Determining Well Test Pore Compressibility from Tidal Analysis

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1 Deemining Well Tes Poe omessibiliy om Tidal Analysis Bob Foulse Deision Managemen Ld 9 Abbey See ene Abbas Dohese Dose DT2 7JQ Unied Kingdom Tel: E Mail: +44 (0) boulse@deisionman.o.uk 1

2 1 Inoduion In well es analysis he oal esevoi omessibiliy is an imoan inu aamee. This is he sum o wo omonens: he eeive luid omessibiliy and he ok oe omessibiliy. One way o esimae he ok oe omessibiliy is o look a he esevoi essue esonse o sea ide loading. The uose o his eo is o ovide a ok mehanis analysis o his. This esuls in a omlex exession o a aio o he oe omessibiliy o he eeive luid omessibiliy involving hee ok oeies o whih esimaes need o be inu. Using eain assumions a vey simle exession o his aio an be odued. Using eesenaive daa o sandsones he esuls o he omlex simliied exessions ae omaed. An ineaive websie ( has been wien o allow esimaes o oe omessibiliy o be made using he ull exessions and allowing o anges o be seiied o he ok daa. 2 Rok Mehanis o Sea Tides 2.1 INTRODUTION Based on a ommonly used basi model o esevoi ok, he ok is egaded as an elasi amewok onaining a newok o luid illed assages. The ok amewok will have maeial oeies ha deend on he way ailes inea o give he bulk maeial sengh o esis alied sesses. We will use he subsi b when disussing oeies o he amewok maeial. In addiion o is bulk behaviou he ok also has omessive oeies deending on he esonse o he minue ailes onsiuing he ok o vaiaions in he luid essue. The subsi is used o denoe he aveaged ok aiulae oeies. Poeies o he luid, illing he oe sae, will be denoed by he subsi. Suesis ae used o disinguish beween he vaious sesses: o he oal sess alied o he ok ame and o he luid sess (essue). Sesses will be osiive when omessive and sains osiive when eduing lengh o volume. 2

3 2.2 NOMENLATURE A Sea ide essue ansmission eiieny oeiien o bulk ok omessibiliy wih ese o hanges b in onining essue oeiien o oe omessibiliy wih ese o hanges in onining essue oeiien o oe omessibiliy wih ese o hanges in luid essue oeiien o oe omessibiliy o be used in esevoi simulaion oeiien o oe omessibiliy wih ese o hanges in onining essue oeiien o ok mineal omessibiliy wih ese o hanges in luid essue Fluid omessibiliy E Young s modulus Radial o-odinae e Inemenal eeive essue e Inemenal onining essue Inemenal luid essue σ, σ, σ omessive eeive sess inemens in he ok in he φφ inile dieions σ Toal omessive sess esuling om sea level vaiaion ε, ε, ε φφ Sain deemens in inile dieions ε Deemenal volumei hange in oe volume sain v Poisson s aio φ Rok oosiy ϑ olaiude ϕ Longiude α Eeive sess oeiien o bulk volumei sain (also known as Bio-Willis oeiien) 2.3 BULK ROK BEHAVIOUR So as o simliy he disussion we assume ha he, ϑ and ϕ dieions ae he inile sess dieions. Assuming he ok is isooi, so ha he inile axes o sain oinide wih he inile axes o sess, hen he heoy o elasiiy elaes sesses o sains via he ollowing equaions: E ε E ε σ σ v σ v σ ϕϕ v σ ϕϕ E ε ϕϕ σ ϕϕ v σ v σ v σ

4 Subjeed o ineased axial sess he ok beomes omessed wih a esuling hange in bulk volume. I an be shown ha he hange in volume an be asibed o he hange in he mean sess given by: e ( σ + + )/ 3 σ σ ϕϕ 4 P e is ommonly alled he eeive essue sine i is aing on he ok abi. I is elaed o he onining essue by: α e 5 Suose he ok is exosed o a veial sess, σ, hen he esonse is govened by how he ok amewok is onsained om esonding. In esevoi aliaions i is nomal o assume ha he ok amewok anno move laeally so ha: ε ϕϕ ε 0 6 These onsains esul in a build u in laeal sesses ha ae ooional o he axial sess, alulaed om equaions 2 and 3, whih ae given by: σ σ ϕϕ ν ν ) σ 7 Subsiuing equaion 7 ino equaion 4 esuls in: P β σ 8 e whee: 1+ β 3 1 v v ( ) RESERVOIR ROK MODEL In his model he onining essue P, and he essue o he luid in he oes P, ae assumed o inluene he ok oe volume. Then aliaion o he hain ule o dieeniaion esuls in: ε 10 4

5 A model is used o elae he omessibiliies in his equaion 1. This onsides equal essue inemens () o he oninemen essue and o he luid essue. The model suoses ha he esuling inease in sess wihin he ok oming he amewok is he same as i oes had been illed wih ok maeial. Thus he hange in sain o he oe volume mus be he same as would have oued had he oe volume been ok mineal maeial. Thus: ε 11 Fom whih: + 12 Subsiuing equaion 12 ino equaion 11 esuls in: 2.5 FLUID BEHAVIOUR ( + ) ε 13 The oe sain ε an be elaed o he oe essue no only hough he models desibed above bu also hough he luid behaviou: ε 14 The luid may be made u o seveal luids. In he simles ase whee hee is no ine-hase mass anse he eeive omessibiliy may be alulaed om: Equaing equaions 13 and 14 esuls in: Si i i 15 ( ) ( + ) TIDAL STRESS AND OMPRESSIBILITY EVALUATION The inease in he veial sess due o he sea level hange ok and he luid in he esevoi: 1 Zimmeman RW. omessibiliy o Sandsone. Elsevie, σ has o be aied by he σ σ + α 17 5

6 17 Subsiuing equaion 8 ino his equaion esuls in: σ βe + α 18 And using equaion 5: ( α ) α Eliminaing he onining essue using equaion 16: σ β + 19 A σ β ( + α( + )) + α( + ) + 20 Eliminaing he unknown eeive essue em using equaions 5 and 16 esuls in he ollowing exession o he ansmission eiieny, A: A α + β ( α ) + α ) ( + ) 1 21 This equaion may be eaanged o give an esimaion o he oe omessibiliy as a aion o he eeive luid omessibiliy: β + α 1 A β ) α β α ) 22 I one makes he aoximaions α 1 and 0 hen A β 23 1 A 2.7 PORE OMPRESSIBILITY FOR WELL TEST INTERPRETATION In well es analysis hee is no geo-mehanial modelling and he oe volume hange is eesened by he simle exession given by: ε 24 6

7 24 Using equaion 10 elaing he sain o ok oeies esuls in: P + In onas o he idal siuaion, duing a well es he esevoi exeienes a onsan ovebuden essue along wih zeo laeal sain. Equaion 18 hen indiaes ha as he luid essue is edued he eeive essue ineases on he ok suue so ha: And using equaion 5: β e + α 0 26 ( 1 β ) 0 And so, using equaion 13, he oe sain behaves as: β + α ε α ( ) ( β ) β 28 O: ε ( α ) α α β ) β β β Equaing his wih equaion 24: α ) α α β ) β + β + β PORE OMPRESSIBILITY FOR WELL TEST INTERPRETATION IN TERMS OF TRANSMISSION EFFIIENY The oe omessibiliy deived above o well es analysis may be exessed in ems o he idal ansmission eiieny by using equaion 22 o eliminae he ok oey esuling in: 7

8 β 1 A + α α β ) β α ) α ) α α β ) β + β + β 31 I one again makes he aoximaions α 1 and 0 hen A 1 A 32 3 Poedue o Finding Exeme Values I is exeed ha he aamees in equaion 31 an be esimaed wihin a ange and ha well es analysis will be ineesed in aouning o he maximum and minimum ossible values o oe omessibiliy onsisen wih he ange esimaes. onsequenly a oedue has been wien o ideniy hese exeme values o omaion omessibiliy. The non-linea onsained oimaion oedue uses he genealised edued gadien mehod wih a Newon line seah. The Newon se emloys analyi evaluaion o he gadien and Hessian o equaion 31. Beause he equaion is linea in he exeme value o he well es omessibiliy will neessaily involve an exeme value o hese aamees. The equaion is no linea in beause o is hidden esene in α. Beause he equaion is monooni in he idal ansmissibiliy eiieny aamee, his will also neessaily ado an exeme value in he oedue. 4 Examle Esimaions o Poe omessibiliy om Tide Daa Equaion 32 ovides an aoximae way o esimaing he oe omessibiliy o well esing om he idal ansmission eiieny. Equaion 31 ovides he moe omlee exession bu i equies esimaes o be available o addiional aamees. These addiionally aamees would nomally be exeed o eine he aoximae evaluaion and his sensiiviy migh iniially be examined based on omaable ok daa. By way o an examle he oedue has been used based on he ollowing inu daa anges: Examle esimaed daa < < si < < si < b < si < ν < 0.3 8

9 0.24 < A < 0.26 The esuls o he alulaions ae shown in he able below: Minimum oe Maximum oe omessibiliy omessibiliy 2.786x x x x x x 10-6 b 2 x x 10-6 ν A By way o omaison he aoximae equaion esuled in: Minimum oe Maximum oe omessibiliy omessibiliy 3.158x x10-6 I is heeoe eviden ha, in his ase, he mos o he uneainy aises om he uneainy in he ansmission eiieny and he eeive luid omessibiliy. In his examle ase he ok gain omessibiliy was assumed o be an ode o magniude geae han ha o Quaz, alie o Feldsa in ode o show a esul ha was sensiive o he deailed aamees and ha esuled in a minimum oe omessibiliy ha did no oesond o only exeme values o he aamees. 9

Lecture 9. Transport Properties in Mesoscopic Systems. Over the last 1-2 decades, various techniques have been developed to synthesize

Lecture 9. Transport Properties in Mesoscopic Systems. Over the last 1-2 decades, various techniques have been developed to synthesize eue 9. anspo Popeies in Mesosopi Sysems Ove he las - deades, vaious ehniques have been developed o synhesize nanosuued maeials and o fabiae nanosale devies ha exhibi popeies midway beween he puely quanum