Modeling of Matrix-Fracture Interaction for Conventional Fractured Gas Reservoirs

Transcription

1 UNIVERSITY OF CALGARY Modling of Marix-Fracur Inracion for Convnional Fracurd Gas Rsrvoirs by Ehsan Ranjbar A THESIS SUBMITTE TO THE FACULTY OF GRAUATE STUIES IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE EGREE OF OCTOR OF PHILOSOPHY EPARTMENT OF CHEMICAL AN PETROLEUM ENGINEERING CALGARY, ALBERTA JANUARY, 04 Ehsan Ranjbar 04

2 Absrac Modling of arix-fracur ransfr funcion is iporan in h siulaion of fluid flow in fracurd porous dia using a dual-porosiy concp. This ransfr funcion is dircly rlad o h shap facor. On of h ain focuss of his sudy is o find h shap facor for h singl-phas flow of coprssibl fluids in fracurd dia using h soluion of nonlinar gas diffusiviy quaion. Th dvlopd shap facor can b usd as an inpu for odling flow of coprssibl fluids in dual-porosiy syss. For a coprssibl fluid, h consqunc of a prssur boundary condiion on h shap facor has no bn invsigad in h prvious sudis. Anohr ajor purpos of his sudy is, hrfor, o invsiga h ffc of h fracur prssur on h shap facor for singl-phas flow of a coprssibl fluid. Mos of h dvlopd odls for fracurd rsrvoirs assu idal arix block siz disribuion. This assupion ay no b valid in raliy for naurally fracurd rsrvoirs and possibly lad o rrors in prdicion of producion fro h naurally fracurd rsrvoirs spcially during arly i producion fro h arix blocks. Th ffc of diffrn arix block siz disribuions on h singl-phas arix-fracur fluid ransfr is sudid using a si-analyical approach. Th proposd odl is abl o siula fluid xchang bwn arix and fracur for coninuous or discr block siz disribuions. In h las par of his sudy w prsn si-analyical soluions for rlas of a singl-phas liquid or gas fro cylindrical wo dinsional flow and sphrical hr dinsional flow arix blocks wih various block siz disribuions and diffrn fracur boundary condiions. This soluion can b siplifid o odl flow of slighly coprssibl fluids lik war or oil in dual-porosiy dia. Th approxia si-analyical odl for h arix fracur ransfr funcion prsnd in his sudy for diffrn cass is vrifid using singl-porosiy, fin-grid, nurical siulaions. This odl can rcovr h shap facor of slighly coprssibl fluids rpord in h liraur. In h proposd si-analyical odl for all cass h prssur variabiliy of viscosiy and isohral coprssibiliy is considrd by solving h nonlinar parial diffrnial quaion of coprssibl fluid flow in h fracurd dia. ii

3 Acknowldgns I would lik o xprss y dps graiud o all hos who providd h opporuniy o copl his hsis. I hav workd wih a gra nubr of popl. I is y plasur o hank h for hir suppors and hlp during y Ph progra a h Univrsiy of Calgary. Firs of all I would lik o xprss dps apprciaion o y suprvisors r. John Chn and r. Hassan Hassanzadh for hir uncondiional suppor during h cours of his sudy. This work would no hav bn possibl wihou hir valuabl cons, ncouragns, guidanc and aids. Thanks r. Hassanzadh for providing his opporuniy o work undr his suprvision and nhus and nrich y growh as a sudn and rsarchr. I rally apprcia r. Hassanzadh for hours of inspiring discussions, produciv suggsions and painc whn h rsuls wr no approaching. Thank you r. Chn for your ousanding suppor, and rusing y abiliis o discovr diffrn houghs. I would lik o b graful r. Mingzh ong and r. Chrisophr Clarkson for srving on y suprvisory coi and hir suppor during y rsarch. I would also lik o hank r. Fanhua Zng and r. Xin Wang as brs of y xaining coi. I wish o hank r. Mhran Pooladi-arvish for aching h advancd rsrvoir nginring cours which was rally hlpful during h priod of his sudy. Financial suppor of NSERC/AERI AIEES/Foundaion CMG and icore AITF Chairs Funds and h parn of Chical and Prolu Enginring a h Univrsiy of Calgary is acknowldgd. Collciv and individual acknowldgns ar also owd o y pas and prsn collagus a h Univrsiy of Calgary and Rsrvoir Siulaion Group. I would lik o hank on of y bs frinds Mr. Haid Eai Mybodi for his hlps during y Ph a h Univrsiy of Calgary. Lasly, and os significanly, I wish o hank y parns, Shasi and Ali who ducad, ros, and advisd. Thy ar y xcpional sourc of ncouragn and oivaion. I would lik o hank y sisr, Arzoo, and y brohr, Oid, for hir hlp and suppor which ar always a par of y lif. iii

4 dicaion I ddica his work o y Mohr for hr uncondiional lov and suppor iv

5 Tabl of Conns Absrac... ii Acknowldgns... iii dicaion... iv Tabl of Conns... v Lis of Tabls... viii Lis of Figurs and Illusraions... ix CHAPTER ONE: INTROUCTION.... Background.... ual-porosiy sys.... Shap facor concp....4 Moivaions and objcivs Coponns and oulin of his sudy... 5 Rfrncs... 8 CHAPTER TWO: EFFECT OF FRACTURE PRESSURE EPLETION REGIMES ON THE UAL-POROSITY SHAPE FACTOR FOR FLOW OF COMPRESSIBLE FLUIS IN FRACTURE POROUS MEIA... 0 Absrac Inroducion.... Mhodology Consan fracur prssur Linarly dclining fracur prssur Exponnially dclining fracur prssur Th approxia analyical soluions..... Consan fracur prssur..... Linarly dclining fracur prssur..... Exponnially dclining fracur prssur Modl vrificaion Rsuls Coparison of odl wih Warrn and Roo odl Linarly dclining fracur prssur Exponnially dclining fracur prssur Conclusions... 8 Nonclaur Rfrncs... 4 Appndix.A: Soluion of h gas diffusiviy quaion for diffrn fracur dplion condiions A: Linarly dclining fracur prssur A: Exponnially dclining fracur prssur CHAPTER THREE: ONE IMENSIONAL MATRIX-FRACTURE TRANSFER IN UAL-POROSITY SYSTEMS WITH VARIABLE BLOCK SIZE ISTRIBUTION Absrac v

6 . Inroducion Marix block siz disribuions Unifor or rcangular disribuion Exponnial disribuion Noral or Gaussian disribuion Linar disribuion Log-noral disribuion Equivaln lnghs for diffrn arix block siz disribuions Equivaln lngh concp iscr arix block siz disribuion Coninuous arix block siz disribuion Mahaical odl for flow of coprssibl fluid in fracurd dia wih variabl block siz disribuion Slighly coprssibl fluids Validaion Rsuls Rcangular, discr, noral and log-noral disribuions Linar disribuion Exponnial disribuion iscussion Conclusions... 8 Nonclaur... 8 Rfrncs Appndix.A: Soluion of diffusion quaion for variabl block siz disribuion Appndix.B: rivaion of dinsionlss ra and dinsionlss cuulaiv producion for diffrn arix block siz disribuion CHAPTER FOUR: SEMI-ANALYTICAL SOLUTIONS FOR RELEASE OF FLUIS FROM ROCK MATRIX BLOCKS WITH IFFERENT SHAPES, SIZES AN EPLETION REGIMES Absrac Inroducion and prvious sudis Approxia analyical soluion Consan fracur prssur Variabl fracur prssur Variabl block siz disribuion ulipl blocks Modl vrificaion Rsuls Effc of fracur prssur dplion rgi Block siz disribuion ffc Coparison of diffrn block goris Conclusions... 5 Nonclaur... 6 Rfrncs... 8 Appndix 4.A: Analyical soluion for cylindrical blocks... 4.A: Consan fracur prssur... 4.A: Variabl fracur prssur... 6 vi

7 4.A: Variabl block siz disribuions... 4 Appndix 4.B: Analyical soluion for sphrical blocks B: Consan fracur prssur B: Variabl fracur prssur B: Variabl Block siz disribuions CHAPTER FIVE: CONCLUSIONS AN RECOMMENATIONS Conclusions Fracur prssur boundary condiion ffc for a slab-shapd block Block siz disribuion ffc for slab-shapd blocks Block gory ffc Rcondaions APPENIX A: COPYRIGHT PERMISSIONS vii

8 Lis of Tabls Tabl.: Sabilizd valus of h shap facor and i a which h prssur disurbanc rachs h innr boundary for diffrn dplion rgis in h fracur Tabl.: Obsrvd frquncy of arix block siz in h soil colun Gwo al., Tabl.: Block siz disribuion for h discr disribuions Tabl.: Valus of dinsionlss quivaln lngh for diffrn arix block siz disribuions Tabl 4.: iffrn probabiliy disribuion funcion and hir quivaln radius.... Tabl 4.: aa usd for si-analyical and nurical odls Tabl 4.: Sabilizd valus of h shap facor basd on his sudy and liraur odls Tabl 4.4: Valus of dinsionlss quivaln radius for diffrn arix block siz disribuions viii

9 Lis of Figurs and Illusraions Figur.: An idalizd dual-porosiy sys Lonnir and Bourbiaux, Figur.: Road ap of his sudy Figur.: Schaic of h arix-fracur odl Figur.: Coparison of h arix-fracur cuulaiv fluid producion obaind fro h approxia analyical soluion and h nurical odl of Eclips for consan fracur prssur Figur.: Coparison of h dvlopd shap facor odl wih liraur odls for slighly coprssibl fluid in h cas of linarly dclining fracur prssur Figur.4: Coparison of h dvlopd shap facor odl wih liraur odls for slighly coprssibl fluid in h cas of xponnially dclining fracur prssur for sall valus of xponn k= Figur.5: Coparison of h dvlopd shap facor odl wih liraur odls for slighly coprssibl fluid in h cas of xponnially dclining fracur prssur k= Figur.6: Coparison of h dvlopd shap facor odl wih liraur odls for slighly coprssibl fluid in h cas of xponnially dclining fracur prssur k=.... Figur.7: Coparison of h dvlopd odl wih nurical and Warrn and Roo odl.... Figur.8: Coparison of h shap facor for linarly dclining and consan fracur prssur Figur.9: Shap facor coparison for diffrn xponns for xponnially dclining fracur prssur Figur.0: Coparison of h dinsionlss shap facor for diffrn prssur dplion rgi in h fracur Figur.: iffrn probabiliy dnsiy funcions Figur.: Illusraion of a arix-fracur sys and is boundary condiions lf and rprsnaion of fracurd rsrvoirs in h cas of non-idal arix block siz disribuion righ Figur.: Coparison of h prsnd odl wih liraur odls Chang 995, Hassanzadh and Pooladi-arvish 006 for slighly coprssibl fluid and idal block siz disribuions ix

10 Figur.4: Coparison of h prsnd si-analyical odl wih h nurical rsuls for h firs and scond cas. Coninuous lins ar prdicions by analyical odl. os and dashs ar fro nurical siulaions Figur.5: insionlss ra vrsus dinsionlss i for idal, rcangular, discr, noral and log-noral disribuions whn F h = Figur.6: Coparison of dinsionlss ra for linarly incrasing =00/8, b=5/8, linarly dcrasing =-00/8, b=45/8 disribuion wih F h =0. and idal arix block siz disribuions Figur.7: Coparison of dinsionlss ra for xponnial xponn valus of -0, -5, 5 and 0 wih F h =0. and idal block siz disribuions Figur.8: insionlss cuulaiv producion vrsus dinsionlss i for diffrn arix block siz disribuions wih F h =0.. Blocks showing h dinsionlss quivaln siz of h arix block L c /L cax Figur.9: insionlss shap facor for diffrn arix block siz disribuions. Blocks showing h dinsionlss quivaln siz of h arix block L c /L cax Figur 4.: Schaic rprsnaion of h probl for a cylindrical block Figur 4.: Coparison of h prsnd odl wih h nurical siulaion for flow cylindrical block approxiaion Figur 4.: Coparison of h prsnd odl wih h nurical siulaion for flow sphrical block approxiaion Figur 4.4: insionlss ra vrsus dinsionlss i for diffrn fracur dplion rgis for a cylindrical block Figur 4.5: insionlss cuulaiv rlas vrsus dinsionlss i for diffrn fracur dplion rgis for a cylindrical block Figur 4.6: insionlss cuulaiv rlas vrsus dinsionlss i for diffrn fracur dplion rgis for a sphrical block Figur 4.7: insionlss cuulaiv rlas vrsus dinsionlss i for diffrn block siz disribuion and cylindrical blocks.... Figur 4.8: insionlss cuulaiv rlas vrsus dinsionlss i for diffrn block siz disribuion and sphrical blocks... Figur 4.9: insionlss cuulaiv rlas vrsus dinsionlss i for diffrn block goris... Figur 4.0: Noralizd cuulaiv fluid rlas vrsus squar roo of scald i... 4 x

11 Chapr On: Inroducion. Background Larg porion of h producion of naural gas occurs fro fracurd foraions including naurally fracurd rsrvoirs NFR, coal bd han CBM, shal and igh fracurd gas rsrvoirs. A naurally fracurd rsrvoir NFR is dfind as a rsrvoir ha conains fracurs crad by naural procsss, such as diasrophis or volu shrinkag. Ths naural fracurs can hav a posiiv or ngaiv ffc on fluid flow Aguilra, 995; Ordonz al., 00. NFRs ar usually hough o consis of an inrconncd fracur nwork, which provids h ain flow pahs fracurs hav high prabiliy and low sorag volu, and h rsrvoir rock or arix, which acs as h ain sourc of h fluid sorag arix blocks hav low prabiliy and high sorag volu Bcknr, 990. On h ohr hand in hs rsrvoirs, h ajor sorag for h rsrvoir fluids is in h arix whras flow priarily occurs in h highly conduciv fracurs Chn, 989. In such a sys, in addiion o h inrinsic propris of h arix and fracur, h inracion bwn h arix and fracurs should b odld accuraly. In gnral, hr ar copuaional challngs in upscaling of fluid flow in fracurd foraions. Upscaling of ranspor and flow parars for porous dia has bn invsigad fro dcads and a rang of upscaling chniqus hav bn inroducd ng al., 00. Flow and ranspor in fracurd porous dia ar ofn dscribd using a dual-porosiy odl.. ual-porosiy sys Th dual-porosiy approach has bn usd in nurical siulaion of groundwar, oil, and gas flow in fracurd porous dia. This odl assus ha h porous diu includs wo diffrn rgions, on rlad wih h acropor or fracur nwork wih high prabiliy and h ohr wih a lss prabl and or porous sys of rock arix blocks. ual-porosiy odls assu ha fluid flow can b dscribd by wo quaions for arix and fracurs, which ar coupld using a r dscribing h xchang of fluid bwn h wo por rgions Grk and van Gnuchn, 99. Figur. shows a schaic rprsnaion of an idal dual-porosiy sys.

12 Chapr. Inroducion Figur.: An idalizd dual-porosiy sys Lonnir and Bourbiaux, 00. In h liraur, wo approachs hav bn widly usd o odl dual-porosiy syss ahaically including non-coupld Chang, 995; Hassanzadh and Pooladi-arvish, 006, Ranjbar and Hassanzadh, 0, and coupld Hassanzadh al., 009 hods. Th quaions ha ar xprssd for coupld dual-porosiy odls ar as follows: p f k f f c f p f q,. c p k p,. q k A V p.. Equaions. and. ar usd o xplain h prssur diffusion of singl-phas slighly coprssibl fluid in fracurd and arix diu rspcivly. In Equaion., q is h sourc r ha sands for h n addiion of fluid o h fracur sys fro h arix blocks, pr uni of oal volu. Equaion. is usd o coupl Equaions. and. and shows h arix-fracur inrflow or arix-fracur ransfr funcion. In hs quaions subscrips f and sand for fracur and arix, rspcivly. Th p f and p rprsn h fluid prssur in h fracur and arix rspcivly, is h fluid viscosiy, is h porosiy of h diu and k is h prabiliy. Laplacian opraor rprsns

13 Chapr. Inroducion h divrgnc of gradin and can b usd for diffrn coordina sys and diffrn arix block goris Carsian, Cylindrical or Sphrical. A coupld dual-porosiy odl can b forulad by assuing ha a ach poin, hr is a arix block wih spcifid shap. Insid ach block h fluid prssur p is i and spac dpndn Equaion.. Marix-fracur inrflow r q dos no appar xplicily in Equaion.. This inrflow is assud o b disribud hrough h fracur dia as a sourc/sink r; h inrflow nrs h arix blocks only a hir boundaris Ziran al., 99. Non-coupld approach is an alrnaiv hod which is usd o drin h arixfracur ransfr ra for fracurd rsrvoirs. In his approach h prssur diffusion is solvd in h arix block and fracur is usd as a boundary for h arix sys. Th soluion of arix diffusion quaion is usd o drin h arix-fracur ransfr funcion. Equaion.4 rprsns h prssur diffusion in h arix block p k c p..4 Th fracur prssur is assud as a boundary condiion for his parial diffrnial quaion PE o drin h arix-fracur ransfr funcion. As w will discuss lar his fracur prssur can b a consan or vary wih i basd on a pr-spcifid i funcion. I should b niond ha Equaion.4 is usd for a slighly coprssibl fluid and is a linar parial diffrnial quaion and his quaion can b solvd by coon hods lik Laplac ransfor or sparaion of variabls.. Shap facor concp Currnly, h dual-porosiy approach is on of h copuaionally fficin and widly usd hods o odl fluid flow in fracurd rsrvoirs. In his approach, h arix and fracur ar sparad ino wo diffrn dia, ach wih is own propris. A ransfr funcion has bn usd o rprsn h arix-fracur inracion and govrn h ass ransfr bwn h arix blocks and h fracurs Barrnbla al., 960. Th ra of ass ransfrrd fro h arix o h fracur is dircly proporional o h shap facor. For odling of naurally fracurd rsrvoirs, an accura valu of h shap facor is

14 Chapr. Inroducion rquird o accoun for boh h ransin and psudo-sady sa bhaviour of h arixfracur inracion and also h gory of h arix-fracur sys. In h dual-porosiy odls and psudo-sady sa condiions h blocks ar rad as a lupd sys. In his approach h arix-fracur ransfr funcion is xprssd as follows: k q p p..5 f In his quaion is calld shap facor. This parar is a funcion of fracur spacing and gory of h blocks and has h dinsion of rciprocal of ara..4 Moivaions and objcivs Sudy of gas flow in fracurd porous dia is iporan in a variy of nginring filds. In hydrology hr xis a larg nubr of ahaical odls nurical, analyical or si-analyical o siula h flow of coprssibl fluids in undrground nvironns and srucurd soils You al., 0. Flow of coprssibl and slighly coprssibl fluids war or oil in fracurd rsrvoirs has bn sudid xnsivly wih applicaions in prdicion of producion ras and wll sing. Thrfor, h flow of coprssibl fluids lik gass and air in fracurd porous dia is iporan in hydrological, nvironnal and prolu nginring. Flow and ranspor in fracurd porous dia ar ofn dscribd using a dual-porosiy odl. I is phasizd ha i is no pracical o odl a larg scal fracurd rsrvoir basd on a fin grid approach du o h rquirn of larg copuaional i. Th prsnd si-analyical odls in his sudy which ar basd on a non-coupld approach can b incorporad ino nurical odls for accura odling of h aoun of ransfrrd fluids bwn arix and fracurs using availabl dual-porosiy forulaion. In ohr words, his sudy is iporan o rduc h copuaional i for larg scal siulaions of gas flow in fracurd porous dia and can b nsd in a nurical odl o rsolv subgridblock scal flows. Th dvlopd odl in his sudy can b usd o siula singl-phas coprssibl fluid flow in h fracurd porous dia. Th prsnd odls can handl h variabl fracur prssur and variabl block siz disribuion for diffrn block goris. I also can b 4

15 Chapr. Inroducion usd for slighly coprssibl fluid wihou rquiring nurical invrsion or infini sris calculaion, in an fficin annr..5 Coponns and oulin of his sudy This is a papr-basd hsis wih ach chapr publishd in a pr rviwd journal. Chaprs ar wrin so ha hy can b followd indpndnly whr ach chapr consiss of spara absrac, inroducion and prvious sudis, arials and hodologis, vrificaion, rsuls, conclusions, nonclaurs, rfrncs and appndics. In Chapr wo w invsiga h ffc of prssur dclin in h fracur on h arixfracur ransfr shap facor for a coprssibl fluid. W sudy h influnc of h fracur prssur as a boundary condiion for h arix block on h shap facor for flow of a coprssibl fluid in a dual-porosiy odl, which has no bn invsigad in h forr works. To drin h valu of h shap facor, diffusiviy quaion for gas flow, which is a nonlinar PE, is solvd using h cobinaion of h ha ingral hod, h hod of ons and uhal s hor and his soluion is usd o valua h shap facor. Chapr hr rprsns a nw si-analyical approach o considr h ffc of variabl arix block siz disribuion on flow of a coprssibl fluid in dual-porosiy dia. In addiion, du o h analyical naur of h proposd odl, i can b usd o odl flow of slighly coprssibl fluids in fracurd dia. Th proposd odl is a nw sianalyical approach ha can handl h variabl arix block siz disribuion for h flow of boh coprssibl and slighly coprssibl fluids in h fracurd dia. In addiion o h coninuous block siz disribuion, h proposd odl is also capabl o odl arixfracur ransfr whn hr is discr block siz disribuion using srucural inforaion of a porous diu. Finally in Chapr four a nw si-analyical odl for diffrn arix block goris cylindrical and sphrical for flow of coprssibl and slighly coprssibl fluids in fracurd porous dia is dvlopd. This approxiaion is usd o driv h arixfracur fluid ransfr for flow or slab-shapd blocks surroundd by wo ss of fracurs and flow or slab-shapd blocks surroundd by hr ss of fracurs. Th 5

16 Chapr. Inroducion odl can handl various block siz disribuions and diffrn prssur rgis in h fracur in h cas of diffrn goris. A h nd in Chapr fiv, conclusions of h hsis ar prsnd lading o h rcondaions for fuur works. Figur. shows h road ap of h sudy. 6

17 Chapr. Inroducion Marix-Fracur Transfr Funcion for Fracurd Gas Rsrvoirs Slab-Shapd Marix Block Cylindrical & Sphrical Marix Block Variabl Fracur Prssur Chapr wo Consan Fracur Prssur Chapr four Variabl Fracur Prssur Chapr four Variabl Block Siz isribuion Chapr hr Variabl Block Siz isribuion Chapr four Fin Grid Nurical Siulaion and Coparison wih Liraur Modl for Validaion Chaprs wo o four Figur.: Road ap of his sudy. 7

18 Chapr. Inroducion Rfrncs Aguilra, R Naurally fracurd rsrvoirs. Tulsa, Oklahoa: PnnWll Prss. Barnbla, G.E. Zhlov, I.P. Kochina, I.N Basic concps in h hory of spag of hoognous liquids in fissurd rocks sraa. J. Appl. Mah. Mch., 0, Bcknr, B.L Iprovd odling of ibibiion arix/fracur fluid ransfr in doubl porosiy siulaors. Ph dissraion, Sanford Univrsiy. Chang, M.M Analyical soluion o singl and wo-phas flow probls of naurally fracurd rsrvoirs: horical shap facor and ransfr funcions. Ph dissraion, Univrsiy of Tulsa. Chn, Z.X Transin flow of slighly coprssibl fluids hrough doubl-porosiy, doubl-prabiliy syss-a sa-of-h-ar rviw. Transp. Porous Md., 4, ng, H. ai, Z. Wolfsbrg, A. Lu, Z. Y, M. Rius, P. 00. Upscaling of raciv ass ranspor in fracurd rocks wih uliodal raciv inral facis. War Rsour. Rs., 46. Grk, H.H. van Gnuchn, M.Th. 99. A dual-porosiy odl for siulaing h prfrnial ovn of war and solus in srucurd porous dia, War Rsour. Rs., 9, 05-9 Hassanzadh, H. Pooladi-arvish, M Effcs of fracur boundary condiions on arix-fracur ransfr shap facor. Transp. Porous Md., 64, 5-7. Hassanzadh, H. Pooladi-arvish, M. Aabay, S Shap facor in h drawdown soluion for wll sing of dual-porosiy syss. Adv. War Rs.,, Lonnir, P. Bourbiaux, B. 00. Siulaion of Naurally Fracurd Rsrvoirs. Sa of h Ar. Par : Marix-Fracur Transfrs and Typical Faurs of Nurical Sudis, Oil & Gas Scinc and Tchnology Rv. IFP, 65, Ordonz, A. Pnula, G. Idrobo, E.A. Mdina C.E. 00. Rcn advancs in naurally fracurd rsrvoir odling. CT&F Cincia, Tcnologia y Fuuro.,, Ranjbar, E. Hassanzadh, H. 0. Marix-fracur ransfr shap facor for odling flow of a coprssibl fluid in dual-porosiy dia. Adv. War Rs., 45, You, K. Zhan, H. Li, J. 0. Analysis of odls for inducd gas flow in h unsaurad zon, War Rsour. Rs., 47, W0455, doi: 0.09/00WR

19 Chapr. Inroducion Ziran, R.W. Chn, G. Bodvarsson, S. 99. A dual-porosiy rsrvoir odl wih an iprovd coupling r. Svnnh Workshop on Gohral Rsrvoir Enginring, Sanford. 9

20 Chapr Two: Effc of Fracur Prssur plion Rgis on h ual-porosiy Shap Facor for Flow of Coprssibl Fluids in Fracurd Porous Mdia Absrac A prcis valu of h arix-fracur ransfr shap facor is ssnial for odling fluid flow in fracurd porous dia by a dual-porosiy approach. Th slighly coprssibl fluid shap facor has bn widly invsigad in h liraur. In a rcn sudy, w hav dvlopd a ransfr funcion for flow of a coprssibl fluid using a consan fracur prssur boundary condiion Ranjbar and Hassanzadh, 0. Howvr, for a coprssibl fluid, h consqunc of a prssur dplion boundary condiion on h shap facor has no bn invsigad in h prvious sudis. Th ain purpos of his chapr is, hrfor, o invsiga h ffc of h fracur prssur dplion rgi 4 on h shap facor for singl-phas flow of a coprssibl fluid. In h currn sudy, a odl for valuaion of h shap facor is drivd using soluions of a nonlinar diffusiviy quaion subjc o diffrn prssur dplion rgis. A cobinaion of h ha ingral hod, h hod of ons and uhal s hor is usd o solv his nonlinar quaion. Th dvlopd soluion is validad by fin-grid nurical siulaions. Th prsnd odl can rcovr h shap facor of slighly coprssibl fluids rpord in h This chapr is an xac copy of: Ranjbar, E. Hassanzadh, H. Chn, Z. 0. Effc of Fracur Prssur plion Rgis on h ual-porosiy Shap Facor for Flow of Coprssibl Fluids in Fracurd Porous Mdia, Advancs in War Rsourcs, Vol. 4, Pag: Th focus of our prvious sudy Ranjbar, E. Hassanzadh, H. 0, Marix-fracur ransfr shap facor for odling flow of a coprssibl fluid in dual-porosiy dia, Advancs in War Rsourcs, 45, pag was o find h shap facor for h singl-phas flow of coprssibl fluids gass in fracurd porous dia for h cas of consan fracur prssur. In his sudy Ranjbar and Hassanzadh, 0, a horical analysis of h consan fracur prssur shap facor for h flow of a coprssibl fluid in fracurd porous dia was prsnd. Th prsnd si-analyical soluion for consan fracur prssur was validad wih fin-grid nurical siulaions. In his chapr w furhr dvlop our prvious sudy o considr h ffc of prssur variaion in h fracur on h arix-fracur shap facor. I is worh noing ha h fracur prssur in his hsis is diffrn han h hydraulic fracur prssur and i iplis h fluid prssur insid h fracur or h boundary condiion iposd on h arix block. 4 In his hsis h fracur dplion rgi iplis h prssur variaions in h fracur which acs as a boundary condiion for h arix block.

21 Chapr. Effc of fracur prssur dplion rgis on h dual-porosiy liraur. This sudy donsras ha in h cas of a singl-phas flow of coprssibl fluid, h shap facor is a funcion of h iposd boundary condiion in h fracur and is variabiliy wih i. I is shown ha such dpndnc can b dscribd by an xponnially dclining fracur prssur wih diffrn dclin xponns. Ths findings iprov our undrsanding of fluid flow in fracurd porous dia.

22 Chapr. Effc of fracur prssur dplion rgis on h dual-porosiy. Inroducion In gnral, hr ar copuaional challngs in upscaling of fluid flow in fracurd foraions. Upscaling of ranspor and flow parars for porous dia has bn invsigad fro dcads and a rang of upscaling chniqus hav bn inroducd ng al., 00. Larg porion of h producd naural gas occurs fro fracurd foraions including naurally fracurd rsrvoirs NFR, coal bd han CBM and igh fracurd gas rsrvoirs. In hs rsrvoirs, h ajor sorag for h rsrvoir fluids is in h arix whras flow priarily occurs in h highly conduciv fracurs Chn, 989. Warrn and Roo 96 sablishd h dual-porosiy odl for odling of a slighly coprssibl fluid flow in h naurally fracurd rsrvoirs. In h dual-porosiy approach, a fracurd rsrvoir is dividd ino wo dia wih coplly diffrn propris: fracur and arix. Th fracur nwork supplis h ain flow pahs and h rsrvoir rock or arix acs as h ajor sourc of h fluid sorag Bcknr, 990. On h ohr hand os of h fluid sorag is in h arix and fluid flows hrough h fracurs as h ain channl. Thrfor, an iprovd dual-porosiy odl should b abl o accuraly accoun for h fracur and arix inracion. A dual-porosiy odl, which is an ffciv and broadly usd approach for odling and upscaling of fluid flow in h fracurd porous dia, assus ha wo disinc yps of porosiy coxis in a rprsnaiv rock volu Chn, 989; Cihan and Tynr, 00; i onao and Blun, 004; Liu and Chn, 990; Ziran al., 996. In gnral, fracur has a low sorag capaciy and high ransissiviy and h adjacn rock arix has a high sorag capaciy and a rlaivly low ransissiviy Kazi and Gilan, 99. fining h ransfr shap facor ha accouns for h inracion aong h arix and fracur is a gra challng in dual-porosiy upscaling. In dual-porosiy odls h arixfracur inracion is odld hrough a shap facor. An quivaln fracur prabiliy, arix-prabiliy, arix-fracur ransfr cofficin shap facor and sauraion funcions for uliphas flow ar ssnial parars for h dual-porosiy approach. Sudis hav bn conducd in h pas o drin h ransfr shap facor for slighly coprssibl fluids in h fracurd rsrvoirs Bourbiaux al., 999; Coas, 989;

23 Chapr. Effc of fracur prssur dplion rgis on h dual-porosiy Kazi al., 976; Quinard and Whiakr, 996; Quinard and Whiakr, 996; Quinard and Whiakr, 998; Thoas al., 98; Uda al., 989. A prcis valu of h shap facor is ssnial o considr ransin and psudo-sady sa prforanc of h arix-fracur inracion and also gory of h arix-fracur sys. I should b nod ha h funcionaliy of h fracur prssur as a boundary o h arix blocks ay also hav a significan ffc on h sabilizd valu of h shap facor for a slighly coprssibl fluid Chang, 995; Hassanzadh and Pooladi-arvish, 006. In radiional dual-porosiy forulaion h flow bwn h arix and h fracur is considrd by a ransfr funcion, which acs as a sourc r in h govrning quaion for fluid flow in h fracurs. arcy s law is usd in his sourc funcion ovr h an pah bwn h arix and h adjacn fracur. In h non-coupld dual-porosiy forulaion, flow in h fracurs acs as a boundary condiion for flow in h arix Kazi and Gilan, 99. This ransfr funcion and h aoun of fluid ha is ransfrrd fro h arix o h fracur ar dircly proporional o h shap facor. Nurical siulaion of naurally fracurd rsrvoirs using a dual-porosiy approach rquirs a prcis valu of h shap facor for h nir priod of h producion i. In gnral, hr ar wo odls o considr h arix and fracur inracion including psudo-sady sa and ransin ransfr. Th forr odl ignors h prssur ransin in h arix whil h lar odl accouns for h prssur ransin in h arix. Th arix-fracur shap facor for a slighly coprssibl fluid can b obaind using h following quaion Li and Aziz, 995: p c,. k p p f whr µ is h fluid viscosiy, c,, and k ar h oal isohral coprssibiliy, porosiy and prabiliy, rspcivly, p shows h avrag prssur of h arix block, p f is h fracur prssur and σ is h arix-fracur ransfr shap facor wih dinsion of L. In a psudo-sady sa odl h arix blocks ar considrd as a

24 Chapr. Effc of fracur prssur dplion rgis on h dual-porosiy lupd sys wih an avrag prssur, find h soluion of h prssur diffusiviy quaion givn by: p, whil in h ransin odl on nds o k p p c,. For a slighly coprssibl fluid, ngligibl variaion of h fluid viscosiy and isohral coprssibiliy wih h prssur lads o a linar flow quaion for h prssur variaion in h arix. This quaion can b solvd by coon analyical or si-analyical hods such as h Laplac ransfor or sparaion of variabls hod Ziran al., 996; Li and Aziz, 995; Hassanzadh al., 009; Shan and Pruss, 005; Ziran al., 99. rinaion of h arix-fracur ransfr shap facor for a slighly coprssibl fluid basd on h psudo-sady sa or ransin ransfr odl has bn sudid in h pas. Invsigaors hav considrd h ffc of fracur boundary condiions on h dualporosiy forulaion and shap facor for h slighly coprssibl fluid Chang, 995; Hassanzadh and Pooladi-arvish, 006; Rangl-Gran al., 00. I has bn shown ha h fracur prssur and is variaion wih i affc h ransin and psudo-sady sa valu of h shap facor. Thr hav bn nw ffors o drin h shap facors for uli-phas flow and hral hods in fracurd porous dia Rangl-Gran al., 00; Civan and Rasussn, 00; van Hl al., 008. Thr hav also bn a fw rpors in h liraur o odl dual-porosiy syss for coprssibl fluids wih diffrn approachs han his sudy Lu and Connl, 007; Pnula al., 00. A or daild rviw of shap facor dvlopns was discussd lswhr Ranjbar and Hassanzadh, 0. Alhough h dual-porosiy approach wih h shap facor concp has so liiaions, i has bn widly usd and wll accpd approach in hydrological scincs and prolu rsrvoir odling. This ay b bcaus of is sipliciy, copuaional fficincy and flxibiliy in applicaion o various fluid flow and ranspor probls. In addiion, lack of or advancd and fficin odls ha can accuraly ak ino accoun h arixfracur inracion hav conribud o xnsiv us of h dual-porosiy odls. Currnly, h ajoriy of corcial flow siulaors us h dual-porosiy approach. 4

25 Chapr. Effc of fracur prssur dplion rgis on h dual-porosiy Howvr, i should b poind ou ha a nw lin of aack on ackling fluid flow and ranspor in fracurd rocks has bn rcnly inroducd basd on discr fracur nwork odls Hoi and Firoozabadi, 005. Hoi and Firoozabadi 005 prsnd a discr fracur odl for singl phas flow of coprssibl fluids in hrognous and fracurd dia. Thy dvlopd a nurical odl by cobining h ixd fini ln and h disconinuous Galrkin hods for uli-coponn gas flow. iscr fracur odl also hav bn usd for uliphas flow and war injcion in fracurd dia Karii- Frad and Firoozabadi, 00; Lonnir and Bourbiaux, 00. I has bn rpord in h liraur ha in h cas a slighly coprssibl fluid h psudo-sady sa valu of h arix-fracur shap facor is a funcion of h prssur dclin rgi in h fracur. Conrary o h slighly coprssibl fluid cas, h variaion of h isohral coprssibiliy and viscosiy wih prssur canno b ignord whn daling wih a coprssibl fluid. This lads o a nonlinar PE. Thrfor, h rpord shap facors for slighly coprssibl fluids canno b applid for coprssibl fluids or hir applicaion has no bn validad in h prvious sudis. In a rcn sudy w drivd h arix-fracur shap facor for a coprssibl fluid in dual-porosiy syss Ranjbar and Hassanzadh, 0. Th ffc of fracur prssur dclin on h coprssibl fluid shap facor has no bn rpord in h prvious sudis. In his sudy, w furhr dvlop our prvious sudy o invsiga h ffc of prssur dclin in h fracur on h arix-fracur ransfr shap facor for a coprssibl fluid. W sudy h influnc of h fracur prssur as a boundary condiion for h arix block on h shap facor for flow of a coprssibl fluid in a dual-porosiy odl which has no bn invsigad in h forr works. To obain h arix-fracur shap facor a nonlinar diffusiviy quaion is solvd using h ha ingral hod and h hod of ons. To considr h ffc of h i variaion of h boundary condiions a odifid rial soluion arly i and uhal s hor la i ar usd o driv h arly and la i shap facors for h dclining fracur prssur cass. Th dvlopd approxia analyical soluion is validad by a nurical odl Ranjbar and Hassanzadh, 0. Th dvlopd shap facor odl can rcovr prdicions fro h shap facor odls availabl in liraur for a slighly coprssibl fluid. This shap 5

26 Chapr. Effc of fracur prssur dplion rgis on h dual-porosiy facor ay find applicaions in dual-porosiy odling of h convnional and unconvnional naurally fracurd gas rsrvoirs such as coalbd han and fracurd igh gas rsrvoirs. This papr is organizd in a annr ha follows a hodology for drivaion of h shap facor for coprssibl fluids. Nx soluion of h nonlinar diffusiviy quaion subjc o a dclining fracur prssur is obaind using h ha ingral and on hods and uhal s hor. Afrwards odl vrificaion and rsuls ar discussd followd by conclusions.. Mhodology In his scion h shap facor for flow of a coprssibl fluid fro a arix block undr diffrn fracur boundary condiions is drivd by aking ino accoun h prssur dpndncy of h viscosiy and isohral coprssibiliy. arcy s law for flow of gas in h porous dia is xprssd as follows: q gsc k A dp,. B dx g whr A is h cross-scion ara and B g is h gas foraion volu facor. Using h dfiniions of h gas foraion volu facor and ral gas psudo-prssur Ikoku, 99 and wriing h arcy s law ovr so characrisics lngh l lads o h following quaion: q sc ktsc A f..4 Tp l sc As shown in Equaion.4, l is a lngh whr h arix prssur is qual o is avrag prssur and his lngh changs wih i during ransin arix producion. For a arix-fracur cobinaion shown in Figur., Equaion.4 is uliplid and dividd by h bulk volu of h arix-block o dfin h ransfr funcion for coprssibl fluids. Using h dfiniion of h shap facor Equaion.5; h final quaion for h arix-fracur ransfr funcion for coprssibl fluids.g. gass, is xprssd as Equaion.6 Ranjbar and Hassanzadh, 0: 6

27 Chapr. Effc of fracur prssur dplion rgis on h dual-porosiy A,.5 l V / b q sc TscVb k f,.6 4 p T sc In his quaion T is h absolu praur, σ is h shap facor, shows h avrag arix block psudo-prssur and f is h fracur psudo-prssur. According o h Warrn and Roo 96 dual-porosiy odl for a slighly coprssibl fluid, h inrporosiy flow ra pr uni volu of h rock can b xprssd in rs of h accuulaion ra in h arix as follows: p qˆ c..7 Th inrporosiy flow ra for coprssibl fluids can b xprssd as follows Ranjbar and Hassanzadh, 0: q sc TscV 4 p b sc c T..8 Cobinaion of Equaions.6 and.8 lads o h following quaion for singl-phas shap facor of coprssibl fluids as follows Ranjbar and Hassanzadh, 0: c..9 k f h Fracur Marix L c Figur.: Schaic of h arix-fracur odl. x 7

28 Chapr. Effc of fracur prssur dplion rgis on h dual-porosiy Thr is anohr alrnaiv o driv h shap facor for coprssibl fluids by ingraing of h diffusiviy quaion ovr h arix-block volu. This hod was usd by Ziran al., 99 o driv h shap facor for slighly coprssibl fluids and lads o Equaion.. By ingraing of h gas diffusiviy quaion ovr half of h arix block volu w rach o h following quaion, k c Adx V /..0 x x b Siplificaion of Equaion.0 lads o h following quaion: c A k.. V / x b Using h Warrn and Roo 96 approxiaion w hav, x f.. l Whr l is h characrisics lngh, which is h disanc fro h arix-fracur boundary whr h arix prssur is qual o is avrag prssur. By subsiuing his quaion in Equaion. w rach o h following quaion: c A l V b / k f.. Using h dfiniion of h shap facor givn by Equaion.5 w rach o h following quaion for h shap facor of coprssibl fluids, which is siilar o h quaion ha was obaind by Ziran al., 99 for slighly coprssibl fluids. I should b nod ha in his quaion psudo-prssur is appard in h final quaion as w ar daling wih coprssibl fluids. c..4 k f This quaion is h sa as Equaion.9. I should b poind ou ha for a coprssibl fluid h viscosiy-isohral coprssibiliy produc is a srong funcion of prssur, in Equaions.9 or.4 h soluion of h nonlinar gas diffusiviy quaion is uilizd o drin h shap facor for diffrn prssur rgis in h fracur. 8

29 Chapr. Effc of fracur prssur dplion rgis on h dual-porosiy.. Consan fracur prssur In his cas i is assud ha a h arix-fracur inrfac, h fracur prssur and hnc h psudo-prssur is a consan. For his cas h dinsionlss variabls ar dfind as follows: i,.5 f i x L x,.6 c,.7..8 L c In Equaion.7 h avrag hydraulic diffusiviy, is givn by Ranjbar and Hassanzadh, 0: p f p i p f p i k k dp c p f p i p f p i dp c..9 Using h dfiniion of dinsionlss variabls Equaions.5,.7 and.8 in h shap facor quaion Equaion.9, h subsqun quaion for h dinsionlss shap facor in h cas of h consan fracur prssur is obaind: h Linarly dclining fracur prssur For a linarly dclining fracur prssur w hav h following quaion for h fracur psudo-prssur as givn by: f i,. whr α is a dclin consan. For his cas h dinsionlss psudo-prssur and h dinsionlss fracur psudo-prssur ar dfind as follows: 9

30 Chapr. Effc of fracur prssur dplion rgis on h dual-porosiy i,. i f,. whr κ is h dinsionlss dclin consan and is dfind as follows:,.4 Applying h xplanaion of h dinsionlss variabls, Equaions.7,.8,. and., in h shap facor quaion Equaion.9 lads o h following quaion for h shap facor in h cas of h linarly dclining fracur prssur: h Exponnially dclining fracur prssur For his cas h fracur prssur dclins xponnially wih i according o h following quaion: f i xp,.6 whr. For an xponnial dclin, h dinsionlss psudo-prssur f and h fracur dinsionlss psudo-prssur ar dfind as follows: i,.7 i f xp,.8 whr κ is h dinsionlss dclin consan and is dfind in Equaion.4. Using h dfiniion of dinsionlss variabls Equaions.7,.8,.7 and.8 in h shap facor quaion Equaion.9 lads o h following quaion for h shap facor in h cas of h xponnially dclining fracur prssur: h 4. xp.9 0

31 Chapr. Effc of fracur prssur dplion rgis on h dual-porosiy. Th approxia analyical soluions Th coprssibl fluid diffusiviy quaion for linar flow can b sad as: c..0 x k Srong prssur dpndnc of h viscosiy and isohral coprssibiliy lads o a nonlinar parial diffrnial quaion PE for coprssibl fluid flow in fracurd porous dia. Soluion of his PE canno b obaind by coon hods lik Laplac ransfor or sparaion of variabls. Equaion.0 in r of h arix hydraulic diffusiviy, η = k /μc is xprssd as follows: p x.. In Equaion., hydraulic diffusiviy is a spac and i dpndn parar. To solv Equaion., w nglc h isohral coprssibiliy-viscosiy produc variaion wih spac and ffc of h spac is considrd by a corrcion facor, β Ranjbar and Hassanzadh, 0; Agarwal, 979. Fin-grid nurical siulaions ar usd o drin his corrcion facor. Sinc h gas coprssibiliy is ordrs of agniud largr han h rock coprssibiliy w ignor h rock coprssibiliy in h soluion Agarwal, 979; Ikoku, 99. Thrfor, w rach h following PE wih h iniial and boundary condiions. x x i,. 0,.a x 0 0,.b x x..c L c f In Equaions. and., β is usd o corrc h ffc of spac on h hydraulic diffusiviy; L c is characrisic lngh of h arix-block which is half of h arix-block hicknss h. Figur. illusras a graphical rprsnaion of h arix-fracur sys.

32 Chapr. Effc of fracur prssur dplion rgis on h dual-porosiy.. Consan fracur prssur A soluion for h consan fracur prssur is givn in our rcn work Ranjbar and Hassanzadh, 0. Sinc his soluion will b usd as a basis for h i-dpndn boundary condiion, h final for of h soluion is givn in h following. Using an ingral hod Finlayson, 97; Goodan, 964; Pooladi-arvish, 994; Ziran and Bodvarsson, 989 h arly i soluion for consan fracur prssur can b found as follow Ranjbar and Hassanzadh, 0: x 4 x, , whr is h avrag psudo-prssur and η is hydraulic diffusiviy of h fracur in dinsionlss for and is xprssd as follows: x k / f c f..6 Th la i soluion can b obaind by h hod of ons Chang, 995; As, 965; Crank, 975 as givn by Ranjbar and Hassanzadh, 0: x,.79xp 0.54xp.5xp 6.75xp 5.686xp 0.489xp x x, xp 0.48xp,..8 4 whr. 486,.9. 8,.40 Subsiuing arly and la i avrag dinsionlss psudo-prssurs Equaions.5 and.8 and hir drivaivs in Equaion.0 lads o h following quaion for h

33 Chapr. Effc of fracur prssur dplion rgis on h dual-porosiy dinsionlss shap facor for flow of a coprssibl fluid fro a arix block subjc o a consan fracur prssur boundary condiion: 4, h.4 4, 0.48xp 0.790xp 4.76xp.964xp 4 h..4 whr parars β and η wr obaind by aching h arly and la i cuulaiv producion fro h arix o h fracur by a nurical flow siulaor Go-Qus, Linarly dclining fracur prssur For h linarly dclining fracur prssur h diffusiviy quaion and is iniial and boundary condiions ar xprssd as follows: x x,.4 0 0,.44a 0 0, x x.44b f x..44c For h arly i soluion of hs quaions w assu ha i has h following for:, x x..45 Whn h boundary condiion changs wih i h pnraion dph in h ha balanc ingral hod HBIM is found by solving h following ordinary diffrnial quaion Michl and Myrs, 00: f f n n n d d..46 In his quaion n is h xponn in h rial soluion, n= for our cas and. In Equaion.46, θ can b found fro h following quaion Michl and Myrs, 00:

34 Chapr. Effc of fracur prssur dplion rgis on h dual-porosiy f n n f.47. n Solving Equaion.46 for a linarly dclining fracur prssur lads o h following pnraion dph for his cas: I should b nod ha Equaion.48 is obaind by assuing θ=0 in Equaion.46 Michl and Myrs, 00. If w do no us h assupion of θ=0 Equaion.46 canno b solvd analyically. Th drivaion of his quaion is shown in Appndix.A in or dails. Our nurical rsuls show ha w can obain a or accura soluion if w us h following quaion for h pnraion dph: Th arly i soluion is valid ill h pnraion dph rachs h innr boundary, so w can find h i a which h prssur disurbanc rachs h boundary as follows: Thrfor, h arly i soluion of h parial diffrnial quaion.4 wih h boundary condiions.44 can b xprssd as follows: x, x, Ingraing of Equaion.5 ovr h arix block volu, lads o Equaion.5 for h arly i avrag dinsionlss psudo-prssur: 9 / k, Th i dpndnc of h boundary condiion for h la i soluion can b considrd using uhal s hor. Whn h fracur psudo-prssur varis wih i Equaion.44c, uhal s hor provids h basis o solv h probl wih variabl boundary condiions basd on h soluion providd for h consan fracur psudo-prssur. Using uhal s hor Chang, 995; Ozisik, 99; Polyanin, 00 4

35 Chapr. Effc of fracur prssur dplion rgis on h dual-porosiy h soluion of parial diffrnial Equaion.4 wih h boundary condiions, Equaions.44b and.44c can b xprssd as: f x d,..5 0 In Equaion.5, wihin h ingral is h soluion whn and on h lfhand sid is h soluion of PE.4 whn h arix-fracur boundary condiion changs wih i. Using uhal s hor lads o h following la i soluion for h cas of h linarly dclining fracur prssur: x x, x x xp x, xp 9 f.54 rivaion of Equaion.54 is shown in Appndix.A in or dails. Th la i avrag arix block psudo-prssur for h linarly dclining fracur prssur is obaind as follows: x, dx xp 0.49 xp, 9.55 Using h avrag psudo-prssur and is drivaiv in h shap facor quaion Equaion.5 rsuls in h following quaions for h arly and la i shap facors in h cas of h linarly dclining fracur prssur: h h xp, 0.49xp 9.56, xp xp 9 I should b nod ha for h linarly dclining fracur prssur, h shap facor for coprssibl fluid is no a funcion of h dinsionlss dclin consan, κ; a siilar 5

36 Chapr. Effc of fracur prssur dplion rgis on h dual-porosiy obsrvaion was rpord by Hassanzadh and Pooladi-arvish 006 for flow of a slighly coprssibl fluid in fracurd porous dia... Exponnially dclining fracur prssur For h xponnially dclining fracur prssur h soluion of h following PE should b usd in Equaion.9 o driv h shap facor for his cas: x x, ,.59a x x 0 0,.59b x xp.59c f Th following funcion is assud for h arly i soluion which saisfis h our boundary condiion: x, x xp..60 Solving h OE of Equaion.46 lads o h following quaion for h pnraion dph in h cas of an xponnially dclining fracur prssur: rf..6 xp xp Th drivaion of his quaion is illusrad in Appndix.A. I should b nod ha in Equaion.6, rfx is h rror funcion dfind as follows: y 0 rf dy..6 Sinc Equaion.6 is obaind basd on an approxiaion of =0 in Equaion.46, our nurical rsuls show ha on can obain a or accura soluion if w us h following quaion for pnraion dph: rf xp xp 6

37 Chapr. Effc of fracur prssur dplion rgis on h dual-porosiy I should b nod ha h ffc of prssur disurbanc will rach h innr boundary whn 0 and for h xponnial dclin w canno obain an xplici quaion for and is drind for any valus of k by aking Equaion.6 qual o zro. Thrfor, h arly i soluion of Equaions.58 and.59 can b xprssd as follows: x, xp.96 xp x xp rf,.64 Ingraing of Equaion.64 ovr h arix block volu, lads o Equaion.65 for h arly i avrag dinsionlss psudo-prssur: xp rf.96,.65 4 uhal s hor and h vrifid soluion of h consan fracur prssur boundary condiion lad o h following quaion for h la i dinsionlss psudo-prssur in h cas of h xponnially dclining fracur prssur: x, xp xp xp xp xp.4874 xp xp xp xp xp xp.4874 xp xp xp xp.4.4x x.4.4x x x x x xp x,.66 Mor dails abou h drivaion of Equaion.66 ar discussd in Appndix.A. Ingraing ovr h arix-block bulk volu rsuls in h following quaions for h avrag dinsionlss psudo-prssur in h cas of h xponnially dclining fracur prssur: 7

38 Chapr. Effc of fracur prssur dplion rgis on h dual-porosiy 8, xp xp xp xp xp xp xp 0.7 xp xp xp xp xp xp xp xp 0.7 xp, x.67 Using h avrag psudo-prssur and is drivaiv in Equaion.9 lads o h following quaions for h arly and la i shap facors in h cas of h xponnially dclining fracur prssur: /, xp.96 4 xp xp xp xp xp rf rf rf rf rf h.68, xp xp xp xp xp xp xp xp 0.7 xp xp xp xp xp xp xp 0.7 xp xp xp xp xp xp xp xp 0.7 xp xp xp xp xp xp 0.7 xp 4 h.69

39 Chapr. Effc of fracur prssur dplion rgis on h dual-porosiy Basd on Equaions.68 and.69, for an xponnially dclining fracur prssur h shap facor for a coprssibl fluid is a funcion of h dclin xponn, k. Siilar obsrvaions hav bn ad in h prvious sudis for a slighly coprssibl fluid Chang, 995; Hassanzadh and Pooladi-arvish, Modl vrificaion Th dvlopd shap facor was validad by a fin-grid singl porosiy odl Eclips 00. Th oal cuulaiv producion fro h arix o h fracur basd on h siulaor was usd o find h corrcion facor β and η and o valida h prsnd odl. Figur. shows h arix-fracur cuulaiv fluid producion vrsus i for a cas wih =0.7, T=9.ºC and prssur drawdown of 45 o.5 MPa. Th valus obaind for h corrcion facor β, h aching parar η, h avrag hydraulic diffusiviy and h dinsionlss fracur hydraulic diffusiviy η ar 0.70, 0.7, and 0.69, rspcivly. In h odl vrificaion sudis, a slab-shapd arix-block wih hicknss h of 4, prabiliy of, and porosiy of 0. ar considrd. W us h sa rsrvoir daa and parars hroughou his papr. As illusrad in Figur. h approxia analyical odl basd on his sudy is in a good agrn wih h fin grid nurical siulaion. Mor dails abou h nurical siulaions and or validaion cass ar discussd lswhr Ranjbar and Hassanzadh, 0. Th dvlopd odl wih us rproduc h shap facor for a slighly coprssibl fluid. For addiional validaion of h dvlopd odl, h shap facor drivd in his sudy is valuad wih h shap facor of h slighly coprssibl fluid. Oucos show ha h odls dvlopd for diffrn boundary condiions can rproduc h slighly coprssibl fluid shap facor for h copl priod of i. Figur. copars h dvlopd shap facor odl for a slighly coprssibl fluid and odls availabl in h liraur Chang, 995; Hassanzadh and Pooladi-arvish, 006 whn h fracur prssur dclins linarly wih i. According o his figur h prsnd odl shows an accpabl ach wih ohr odls. 9

40 Chapr. Effc of fracur prssur dplion rgis on h dual-porosiy 700 Cuulaiv Fluid ProducionS Nurical Coprssibl Fluid Modl Ti sc Figur.: Coparison of h arix-fracur cuulaiv fluid producion obaind fro h approxia analyical soluion and h nurical odl of Eclips for consan fracur prssur. 000 insionlss Shap facor h 00 0 Coprssibl Fluid Modl Hassanzadh & Pooladi-arvish Modl Chang Modl insionlss Ti Figur.: Coparison of h dvlopd shap facor odl wih liraur odls for slighly coprssibl fluid in h cas of linarly dclining fracur prssur. 0

41 Chapr. Effc of fracur prssur dplion rgis on h dual-porosiy Figurs.4,.5 and.6 donsra h coparisons bwn h prsnd shap facor odls in his sudy wih h liraur odls for a slighly coprssibl fluid whn h fracur prssur dclins xponnially wih i for diffrn valus of h dclin xponn. Ths figurs donsra ha h prsnd odl can rproduc h slighly coprssibl fluid shap facor wih an accpabl accuracy. 000 insionlss Shap Facor h 00 0 Coprssibl Fluid Modl Hassanzadh & Pooladi-arvish Modl insionlss Ti Figur.4: Coparison of h dvlopd shap facor odl wih liraur odls for slighly coprssibl fluid in h cas of xponnially dclining fracur prssur for sall valus of xponn k=0.000.

42 Chapr. Effc of fracur prssur dplion rgis on h dual-porosiy 000 insionlss Shap Facor h 00 0 Coprssibl Fluid Modl Chang Modl insionlss Ti Figur.5: Coparison of h dvlopd shap facor odl wih liraur odls for slighly coprssibl fluid in h cas of xponnially dclining fracur prssur k= insionlss Shap Facor h 00 0 Coprssibl Fluid Modl Hassanzadh & Pooladi-arvish Modl insionlss Ti Figur.6: Coparison of h dvlopd shap facor odl wih liraur odls for slighly coprssibl fluid in h cas of xponnially dclining fracur prssur k=.

43 Chapr. Effc of fracur prssur dplion rgis on h dual-porosiy.5 Rsuls In his scion a coparison of h dvlopd odl wih h Warrn and Roo odl 96 is prsnd and hn h bhaviour of h shap facor for diffrn fracur prssur dplion rgis for flow of a coprssibl fluid hrough dual-porosiy dia is dscribd..5. Coparison of odl wih Warrn and Roo odl Warrn and Roo 96 usd a psudo-sady sa approach o driv h shap facor for a slighly coprssibl fluid in h fracurd dia. Thy drivd h following shap facor for slighly coprssibl fluid for diffrn ss of fracurs, 4n n..70 h whr n is h nubr of ss of fracurs. For on s of fracur h drivd valu of h shap facor is /h. Figur.7 shows h coparison of h arix-fracur cuulaiv fluid producion basd on h Warrn and Roo shap facor, prsnd si-analyical odl i dpndn shap facor and nurical rsuls. 800 Cuulaiv Fluid ProducionS Nurical Coprssibl Fluid Modl Warrn & Roo Ti sc Figur.7: Coparison of h dvlopd odl wih nurical and Warrn and Roo odl.

44 Chapr. Effc of fracur prssur dplion rgis on h dual-porosiy Basd on Figur.7 i can b concludd ha using a consan slighly coprssibl fluid shap facor Basd on Warrn and Roo odl for flow of coprssibl fluids lads o larg rror in prdicion of h cuulaiv producion fro h arix..5. Linarly dclining fracur prssur Figur.8 shows h shap facor for h linarly dclining fracur prssur drivd fro Equaions.56 and.57. Th shap facor for a consan fracur prssur is also shown in his figur for coparison. As illusrad in his figur, for h linarly dclining fracur prssur, h ransin priod for h linar dclin is longr han ha of h consan fracur prssur and h shap facor is sabilizd a valu of 0.8 whn is abou.8. For h cas of h consan fracur prssur h sabilizd valu of h shap facor is 8.57 whn dinsionlss i is abou 0.7. Th sa bhaviour has bn rpord for a slighly coprssibl fluid by Chang 995 and Hassanzadh and Pooladi-arvish 006 in h cas of a slighly coprssibl fluid. Fro his figur i can b concludd ha h ransin and psudo-sady sa valus of h shap facor for a linar dclin is largr han hos of h consan fracur prssur. I should b nod ha lik slighly coprssibl fluids h valu of h dplion ra κ has no ipac on h ransin and sabilizd valu of h shap facor. 4

45 Chapr. Effc of fracur prssur dplion rgis on h dual-porosiy 000 insionlss Shap Facor h 00 0 Linar dclin Fracur Prssur Consan Fracur Prssur insionlss Ti Figur.8: Coparison of h shap facor for linarly dclining and consan fracur prssur..5. Exponnially dclining fracur prssur Figur.9 shows h ffc of h xponn of h xponnial dclin on h shap facors basd on Equaions.68 and.69. iffrn valus of h dclin xponn ranging fro o,000 ar usd. A larg dclin facor iplis fas prssur dplion in h fracur whil a sall valu rprsns a slow dplion. As illusrad in Figur.9 for sall valus of κ κ<0., h shap facor bgins a larg valus and subsqunly convrgs o a sabilizd valu of 0.8 as copard o 8.57 for a consan fracur prssur cas. As h valu of h xponn incrass fas dplion h ransin and psudo-sady sa valus of h shap facor nd o hos of h consan fracur prssur boundary condiion. Whn κ>0 h consan fracur prssur and h xponnially dclining fracur prssur hav h sa valu of h sabilizd shap facor. Siilar bhaviour was rpord for a slighly coprssibl fluid and xponnially dclining fracur prssur by Chang 995 and Hassanzadh and Pooladi-arvish

46 Chapr. Effc of fracur prssur dplion rgis on h dual-porosiy 000 insionlss Shap facor h 00 0 k=0.000 k=0.4 k=00 k=000 σh =0.8 σh =8.57 σh = insionlss Ti Figur.9: Shap facor coparison for diffrn xponns for xponnially dclining fracur prssur. Figur.0 shows h dinsionlss shap facor for flow of a coprssibl fluid in a dual-porosiy diu for various prssur dplion rgis in h fracur. Rsuls show ha h sabilizd valus of h shap facor vary fro 8.57 for h consan fracur prssur o 0.8 for h linarly dclining fracur prssur. For h xponnially dclining fracur prssur h sabilizd valus vary bwn hs wo liis. For a vry sall valu of h xponn slow dplion h sabilizd valu is h sa as ha for h linarly dclining fracur prssur. Howvr, as h valu of h xponn incrass, h sabilizd valus of h shap facor for h xponnial dclin shif o h consan fracur prssur valu. A a larg valu of h xponnial dclin fas dplion, h consan and xponnially dclining fracur prssurs hav h sa sabilizd valu of 8.57 for h shap facor. 6

47 Chapr. Effc of fracur prssur dplion rgis on h dual-porosiy 000 insionlss Shap facor h 00 0 Exponnialk=0.000 Exponnialk=0.4 Exponnialk=000 Linar dclin Consan Fracur Prssur σh =0.8 σh =8.57 σh = insionlss Ti Figur.0: Coparison of h dinsionlss shap facor for diffrn prssur dplion rgi in h fracur. Th abov rsuls show ha boh h ransin and psudo-sady sa valus of h singlphas shap facor dpnd on how h fracur prssur changs wih i. I should b nod for a linarly dclining fracur prssur h sabilizd valu of h shap facor is indpndn of h dclin ra. On h ohr hand, for an xponnial dclin h sabilizd valu of h shap facor dpnds on h dclin xponn. Furhror, h i dpndnc of h fracur boundary condiion on h sabilizd valu of h shap facor can b dscribd by using an xponnially dclining rgi wih diffrn dclin xponns. In such cass, h sall dclin xponns rplica h linar prssur dclin in h fracur whras a larg dclin xponn rproducs h consan fracur prssur boundary condiion. Tabl. shows h sabilizd valus of h singl-phas shap facor and h i a which h ffc of prssur disurbanc rachs h innr boundary for diffrn prssur dplion rgis in h fracur. I should b poind ou ha h dvlopd odl is applicabl for singl-phas flow of a coprssibl fluid in h fracurd dia. 7

48 Chapr. Effc of fracur prssur dplion rgis on h dual-porosiy Tabl.: Sabilizd valus of h shap facor and i a which h prssur disurbanc rachs h innr boundary for diffrn dplion rgis in h fracur. plion rgi in h fracur Sabilizd valu of h dinsionlss shap facor Linar clin Exponnial clin k= Exponnial clin k= Exponnial clin k= Consan Fracur Prssur Conclusions Th following ajor conclusions ar ad as a rsul of his sudy: Th arix-fracur shap facor for singl-phas flow of coprssibl fluids illusras a ransin priod and hn sabilizs o a sabl valu hroughou psudosady sa ransfr. Th approxia analyical soluion prsnd rvald ha h arix-fracur ransfr shap facor for singl phas flow of a coprssibl fluid in h dualporosiy dia is a funcion of h prssur dplion rgi in h fracur. Basd on h prssur dplion rgi in h fracur h sabilizd valu of h shap facor varis bwn wo liis. Th uppr lii is obaind for a linarly dclining fracur prssur which corrsponds o a slow prssur dplion rgi. Th lowr lii is drivd for h consan fracur prssur boundary condiions whr dplion aks plac fasr. Whn h fracur prssur dpls xponnially wih i, h sabilizd valu of h shap facor falls bwn hos valus of h consan fracur prssur and linarly dclining fracur prssur. This sabilizd valu is a funcion of h xponn κ. For sall xponn valus h sabilizd shap facor has h sa valu as ha for h linarly dclining fracur prssur. For larg xponn valus, 8

49 Chapr. Effc of fracur prssur dplion rgis on h dual-porosiy h sabilizd valu of h shap facor is qual o ha for a consan fracur prssur. Th psudo-sady sa i sabilizaion i of h shap facor incrass as h fracur boundary condiion changs fro a fas dplion rgi oward a slow dplion rgi. 9

50 Chapr. Effc of fracur prssur dplion rgis on h dual-porosiy Nonclaur A Cross-scional ara [L ] Gas foraion volu facor Marix oal coprssibiliy [LT /M] h =L c Marix block lngh [L] k Marix prabiliy [L ] B g c l Ti dpndn lngh whr prssur is avrag prssur [L] L c n Marix block characrisic lngh [L] Exponn in h polynoial rial soluion using HBIM p Marix-block prssur [M/LT ] p Avrag arix-block prssur [M/LT ] p f Fracur prssur [M/LT ] q Marix-fracur fluid ransfr [L /T] sc qˆ Inrporosiy flow ra pr uni volu of rock [/T] Ti [T] insionlss i a which h prssur disurbanc rach o h boundary T insionlss i Rsrvoir praur [K] V b Marix-block volu [L ] x insionlss disanc Grk Sybols β clin consan [/T] Spac corrcion facor Gas spcific graviy Pnraion dph Proporionaliy consan for pnraion dph Marix hydraulic diffusiviy [L /T] Avrag hydraulic diffusiviy [L /T] insionlss hydraulic diffusiviy insionlss fracur hydraulic diffusiviy insionlss dclin consan insionlss xponn of soluion of gas diffusiviy quaion using on hod Fluid viscosiy [M/LT] Shap facor [/L ] uhal s variabl 40

51 Chapr. Effc of fracur prssur dplion rgis on h dual-porosiy Porosiy Avrag arix-block psudo-prssur [M/LT ] f Fracur psudo-prssur [M/LT ] i Iniial psudo-prssur [M/LT ] insionlss psudo-prssur insionlss fracur psudo-prssur f Avrag dinsionlss psudo-prssur Fracur psudo-prssur whn nds o infiniy [M/LT ] Subscrips f g i SC insionlss Fracur Gas Iniial condiion Marix Sandard condiions 4

52 Chapr. Effc of fracur prssur dplion rgis on h dual-porosiy Rfrncs Agarwal, R.G Ral gas psudo-i-a nw funcion for prssur build-up analysis of MHF gas wlls. SPE papr 879. As, W.F Nonlinar parial diffrnial quaions in nginring. Nw York: Acadic Prss. Bcknr, B.L Iprovd odling of ibibiion arix/fracur fluid ransfr in doubl porosiy siulaors. Ph dissraion, Sanford Univrsiy. Bourbiaux, B. Gran, S. Landrau, P. Noingr, B. Sarda, S. Sabahir, J.C Scaling up arix-fracur ransfr in dual-porosiy odls: hory and applicaion. SPE Papr Chang, M.M Analyical soluion o singl and wo-phas flow probls of naurally fracurd rsrvoirs: horical shap facor and ransfr funcions. Ph dissraion, Univrsiy of Tulsa. Chn, Z.X Transin flow of slighly coprssibl fluids hrough doubl-porosiy, doubl-prabiliy syss-a sa-of-h-ar rviw. Transp. Porous Md., 4, Cihan, A. Tynr, J.S radial analyical soluions for solu ranspor in a dualporosiy diu. War Rsour. Rs., 474, doi: 0.09/009WR Civan, F. Rasussn, M.L. 00. Analyical hindrd-arix-fracur ransfr odls for naurally fracurd prolu rsrvoirs. SPE papr Coas, K.H Iplici coposiional siulaion of singl-porosiy and dual-porosiy rsrvoirs. SPE papr 847. Crank, J Th ahaics of diffusion. Oxford: Clarndon Prss. ng, H. ai, Z. Wolfsbrg, A. Lu, Z. Y, M. Rius, P. 00. Upscaling of raciv ass ranspor in fracurd rocks wih uliodal raciv inral facis. War Rsour. Rs., 46, W0650, doi: 0.09/009WR0086. i onao, G. Blun, M.J Sralin-basd dual-porosiy siulaion of raciv ranspor and flow in fracurd rsrvoirs. War Rsour. Rs., 40, W040, doi: 0.09/00WR

53 Chapr. Effc of fracur prssur dplion rgis on h dual-porosiy Finlayson, B.A. 97. Th hod of wighd rsiduals and variaional principls. Nw York: Acadic Prss. Go-Qus, 009. Eclips 00 chnical dscripions Go-Qus, Schlubrgr. Goodan, T.R Applicaion of ingral hods o ransin nonlinar ha ransfr. Advancs in Ha Transfr,, 5-, San igo, CA: Acadic. Hassanzadh, H. Pooladi-arvish, M Effcs of fracur boundary condiions on arix-fracur ransfr shap facor. Transp. Porous Md., 64, 5-7. Hassanzadh, H. Pooladi-arvish, M. Aabay, S Shap facor in h drawdown soluion for wll sing of dual-porosiy syss. Adv. War Rs.,, Hoi, H. Firoozabadi, A Mulicoponn fluid flow by disconinuous Galrkin and ixd hods in unfracurd and fracurd dia. War Rsour. Rs., 4, W4, doi: 0.09/005WR0049. Ikoku, C.U. 99. Naural gas rsrvoir nginring. Florida: Krigr Prss. Karii-Fard, M. Firoozabadi, A. 00. Nurical Siulaion of War Injcion in Fracurd Mdia Using h iscr-fracur Modl and h Galrkin Mhod. Soc. P. Eng. J., 6, 7-6. Kazi, H. Gilan, J.R. 99. Muliphas flow in fracurd prolu rsrvoirs. In: J. Bar, C. F. Tsang, and G. d Marsily ds., Flow and Conainan Transpor in Fracurd Rock. Acadic Prss, San igo, 67-. Kazi, H. Mrrill, L.S. Porrfild, K.L. Zan, P.R Nurical siulaion of war-oil flow in naurally fracurd rsrvoirs. Soc. P. Eng. J., 66, 7-6. Lonnir, P. Bourbiaux, B. 00. Siulaion of naurally fracurd rsrvoirs. Sa of h ar. Par : Physical chaniss and siulaor forulaions. Oil and Gas Scinc and Tchnology, 65, 9-6. Li, K.T. Aziz, K Marix-fracur ransfr shap facors for dual-porosiy siulaors. J. P. Sci. Eng.,, Liu, M.X. Chn, Z.X Exac soluion for flow of slighly coprssibl fluids hrough ulipl-porosiy, ulipl-prabiliy dia. War Rsour. Rs., 67,

54 Chapr. Effc of fracur prssur dplion rgis on h dual-porosiy Lu, M. Connl, L A dual-porosiy odl for gas rsrvoir flow incorporaing adsorpion bhavior-par I. Thorical dvlopn and asypoic analysis. Transp. Porous Md., 68,5-7. Michl, S.L. Myrs, T.G. 00. Iproving h accuracy of ha balanc ingral hods applid o hral probls wih i dpndn boundary condiions. In. J. Ha Mass Transfr, 5, Özisik, M.N. 99. Ha Conducion. Unid Sas: John Wily & Sons Inc. Pnula, G. Civan, F. Hughs, R.G. Wiggins, M.L. 00. Ti-dpndn shap facors for inrporosiy flow in naurally fracurd gas-condnsa rsrvoirs. SPE papr Polyanin, A.. 00 Handbook of linar parial diffrnial quaions for nginrs and sciniss. Unid Sas: CRC Prss. Pooladi-arvish, M. Torik, W.S. Farouq Ali, S.M Non-isohral graviy drainag undr conducion haing. Prolu Sociy of CIM and AOSTRA. Papr NO Quinard, M. Whiakr, S. 996 Transpor in chically and chanically hrognous porous dia. I: horical dvlopn of rgion-avragd quaions for slighly coprssibl singl-phas flow. Adv. War Rs., 9, Quinard, M. Whiakr, S Transpor in chically and chanically hrognous porous dia II: coparison wih nurical xprins for slighly coprssibl singl-phas flow. Adv. War Rs., 9, Quinard, M. Whiakr, S Transpor in chically and chanically hrognous porous dia III: Larg-scal chanical quilibriu and h rgional for of arcy's law. Adv. War Rs., 7, Rangl-Gran, E. Kavsck, A.R. Akin, S. 00. Ti-dpndn shap facors for unifor and non-unifor prssur boundary condiions. Transp. Porous Md., 8, Ranjbar, E. Hassanzadh, H. 0. Marix-fracur ransfr shap facor for odling flow of a coprssibl fluid in dual-porosiy dia. Adv. War Rs., 45, Shan, C. Pruss, K An analyical soluion for slug racr ss in fracurd rsrvoirs. War Rsour. Rs., 4:W0850, doi:0.09/005wr

55 Chapr. Effc of fracur prssur dplion rgis on h dual-porosiy Thoas, L.K. ixon, T.N. Pirson, R.G. 98. Fracurd rsrvoir siulaion. Soc. P. Eng. J.,, Uda, Y. Muraa, S. Waanab, Y. Fanasu, K Invsigaion of h shap facor usd in h dual-porosiy rsrvoir siulaor. SPE Papr van Hl, A.P.G. Borrigr, P.M. van orp J.J. Thral and hydraulic arix-fracur inracion in dual prabiliy siulaion. Soc. P. Eng. J. rsrvoir valuaion and nginring, 4, Warrn, J.E. Roo, P.J. 96. Th bhavior of naurally fracurd rsrvoirs. Soc. P. Eng. J.,, Ziran, R.W. Bodvarsson, G.S An approxia soluion for on-dinsional absorpion in unsaurad porous dia. War Rsour. Rs., 56, Ziran, R.W. Chn, G. Hadgu, T. Bodvarsson, G.S. 99. A nurical dualporosiy odl wih si-analyical ran of fracur/arix flow. War Rsour. Rs., 97, 7-7. Ziran, R.W. Hadgu, T. Bodvarsson, G.S A nw lupd-parar odl for flow in unsaurad dual-porosiy dia. Adv. War Rs., 95,

56 Chapr. Effc of fracur prssur dplion rgis on h dual-porosiy 46 Appndix.A: Soluion of h gas diffusiviy quaion for diffrn fracur dplion condiions.a: Linarly dclining fracur prssur In his cas h following PE wih iniial and boundary condiions Equaions.A.a-.A.c should b solvd: x x,.a. 0 0,.A.a 0 0, x x.a.b f x..a.c For h arly i soluion w us h following rial soluion:, x x..a. Whn h boundary condiion changs wih i h pnraion dph in h ha balanc ingral hod HBIM is found by solving h following ordinary diffrnial quaion Michl and Myrs, 00: f f n n n d d..a.4 In his quaion n is h xponn in h rial soluion n= for our cas and. In Equaion.A.4, θ can b obaind by using h following quaion: n n n f f..a.5 Assu ha and subsiuing his quaion in h OE quaion of.a.4 lads o following OE for h linarly dclining fracur prssur: d n n d 0..A.6

57 Chapr. Effc of fracur prssur dplion rgis on h dual-porosiy I should b nod ha his OE is obaind by assuing θ=0 in Equaion.A.4 Michl and Myrs, 00. Ingraing of Equaion.A.6 lads o h following quaion for ε: n n n 8..A.7 So w rach h following quaion for pnraion dph in h cas of a linar PE: 8 8..A.8 In h cas of a nonlinar PE w hav h following quaion for h pnraion dph: 8..A.9 As was illusrad in h odl vrificaion scion w can obain a or accura soluion if w us h following quaion for h pnraion dph: 9..A.0 Thrfor, h arly i soluion of h parial diffrnial Equaion.A. wih h boundary condiions.a. can b xprssd as follows: x, x,..a. 9 9 Ingraing of Equaion.A. ovr h arix block volu, lads o h following quaion for h arly i avrag dinsionlss psudo-prssur: 9, / k..a. 4 9 For h la i soluion in h cas of h linarly dclining fracur prssur h following PE should b solvd. I should b nod ha h iniial condiion for h la i soluion cos fro h arly i soluion Equaion.A.. x x,.a. x,.a.4a 9 9 x 0 0, x.a.4b 47

58 Chapr. Effc of fracur prssur dplion rgis on h dual-porosiy x..a.4c f Th i dpndnc of h boundary condiion for h la i soluion can b considrd by uhal s hor. Whn h fracur psudo-prssur varis wih i Equaion.A.4c, uhal s hor provids h basis o solv h probl wih variabl boundary condiions basd on h soluion providd for h consan fracur psudo-prssur. Using uhal s hor Chang, 995; Özisik, 99; Polyanin, 00 h soluion of PE.A. wih condiions.a.4 can b xprssd as: x, d,..a.5 f 9 0 In his quaion, wihin h ingral is h soluion whn Equaion.A.6 and on h lf-hand sid is h soluion of h PE whn h arix-fracur boundary condiion changs wih i. x,.4 xp xp.4 xp.086 xp x.a.6 xp xp x. rivaion of Equaion.A.6 has bn shown in our prvious sudy Ranjbar and Hassanzadh, 0. Using uhal s hor and subsiuing Equaions.A.6 and.a.4c in Equaion.A.5 lad o h following la i soluion for h cas of h linarly dclining fracur prssur: f x, xp xp C xp xp x xp xp x, 9.A.7 48

59 Chapr. Effc of fracur prssur dplion rgis on h dual-porosiy Th iniial condiion is usd o find and as follows: C 0,.A Solving h sys of Equaions.A.8 lads o h following valus for and.55790, A.9 0 Using hs valus in Equaion.A.7 and so siplificaions lads o h following la i psudo-prssur for h linarly dclining fracur prssur: x x, x x xp x, xp 9.A.0 Th la i avrag arix block psudo-prssur for h linarly dclining fracur prssur is obaind as follows: x 0.49 xp, dx 0, xp.a: Exponnially dclining fracur prssur 9.A. Whn h fracur prssur changs xponnially wih i w hav h following PE wih hs iniial and boundary condiions: x x,.a. 0 0,.A.a x x 0 0, x.a.b xp..a.c f For h arly i h following rial soluion is suggsd which saisfis h our boundary condiion: x, xp x,.a. 49

60 Chapr. Effc of fracur prssur dplion rgis on h dual-porosiy Whn h boundary condiion chang wih i, h pnraion dph is found by solving h following OE Michl and Myrs, 00: d f f n..a.4 d n n whr: f n n f.a.5. n By assuing θ=0 and and subsiuing Equaion.A.c in Equaion.A.4 w rach h following OE: xp n xp d..a.6 n Solving his OE for ε, subsiuing i in h pnraion dph quaion and aking n= lad o h following quaion for h pnraion dph for h linar parial diffrnial quaion: rf..a.7 xp xp And for h nonlinar parial diffrnial quaion Equaions.A. and.a. w rach h following quaion for h pnraion dph: rf..a.8 xp xp Our nurical rsuls show ha w can incras h accuracy of h soluion if w us h following quaion for h pnraion dph: rf.96..a.9 xp xp Thrfor, h arly i soluion in Equaions.A. and.a. can b xprssd as follows: 50

61 Chapr. Effc of fracur prssur dplion rgis on h dual-porosiy x, xp.96 xp x rf xp,.a.0 Ingraing of Equaion.A.0 ovr h arix block volu, lads o h following quaion for h arly i avrag dinsionlss psudo-prssur: xp rf.96,.a. 4 For h la i soluion in h cas of h xponnially dclining fracur prssur h following PE should b solvd. I should b nod ha h iniial condiion for h la i soluion cos fro h arly i soluion Equaion.A.0. x x,.a. xp x,.a.a x 0 0, x.a.b x xp..a.c f Using uhal s hor Equaion.A.5 and h soluion of h consan fracur prssur Equaion.A.6 lads o h following la i soluion for h cas of h xponnially dclining fracur prssur: 5

62 Chapr. Effc of fracur prssur dplion rgis on h dual-porosiy 5, xp xp xp xp xp.086 xp.4 xp xp xp.4 xp, C C x x x.a.4 Th iniial condiion is usd o find and as follows: xp 0, C.A.5 Solving h sys of Equaions.A.5 lads o h following valus for and xp xp xp.4874 xp xp xp A.6 Using hs valus in Equaion.A.4 and siplifying lad o h following la i psudo-prssur for h xponnially dclining fracur prssur:

63 Chapr. Effc of fracur prssur dplion rgis on h dual-porosiy 5, xp xp xp xp, x x x x x x x x x.a.7 Ingraing ovr h arix block volu rsuls in h following quaions for h avrag dinsionlss psudo-prssur in h cas of h xponnially dclining fracur prssur:, xp xp xp 0.7 xp, x.a.8

64 Chapr Thr: On insional Marix-Fracur Transfr in ual-porosiy Syss wih Variabl Block Siz isribuion Absrac Mos of h dvlopd odls for fracurd rsrvoirs assu idal arix block siz disribuion. This assupion ay no b valid in raliy for naurally fracurd rsrvoirs and possibly lad o rrors in prdicion of producion fro h naurally fracurd rsrvoirs spcially during a ransin priod or arly i producion fro h arix blocks. In his sudy, w invsiga h ffc of variabl block siz disribuion VBS on on dinsional flow of coprssibl fluids in fracurd rsrvoirs. Th ffc of diffrn arix block siz disribuions on h singl phas arix-fracur ransfr is sudid using a rcnly dvlopd si-analyical approach. Th proposd odl is abl o siula fluid xchang bwn arix and fracur for coninuous or discr block siz disribuions using probabiliy dnsiy funcions or srucural inforaion of a fracurd foraion. Th prsnd si-analyical odl donsras a good accuracy copard o h nurical rsuls. Thr hav bn rcn aps o considr h ffc of variabl block siz disribuion in naurally fracurd rsrvoir odling for slighly coprssibl fluids wih a consan viscosiy and coprssibiliy. Th ain objciv of his sudy is o considr h ffc of variabl block siz disribuion on a on dinsional arix-fracur ransfr funcion for singl phas flow of a coprssibl fluid in fracurd porous dia. In h proposd si-analyical odl h prssur variabiliy of viscosiy and isohral coprssibiliy is considrd by solving h nonlinar parial diffrnial quaion of coprssibl fluid flow in h fracurd dia. Th closd for soluion providd can b applid o flow of coprssibl fluids wih variabl arix block siz disribuion in naurally fracurd gas rsrvoirs. This chapr is an xac copy of: Ranjbar, E. Hassanzadh, H. Chn, Z. 0. On insional Marix- Fracur Transfr in ual-porosiy Syss wih Variabl Block Siz isribuion, Transpor in Porous Mdia, Vol. 95, Pag: 85-.

65 Chapr. On dinsional arix-fracur ransfr in P syss wih VBS. Inroducion On of h os coon approachs for h odling of naurally fracurd rsrvoirs is hrough dual-porosiy, which ras h fracur and arix syss as spara doains. In his approach arix ain doain for fluid sorag and fracur ain doain for fluid flow ar conncd wih a cross-flow ransfr funcion Chn, 995. Th ransfr r, which is dircly proporional o a shap facor, is on of h ky parars in a prssur ransin analysis of dual-porosiy rsrvoirs. This parar considrs h ffc of h fracur spacing disribuion ino h prssur ransin rspons quaions. Sudis hav bn don o valua h shap facor for flow of slighly coprssibl fluids in h fracurd dia.g., Warrn and Roo, 96; Kazi al., 976; Li and Aziz, 995; Chang, 995; Ziran al., 99, 996; Civan and Rasussn, 00; Pnula al., 00; Hassanzadh and Pooladi-arvish, 006; Lu and Connl, 007; van Hl al., 008; Hassanzadh al., 009; Rangl-Gran al., 00. In our prvious sudis w hav dvlopd a odl for a coprssibl fluid shap facor and considrd h ffc of fracur prssur dplion rgis on his shap facor Ranjbar and Hassanzadh, 0; Ranjbar al., 0. On of h ain difficulis wih dual-porosiy odls proposd by Warrn and Roo 96 is ha all fracurs ar assud o b vnly spacd hroughou h rsrvoir. Masurns of fracur spacing in gological foraions hav shown ha foraions hav a broad disribuion of arix block sizs du o lihologic variaions, variabl rock srsss, and diagnic procsss Ris, 998. Thrfor, for accura prdicion of ass ransfr bwn h arix and fracur syss h ffc of variabl block siz disribuion should b considrd. Thr ar rcn sudis on h ffc of variabl block siz disribuion in h fracurd rsrvoir odling for slighly coprssibl fluids. This variabiliy in h arix block siz disribuion influncs h hydrocarbon rcovry spcially during arly i producions fro arix blocks Rodriguz al., 00. Thrfor, for accura prdicion of producion, i is ssnial o characriz and odl fracurd porous dia by considring variabl arix block siz faurs Blani, 988. Consqunly, dual-porosiy odls such as Barnbla al., 960 and Warrn and Roo s 96 ar an ovr-siplificaion of ral fracurd foraions for fluid flow in 55

66 Chapr. On dinsional arix-fracur ransfr in P syss wih VBS fracurd porous dia and canno b considrd as a coprhnsiv odl for h fracurd rsrvoirs. Th ain objciv of his sudy is o invsiga h ffc of variabl arix block siz disribuion on h arix-fracur fluid ransfr for coprssibl fluids in fracurd dia. Th si-analyical odl prsnd dos no rquir nurical invrsion or infini sris calculaions and ay find applicaions for odling h flow of slighly coprssibl fluids in fracurd porous dia wih variabl arix block siz disribuion. uring h las wo dcads, h odling of variabl arix block siz disribuion in naurally fracurd rsrvoirs has bn h focus of nurous invsigaions Cinco-Ly al., 985; Abdassah and Ershaghi, 986; Blani and Jalali-Yazdi, 988; Johns and Jalali- Yazdi, 99; Jlr, 995; Ris, 998; Gwo al., 998. A brif rviw of so of h dvlopd odls ha considr h ffc of h variabl block siz disribuion on h rcovry fro arix blocks and h shap facor in h cas of a slighly coprssibl fluid is discussd in his scion. Ziran and Bodvarsson 995 odld collcions of arix blocks of diffrn sizs wih a singl quivaln block wih an quivaln avrag volurically-wighd radius. Thy showd ha h collcion of arix blocks prfors lik a singl arix block wih an quivaln radius. Thir rsuls wr validad during arly is. Howvr, in h long i lii, no ffciv radius was rigorously dfind ha can b usd as an quivaln radius. Rodriguz al., 00 prsnd a variabl arix block siz odl for flow of slighly coprssibl fluids by considring unifor, xponnial and noral disribuion of arix block siz disribuions. Thy obaind hir soluion in h Laplac spac involving an ingral, which was nurically valuad. Thr ar also rcn aps o considr block siz disribuion for gas rsrvoirs wih approachs diffrn fro h si-analyical approach prsnd in his sudy Fahi and Akkulu, 009; Lu and Connl, 0. In addiion discr fracur rprsnaion has bn usd for rando fracur nwork wih various fracur dnsiis and conduciviis Bogdanov al., 00; Mourznko al., 0. 56

67 Chapr. On dinsional arix-fracur ransfr in P syss wih VBS According o h prvious sudis i can b concludd ha h ransin prssur rspons for h variabl arix block siz disribuion is diffrn fro ha of h convnional sugar cub odl of Warrn and Roo for slighly coprssibl fluids. I should also b poind ou ha os of h prsnd odls for variabl arix block siz disribuion in h liraur do no provid a sipl prdicion for h prssur ransin rspons. Mos of hs odls provid ihr a soluion in h Laplac spac ha rquirs a nurical invrsion ino h i doain, or hy rquir h us of infini sris calculaions. Th ffc of variabl block siz disribuion on flow of coprssibl fluid, which lads o a nonlinar parial diffrnial quaion, has no bn widly invsigad in h pas. Th ain goal of his sudy is o prsn a nw si-analyical approach o considr h ffc of variabl arix block siz disribuion on flow of a coprssibl fluid in dual-porosiy dia. In addiion, du o h analyical naur of h proposd odl, i can b usd o odl flow of slighly coprssibl fluids in fracurd dia. Th proposd odl in his sudy is a nw si-analyical approach ha can handl h variabl arix block siz disribuion for h flow of boh coprssibl nonlinar parial diffrnial quaion and slighly coprssibl fluids linar parial diffrnial quaion in h fracurd dia. In addiion o h coninuous block siz disribuion, h proposd odl is also capabl o odl arix-fracur ransfr whn hr is discr block siz disribuion using srucural inforaion of a porous diu. In h prsnd odl a cobinaion of a ha ingral hod, h hod of ons and a concp of quivaln lngh ar usd o considr h ffc of variabl block siz disribuion on h arix-fracur fluid ransfr. This nw odl can b siply prograd ino a spradsh for quick us. Rsuls of his sudy show ha h variabiliy of arix block siz disribuion has a significan ffc on h ransin valus of h arix-fracur fluid ransfr and shap facor for coprssibl and slighly coprssibl fluids. This papr is organizd as follows: Firs, a shor sudy on h diffrn arix block siz disribuions is prsnd. In h nx sp, h dfiniion of quivaln lngh is usd o driv h quivaln lngh for diffrn arix block siz disribuions. Afr ha, a ahaical odl for variabl arix block siz disribuion of coprssibl fluid flow 57

68 Chapr. On dinsional arix-fracur ransfr in P syss wih VBS in fracurd dia is dvlopd. Subsqunly, h dvlopd odl is applid o slighly coprssibl fluids. Afrwards, validaion of h proposd odl is discussd. Finally, h rsuls and discussions ar prsnd lading o conclusions.. Marix block siz disribuions Gological and wll logging sudis hav shown ha fracurd foraions hav a wid disribuion of arix block sizs du o lihologic variaions, variabl rock srsss, and diagnic procsss. To iprov h odling of fluid flow in fracurd foraions on nds o considr variaion in h arix block siz disribuion. Alhough hs parars ar pracically ipossibl o asur dircly bcaus of h shorag of hr-dinsional accss o rprsnaiv rocks, hy can b siad fro fracur spacing asurd on orhogonal fracur ss fro oucrops. Fracur spacing is classically asurd using a scanlin along an oucrop or a cor, whr h succssiv spacing bwn fracurs is rcordd as hy inrsc h scanlin Ris, 998. In gnral, variabl arix block siz disribuion is rpord basd on wo approachs including discr block siz disribuion and coninuous block siz disribuion. In h cas of discr block siz disribuion srucural inforaion of a fracurd rock is rpord basd on h obsrvd fracur spacing and h frquncy of h arix block siz disribuion for a fini nubr of block sizs Gwo al., 998. In h coninuous block siz disribuion h acual gological foraion is rpord basd on a probabiliy dnsiy funcion. Exponnial and linar disribuions ar wo os probabl coninuous block siz disribuions xising in oucrops Sgal, 98. For a discr disribuion w hav h following quaion: N i f i L ci. whr L c is h arix block characrisic lngh half of h arix block hicknss for a slab-shapd arix block, h, f i L ci is h fracion of h block volu of siz L ci and N is h nubr of block sizs, which is fini. Coninuous block siz disribuion is rpord using a probabiliy dnsiy funcion PF, which is a posiiv funcion. Th ara undr h curv of a PF as illusrad in Equaion. is qual o on. 58

69 Chapr. On dinsional arix-fracur ransfr in P syss wih VBS Lc ax Lc in f L c dl c. In Equaion., fl c is h probabiliy dnsiy funcion wih h rando variabl L c and dscribs h disribuion of h block siz. Sinc h disribuion funcions ar usually xprssd in dinsionlss for w us h following dinsionlss variabls o xprss h PFs in dinsionlss for: L c L. Lc ax f L L f L.4 c ax c L c in Fh.5 Lc ax In h following, a brif dscripion of diffrn probabiliy dnsiy funcions is prsnd... Unifor or rcangular disribuion In his cas h following probabiliy dnsiy funcion is usd o dscrib h arix block siz disribuion: f Lc L L.6 c ax c in Whn nough daa abou arix block siz disribuion dos no xis and h arix block siz disribuion is unknown his disribuion ay b usd. Equaion.6 in h dinsionlss for can b xprssd as follows: f L.7 F h.. Exponnial disribuion Th corrsponding quaion for an xponnial probabiliy dnsiy funcion is xprssd as follows: Lc f Lc.8 Lc in Lc ax whr α is an xponnial disribuion consan. Using Equaions. o.5 in Equaion.8 lads o h following dinsionlss xponnial probabiliy disribuion funcion: 59

70 Chapr. On dinsional arix-fracur ransfr in P syss wih VBS f a L a L.9 a Fh a In Equaion.9, F h dfins h rang of block sizs for xponnial funcions and a is a dinsionlss xponnial disribuion consan. Th liraur valus for h dinsionlss xponnial consans ar -0, -5, 5, and 0 Rodriguz al., 00. Th posiiv valus of a.g., 0 or 5 rprsn disribuions wih a high fracion of sall blocks or high dnsiy of fracurs. For sparsly fracurd disribuion or a high fracion of larg blocks h valus of h xponnial consan ar sall a= -0 or Noral or Gaussian disribuion Th following quaion shows h noral or Gaussian funcion: f L xp L M.0 In Equaion.0, M is h an and σ is h varianc of h disribuion. Using h valus for an and varianc in h Gauss funcion lads o h following noral arix block siz disribuion Rodriguz al., 00: f 6 8 F xp [ L ] F Fh h h L..4 Linar disribuion. For h linar block siz disribuion w hav h following dinsionlss quaion o dscrib arix block siz disribuion: f L L b. In Equaion., >0 ans a low fracur dnsiy or larg arix block sizs. For a high proporion of sall arix blocks is lss han zro. In gnral, wo cass for h linar PF is considrd in h liraur as linar dcrasing disribuion =-00/8; b=45/8 and linar incrasing disribuion =00/8; and b=5/8 Rodriguz al., 00. I should b nod ha linar disribuion changs o h unifor disribuion if w us =0 and b=0/9, whn F h =0. Rodriguz al., Log-noral disribuion A log-noral disribuion is a probabiliy disribuion which is dfind basd on h following quaion, 60

71 Chapr. On dinsional arix-fracur ransfr in P syss wih VBS [ln L M ln ] f L xp. L ln ln iffrn valus for varianc and avrag arix block siz lngh has bn usd in h liraur for log-noral disribuion Lung and Ziran, 0; Ris, 998. In his sudy w usd h valu of 0.5 for sandard dviaion of spacing logarih and -.0 for h an of spacing logarih. Figur. copars h diffrn disribuion funcions for spcific valus of sandard dviaion, an,, b, and a whn F h =0.. 6 f L 5 4 Rcangular Linar Incrasing =00/8 Linar crasing =-00/8 Exponnal a= -0 Exponnial a=-5 Exponnial a=5 Exponnial a=0 Noral Log-Noral L Figur.: iffrn probabiliy dnsiy funcions. 6

72 Chapr. On dinsional arix-fracur ransfr in P syss wih VBS. Equivaln lnghs for diffrn arix block siz disribuions.. Equivaln lngh concp I has bn shown ha h collcion of arix blocks wih diffrn disribuions prfors lik a singl arix block wih an quivaln lngh Ziran and Bodvarsson, 994. For h cas of discr arix block siz disribuion h quivaln half arix block siz is dfind as follows Gwo al., 998: L c N N L N N L i ci i ci N N i i i N Lci fi Lci N i N i Ni i N L.4 whr N is h oal nubr of arix blocks and f i L ci is h probabiliy of h arix block siz of L ci. Equaion.4 using Equaion. can b xprssd in h dinsionlss for as follows: N L f L L.5 i i ci i For h cas of coninuous block siz disribuion h following quaion is usd o find h half arix block quivaln lngh: L c ax L L f L dl.6 c L c in c c c Using Equaions. o.5 in Equaion.6 lads o h following quaion for h dinsionlss quivaln lngh in h cas of coninuous arix block siz disribuion: L L f L dl.7 F h.. iscr arix block siz disribuion Equaion.5 is usd o calcula h quivaln lngh for h cas of discr arix block siz disribuion whn h obsrvd frquncy of arix block sizs is availabl for a fini nubr of arix blocks. Tabl. shows h obsrvd fracur spacing and h frquncy of arix block siz disribuion rproducd by Gwo al., 998 for fracurd Saproli. ci 6

73 Chapr. On dinsional arix-fracur ransfr in P syss wih VBS To calcula h quivaln lngh using h srucur of porous dia, h dinsionlss lngh L and probabiliy of arix blocks fl ci ar calculad as illusrad in Tabl.. Using Equaion.5 h calculad valu for h quivaln lngh of h daa providd in Tabls. is This hod can b usd o find h quivaln lngh for any obsrvd frquncy of arix block sizs. Tabl.: Obsrvd frquncy of arix block siz in h soil colun Gwo al., 998 Block Siz h, Block siz characrisic Avrag frquncy op r lngh L c, r and boo of soil N i L =L c /L cax fl ci =N i /N Coninuous arix block siz disribuion Subsiuing h probabiliy dnsiy funcions for rcangular Equaion.7, xponnial Equaion.9, Gaussian or noral Equaions.0 or., linar Equaion. and log-noral Equaion. disribuions in h quivaln lngh quaion Equaion.7 lads o h following quaions for h quivaln lngh in h cas of diffrn probabiliy dnsiy funcions. Rcangular disribuion : L Exponnial disribuion : L Noral disribuion : L M F M rf h F.8 afh a afh a afh a.9 a M rf M Fh M h.0 b Linar disribuion : L Fh Fh. 6

74 Chapr. On dinsional arix-fracur ransfr in P syss wih VBS Log-noral disribuion: ln M ln M ln Ln ln Fh M ln ln L rf [ ] rf [ ]. ln ln Ths drivd quivaln lnghs for discr Equaion.5 and coninuous Equaions.8-. block siz disribuions should b rplacd in diffusiviy quaion and shap facor quaion o calcula h avrag arix block prssur or arix-fracur ransfr shap facor for diffrn arix block siz disribuions in h cas of coprssibl fluid flow in fracurd dia. This will b discussd in h following scions..4 Mahaical odl for flow of coprssibl fluid in fracurd dia wih variabl block siz disribuion In his scion h ahaical odl for h arix-fracur ransfr funcion for a coprssibl fluid is drivd by considring h variabl arix block siz disribuion. In his drivaion h dpndnc of h coprssibl fluid hydraulic diffusiviy on prssur is considrd by solving a nonlinar flow quaion in porous dia. Th diffusiviy quaion for a slab shap arix can b xprssd as: p x. In Equaion., η is arix hydraulic diffusiviy and is h raio of arix prabiliy o produc of arix porosiy, fluid viscosiy and arix fluid and rock coprssibiliy. I should b poind ou ha η is srongly prssur-dpndn; sinc prssur is spac- and i-dpndn; h arix hydraulic diffusiviy also varis wih i and spac. For solving his PE, w suppos ha h hydraulic diffusiviy is i-dpndn and h ffc of spac dpndnc is accound for by inroducing a corrcion facor, β. This corrcion facor is hn obaind using fin-grid nurical siulaions Ranjbar and Hassanzadh, 0. Thrfor, in h cas of variabl arix block siz disribuion h following PE wih h iniial and boundary condiions should b solvd..4 x x 0.5a i 64

75 Chapr. On dinsional arix-fracur ransfr in P syss wih VBS x 0 0.5b x x.5c L c f I should b nod ha in Equaion.5c, L c is quivaln lngh which is no a consan and is a funcion of block siz disribuion Equaions.4 and.6. Figur. lf shows a arix-fracur sys and is boundary condiions. A schaic of h odl for variabl block siz disribuion is illusrad in Figur. righ. This figur donsras a discr fracur odl wih non-unifor block siz disribuion and boundary condiions for h blocks. f x 0 Figur.: Illusraion of a arix-fracur sys and is boundary condiions lf and rprsnaion of fracurd rsrvoirs in h cas of non-idal arix block siz disribuion righ. For solving Equaion.4 wih h boundary condiions in Equaion.5, w dfin h avrag hydraulic diffusiviy ovr h arix-fracur prssur drawdown as follows: p f p i p f p i k k dp c p f p i p f p i dp c.6 I should b poind ou ha in his quaion k, c and ar arix prabiliy, coprssibiliy and porosiy rspcivly, µ is fluid viscosiy and p f and p i ar fracur and iniial prssur, rspcivly. Equaions.4 and.5 ar ad dinsionlss by dfining h following dinsionlss variabls: 65

76 Chapr. On dinsional arix-fracur ransfr in P syss wih VBS i.7 f i x x.8 L c ax.9.0 L c ax L c L. Lc ax Using Equaions.7 o. in Equaions.4 and.5, w obain h following dinsionlss PE wih h iniial and boundary condiions Equaions.a,.b,.c: x x. 0 0.a x 0 0 x.b x.c L In Equaion.c, L is usd o considr h ffc of variabl block siz disribuion; his quivaln lngh is calculad basd on Equaions.5 or.7. A ha ingral hod Goodan, 964; Finlayson, 97; Pooladi-arvish al., 994 and h hod of ons As, 965; Crank, 975 ar usd o find h arly and la i soluions of his nonlinar PE, rspcivly. Th following quaions giv h soluion of h PE: L x L 4,

77 Chapr. On dinsional arix-fracur ransfr in P syss wih VBS x,.79 xp L 0.54 xp L.5xp 6.75 xp L xp L 0.489xp x x, L 4.5 Mor dails on h drivaion of hs wo quaions ar shown in Appndix.A. Ingraing ovr h bulk volu of a arix block lads o h following quaion for h arly and la i avrag dinsionlss psudo prssurs: L L L 4 L 4 x L dx 4 4L 4.6 L L dx 0.790xp 0.48xp L In hs quaions h fracur dinsionlss hydraulic diffusiviy, η, λ and λ ar dfind as follows, rspcivly: x k / f c f L L In our prvious sudis h following quaion was drivd for h dual-porosiy shap facor of coprssibl fluid in fracurd dia Ranjbar and Hassanzadh, 0; Ranjbar al., 0: c.40 k f Using h dfiniion of dinsionlss variabls Equaions.7 o. in his quaion lads o h following dinsionlss shap facor for variabl block siz disribuion: Lc ax.4 In his quaion, η is no a consan and is a funcion of prssur, spcific graviy and praur sinc h viscosiy-coprssibiliy produc is dpndn on hs parars. 67

78 Chapr. On dinsional arix-fracur ransfr in P syss wih VBS Nurical siulaion is usd o find h dinsionlss hydraulic diffusiviy Ranjbar and Hassanzadh, 0: Subsiuing Equaion. ino Equaion.4 lads o h following dinsionlss shap facor for a cas wih variabl arix block siz disribuion: L c L or h 4L.4 In Equaion.4, h is h quivaln arix block lngh hicknss. Subsiuing Equaions.6 and.7 and hir drivaivs in Equaion.4 lads o h following quaion for h arly and la i dinsionlss shap facors in h cas of variabl arix block siz disribuion, rspcivly: h h L, L xp L xp L xp L xp L L, L 4.44 I should b nod ha for diffrn arix block siz disribuion L is calculad using Equaions.6 and Equaions.8 o. and subsiud in Equaions.6,.7,.4 and.44 o calcula h avrag psudo-prssur and shap facor for ach disribuion..5 Slighly coprssibl fluids Th abov prsnd odl can b also usd o considr h arix block siz disribuion for flow of a slighly coprssibl fluid in fracurd porous dia. I should b poind ou ha in h cas of a slighly coprssibl fluid flow in fracurd dia du o h consan viscosiy and coprssibiliy of such a fluid, h hydraulic diffusiviy is consan and h arix flow quaion is a linar parial diffrnial quaion as follows: 68

79 Chapr. On dinsional arix-fracur ransfr in P syss wih VBS p p i.45 x p 0 p p i, x 0 0 x x L p p.46, c f In his cas, Ranjbar and Hassanzadh, 0. Using hs valus in h coprssibl fluid odl, h soluion for slighly coprssibl fluid flow in fracurd dia wih variabl block siz disribuions can b obaind. Using h sa procdur as abov, h following quaions for h avrag dinsionlss arix block prssur and arix-fracur ransfr shap facor ar obaind in a closd for wihou rquiring a nurical invrsion ino h i doain or h us of an infini sris valuaion: p p 4 L.47 4L L 0.790xp 0.48xp.48 L L 4 L 6 L h L xp 4.76xp L L L h xp 0.48xp 4 L L In hs quaions h quivaln lngh drivd in Scion. for diffrn discr and coninuous arix block siz disribuion is usd o driv h avrag dinsionlss prssur and dinsionlss shap facor for diffrn arix block siz disribuions for slighly coprssibl fluids..6 Validaion In ordr o valida h prsnd odl by h quivaln lngh concp, h soluion for arix blocks wih idal block siz is drivd, and i is shown ha h prsnd odl can rproduc h liraur odl for h idal block siz disribuion. A h nd of his 69

80 Chapr. On dinsional arix-fracur ransfr in P syss wih VBS scion h dvlopd si-analyical odl for wo diffrn discr block siz disribuions in h cas of coprssibl fluid is copard wih h fin grid nurical rsuls, which show an accpabl accuracy. In h cas of h idal cubic sugar of h Warrn and Roo odl, h disribuion funcion can b xprssd using a irac dla funcion which is dscribd as follows: 0 L f L L.5 L Th irac dla funcion has h following propry: f L L a dl f a L L L dl.5 F h Subsiuing h quivaln lngh bing qual o uniy in h shap facor quaions Equaions.4 and.44 can rcovr h proposd odl for h shap facor of coprssibl fluid flow in fracurd porous dia Ranjbar and Hassanzadh, 0. Figur. copars h slighly coprssibl fluid shap facor basd on h shap facor drivd in his sudy wih quivaln lngh of on Equaions.49 and.50 and h prvious odls in h liraur Chang, 995; Hassanzadh and Pooladi-arvish, 006. As illusrad in his figur hr is an accura ach bwn h prsnd odl and h prvious odls. I should b poind ou ha h sabilizd valu of h slighly coprssibl fluid shap facor basd on h prsnd odl is 9.944, which is vry clos o as was rpord by Li and Aziz

81 Chapr. On dinsional arix-fracur ransfr in P syss wih VBS 000 insionlss Shap Facor 00 0 Prsnd Modl Prvious Modls insionlss Ti Figur.: Coparison of h prsnd odl wih liraur odls Chang 995, Hassanzadh and Pooladi-arvish 006 for slighly coprssibl fluid and idal block siz disribuions. In his par w prsn h coparison of our odl wih a fin-grid, singl porosiy nurical odl Eclips 00 for wo diffrn discr block siz disribuions. For raching his goal w drin h cuulaiv fluid xchang as a funcion of ral i basd on h dvlopd odl Equaion.B.5 in Appndix.B and copar his wih h nurical rsuls. W hav considrd h gas spcific graviy of 0.7, praur of 9. C and prssur draw-down of 45 o.5 MPa for hs wo arix block siz disribuions. Th valus usd for β and η ar 0.7 and 0.7, rspcivly. I should b poind ou ha in our prvious sudis Ranjbar and Hassanzadh, 0 and Ranjbar al., 0 for h coprssibl fluid flow w did no considr h arly i cuulaiv producion, sinc i was a shor i priod abou 7 sconds. Bu in his sudy w considr h arly i cuulaiv producion, which rsuls in a chang in η fro 0.7 o 0.7. All h ohr propris ar h sa as h bas cas in our prvious sudy Ranjbar and Hassanzadh, 0. Now w will considr wo discr odls and copar our rsuls wih h nurical rsuls. Following abl shows h discr block siz disribuion for hs wo cass. Th iniial prssur is 45 MPa and fracur prssur is.5 MPa. All ohr propris hav h sa valu as was rpord by Ranjbar and Hassanzadh 0. 7

82 Chapr. On dinsional arix-fracur ransfr in P syss wih VBS Tabl.: Block siz disribuion for h discr disribuions. Block Siz h, r Block Siz h, r N i fh i Firs cas Scond cas N i fh i 4 / 4 0. / 0.4 / Th arihic quivaln lngh of h arix blocks and rsrvoir lngh for cas on and wo ar calculad basd on h following quaions rspcivly: N h h fi hi hi, Lc.5, LR 4 9, Lc ax i.5 N h h fi hi hi.5, Lc.5, LR 4.5.5, Lc ax i Figur.4 donsras h coparison of h prsnd si-analyical odl Basd on Equaion.B.5 wih h nurical rsuls for hs wo discr disribuions. As illusrad in his figur h prsnd si-analyical odl can prdic h bhaviour of h coprssibl fluid flow for diffrn block siz disribuions wih an accpabl accuracy..54 7

83 Chapr. On dinsional arix-fracur ransfr in P syss wih VBS Cuulaiv Fluid Exchang S Cas Cas Ti Sc. Figur.4: Coparison of h prsnd si-analyical odl wih h nurical rsuls for h firs and scond cas. Coninuous lins ar prdicions by analyical odl. os and dashs ar fro nurical siulaions..7 Rsuls In h following wo scions w focus on h dinsionlss ra, dinsionlss cuulaiv producion and dinsionlss shap facor for diffrn arix block siz disribuions in h cas of singl-phas coprssibl fluid flow in fracurd porous dia. Th dinsionlss shap facor is calculad basd on Equaions.4 and.44. In hs scions a bas cas wih h gas spcific graviy of 0.7, praur of 9. C and prssur draw-down of 45 o.5 MPa for diffrn arix block siz disribuion is considrd. Th valus usd for diffrn parars ar rpord lswhr Ranjbar and Hassanzadh, 0 xcp for η ha has changd fro 0.7 o 0.7 du o considraion of arly i cuulaiv producion. Th dinsionlss ra and dinsionlss cuulaiv producion for diffrn arix block disribuions vrsus 7

84 Chapr. On dinsional arix-fracur ransfr in P syss wih VBS dinsionlss i for coprssibl fluid flow can b calculad using h following quaions: q 4 8 L L, L 4.964xp 4.76xp, 4 L.55 Q 4 4 L, L xp 0.48xp, 4 quaions. I is worh noing ha L cax is usd as h arix block characrisic lngh o scal h i, h producion ra and h cuulaiv producion, which is indpndn of h arix block siz disribuion. Thrfor, i is possibl o copar h dinsionlss ra and h cuulaiv producion for diffrn arix block siz disribuions as prsnd in h following subscions..7. Rcangular, discr, noral and log-noral disribuions Figur.5 copars h dinsionlss ra vrsus dinsionlss i on h log-log plo for h discr fracur, rcangular, noral and log-noral disribuions and an idal block siz disribuion wih h quivaln lngh of 0.489, 0.550, and 0.44 and.000, rspcivly. According o his figur a h arly i of h producion h salls quivaln lngh disribuion log-noral producs wih h largs dinsionlss producion ra and h rsrvoir is dpld or quickly han h ohr disribuions. I should b poind ou ha a h arly i of h producion h dinsionlss ra is invrsly proporional o h squar roo of h dinsionlss i, which is characrisic of arly i prssur diffusion. According o his figur basd on h quivaln lnghs of 74 L.56 In hs wo quaions κ is h raio of L R o L cax for h disribuions. In h rsuls and discussion scion w hav usd h valu of 6.6 for κ, which is h cas for h discr fracur odl Tabl.. rivaion of Equaions.55 and.56 is discussd in Appndix.B in or dails. In h cas of a slighly coprssibl fluid h producion ra and cuulaiv producion can b calculad by puing in hs wo

85 Chapr. On dinsional arix-fracur ransfr in P syss wih VBS h disribuions h arly i arix-fracur fluid ransfr is lows for h highs quivaln lngh. Rsuls show ha h arly i fluid ransfr incrass as h quivaln lngh dcrass and h salls arly i producion ra blongs o h idal disribuion larg quivaln lngh L =. Th arix blocks ar dpld or quickly for disribuion wih sallr quivaln lnghs q Idal Rcangular iscr Noral Log-Noral 00 insionlss Ra insionlss Ti Figur.5: insionlss ra vrsus dinsionlss i for idal, rcangular, discr, noral and log-noral disribuions whn F h = Linar disribuion Two yps of linar disribuions including incrasing linar disribuion and dcrasing linar disribuion ar dfind basd on h diffrn valus of and b as was discussd in Scion..4. Th obaind valus for h quivaln lngh for linarly incrasing and linarly dcrasing ar 0.65 and 0.475, rspcivly. I should b nod ha h quivaln 75

86 Chapr. On dinsional arix-fracur ransfr in P syss wih VBS lngh for rcangular disribuion is h avrag of h quivaln lngh for linarly incrasing and linarly dcrasing disribuions. Figur.6 copars h dinsionlss ras for diffrn valus of h dinsionlss i for h linar probabiliy dnsiy funcion. Figur.6 shows ha a h spcifid valus of h arly dinsionlss i h linarly incrasing disribuion largr block sizs producs wih a sallr producion ra han h linarly dcrasing sallr block sizs disribuion. Howvr, as i incrass h producion ra for linarly incrasing bcos highr han h linarly dcrasing disribuion and h dplion i for h linarly incrasing disribuion is largr han ha for h linarly dcrasing disribuion. In gnral, h linar disribuion shows a largr arly producion ra han h idal arix block siz disribuion. In addiion, arix blocks wih linar disribuion ar dpld or quickly han h arix blocks wih idal disribuion Idal isribuion Linar Incrasing Linar crasing 00 insionlss Ra insionlss i Figur.6: Coparison of dinsionlss ra for linarly incrasing =00/8, b=5/8, linarly dcrasing =-00/8, b=45/8 disribuion wih F h =0. and idal arix block siz disribuions. 76

87 Chapr. On dinsional arix-fracur ransfr in P syss wih VBS.7. Exponnial disribuion As was niond in h prvious scions four diffrn coon valus for xponn, a, in h liraur wr rpord: 0, 5, -5 and -0. Th obaind valus for quivaln lngh for hs valus of a, ar 0.50, 0.90, 0.80 and 0.950, rspcivly. Figur.7 copars h dinsionlss ra vrsus dinsionlss i for h idal block siz disribuion and h xponnial disribuion wih diffrn valus of a. Ths rsuls show ha whn h probabiliy of sall blocks is highr a>0, h arly i producion ra is largr han ha of h xponnial disribuion wih largr arix blocks sizs a<0. As h probabiliy of larg block sizs incrass i.., a=-0 h producion ra dcrass and nds oward ha of h idal block siz disribuion. In gnral, w can sa ha as h probabiliy of largr arix block sizs incrass h arly i producion ra dcrass and i approachs ha of h idal block siz disribuion. As a consqunc w can conclud ha for any disribuions as h quivaln lngh dcrass i.. h proporion of sall blocks incrass, h arly-i producion ra incrass and h blocks ar dpld or quickly insionlss Ra Idal isribuion Exponnial isribuiona=5 Exponnial isribuiona=-5 Exponnial isribuiona=0 Exponnial isribuiona= insionlss Ti Figur.7: Coparison of dinsionlss ra for xponnial xponn valus of -0, -5, 5 and 0 wih F h =0. and idal block siz disribuions. 77

88 Chapr. On dinsional arix-fracur ransfr in P syss wih VBS.8 iscussion In gnral, i can b concludd ha during arly i of producion h dinsionlss ra for diffrn arix block siz disribuions is invrsly proporional o h squar roo of h dinsionlss i. All of h disribuions donsra a largr producion ra in coparison wih h idal block siz disribuion a h arly i. As h probabiliy of largr arix block sizs incrass hr is a dcras in h dinsionlss producion ra and h blocks ar dpld or gradually for largr arix block sizs. I should b nod ha h prsnd odl is usful for diffusion in a collcion of slab on dinsional shapd arix blocks and hr igh b diffrncs in h prsnd soluion whn w ar daling wih wo dinsional cylindrical or hr dinsional sphrical fluid flow in h cas of variabl block siz disribuion. Figur.8 copars h dinsionlss cuulaiv producion vrsus dinsionlss i for diffrn arix block siz disribuions basd on Equaion.56. Th rsuls show ha h arly i cuulaiv producion and h i of sabilizaion of cuulaiv producion is a funcion of arix block siz disribuion. According o Figur.8, as h probabiliy of h sallr blocks incrass h blocks ar dpld or quickly. Ths rsuls show ha h spcifid discr arix block siz disribuion has a cuulaiv producion vry clos o ha of h linarly dcrasing arix block siz disribuion. For h daa usd in his sudy, h rcangular and Gaussian disribuions hav h sa cuulaiv producion. I should b poind ou ha h changs in cuulaiv producion ar rlad o h ara undr h dinsionlss ra curv Figurs

89 Chapr. On dinsional arix-fracur ransfr in P syss wih VBS 50 insionlss Cuulaiv Producion Idal Exponnial a=-0 Linar Incrasing =00/8, b=5/8 Rcangular Noral M=0.55 iscr Linar crsing =-00/8, b=45/8 Exponnial a=0 Log-Noral =0.5, M= insionlss Ti Figur.8: insionlss cuulaiv producion vrsus dinsionlss i for diffrn arix block siz disribuions wih F h =0.. Blocks showing h dinsionlss quivaln siz of h arix block L c /L cax. Th variaions in ransin dinsionlss cuulaiv producion aong various block siz disribuions can b xplaind by h valus of h dinsionlss quivaln lngh. Tabl. donsras h valus of h dinsionlss quivaln lngh for diffrn arix block siz disribuions. According o his abl h dinsionlss quivaln lngh for h idal block siz disribuion has h axiu valu and his ans ha for his disribuion h ransin i is largr han ha for h ohr disribuions. For xapl, in h cas of xponnial disribuion wih a larg posiiv xponn sall arix blocks or highly fracurd foraions h dinsionlss quivaln lngh has h lows valu and h cuulaiv producion rachs is plaau or quickly. Th i for h cuulaiv producion o rach is plaau ss proporional o h quivaln lngh. 79

90 Chapr. On dinsional arix-fracur ransfr in P syss wih VBS Figur.9 shows h dinsionlss shap facor vrsus dinsionlss i for diffrn arix block siz disribuion for flow of coprssibl fluids in fracurd dia. Th rsuls show ha h sabilizd valu of h shap facor is no a funcion of arix block siz disribuion and diffrn arix block siz disribuions hav h sa valus of 8.05 for h sabilizd valu of h shap facor. I should b niond ha in our prvious sudy Ranjbar and Hassanzadh, 0 h rpord valu for h sabilizd shap facor was 8.59 wihou considring of h arly i cuulaiv producion o ach wih h nurical rsuls. In his sudy w hav iprovd our prvious sudy and considrd h arly i cuulaiv producion and his caus a chang in h aching parar fro 0.7 o 0.7 and also h sabilizd valu of h shap facor fro 8.59 o Ths rsuls show ha h ransin valus and h i of sabilizaion of h shap facors ar a srong funcion of h arix block siz disribuion. An xponnial arix block siz disribuion wih a larg posiiv xponn sall arix blocks or highly fracurd foraions has h salls valus for h dinsionlss ransin shap facor and dinsionlss i of sabilizaion. Th dinsionlss shap facor for h idal block siz disribuion is largr han ha of ohr disribuions and sabilizs a a largr dinsionlss i. An xponnial disribuion wih a larg ngaiv xponn sparsly fracurd foraion has a largr ransin dinsionlss shap facor and dinsionlss i of sabilizaion han h linar, noral, log-noral, rcangular and discr disribuions. Th ransin valus of h dinsionlss shap facor and sabilizd dinsionlss i for linar, noral, log-noral, rcangular and discr disribuions ar clos o ach ohr. Basd on Figur.9 h bhaviour of h dinsionlss shap facor for noral and rcangular disribuions is siilar o ach ohr. 80

91 Chapr. On dinsional arix-fracur ransfr in P syss wih VBS 000 insionlss Shap Facor 00 0 Idal isribuion Rcangular isribuion Linar Incrasing isribuion Linar dcrasing disribuion Exponnial isribuion a=0 Exponnial isribuion a=-0 iscr isribuion Noral isribuion Exponnial isribuion a=5 Exponnial isribuion a=-5 Log-Noral isribuion =0.5, M= insionlss Ti Figur.9: insionlss shap facor for diffrn arix block siz disribuions. Blocks showing h dinsionlss quivaln siz of h arix block L c /L cax. Tabl.: Valus of dinsionlss quivaln lngh for diffrn arix block siz disribuions. Block Siz isribuion insionlss Equivaln Lngh L Idal disribuion.000 Exponnial disribuion a= Linar incrasing disribuion 0.65 Rcangular disribuion Noral disribuion iscr disribuion Linar dcrasing disribuion Log-noral disribuion 0.44 Exponnial disribuion a=

92 Chapr. On dinsional arix-fracur ransfr in P syss wih VBS.9 Conclusions Th following ajor conclusions ar ad as a rsul of his sudy: Th rsuls of his sudy show ha variabl block siz disribuion has a significan ffc on h arix producion profil during ransin sa. Th rsuls show ha as h probabiliy of largr blocks incrass or h fracur dnsiy dcrass h producion profil bcos closr o ha for h idal block siz disribuion Warrn and Roo s odl. Aong diffrn disribuions h idal block siz disribuion and h xponnial disribuion wih vry sall xponn sparsly fracurd sys hav h salls ransin cuulaiv producion and hir cuulaiv producion rachs is plaau or gradually. Th larg xponn xponnial disribuion innsivly fracurd sys has h highs ransin arix-fracur fluid ransfr and dpl or quickly han ohr disribuions. Th ransin valus of h dinsionlss shap facor and h dinsionlss i of sabilizaion ar a funcion of arix block siz disribuion. 8

93 Chapr. On dinsional arix-fracur ransfr in P syss wih VBS Nonclaur a insionlss xponnial disribuion consan A Cross-scional ara [L ] A Firs cofficin of h rial soluion b Inrcp for linar arix block siz disribuion B Scond cofficin of h rial soluion C Third cofficin of h rial soluion c Marix coprssibiliy [LT /M] Fourh cofficin of h rial soluion f i L ci Fracion of h block volu of siz L ci fl c Probabiliy dnsiy funcion f L insionlss probabiliy dnsiy funcion F h Raio of iniu block siz o h axiu block siz characrisic lngh h =L c Marix block lngh [L] k Marix prabiliy [L ] L c L L L R M N N Marix block characrisic lngh [L] insionlss lngh insionlss quivaln lngh Rsrvoir Lngh Man of h disribuion Slop of linar arix block siz disribuion Nubr of arix block sizs Toal nubr of arix blocks p Marix block prssur [M/LT ] p Avrag dinsionlss arix block prssur p f Fracur prssur [M/LT ] p i Iniial prssur [M/LT ] q insionlss arix-fracur fluid ransfr q sc Marix-fracur fluid ransfr [L /T] Q insionlss cuulaiv fluid xchang Q Cuulaiv fluid xchang [L ] R Rsidual in h hod of ons Ti [T] T insionlss i Rsrvoir praur [K] V b Marix block volu [L ] x insionlss disanc 8

94 Chapr. On dinsional arix-fracur ransfr in P syss wih VBS Grk Lrs α Exponnial disribuion consan [/L] Corrcion facor Unaffcd hicknss of arix block during h arly i Marix hydraulic diffusiviy [L /T] Avrag hydraulic diffusiviy [L /T] insionlss hydraulic diffusiviy insionlss fracur hydraulic diffusiviy κ Raio of L R o L cax insionlss xponn of soluion of gas diffusiviy quaion using h on hod Fluid viscosiy [M/LT] Shap facor [/L ] Varianc of h disribuion Porosiy Marix psudo-prssur [M/LT ] Avrag arix block psudo prssur [M/LT ] f Fracur psudo prssur [M/LT ] i Iniial psudo prssur [M/LT ] insionlss psudo prssur Avrag dinsionlss psudo prssur Subscrips f g i ln in ax R sc insionlss Equivaln Fracur Gas Iniial condiion Log-noral Marix Miniu Maxiu Rsrvoir Sandard condiions 84

95 Chapr. On dinsional arix-fracur ransfr in P syss wih VBS Rfrncs Abdassah,. Ershaghi, I Tripl-porosiy syss for rprsning naurally fracurd rsrvoirs. Soc. P. Eng. J.,, -7. As, W.F Nonlinar parial diffrnial quaions in nginring. Nw York: Acadic Prss. Barnbla, G.E. Zhlov, I.P. Kochina, I.N Basic concp on h hory of hoognous liquids in fissurd rocks. J. Appl. Mah. Mch., 0, Blani, A.K Esiaion of arix block siz disribuion in nauarally fracurd rsrvoirs. MSc Thsis, Sanford Univrsiy. Blani, A.K. Jalali-Yazdi, Y Esiaion of arix block siz disribuion in naurally fracurd rsrvoirs. Papr SPE 87. Bogdanov, I.I. Mourznko, V.V. Thovr, J.F. Adlr, P.M. 00. Prssur drawdown wll ss in fracurd porous dia. War Rsour. Rs., 9, 0. Chang, M.M Analyical soluion o singl and wo-phas flow probls of naurally fracurd rsrvoirs: horical shap facor and ransfr funcions. Ph dissraion, Univrsiy of Tulsa. Chn, Z.X Transin flow of slighly coprssibl fluids hrough doubl-porosiy, doubl-prabiliy syss-a sa-of-h-ar rviw. Transp. Porous Md., 4, Cinco-Ly, H. Saanigo, V. Kucuk, F Th prssur ransin bhavior for naurally fracurd rsrvoirs wih ulipl block siz. Papr SPE 468. Civan, F. Rasussn M.L. 00. Analyical hindrd-arix-fracur ransfr odls for naurally fracurd prolu rsrvoirs. Papr SPE Crank, J Th ahaics of diffusion. Oxford: Clarndon Prss. Fahi, E. Akkulu, I.Y Marix hrogniy ffcs on gas ranspor and adsorpion in coalbd and shal gas rsrvoirs. Transp. Porous Md., 80, Finlayson, B.A. 97. Th hod of wighd rsiduals and variaional principls. Nw York: Acadic Prss. Goodan, T.R Applicaion of ingral hods o ransin nonlinar ha ransfr. Advancs in Ha Transfr,, 5-, San igo, CA: Acadic. 85

96 Chapr. On dinsional arix-fracur ransfr in P syss wih VBS Gwo, J.P. O Brin, R. Jardin, P.M Mass ransfr in srucurd porous dia: bdding soscal srucur and icroscal hydrodynaics in a wo-rgion odl. J. Hydrol., 08, 04-. Hassanzadh, H. Pooladi-arvish, M Effcs of fracur boundary condiions on arix-fracur ransfr shap facor. Transp. Porous Md., 64, 5-7 Hassanzadh, H. Pooladi-arvish, M. Aabay, S Shap facor in h drawdown soluion for wll sing of dual-porosiy syss. Adv. War Rs.,, Jlr, T.A Th ffc of a disribud block lngh funcion on doubl porosiy ransiions during linar flow. J. P. Sci. Eng.,, Johns, R.T. Jalali-Yazdi, Y. 99. Coparison of ransin rspons in innsly and sparsly fracurd rsrvoirs. Soc. P. Eng. J., 64, Kazi, H. Mrrill, L.S. Porrfild, K.L. Zan, P.R Nurical siulaion of war-oil flow in naurally fracurd rsrvoirs. Soc. P. Eng. J., 66, 7-6. Kryszig, E Advancd nginring ahaics. Unid Sas: John Wily & Sons Prss. Lung, C.T.O. Ziran, R.W. 0. Esiaing h hydraulic conduciviy of wodinsional fracur nworks using nwork goric propris. Transp. Porous Md., 9, Li, K.T. Aziz, K Marix-fracur ransfr shap facors for dual-porosiy siulaors. J. P. Sci. Eng.,, Lu, M. Connl, L A dual-porosiy odl for gas rsrvoir flow incorporaing adsorpion bhavior-par I. Thorical dvlopn and asypoic analysis. Transp. Porous Md., 68,5-7. Lu, M. Connl, L.. 0. A saisical rprsnaion of h arix fracur ransfr funcion for porous dia. Transp. Porous Md., 86, Mourznko, V.V. Bogdanov, I.I. Thovr, J.-F. Adlr, P.M. 0. Thr-dinsional nurical siulaion of singl-phas ransin coprssibl flows and wll-ss in fracurd foraions. Mah..Copu. Siul., 8, Pnula, G. Civan, F. Hughs, R.G. Wiggins, M.L. 00. Ti-dpndn shap facors for inrporosiy flow in naurally fracurd gas-condnsa rsrvoirs. SPE Papr

97 Chapr. On dinsional arix-fracur ransfr in P syss wih VBS Pooladi-arvish, M. Torik, W.S. Farouq Ali, S.M Non-isohral graviy drainag undr conducion haing. Prolu Sociy of CIM and AOSTRA, Papr NO Ranjbar, E. Hassanzadh, H. 0. Marix-fracur ransfr shap facor for odling flow of a coprssibl fluid in dual-porosiy dia. Adv. War Rs., 45, Ranjbar, E. Hassanzadh, H. Chn, Z. 0. Effc of fracur prssur dplion rgis on h dual-porosiy shap facor for flow of coprssibl fluids in fracurd porous dia. Adv. War Rs., 4, Rangl-Gran, E. Kavsck, A.R. Akin. S. 00. Ti-dpndn shap facors for unifor and non-unifor prssur boundary condiions. Transp. Porous Md., 8, Ris, J.C Effc of fracur spacing disribuion on prssur ransin rspons in naurally fracurd rsrvoirs. J. P. Sci. Eng., 0, -47. Rodriguz, N.R. Cinco-Ly, H. Saanigo, V.F. 00. A variabl block siz odl for h characrizaion of naurally fracurd rsrvoirs. Papr SPE Sgall, P. 98. Th dvlopn of joins and fauls. Ph dissraion Sanford Univrsiy, Sanford, California. van Hl, A.P.G. van orp, J.J. Borrigr, P.M Havy oil rcovry by sa injcion in fracurd rsrvoirs. SPE Papr 46. Warrn, J.E. Roo, P.J. 96. Th bhavior of naurally fracurd rsrvoirs. Soc. P. Eng. J.,, Ziran, R.W. Bodvarsson, G.S Effciv block siz for ibibiions or absorpion in dual-porosiy dia. Gophys. Rs. L.,, Ziran, R.W. Chn, G. Hadgu, T. Bodvarsson, G.S. 99. A nurical dualporosiy odl wih si-analyical ran of fracur/arix flow. War Rsour. Rs., 97, 7-7. Ziran, R.W. Hadgu, T. Bodvarsson, G.S A nw lupd-parar odl for flow in unsaurad dual-porosiy dia. Adv. War Rs., 95,

98 Chapr. On dinsional arix-fracur ransfr in P syss wih VBS Appndix.A: Soluion of diffusion quaion for variabl block siz disribuion Ingral hods and h hod of ons ar usd o driv a closd for approxia soluion o nonlinar diffusion quaions. In his appndix a ha ingral hod arly i soluion and h hod of ons la i soluion ar xplaind for solving h diffusion quaion for coprssibl fluids in h fracurd porous dia in or dails. Th ingral hod is usd o driv h arly i soluion Equaion.4 of h nonlinar diffusion quaion as follows: x x.a. 0 0.A. x.a. x L A.4 x x Th following rial soluion is usd in h ha ingral hod: L x.a.5 L Using his rial soluion in h ingral for of Equaion.A. lads o h following ordinary diffrnial quaion: L d L d 0 0 L.A.6 Solving his ordinary diffrnial quaion lads o h following quaion for δ:.a.7 L 4 Th arly i soluion is valid unil 0 so L L 4 0.A.8 4 Thrfor, h final soluion for h arly i has h following for which is h sa as Equaion.4 in h x: L x L.A

99 Chapr. On dinsional arix-fracur ransfr in P syss wih VBS Th la i soluion Equaion.5 of h nonlinar parial diffrnial quaion PE using h hod of ons is xplaind in or dails in his par. x x L 4 x 0 0 x L x.a.0.a.a.a.b x.a.c L Th hod of ons is usd o obain h la i soluion of his PE by rconding a hird ordr rial soluion and h rsidual R as follows: x, R L A B x C x x,.a. 4 x x.a. Th unknown cofficins A, B, C and in Equaion.A. ar found by using h abov boundary condiions and aking h zro and firs ons of R vanish by nforcing h following condiions: L L 0 Rdx 0 0 x dx x 0.A.4 L L x x xrdx 0 x dx A.5 Fro h firs boundary condiions, w can conclud ha B = 0; h scond boundary condiion Equaion.A.c lads o: A.A.6 CL L Solving Equaions.A.4 and.a.5, cobining h rsuls wih Equaion.A.6 and siplifying, w obain h following sys of ordinary diffrnial quaions OEs: dc 48 C.A.7 d L L 89

100 Chapr. On dinsional arix-fracur ransfr in P syss wih VBS d d C.A.8 L L Th ignvalus of his sys of OEs ar obaind as follows Kryszig, 999:.486 and.8.a.9 L L Th corrsponding ignvcors ar hn obaind, and, finally, h following quaions for h unknown cofficins of C and ar drivd. C.4L xp.086l xp.a.0 xp xp.a. Whn L / 4 w hav x L /. Thrfor, and ar obaind using iniial condiion for h la i as follows: A. L L Thrfor, h rial soluion of h nonlinar PE for h la i bhaviour is obaind as follows: x,.79 xp L 0.54 xp L.5xp 6.75 xp L xp L 0.489xp x x. L 4.A. Appndix.B: rivaion of dinsionlss ra and dinsionlss cuulaiv producion for diffrn arix block siz disribuion In his appndix h drivaion of Equaions.55,.56 and cuulaiv fluid xchang as a funcion of i is xplaind in or dails. Th quaion for h arix-fracur ransfr funcion was drivd as follows Ranjbar and Hassanzadh, 0: q sc TscVb k f.b. 4 p T sc For h cas of variabl block siz disribuion Equaion.B. can b xprssd as follows: 90

101 Chapr. On dinsional arix-fracur ransfr in P syss wih VBS q sc TscVb kh f.b. 4 p Th sc In Equaion.B., T sc and p sc ar sandard praur and prssur rspcivly, V b is h bulk volu of a arix block, T is rsrvoir praur, k is arix prabiliy, σ is h shap facor, h is h quivaln arix block hicknss, is h avrag arix block psudo-prssur and f is h fracur psudo-prssur. For h cas of variabl block siz disribuion V b is dfind as follows: V A.B. b h Subsiuing Equaion.B. in Equaion.B. and using h dfiniion of h dinsionlss psudo-prssur lad o h following ransfr funcion for a rprsnaiv arix block in h cas of variabl block siz disribuion: q sc T sc Ak f i h 8 p TL sc c.b.4 In h cas of block siz disribuion h rsrvoir bulk volu conains h L R /L c nubr of hs blocks and w hav h following quaion for h producion ra in h bulk volu of h rsrvoir: Tsc Ak f i q scr h 8 psctlc L L R c Saring wih arcy s law w hav h following quaion for axiu fluid ransfr: q sc ax sc f i psct Lc ax.b.5 k AT.B.6 ividing boh sids of Equaion.B.5 by h axiu fluid ransfr Equaion.B.6 lads o h following quaion for h dinsionlss ra for h rsrvoir in h cas variabl block siz disribuion: q h L L h.b.7 R c ax 4 Lc 4 L I should b niond ha in Equaion.B.7, κ is h raio of L R o L cax for ach disribuions. By subsiuing h quaions for h dinsionlss shap facor Equaions.4 and.44 and dinsionlss avrag psudo-prssur Equaions.6 and 9

102 Chapr. On dinsional arix-fracur ransfr in P syss wih VBS.7 in h Equaion.B.7, for h nir rsrvoir h following quaion is drivd o calcula h dinsionlss arix-fracur fluid ransfr for diffrn block siz disribuions. q 4 8 L L, L 4.964xp 4.76xp, 4 Th cuulaiv fluid producion is dfind basd on h following quaion: L.B.8 Q sc 0 q sc d.b.9 Th dfiniion of h dinsionlss ra and dinsionlss i is usd in his quaion o find dinsionlss cuulaiv producion as follows: Q sc q L sc ax c ax q 0 d.b.0 Using h dfiniion of axiu cuulaiv producion Equaion.B.0 for any disribuion in Equaion.B.9 lads o h following quaion for h dinsionlss cuulaiv producion: Lc ax Qsc ax qsc ax.b. Q 0 q d.b. For h prsnd soluion h dinsionlss cuulaiv fluid producion for h nir rsrvoir is drind by using h following quaion: 9

103 Chapr. On dinsional arix-fracur ransfr in P syss wih VBS Q Q Q d, 8 L L 4 L 4 L 4.964xp 4.76xp, d L L Ingraing of Equaion.B. lads o h following quaion for dinsionlss cuulaiv producion as a funcion of dinsionlss i:.b. Q 4 4 L, L xp 0.48xp, 4 Subsiuing for dinsionlss variabls Equaions.5,.7,.B.6 and.b. and using h dfiniion of κ in Equaion.B.4 lads o h following quaion for h cuulaiv producion as a funcion of ral i. This quaion was usd o copar h rsuls of his sudy wih h nurical rsuls for vrificaion. L.B.4 T T Q sc sc Ak 8 p sc f i L T L Ak f i L p T sc c R R 4, xp L c, xp Lc Lc 4 Lc 4.B.5 9

104 Chapr Four: Si-analyical soluions for rlas of fluids fro rock arix blocks wih diffrn shaps, sizs and dplion rgis Absrac ual-porosiy odls hav bn xnsivly usd o siula h flow of fluids war or gas in aggrga soils and fracurd porous dia. Th fluid xchang bwn h rock arix blocks and h fracur nwork is vry iporan in dual-porosiy odls. In his sudy w prsn si-analyical soluions for rlas of a singl-phas liquid or gas fro cylindrical and sphrical arix blocks wih various block siz disribuions and diffrn prssur dplion rgis in h fracur. Th nonlinar prssur diffusiviy quaions for flow of gas and air ar solvd analyically using an approxia ingral hod. I is shown ha his soluion can b siplifid o odl flow of slighly coprssibl fluids lik war or oil in dual-porosiy dia. Th ffc of variabl block siz disribuion on h rlas ra for diffrn block goris is sudid. Pracically i is no fasibl o odl a larg scal fracurd rsrvoir basd on a fin grid approach du o h rquirn of larg copuaional i. Th prsnd sianalyical odl can b incorporad ino nurical odls for accura odling of h aoun of ransfrrd fluid bwn arix and fracurs using a dual-porosiy approach. I is shown ha h rsuls calculad by h dvlopd odl ach wll wih hos fro fin grid nurical siulaions. Furhror, h dvlopd odl can rcovr h availabl soluions in h liraur for slighly coprssibl fluids such as war or oil. I can b usd o calcula wo- or hr-dinsional flows in arix blocks boundd by wo or hr ss of fracurs, rspcivly. This chapr is an xac copy of: Ranjbar, E. Hassanzadh, H. Chn, Z. 0. Si-analyical soluions for rlas of fluids fro rock arix blocks wih diffrn shaps, sizs and dplion rgis, War Rsourcs Rsarch, Volu 49 4, Pag:

105 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks 4. Inroducion and prvious sudis Th fluid flow and ranspor in fracurd rocks is of gra significanc in any groundwar nvironns Novakowski and Lapcvic, 994. Flow and ranspor in fracurd porous dia ar ofn dscribd using a dual-porosiy odl. This odl assus ha h porous diu includs wo diffrn rgions, on rlad wih h acropor or fracur nwork wih high prabiliy and h ohr wih a lss prabl and or porous sys of soil aggrgas or rock arix blocks. ual-porosiy odls assu ha boh war flow and solu ranspor can b dscribd by wo quaions for arix and fracurs, which ar coupld using a r dscribing h xchang of fluid or solus bwn h wo por rgions Grk and van Gnuchn, 99. Th dual-porosiy approach has bn usd in nurical siulaion of groundwar, oil, and gas flow in fracurd porous dia. Howvr, for larg-scal siulaions h us of his approach is liid by h hug nubr of grid-blocks and sall i sps ha ar ofn ndd o accuraly siula h flow and ar xpnsiv in rs of copuaional i. Thrfor, dvloping analyical and si-analyical approachs ha can handl fracurd porous dia or srucurd soils wih lss copuaional i is iporan Ziran and Bodvarsson, 989. Sudy of gas flow in unsaurad soils and fracurd porous dia is significan in a variy of nginring filds. In agriculural nginring, airflow in h roo zon du o baroric prssur variaion is iporan o plan growh. Rcn sudis hav shown ha airflow can b applid o cra a "dry barrir" for was disposal faciliis Shan, 995. Coplx gaswar procsss in fracur-arix syss ar h ain procsss in a rang of nvironnal nginring syss, changing fro CO sorag in dp gological foraions Alvog and Clia, 004; Chn and Zhang, 00 ovr radioaciv was disposal in cavrns o vaporaion procsss in h unsaurad zon Nusk al., 00. As an xapl, soil vapor xracion is a widly usd chniqu o liina volail organic conainans fro h unsaurad zon Fala, 995. In his procss gas is inducd by vapor xracion for claning up vados zon conainaion of volail organic 95

106 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks chicals. Th gas flow is also iporan in h unsady flow of air in an anisoropic layr of snow Fan and Yn, 968. Anohr iporan aspc of gas flow in h fracurd dia is air injcion ss o drin h hydrologic propris and parars of h fracur nworks Huang al., 999. Illan and Nuan 00 dvlopd prssur and prssur drivaiv yp curvs for singl-phas air flow. Illan 005 analyzd h singl-hol pnuaic injcion and rcovry ss for drining prabiliy, porosiy, and skin. Shan 995 dvlopd analyical soluions for ransin, on-dinsional gas flow causd by baroric puping, and applid hs soluions o sia h air prabiliy of h vados zon. In gohral rsrvoirs war and vapor sa ranspor occurs in fracurd porous dia. Thr ar svral sudis rlvan o flow of fluids in gohral rsrvoirs, which is anohr iporan applicaion of coprssibl fluids vapor flow in h fracurd dia Pruss, 98; Fizgrald and Woods, 998. Schrauf and Evans 986 and Parkr al., 006 sudid h flow of gas in a singl naural fracur and porous dia, rspcivly. In hydrology hr xis a larg nubr of ahaical odls nurical, analyical or si-analyical o siula h flow of coprssibl fluids in undrground nvironns and srucurd soils You al., 0. McWhorr 990 prsnd an xac si-analyic soluion for ransin radial gas flow and applid i for siaing h gas prabiliy using puping s daa. Wang and ussaul 99 obaind analyical soluions for coprssibl fluids flowing hrough a saurad porolasic diu by solving a nw dnsiy diffusion quaion. Fala 995 dvlopd an analyical soluion for ransin and sady sa coprssibl gas flow o a pair of horizonal wlls in an unsaurad zon. Shan al., 999 prsnd analyical soluions for ransin, wo-dinsional gas flow in a vrical vados unsaurad zon scion, and prsnd chniqus for approxiaing h air prabiliy of a vrical laky faul. Shan 006 dvlopd an analyical soluion for ransin gas flow in a uliwll sys. In h cas of gas-war displacn, Thunvik and Brasr 990 analyzd h displacn of gas-war in fracur nworks wih diffrn prabiliy and diffrn inclinaion by focusing ainly on h gas brakhrough i. Brgr and Brasr 000 prsnd a ahaical odl for displacn of gas-war in h fracur nworks 96

107 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks wih a nonlinar sys of parial diffrnial quaions and solvd his sys nurically using an iraiv approach. Flow of coprssibl and slighly coprssibl fluids war or oil in fracurd rsrvoirs has bn sudid xnsivly wih applicaions in prdicion of producion ras and wll sing Warrn and Roo, 96; Kazi al., 976; Ziran al., 99, 996; Civan and Rasussn, 00; Pnula al., 00; Bogdano al., 00; Lu and Connl, 007; van Hl al., 008; Mora and Wanbargr, 009; Hassanzadh al., 009; Ranjbar and Hassanzadh, 00; Ranjbar al., 0; Mourznko al., 0; Ranjbar al., 0; Y and Ayala, 0. As an xapl, Hoi and Firoozabadi 005 dvlopd a discr fracur odl o siula h flow of coprssibl fluids in hoognous, hrognous and fracurd porous dia. Thr ar also a nubr of sudis ha hav considrd h ffc of block goris on dual-porosiy fluid flow forulaion Barkr, 985; van Gnuchn and alon, 986; Wuhicharn and Ziran, 0. As an xapl, Ziran al., 990 dvlopd a dual-porosiy odl basd on an approxia soluion for absorpion of war ino slabshapd, cylindrical and sphrical blocks. According o h aforniond works h flow of coprssibl fluids lik gass and air in porous dia is iporan in hydrological, nvironnal and prolu nginring. In our prvious sudis si-analyical odls for a slab-shapd on-dinsional arix block was dvlopd and h ffc of fracur prssur dplion rgis and arix block siz disribuion was invsigad Ranjbar and Hassanzadh, 0; Ranjbar al., 0; Ranjbar al., 0. Th ain objciv of his sudy is o dvlop a nw si-analyical odl for diffrn arix block goris cylindrical and sphrical for flow of coprssibl and slighly coprssibl fluids in fracurd porous dia. I is phasizd ha i is no pracical o odl a larg scal fracurd rsrvoir basd on a fin grid approach du o h rquirn of larg copuaional i. Th prsnd si-analyical odl can b incorporad ino nurical odls for accura odling of h aoun of ransfrrd fluids bwn arix and fracurs using availabl dual-porosiy forulaion. In ohr words, his sudy is iporan o rduc h copuaional i for larg scal siulaions 97

108 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks of gas flow in fracurd porous dia and can b nsd in a nurical odl o rsolv subgridblock scal flows. In h cas of coprssibl fluids h prsnd odl ay find applicaions for soil vapor xracion, gological CO squsraion, hydrological drinaion of fracur propris by air injcion, and flow of gas in dual-porosiy rsrvoirs. For slighly coprssibl fluids h prsnd odl is capabl o odl flow of war or oil in fracurd dia for diffrn block goris and diffrn block siz disribuions. In h prsnd sudy h odls for rlas of fluids fro cylindrical rprsnaiv of wo ss of fracurs or and sphrical rprsnaiv of hr ss of fracurs or blocks ar drivd. Th prsnd odls can b xprssd xplicily in rs of i and dos no rquir h nurical invrsion or infini sris calculaions. This soluion approach can significanly dcras h copuaional i of dual-porosiy odls wih an accpabl accuracy. 4. Approxia analyical soluion A ransfr funcion is uilizd o characriz h arix-fracur inracion and drin h ass ransfr bwn h arix blocks and h fracurs. Th ra of ass ransfrrd fro h arix o h fracurs is dircly rlad o h shap facor. For odling of naurally fracurd rsrvoirs, an xac valu of h shap facor is rquird o accoun for boh h ransin and psudo-sady sa bhaviour of h arix-fracur inracion and also h gory of h arix-fracur sys Ranjbar and Hassanzadh, 0. In his scion h dual-porosiy arix-fracur ransfr funcion for cylindrical and sphrical blocks is drivd. Afr ha an approxiaion is usd o driv h arixfracur fluid ransfr for flow cylindrical arix blocks approxiaion or slabshapd blocks surroundd by wo ss of fracurs and flow sphrical arix blocks approxiaion or slab-shapd blocks surroundd by hr ss of fracurs. A siilar approach has bn usd in h liraur Ziran al., 990; Li and Aziz, 995; Hassanzadh and Pooladi-arvish, 006. To driv h ransfr funcion for dual-porosiy cylindrical and sphrical blocks w us arcy s law as givn by: q sc k A dp. 4. B dr g 98

109 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks In his quaion, A is h surfac ara, r is h disanc fro h cnr of h cylindr or sphr and B g is h gas foraion volu facor h volu of fluid a undrground condiions dividd by h volu of fluid a sandard surfac condiions. Th following quaions ar usd o drin h ara for cylindrical and sphrical blocks, rspcivly: A Cylindr rh, 4. A Sphr 4 r. 4. Using h dfiniion of a coprssibl fluid foraion volu facor and h surfac ara of h cylindr w hav: R R r dr r kh T Tq p sc sc sc p f p p dp. 4.4 Z In Equaion 4.4, r is a i-dpndn radius whr h arix prssur is qual o is avrag prssur, h is h high of h cylindr and µ and Z ar h gas viscosiy and coprssibiliy facor, rspcivly. In gas rsrvoirs psudo-prssur ransforaion is usd o accoun for h variabiliy of prssur wih viscosiy and gas coprssibiliy facor. Th psudo-prssur ransforaion, which is siilar o h Kirchhoff ransforaion Tarakovsky al., 999, is givn by: p p dp. 4.5 z p b whr p b is a rfrnc or bas prssur. Ingraing of Equaion 4.4 and using h dfiniion of h ral gas psudo-prssur Equaion 4.5 lads o h following quaions for cylindrical and sphrical blocks, rspcivly: q q k T h f R ln R r sc sc Cylinrical, Tpsc sc 4.6 ktsc 4R R r f Sphrical. 4.7 Tp r sc Th shap facor for conducion of ha hrough a cylindrical wall or sphrical block is dfind as follows, rspcivly Holan, 00: 99

110 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks h S Cylindr, ln 4.8 S Sphr 4r r. 4.9 r r whr and ar h diars of h innr and h our cylindrs wih highr and lowr praurs, rspcivly, and r is h largr sphr radius wih lowr praur. Using h sa noion for h cas of prssur diffusion, h following quaions ar usd o dfin h prssur diffusion shap facor for cylindrical and sphrical blocks, rspcivly. This parar is on of h os iporan parars in dual-porosiy odling of fracurd rsrvoirs. Th aoun of fluid ha is ransfrrd fro arix o fracurs is dircly proporional o h shap facor: h Cylindr, R ln Vb R r 4 R R r 4.0 Sphr. 4. rvb In h shap facor quaion, V b is h bulk volu of a arix block. Using h dfiniion of h shap facor Equaions 4.0 and 4. in Equaions 4.6 and 4.7 lads o h following quaion for h ransfr funcion of a cylindrical and sphrical block in rs of h shap facor: q sc ktscv b f. 4. Tp sc Th quaion for fluid ransfr in rs of i drivaiv can b xprssd as follows Ranjbar and Hassanzadh, 0: q sc TscV p b sc c T. 4. Solving Equaions 4. and 4. for h shap facor rsuls in h following quaion: c. 4.4 k f 00

111 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks Soluion of h diffusiviy quaion is usd in Equaions 4. and 4.4 o driv h arix-fracur fluid ransfr and shap facor for coprssibl or slighly coprssibl fluids. Th diffusiviy quaion for flow of coprssibl fluid for diffrn goris can b xprssd as follows: I r r r I c r k, 4.5 whr for a cylindrical block I= and for a sphrical block I=. I should b niond ha Equaion 4.5 is a nonlinar parial diffrnial quaion PE. This nonlinariy is du o h prssur dpndnc of viscosiy and coprssibiliy of h coprssibl fluids lik air or gas. Th diffusiviy quaion for a cylindrical and sphrical block in rs of hydraulic diffusiviy, η, raio of arix prabiliy o h produc of gas viscosiy, arix coprssibiliy fluid and rock and arix porosiy can b xprssd as follows: I I r r. 4.6 r r A h iniial condiion a psudo-prssur in h arix block can b obaind fro h iniial prssur. A h cnr of h cylindrical or sphrical arix block w hav no flow boundary condiion and a h our boundary, h prssur is qual o h fracur prssur, which ay b a consan or varis wih i. Figur 4. is a schaic rprsnaion of h probl for a cylindrical block surroundd by fracurs. For a sphrical block h probl is h sa bu insad of a cylindr w ar daling wih a sphrical block. 0

112 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks 0 Figur 4.: Schaic rprsnaion of h probl for a cylindrical block. To solv his quaion h avrag hydraulic diffusiviy ovr h arix-fracur prssur drawdown is dfind as follows Ranjbar and Hassanzadh, 0: f i f i p p i f p p i f c dp p p k dp c k p p. 4.7 To xprss h diffusiviy quaion in dinsionlss for h following dinsionlss variabls ar dfind: i f i, 4.8 R r r, 4.9, 4.0 R. 4. Using hs dinsionlss variabls in Equaions 4.4 and 4.6 lads o h following quaions for h dinsionlss shap facor and diffusiviy quaion: f R, 4. R 0 r f Marix Fracur

113 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks I I r, r r. 4. r r In his quaion h hydraulic diffusiviy is a funcion of h dinsionlss radius and dinsionlss i. For solving his quaion w assu ha η is only a funcion of h dinsionlss i. To considr h ffc of h spac w uliply η by a corrcion facor β and nurical siulaion is usd o find his corrcion facor Ranjbar and Hassanzadh, 0. Thrfor, w hav h following dinsionlss quaion for h diffusiviy quaion of h cylindrical or sphrical block: r I I r r. 4.4 r In his quaion, I= is usd for h cylindrical block and I= is usd for h sphrical block. This quaion is solvd for diffrn fracur boundary condiions and diffrn block siz disribuions o driv h shap facor and arix-fracur fluid ransfr for flow of coprssibl and slighly coprssibl fluids in fracurd porous dia or aggrga and srucurd soils. I should b niond ha h prsnd soluion in his sudy calculas h prssur disribuion in h arix block by assuing fracurs as a boundary condiion. 4.. Consan fracur prssur In his cas h fracur prssur a h arix-fracur inrfac is a consan and w ar daling wih a singl block. For h consan fracur prssur w hav h following iniial and boundary condiions for h non-linar diffusiviy quaion Equaion 4.4: 0 0, 4.5 r 0 0, 4.6 r r. 4.7 f An ingral hod Goodan, 964; Ziran and Bodvarsson, 989; Pooladi-arvish al., 994 and h hod of ons As, 965 ar usd o find h arly and la i soluions of his quaion rspcivly. Afr solving h diffusiviy quaion and 0

114 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks ingraion ovr h bulk volu of h cylindrical arix block w obain h arly and la i avrag dinsionlss psudo-prssurs as follows: 48,, ,, whr ω and ω ar dfind as follows for a cylindrical block: Using h sa approach lads o h following arly and la i avrag dinsionlss psudo-prssurs for a sphrical block: 6 5 7, ,, ,, whr ω and ω for a sphrical block ar dfind as follows: In hs quaions η is h fracur dinsionlss hydraulic diffusiviy. rivaion of hs quaions is shown in Appndixs 4.A and 4.B in or dails. In Equaion 4., δ is h i-dpndn pnraion dph for a sphrical block and Equaion 4. is usd o rla h pnraion dph o h dinsionlss i. Equaions 4.8 and 4.9 for a cylindrical block and for a sphrical block and hir drivaivs wih rspc o i ar subsiud ino Equaions 4. and 4. o driv h arix-fracur ransfr ra rlas ra and shap facor for h cylindrical and sphrical arix blocks. To xprss h ra in h dinsionlss for w wri arcy s law ovr h whol doain of h cylindr o rach h following quaion: ktscvb qg f i. 4.5 Tp R sc 04

115 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks Using h dfiniion of h dinsionlss psudo-prssur in h ransfr funcion Equaion 4. lads o h following quaion: q sc TscV p b sc k f i T. 4.6 ividing Equaion 4.6 o Equaion 4.5 lads o h following quaion for h dinsionlss rlas ra for h cylindrical and sphrical blocks: q q q sc g R I I. 4.7 Ingraing of hs quaions lads o h dinsionlss cuulaiv fluid rlas fro a cylindrical or sphrical block wih a consan fracur prssur, rspcivly: Q qd. 4.8 I 0 Th arly and la i soluions of h diffusiviy quaion Equaions 4.8 and 4.9 and will b usd in Equaions o drin h dinsionlss ra and h cuulaiv fluid rlas for h cylindrical and sphrical blocks whn h fracur prssur is consan. I should b poind ou ha h prsnd odl can b usd o odl flow of slighly coprssibl fluids lik war or oil if w s. 4.. Variabl fracur prssur In his cas w assu ha h our boundary condiion changs wih i. In his scion w considr h ffc of linarly and xponnially dclining fracur prssur on h dualporosiy forulaion of coprssibl and slighly coprssibl fluids. For h variabl fracur prssur h our i-dpndn boundary condiion for h diffusiviy quaion Equaion 4.4 can b linar or xponnial as follows, rspcivly: i f, 4.9 i i f xp i I should b nod ha h iniial condiion and h innr boundary condiion ar h sa as Equaions 4.5 and 4.6, rspcivly. Using a rcnly dvlopd si-analyical 05

116 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks 06 hod o solv h quaion for a variabl fracur prssur Michl and Myrs, 00; Ranjbar al., 0 w rach h following quaions for h avrag dinsionlss psudo-prssur for h linarly dclining fracur prssur of a cylindrical block: 6, 6 k, 4.4 6, , 4.4 whr ω and ω ar drind basd on Equaion 4.0 in h cas of a cylindrical block. For a sphrical block wih linarly dclining fracur prssur h following soluions ar obaind for h arly and la i avrag dinsionlss psudo-prssur: or , 4.4, In h cas of an xponnially dclining fracur prssur w hav h following quaions for h avrag psudo-prssur of a cylindrical block:, 4 rf, 4.45, In h cas of a sphrical block and xponnially dclining fracur prssurs h following quaions ar obaind for h avrag dinsionlss psudo-prssur:

117 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks xp rf, 4.47, , 4.48 whr rf is h rror funcion and is h i a which h ffc of prssur disurbanc will rach h arix boundary. rivaion of hs quaions is shown in Appndics 4.A and 4.B in or dails. To drin h dinsionlss ra and h cuulaiv fluid rlas for variabl fracur prssur h sa approach as dscribd in Scion 4.. is usd. Th following quaion rprsns h dinsionlss rlas ra for h variabl fracur prssur of a cylindrical or sphrical block: f I I R q Ths quaions ar ingrad o drin h cuulaiv rlas fro a cylindrical or sphrical block wih i-dpndn fracur prssur. I Q Equaions 4.4 o 4.48 ar subsiud ino hs quaions Equaions 4.49 and 4.50 o drin h arly and la i dinsionlss rlas ras and h cuulaiv rlas of fluids for diffrn fracur prssur dplion rgis. 4.. Variabl block siz disribuion ulipl blocks In h cas of ulipl blocks w subsiu h collcion of blocks wih a singl block wih an quivaln radius Ziran and Bodvarsson, 995; Ranjbar al., 0. To odl flow of fluids in fracurd dia or aggrga soils wih ulipl cylindrical or sphrical blocks of variabl block siz disribuions h following iniial and boundary condiions can b wrin:

118 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks I r r r I r, 4.5 0, 4.5 i r 0 0, 4.5 r r R f I should b nod ha in Equaion 4.54, R is h quivaln cylindr or sphr radius and is a funcion of block siz disribuion. Th following quaions ar usd o find h quivaln radius for discr and coninuous block siz disribuion, rspcivly Gwo al., 998; Ranjbar al., 0: R N N R i i N N N i i N NiRi Ri fi Ri N i i N i Ni i N R i, 4.55 R ax R R f R dr R in In Equaion 4.56, fr is h probabiliy dnsiy funcion PF, which is usd o rprsn h probabiliy of h blocks as a funcion of block sizs. In h cas of ulipl blocks of variabl block siz disribuion in addiion o Equaions 4.8 and 4.0 h following dinsionlss variabls ar dfind: r r, 4.57 R ax, 4.58 R ax R R, 4.59 R ax f R R f R, 4.60 ax R in Fh. 4.6 R ax 08

119 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks For h ulipl blocks R ax, which is indpndn of h block siz disribuion, is usd o scal h i and radius. Using Equaions 4.8, 4.0 and in h diffusiviy quaion and h shap facor quaion Equaion 4.4 for h variabl block siz disribuion, w rach h following quaions for h diffusiviy quaion, h quivaln radius and h dinsionlss shap facor: r I I r r, 4.6 r 0 0, 4.6 r 0 0, 4.64 r r, 4.65 R N i i R f R R, 4.66 i i R R f R dr, 4.67 R F h R To solv h diffusiviy quaion of ulipl cylindrical blocks, h ingral hod and hod of ons ar usd o find h soluion Ranjbar al., 0. Th following quaions giv h arly and la i avrag dinsionlss psudo-prssur for ulipl cylindrical blocks wih block siz disribuion: 48,, R R R ,, whr ω and ω ar dfind as follows: R R rivaion of hs quaions is shown in Appndix 4.A in or dails. 09

120 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks Th following quaions giv h arly and la i avrag dinsionlss psudoprssur for ulipl sphrical blocks wih block siz disribuion using h ingral hod and hod of ons: 6R 5R 0R 4 6R 0R 70 R, 7R 70, R ,, whr ω and ω ar dfind as follows: R R rivaion of hs quaions is shown in Appndix 4.B in or dails. Tabl 4. shows h quivaln radius and h PF for diffrn coninuous block siz disribuions. For h cas of discr block siz disribuion Equaion 4.66 is usd o find h quivaln lngh. Mor dails abou hs disribuions and h corrsponding quivaln radius ar discussd in Ranjbar al., 0. 0

121 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks Tabl 4.: iffrn probabiliy disribuion funcion and hir quivaln radius. isribuion isribuion Funcion Equivaln Radius Exponnial a a F a R a R f h a af a af h a a af R h h Noral M R R f M F M h M rf F M rf M R h Linar b R R f h h F b F R Log-noral ln ln ] [ln ln M R R R f ] [ ] ln [ ln ln ln ln ln ln ln ln M rf F M rf R h M For h cas of variabl block siz disribuion h following quaion is usd o drin h rlas ra for h oal volu of h rsrvoir: R i f sc b sc scr R R T k p V T q In his quaion, R R is h oal radius of h rsrvoir. To drin h dinsionlss ra Equaion 4.75 is dividd by arcy s ra, which is xprssd as follows: ax ax i f sc sc b g R TR p V T k I q Th following quaions ar usd o drin h dinsionlss ra of rlas and h dinsionlss cuulaiv rlas for h variabl block siz disribuion for a cylindrical or sphrical block: I R I R q, 4.77

122 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks Q I I should b niond ha in hs wo quaions χ is h raio of h rsrvoir radius o h radius of h axiu block. In h rsul scion w assu ha his raio is 0. Th avrag psudo-prssurs Equaions and hir drivaivs ar rplacd in Equaions 4.77 and 4.78 o drin h dinsionlss ra of rlas and h dinsionlss cuulaiv rlas for cylindrical and sphrical blocks wih variabl block siz disribuion. Th quaions drivd for h cylindrical sphrical block can b usd for h wo-dinsional hr-dinsional flow in h slab-shapd block if h wo yps of blocks hav h sa volu. For xapl, if w s R / R /4 in h h hs quaions h soluion can b applid for wo-dinsional hr-dinsional blocks. 4. Modl vrificaion To drin h validiy of h prsnd odl w copar our rsuls wih h fin grid nurical siulaion. Cuulaiv rlas of fluids fro arix o fracurs is usd o valua h accuracy of h prsnd odl. In addiion, coparison of h prsnd odl wih odls availabl in h liraur for slighly coprssibl fluids will b usd o valida h dvlopd odl. To show h accuracy of h rsuls in h cas of and flow wo and hr ss of fracurs h obaind soluion for h avrag psudo-prssur is usd in h ransfr funcion quaion Equaion 4.. Changing all h variabls ino h dinsional for and ingraing his quaion ovr i h following quaion is obaind for h cuulaiv rlas of fluids for wo and hr ss of fracurs, rspcivly: k Q h 48, V btsc i f p scth h h h h h 0.67, 48

123 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks k Q 0.84h 6 5, T 0 scvb i f p sct h h, h h 70 I should b poind ou ha Equaions 4.79 and 4.80 ar obaind fro h cylindrical and sphrical blocks odls wih h assupion of having h sa volu wih blocks ford by wo and hr ss of fracurs, rspcivly. For xapl, in h cas of a hrdinsional block a arix block ford by hr ss of fracurs i is assud ha h sphr and h cub hav h sa volu Li and Aziz, 995. Figurs 4. and 4. copar h cuulaiv rlas of gas basd on h prsnd sianalyical odl Equaions 4.79 and 4.80 and h nurical rsuls Eclips 00 for wo- and hr-dinsional flow, rspcivly. Basd on hs figurs h prsnd sianalyical odl is in a good agrn wih h fin grid nurical siulaions. Tabl 4. shows h daa ha has bn usd in h fin grid nurical siulaions and h prsnd si-analyical odl. I should b niond ha h daa for h sianalyical odl and h aching parars β and η ar h sa as hos in our prvious sudis Ranjbar and Hassanzadh, 0; Ranjbar al., 0. This shows ha hs parars ar no arix block gory dpndn. 4.80

124 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks QS Si-Analyical Modl Nurical Siulaion Tiscond Figur 4.: Coparison of h prsnd odl wih h nurical siulaion for flow cylindrical block approxiaion QS Si-Analyical Modl Nurical Siulaion Tiscond Figur 4.: Coparison of h prsnd odl wih h nurical siulaion for flow sphrical block approxiaion. 4

125 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks Tabl 4.: aa usd for si-analyical and nurical odls. aa for fin-grid odl Grid insion: 8 grids for arix and 6 grids for fracur 8 Grid Spacing: Δx: 000,0,5,0.005,0.0,0.5,0.,0.5,0.,0.95,0.5,0.5,0.95,0.,0.5,0.,0.5,0.0,0.005,5,0,000 Δy: Th sa as Δx Δz = 4 for, for siulaion cas h sa grid spacing as Δx is usd in z-dircion. Fracur Porosiy = Fracur Prabiliy = 4,000 Coon daa for si-analyical and nurical odls Gas Spcific Graviy = 0.7 Marix Prabiliy = = Marix Porosiy = 0. Iniial Prssur = 45 MPa Fracur Prssur =.5 MPa Rsrvoir Tpraur = 66.45K h = 4 aa for si-analyical odl β =0.7 η =0.7 p sc =0.5 kpa T sc =88.7K η = Th prsnd si-analyical odl can rcovr h shap facors rpord in h liraur for slighly coprssibl fluids. Tabl 4. shows h sabilizd valus of h shap facor basd on his sudy and h liraur odls Li and Aziz, 995; Hassanzadh and Pooladi-arvish, 006 for diffrn boundary condiions and diffrn ss of fracurs and. As illusrad in his abl hr is an accpabl accuracy bwn h sabilizd valus of h shap facor basd on h prsnd si-analyical odl and h liraur odls. 5

126 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks Tabl 4.: Sabilizd valus of h shap facor basd on his sudy and liraur odls. Prssur dplion rgi Hassanzadh and Pooladi-arvish Li and Aziz This Sudy Flow cylindrical approxiaion Consan Fracur Prssur Linar dclin Exponnial clin sall xponn Exponnial clin larg xponn Flow sphrical approxiaion Consan Fracur Prssur Linar dclin 9-8. Exponnial clin sall xponn Exponnial clin larg xponn Rsuls In h following h ffc of fracur prssur dplion rgis, block siz disribuion and gory ar sudid, rspcivly Effc of fracur prssur dplion rgi Thr diffrn dplion rgis in h fracur ar considrd including consan fracur prssur, linar and xponnial dclin. Rsuls of his sudy show ha h fracur prssur dplion rgi will affc h ra of rlas. In fas dplion rgis lik consan fracur and larg xponn xponnial dclin h arix dpls or rapidly han ha in h cas of h linar dclin and h sall xponn xponnial dclin. Figur 4.4 copars h dinsionlss rlas ra vrsus h dinsionlss i for diffrn dplion rgis in h fracur for a cylindrical block. In h dplion rgi of xponnial dclin wih a sall xponn, h dinsionlss gas rlas ra is proporional o h squar roo of h i a h arly i. A h iddl i, h ra sabilizs o a consan valu. Th ra vnually drops o zro a h la i. Th gas rlas ra for linarly dclining fracur prssur also has h sa bhaviour as h xponnially dclining fracur prssur wih a sall xponn a h arly and iddl 6

127 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks is. For h linar dclin, sinc h dclin i is liid o </k, h prssur dclin in h fracur is no copl and h rlas ra dos no fall o zro. For h linar dclin h ra of dplion of h arix incrass as h dclin ra k incrass. As illusrad in Figur 4.4 for h xponnial dclin, as h valu of h xponn incrass, h arly i rlas ra incrass and h blocks ar dpld or quickly. Th xponnial dclin wih a larg xponn and consan fracur dplion rgis bhav in h sa way and h block is dpld fasr han ohr dplion rgis. For h fas dplion rgis consan fracur prssur and xponnial dclin wih a larg xponn h arly i dinsionlss ra varis invrsly proporionally o h squar roo of h i. A siilar obsrvaion has bn rpord for h shap facor in h cas of slighly coprssibl and coprssibl fluids Hassanzadh and Pooladi- arvish, 006; Ranjbar al., 0. I should b niond ha du o h approxia naur of h prsnd soluion hr is a disconinuiy in h flux spcially for h xponnial dclin wih a larg xponn. This disconinuiy occurs a whr h arly and la i soluions coincid. This disconinuiy is ainly du o h chang in h slop of h prssur a h arly and la i prssurs ar qual bu hr is a lil diffrnc in hir drivaivs bcaus of h approxia naur of h soluion. Siilar bhaviour is rpord by Ziran al., 990 using h ingral approxia soluions. 7

128 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks Consan fracur prssur Linar clink=0.000 Exponial clin k=0.000 Exponial clin k= Exponial clin k=000 Linar clink= q Figur 4.4: insionlss ra vrsus dinsionlss i for diffrn fracur dplion rgis for a cylindrical block. Figurs 4.5 and 4.6 copar h dinsionlss cuulaiv gas rlas for diffrn fracur prssur dplion rgis for cylindrical and sphrical blocks, rspcivly. Th rsuls show ha h i rquird for h cuulaiv gas rlas fro a arix block o rach is plaau dpnds on h dplion rgi in h fracur. Fas dplion rgis lik consan fracur prssur and xponnial dclin wih a larg xponn rach hir plaau or rapidly han hos wih h linar dclin and h xponnial dclin wih a sall xponn. On h ohr hand h xponnial dclin wih a sall xponn and linar dclin donsra a prolongd rlas priod as copard o h ohr dclins. 8

129 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks Q Consan fracur prssur Linar clink=0.000 Exponial clin k=0.000 Exponial clin k= Exponial clin k=000 Linar clink= Figur 4.5: insionlss cuulaiv rlas vrsus dinsionlss i for diffrn fracur dplion rgis for a cylindrical block Q Consan fracur prssur Linar clink=0.000 Exponial clin k=0.000 Exponial clin k= Exponial clin k=000 Linar clink= Figur 4.6: insionlss cuulaiv rlas vrsus dinsionlss i for diffrn fracur dplion rgis for a sphrical block. 9

130 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks 4.4. Block siz disribuion ffc In his scion h ffc of diffrn block siz disribuion on h cuulaiv gas rlas is invsigad. Tabl 4.4 illusras h valus of h quivaln radius for diffrn block siz disribuion. Equaions 4.66 and 4.67 and Tabl 4. ar usd o drin h valus of h quivaln radius for diffrn disribuions. Mor dails abou h disribuion and h diffrn valus in ach disribuion ar discussd lswhr Ranjbar al., 0. I should b niond ha h prsnd odl can also b usd for discr block siz disribuion as usd in our prvious sudy Ranjbar al., 0. Tabl 4.4: Valus of dinsionlss quivaln radius for diffrn arix block siz disribuions. Block Siz isribuion insionlss Equivaln Radius R Idal disribuion.000 Exponnial disribuion a= Linar incrasing disribuion 0.65 Noral disribuion Linar dcrasing disribuion Log-noral disribuion 0.44 Exponnial disribuion a= Figurs 4.7 and 4.8 copar h cuulaiv rlas for diffrn disribuions wih cylindrical and sphrical blocks, rspcivly. As donsrad in hs figurs h i o rach h ulia cuulaiv rlas is proporional o h quivaln radius of h disribuion. Basd on hs figurs h xponnial disribuion wih a larg xponn is dpld fasr han h ohr disribuions. Th idal disribuion and xponnial disribuion wih a sall xponn bhav siilarly and dpl or gradually han ohr disribuions. 0

131 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks Q Idal is. Exponnial is.a=-0 Linar Incrasing is. Noral is. Linar crasing is. Log-Noral is. Exponnial is.a= Figur 4.7: insionlss cuulaiv rlas vrsus dinsionlss i for diffrn block siz disribuion and cylindrical blocks. 0 8 Q 6 4 Idal is. Exponnial is.a=-0 Linar Incrasing is. Noral is. Linar crasing is. Log-Noral is. Exponnial is.a= Figur 4.8: insionlss cuulaiv rlas vrsus dinsionlss i for diffrn block siz disribuion and sphrical blocks

132 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks 4.4. Coparison of diffrn block goris In his scion w copar h cuulaiv rlas for a slab-shapd arix block wih h half hicknss of α, a cylindr and a sphr wih h radius of α. For h slab-shapd block h following quaion is drivd o calcula h cuulaiv rlas: Q. 4.8 In our prvious sudy Ranjbar and Hassanzadh, 0 h following quaions wr drivd for arly and la i avrag psudo-prssur in a slab-shapd arix block: 4,, , Equaions 4.8 and 4.8 ar subsiud ino Equaion 4.8 o drin h cuulaiv fluid rlas. For h cylindrical and h sphrical block Equaion 4.8 I= for cylindr and I= for sphr is drivd o drin h cuulaiv rlas. Coparison is prford for diffrn arix block goris wih h sa characrisic lngh availabl for rlas of fluid. Figur 4.9 copars h dinsionlss cuulaiv rlas for diffrn arix block goris whn h fracur prssur is consan. As illusrad in his figur h slab-shapd block has h axiu valu of h cuulaiv rlas and h blocks ar dpld or slowly han h cylindrical and sphrical blocks. Th sphrical block is dpld fasr wih h salls valu of h final cuulaiv rlas. I should b niond ha using h sa characrisic lngh availabl for rlas for diffrn goris rsuls in blocks wih diffrn volus o h surfac ara raio i.., V/A slab =V/A cylindr =V/A sphr. Thrfor, h diffrnc obsrvd in h ulia cuulaiv rlas shown in Figur 4.9 can b xplaind basd on h raio of h block volu o h block surfac ara for diffrn goris wih h sa characrisic lngh. I should b nod ha in Figur 4.9 h dinsionlss i is scald basd on characrisic lngh L c for slab and R for cylindr and sphr.

133 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks Thrfor, diffrn goris should hav h sa characrisic lngh, which rsuls in diffrn raios of V/A. Figur 4.9 shows ha h sabilizd valus of h dinsionlss cuulaiv rlas ar.00,.50 and.00 for h slab-shapd, h cylindrical and h sphrical blocks, rspcivly. Ths valus can b xplaind basd on h raio of h volu o h surfac ara of h blocks for diffrn goris. Th raio of h volu o h surfac ara for h slab-shapd block is α, for h cylindrical block i is α/ and for h sphrical block i is α/ Ziran al., 990. Thrfor, h ulia cuulaiv rlas is proporional o h raio of h volu o h surfac ara for diffrn goris Assuing h sa characrisic lngh, α, for diffrn goris. For xapl, h raio of V/A of h slab o h raio of V/A of h cylindr is wo and h ulia cuulaiv rlas for h slab is wo is grar han ha for h cylindr Slab-Shapd block Cylindrical block Sphrical block Q Figur 4.9: insionlss cuulaiv rlas vrsus dinsionlss i for diffrn block goris To noraliz h cuulaiv rlas for hr diffrn goris w scal h dinsionlss i by / V / A and xprss h noralizd cuulaiv rlas Q /Q vrsus h squar roo of h scald i τ Ziran al., 990 whr

134 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks Q is h ulia cuulaiv rlas for any gory. As an xapl, h i scal for a sphr of radius α is dfind as follows: / Using h sa approach w obain 4 and for h cylindrical and slab-shapd arix blocks, rspcivly. Figur 4.0 donsras h noralizd cuulaiv rlas vrsus h squar roo of h scald i. As a rsul of h scaling, h rlas curvs bco closr o ach ohr for diffrn goris as illusrad in Figur 4.0. A siilar obsrvaion has bn ad by Ziran al., 990 for absorpion curvs for diffrn goris. This scaling law ay find applicaions for irrgular shapd blocks, which is byond h scop of his sudy Slab-Shapd block Cylindrical block Sphrical block Q / Q SQRT[ Figur 4.0: Noralizd cuulaiv fluid rlas vrsus squar roo of scald i 4

135 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks 4.5 Conclusions An ingral approxiaion hod has bn usd o driv h soluions for nonlinar prssur diffusion in blocks wih diffrn goris including cylindrical and sphrical blocks. Th prsnd soluions hav considrd h ffc of fracur prssur dplion rgis and h variabl block siz disribuions or ulipl blocks. Th rsuls calculad by h approxia soluions ar in good agrn wih hos calculad by h fin grid nurical odls. I has bn shown ha h dplion i of a arix block is a funcion of h fracur prssur dplion rgis. In h cas of consan fracur prssur or xponnial dclin wih a larg xponn h block is dpld fasr han ha in h linar dclin and h xponnially dclin wih a sall xponn. For h linar dclin and xponnial dclin wih a sall xponn h arly i dinsionlss rlas ra incrass proporionally o h squar roo of h dinsionlss i, hn sabilizs a a consan ra and finally falls o zro. Block siz disribuion is anohr iporan parar ha affcs h arix producion during h ransin sa. Th blocks wih a sallr quivaln radius ar dpld or quickly. For a larg quivaln radius, disribuions lik idal and xponnial wih a sall xponn i.., a=-0 h ransin priod is longr han ha for h ohr disribuions. Finally, h noralizd cuulaiv rlas fro all arix block goris is xprssd as a funcion of h squar roo of h scald dinsionlss i, which ay find applicaion for irrgular shap arix blocks. 5

136 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks Nonclaur a insionlss xponnial disribuion consan A Cross-scional ara [L ] A Firs cofficin of h rial soluion b Inrcp for linar arix block siz disribuion B Scond cofficin of h rial soluion C Third cofficin of h rial soluion c Marix coprssibiliy [LT /M] Fourh cofficin of h rial soluion f i R i Fracion of h block volu of siz R i fr Probabiliy dnsiy funcion f R insionlss probabiliy dnsiy funcion F h Raio of iniu block siz o h axiu block radius h =L c Marix block hicknss [L] k Marix prabiliy [L ] M Man of h disribuion Slop of linar arix block siz disribuion N Nubr of arix block sizs N Toal nubr of arix blocks p prssur [M/LT ] q Marix-fracur fluid rlas [L /T] Q Cuulaiv fluid rlas [L ] Q R R R S T insionlss ulia cuulaiv rlas Marix block radius [L] Equivaln arix block radius [L] Rsidual in h hod of ons Ha conducion shap facor [L] Ti [T] Ti which h ffc of prssur rachs o h innr arix boundary Rsrvoir praur [K] V b Marix block volu [L ] r Grk Sybols insionlss radius α β δ Radius of cylindr, sphr or half hicknss of slab [L] Corrcion facor Pnraion dph 6

137 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks ε η η κ Un-pnrad dph for sphrical block Marix hydraulic diffusiviy [L /T] Avrag hydraulic diffusiviy [L /T] insionlss fracur hydraulic diffusiviy insionlss xponn and slop in fracur dplion rgis µ Fluid viscosiy [M/LT] σ iffusion shap facor [/L ] σ τ χ Varianc of h disribuion insionlss scal i Porosiy Raio of rsrvoir radius o h axiu block radius ψ psudo-prssur [M/LT ] ω insionlss xponn of soluion of gas diffusiviy quaion using h on Subscrips f g i ln in ax R sc insionlss Equivaln Fracur Gas Iniial condiion Log-noral Marix Miniu Maxiu Rsrvoir Sandard condiions 7

138 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks Rfrncs Alvog, A.S. Clia, M.A Nurical odling of carbon dioxid in unsaurad soils du o dp subsurfac lakag. War Rsour. Rs., 40, W0509. As, W.F. 965, Nonlinar parial diffrnial quaion in nginring. Nw York, Acadic Prss. Barkr, J.A Block-gory funcions characrizing ranspor in dnsly fissurd dia. J. Hydrol., 77, Brgr,. Brasr, C Gas-war displacn hrough fracur nworks, War Rsour. Rs., 6, Bogdanov, I.I. Mourznko, V.V. Thovr, J.-F. Adlr, P.M. 00. Prssur drawdown wll ss in fracurd porous dia. War Rsour. Rs., 9, 0. Chn, C. Zhang,. 00. Por-scal siulaion of dnsiy-drivn convcion in fracurd porous dia during gological CO squsraion, War Rsour. Rs., 46, W57, doi: 0.09/00WR Civan, F. Rasussn, M.L. 00. Analyical hindrd-arix-fracur ransfr odls for naurally fracurd prolu rsrvoirs, Papr SPE Fala, R.W Analyical soluions for gas flow du o gas injcion and xracion fro horizonal wlls, Ground War,, 5-46 Fan, S.S.T. Yn, Y.C Nonsady coprssibl flow hrough anisoropic porous dius wih paricular rfrnc o snow, War Rsour. Rs., 4, Fizgrald, S.. Woods, A.W Insabiliis during liquid igraion ino suprhad gohral rsrvoirs, War Rsour. Rs., 49, Grk, H.H. van Gnuchn, M.Th. 99. A dual-porosiy odl for siulaing h prfrnial ovn of war and solus in srucurd porous dia, War Rsour. Rs., 9, Goodan, T.R Applicaion of ingral hods o ransin nonlinar ha ransfr. Advancs in Ha Transfr,, 5-, San igo, CA: Acadic. Gwo, J.P. O Brin, R. Jardin, P.M Mass ransfr in srucurd porous dia: bdding soscal srucur and icroscal hydrodynaics in a wo-rgion odl, J. Hydrol., 08,

139 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks Hassanzadh, H. Pooladi-arvish, M Effc of fracur boundary condiions on arix-fracur ransfr shap facor, Transp. Porous Md., 64, 5-7. Hassanzadh, H. Pooladi-arvish, M. Aabay, S Shap facor in h drawdown soluion for wll sing of dual-porosiy syss, Adv. War Rs.,, Holan, J.P. 00. Ha Transfr. Mc Graw Hill Prss, Tnh Ediion. Hoi, H. Firoozabadi, A Mulicoponn fluid flow by disconinuous Galrkin and ixd hods in unfracurd and fracurd dia, War Rsour. Rs., 4, W4. Huang, K. Tsang, Y.W. Bodvarsson, G.S Siulanous invrsion of air-injcion ss in fracurd unsaurad uff a Yucca Mounain, War Rsour. Rs., 58, Illan, W.A Typ curv analyss of pnuaic singl-hol ss in unsaurad fracurd uff: irc vidnc for a porosiy scal ffc, War Rsour. Rs., 4, W0408. Illan, W.A. Nuan, S.P. 00. Typ curv inrpraion of a cross-hol pnuaic injcion s in unsaurad fracurd uff, War Rsour. Rs., 7, Kazi, H. Mrrill, L.S. Porrfild, K.L. Zan, P.R Nurical siulaion of war-oil flow in naurally fracurd rsrvoirs. Soc. P. Eng. J., 66, 7-6. Li, K.T. Aziz, K Marix-fracur ransfr shap facors for dual-porosiy siulaors. J. P. Sci. Eng.,, Lu, M. Connl, L A dual-porosiy odl for gas rsrvoir flow incorporaing adsorpion bhavior-par I. Thorical dvlopn and asypoic analysis. Transp. Porous Md., 68,5-7. McWhorr,.B Unsady radial flow of gas in h vados zon, J. Cona. Hydrol., 5, Michl, S.L. Myrs, T.G. 00. Iproving h accuracy of ha balanc ingral hods applid o hral probls wih i dpndn boundary condiions, In. J. Ha Mass Transfr, 5, Mora, C.A. Wanbargr, R.A Analysis and vrificaion of dual-porosiy and CBM shap facors, J. Can. P. Tchnol., 48, 7-. 9

140 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks Mourznko, V.V. Bogdanov, I.I. Thovr, J.F. Adlr, P.M. 0. Thr-dinsional nurical siulaion of singl-phas ransin coprssibl flows and wll-ss in fracurd foraions. Mah. Copu. Siul., 8, Novakowski, K.S. Lapcvic, P.A Fild asurn of radial solu ranspor in fracurd rock, War Rsour. Rs., 0, Nusk, P. Faigl, B. Hlig, R. Nissnr, J. Nuwilr, I. 00. Modling gas-war procsss in fracurs wih fracur flow propris obaind hrough upscaling, War Rsour. Rs., 46, W0958, doi: 0.09/009WR Parkr, L. Yarwood, R. Slkr, J Obsrvaions of gas flow in porous dia using a ligh ransission chniqu, War Rsour. Rs., 4, W0550, doi: 0.09/005WR Pnula, G. Civan, F. Hughs, R.G. Wiggins, M.L. 00. Ti-dpndn shap facors for inrporosiy flow in naurally fracurd gas-condnsa rsrvoirs. SPE Papr Pooladi-arvish, M. Torki, W.S. Farouq Ali, S.M Sa haing of fracurd foraions conaining havy oil: basic priss and a singl-block analyical odl, SPE Papr 864. Pruss, K. 98. Ha ransfr in fracurd gohral rsrvoirs wih boiling, War Rsour. Rs., 9, Ranjbar, E. Hassanzadh, H. 0. Marix-fracur ransfr shap facor for odling flow of a coprssibl fluid in dual-porosiy dia. Adv. War Rs., 45, Ranjbar, E. Hassanzadh, H. Chn, Z. 0. Effc of fracur prssur dplion rgis on h dual-porosiy shap facor for flow of coprssibl fluids in fracurd porous dia, Adv. War Rs., 4, Ranjbar, E. Hassanzadh, H. Chn, Z. 0. On-insional arix-fracur ransfr in dual porosiy syss wih variabl block siz disribuion. Transp. Porous Md., 95, 85-. Schrauf, T.W. Evans, Laboraory sudis of gas flow hrough a singl naural fracur. War Rsour. Rs., 7, Shan, C Analyical soluions for drining vrical air prabiliy in unsaurad soils. War Rsour. Rs., 9,

141 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks Shan, C An analyical soluions for ransin gas flow in a uliwll sys. War Rsour. Rs., 4, W040, doi: 0.09/005WR Shan, C. Javandl, I. and P. A. Whirspoon 999, Characrizaion of laky fauls: sudy of air flow in fauld vados zons, War Rsour. Rs., 57, Tarakovsky,.M. Nuan, S.P. Lu, Z Condiional sochasic avraging of sady sa unsaurad flow by ans of Kirchhoff Transforaion. War Rsour. Rs., 5, Thunvik, R. Brasr, C Gas igraion in discr fracur nworks, War Rsour. Rs., 60, van Gnuchn, M.Th. alon, F.N Modls for siulaing sal ovn in aggrgad fild soils, Godraa, 8, van Hl, A.P.G., van orp, J.J. Borrigr, P.M Havy oil rcovry by sa injcion in fracurd rsrvoirs, SPE Papr 46. Wang, Y. ussaul, M.B. 99. Th ffc of quadraic gradin rs on h borhol soluion in porolasic dia, War Rsour. Rs., 7, 5-. Warrn, J.E. Roo, P.J. 96. Th bhavior of naurally fracurd rsrvoirs. Soc. P. Eng. J.,, Wuhicharn, K. Ziran, R.W. 0. Shap facors for irrgularly arix blocks. SPE Papr Y, P. Ayala, L.F. 0. A dnsiy-diffusiviy approach for h unsady sa analysis of naural gas rsrvoirs. J. Na. Gas Sci. Eng, 7, -4. You, K. Zhan, H. Li, J. 0. Analysis of odls for inducd gas flow in h unsaurad zon. War Rsour. Rs., 47, W0455, doi: 0.09/00WR Ziran, R.W. Bodvarsson, G.S Ingral hod soluion for diffusion ino a sphrical block. J. Hydrol.,, -4. Ziran, R.W. Bodvarsson, G.S. Kwicklis, E.M Absorpion of war ino porous blocks of various shaps and sizs. War Rsour. Rs., 6, Ziran, R.W. Chn, G. Hadgu, T. Bodvarsson, G.S. 99. A nurical dualporosiy odl wih si-analyical ran of fracur/arix flow. War Rsour. Rs., 97, 7 7.

142 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks Ziran, R.W. Bodvarsson, G.S Effciv block siz for ibibiions or absorpion in dual-porosiy dia. Gophys. Rs. L.,, Ziran, R.W. Hadgu, T. Bodvarsson, G.S A nw lupd-parar odl for flow in unsaurad dual-porosiy dia. Adv. War Rs., 95, 7 7.

143 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks Appndix 4.A: Analyical soluion for cylindrical blocks In his Appndix h soluion of nonlinar gas prssur diffusion in a cylindrical arix block is discussd in or dails. 4.A: Consan fracur prssur In his cas as niond in h x h diffusiviy quaion and is condiions ar basd on Equaions 4.4 wih I=, 4.5, 4.6 and 4.7. Th ingral hod is usd o find h arly i soluion by dfining h i-dpndan pnraion dph, δ, in which h prssur disurbanc has rachd ha dph. For h arly i soluion w hav h following boundary condiions: r, 4.A. r 0, 0. 4.A. r Sinc h xac soluion for h cylindrical block is in h for of firs kind, h ordr zro of Bssl s funcion J 0 r w suggs h following polynoial rial soluion in h ingral hod Pooladi-arvish al., 994: A. 4.A. 4 B r C r Using acual and auxiliary boundary condiions in h rial soluion lads o h following quaion: [ r ]. 4.A.4 [ ] Using his rial soluion in h ingral for of h diffusiviy quaion Equaion 4.4, wih I= lads o h following soluion for arly i psudo-prssur: [ r ], A.5 I should b nod ha η is h dinsionlss hydraulic diffusiviy of h fracur a h our boundary, which is dfind as follows: x x 4.A.6 c f f

144 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks Ingraing ovr h bulk volu of h cylindrical block is usd o find h avrag dinsionlss psudo-prssur as follows: dv r dr 4.A.7 V 0 V Subsiuing Equaion 4.A.5 ino Equaion 4.A.7 lads o h following quaion for h arly i avrag dinsionlss psudo-prssur for h cylindrical block in h cas of consan fracur prssur: 48 r dr, A.8 Equaion 4.A.8 is h sa as Equaion 4.8 in h ain x. Th la i soluion of h nonlinar PE using h hod of ons is xplaind in or dails in his par. For h la i soluion h following quaion is usd as h iniial condiion, which cos fro h arly i soluion: 48 r 4. 4.A.9 Th innr and our boundary condiions and h diffusiviy quaion ar h sa as Equaions 4.6, 4.7 and 4.4 wih I=, rspcivly. Th hod of ons is usd o find h la i soluion of his PE by suggsing h following rial soluion and h rsidual R as follows: r R r,, 4 A B r C r r, 4.A.0 48 r r r. 4.A. Th unknown cofficins A, B, C and in Equaion 4.A.0 ar found by using h boundary condiions and aking h zro and firs ons of R vanish by nforcing h following condiions: 0 Rdr 0 r r dr 0 r r 0, 4.A. 4

145 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks r Rdr 0 r r r dr 0. 4.A. 0 0 r r Fro h innr boundary condiions, w can conclud ha B = 0; h our boundary condiion Equaion 4.7 lads o: A C. 4.A.4 Solving Equaions 4.A. and 4.A., cobining h rsuls wih Equaion 4.A.4 and so siplificaion lad o a sys of ordinary diffrnial quaions as follows: dc d d d 0 C 96, 4.A.5 C A.6 Solving h sys of h ordinary diffrnial quaions lads o h following quaions for unknown cofficins C and : C.75 xp.7 xp, 4.A.7 xp xp, 4.A.8 whr ω and ω ar h ignvalus of h sys of h ordinary diffrnial quaions and ar dfind basd on Equaion 4.0 in h ain x. So w hav h following quaion for h la i dinsionlss psudo-prssur: r,.7.75 r 0.7 r 4,.75 48, 4.A.9 whr and ar obaind by using h iniial condiion of Equaion 4.A.9, and h rial soluion of h nonlinar PE for h la i bhaviour is obaind as in Equaion 4.A.0. Afr ha, Equaion 4.A.0 is ingrad Equaion 4.A.7 ovr h arix block volu o obain h avrag arix block psudo-prssur Equaion 4.9 in h ain x. r, r r 4, A.0 5

146 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks 4.A: Variabl fracur prssur For h linarly dclining fracur prssur h our boundary condiion fracur prssur varis linarly wih h i basd on h following quaion: r R f i,. 4.A. In his cas h dinsionlss psudo-prssur and fracur dinsionlss psudo-prssur ar dfind as follows: i, 4.A. i, f. 4.A. For h linarly dclining fracur prssur h diffusiviy quaion and is iniial and innr boundary condiions ar h sa as Equaions 4.4 wih I=, 4.5 and 4.6, rspcivly. Equaion 4.A. is usd as h our boundary condiion. In his cas h shap facor quaion has h following for for a cylindrical block: R. 4.A.4 For h arly i soluion of h linar dclin of a cylindrical block w assu ha h rial soluion has h following for: r r,. 4.A.5 In Equaion 4.A.5 h rs in h brack is h soluion of h consan fracur prssur cas. In h cas of variabl fracur psudo-prssur h pnraion dph nuraor in Equaion 4.A.5 is found by solving h following ordinary diffrnial quaion Michl and Myrs, 00; Ranjbar al., 0: d d f n n n f. 4.A.6 6

147 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks In his quaion, n is h rial soluion xponn n=4 for h cylindrical block and θ is obaind basd on h following quaion: f n n f 4.A.7. n To find a si-analyical soluion for h pnraion dph w assu ha θ=0 in Equaion 4.A.6 as assud by Michl and Myrs 0. For solving Equaion 4.A.6 w us h following subsiuion: z. 4.A.8 Solving Equaion 4.A.6 for h linar dclin lads o h following quaion for h pnraion dph: A.9 I should b niond ha a or accura soluion can b obaind if h following quaion is usd in h rial soluion in h cas of h linarly dclining fracur prssur for h cylindrical arix block:. 4.A.0 6 In h dvlopd soluion, Equaion 4.A.0 is usd for h i dpndn pnraion dph. I should b nod ha his soluion is valid ill -δ =0. Subsiuing h pnraion dph quaion in h dinsionlss psudo-prssur and ingraing ovr h bulk volu of h arix block lads o h following quaions for h arly i dinsionlss psudo-prssur and h avrag dinsionlss psudo-prssur, rspcivly: [ r 6 6 ], 6, 4.A. 6, 6. 4.A. uhal s hor is usd o find h la i soluion of h diffusiviy quaion whn h fracur boundary condiion changs wih i as follows: 7

148 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks f r, d,. 4.A. 6 0 In his quaion, wihin h ingral is h soluion whn Equaion 4.A.9 and on h lf-hand sid is h soluion of h PE whn h arix-fracur boundary condiion changs wih i. Subsiuing Equaions 4.A.9 and 4.A. ino uhal s quaion and using h iniial condiion a w hav 6 4 r lads o h following la i soluion for h dinsionlss psudo- 6 prssur for h linarly dclining fracur prssur and h cylindrical arix block: r, r r r 4, r 4 6 f. 4.A.4 Ingraing his quaion ovr h bulk volu of h cylindrical arix block wih linarly dclining fracur prssur Equaion 4.A.7 lads o h following quaion: , 6. 4.A.5 For xponnially dclining fracur prssur w hav h sa PE wih h sa iniial and innr boundary condiions wih h following our boundary condiion: r R xp. 4.A.6 f i In his quaion, ψ is h fracur psudo-prssur whn i nds o h infiniy. In his cas h dinsionlss psudo-prssur and h dinsionlss fracur psudo-prssur ar dfind as follows: i, 4.A.7 i xp. 4.A.8 f In his cas h diffusiviy quaion and is iniial and boundary condiions ar h sa as Equaions 4.4 wih I=, 4.5 and 4.6, rspcivly, wih Equaion 4.A.8 as h 8

149 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks 9 our boundary condiion. Th dinsionlss shap facor quaion has h following for for a cylindrical block and xponnially dclining fracur prssur: xp R. 4.A.9 For h arly i soluion of h xponnial dclin w assu ha h rial soluion has h following for: xp, r r. 4.A.0 Using Equaions 4.A.6 and 4.A.8 and coparison wih h liraur odl Hassanzadh and Pooladi-arvish, 006 lads o h following quaion for pnraion dph in h cas of xponnially dclining fracur prssur: 4 rf. 4.A. I should b nod ha h ffc of prssur disurbanc will rach h innr boundary whn -δ =0 and for h xponnial dclin w canno obain an xplici quaion for and i is drind for any valus of k by aking Equaion 4.A. qual o zro. Thrfor, h arly i soluion for h xponnially dclining fracur prssur and cylindrical arix block can b xprssd as follows:, 4 4, rf rf r r. 4.A. Th iniial condiion for h la i soluion cos fro h arly i soluion Equaion 4.A. as follows: 4 xp r. 4.A.

150 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks 40 Th diffusiviy quaion and is boundary condiions ar h sa as bfor Equaions 4.4, wih I=, 4.6 and 4.7. Using uhal s hor Equaion 4.A. and h soluion of h consan fracur prssur Equaion 4.A.9 lads o h following la i soluion for h cas of h xponnially dclining fracur prssur: 4, , r r r C 4.A.4 Th iniial condiion is usd o find and as follows: xp 0, C. 4.A.5 Solving h sys of Equaions 4.A.5 lads o h following valus for and : A.6 Using hs valus in Equaion 4.A.4 and siplifying lad o h following la i psudo-prssur for h xponnially dclining fracur prssur:, , r r r r r r r r r. 4.A.7

151 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks 4 Ingraing of h psudo-prssur quaions ovr h bulk volu of h cylindrical arix block Equaion 4.A.7 lads o h following quaions for arly and la i avrag dinsionlss psudo-prssur whn h fracur prssur dclins xponnially wih i for h cylindrical arix block:, 4 rf. 4.A.8, A.9 4.A: Variabl block siz disribuions In his scion h soluion of PE Equaion 4.6, wih I= wih is iniial and boundary condiions Equaions 4.6 o 4.65 for variabl block siz disribuion is discussd in or dails. Th ingral hod is usd o driv h arly i approxia soluion of h diffusiviy quaion for flow of coprssibl and slighly coprssibl fluids in h cylindrical arix block for diffrn block siz disribuions. For h arly i soluion, in addiion o h diffusiviy quaion Equaion 4.6, wih I= and our boundary condiion Equaion 4.65, w hav h following condiion a h radius whr h prssur disurbanc has rachd: 0. 0, r R r 4.A. Siilar o h singl arix block w suggs a fourh-ordr polynoial rial soluion Equaion 4.A. o b usd in h ingral hod. Using h acual and auxiliary boundary condiions Equaions 4.65 and 4.A. in h rial soluion lads o h following quaions for A, B and C cofficins: 4 ] [ ] [ R R R A, 4.A.

152 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks [ R ] B, 4.A. [ R R ] C. 4.A.4 [ ] R R Subsiuing hs quaions for i-dpndn cofficins and so siplificaion lads o h following quaion: [ r [ R R R ] ]. 4.A.4 Using his rial soluion Equaion 4.A.4 in h ingral for of h diffusiviy quaion fro R -δ o R lads o h following ordinary diffrnial quaion OE: 6 d d 4 R R R R R. 4.A.5 Solving his OE lads o h following quaion for pnraion dph and arly i psudo-prssur: R R R [ r R R 48R 48 and R, 4.A.6 48 ], R A.7 Th arly i avrag dinsionlss psudo-prssur is obaind by ingraing ovr h bulk volu of h arix block: R R r R [ r R R 48 48R ] dr 48 R, R 48 4.A.8 For h la i soluion w hav h sa PE Equaion 4.6, wih I= wih h sa boundary condiions Equaions 4.64 and 4.65 and h following iniial condiion: R 48 R 4 r 4. 4.A.9 Th hod of ons is usd o find h la i soluion of his PE by suggsing a forh-ordr rial soluion Equaion 4.A.0 and h rsidual R as Equaion 4.A.. Th unknown cofficins A, B, C and in Equaion 4.A.0 ar found by using h 4

153 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks boundary condiions and aking h zro and firs ons of R vanish by nforcing h following condiions: R R 0 Rdr 0 0 r r r r dr 0, 4.A.0 R R r Rdx 0 r r r dr 0. 4.A. r r 0 0 Fro h innr boundary condiions, w can conclud ha B = 0; h our boundary condiion Equaion 4.65 lads o: A. 4.A. 4 CR R Solving Equaions 4.A.0 and 4.A., cobining h rsuls wih Equaion 4.A. and so siplificaion lad o a sys of OEs as follows: dc d d d 0 C 96, 4.A. R 84 C. 4 4.A.4 R R Solving h sys of ordinary diffrnial quaions lads o h following quaions for h unknown cofficins C and : C.75R xp.7r xp, 4.A.5 xp xp, 4.A.6 whr ω and ω ar h ignvalus of h sys of h ordinary diffrnial quaion and ar dfind basd on Equaion 4.7 in h ain x. Th iniial condiion Equaion 4.A.9 is usd o find and. Thrfor, and ar found by solving h following sys of quaions:.75r R R xp.7r xp 0, 4.A R R xp xp. 4.A R 4 Solving his sys of quaions for and and subsiuing in h i-dpndn cofficins of h rial soluion h la i bhaviour is obaind as follows: 4

154 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks r, 6.6 R.5 r R R.067 R r 4, R 48 4.A.9. Afr ha, Equaions 4.A.9 is ingrad ovr h arix block volu o obain h la i avrag arix block psudo-prssur as follows: R r dr R 48 R 0. 4.A.0 Appndix 4.B: Analyical soluion for sphrical blocks In his appndix h soluion of nonlinar gas prssur diffusion in a sphrical arix block is discussd in or dails. 4.B: Consan fracur prssur In his scion h soluion of PE Equaion 4.4, wih I= wih h iniial and boundary condiions Equaions 4.5 o 4.7 is prsnd. Th ingral hod Ziran and Bodvarsson, 989 is usd o find h arly i soluion of his quaion by dfining h i dpndan pnraion dph, δ, in which h prssur disurbanc has rachd. For h arly i soluion, in addiion o h our boundary condiion Equaion 4.7, w hav h following auxiliary boundary condiions: r 0, 0, 0. 4.B. r r Th following hird-ordr polynoial rial soluion is suggsd o b usd in h ingral hod: A. 4.B. B r C r r Using h acual and auxiliary boundary condiions in h rial soluion lads o h following quaion for dinsionlss psudo-prssur: r. 4.B. 44

155 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks Using his rial soluion in h ingral for of h diffusiviy quaion Equaion 4.4, wih I= lads o h following OE: d d 80 d [ ] 60 d[ 6 0] Solving his OE lads o h following quaion:. 4.B B.5 70 This soluion is valid ill ε=0 or Equaion 4.B. is ingrad ovr h bulk volu of h arix block o drin h avrag dinsionlss psudo-prssur as follows: r dr r r dr 6 0, B.6 Equaions 4.B.5 and 4.B.6 in rs of pnraion dph δ can b xprssd as follows: 6 5 7,, 4.B B.8 70 For h la i soluion w hav h following iniial condiion h diffusiviy quaion and boundary condiions ar h sa as Equaions 4.4, 4.6 and 4.7, rspcivly: 7 70 r. 4.B.9 Th hod of ons is usd o find h la i soluion of his PE by suggsing a hird ordr rial soluion as Equaion 4.B. and h rsidual R as follows: R r r r. 4.B.0 r 45

156 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks Th unknown cofficins A, B, C and in Equaion 4.B. ar found using h boundary condiions and aking h zro and firs ons of R vanish by nforcing h following condiions: 0 Rdr 0 r r dr r r 0 0, 4.B. r Rdr 0 r r r dr 0. 4.B. 0 0 r r Fro h firs boundary condiions, w can conclud ha B = 0; h scond boundary condiion Equaion 4.7 lads o: A C. 4.B. Solving Equaions 4.B. and 4.B., cobining h rsuls wih Equaion 4.B. and so siplificaion lads o a sys of OEs as follows: dc d d d 90 C 98, 4.B.4 84 C B.5 Solving h sys of ordinary diffrnial quaions lads o h following quaions for unknown cofficins C and : C.9884 xp xp, 4.B.6 xp xp, 4.B.7 whr ω and ω ar h ignvalus of h sys of OEs and ar dfind basd on Equaion 4.4 in h ain x. So w hav h following quaion for h la i dinsionlss psudo-prssur: r, r 4.B.8 Th iniial condiion of Equaion 4.B.9 is usd o find and. Thrfor, h rial soluion of h nonlinar PE for h la i bhaviour is obaind as follows: 46

157 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks r, r 4.B.9 Afr ha, Equaions 4.B.9 is ingrad Equaion 4.B.6 ovr h arix block volu o obain h avrag arix block psudo-prssur as follows: ,. 4.B B: Variabl fracur prssur In h cas of linar dclin for h sphrical block h our boundary condiion is givn in Equaion 4.A.. Th PE and iniial and innr boundary condiions ar h sa as Equaions 4.4 wih I=, 4.5 and 4.6, rspcivly. For h arly i soluion of h linar dclin w assu ha h rial soluion has h following for: r r. 4.B. In h cas of variabl fracur psudo-prssur h pnraion dph is found by solving h following OE Michl and Myrs, 00; Ranjbar al., 0: d f f n d n n. 4.B. Using h sa procdur as dscribd in Scion 4.A for h cylindrical block and coparing wih h liraur odl lads o h following quaion for arly i dinsionlss psudo-prssur: r,. 4.B. Ingraing ovr h bulk volu of h arix block lads o h following quaion for arly i avrag dinsionlss psudo-prssur: , or 4.B.4 60 uhal s hor Equaion 4.A. is usd o find h la i soluion of h diffusiviy quaion in fracurd dia whn h fracur boundary condiion changs wih 47

158 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks 48 i. Subsiuing Equaions 4.B.8 and 4.A. in uhal s quaion and using h iniial condiion a w hav r lads o h following la i soluion for dinsionlss psudo-prssur for h linarly dclining fracur prssur and sphrical arix block: r r. 4.B.5 Ingraion of his quaion ovr h bulk volu of h arix block Equaion 4.B.6 fro zro o on lads o h following quaion for h avrag dinsionlss psudoprssur of a sphrical block whn h fracur prssur dclins linarly wih i:, B.6 For xponnially dclining fracur prssur h sa approach is usd and h following quaions ar obaind for arly and la i dinsionlss psudo-prssur:, 8 8, rf rf r r. 4.B.7

159 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks 49, , r r r r r r r r r. 4.B.8 4.B: Variabl Block siz disribuions In h cas of ulipl sphrical blocks wih variabl block siz disribuion h diffusiviy quaion Equaion 4.6, wih I= wih h iniial and boundary condiions Equaions 4.6 o 4.65 should b solvd. Th ingral hod is usd o find h arly i soluion wih h following auxiliary quaion: 0. 0, 0, r r R r 4.B. Th hird-ordr rial soluion as Equaion 4.B. is suggsd o b usd in h ingral hod. Using h acual and auxiliary boundary condiions in h rial soluion lads o h following quaions for dinsionlss psudo-prssur: ] [ ] [ ] [ R r R r. 4.B. Using his rial soluion Equaion 4.B. in h ingral for of Equaion 4.6, wih I= Fro ε o R and so siplificaion lads o h following OE: R R R R R d d. 4.B. Solving his OE lads o h following quaion: R R R R. 4.B.4 This quaion in rs of pnraion dph can b xprssd as follows:

160 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks 4 6R 0R. 4.B.5 70 R Th arly i soluion is valid ill δ<r or 7R. 70 Th arly i avrag dinsionlss psudo-prssur is obaind by ingraing ovr h bulk volu of h arix block: R R r r [ R ] dr 6R 5R 0R, 7R 70, 4.B.6 whr Equaion 4.B.5 is usd o rla h pnraion dph o h dinsionlss i. For h la i soluion w hav h following iniial condiion and h innr and our boundary condiions ar h sa as Equaions 4.64 and 4.65: 7R 70 R r. 4.B.7 Th hod of ons is usd o find h la i soluion of his PE by suggsing a hird-ordr rial soluion Equaion 4.B. and h rsidual R as Equaion 4.B.0. Th unknown cofficins A, B, C and in h rial soluion of Equaion 4.B. ar found by using h boundary condiions and aking h zro and firs ons of R vanish by nforcing h following condiions: R R 0 Rdr 0 0 r r r dr r 0, 4.B.8 R R r Rdr 0 r r r dr 0. 4.B.9 r r 0 0 Fro h firs boundary condiions, w can conclud ha B = 0; h scond boundary condiion Equaion 4.65 lads o: A. 4.B.0 4 CR R Solving Equaions 4.B.8 and 4.B.9, cobining h rsuls wih Equaion 4.B.0 and so siplificaion lads o a sys of OEs as follows: dc C, 4.B. d R R 50

161 Chapr 4. Si-analyical soluions for rlas of fluids fro rock arix blocks d d C. 4.B. R R Solving h sys of OEs and using h iniial condiion for h la i soluion Equaion 4.B.7 lads o h following quaion for la i psudo-prssur for ulipl sphrical blocks: R R r, R 7R R r 4.B. whr ω and ω ar dfind basd on Equaion 4.74 in h ain x. Afr ha, Equaion 4.B. is ingrad ovr h arix block volu fro zro o on o obain h la i avrag arix block psudo-prssur as Equaion 4.7 in h ain x. 5

162 Chapr Fiv: Conclusions and Rcondaions 5. Conclusions This sudy has prsnd a validad si-analyical odl for fracurd gas rsrvoirs. Th dvlopd ahaical odl can b usd for odling of arix-fracur ransfr funcion for fracurd gas rsrvoirs using dual-porosiy approach. Afr drivaion of arix-fracur ransfr funcion and shap facor for dual-porosiy gas rsrvoirs h ffc of fracur prssur boundary condiion, block siz disribuion and block gory was invsigad. u o h analyical naur of h prsnd soluion i can b nsd in singl-phas dual-porosiy siulaion of fracurd dia which is fficin in rs of copuaional i. In h prsnd horical analysis h nonlinar prssur diffusion quaion for dualporosiy gas rsrvoirs was solvd analyically. Thn, h prsnd si-analyical odl was xndd o considr h ffc of fracur prssur boundary condiion, block siz disribuion and block gory. Th prsnd odl can also b usd for singl-phas flow of slighly coprssibl fluids in h fracurd dia. Th following ajor conclusions ar ad as a rsul of his sudy. 5.. Fracur prssur boundary condiion ffc for a slab-shapd block In h firs par of his sudy, a horical analysis of h shap facor for h flow of a coprssibl fluid in fracurd porous dia is prsnd. Cobinaion of h ha ingral hod, h hod of ons and uhal s hor ar usd o solv h nonlinar diffusion quaion rsuling fro h flow of a coprssibl gas in a dual-porosiy sys. Th approxia si-analyical soluion is validad wih fin-grid nurical siulaions. Siilar o h flow of a slighly coprssibl fluid, h shap facor donsras a ransin bhaviour and hn convrgs o a consan valu during h psudo-sady sa priod for a coprssibl fluid. Th approxia analyical soluion prsnd rvald ha h arix-fracur ransfr shap facor for singl-phas flow of a coprssibl fluid in h dual-porosiy dia is a funcion of h prssur boundary condiion in h fracur. Basd on h prssur boundary condiion in h fracur h sabilizd valu of h shap facor varis bwn wo liis. Th uppr lii is obaind for a linarly dclining fracur prssur, which

163 Chapr 5. Conclusions and rcondaions corrsponds o a slow prssur dplion rgi. Th lowr lii is drivd for h consan fracur prssur boundary condiions whr dplion aks plac fasr. Whn h fracur prssur dpls xponnially wih i, h sabilizd valu of h shap facor falls bwn hos valus of h consan fracur prssur and linarly dclining fracur prssur. This sabilizd valu is a funcion of h dclin xponn, κ. For sall dclin xponns h sabilizd shap facor has h sa valu as ha for h linarly dclining fracur prssur. For larg xponns, h sabilizd shap facor is qual o ha of a consan fracur prssur. Th psudo-sady sa i sabilizaion i of h shap facor, incrass as h fracur boundary condiion changs fro a fas dplion rgi oward a slow dplion rgi. 5.. Block siz disribuion ffc for slab-shapd blocks Variabl block siz disribuion has a significan ffc on h arix producion profil during ransin sa. Th rsuls show ha as h probabiliy of largr blocks incrass or h fracur dnsiy dcrass h producion profil bcos closr o ha for h idal block siz disribuion Warrn and Roo s odl. Aong diffrn disribuions h idal block siz disribuion and h xponnial disribuion wih vry sall xponn sparsly fracurd sys hav h salls ransin cuulaiv producion and hir cuulaiv producion rachs is plaau or gradually. Th larg xponn xponnial disribuion innsivly fracurd sys has h highs ransin arix-fracur fluid ransfr and dpl or quickly han ohr disribuions. Th ransin valus of h dinsionlss shap facor and h dinsionlss i of sabilizaion ar a funcion of arix block siz disribuion Block gory ffc An ingral approxiaion hod has bn usd o driv h soluions for nonlinar prssur diffusion in blocks wih diffrn goris including cylindrical and sphrical blocks. Th prsnd soluions hav considrd h ffc of fracur prssur boundary condiion and h variabl block siz disribuions or ulipl blocks. Th rsuls calculad by h approxia soluions ar in good agrn wih hos calculad by h fin grid nurical odls. 5

164 Chapr 5. Conclusions and rcondaions I has bn shown ha h dplion i of a cylindrical or sphrical arix block is a funcion of h fracur prssur boundary condiions. In h cas of consan fracur prssur or xponnial dclin wih a larg xponn h block is dpld fasr han ha in h linar dclin and h xponnially dclin wih a sall xponn. For h linar dclin and xponnial dclin wih a sall xponn h arly i dinsionlss rlas ra incrass proporionally o h squar roo of h dinsionlss i, hn sabilizs a a consan ra and finally falls o zro. Block siz disribuion is anohr iporan parar ha affcs h arix producion during h ransin sa for cylindrical and sphrical blocks. For h cylindrical or sphrical blocks h rsuls for diffrn block siz disribuion ar h sa as ha for slab-shapd blocks. 5. Rcondaions In his sudy w hav considrd a non-coupld approach for dual-porosiy odling of coprssibl fluids, xnsion of his odl using a coupld approach is rcondd for fuur works. Also, w dvlopd a singl-phas odl of gas flow in fracurd dia and incorporaing uli-phas flow including gas and war in fracurd dia is an iporan ara for fuur sudis. In h cas of uliphas flow graviy can hav a significan ffc and should b considrd in h odling of such cass. Adsorpion plays an iporan rol in unconvnional fracurd gas rsrvoirs such as CBM and shal gas rsrvoirs. Considring h adsorpion in dual-porosiy odling of coprssibl fluid is rcondd for fuur works. 54

165 Appndix A. Copyrigh prission APPENIX A: Copyrigh Prissions 55

166 Appndix A. Copyrigh prission 56