19

Physical modelling of the electromagnetic heating of oil sand and other earth-type and biological materials

By F. E. Vermeulen, MIEEE, F. S. Chute, MIEEE, andM. R. Cervenan, Department of Electrical Engineering, University of Alberta.

Maxwell's equations and the thermal equation for heat flow are examined and scaling criteria are developed which show that it is possible to construct scaled physical models in which the electromagnetic and thermal phenomena of the full scale system can be modelled simultaneously. Simultaneous modelling is made possible by simulating electromagnetic frequency and thermal events on different time scales. It is also shown that simultaneous modelling of electromagnetic and thermal phenomena can be carried out when the electrical conductivity of the medium of the full scale system is temperature dependent. In this case electromagnetic frequency and thermal events are still modelled on different time scales; the time variation of the envelope of the electromagnetic field distribution, however, is modelled on the same time scale as thermal events. The electrical and thermal properties of oil sand and the general problem of in-situ recovery of oil therefrom have been used to guide the development of the modelling criteria. The results obtained can be applied to a large class of other problems as well, such as electromagnetic heating of earth-type materials in the mining and construction industry, as well as the electromagnetic heating of food stuffs and biological tissue.

On trouvera dans cet article un examen des equations de Maxwell et de l'equation thermique pour le chauffage des fluides et une mise au point de criteres de reduction d'echelle qui demontrent qu'il est possible de construire des modeles physiques d'echelle reduite dans lesquels les phenomenes electromagnetiques et thermiques du systeme de grandeur reelle peuvent etre modelises simultanement. La modelisation simultanee est rendue possible grace a une simulation des phenomenes thermiques et de la frequence electromagnetique selon des echelles de temps differentes. On y demontre egalement que la modelisation simultanee des phenomenes electromagnetiques et thermiques peut se faire lorsque la conductivite du medium de grandeur reelle est fonction de la temperature. Dans cette hypothese, la frequence electromagnetique et les phenomenes techniques sont toujours modelises selon des echelles de temps differentes; cependant, la variation dans le temps de la distribution du champ magnetique est modelisee selon la meme echelle de temps que les phenomenes thermiques. Les proprietes thermiques et electriques des sables bitumineux et le probleme general de la recuperation in-situ du petrole a servi de guide pour la mise au point des criteres de modelisation. II est possible d'appliquer les resultats obtenus a une large categorie d'autres problemes tels que chauffage electromagnetique de materiaux de type terreux dans l'industrie miniere et de la construction et le chauffage electromagnetique de produits alimentaires et de tissu biologique.

Introduct ion

T h e very la rge oil sand deposi ts in C a n a d a , V e n e z u e l a and the U . S . S . R . are es t imated to conta in s o m e 2 1 0 0 b i l l ion bar re l s of oil in p l ace , r epresen t ing near ly as m u c h as the w o r l d ' s total d i s c o v e r e d m e d i u m and light gravi ty oil in p l ace . Near ly half of these oil s and r e sou rces , or about 9 5 0 bi l l ion ba r re l s , a re located in Albe r t a in the A t h a b a s c a , Co ld L a k e , W a b a s c a , and Peace River depos i t s . H o w ­eve r , on ly about 7 4 bi l l ion barre ls or 8 % of the Alber ta depos i t s are cove red by less than 50 m of ove rburden and are cons ide red su i tab le for e c o n o m i c surface min ing . Abou t 135 bi l l ion barre ls or 1 4 % are cove red by 50 to 150 m of overburden and it is unc lear at present whe the r this part of the deposi ts will b e recovered by s o m e form of min ing or by in-situ t e chn iques . T h e r ema inde r of about 741 bi l l ion barrels or 7 8 % , by far the largest part of the Alber ta d e p o s i t s , is benea th m o r e than 150 m of ove rburden and mus t b e r ecove red by in-situ m e t h o d s . 1 , 2

T h e p rob lem of in-situ recovery of oil from oil sand depos i t s rece ived at tent ion as early as 1920 . 3 S ince then , par t icular ly in the last t w o d e c a d e s , a var iety of in : s i tu recovery t echn iques h a v e b e e n s tud ied , inc luding such me thods as the u n d e r g r o u n d inject ion of s t e am, hot wate r and hot g a s , ignit ion of the oil wi th in the fo rmat ion and u n d e r g r o u n d a tomic e x p l o s i o n s . 4 , 5 A c o m m o n goal of t he se t echn iques is to transfer heat to the oil sand format ion to ra i se the t empera tu re of the very v iscous oil sufficiently a b o v e the in-si tu t empera tu re of 5 0 to 60°F so that the oil can flow and b e s w e p t f rom the hos t format ion by a sui table gas or fluid dr iv ing agen t . S ince the format ion is qui te i m p e r m e a b l e and has very low the rmal conduc t iv ­ity, heat t ransfer by conduc t ion and convec t ion , as in t he fo rego ing m e t h o d s , is a very s low p roces s . M o r e o v e r , cont ro l of t he m o v e m e n t of the injected hea t ing fluid wi thin the format ion is difficult o r i m p o s ­sible so that a major unso lved p rob lem of in-situ t echno logy is that of d i rec t ing t he fluid to t he reg ion w h i c h is to b e hea t ed , th is r eg ion be ing genera l ly the v o l u m e of the format ion b e t w e e n a sy s t em of injection and p roduc t ion we l l s .

A qui te different me thod of lower ing the viscosi ty of in-si tu oil wh ich is a t t ract ing increas ing at tent ion is the d e p l o y m e n t of e lec t ro ­magne t i c energy to heat the reservoir . S ince ' e l ec t romagne t i c ene rgy heats from wi th in , this m e t h o d is relat ively independen t of the low the rmal conduc t iv i ty of oil sand and is unaffected by the oil s a n d ' s lack of pe rmeab i l i t y . W h i l e the cos t per unit of heat i n t roduced in to Can. Elec. Eng. J. Vol 4, No 4 , 1979

the fo rmat ion by e l ec t romagne t i c t echn iques is p robab ly h ighe r than the cos t of a s imi lar a m o u n t of hea t in t roduced b y m o r e conven t iona l m e a n s such as s t e am, e l ec t romagne t i c hea t ing offers t he dist inct a d v a n t a g e that it can b e d i rec ted so that the reg ion to b e hea ted can b e pre-speci f ied . Th i s ensures min ima l was t age of the ene rgy suppl ied to the fo rma t ion , so that i t m a y very wel l b e poss ib le that a g iven t empe ra tu r e d is t r ibut ion wi th in the fo rmat ion can b e es tab l i shed wi th fewer uni t s of ene rgy if this ene rgy is e l ec t romagne t i c ra ther than in t he form of, say, s t e a m . O the r impor tan t cons idera t ions are that e l ec t romagne t i c energy can b e eas i ly con t ro l led and that it can b e c o n d u c t e d to the oil bea r ing format ion via boreho les wi th min ima l l o s se s , the lat ter not be ing the ca se w h e n hea t ing wi th s t eam w h e r e significant the rmal losses m a y occur as the s t eam t ravels th rough the o v e r b u r d e n . Fur the r , as present ly env i saged for mos t app l i ca t ions , e l ec t romagne t i c ene rgy w o u l d se rve pr imar i ly to preconctftion a se lec ted v o l u m e of the depos i t to c rea te a preferent ia l channe l b e ­t w e e n an inject ion and a r ecovery wel l w h i c h can then b e pene t ra ted wi th re la t ive ea se by a subsequen t ly injected dr iv ing fluid; this d r iv ing fluid w o u l d in t roduce a substant ia l a m o u n t of addi t ional thermal ene rgy at re la t ively low cost to further heat the oil bea r ing depos i t . T h e m o r e cos t ly e l ec t romagne t i c energy w o u l d thus b e used to supply on ly a fract ion of the the rmal ene rgy r equ i r emen t of the oil r ecovery p r o c e s s .

P r o p o s e d s c h e m e s for the e l ec t romagne t i c p rehea t ing of oil sand depos i t s r a n g e in f requency from a fraction of a Her tz to several G H z . A br ief o v e r v i e w of these s c h e m e s is g iven be low. All of these m e t h o d s are fundamenta l ly capab le of in t roduc ing heat in-s i tu . M u c h w o r k at bo th the fundamenta l and pract ical level r ema ins to b e d o n e , h o w e v e r , to identify the economica l ly m o s t feasible m e t h o d s .

A s e q u e n c e of r ecen t C a n a d i a n and U . S . pa ten ts var ious ly i ssued to P e t r o - C a n a d a , 6 t he P . C . E . g r o u p 7 and At lan t ic Richfield C o m ­p a n y 8 , 9 * 1 0 dea ls specifically wi th conduc t ion hea t ing at l o w fre­q u e n c y . T h e p r o p o s e d p rocesses invo lve s inking a p lura l i ty of wel l s in to the fo rmat ion wi th a typical spac ing of pe rhaps 3 0 m to 100 m . E lec t rodes a re l o w e r e d in to these wel l s so as to con tac t t he format ion bu t r e m a i n insu la ted from the cas ing in the o v e r b u r d e n r eg ion . A n a l te rna t ing e lect r ic potent ia l is appl ied to the e lec t rodes f rom a sou rce at t he surface t h rough insula ted c o n d u c t o r s , and cur ren t is forced to flow th rough the fo rmat ion . T h e f requency of exc i ta t ion is nomina l ly 6 0 H z , bu t m a y b e as low as 0 .1 H z in o rde r to r e d u c e magne t i c hys teres is and e d d y cur ren t p o w e r losses in the steel wel l c a s i n g s . 7

20 C A N . E L E C . E N G . J. V O L 4 N O 4 , 1979

Conduc t ion is pr imar i ly electrolyt ic in nature as the conduc t ing pa th is th rough the conna t e wate r he ld in the dev ious pore spaces pene t ra t ing the depos i t . This conna te water is gradual ly hea ted and by the rmal conduc t ion the format ion t empera tu re as a w h o l e is inc reased . T h e rate of .power d iss ipa t ion is control led so that the po re wate r is not vapor i zed and a conduc t ive pa th th rough the format ion is ma in t a ined . After a per iod of t ime that varies from a few mon ths to severa l years in typical app l ica t ions , the format ion t empera tu re will b e increased such that the viscosi ty of the heavy hyd roca rbon is r educed to the point permi t t ing recovery by, for ins tance , a s team dr ive .

T h e s imples t e m b o d i m e n t of this s cheme involves t w o wel l s and t w o e lec t rodes as s h o w n in Figure 1. A more e labora te e m b o d i m e n t for hea t ing a m o r e ex tens ive por t ion of the depos i t w o u l d cons is t of a relat ively la rge n u m b e r of wel ls and e lec t rodes pos i t ioned in a p r e ­de t e rmined grid type pa t te rn . Each individual wel l and e lec t rode wou ld contac t the format ion as indicated in F igure 1, and the sys tem of e lec t rodes w o u l d b e dr iven by a mul t iphase source so that cur ren t pa ths wou ld b e cont inual ly swi tched b e t w e e n the mul t ip l ic i ty of e lec t rodes to at tain more near ly uniform heat ing of the sub te r ranean fo rmat ion . 8 T h e foregoing concepts will soon b e tes ted in a s equence of pilot projects in the Athabasca oil sands . T h e tests will b e con­duc ted by Pe t ro -Canada and Japanese Oil Sands Alber ta L t d . , wi th the latter conce rn cont r ibut ing to this project a m i n i m u m of $ 7 4 . 8 mi l l ion over a per iod of 15 yea r s .

O V E R B U R D E N

O I L S A N D

L I N E S OF C U R R E N T F L O W

U N D E R B U R D E N

Figure 1: Full scale in-situ electromagnetic heating configuration.

At h ighe r f requencies proposa ls h a v e been put forward for in-si tu hea t ing by induc t ion . A sequence of recent patents env i sages the cons t ruc t ion of an unde rg round coil sur rounding the oil bea r ing fo rmat ion , this coil to b e formed by th read ing heavy conduc t ing cab le th rough mul t ip le vert ical and hor izonta l bore holes dr i l led t h rough the f o r m a t i o n . 1 1 , 1 2 H igh currents flowing through such a coi l w o u l d in­d u c e eddy currents in the v o l u m e enc losed by the coil and the reby heat the oil s and . S c h e m e s have a lso been sugges ted for in-si tu hea t ing at r ad io f r e q u e n c i e s . 1 3 , 1 4 O n e of these s tud ies , largely c o n c e p ­tual in na ture and not referr ing to any specific u n d e r g r o u n d rad ia t ing d e v i c e , involves the lower ing of a radia tor into the format ion and feeding it by coaxia l cab le from the su r f ace . 1 3 Tha t s tudy has s h o w n that such a sys tem can p rov ide more rap id and m o r e un i form hea t ing than w h e n an equiva len t amoun t of power is used to heat the fo rma­t ion by the rmal conduc t ion from the outer surface of a wel l bo re wh ich itself is hea ted e i ther by injected s team or d o w n ho le e lectr ical res i s tance hea te r s . In another m o r e specific p roposa l , long cyl indr ica l e lec t rodes are p laced into a sys tem of relat ively c losely spaced b o r e holes dr i l led in to the format ion . It is p red ic ted that R F energy gene r ­ated above g round and in t roduced into the format ion th rough these e lec t rodes wil l p r o d u c e spatial ly m u c h m o r e un i fo rm and m o r e rap id hea t ing than cou ld b e a t ta ined wi th me thods opera t ing at p o w e r f requencies , 1 5

T h e au thors h a v e unde r t aken a sys temat ic s tudy to e x a m i n e the fundamenta l feasibil i ty of the e lec t romagne t ic hea t ing of oil sand

in-si tu o v e r the r ange of f requencies f rom 5 0 H z to 10 9 H z . A typica l e l e c t romagne t i c hea t ing opera t ion migh t p rehea t a r eg ion ex t end ing ove r 3 0 m or m o r e in every d i rec t ion and loca ted h u n d r e d s of mete rs u n d e r g r o u n d , and b e of dura t ion of severa l days to severa l yea r s . For these r ea sons it is c lear ly imprac t i ca l , init ially at leas t , to unde r t ake field tests to eva lua te or op t imize a great var ie ty of in-situ hea t ing s c h e m e s . A n u m b e r of au thors h a v e used the m o r e feasible app roach of a n a l y t i c a l 1 3 , 1 4 and c o m p u t e r numer i ca l c a l c u l a t i o n s 1 6 to eva lua te the p e r f o r m a n c e of specific e l ec t romagne t i c hea t ing a r r a n g e m e n t s . W h i l e these app roaches can yie ld very i l lumina t ing in fo rmat ion , they d o h a v e l imi ta t ions . Genera l ly s p e a k i n g , the analyt ical a p p r o a c h can so lve the idea l ized p r o b l e m but genera l ly canno t accoun t for the e l ec t romagne t i c b o u n d a r y condi t ions at bo th the e lec t rodes and at larger d i s t a n c e s , for ins tance at the interface b e t w e e n an oil bea r ing fo rmat ion and o v e r b u r d e n and u n d e r b u r d e n . T h e numer i ca l a p p r o a c h is m o r e versa t i le and is re la t ively s t ra ight forward w h e n the p r o b l e m is o n e or t w o d i m e n s i o n a l . A numer ica l so lu t ion for t h e mos t genera l p r o b l e m , w h i c h will b e in three d i m e n s i o n s , will h o w e v e r b e exceed ­ingly c o m p l e x and requ i re a very, l a rge inves tmen t in c o m p u t i n g t ime every t i m e such a p r o g r a m is run to exp lo re the c o n s e q u e n c e s of s o m e smal l c h a n g e in o n e of the sys t ems p a r a m e t e r s . T o the bes t k n o w l e d g e of the au thors n o such p r o g r a m has b e e n repor ted in the o p e n li tera­tu re .

A n a l te rna t ive t e chn ique w h i c h can p rov ide c o m p l e t e so lu t ions to very c o m p l e x in-s i tu hea t ing conf igura t ions wi th re la t ive e a s e is the u s e of smal l sca le phys ica l m o d e l s to s imula te the full scale sys t em. Th i s a p p r o a c h p rese rves the essent ia l pa rame te r s of t he full scale sy s t em and p rov ides g o o d insight into the la rge sca le p h e n o m e n a at a re la t ive ly smal l i nves tmen t of t ime and m o n e y . It a l so offers fine cont ro l ove r expe r imen ta l cond i t ions . Phys ica l mode l l i ng has been used successful ly for a n u m b e r of years in geophys ica l e l ec t romag­net ic su rveys and p r o s p e c t i n g , 1 7 and a l so in the field of c o m m u n i c a ­t ions in the des ign of an tennas for ships and a i r c ra f t . 1 8

T h e specific object of this pape r is to exp lo re the viabi l i ty of phys ica l mode l l i ng for so lv ing the genera l in-si tu e l ec t romagne t i c hea t ing p r o b l e m . This p r o b l e m presents itself and is addressed in t w o pa r t s . It en t a i l s , firstly, a s tudy of the feasibi l i ty of es tab l i sh ing e l ec t romagne t i c s imi l i tude b e t w e e n a full sca le sy s t em and a labora­tory m o d e l at all f requencies to b e exp lo red . W h e t h e r or not such s imi l i tude c a n b e rea l ized is crucia l ly d e p e n d e n t u p o n the electr ical p a r a m e t e r s of oil s and . T o this end the au thors h a v e c o m p l e t e d an ex t ens ive s e q u e n c e of labora tory m e a s u r e m e n t s o n a w i d e var ie ty of oil s and s a m p l e s from the A t h a b a s c a depos i t ove r t he r a n g e of fre­quenc i e s f rom 5 0 H z to 10 9 H z to es tab l i sh the d e p e n d e n c e of e lectr i ­cal conduc t iv i ty and die lect r ic cons tan t o n f r equency , t e m p e r a t u r e and wa te r con ten t . In p repar ing samples for m e a s u r e m e n t grea t efforts w e r e t a k e n t o a t t empt t he s imula t ion of cond i t ions p reva i l ing in-s i tu . Resul t s of this s tudy are r epor ted e l s e w h e r e . 1 9 S imula t ion of the e l ec t romagne t i c p r o b l e m c a n , h o w e v e r , not b e unde r t aken in isolat ion of the the rma l p r o b l e m . T h e electr ical p roper t i es of oil sand are h ighly t e m p e r a t u r e d e p e n d e n t and t h e e l ec t romagne t i c p r o b l e m to b e so lved con t inua l ly c h a n g e s as the oil sand fo rmat ion increases nonun i fo rmly in t e m p e r a t u r e and b e c o m e s electr ical ly p rogress ive ly m o r e i n h o m o -g e n e o u s . Th i s then leads to t he second par t of the gene ra l p r o b l e m , w h i c h dea l s wi th es tab l i sh ing the rmal s imi l i tude b e t w e e n a full scale sy s t em and a mode l a n d , most importantly, wi th s h o w i n g that e lec­trical and the rmal s imi l i tude are cons is ten t and can b e s imul t aneous ly p resen t in a s ingle mode l u n d e r these ra ther c o m p l e x c o n d i t i o n s .

T h e d e v e l o p m e n t of the mode l l i ng cr i ter ia is g u i d e d by the proper­ties of oil s and . T h e resul ts w h i c h are de r ived are of a genera l na tu re , h o w e v e r , and can b e appl ied to m a n y o ther e l ec t romagne t i c heat ing p r o b l e m s w h e r e the m e d i u m to b e hea ted is s o m e t h i n g o ther than oil s and .

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

A c o m p r e h e n s i v e t r ea tmen t of the e l ec t romagne t i c theory of simili­t ude and the cond i t ions unde r w h i c h a m o d e l wil l accura te ly s imula te the e l ec t romagne t i c p h e n o m e n a in a ful l-scale s y s t e m has b e e n given

V E R M E U L E N / C H U T E / C E R V E N A N : E L E C T R O M A G N E T I C H E A T I N G 21

by S inc l a i r . 1 8 His results p rov ide a conven ien t s tar t ing po in t for t h e fo l lowing d e v e l o p m e n t of the scal ing cri teria appropr ia te for e s t ab ­l ishing c o m p o s i t e e lec t romagne t ic - thermal m o d e l s for oil sand sys ­t e m s . It will b e initially a s sumed that all med ia electr ical p roper t ies are t empera tu re independen t . T h e extens ion of the mode l l i ng cr i ter ia for cases w h e r e the proper t ies do vary wi th the t empera tu re is unde r ­taken in a later sect ion.

T h e e l ec t romagne t i c behavior of a sys tem is desc r ibed by M a x w e l l ' s differential equa t ions wh ich are l inear and a l low l inear scal ing so long as non- l inear med ia are exc luded . T h e electr ical pa ramete r s of oil s and , as far as is k n o w n , are l inear a l though nonl inear cur ren t conduc t ion may occur at the b o u n d a r y b e t w e e n oil sand and a meta l e l ec t rode , these be ing respect ive ly ionic and e l ec ­tronic c o n d u c t o r s . H o w e v e r , measu remen t s wi th var ious e lec t rodes e m b e d d e d in oil sand indicate that t he re la t ionship b e t w e e n vo l t age and cur ren t is substant ia l ly l inear for current densi t ies of interest in in-situ hea t ing .

Fo l lowing S inc la i r , let any point in the full scale sy s t em b e loca ted by the rec tangula r coord ina tes x , y, z . T h e n the co r r e spond ing po in t in the mode l is located at x ' , y ' , z ' in a co r respond ing m o d e l coo rd i ­nate sy s t em. T h e t w o coord ina te sys tems are rela ted b y the t rans ­format ion

p x , y = p y , z = p z (1)

where p is the mechan ica l scale factor. T h e coord ina tes ( x , y , z) and (x ' , y ' , z ' ) are measu red in te rms of a unit of length and the s a m e uni t of length mus t b e used for bo th sys t ems .

T h e fo l lowing further sca l ing condi t ions are n o w added to those of equa t ion ( 1 ) ,

t = yt' or co' = yco E ( x , y , z , t ) = a E ' ( x ' , y ' , z ' , t ' )

H(x, y, z , t) = /3H'(x', y \ z \ t ' ) (2)

where u n p r i m e d quant i t ies refer to the full scale sys tem and p r i m e d quant i t ies to the mode l sys tem and

t = t ime E = electr ic field intensi ty H = magne t i c field intensi ty co = angular f requency

and w h e r e y, a , /3 are the respec t ive scal ing factors .

It is n o w readi ly s h o w n that M a x w e l l ' s equa t ions in the full scale sys tem,

aH at

curl E = - ( J L -

curl H = o -E + fi­at

(3)

will b e invar iant u n d e r t ransformat ion and in the m o d e l s y s t e m become

,aH'

(4)

cu r l ' E ' =

c u r l ' H ' = o - ' E ' + e ' ^ '

at' provided that the electr ical pa ramete r s of the mode l and full sca le system are re la ted by

( 5 a )

(5b)

(5c )

where cr is the electr ical conduct iv i ty , e is the dielectr ic cons tan t a n d fi is t he magne t i c pe rmeab i l i ty . N o t e that there has b e e n no res t r ic t ion on t h e spat ia l var ia t ion of these pa rame te r s .

cr pa cr T

e' p a

e

Hi. = M r1 ay

W h e n the four sca le factors p , a, /3 and y are k n o w n quant i t ies then the re la t ionsh ips b e t w e e n all o ther e l ec t romagne t i c quant i t ies in t he full sca le sy s t em and the mode l are fixed. S u c h a m o d e l is an abso lu te m o d e l in the sense that not only are the geomet r i ca l conf igura t ions of the l ines of force of the e l ec t romagne t i c field mode l l ed bu t so a l so are the r e spec t ive p o w e r l eve l s . H o w e v e r , arbi trar i ly se lect ing these pa rame te r s c an resul t in non- rea l i zab le va lues for the necessa ry e l ec ­trical proper t ies of the m o d e l m e d i u m . T h u s , t he se lec t ion of t he sca le factor is not a lways s t ra ight forward and canno t b e unde r t aken wi thou t re fe rence to the exac t na tu re of the sys t em to b e m o d e l l e d .

In p r ac t i c e , ava i lab le mode l med ia h a v e ra ther l imi ted ranges of /x,', e ' and cr' and a r ea sonab l e app roach to the p r o b l e m is to e x a m i n e the sca l ing cr i ter ia exp res sed by equa t ions (5) subject to t he a s s u m p t i o n that t he ra t ios yJ\\x, s'/s and cr'/cr are g iven . T h e n f rom equa t ions (5) it is readi ly s h o w n that

P = £ j ^ (6)

T h e a b o v e re la t ionsh ip is a significant cons t ra in t that states that t he m e c h a n i c a l sca le factor canno t b e independen t ly se lec ted o n c e the m o d e l m e d i u m has b e e n specif ied. T h e ach ievab le mechan i ca l sca le factor is fixed by t he cho ice of the mode l m e d i u m . M o r e o v e r it is apparen t that to rea l ize any reduc t ion in s ize the m o d e l m e d i u m canno t b e the s a m e as that of the full sca le s y s t e m . In o ther w o r d s , it-is not genera l ly poss ib le to mode l an e l ec t romagne t i c hea t ing s c h e m e in oil sand wi th a m o d e l us ing oil sand . If, on the o ther h a n d , an artificial m e d i u m is used wi th a va lue of conduc t iv i ty such tha t cr'/cr i s , for e x a m p l e , 5 0 , and a va lue of yJ and e ' such that fx'/fx and e'/e a re substant ia l ly uni ty , it is then poss ib le to m o d e l wi th a mechan ica l sca le factor h a v i n g a va lue of 5 0 as we l l .

E x a m i n a t i o n of equa t ions (5a) and (5b) a lso shows that the scal ing factor y is g iven by

co' cr'e y = — = ; (7)

co cr e T h u s , in o rde r to h a v e e l ec t romagne t i c s imi l i tude it is necessa ry to ope ra t e t h e mode l at a f requency or'si ere' t imes the in tended oper­a t ing f requency of t he full scale sy s t em.

E q u a t i o n (7) m a y b e r ea r r anged to the form

a' = cr co'e' coe

w h i c h expresses the fact that the cho ice of m o d e l f requency is s u c h as to ensu re that t he ra t io of conduc t ion cur ren t to d i s p l a c e m e n t cur ren t in t he m o d e l is the s a m e as the co r r e spond ing ra t io in the full sca le s y s t e m . S imi la r ly it c a n b e s h o w n that this cho ice of f requency ensures tha t t he ra t ios of b o t h w a v e l e n g t h and depth, of pene t ra t ion to the phys ica l s ize of the m o d e l are equa l to the co r r e spond ing ra t ios in the full sca le sy s t em. In o ther w o r d s

X' X , A' A

I7 = L A N D L ' = l w h e r e the w a v e l e n g t h , A., is g iven b y 2 0

and w h e r e the dep th of pene t ra t ion , A , is g i v e n b y

A" • 4?] [>/• • <=H B o t h X and A refer t o un i fo rm p lane w a v e p ropaga t ion t h rough the m e d i u m . T h e d e p t h of pene t ra t ion is defined as the d i s tance the w a v e t rave ls be fo re its ampl i t ude is a t tenuated b y t he factor 1/e, w h e r e e is the N a p e r i a n b a s e . L is any charac ter is t ic phys ica l d i m e n s i o n of the s y s t e m .

22 C A N . E L E C . E N G . J . V O L 4 N O 4 , 1979

Also from equa t ions (5b) and (5c) the rat io of a/p can b e deter ­mined as

a

1

(8)

Note that it is not poss ib le to de te rmine a or /3 from equa t ions (5 ) , but on ly the ra t io of these t w o quant i t ies . T h u s , to fully d e t e r m i n e the mode l it is necessary to choose arbitrarily a va lue for e i ther a or /3 and to use equa t ion (8) to find the r ema in ing scale factor . For e x a m ­p le , an arbi trary value of a or f3 could b e selected by sui tably adjust ing the rat ios of terminal vo l tages , te rminal currents or p o w e r s in the m o d e l and full scale sy s t ems . H o w e v e r , as will b e s e e n shor t ly , the cho ices of a and ft are not entirely arbi trary w h e n the the rmal cri teria necessary for a compos i t e e lec t romagne t ic - the rmal mode l are in t roduced . Indeed , it is fortunate that the e lec t romagne t i c sca l ing cri teria dic ta te only the rat io and not a par t icular va lue for e i ther a or j8, for o the rwise e lec t romagnet ic and thermal s imi l i tude cou ld not s imul taneous ly exist .

T h e mode l l ing of thermal p h e n o m e n a

T h e ins tan taneous rate Q at wh ich e lec t romagne t ic energy is c o n ­ver ted to heat in the p resence of an electric field E , in unit v o l u m e of a m e d i u m of conduct iv i ty o~, is

Q = o - E - E

T h e rate Q and the ra te at which tempera ture rises vary dur ing the course of each cycle of the e lec t romagnet ic field. O n e might thus b e t empted to use the previous ly in t roduced scale factor y to scale the thermal t imes be tween the full scale sys tem and the m o d e l . Th i s cho ice wou ld b e correct in p r inc ip le . In p rac t i ce , h o w e v e r , this cho ice wou ld in t roduce constraints wh ich wou ld lead to phys ica l ly unrea l iz ­able mode l l ing r equ i r emen t s . Indeed , from what fo l lows it will b e apparen t that such a cho ice wou ld lead to the r equ i remen t that the electr ical and thermal propert ies b e related as

<T FX

cr FX

p ' c ' k

p c k '

whe re p is the densi ty , c is the thermal capaci ty and k is the the rma l conduc t iv i ty . Such a restr ict ion is general ly unrea l izable wi th ava i l ­able mode l med ia .

Rap id smal l scale fluctuations in t empera tu re , such as t h o s e c a u s e d by changes in Q dur ing each cycle of the e lec t romagne t ic field, a r e , h o w e v e r of secondary interest . Of impor tance is the large scale relat ively s lowly occur r ing t empera tu re r ise T, w h i c h d e p e n d s on the average hea t ing rate on ly , wh ich for t ime ha rmon ic fields is i ndepen­dent of t ime and is g iven by

F 2 R\ P Qave = O- —

w h e r e E p is the peak va lue of the per iodical ly vary ing electr ic field. Bear ing this in mind the authors have avoided the a fo remen t ioned mode l l ing difficulties by in t roducing a new scal ing factor , T, inde­penden t of y , to scale thermal t imes be tween the full scale sys tem and the m o d e l . In o ther w o r d s , for t imes relat ing to thermal p h e n o m e n a

t = I Y (9)

A l s o , the t empera tu re in the mode l is chosen to b e re la ted to the t empera tu re in the full scale sys tem as

(10) T ( x , y , z , t ) = £ T ' ( x ' , y ' , z ' , t ' )

w h e r e £ is the t empera ture scal ing factor. T h e t empera tu re T is defined as the difference be tween the ins tantaneous t empera tu re of the full sca le sys tem at t ime t and at a point x , y , z , and the initial t empera tu re of the full scale sys tem at this po in t . S imi lar ly T ' is defined as the difference be tween the ins tantaneous t empera tu re of the mode l at the co r respond ing t ime t ' , and at the co r r e spond ing poin t x ' , y ' , z ' , and the initial t empera ture of the mode l at this po in t . T h e initial t empera tures of the model and full scale med ia m a y wel l be different. Equa t ion ( 1 0 ) , t hen , indicates that for an obse rved differ­

ence b e t w e e n the initial mode l t empera tu re and the mode l t empe ra tu r e at a par t icu la r po in t in t h e mode l at a t i m e t ' , t h e full sca le sy s t em t e m p e r a t u r e at the co r r e spond ing point w o u l d c h a n g e f rom the initial full sca le t empe ra tu r e by an a m o u n t f t imes the obse rved c h a n g e in the m o d e l , bu t at a t ime t = I Y .

N o t e that the initial t empera tu re d is t r ibut ions in the full s ca l e ' sys t em and the mode l m a y b e spatial ly non -un i fo rm . In such a case equa t ion (10) impl ies tha t , ini t ia l ly , the t e m p e r a t u r e difference b e ­t w e e n any t w o points in the full scale sy s t em mus t be f t imes the t empe ra tu r e di f ference b e t w e e n the co r r e spond ing t w o points in the m o d e l .

It is n o w readi ly s h o w n that the heat equa t ion in the full scale sy s t em

dT o - E p

2

p c — = — - + k V 2 T (11) ^ a t 2

is invar iant unde r t ransformat ion and in the mode l sys t em b e c o m e s

,BT = o ^ 2

+ k , v , 2 T , a t ' 2

p rov ided that

r = P 2 ^ A ( 1 3 a ) k p c

and

^ = ^ E ^ ' ( 13b) (T f P C

Equa t ions (13a) and (13b) insure in va r iance unde r t r ans fo rmat ion as wel l w h e n equa t ion (11) incorpora tes spatial var ia t ion of p , c and k. It is of interest to no te that the above stated the rmal scal ing re la t ions express the fact that the rat io of the ra te at w h i c h hea t is de l ivered to a g iven reg ion by conve r s ion of e l ec t romagne t i c ene rgy wi th in that r eg ion to the ra te at w h i c h hea t is r e m o v e d from the r eg ion by the rmal c o n d u c t i o n is the s a m e for bo th the mode l and the full scale s y s t e m s .

T h e r m a l m o d e l l i n g , as d i scussed h e r e , is res t r ic ted in the s ense that there is no fluid flow th rough the m o d e l . H e n c e , the scal ing of such pa rame te r s as v i scos i ty , pe rmeab i l i t y , po ros i ty , capi l lar i ty and gravi ty is not c o n s i d e r e d . 2 1

A s s u m i n g that an ava i lab le mode l m e d i u m has been se lec ted and va lues of k ' / k , p ' / p and c ' / c are k n o w n and that p is as d e t e r m i n e d by equa t ion (6 ) , then equa t ion (13a) g ives the thermal t ime sca le factor T. In o ther w o r d s the t empera tu re d is t r ibut ions obse rved in the mode l at t ime t ' will exis t in the full scale sys tem at t ime t = I Y , p r o v i d e d of cou r se that the r e m a i n d e r of the scal ing cri teria are satisfied.

F r o m equa t ions (13) it is a l so n o w apparen t that the t empera tu re sca l ing factor f and the electr ic field scal ing factor a mus t b e related as

T - i n ? ( 1 4 )

£ p o~ k N o t e , h o w e v e r , that s imi lar to the ca se involv ing the ra t io a i t is not poss ib le to un ique ly de t e rmine e i ther a o r £ f rom this re la t ionship . H o w e v e r , as will b e c o m e apparen t in the d i scuss ion of the t empera­ture d e p e n d e n c e of the mater ia l p rope r t i e s , an exped ien t cho i ce for the t empe ra tu r e sca l ing factor will b e f = 1. T h e r e m a y b e s o m e si tuat ions w h e r e a l te rna t ive va lues of f are se lec ted but in any case o n c e f is chosen the va lue of a is un ique ly de t e rmined from equa t ion (14) as

( i 5 )

T h e va lue of a hav ing thus been d e t e r m i n e d , equa t ion (5a) now un ique ly de t e rmines the magne t i c field sca l ing factor /3 as

V E R M E U L E N / C H U T E / C E R V E N A N : E L E C T R O M A G N E T I C H E A T I N G 23

In summary, once a suitable and realizable model medium has been selected as well as a value for f equations (6), (7), (13a), (15) and (16) uniquely define a composite electromagnetic-thermal model from which values of the temperature and electromagnetic field distribution of the corresponding full scale system may be determined. M o r e o v e r , once f, p , y , T, a and /3 are de te rmined the vo l t age , current imped ­ance and p o w e r levels of the full scale sys tem may also b e ca lcu la ted . T h e necessary re la t ionships fol low direct ly from the respec t ive definit ions and the scal ing cri teria deve loped a b o v e . T h u s , it is eas i ly s h o w n that

P = P'cxPp2 = P ' p £ ^

z = z>« = z>±zl P p o -

V = V ' « p = V y JL (17)

p p V °" k

where P is the p o w e r de l ivered to the sys t em, Z is the input imped­ance , V and I are the te rminal vol tages and cur ren t s , and J is the current dens i ty in the m e d i u m .

As noted p rev ious ly , in order to ach ieve a reduct ion in physica l s ize an artificial mode l m e d i u m mus t b e used . T h e select ion of a su i table model m e d i u m is not s t ra ightforward and previous solut ions for purely e l ec t romagne t i c mode l s have included electrolyte and agar-agar so lu t ions , ca rbon impregna ted epoxy resins and the l i k e . 1 7 , 2 2 ' 2 3

Choos ing a m e d i u m for a compos i t e e lec t romagne t ic - thermal mode l is m a d e even m o r e difficult by the desirabil i ty of us ing a mode l m e d i u m with thermal proper t ies that are s imi lar to those of oil sand . A s w e l l , it p roves impor tan t that the changes of these thermal proper t ies and a l so of e lectr ical proper t ies wi th t empera tu re m u s t b e s imilar for t he full scale and mode l s y s t e m s .

E x a m i n a t i o n of the typical behav ior of the electrical proper t ies of oil sand wi th f requency and mois ture content p rovides s o m e insight into h o w oil sand itself m a y b e used as a mode l m e d i u m . Represen ta ­tive va lues of cr and relat ive dielectr ic constant e R for labora tory samples of oil sand from the A thabasca depos i t , for wide ly r ang ing values of moi s tu re and b i t umen conten t , are s h o w n in F igure 2 . (H igh values of re la t ive die lect r ic cons tant at low f requenc ies , s imi lar to those in F igure 2 , are c o m m o n to m a n y ear th- l ike mater ia ls and are thought t o b e d u e to interfacial po la r iza t ion , e lec t rochemica l po la r iza­tion and ion s ieving p o l a r i z a t i o n . 1 9 ) S h o w n also is the rat io of c o n d u c ­tion cur ren t to d i sp l acemen t cur ren t wh ich has been c o m p u t e d from the numer i ca l values of or and e R .

Cons ide r n o w , for ins tance , a full scale sys tem that is to b e opera ted at a f requency of 10 M H z for the in-situ heat ing of a body of oil sand of type B . T h e values of cr, e R , and fi are respect ively 2 .5 x 1 0 " 3

S / m , 7, and 4tt x 1 0 - 7 hen r i e s /m. Fur ther , n o w suppose that oil sand A is to b e used as the mode l m e d i u m . A poss ib le mode l f requency found after s o m e trial and e r ror , is 150 M H z w h e r e cr' = 5 4 x 1 0 ~ 3

S/m, £ R ' = 10 and yJ = 47R x 1 0 ~ 7 h e n r i e s / m . This mode l f requency cor responds to a va lue of y = 1 5 0 / 1 0 = 15 which proper ly satisfies the cons t ra in t of equa t ion (7) . F r o m equa t ion (6) it is apparen t that a mechanica l scale r educ t ion of p = 18 has been ach ieved . Here the ratios of k ' / k and p ' c ' / p c are essent ial ly unity wi th typical va lues of k = 1 w a t t / m ° C and p c = 0 . 4 5 x 1 0 7 j o u l e s / m 3 ° C .

Initial expe r imen t s in the au tho r s ' laboratory indicate that m o d e l s can b e rea l ized in a s imi lar fashion by us ing mixtures of sand and saline wa te r . M e d i a wi th electr ical conduct iv i t ies several h u n d r e d times that of oil sand can b e ob ta ined wi th dielectr ic and the rma l propert ies substant ia l ly equal to those of oil sand . Such m e d i a ex tend the r ange of ach ievab le mechan ica l scale factors cons ide rab ly . H o w ­ever , genera l ly s p e a k i n g , as indicated by equat ions (6) and (7 ) , t he

10 100 .IK I0K I00K IM I0M I 0 0 M IG I0G

FREQUENCY (Hz)

Figure 2: Conductivity cr, relative dielectric constant e R and loss tangent tr/(oe for oil sand as a function of frequency at a temperature of24°C.

r equ i red va lue of y, and h e n c e of mode l f r equency , increases to inconven ien t ly h igh va lues as the ach ievab le sca le factor is inc reased . Indeed w h e n y'ljx ~ 1 and sis ~ 1 it is ev iden t that p « y. In such a c a s e , a ch i ev ing a sca le r educ t ion of, say, 5 0 for a sys t em opera t ing at 1 G H z w o u l d requ i re a mode l f requency of 5 0 G H z . L imi t ed phys ica l r eg ions of a full sca le s y s t e m , a m e n a b l e to use of smal le r mechan i ca l sca le factors cou ld b e m o d e l l e d , h o w e v e r , wi thou t the need for undu ly h igh mode l f requenc ies .

Str ict ly s p e a k i n g , w h e n m o d e l l i n g , the exac t forms of M a x w e l l ' s equa t ions and of the t he rma l equa t ion mus t b e used at all t i m e s . In p r ac t i c e , h o w e v e r , d e p e n d i n g o n the f requency , hea t ing ra tes a n d o n the electr ical and the rma l proper t ies of the full scale m e d i u m , o n e or m o r e t e rms in these equa t ions m a y b e neg l ig ib le . If this is the c a s e , it is t hen poss ib l e to m o d e l by us ing var ious a p p r o x i m a t e fo rms of the e q u a t i o n s . In the fo l lowing sect ions the na ture of s o m e of these a p p r o x i m a t i o n s will b e e x a m i n e d .

M o d e l s b a s e d o n the induc t ion a p p r o x i m a t i o n

T h e ra t io of conduc t i on cur rent to d i sp l acemen t cur ren t , as g iven by or I cos, is s h o w n as a funct ion of f requency for typical oil s ands in F igure 2 . It is ev iden t that at f requencies b e l o w about 1 M H z the d i s p l a c e m e n t cur ren t usual ly can b e cons ide red neg l ig ib le . A sui table a p p r o x i m a t i o n to M a x w e l l ' s equa t ions in such a case is

curl E

c u r l H

6 H

' a t (18) crE

T h e s e equa t ions can b e used w h e n e v e r field so lu t ions are an t ic ipa ted that h a v e i nduced conduc t i on currents w h i c h are p r o d u c e d by the e l ec t romot ive force that is gene ra t ed by a t ime-va ry ing magne t i c field, and in m e d i a w h e r e c o n d u c t i o n currents are m u c h larger than d i s ­p l a c e m e n t cu r r en t s . T h i s app rox ima t ion is often refer red to as the induc t ion a p p r o x i m a t i o n .

T h e per t inen t e l ec t romagne t i c sca l ing cri ter ia for this c a s e are easi ly s h o w n to b e

cr _ p a

H ay

( 5 a )

(5c )

If it is assumed that the model medium has been selected and hence values of cr'/cr and fi'lfi are g iven then the above equations permit

24 C A N . E L E C . E N G . J . V O L 4 N O 4 , 1979

the arbitrary cho ice of one of the paramete r s p , y or alp. Th i s addi t ional deg ree of f reedom is direct ly re la ted to the fact that t he d i sp lacemen t current te rm has been neglec ted in the p resen t fo rmula ­t ion . T h e r e m a y b e occas ions w h e r e avai labi l i ty of e q u i p m e n t d ic ta tes a par t icular cho ice of m o d e l f requency and h e n c e of y, or of t h e m o d e l i m p e d a n c e level and hence of alp. M o r e of ten, h o w e v e r , it is adv i s ­able to select the mechan ica l scale factor p to p rov ide a conven i en t physica l size for the laboratory m o d e l . W h e n p is selected t hen f rom equa t ions (5a) and (5c ) , y and alp are de te rmined as

a

|3

(X /JL

or jJL

1 Z. P ( T

(19a)

(19b)

As be fo re , a is de te rmined by equat ion (15) w h i c h s t ems f rom the requ i remen t of thermal s imi l i tude . It then fol lows from equa t ion (19b) that P is g iven by equa t ion (16) .

When using the induction approximation a unique composite electromagnetic-thermal model is obtained by selecting values for p , f and the model medium. Moreove r , equat ions (17) for the v o l t a g e s , currents and p o w e r are equal ly appl icable in this c a s e .

N o t e that s ince p can b e arbitrari ly selected it is not necessa ry to use an artificial mode l m e d i u m in order to ach ieve a r educ t ion in the physical size of the m o d e l . I ndeed , oil sand itself m a y b e used as the m o d e l m e d i u m thereby min imiz ing the p rob l ems assoc ia ted w i th rea l iz ing a mode l m e d i u m with thermal proper t ies s imi lar to the full scale sys t em. H o w e v e r , in this case y equa ls p 2 and for r ea sonab l e scale factors of b e t w e e n 100 and 300 mode l f requencies m a y b e c o m e so h igh that the induct ion approx imat ion is n o longer va l id in the m o d e l . It fo l lows from Figure 2 that , if oil sand is to b e used as a mode l m e d i u m , the highest mode l f requency mus t b e n o m o r e than of the o rder of 1 M H z if d i sp lacement current is to b e neg l ig ib le . T h u s , the largest a t ta inable va lue of p for a full scale sys tem opera t ing a t , say, 1 K H z w o u l d b e about V l M H z / 1 K H z , or about 3 2 . Ful l sca le sys t ems opera t ing at frequencies h igher than 1 K H z w o u l d h a v e to b e mode l l ed wi th e v e n smal ler mechan ica l sca le fac tors ; for ins tance a full scale sys tem opera t ing at 100 K H z cou ld b e s imula ted by a m o d e l opera t ing at 1 M H z wi th a mechanica l scale factor of 3 . 2 . T h e s e mechan ica l scale factors are insufficient for m a n y prac t ica l hea t ing conf igura t ions . T o significantly increase the r a n g e of m e c h a n i c a l sca le factors the sand and sal ine water mixtures prev ious ly refer red to m a y b e used . Such m e d i a ex tend the r ange of usab le m e c h a n i c a l sca le factors on t w o accoun t s . First ly, for a fixed f requency ra t io y the va lue of p varies as V c r ' / c r . Second ly , due to the increased va lue of m o d e l conduct iv i ty the uppe r f requency l imit of the induc t ion r a n g e in the mode l m e d i u m is also increased , thereby permi t t ing a g iven full sca le sys tem to b e mode l l ed at m u c h h igher f requencies than w o u l d b e poss ib le if the m o d e l m e d i u m were oil sand. T h e la rger va lues of y in turn lead to larger va lues of p . T h u s , the use of artificial s and and wate r m e d i a of sui tably h igh conduct iv i ty will pe rmi t t he s imula t ion of any full scale sys tem opera t ing in oil sand at f requencies b e l o w about 1 M H z .

It is of interest t o note that select ion of the m o d e l f requency accord ing to equa t ion (19a) cor responds to

k' k , A' A L' = L a n d V = Z

as be fo re , w h e r e k = 27rA

and A =

This express ion for the dep th of pene t ra t ion is de r ivab le f rom the exact express ion for A quo ted earl ier unde r the a s s u m p t i o n tha t (J 10)8 > 1.

M o d e l s b a s e d o n a quas i - s ta t i c a p p r o x i m a t i o n

A poss ib l e quas i -s ta t ic app rox ima t ion of M a x w e l l ' s equa t ions is

cur l E = 0

c u r l H = o-E (20)

A l t h o u g h these equa t ions are strictly va l id for stat ic d is t r ibut ions of c h a r g e and current on ly , they m a y b e appl ied to t ime va ry ing sys tems w h e r e the f requency is l o w e n o u g h that e lectr ic fields assoc ia ted wi th t ime va ry ing m a g n e t i c fields are not impor t an t . In such cases these equa t ions lead to e lect r ic and magne t i c field so lu t ions w h o s e electr ic and m a g n e t i c l ines of force h a v e the geomet r i ca l conf igura t ion of a c o r r e s p o n d i n g non - t ime vary ing p r o b l e m and that are val id p r o v i d e d the w a v e l e n g t h , \ , in the mater ia l m e d i u m of a s y s t e m is m u c h larger than any phys ica l d i m e n s i o n of that s y s t e m . 2 4 F u r t h e r m o r e , no te that the equa t ions a re val id only if the d i sp l acemen t cur ren t is neg l ig ib l e .

10*

X(m) and io

Mm)

I

.01

:

^ 1 1 1 MH1

^ \

1 1 MM11 1 1 1 1 l l l l 1 1 1 Mil l 1 1 T i l l -

:

^ 1 1 1 MH1

^ \

1 1 MM11

OILSAND WATER (% wt)

A 5.8 B 1.3

E ^ >

OILSAND WATER (% wt)

A 5.8 B 1.3 -

: -

E ^ >

- Quasi -approx

static motion

I

- Inductu >n appro (imation M( Exact fc

jxwell's e rm of quations ' 1

1 Mill

— i i 11 D in—i i i n u n — i i i n u n — i i 11 inn i i i m i l , , ,,,,,„ , , ,,,,,„ , , , ,„

100 I0K I00K

FREQUENCY

IM

(Hz)

I0M IOOM IG I0G

Figure 3 : Wavelength k and depth of penetration A for oil sand as a function of frequency at a temperature of24°C.

Figure 3 s h o w s w a v e l e n g t h and dep th of pene t ra t ion in oil sand as a funct ion of f requency . T h e va lues s h o w n h a v e been de r ived us ing the expres s ions for k and A and the exper imen ta l ly ob ta ined values of o-and e R s h o w n in F igu re 2 . E x a m i n a t i o n of F igures 2 and 3 s h o w s that this quas i -s ta t ic app rox ima t ion migh t b e app l icab le in oil sand hea t ing s i tua t ions at f requencies b e l o w abou t 1 K H z if typ ica l full sca le sy s t em d i m e n s i o n s a re a s s u m e d to b e less than abou t 1 0 0 - 3 0 0 m , i . e . m u c h less than the w a v e l e n g t h k. I n d e e d , full scale sy s t ems opera t ing at f requencies less than approx ima te ly 1 K H z m a y , d e p e n d i n g o n the par t icu lar s i tua t ion , b e desc r ibed by ei ther this quas i -s ta t ic a p p r o x i m a ­t ion or b y the induc t ion app rox ima t ion . For in s t ance , t he t w o e lec­t rode hea t ing p r o b l e m dep ic t ed in F igure 1 w o u l d b e m o d e l l e d as a quas i -s ta t ic p r o b l e m , w h e r e a s a la rge u n d e r g r o u n d coi l e m b e d d e d in the oil s and fo rmat ion and opera t ing at the s a m e f requency w o u l d b e m o d e l l e d as an induc t ion p r o b l e m .

T h e r e is on ly o n e e l ec t romagne t i c sca l ing cr i te r ion in this ca se and it is g iven by

cr

cr

p a

~P (5a)

N o t e that y doe s not appear in this exp res s ion . I n d e e d , as long as the cond i t ions u n d e r w h i c h the quasi -s ta t ic app rox ima t ion is va l id are satisfied for b o t h the m o d e l and the full scale s y s t e m there is no res t r ic t ion o n y. A logica l and conven ien t cho ice is to set y = 1. This ex t ra d e g r e e of f reedom is direct ly re la ted to neg lec t ing the electric fields gene ra t ed b y the t i m e var ia t ion of t h e m a g n e t i c f ields.

A s s u m i n g that a m o d e l m e d i u m has b e e n se lec ted g ives a va lue of a'la. T h e n , e i ther p or alp m a y b e c h o s e n arbi t rar i ly , a n d , as in the induc t ion c a s e , it is usua l ly conven ien t to select p . M o r e o v e r , a is set b y the t he rma l cons t ra in t s , g i v e n b y equa t ions ( 1 3 ) , o n c e a va lue for f is se lec ted . T h e va lue of a is thus g i v e n by equa t ion (15) a n d the value o f p b y equa t ion ( 1 6 ) , de r ived from equa t ions (5a ) a n d ( 1 5 ) .

p 2

y = —r-,

V E R M E U L E N / C H U T E / C E R V E N A N : E L E C T R O M A G N E T I C H E A T I N G 25

p = ipp

Z = -Z'

i = pV?r j = - v f j ' p (21)

In the quasi-static case then, a unique composite electromagnetic-thermal model is obtained by selecting values for p , f, y and the model medium.

It is ev ident that wi thin the quasi-s tat ic r e g i m e , for ins tance for in-si tu hea t ing at 6 0 H z , it is conven ien t t hough not essent ia l to u s e oil sand as the mode l m e d i u m . T h e n , w h e n p , f, y h a v e b e e n se lec ted

r = p 2

V? a =

P P

p = V£ v = V | v (No te that s ince y can b e selected independent ly there is in this c a s e n o p r o b l e m wi th a requ i red mode l f requency that migh t b e ou t s ide the l imits for wh ich the quasi-s ta t ic approx imat ion is va l id . ) T h u s , for ins t ance , if oil sand is t he mode l m e d i u m and p = 2 0 0 , then a full scale sys tem ex tend ing o v e r , say, 100 m in every d i rec t ion and subjec ted to electr ical hea t ing for a per iod of, say, t w o y e a r s , c o u l d b e mode l l ed in the labora tory wi th a mode l of d imens ions of 0 . 5 m in eve ry d i rec t ion and in a t ime jus t under thirty m i n u t e s . F o r equa l t empera tu res in the full scale sys tem and the mode l ( f = 1), full sca le p o w e r and cur rent wou ld b e 200 t imes those of the m o d e l , vo l t age levels wou ld b e ident ica l , and full scale i m p e d a n c e levels and cur ren t densi t ies wou ld be 1/200 of the mode l va lues .

It m a y a l so be des i rab le to mode l the p resence of ove rbu rden and u n d e r b u r d e n . In this ins tance , for cons i s tency , ove rbu rden and unde r -b u r d e n mate r ia l s in the m o d e l wou ld consis t of s labs of the ac tual full scale mater ia l appropr ia te ly scaled d o w n in s ize . C a r e w o u l d h a v e to b e exe rc i sed in apply ing the scal ing cri ter ia . For in s t ance , wh i l e at a par t icular f requency the e lec t romagne t ic p h e n o m e n a wi th in oil s and might fall wi th in the quasi-s ta t ic r e g i m e , such migh t not necessa r i ly b e the c a s e for the e l ec t romagne t i c p h e n o m e n a in the o v e r b u r d e n and u n d e r b u r d e n . In par t icular , it might b e necessary to inc lude the effect of d i s p l a c e m e n t cur ren t , even at very low f requenc ies , to es tab l i sh the cor rec t field dis t r ibut ion in a low conduct iv i ty ove rburden or under ­b u r d e n . T h e appropr ia te equa t ions cou ld then b e c o m e

curl E = 0

curl H = crE + e —

at and the appropr ia te e lec t romagne t ic scal ing cri teria w o u l d b e

and the appropr ia te sca l ing cr i ter ia b e c o m e

pa;

Jy

(22)

( 5 a )

(5b)

In this c a s e , for a g iven mode l m e d i u m , the mode l f requency can n o longer b e arbitrari ly se lec ted and the above equa t ions d ic ta te that y b e g iven by equa t ion (7 ) . T h u s , ei ther the ra t io of alp o r the va lue of p can b e se lec ted arbi t rar i ly . Usual ly the mechan ica l scale factor p is se lec ted to p rov ide a convenien t ly sized labora tory m o d e l . T h e n , s ince the s a m e the rmal const ra ints exist the va lue of a is d e t e r m i n e d as in equa t ion (15) and from equat ions (5a) and (15) the va lue of ft is as g iven by equa t ion (16 ) . N o t e that the only difference b e t w e e n the or iginal quas i -s ta t ic approx ima t ion and this approx ima t ion is tha t he re y is fixed whi le in the former y was selected for c o n v e n i e n c e .

M o d e l s b a s e d o n p r o p a g a t i o n t h r o u g h a g o o d die lectr ic

Examir ia t ion of F igure 2 shows that at f requencies a b o v e a few h u n d r e d m e g a h e r t z the rat io of conduc t ion current to d i sp l acemen t cur ren t is neg l ig ib le . In o ther w o r d s , d i sp lacement current p r e d o m i ­nates and e lec t romagne t i c p h e n o m e n a in the oil sand can b e desc r ibed by

c u r l E = - u ^ at

curl H = e aE* at

(23)

6 _ p a

e (3y

El = l!L /x ay

(5b)

(5c)

It is apparen t that in this c a s e , for a g iven mode l m e d i u m , the a b o v e equa t ions pe rmi t only the ra t ios of ply and a/p to b e de t e rmined . It is usua l ly conven ien t to select p in w h i c h case

7 = P (24)

D e t e r m i n i n g a va lue for y f rom equa t ion (24) s imply co r r e sponds to se lect ing a m o d e l f requency such that the ra t io of w a v e l e n g t h to phys ica l s ize is the s a m e in t he mode l as in the full sca le sy s t em.

T h e appropr i a t e the rmal cons t ran ts are g iven by equa t ions (13) and o n c e £ is se lec ted they de t e rmine the va lue of a as g iven by equa t ion ( 1 5 ) . S ince from the scal ing cri teria s tated a b o v e

a _ m e '

P Vp ' e the va lue of p is g iven b y

/JL'E cr'k

p e'er k 7* (25)

U s e of this app rox ima t ion for the c o m p o s i t e e l ec t romagne t i c - the rma l m o d e l co r r e sponds to ca lcula t ing the hea t ing ra te Q , not wi th the e lect r ic field that actual ly exists in the m e d i u m , but wi th the e lect r ic field tha t w o u l d exis t if the m e d i u m w e r e loss less . Such an app roach is just i f iable if the phys ica l s ize of the full sca le sys t em is less t h a n , say, the dep th of pene t ra t ion . H o w e v e r , at f requencies for w h i c h the conduc t i on cur ren t is neg l ig ib le in oil sand the dep th of pene t ra t ion usua l ly does not e x c e e d o n e me te r , and it is to b e expec ted that this a p p r o x i m a t e mode l is of l imi ted usefu lness .

I n d e e d , it m igh t b e a rgued that hea t ing at these f requencies at all is imprac t i ca l . H o w e v e r , it is of interest to no te that even t hough the dep th of pene t ra t ion is sma l l , hea t ing of the m e d i u m is not necessar i ly res t r ic ted to reg ions c lose to the e l ec t romagne t i c sou rce . Init ial ly the hea t ing is res t r ic ted in this m a n n e r but subsequen t evapora t ion of mo i s tu re f rom the hea ted reg ion w o u l d r ende r this r eg ion qui te loss­less and the e l ec t romagne t i c energy w o u l d p ropaga te th rough it wi th little a t tenuat ion and w o u l d then hea t r eg ions of the oil sand that are further r e m o v e d . Mode l l i ng of such a s i tua t ion , h o w e v e r , w o u l d b e c o m p l i c a t e d by the loss of mois tu re f rom the sys tem as hea t ing takes p l a c e .

T h e resul ts for the var ious mode l l i ng app roaches are s u m m a r i z e d in Tab le 1. In each c a s e it is a s s u m e d that a mode l m e d i u m has been se lec ted and a lso a va lue of f.

A n a p p r o x i m a t e t h e r m a l m o d e l

T h e the rmal conduc t iv i ty of oil sand is re la t ively l o w and for m a n y c a s e s , espec ia l ly dur ing the initial s tages of hea t ing w h e n the rmal g rad ien ts are s m a l l , t he ra te at w h i c h hea t is de l ive red to a r eg ion by conve r s ion of e l ec t romagne t i c ene rgy great ly e x c e e d s the ra te at w h i c h hea t is de l ive red or r e m o v e d from that r eg ion by the rmal c o n d u c t i o n . W h e n these condi t ions exis t t he t empera tu re d is t r ibut ion as a funct ion of t ime can b e pred ic ted on the bas is of the a p p r o x i m a t e the rma l equa t ion

aT p c — ^ at

o r E p

2

In this ca se the appropr ia te sca l ing factor b e c o m e s

q 2 T p ' c ' cr

f p c

(26)

(13b)

/ p V

E'

E

cr' a

26 C A N . E L E C . E N G . J . V O L 4 N O 4 , 1979

TABLE 1 Summary of scaling criteria for constructing physical models when using the exact form

and various approximate forms of Maxwell's equations

or , r ea r r ang ing ,

a2T _ c r 'p c

£ o - p ' c '

As in p rev ious cases it is expedien t to select f = 1 in w h i c h case a2T is de t e rmined . H o w e v e r , ei ther a or V can b e se lec ted arbi trar i ly s ince there is n o w n o expl ic i t restr ict ion on the thermal t ime scale factor . T h u s , usua l ly T and f are selected and

£ Q-'P c T cr p ' c '

(27)

This va lue of a can then be used to de t e rmine /3 for any of the p rev ious ly d i scussed m o d e l s .

T h e s ignif icance of this ext ra degree of f reedom assoc ia ted wi th arbi trar i ly select ing T is m o r e easi ly apprec ia ted if the express ion for p o w e r is e x a m i n e d . T h u s , from equat ions (17)

P = P ' a / 3 p 2

w h i c h , a s suming f = 1 and a mode l m e d i u m wi th the s a m e the rma l proper t ies as the full sca le sys t em, b e c o m e s

P = p 3 P '

r (28)

It is n o w apparen t that the behav io r of a full sca le s y s t e m opera t ing at a p o w e r level of P wat t s can b e r e p r o d u c e d in a labora tory m o d e l at an arbi t rary p o w e r level P ' b y sui tably se lect ing the the rmal scal ing factor f. T h u s , for e x a m p l e , the r e s p o n s e of a 1 M W s y s t e m , wi th a hea t ing t ime of 1000 days cou ld b e r e p r o d u c e d in a sca le m o d e l , wi th p = 100 , hea t ed for 10 days at 100 wat ts or 1 day at 1000 wat t s or 0 .1 days at 10 K W .

In c o m p a r i s o n , w h e n the rma l conduc t iv i ty m u s t b e cons ide red the mode l p o w e r is specif ied di rect ly s ince for the case w h e r e the thermal p a r a m e t e r s are the s a m e in the m o d e l and in the full sca le s y s t e m , and w h e r e f = 1, it fo l lows direct ly f rom equa t ions (17) that

P = p P '

T h e n , for the e x a m p l e c i ted a b o v e the mode l cou ld on ly b e p o w e r e d at a level of 10 K W .

M o d e l l i n g w i t h t h e inc lus ion o f t h e t e m p e r a t u r e d e p e n d e n c e o f oi l s a n d e lectr ica l conduc t iv i ty

T h e e lec t r ica l conduc t iv i ty and the re la t ive die lect r ic cons tan t of oil s and are t empe ra tu r e d e p e n d e n t . Th i s t empe ra tu r e d e p e n d e n c e , which

Note: or'lcr, p ' / p , e'/e and £ are se lected

A p p r o p r i a t e E q u a t i o n s P y a r

dH curl E = - u —

^ at curl H = c rE + e —

at cr Ui'e

cr V fx e' cr' e cr s' P >

P 2 P , C , K ' v p ' c ' k

p c £ E = k V 2 T + ^ l r at 2

curl E = - a — at selected

for conven ience

p 2

cr 'p , ' cr fx

curl H = c rE

p c — = k v 2 T + — — H at 2

selected for

conven ience

p 2

cr 'p , ' cr fx P ^

v p'c'k

curl E = 0

curl H = c rE

aT , U 2 T ^ c r E 2

p c — = k v i T + — — K at 2

selected for

conven ience

se lec ted for

c o n v e n i e n c e

curl H = c rE

aT , U 2 T ^ c r E 2

p c — = k v i T + — — K at 2

selected for

conven ience

se lec ted for

c o n v e n i e n c e P ^

P 2 P , C , K ' v p ' c ' k

curl E = 0

c u r l H = o-E + e — at

selected for cr' e i

V o - ' k ' e

2 P C k ' v p ' c ' k

p c — = k v 2 T H — at 2

conven ience cr e ' P 1 V o - ' k ' e

2 P C k ' v p ' c ' k

curl E = -ix — * at

curl H = e — at

at 2

selected for

conven ience

l/x s p 1

l jfi'e <r'k

p V M e'ff k' V p ' c ' k

a =

V E R M E U L E N / C H U T E / C E R V E N A N : E L E C T R O M A G N E T I C H E A T I N G 27

has been ignored up to this po in t , is n o w e x a m i n e d to d e t e r m i n e the effect w h i c h it has u p o n mode l l ing .

Dielect r ic cons tan t is expec ted to b e only marg ina l ly d e p e n d e n t on t empera tu re at h igh frequencies s ince he re the p r imary effects c o n ­t r ibut ing to polar iza t ion are a tomic in or igin and are no t the rmal ly exc i t ed . At low f requenc ies , h o w e v e r , w h e r e polar iza t ion p h e n o m e n a marked ly r e s p o n d to t empera tu re , substant ial t empera tu re d e p e n d e n c e m a y b e p r e s e n t . 2 3 At these low f requenc ies , h o w e v e r , conduc t i on current p r e d o m i n a t e s and dielectr ic cons tant p lays n o role in m o d e l ­l ing. T h u s , t empera tu re dependence of the re la t ive dielectr ic cons tan t of oil sand and of any mode l m e d i u m in wh ich s imi lar po la r iza t ion m e c h a n i s m s exis t , is d e e m e d relat ively un impor tan t and is cons ide red no further.

Labora to ry tests by the a u t h o r s 1 9 over the t empera tu re r a n g e 4 ° C to 150°C of oil sand samples of wide ly differing mois tu re con ten ts h a v e es tab l i shed tha t , for all s a m p l e s , oil sand conduct iv i ty var ies ap ­p rox ima te ly l inearly wi th t empera tu re and increases app rox ima te ly five fold b e t w e e n 4 ° C and 100°C. T h e s e m e a s u r e m e n t s w e r e c o n ­duc ted ove r the f requency r ange of 50 Hz to 1 G H z .

If t empera tu re changes s lowly in bo th full scale sys t em and m o d e l , i .e . dur ing t ime intervals wh ich are very m u c h larger than the r e s p e c ­t ive per iods of osci l la t ion of the e lec t romagne t ic f ields, then field solut ions will b e essent ial ly h a r m o n i c , but wi th ampl i tudes that c h a n g e s lowly wi th t i m e . T h e changes in field ampl i tudes k e e p p a c e wi th the c h a n g i n g electr ical conduct iv i ty of the mater ia l m e d i u m . If in M a x w e l l ' s e q u a t i o n s , equat ions (3 ) , magne t i c permeabi l i ty p is as ­s u m e d to be t ime invar iant , and if t empera tu re d e p e n d e n c e of c o n d u c ­tivity and die lect r ic cons tant are wri t ten as c r ( x , y , z , T ) a n d e ( x , y , z , T ) , and if the t e rm

Clear ly , the cond i t ion that

E ( x , y , z , t ) d e ( x , y , z , T ) dT

dT at

and its counte rpar t in the mode l sys tem are neg lec ted , then M a x w e l l ' s equa t ions are invar iant unde r t ransformat ion from the full sca le sys ­tem to the mode l sys tem if

c r ' ( x ' , y ' , z ' , T ' )

or(x, y, z , T ) p a

J (29)

This equa t ion states that a l though the electr ical conduc t iv i ty of bo th the full scale sys tem and the mode l changes wi th t empera tu re and hence wi th t i m e , s imi l i tude be tween full scale sys tem and m o d e l exists t h roughou t t ime p rov ided that the rat io of these quant i t ies is cons tant at co r r e spond ing t imes t and t ' , related th rough the the rmal t ime scale factor f. If full scale and mode l m e d i a are the s a m e and if the t empera tu re scale factor £ is unity,, then equa t ion (29) is a l w a y s satisfied s ince the t empera tu re at any point in the full scale sys t em at t ime I Y is a lways equal to the t empera tu re at the co r re spond ing po in t in the mode l sys tem at t ime t ' . H e n c e , any t empera tu re induced changes in the conduct iv i t ies at these t w o cor responding points are equal and the ra t io cr'/cr for these t w o points r emains cons tan t .

T h e s i tuat ion is not qui te so s imple w h e n f ^ 1 and , o r , w h e n an artificial mode l m e d i u m is used . Let the t empera tu re d e p e n d e n c e of oil sand b e expressed a s 1 9

o-(T) = cr 0(l + aT)

where a is the t empera tu re coefficient of oil sand electr ical conduc t iv ­ity and cr 0 is the oil sand electr ical conduct iv i ty at the initial fo rmat ion t empera tu re . S imi la r ly , let the t empera tu re d e p e n d e n c e of the m o d e l m e d i u m , w h i c h m a y or m a y not b e oil s and , b e expressed as

o- ' (T ' ) = o - 0 ' ( l + a ' T ' )

where a' is the t empera tu re coefficient of the mode l m e d i u m electr ical conduct iv i ty and cr0' is the mode l m e d i u m electr ical conduc t iv i ty at the initial m o d e l t empera tu re . T and T ' are co r respond ing full sca le and mode l t empera tu res and are re la ted th rough equa t ion ( 1 0 ) , i . e .

T = f T

o- 'cn o-(T)

p a

J cons tan t

is m e t if a ' = a for the c a s e w h e r e the t empera tu re sca le factor is f = 1, and if a ' = f a for all o ther c a se s . As p rev ious ly m e n t i o n e d , the au thors h a v e e x a m i n e d the electr ical conduct iv i t ies of artificial mode l m e d i a cons t ruc ted of mix tures of sand and sa l ine wa te r . W h i l e the e lectr ical conduct iv i t i es of such m e d i a can b e severa l o rders of m a g n i ­tude larger than electr ical conduct iv i t ies of oil s and , t he t empera tu re coefficients of e lectr ical conduc t iv i ty of these t w o different m e d i a are vir tual ly the s a m e . This is to b e expec ted s ince a s imi lar m e c h a n i s m , ionic c o n d u c t i o n , accounts for electr ical conduc t iv i ty in bo th m e d i a . T h u s , a l t hough oil sand electr ical conduc t iv i ty is h igh ly t empe ra tu r e d e p e n d e n t , mode l l i ng w h e n the t empera tu re sca l ing coefficient is f = 1 is p o s s i b l e , us ing e i ther oil sand or an artificial m e d i u m as the mode l l i ng ma te r i a l .

M o d e l m e d i a wi th sui table values of e lectr ical conduc t iv i ty , ye t wi th t e m p e r a t u r e coefficients of electr ical conduc t iv i ty that are re la ted t o the t empera tu re coefficients of e lectr ical conduc t iv i ty of oil sand by a subs tant ia l cons tan t factor a r e , to the bes t k n o w l e d g e of the a u t h o r s , no t ava i l ab le . W h e n mode l l i ng cases w h e r e the t e m p e r a t u r e sca l ing coefficient is f ^ 1, an imperfec t mode l m e d i u m w o u l d thus h a v e to b e u s e d , and the mode l l i ng resul ts ob ta ined w o u l d h a v e to b e inter­p re ted wi th cau t ion , par t icular ly if la rge t empera tu re changes a re p re sen t .

In p r inc ip l e , a p r o b l e m s imi lar to that of the t empe ra tu r e d e p e n ­d e n c e of e lectr ical conduc t iv i ty of oil sand exists wi th respec t to the rmal conduc t iv i ty and the rmal capaci ty . At p resen t vir tual ly n o da ta is ava i lab le on the t empera tu re d e p e n d e n c e of thermal conduc t iv ­ity of oil s and . Publ i shed resul ts for re la ted ma te r i a l s , oil sa tura ted s a n d s t o n e s , 2 5 s h o w that these mater ia l s exhib i t re la t ively smal l c h a n g e s in the rmal conduc t iv i ty wi th t e m p e r a t u r e . T h e s e da ta w o u l d s e e m to sugges t that changes in the rmal conduc t iv i ty of oil sand can b e neg lec ted ove r the t empera tu re r a n g e of interest in mos t e lectr ical hea t ing . T h e r m a l capaci t ies of mix tures can b e es t ima ted sat isfac­tori ly f rom the co r r e spond ing values of the cons t i tuent ma te r i a l s . T h e s e va lues are a lso substant ia l ly t empera tu re i ndependen t . H e n c e , it is to b e expec t ed that k ' / k and c ' / c for oil sand and sand-sa l ine wa te r mix tu res wil l a l so b e substant ia l ly independen t of t e m p e r a t u r e , at least b e t w e e n 0 ° C and 100°C.

C o n c l u s i o n

T h e au thors h a v e s h o w n in pr inc ip le that phys ica l mode l l i ng is a v iab le app roach to so lve in-si tu e l ec t romagne t i c hea t ing p r o b l e m s in oil sand fo rma t ions . It has b e e n s h o w n that mode l l i ng of the e lec t ro­magne t i c a n d of the the rmal p h e n o m e n a can be car r ied out s imul ta ­n e o u s l y , p r o v i d e d that the t ime scale factors w h i c h are used for the mode l l i ng of these t w o p h e n o m e n a are different . B a s e d on exper i ­menta l ly d e t e r m i n e d va lues of oil s and electr ical conduc t iv i ty and die lec t r ic cons t an t , mode l l ing cr i ter ia and f requency r anges h a v e been de r ived w h e r e the in-si tu e l ec t romagne t i c p r o b l e m can b e m o d e l l e d b y us ing a quas i -s ta t ic a p p r o x i m a t i o n , an induct ion a p p r o x i m a t i o n , a l o w loss d ie lec t r ic a p p r o x i m a t i o n , o r the exac t form of M a x w e l l ' s e q u a ­t ions . It has a l so b e e n s h o w n that wi th a sui table m o d e l m e d i u m , w h i c h m a y b e oil sand itself or an artificial m e d i u m , mode l l i ng can b e car r ied ou t in spi te of the fact that oil sand conduc t iv i ty is h igh ly t empe ra tu r e d e p e n d e n t .

T h e au thors h a v e c o n d u c t e d several p re l iminary labora tory exper i ­men t s wi th s imi la r m o d e l s differing in mechan ica l s ize and us ing m e d i a of different conduc t iv i t i es to subs tant ia te the sca l ing cr i ter ia d e v e l o p e d for quas i - s ta t ic m o d e l s . C o m p a r i s o n s b e t w e e n m o d e l s w e r e m a d e o n the bas is of m e a s u r e m e n t s of vo l t age , cur ren t and i m p e d a n c e l e v e l s , and t e m p e r a t u r e d is t r ibut ions wi th in the m o d e l s . E x p e r i m e n t s at h ighe r f requencies are in p rog re s s . C o m p u t e r numer i ca l s tudies h a v e a l so b e e n c o n d u c t e d to verify s imi l i tude of quas i -s ta t ic m o d e l s wi th cy l indr ica l s y m m e t r y , and hav ing different mechan ica l s i ze s , different m e d i a conduc t iv i t i es and the rmal sca le factors equal to un i ty and o ther than uni ty .

28 C A N . E L E C . E N G . J. V O L 4 N O 4 , 1979

W h i l e the proper t ies of oil sand have been used as a gu ide in

deve lop ing the foregoing s imul taneous e lec t romagne t i c and the rma l

mode l l ing cr i ter ia , the results ob ta ined are genera l ly app l icab le in

o ther areas w h e r e e lec t romagne t i c hea t ing is of i m p o r t a n c e . S u c h

areas include,* for e x a m p l e , the e lec t romagnet ic hea t ing of b io log ica l

t i s sue . A par t icular p rob lem might b e , for ins tance , the hea t ing of

h u m a n o r g a n s , such as the e y e , in a haza rdous e l ec t romagne t i c en ­

v i ronmen t . He re it is of interest to measu re p o w e r absorp t ion for

k n o w n levels of incident p o w e r dens i ty . In this c a s e it w o u l d b e

poss ib le to mode l by us ing a mechanica l scale factor less than uni ty so

that t empera tu re measu remen t s could b e under taken on a mode l m a n y

t imes the size of the actual b iological sys t em. Other appl ica t ions of

s imul taneous e lec t romagne t ic and thermal mode l l ing cou ld b e in

s imula t ing the use of e lec t romagne t ic energy in the t h a w i n g of p e r m a ­

frost, the thawing of soil in winter cons t ruc t ion , the cur ing of c e m e n t ,

rock break ing in min ing opera t ions , dry ing of w o o d , g ra in , or o ther

ma te r i a l s , and in large scale cook ing .

A c k n o w l e d g e m e n t

T h e authors w i sh to express their thanks to the Alber ta Oil Sands

T e c h n o l o g y and Resea rch Author i ty and the Nat iona l Sc iences and

Eng inee r ing Resea rch Counc i l for financial suppor t .

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