A Model for Water Injection Into Frac-Packed Wells

Transcription

1 A Model for Waer Injecion Ino Frac-Packed Wells Ajay Suri, SPE, and Mukul M. Sharma, SPE, Universiy of Texas a Ausin Summary Frac packs are increasingly being used for sand conrol in injecion wells in poorly consolidaed reservoirs. This compleion allows for large injecion raes and longer injecor life. Many of he large offshore developmens in he Gulf of Mexico and around he world rely on hese compleions for waerflooding and pressure mainenance. The performance of hese injecors is crucial o he economics of he projec because well inervenion laer in he life of he field is expensive and undesirable. For he firs ime, we presen a model for waer injecion in frac-packed wells. The frac pack and he formaion are plugged because of he deposiion of paricles from he injeced waer, and heir effecive permeabiliy o waer is coninuously reduced. However, as he boomhole pressure (BHP) reaches he frac-pack widening pressure, he frac-pack widh increases and a channel ha accommodaes addiional injeced paricles is creaed. Injeciviy depends on he inersiial velociy of he injeced waer in he frac pack, volume concenraion of he solids in he injeced waer, injecion rae, injecion-waer emperaure, size of proppans in he frac pack, widh and lengh of he frac pack, and he iniial minimum horizonal sress. In case of frac packs wih large proppan size and high injecion raes, he plugging of he frac pack is found o be negligible excep in he building of a filer cake a he frac-pack walls. In he case of narrow frac packs wih small proppan, significan plugging is expeced, which leads o sharp permeabiliy decline of he frac pack and a rapid rise in he BHP. The long-erm injeciviy of a frac-packed injecor depends primarily on he filraion coefficien value of he frac pack, solids concenraion in he injeced waer, and he injecion rae. Frac packs are expeced o mainain higher injeciviies compared o any oher compleions such as openhole, cased-hole, perforaed, or gravel packs. Background and Problem Descripion Injecion waer always conains suspended paricles, which can poenially plug he formaion around a well. The suspended paricles deposi on he grains and reduce he porosiy and permeabiliy of he near-wellbore region. The reduced near-wellbore permeabiliy causes a rise in he injecion pressure if a consan injecion rae is mainained. When he injecion pressure exceeds he formaion-breakdown pressure, fracures are iniiaed. Subsequenly, he face of he fracure also becomes plugged wih coninued injecion, and he pressure a he fracure ip exceeds he fracure-propagaion pressure. This causes he fracure o exend furher (Suárez-Rivera e al. 00; Saripalli e al. 999). I is well esablished ha coninued paricle plugging and fracure propagaion are coupled. In addiion, if he injeced fluid is colder han he reservoir, hermal sresses are induced, which lower he fracure-propagaion pressure and may cause hermally induced fracures wihou even plugging of he fracure. Mosly, i is he combinaion of boh of hese effecs in injecion wells ha causes injecion-induced fracures. In poorly consolidaed formaions, sand conrol mus be esablished by using prepack screens, gravel packs, or frac packs. These compleions are also suscepible o pariculae plugging. There Copyrigh 00 Sociey of Peroleum Engineers This paper (SPE 0084) was acceped for presenaion a he SPE Annual Technical Conference and Exhibiion, Anaheim, California, USA, 4 November 007, and revised for publicaion. Original manuscrip received for review Augus 007. Revised manuscrip received for review 6 Ocober 009. Paper peer approved 30 Ocober 009. are no models and lile field published on he performance of frac-packed injecors daa (McCary e al. 006). Frac packing is widely used in producion wells, bu recenly many waer injecors are using frac-pack compleion echnology also. Frac packing of injecion well: () provides a resilien and robus long-erm compleion wih minimum inervenion, especially for subsea wells; () provides sand conrol if he formaion is poorly consolidaed; (3) minimizes rae of plugging of he injecion well by providing a larger permeabiliy and flow area in comparision o perforaed or gravel-pack compleions; and (4) allows generally larger injecion raes for longer periods. McCary e al. (006) have documened he use of frac-pack compleion echnology for waer injecors in sand-conrol environmens. They observed no sand-conrol failures in frac-packed injecors for more han 5 years of injecion. Model Formulaion A model for waer injecion ino frac-packed wells is developed ha includes he following basic feaures: The pressure profile and flow field in and around a fracpacked well are calculaed using a mix of analyical (Pras 96) and numerical mehods (flow resisors in series and parallel). This flow field is coupled wih a model for paricle deposiion of solids, baceria, and/or oil paricles from he injeced waer ono he formaion grains and he proppan in he frac pack and for he corresponding permeabiliy decline. The change in minimum horizonal sress induced by he change in formaion emperaure and pore pressure from he injecion waer is calculaed as a funcion of ime. The widening of he exising frac pack is modeled in response o changes in he BHP and in-siu minimum horizonal sress over ime (ne pressure). Finally, he preceding model elemens are coupled explicily for esimaing he injeciviy of he frac-packed injecor wih ime. Deails of hese models are presened in subsequen secions. A reader primarily ineresed in he resuls may urn o he Resuls and Discussion secion direcly. In developing he model, he following assumpions are made:. The reservoir is single-layered and isoropic. No verical flow is considered wihin he layer.. The frac pack is assumed o have consan widh in he verical direcion and is exended hroughou (bu confined wihin) he enire formaion heigh. Is widh is correlaed wih pressure using he Perkins, Kern, and Nordgren (PKN) fracure model. 3. Boh he injeced fluid and he reservoir fluids are assumed incompressible (consan densiy). 4. Darcy s law (Reynold s number ) is assumed valid boh in he reservoir and in he frac pack. 5. Fluid is injeced a a consan rae ino he well, which is hen disribued ino he frac pack and he formaion according o a flow-resisance model. The flow is assumed o be bilinear (linear ino he frac pack and perpendicular and linear from he frac pack ino he formaion). 6. The displacemen of fluids is pison-like. The oil sauraion is residual in he waerflooded region and remains a he iniial oil sauraion ahead of he waerflood fron. 7. Hea conducion o overlying and underlying formaions is negleced. 8. The gravel size in he frac pack and he formaion grains are represened by a mean size. June 00 SPE Reservoir Evaluaion & Engineering 449

2 0.9 a=infiniy a= a= Flow lines Equipressure lines 0.35 a=0. a=0 Well 0 0. Frac-Pack (x/l ) 0.8 Fig. Equipressure lines jus around he frac pack for five values of a (0, 0.,, 0, and infiniy) and r e /L = An equipressure ellipse encloses he frac pack whose major and minor axes are obained from soluions presened by Pras (96). Pressure Profile Around a Frac-Packed Well. A model for pressure profile and flow disribuion in and around a frac-packed well is formulaed using a resisor model wih bilinear flow bounded by an equipressure ellipse. Pras (96) presened a seady-sae soluion for pressure disribuion around a fracured well. Equipressure lines were deermined by a parameer, a, and a dimensionless lengh, r e /L. The parameer a is slighly modified from Pras definiion for singlephase flow o wo-phase flow for waer-injecion wells: k L a = eff, () kiwi where k eff is he formaion s effecive permeabiliy o waer (k i k o ), rw L is he lengh of he frac pack (disance from he wellbore o he frac-pack ip), k i is he iniial permeabiliy of he frac pack, and w i is he iniial widh of he frac pack. Fig. shows he equipressure lines around he frac pack for five values of a (0, 0.,, 0, and infiniy) and for r e /L =.4. The exreme values for a are (a = 0), which mimics an infiniely conducive fracure, and (a = infiniy), which mimics a fracure wih zero widh. A ypical value of a will be less han for frac packs in low-permeabiliy formaions. However, in highly permeable formaions, he value of a could be significanly larger. The pressure disribuion for a > 00 will be nearly he same as ha for radial flow. For (a = 0), here is no pressure drop in he fracure and he equipressure lines away from he fracure are confocal ellipses (wih foci = L and L ). Comparison of Bilinear-Flow Model Wih Pras Soluions. Fig. a shows he assumed bilinear-flow model for he frac pack in he paper compared o he acual ellipical flow (Fig. b). This is an approximaaion o he acual flow, which is neiher bilinear nor radial. Wih he bilinear-flow assumpion, flow from he wellbore divides ino wo: () flow going ino he frac pack and () flow going ino he formaion perpendicular o he frac pack. The flow going ino he frac pack ges furher divided ino () flow perpendicular o he frac pack ofen ermed as leakoff and () flow coninuing hrough he frac pack. Fig. 3 shows a comparision beween Pras soluion and he assumed bilinear-flow model for four values of a (0, 0.,, and 0). The difference in he oal pressure drop in he fracure obained from he Pras soluion and from our bilinear flow model for (a = 0, 0., and ) was found o be < 5% of he oal pressure drop in he reservoir. For a = 0, he error was 4%. Therefore, up o a =, he assumpion of bilinear flow is quie accurae, and up o a = 0 is a reasonable engineering approximaion; above ha, he flow is essenially radial. The equipressure ellipse-axes values for he differen a values were obained for r e /L =.4 because almos no dependence on his dimensionless number was found by Friehauf e al. (009) for he minor-axes values. So, even in cases wih oher r e /L raios, he minor axis of he equipressure is sill inerpolaed from Fig.. Filraion and Permeabiliy Decline Model. The following expression shows a D mass balance of suspended paricles in an incompressible fluid flowing hrough a porous medium: d C dc d u C d + + =0, () d d dx where C is he volume of suspended paricles per uni volume of fluid injeced, u is he fluid volumeric rae per uni area, is he porosiy of he porous medium, and C d is he volume of paricles deposied per uni bulk volume of he porous medium. We assume Iwasaki s (937) empirical model for paricle deposiion, as given by dc d = Cu, (3) d where is he filraion coefficien obained in our model from a semianalyical equaion presened by Rajagopalan and Tien (976). The filraion coefficien equaion is discussed exensively by Sharma e al. (997) and Saripalli e al. (999). The suspended-paricle concenraion has been observed o become seady and exponenial wih disance for D linear-flow geomeries a consan injecion rae. This simple soluion allows us o easily apply he filraion soluion o our bilinear-flow model, boh ino and perpendicular o he frac pack. Eq. 4 shows he suspended-paricle concenraion in he frac pack. I is assumed ha he superficial/darcy velociies 450 June 00 SPE Reservoir Evaluaion & Engineering

3 (a) (b) Fig. (a) Top view of equipressure lines and flow direcion around a frac-packed injecor, (b) equivalen recangular wellbore wih bilinear-flow assumpion inside he inner equipressure ellipse. in he formaion and in he frac pack do no change significanly wih disance from he well because in deriving he simple exponenial soluion for filraion, linear flow wih consan superficial velociies was assumed. C = x Ciexp( gx ), (4) fracpack where C i is he concenraion of suspended paricles in he well, g is he filraion coefficien for he proppan in he frac pack, and x is he disance in he frac pack from he wellbore face. The formaion around he wellbore and adjacen o he frac pack is also damaged because of paricle deposiion. The leakoff from he frac pack ino he formaion causes paricle deposiion near he frac-pack faces, while he flow from he well direcly ino he formaion causes damage o ha region. The concenraion of paricles in he formaion adjacen o he frac pack and he wellbore is given as C, = C exp( y), (5) xy formaion xfracpack f Normalized ΔP from well Pras Soluion Bilinear Model a=0 0 3 Dimensionless disance from he well in he fracure direcion (x/lf) Normalized ΔP from well a=0. Pras Soluion Bilinear Model 0 3 Dimensionless disance from he well in he fracure direcion (x/lf) Normalized ΔP from well a= Pras Soluion Bilinear Model 0 3 Dimensionless disance from he well in he fracure direcion (x/lf) Normalized ΔP from well a=0 Pras Soluion Bilinear Model 0 3 Dimensionless disance from he well in he fracure direcion (x/lf) Fig. 3 Comparison of resuls obained from he Pras (96) ellipical model for fracured wells and from he bilinear-flow model. June 00 SPE Reservoir Evaluaion & Engineering 45

4 Fig. 4 Concenraion profile of suspended paricles in he frac pack and adjacen formaion. where C x is given by Eq. 4, f is he filraion coefficien of he formaion, and y is he disance of he formaion perpendicular o he frac pack. The disance x is se o 0 in Eq. 4 for calculaing concenraion of suspended paricles in he formaion jus around he wellbore. For he formaion adjacen o he frac pack, he concenraion of suspended paricles in he frac pack (a a disance x away from he wellbore) is he inle boundary condiion for he paricles suspended in he formaion a a disance y. The paricle concenraion ahead of he frac-pack ip is also calculaed using Eq. 5, where y is hen he disance beween he frac-pack ip and he locaion in he formaion ahead of he frac-pack ip. Fig. 4 presens a schemaic of he concenraion profile of he suspended paricles in he frac pack and he formaion adjacen o i. I also shows a schemaic of filraion of paricles in a porous medium wih an inernal and exernal filer cake. Because he frac pack is filled wih proppan ha have larger pore-hroa radii han he formaion, i has a very low filraion coefficien. This means ha he paricle-rapping capaciy of he frac pack is much smaller han ha of he formaion sand and injeced paricles can ravel relaively far along he lengh of he frac pack. The filraion coefficien for he formaion, however, will be large because of he smaller formaion-grain size. Therefore, he paricles will be rapped wihin a relaively shor disance (inches) in he formaion adjacen o he well and he frac pack. Permeabiliy-Reducion Model. The deposied paricles will reduce he porosiy and permeabiliy of he proppan pack and he formaion. The reducion in permeabiliy is calculaed from an empirical equaion proposed by Sharma e al. (000). They calculaed he reduced permeabiliy because of paricle deposiion hrough hree erms of () reduced porosiy, () increased surface area, and (3) increased oruosiy, as follows: k / k0 = kdpkdskd, (6) where k is he reduced permeabiliy because of damage caused by deposied paricles and k i he iniial undamaged permeabiliy, k dp is he decline associaed wih decrease in porosiy, k ds is he decline associaed wih he increase in surface area, and k d is he decline associaed wih increased oruosiy. We have assumed k d o be equal o zero in our model. The reduced porosiy a any ime is given as = 0 C d. The equaions for C d, he deposied-paricle concenraion in he frac pack and he formaion wih ime, are presened in Appendix A. When he porosiy a he formaion/wellbore inerface, formaion/ frac-pack inerface, and/or frac-pack/wellbore inerface reaches a criical porosiy (defined as he produc of iniial porosiy and he filer-cake porosiy) no more paricles ener hrough ha inerface and an exernal filer cake is assumed o sar forming. This ransiion-ime model is discussed horoughly and applied o decline in injecion wells by Sharma and Wennberg (997). Exernal-Filer-Cake Porosiy and Permeabiliy. The packing of equal-sized spheres can ideally be arranged in cubic, hexagonal, and rhombohedral paerns. The corresponding porosiies for hese packings are 0.48, 0.4, and 0.6. However, because of he nonuniform size of paricles in he acual filer cake ha would form from he solids in he injecion waer and he compressible naure of he filer cake, he porosiy could be much less han 0.6. Khaib (994) has presened he porosiy and permeabiliy of filer cakes using a compression-permeabiliy cell as a funcion of applied pressure, solids ype, and he presence of oil. The solids invesigaed were FeS (5 m), Fe (OH) 3 ( m), CaCO 3 ( m), CaSO 4 (5 m), produced sil, and clay (6 m). Empirical correlaions beween permeabiliy and porosiy were presened ha showed ha he permeabiliy of he filer cake could range beween and md depending on he porosiy, overbalance, and solid ype. The resuls presened in his paper assume a consan porosiy and permeabiliy of he filer cake. Flow-Resisance Model. A flow-resisance model is used o calculae he pressure difference beween he equipressure ellipse around he frac pack and he wellbore. The flow resisances are calculaed assuming he bilinear-flow field around he frac pack. Fig. 5 shows a discreized half of he horizonal plane beween he well and he equipressure ellipse around he frac pack. The frac pack is discreized ino equal segmens along is lengh. The formaion near he well and he frac pack is discreized ino equal bu much smaller segmens for capuring permeabiliy decline, which occurs in he formaion adjacen o he wellbore and he frac pack. By definiion, he flow resisance is defined as he raio of differenial pressure o he flow rae across a permeable medium and is given as P R = (7) q The flow resisance can be calculaed easily using Darcy s equaion for linear flow, as follows: ( x) x Rx ( ) = d, (8) keff ( x) A 45 June 00 SPE Reservoir Evaluaion & Engineering

5 Fig. 5 Discreized frac pack and he reservoir wihin he equipressure ellipse and bilinear-flow resisors. where A is he cross-secional area, (x) is he viscosiy of he injeced fluid ha can be a funcion of disance x, and k eff (x) is he effecive permeabiliy. Noe ha he Reynold s number for ypical flow raes in he frac pack can be on he order of 00. This implies ha non-darcy flow in he frac pack can be imporan and will resul in an addiional flow resisance along he frac pack. To include non- Darcy effecs, he Forchheimer equaion should be used insead of Darcy s equaion o calculae he flow resisance in he frac pack. All he flow resisances for all he resisors in Fig. 5 are based on Eq. 8. The deails of each of he flow resisances (along he frac pack, perpendicular o he frac pack, and perpendicular o he wellbore) up o he equipressure ellipse are discussed in deail nex. Flow Resisance Along he Frac Pack. The flow resisance of a frac-pack segmen beween x and x+ x disance from he wellbore face is calculaed as w Tw x R = ( ), (9) i k (,) i wih (,) where w(i,)h is he area of he ih frac-pack segmen for flow in he frac pack, i = L /N s, and N s is he oal number of frac-pack segmens. Noe ha he facor is used o include he wo sides of he frac pack. x is he lengh of he frac-pack segmen. A =0, k (i,) = k i and w (i,) = w i. From >0 unil he frac-pack segmen widens, k (i,) is calculaed from Eq. 6 wih C d obained from Eq. A-and A- in he Appendices. Noe ha Eq. A- needs he flow rae in he frac pack o calculae he reduced permeabiliy. This is explicily provided from he previous-imesep flow rae. Once he frac pack sars o widen (discussed in he subsecion Frac-Pack Widening and Channeling), Eqs. 30 and 36 are used for k (i,) and w (i,). Similarly, he flow resisance of any exernal filer cake formed a he well/frac-pack inerface is calculaed as R = ( T ) h cake kc w w w c i, (0) h where h c is obained from Eq. A-3 and k c is he permeabiliy of he exernal filer cake, assumed o be a consan. We have assumed ha if he frac pack widens (up o a lengh defined as he channel lengh), he injeced-paricles concenraion becomes equal o C i all along he channel lengh in he frac pack and Eqs. 4 and 5 are modified accordingly for calculaing he suspended-paricle concenraion for boh he frac pack and he adjacen formaion. Flow Resisance Perpendicular o he Frac Pack. The flow resisance perpendicular o he frac pack (i.e., leakoff) shown in Fig. 5 is calculaed as a sum of he following: () any exernal filer cake a he frac-pack/formaion inerface, () damaged formaion wih reduced permeabiliy adjacen o he frac-pack face, and (3) undamaged formaion up o he equipressure ellipse around he frac pack. The hree disinc regions in he reservoir around he frac pack are modeled as. Cool injecion zone: he region closes o he wellbore and he frac pack wih only injecion waer as mobile phase, oil a residual sauraion (S w = S or and S o =S or ), and emperaure equal o he injeced-waer emperaure. The moving fron of his zone is known as cool/hermal fron.. Warm injecion zone: he region ahead of he cool fron, wih boh injecion waer and he displaced connae waer as mobile phase and oil a residual sauraion S w = S or and S o =S or, bu a he iniial reservoir emperaure. The moving fron of his zone is known as waerflood fron. 3. Oil zone: he region ahead of he waerflood fron in which boh oil and waer are mobile, wih S wi and S o equal o he iniial fluid sauraions and emperaure equal o he iniial reservoir emperaure. Fig. 6 shows confocal ellipses wih he waerflood and hermal frons. Each of hese five flow resisances perpendicular o he ih frac-pack segmen up o he equipressure ellipse are summed and calculaed as: R ( T ) h ( i) w( Tw) y = + k ( 4h x) ( 4h x) j = k (, i j,) o w w co i co Ns { } w( Tw) bc i ( Ns ) + o ( 4h x) krwki { + } 05. w( TR) ( bwf bc ) i ( Ns ) + o ( 4h x) krwki { + } 05. o( TR) ( b bwf) i ( Ns ) +. o ( 4h x) k k f ro i () June 00 SPE Reservoir Evaluaion & Engineering 453

6 T R S wi + S o = T R T w S w + S or = S w + S or = P ~ P ip Thermal fron Waerflood fron Drainage boundary (P R ) Fig. 6 A schemaic of hermal and waerflood fron as confocal ellipses afer hey have crossed he equipressure ellipse. The subscrip i denoes he ih frac-pack segmen. The produc (4h x) is he area of frac-pack segmens for flow perpendicular o he four faces of he frac pack, and h co and k co are he hickness and he permeabiliy, respecively, of he exernal cake formed a he face of he ih segmen of he frac pack and are nonzero only afer he inernal filraion is complee; oherwise, hey are se o zero. The hickness of any exernal filer cake formed a he inerface beween he frac pack and he formaion is calculaed using Eq. A-6 in he Appendix A. The flow resisance is calculaed for i = o N s + (i.e., leakoff from all he frac-pack segmens including he frac-pack ip). Noe ha he flow resisance from he ip (i = N s +) up o he equipressure ellipse is in he frac-pack direcion and is calculaed by changing he segmen area (4h x) in Eq. and Eq. A-6 by (wh). Eq. assumes b wf < b (i.e., boh he hermal and waerflood frons are inside he equipressure ellipse around he frac pack). For he case of b wf b (i.e., afer he waerflood fron has crossed he equipressure ellipse around he frac pack), b wf is made equal o b in Eq., which makes he las erm equal o zero. This is done because Eq. is calculaing he flow resisance only up o he equipressure ellipse around he frac pack. The flow resisance ahead of he equipressure ellipse is discussed in secion Pressure Profile in he Reservoir and he Frac Pack. When b c b (i.e., when he cool fron has also crossed he equipressure ellipse), b c is also made equal o b, and he las wo erms become zero. Noe he damaged formaion flow resisance (second erm) also has N s discreized segmens wih oal damage lengh equal o N s y. The k f (i,j,) is calculaed where j refers o he number of segmen perpendicular o he frac pack and i refers o he number of he fracpack segmens. Eq. 6 is used o calculae he reduced permeabiliy wih C d obained from Eq. A-4 and A-5 in he Appendix A. We also assume ha he damage lengh is negligible compared o he hermal-fron lengh from he frac pack. The hird erm (referred o as he hermal zone) has injeced waer wih viscosiy calculaed a injecion-waer emperaure w (T w ), wih b c as he minor axis of he hermal fron (discussed in Appendix C). The fourh erm is he flow resisance beween he hermal fron and he waerflood fron. The injeced-waer viscosiy in his zone is calculaed a reservoir emperaure w (T R ), and b wf is he minor axis of he waerflood fron (Appendix C). The fifh erm is he flow resisance beween he waerflood fron and he equipressure ellipse for when he waerflood fron has no crossed he equipressure ellipse. The major flowing phase is oil wih viscosiy a reservoir emperaure o (T R ), wih b as he minor axis of he equipressure ellipse. Flow Resisance Beween Wellbore and he Equipressure Ellipse. The flow resisance beween he wellbore and he equipressure ellipse is given by R w s w( Tw) hcw w( Tw) y = + k ( r h) ( r h) k ( 0, j, ) cw w w N j= w( Tw) bc w( TR)( bwf bc ) + + o o, ( rh w ) krwki ( rwh) krwki o( TR)( b bwf ) + o ( rh w ) krok i f () where ( rw h) is he cross-secional area of he wellbore for flow in he direcion perpendicular o he frac pack (assuming ha he widh of he frac-pack flow area a he wellbore is negligible compared o he flow area of he wellbore). The res of he erms are similar o hose defined for Eq.. The firs erm is for he flow resisance caused by any exernal filer cake formed a he wellbore, wih h cw and k cw as he hickness and he permeabiliy, respecively of he exernal cake formed. h cw is calculaed from Eq. A-9. The second erm is he flow resisance of he near-wellbore region, wih reduced permeabiliy up o he damage lengh; k f (0, j, ) 454 June 00 SPE Reservoir Evaluaion & Engineering

7 is he permeabiliy of he formaion near he wellbore in he direcion perpendicular o he frac pack (calculaed using Eq. 6 wih C d obained from Eq. A-7 wih he condiional equaion given by A-8). The hird, fourh, and fifh erms become zero once he corresponding fron crosses he equipressure ellipse (as described in Eq. ). Toal Flow Resisance up o he Equipressure Ellipse. All he flow resisances as shown in Fig. 5 are calculaed a each imesep using Eqs. 9, 0,, and. The flow resisances mosly increase because of permeabiliy reducion and any exernal filer cake bu may also reduce if he mobiliy raio of he injeced waer o oil is >>. The oal flow resisance (R T ) beween he wellbore and he equipressure ellipse is calculaed as wo flow resisances in parallel: () an equivalen flow resisance (R eq ) by he frac pack and is adjacen formaion and () flow resisance by he formaion around he wellbore (R w ) up o he equipressure ellipse. The equivalen flow resisance of he frac pack and is adjacen formaion (R eq ) is calculaed by ieraing Eq. 3 as follows: (sar wih i=ns, reduce i by, unil i = ): = + R R + R R where R eq i eq i + i o i, (3) = R o (4) Ns eq Ns+ + R o and R are calculaed using Eqs. 9 and. To R eq i=, we add any exernal-filer-cake flow resisance from Eq. 0, as follows: R = R + R eq eq i = cake (5) Finally, he oal flow resisance (R T ) beween he wellbore and he equipressure ellipse is calculaed by adding he well-flow resisance (R w ) and he frac-pack-flow resisance (R eq ) in parallel: = (6) RT Rw Req Bilinear-Flow Disribuion Model. The oal injecion rae is also divided ino wo: () flow rae from he well ino he formaion (Q w ) and () flow rae from he well ino he frac pack (Q ): Q Q R T w = (7) T R w Q Q R T =, (8) T Req where Q T is he oal injecion rae ino he well (sum of he wo flow raes, Q w and Q ). The flow rae perpendicular o he frac pack (leakoff) is also calculaed for all he segmens, as follows: Q Q R eq o = i, (9) i i R o i where Q i= is he oal flow rae enering ino boh sides of he firs frac-pack segmen (from he wellbore) and is equal o Q (Eq. 8). Q o is he oal flow rae leaking off from a frac-pack segmen (where subscrip i denoes he segmen number). Noe ha he leakoff rae includes boh he frac-pack wings (i.e., all four fracpack faces). The subscrip i is varied from o N s + in Eq. 9 o calculae he leakoff raes from all he frac-pack segmens. Noe i = N s + denoes he leakoff rae from he frac-pack ip along he frac-pack direcion differen from he leakoff rae in he direcion perpendicular o he las frac-pack segmen (N s ). The flow rae ino any subsequen frac-pack segmen is equal o he flow rae in he previous segmen minus he leakoff rae from ha segmen and is calculaed as follows: Q = Q Q + i i o i where subscrip i is varied from o N s., (0) Pressure Profile in he Reservoir and he Frac Pack. The P beween he well and he equipressure ellipse around he frac pack is calculaed as follows: P = Q R () ep T T The BHP in he well is calculaed as follows: P = P + P+ P + P3 + P, () wf R ep where P, P, and P 3 are pressure differences beween he reservoir boundary and he equipressure ellipse wih he hree disinc regions. The following hree cases arise: () A early injecion ime, boh he waerflood fron and he hermal fron are wihin he equipressure ellipse around he frac pack (i.e., he minor axis of he waerflood fron), b wf < b (he minor axis of he equipressure ellipse). In his case, he approximae pressure difference beween he reservoir boundary and he equipressure ellipse is calculaed as follows: q o r e P = ln o, (3) kk i roh a + b wf and P and P 3 are equal o zero. () A some laer ime, he waerflood fron crosses he equipressure ellipse (i.e., b wf > b ) bu he hermal fron sill remains wihin he equipressure ellipse, (i.e., b c < b ). For his case, P and P are calculaed as follows: q o r e P = ln o, (4) kk i roh a + b wf wf q a b w wf + wf P = ln o, (5) kk i rwh ac + b and P 3 is equal o zero. (3) Finally boh he waerflood fron and he cool fron cross he equipressure ellipse (i.e., boh b wf and b c are > b ). For his case, P is he same as Eq. 4, while P and P 3 are calculaed as follows: q a b wr wf + wf P = ln o kk h a + b, (6) i rw c c q w ac + b c P3 = ln o (7) kk i rwh l + b The pressure a he frac-pack ip is also calculaed as follows: PR P P Pip = P + R 3 o Ns+ Q o Ns +, (8) where P and P 3 values would depend upon he fron s locaion, as described before. The pressure in any frac-pack segmen is also calculaed as follows: June 00 SPE Reservoir Evaluaion & Engineering 455

8 Widening pressure a well Noe: The filraion coefficiens are of he frac-pack. /m 5 /m /m 0. /m 0.0 /m 0.00 /m Fig. 7 Rise in BHP as a funcion of differen filraion coefficiens for he frac pack. P = Pip + R Q (9) i Ns i= i i i Frac-Pack Widening and Channeling. Fig. 7 is a ypical plo showing he increase in BHP in a frac-packed well for a range of filraion coefficiens. For small filraion coefficiens, he rise in BHP is gradual, while for large filraion coefficiens, he rise in BHP is seep. Noe ha Fig. 7 has a decreasing frac-pack widening pressure wih ime required o widen he frac-pack because of reducion in he in-siu sresses caused by cold-waer injecion. Noe ha he assumed iniial widh of he frac pack is creaed by high fluid pressures generaed during hydraulic fracuring. The pressure in he frac pack afer hydraulic fracuring sabilizes back o he iniial reservoir pressure, and he frac-pack widh is mainained by he proppans placed in he fracure. The injecion of waer causes an increase in he BHP which reduces he sress (minimum horizonal sress) on he proppan. However, once he fluid pressure in he frac pack increases o a criical pressure, he frac pack can furher widen from is iniial widh. The widh pressure relaion in he frac pack afer he pressure in i has reached a criical pressure is assumed o follow he well-known relaion of a D PKN consan-heigh fracure. The maximum widh of a PKN fracure is given by (Schecher 99): * [ P( x, ) h,min ( )]( ) h wx (, ) =, (30) E where P is he fluid pressure in he frac pack, ν is Poisson s raio, E is Young s modulus, h is he frac-pack heigh, and hmin is he minimum horizonal sress perpendicular o he frac pack. In he PKN model, he srain is assumed o be zero along he lengh of he frac pack. We undersand ha he widh/pressure relaion is an approximae soluion for an acual 3D fracure, bu we believe ha i is a good engineering approximaion. By rearranging Eq. 30, he criical pressure beyond which furher widening of he frac pack will occur is calculaed as follows: w E P * i ( x, ) = h,min ( ) +, (3) ( v ) h where w i is he iniial frac-pack widh. Coninued paricle deposiion in he frac pack will cause he pressure in he frac pack o keep rising and may lead o fracure widhs larger han he original frac pack. This widening of he frac pack will resul in a high-permeabiliy channel hrough he proppan pack. The lengh of he channel is he disance up o which he widening has aken place. The channel widh is he addiional widh in a frac-pack segmen compared o he iniial widh and is calculaed as follows: * [ P( x, ) P( x, )]( ) h wx (, ) wi = (3) E The permeabiliy of he channel in he widened frac pack is calculaed assuming slo flow as follows: k ch ( x, ) = { } wx (, ) max hco( x, ), w i (33) The permeabiliy of a frac-pack segmen is affeced by: () any filer cake forming a he face of he frac pack, () permeabiliy reducion of he proppan pack caused by deposiion of paricles, and (3) permeabiliy enhancemen caused by any widening of he segmen if he pressure is above he iniial fracuring pressure. The gravel/proppan permeabiliy iniially for all he frac-pack segmens is he same as he iniial frac-pack permeabiliy, k g (x, 0)=k i. However he gravel/proppan permeabiliy is recalculaed every imesep from Eqs. 6 and 35. Wih ime, h co will increase in every segmen wih a simulaaneous increase in P. The gravel permeabiliy in a frac-pack segmen is also affeced by any formaion of he filer cake a he frac-pack face, given by Eq. A-6, and is recalculaed a every imesep as follows: k g ( x, )= { kg ( x, ) wi hco ( x, ) + kco hco x, },.... (34) w i ( ) if h ( x, ) w hen k ( x, ) = k o (35) co i g c Depending on he exernal-filer-cake hickness a he face of he frac-pack segmen, widh of he frac pack in he previous imesep, and he reduced permeabiliy of he proppan pack, he effecive permeabiliy of he segmen can be calculaed: k { i } kg( x, ) wi + kch ( x, ) w( x, ) max hco( x, ), w ( x, ) = wx (, ) (36) The w(x,) and k (x,) are ieraed unil Darcy s flow equaion is also saisfied wih flow rae Q assumed equal o he previousimesep flow rae. P wq( x, ) ( x, ) = k x w x h x + P ( x+ x,) (37) (, ) (, ) eff. 456 June 00 SPE Reservoir Evaluaion & Engineering

9 Fig. 8 A op view of he widened frac pack wih a high-permeabiliy channel. Channel permeabiliy (Darcy) Channel permeabiliy Channel widh Δp above criical pressure for frac-pack widening (psi) Fig. 9 Channel widh and permeabiliy as a funcion of pressure in he frac pack above he criical pressure. The pressure in he segmen ahead of he channel P (x+ x), where x=channel lengh, is calculaed from Eqs. 36 and 37 wih w(x,)=w i because ha segmen does no need o saisfy he widh/pressure relaion (Eq. 3). However he frac-pack segmens ha have pressure above he criical widening pressure have o saisfy Eq. 3 wih w(x,) as an addiional unknown. All hree unknowns (pressure, widh, and permeabiliy) are calculaed ieraively up o he channel ip by simulaneously saisfying Eqs. 3, 36, and 37. Fig. 8 shows a schemaic of he widened frac pack wih pressures, widhs, and he permeabiliies of he frac-pack segmens. Fig. 9 shows a plo of channel/slo permeabiliy and slo widh as a funcion of pressure above he criical pressure for h=00 f, E=4E+6 psi, and ν=0.8. Frac-Pack Lenghening. The frac pack will exend in lengh if he pressure a he ip becomes larger han he fracure-propagaion pressure obained from Perkins and Gonzalez (985): Channel widh (mm) UE Pfracure = h +,min, (38) ( ) h where P fracure is he fracure-propagaion pressure needed a he ip of he frac pack for frac-pack exension, h,min is he Earh s minimum sress perpendicular o he frac-pack faces, and U is he specific surface energy. The exension of he frac pack can occur eiher because of a large reducion in he minimum horizonal sress because of injecion of cold waer (discussed in Appendix B) or because of rapid plugging and channeling in he frac pack. Wheher he frac pack will grow in lengh will depend on he pressure a he ip of he frac pack (Eq. 8). If P ip becomes > P fracure, hen he frac pack will exend in lengh. Well Injeciviy The model presened in he paper is implemened in an injecionwell simulaor. The well injeciviy, I, is a performance indicaor for waer-injecion wells and is defined as QT I =, (39) Pwf PR where Q T is he oal injecion rae a he well boomhole, P wf is he boomhole flowing pressure, and P R is average reservoir pressure a a circular boundary (r e ). Resuls and Discussion Table shows inpu parameers for hree simple case sudies wih injecion in () A frac-packed well wih significan solids rapping in he frac-pack modeled wih filraion-coefficien-values esimaes from he Rajagopalan and Tien (976) model wih no correcion for high flow velociies. () A frac-packed well wih very lile solids rapping in he frac-pack modeled wih filraion-coefficien-values esimaes from he Rajagopalan and Tien (976) model wih correcion a for high TABLE INPUT PARAMETERS FOR SIMULATION OF WATER INJECTION INTO A FRAC-PACKED AND AN UNFRACTURED WELL k i = k h 00 md S wi 0.5 r e 500 m S or 0.48 r w 3.6 in. k rw o Q T 5,000 bpd k ro o dp 5 µm µ w cp C i 5 ppm µ o cp h 00 f φ f 0. P R,00 psi φ 0.38 hmin 4,000 psi φ c k c 0.00 md E psi d formaion grains 00 µm T R = T w 50 F d gravels 500 µm k i = k g (=0) 500 Darcy w i 5.08 mm L 00 f June 00 SPE Reservoir Evaluaion & Engineering 457

10 Filraion coefficien (/cm) Experimenal Model Mismach for velociies in he fracpack Darcy velociy (cm/s) Fig. 0 Comparison of filraion-coefficien values obained from model (Rajagopalan and Tien 976) and experimens for a range of Darcy velociies. Noe he deviaion a higher velociies (model predicing higher values). flow velociies from experimens and values abulaed by Maroudas e al. (965a, 965b). (3) A well ha is iniially unfracured (openhole or cased-hole/ perforaed/gravel-packed compleion). All he cases assume an injecion rae of 5,000 BWPD wih 5 ppm of solids. The reservoir in boh cases is a single-layered reservoir wih k i = k h = 00 md and wih k o rw = 0.. The lengh of he frac pack is assumed o be 00 f, iniial widh is 0. in., diameer of proppan is.5 mm, and frac-pack permeabiliy is 500 darcies. The a value for he frac-packed well is calculaed using Eq. and is equal o 0.75, which is wihin he range where he bilinear-flow approximaion is valid. Noe ha he corresponding fracure conduciviy, F CD (used in fracuring lieraure), is equal o 0.4, which is considered low. This is primarily because he formaion permeabiliy is relaively high (00 md) compared wih hose encounered in ypical hydraulic-fracuring applicaions bu is ypical for poorly consolidaed formaions. The injecion-waer emperaure is assumed o be equal o he reservoir emperaure for simpliciy (i.e., no hermal sresses are considered in hese examples). The filraion coefficien in he frac pack, g, deermines he degree of paricle reenion in he frac pack. When using an esimae of g based on correlaions derived from he filraion lieraure [Rajagopalan and Tien (976)] a value of he filraion coefficien for he frac pack is calculaed o be approximaely 0. /m. However, he acual value could be much smaller han his esimaed value because of he large flow velociies in he frac pack. The model maches beer wih he filraion experimens a Fig. Widh of he frac pack wih uncorreced model filraion coefficien values for high flow velociies. Fig. Pressure profile in he frac pack wih uncorreced model filraion-coefficien values for high flow velociies. low Darcy velociies wih a maximum of up o cm/s and sars o deviae significanly a higher Darcy velociies (Fig. 0). The examples presened here have flow velociies higher han cm/s in he frac pack, wih a value 3.9 cm/s a he frac-pack enrance from he well. Resuls Using Filraion Coefficiens Uncorreced for High Flow Velociies in he Frac Pack. Fig. shows how he pressure profile in he frac pack varies wih ime using he filraion-coefficien values obained from he Rajagopalan and Tien (976) empirical correlaions and wihou correcing hem for high flow velociies. A pressure drop of 73 psi is compued along he frac pack from he wellbore o he frac-pack ip on Day 0 since he frac pack is clean iniially (paricle deposiion has no sared). As injecion proceeds, he pressure in he frac pack near he wellbore rises because of paricle deposiion in he frac pack. If no correcion for he high flow velociies in he frac pack is applied, he frac-pack permeabiliy decreases from 500 darcies o approximaely 0 darcies because of coninued deposiion of paricles. As he permeabiliy of he frac pack decreases, he pressure in he frac pack quickly reaches he pressure needed for widening i beyond is iniial widh (in approximaely 70 days). The widening of he frac pack a pressures above 4,378 psi resuls in a channel wihin he frac pack, which does no allow he permeabiliy of he frac pack o be reduced much furher. As he pressure a he channel ip rises above he fracure widening pressure, he frac pack widens furher and he high-permeabiliy channel propagaes oward he frac-pack ip. Fig. shows he widh of he frac pack as a funcion of ime and disance up o he frac-pack ip. The frac pack widens and becomes ellipical in shape wih an exernal filer cake building up a he faces of he frac pack wih a higher hickness near he well. Noe ha even hough he frac-pack widh is increasing wih ime, he ne permeabiliy of he frac pack wih he channel may be less han he iniial frac-pack permeabiliy. This is caused by he combined effec of () reduced proppan permeabiliy because of deposied injeced paricles and () buildup of an exernal filer cake wih very low permeabiliy a he faces of he frac pack. This filer cake may even fill up he addiional channel creaed because of he widening of he frac pack. The ne permeabiliy of he frac pack, wih proppans wih deposied paricles, filer cake a he frac-pack walls, and any channel creaed in he widened frac-pack region, is calculaed using Eq. 36. The pressure is found o increase quickly, wihin 00 days. This example provides one limiing case in which paricles are reained in he frac pack. Resuls Using Filraion Coefficiens Correced for High Flow Velociies in he Frac Pack. Maroudas e al. (965a) conduced experimens a high flow velociies and showed ha above a cerain inersiial velociy here is virually no paricle reenion (very low values of g ). They conduced filraion experimens using dilue suspensions of spherical and angular paricles (0 00 μm) in 458 June 00 SPE Reservoir Evaluaion & Engineering

11 Criical inersiial velociy (cm/s) V cr ( dg dp) ( dg d p ).74 ( ) ( ) = e <= 00 V = d d d d > 00 cr g p g p dg/dp Fig. 3 Criical inersiial velociies (above which here is no paricle deposiion) as a funcion dg/dp (raio of gravel diameer o mean injeced-paricle diameer). hree differen ypes of filer media: a ransparen D model bed, a packed bed of glass spheres, and a sinered disc. Two differen modes of channel obsrucions were observed: a gradual consricion for he spherical paricles or a rapid blocking for he angular paricles. The runs erminaed in wo differen saes: () complee blocking leading o cake formaion or () nonreenion of paricles. The laer was observed o occur when he inersiial velociy exceeded a criical value above which paricle deposiion ceased. Sufficien quaniaive measuremens were made wih packed spheres ha closely approximae a frac pack. The packed bed consised of a.5-cm inside diameer by 0-cm-long Perspex /Plexiglas (a ransparen hermoplasic acrylic resin used insead of glass) ube ighly filled wih -mm glass spheres, vibraion packed o a consisen porosiy of The pore-hroa diameers were 300 μm wih angular quarz paricles wih wo sizes (<0 μm and <50 μm) and polysyrene paricles wih size of 65 5 μm. The average inersiial velociies were varied beween 0.5 and 36 cm/s. The experimenal filraion-coefficien values were compared wih he filraion-coefficien values generaed from he Rajagopalan and Tien (976) model ha is used in he model here, and i was found ha he experimenal filraion coefficien values were much smaller (nearly an order of magniude) han he model values. Moreover, he model always yielded finie filraion-coefficien values even a high inersiial flow velociies as agains he experimens, which sugges a value of zero for he filraion coefficien a inersiial velociies above a criical value. We have modified he Rajagopalan and Tien filraion-coefficien model o accoun for less reenion or no reenion of paricles a high inersiial flow velociies. The granular-bed experimens conduced by Maroudas very well represen waer injecion ino frac-packed wells because he gravel size used in frac packing is usually beween and mm compared o mm glass spheres used in he experimens. The solids in an injecion well usually size beween 0 and 00 μm, which is comparable o paricles size of 0 and 5 μm used in he granular-bed experimens. The inersiial velociies in a frac-packed well range beween 0 and 30 cm/s, which is comparable o experimenal inersiial velociy range of cm/s. However, here were wo major differences beween he experimenal seup and a ypical frac-packed well: () he paricle concenraions in he experimens were beween,000 and,000 ppm while in a ypical injecion well he waer qualiy is usually kep beween 00 ppm, and () he run ime for he experimens was on he order of hours while a ypical injecion well runs for years. Even wih hese wo differences, Maroudas experimens are he bes ha are available o mimic injecion ino frac packs. Addiional experimenal work needs o be conduced in his area in he fuure. Fig. 3 shows he criical inersiial velociies obained from Maroudas e al. (965a) for granular-bed experimens. Two empirical equaions (shown on Fig. 3) are used o fi he daa. Five d g /d p daa ses wih repeaed experimens resuled in a plo of criical inersiial velociies shown in Fig. 3. For lower d g /d p values, a higher criical inersiial velociy is needed beyond which here would be no paricle reenion (e.g., for d p = 0 μm and d g = 000 μm, V cr = cm/s). Therefore, a inersiial velociies higher han cm/s here will be no reenion of paricles beween he proppans of he frac pack if ha is he case. However, because he formaion grains will mosly be much smaller han 000 μm, he injeced paricles will be reained a he frac-pack walls and build a filer cake ha will grow ino he frac pack and will evenually fill up he frac pack. Table shows he acual daa in abular form wih experimenal filraion-coefficien values a velociies lower han criical velociies compared agains model filraion-coefficien values. The model filraion-coefficien values are nearly one order of magniude higher han he experimenal values. Therefore, even a velociies lower han criical velociies a correcion is needed o he filraion-coefficien model. This correlaion was developed on he basis of he experimenal daa. Simulaion resuls are shown here for boh he smaller and he larger value of he frac-pack filraion TABLE CRITICAL INTERSTITIAL VELOCITIES BEYOND WHICH THERE IS NO RETENTION OF INJECTED PARTICLES (MAROUDAS ET AL. 964, 965) Type of Porous Media Run No. Paricle ype d p (microns) Porosiy C i (ppm) d g (m) Darcy velociy (cm/s) λ experimen (/cm) λ model (/cm) d g/d p V cr (cm/s) 48 Angular quarz Angular quarz Angular quarz Angular quarz Angular polysrene Granular Bed 35 Angular polysrene Angular polysrene Angular polysrene Angular polysrene Angular polysrene Spherical 47 polysrene June 00 SPE Reservoir Evaluaion & Engineering 459

12 Flowrae in he frac-pack (bpd) Fig. 4 Pressure profile in he frac pack wih correced filraion coefficien for high flow velociies in he frac pack. Disance in he frac-pack from he well (f) Fig. 5 Flow rae along and in he frac pack wih injecion days. coefficien. For high-rae injecors, i is expeced ha he smaller value of he filraion coefficien will be more represenaive. A velociy-correcion facor (VCF) is developed o improve he filraion coefficien and is shown as follows: VCF = ( Vc u/ )/ Vc u V c, (40) VCF = 0 u > V c, (4) c = VCF, (4) where VCF is he velociy-correcion facor, is he iniial filraion coefficien calculaed from he Rajagopalan and Tien (976) model, c is he correced filraion coefficien, V c is he criical velociy beyond which here is no paricle rapping, and u is he Darcy velociy. Using he correced filraion-coefficien values, he example was rerun and he resuls were generaed using he waer-injecion model presened for he frac-packed wells. Fig. 4 shows he pressure profile in he frac pack as a funcion of injecion ime. The pressure rises much more slowly wih he correced filraion coefficien compared wih he previous resuls because he correced filraion-coefficien values were zero a mos locaions up o he frac-pack ip because he inersiial velociies were higher han criical velociy for he example wih d g /d p = 300. Even afer,000 days of injecion, he pressure in he frac pack is below he minimum horizonal sress of 4,000 psi. This resul is significanly differen from he previous resuls where wihin 00 days he pressure in he frac pack rose o 4,000 psi. The difference in he resuls shows he imporance of esimaing a correc filraion-coefficien value for he frac packs. Fig. 5 shows he flow rae in he frac pack as a funcion of injecion ime. The flow rae decreases linearly as par of he injeced fluid leaks off ino he formaion. Fig. 6 shows he leakoff rae as a funcion of injecion ime. Noe ha he leakoff rae is slighly higher near he ip of he frac pack, as observed in experimens by Suarez e al. (00). Fig. 7 shows he inersiial velociies in he frac pack and perpendicular (leakoff) o he frac-pack walls. The leakoff inersiial velociies are significanly smaller (0.005 cm/s), while he frac-pack inersiial velociies are much larger, wih mos above he criical inersiial velociy of 0.36 cm/s for he frac pack wih d g /d p = 300. The correced filraion coefficien for he frac pack is equal o zero from Eq. 4, and as a resul, no injeced paricles are deposied in he frac pack. Fig. 8 shows he average reservoir permeabiliy up o a disance of 0 mm from he frac pack. I is clear ha he permeabiliy decreases wih ime because of he deposiion of injeced paricles. Once he ransiion ime is reached, which is less han 40 days in his example [i.e., for he porosiy of he formaion (0.) a he frac-pack walls o reach he criical porosiy (0.0], no more paricles can ener he formaion from he frac pack. Thereafer, all he paricles will remain in he frac pack and sar building a filer cake a he walls. The average permeabiliy of he formaion from he frac-pack walls down o 0 mm ino he formaion reaches a consan damage-permeabiliy value of 5 md, from 00 md. Fig. 9 shows he filer-cake hickness a he frac-pack walls as a funcion of injecion ime. Noe ha even a,000 days of injecion, he filer cake has no filled he frac pack (filer-cake hickness < mm while half of he frac pack widh is.54 mm). Because Leakoff rae (bpd) Inersiial vel. in he frac-pack (cm/s) Inersiial vel. perp. o he frac-pack (cm/s) Disance in he frac-pack from he well (f) Fig. 6 Flow rae perpendicular o he frac-pack walls (leakoff rae) wih injecion days. Disance in he frac-pack from he well (f) Fig. 7 Inersiial velociies in and perpendicular o he frac pack. 460 June 00 SPE Reservoir Evaluaion & Engineering