

inhomogeneous models This method is used for calculating electric field of the

generating electrodes in a medium with given specific conductivity distribution

and electric field of the secondary charge electrics at inhomogeneous points of

the medium The method enables us to derive directly the expressions for

secondary charge density, electric field, electric potential at measuring points

From these results, the apparent resistivity curves of the different electrodes

arrays are drawn Test of this method by several different models and some of

which are demonstrated in this paper, shows that the method gives reliable

information

1 I n t r o d u c t i o n

The m ethods of in te g ra l eq u atio n s or differential e q u a tio n s using of calculating for th e 2-D a n d 3-D m edium s req u ire th e disconnection of whole

m e d i u m i n t o v o l u m e e l e m e n t s a n d l e a d t o v e r y l a r g e s y s t e m o f a l g e b r a i c equations The in creasin g of effectivem ent of these m ethods is re la te d to th e im p ro v em en t of algorithm s for solution of system of eq u atio n s and to th e disconnected process Beside, convergence of th ese m ethods strongly depends on physical d istrib u tio n

c h a ra c te r of calcu lated model; convergence is very low for com plicated conductivity

d istribution

A p a rt from above m entioned two methods, th e r e is a seco n d ary c h a rg e method proposed by Alpin [1] for solution of tw o-dim ensional a n d th ree-d im en sio n al problems

2 T h e o r e t i c a l o u t l i n e o f s e c o n d a r y c h a r g e m e t h o d

W hen a c u rr e n t is g e n e ra te d in a m edium w ith c e rta in conductivity Ỵ = 1/p, at points of m edium , w here inhom ogeneity is caused by th e g e n e ra to r th en the secondary charge electrics are created The aim of th is m ethod is to d e te rm in e the

secondary field E tc as a sum of th e fields d E tc g e n e ra te d by seco n d ary charge elem ents From th is q u a n tity we are able to d e te rm in e U tc and derive values p at

observation points

At an observation point p , sum po ten tial a n d sum electric field are:

20

U(P) =Utc(P) + U0(P)

where: U()(P) is th e po ten tial of p rim a ry field; E Q( P ) is the s tr e n g th of the p rim ary electric field, which are c reated by g e n eratin g electrodes; Utc(P) and E tc( P ) are

respectively th e po ten tial and the s tre n g th of the secondary electric field a t the

point P.

The p o ten tial a n d th e str e n g th of the p rim a ry electric field are calculated by:

where: rQp is th e distan ce from th e g e n eratin g electrode A q a t th e point Q to observation po in t P: Cq is charge of point g e n eratin g electrode A q :

47tyQ 4 71

where: Yọ is v alu e for conductivity y in vicinity of th is point; pQ is th e real resistivity

in vicinity of th is point Q, ỈQ is th e c u rre n t g e n erate d by g e n e ra tin g electrode placed a t point Q.

If given electrode A q placed a t point Q goes th ro u g h s e p a r a te d surface, the

Eq (2.3) sh o u ld be replaced yQ by average a n g u la r conductivity:

y W: V:

w h e r e : w I i s t h e v o l u m e a n g l e w i t h p e a k a t v i c i v i t y o f t h e e l e c t r o d e i n t h e m e d i u m

w ith conductivity y, (i = 1,2) a n d w, = W2 = 2n In case one se p a r a te d surface is

p la n a r th e n yglb = — If th e m edium on one side of th e s e p a r a te d surface is

Y

non-conducting th e n Ỵgtb = —

2

U sin g Eq (2.3), Eq (2.2) becomes:

I q P q

Q 4tu ' q p 4 71 Q r ậ p

In case th e m edium is of p a rtia l homogeneity, th e source for the secondary

is th e surface ch arg es es a p p ea rin g on se p a ra te d surface w here function Y is discrete At point p on diviseve surface S ap betw een th e m edium in which 7 equals

to y a n d Yp , charge d e n sity is d e term in ed by:

( 2 6 )

where: E “(P) and E„(P) are electric field s tr e n g th s a t point p along direction of

two norm al vectors of diviseve surface

E ^ b (P) is th e algebraic average of th ese two values:

In case the function Ỵ is continuous (g rad ien t m edium ), th e source for the

secondary field is th e volume ch arg es ev g e n erate d in th e m edium lim ited by the volume Vrg h a v in g g r a d y not equal to zero, i.e th e con tin u o u s function Ỵ is charged

a t different points At point p in th e medium Vrg , th e d e n sity of th ese ch arg es is:

In general case, function Ỵ can be ch arged discretely or continuously in medium a t the sam e tim e, th erefo re source for the secondary field can be both

surface ch arg es es and volume charges ev T hen p o ten tial utc an d electric field

s tre n g th E tc are d e te rm in e d by:

where: O q is the surface charge density in the m edium a t point Q of an e lem en t dSq

of the diviseve surface;

5Q is th e volume charge den sity in the m edium a t point Q of an e lem en t d V Q

a t which Vy h as a finite value;

ĨQP is th e distan ce from point Q to point P;

des = OqdSq and dev = ỏọdVọ are respectively th e ch arg es of surface and

volume elem en ts a t Q.

The surface in te g ra l in (2.10) is over all diviseve surface, and volume

in teg ral is over all volumes w here Ỵ is charged continuously

For d e te rm in a tio n of (2.1) and (2.10) we need to know Ơ Q a n d ỗQ (Q site)

These functions are derived by in te g ral equation:

(2.7)

K up(P) is th e coefficient of contacting surface a t p :

(2 8)

(2.9)

Utc(P)= ị ^ - d S Q+ Ị ^ - d V q

Sop r»p V, rQp

(2.10)

K Ạ P )

2 k

o p =

4ny

Ị ¥ - ( f ọ p V y , K + f - ^ - f e p v r p K + f S “ (P)VT

( 2 11)

(2 1 1’)

where:

W ) = ^ I - Y - ( r p Q - " p )

4 7 1 „ , 3

« YrPQ

B.r<p)VTP = Ị - 1 - T

471 Q y r

(rpQ.Vyp)

(2 12)

(2 12’)

PQ

Thus, d e te rm in a tio n in case of complicated m edium is lim ite d to deriving the

functions ơp a n d Sp by solving of system of eq u atio n s (2.11) a n d (2.11’).

3 A p p l i c a t i n g o f s e c o n d a r y c h a r g e m e t h o d f o r s o l v i n g f a v o u r a b l e

p r o b l e m i n p a r t i a l l y i n h o m o g e n e o u s m o d e l s

In th is p ap er, we apply th e secondary charge m ethod only for th e partially inhom ogeneous models In th is case, th e source for th e secondary field IS th e surface

c h a r g e s es , a p p e a r i n g o n d i v i s e v e s u r f a c e a t w h i c h f u n c t i o n Y i s d i s c r e t e

As it is known, th e field a t point p is th e field g e n e ra te d by th e gen eratin g

electrodes a n d th e field of th e secondary charges a p p e a rin g on diviseve surface Therefore:

(3.1) (3.2) Where:

E n(P) = E™ (p ) = E ‘nc (p ) + E„(P)

E ?(P)= j d E ' c(P)

sL

with d E tc(P) is the field a t p caused by a secondary charge e le m e n t a t a point Q on

diviseve surface:

r q p H p

d E i: { P ) = d^ ~

(3.3)

Where: rQP is th e d istan c e from Q to p,

rip is n o rm al vector a t point p of diviseve surface S ap,

d E tcQ is th e secondary charge of the surface e le m e n t d S Q a t point Q on

se p a ra te d surface:

From (3.3) a n d (3.4) we have

dEtc( P ) - —Q- ịrqp.ìip \ d S t

rQP

Using (3.5), (3.2) be comes:

K ( P ) ‘ \ ^ - { r QP.nP}lS

o rX r>

(3,5

(3.6)

s„p 'QP

The p rim a ry field E°n(P) a t point p is d ete rm in e d by:

4 7 1 Q y Q r Qp

W ith (3.6), (3.7) a n d (3.1) become:

E n { P ) = -f ^ ( r QP - n p )

(3.7

Q

(3.8)

Com bining (3.8) with expression (2.6), we d e te rm in e th e surface charge

density a t point p as:

From th ese r e s u lts we are able to derive th e following expression for potential

and stre n g th of the s e c o n d a r y field a t o b s e r v a t i o n P:

Utc(P)= I — dSp ; Ẻ"(P) = Ị Ĩ ± ( r ọp.np )dS

111 p r a c t i c a l f i e l d , w i t h i n s t a l l a t i o n s o f d i f f e r e n t GỈGCtrodG c i r r a y s n i G c i s u r i n g

g e n e r a t i n g c u r r e n t I a n d p o t e n t i a l d i f f e r e n c e AƯ b e t w e e n r e c e i v i n g e l c c t r o d e s w e

c a n c a l c u l a t e t h e a p p a r e n t r e s i s t i v i t y b y t h e e x p r e s s i o n :

I

where: K(r) is coefficient of electrode array.

4 S o m e r e s u l t s

We have developed a softw are by MATLAB lan g u a g e [2] for PC to solve

favourable problem by u sin g above - m entioned secondary charge method We

applied it to some models of p a rtia lly inhom ogeneous m edium Below we show

some obtained resu lts

The t w o - l a y e r e d g e o - e l e c t r i c a l m o d e l

Figure 1 shows the a p p a r e n t resistiv ity curves in the two-laycred geo-electrical

model (p ] = 1, h ] = 10, p2 — 10) corresponding to t w o - e l e c t r o d e a rra y , calculated by Rưgiỏp a lg o rith m (dashed line) and by secondary charge m ethod (solid line) As can

be seen in th is figure 1, two curves are very close to each other

10* r .

10°

t o '

1 0 *

Fig 1 The a p p a re n t resistivity curves calculated by secondary charge method (solid

line) and by Rưgiôp algorithm (dashed line)

M od e l o f one v e r t i c a l s e p a r a t e d su rf a c e

30 40 50

p 2 = 10 Q m

p, = 100 Qm

Fig 2 The a p p a re n t resistivity curves calculated by secondary charge method (solid line) and by electric image method (dashed line) (pj = 100 Qm, p2 = 10 Qm).

Mo d e l o f one o b l i q u e 30° b o u n d a r y s u r f a c e

Fig 3 The a p p aren t resistivity curve in model of one oblique 30° boundary surface,

two-electrodes array of 10 m spacing

M od e l o f one o b l i q u e 150° b o u n d a r y s u r f a c e

Fig 4 The a p p aren t resistivity curve in model of one oblique 150° boundary surface,

two-electrodes array of 10 m spacing

R esu lts show n in Fig.3 and Fig.4 fully correspond to s ta n d a r d p a le ts for geo­ electrical slices w ith vertical contacting boundary surface [3]

5 C o n c l u s i o n s

By in v e s tig a tin g theoretically the secondary charge m ethod and applying it for calculation in some models of p a rtia l homogeneity, we m ay d raw th e following rem ark s:

- This m eth o d in com bination of modem calculation m ethods can well be applied for model of any m edium

- T est of th is m ethod by several different models a n d some of which are

d e m o n s tra te d above, shows t h a t th e m ethod gives reliable inform ation

We will co n tin u e stu d y in g f u rth e r for grid of th e secondary charge elem ents corresponding to c h a ra c te ristic s of each type of m edium to h a v e a n aly sin g re s u lts of geophysics d o c u m e n t are in progress, as well as in o rd er to have practical applications in th e n e a r future

R e f e r e n c e s

1 A n b n n H Jl M , M e T O A B T o pM H H b i x 3 a p f l A O B , n p u K J ia d H a n 2 e o c p u 3 U K a B b in , M , 9 9

(1981)

2 E tte r D.M., Engineering Problem Solvi ng with M a t l a b , P re n tic e In te rn a tio n a l,

Inc U n iv e rsity of Colorado Boulder, 1999, 423 p

3 Lâm Q u a n g Thiệp, Nguyễn Kim Quang, Giải bài toán t h u ậ n đo sâu điện trên

môi trườ ng b ấ t đồng n h ấ t b ấ t kỳ Tạp chí Đại học Tông hợp Hà N ộ i , (1995).