Nargozy T. Danayev*, Darkhan Zh. Akhmed-Zaki* THE USAGE OF MATHEMATICAL MLT MODEL FOR THE CALCULATION OF THERMAL FILTRATION

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1 WIERTNICTWO NAFTA GAZ TOM 3/ 6 Nargozy T. Daayev*, Darka Z. Akmed-Zak* THE USAGE OF MATHEMATICAL MLT MODEL FOR THE CALCULATION OF THERMAL FILTRATION Durg te reearc we ued a well-kow matematcal MLT model Ìukat Leverett termal for te umercal calculato of te termal fltrato of a bpac lqud a porou pace. Te matematcal teory of termal fluece o fltrato of te bpac lqud for MLT model wa offered by V.N. Moakov ad O.B. Bocarov (ee [, ]). Te fluece of temperature o te caracter of lqud moto poble troug te cage of vcoty ad capllary properte of dfferet lqud compoet wc deped o ter temperature ad temperature of keleto a porou pace. Te caractertc feature of te termal model of bpac fltrato all equalzato wc are eceary for makg bpac fltrato, ecept te law of Darcy ad Lapla, are te reult of avg law of te cotuou evromet mecac. I addto, a ÌLÒ-model tecologcal ee tat for t decrpto oly te fucto-parameter are ued, wc are defed epermetally. Te bac caractertc of movg lqud are: p v pae aturato (.e. cocetrato + =, water, ol), dety, preure, vcoty depedet o temperature, Darcy velocte of pae (carge), v v v Darcy velocte of a fltrato of a m. Te o-uform aotro porou evromet caracterzed by poroty m(), K () abolute permeablty ad relatve pae permeablty k ( ), = m ( =, ), 3 = m. * Te Kazak atoal uverty of a amed after al-farab, Almaty, Kazakta 49

2 MLT model equato look lke were: m ( ) dv ( v) t v K k () p g ( ) ( ) (,, ) p p m ( ) (,, ) ( )co ( ) () K ( ) J dv( v ), t () p c (,, ) capllary preure, (,, geeralzed factor of eat coductvty (fucto p c (,, ) ad are gve, equlbrum temperature). Te reolvablty of MLT model wt correpodg boudary codto for a o- -tatoary cae a bee codered work [], ad for a tatoary cae work [3]. By aalogy wt [] we all eter p average preure of a m p p k k k ( (, d () After correpodg traformato of MLT equato t wll be traformed to te followg ytem were: m t dv K a a f b v [ ( ) dvk ( p f a ) 3 dv[ (,, ) v ) t p (,, ) p p ( )co ( ) c m ( ) () k ( ) J (3) cot reulted aturato of a moteg pae,, *, redual aturato, mm ( ) effectve poroty. 5

3 b k ( k k ) (, ) K K ( k k ); a kk ( k k ) ; a a; a a; f [ ( ) g] a; a3 b b ; f b p 3 c p c bd b g ( ) ; k(, ) k() ( ); v v v; ( ccp ) ; m ;, ; 3 m. Factor of ytem (3) are epreed obvouly troug fuctoal parameter te tal MLT model ad vew of pycal poe ave followg properte: a ( >, (, ); (, t) m; (a (, ), (, )) C (Q), Q ={, < <, * < < * }. Furter te opportuty of te umercal deco tatoary MLT model for oe-dmeoal otro porou evromet codered ad te algortm of te deco offered. [ K( a a ) bv] [ K( p a3 )] (4) [ v] v K( p a3 ) Wt boudary codto: ; L L ; ( ) (5) L ev p p p p were te factor of eat ecage. ; L L ( Ka ) ( Ka ) ( b v) ( Kp ) ( Ka ) 3 ( ) ( v) v K( p a ) 3 (6) 5

4 Let eter a uform grd = +;, N, were = cot a legt of te grd block. Te we all wrte dow dfferece type of te equato (6) for a cotat tep of te grd ( Ka ) ( Ka / ( Ka ) / ) bv ( Ka ) ( ) / ( bv ) / / / v / / / v / (7) K p p p p / K / Ka ( ) v p p a3 / K /, / 3 / ( Ka3 ) / Te equato from te frt to te trd of te ytem (7) ave accuracy of appromato of te order O( ), ad te equato for peed of a fltrato of a m a appromato of te order O(). Sytem of te equato (7) olved by a mplct teratve metod p p p 3L ( p v,,, ) v L v / (,, p, v ) L (,, v ) L p v (,,, ) (8) were factor at dfferece of aalogue of dervatve calculated o te layer. Tu, te umercal deco of te tal ytem of te olear equato (6) tur out wt coecutve olve of te equato of te ytem (8). Factor,, 3 ytem of te equato (8) are teratve parameter for te equato of a aturato, temperature ad preure accordgly. Boudary codto are appromated a follow: ev N L N ; N ; ; ( ) N N ev (9) L N N p p ; p p ; v v ; v v L 5

5 Te receved ytem (8) olved a metod of prorace for eac equato eparately. Furter collectg proracg factor we are covced, tat tey atfy to a codto of dagoal prevalece, provdg tablty of a metod of prorace. Ifluece of a temperature feld o caracter of proce of a fltrato t wa vetgated troug decreae vcoty wt growt of temperature tat a ow creae oly te fal petrofeedback of te layer. Tu t wa uppoed, tat te temperature of a frm keleto ad lqud pae leveled tatly. Numercal calculato a ow advatage of ue of a o-uform grd for te deco of ytem (4) wc caued by fat covergece of deco at gfcat umber of terato ad te mot adequate coformty of te receved reult wt pyc of codered procee. Furter te gve work te reearc problem of factor of ytem of te equato (3) codered at te gve tatemet of problem (4) (5) for tetg te offered algortm (8) (9). By mea of computg epermet te aaly of fluece of factor a(, ) (a, a, a, a 3 ) ad (, ) ytem of te equato (3) o caracter of coure of reearced procee lead. Ital dtrbuto of aturato, preure ad temperature wll be eceary for te orgazato of teratve proce of te accout. Hece a tal dtrbuto we all put for aturato =.37, temperature = 75 C, preure p =.5 MPa. Reult of calculato a cae we =.6mPa, =3.5mPa,m =.5, K = 4 md, =.8, L 36. =.,T ev = 7 C, p c (,, )=. ( 3 ) 39. ( )co ( ) 3 m K are reulted Fgure. a) b) c) Fg.. Reult of calculato: a) aturato S, S ; b) temperature; c) preure 53

6 Te computatoal aalye of te fluece of termal procee of jot moto of may-fazed lqud allow to decrbe te proce of fltrato of two ad more lqud wt dfferet pycal properte a porou evromet adequately, wc repreet a large teret plag ad developmet of ga- ad ol feld. REFERENCES [] Atotev S.N., Kazkov A.V., Moakov V.N.: Te boudary problem of o- -uform lqud mecac. Novobrk, Nauka 983, 39 [] Bocarov O.B., Moakov V.N.: Te boudary problem of o-otermal bpac fltrato porou meda. Te dyamc of cotuou evromet. Novobrk, 988, [3] Abylkarov U.U., Akmed-Zak D.Z.: Te reolvablty of a tatoary problem of a termal bpac fltrato. Te Iteratoal Scetfc Coferece Actual problem of mecac ad mecacal egeerg, Almaty, 5, 4 [4] Abylkarov U.U., Daayev N.T., Akmed-Zak D.Z.: Te calculato of a termal fltrato o te ba of matematcal model MLT. Te Iteratoal coferece Problem of moder matematc ad mecac, Almaty, 5,