SPE Modeling of Performance Behavior in Gas Condensate Reservoirs Using a Variable Mobility Concept Objectives Fig.

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1 SPE Moeling of Pefomance Behavio in Gas Conensae Resevois Using a Vaiale Moiliy Conce B.W. Wilson, Bae Alas, R.A. Ache, Univesiy of Auclan, an T.A. Blasingame, Texas A&M U. Coyigh 3, Sociey of Peoleum Enginees Inc. This ae was eae fo esenaion a he SPE Annual Technical Confeence an Exhiiion hel in enve, CO., 5-8 Ocoe 3. This ae was selece fo esenaion y an SPE Pogam Commiee following eview of infomaion conaine in an asac sumie y he auho(s). Conens of he ae, as esene, have no een eviewe y he Sociey of Peoleum Enginees an ae sujec o coecion y he auho(s). The maeial, as esene, oes no necessaily eflec any osiion of he Sociey of Peoleum Enginees, is offices, o memes. Paes esene a SPE meeings ae sujec o ulicaion eview y Eioial Commiees of he Sociey of Peoleum Enginees. Eleconic eoucion, isiuion, o soage of any a of his ae fo commecial uoses wihou he wien consen of he Sociey of Peoleum Enginees is ohiie. Pemission o eouce in in is esice o an asac of no moe han 3 wos; illusaions may no e coie. The asac mus conain consicuous acnowlegmen of whee an y whom he ae was esene. Wie Liaian, SPE, P.O. Box , Richason, TX , U.S.A., fax Asac This wo ovies a conce fo moelling well efomance ehavio in a gas conensae esevoi using an emiical moel fo he gas moiliy funcion. This moel is given y: - min ( max min ) - ex α This conce moel eesens he minimum gas emeailiy (o moiliy) nea he welloe an he maximum (o oiginal) gas emeailiy (o moiliy) in he "y gas" oion of he esevoi, as well as he ansiion egime. This moel was consuce ase on osevaions eive fom numeical simulaion esuls whee he sauaion, effecive emeailiy, an gas moiliy ae esene as funcions of isance in he esevoi. The uiliy of his conce is ha i can e use o evelo a essue soluion fo he ehavio of he gas hase ouce fom a gas conensae esevoi. This new soluion is valiae agains numeical simulaion an has een esene gahically fo use in well es analysis (in he fom of "ye cuves"). The avanage of his soluion ove he convenional aial comosie esevoi soluions is ha he evoluion of he conensae zone can e eesene an evaluae as i occus in ime. The ovious limiaion is he simlifie fom of he g ofile as a funcion of aius an ime, as well as he eenence/aoiaeness of he "α" coefficien. Alicaion of his new essue soluion o well es analysis is oose an comaisons o he aial comosie (an ohe esevoi moels) ae also esene. Ou goal is o emonsae ha he oose soluion has oenial uiliy in he analysis an ineeaion of esevoi efomance aa (mos liely, essue awown an essue uilu es aa). We ecognize ha he simliciy of his aoach may have acical limiaions fo examle, we consie a aiallyvaying moiliy ofile, u we also assume a consan iffusiviy his is a oenial shocoming ha shoul e consiee in fuue wo. Ojecives The imay ojecive of his wo is: 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 he moiliy ofile oseve fom numeical simulaion fo a aial gas conensae esevoi sysem). 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 so ha he new soluion can e comae o ohe soluions (e.g., he -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 new moel o evelo soluions which inclue welloe soage an sin effecs fo moeling he essue o an essue o eivaive funcions in ime. To oose alicaions in he analysis of well es aa acquie fom essue awown o essue uilu ess. Saemen of Polem This wo is focuse on he conce of using a funcional fom o eesen a escie moiliy ofile (i.e., /μ) an o incooae his emiically-eive moel ino he igoous iffusiviy equaion fo he liqui case. The goal is o use his conce an he esuling flow moel o eesen he gas conensae case. We ae eaing his case as a "liqui equivalen" olem whee non-iealiies (e.g., essueeenen PVT funcions) ae aesse 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 as

2 B.W. Wilson, R.A. Ache, an T.A. Blasingame SPE 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 Roussenac offe an aoiae saing oin as hese cases ae well caliae an veifie. Using he esuls esene y Roussenac (see Fig. ); we have esalishe he following conceual moel fo eesening he emeailiy as a funcion of aius an essue: - min ( max min ) - ex... () α The aoiae fom of he iffusiviy equaion fo his wo is given as: c φμ (Fiel unis)... ().637 Recall ha we have esume a "liqui equivalen" case (i.e., hee ae no essue eenen oeies). As such, Eq. is use as he emeailiy moel an is coule wih he aial flow iffusiviy equaion fo his case (Eq. ) hen solve fo he case of a well ouce a a consan ae in an infinieacing esevoi. The elevan foms of his soluion ae eive in Aenix A an esene elow. "Pessue Soluion" α [ / ] 4 x min max 4 4 α α ex ln[ e 4 α ( min / max 4 )] α... (3) "Pessue eivaive in Time" α 4 max [ / ] min 4 4 α α x ex ln[ e 4 α 4 ( min / max 4 )] α 4... (4) "Pessue eivaive in Raial isance" α 4 max [ / ] min 4 4 α α x ex ln[ e 4 α 4 ( min / max 4 )] α 4... (5) The mos imoan issue o consie in evaluaing Eqs. 3-5 is ha no limiing assumions ae mae (ohe han he aiional soluion aoach using he Bolzmann ansfom) an each fomulaion is esume o e unique. We will noe ha Eq. 3 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. 4 an 5 ae close fom esuls ha ae essenially ienical in fom, we noe ha comaison of Eqs. 4 an 5 yiel he following ieniy:... (6) Whee (as noe in Aenix A), Eq. 6 is uniquely vali fo his case as well as he homogeneous esevoi soluion. Fo loing he essue eivaive funcions in oh ime an sace we have efine he following efiniions: (which ae eive y insecion of Eqs. 4 an 5)... (7)... (8) Pesenaion of Resuls Pessue Soluion fo he Case of a Pemeailiy Pofile ha Vaies wih Time an Raial isance The new soluion is esene in he following fomas: vesus α / : (Fig. ) This foma ovies esecive on "ealy" an "lae" ehavio, whee fo a aicula value of α we can oseve he evoluion of he emeailiy ofile in he essue eivaive esonse. The genealize x-axis loing funcion emis us o view all ossile scenaios of he α - aamee on he same lo. ( ) vesus /(α ): (Fig. 3a) This foma illusaes he aial isiuion of imensionless essue (analogous o essue o). We immeiaely noe ha he highes values of essue o (o imensionless essue, ) occu nea he well (i.e., a small values)). Fo comaison, we noe ha, nea he well, he lowes values of occu fo he homogeneous esevoi case (i.e., min / max ) his confims ou conce ha he evolving emeailiy ofile acs o euce flow nea he well. vesus /(α ): (Fig. 3) We noe ha, in he fomas use in hese los, Figs. an 3 ae essenially "mio images" which is he ehavio we exec when we consie he ieniy given y Eq. 9. Fig. 3 (as Fig. 3a) has lile acical alicaion (we o no measue essues in he esevoi) howeve, we will use oh Figs. 3a an 3 o evaluae/valiae a numeical simulaion whee essue is ovie as a funcion of aius. Valiaion Pessue Behavio in Raial isance A cusoy valiaion of he new soluion is achieve y comaison of he soluion (secifically, he an fomulaions) wih esuls oaine fom he lieaue (Roussennac (ef. )). In Fig. 4a we esen he Δ (i.e., Δ i - ) mach fo his case an he (/) mach is shown in Fig. 4. Wih he exceion of he ehavio nea he well (i.e., fo small -values), we noe an excellen mach of he aa wih he oose soluion.

3 SPE Moeling of Pefomance Behavio in Gas Conensae Resevois Using a Vaiale Moiliy Conce 3 The ehavio a small values of is ominae y he "sin zone" acually a egion of secifie emeailiy use o ovie he effec of nea-well amage. We o no consie he exisence of a "nea-well amage" zone, we simly moel a oagaing emeailiy ofile as shown in Fig.. In Fig. 5 we esen he Δ an (/) aa along wih he Δ an (/) funcions comue using ou new esevoi moel. Using Fig. 5 we have aeme o ienify/classify he flow egimes which wee oseve uing his simulaion. We noe ha his comaison of aa an ou oose soluion ovies a song valiaion of he oose soluion. Pessue Behavio in Time Ou goal is o ovie a qualiaive comaison of he new oose soluion (ime foma esul) 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. In Fig. 6 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. 7 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) 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. In Fig. 8 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 in he esuls shown in Fig. 8 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 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. 8, 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. Aiion of Welloe Soage Effecs In ou wo so fa we have only consiee he case of an ieal well in an infinie-acing esevoi wih a oagaing emeailiy ofile whee he well is ouce a a consan ae. 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 convoluion shoul e vali fo his olem 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 )... (9) τ Whee he q funcion (imensionless sanface ae ofile) is given as follows fo he welloe soage moel: qsf q ws C [ w ]... () qsu An: s s... () Eqs. 9 an can e comine o yiel a "ecusion elaion" fo he welloe soage imensionless essue, w, u his aoach is eious an one o eo oagaion. Tyical imlemenaions of Eqs. 9 an involve he use of he Lalace ansfomaion unfounaely, ou oose soluion (Eq. 3) is no suie o he use of he Lalace ansfom, an, as such, we mus eso o anohe aoach. Fo convenience we emloy he meho y Blasingame, e al. 3 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. 9a an 9 as valiaions fo he Blasingame, e al. meho secifically fo he case of an infinieacing 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 Aenix C (whee we noe ha we have use a olynomial egession (a 3-oin fomula) o calculae he w funcion. We noe excellen ageemen eween he "exac" soluions (i.e., he numeical invesion soluion) an he aoximae soluions

4 4 B.W. Wilson, R.A. Ache, an T.A. Blasingame SPE ovie y he mehos given in ef. 3. 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. a-f we ovie a sequence of soluions fo he secific case of C x 3 an min / max vaying fo each case fom x o x -3. Iniviual los consie a single value of α, an he following cases of α x, -, -, -3, - 4, -5 ae consiee (Figs. a-f, esecively). In Figs. a-f we noe he "evolving" effecs of he α -aamee, an we commen ha non-unique 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, α, ec.)). Mos of he cases in Figs. a- f shoul e escie as unique (alhough Figs. an c o aea o e vey simila). Ou ojecive in his secion is wo-fol fis we wane o esen he evelomen of welloe soage soluions using ou new aially oagaing emeailiy esul. Secon, we wan o esalish he geneal chaace/ehavio of such esuls. In Fig. we esen a "comosie" lo of all w ens geneae fo C x 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. we esen he same suie of soluions fo C x. The mos ovious commen ha we can mae is ha viually all of he ens geneae fo he C x case ae ominae y welloe soage effecs i.e., he α -aamee has viually no influence on he esonse of he soluion fo he C x case. The Pessue Builu Case In his wo we esume ha he oose soluion is "linea" in ha hee ae no essue-eenen coefficiens in he govening iffeenial equaion. As lineaiy is esume, we can use he heoem of sueosiion o accoun fo he vaiaion of ae which occus in a essue uilu es (we mae he convenional assumion ha a "essue uilu" is a eio of zeo ae, ecee y a eio of oucion,, a a consan flowae. The mahemaical exession of sueosiion fo his case is given in he fom of imensionless essue an ime as follows: 4 w, BU maxh ( 4. qbμ w ( Δ ws ) (Δ) w ( wf Δ (Δ )) ) w ( Δ... (9) We have use Eq. 7 o geneae a seies of cases whee he α an C aamees ae hel consan (α x -3, an C x 3 ), whee an an min / max ae vaie. The vaiance in he min / max aamee ovies a "san" of oucomes which illusae he influence of he vaiale moiliy funcion. In aiion, only a single value of he α an C aamees is use in oe o caue he secific influence of he an min / max aamees. Figs. 3a-3 illusae cases fo x 3, 6, 9, esecively, whee oh he w an w funcions ae ) loe vesus Δ /C. Fo he x 3 case (Fig. 3a) we noe an exeme influence of he oucing ime ( ). In Fig. 3 ( x 6 ) we noe ha oucing ime ( ) oes have a song influence, u a leas a of he esonse is unaffece y effecs a ealy imes (small Δ /C values). Fo he case of x 9 (Fig. 3c) we fin ha effecs ae only evien a lae imes (lage Δ /C values) his is exece since (Δ /C ) max C Δ,max x 9. As such, Δ,max /, an we woul exec some influence of oucing ime effecs (y analogy wih he homogeneous esevoi soluion). Ou final comaison, Fig. 3, fo he case of x we fin no evience of oucing ime effecs which is exece since (Δ /C ) max C Δ,max x 9 an Δ,max /. (he "awown" eio is en imes he uaion of he "uilu" eio). In Fig. 4 we esen a comaison of he ime eivaive funcions ( w ) vesus Δ /C fo all of he essue uilu soluions as well as he essue awown soluion. This comaison is useful o gauge he influence of elaive o he ase soluion (i.e., he awown soluion). We can assess he qualiaive influence of as eing none ( x ), o moeae ( x 9 ), o significan ( x 6 ), o exeme ( x 3 ). We also consiee cases whee no welloe soage effecs in-fluence he essue uilu soluion an we woul noe ha simila commens egaing he influence of he -aamee can e mae. We elece o focus on cases which o inclue welloe soage (an sin) effecs in oe o assess he acical influence of he -aamee. We will noe ha, as wih he homogeneous esevoi case, oucing ime effecs can e aesse (a leas aoximaely) using he "effecive ime" coecion oose y Agawal. 4 Ou final commens wih ega o he essue uilu case ae ha his emains a oic fo aiional suy. We elieve ha we have valiae (a leas conceually) ha he vaiale moiliy moel may eesen he ehavio of some cases of esevoi efomance in gas conensae esevois. Howeve, we elieve ha ou conce of using a escie emeailiy ofile may e inaequae an ha we mus also emloy a escie iffusiviy ofile. In sho, we elieve ha we mus consie he iffusive effecs of he conensae an in aiion o he emeailiy (o effecive/elaive emeailiy) ofile. Again, his is a ecommenaion fo fuue wo. Summay an Conclusions. New Soluion: We have oose, eveloe, an veifie new soluions (fo essue an he essue eivaive funcions in ems of aial isance an ime) fo he case of a well oucing a a consan flowae fom an infinie-acing aial flow sysem whee he emeailiy vaies in aial isance an ime (see Eq. fo he moel emloye in his wo). Eq. is oose ase on osevaions of well efomance ehavio fom numeical simulaion of he gas hase fo a aial gas conensae esevoi sysem.

5 SPE Moeling of Pefomance Behavio in Gas Conensae Resevois Using a Vaiale Moiliy Conce 5 The elevan esuls in his wo ae given y Eqs. 3-5 an we noe ha Eq. 3 canno e esolve eyon he inegal fomulaion as esene. As such, all esuls fo Eq. 3 ae geneae using numeical inegaion efome in Mahemaica. The eivaive fomulaions given y Eqs. 4 an 5 ae close fom esuls an ae comuaionally efficien as well as convenien funcional foms.. Comaison/Valiaion: The oose soluion is esene in comaison o numeical simulaion esuls (fo he / fomulaion). The / fomulaion is comae o he -zone aial comosie moel as well as simlifie cases of "sealing fauls" he comaions inicae ha he oose soluion oes ouce simila feaues an suggess he moel woul e an effecive ineeaion ool fo well es analysis. Ou esenaion of he w ( ) an w ( ) funcions (which inclue welloe soage an sin effecs) inicae ha he influence of he α (o α ) aamee an he min / max aio is susanive an unique fo ceain cases (e.g., low values of C ), while fo highe values of C welloe soage effecs ominae he esonse. This is analogous o say, he case of well efomance in a ual oosiy/naually facue esevoi. 3. Pessue Builu Case: This is a case fo fuue invesigaion. The convenional essue uilu fomulaion mimics he essue awown case fo he escie emeailiy ofile. This is no an unexece ehavio fo his fomulaion an we sugges ha his aoach mus e exene o inclue a vaiale "iffusiviy" ofile ((φμ g c g )/ g ) as well as a vaiale moiliy ofile ( g /μ g ) in oe o caue he unique signaue esene y he essue uilu case. Nomenclaue Fiel Vaiales (Pessue, Fomaion, an Flui Poeies) B Fomaion volume faco, RB/STB c g Gas comessiiliy, si - c Toal comessiiliy, si - h Ne ay hicness, f o g Effecive emeailiy o gas, m max Maximum effecive emeailiy o gas, m min Minimum effecive emeailiy o gas, m i Iniial esevoi essue, sia wf Flowing essue, sia ws Shu-in essue, sia q Flowae, STB/ q sf "Sanface" flowae, STB/ q su "Suface" flowae, STB/ w Welloe aius, f Raial isance, f Time, h Poucion ime, h Δ Shu-in Time, h α Scaling em fo essue ehavio, (c-si - )/m Bolzmann ansfom vaiale ( /(4)) μ Viscosiy, c imensionlessvaiales C imensionless welloe soage coefficien imensionless emeailiy funcion imensionless essue (geneic) imensionless essue eivaive funcion in aial isance (Eq. 8) imensionless essue eivaive funcion in ime (Eq. 7) s s, imensionless essue wih sin effecs w imensionless essue wih welloe soage an sin effecs w imensionless essue eivaive funcion in ime incluing welloe soage an sin effecs q ws imensionless flowae fo welloe soage imensionless aius imensionless ime imensionless oucion ime Δ imensionless shu-in ime imensionless Bolzmann ansfom vaiale a imensionless emiical scaling em fo essue ehavio s Sin faco, imensionless Refeences. Roussenac, B.: Gas Conensae Well Analysis, M.S. Thesis, Sanfo Univesiy, June.. Mahemaica (sofwae) Ve. 4., Wolfam Reseach, Chamaign-Uana, IL (). 3. Blasingame, T.A., Johnson, J.L., an Lee, W.J.: "Avances in he Use of Convoluion Mehos in Well Tes Analysis," ae SPE 86 esene a he 99 Join Rocy Mounain Regional/Low Pemeailiy Resevois. 4. Agawal, R.G.: "A New Meho o Accoun fo Poucing Time Effecs When awown Tye Cuves ae use o Analyze Pessue Builu an ohe Well Tes aa, ae SPE 989 esene a he 98 SPE Annual Technical Confeence an Exhiiion, allas, TX, -4 Seeme 98. 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) In his Aenix, we eive wo exessions fo he essue eivaive (ime an aial isance fomulaions) ha consie he changing effecive (o elaive) emeailiy of he eogae gas as conensae evolves wih eceasing essue. This eivaion egins wih he ase iffusiviy equaion i.e., he aial iffeenial equaion which escies he flow of a single hase flui in a oous meium wih esec o ime an isance. The effecive emeailiy o gas in such cases will no e consan, u is eenen on he PVT an oc-flui oeies. The imay coniuion of his wo is he evelomen of a close fom analyical soluion fo he case of a aially vaying moiliy (o effecive emeailiy) funcion in a esevoi sysem. The suoinae coniuion (which is, in some ways, moe imoan han he soluion) is

6 6 B.W. Wilson, R.A. Ache, an T.A. Blasingame SPE ou oosal of a simle funcional elaionshi o eesen he ime an sace-eenency of he gas moiliy (o emeailiy) funcion. The ase fom of he iffusiviy equaion which consies a vaying emeailiy wih esec o aius is given as: c φμ (Fiel unis)... (A.). 637 As menione aove, we have oose a geneal moel fo he ehavio of he emeailiy o gas as a funcion of ime an aius. Ou oose moel is given in is mos asic foms as: - min ( max min ) - ex... (A.a) α - max ( max min) ex... (A.) α We noe ha he α-aamee in Eqs. A.a an A. is an emiical consan, mos liely elae o he PVT chaaceisics of he esevoi flui, as well as he oc-flui oeies. Ou goal is no o assess he naue of he α-aamee, u ahe, o use his as a mechanism o eesen a comlex ocess wih a simle moel. Eq. A. is he imay fom use in his Aenix, an we will noe ha Eqs. A.a an A. have een valiae conceually via comaison wih simulae efomance fo a gas conensae esevoi sysem. We consie he Bolzmann ansfom, which allows us o elae imensionless ime an isance:... (A.3) 4... (A.4) α α Susiuing Eqs. A.3 an A.4 ino Eq. A. yiels: - 4 max ( max min ) ex... (A.5) α efining a "imensionless" emeailiy,, we have: / max... (A.6) o, solving Eq. A.6 fo he emeailiy, we oain: max We noe ha we will use he ems "emeailiy" an "effecive emeailiy" inechangealy in his eivaion howeve, he vaiale in quesion is always effecive emeailiy. Susiuing he efiniion of "imensionless" emeailiy (i.e., Eq. A.6) ino Eq. A.5 gives us: min 4 ex... (A.7) max α We nee o ansfom Eq. A. ino imensionless fom hence, we sae he imensionless vaiales use in his wo ae as follows: (Fiel unis fomulaion) imensionless Pessue: h max ( i )... (A.8) 4. qbμ imensionless Time:. max (A.9) φμcw imensionless Raius:... (A.) w Susiuing Eqs A.9 an A. ino Eq. A.4 an solving fo he α aamee, we have: φμc w α α α.637 w max O, φμc α α... (A.).637 max Fom Eq. A. we noe ha he α-aamee has he unis of invese iffusiviy (i.e., iffusiviy (/(φμc )) has he unis of (m/(c-si - ) fiel unis fomulaion) heefoe, α also has he unis of m/(c-si - ). Physically, we assign he oeies of he flui an oc-flui ineacion o he α- aamee howeve, we consie α o e an emiical aamee, an, as such, we shoul no aem o quanify he comonens of α, u ahe, we shoul simly use α o qualify he influence of he flui on he emeailiy ofile. Susiuing Eqs A.6, A.8-. ino Eq. A. an eaanging yiels he iffusiviy equaion in imensionless fom:... (A.) We noe ha fo he case whee max, Eq. A. eves o he convenional iffusiviy equaion fo a consan emeailiy. We also noe ha we have assume a slighly comessile flui (i.e., a liqui) in he eivaion of he iffusiviy equaion fo aial flow (i.e., Eq. A.). The assumion of a "liqui" may seem incomaile wih he conce of a gas case howeve, we ae eiving a fomulaion fo a "liqui" ha will, in un, e use fo gases whee he convenional gas seuofuncions will e emloye (i.e., seuoessue an seuoime). Simly u, his case eesens an "equivalen" liqui, moificaions will e aesse using seuofuncions ha "conve" he case in quesion o he "equivalen" liqui case. Uilizing he Bolzmann ansfom we eive a elaionshi fo essue wih esec o ime an aius which inclues he escie vaying emeailiy moel (i.e., Eq. A. o A.7). Fo convenience, we efine he consans a an as follows: a ( min / max )... (A.3) 4... (A.4) α Susiuing Eqs. A.3 an A.4 ino Eq. A.7 yiels: a ex[ ]... (A.5) Alying he ouc ule o he lef-han-sie (LHS) of Eq. A. we have:

7 SPE Moeling of Pefomance Behavio in Gas Conensae Resevois Using a Vaiale Moiliy Conce 7 Mulilying hough he lef-han-sie y / gives: Collecing lie ems an consoliaing he ems:... (A.6) Uilizing he Bolzmann vaiale,, o ansfom Eq. A.6 fom an ino, we have:... (A.7) A his oin we ecognize ha Eq. A.7 is he funamenal govening elaion fo flui flow in ou sysem whee he moiliy/emeailiy funcion is emie o vay as a funcion of ime an isance. Eq. A.7 is a comleely geneal esul no assumions have een mae a his oin. Ou goal is o solve Eq. A.7 fo an aoiae se of iniial an ounay coniions. The aicula case whee he Bolzmann ansfom alies is he case of a unifom iniial essue ofile in he esevoi (i.e., (, ) ) an he case of an "infinie-acing" oue ounay (i.e., (, ) ). Recalling he efiniion of he Bolzmann ansfom vaiale,, we have: 4... (A.3) The iniial an oue ounay coniions ae exesse in ems of,, an as follows: Iniial Coniion: ) (, fo ), ( Oue Bounay Coniion: ) (, fo ), ( We noe ha in using he Bolzmann ansfomaion, he iniial an oue ounay coniions collase o a single elaion: ) (... (A.8) This esul is a unique ouc of he Bolzmann ansfomaion fo his aicula case. We will ocee wih his esul an nex we consie he case of a consan flowae a he well. Inne Bounay Coniion: (Consan Rae) B h q μ (A.9) Whee Eq. A.9 is wien iecly fom acy's law fo a aial flow geomey. Isolaing he (/) em, we have: h qb μ (A.) Susiuing he efiniions of imensionless essue, aius, an emeailiy (i.e., Eqs. A.6, A.8, an A.) ino Eq. A., an eaanging gives us he following esul fo he ehavio a : (i.e., he "line souce" fomulaion)... (A.) Tansfoming Eq. A. using he Bolzmann vaiale,, gives us:... (A.) In oe o evelo a soluion fo Eq. A.7 we will uilize a "vaiale of ansfomaion" ha euces he iffeenial equaion o a moe convenien fom. A his oin we noe ha Eqs. A.7, A.8, an A. ae only a funcion of he Bolzmann ansfom vaiale,. As Eq. A.7 is a secon oe oinay iffeenial equaion, we can sumise ha a soluion can e oaine y wice inegaing his iffeenial equaion (analogous o he homogeneous esevoi case). This will e ou ah, u we will also use a vaiale of ansfomaion o euce he comlexiy fo he inegaion of Eq. A.7. Ou "vaiale of ansfomaion," ν, is given y: v whee v Maing hese susiuions ino Eq. A.7 gives he following "comac fom" of he iffeenial equaion we hen will solve his elaion y inegaion fo he ν-vaiale. v v... (A.3) Using Eq. A.5 in he em fom Eq. A.3: ] ex[ ] ex[ a a... (A.4) Susiuing Eq.A.4 ino Eq. A.3 yiels, ] ex[ ] ex[ v a a v Isolaing/seaaing he elevan ems we have: a a v v ] ex[ ] ex[... (A.5)

8 8 B.W. Wilson, R.A. Ache, an T.A. Blasingame SPE Seing u he inegaion of Eq. A.6 gives us: a ex[ ] v v a ex[ ] Exaning he igh-han-sie inegal gives: a ex[ ] v v a ex[ ] Comleing he inegaion, we have: a ex[ ] ln[ v] ln[ ] β... (A.6) a ex[ ] We noe ha he β-em in Eq. A.6 is a consan of inegaion which esuls fom he inefinie inegaion. The inegal ha can no e esolve iecly in Eq. 6 mus e aesse using ales of inegals, susiuion mehos, o a symolic inegaion ouc (in his case, we use Mahemaica (ef. )). Fom Mahemaica we oaine he following esul fo he emaining inegal: I a ex[ ] a ex[ ] ( ) ln[ e a] Susiuing his esul ino he soluion, we have: ( ) ln[ v] ln[ ] ln[ e a] β Exoneniaing he soluion, we oain: ex[ β ] ( ) v ex ln[ e a] efining ou consan of inegaion as c ex[β], an susiuing his esul ino he soluion, along wih he efiniion ν /, we have: ( ) c ex ln[ e a]... (A.7) Ou nex as is o eemine he consan of inegaion, c, whee his can e accomlishe using he inne ounay coniion (i.e., Eq., A.). Mulilying hough Eq. A.7 y he Bolzmann ansfom vaiale,, we have: ( ) c ex ln[ e a]... (A.8) Solving fo he consan of inegaion, c, we have: c a ( )... (A.9) Susiuion of he consan of inegaion, c, (Eq. A.9) ino he geneal soluion (Eq. A.8) gives: ( ) ( a) ex ln[ e a]... (A.3) efinie inegaion of Eq. A.3 (using his iniial coniion) yiels he soluion in ems of ( ) his esul is given as: ( a) ( ) ex e a ln[ ]... (A.3) Revesing he limis of inegaion in Eq. A.37 eliminaes he (-) sign an us he esul ino a moe aiional fom. ( a) ( ) ex e a ln[ ]... (A.3) Unfounaely, Eq. A.3 can only e inegae numeically we have also emloye Mahemaica as he mechanism o comue he numeical inegaion of Eq. A.3 fo he cases consiee in his wo. Solving Eq. A.3 fo he ime an aial isance eivaive foms, we have: ( ) 4 ( a) ex ln[ e a] 4... (A.33) An, ( ) 4 ( a) ex ln[ e a] 4... (A.34) Whee we ecognize ha he igh-han-sies (RHS) of Eqs. A.33 an A.34 ae ienical (exce fo he / mulilie in Eq. A.33) as such, equaing Eqs. A.33 an A.34 gives he following ieniy:... (A.35) We noe ha he ieniy given y Eq. A.35 is also oaine fo he "homogeneous" case whee. In fac, fo he case of, he enie sequence of esuls eves o he "aiional" line souce soluion fo a homogeneous, infinie-acing esevoi. We also noe ha Eq. A.34 is yically alie as he asolue value of his esul fo comaaive/illusaive los (oviously he / em is negaive). Aenix B An Aoximae Technique fo he iec Aiion of Welloe Soage an Sin Effecs In his Aenix, we esen a simle, aoximae echnique fo aing welloe soage o imensionless essue soluions. This esul is aen fom Blasingame, e al. 3

9 SPE Moeling of Pefomance Behavio in Gas Conensae Resevois Using a Vaiale Moiliy Conce 9 The equie esul fom (ef. 3) is given as: ψ θ w (- ex[-ω ]) (ex[-ω ] -) ω... (B.) ω ω Whee he ω, θ, an ψ aamees in Eq. B. ae given y: C ˆ ω... (B.) Câ ψ... (B.3) C ˆ θ... (B.4) âc Whee he â an ˆ aamees in Eqs. B. an B.4 ae given y: â s s... (B.5) ˆ s... (B.6) An, s s... (B.7) Eq. B. shoul ovie esuls which ae accuae o wihin - ecen of he exac soluion fo he w an he w funcions (whee w ( w / )). Aenix C A Quaaic Fomula fo Numeical iffeeniaion Pesuming a geneal quaaic olynomial, we have: y a a a... (C.) The coefficiens of an ineolaing LaGange collocaion olynomial ae sae as follows: c y( i )... (C.) y( i y c ) ( i)... (C.3) i i y( ) ( ) i c c i c i... (C.4) ( i i )( i i) Whee we noe ha we have use a "acwa" samling fo he coefficiens (i.e., in ems of i, i-, an i- ) his is fo convenience in ou esen wo. Alenaively, we coul use fowa o cenal samling wih no loss in genealiy. The a, a, an a coefficiens fo Eq. C. ae efine in ems of he coefficiens of he collocaion olynomial as follows: a c c i ci i... (C.5) a c c ( i i )... (C.6) a c... (C.7) The eivaive of Eq. C. is: y a a... (C.8) Given a ale of an y() values, Eqs. C.-C.7 ae use o comue he equie coefficiens. Eq. C.8 is use o comue he esie eivaive.

10 B.W. Wilson, R.A. Ache, an T.A. Blasingame SPE 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). Figue "Tye cuve" eesenaion of he new moel ( ( / ) fomulaion (Eq. 4)). Soluion is loe vesus he invese of he moifie Bolzmann ansfom vaiale ((α )/ ).

11 SPE Moeling of Pefomance Behavio in Gas Conensae Resevois Using a Vaiale Moiliy Conce Figue 3a "Tye cuve" eesenaion of he new moel ( ( ) fomulaion (Eq. 3)). Soluion is loe vesus he moifie Bolzmann ansfom vaiale ((α )/ ). Figue 3 "Tye cuve" eesenaion of he new moel ( ( / ) fomulaion (Eq. 5)). Soluion is loe vesus he moifie Bolzmann ansfom vaiale ((α )/ ).

12 B.W. Wilson, R.A. Ache, an T.A. Blasingame SPE Figue 4a Mach of Roussenac aa (igiize) an he new vaiale moiliy moel (ye cuve mach) ( ) vesus (α )/ foma. Figue 4 Mach of Roussenac aa (igiize) an he new vaiale moiliy moel (ye cuve mach) ( / ) vesus (α )/ foma. Figue 5 Mach of Roussenac aa (igiize) an he new vaiale moiliy moel (aa/moel mach) Δ an / vesus foma. Noe he excellen mach of he aa an moel funcions houghou he esevoi exce fo he "sin zone" nea he welloe.

13 SPE Moeling of Pefomance Behavio in Gas Conensae Resevois Using a Vaiale Moiliy Conce 3 Figue 6 Pessue eivaive ye cuve fo a veical well oucing a a consan ae nea a sealing faul in a homogeneous, infinie-acing esevoi. Figue 9a 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. 3 an geneae using Mahemaica. Figue 7 Pessue eivaive ye cuve fo a veical well oucing a a consan ae in a comosie aial sysem, vaious moiliy (λ)/soaiviy (ω) cases. Figue 9 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. 3 an geneae using Mahemaica. Figue 8 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.

14 4 B.W. Wilson, R.A. Ache, an T.A. Blasingame SPE Figue a Tye cuve lo ( w an w ' vesus /C ) C x 3, α x, vaious min / max cases. Figue Tye cuve lo ( w an w ' vesus /C ) C x 3, α x -3, vaious min / max cases. Figue Tye cuve lo ( w an w ' vesus /C ) C x 3, α x -, vaious min / max cases. Figue e Tye cuve lo ( w an w ' vesus /C ) C x 3, α x -4, vaious min / max cases. Figue c Tye cuve lo ( w an w ' vesus /C ) C x 3, α x -, vaious min / max cases. Figue f Tye cuve lo ( w an w ' vesus /C ) C x 3, α x -5, vaious min / max cases.

15 5 B.W. Wilson, R.A. Ache, an T.A. Blasingame SPE Figue awown ye cuve lo ( w ' vesus /C ) C x 3, α x, -, -, -3, -4, - 5, vaious min / max x, -, -, -3. Noe he unique influence of oh he α an he min / max aamees his comaison inicaes ha he oose moel has a song chaaceisic ehavio fo C x 3. Figue awown ye cuve lo ( w ' vesus /C ) C x, α x, -, -, -3, -4, -5, vaious min / max x, -, -, -3. Noe ha he α aamee oes no exe a song influence on he chaaceisic ehavio fo C x.

16 6 B.W. Wilson, R.A. Ache, an T.A. Blasingame SPE Figue 3a Tye cuve lo ( w an w ' vesus Δ /C ) x 3, C x 3, α x -3, vaious min / max cases. Figue 3 Tye cuve lo ( w an w ' vesus Δ /C ) x, C x 3, α x -3, vaious min / max cases. Figue 3 Tye cuve lo ( w an w ' vesus Δ /C ) x 6, C x 3, α x -3, vaious min / max cases. Figue 4 Builu ye cuve lo ( w ' vesus Δ /C ) x 3, 6, 9,, C x 3, α x -3, vaious min / max cases. Noe he song influence of he - aamee (analogous o he homogeneous esevoi case). Figue 3c Tye cuve lo ( w an w ' vesus Δ /C ) x 9, C x 3, α x -3, vaious min / max cases.