A New Correlating Parameter to Quantify Productivity of Extended Hydraulic Fractured Wells

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1 Inernaional Journal o Applied Engineering Research ISSN Volume, Number 5 (6) pp A New Correlaing Parameer o Quani Producivi o Eended Hdraulic Fracured Wells T. Marhaendrajana* Associae Proessor, Peroleum Engineering, Insiue Technolog Bandung (ITB) Bandung dan Sekiarna, Jawa Bara, Indonesia. S. Parsaulian Senior Reservoir Engineer, PT PERTAMINA EP, Indonesia. Absrac A modiied model o eended heigh racured well is presened o accoun or a more realisic case ha allows luid low rom racure o he wellbore onl hrough he reservoir hickness inerval (commonl peroraed). One o he resuls, his work proposes a new correlaing parameer o eecive racure conducivi or eended heigh racured well. This correlaing parameer enable ones o deermine pseudo-skin acor which hen can be used o compue well producivi. Two oher imporan resuls are presened, he are: (i) Eending racure heigh beond he reservoir hickness a consan racure lengh increases racure conducivi unil i reaches a maimum value a an opimum racure heigh. (ii) The increase o racure conducivi b eending racure heigh greaer han reservoir hickness is also rue or he case o consan racure volume provided ha racure heigh is less han /.77 imes racure hal-lengh ( ). These are dieren perspecives rom Benne s proposal saing ha a consan racure lengh, increasing racure heigh alwas increases eecive racure conducivi, and a consan racure volume, eending racure heigh beond reservoir hickness lower eecive racure conducivi. Kewords: Well Producivi, Eended Fracure Heigh, Hdraulic Fracure, Pseudo-skin Facor, Fracure Conducivi. Inroducion Hdraulic racuring is a mean o improve low connecivi beween well and reservoir b injecing pressurized luid ino he reservoir o creae cracks in he argeed reservoir. I is hen epeced o increase well producivi [], or o increase eecive wellbore radius [, ], or o lower pseudo-skin acor [, 5]. These producivi, eecive wellbore radius and pseudo-skin acor are relaed each oher and he correlae wih racure conducivi. The available published resuls correlaing racure conducivi wih eiher o hose hree parameers are developed or case o racure heigh equal o reservoir hickness. Fracure propagaes boh a laeral and verical direcions. Is laeral and verical dimensions are iner-dependen aeced b conras o sress proile in he ormaion, racuring luid properies, and rock properies. I is hen almos impossible o mainain is size in one and change he oher. Thereore, i is no uncommon i racure heigh eend beond reservoir hickness o achieve cerain and desired racure lengh. Fracure grow perpendicular o he direcion o leas principal sress, or in he direcion o maimum sress. The deeper he wells, i is more likel ha principal racure orienaion is epeced o be verical. Fisher [6] presened verical growh o hdraulic racures rom micro-seismic and il-meer daa, and he racure could grow, 5 vericall a he deph greaer han,. The analical soluion [5, 7], and numerical soluion [8] were alread presened in he lieraures or he case o racure heigh equal o reservoir hickness. Benne [9] proposed a numerical model o ake ino accoun he racure heigh greaer han reservoir hickness. In ha model, luid low rom racure oward he well hrough he enire inerval o racure heigh. This work, on he oher hand, considered luid low rom racure ino he wellbore is limied hrough he reservoir inerval. The new correlaing parameer or eecive racure conducivi was developed and i can be used o compue pseudo-skin acor or well producivi calculaion. aa and Mehods A Compuaional model was developed and was solved using numerical reservoir simulaor (Figure ). A well is locaed a he cener o a square drainage area. However, he boundar eec is no considered and i is no presened in his sud. The reservoir is isoropic and uniorm wih consan hickness. The racure heigh, h, was consan along he racure lengh,, and was allowed o be equal o or greaer han pa (reservoir) hickness, h, peneraing he impermeable zones above and below. The racure is assumed o be illed in wih proppan wih uniorm permeabili. Boh racure wings are smmerical. The peroraion inerval was se or he whole pa hickness. I assumes ha peroraion densi is high enough so ha pressure loss is negligible. A his condiion, here will be converging low rom racure o well when racure heigh eends beond he reservoir hickness. This diers rom model presened b Benne [9], in which he enire racure heigh is open o he well. 979

Inernaional Journal o Applied Engineering Research ISSN 97-56 Volume, Number 5 (6) pp 979-985 The grid size is smaller near he well and near he racure ip as hese regions subjec o higher lu densi [5].

2 Inernaional Journal o Applied Engineering Research ISSN Volume, Number 5 (6) pp The grid size is smaller near he well and near he racure ip as hese regions subjec o higher lu densi [5]. To avoid alse wellbore sorage response, he eecive well-block radius was compued using Peaceman ormula []; Equaion or isoropic and Equaion or no isoropic. The discreizaion o he reservoir model is shown in Figure. Reservoir and racure are assumed o be illed wih slighl compressible and single-phase luid o consan viscosi. Gravi eec is negleced and wellbore pressure is independen o deph. r o r o k. () / / / k k / k k / k / k / k / /.8 () Grid was consruced in smmerical manner, and a well in he cener o reservoir. The dimension o grids were calculaed b using equaions below. A locaion ha is awa rom he well a -direcion: /( N ) i e a; where a i w In he ormula, e is reservoir dimension a -direcion; w is well grid block size a -direcion; and N is number o grid blocks. A -direcion and awa rom well (direcion o racure), grid blocks were compued similarl wih ecepion ha i was divided ino several secions; (i) well-o-cener o racure, (ii) cener o racure-o-racure ip, (iii) racure ip-o-reservoir limi ( e/). z Fracure heigh, h Producion inerval w / verical racure reservoir Figure : Reservoir Model wih Eended Verical Fracure Well pressure response as uncion o ime was generaed a speciied consan rae or various racure conducivi, hickness raio, racure lengh and racure heigh. Wellbore sorage is no included in his model. The soluions are presened in he dimensionless orm, in which he dimensionless variables are deined as below. imensionless well pressure drop, khpi pw pw ().qb imensionless ime,.67k () c h imensionless racure conducivi, k w Fc (5) k imensionless racure heigh, and h H (6) Thickness raio, h h (7) h Where p i is iniial pressure (psi), p w is wellbore boom hole pressure (psi), q is surace lowrae (sb/d), B is luid ormaion volume acor (v/v), is luid viscosi (cp), c is oal compressibili (/psi), k is reservoir (m), is porosi (racion), h is reservoir hickness, k is racure permeabili (m), w is racure widh, is hal racure lengh, h is racure heigh, and is producing ime. Verical well Fracure ip Figure : Model o Grid Block Size and Geomer aa used o generae numerical soluion or eended racure heigh is lised in Table. e Table. Rock, Fluid and Fracure aa Parameers Uni Value Reservoir Permeabili, k m 5 Reservoir Thickness, h Porosi, racion. Rock Compressibili, cr /psi -6 Iniial Reservoir Pressure, psia 5 pi Fluid Formaion Volume v/v. Facor, B Fluid Viscosi, cp.8 Fluid Compressibili, co /psi 6-6 imensionless Fracure Heigh, H=h/ imensionless.;.5;.;.5 Thickness Raio, h=h/h imensionless ; ; 5; imensionless Fracure Conducivi, Fc imensionless Surace Flow Rae, q Sb/da Wellbore iameer in 6 e.;.5;.5;.75; ;.5; 5; 7.5; ; 5; 5; 75; ; 98

Inernaional Journal o Applied Engineering Research ISSN 97-56 Volume, Number 5 (6) pp 979-985 To validae he model seup (i.e. grid size and orienaion), a special case condiion o racure heigh equal o reservoir hickness was seleced.

Flu rae along he racure ace (or-< <, and >) is calculaed b: C n q Where: n n ', cos n q ; k c k c ep n ep n n ', cosn ( ' ) ; C ' n ; q ep n d' d (8) d' d n ( ) w c c q ', ', q Wellbore pressure is

beond he reservoir hickness. Resuls and iscussion The resuls are ploed in orms o dimensionless variables deined in Equaions o 7.

3 Inernaional Journal o Applied Engineering Research ISSN Volume, Number 5 (6) pp To validae he model seup (i.e. grid size and orienaion), a special case condiion o racure heigh equal o reservoir hickness was seleced. The wellbore pressure rom numerical soluion was compared wih he analical soluion ha was compued using ormula proposed b Cinco e.al. [5] or consan lowrae case (Equaions 8 o 9). Flu rae along he racure ace (or-< <, and >) is calculaed b: C n q Where: n n ', cos n q ; k c k c ep n ep n n ', cosn ( ' ) ; C ' n ; q ep n d' d (8) d' d n ( ) w c c q ', ', q Wellbore pressure is hen compued b: p w ' ep d' d (9) q ', When he wo soluions (numeric and analic) are in good mach, an analogous numerical model seup is used o generae consan lowrae soluion or racure heigh eending beond he reservoir hickness. Resuls and iscussion The resuls are ploed in orms o dimensionless variables deined in Equaions o 7. I can be seen rom Figure ha he numerical soluion mimic he analical soluion [5] ver well or various racure conducivi o.,,,, and. The resuls were saisacor enough so as oundaion o use similar numerical model seup o generae consan lowrae soluion or racure heigh eending beond he reservoir hickness. For his case, Benne [9] published numerical model, bu he enire racure heigh is open o he wellbore. On he oher hand, his paper limis he low connecion o racure and wellbore onl a reservoir hickness inerval. I is more appropriae or general pracice since, when simulaed b hdraulic racuring, wells are commonl cased and are peroraed a pa zone. w ; Figure : Comparison beween Numerical and Analical Soluions or Fracure Heigh Equal To Reservoir Thickness As he conac area beween well and racure is onl limied a he producing zone, he well pressure drop o he proposed model is higher han Benne s model (Figure ). A his case he eended racure heigh is hal o racure hal-lengh. The dierences are so signiican ha i canno be ignored especiall or low dimensionless racure conducivi, F c=.. For a air dimensionless racure conducivi, F c =, he dierence beween wo models become smaller during radial low ( >). When racure heigh is much less han racure hal-lengh (h / =.), he limied enr has lile eec on he pressure drop as seen in Figure 5, a boh F c =. and F c =. Furhermore, he eended racure heigh soluions coincide wih he Cinco soluion (h =h) a F c, app = h.f c. For eample, he soluion o eended racure heigh a h =5 and F c=. coincides wih Cinco soluion a F c=.5. Likewise, he soluion o eended racure heigh a h =5 and F c= coincides wih Cinco soluion a F c=5. Figure : imensionless Wellbore Pressure o Eended Heigh Fracured Well, Case h / =.5 and h /h = 5 98

Inernaional Journal o Applied Engineering Research ISSN 97-56 Volume, Number 5 (6) pp 979-985 I can be said ha eending racure heigh beond he producing zone inerval improves he well perormance b

This correlaing parameer were irs inroduced b Benne. However, resuls rom his paper indicae ha i onl applies i h <<.

4 Inernaional Journal o Applied Engineering Research ISSN Volume, Number 5 (6) pp I can be said ha eending racure heigh beond he producing zone inerval improves he well perormance b reducing well pressure drop. Anoher hing can be drawn rom Figure 5 is ha Cinco soluion can sill be used or eended racure heigh provided F c, app = h.f c is used as correlaing parameer. This correlaing parameer were irs inroduced b Benne. However, resuls rom his paper indicae ha i onl applies i h <<. For radial and horizonal low o a ull peneraed verical well, producivi inde (PI) is compued rom arc s equaion and i is wrien as: q kh PI () p re.b ln rw For eended racured well, he epression o producivi inde can use eecive/apparen wellbore radius concep (Pras, 96) or pseudo-skin acor concep (Cinco, 978) or vericall racured well. kh PI or re.b ln rwa kh PI () r e.b ln s rw Where r wa is eecive wellbore radius; and s is pseudo-skin acor. The producivi increase o eended racured well is hen measured b producivi raio. re re ln PI r ln w r w () PI re re ln s r ln wa r w Equaion will alwas be posiive because r w and r wa canno be greaer han r e. Hence [ln(r e/r w) + s ] will alwas be posiive. The pseudo-skin acor is deermined b subracing he eended racure heigh soluion b ull peneraed verical well soluion (Equaion ). Pseudo-skin acor or case o H =. and h =5 is shown in Figure 6. kh s ps pw, pw, v ().qb In Figure 6, pseudo-skin acor o eended racure is alwas negaive, and i decreases wih ime or all dimensionless racure conducivi, F c. The value becomes sable (approaching a consan value) during pseudo-radial low period (or his case he ime is approimael greaer han das). This consan value, s, is used in he producivi inde (Equaion ). Value o s depends on dimensionless racure conducivi, F c, and i decreases (becomes more negaive) as F c increases. For racure heigh is equal o reservoir hickness, h /h=, Reerence [5] proved ha group o parameer s +ln( /r w), called pseudo-skin uncion, correlaes wih F c in a single curve. To validae our pseudo-skin acor compuaion, he numerical soluion o eended racure or h /h = are ploed and are compared wih daa rom Reerence [5]. Figure 7 shows ha all daa o pseudo-skin uncion or various H lie on a single curve. I is seen he curve approach o a consan value a abou F c > and his value is equal o.7. Figure 5: imensionless Wellbore Pressure o Eended Heigh Fracured Well, Case h / =. and h /h = 5 Figure 6: Pseudo-Skin Facor o Eended Heigh Fracured Well, Case h / =. and h /h = 5 Figure 7: Plo o Pseudo-skin Funcion a h = or various value o H 98

5 Inernaional Journal o Applied Engineering Research ISSN Volume, Number 5 (6) pp For h /h >, he curves o pseudo-skin uncion deviae rom h /h= case and he pseudo-skin acor decreases as h /h increases (Figures 8 o ), indicaing improvemen o well producivi. Correlaing parameer F c, app = h.f c was aemped o plo pseudo-skin uncion as he are depiced b Figures and. All daa or h >= lies on he same curve (Figure ) or h / =. (h << ). Furhermore when h / >. as is shown in Figures o 5, some o daa or h > deviae rom hose o h =. Neverheles, a F c, app > all daa remain ollowing a single curve and converge o a value o.7. B observing hose igures, H plas a role in spreading hose curves or F c, app <. Because o ha, a new correlaing parameer was developed o include variable H. Two simple orm o equaions ma be used, he are polnomial and power uncions. Polnomial orm: Figure 9: Plo o Pseudo-skin Funcion a H =.5 or various value o h F F h ( H ) () c, app c Power orm: F ( H ) c, app Fch (5) Those wo ormulaion saisies condiion ha as H is ver small hen F c, app = F ch has been eercised and been observed. The second crierion is ha F c, app mus be posiive, and Equaion 5 saisies his condiion. Regresion mehod was used o obained coeicien in Equaion 5. I was ound ha =.77 wih R =.988. Hence he correlaing parameer F c, app is wrien b Equaion 6. Figure 6 implies ha he original curve proposed b Cinco e al. or pseudoskin uncion can sill be used provided F c, app = F ch ^(-.77 H ) is used replacing F c or eended heigh racured well. (.77H ) F F h (6) c, app c Figure : Plo o Pseudo-skin Funcion a H =. or various value o h Figure 8: Plo o Pseudo-skin Funcion a H =. or various value o h Figure : Plo o Pseudo-skin Funcion a H =.5 or various value o h 98

6 Inernaional Journal o Applied Engineering Research ISSN Volume, Number 5 (6) pp H=. h= h= h=5 h= Cinco aa H=.5 h= h= h=5 h= Cinco aa S+ln(/rw) S+ln(/rw). Fc,app=Fc*h Figure : Plo o Pseudo-skin Funcion a H =.5 or various value o h. Fc,app=Fc*h Figure 5: Plo o Pseudo-skin Funcion a H =.5 or various value o h using correlaing parameer F c, app = F c*h H=.5 h= h= h=5 h= Cinco aa S+ln(/rw). Fc,app=Fc*h Figure : Plo o Pseudo-skin Funcion a H =.5 or various value o h using correlaing parameer F c, app = F c*h S+ln(/rw) H=.. Fc,app=Fc*h h= h= h=5 h= Cinco aa Figure : Plo o Pseudo-skin Funcion a H =. or various value o h using correlaing parameer F c, app = F c*h Figure 6: Plo o Pseudo-skin Funcion using a new correlaing parameer F c, app= F c*h ^(-.77 H ) or various h and H Equaion 6 can be eamined urher o deermine he inluence o increasing racure heigh on he eecive (apparen) racure conducivi, ha is increasing producivi b lowering skin acor. A a given reservoir hickness and keeping he racure lengh consan, eecive racure conducivi is increased b increasing racure heigh. I is unil reaching a maimum poin a which he racure heigh is a opimum (Figure 7). Aer his poin, urher increasing racure heigh lower eecive racure conducivi and evenuall i becomes disadvanageous as F c, app/f c below. This resul is considered imporan and i was no observed b previous lieraures. This work also produce resuls wih dieren inding when racure heigh is greaer han reservoir hickness a consan racure volume. Unlike Benne s conclusion saed ha he opimum condiion or his case is a h = h, his work shows ha eecive racure conducivi can increase a h > h unil i reaches a maimum provided ha h < /.77. When h > /.77, increasing racure heigh becomes derimenal since eecive racure conducivi is less han case o h =h. 98

7 Inernaional Journal o Applied Engineering Research ISSN Volume, Number 5 (6) pp Fc,app/Fc Fc,app /F c h/ =. h/ =.5 h/ =.5 h/ =. h/ =.5 h/ =. h/ =.5 h Figure 7: Eec o racure heigh on eecive racure conducivi a various value o h/ h = h / h h = h = /.77 A = 5 sq- A = sq- A = 5 sq- A = sq- A = sq- A = sq- Figure 8: Eec o racure heigh on eecive racure conducivi wih racure volume is consan Conclusion This work presens a modiied model or eended heigh racured well, where luid enering o he well rom racure on hrough he reservoir hickness inerval. This model is considered o be more realisic compared o he laes model b Benne. This work proposes a new correlaing parameer o eecive racure conducivi or eended heigh racured well. This correlaing parameer, F c, app = F ch ^(-.77H ), enable ones o deermine pseudo-skin acor using he relaionship beween pseudo-skin uncion versus F c previousl published b Cinco e al. which hen can be used o compue well producivi. Two imporan indings are oered in his work: (i) eending racure heigh beond he reservoir hickness a consan racure lengh increases racure conducivi unil i reaches a maimum a opimum racure heigh. Eending racure heigh urher lower he racure conducivi and i becomes disadvanageous when F c, app/f c <, (ii) The increase o racure conducivi b eending racure heigh greaer han reservoir hickness is also rue or he case o consan racure volume provided ha h < /.77. When h > /.77, eecive racure conducivi is no beer han case o h =h. Reerences [] W.J. McGuire, and V.J. Sikora, The Eec o Verical Fracures on Well Producivi, Journal o Peroleum Technolog, pp. 7-7, Ocober 96. hp://d.doi.org/.8/68-g. [] M. Pras, P. Hazebroek, and W.R. Srickler, Eec o Verical Fracures on Reservoir Behavior-- Compressible-Fluid Case, Socie o Peroleum Engineers Journal, pp. 87-9, June 96. hp://d.doi.org/.8/98-pa [] M. Pras, Eec o Verical Fracures on Reservoir Behavior--Incompressible Fluid Case, Socie o Peroleum Engineers Journal, pp. -8, June 96. hp://d.doi.org/.8/575-g. [] A.C. Gringaren, H.J. Rame, Jr., and R. Raghavan, Applied Pressure Analsis or Fracured Wells, Journal o Peroleum Technolog, pp Jul 975. hp://d.doi. org/.8/596-pa. [5] L.H. Cinco, V.F. Samaniego, and A.N. ominguez, Transien Pressure Behavior or a Well wih a Finie-Conducivi Verical Fracure, Socie o Peroleum Engineers Journal, pp. 5-6, Augus 978. hp://d.doi.org/.8/6-pa. [6] K. Fisher, and N. Warpinski, Hdraulic Fracure Heigh Growh: Real aa, Journal SPE Producion & Operaions, pp. 8-9, Februar. hp://d.doi.org/.8/68-jpt [7] H. Cinco-Le, and F. Samaniego-V., Transien Pressure Analsis or Fracured Wells, Journal o Peroleum Technolog, pp , Sepember 98. hp://d.doi.org/.8/79-pa. [8] R.G. Agarwal, R.. Carer, and C.B. Pollock, Evaluaion and Perormance Predicion o Low- Permeabili Gas Wells Simulaed b Massive Hdraulic Fracuring, Journal o Peroleum Technolog, pp. 6-7, March 979. hp:// d.doi.org/.8/688-pa. [9] C.O. Benne, A.C. Renolds, R. Raghavan, and J.L. Elbel, Perormance o Finie-Conducivi, Vericall Fracured Wells in Single Laer Reservoirs, Journal SPE Formaion Evaluaion, pp. 99-, Augus 986. hp://d.doi.org/.8 /9-PA. [].W. Peaceman, Inerpreaion o Well-Block Pressures in Numerical Reservoir Simulaion wih Non-square Grid Blocks and Anisoropic Permeabili, Socie o Peroleum Engineers Journal, pp. 5-5, June 98. hp://d.doi.org/.8/58-pa. 985

PET467E-Analysis of Well Pressure Tests/2008 Spring Semester/İTÜ Midterm Examination (Duration 3:00 hours) Solutions

M. Onur 03.04.008 PET467E-Analysis of Well Pressure Tess/008 Spring Semeser/İTÜ Miderm Examinaion (Duraion 3:00 hours) Soluions Name of he Suden: Insrucions: Before saring he exam, wrie your name clearly