MODELING OF PERFORMANCE BEHAVIOR IN GAS CONDENSATE RESERVOIRS USING A VARIABLE MOBILITY CONCEPT

Transcription

1 MOELING OF PERFORMANCE BEHAVIOR IN GAS CONENSATE RESERVOIRS USING A VARIABLE MOBILITY CONCEPT A Thesis y BENTON WAE WILSON Sumie o he Office of Gauae Suies of Texas A&M Univesiy in aial fulfillmen of he equiemens fo he egee of MASTER OF SCIENCE eceme 003 Majo Sujec: Peoleum Engineeing

2 MOELING OF PERFORMANCE BEHAVIOR IN GAS CONENSATE RESERVOIRS USING A VARIABLE MOBILITY CONCEPT A Thesis y BENTON WAE WILSON Sumie o he Office of Gauae Suies of Texas A&M Univesiy in aial fulfillmen of he equiemens fo he egee of MASTER OF SCIENCE Aove as o syle an conen y: Thomas A. Blasingame (Chai of Commiee W. John Lee (Meme Roe R. Beg (Meme Hans C. Juvam-Wol (Ineim Hea of eamen eceme 003 Majo Sujec: Peoleum Engineeing

3 iii ABSTRACT Moeling of Pefomance Behavio in Gas Conensae Resevois Using a Vaiale Moiliy Conce. (eceme 003. Benon Wae Wilson, B.S., Geogia Insiue of Technology Chai of Avisoy Commiee:. Thomas A. Blasingame The oose wo ovies a conce fo eicing well efomance ehavio in a gas conensae esevoi using an emiical moel fo gas moiliy. The oose moel eics he ehavio of he gas emeailiy (o moiliy funcion in he esevoi as conensae evolves an he gas emeailiy is euce in he nea-well egion ue o he "conensae an". The oose moel is ase on osevaions of simulae esevoi efomance an eics he ehavio of he gas emeailiy ove ime an aial isance. This moel is given y: - min ( max min - ex The oose conce has oenial alicaions in he evelomen of a essue-ime-aius soluion fo gas conensae esevois exeiencing his ye of moiliy ehavio. We ecognize ha he oose conce (i.e., a aially-vaying gas emeailiy is ovesimlifie, in aicula, i ignoes he iffusive effecs of he conensae (i.e., he viscosiy-comessiiliy ehavio. Howeve, we have effecively valiae he oose moel using lieaue esuls eive fom numeical simulaion. This new soluion is esene gahically in he fom of "ye cuves." We oose ha he "ime" fom of his soluion e use fo alicaions in well es analysis. Pevious evelomens use fo he analysis of well es aa fom gas conensae esevois consie he aial comosie esevoi moel, which uilizes a "se change" in emeailiy a some aial isance away fom he welloe. Using ou oose soluion we can visualize he effec of he vaying gas emeailiy in ime an aius (a suie of (imensionless aius an ime foma los ae ovie. In sho, we can visualize he evoluion of he conensae zone as i evolves in ime an aial isance. A limiaion is he simlifie fom of he g ofile as a funcion of aius an ime as well as he eenence/aoiaeness of he -aamee. While we susec ha he -aamee eesens he influence of oh flui an oc oeies, we o no examine how such oeies can e use o calculae he -aamee.

4 iv EICATION My wife Moya, an hee chilen, Biany, Colon, an Coly Than you fo he suo an unconiional love My family, as an esen mos all who have eihe woe in he oilfiel o have een maie o i. My fahe, Jac Wilson. an he many als we have ha "comleing a well".

5 v ACKNOWLEGEMENTS I woul lie o exess my aeciaion an sinceely han he following:. Tom Blasingame, chai of my avisoy commiee fo his unique mix of inellec an acicaliy in his wo, which I highly amie. Also, my ee gaiue fo his fienshi, aience an willingness o shae his nowlege, ime, an exeience wih myself an ohes. Than you fo he song commimen o ou eseach.. Rosalin Ache fo enhusiasm in he eaching, as well as he aience an ailiy o mae ifficul oics much cleae. My sincees gaiue an amiaion fo he ligh-heae esonaliy an he exensive nowlege. Than you fo he valuale assisance wih his eseach.. John Lee fo his shaing of exeience an nowlege hough his eaching an ulishe wos, an fo seving as a meme on my avisoy commiee.. Bo Beg fo seving as a meme of my avisoy commiee The Nuclea Technology gou a Bae Hughes, aiculaly. ayl Tca,. Allen Gilchis, an. Alan McFall, as well as. on Olive an. Meha Micael, oh who have usue ohe ahs fom Bae Hughes fo suo in my caee, aience, an elief in my ailiies. Bae Hughes, Inc. fo oviing he suo, ime, an financial acing o mae his eneavo ossile.

6 vi TABLE OF CONTENTS Page CHAPTER I INTROUCTION.... Reseach Polem.... Reseach Ojecives....3 Saemen of he Polem an Summay of a Poose Soluion Ouline of Thesis... 0 CHAPTER II LITERATURE REVIEW.... Raial Comosie Resevoi Sysem.... Geneal Conces (, Pefomance in Gas Conensae Resevois Ohe Soluions/Consieaions... CHAPTER III CHAPTER IV CHAPTER V EVELOPMENT OF AN ANALYTICAL PRESSURE SOLUTION FOR THE CASE OF A PERMEABILITY PROFILE THAT VARIES IN TIME AN RAIAL ISTANCE Conce of a Moiliy Pofile ha Vaies in Time an Raial isance Alicaion of he Bolzmann Tansfomaion o he Raial iffusiviy Equaion evelomen of Pessue eivaive Soluions Welloe Soage an Sin Effecs... 3 VALIATION OF AN ANALYTICAL PRESSURE SOLUTION FOR THE CASE OF A PERMEABILITY PROFILE THAT VARIES IN TIME AN RAIAL ISTANCE GAS CONENSATE RESERVOIRS Comaison of he New Soluion an Soluions fo he Sealing Fauls an Raial Comosie Resevoi Cases Valiaion of he New Soluion Lieaue aa... 3 SUMMARY, CONCLUSIONS, AN RECOMMENATIONS FOR FUTURE WORK Summay Conclusions Limiaions an Recommenaions... 7 NOMENCLATURE... 8 REFERENCES... 50

7 vii APPENIX A APPENIX B Page ERIVATION OF THE PRESSURE ERIVATIVE FUNCTIONS WITH RESPECT TO TIME AN RAIUS FOR THE CASE OF A RAIALLY VARYING PERMEABILITY PROFILE (EQUIVALENT LIQUI CASE... 5 AN APPROXIMATE TECHNIQUE FOR THE IRECT AITION OF WELLBORE STORAGE AN SKIN EFFECTS APPENIX C A QUARATIC FORMULA FOR NUMERICAL IFFERENTIATION VITA... 68

8 viii LIST OF FIGURES FIGURE Page. Schemaic iagam of gas conensae (liqui ehavio as a funcion of isance in he esevoi (afe Roussennac Gas moiliy ofiles fo a gas conensae esevoi sysem (as a funcion of ime an aius (aae fom Roussennac "Tye cuve" eesenaion of he new moel ( ( fomulaion (Eq..5. Soluion is loe vesus he moifie Bolzmann ansfom vaiale /( (. "Tye cuve" eesenaion of he new moel ( / fomulaion (Eq..6. Soluion is loe vesus he moifie Bolzmann ansfom vaiale /( (.5 "Tye cuve" eesenaion of he new moel ( ( / fomulaion (Eq..6. Soluion is loe vesus he invese of he moifie Bolzmann ansfom vaiale (( / imensionless essue ye cuve fo aial flow ehavio incluing welloe soage an sin effecs ( w vesus /C foma. This lo esens a comaison of he soluion geneae using numeical invesion (as a suogae fo he exac soluion an he aoximae soluion echnique oose in ef. 7 an geneae using Mahemaica imensionless essue eivaive ye cuve fo aial flow ehavio incluing welloe soage an sin effecs ( w ' vesus /C foma. This lo esens a comaison of he soluion geneae using numeical invesion (as a suogae fo he exac soluion an he aoximae soluion echnique oose in ef. 7 an geneae using Mahemaica Pessue eivaive ye cuve fo a veical well oucing a a consan ae nea a sealing faul in a homogeneous, infinie-acing esevoi Pessue eivaive ye cuve fo a veical well oucing a a consan ae in a comosie aial sysem, vaious moiliy (λ/soaiviy (ω cases Gas moiliy ofiles fo a gas conensae esevoi sysem (as a funcion of ime an aius (aae fom Roussennac noe he comaison of he simulae efomance an he oose moels (i.e., he ex(x an he ef (x moiliy moels.... 8

9 ix FIGURE 3. "Tye cuve" eesenaion of he ( soluion (Eq. 3.. Soluion is Page loe vesus he moifie Bolzmann ansfom vaiale /(.... ( 3.3 "Tye cuve" eesenaion of he / soluion (Eq. 3.. Soluion is loe vesus he moifie Bolzmann ansfom vaiale /(.... ( 3. "Tye cuve" eesenaion of he new moel ( / fomulaion (Eq..6. Soluion is loe vesus he moifie Bolzmann ansfom vaiale /(.... ( 3.5 "Tye cuve" eesenaion of he new moel ( ( / fomulaion (Eq..6. Soluion is loe vesus he invese of he moifie Bolzmann ansfom vaiale (( / a imensionless essue ye cuve fo aial flow ehavio incluing welloe soage an sin effecs ( w vesus /C foma. This lo esens a comaison of he soluion geneae using numeical invesion (as a suogae fo he exac soluion an he aoximae soluion echnique oose in ef. 7 an geneae using Mahemaica imensionless essue eivaive ye cuve fo aial flow ehavio incluing welloe soage an sin effecs ( w ' vesus /C foma. This lo esens a comaison of he soluion geneae using numeical invesion (as a suogae fo he exac soluion an he aoximae soluion echnique oose in ef. 7 an geneae using Mahemaica a Tye cuve lo ( w an w ' vesus /C C x0 3, x0 0, vaious min / max cases Tye cuve lo ( w an w ' vesus /C C x0 3, x0 -, vaious min / max cases c Tye cuve lo ( w an w ' vesus /C C x0 3, x0 -, vaious min / max cases Tye cuve lo ( w an w ' vesus /C C x0 3, x0-3, vaious min / max cases e Tye cuve lo ( w an w ' vesus /C C x0 3, x0 -, vaious min / max cases f Tye cuve lo ( w an w ' vesus /C C x0 3, x0-5, vaious min / max cases... 9

10 x FIGURE Page 3.8 awown ye cuve lo ( w ' vesus /C C x0 3, x0 0, 0 -, 0 -, 0-3, 0 -, 0-5, vaious min / max x0 0, 0 -, 0 -, awown ye cuve lo ( w ' vesus /C C x0 0, x0 0, 0 -, 0 -, 0-3, 0 -, 0-5, vaious min / max x0 0, 0 -, 0 -, Pessue eivaive ye cuve fo a veical well oucing a a consan ae nea a sealing faul in a homogeneous, infinie-acing esevoi Pessue eivaive ye cuve fo a veical well oucing a a consan flowae in a -zone aial comosie esevoi sysem, vaious moiliy (λ/soaiviy (ω cases Comine essue eivaive ye cuve fo he following cases: sealing fauls, a single aial comosie egion, an he oose moel fo a aially-vaying moiliy ofile a z-faco ofiles fo he Roussennac aa case (Mix inclues Roussennac EOS (simulaion aa an aa fom y gas coelaion Gas viscosiy ofiles fo he Roussennac aa case (Mix inclues Roussennac EOS (simulaion aa an aa fom y gas coelaion c Pseuoessue en fo he Roussennac aa case (Mix geneae using he z-faco an gas viscosiy aa oaine fom he y gas coelaions, an he seuoessue efiniion given y Eq vesus fo he Roussennac case of Fig..7 he essue aa wee igiize an lae convee o seuoessue ( using he ansfomaion shown in Fig..c Examle case of vesus fo liqui flow esene y Lee (ef Δ an η Δ /η funcions vesus η fo Roussennac Fig..7 (η /. The scae in he η Δ /η funcion is ue o he Δ aa no eing uniquely "line souce" in chaace (i.e., hese ae numeical simulaion esuls, an ae no oun o he "line souce" cieion Δ an η Δ /η funcions vesus η (η / (aa fom Roussennac Fig..7, moel ens ae eive fom he new moel fo a vaying moiliy ofile a Mach of Roussenac aa (igiize an he new vaiale moiliy moel (ye cuve mach ( vesus ( / foma Mach of Roussenac aa (igiize an he new vaiale moiliy moel (ye cuve ( / vesus ( / foma...

11 xi FIGURE Page.0 Mach of Roussenac aa (igiize an he new vaiale moiliy moel (aa/moel mach Δ an / vesus foma....a z-faco ofiles fo he Vo 9 aa case (Flui Vo EOS (simulaion aa.... Gas viscosiy ofiles fo he Vo 9 aa case (Flui Vo EOS (simulaion aa....c Pseuoessue en fo he Vo 9 aa case (Flui geneae using he z-faco an gas viscosiy aa oaine fom Vo EOS calculaions, an he seuoessue efiniion given y Eq Mach of Vo 9 aa (igiize an he new vaiale moiliy moel (aa/moel mach Δ an Δ / vesus η (η / foma. (s0...

12 CHAPTER I INTROUCTION. Reseach Polem uing he oucion of a gas esevoi, he associae essue hisoy can e use o esimae esevoi oeies an ovie insigh ino well efomance vesus execaions. This essue hisoy, howeve, may e ifficul o caegoize. No only oes he efomance of a gas esevoi (an aiculaly, a gas conensae esevoi exhii vaious yes of eleion efomance, u geological comlexiies (such as fauls an emeailiy vaiaion also yiel vaiaions in he oucion-essue hisoy. When comine, such effecs ae vey ifficul o "uncoule" an may acually e inisinguishale fom one anohe (e.g., he soluions fo a aial comosie sysem an a sealing faul(s can e vey simila o one anohe (even inisinguishale in exeme cases (such as a single sealing faul. Well esing is he imay means fo esalishing he esence of such feaues as gas conensae efomance effecs, geological sucues, ec howeve, we mus ecognize ha he olem of "uniqueness" is ehas he mos ifficul o ovecome, an convenional analysis/ineeaion echniques may no e sufficien o oely chaaceize such effecs. Hence, i is he moivaion fo his wo ha we esalish a new soluion fo he ansien awown efomance of gas conensae esevois. We noe ha he cuen aoach of using esevoi simulaion o esolve such issues is moe flexile han he aiional well es analysis mehos howeve, he eail a which esevoi simulaion is efome may no aess he hysical henomena eing oseve in he efomance aa. Simulaion can e scale as finely o as coasely as esie u how oes one "caliae" he numeical moel o he hysical olem wihou maing limiing assumions? On he ohe han, an analyic (o semianalyic soluion is also simlifie o fi coniions whee i can e solve, u such soluions ovie insigh ino he chaaceisic ehavio of he sysem. The ehavio of gas conensae esevoi sysems can e ifficul o moel an eic. Secifically, in many aeas (e.g., he Noh Sea he quesion ofen aises as o whehe an unexece ecline in gas oucion is a esul of eleion, o if his is a esul of conensae aning. Liqui conensae evelos as esevoi essue eclines elow he ewoin essue an he egee o which his occus eens on many facos such as he comosiion of he gas, an he esevoi coniions. Liqui conensae will imee he flow of he gas hase, esicing oucion flowae an avesely affecing ecovey. This hesis follows he syle an foma of he SPE Jounal.

13 Examinaion of iffeenial essue aa loe wih esec o aial isance fom he welloe (geneae using numeical simulaion will inicae he ossiiliy of conensae aning. Roussennac ooses ha hee egions (o zones yically exis in a gas conensae esevoi sysem hese egions ae escie y Roussennac (an Fevang, as efeence y Roussennac as follows: Region Conensae Ban: By efiniion, his egion nea he welloe has a conensae (oil sauaion ha is high enough o emi he conensae flui o flow. Oviously, he esevoi essue in his aea is he lowes of he hee egions. As noe y Roussennac, he oveall comosiion of he flowing mixue in his egion is essenially consan in his aea (as inicae y a nea consan GOR an is aoximaely he same comosiion as he single hase gas a he ounay of Region an Region. The secific cieia use o chaaceize he conensae hase is ha hee is conensae flow in Region (alhough he evoluion of his "moile" conensae is hough o e he ounay secion fo Regions an. As shown schemaically in Fig.., he oil sauaion eceases as aial isance fom he welloe inceases ha is, he isiuion of fluis nea he well is elaively sale "ying" o essenially he oiginal y gas a isances fom he welloe. Fig.. suggess ha hee is a "conensae" gaien in he nea-well egion, u ha he gaien in his egion is susanially less han he one exeience in "Region " (i.e., he "conensae uilu" o "ansiion" zone. This conce (valiae y numeical simulaion suggess ha Region can e eae as a simle wohase egion wih consan hase moiliies. On he ohe han, Region is seen as a egion of ai change in conensae sauaion. Region Conensae Builu Zone: This egion iffes fom Region in ha he conensae is elieve o have a low moiliy an while i will esalish a gaien o ansiion zone, he conensae will no en o flow. The oue ege of Region is he oin some aial isance fom he well whee he fis oles of liqui evolve fom he gas hase heefoe, he essue a his aicula isance (which oes coninue o oagae is he ewoin essue of he oiginal esevoi gas. As noe y Roussennac, he gas hase comosiion "leans ou" in Region, wih he heavie comonens eing evolve as conensae. This henomenon coninues as we aoach he welloe an he gas "leans ou" o a minimum ichness a he welloe. I is woh noing ha he conensae sauaion is susanially lowe in Region han Region, which oes (conceually emi us o consie Region o e a single-hase gas egion fo he uose of well esing (in some cases. Roussennac (an ohes have uilize he 3-egion conce fo he analysis of well es aa fom gas conensae esevois wih he ojecive of chaaceizing each egion using a 3-zone aial comosie esevoi moel. Thee ae vaying egees of success wih his conce, an many analyss efe using only a -zone moel, while ohe analyss insis ha he 3-zone moel is moe aoiae.

14 3 Finally, we eea he emise ha only gas is flowing in Region heefoe, he inemeiae an heavie comonens evolve as conensae nea he ounay of Regions an. This ovies he conensae which foms he "an" in Region. Region 3 Oiginal y Gas Region: By efiniion, no conensae exiss in his egion only he gas hase is esen (i.e., he essue is geae han he ewoin essue. We noe ha he evailing wisom is ha all hee egions exis in a yical gas conensae esevoi. Region is liely o exis when es < ew an Region will always exis if Region exiss (i.e., hee mus e a conensae gaien egion. Region 3 exiss uing ansien flow ehavio, an if oue ounaies ae encounee hen he esevoi essue may o elow ew, an Region 3 (i.e., he oiginal y gas sae may no exis. Roussennac suggess ha Region may ecome negligile fo he case of a vey ich gas o nea ciical gas conensae fluis. This henomenon can e moele wih PVT exeimens howeve, if his ehavio exise, i woul e ifficul o isinguish fom ohe coniions. We elieve ha he "conce" of 3 egions is elevan (an ehas aoiae fo many cases. Roussennac has oose a schemaic iagam fo his ocess (see Fig.. an we agee wih his oosal as fo as he esevoi ocesses, we ae less ceain egaing well es analyses u we acnowlege ha, conceually, Fig.. valiaes he alicaion of he o 3-zone aial comosie esevoi moel fo he analysis of well es aa oaine fom gas conensae esevois. Figue. Schemaic iagam of gas conensae (liqui ehavio as a funcion of isance in he esevoi (afe Roussennac. We also elieve ha alicaions in acual esevois will iffe somewha fom he esuls of such "iealize" suies in aicula, a gas conensae esevoi may no exhii he exece conensae

15 an an/o hee coul e ohe esevoi chaaceisics (e.g., geologic feaues ha imai o comlicae he analysis/ineeaion of esevoi efomance aa fom gas conensae esevois. The economic asecs of his siuaion ae elevan as well fuue evelomen saegies een on a eesenaive chaaceizaion of he esevoi in quesion. This is one asec of ou moivaion o aess he olem of vaiale moiliy ofile iecly using a soluion which exlicily incooaes his ehavio. Ou oose soluion involves he ienificaion of (liqui conensae evelomen wih esec o ime an isance fom he well. We esen isincive soluions in he fom of "ye cuves" ha can e use o visually ienify conensae evoluion in ems of moiliy ehavio wih esec o ime an aius. These ye cuves ae eveloe using a new soluion fo he case of a changing effecive emeailiy (o moiliy as a funcion of imensionless ime an imensionless aius an ae esene in hee fomas: a unifie vaiale ase on he imensionless Bolzmann ansfom vaiale ( /(, imensionless aius (, an imensionless ime(. As we canno measue essue in he esevoi, he only acical ool fo well es analysis is he family of ye cuves given in he imensionless ime ( foma.. Reseach Ojecives The imay ojecives of his wo ae: To evelo an analyical eesenaion of he essue ehavio in ime an sace fo a esevoi sysem wih a vaying moiliy ofile (see Fig.. fo a schemaic of a vaying moiliy ofile fo a gas conensae esevoi sysem. The conce is ase on an emiical moel fo he gas moiliy funcion. The moel consies a vaying gas emeailiy ha assigns he maximum gas emeailiy fo he coniion whee only gas (no conensae is esen in he esevoi. The minimum gas emeailiy is he value a he coniion whee he moiliy of he gas has een imee y maximum conensae oou. This moiliy moel is given as: - min ( max min - ex...(. The conce is ase on he osevaion of minimum gas emeailiy (o moiliy nea he welloe an he maximum (oiginal gas emeailiy in he "y gas" oion of he esevoi. The moel eics he emeailiy ehavio uing he ansiion egime eween he wo exeme maximum an minimum emeailiy values. The moel was consuce afe consieing osevaions mae fom numeical simulaion esuls whee sauaion, effecive emeailiy, an gas moiliy ae esene as funcions of isance in he esevoi.

16 5 The seconay ojecives of his wo ae: To uilize his new moel as a mechanism o evelo gahical soluions fo he essue eivaive in ime an aial isance. This soluion can e comae o ohe soluions (e.g., he (o 3- zone aial comosie esevoi moel an vaious cases of he sealing faul moel (ime eivaive, as well as he essue an essue eivaive (aial eivaive as a funcion of aial isance eive fom numeical simulaion. To use his moel fo he analysis of well es aa fom gas conensae esevois wih he inenion of eveloing soluions which inclue welloe soage an sin effecs. To oose alicaions fo he analysis of well es aa acquie fom essue awown o essue uilu ess..3 Saemen of he Polem an Summay of a Poose Soluion This wo is focuse on he conce of using a funcional fom fo he gas moiliy ofile (i.e., /μ an incooaing an emiically-eive moel ino he igoous iffusiviy equaion fo he liqui case. Figue. Gas moiliy ofiles fo a gas conensae esevoi sysem (as a funcion of ime an aius (aae fom Roussennac noe he comaison of he simulae efomance an he oose moels (i.e., he ex(x an he ef (x moiliy moels.

17 6 We wish o use his conce an he esuling flow moel o eesen he essue ehavio of he gas conensae case wih esec o ime an aial isance fom he welloe. We ea his case as "liqui equivalen," whee we consie non-iealiies (e.g., essue-eenen PVT funcions y using he convenional seuofuncions (i.e. seuoessue an seuoime. We have use he simulaion cases esene y Roussennac as a saing oin fo esalishing a moel fo gas moiliy ehavio as a funcion of aius an ime fo a gas conensae esevoi. We ecognize ha simulae ofiles ae olemaic (i.e., a iffeen se of inu aa may yiel a iffeen ofile, u we elieve ha he cases esene y Roussennac offe an aoiae saing oin as hese cases ae well caliae an veifie Using he esuls esene y Roussennac (see Fig.., we have esalishe he following conceual moel fo eesening he gas emeailiy as a funcion of aius an essue: min ( max min - - ex ("exonenial" o ex(x moel...(. We also comae he exonenial moel wih he following ef(x moel: min ( max min ef ("eo funcion" o ef(x moel...(.3 Fo he uoses of his wo we will use he fom given y Eq.. (i.e., he ex(x moel an esume a "liqui equivalen case" (i.e., is simly a funcion of aius an ime (no exlicily a funcion of es-sue. The efiniion of he iffusiviy equaion fo his case is given as: c φμ (Fiel unis...( Eq.. is use as he emeailiy moel, an is coule wih he aial flow iffusiviy equaion fo his case (i.e., Eq... We assume a well in an infinie-acing aial flow sysem ouce a a consan flowae, an, as noe ealie, we secifically assume ha emeailiy is an exlici funcion of aius an ime f(,. In oe o solve he esuling iffeenial equaion, we use he Bolzmann ansfomaion (ase on he aoiae efiniion of imensionless vaiales (see Aenix A. We ovie iffeen foms of he soluion foms in ems of he Bolzmann vaiale ( /, as well as he imensionless essue an he imensionless essue eivaive funcions in ems of he imensionless aius an ime vaiales. These foms will ove useful fo iffeen alicaions he aial isance foms ae useful fo valiaion of he new soluion wih esevoi simulaion esuls, while he ime foms of he soluion will have uiliy in he analysis of well es an oucion aa.

18 7 "Pessue Soluion" [ ] e ] / ( ln[ ex / max min max min * (.5 "Pessue eivaive in Time" [ ] e ] / ( ln[ ex / max min max min *...(.6 "Pessue eivaive in Raial isance" [ ] e ] / ( ln[ ex / max min max min *...(.7 The mos imoan issue o consie in evaluaing Eqs is ha we have mae no limiing assumions in his evelomen we have simly use he aiional soluion aoach ase on he Bolzmann ansfom. We noe ha Eq..5 canno e exesse analyically an mus e evaluae numeically. In ou case we have uilize he sofwae Mahemaica, which is comuaionally flexile, as well as caale of geneaing "nea exac" esuls. Eqs..6 an.7 ae "close fom" esuls which ae essenially ienical in fom. We noe ha comaison of Eqs..6 an.7 yiel he following ieniy:...(.8 As noe in Aenix A, Eq..8 is uniquely vali fo his case, as well as he homogeneous esevoi soluion (his esul is a isinc ieniy fo he case of an infinie-acing esevoi. The aamee is he imensionless fom of he emiical -aamee given in Eqs.. an.3. A hysical efiniion o exlanaion of canno e mae iecly; an, fo he uose of his wo, we ea he -aamee simly as a moel aamee in a simila fashion as emeailiy, sin faco, ec. We elieve ha he -aamee eesens he aggegae ehavio of he elaive emeailiy funcions an he flui oeies (oaly oh gas an gas conensae.

19 8 Using he efiniions of he imensionless vaiales, we efine in ems of as: φμc (convenional oilfiel unis...( Fo loing he essue eivaive funcions in oh ime an sace we have efine he following efiniions: (which ae eive y insecion of Eqs..6 an.7...(.0...(. In Fig..3 we esen a log-log foma lo of he ( funcion loe vesus he moifie Bolzmann ansfom vaiale ( /(. This lo equies some oienaion fo examle, we can use his lo o consie he essue o as a funcion of isance fo a "snasho" in ime. aa fom numeical simulaion can e comae o his lo as a mechanism o valiae he analyical soluion (as we will show in a lae secion. This lo coul also e use o consie aa esene in ems of ime howeve, he "/" fom given y he moifie Bolzmann ansfom vaiale oes no mae Fig..3 aiculaly convenien fo he analysis/ineeaion of essue-ime aa. The "/" foma is igoous, u he cuen convenion of using ime (o woul mae his lo less liely o e use in acice. Figue.3 "Tye cuve" eesenaion of he new moel ( ( fomulaion (Eq..5. Soluion is loe vesus he moifie Bolzmann ansfom vaiale ( /(.

20 9 Figue. "Tye cuve" eesenaion of he new moel ( / fomulaion (Eq..6. Soluion is loe vesus he moifie Bolzmann ansfom vaiale /(. ( Figue.5 "Tye cuve" eesenaion of he new moel ( ( / fomulaion (Eq..6. Soluion is loe vesus he invese of he moifie Bolzmann ansfom vaiale (( /.

21 0 In Fig.. we esen he aial eivaive funcion ( funcion loe vesus he moifie Bolzmann ansfom vaiale ( /(. The ( fomulaion eesens he change in essue o wih esec o aius as we move ou ino he esevoi clealy hee ae seaae facos a issue he ehavio of he ( / (o (/ funcions show he influence of he oagaing emeailiy ofile. In aicula, his fomulaion shows how essue gaien eceases wih isance in he esevoi (as woul e exece, u i clealy illusaes he "nea well" an "esevoi" ehavio of he essue gaien funcion. We will uilize Fig.. as a "valiaion lo" fo aa geneae fom numei-cal simulaion. In aicula, we will mach simulae efomance o he oose esevoi moel. Fig..5 esens he ime eivaive funcion ( funcion (Eq..0 loe vesus he invese of he moifie Bolzmann ansfom vaiale (( /. In his lo we noe ha he essue eivaive efomance is amaically influence y he evolving aial isiuion of emeailiy. I is ifficul o mae an analogy wih his ehavio wihou efeencing a aicula esevoi moel, u he efomance oes aea o eesen some so of flow aie/imeimen a some aial isance. In a lae secion of his wo we will comae he ens shown on Fig..5 wih he esonses fom seveal iffeen esevoi moels in aicula: he -zone aial comosie moel as well as a sequence of sealing faul moels. I is no suise ha hese moels (i.e., -zone aial comosie moel/sealing faul moels ae ofen use in he ineeaion of well es aa oaine fom gas conensae esevois i is ou goal o esalish he oose wo as he aoiae sana fo he analysis of such aa.. Ouline of Thesis Chae I Inoucion Reseach Polem Reseach Ojecives Summay Chae II Lieaue Review Raial Comosie Resevoi Sysem Geneal Conces (, Pefomance in Gas Conensae Resevois Ohe Soluions/Consieaions Chae III evelomen of an Analyical Pessue Soluion fo he Case of a Pemeailiy Pofile ha Vaies in Time an Raial isance Conce of a Moiliy Pofile ha Vaies in Time an Raial isance Alicaion of he Bolzmann Tansfomaion o he Raial iffusiviy Equaion evelomen of he Pessue eivaive Soluions Welloe Soage an Sin Effecs

22 Chae IV Valiaion of an Analyical Pessue Soluion fo he Case of a Pemeailiy Pofile ha Vaies in Time an Raial isance Gas Conensae Resevois Comaison of he New Soluion an Soluions fo he Sealing Fauls an Raial Comosie Resevoi Cases Valiaion of he New Soluion Lieaue aa Chae V Summay, Conclusions, an Recommenaions fo Fuue Wo Summay Conclusions Limiaions an Recommenaions Nomenclaue Refeences Aenices Aenix A eivaion of he Pessue eivaive Funcions wih Resec o Time an Raius fo he Case of a Raially-Vaying Pemeailiy Pofile (Equivalen Liqui Case Aenix B An Aoximae Technique fo he iec Aiion of Welloe Soage an Sin Effecs Aenix C A Quaaic Fomula fo Numeical iffeeniaion Via

23 CHAPTER II LITERATURE REVIEW. Raial Comosie Resevoi Sysem Hisoically, much wo has een efome in he eoleum inusy egaing he suy of efomance ehavio in a esevoi as his efomance elaes o essue as a funcion of ime an isance fom a veical well. An accuae unesaning of how a esevoi will efom ove ime in ems of essue an flowae is essenial fo maing oimal ecisions egaing invesmen, exloaion, an evelomen. In aicula, he unesaning of gas conensae esevois has een vey challenging an suy in his aea has given he inusy insigh ino his oic, u ecen avances in aa acquisiion an moelling have aise many quesions egaing he analysis an ineeaion of well efomance aa oaine fom gas conensae esevois. Ealy wo in he eoleum inusy focuse on he analogy of lamina flui flow hough oous maeial wih he conucion of hea in solis. The ehavio of fluis unegoing acy (lamina flow in a aial flow geomey is govene y he aial iffusiviy equaion (Eq... Many soluions of he iffusiviy equaion fo flow in oous maeials can e oaine fom analog cases in hea conucion (Caslaw an Jaege 3, an some cases of "non-unifom" esevoi oeies have aleay een oose in he hea conucion lieaue. In 970, Ramey esene wo which summaize effos o ae fo eveloing acical soluions fo flui flow in an 3-zone aial comosie esevoi sysems. This wo came a a ime of inense inees in eveloing useful an acical soluions fo he case of wae injecion in oil esevois (in aicula, he evelomen of soluions fo injecion/falloff ess in such esevois. I was Ramey's inen (as imlie in his inoucion o eive a class of soluions which conaine only elemenay funcions so ha hese soluions coul e use fo he uose of analysis/ineeaion of well es aa. Ramey use he aial iffusiviy equaion as a saing oin, an hen ae he consain of wo (o moe iscee zones (o "egions" of consan moiliy (/μ an hyaulic iffusiviy (/(φμc. This aoach gives each iscee egion a consan emeailiy, viscosiy, oosiy, an comessiiliy whee hese oeies can vay fom egion o egion. Each egion is homogeneous an isooic, an he change in oeies fo a aicula zone occus auly a he zonal ounay(s. While he hysical conce of concenic aial "ings" of iffeing esevoi oeies can e eae, we will noe ha his soluion has een shown o eesen a emaaly lage nume of fiel cases.

24 3. Geneal Conces (, Pefomance in Gas Conensae Resevois In 985, Jones 5 ulishe a Ph.. isseaion ha esene a unifie heoy fo he esing of gas conensae esevoi sysems, whee his wo was ase on heoy of flow fo a slighly comessile liqui as a moel fo mulihase flow ehavio (i.e., he "equivalen liqui" conce. Pseuofuncions (i.e., seuoessue an seuoime wee eive fo essue eenen aamees an in his wo Jones eveloe he "esevoi inegal" an "sanface inegal" conces fo mulihase flow in oous meia. These inegals wee aae fom seay-sae heoy an wee he esul of eucing he heoeical inegals aen ove sace an ime o inegals aen ove essue. This wo ovie a efiniion of seuoessue ha has since shown vey goo efomance in esimaing emeailiy an sin fom well ess efome in gas conensae esevoi sysems. In 989, Bǿe, e al. 6 oose a heoeical asis fo he analysis of well es aa oaine fom soluion gas an gas conensae esevoi sysems uing he infinie-acing flow eio. Bǿe, e al. iscuss he analysis an ineeaion of essue ansien es aa using soluions ase on he liqui analogy (i.e., he case of a single hase liqui wih a small an consan comessiiliy an consan viscosiy. Al-Hussainy, e. al 7 sugges ha gas ess can e ineee wih his liqui analogy y he use of a seuoessue funcion (alhough as Agawal 8 (seuoime lae showe, a seuoime funcion is also equie fo he analysis of essue uilu ess conuce in gas wells. Bǿe, e al. uilize a seuoessue fomulaion as well as he Bolzmann ansfom whee we noe ha he Bolzmann ansfom is secifically vali fo he infinie-acing eio. When he Bolzmann ansfom is violae y he ounay coniions (i.e., os-ansien flow coniions exis, he oose soluion eviaes fom he liqui analogy (shown in ef. 6. Bǿe, e al. sugges ha as long as infinieacing flow ehavio is oseve, he seuoessue funcion can e evaluae using he coec essue/ sauaion elaion a he welloe. Ou wo is somewha comaale in heme o ha of Bǿe, e al. in ha we evelo a Bolzmann ansfom soluion an hen valiae he soluion using numeical simulaion (oviously, he sucue of ou olem is iffeen, u ou aoach is simila o ha of Bǿe, e al.. In 999, Xu an Lee 9 invesigae he conensae gas olem wih he inenion of imoving evious soluions ha consiee he wo-zone, aial comosie case. Pevious wo y Raghavan, Chu, an Jones 0 consiee seay-sae flow in a wo-zone comosie moel an foun ha hei oose soluions woe well in cases whee he esevoi essue was susanially highe han ewoin essue, an oomhole flowing essue in he well was much lowe han he ewoin essue. Howeve, wih he esence of a significan mile zone, whee conensae evelos, u has no eache ciical sauaion (i.e., is immoile, he Raghavan, e al. soluions ae no as accuae. The seay-sae flow assumion yiels a elaionshi eween conensae sauaion in he esevoi an essue whee his elaionshi is no vali when flow is imee y he immoile conensae oou yically foun in Region (i.e., he conensae o-ou zone.

25 The wo y Xu an Lee 9 consies a hee-zone aial comosie esevoi moel. The fis zone nea he well assumes seay sae wo hase flow. The secon zone assumes an immoile conensae sauaion, u a moile gas hase. The hi zone assumes only y gas exiss in his egion. The ehavio of hese zones is he same as escie in he wo y Roussennac (whee some of Roussennac's commens confime he osevaions u foh y Fevang an Whison. Fo hese yes of analyses (Fevang an Whison, Xu an Lee, Roussennac, ec., elaionshis ha o e eveloe o eesen sauaion an essue fo he calculaion of he seuoessue funcion. The meho equies Consan Volume eleion (CV aa (fo use in moeling Region, gas an conensae elaive emeailiy, oucing GOR (Region, an essue ansien aa. Fevang an Whison oose he following elaionshi fo Region which elaes he gas an conensae elaive emeailiy aio wih essue: o g ( s R μob ( ( R Rs μg B o g...(. Using seuoessue funcions o aoximae he esevoi inegal eveloe y Jones an Raghavan as well as Jones, Vo, an Raghavan 3 ; Xu an Lee 9 comue hese seuoessue funcions using he essue-sauaion elaionshis fo Regions,, an 3 (fom whaeve souce(s hese aa may e eive (in mos cases numeical simulaion. This oceue (analogous o mehos in efs. 7,, an 5, allows fo he esimaion of iniial esevoi essue, fomaion emeailiy, an sin faco..3 Ohe Soluions/Consieaions Welloe Soage Effecs One of ou goals in his wo is o geneae soluions ha can e use fo he analysis an ineeaion of well es aa. In aicula, we have chosen o evelo "ye cuve" soluions fo he case of welloe soage effecs. Using he liqui analogy, we emloye sueosiion o inclue he effecs of welloe soage on awown soluions (ecall ha sueosiion is only vali fo he case of linea aial iffeenial equaions whee we have esume ha Eq.. mees hese cieia. Blasingame, e al 6 ovie he eivaion an valiaion of analyical aoximaions fo he case of "aing" welloe soage effecs via he Lalace ansfomaion. We have use he mehos in ef. 6 o geneae he welloe soage soluions esene in his wo (he secific eails ae esene Aenix B.

26 5 Fo his wo we have use "Case " esene in ef. 6 (i.e., he case whee he s ( funcion is esume o e linea nea a aicula ime of inees. This esul is given y: w ψ θ (- ex[-ω ] (ex[-ω ] - ω...(. ω ω The coefficiens ae ω, ψ an θ ae eive using values of he s ( funcion as escie in Aenix B. In summay, we ae saisfie ha he aoach given in ef. 6 is ous an sufficienly accuae fo ou esen wo see Figs.. an., which ae valiaions of Eq.. eae in his wo fo he case of an unfacue well oucing in an infinie-acing esevoi. Figue. imensionless essue ye cuve fo aial flow ehavio incluing welloe soage an sin effecs ( w vesus /C foma. This lo esens a comaison of he soluion geneae using numeical invesion (as a suogae fo he exac soluion an he aoximae soluion echnique oose in ef. 6 an geneae using Mahemaica. Pessue Behavio in Time (sealing fauls an he -zone aial comosie soluion Ou new oose soluion fo a vaying moiliy aio mus e comae o exising soluions use fo he analysis of essue ansien ehavio fo gas conensae esevois. As such, we consie wo cases which ae ofen emloye in such analyses he sealing fauls moel (vaious cases an he -zone aial comosie esevoi moel.

27 6 Figue. imensionless essue eivaive ye cuve fo aial flow ehavio incluing welloe soage an sin effecs ( w ' vesus /C foma. This lo esens a comaison of he soluion geneae using numeical invesion (as a suogae fo he exac soluion an he aoximae soluion echnique oose in ef. 6 an geneae using Mahemaica. Figue.3 Pessue eivaive ye cuve fo a veical well oucing a a consan ae nea a sealing faul in a homogeneous, infinieacing esevoi. (Soluion fom ef.

28 7 The "sealing fauls" moel is no inuiively alicale fo he case of gas conensae esevoi sysems howeve, he conce of a "flow consicion" o "flow aie" has een suggese as an analog o he gas conensae case. We o no avocae he use of he "sealing fauls" moels fo he analysis an ineeaion of well efomance aa fom gas conensae esevoi sysems; we simly noe ha some analyss have suggese similaiy in he efomance of he sealing fauls moels an he oseve efomance fom gas conensae esevoi cases. In Fig..3 we esen he soluion fo a well in he viciniy of one o moe sealing fauls his esenaion clealy inicaes ha he oienaion an nume of fauls amaically affecs he ehavio of he funcion. This soluion was oaine fom Sewa, e al.. In Fig.. we esen he "unifie" lo ( funcion fo mulile cases of he aial comosie esevoi soluion (efs. 5, 7. The mos imoan (an mos elevan issue is ha he aial comosie soluion has fixe moiliy an iffusiviy aios (fo he inne an oue zones y conas o ou soluion which uses a emeailiy ofile in aius an ime, u only a single value of iffusiviy fo he enie esevoi. As such, we will only comae cases fo he aial comosie esevoi moel whee he iffusiviy aio is uniy. Figue. Pessue eivaive ye cuve fo a veical well oucing a a consan ae in a comosie aial sysem, vaious moiliy (λ/soaiviy (ω cases. (Soluion fom efs. 5,7

29 8 CHAPTER III EVELOPMENT OF AN ANALYTICAL PRESSURE SOLUTION FOR THE CASE OF A PERMEABILITY PROFILE THAT VARIES IN TIME AN RAIAL ISTANCE 3. Conce of a Moiliy Pofile Tha Vaies in Time an Raial isance As noe ealie, we have use he osevaion of he moiliy/effecive emeailiy ofiles in aius an ime oaine fom numeical simulaion fo he case of a gas conensae esevoi as he asis fo ou oose moel fo gas moiliy as a funcion of aius an ime. The oiginal asis fo his oose moel was eveloe using he esuls ulishe y Roussennac. A samle case aae fom he Roussennac wo is shown elow in Fig. 3.. Figue 3. Gas moiliy ofiles fo a gas conensae esevoi sysem (as a funcion of ime an aius (aae fom Roussennac noe he comaison of he simulae efomance an he oose moels (i.e., he ex(x an he ef (x moiliy moels.

30 9 In Fig. 3. we esen he following moels fo eesening he gas emeailiy as a funcion of aius an essue he "ex(x" moel is given as: min ( max min an he "ef(x" moel is given y: - - ex ("exonenial" o ex(x moel...(3. min ( max min ef ("eo funcion" o ef(x moel...(3. 3. Alicaion of he Bolzmann Tansfomaion o he Raial iffusiviy Equaion evelomen of he Pessue eivaive Soluions In his wo we efe he "ex(x" moel (i.e., Eq. 3. imaily ecause of he mahemaical simliciy of his moel (i.e., his moel is eaily aae o he Bolzmann ansfomaion aoach ha is use o evelo he soluion fo his case. As noe in Chae I, he efiniion of he iffusiviy equaion fo liqui flow (ou ase assumion is given y: c φμ (Fiel unis...( Assuming ha he emeailiy is a funcion of aius an ime f(,, we oain he following genealize imensionless fom of he iffusiviy equaion ase on he Bolzmann ansfomaion (eails in Aenix A: 0... (3. Given Eq. 3. in imensionless fom we noe he efiniions of he elevan imensionless vaiales: / (imensionless Bolzmann ansfom aamee... (3.5 / max (imensionless emeailiy (noe in ems of max... (3.6 maxh ( i. qbμ (imensionless essue... (3.7 max (imensionless ime... (3.8 φμc w / (imensionless aius... (3.9 w Using hese efiniions, he imensionless fom of he emeailiy funcion (i.e., Eq., 3. is given y: min ex... (3.0 max Fo convenience, we efine he consans a an as follows: a min /... (3. ( max /... (3.

31 0 Susiuing Eqs. 3. an 3. ino Eq. 3.0 yiels: ] ex[ a... (3.3 Susiuing Eq. 3.3 ino Eq. 3. an comleing he soluion (using he Bolzmann ansfomaion ocess, we oain he following soluions: (again, he eails of his eivaion ae given in Aenix A e / ( ln ( ex / ( max min max min ("eivaive" fom... (3. e / ( ln ( ex / ( ( max min max min ("essue" fom... (3.5 Alenae foms of Eq. 3., wien in ems of an ae given as: e / ( ln ( ex / ( max min max min (" eivaive" fom... (3.6 e / ( ln ( ex / ( max min max min (" eivaive" fom... (3.7 Insecion of Eq. 3. (he "essue" fom of he soluion leas us o ecognize ha Eq. 3. can no e esolve as a close fom soluion his esul can only e evaluae numeically. As such, we have elece o use Mahemaica o comue he an / soluions. We esen a vaiey of soluions fo he an / funcions in Figs. 3. an 3.3 whee he funcion is shown in Fig. 3. an he / funcion is esene in Figs. 3.3.

32 Figue 3. "Tye cuve" eesenaion of he ( soluion (Eq. 3.. Soluion is loe vesus he moifie Bolzmann ansfom vaiale /(. ( In Fig. 3. we can view hese ens as eing he essue o (in imensionless fom aen as isance inceases away fom he welloe as we hol ime consan. We noe he effec of he min / max aio on he efomance of he soluion an we oseve ha he -aamee is as a scaling mechanism in he moifie Bolzmann ansfom vaiale /(. We woul escie his siuaion ( hysically as a eceasing essue o as we ogess ino he esevoi, noing ha he min / max aio conols he essue o nea he well an ha he cominaion of he min / max aio an he - aamee conol he ansiion an "fa fiel" essue soluions. Fig. 3.3 esens he / soluion fo he same min / max an cases shown in Fig. 3.. In his case we noe he isinc similaiy of he / soluion wih he / soluion (Fig..5 his is ecause hese funcions ae simly "escale" (i.e., / / (comaing Eq..7 wih Eq. 3.. Fig. 3.3 is of elaively lile acical uiliy unless we wo in ems of he vaiale η / (we noe ha we o esen seveal comaisons in ems of η, u fo acical alicaions soluions in (ime ae of moe iec use.

33 Figue 3.3 "Tye cuve" eesenaion of he / soluion (Eq. 3.. Soluion is loe vesus he moifie Bolzmann ansfom vaiale /(. ( Figue 3. "Tye cuve" eesenaion of he new moel ( / fomulaion (Eq..6. Soluion is loe vesus he moifie Bolzmann ansfom vaiale /(. (

34 3 Figue 3.5 "Tye cuve" eesenaion of he new moel ( ( / fomulaion (Eq..6. Soluion is loe vesus he invese of he moifie Bolzmann ansfom vaiale (( /. We also esen he essue eivaive soluions loe in ems of vaiales elae o imensionless aius an ime in Figs. 3. an 3.5 (we coninue o use a moificaion of he imensionless Bolzmann ansfom vaiale. Fig. 3. (he aius foma lo will e use o valiae simulae efomance aa whee we will have essue values a a secific saial gi as geneae y numeical simulaion. We coul use Fig. 3.5 as an analysis mechanism fo essue ansien aa howeve, as we iscuss in he nex secion, fo acical alicaions, he soluion mus e moifie o inclue welloe soage effecs. Fig. 3.5 will also e use in he valiaion oion of his wo o comae agains he exising soluions which ae ofen uilize in he analysis of essue ansien es aa oaine fom gas conensae esevoi sysems (namely, he -zone aial comosie esevoi case an he "sealing fauls" cases. 3.3 Welloe Soage an Sin Effecs Aiion of Welloe Soage Effecs awown Cases (Base Comaisons So fa in his wo we have only consiee he case of an ieal well oucing in an infinie-acing esevoi wih a oagaing emeailiy ofile whee he well is ouce a a single-consan flowae. In his secion we ovie a mechanism fo aing welloe soage effecs o ou new soluion fo a oagaing emeailiy ofile. Welloe soage is yically "ae" o he ase essue soluion using convoluion (o sueosiion whee we elieve ha convoluion shoul e vali fo his olem

35 ecause we have assume ha ae no non-lineaiies in he govening iffeenial equaion (Eq.. As such, he convoluion fo welloe soage is wien as: w [ qws( τ ] s ( τ...(3.6 0 τ Whee he q funcion (imensionless sanface ae ofile is given as follows fo he welloe soage moel: q ws qsf C [ w ]...(3.7 q su An he efiniion of he imensionless essue funcion which inclues sin effecs is given as: s s...(3.8 Eqs. 3.6 an 3.7 can e isceize an comine o yiel a "ecusion elaion" fo he welloe soage imensionless essue, w howeve, his aoach is eious an one o eo oagaion. Tyical imlemenaions of Eqs. 3.6 an 3.7 involve he use of he Lalace ansfomaion unfounaely, ou oose soluion (Eq. 3. is no suie o he use of he Lalace ansfom (i.e., his soluion can no e inegae analyically, an, as such, we mus eso o anohe aoach. Fo convenience we emloy he meho y Blasingame, e al. 6 fo geneaing essue soluions which inclue welloe soage an sin effecs he soluion use in his wo is given in Aenix B. We ovie Figs. 3.6a an 3.6 as valiaions fo he Blasingame, e al. meho secifically fo he case of well oucing in an infinie-acing homogeneous esevoi. The w funcion is comue using he oceues given in Aenix B an he w funcion is comue using he oceues given in

36 5 Figue 3.6a imensionless essue ye cuve fo aial flow ehavio incluing welloe soage an sin effecs ( w vesus /C foma. This lo esens a comaison of he soluion geneae using numeical invesion (as a suogae fo he exac soluion an he aoximae soluion echnique oose in ef. 6 an geneae using Mahemaica. Figue 3.6 imensionless essue eivaive ye cuve fo aial flow ehavio incluing welloe soage an sin effecs ( w ' vesus /C foma. This lo esens a comaison of he soluion geneae using numeical invesion (as a suogae fo he exac soluion an he aoximae soluion echnique oose in ef. 6 an geneae using Mahemaica.

37 6 Aenix C (we noe ha we have use a olynomial egession (a 3-oin fomula o calculae he w funcion. Excellen ageemen exiss eween he "exac" soluions (i.e., he numeical invesion soluion an he aoximae soluions ovie y he mehos given in ef. 6. By exension, we will aly he oceues given in Aenices B an C o ou new soluion fo a aially oagaing emeailiy funcion. In Figs. 3.7a-3.7f we ovie a sequence of soluions fo he secific case of C x0 3 an fo cases whee he min / max vaies fom x0 0 o x0-3. Iniviual los consie a single value of he - aamee, an he following cases of x0 0, 0 -, 0 -, 0-3, 0 -, 0-5 ae consiee (Figs. 3.7a-3.7f, esecively. Figs. 3.7a-3.7f illusae he "evolving" effecs of he -aamee, an we noe ha nonunique effecs ae ossile (i.e., a aicula case o en which aeas simila o anohe case, alhough hese cases have susanially iffeen ase oeies (e.g., min / max,, ec.. Mos of he cases in Figs. 3.7a-3.7f shoul e escie as unique (alhough Figs. 3.7 an 3.7c o aea o e vey simila. Aiion of Welloe Soage Effecs awown Cases ( /C Foma Plos Anohe ojecive fo wo in his secion is o esalish he geneal chaace/ehavio of such esuls. In Fig. 3.8 we esen a "comosie" lo of all w ens geneae fo C x0 3. We noe isinc ehavio fo each case an we sugges ha he chaace in hese welloe soage soluions (fo his aicula case is oh accuae an isinc. Similaly, in Fig. 3.9 we esen he same suie of soluions fo C x0 0. The mos ovious commen we can mae is ha viually all of he ens geneae fo he C x0 0 case ae ominae y welloe soage effecs i.e., he -aamee has viually no influence on he esonse of he soluion fo he C x0 0 case.

38 7 Figue 3.7a Tye cuve lo ( w an w ' vesus /C C x0 3, x0 0, vaious min / max cases. Figue 3.7 Tye cuve lo ( w an w ' vesus /C C x0 3, x0 -, vaious min / max cases.

39 8 Figue 3.7c Tye cuve lo ( w an w ' vesus /C C x0 3, x0 -, vaious min / max cases. Figue 3.7 Tye cuve lo ( w an w ' vesus /C C x0 3, x0-3, vaious min / max cases.

40 9 Figue 3.7e Tye cuve lo ( w an w ' vesus /C C x0 3, x0 -, vaious min / max cases. Figue 3.7f Tye cuve lo ( w an w ' vesus /C C x0 3, x0-5, vaious min / max cases.

41 30 Figue 3.8 awown ye cuve lo ( w ' vesus /C C x0 3, x0 0, 0 -, 0 -, 0-3, 0 -, 0-5, vaious min / max x0 0, 0 -, 0 -, 0-3. Figue 3.9 awown ye cuve lo ( w ' vesus /C C x0 0, x0 0, 0 -, 0 -, 0-3, 0 -, 0-5, vaious min / max x0 0, 0 -, 0 -, 0-3.

42 3 CHAPTER IV VALIATION OF AN ANALYTICAL PRESSURE SOLUTION FOR THE CASE OF A PERMEABILITY PROFILE THAT VARIES IN TIME AN RAIAL ISTANCE GAS CONENSATE RESERVOIRS. Comaison of he New Soluion an Soluions fo he Sealing Fauls an Raial Comosie Resevoi Cases Pessue Behavio in Time Ou goal is o ovie a qualiaive comaison of he new oose soluion (he esul given in ems of ime an he -zone aial comosie esevoi moel whee we noe ha he aial comosie moel is he mos commonly use esevoi moel fo he ineeaion an analysis of well es aa fom gas conensae esevois. We also esen a comaison of he oose moel wih he moel fo a well in he viciniy of one o moe "sealing fauls" whee ou goal is o simly comae he influence of ou new moel as a "flow consicion" o "flow aie." We ae no avocaing he use of he "sealing fauls" moels fo he analysis an ineeaion of well efomance aa in gas conensae esevoi sysems; we ae simly maing a qualiaive (gahical comaison of he soluions. Figue. Pessue eivaive ye cuve fo a veical well oucing a a consan ae nea a sealing faul in a homogeneous, infinie-acing esevoi.

43 3 In Fig.. we esen he soluion fo a well in he viciniy of one o moe sealing fauls his esenaion clealy inicaes ha he oienaion an nume of fauls amaically affecs he ehavio of he funcion. In Fig.. we esen he "unifie" lo ( funcion fo mulile cases of he aial comosie esevoi soluion. The mos imoan, an mos elevan issue is ha he aial comosie soluion has fixe moiliy an iffusiviy aios (fo he inne an oue zones. This use of fixe moiliy an iffusiviy aios is in iec conas o ou soluion which uses a emeailiy ofile in aius an ime, u only a single value of iffusiviy fo he enie esevoi. As such, we will only comae cases fo he aial comosie esevoi moel whee he iffusiviy aio is uniy. Figue. Pessue eivaive ye cuve fo a veical well oucing a a consan flowae in a -zone aial comosie esevoi sysem, vaious moiliy (λ/soaiviy (ω cases. In Fig..3 we esen a comine lo of all hee esevoi cases: he sealing fauls case, he -zone aial comosie esevoi case, an ou oose esevoi moel fo a emeailiy ofile which vaies in ime an aial isance. We noe suising similaiy fo he esuls shown in Fig..3 esie he fac ha he esevoi moels shown have lile in common. One ineeaion coul e ha his ehavio is a cause fo concen since he moels ae isincly iffeen ye ouce simila ehavio. Anohe ineeaion coul e ha he -zone (fixe aial comosie esevoi moel an he new oagaing

44 33 emeailiy ofile moel have, a leas in conce, a common enominao of ominan emeailiies (i.e., he "nea well" an "esevoi" emeailiies. In fac, as we noe fom Fig..3, he aial comosie an oagaing emeailiy soluions convege a "lae imes," i.e., when he esevoi emeailiy ominaes he essue esonse. This is an imoan valiaion as he moels o agee uniquely a lae imes. We conclue ha his comaison suggess uiliy of ou new moel fo he analysis of well es aa in gas conensae esevois wih he cavea ha we noe ealie egaing he fac ha ou oose moel uses a single value of iffusiviy, an he -zone comosie esevoi moel uses isinc iffusiviies (i.e., he "nea well" an he "esevoi" iffusiviies. The issue of he "sealing fauls" moel is somewha moe comlex we will simly sugges ha a "flow aie" (i.e., a sealing faul an a flow conas (i.e., he -zone aial comosie esevoi moel an he oagaing emeailiy moel have simila (hough no ienical ehavio ecause he flow aie/ conas affecs he essue ehavio in a simila fashion. This conclusion is somewha inucive, u we elieve i is oh lausile an elevan. Figue.3 Comine essue eivaive ye cuve fo he following cases: sealing fauls, a single aial comosie egion, an he oose moel fo a aially-vaying moiliy ofile.