AGH DRILLING, OIL, GAS Vol. 3 No. 3 204 http://dx.doi.org/0.7494/drill.204.3.3.43 Vasyl Moisyshy*, Bogda Borysevych*, Oleg Vytyaz*, Yuriy Gavryliv* DEVELOPMENT OF THE MATHEMATICAL MODELS OF THE INTEGRAL DRILLING INDICES BASED ON THE diesioal aalysis. THEME TOPICALITY AND PROBLEM STATE It is ecessary to have atheatical odels of the itegral drillig idices, i.e. echaical speed, value of torque, ad eergy cosuptio for the rock breakig process, to optiize the ode cotrol of the drillig bit ru. The cotrolled paraeters of the greater part of the kow odels are the axial load o the drillig bit F st ad its rotatio speed d. The circulatio rate Q for ost drillig rigs is cosidered to be a partially cotrolled paraeter. It is defied discretely ad believed to be sufficiet if the further icrease i the fluid effective ijectio rate does ot lead to a icrease i the drillig speed. It was deteried with the help of [] that the utilizatio of the tools for stiffess chage C ad dapig β i the botto-hole assebly ca both iprove ad worse the itegral drillig idices. At the preset there are o studies of the ifluece of the drillig tool paraeters C ad β o the echaical speed V ech, value of torque T d (average value o the bit), ad eergy cosuptio for rock breakig W R o the well botto-hole. Takig ito accout the above etioed, there arises urgecy for developig atheatical odels of the itegral well drillig idices. The objective of the study is to estiate the ifluece of the ode paraeters F st ad d ad drillig tool perforace idices C ad β o the itegral drillig idices o the basis of the diesioal aalysis, as well as to develop their atheatical odels i accordace with the results of the bech tests. 2. DEVELOPMENT OF THE MATHEMATICAL MODELS OF THE MECHANICAL DRILLING SPEED I accordace with [2] the diesios of ay physical agitude ay be preseted as a product of the raised-to-power diesios of the ai physical agitudes. * Ivao-Frakivsk Natioal Techical Uiversity of Oil Ad Gas, The Petroleu Egieerig Faculty, Ukraie 43
The followig forula is true for echaical pheoea: where: L legth [], T tie [s], M ass [kg], α, β, ad γ expoets. [X] L α T β M γ () The three siilarity theores are a theoretical basis for deteriig the coditios of trasitio fro a odel to a real process. The first ad secod theore, which is called a π-theore, deterie the relatios betwee the paraeters of the drillig process odelig o the test stad ad i the idustrial coditios. The ethods for fidig out the siilarity ad ways of the siilarity realizatio whe developig the odels are deteried by the third theore. I accordace with the first theore the drillig processes siilarity o the test stad ad i real coditios is characterized by the equality of all the siilarity criteria that are diesioless coplexes of the physical agitudes, coposed of geeralized coordiates ad paraeters, ad they are described by the siilar fuctioal depedecies. Let s cosider the echaical drillig speed to be a geeralized coordiate ad the let s describe the process of its chace with the help of the followig depedece: V ech f (F st ω d C β D p Q) (2) where: F st axial static drillig bit load [N], ω d agular drillig bit rotatio speed [s ], C ad β stiffess [N/] ad drillig tool dapig [N s/] respectively, D drill bit diaeter [], p rock hardess accordig to the ark [Pa], Q voluetric drillig ud flow rate [ 3 /s]. Let s deterie the diesios of the show above physical agitudes with the help of the diesios of the ai easureet uits of the SI Syste: [ Vech ] L T ; ;, s ;, 432 3 Fst ; Q L T A s 3. I these forulas L drillig tool deforatio uder the static load ifluece []; F res resistace force of the hydraulic vibratio daper [N]; V pisto speed of the hydraulic vibratio daper; A ark basis area [ 2 ].
The expoets α, β, ad γ of the physical agitudes that deterie the well drillig process are show i Table. Table Expoets α, β, ad γ of physical quatities that deterie the process of drillig wells Size L T M α i β i γ i V ech 0 F st 2 ω 0 0 C 0 2 β 0 D 0 0 p 2 Q 3 0 Check the coditio of idepedece of paraeters D, p, Q, that is, covice ourselves that the deteriat of the atrix of their diesios is ot equal to zero: D 0 0 p 2 Q 3 0 ( 2) 0+ 0 ( ) ( ) + 0 3 0 ( 2) 3 ( ) 0 0 ( ). Make use of these idepedet paraeters for fidig the siilarity criteria. The first siilarity criterio has the for: We express this criterio through the diesios of the relevat quatities (see Tab. ). (3) (4) Fro the coditio of expoets equality of powers at L, T ad M i the uber at or ad deoiator we obtai the syste of equatios: + 3p p 0 (5) 433
By substitutig the value 2, 0, p i (3), write dow the first criterio of siilarity: Π Vech (6) D -2 Q The secod siilarity criterio has the for: Π 2 Fst D p Q 2 2 p2 S Expressig it through the diesios of the relevat quatities (see Tab. ), we obtai: (7) 2 Π 2 2 3 ( L) ( ) ( L T ) 2 2 p2 (8) Fro the coditio of equality of the expoet sat L, T ad M i the uerator ad deoiator we obtai the syste of equatios: + 3p 2 2 2 p 2 2 2 2 (9) By substitutig the value 2 0, 2, p 2 0 i (7), we write the secod criterio of siilarity: Π 2 Fst (0) p We write the third criterio of siilarity: Π 3 () By subittig it through the diesio of the relevat quatities (see Tab. ), we fid: 0 0 Π 3 2 3 ( L) ( ) ( L T ) 3 3 p3 + 3p 0 3 3 3 p 3 3 3 0 (2) (3) By substitutig the value 3 3, 3 0, p 3 i (), we record: Π 3 (4) 434
The fourth siilarity criterio has the for: Π 4 (5) Expressig its diesios through relevat quatities (see Tab. ), we obtai: Π 4 0 2 ( L) 2 ( ) 3 ( L T ) 4 4 p4 (6) + 3p 0 4 4 4 p 2 4 4 4 By substitutig 4, 4, p 3 0 i (5), we record: (7) Π 4 (8) Fially, the fifth siilarity criterio has the for: Π 5 (9) Presetig through the diesio of the relevat quatities (see Tab. ) we obtai: Π 5 0 ( L) 2 ( ) 3 ( L T ) 5 5 p5 (20) + 3p 0 5 5 5 p 5 5 5 (2) By substitutig 5, 5, p 3 i (9), we record the fifth criterio of siilarity: Π 5 (22) Let s give the desired fuctioal depedeces: (23) Based o the results of stad experietal studies [] coducted accordig to a classic pla, the depedeces V ech f (F st ), V ech f (ω d ), V ech f (C) ad V ech f (β) are satisfactorily iterpreted by a power fuctio, which is a idicator for the depedece V ech f (F st ) α, for the depedece V ech f (ω d ), α 2, for V ech f (C) ad V ech f (β) respectively by «α 3» ad «α 4». 435
Let s give the desired fuctioal depedeces: (24) Where k, α, α 2 α 3 α 4 epirical coefficiets is to be deteried by the results of experietal studies coducted i accordace with the plaed experiet. Durig these studies drillig a well was perfored with bit 93 Т at the fixed flow rate of drillig fluid (water) 7 Q 7 0 3 3 /s, ad hardess accordig to the ark of sadstoe iterlayers of Vorotyscheska strata p 440 MPа 440 0 6 Pа ad p 2050 MPа 2050 0 6 Pа. The calculatio for the iterlayer with hardess p 440 MPа give: D 2 Q (93 0 3 ) 2 7 0 3 0,8093; D 3 Q (93 0 3 ) 3 7 0 3 8,7026; D p 93 0 3 440 0 6,3392 0 8 ; D 4 p Q (93 0 3 ) 4 440 0 6 (7 0 3 ) 5388495. Gratig this: (25) For iterlayer hardess р s 2050 MPа we will have: D p 93 0 3 2050 0 6,9065 0 8 ; D 4 p Q (93 0 3 ) 4 2050 0 6 (7 0 3 ) 2907232. (26) 3. MATHEMATICAL MODELS DEVELOPMENT OF MOMENT CAPACITANCE FOR DRILLING PROCESS Takig for the geeralized coordiate the average value of torque o the bit, let s describe the process of its chage as the fuctioal depedece: 436 T d f (F st ω d C β D p Q) (27)
The diesio of the oet capacitace T d N kg 2 M L T 2. 2 s Idepedet paraeters, accordig to which we deterie the siilarity criteria are the sae as i the defiitio of the fuctioal depedece siilarity criteria of the echaical drillig speed. Accordig to this siilarity criteria 2 5 do t chage. The first siilarity criterio has the for: Π Td D p Q p Expressio through the criterio diesio correspodig values: 2 2 Π 2 3 ( L) ( ) ( L T ) p (28) (29) Fro the coditio of expoets equality at L, T ad M i the uerator ad deoiator we obtai the syste of equatios: + 3p 2 p 2 By substitutig 3,, p 0, i (28), write dow the first criterio of siilarity: (30) Π Td D p 3 Let s give the desired fuctioal depedece as: (3) (32) Based o the results of the stad experietal studies [] coducted due to a classic pla, depedig ces T d f (F st ), T d f (ω d ), T d f (C) are satisfactorily iterpreted by a power degree fuctio the expoet of which for the depedece T d f (F st ) «α 9», ad for the depedece T d f (ω d ) «α 0», T d f (C) «α». Depedeces T d f (β) are satisfactorily iterpreted by a liear fuctio T d a b β. Let s give the desired fuctioal depedece as: (33) Where k 3, α 9, α 0, α, a, b epirical coefficiets to be deteried by the results of the experietal studies coducted for the plaed experiet. 437
The calculatio for the iterlayer hardess p 440 MPа we have: Q/D 7 0 3 /93 0 3 0,0753; D 3 Q (93 0 3 ) 3 7 0 3 8,7026. D p 93 0 3 440 0 6,3392 0 8. Give the above: (34) For iterlayer hardess р 2050 MPа we will have: D p 93 0 3 2050 0 6,9065 0 8 (35) 4. DEVELOPMENT OF MATHEMATICAL MODELS OF POWER CONSUMPTION OF ROCK DESTRUCTION PROCESS IN THE BOTTOMHOLE Takig for the geeralized coordiate the average torque value o the bit, the process of its chage will be described as fuctioal depedece: W P f (F st ω d C β D p Q) (36) 3 kg The diesio of the oet capacitace [ W P ] N / 2 3 s 2 M L T. Idepedet paraeters, accordig to which we deterie the siilarity criteria are the sae as i the defiitio of the fuctioal depedece siilarity criteria of the echaical drillig speed. Accordig to this siilarity criteria 2 5 are ot chaged. The first siilarity criterio has the for: Π W P p D p Q. (37) We express this criterio through the diesios of correspodig values: 2 2 Π 2 3 ( L) ( ) ( L T ) p. (38) 438
Fro the coditio of equality of expoets at L, T ad M i the uerator ad deoiator we obtai the syste of equatios: + 3p p 2 By substitutig the value 0,, p 0 i (28), we write dow the first criterio of siilarity: P Π p (39) W. (40) Let s give the desired fuctioal depedece as: (4) Based o the results of bech experietal studies [] coducted due to a classic pla, the depedeces W P f (F st ), W P f (ω d ) are satisfactorily iterpreted by a power fuctio, which is a idicator for the depedece W P f (F st ) «α 5», ad for the depedece W P f (ω d ) «α 6». Depedece W P f (C), which has a local iiu, is satisfactorily iterpreted by the polyoial of the secod degree W P A A 2 C + A 3 C 2. Depedece W P f (β) which has a local axiu is satisfactorily iterpreted by a polyoial of the secod degree W P A 4 + A 5 β A 6 β 2. Let s preset the required fuctioal depedece as: (42) Where k 5, α 5, α 6, A... A 6 epirical coefficiets to be deteried by the results of experietal studies coducted accordig to the plaed factorial experiet. The calculatio for the iterlayer hardess p 440 MPа give: Q 5 D p 7 0 3 ( ) 93 0 440 0 3 5 6 6, 9875 0 7 D 3 Q (93 0 3 ) 3 7 0 3 8,7026. 439
Ilight of outlied: (43) For iterlayer hardess р 2050 MPа we have: Q 5 D p 7 0 3 ( ) 93 0 2050 0 3 5 6 4 9083 0 7,. (44) 5. Coclusios For the first tie o the basis of diesioal aalysis is usig siilar criteria we obtaied a ultifactorial atheatical odel of the echaical speed (25), (26), oet capacitace (33), (34) ad eergy itesity (43), (44) of drillig with roller-coe bits the sadstoe iterlayers of Vorotyscheska strata with hardess 440 ad 2050 MPa, respectively, takig ito accout the drive paraeters ad the stiffess ad dapig of the drillig tool. REFERENCES [] Moysyshy V.M., Borysevych B.D., Havryliv Yu.L., Zicheko S.A.: Stiykist i kolyvaya buryl oyi koloy. Lileya-NB, Ivao-Frakivsk 203. [2] Myslyuk M.A., Zarubi Yu.O.: Modelyuvaya yavyshch i protsesiv u aftohazoproysloviy spravi. Ekor, Ivao-Frakivsk 999.