PRESSURE DRAWDOWN EQUATIONS FOR MULTIPLE-WELL SYSTEMS IN CIRCULAR-CYLINDRICAL RESERVOIRS

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1 VOL. 8, NO. 7, JULY 013 ISSN ARPN Jounal of Engineeing and Applied Sciences Asian Reseac Publising Netwok (ARPN). All igts eseved. PRESSURE RAWOWN EQUATIONS FOR MULTIPLE-WELL SYSTEMS IN CIRCULAR-CYLINRICAL RESERVOIRS Jing Lu 1, Tao Zu 1, An in, jebba Tiab 3 and Feddy H. Escoba 4 1 epatment of Petoleum Engineeing, te Petoleum Institute, Abu abi, United Aab Emiates Sclumbege, Midland, TX 3 Te Univesity of Oklaoma, Noman, OK, USA 4 Univesidad Sucolombiana/CENIGAA, Av. Pastana-Ca 1, Neiva, Huila, Colombia fescoba@usco.edu.co ABSTRACT Cuently, in te oil industy, a field usually contains seveal wells poducing fom te same dainage domain, and eac well will ave an effect on te pessue at ote wells. Fo an infinite-acting multiple-wells system, pessue dawdown equation is aleady obtained by using supeposition pinciple. Tis pape pesents pessue dawdown equations of a multiple-wells system in a cicula cylindical esevoi wit constant pessue oute bounday. Te poposed equations povide fast analytical tools to evaluate te pefomance of multiple wells, wic ae located abitaily in a cicula cylinde esevoi, and ae poducing at diffeent ates. Hee, it is also examined te pessue dawdown esponse of a specific well located in a system of poducing wells. Te intefeence effects of neaby poducing wells on te pessue dawdown esponse ae investigated. Te poposed equations ae illustated by numeical examples. It is concluded tat, fo a given multiple-wells system in a cicula cylindical esevoi, well patten, well spacing, skin facto, flow ates and well off-cente distances ave significant effects on single well pessue dawdown beavio. Because te esevoi is unde edge wate dive, te oute bounday is at constant pessue, wen poducing time is sufficiently long; steady-state is definitely eaced. Keywods: multiple-wells, supeposition pinciple, exponential function, esevoi sape, pessue dop. 1. INTROUCTION It is ae to find a esevoi being poduced fom only a single well. A field usually contains seveal wells poducing fom te same dainage domain, and eac well will ave an effect on te pessue at ote wells. If we ave one well poducing at a constant ate, te flowing bottomole pessue in tat well is a function of its own poduction as well as te poduction fom suounding wells. Fo an infinite-acting multiple-wells system, pessue dawdown equation is aleady obtained by using supeposition pinciple (Lee, et al., 003). It is not unusual duing dilling, completion, o wokove opeations fo mateials suc as mud filtate, cement sluy, o clay paticles to ente te fomation and educe te pemeability aound te wellboe. Tis effect is commonly efeed to as "wellboe damage" and te egion of alteed pemeability is called te "skin zone." Tis zone can extend fom a few inces to seveal feet fom te wellboe and causes an additional pessue loss in te fomation. Because te adius of skin zone is small, we can always assume steady-state adial flow in tis alteed zone and te steady-state adial flow equations fo te pessue dops ae applicable in tis egion. As given by Lee, Rollins, and Spivey, (003), if we efe to te pessue dop in te skin zone as P S, it yields, qµ B Ps = s (1) π k If a fully penetating off-cente vetical well is located in an anisotopic cicula cylinde esevoi wit constant pessue oute bounday, and te off-cente distance is o, te dainage adius is e, in te constant pessue oute bounday dominated flow egime, te dimensionless flowing bottom ole pessue is given by Owayed and Lu (009): e w e [exp P e w = + 4s exp e 4 t Wee is exponential integal function, S is mecanical skin facto. Te definitions of dimensionless vaiables ae given in Appendix A. Reaange Equation (), te flowing bottomole pessue dawdown equation is obtained, e w e 4 qµ B t Pini Pw = 4π k exp e exp e + s wee Equation (1) is used, and P ini is initial esevoi pessue. Te goal of tis study is to pesent pessue dawdown equations of a multiple-wells system in a () (3) 459

2 VOL. 8, NO. 7, JULY 013 ISSN ARPN Jounal of Engineeing and Applied Sciences Asian Reseac Publising Netwok (ARPN). All igts eseved. cicula cylinde esevoi wit constant pessue oute bounday. Taking fully penetating vetical wells as unifom line sinks, easonably accuate pessue dawdown equations ae obtained fo a multiple-wells system in te bounday dominated flowing peiod. Te intefeence effects of neaby poducing wells on a specific well esponse ae investigated.. WELL AN RESERVOIR MOELS Figue-1 is a scematic of a multiple-wells system, a numbe of poducing wells dain an anisotopic cicula cylinde esevoi wit eigt and adius e, and te j-t well is located j away fom te cente of te dainage cicle, in pola coodinate system, te j-t well is located ( j, θ j ). Te following assumptions ae made: a) At time t = 0, pessue is unifomly distibuted in te esevoi, equal to te initial pessue P ini, all wells begin to poduce at time t = 0, and te bounday dominated flow is eaced at te same time fo eac well. b) Te wells ae paallel to te z diection wit a poducing lengt equal to tickness, te top and bottom esevoi boundaies ae impemeable. Te oute bounday pessue P e is always equal to initial pessue P ini. c) A single pase fluid, of small and constant compessibility c f, constant viscosity µ, and fomation volume facto B, flows fom te esevoi to te wells. Fluid popeties ae independent of pessue. Gavity foces ae neglected. d) Only te mecanical skin facto due to fomation damage o stimulation is consideed, ote additional pessue dops ae ignoed. e) Te esevoi as constant adial pemeability k, vetical pemeability k v, and tickness. In any given time inteval, te numbe of wells, n, tei locations ( j, θ j ), and te skin factos, s j, ae consideed constant, and te wellboe adio, R wj ae identical fo eac well..1. Equations fo initial condition and bounday conditions Te initial esevoi pessue is constant, P t 0 P ini = = (4) And te top and bottom esevoi boundaies ae impemeable., P P z z= 0 = 0 z z= = 0 (5) Te oute bounday pessue is always equal to initial esevoi pessue P ini... imensionless equations fo multiple-wells system In a multiple-wells system as sown in Figue-1, te numbe of wells is n, assume te wellboe adii ae identical, ten te dimensionless flowing bottomole pessue of te obsevation well (assume it is k-t well) is n Pw, k = Wkj, ( k = 1,,3,..., n) j= 1 wee (7) q j τ j =, ( j = 1,,3,..., n) (8) qef Wen j = k, ten e k w k e 4 t k Wkk = τ k + 4sk exp k e exp e wee s k is mecanical skin facto of k-t well (te obsevation well), and Wen j k, ten Wkj = τ jk exp[ i( θj θk )] e RE exp e tχ ( j + k e) j exp[ χ exp[ i( θj θk)]] 1 RE exp[ i( θj θk)] [ j + k jk cos( θj θk )] (9) (10) Wee i = -1, θ j is te wellboe location angle of j-t well, j is off - cente distance of j-t well, RE is eal pat opeato, (if a 1, a, b 1, b ae eal numbes, ten z = (a 1 + b 1 i) (a + b i) is a complex numbe, RE (z) = a 1 a - b 1 b ) and R jrk χ = (11) Te definitions of dimensionless vaiables in te above and following equations ae te same as given in Appendix-A. P( = e) = Pini (6) 460

3 VOL. 8, NO. 7, JULY 013 ISSN ARPN Jounal of Engineeing and Applied Sciences Asian Reseac Publising Netwok (ARPN). All igts eseved. exp[ i( θ j θk )] e RE cos( θ j θk ) Wkj = τ j{ t exp cos( θ j θk) e exp e cos[( θ ) j θk sin ( θ j θk)] cos( θ j θk) (17) Figue-1. Multiple-wells system. y At ealy time, (t 5) te flowing bottomole pessue of te obsevation well is only a function of its own poduction as if suounding wells do not exist; and because e e k 0, k e exp k e exp e 0 k (1) o o x Tus, Equation (7) can be simplified below: 4 P, w w k = τk + 4sk τk ln( t) + ln R w 4sk = τ k[ ln ( t) ln ( w) ] + 4sk (13) If τ k = 1, te dimensionless pessue deivative at ealy time is, dpw, k dpw, k = t = (14) d(ln t) dt If all wells ave te identical off-cente distance, and if n 3, te wells ae located at vetexes of a egula polygon inside a cicula esevoi, as sown in Figue-, we ten assume: j = o, ( j = 1,,3,..., n ) (15) Tus wen j = k, Equation (9) educes to e w e 4 t Wkk = τ k + 4sk exp( e ) exp e And wen j k, Equation (10) educes to (16) Figue-. Multiple wells located at vetexes of a egula polygon..3. imensionless equations fo two-wells system If only two wells ae located in te esevoi wit identical off-cente distance o, witout losing geneality, we may always assume te fist well is te obsevation well, wic is located at ( o, 0); te second well is located at ( o, θ), ten e w,1 P e w = τ1 exp + 4s1 e exp e 4 t exp( iθ ) RE e cos( θ ) t + τ e exp cos( θ ) e exp cos θ sin ( θ) cos( θ) (18) 461

VOL. 8, NO. 7, JULY 013 ISSN 1819-6608 ARPN Jounal of Engineeing and Applied Sciences 006-013 Asian Reseac Publising Netwok (ARPN). All igts eseved. www.apnjounals.

4 VOL. 8, NO. 7, JULY 013 ISSN ARPN Jounal of Engineeing and Applied Sciences Asian Reseac Publising Netwok (ARPN). All igts eseved. Wee s 1 is te mecanical skin facto of te fist well (obsevation well). If θ = π /, ten Equation (18) educes to e w Pw,1 = τ 1 + 4s1 e e e exp exp i e RE t + τ e exp e sin (19) And if te two wells ae located symmetic wit espect to te cicula cente, i.e., te two wells ae on a diamete wit identical off-cente distance o, (θ = π) as sown in Figue-3, ten Equation (18) educes to e w e e e exp exp } + 4s 1 Pw,1 = τ 1 + e + τ { t e exp + + e exp e 4 t e Well 1 Well o y o x (0) e ln e, 4 t e exp e exp e 0 () Tus, if τ 1 = τ, wen late is vey long, Equation (0) educes to, 4 4,1 ln e P w = 4 w (3) 3. PRESSURE RAWOWN ANALYSIS In eac case of te following examples, we always assume all te wells ae fully penetating wit identical wellboe adius w, ave identical off-cente distance o, and identical mecanical skin facto s, and if n 3, te wells ae located at vetexes of a egula polygon as sown in Figue-. In Examples 1, and 4, te well flow ates ae identical, tus evey well as te same pessue beavio in eac case, any well can be cosen as te obsevation well. Example-1: w = 0.01, = 10, e = 0, s =, τ = 1 Figue-4 sows tat te numbe of wells as significant effects on dimensionless flowing bottomole pessue wen all te wells ae poducing at te identical flowing ates, (τ = 1). At ealy times, te flowing bottomole pessue of te obsevation well is only a function of its own poduction as if suounding wells do not exist; wen poducing time is long, te influence fom suounding wells appeas. At a given time t, if te numbe of wells n inceases, P w also inceases, i.e., te wellboe pessue dop P i - P w (t) at te obsevation well inceases wit n, wic indicates te intefeence effect fom suounding wells is moe ponounced. Because te oute bounday is at constant pessue, wen te poducing time is sufficiently long, steady state will be eaced, P w becomes a constant, i.e., te flowing bottomole pessue P w (t) eaces a constant value, does not cange wit time. Figue-3. Symmetic two-wells system. At late time, (t 100), it is obtained, e w ln e, 4 t w e exp e exp e 0 (1) Figue-4. Te effects of numbe of wells on P w. 46

i.e., te wellboe pessue dop P i - P w (t) at te obsevation well deceases wen o inceases, wic indicates te effect fom constant pessue oute bounday is moe ponounced.

If two wells wit identical off-cente distance R o ae poducing at te identical flowing ates, (τ = 1), te fist well is at ( o, 0 ), te second well is at (R o, θ).

5 VOL. 8, NO. 7, JULY 013 ISSN ARPN Jounal of Engineeing and Applied Sciences Asian Reseac Publising Netwok (ARPN). All igts eseved. Example-: w = 0.01, e = 0, s =, τ = 1 Figue-5 sows tat fo a given numbe of wells and at a given time t, all te wells ae poducing at te identical flowing ates, (τ = 1), if off-cente distance o inceases, P w deceases, i.e., te wellboe pessue dop P i - P w (t) at te obsevation well deceases wen o inceases, wic indicates te effect fom constant pessue oute bounday is moe ponounced. At a given off-cente distance o and at a given time t, if te numbe of wells n inceases, P w also inceases. If two wells wit identical off-cente distance R o ae poducing at te identical flowing ates, (τ = 1), te fist well is at ( o, 0 ), te second well is at (R o, θ). Figue-7 sows tat at a given time t and at given off-cente distance o, if θ deceases, P w inceases, i.e., te wellboe pessue dop P i - P w (t) at te fist well inceases wen θ deceases, wic indicates te intefeence effects fom te second well is moe ponounced. Figue-7. Te effects of well location angle on P w. Figue-5. Te Effects of off-cente distance on P w. Example-3: w = 0.01, e = 0, s =, θ = π / If two wells wit identical off-cente distance o, te fist well is at ( o, 0 ), te second well is at ( o, π / ), and define flow ate atio τ = q / q 1. Figue-6 sows tat at a given time t and at given off-cente distance o, if τ inceases, P w also inceases, i.e., te wellboe pessue dop P i - P w (t) at te fist well inceases wit τ, wic indicates te intefeence effect fom te second well is moe ponounced wen its flow ate inceases. At a given flow ate atio τ and at a given time t, if te off-cente distance o inceases, P w deceases. CONCLUSIONS Assuming no wate encoacment, and no multipase flow effects, and taking edge wate as constant pessue bounday, and assuming top and bottom esevoi boundaies ae impemeable, te following conclusions ae eaced: a) Te poposed pessue dawdown equations povide a easonably accuate and fast analytical tool to evaluate te pefomance of multiple-wells system; b) Fo a given numbe of wells, pemeability, well patten, well spacing, skin facto, flow ates and well off-cente distances ave significant effects on single well pessue tansient beaviou; c) Wit constant pessue oute bounday, wen poducing time is sufficiently long, steady-state is definitely eaced; d) If all te wells ae fully penetating wit identical mecanical skin facto S, and ae located at vetexes of a egula polygon, at a given time t, if te numbe of wells n inceases, P w also inceases; if off-cente distance o inceases, P w deceases; fo te two-wells system, if flow ate atio τ inceases, P w inceases, and if wellboe location angle θ deceases, P w inceases. Nomenclatue Figue-6. Te effects of flow ate atio on P Wd. Example-4: w = 0.01, e = 0, = 10, S = B = fomation volume facto, L 3 / L 3 = payzone tickness, L i = -1 k = effective pemeability, L n = numbe of wells. P e = esevoi oute bounday pessue, m/ (Lt ) 463

6 VOL. 8, NO. 7, JULY 013 ISSN ARPN Jounal of Engineeing and Applied Sciences Asian Reseac Publising Netwok (ARPN). All igts eseved. P ini = initial esevoi pessue, m/ (Lt ) P w,j = flowing bottomole pessue of j-t well in te multiple wells system, m/(lt ) q j = flow ate of j-t well in te multiple wells system, L 3 / t q ef = efeence flow ate, L 3 / t w = wellboe adius, L e = adius of te cicula cylinde esevoi, L j = off - cente distance of j-t well in te multiple wells system, L RE = eal pat opeato. S = mecanical skin facto, dimensionless. t = time, t Geeks θ j = wellboe location angle of j-t well in te multiple wells system, adians. µ = fluid viscosity, m/ (Lt) x = a function defined by Equation (11). τ = well flow ate atio defined by Equation (8) APPENIX-1. EFINITION OF IMENSION-LESS VARIABLES y x x =, y =, z k z =, = kv j o =, j =, e, w e = H w = t 4kt t k kv (A-1) (A-) = (A-3) φµc 8 π kp ( ini Pw) Pw = (A-4) qef µ B Suffices = dimensionless ini = initial j,k = well index = adial t = total v = vetical w = well x, y, z = coodinate indicatos ACKNOWLEGEMENTS Te autos would like to tank te senio management of ANOC and Te Petoleum Institute fo tei continuous suppot and fo pemission to publis tis wok. REFERENCES Lee J., Rollins J. B. and Spivey J. P Pessue Tansient Testing. SPE Textbook Seies, Vol. 9, Society of Petoleum Enginees, Ricadson, Texas, USA. Owayed F. and Lu J Pessue op Equations fo a Patially Penetating Vetical Well in a Cicula Cylinde ainage Volume. Matematical Poblems in Engineeing. Mac. pp Lu J. and Tiab Poductivity Equations fo an Off - Cente Patially Penetating Vetical Well in an Anisotopic Resevoi. Jounal of Petoleum Science and Engineeing. 60:

ANALYSIS OF PRESSURE VARIATION OF FLUID IN AN INFINITE ACTING RESERVOIR

ANALYSIS OF PRESSURE VARIATION OF FLUID IN AN INFINITE ACTING RESERVOIR Nigeian Jounal of Technology (NIJOTECH) Vol. 36, No. 1, Januay 2017, pp. 80 86 Copyight Faculty of Engineeing, Univesity of Nigeia, Nsukka, Pint ISSN: 0331-8443, Electonic ISSN: 2467-8821 www.nijotech.com