A NOVEL 3-D MODEL FOR THE WATER CRESTING IN HORIZONTAL WELLS *

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79 008,0(6:79-755 A NOVEL 3- MOEL FOR THE WATER CRESTNG N HORZONTAL WELLS * LUO Wa-jig, ZHOU Yig-fag, WANG Xiao-dog School of Ergy Rsourcs, Chia Uirsity of Goscics, Bijig 00083, Chia, E-mail:

Thrfor, hydrocarbo productio from a ll is limitd by th critical flo rat. Th mai purpos of this study is to istigat th brakthrough tim ad th critical rat by usig a ol 3- horiotal ll modl.

ifiit coductiity li sik ad th a 3- stady-stat flo modl of th horiotal ll is st up. A poit sik prssur solutio ca b obtaid ith th Fourir trasform.

Accordig to th stabl atr crstig thory, this study cofirms th stabl hight of atr crstig ad th critical rat. Mahil, it ca r-cofirm th brakthrough tim at a spcific rat.

1 79 008,0(6: A NOVEL 3- MOEL FOR THE WATER CRESTNG N HORZONTAL WELLS * LUO Wa-jig, ZHOU Yig-fag, WANG Xiao-dog School of Ergy Rsourcs, Chia Uirsity of Goscics, Bijig 00083, Chia, luoajig@6.com (Rcid Sptmbr 5, 007, Risd April 9, 008 Abstract: th prsc of bottom atr, a drop i th rsroir prssur du to fluid productio causs th aquifr atr to xpad ad to flo ito th rsroir. Thrfor, hydrocarbo productio from a ll is limitd by th critical flo rat. Th mai purpos of this study is to istigat th brakthrough tim ad th critical rat by usig a ol 3- horiotal ll modl. Basd o th hypothsis that th horiotal ll is locatd i ay positio of a circular rsroir ith o-flo boudary o th top of th rsroir ad costat prssur boudary at th bottom, th horiotal ll has b rgardd as a ifiit coductiity li sik ad th a 3- stady-stat flo modl of th horiotal ll is st up. A poit sik prssur solutio ca b obtaid ith th Fourir trasform. Th rsult of th prssur distributio of th uiform flux horiotal ll ca b prstd by mas of th pricipl of suprpositio. Accordig to th stabl atr crstig thory, this study cofirms th stabl hight of atr crstig ad th critical rat. Mahil, it ca r-cofirm th brakthrough tim at a spcific rat. Th output of a compariso bt this 3- modl ad th rsroir umrical simulator (Eclips shos th mthod prstd hr ca b applid to istigat th bhaior of a atr crstig ad to prdict th brakthrough tim at th bottom atr drir rsroir. y ords: horiotal ll, bottom rsroir, critical rat, brakthrough tim, 3- modl fiitio dimsiolss ariabls P r khp h ( i P L k, L qb h k, h, x x L, y y L, r L, ( o gh h,, t b k( o t ( s s h h Thickss of th rsroir L Half of horiotal ll lgth * Biography: LUO Wa-jig (980-, Mal, Ph.. Cadidat c h or b P Prssur r Radial distac g Graity sity g Graitatioal costat prmability q Productio rat of th horiotal ll B Oil FVF, dimsiolss Oil iscosity x istac i th x -dirctio y istac i th y-dirctio istac i th -dirctio Vlocity pottial-oil Porosity, fractur sc Coat atr saturatio, fractio sor rrducibl oil saturatio, fractio t Brakthrough tim

750 0, 0 0 ordr modifid Bssl fuctio, ordr modifid Bssl fuctio Subscripts b brakthrough C critical dimsiolss xtral boudary h horiotal dirctio i iitial o oil rtical dirctio llbor or atr.

2 750 0, 0 0 ordr modifid Bssl fuctio, ordr modifid Bssl fuctio Subscripts b brakthrough C critical dimsiolss xtral boudary h horiotal dirctio i iitial o oil rtical dirctio llbor or atr. troductio Th applicatio of th horiotal ll tchology has b idly usd i may coutris to hac oil rcory for th bottom dri rsroirs [-5,6]. urig th past 30 yars, a larg umbr of modls dscribig th atr croachmt i a horiotal ll ha mrgd. Gigr [7] stablishd a aalytic - modl of atr crstig bfor brakthrough for th horiotal ll ad got th critical flo rat of latral dg dri, gas cap dri ad bottom atr dri rsroir. Th output of a compariso bt th critical rats of horiotal lls ad rtical lls i aisotropic formatio by Chaparo [8] shod that a critical co com closr to th horiotal ll tha to th rtical ll. Oka ad Raghaa [9] prstd a approximat aalytical mthod to istigat th bhaior of a atr crstig ad to prdict th brakthrough tim. This modl is simplr tha th xistig aalytical mthods ad ca proid quick ad approximat girig stimatio for th brakthrough tim bfor dtaild simulatio studis. Liu ad Wag [0] usd th pottial thory for - flo to study th statioary atr crstig toards a horiotal ll locatd at ay positio i a rtical cross-sctio i a oil rsroir. Th aalytical solutio of quasi-liar hyprbolic partial diffrtial quatio for th - flo of th -phas fluid has b obtaid by Guo [] ho usd th mthod of charactristic. Wiboo t al. [] studid th bhaior of atr crstig ad productio prformac of th horiotal ll i bottom atr dri rsroir ith xprimt. Liu t al. [3] prstd a 3- fluid quatio i th spatial domai hich had chals, rtical ad horiotal lls. Yi t al. [] adoptd a quialt filtratio rsistac mthod ad bottom atr driig rtical critical spd to dri th calculatio modl of critical productio prssur diffrc. This mthod did ot tak th lgth of horiotal ll ito accout. Hou t al. [5] prstd th aalytic solutio of atr crstig hight ad latral distac from llbor udr stady flo by mas of imag thory ad pottial suprpositio pricipl. A modl cosidrig th ffct of all frictio ad acclratio o th prssur drop i horiotal ll has b st up by Li t al. [6]. Bsids Oka ad Raghaa s approximat 3- modl, all studis o th atr crstig of horiotal ll i th past coctratd o th - flo modl. this articl, a ol 3- stady-stat horiotal ll modl proids a rliabl aalytical solutio to th prssur distributio of th llbor. Basd o this, th critical rat ad th brakthrough tim ca b obtaid.. scriptio of flo quatios Figur prsts a idalid illustratio of th rsroir systm i this articl. Th bottom atr layr hich is blo th rsroir is ot shod i Fig.. Th rsroir layr is of thickss h, horiotal prmability k, rtical prmability k, ad radius h, horiotal ll is of lgth L, radius r, positio, ad costat yild q. r Fig. Th sktch of horiotal ll i a circular rsroir Th 3- dimsiolss partial diffrtial quatio gorig th flo i th rsroir-aquifr systm is gi by P P r r r r L 0 ( Th o-flo boudary is applid at th top of th rsroir P( r, 0 ( Ad th bottom of th rsroir is coctd ith a costat prssur aquifr

3 75 P ( r,00 (3 Th horiotal outr o-flo boudary is P( r, 0 ( Th ir boudary coditio (li sik ith dimsiolss rtical lgth, poit sik solutio ca obtai h 0 is dp d P Substitutio of Eq.(9 ito Eq.(6 rsults i [si ( C cos ( C ] [si ( cos ( ] C C (9 P lim r 0, r r 0 r 0, P lim r, r (5 Systms of partial diffrtial quatios, ir ad outr boudary coditios r sold by usig th Fourir trasform tchiqu, th pricipl of suprpositio, th trigoomtric fuctio trasformatio, ad th modifid Bssl fuctios. Th solutio of Eqs. ( to (5 is gi: (s Appdix A for driatios {l[ta ( C ] C l[ta ( C ]} (0 Th ll positio gi. C is calculatd ith diffrt horiotal ad th figurs for ~ C is ta ( P (,0, l x L ta ( L (6 3. Calculatio of critical atr crstig hight at th critical rat fiig th pottial fuctio 6 ( Pi P0 o (7 Fig. Plot of dimsiolss C s. dimsiolss t ca b s from th Fig. that ith icrasig C. icrass. Calculatio of th critical rat of horiotal lls Th dimsiolss critical rat of th horiotal ll is gi as th coditio of stabl atr crstig is gi by [0] (8 q C q ta ( C [ l C L ta ( fiig th dimsiolss q ad P (s fiitio dimsiolss ariabls, ca obtai L C ( o ] C (

4 75 Usig th Nto itratio mthod to calculat 3 3 Eq.( ad lttig 0 kg/m, 800kg / m 3 0, ca obtai Fig.3 Fig.3 Plot of dimsiolss qc s. dimsiolss for arious L As ca b s from Fig.3, q C has a icrasig trd as gos up, but th slop of th cur bcom smallr. Wh th horiotal ll is locatd i th rsroir, th logr horiotal ll bcoms, th gratr th critical rat dos. Espcially, a horiotal ll ca b tak as a rtical ll h th lgth is qual to th radius of llbor r. Hr th critical rat drops sigificatly. This is th raso that mor ad mor horiotal lls ha b usd to dlop th bottom atr drir rsroir. 5. Calculatio of th atr brakthrough tim Usig th mthod of atr brakthrough tim calculatio dlopd by Oka [9], ca obtai th dimsiolss atr brakthrough tim q b L 0 t { [si ( cos ( ] [si ( q r cos ( ] } d L o ( Th trapoidal itgral ca b usd to calculat th dimsiolss atr brakthrough tim t b. Th 3 3 plot of th tb s q /L h 0 kg/m, 3 800kg / m 0 ad L is gi i Fig.. t is clar that th gratr q /L is, th Fig. Plot of dimsiolss for arious t s dimsiolss q /L smallr tb is h thy ar o th sam. Each cur has a critical poit. f th q /L is lor tha th critical poit ( t b ~ q /L plot ill b ry stp, ad as tb, atr crstig ill r croach th llbor ad ill stay at a stabl poit i th rsroir. f tak masur to mak q smallr or L gratr, it ill bfit to hac th oil rcory at lo atr cut. 6. Exampl By usig a umrical simulator (Eclips, a tst has b simulatd for th cas of a horiotal ll producig at th bottom dri rsroir. Th horiotal ll ith th lgth of 00 m is locatd o th top of rsroir (. Othr paramtrs ar gi as follos: h is 0 m, kh is 0., k is qual to 0.00, ad r is 300 m, coat atr saturatio ( s c is 0.35, irrducibl oil saturatio ( s or is 0.3, oil iscosity is 5 mpas ad porosity of formatio is 0.3, th dsity of oil ad atr ar qual to 800 kg/m 3 ad 000 kg/m 3 rspctily. Mahil, th rsult of th tst is dtrmid by usig th folloig stp-by-stp procdur: Stp : Calculatio of dimsiolss paramtr L, usig th dimsiolss dfiitio. Stp : From th plot of ( qc rsus (, ca obtai th q C (hich is 3.95 at th spcific L ad. Stp 3: R-usig th dimsiolss dfiitio, th critical rat is 8.58 m 3 /d. Stp : For a gi productio q, ca calculat q /L. from th plot of ( t b rsus ( q /L, ca b rad dirctly from Fig.. t b b

5 753 Tabl Compariso of brakthrough tim obtaid by Eclips ad by 3- solutio for arious horiotal ll productios Horiotal ll productio q (m 3 /d q q /L t b Brakthrough tim ( by 3- solutio(d Brakthrough tim (by Eclips(d Fractioal rror (% Stp 5: Usig th dimsiolss dfiitio, th brakthrough tim ca b obtaid at th dimsiolss tim t b. As ca b sho from Tabl, th output of this 3- modl is clos to th rsult of rsroir umrical simulator (Eclips ad th fractioal rror is blo 0%. 7. Coclusios this study, a ol 3- modl to istigat th bhaior of a atr crstig has b built. Basd o th modl, th prssur distributio of th llbor, th critical rat, th atr crstig hight ad th atr brakthrough tim at a spcific productio ca b obtaid. Combiig th rsults of this ork, th fial coclusios ar rachd as follos: ( A horiotal ll is of hlp to icras th brakthrough tim tha a rtical ll. So horiotal lls ar suitabl to xploit th bottom dri rsroir. ( Th critical rat icrass as ad L gos up. Thrfor, it is usful to mak th horiotal ll far aay from th itrfac of th aquifr ad rsroir or to icras th lgth of horiotal lls. (3 Th output of a compariso bt this 3- modl ad th rsroir umrical simulator (Eclips shos that th mthod of this rsarch ca b applid to istigat th bhaior of a atr crstig ad to prdict th brakthrough tim at th bottom atr drir rsroir rapidly ad rliably. Rfrcs [] SERRA. W., BRCE B. W. ad MACONAL. G. Applicatio of horiotal ll at Prudho Bay[J]. JPT, 987, 7-5. [] HANSEN. L., VERHYEN M. Horiotal ll sols atr coig[j]. Ptrolum Egir tratioal, 99, 6-5. [3] TARGET P. T. Th ha oil fild: lopmt of a tiy margial fild ith horiotal lls[j]. JPT, 99, [] MURPHY P. J. Prformac of horiotal ll i th Hldr fild[j]. JPT, 990, [5] PENG C. P., YEN N. Rsroir girig aspcts of horiotal lls-applicatio to oil rsroir ith gas of atr coig problms[c]. SPE9958. Bijig, Chia, 995, -7. [6] BE Bi-, WANG Shi-ju ad XAO Xi-hag t al. Applicatio of horiotal ll tch i diggig pottial to oil rsroir ith hypo-produc bottom atr ad positi gradus thick oil layr[j]. Wll Tst, 007, 6(: 65-66(i Chis. [7] GGER F. Aalytic modls of atr crstig bfor brakthrough tim for horiotal ll[c]. SPE N Orlas, USA, 986, -3. [8] CHAPERON. Thortical study of coig toard horiotal ad rtical lls i aisotropic formatio: Subtropical ad critical rats[c]. SPE N Orlas, USA, 986, -9. [9] OZAN E., RAGHAVAN R. A brakthrough tim corrlatio for coig toard horiotal lls[c]. SPE 096. Hagu, Th Nthrlads, 990, [0] LU Ci-qu, WANG Xiao-dog. Critical productio rat i a horiotal ll for th formatio of a atr crst [J]. Acta Ptroli Siica, 993, (: 59-6(i Chis. [] GUO a-li. Th atr co problm of a horiotal ll i bottom atr rsroir[j]. Joural of Southstr Ptrolum stitut, 995, 7(: 9-3(i Chis. [] WBOWO W., MARSEWOJO P. ad SUARNO P. Bhaior of atr crstig ad productio prformac of horiotal ll i bottom atr dri rsroir[c].spe uala, Malaysia, 00, -0. [3] LU Fu-pig, WANG Xu-sog ad WANG Ju. A umrical simulatio of chal rsroirs cotaiig rtical horiotal lls[j]. Joural of Hydrodyamics, Sr. B, 006, 8(5: (i Chis. [] YN Gui-qi, ZHANG Gog-sh ad LU Zhi-ju t al.

6 75 Rsarch o horiotal ll critical productio prssur i bottom atr rsroir [J]. Ptrolum Gology ad Egirig, 006, 0(5: 0-3(i Chis. [5] HOU Ju, CHEN Sog-li ad L Chu-la. Calculatio of atr crstig hight for horiotal ll i bottom atr [J]. Joural of Hydrodyamics, Sr. A, 006, (3: 37-38(i Chis. [6] L Xiao-pig, ZHANG Li-hui, L Yu t al. ariatio las of prssur ad productio rat i horiotal ll[j]. Joural of Hydrodyamics, Sr. A, 005, 0(: 9-97(i Chis. Appdix A Th Fourir trasforms ar dfid as follos: 0 P ( r P ( r, si( u d, si( u P ( r, P( u, r (A- N ( Th poit siks solutio of Eqs.(-(5 is r P 0r 0r N r si( u si( u (A- Th prssur distributio of ay poit i th rsroir ca b obtaid by itgratig th xprssio (A- alog th horiotal ll lgth (,, { 0[ ( ] P x y x y r d [ ( ]d } 0 x y r si( u si( u (A-3 Th llbor prssur distributio is gi blo h 0, y tgratig xprssio (A- from x to x, obtai th prssur distributio of uiform flux horiotal ll (, { 0[ ( ] P x x ( r dd x 0[ ( x ] ( r d d x }si( u si( u (A-5 Usig th progrssi itgral i th ual tgral i xprssio (A-5, ha hr x a a 0[ ( ]d d x [ i( ( ] (A-6 ( [ ( ] ad ( [ ( ] i 3/ 0[ ( x a ]dad x hr i (A-6-a (A-6-b [ ( ( ] (A-7 (,0, { 0[ ( ]d P x x ( [ ( ] (A-8-a r 0 x r si( u si( u [ ( ]d } (A- ( [ ( ] i 3/ (A-8-b Combiatio of Eqs.(A-6, (A-7, ad (A-8 yilds

7 755 P ( x,0, { [ i ( ul ( ul ul L L ( r ( ] [ i( ( r ( L (A-6 ( ]}si( u si( u (A-9 ad Equatio (A-9 is difficult to mak th calculatio ith th computr program. Applyig th trigoomtric fuctio trasformatio to Eq.(A-9: cos x x l(si (A-0 cos x x x (A- 6 cos( x x l(ta (A- cos( x ( x (A-3 ( r i( ( r ( r ( r ( (A-7 Wh r, Eq.(A-7 ca b ritt ito r i( ( r ( r W ca obtai Eqs.(A- ad (A-5 si( u cos( u ul L ( r ta ( l (A- ta ( si( u cos( u (A-5 u L L Espcially h L / (i fact, it is asy to mt th coditio for fild applicatio ( 0.0 (A-8 Combiatio of Eqs.(A-9 to (A-8 yilds a approximat prssur distributio alog th llbor ta ( P (,0, l x L ta ( i ( ( si( u si( u L (6

MONTGOMERY COLLEGE Department of Mathematics Rockville Campus. 6x dx a. b. cos 2x dx ( ) 7. arctan x dx e. cos 2x dx. 2 cos3x dx

MONTGOMERY COLLEGE Department of Mathematics Rockville Campus. 6x dx a. b. cos 2x dx ( ) 7. arctan x dx e. cos 2x dx. 2 cos3x dx MONTGOMERY COLLEGE Dpartmt of Mathmatics Rockvill Campus MATH 8 - REVIEW PROBLEMS. Stat whthr ach of th followig ca b itgratd by partial fractios (PF), itgratio by parts (PI), u-substitutio (U), or o of