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

I VN U JO U R N A L O F S C IE N C E , M a th e m a tic s - P hysics, T.xx, N02, 2004 A P P L IC A T IN G O F SE 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 IN G F A V O U R A B L E P R O B L E M IN P A R T IA L L Y IN 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 this paper, we present a theoretical basis of a secondary charge 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 Abstract In I n t r o d u c t i o n The m ethods of integral equations or differential eq u atio n s using of calculating for th e 2-D and 3-D m edium s require the 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 increasing of effectivem ent of these m ethods is re la te d to the im provem ent of algorithm s for solution of system of equations and to th e disconnected process Beside, convergence of these m ethods strongly depends on physical distribution c h aracter of calculated model; convergence is very low for complicated conductivity distribution A part from above m entioned two methods, th e re is a secondary charge method proposed by Alpin [1] for solution of two-dim ensional and three-dim ensional problems 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 When a c u rre n t is g enerated in a medium w ith c ertain conductivity Ỵ = 1/p, at points of medium, w here inhomogeneity is caused by th e g e n erato r then the secondary charge electrics are created The aim of this m ethod is to determ in e the secondary field E tc as a sum of the fields d E tc g e n erated by secondary charge elements From th is q u a n tity we are able to determ ine U tc and derive values p at observation points At an observation point p , sum potential and sum electric field are: 20 A p p l ic a t in g o f se condary charge method for 21 U(P) =Utc(P) + U0(P) (2.1) Ẻ ( P ) = Ẻ tc( P) + Ẽ 0( P) where: U()(P) is th e potential of prim ary field; E Q(P ) is the stre n g th of the prim ary electric field, which are created by generating electrodes; Utc(P) and E tc(P ) are respectively the potential and the stre n g th of the secondary electric field at the point P The p o tential and the stre n g th of the prim ary electric field are calculated by: Q rQP Q rQP where: rQp is the distance from the generating electrode A q a t the point Q to observation point P: Cq is charge of point generating electrode A q Iq eQ Pq I q A -4 71 47tyQ : (2-3) where: Yọ is value for conductivity y in vicinity of this point; pQ is the real resistivity in vicinity of this point Q, ỈQ is the c u rre n t generated by g en eratin g electrode placed at point Q If given electrode A q placed at point Q goes through se p a te d surface, the Eq (2.3) should be replaced yQ by average a n g u lar conductivity: w y W: V: f f (2-4) w h e r e : w I is 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 ic iv ity of t h e e l e c t r o d e in t h e m e d i u m w ith conductivity y, (i = 1,2) and w, = W2 = 2n In case one se p a te d surface is p lan a r th e n yglb = — If th e m edium on one side of th e se p a te d surface is Y non-conducting th en Ỵgtb = — U sing Eq (2.3), Eq (2.2) becomes: u 0{ P ) = ỵ ^ s - ; E 0ị P ) = j - ỵ Q tu 'q p 71 Q I q Pq , QP (2.5) rậ p In case the m edium is of partial homogeneity, th e source for the secondary is the surface charges es appearing on se p ara ted surface where function Y is discrete At point p on diviseve surface S ap betw een the m edium in which equals to y a n d Yp , charge density is determ ined by: 22 Vu D u e Mirth ( ) where: E “(P) and E„(P) are electric field stre n g th s a t point p along direction of two normal vectors of diviseve surface E ^ b (P) is the algebraic average of these two values: (2.7) K up(P) is the coefficient of contacting surface at p : (2 8) In case the function Ỵ is continuous (gradient medium), the source for the secondary field is the volume charges ev generated in the m edium lim ited by the volume Vrg having g r a d y not equal to zero, i.e the continuous function Ỵ is charged at different points At point p in the medium Vrg , the density of these charges is: (2.9) In general case, function Ỵ can be charged discretely or continuously in medium at the sam e tim e, therefore source for the secondary field can be both surface charges es and volume charges ev T hen potential and electric field utc stre n g th E tc are determ in ed by: Utc(P)= ị ^ - d S Q+ Ị ^ - d V q Sop r»p V, rQp ( 10 ) where: Oq is the surface charge density in the m edium a t point Q of an elem ent dSq of the diviseve surface; 5Q is the volume charge density in the m edium at point Q of an elem ent d V Q a t which Vy has a finite value; ĨQP is the distance from point Q to point P; des = OqdSq and dev = ỏọdVọ are respectively the charges of surface and volume elem ents a t Q The surface in tegral in (2.10) is over all diviseve surface, and volume integral is over all volumes w here Ỵ is charged continuously For determ in atio n of (2.1) and (2.10) we need to know These functions are derived by integral equation: ƠQ and ỗQ (Q site) 23 A p p lic a tin g of se condary charge method for KẠP) 2k op = 4ny J c rQp ( 11 ) S p K +K (P ) J rQp Ị¥ -(fọ p V y , K i + f - ^ - f e p v r p K + f S “ (P)VT V, r«p (2 11’) where: W (2 12) ) =^ I-Y -(rp Q -"p ) « YrPQ „, 471 Q y r ( 12 ’) (rpQ.Vyp) B.r