Water-drive gas reservoir: sensitivity analysis and simplified prediction

Transcription

1 Lousana State Unversty LSU Dgtal Commons LSU Master's Theses Graduate School 00 Water-drve gas reservor: senstvty analyss and smlfed redcton Juneng Yue Lousana State Unversty and Agrcultural and Mechancal College, Follow ths and addtonal works at: htts://dgtalcommons.lsu.edu/gradschool_theses Part of the Petroleum Engneerng Commons Recommended Ctaton Yue, Juneng, "Water-drve gas reservor: senstvty analyss and smlfed redcton" (00). LSU Master's Theses. 7. htts://dgtalcommons.lsu.edu/gradschool_theses/7 Ths Thess s brought to you for free and oen access by the Graduate School at LSU Dgtal Commons. It has been acceted for ncluson n LSU Master's Theses by an authorzed graduate school edtor of LSU Dgtal Commons. For more nformaton, lease contact gradetd@lsu.edu.

2 WATER-DRIVE GAS RESERVOIR: SENSITIVITY ANALYSIS AND SIMPLIFIED PREDICTION A Thess Submtted to the Graduate Faculty of the Lousana State Unversty and Agrcultural and Mechancal College n artal fulfllment of the requrements for the degree of Master of Scence n Petroleum Engneerng n The Deartment of Petroleum Engneerng by Juneng Yue B.S., Unversty of Petroleum (East Chna), 994 M.S., Unversty of Petroleum (East Chna), 997 December 00

3 DEDICATION To my arents.

4 ACKNOWLEDGEMENTS I would lke to gve artcular thanks to Dr. Chrstoher Whte for hs atence, gudance and mentorng. I benefted from each dscusson wth hm n the ast two years. Wthout hs drecton, suggestons and encouragements, ths work would have been mossble. To Dr. Zak Bassoun, for hs serve on the eamnng commttee and gvng good suggestons. To Dr. Andrew Wojtanowcz, for hs serve on the eamnng commttee and gvng good suggestons.

5 TABLE OF CONTENTS ACKNOWLEDGEMENTS... ABSTRACT...v CHAPTER. INTRODUCTION... CHAPETR. RESPONSE SURFACE METHODOLOGY Aromatng Resonse Functons Buldng Emrcal Models Lnear Regresson Models Model Fttng Hyothess Testng n Multle Regresson Test for Sgnfcance of Regresson Tests on Indvdual Regresson Coeffcents Eermental Desgn Two-level Factoral Desgns Confoundng Two-Level Fractonal Factoral Desgns Desgn Resoluton The Bo-Behnken Desgn... 9 CHAPETR 3. MATERIAL BALANCE Water-drve Gas Reservor Materal Balance Fetkovtch Aqufer Model Need for A Well Flow Model The Al-Hussany, Ramey, Crawford Soluton Technque CHAPTER 4. SENSITIVITY ANALYSIS Defnng Resonses Aqufer Productvty Inde Water Produced Cumulatve Gas Producton Cut-off lne Model Descrton Reservor Geometry and Proertes Grd Descrton Reservor Factor Ranges Smulaton Desgn Senstvty Analyss Matchng Aqufer Productvty Inde Gas Producton Factor Gas Recovery Swee effcency... 0 v

6 4.4. Water Breakthrough Other Resonses... CHAPTER. APPLICATIONS.... Smlfed Predcton Usng Resonse Models.... Smlfed Predcton for /z Curves... CHAPTER. DISCUSSIONS AND CONCLUSIONS Dscussons Conclusons... 9 REFERENCES... APPENDIX: RESPONSE MODELS... VITA... 7 v

7 ABSTRACT Water nflu and well comletons affect recovery from water-drve gas reservor. Materal balance, aqufer models and well nflow equatons are used to eamne and redct the ressure deleton, water nflu, and roducton rates of water-drve gas reservors. The arameters of these smle, lumed models are estmated from smulaton results usng resonse surfaces and eermental desgns for eght varyng geologc and engneerng factors. Eleven smulated resonses (ncludng mamum gas rate, aqufer and well constants, and water breakthrough) are analyzed usng ANOVA and resonse models. A senstvty analyss of aqufer roductvty nde, gas roducton factor, and swee effcency reveals that ermeablty s the domnatng factor. In contrast to earler nvestgatons, ths study ndcates that water-drve gas recovery s often hgher for hgher ermeablty water-drve gas reservors. The hgh gas moblty more than offsets the hgh aqufer moblty. The other seven factors are statstcally sgnfcant for many resonses, but much less mortant n determnng reservor behavor. The roosed aroach combnes smle analytc eressons wth more comlete but dffcult-to-use reservor smulaton models. The resonse models can be used to make quck, accurate redctons of water-drve gas reservors that nclude the effects of changng geologc and engneerng varables. These smle, aromate models are arorate for rosect screenng, senstvty analyss and uncertanty analyss. v

8 CHAPTER. INTRODUCTION Predcton of gas roducton s an mortant art of reservor develoment and management, elne and dstrbuton management, and economc evaluaton. The roducton of gas reservors that have no assocated aqufers s relatvely smle to redct and recovery effcency s usually hgh (Lee et al., 99). However, gas recovery from water-drve reservors may decrease because water nflu may tra gas. The gas s traed as an mmoble, mmscble hase wthn the orton of the reservor nvaded by water. At hgher abandonment ressure, the amount of traed wthn the water-nvaded ore sace s hgher. Efforts to redct water-drve gas reservor erformance have focused on materal balances. Materal balances are a fundamental reservor engneerng tool that descrbe and redct the relaton between flud wthdrawal, eanson, nflu and ressure. Materal balances rovde a smle but effectve alternatve to volumetrc methods based on soach mas. Materal balances can redct orgnal gas n lace and gas reserves at any stage of reservor deleton (Craft and Hawkns, 99). For a constant-volume (or volumetrc) gas reservor wthout water nflu, the z versus cumulatve gas roducton lot can redct the gas reservor behavor. If the rock and water comressblty are small, the z versus cumulatve gas roducton G lot s a straght lne (Craft and Hawkns, 99). For a water-drve gas reservor, the aqufer affects the reservor behavor. The vs. z G lots for these water-drve gas reservors are no longer straght lnes (Bruns, 9). The devaton from a straght

9 lne s determned by the aqufer roertes, sze and the roducton means. Materal balance and related models are dscussed n Chater 3. Water-drve gas materal balances nclude aqufer models. The aqufer water nflu can be estmated usng the Schlthus (93) steady-state method, Hurst modfed steady-state method (Prson, 98), and varous unsteady-state methods such as those of van Everdngen and Hurst (949), Hurst (98), and Carter and Tracy (90). The unsteady state nflu theory of Hurst and van Everdngen s the most rgorous method for radal and lnear aqufers. Unfortunately, ths method requres awkward, tme-consumng sueroston calculatons. Ths drawback s eacerbated by the reetton n most nflu calculatons when hstory matchng. Because of ths, engneers have sought a more drect method of water nflu calculaton that dulcates results obtaned wth the Hurst and van Everdngen method wthout requrng sueroston (Dake, 978). The most successful of the methods was roosed by Fetkovtch (97). Chater 3 detals the Fetkovtch method. The aqufer roductvty nde n the Fetkovtch aroach s one mortant arameter used to redct the water nflu. It s determned by the reservor roertes, reservor geometry, and flud roertes. The smle mechanstc model for the relatonsh between aqufer roductvty nde and those factors s avalable (Dake, 978). But for secfc cases, when there ests a d or the reservor s n secal shae or more comle, how these factors nteract n the model make t dffcult to use those smle models. In ths stuaton, the researchers can aromate the mechanstc model wth an emrcal model. Ths emrcal model

10 s called a resonse surface model. Resonse surface methodology s often realzed n combnaton wth eermental desgn method. In ths study, the -level fractonal factoral desgns were used. Chater dscusses resonse surface methodology and eermental desgns. In addton to the aqufer roductvty nde, the emrcal model for swee effcency and gas roducton factor were also derved usng resonse surface methodology. These derved resonses are dscussed n Chater 4. A smle rectangular reservor model was used to study the water nflu. The model s descrbed n Chater 4. Eght factors, ncludng ntal reservor ressure, ermeablty, reservor wdth, reservor thckness, aqufer sze, tubng sze, tubng head ressure and reservor d, were selected to do -level half-fracton factoral desgn; 8 smulaton runs are requred comared to runs for a two-level full factoral desgn. A frst-order resonse surface model wth two-term nteractons was derved to do senstvty analyss. These models were also used to do smlfed redcton. Ths study rovdes resonse-surface based methods for quck reserve estmates and erformance redcton. The objectves of ths study are to understand roducton senstvtes and to formulate the aqufer roductvty nde, gas roducton factor, ntal mamum gas roducton and swee effcency. 3

11 CHAPTER. RESPONSE SURFACE METHODOLOGY In ths chater, the Resonse Surface Methodology and eermental desgn are ntroduced. Resonse surface methodology (RSM) s a collecton of statstcal and mathematcal technques to develo, mrove, and otmze rocesses (Myers and Montgomery, 99). The most etensve alcatons of RSM are n the ndustral world, artcularly n stuatons n whch several nut varables otentally nfluence some erformance measure or qualty characterstcs of roducts or rocesses. Ths erformance measure or qualty characterstcs s called the resonse. The nut varables are called ndeendent varables or factors.. Aromatng Resonse Functons In some systems the nature of the relatonsh between resonse y and the nut varables,,, mght be known eactly, based on the underlyng, 3 k engneerng, chemcal, or hyscal rncles. Then we could wrte a model of the form y g(,,, k ), where the term n ths model reresents the error n the system. Ths tye s often called a mechanstc model. If the underlyng mechansm s not fully understood, the eermenter must aromate unknown functon g wth an aromate emrcal model y f,,, ). Such ( k emrcal models are called a resonse surface model (Myers and Montgomery, 99). In some stuatons, the mechanstc model ests, but t s dffcult to comute or use. Researchers can also use emrcal resonse models to aromate the mechanstc model. 4

12 Usually the functon f s a frst-order or second order olynomal, and s the term that reresents other sources of varablty not accounted for n f. Thus ncludes effects such as measurement error on the resonse, other sources of varaton that are nherent n the rocess or system, the effect of other varables, and so on. s treated as a statstcal error, and often t s assumed to have a normal dstrbuton wth mean zero and varance and Montgomery, 99) y Ef,,, E. If the mean of s zero, then (Myers E k (.) f,,, k (.) The varables,,, k n Equaton (.) and (.) are usually called natural varables, because they are eressed n the natural unts of measurement. In much RSM work t s convenent to transform the natural varables to coded varables,,, k, where these coded varables are usually defned to be dmensonless wth center ont value zero and the same sreads around the center ont, usually eressed wth and. Chater 4 dscusses some codng functons. In terms of the coded varables, the true resonse functon (.) s now wrtten as (Myers and Montgomery, 99) k f,,, (.3) Because the form of the true resonse functon f s unknown, t needs to be aromated. RSM deends uon a sutable aromaton for f. Usually, a loworder olynomal n some relatvely small regon of the ndeendent varable sace

13 s arorate. In many cases, ether a frst-order or a second-order model s used. In general, the frst-order model s (Myers and Montgomery, 99) 0 (.4) and the second-order model s (Myers and Montgomery, 99) k k k k k k 0 j j jj j j j j j (.) The form of the frst-order model n Equaton (.4) s sometmes called a man effect model, because t ncludes only the man effects of the varables,,,. j k The nteracton between varables j j can be added to the model. The frst-order model (even wth nteracton term ncluded) cannot descrbe the curvature n resonses. A second-order model s requred n these stuatons. The second-order model s wdely used n RSM for several reasons. Frst, the second-order model s very fleble. It can take on a wde varety of functonal forms, so t wll often work well as an aromaton to the true resonse surface. Contour lots are good alcaton of the second-order model. Second, t s easy to estmate the arameters n the second-order model. The method of least squares dscussed later n ths chater can be used. Thrd, there s consderable ractcal eerence ndcatng that second-order models work well n solvng real resonse surface roblems (Narayanan, 999). Fourth, the second-order model can be used for otmzaton whereas frst-order models cannot. In some stuatons, aromatng olynomals of order greater than two are used. The general motvaton for a olynomal aromaton for the true resonse

14 functon s based on the Taylor seres eanson around the ont,. f 0, 0, k 0 For eamle, the frst-order model s develoed from the frst-order Taylor seres eanson (Myers and Montgomery, 99) f f 0, 0,, f f f 30 (.) 0 0 k 0 and the hgher-order models can be derved smlarly.. Buldng Emrcal Models Multle regresson s a collecton of statstcal technques useful for buldng the tyes of emrcal models requred n resonse surface methodology... Lnear Regresson Models A frst-order resonse surface model descrbed as (Myers and Montgomery, 99) y 0 s a multle lnear regresson model wth two varables or regressors. In general, the resonse varable y may be related to k regressor varables. The model y 0 k k s called a multle lnear regresson model wth k regressor varables. The arameters j, j 0,,, k, are called the regresson coeffcents. Ths model descrbes a hyerlane n the k -dmensonal sace of the regressor varables j, j 0,,, k. The arameter reresents the eected change n resonse y j 7

15 er unt change n when all the remanng ndeendent varables j are held constant. j The varable j could be a functon of other varables, such as, 3 or other forms. At ths stuaton, we can let j or j. In general, any 3 regresson model that s lnear n the arameters, j j 0,,, k s a lnear model, regardless of the shae of the resonse surface that t generates... Model Fttng The methods for estmatng the arameters n multle lnear regresson models are often called model fttng. The least squares method s tycally used to estmate the regresson coeffcents from multle lnear regresson. Suose that n k observatons on the resonse varable are avalable, say y, y,, y n. Along wth each observed resonse y, we wll have an observaton on each regressor varable, and let j denote the th observaton of. The data wll aear as n Table.. j Classcally, we assume that the error term n the model has Var ( ) and that the { } are uncorrelated random varables. Table. Data for Multle Lnear Regresson E 0 and y y y y n n n k k k nk In terms of the observatons n Table. the model equaton may be wrtten as y 0 k k 8

16 n k j j j,,,, 0 (.7) The method of least squares chooses the s n Equaton (.7) so that the sum of the squares of the errors,, are mnmzed. The least squares functon s (Myers and Montgomery, 99) n L 0 n k j j j y (.8) The functon L s to be mnmzed wth resect to. The least squares estmators, say b, must satsfy k,..., 0, b,b k,, 0 0 0,..., 0 0 n k j j j b b b b b y L k (.9a) and k j b b y L j n k j j j b b b j k,,, 0, 0,,... 0 (.9b) Equaton (.7) may be wrtten n matr notaton as ε Xβ y where y n y y y, nk n n k k X 9

17 0 β, and ε k n where s a constant (or mean) term. In general, y s an (n) vector of the 0 observed resonses, X s an (n) matr of the levels of the ndeendent varables, β s a () vector of the regresson coeffcents, and ε s an (n) vector of random errors. To fnd the vector of least squares estmators, b, that mnmzes n L εε y Xβ y Xβ yy βxy yxβ βxxβ yy βxy βxxβ (.0) snce βxy s a () matr, or a scalar, and ts transose βx y yxβ s the same scalar. The least squares estmators must satsfy L β b Xy XXb 0 whch smlfes to X Xb Xy (.) Equaton (.) s the set of least squares normal equatons n matr form. To solve the normal equatons, multly both sdes of Equaton (.) by the nverse of XX. Thus, the least squares estmator of β s b XX Xy (.) 0

18 The ftted regresson model s yˆ Xb (.3) In scalar noton, the ftted model s yˆ k b0 b j j,,,, The dfference between the observatons y and the ftted value ŷ s a resdual, say n e y yˆ. The (n) vector of resduals s denoted by e y yˆ (.4).3 Hyothess Testng n Multle Regresson In multle lnear regresson roblems, certan tests of hyothess about the model arameters hel n measure the usefulness and sgnfcance of the model. In ths secton, several hyothess-testng rocedures are descrbed. These rocedures requre that the errors n the model be normally and ndeendently dstrbuted wth mean zero and varance, abbrevated ~ NID (0, ). As a result of ths assumton, each observaton y s normally and ndeendently dstrbuted wth k mean 0 j j and varance. j.3. Test for Sgnfcance of Regresson The test for sgnfcance of regresson s a test to determne f there s lnear relatonsh between the resonse y varable and a subset of the regressor varables,,, k. The arorate hyotheses are H 0 : k 0 H : 0 for at least one j (.) j

19 Rejecton of H 0 : n (.) mles that at least one of the regressor varables,,, k contrbutes sgnfcantly to the model. The test rocedure nvolves arttonng the total sum of squares S yy nto a sum of squares due to the model (or to the regresson) and a sum of squares due to resdual (or error), say S yy SS SS (.) R E The regresson sum of squares s SS R n y b Xy (.7) n and the error sum of squares s SS yy bxy (.8) E The test rocedure for H 0 : k 0 s to comute F SS / k MS R R 0 (.9) SS E / n k MS E H 0 F0 F, k, nk and to reject f eceeds where s the confdence level. Alternatvely, one could use the P-value aroach to hyothess testng and reject H 0 f the P-value for the statstc F0 s less than. The test rocedure s called an analyss of varance or F-test. The coeffcent of multle determnaton R s defned as R SS S R yy SS S E yy (.0)

20 R s a measure of the amount of reducton n the varablty of y obtaned by usng the regressor varables,,, n the model. 0 R. However, a k large value of R does not necessarly mly that the regressons model s a good one. Addng a varable to the model wll always ncrease R, regardless of whether the addtonal varable s statstcally sgnfcant or not. Because R always ncreases as terms are added to the model, some regresson model bulders refer to use an adjusted statstc defned as R adj n R (.) n k In general, the adjusted R adj statstc wll not always ncrease as varables are added to the model: f unnecessary terms are added, the value of R adj may decrease..3. Tests on Indvdual Regresson Coeffcents Indvdual regresson coeffcents may be tested to determne the mortance of the regressor varables n the regresson model. For eamle, the model mght be more effectve wth the ncluson of addtonal varables, or erhas wth the deleton of one or more of the varables already n the model. If some regressor varable s not mortant and deleted, t s not necessary to measure any j more, whch can make the eerments less eensve. In the contet of reservor modelng, ths may be a very mortant result: eensve core measurements mght be susended f ther results were shown not to affect mortant rocess measures. The hyotheses for testng the sgnfcance of any ndvdual regresson j coeffcent, say j, are 3

21 H : 0 j H : j If H 0 : j 0 s not rejected, then ths ndcates that j can be deleted from the model. The test statstc for ths hyothess s 0 0 t 0 b j (.) ˆ C jj where C s the dagonal element of XX corresondng to b. The null jj j hyothess H 0 : 0 s rejected f t 0 t /,. Note that ths s really a artal j nk or margnal test, because the regresson coeffcent b deends on all the other j regressor varables j that are n the model. Resonse surface methodology s often used wth eermental desgn. Eermental desgn allows us to select a small set of smulatons to run from the large set that we could run. By choosng an arorate desgn, we mnmze the number of runs that need to be made to obtan the requred results..4 Eermental Desgn Eermental desgn has been used n reservor engneerng alcatons ncludng erformance redcton (Chu, 990), uncertanty modelng (Damsleth et al., 99, van Elk et al., 000, Fredmann et al., 00), senstvty studes (Wlls and Whte, 000), uscalng (Narayanan and Whte, 999), hstory matchng (Ede et al., 994) and develoment otmzaton (Dejean and Blanc, 999). The smlest eermental desgns are factorals. The most common desgns are two-level desgn. 4

22 .4. Two-level Factoral Desgns Factoral desgns are wdely used n eerments nvolvng several factors to nvestgate jont effects of factors on a resonse. Jont factor effects are man effects and nteractons. A case of the factoral desgn s that where each of the k factors of nterest has only two levels. Because such a desgn has eactly k eermental trals or runs, these desgns are called k factoral desgns..4. Confoundng The k factoral desgns are smle to use. However, n many stuatons, t s mossble to erform a comlete factoral desgn n one block (usually n agrculture and ndustry). Also, when there are many factors the number of eerments requred becomes large (for 0 factors, more than one mllon eerments would be needed). To reduce the number of eerments requred, statstcans have formulated a number of strateges ncludng artal factorals or confoundng. Confoundng s a desgn technque for arrangng a comlete factoral eerment n blocks, where the block sze s smaller than the number of treatment combnatons n a comlete factoral. The technque causes nformaton about certan treatment effects (usually hgh-order nteractons) to be ndstngushable from, or confounded wth, blocks. Confoundng reduces the ower or resoluton of the desgn but greatly decreases the cost. Below, we wll dscuss how the deas of confoundng and blockng can be used to create two-level factoral desgns that requre fewer eerments.

23 .4.3 Two-Level Fractonal Factoral Desgns A comlete relcate of the desgns requres 4 runs. In ths desgn only of 3 degrees of freedom are used to estmate the man effects, and only degrees of freedom are used to estmate the man two-factor nteractons. The remanng 4 degrees of freedom (one s used to estmate the mean) are assocated wth threefactor and hgher nteractons. If the eermenters can assume that these hgh-order nteractons are neglgble, then the man effects and low-order nteractons may be estmated from only a fracton of the comlete factoral eerment. These fractonal factoral desgns are among the most wdely used tyes of desgn. Consder the stuaton n whch three factors are of nterest, but the eermenters do not wsh to run all 3 =8 treatment combnatons. Suose they consder a desgn wth four runs. Ths suggests a one-half fracton of the 3 desgns. Because the desgn contans 3- =4 treatment combnatons, a one-half fracton of the 3 desgns s often called a 3- desgn. The table of and + sgns for 3 desgns s shown n Table.. Suose to select the four treatment combnatons a, b,c and abc as the one-half fracton. These runs are shown n the to half of Table.. Table. + and Sgns for the 3 Factoral Desgn Treatment Factoral Effect Combnaton I A B C AB AC BC ABC A B C ABC AB AC BC ()

24 Notce that 3- desgns are formed by selectng only those treatment combnatons that have a lus n the ABC column. Thus, ABC s called the generator of ths artcular fracton. Furthermore, the dentty column I s also always lus, so I=ABC s called the defnng relaton of the desgn. In general, the defnng relaton for a fractonal factoral wll always be the set of all columns that equal to the dentty column I..4.4 Desgn Resoluton The recedng 3- desgn s called a resoluton III desgn. In such a desgn, man effects are alased wth two-factor nteractons. The alas structure for ths desgn may be easly determned by usng the defnng relaton I=ABC. Multlyng any column by the defnng relaton yelds the alases for that effect. In ths eamle, ths yelds as the alas of A A I A ABC A BC or, because the square of any column s just the dentty I, A BC Smlarly, the alases of B and C as B I B ABC B AB C AC and C I C ABC C ABC AB 7

25 Consequently, t s mossble to dfferentate between A and BC, B and AC and C and AB. In fact, when we estmate A, B, and C, we are really estmatng A BC, B AC, and C AB. These desgns are often descrbed usng a notaton such as k R where k s the number of factors, s the fracton of the factoral, and R s the resoluton. The number of runs requred n a fractonal factoral s smaller than a full factoral by. Usually a Roman numeral subscrt s emloyed to denote desgn resoluton, thus the one-half fracton of the 3 desgn wth the defnng relaton I=ABC s a 3 III desgn. Desgns of resoluton III, IV, V are wdely used. Table.3 gves defntons of these resolutons. Table. 3. Defntons of Resoluton III, IV, and V Resoluton Man Effects Two-factor Interactons Eamles III Not alased wth each May be alased 3 III : I=ABC, other, but are alased wth each other. A=BC, B=AC, C=AB. wth two-factor nteracton. IV Not alased wth each May be alased 4 IV : I=ABCD, other or two-factor wth each other. A=BCD, AB=CD, nteractons. V Not alased wth each other, two-factor, or three factor nteractons. Not alased wth each other; may be alased wth threefactor nteractons. AC=BD, AD=BC. V : I=ABCDE, A=BCDE, AB=CDE, AC=BDE, AD=BCE. These two-level desgns can estmate frst-order effects and nteracton only. To consder quadratc effects, a thrd level must be ntroduced nto the desgn. The most straghtforward way to do ths s wth a three-level factoral, n whch factors are set to mnmum, center, or mamum values. Full three-level desgns requre 3 k eerments. It s very eensve f the factor number becomes large. 8

26 . The Bo-Behnken Desgn Bo and Behnken (90) develoed a famly of effcent three-level desgns for fttng second-order resonses. The Bo-Behnken desgn s a fractonal desgn wth addtonal runs on the edges of the faces of the hyercube and at the center. Comared wth three-level full factoral desgn, a Bo-Behnken desgn reduces the number of requred eerments by confoundng hgher-order nteractons. Ths reducton become more sgnfcant as the number of factors ncreases. For 7 factors a Bo-Behnken desgn requres 7 eerments comared to 87 eerments requred for a full 3-level factoral and 8 for a full -level factoral. Bo-Behnken desgns have the desrable qualtes of beng nearly orthogonal and rotatable for many cases (Bo and Behnken, 90). Bo-Behnken desgns allow estmaton of quadratc terms and do not mly constant senstvtes of resonses to factors. Most two-level desgns do not nclude eerments at the desgn centeront. By ncludng the center ont, Bo-Behnken desgns reduce estmaton error for the most lkely resonses. The recedng dscussed the resonse surface methodology and eermental desgn. Researchers have aled the RSM n ol ndustry. Wang (00) llustrated some researchers alcatons. Besde those alcatons, Gerbaca et al. (980) conducted eerments to study the effects of the fracton of hgh-equvalent-weght sulfonate, the cosurfactant HLB (Hydrohle-lohle balance) and the weght rato of cosurfactant to sulfonate on ol recovery and nterfacal tenson. He evaluated the data statstcally, obtanng otmal formulaton for ths data sace and develoed a hgh crude ol recovery formulaton for that crude ol recovery. Aanonsen et al. 9

27 (99) otmzed well locaton under reservor geometry and etrohyscal arameter uncertantes. Wang and Whte (00) aromated the relatonsh between gas recovery resonses and reservor and roducton arameters, and generated qualty mas to choose otmal well locatons for roducton. In ths study, eermental desgn and resonse surfaces were used n redctng the materal balance shae, aromatng water nflu, and nflow gas erformance. Chater 3 ntroduces the gas reservor materal balance method, water nflu redcton method and gas nflow erformance. 0

28 CHAPTER 3. MATERIAL BALANCE Water-drve gas recovery ncreases wth decreasng ermeablty, traed gas saturaton, and ncreasng wthdrawal rates (Agarwal et al., 9). Gas recovery decreases wth ncreasng aqufer sze (Al-Hashm et al., 988). Gas recovery under water drve deends on geologc uncertantes and engneerng factors, whch are all nterrelated and comlcate the analyss. These arameters determne the shae of the z erformance curves for the reservor. The z method (volumetrc materal balance) s a common rocedure used n an attemt to descrbe and redct the behavor of a etroleum reservor. It can be used to redct the ultmate gas recovery. 3. Water-drve Gas Reservor Materal Balance Agarwal (9) demonstrated the effect water nflu has on z versus cumulatve gas roduced for a gas reservor usng a materal balance equaton for the reservor and a water nflu equaton for the aqufer. Smultaneous soluton rovdes the cumulatve water nflu and reservor ressure. If water and rock comressblty are neglected, a general form of the materal balance for a water-drve dry gas reservor s (Agarwal, 9) GB g G G B g We W Bw (3.) Equaton (3.) can be rearranged to

29 z G sc G z G zsc Tsc W e W B z T w (3.) where G = orgnal gas n lace G = cumulatve gas roduced Bg = gas formaton volume factor at ntal reservor ressure B = gas formaton volume factor at reservor ressure B w = water formaton volume factor = cumulatve water nflu W e W = cumulatve water roduced = ntal reservor ressure = standard condton ressure sc = reservor ressure T = reservor temerature T sc = standard condton temerature z = gas devaton factor at ressure z sc = gas devaton factor at ressure sc Agarwal (9) used the Carter-Tracy method (Carter and Tracy, 90) to aromate water nflu. Then z s related to the gas roduced G at any tme.

30 Agarwal (9) derved one further equaton to set the end ont, or abandonment condton a materal balance whch states that the mamum gas recovery s equal to the ntal gas n lace, less gas traed as resdual gas n the watered regon, less gas regons not swet by water, but unavalable to roducton because of breakthrough of water nto all estng roducng wells. The end-ont equaton s (Agarwal, 9) G S gr E z G E (3.3) S g E z where, S gr E = resdual gas saturaton = volumetrc nvason effcency (also swee effcency) = ntal gas saturaton S g Equaton (3.3) can be rearranged to z E G z G S gr E S g E (3.4) Equaton (3.4) eresses the end-ont z as a lnear functon of the ultmate gas recovery, and that the lne asses through the ont G, ntal gas n lace, at a zero value of z. The lne n Equaton (3.4) s referred as to cut-off lne. 3

31 Equatons (3.) and (3.4) suggest a grahcal soluton of the water flu gas reservor erformance roblem (Agarwal, 9). If z vs. G can be estmated, the ntersecton of z vs. G (Equaton (3.)) and Equaton (3.4) s the estmated ultmate gas recovery (Agarwal, 9, Fgure 3.). /z Cut-off 0MMSCF/D 30MMSCF/D Infnte Rate G Fgure 3. / z vs. G (from Agarwal, 9) Agarwal (9) estmated the reservor erformance for ranges of aqufer ermeabltes, reservor roducton rates, ntal formaton ressures, resdual gas saturatons, and water nflu reservor effcences. Performance for a water-drve gas reservor was comuted for a reservor of 000 acres n area surrounded by an nfntely large aqufer. Agarwal dd not vary other arameters ncludng reservor d, thckness, wdth, aqufer comressblty. Gas recovery for Agarwal s case deends uon roducton ractces. A hgh roducton rate draws down reservor ressure before water nflu comletely engulfs the reservor (Fgure 3.). Gas recovery effcency s lower at a gven roducton rate for hgh-ressure reservors. Gas recovery s less senstve to roducton rates as aqufer ermeablty ncreases. Water nflu resonds to ressure 4

32 changes n hgh-ermeablty gas reservor so quckly that there s no beneft from ncreased roducton rate. In the lmt, aqufer erformance aroaches a full water drve as ermeablty ncreases for suffcently large aqufers (Agarwal, 9). Al-Hashm et al. (988) researched the effect of aqufer sze on artal waterdrve gas reservors. They concluded that f r a / r g, the effect of the aqufer on the erformance of the gas reservor can be neglected. Gas recovery s senstve to ntal reservor ressure and the aqufer sze f r r. As r / r and the ntal a / g reservor ressure ncrease, gas recovery decreases. Saleh (988) establshed a model for develoment and analyss of gas reservors wth artal water drve. Hower et al. (99) establshed an analytcal model to redct the erformance of gas reservors roducng under water-drve condtons. All these studes used artcular methods to calculate the water encroachment (Chater ). The theory of Fetkovtch (97) for fnte aqufers to aromate water nflu s used n ths study. 3. Fetkovtch Aqufer Model In ths aroach the flow of aqufer water nto a hydrocarbon reservor s modeled n recsely the same way as the seudosteady flow of ol from a reservor nto a well. An nflow equaton of the form a g q w dwe J w ( a ) (3.) dt s used where J w = aqufer roductvty nde q w = water nflu rate

33 = reservor ressure,.e. ressure at the ol or gas water contact a = average ressure n the aqufer W e = water nflu The latter s evaluated usng the smle aqufer materal balance W e cw (3.) a where n whch W = ntal volume of water n the aqufer and s therefore deendent uon aqufer geometry c = total aqufer comressblty s the ntal ressure n the aqufer and reservor. Ths balance can be alternatvely eressed as W e W e a (3.7) cw We where W cw s defned as the ntal amount of encroachable water and e reresents the mamum ossble eanson of the aqufer. Dfferentatng equaton (3.7) wth resect to tme gves dw dt e We da (3.8) dt and substtutng equaton (3.8) nto equaton (3.) and searatng the varables gves d a a J w W e dt ths equaton can be ntegrated for the ntal condton that at t 0, W 0 and. There s a ressure dro mosed at the e a

34 reservor boundary. Furthermore, the boundary ressure remans constant durng the erod of nterest so that ln w C a J W t e where C s an arbtrary constant of ntegraton whch can be evaluated from the ntal condtons as C ln, and therefore a J w t We e / (3.9) whch on substtutng n the nflow equaton (3.) gves dw e J w t We J w e / (3.0) dt Fnally, ntegratng equaton (3.0) for the stated ntal condtons yelds the followng eresson for the cumulatve water nflu W e W J w t We e / e (3.) As t tends to nfnty, then W e W e cw whch s the mamum amount of water nflu that could occur once the ressure dro has been transmtted throughout the aqufer. As t stands, equaton (3.) s not artcularly useful snce t was derved for a constant nner boundary ressure. To use ths soluton n the ractcal case, n whch the boundary ressure s varyng contnuously as a functon of tme, t should agan to aly the sueroston theorem. Fetkovtch has shown, however, that a dfference form of equaton (3.) can be used whch elmnates the need for 7

35 sueroston. That s, for nflu durng the frst tme ste t, equaton (3.) can be eressed as W W J w t / We e e e (3.) where s the average reservor boundary ressure durng the frst tme nterval.. s the reservor boundary ressure at the end of the frst tme nterval. For the second nterval W W t J w t / We e e e a (3.3) where a s the average aqufer ressure at the end of the frst tme nterval and s evaluated usng equaton (3.7) as We a (3.4) We In general for the n th tme erod, where W W J w tn / We e e en an n (3.) an n Wej j (3.) We The values of n, the average reservor boundary ressure, are calculated as 8

36 n n n (3.7) Fetkovtch has demonstrated that usng equatons (3.) and (3.7), n stewse fashon, the water nflu calculated for a varety of dfferent aqufer geometres matches closely the results obtaned usng the unsteady state nflu theory of Hurst and van Everdngen (949) for fnte aqufers. Values of the aqufer roductvty nde J w deend both on the geometry and flowng condtons, and are tabulated n the book Fundamentals of Reservor Engneerng (Dake, 978). Materal balance Equaton (3.) and water nflu Equaton (3.) can be jontly used to redct the reservor erformance. However all these deend on the deleton erformance of the well. 3.3 Need for a Well Inflow Model Cumulatve gas roducton G at any tme needs be calculated. Two methods can be used to eress gas flow aromately. One s the Russell, Goodrch et al. formulaton (Russell et al., 9), the other s the Al-Hussanny, Ramey and Crawford real gas seudo-ressure m formulaton (9). Ths study uses real gas seudo ressure. The reasons for adotng ths aroach are (Dake, 978):. It s theoretcally the better method and n usng t one does not have to be concerned about the ressure ranges n whch t s alcable, as s the case when usng the formulaton.. It s techncally the more smle method to use because the basc relatonsh for m as a functon has been avalable. 9

37 3. The necessty for teraton n solvng the nflow equaton for bottom hole ressure wf s avoded. 3.4 The Al-Hussany, Ramey, Crawford Soluton Technque The basc equaton for the radal flow of flud n a homogeneous orous medum s derved as k r c r r r t (3.8) For real gas flow, ths equaton s non-lnear because the coeffcents on both sdes are themselves functons of the deendent varable ressure. Al-Hussany et al. lnearzed the equaton (3.8) usng an ntegral transformaton (Dake, 978) m d (3.9) Z b whch s the real gas seudo ressure. The lmts of ntegraton are between a base ressure and the ressure of nterest. The value of the base ressure s b arbtrary snce n usng the transortaton only dfferences n seudo ressures are consdered.e. m mwf b d Z wf b d Z wf d Z Al-Hussany et al. relaced the deendent varable ressure m n the followng manner. Because and m r m r by the real gas seudo 30

38 Z m then r Z r m (3.0) and smlarly t Z t m (3.) Substtutng for r and t n Equaton (3.8) and usng Equaton (3.0) and (3.) gves t m Z c r m Z r k r r (3.) Fnally, usng the equaton of state for a real gas ZRT M and substtutng ths eresson for n Equaton (3.) and cancelng some terms gves the smlfed eresson t m k c r m r r r (3.3) Equaton (3.3) has recsely the same form as the dffusvty equaton t k c r r r r m (Dake, 978) ecet that the deendent varable has been relaced by. 3

39 Note that n reachng ths stage t has not been necessary to make any restrctve assumtons about the vscosty beng ndeendent of ressure. c The dffusvty n Equaton (3.3) s not a constant snce for a real gas k both vscosty and comressblty are hghly ressure deendent. Equaton (3.3) s therefore, a non-lnear form of the dffusvty equaton. Contnung wth the above rocedure, n order to derve an nflow equaton under sem-steady state flow condtons, then alyng the smle materal balance for a well dranng a bounded art of the reservor at a constant rate cv t V t q (3.4) and for the dranage of radal volume element t q r hc e (3.) Also, usng Equaton (3.) m t Z t q Z r hc e (3.) and substtutng Equaton (3.) n (3.3) gves or m c q r r r r k Z re hc r m r r r q r kh Z e reservor (3.7) Furthermore, usng the real gas equaton of sate, 3

40 q Z reservor sc q sc T T sc Equaton (3.7) can be eressed as r m r r r q r kh sc e sc T T sc (3.8) For sothermal reservor deleton, the rght hand sde of equaton (3.8) s a constant, and the dfferental equaton has been lnearzed. A soluton can now be obtaned usng recsely the same technque aled for lqud flow. If n addton, feld unts are emloyed then the resultng sem-steady state nflow equaton s m 4qT e m S kh r wf ln w 4 r 3 (3.9) A generalzed eresson consderng reservor geometry and well asymmetry s m 4qT m S kh C r wf ln A w 4A (3.30a) Equaton (3.30a) can be rearranged to q 4T ln kh 4A C r A w S m m wf (3.30b) The recedng dscussed the materal balance, water nflu redctng aroach and gas nflow erformance method. These aroaches can be combned to use to redct the reservor behavor. Ther alcatons are dscussed n the net chater. 33

41 CHAPTER 4. SENSITIVITY ANALYSIS Eermental desgns and resonse surface methods are aled n a senstvty analyss of water-drve gas reservors. Resonses analyzed nclude aqufer roductvty nde, water swee effcency, gas roducton factor, total water nflu, ntal mamum gas roducton, and gas recovery. As dscussed n Chater, all eght factors san a range (mamum to mnmum). They are transformed to (-, ) usng codng functons (Reservor Factor Ranges, below). A 8- factoral desgn was used to reduce the number of smulaton runs. Reservor smulatons were used to estmate the aqufer roductvty nde, gas roducton factor and swee effcency. These resonses were related to the eght factors. Multle regressons ft emrcal models ncludng man effect and twoterm nteractons. These are the resonse models. Smulaton can be used to model comle reservor models, such as heterogeneous or rregularly shaed reservors. In ths study, t was used to study a smle rectangular reservor model. 4. Defnng Resonses Analytc water nflu redctng methods are dscussed n chater 3. From Equaton (3.), f G,W, W can be estmated, the reservor erformance e can be z estmated at any tme. z G sc G z G zsc Tsc W e W B z T w (3.) 34

42 4.. Aqufer Productvty Inde Aqufer roductvty nde n the theory of Fetkovtch for fnte aqufers to aromate water nflu s determned by the flud vscosty, reservor ermeablty, and reservor geometry. It s defned as the frst resonse. Referrng to Equatons (3.0) to (3.), J w f,,,... (4.), 3 4 q w f t / W f e e ( ) (4.) W e q dt w f ( ) e f t / We dt W e W e f t / We e (4.3) In the above equaton,, 3, 4,... are coded varables for ndeendent varables ncludng ntal ressure gradent, ermeablty, reservor wdth, and aqufer sze. These factors wll be dscussed n Secton Water Produced Equaton (3.) ncludes water roduced, W. Before water breakthrough, the water roducton s zero. After breakthrough, the water roducton s ncreasng wth the gas roducton. Water breakthrough tme t bt s controlled by roducton means, reservor roertes and reservor geometry. It s defned as the second resonse and, t bt f,,,... (4.4), 3 4 3

43 Ths research assumes the water roducton rate after breakthrough can be aromated usng a second-order olynomal, q at t bt t arameters a and b are defned the thrd and fourth resonse. w bt bt. The a f,,,... 3, 3 4 (4.) b f,,,... (4.) 4, 3 4 t t f t t W f 3 bt 4 bt dt (4.7) 4..3 Cumulatve Gas Producton Ths study stulates that the gas s roduced at a constant tubng head ressure. Thus, gas roducton wll decrease wth the reservor ressure deleton, and the bottom hole ressure also changes wth gas roducton. To use Equaton (3.9) and (3.30), semsteady flow and constant gas roducton rate are assumed n a short tme nterval; that s, the reservor s assumed to ass through a successon of semsteady states. The cumulatve gas roducton can be aromated usng Al- Hussany, Ramey, Crawford Soluton Technque and the nflow erformance can be eressed as q 4T ln kh 4A C r A w S m m wf (3.30b) where non-darcy effects are neglected and sem-steady state flow s assumed n the above model. Equaton (3.30b) can be revsed to where m q C m (4.8) wf 3

44 G kh C 4A 4T ln C Ar Qdt C m w S m wf dt (4.9) C s determned by reservor roertes, geometry and skn. C s referred to as the gas roducton factor. Also, C s defned the ffth resonse and C f,,,... (4.0), 3 4 For a volumetrc reservor wthout water nflu, C s a constant. Unfortunately, aqufer water wll flow nto the gas zone and therefore the gas effectve ermeablty decreases wth the water nflu. Hence C wll usually decrease wth the water nflu and thus vary n tme. Ths comlcaton s neglected n the current study. Once these resonses are derved, they can be used to redct the reservor ressure at any tme for smlar reservors. Then the materal balance lot descrbed usng these resonses s as follows. In general for the n th tme erod, before water breakthrough: z n n G n z z T sc z sc n n t sc T t n n f n W e m m ( n G an n wf, n ) e dt f tn / We (4.) wf s the flowng bottom hole ressure and can be estmated usng vertcal flow erformance curves. 37

45 after water breakthrough: n n n W t f n n a e sc sc sc n n t t n wf n n n W e W T T z z G G dt m m f z z e n n n /,, ) ( (4.) where dt t t f t t f W bt n bt n n Cut-off lne The swee effcency s determned by reservor roertes and roducton condtons. From equaton (3.4), the cut-off lne s a straght lne, and the sloe and ntercet are determned by the resdual gas saturaton and swee effcency. In ths research, the swee effcency s defned as the sth resonse.,...,,, 4 3 f E (4.3) Once the resonse of effcency s derved, the cut-off lne at any cases s determned as f f S S f G G z z g gr (4.4) 4. Model Descrton Reservor smulaton was used to model water nflu nto gas reservors. The reservor can be dvded nto two regons, the gas zone and the aqufer. The water volume change n the regon aqufer s calculated wth the roducton of gas. 38

46 4.. Reservor Geometry and Proertes In ths study, a smle rectangular reservor model s used. The reservor length, wdth and thckness can be vared to dfferent levels for the smulaton desgns. In ths study, the gas zone length was set to 70 feet (when the reservor d was zero), and was not one of the eght factors eamned. The reservor d can be modeled by rotatng the reservor, keeng the reservor thckness unchanged. The gas zone ore volume was ket constant before and after the rotatng. The aqufer sze can be vared through addng to the gross model length. The gross model length s eanded and the eanded zone contans only water. A sketch of the smle gas-water system s llustrated n Fgure 4.. Gas WGC Addng Aqufer Fgure 4. Sketch of the Smle Rectangular Reservor Model The center elevaton of the gas zone was set at 000 feet. The ground surface temerature was set to 0 F and the temerature gradent was set to. F er 00 feet. The reservor temerature s 0 F. 39

47 The orosty was set to %. The rreducble water saturaton was set to 30% and the resdual gas saturaton was set to 0%. The vertcal ermeablty was set to 0% of the horzontal ermeablty. The gas and water roertes were estmated usng correlatons. The gas secfc gravty was set to 0. wthout consderng CO, H S and N. The gas vscosty was estmated usng the correlaton develoed by Lee et al. (9) and etended by Gonzalez et al. (98). The gas devaton factor was estmated usng correlatons resented by Dranchuk (Dranchuk et al., 974). The water secfc gravty was set to and the water vscosty was estmated usng correlatons ublshed by Numbere et al. (977). McMullan (000) also cted these methods. The gas-water two-hase relatve ermeabltes resented by McMullan (000) were used. Callary ressure was gnored n ths study. Ths study consdered only one roducng well. The well was drlled at the center of the gas zone. Ths study stulated the well roduces at constant tubng head ressure. The tubng head ressure was related to the bottom hole ressure usng Gray method (Eclse Reference Manual, 000A). 4.. Grd Descrton Block-centered grd would have worked fne for ths constant-thckness rectangular reservor. At the begnnng of ths study, consderng the aqufer zone could be grdded nto wedge-shaed zone, corneronts grd was selected. Smulaton runs usng corneronts grds have demonstrated the mortance of accurately reresentng the geometry of rock roerty varatons. Corner ont grd 40

48 was used n ths study. The reservor was grdded nto fve layers n the vertcal drecton Reservor Factor Ranges Prelmnary work determned ranges of ntal reservor ressure, aqufer ermeablty, and gas roducton rate to the materal balance (Agarwal, 9). In ths study, eght factors were selected wth varyng levels. The eght factors were llustrated n Table 4.. The ntal reservor ressure gradent PIDZ= / Deth ( Deth s the center elevaton of the reservor.) was used to arameterze the ntal reservor ressure. Its ranges were set to 0.7 to 0.9. It s transformed usng codng functon PIDZ Permeablty functon log k. k was set to varyng from 0 to 000 md. It s transformed usng Table 4. Factors Consdered for Materal Balance Factors Varables Coded Levels Varables - 0 Reservor ressure PIDZ gradent Permeablty K Reservor wdth W Aqufer sze AQ Reservor thckness H 7 7 Tubng sze DT Tubng head ressure PTH Reservor d DIP The aqufer sze AQ= L / (L 70 ) were vared from 3 to 8, where a L g g L a s the effectve aqufer zone length. It s defned as 4

49 L a V a WH where V s the bulk volume of the aqufer zone. Aqufer sze s transformed usng a functon AQ 0. Those fve other factors were transformed usng the same format wth reservor ressure gradent and aqufer sze. 4.3 Smulaton Desgn A full two-level factoral desgn for eght factors wll have 8 () runs. The artal factoral desgn s an effectve method to reduce the number of runs. In ths study, a 8- artal factoral desgn was used, then only 8 runs are requred n ths knd of desgn. One desgn generator secfes a 8- artal factoral desgn. The generator s ABCDEFGH and the defnng relaton s I=ABCDEFGH. The results after confoundng wll be: The seven-factor nteractons are alased wth the man effect; The s-factor nteractons are alased wth two-factor nteractons; The fvefactor nteractons are alased wth three-factor nteractons; The four-factor nteracton are alased wth each other. They are shown as follows. A I A ABCDEFGH A BCDEFGH AB I AB ABCDEFGH AB CDEFGH ABC I ABC ABCDEFGH ABC DEFGH 4

50 ABCD ABCD ABCDEFGH ABCD EFGH 4.4 Senstvty Analyss 8 smulatons were run usng Eclse 00, and the necessary oututs ncludng reservor ressure and water change n the aqufer were wrtten nto summary fles. These summary fles were used to calculate the aqufer roductvty nde, gas roducton factor, swee effcency and gas recovery Matchng Aqufer Productvty Inde Equaton (3.) was used to calculate the water roductvty nde. One-half year was set as the dfference tme ste. Average ressures were calculated usng Equaton (3.) and (3.7). For each run, the aqufer roductvty nde was calculated usng nonlnear regresson (Mcrosoft Ecel 000, solver) to match the water nflu calculated usng Fetkovtch theory wth the smulaton results. The water roductvty nde for all 8 smulatons was then set as the deendent varable or resonse. A multle lnear regresson was run to relate the water roductvty nde to the eght factors. The frst-order olynomal model consderng the two-term nteracton between the eght factors was used n the multle regressons. Ths regresson results are llustrated n Table 4.3. Table 4.3a Analyss of Varance for Aqufer Productvty Inde Source df SS MS F Sgnfcance F Model <0.000 Error Total

51 Table 4.3b Regresson Statstcs - Aqufer Productvty Inde Regresson Statstcs R Square Adjusted R Square Table 4.3c Parameter Estmates for Water Productvty Inde (For Sgnfcant Terms Only) Varable DF Parameter Estmate Standard Error t Value Pr > t Intercet <.000 K <.000 W <.000 H <.000 DIP K*W <.000 K*H <.000 K*DIP W*H <.000 H*DIP A 0% sgnfcance level was set. The coeffcents statstcally sgnfcantly ( n) n dfferent from zero are llustrated n Table 4.3c. Of the ma 37 ossble terms n the lnear model wth two-term nteractons, only 0 are sgnfcant. Fgure 4. shows whch terms are sgnfcant and whch are not. Insgnfcant terms can be deleted from the frst order model. Stewse regresson was further done to these sgnfcant factors. The stewse regresson results wll be used as the resonse model. The derved frst-order model wth nteracton for aqufer roductvty nde s J w (4.) 44