CHAPTER-II WATER-FLOODING. Calculating Oil Recovery Resulting from Displ. by an Immiscible Fluid:

Transcription

1 CHAPTER-II WATER-FLOODING Interfacal Tenson: Energy requred ncreasng te area of te nterface by one unt. Te metods of measurng IFT s nclude a rng tensometer, pendant drop and spnnng drop tecnques. IFT s between ol and water are usually 0 to 30 dyne/cm at 25 o C. Te IFT between a lqud and ts vapor s referred to as surface tenson. Wettablty: Wen two mmscble pases are placed n contact wt a sold surface, one of te pases s usually attracted to te surface more strongly tan te oter pase. Ts pase s dentfed as te wettng pase wle te oter pase s te non-wettng pase. Wettablty s explaned quanttatvely by examnng te force balances between two mmscble fluds at te contact lne between te two fluds (water and ol) and te sold. Contact angle, θ, s te prncpal measure of wettablty for a smoot, omogeneous surface. Capllary Pressure: Te concept of capllary pressure as caracterstcs of a porous roc evolved from te representaton of capllary penomena n capllary tubes. Te capllary penomena are predctable from te analyss of te forces at te contact lne between te nterface and te sold surface. Calculatng Ol Recovery Resultng from Dspl. by an Immscble Flud: Te dsplacement of one flud by anoter flud s an unsteady state process because te saturatons of te fluds cange wt tme. Ts causes canges n relatve permeabltes and eter pressure or pase veloctes. In te ntal case, water and ol saturatons are unform. Inecton of water at flow rate qt causes ol to be dsplaced from te reservor. A sarp water saturaton gradent develops. Water and ol flow

2 smultaneously n te regon bend te saturaton cange. Tere s no flow of water aead of te saturaton cange because te permeablty to water s essentally zero. Eventually, water arrves at te end of te reservor. Ts pont s called breatroug. After breatroug, te formaton of water n te effluent ncreases as te remanng ol s dsplaced. Tere are bascally two metods to predct dsplacement performance. Te frst metod s te Bucley-Leverett or Frontal Advance Model, wc can be solved easly wt grapcal tecnque. Te second metod s te generalzed treatment of two-pase flow leadng to set partal dfferental equatons tat can be solved on a computer wt numercal tecnques. Bucley-Leverett One Dmensonal Dsplacement: Te Bucley-Leverett model was developed by applcaton of te law of conservaton of mass to te flow of two fluds (ol and water) n one drecton (x). Wen ol s dsplaced by water from a lnear system, a lnear volume element can be used for dervaton. Te law of conservaton of mass, wrtten n terms of rates s descrbed as: mass n mass out mass of accumulaton In a Bucley-Leverett model, water and ol are consdered ncompressble and tus ρ ol and ρ water are constant. Porosty s also constant. Fractonal Flow Equaton: Let us assume tat te coordnate axs for te x-drecton to be at an angle of α, above te orzontal plane. For te defnton of fractonal flow: q w f w. q t q o f o. q t q o (-f w ). q t

3 Wen te fractonal flow of water s assumed to be only a functon of water saturaton. If f w f w (S w ), and apply te can rule, te fnal form of te equaton wll be: (dx /dt) Sw (q t / Φ A). (σ fw / σ fw ) t wc, s Bucley-Leverett equaton. Te velocty of a plane of constant water saturaton s drectly proportonal to te dervatve of te fractonal flow equaton evaluated for tat saturaton. Tree assumptons were made n developng te above equaton. - ncompressble flow, - fractonal flow of water s a functon of only te water saturaton, - no mass transfer between pases... - (σ fw / σ l ) ( Φ A / q t ). (σ Sw / σ t ) To develop a soluton for ts equaton, t s necessary to obtan an equvalent form of equaton wc nvolves one dependent varable (eter fw or Sw). In te Bucley Leverett model, an expresson for (σ Sw / σ t ) s obtaned followng te can rule of dfferentaton. On te oter and, saturaton So and Sw vary wt dstance x. However, because ol and water are assumed to be ncompressble, te total volumetrc flow rate, at any tme t, s constant for every poston x n te lnear system. Te fractonal flow of a pase, f s defned as te volume fracton of te pase tat s flowng at x,t. For ol and water pases, f o q o / q t q o / (q o q w ), and, f w q w / q t q w / (q o q w ), and f o f w f w / [ ( ro. µ w / rw µ o )]

4 Example- A waterflood s under consderaton for a narrow reservor. It s proposed to drll a row of necton wells at one end of te reservor. Te reservor s orzontal. W 300 ft, 20 ft; L 000 ft, Φ 5 %, S w 0.363; µ o 2 cp ; Μ w cp; q n 338 bbl/day; S or 0.205; B o B w ; S w values are: S WD (S w S w ) / (-S or S w ) ro ( S WD ) 2.56 rw S WD 3.72 f w / [ ( ro. µ w / rw µ o )] Soluton: -For S w 0.56: S WD ( ) / ( ) ro ( ) rw f w / [ (0.2 x / x 2)] For S w 0.72: S WD ( ) / ( ) ro ( ) rw f w / [ (0. x / x 2)] 0.985

5 So construct te table: Sw ro rw fw Draw a tangent from Sw tat ntersects te fractonal flow curve at Sw Tus te stablzed zone ncludes all water saturatons from Sw to Sw Te average saturaton at breatroug s estmated as Te cumulatve ol recovered at breatroug s determned as follows: Q bt (S wf S w ) Q bt ( ) On te oter and, ol recovery at breatroug may be computed usng te average water saturaton as:

6 N p V p (S w S w ) Vp (W x x L x Φ) / 5.65, bbl Vp (300 x 20 x 000 x 0.5) / bbl N p ( ) 5406 bbl So tme to reac breatroug s estmated as: t Q bt x V p / q t, days t x / days Also determne water-ol-rato wc s equal to te volume of water per volume of ol produced. F WOR f w2 / f o2 F WOR / ( 0.899) 8.9 -Performance after breatroug was found by selectng about nne values of S w2 and determnng f w2 and f Sw2 for eac S w2 by drawng tangents to te expanded fractonal flow curve. S w S w2 f w2 Q (PV) Tme,days N p,stb q o,bbl/day WOR

7 Areal Sweep Effcency, 5-Spot Pattern In water floodng, water s nected nto some wells and produced from eter wells. In an areal sense, necton and producton taes place at ponts. As a result, pressure dstrbutons and correspondng stream lnes are developed between necton and producton wells. At te tme of water breatroug, some porton of te reservor s lyng between necton and producton wells are contacted by water. Ts contacted fracton s te pattern area sweep effcency at breatroug, E abt. In an example of fve-spot pattern, pattern conssts of a producton well surrounded by four necton wells. An nverted pattern as an necton well, surrounded by four producton wells. Models of dsplacement performance refer to an deal fve-spot pattern, were tere are four necton wells and one producton well. It s assumed tat te necton rates are equal to te producton rates. Tus flow s symmetrc around eac necton well wt 0.25 of te necton rate from eac well confned to te pattern. In a omogeneous reservor were necton and producton rates are equal, te boundares also represent no flow boundares. Terefore analyss of a fve-spot pattern n a reservor can be smplfed by examnng te beavor of a sngle fve-spot pattern. -Caudle and Wtte Approac Ts metod uses two fgures for te performance calculatons: Tese are: ) Area swept vs. Moblty rato as a functon of fw ) Area swept vs. Moblty rato as a functon of V / V D

8 Example-2 Well spacng 40 acres; Bo.25 rbbl/day; 5 ft; Φ 20 %, ro 0.75 rw 0.25; S or 0.30; µ o 5 cp ; µ w 0.8 cp; q n 200 bbl/day; S o 0.70; a) Calculate breatroug recovery and breatroug tme b) Calculate recovery at.5 dsplaceable pore volume nectons c) Calculate ol recovery at producng water-cut 0.60 Soluton: a) MR ( rw / µ w ) / ( ro / µ o ) MR (0.25 / 0.8) / (0.750 / 5) 2.08 From Fgure (2.24) Es 0.58, breatroug areal sweep effcency, so te recovery at breatroug, N p, s : N p (E s x V Do ) / B o, bbl were: V Do 7758 x A x x Φ x (S o S or ) V Do 7758 x 40 x 5 x 0.20 x ( ) bbl N p (0.58 x ) / bbl Snce te necton rate s constant, tme to breatroug s; t b (E s x V Do ) / q t, days t b (0.58 x ) / days b) MR ( rw / µ w ) / ( ro / µ o ) MR (0.25 / 0.8) / (0.750 / 5) 2.08 From Fgure (2.25) V / V D nected water volume / dsplaceable volume V / V D.5 and MR 2.08 So, E s 0.89

9 V Do 7758 x 40 x 5 x 0.20 x ( ) bbl N p (0.89 x ) / bbl Wen V / V D s gven t b ((V / V D x V Do ) / q t, days t b (.5 x ) / days c) f w 0.60 and MR (0.25 / 0.8) / (0.750 / 5) 2.08 From Fgure (2.24) E s 0.73 V Do 7758 x 40 x 5 x 0.20 x ( ) bbl N p (0.73 x ) / bbl Es 0.73 and MR 2.08 so V / V D 0.82 t b ((V / V D x V Do ) / q t, days t b (0.82 x ) / days CONDUCTANCE RATIO: If te dsplacng flud s nected at constant rate, te necton pressure wll vary durng te producton performance of te feld. If te water vscosty s low and ol vscosty s ger, te pressure drop troug te flooded out zone wll be less tan te pressure drop experenced n te source lengt, but were only ol s flowng. Ts cange s termed as Conductance Rato (γ). γ (q / P) / (q / P) ntal q total flow rate P pressure drop between nector and producer

10 Example-3 Well spacng 40 acres; Bo.25 rbbl/day; 5 ft; Φ 20 %, ro 0.75 rw 0.25; S or 0.30; µ o 5 cp ; µ w 0.8 cp; q n 200 bbl/day; S o 0.70; Intal P 000 psa and Intal q 200 bbl/day Soluton: a) MR ( rw / µ w ) / ( ro / µ o ) MR (0.25 / 0.8) / (0.750 / 5) 2.08 From Fgure (2.24), Es 0.58 From conductance rato vs. M wt dfferent Es values: (From Fg. 2-2), γ.4 γ (q / P) / (q / P) ntal P P nt. / γ 000 /.4 74 psa b) At V / V D.5 (dsplaceable PV nected), Es 0.89 γ.8 γ (q / P) / (q / P) ntal P P nt. / γ 000 / psa

11 Dsplacement of Ol by Water n Stratfed Reservors Stles Metod Te mostly encountered type of reservor eterogenety s te stratfcaton wtn te producng formaton. Tese strata may dffer n permeablty, porosty, ntal and resdual saturatons, and te relatve permeabltes wc control te moblty ratos. Te strata may be separated by mpermeable layers. In order to smplfy te calculatons, te followng assumptons were used n Stles metod: -Te formaton s composed of a number of strata of constant tcness contnuous between all wells. -No segregaton of fluds n te layers or cross-flow between te layers occurs. -Te dsplacement occurs n a pston-le manner wt no ol produced bend te dsplacement front. -Te system s lnear, as constant porosty, constant flud saturatons and te same relatve permeablty to ol a ead of te front and to front bend te front n all layers. -Te poston of te floodng front n any gven layer s drectly proportonal to te absolute permeablty of tat layer. -Te producng water-cut depends upon te total reservor capacty () wc as experenced breatroug. Te procedure of calculaton s; -Dvde te strata nto layers of constant tcness,, 2, 3,.. n -Correspondng absolute permeabltes, 2,.. n -Arrange tem n order of decreasng permeablty.

12 Wen te t bed as ust experenced water breatroug at te producng well, ten te fracton of te recoverable ol tat as been produced wll be equvalent to te fracton of te pay secton flooded out, plus te flooded parts of te strata stll contrbutng to te producton of ol. t n n t ı ı t t R... { } n n ı ı t t R... dfference : md-ft stll producng ol at te outflow end. { } t t t C C R C were; C t total capacty of te formaton n md-feet C md-ft wc ave been completely flooded t total tcness of strata beng flooded ol recovered from te t bed.... 2, te total tcness of formaton wc as been flooded out wt water.

13 Example-4 Use te metod of STILES to predct te rates of ol producton and water-cut for te followng reservor and flud property data. Core data: S awr 22 %, φ 25 %, S o 60 %, S or 7 % B o.2 rbbl/stb, µ w 0.9 cp, µ o.6 cp Area (5-spot pattern) 0 acres, S wc 22 % Inecton rate per fve-spot 200 bbl/day Te net ol pay s present between ft n te wells of te pattern. Ts permeablty values are trougout to be vald averages for all wells n te 5-spot pattern area. Dept (ft) : Permeablty (md) : S op saturaton at te start of te flood ft () (ft) (2) (md) (3) (md-ft) (4) Σ (md-ft) (5) R (fracton) t C t R Ct t n R (45 x 2 ( ))/(20 x 45) 0.46 R 2 (480 ( ))/(20 x20) 0.537

14 Te orgnal ol n place (OOIP) at surface condtons; N 7758 x φ x V x (-S w ) / B o V 0 acres x 20 ft 200 acre-ft N 7758 x 0.25 x 200 x 0.60 / stb ol Te saturaton at te start of te flood or after prmary depleton, S op, s assumed as 59 %. Te recoverable ol s 7758 x φ x V x {(S op B o x S awr ) / B o } 7758 x 0.25 x 200 {( x 0.22) /.2} STB ol If we tae an areal sweep effcency of 95 %. Te recoverable ol wll ten be 95 % of STB or STB (6) N p 3050 x (5) (STB) (7) N p (STB) (8) f w (at res. Cond.) (9) q sc (STB/DAY) Te water-cut at te producng well at reservor condtons s gven n Column (8) and s calculated from, f w Mc Mc ( Ct C )

15 M ( rw µ w ) ( µ ) ro o If te relatve permeablty of te water bend te dsplacng front s 0.20 and te relatve permeablty of te ol aead of te front s M (0.20 / 0.9) / (0.80 /.60) Te surface ol producton rate can be readly determned from te followng equaton; f * q o n w q scol * Bo Bo f q n (-0.09) /.2 * stb/day. If te ol saturaton after prmary depleton s 59 % and te connate water saturaton 22 % s te remanng pore space wc must be occuped by gas. Te gas space n te reservor must be replaced or flled up by water before te producton wells respond fully to te water necton. Ten te fll-up tme for ts unt t 7758xφxVxS q w g * f were; f fracton of gas space fll-up at frst ol producton ncrease. Let s f 0.37 t (7758x0.25x200x0.9) / t 36 days To calculate surface water-cut and te nected amount of water, we use te followng equatons: W 200 bbl/day x t

16 f ' w AC AC ( C C ) t were; A rw ro µ o B µ w B o w (7) / (9) (days) (0) t36 Σ(7)/(9) (days) () W (bbl at surface) (2) f w (Surf. Cond.) Fgure 2.27 represents te predcted performance of a fve-spot well pattern for te data used were te Stles Metod s employed. Coverage : fracton of te vertcal cross secton swept to S or by water: C n n number of layers last layer flooded out Te water-ol rato (WOR); correspondng to te flood out of layer ;

17 Dystra-Parsons Metod: Dystra and Parsons ave presented a waterflood performance predctve metod n stratfed reservors. Te metod s applcable for a wde range of moblty rato values. Tey constructed graps of Coverage (C) as a functon of permeablty varaton and moblty ratos for producng WOR. Jonson as presented a smplfed grapcal treatment of te Dystra-Parsons metod wc s useful were quc estmates are requred. Assumptons : -Reservor s made up of solated layers. Impermeable layer exsts between stratfcatons. -Pston-le dsplacement only one pase flows aead of front and bend te front. -Flow s lnear and steady -Incompressble flow - P across eac layer s te same -Except for () permeablty. All oter flud and roc propertes are unform n te reservor. Te tecnque s sem-emprcal and s based upon te correlaton for four varables. -Te vertcal permeablty varaton, ν -Te ntal water saturaton, S w -Te water to ol moblty rato, M -Te fractonal recovery of te ol-n-place at a specfed WOR. -Cumulatve frequency dstrbuton on log-normal probablty paper can be approxmated by a stragt lne.

18 Permeablty data are arranged n descendng order and cumulatve frequency dstrbuton, te fracton of te samples wt permeablty greater tan te partcular sample s obtaned. -A stragt lne s drawn troug te data -Te md pont of te permeablty dstrbuton s te log mean permeablty ( 50 ). -On te oter and, n normal dstrbuton, te standard devaton of samples, ν s 5.9 %. Tere, 5.9 % of samples sow values less tan 50 - ν, and 84. % sow values ger tan 50 ν. Permeablty varaton υ ν ranges from 0 to and 0 ndcate omogeneous system. -From te dstrbuted graps determne C (coverage) for a WOR equal to, 5, 25 and 00. -Calculate te ol recovery, N p ; N p 7758* A* * φ * C *( So S B o -Plot N p vs WOR -Integrate N p vs WOR curve grapcally to get te volume of produced water, W p. -Calculate water nected, W I W N B W p o p -Fnd te lfe n years, t, (q daly necton rate, bbl/day) or )

19 t ( W ) /( q *365) -Calculate ol producton rate by dvdng te dfferences n recovery by te correspondng dfferences n tme. Example-5 Use te Dystra-Parsons metod to obtan te fractonal ol recovery wc would be expected for te reservor. Te relatve permeablty to ol aead of te dsplacement front s 0.92, wle te relatve permeablty to water at te rreducble ol saturaton s Assume tat te economc lmt wll be reaced at a producng WOR of 3.0. Addtonal Data: S wr 0.22 B o.2 rbbl/stb φ 0.25 Area of te enclosed 5-spot pattern 0 acres S o 0.60 µ w 0.9 cp µ o.6 cp S or 0.7 Inecton rate 200 bbl/day ROIP stb

20 Layer K (md) Soluton: (In descendng order) (md) Cum % > 45 0 N 0 (number of data) (2 / 0) x % (8 / 0) x % 2 90 From (Cum % >) vs log md were 50 % 28 md were 84. % υ

21 υ M υ rw ro µ µ w o (0.40 / 0.92) (.6 /0.9) 0.77 WOR C N p (STB) N p (rbbl) N p (rbbl) N p C * ROIP, stb N p N p (stb) x B o, rbbl Average WOR W p N p x Avg.WOR (rbbl) W W p N p (rbbl) t W /q (days) q o N p / t (rbbl/day) (0 )/ ( 5)/ (5 0)/ (025)/ Dystra and Parsons presented a model for estmaton of te dsplacement performance of a layered reservor by subdvson of te reservor nto (n) no communcatng layers wt no cross flow between layers. A constant pressure drop s mantaned across te layers. Pstonle dsplacement was assumed for te waterflood. Tus, as eac layer s flooded out, producton goes from 00 % ol to 00 % water.

22 Te WOR (water-ol rato) s a drect measure of te fracton of te vertcal cross secton tat s flooded. Te term coverage, denoted by C, was defned as te fracton of te reservor depleted by water. Te followng equaton s an expresson for te coverage wen te tcness of eac layer s. n C α Te (n) layers are arranged n order of descendng permeablty wt layer () representng te most recent layer flooded out. Te term α I s gven by ) ( 2 2 M M M M α were: M s te moblty rato based on end ponts of te relatve permeablty curves. Te WOR (water-ol rato) correspondng to te flood out of layer s gven by ( ) n wo M M F 2 2

23 In 949, Stles proposed one of te frst metods of predctng te vertcal sweep effcency of a lnear waterflood operatng at constant pressure drop n a layered reservor wen tere was no cross flow between layers. Te Stles model assumes pston-le dsplacement wt unt moblty rato. Wt tese assumptons, te coverage (fracton of te vertcal cross secton swept to S or by water) s gven by C n Were te subscrpt represent te last layer flooded out and te ndex, n,s te number of layers. Te WOR (water-ol rato) correspondng to te flood out of layer s gven by F wo n