Oil & Gas Reseach ISSN: 47-0518 Oil & Gas Reseach Gaal Rezk, Oil Gas Res 016, :3 DOI: 10.417/47-0518.100011 Reseach Aicle Aicle Open Open Access Analysis o Pessue Tansien Tess in Naually Facued Resevois Gaal Rezk M* Gaza Univesiy, Caio, Egyp Absac Pessue ansien ess in naually acued esevois oen ehibi non-unio esponses. Dieen echniques can be used o analyze he pessue behavio in dual poosiy esevois in an aep o coecly chaaceize esevoi popeies. In his pape, he pessue ansien ess in naually acued esevois ee analyzed using convenional sei-log analysis, ype cuve aching (using coecial soae and Tiab s diec synhesis (TDS echnique. In addiion, he TDS ehod as applied in case o a naually acued oaion ih a veical hydaulic acue. These echniques ee applied o a single laye naually acued esevoi unde pseudoseady sae ai lo. By sudying he unique chaaceisics o he dieen lo egies appea on he pessue and pessue deivaive cuves, vaious esevoi chaaceisics can be obained such as peeabiliy, skin aco, and acue popeies. Fo naually acued esevois, a copaison beeen he esuls sei-log analysis, soae aching, and TDS ehod is pesened. In case o ellboe soage, ealy ie lo egie can be obscued ha lead o incoplee sei-log analysis. Fuheoe, he ype cuve aching usually gives a non-uniqueness soluion as i needs all he lo egies o be obseved. Hoeve, he diec synhesis ehod used analyical equaion o calculae esevoi and ell paaees ihou ype cuve aching. Fo naually acued esevois ih a veical acue, he pessue behavio o ells cossed by a unio lu and ininie conduciviy acue is analyzed using TDS echnique. The dieen lo egies on he pessue deivaive cuve ee used o calculae he acue hal-lengh in addiion o ohe esevoi popeies. The esuls o dieen cases shoed ha TDS echnique oes seveal advanages copaed o sei-log analysis and ype cuve aching. I can be used even i soe lo egies ae no obseved. Diec synhesis esuls ae accuae copaed o he available coe daa and he soae aching esuls. Keyods: Naually Facued Resevois; Pessue Tansien Analysis; Veical Facue; Unio Flu Facue; Ininie Conduciviy Facue Noenclaue B: Foaion volue aco, es bbl/sb C : Toal copessibiliy, psi -1 C: Wellboe soage coeicien, bbl/psi C A : Shape aco C d : Diensionless ellboe soage h: Toal oaion hickness, K : Bulk acue peeabiliy, d p: Pessue, psi P D : Diensionless pessue P D : Diensionless boo-hole pessue P in : Iniial pessue, psi P : Boo-hole pessue, psi PʹD: Diensionless pessue deivaive PʹD : Diensionless boo-hole pessue deivaive ΔP: Pessue dieence, psi q : Flo ae, sbd e : Resevoi oue adius, : Wellboe adius, S: Skin aco : Tes ie, h D : Diensionless ie e : Hal side o ecangle in -ais, : Facue hal lengh, y e : Hal side o ecangle in y-ais, Geek Sybols : Inepoosiy lo paaee ω: Diensionless soage coeicien : Viscosiy, cp φ: Poosiy Subscips b1: Beginning o is adial lo line b: Beginning o second adial lo line BR: Bi-adial D: Diensionless *Coesponding auho: Gaal Rezk M, Assian Poesso, Gaza Univesiy, Caio, Egyp, Tel+ +0100058903, +07039306; E-ail: ohaedgaalezk@gail.co Received Ocobe 15, 016; Acceped Ocobe 19, 016; Published Ocobe 5, 016 Ciaion: Gaal Rezk M (016 Analysis o Pessue Tansien Tess in Naually Facued Resevois. Oil Gas Res : 11. doi: 10.417/47-0518.100011 Copyigh: 016 Gaal Rezk M. This is an open-access aicle disibued unde he es o he Ceaive Coons Aibuion License, hich peis unesiced use, disibuion, and epoducion in any ediu, povided he oiginal auho and souce ae cedied. ISSN: 47-0518 Volue Issue 3 100011
Ciaion: Gaal Rezk M (016 Analysis o Pessue Tansien Tess in Naually Facued Resevois. Oil Gas Res : 11. doi: 10.417/47-0518.100011 Page o 10 e: Oue bounday e1: End o is adial lo line : Facue : Mai L: Linea o: Oil PSS: Pseudoseady sae R: Radial : Toal Inoducion The analysis o pessue daa eceived duing a ell es in dual poosiy oaion has been idely used o esevoi chaaceizaion. Convenional sei-log analysis and log-log ype cuve ehods ae he ealy echniques used o analyze pessue ansien daa. Hoeve, boh ehods need ceain cieia o give accuae esuls, such as; all lo egies us be ideniied in he pessue and pessue deivaive plo. In case soe lo egies ae no ideniied, ype cuve aching ill give a non-uniqueness soluion and is essenial ial and eo, and sei-log analysis canno be copleed. Tiab [1] used a ne ehod o analyze pessue ansien ess, called Diec Synhesis Technique. This ehod can calculae dieen esevoi paaes ihou ype cuve aching by using pessue and pessue deivaive loglog plos. In 1994, Tiab [] eended he ok o veically acued ells in closed syse. Engle and Tiab [3] developed diec synhesis ehod o analyze pessue ansien ess in dual poosiy oaion ihou using ype cuve aching. They used analyical and epiical coelaions o calculae he naually acued esevoi paaees. Jalal [4] discussed he analyical soluions o ells in dual poosiy esevois ih a veical acue. The diec synhesis ehod oes anys advaages in analyzing pessue ansien ess. The objecive o his pape is o analyze pessue ansien in naually acued esevois using: convenional sei-log analysis, ype cuve aching (using coecial soae, and Tiab s diec synhesis ehod o coecly chaaceize he esevoi popeies. These echniques ee applied o naually acued esevois, ih and ihou hydaulic (veical acue. Popeies o Dual Poosiy Foaion The Dual poosiy esevoi consiss o piay and seconday poosiy hich ae he ai and acues. Waen and Roo [5] deined he acued esevois by o key paaees, ω and. These diensionless paaes ae deined as ollos: The elaive soaiviy, ( φc ω (1 ( φc + ( φc The inepoosiy lo paaee, K α ( K Whee he shape aco α, -, depends on he ai block geoey (hoizonal slab o spheical ai block. By assug ha he esevoi is ininie acing and poducing a single phase, slighly copessible luid ih pseudoseady sae ai lo, he pessue soluion is given by [6]: 1 D D PD lnd + 0.80908 + Ei Ei + S ω( 1 ω ( 1 ω Convenional sei-log analysis Naually acued esevois give o paallel sei-log saigh lines in plo o dadon and build-up ess as shon in Figue 1. Peeabiliy hickness poduc: The peeabiliy hickness poduc o he oal syse (acually o he acues as he ai peeabiliy can be negleced can be calculaed o he slope o he iniial o inal saigh line,. ( Kh 16.6 QB (4 1. The elaive soaiviy ω can be calculaed o he pessue dieence, ΔP, beeen he iniial and inal saigh lines hen boh o he can be ideniied. ω 10. By daing a hoizonal line hough he iddle o ansiion peiod o inesec ih boh sei-log saigh lines, he ies o inesecion ih he is and he second sei-log saigh lines ae donaed by 1 and, especively. The soaiviy aio also can be deeed as ollos [7]: ω (6 1 3. The inepoosiy lo coeicien,, can be calculaed by [8]: Fo dadon ess: ω ( φhc ( ( (7 1 ω 1.781 k 1 o 1 ( φ hc ( ( (8 1 ω 1.781 k Fo build-up ess ω ( φ hc p + ( ( ( 1 (9 1 ω 1.781 k p Figue 1: A Build-up Sei log plo o a dual Pioiy syse. (3 (5 ISSN: 47-0518 Volue Issue 3 100011
Ciaion: Gaal Rezk M (016 Analysis o Pessue Tansien Tess in Naually Facued Resevois. Oil Gas Res : 11. doi: 10.417/47-0518.100011 Page 3 o 10 o ( φ 1 hc p + ( ( ( 1 ω 1.781 k Diec synhesis echnique p (10 Diec synhesis ehod uses a log-log plo o pessue and pessue deivaive daa vesus ie o calculae vaious esevois and ell paaees. I uses he pessue deivaive echnique o ideniy esevoi heeogeneiies. In his ehod, he values o he slopes, inesecion poins, and beginning and ending ies o vaious saigh lines o he log-log plo can be used in eac analyical equaions o calculae dieen paaees as i is shon in he olloing pocedues [6]: Ininie Acing Resevoi ihou Wellboe Soage 1. Facue peeabiliy: The acue peeabiliy, K, can be deeed using ealy o lae ie ininie acing adial lo lines (only one o he o deivaive segens needs o be obseved 70.6 K qbo h( P '* (11. Relaive soaiviy: The aio beeen iu and adial pessue deivaive values can be used in equaion o calculae ω ( * ( * ω 0.15866 + 0.54653 (1 ( P * ( P * The elaive soaiviy can also be calculaed by using he chaaceisic ies as in he olloing equaions: 1 ln ω (13 1 ω 50 e1 1 ωln ω 5 1 ω ω e b 1 ( 0.4383 0.93 50e1 5 5 b b ω 0.1911 + 0.80678 (14 (15 (16 Whee e1 is he end ie o he ealy ininie acing adial lo line, b is he beginning ie o he lae ininie acing adial lo line and is he iu ie. 3. The inepoosiy lo paaee: The inepoosiy lo paaee can be also obained o he chaaceisic ies as olloing: S T ωln1 / ω (17 0.000637k S T ω(1 ω (18 0.000637k 50 b e1 ST 5(1 ω (19 0.000637k Whee S T is he poduc o he aveage bulk poosiy (o coes o logs and he aveage copessibiliy. can be also calculaed o he iu coodinaes: 4.5hS * T (0 qb o In case ω less han 0.05, lae ansiion peiod uni slope saigh line is ell obseved. The inepoosiy lo paaee can be calculaed o: S T 1 0.000637k (1 us, i Whee us,i he inesecion o he ansiion peiod uni slope line ih he ininie acing adial lo line. 4 Skin aco: The skin aco can be calculaed o he ealy o lae ie adial lo pessue and pessue deivaive daa by using he olloing equaions: 1 k 1 1 S ln 7.43 + ( * S 1 T ω 1 k S ln 7.43 + (3 * S T Whee 1 is any poin on he ealy hoizonal adial lo line and is any poin on he lae hoizonal adial lo line. Ininie Acing Resevoi ih Wellboe Soage Wellboe soage eecs can obscue ealy lo egies on log-log plo o pessue and pessue deivaive vesus ie. I is epesened by ealy ie uni slope saigh line on he log-log plo. This uni slope peiod is olloed by a peak on he pessue deivaive cuve as shon in Figue. The eec o ellboe soage can aec he iu coodinaes o he pessue deivaive cuve and cause he appeaance o pseudo-iu coodinaes. Theeoe, he eec o ellboe soage should be invesigaed pio o he analysis o kno hehe he obseved iu is he eal iu o he pseudo-iu. Fo ( d /( d 10, he ellboe soage doesn aec he iu coodinaes. [( d is he diensionless ie o he peak poin, [6]. In case he iu coodinaes ae no aeced by ellboe soage, calculae he esevoi paaees using he olloing pocedue [6]: 1-Deee he acue peeabiliy using he lae ie adial lo line. Figue : The eec o Wellboe soage on iu coodinaes. ISSN: 47-0518 Volue Issue 3 100011
Ciaion: Gaal Rezk M (016 Analysis o Pessue Tansien Tess in Naually Facued Resevois. Oil Gas Res : 11. doi: 10.417/47-0518.100011 Page 4 o 10 -Calculae he ellboe soage coeicien o he ealy ie uni slope using he olloing equaions: qbo C (4 4 p Whee, Δp ae any poin on he uni slope line. (Δpp i - p o dadon and Δpp s - p (Δ0 o buildup ess qbo C (5 4 p * The ellboe soage coeicien can also be calculaed o he inesecion ie o he ealy ie uni slope ih he ininie acing adial lo line ( i. k hi C (6 1695 3-Deee he ω and as oulined beoe. 4-Deee skin aco o he lae ie adial lo pessue and pessue deivaive aio. I he iu coodinaes ae inluenced by ellboe soage, he inepoosiy lo paaee and he elaive soaiviy can be calculaed using he olloing equaions: Deee o he peak o iu ie aio o o he peak o adial pessue deivaive aio: 10 1 1 log 5.565 (7 C d, o 5.6146 C Whee C d (8 π φc h (C: bbl/psi 1.0845 1 log 1.94 ( p * 1 0.88log S ( p * e (9 (30 Calcuale ω o he peak o beginning o second adial lo line Sei-log analysis Hone plo is shon in Figue 3. This igue depics he ealy poins ha ae aeced by ellboe soage, hoeve, he is saigh line can be obseved clealy. The igue shos o paallel saigh lines ha poves he dual poosiy behavio. Theeoe, he convenional sei-log analysis can be used o esiae esevoi paaees. The acue peeabiliy can be calculaed o he slope o he second saigh line ( o give: 16.6 QB o o o 16.6* 300*1.35* 0.68 Kh 4.94 156.3 d. 156.3 Theeoe, K 5.45 d 80 The soaiviy aio (ω can be calculaed o he veical disance beeen he o saigh lines (Δp and he slope (: 130 4.94 ω 10 10 0.64 A hoizonal saigh line hough he iddle o he ansiion egion is dan o inesec ih he o sei-log saigh lines. Read he coesponding ies and calculae he inepoosiy lo coeicien (: p + ( 1 17, p + ( 4.3 ω ( φ h C p + 1 1 ω 1.781 k p 1 ( 5 0.64 0.15* 80*1.5*10 * 0.68* 0.81 7 * *17.955*10 1 0.64 1.781* 5.45* 7 1 ( φ h C p + 1 ω 1.781 k p 5 1 0.15* 80*1.5*10 * 0.68* ( 0.81 * * 4.3.88*10 1 0.64 1.781* 5.45* 7 7 ie aio: 1 C log 5(1 d ω Case 1 b (31 This case pesens an oil ield in Ian. A build-up es is conduced on a ell o naually acued esevoi. The aveage coe peeabiliy eceived o he Ianian oil copany anges o 4 o 6 d. The ell as loing o 7 hous ih q300 STB/day beoe shu-in o a build-up es. The build-up daa ae given in Table 1. The olloing esevoi and ell daa ae also knon: h80 P 7 hs 0.81 Bo1.35 bbl/stb q300 STB/day 0.68 cp P (Δ0881 psi a φ0.15 C 1.5*10-5 psi -1 Figue 3: Sei log plo o he build up es daa o case 1. ISSN: 47-0518 Volue Issue 3 100011
Ciaion: Gaal Rezk M (016 Analysis o Pessue Tansien Tess in Naually Facued Resevois. Oil Gas Res : 11. doi: 10.417/47-0518.100011 Page 5 o 10 Δ, h Ps, psia Δ, h Ps, psia Δ, h Ps, psia Δ, h Ps, psia Δ, h Ps, psia Δ, h Ps, psia 0.0083 881.36 0.6958 33.181 3.5083 3393.13 49.6083 3489.319 53.7333 3493.64 57.85831 3497.6 0.036 901.811 0.7111 335.39 3.6917 3395.58 49.7 3489.4 53.85 3493.73 58.0417 3497.8 0.0389 916.59 0.764 338.541 3.875 3397.719 49.7917 3489.53 53.9167 3493.8 58.1333 3497.9 0.054 930.591 0.7417 341.54 4.0583 3399.63 49.8833 3489.64 54.0083 3493.9 58.501 3498.01 0.0694 943.96 0.7569 344.51 4.417 3401.311 49.975 3489.73 54.1 3494 58.3167 3498.09 0.0847 957.04 0.77 347.36 4.45 340.97 50.0667 3489.83 54.1917 3494.09 58.4089 3498.18 0.1 969.7 0.7875 350.16 4.6083 3404.551 50.1589 3489.9 54.839 3494.18 58.5 3498.6 0.1153 981.93 0.808 35.86 4.7917 3406.05 50.5 3490.0 54.375 3494.8 58.59171 3498.35 0.1306 993.64 0.8181 355.5 4.975 3407.46 50.34171 3490.1 54.46671 3494.38 58.6833 3498.46 0.1458 3004.94 0.8333 358.031 5.1583 3408.77 50.4333 3490. 54.5583 3494.48 58.77499 3498.55 0.1611 3015.88 0.8486 360.509 5.3417 3410.051 50.55 3490.3 54.65 3494.55 58.8667 3498.611 0.1764 306.45 0.8639 36.871 5.55 3411.7 50.6167 3490.39 54.7417 3494.66 58.9583 3498.68 0.1917 3036.69 0.879 365.1 5.7083 341.4 50.7083 3490.5 54.8333 3494.79 59.05 3498.74 0.069 3046.611 0.8944 367.4 5.8917 3413.56 50.8 3490.61 54.95 3494.89 59.14169 3498.83 0. 3056.19 0.9097 369.49 6.3417 3415.769 50.8917 3490.7 55.0167 3494.909 59.3331 3498.93 0.375 3065.5 0.95 371.59 7.583 340.139 50.9833 3490.8 55.1083 3494.99 59.35 3499.03 0.58 3074.41 0.9403 373.47 8.175 343.5 51.075 3490.9 55. 3495.06 59.4167 3499.13 0.681 3083.03 0.9556 375.35 9.0917 346.319 51.1667 3490.99 55.917 3495.13 59.5083 3499. 0.833 3091.33 0.9708 377.17 10.0083 348.771 51.583 3491.07 55.3833 3495.4 59.60001 3499.3 0.986 3099.31 0.9861 378.931 11.8417 343.46 51.35 3491.18 55.475 3495.35 59.6917 3499.4 0.3139 3106.99 1.0014 380.63 13.675 3436.14 51.4417 3491.7 55.5667 3495.46 59.7839 3499.471 0.39 3114.41 1.0583 385.9 16.0083 3439.83 51.5339 3491.35 55.6589 3495.55 59.875 3499.579 0.3444 311.581 1.15 394.58 19.675 3446.4 51.65 3491.44 55.75 3495.65 59.96671 3499.669 0.3597 318.46 1.417 330.33 3.3417 3453.7 51.71671 3491.53 55.84171 3495.74 60.14999 3499.859 0.375 3135.1 1.3333 3309.449 8.5083 346.1 51.8083 3491.64 55.9333 3495.8 60.417 3499.9 0.3903 3141.47 1.45 3316.091 34.0083 3470.19 51.9 3491.7 56.0499 3495.9 60.3333 3499.98 0.4056 3147.66 1.5167 33.179 39.5083 3477.331 51.9917 3491.81 56.1167 3496.01 60.45 3500.049 0.408 3153.619 1.6083 337.91 45.0083 3483.611 5.0833 3491.93 56.083 3496.09 60.51669 3500.149 0.4361 3159.34 1.7 3333.5 48.05 3487.63 5.175 349.04 56.3 3496.18 60.60831 3500.5 0.4514 3164.89 1.7917 3338.809 48.1417 3487.7 5.667 349.1 56.39169 3496.41 60.7 3500.339 0.4667 3170.31 1.8833 3343.91 48.333 3487.81 5.3583 349. 56.48331 3496.3 60.7917 3500.44 0.4819 3175.35 1.975 3349.4 48.35 3487.91 5.45 349.3 56.575 3496.39 60.8833 3500.519 0.497 3180.36.0667 3354.88 48.4167 3488.0 5.5417 349.4 56.6667 3496.49 60.97501 3500.611 0.515 3185.159.1583 3359.47 48.5083 3488.14 5.6333 349.5 56.7583 3496.6 61.0667 3500.701 0.578 3189.819.5 3363.501 48.6 3488.51 5.75 349.59 56.85001 3496.69 61.1589 3500.77 0.5431 3194.3.3417 3367.15 48.6917 3488.34 5.8167 349.66 56.9417 3496.78 61.5 3500.87 0.5583 3198.67.4333 3370.341 48.7839 3488.43 5.9089 349.75 57.0339 3496.87 61.34171 3500.96 0.5736 30.86.55 3373.5 48.875 3488.5 53 349.85 57.15 3496.96 61.4333 3501.04 0.5889 306.909.6167 3375.851 48.96671 3488.61 53.09171 349.96 57.1671 3497.04 61.5499 3501.1 0.604 310.85.7083 3378.9 49.0583 3488.731 53.1833 3493.08 57.3083 3497.13 61.6167 3501.179 0.6194 314.66.8 3380.48 49.15 3488.839 53.75 3493.18 57.39999 3497. 61.7083 3501.4 0.6347 318.36.8917 338.509 49.417 3488.99 53.3667 3493.6 57.4917 3497.3 61.8 3501.31 0.65 31.94.9833 3384.409 49.3333 3489.09 53.4583 3493.36 57.5833 3497.4 61.89169 3501.39 0.6653 35.449 3.1417 3387.69 49.45 3489.11 53.55 3493.45 57.675 3497.471 61.98331 3501.479 0.6806 38.84 3.35 3390.44 49.5167 3489.19 53.6417 3493.56 57.76669 3497.55 6.1667 3501.659 Table 1: Pessue build up es Daa o Case 1. Diec Synhesis Technique The log-log plo o pessue and pessue deivaive shon in Figue 4. I is clea ha hee is a ellboe soage ih an ealy ie uni slope and he ealy adial lo peiod is ell deined. Hoeve, he lae adial lo peiod no las o long ie. The daa ehibi a unique behavio hich is indicaive o a naually acued esevoi. Fo Figue 4: (*ΔPʹ 99.7 psia 3 h, ΔP 614 psia (*ΔPʹ 7.56 psia, 8.8 h us 0.054 h, ΔP us 49.6 psia 0.4 h, b 8 h (*ΔPʹ US 49.6 psia, The eec o he WBS on he iu deivaive coodinaes can be deined by calculaing he aio ( d / ( d / 8.8/0.40.95 (>10. Theeoe he iu deivaive coodinaes ae he eal iu and no aeced by ellboe soage. The acue peeabiliy can be calculaed o he lae ie ininie acing adial lo line: 70.6qBo 70.6* 300* 0.68*1.35 K 5.339 d h P * 80*99.7 ( ISSN: 47-0518 Volue Issue 3 100011
Ciaion: Gaal Rezk M (016 Analysis o Pessue Tansien Tess in Naually Facued Resevois. Oil Gas Res : 11. doi: 10.417/47-0518.100011 Page 6 o 10 Figue 4: Pessue and Pessue deivaive plo o case 1. Wellboe soage coeicien is calculaed by: qbo 300* 0.68 0.054 C * 0.14085 bbl / psi 4 p 4 49. 6 Skin aco o he lae ie pessue and pessue deivaive daa: S 1 k ln + 7.43 * S T 1 614 5.339* 3 ln 7.43 3.74 6 + 99.7.5*10 * 0.68* 0.81 The diensionless soage coeicien (ω: ( ( ( ( P * P * ω 0.15866 + 0.54653 P * P * 7.56 7.56 0.15866 + 0.54653 0.0856 99.7 99. 7 The inepoosiy lo paaee (: S T 0.000637k ωln1 / ω 6.5*10 * 0.68* 0.81 0.0856* ln1/ 0.0856 0.000637 * 5.339 8.8.051 * 10-6 Naually acued esevois ih a veical acue The pessue behavio o a dual poosiy oaion ineseced by unio lu and ininie conduciviy acue can be invesigaed using log log plos o pessue and pessue deivaive uncions. The diec synhesis echnique can be used o calculae esevoi paaees such as skin, ellboe soage, peeabiliy, inepoosiy lo paaee, elaive soaiviy and hal-acue lengh ihou ype cuve aching. The applied assupions ae: he esevoi is isoopic, hoizonal, and has consan hickness and acue peeabiliy. The acued ell is poducing a consan ae ih consan viscosiy, slighly copessible luid. In addiion, he acue ully peneaes he veical een o he oaion and has he sae lengh in boh sides o he ell. A pseudoseady sae inepoosiy lo beeen he ai and he acue syse is also assued. Unio Flu Facue Figue 5 shos he pessue deivaive plos o vaious values o e / aios, in a single laye squae, dual poosiy esevoi ih pseudoseady sae inepoosiy lo. Thee lo egies ae shon in hese igues: he linea lo egie, ininie acing adial lo egie, and pseudoseady seady sae lo egie [9]. 1 Linea lo peiod: The linea lo peiod occus a ealy ie. Duing his peiod, he lo esuled o he epansion o he luid ihin he acue neok as he ai eec is negligible. The linea lo peiod can be ideniied by a saigh line o slope 0.5. This saigh line is used o calculae he acue hal lengh. The equaion o pessue deivaive duing his lo egie is: π e DA * P' D DA ω By aking logaih o boh sides o he equaion gives: Paaee Convenional sei-log Diec synhesis Soae aching K (d 5.45 5.339 5.375 ω 0.64 0.0856 0.0865.955*10-7 -.88*10-7.051*10-6 -.13*10-6.01*10-6 S - 3.74-3.71 C (bbl/psi 0.14085 0.1453 Table : Copaison o he Resuls o he Case 1. (3 Fo veiicaion 4.5hS qb o * T 6 4.5* 80*.5*10 * 0.81 7.56.13 * 10-6 300*1.35 8.8 Copaison o he esuls o convenional sei-log analysis, diec synhesis echnique, and ype cuve aching is shon in Table. The esuls o he sei-log analysis ae only aching ih he diec synhesis and soae esuls in peeabiliy. Hoeve, he soage coeicien and he inepoosiy lo paaee ae inaccuae. On he ohe side, he diec synhesis echnique and he soae esuls sho an ecellen ach in all esevoi paaees. Figue 5: Pessue deivaive esponse in a single-laye squae, naually acued esevoi ih pseudo seady sae ine poosiy lo. Boh WBS and Skin ae ignoed. ISSN: 47-0518 Volue Issue 3 100011
Ciaion: Gaal Rezk M (016 Analysis o Pessue Tansien Tess in Naually Facued Resevois. Oil Gas Res : 11. doi: 10.417/47-0518.100011 Page 7 o 10 ( ( log * P log + log( L (33.034qB L (34 ωh ( ϕc K ' 1 Based on Eq. (34 he log-log plo o pessue deivaive vesus ie gives hal slope saigh line duing he linea lo peiod. The acue hal-lengh can be calculaed by:.03qb (35 (* p K ( L1 ωh ΦC hee (* p L1 is he value o pessue deivaive a 1h on he linea lo line. Pseudoadial lo peiod: The ininie acing adial lo peiod is doaed only o ( e / > 8, as shon in Figue 5. This lo egie is ideniied by a hoizonal saigh line on he pessue deivaive plo and can be used o calculae peeabiliy and skin [4]. The pessue deivaive equaion duing his lo egie is: * P ' 0.5 (36 DA D The above equaion in diensional o yields: ' 70.6qB (* P R (37 kh R sands o adial lo. Solving he above equaion o peeabiliy gives: 70.6qB k (38 ' (* P h R The skin can be deeed by: ( p K R ( R * p ( ΦC S 0.5 ln + 7.43 (39 R 3 Pseudoseady sae lo peiod: In case o a veically acued ell inside a closed syse, a hid saigh line o uni slope appeas. This line coesponds o he pseudoseady sae lo egie is used o calculae he dainage aea and shape aco. The pessue deivaive equaion descibing his lo peiod is: * P' π (40 DA D DA By aking logaih o boh sides o he above equaion, he diensional o is: ' qb log ( * P log( + log( (41 4.7 h( ΦC A By subsiuing 1h, he dainage aea can be calculaed using he olloing equaion: qb A (4 ' 4.7 ( * P h ΦC ( PSS1 ' Whee (* P PSS1 sands o pseudoseady sae lo peiod a ie equal 1 h. The shape aco, C A, can be calculaed by he olloing equaion: ( p ( * 0.00057K e PSS pss CA.458 ep 1 (43 A( C p Φ pss 4 Tansiion peiod: The ansiion can occu duing he ininie acing adial lo as shon in Figue 5. In his case, he elaive soaiviy, ω, and he inepoosiy lo paaee,, can be esiaed by seveal ehods as peviously descibed in he pevious secion. I he ansiion akes place duing he linea lo peiod as shon in Figue 6, o paallel saigh lines o slope equal 0.5 can be obseved. The is line epesens he epansion o he acue neok, his lo peiod is called acue soage doaed lo peiod. While he second line appeas duing he oal syse doaed lo peiod (o his peiod ω1. Also, a saigh line o uni slope is obseved duing lae ansiion peiod. The inesecion ie o he saigh lines o dieen lo egies have been used in seveal equaions o calculae esevoi paaees in case one o he lo egies is issing o o veiicaion puposes. These equaions ae pesened in he olloing pocedue: Sep 1 - Plo he pessue dieence ΔP and he pessue deivaive (*ΔPʹ vesus ie on log-log plo and ideniy dieen lo egies. Sep - Calculae he acue peeabiliy o Eq. (38. Sep 3 - Calculae ω and as oulined beoe. Sep 4 - I he ansiion occu duing linea lo egie and he o paallel saigh lines o slope 0.5 obseved, veiy ω using he olloing o equaions: ω (* p L1 (* p L1 (44 hee L1 sands o he linea lo a he oal syse doaed egie, and L1 sands o acue soage doaed lo egie. LUSi ω LUSi (45 hee LUSi sands o he inesecion poin beeen he lae ansiion peiod uni slope line and oal syse doaed lo line, and LUSi sands o he inesecion poin beeen he lae ansiion peiod uni slope line and he acue soage doaed lo peiod. Sep 5 - Read he value o (*Δpʹ a ie 1 h o he linea lo line (eapolaed i necessay, (*Δpʹ L1. Sep 6 - Calculae he acue hal-lengh,, o he linea lo saigh line (Eq. 35. Figue 6: Pessue deivaive esponse in a veical acued esevoi ih pseudo seady sae ine poosiy lo. The ansiion occus duing he linea lo peiod. ISSN: 47-0518 Volue Issue 3 100011
Ciaion: Gaal Rezk M (016 Analysis o Pessue Tansien Tess in Naually Facued Resevois. Oil Gas Res : 11. doi: 10.417/47-0518.100011 Page 8 o 10 I he linea lo no obseved (due o ellboe soage o noise, hen acue hal-lengh can be calculaed o he hal slope pessue Δp insead o pessue deivaive as ( p *( * p. so, L1 L1 4.064qB ( p ωh K ( ΦC L1 (46 hen da a saigh line o slope 0.5 paallel o he pessue saigh line o coss he (*Δpʹ L1. Sep 7 - Deee he inesecion beeen he linea and adial lo line LRi o he log- log plo o he deivaive (*Δpʹ cuve. Sep 8 - Calculae he aio o equaion: (o squae geoey A k e LRi k 107.1( Φc ω (47 Copae his aio ih he peviously calculaed values o and K. I he o aios ae nealy equal, hen he values ae coec. I hey ae dieen, shi one o boh saigh lines hen epea he pevious seps unil hei values appoach. Sep 9 - Deee he value o (*Δpʹ PSS1 o he pseudoseady sae line and ind he dainage aea A o: qb A (48 4.7( * p h( ΦC pss1 Sep 10 - Read he inesecion ie o he ininie acing line and he pseudoseady sae line ( RPSSi o he plo and calculae he dainage aea A: K RPSSi A (49 301.77( ΦC Aeas o seps 10 and 11 should be equal. I hey ae no equal, shi he lines le o igh and epea he calculaions. Sep 11 - Deee he inepoosiy lo paaee ae siulaion ( o: (50 Sep 1 - Veiy ( using he lae ansiion peiod uni slope line by he olloing equaions: ( ΦC (51 0.000637K RUSi hee RUSi sands o he inesecion ie beeen he lae ansiion peiod uni slope line and he ininie acing line. ( Φ Π C ω 0.000637K LUSi 0.5 (5 hee LUSi sands o he inesecion ie beeen he lae ansiion peiod uni slope line and he acue soage doaed linea lo peiod. 0.5 Π( ΦC (53 0.000637K LUSi hee LUSi sands o he inesecion poin beeen he lae ansiion peiod uni slope line ih he oal syse doaed linea lo peiod. Sep 13 - Calculae skin using Eq. (39. Sep 14 - Calculae he shape aco o he value o Δp and (*Δpʹ coesponding o any convenien ie duing he pseudoseady sae lo egie using Eq. (43. Ininie Conduciviy Facue The pessue and pessue deivaive obained o ininie conduciviy acue ae he sae as he unio lu acue ecep o a ouh doaed lo egie called bi-adial lo. This lo egie can be ideniied by a saigh line o slope 0.36. I coesponds o he ansiion peiod beeen he ealy ie linea lo egie and he ininie acing adial lo egie 4. The chaaceisics o he linea, adial, and pseudoseady sae lo peiods ae he sae as illusaed ealie in he case o unio lu acue. The chaaceisics o he bi-adial lo egie ae as olloing: Bi-adial Flo Peiod: The bi-adial lo egie can be ideniied o he pessue deivaive uncion by a saigh line o slope 0.36. Hoeve, i canno be ideniied o he pessue uncion. In case he linea lo line is no obseved, he bi-adial lo line is used o deee he acue hal lengh. The pessue deivaive equaion descibing he bi-adial lo peiod is [10]: e 0.7 DA 0.36 DA * P' D 0.769 ( ( ω (54 By aking logaih o boh sides o he above equaion, he diensional o is: ' ( ( log * P 0.36log + log( BR (55 hee 5.589qB K e 0.7 0.36 BR ( ( 0.36 (56 ω Kh ( ΦC A Solving o he acue hal-lengh, 1.389 0.5 e qb K 10.914 (57 0.5 ω K h(* p' BR1 ( ΦC A I all he lo egies ha ee ound in he case o unio lu acue ae available, use he pevious pocedue o he unio lu acue o analyze he pessue es o ininie conduciviy acue. Hoeve, in case he linea lo line is eihe oo sho o no obseved, he olloing pocedue 4 can be used: Sep 1 - Plo he pessue dieence ΔP and he pessue deivaive (*ΔPʹ vesus ie on log-log plo and ideniy dieen lo egies. Sep - Deee he value o (*ΔPʹ R o he ininie-acing adial lo line. Sep 3 - Deee he acue peeabiliy as discussed beoe in unio lu acue. Sep 4 - Calculae ω and as oulined beoe. Sep 5 - Veiy ω as discussed beoe in he case o he unio lu acue. Sep 6 - Read he value o (*Δpʹ PSS1 coesponding o he pseudoseady sae line and deee he dainage aea (A. Sep 7 - Read he inesecion ie o he ininie acing line and he pseudoseady sae line ( RPSSi o he plo o deee A (Eq. 49. ISSN: 47-0518 Volue Issue 3 100011
Ciaion: Gaal Rezk M (016 Analysis o Pessue Tansien Tess in Naually Facued Resevois. Oil Gas Res : 11. doi: 10.417/47-0518.100011 Page 9 o 10 Aeas o sep 6 and 7 should be equal, i no shi le o igh and epea he seps. Sep 8 - Read he value o (*Δpʹ a ie 1h o he bi-adial lo line, (*Δpʹ BR1. The bi-adial lo line can be eapolaed i necessay. Sep 9 - Deee he acue hal lengh,, o Eq. (57 Sep 10 - Calculae he aio using he values o sep 3 and 9. k Sep 11 - Fo he plo, ead he ie o inesecion o he adial lo and he bi-adial lo line, RBRi, hen deee he aio o 4 : (o squae geoey e Y e k RBRi k 1147 C ω ( Φ (58 I Sep 10 and 11 ae he sae, hen and K ae coec and i hey no he sae, shi one o boh lines (bi-adial and ininie acing and epea all he pocedue. Sep 1 - Deee he inepoosiy lo coeicien ae siulaion ( by Eq. (50. Sep 13 - Veiy ( as peviously oulines. Sep 14 - Calculae he skin aco by Eq. (39. Sep 15 - Calculae he shape aco o he value o Δp and (*Δpʹ coesponding o any convenien ie duing he pseudoseady sae lo peiod by Eq. (43. Case Bi e al. [11] inepeed he pessue all o es peoed on a ell. The ell has been acidized seveal ies beoe acue siulaing in Novebe 198. The es as peoed using don hole shu-in device and pessue gauges. The pessue all o es daa ae shon in Table 3 and he olloing esevoi and luid daa ae knon: h135 C 9.5*10-6 psi -1 0.5 Bo1 esbbl/stb Δ, h Ps, psia Δ, h Ps, psia 0 1183 0.066358 117.6 0.000556 1171.7 0.08385 111.6 0.00111 1170.4 0.3887 1073 0.001667 1169.4 0.7131 1060 0.00 1168.6 0.87794 1049.9 0.005 1168.3 1.044 1041.7 0.003055 1167.4 1.3701 1034.7 0.003611 1166.6 1.859 103.4 0.004167 1166.3445 1010.3 0.0047 1165..867 1000 0.00578 1164.6 3.3056 991.6 0.005555 1164. 3.7813 984.4 0.006111 1163.7 4.538 978.4 0.006667 1163 5.1893 97.7 0.0075 116 6.113 963.1 0.0083 1161.3 1.87 955 0.0157 1155.1 19.559 915.4 0.016668 1154 3.884 89.7 0.019997 1151.6 7.65 881 0.017 1150 9.04 873.1 0.03335 1143. 31.079 868.7 0.049703 1134.7 33.485 859.5 Table 3: Pessue all o es daa o Case. q1050 bbl/day 0.7 cp K3.33 d φ0.085 Figue 7 shos he log-log plo o pessue and pessue deivaive vesus ie. I is clea o he log-log plo ha he ansiion peiod occus ealy duing he linea lo egie. Consequenly, o paallel saigh lines o hal-slope appea. The is line esuled o he epansion o he acue neok, hile he second line epesens he oal syse behavio. Also, he ininie acing adial lo line is deeced bu no long enough. The pessue deivaive plo ehibi a unique behavio o a hydaulically acued ell in a naually acued esevoi. Bi e al. [11] analyzed he pessue behavio o his ell by using he ype cuves o hoogeneous esevois ih hydaulic acue. Theeoe, hey canno esiae he values o he elaive soaiviy and he inepoosiy lo coeicien. Fo Figue 7: (*ΔPʹ 110 psi a 3 h ΔP 90 psi a (*ΔPʹ L1 1 psi a (*ΔPʹ L1 100 psi a Diec Synhesis echnique is used o esiae he esevoi paaees as olloing: The acue peeabiliy can be calculaed o he ininie acing adial lo line: 70.6qBo 70.6*1050* 0.7 *1 K 3. 49 d h( P * 135*110 The elaive soaiviy can be calculaed o he o paallel hal slope saigh lines: ω (* p L1 1 ( * p L1 100 0.0441 Calculae he acue hal-lengh o he linea lo saigh line:.03qb p ωh K ( ΦC ( * 1 L.03*1050*1 0.7 375 6 100* 0.0441*135 3.49* 0.085* 9.5*10 The acue hal-lengh can also be calculaed o he oal syse doaed lo peiod: Figue 7: Pessue and pessue deivaive daa vs. o case. ISSN: 47-0518 Volue Issue 3 100011
Ciaion: Gaal Rezk M (016 Analysis o Pessue Tansien Tess in Naually Facued Resevois. Oil Gas Res : 11. doi: 10.417/47-0518.100011 Page 10 o 10.03qB * p h K ( ΦC ( 1 L.03*1050*1 0.7 135* 1 3.49*0.085*9.5*10 6 375 The inepoosiy lo paaee ae siulaion ( can be esiaed o he oiginal inepoosiy lo paaee: Whee, S T 0.000637k ωln1 / ω 6 0.085* 9.5*10 * 0.5 0.044* ln1/ 0.044 0.000637 * 3.49 1.34 5.6*10-6 Theeoe, 6 375 (5.6*10 * 1.6 0.5 Calculae skin by eading any convenien poin duing he ininie acing peiod: ( p K R ( R * p ( ΦC S 0.5 ln + 7.43 R 90 3.49* 3 0.5 ln 7.43 5.7 6 + 110 0.085* 9.5*10 * 0.7 *0.5 The dieences beeen he esuls o he ype cuve aching obained by Bi e al. [11] and he esuls o diec synhesis echnique ae shon in Table 4. The acue peeabiliy o ype cuve aching nealy he sae as ha o diec synhesis, hile he acue hal lenghs ae vey close. Conclusions 1. The use o pessue deivaive plos ipoved he analysis o ell es daa. Dieen lo egies can be ideniied on he deivaive log-log plos. Type cuve aching can give good esuls in case all o he lo egies ae ideniied.. In his sudy, Tiab diec synhesis echnique as shon o be accuae and siple. I gave diec esiaes o esevoi paaees and acue chaaceisics by using a log-log plo o pessue and pessue deivaive daa ihou ype cuve aching. 3. In case o high ellboe soage, he convenional sei-log analysis gives inaccuae esuls and canno esiae all naually acued esevoi paaees. Paaee Type cuve aching by Bi e al. Diec synhesis echnique K, d 3.33 3.49, 44 375 ω 0.0441 5.6*10-6 1.6 Skin -5.7 Table 4: Copaison o Resuls o Case. 4. When no all he lo egies ae ideniied, ype cuve aching gives non-unique soluion. Hoeve, he diec synhesis echnique gives accuae esuls o he naually acued esevoi paaees and acue popeies. 5. The diec synhesis ehod, shoed accuae esuls copaed o coecial soae aching. I can be used o calculae he esevoi and acue popeies in case o a ell cossed by a unio lu o ininie conduciviy acue. 6. In case o naually acued esevois ih a veical acue, i he ansiion peiod occus duing he linea lo, o paallel saigh lines o slope 0.5 appea on he pessue deivaive plo. This pessue deivaive behavio can be used in calculaing dieen esevoi paaees. Reeences 1. Tiab D (1989 Diec Type-Cuve Synhesis o Pessue Tansien Tess. SPE 1899 pepaed o pesenaion a he SPE Join Rocky Mounain Regional/ Lo Peeabiliy Resevois, Syposiu and Ehibiion held in Denve Coloado.. Tiab D (1994 Analysis o Pessue and Pessue Deivaive ihou Type Cuve Maching; Veically Facued Wells in Closed Syses. Jounal o Peoleu Science and Engineeing, pp: 33-333. 3. Engle T, Tiab D (1996 Analysis o Pessue and Pessue Deivaive ihou Type Cuve Maching; Naually Facued Resevoi. Jounal o Peoleu Science and Engineeing, pp: 17-138. 4. Jalal F (000 Applicaion o Tiab s diec synhesis echnique o uli-laye naually acued esevois. PhD disseaion Noan Oklahoa. 5. Waen JE, Roo PJ (1963 The behavio o naually acued esevois. Soc Pe Eng J 63 3: 45-55. 6. Engleq T (1995 Inepeaion o pessue ess in naually acued esevois by he diec synhesis echnique. PhD disseaion Noan Oklahoa. 7. Boude D, Gingaen AC (1980 Deeaion o Fissue Volue and Block Size in Facued Resevois by Type-Cuve Analysis. SPE Pape 993 pesened a he Annual Technical Coneence and Ehibiion Dallas T Sepebe, pp: 1-4. 8. Ahed T (010 Resevoi engineeing handbook (4hedn Aseda: Gul Poessional Pub. 9. Liu Ci, Yang J (1990 Tansien Pessue Behavio Wih Regad To Wellboe Soage and Skin Eecs o A Veically Facued Well In Dual-Poosiy Syse SPE. 10. Tiab D (1994 Analysis o Pessue and Pessue Deivaive Wihou Type Cuve Maching; Veically Facued Wells In Closed Syses. Jounal o Peoleu Science and Engineeing, 11: 33-333. 11. Bi K, Benne CO (1985 Deeaion o Facue Conduciviy in Modeae- Peeabiliy esevois using he Bilinea lo conceps. SPE 14165 This pape as pesened a he 60* Annual Technical Coneence and Ehibiion o he SPE Las Vegas NV Sepebe pp: -5. Ciaion: Gaal Rezk M (016 Analysis o Pessue Tansien Tess in Naually Facued Resevois. Oil Gas Res : 11. doi: 10.417/47-0518.100011 ISSN: 47-0518 Volue Issue 3 100011