A New Analytical Method for Analyzing Linear Flow in Tight/Shale Gas Reservoirs: Constant-Flowing-Pressure Boundary Condition

Transcription

1 A New Analytcal Method for Analyzng Lnear Flow n Tght/Shale Gas Reservors: Constant-Flowng-Pressure Boundary Condton Morteza Nobakht, * SPE, Unversty of Calgary and Fekete Assocates, and C.R. Clarkson, SPE, Unversty of Calgary Summary Many tght/shale gas wells exhbt lnear flow, whch can last for several years. Lnear flow can be analyzed usng a square-root-oftme lot, a lot of rate-normalzed ressure vs. the square root of tme. Lnear flow aears as a straght lne on ths lot, and the sloe of ths lne can be used to calculate the roduct of fracture half-length and the square root of ermeablty. In ths aer, lnear flow from a fractured well n a tght/shale gas reservor under a constant-flowng-ressure constrant s studed. It s shown that the sloe of the square-root-of-tme lot results n an overestmaton of fracture half-length, f ermeablty s known. The degree of ths overestmaton s nfluenced by ntal ressure, flowng ressure, and formaton comressblty. An analytcal method s resented to correct the sloe of the squareroot-of-tme lot to mrove the overestmaton of fracture halflength. The method s valdated usng a number of numercally smulated cases. As exected, the square-root-of-tme lots for these smulated cases aear as a straght lne durng lnear flow for constant flowng ressure. It s found that the newly develoed analytcal method results n a more relable estmate of fracture half-length, f ermeablty s known. Our aroach, whch s fully analytcal, results n an mrovement n lnear-flow analyss over revously resented methods. Fnally, the alcaton of ths method to multfractured horzontal wells s dscussed and the method s aled to three feld examles. Introducton The domnant flow regme observed n most fractured tght/shale gas wells s lnear flow, whch may contnue for several years. The square-root-of-tme lot, normalzed ressure vs. square root of tme, s robably the most mortant lot for analyzng lnear flow (Anderson et al. 2). Lnear flow aears as a straght lne on the square-root-of-tme lot. The sloe of ths lne can be used to calculate the roduct of fracture half-length and the square root of ermeablty. Ths means that the fracture half-length can be derved from lnear-flow analyss, gven that the ermeablty s known. The y- ntercet on the square-root-of-tme lot reresents an aarent skn. It s reorted n the lterature that usng the sloe ffffof the square-root-of-tme lot leads to an overestmaton of x f k (Ibrahm and Wattenbarger 25, 26; Nobakht et al. 2), where x f s the fracture half-length and k s the ermeablty. Ibrahm and Wattenbarger (25, 26) ntroduced a correcton factor that corrects ffffthe sloe of the square-root-of-tme lot and mroves the x f k values obtaned from the square-root-of-tme lot. They develoed the followng emrcal equaton to estmate the correcton factor under a constant-flowng-ressure condton: f CP ¼ :852D D :857D 2 D ; * Now wth Encana Cororaton Coyrght VC 22 Socety of Petroleum Engneers... Ths aer (SPE 43989) was acceted for resentaton at the Amercas Unconventonal Gas Conference, The Woodlands, Texas, USA, 2 6 June 2, and revsed for ublcaton. Orgnal manuscrt receved for revew 5 May 2. Revsed manuscrt receved for revew 3 January 22. Paer eer aroved 7 March 22. ðþ where f CP s the correcton factor and D D s the drawdown arameter, whch s related to seudoressure at ntal ressure,, and seudoressure at flowng ressure, wf, usng Eq. 2, D D ¼ wf :... ð2þ In ths aer, the correcton of the sloe of the square-root-oftme lot for constant flowng ressure s analytcally derved. The method s then valdated by comarng ts results aganst test cases that are bult usng numercal smulaton. The effects of ntal ressure, flowng ressure, ermeablty, and formaton comressblty are nvestgated. It s found that n general, the correcton factors obtaned analytcally usng the methodology ntroduced n ths aer are more relable than those obtaned usng Eq.. Dervaton The base reservor geometry that s used ffffto analytcally develo the correcton factor for calculatng x f k from the sloe of the square-root-of-tme lot s shown n Fg.. Ths base geometry was chosen because n tght/shale gas reservors, t s reasonable to assume that dranage beyond the stmulated regon s nsgnfcant (Carlson and Mercer 989; Mayerhofer et al. 26; Bello and Wattenbarger 28). It s also assumed that the fracture has nfnte conductvty and that there s no skn. In ractce, tght gas and shale gas wells are roduced under hgh drawdown to maxmze roducton. Therefore, the equatons resented here are based on the assumton of a constant flowng ressure at the well. Lnear-flow theory (Wattenbarger et al. 998; El-Banb and Wattenbarger 998) ndcates ff that at a constant flowng ressure, a lot of =q vs. t on Cartesan coordnates s ffff a straght lne. The sloe of ths lne can be used to calculate x f k for gas as ffff 35:4T x f k ¼ qffffffffffffffffffffffffffffffff h ð/l g c t Þ wf m :... ð3þ In ths equaton, x f s the fracture half-length; k s the reservor ermeablty; T s the reservor temerature; h s the net-ay thckness; / s the reservor orosty; l g s gas vscosty; c t s total comressblty (subscrt refers to ntal condtons); and wf are seudoressures at ntal ressure and flowng ff ressure, resectvely; and m s the sloe of the =q vs. t lot. As mentoned ffff earler, usng ths sloe n Eq. 3 causes an overestmaton of x f k. Ths s because Eq. 3 s based on lqud-flow theory and the use of seudoressure to account for gas. However, ths substtuton alone s not suffcent. Pseudotme should also have been ncororated to account for changng gas comressblty (Fram and Wattenbarger 987; Agarwal et al. 999; Anderson and Mattar 25), rovded that the correct reference ressure ffff s ncluded. In other words, ffffff to obtan the correct value of x f k, the sloe of =q vs. t a should be used n Eq. 3, where ta s seudotme and s defned as follows: ð t dt t a ¼ðl g c t Þ : l g c t... ð4þ 37 June 22 SPE Reservor Evaluaton & Engneerng

2 Z ¼ Z G :... ðþ G 2y Fracture y e Here, Z and Z ** are modfed Z-factors [ntroduced by Moghadam et al. (2)] at average ressure n the regon of nfluence and ntal ressure, resectvely. Substtutng G and G from Eq. 6 and Eq. 9, resectvely, nto Eq. leads to 2 qffffffffffffffffffffffffffffffff 3 Z ¼ 2; B g ð/l g c t Þ 4 Z ffff5:... ðþ 4 :59mh/S g x f k Here, l g and c t are gas vscosty and total comressblty at the average reservor ressure, resectvely. Tradtonally, the average ressure n the whole reservor s beng used to calculate the seudotme (Fram and Wattenbarger 987; Agarwal et al. 999). However, as ndcated by Anderson and Mattar (25), ths ntroduces serous errors, esecally n low-ermeablty reservors. Anderson and Mattar (25) suggested usng the average ressure n the regon of nfluence to calculate seudotme. The seudotme calculated usng average ressure n the regon of nfluence s called corrected seudotme (Anderson and Mattar 25). In ths study, the corrected seudotme s beng used for dervaton. Because t s assumed that there s no skn effect, the lnear flow can be reresented by q ¼ m ff t:... ð5þ Usng Eq. 5, the cumulatve roducton at tme t s G ¼ ; ð t qdt ¼ x e Fg. A hydraulcally-fractured vertcal well n the center of a rectangular reservor. 2; ff t : m... ð6þ The unt of q n Eq. 5 s Mscf/D, and a converson factor of, s used n Eq. 6 to convert G to standard cubc feet (scf). When a well s roducng under constant flowng ressure, the dstance of nvestgaton, y, can be obtaned from the followng equaton durng the lnear-flow erod (Wattenbarger et al. 998): sffffffffffffffffffffffffffffffff kt y ¼ :59 :... ð7þ ð/l g c t Þ The contacted gas n lace (.e., gas n lace n the regon of nfluence) s G ¼ Ah/S g ;... ð8þ B g where A s the area of the regon of nfluence, S g s ntal gas saturaton, and B g s ntal gas formaton volume factor. Usng the defnton of area of the regon of nfluence, A ¼ 2x e y ¼ 4x f y, and relacng y from Eq. 7, results n G ¼ 4 :59h/S ffff gx f k ff qffffffffffffffffffffffffffffffff t :... ð9þ B g ð/l g c t Þ The unt of G n ths equaton s scf. The average ressure n the regon of nfluence,, can be calculated usng the followng equaton (Moghadam et al. 2): Ths equaton shows that for constant-flowng-ressure roducton, the average ressure n the regon of nfluence s not tme deendent durng lnear flow, and more mortantly, t s not ntal ressure or average reservor ressure. Because the average ressure n the regon of nfluence s constant, and by usng Eq. 4, the corrected seudotme, t a, becomes t a ¼ ðl gc t Þ t:... ð2þ l g c t Ths means that the corrected seudotme has a lnear relatonsh ffffff wth tme. Eq. 2 also shows that the ff sloe of the =q vs. t a lot, m, and the sloe of the =q vs. t lot, m, have the followng relatonsh: sffffffffffffffffffffffffff m ¼ m ðl g c t Þ :... ð3þ l g c t As ffff mentoned earler, n order to obtan the correct value for ffffff x f k when gas s beng analyzed, the sloe of the =q vs. t a lot should be used n Eq. 3. Therefore, usng ffff Eq. 3, the followng equaton ff can be used to calculate x f k from the sloe of the =q vs. t lot, m: sffffffffffffffffffffffffff ffff 35:4T x f k ¼ qffffffffffffffffffffffffffffffff h ð/l g c t Þ wf m ðl g c t Þ :... ð4þ l g c t Usng Eq. 4, ffff the correcton factor f CP that s used to mrove ff the values of x f k calculated from the sloe of the =q vs. t lot can be defned as sffffffffffffffffffffffffff ðl g c t Þ f CP ¼ :... ð5þ l g c t Ths equaton ndcates that the correcton factor s related to the average ressure ffff n the regon of nfluence and ntal ressure. Substtutng x f k from Eq. 4 and Bg ¼ :282Z T= nto Eq. results n " Z ¼ Z :28 ðzl sffffffffffffffffffffffffff # gc t Þ ð wf Þ l g c t : S g ðl g c t Þ ð6aþ Eq. 6a shows that the average ressure n the regon of nfluence deends on ntal ressure, flowng ressure, and gas roertes. Eq. 6a can be solved to obtan average ressure n the regon of nfluence, and then the correcton ffff factor, f CP, can be calculated usng ffff Eq. 5. To mrove x f k obtaned from lnear-flow analyss, x f k calculated from Eq. 3 can be multled by fcp. Note that Eq. 6a contans the average ressure n the regon of nfluence and the gas roertes (modfed Z-factor, vscosty, and total comressblty) at ths average ressure as well. To solve ths equaton, g () defned n Eq. 6b s lotted vs. ressure to fnd the ressure at whch g () becomes zero; gðþ ¼ Z Z " :28 ðzl sffffffffffffffffffffffffff # gc t Þ ð wf Þ l g c t S g : ðl g c t Þ ð6bþ June 22 SPE Reservor Evaluaton & Engneerng 37

3 TABLE INPUT PARAMETERS USED FOR NUMERICAL SIMULATION FOR DIFFERENT CASES USED TO VALIDATE CORRECTION FACTORS ANALYTICALLY DERIVED IN THIS STUDY Case (s) wf (s) k (md) c f (s ), 5, , 5 3,. 6, 7 2, 8 4, 9 6, 7, 8, 2 2, 2 3 2, 5 4 2,, 5 2,,5 6 2,,75 7, , , 5 6 2, , , , , , , , , , , , , * The blank cells ndcate that the value for that arameter s the same as that of Case. Valdaton To valdate the correcton factor analytcally derved n ths study, a number of test cases were bult for a year of numercally smulated roducton rofle obtaned from a black-ol smulator for a sngle-orosty reservor. The common arameters among all the test cases are as follows: T ¼ 2 F, h ¼ ft, / ¼ %, S g ¼ %, c g ¼ 5, x f ¼ 25 ft, x e ¼ 5 ft, and y e ¼ 5, ft. The nut data for ntal ressure, flowng ressure, ermeablty, and formaton comressblty for the numercal-smulaton cases are gven n Table. The blank cells n ths table ndcate that the value for that arameter s the same as that of Case. To model the hydraulc fracture n the numercal smulaton, we added a hgh-ermeablty grd n the x-drecton at the center of the reservor. The ermeablty of ths grd was chosen to be large enough to allow a fracture conductvty of F CD ¼ 4. In other words, the hydraulc fracture was assumed to have nfnte conductvty n these smulated cases (.e., neglgble ressure dro along the fracture). Logarthmc grddng was used to model ressure transents accurately. The sgnatures of exected flow regmes for the hydraulcally fractured well shown n Fg. were observed on the semlog dervatve lot (not shown here), suggestng the grddng used was adequate. Frst, and for each case, the average ressure n the regon of nfluence was calculated from Eq. 6a and the correcton factor ffff was analytcally calculated usng Eq. 5. ff Then, x f k was calculated usng the sloe of the =q vs. t lot ffff n Eq. 3, and t was comared wth the exected value for x f k (.e., entered nto numercal smulaton). Fnally, the exected ffff correcton factor ffff was calculated as the rato of exected x f k to calculated xf k. The comarson between analytcally calculated and exected correcton factors for dfferent values of formaton comressbltes s shown n Fgs. 2 through 4. For c f ¼ and c f ¼ 5 6 s (Fgs. 2 and 3, resectvely), the analytcal method resented n ths study underestmates the correcton factor, whereas Eq. overestmates the correcton factor. However, the average of analytcally obtaned correcton factors and those obtaned usng Eq. agrees well wth the exected correcton factors. Fg. 4 shows that for c f ¼ 5 5 s, n general, both methods underestmate the correcton factor. However, analytcally obtaned correcton factors are n better agreement wth exected correcton factors comared wth those obtaned from Eq.. Ths s because the correlaton develoed by Ibrahm and Wattenbarger (25, 26) does not nclude the effect of formaton comressblty, whereas the analytcal method resented n ths study consders the effect of formaton comressblty through the total comressblty. Imact of the Dstance-of-Investgaton Equaton Although we have rovded ff an analytcal method for correctng the sloe of the =q ffff vs. t lot, the new method stll leads to underestmatng x f k. In ths secton, the cause of ths underestmaton s nvestgated. Accordng to Eq. 7, the end of lnear flow s gven by sffffffffffffffffffffffffffffffffffff y e 2 ¼ :59 kt elf ;... ð7þ ð/l g c t Þ where y e s reservor length and t elf s the duraton of lnear flow. Eq. 7 can be rewrtten as follows: 2 qffffffffffffffffffffffffffffffff32 y e ð/l g c t Þ t elf ¼ 4 2 :59 ffff 5 :... ð8þ k Case 5 (where ¼, s, wf ¼ 3, s, k ¼. md, and c f ¼ ) s chosen to evaluate Eq. 8. The end of lnear flow s estmated to be 33 days for Case 5 usng Eq. 8. Fg. 5 shows the ressure dstrbuton n the reservor for ths case after 33 days. Although Eq. 8 redcts that the end of lnear flow (or start of boundary-domnated flow) s after 33 days, ths fgure clearly ndcates that the ressure roagaton reaches the boundares before 33 days. Fg. 6 shows the semlog dervatve vs. tme lotted on log-log coordnates. Ths fgure also shows that the lnear flow ended before 33 days n Case 5. Therefore, there s a ossblty that Eq. 7 underestmates the dstance of nvestgaton, whch results n underestmatng the average ressure n the regon of nfluence. Ths can exlan underestmatng the correcton factor usng the method ntroduced n ths study. To evaluate Eq. 7 for calculatng the dstance of nvestgaton, a number of test cases were bult usng numercal smulaton. The nut data for ntal ressure, flowng ressure, and ermeablty for these cases are gven n Table 2. The formaton comressblty s zero, and the other arameters not lsted n Table 2 are the same as n revously resented cases. Frst, and for each case, the end of lnear flow was calculated usng Eq. 8. Then, the end of lnear flow was obtaned usng the ressure-dstrbuton rofle n the reservor. The end of lnear flow n ths method s defned as the tme at whch the ressure dro at the uer and lower reservor boundares reaches % of the maxmum ressure dro [.e., D ¼.( wf )]. For examle, for ¼, s and wf ¼ 3, s, the end of lnear flow s assumed to be reached when the ressure at the boundares reaches 9,3 s. Fg. 7 shows a lot of the end of lnear flow, obtaned usng the ressure rofle n the reservor, (t elf ) o, vs. the end of lnear flow calculated 372 June 22 SPE Reservor Evaluaton & Engneerng

4 5 Ibrahm and Wattenbarger (25, 26) Ths study y=x Calculated f CP Exected f CP Fg. 2 Comarson between calculated correcton factors calculated from ths study (combnaton of Eqs. 5 and 6a) and Ibrahm and Wattenbarger (25, 26) method wth exected correcton factor for c f 5. usng Eq. 8, (t elf ) c. On the bass of ths data, (t elf ) o s correlated to (t elf ) c by alyng the lnear regresson ðt elf Þ o ¼ :63ðt elf Þ c :... ð9þ Usng Eq. 9, the followng equaton can be obtaned to calculate the dstance of nvestgaton: sffffffffffffffffffffffffffffffff kt y % ¼ :23 ;... ð2þ ð/l g c t Þ where y % s the dstance at whch ressure dro s % of the maxmum ressure dro. Usng Eq. 2 nstead of Eq. 7 for dervaton resented n the revous sectons, Eq. 6a wll be changed to the followng equaton: 5 Ibrahm and Wattenbarger (25, 26) Ths study y=x Calculated f CP Exected f CP Fg. 3 Comarson between calculated correcton factors calculated from ths study (combnaton of Eqs. 5 and 6a) and Ibrahm and Wattenbarger (25, 26) method wth exected correcton factor for c f s. June 22 SPE Reservor Evaluaton & Engneerng 373

5 .5 Ibrahm and Wattenbarger (25, 26) Ths study y=x Calculated f CP Exected f CP Fg. 4 Comarson between calculated correcton factors calculated from ths study (combnaton of Eqs. 5 and 6a) and Ibrahm and Wattenbarger (25, 26) method wth exected correcton factor for c f s. Z ¼ Z " :22 ðzl ffffffffffffffffff# gc t Þ ð wf Þ l g c t S g ðl g c t Þ :... ð2þ The comarson between analytcally calculated correcton factors usng Eq. 2 nstead of Eq. 6a and exected correcton factors for dfferent values of formaton comressbltes s shown n Fgs. 8 through. It can be seen that usng Eq. 2 sgnfcantly mroves the analytcally calculated correcton factors. Aroxmate Soluton In ths secton, the smlfed form of Eq. 2 s obtaned usng the followng assumtons:. The gas s deal (Z ¼ ). Usng the defnton of gas comressblty, ths assumton leads to c g ¼ dz Z d ¼ :... ð22þ y e =5, ft Hydraulc fracture, 9,3 8,6 7,9 7,2 6,5 5,8 5, 4,4 3,7 3, x e =5 ft Fg. 5 Pressure dstrbuton n the reservor after 33 days for Case 5 ( 5, s, wf 5 3, s, k 5. md, and c f 5 ). 374 June 22 SPE Reservor Evaluaton & Engneerng

6 Semlog Dervatve, ( 6 s 2 /c)/(mmscf/d) t =8 days t =33 days Tme, Days Fg. 6 Semlog dervatve lot for Case 5 ( 5, s, wf 5 3, s, k 5. md, and c f 5 ). 2. Gas vscosty s not changng wth ressure. Usng the defnton of seudoressure, ð ¼ 2 l g Z d ¼ 2 ð d ¼ 2 :... ð23þ l g l g 3. Total comressblty s domnated by gas comressblty, that s, c t ¼ S g c g :... ð24þ Eq. 25 s the smlfed form of Eq. 2, whch s obtaned by combnng Eqs. 2 through 24 as follows: ¼ :22 rffffffff wf :... ð25þ TABLE 2 INPUT PARAMETERS USED FOR NUMERICAL SIMULATION FOR DIFFERENT CASES TO EVALUATE EQ. 7 (s) wf (s) k (md),,. 3,. 5,. 5,.25 5,. 5,.5 5,,.,.,.25, 2, Because gas vscosty s assumed to be constant and gas comressblty s assumed to be nversely roortonal to ressure, Eq. 5 smlfes to sffffffffffffffffffffffffff rffffffff rffffffff ðl g c t Þ f CP ¼ c t ¼ ¼ :... ð26þ l g c t c t Combnng Eqs. 25 and 26, we wll end u wth the followng equaton: f 3 CP f CP þ :22D D ¼ ;... ð27þ where D D s the drawdown arameter defned n Eq. 2. Eq. 27 shows that under the assumtons resented revously, the correcton factor, f CP, deends only on the drawdown arameter defned by Ibrahm and Wattenbarger (25, 26). Fg. shows the comarson between the correcton factors obtaned from Eq. and Eq. 27. Ths fgure shows that the correcton factors obtaned usng Eq. 27 and the emrcal values calculated from the correlaton develoed by Ibrahm and Wattenbarger (25, 26) are very close for an deal gas wth constant vscosty when gas comressblty domnates the total comressblty. Dfferences Between Constant-Flowng-Pressure and Constant-Rate-Producton Lnear Flow Nobakht and Clarkson (22) resented a method to analyze lnear flow n gas reservors roducng wth constant-rate roducton. Comarng the fndngs n that work and those n ths study, the followng dfferences are observed between constant-flowng-ressure and constant-rate-roducton lnear flow n gas reservors:. The average ressure n the regon of nfluence s constant (Eq. ) for constant flowng ressure, whereas the average ressure n the regon of nfluence s tme deendent and decreases wth tme for constant-rate roducton. 2. The corrected seudotme has a lnear relatonsh wth tme (Eq. 2) for constant-flowng-ressure roducton. However, for constant-rate roducton, corrected seudotme s almost equal June 22 SPE Reservor Evaluaton & Engneerng 375

7 (t elf ) o, Days (t elf ) c, Days Fg. 7 The end of lnear flow obtaned usng the ressure rofle n the reservor, (t elf ) o, vs. the end of lnear flow calculated usng Eq. 8, (t elf ) c. to tme at early tme and becomes smaller than tme as tme ncreases. In other words, the relatonsh between corrected seudotme and tme s not lnear for constant-rate roducton. 3. For constant-flowng-ressure roducton, the square-rootof-tme lot s a straght lne; however, for constant-rate roducton, the square-root-of-tme lot may not be a straght lne f corrected seudotme s not used, and ts shae deends on gasroducton rate. 4. For constant-rate roducton, the roosed analyss method s teratve, whereas for constant-flowng-ressure roducton, the 5 5 Calculated f CP Exected f CP Fg. 8 Comarson of calculated correcton factors calculated from ths study (combnaton of Eqs. 5 and 2) wth exected correcton factor for c f June 22 SPE Reservor Evaluaton & Engneerng

8 5 5 Calculated f CP Exected f CP Fg. 9 Comarson of calculated correcton factors calculated from ths study (combnaton of Eqs. 5 and 2) wth exected correcton factor for c f s. rocedure resented s not teratve and t nvolves only fndng a correcton factor. Alcaton to Multfractured Horzontal Wells Comleted n Ultralow-Permeablty Reservors Multfractured horzontal wells are the most commonly used method for exlotng shale gas reservors. Because massve hydraulc fractures are tycally created, the domnant flow regme observed n these wells s lnear flow to fractures that may last for a long tme because of extremely low ermeablty of shale gas reservors. Therefore, t s of ractcal nterest to valdate the methodology resented n ths study for alcaton n multfractured horzontal wells. Lnear-flow analyss results n the roduct of total fracture length (or surface area to flow) and square root of ermeablty. If.5 Calculated f CP Exected f CP Fg. Comarson of calculated correcton factors calculated from ths study (combnaton of Eqs. 5 and 2) wth exected correcton factor for c f s. June 22 SPE Reservor Evaluaton & Engneerng 377

9 Eq. Eq Correcton Factor Drawdown Parameter Fg. Comarson between correcton factors obtaned usng Eq. and those obtaned usng Eq. 27. the ermeablty s known, the total fracture length can be obtaned. Ths means that when analyzng lnear flow for a hydraulcally fractured vertcal well, the length of the fracture can be obtaned, whereas lnear-flow analyss for a multfractured horzontal well results n total fracture length (.e., the summaton of lengths of all fractures). Ths s essentally the dfference between analyzng lnear flow for hydraulcally fractured vertcal wells and horzontal wells wth multle fractures. On the bass of ths dscusson, one can lot nverse gas rate vs. square root of tme for a multfractured horzontal well and use the sloe of ths lot n Eq. 3 to calculate the roduct of total fracture length and square root of ermeablty. Then, ths value can be multled by the correcton factor calculated usng the method resented n ths study or that calculated from Eq.. The ermeablty for the test cases resented n Table s more arorate for tght gas reservors. To valdate the alcablty of the method resented n ths study for shale gas reservors, a new test case was bult smlar to Case 24 n Table, excet c f ¼ and the ermeablty was reduced to k ¼ nd. Usng the sloe of nverse gas rate vs. the square root of tme lot (not shown here) n Eq. 3 and k ¼ nd, fracture half-length was calculated to be 325 ft. The correcton factor obtaned usng the L e 2y f Fg. 2 Schematc of the multfractured horzontal well used to valdate the alcablty of the methodology resented n ths study when fractures have fnte conductvty. method resented n ths study (.e., Eq. 5) for ths case was.77. Therefore, the mroved fracture half-length was ¼ 25 ft, whch was the same as the exected value of 25 ft (.e., the nut to numercal smulaton). Ths demonstrates the valdty of the method resented n ths study n ermeablty range for shale gas reservors. For comarson, the correcton factor usng the Ibrahm and Wattenbarger (25, 26) method (calculated from Eq. ) was 3, whch corresonds to a fracture halflength of 27 ft. Dscusson The dervaton resented n ths study s develoed assumng nfnte-conductvty fracture(s) and no skn. However, as mentoned by Nobakht and Mattar (22), shale gas roducton often exhbts lnear flow wth a sgnfcant aarent skn. The aarent skn can be caused by flow convergence n a horzontal well (Bello and Wattenbarger 2), fnte conductvty of the fractures (Anderson et al. 2), and/or multhase flow n the reservor (Nobakht and Mattar 22). As mentoned by Nobakht and Mattar (22), all these effects create an extra ressure dro, whch for all ractcal uroses, can be treated as an aarent skn effect. The methodology resented n ths study can be aled when dealng ffff wth fractures wth fnte conductvty. In other words, x f k calculated from Eq. 3 can be multled by the correcton factor to mrove the lnear-flow analyss. To verfy ths, a smulaton case was created for the multfractured horzontal well shown n Fg. 2. The reservor ermeablty was assumed to be nd, the length of the horzontal well was 2,5 ft, and there were fve hydraulc fractures, each of whch had a 52-ft halflength and dmensonless fracture conductvty F CD of 5. A smulated gas-rate rofle was created usng real flowng-ressure data obtaned from a multfractured horzontal well that was roducng under hgh drawdown. Fg. 3 shows a lot of nverse gas rate vs. square root of tme for ths case. Usng the sloe of ths lne n Eq. 3 and k ¼ nd, the total fracture half-length becomes 3,485 ft. Therefore, the half-lengths of ndvdual fractures y f were calculated to be 697 ft, whch s 35% greater than the exected value of 52 ft (.e., nut to smulaton). Usng the correcton 378 June 22 SPE Reservor Evaluaton & Engneerng

10 2 Inverse Gas Rate Lnear Flow Lne.6 Inverse Gas Rate, /(MMscf/D) Square Root of Tme, Day.5 Fg. 3 Inverse gas rate vs. the square root of tme lot for the test case used to valdate the methodology resented n ths study when fractures have fnte conductvty. factor resented n ths study mroves y f to 54 ft. It should be mentoned that f we use the correcton factor from Eq., y f becomes 585 ft. Fg. 3 shows that the ntercet of the lne s ostve, whch s an ndcaton of aarent skn (Nobakht and Mattar 22), caused n ths case by fnte conductvty of fractures. Feld Examles Case. Ths case study s for a multfractured horzontal well drlled n a Barnett-shale gas reservor wth ¼ 3, s, T R ¼ 6 F, h ¼ 26 ft, / ¼ 4%, S g ¼ 9%, S w ¼ %, and c f ¼ 7 6 s. Fg. 4 shows the drawdown arameter, D D, vs. tme for Drawdown Parameter Tme, Days Fg. 4 Drawdown arameter vs. tme lot for Feld Examle. June 22 SPE Reservor Evaluaton & Engneerng 379

11 .5 Inverse Gas Rate Lnear Flow Lne.4 Inverse Gas Rate, /(MMscf/D) Square Root of Tme, Days.5 Fg. 5 Inverse gas rate vs. the square root of tme lot for Feld Examle. ths well. Ths fgure shows that the well s roducng under hghdrawdown condton, and therefore the methodology resented n ths study for constant flowng ressure can be aled. Fg. 5 shows the nverse gas rate vs. the square root of tme lot for ths case. Clearly, the data form a straght lne that ndcates that lnear flow s the domnant flow regme. Usng the sloe of the lot n Eq. 3 and assumng k ¼ nd, the total fracture half-length s calculated to be 3,29 ft. For ths case, the correcton factor calculated from Eq. [.e., Ibrahm and Wattenbarger (25, 26)] s aroxmately 4 and the correcton factor calculated from the method resented n ths study (.e., combnaton of Eqs. 5 and 2) s aroxmately 6. Therefore, the total fracture halflengths calculated usng these two methods are 2,76 and 2,83 ft, resectvely. In ths examle, the correcton factors calculated from Eq. and Eq. 5 are very close. Case 2. The varaton of drawdown arameter wth tme for ths multfractured horzontal well s shown n Fg. 6. Ths fgure ndcates that, for the most art, ths well s roducng under hghdrawdown condton, and therefore the methodology resented n ths study for constant flowng ressure can be aled. Fg. 7 shows the nverse gas rate vs. the square root of tme lot for ths well. Clearly, the data form a straght lne, ndcatng that lnear flow s the domnant flow regme. Usng the sloe of the lot n Eq. 3 and assumng k ¼ nd, the total fracture half-length s calculated to be 2,57 ft. For ths case, the correcton factors calculated from Eq. and Eq. 5 are aroxmately 5 and.74, resectvely. Therefore, the total fracture half-lengths calculated usng these two methods are 2,8 and,9 ft, resectvely. Case 3. Ths case, whch s a 5,-ft horzontal well wth equally saced fractures along the horzontal well, s resented to show the alcablty of the method resented n ths study to real data when dealng wth fnte-conductvty fractures. The roertes of ths reservor (Marcellus-shale gas) are as follows: ¼ 5,3 s, T R ¼ 5 F, h ¼ ft, / ¼ 8%, S g ¼ 76%, S w ¼ 24%, and c f ¼ 5 6 s. The lot of drawdown arameter vs. tme for ths well s shown n Fg. 8. Ths fgure shows that, as wth Case and Case 2, ths well s also roducng under hgh drawdown (between 95 and %). Fg. 9 shows the nverse gas rate vs. the square root of tme lot for ths well. The data form a straght lne, ndcatng exstence of lnear flow throughout the roducton to date. Ths fgure also shows that the ntercet of the lne s ostve because of a sgnfcant amount of skn (Nobakht and Mattar 22). In the next ste, a numercal model s used to hstory match the roducton data. Because lnear flow was the only regme avalable, we consdered only the stmulated reservor volume for modelng (.e., schematc smlar to Fg. 2 wth fractures). Fg. 2 shows the comarson between rates obtaned from numercal smulaton and hstorcal data. The matrx ermeablty, dmensonless fracture conductvty, and half-length of each fracture s estmated to be k ¼ 9 nd, F CD ¼ 25, and y f ¼ 52 ft, resectvely. From the sloe of the lne n Fg. 9, the roduct of the total fracture half-length and the square root of the reservor (or matrx) ermeablty s estmated ffff to be md /2 ft from Eq. 3, whch corresonds to y f k of md /2 ft. Usng k ¼ 9 nd (obtaned from hstory matchng), y f s calculated to be 67 ft, whch s almost 3% greater than the value obtaned from numercal smulaton. Alyng the correcton factor resented n ths study, the half-lengths of each of the fractures become 529 ft, whch agrees very well wth the data obtaned from numercal smulaton. Usng the correcton factor from Eq., the mroved fracture half-length becomes 565 ft. Conclusons Ths aer resented an analytcal ffff method to determne a correcton factor for calculatng x f k from the sloe of nverse gas rate vs. the square root of tme lot for constant-flowng-ressure roducton. It was shown that for the reservor geometry shown n Fg., the average ressure n the regon of nfluence s constant 38 June 22 SPE Reservor Evaluaton & Engneerng

12 Drawdown Parameter Tme, Days Fg. 6 Drawdown arameter vs. tme lot for Feld Examle 2. durng lnear flow. The method was then valdated by comarng ts results aganst numercally smulated cases. It was found that, n general, the correcton factors obtaned usng the method roosed n ths study are lower than those exected. It was also found that Eq. 7 underestmates dstance of nvestgaton for lnear flow. Ths equaton was then modfed to match the dstance of nvestgaton observed n smulaton. Usng the modfed equaton for dstance of nvestgaton, the analytcally calculated.5 Inverse Gas Rate Lnear Flow Lne.4 Inverse Gas Rate, /(MMscf/D) Square Root of Tme, Days.5 Fg. 7 Inverse gas rate vs. the square root of tme lot for Feld Examle 2. June 22 SPE Reservor Evaluaton & Engneerng 38

13 8 Drawdown Parameter Tme, Days Fg. 8 Drawdown arameter vs. tme lot for Feld Examle 3. correcton factors were n good agreement wth those exected. Then, t was shown that for a case of deal gas wth constant vscosty, f total comressblty s domnated by gas comressblty, the correcton factor deends only on the drawdown arameter defned n Eq. 2. Fnally, the alcaton of the method to multfractured horzontal wells was dscussed, and the method was aled to three multfractured horzontal wells drlled n dfferent shale lays..5 Inverse Gas Rate Lnear Flow Lne.2 Inverse Gas Rate, /(MMscf/D) Square Root of Tme, Day.5 Fg. 9 Inverse gas rate vs. the square root of tme lot for Feld Examle June 22 SPE Reservor Evaluaton & Engneerng

14 4.5 4 Hstory Gas Rate Synthetc Gas Rate 3.5 Gas Rate, MMscf/D Tme, Days Fg. 2 Comarson between numercally smulated rates and hstorcal data for Feld Examle 3. Nomenclature A ¼ area of regon of nfluence, ft 2 B g ¼ ntal gas formaton volume factor, ft 3 /scf c f ¼ formaton comressblty, s c g ¼ gas comressblty, s c t ¼ total comressblty, s D D ¼ drawdown arameter defned n Eq. 2, fracton f CP ¼ correcton factor F CD ¼ dmensonless fracture conductvty, dmensonless G ¼ gas n lace n the regon of nfluence, scf G ¼ cumulatve gas roducton, scf h ¼ net-ay thckness, ft k ¼ ermeablty, md L e ¼ horzontal-well length, ff ft m ¼ sloe of the =q vs. t lot, day /2 /Mscf m ffffff ¼ sloe of the =q vs. t a lot, day /2 /Mscf ¼ ntal ressure, s ¼ seudoressure at ntal ressure, s 2 /c wf ¼ seudoressure at flowng ressure, s 2 /c q ¼ gas rate, Mscf/D S g ¼ ntal gas saturaton, fracton S w ¼ ntal water saturaton, fracton t ¼ tme, days t a ¼ seudotme, days t a ¼ corrected seudotme, days t elf ¼ end of lnear flow, days (t elf ) c ¼ end of lnear flow calculated usng Eq. 8, days (t elf ) o ¼ end of lnear flow obtaned usng the ressure rofle n the reservor, days T ¼ reservor temerature, R x e ¼ reservor wdth, ft x f ¼ fracture half-length n x-drecton, ft y ¼ dstance of nvestgaton, ft y % ¼ dstance at whch ressure dro s % of the maxmum ressure dro, ft y e ¼ reservor length, ft y f ¼ fracture half-length n y-drecton, ft Z ¼ gas-comressblty factor c g ¼ reservor gas secfc gravty (ar ¼ ) l g ¼ gas vscosty, c / ¼ orosty, fracton Acknowledgments The authors would lke to thank ConocoPhlls for ther suort of ths research. Chrs Clarkson would lke to acknowledge Encana for suort of hs char oston n Unconventonal Gas at the Unversty of Calgary, Deartment of Geoscence. Fnally, both authors would lke to thank Fekete Assocates, artcularly Lous Mattar and Mehran Poolad-Darvsh, for frutful dscussons on the subject of rate-transent analyss. References Agarwal, R.G., Gardner, D.C., Klensteber, S.W., and Fussell, D.D Analyzng Well Producton Data Usng Combned-Tye-Curve and Declne-Curve Analyss Concets. SPE Res Eval & Eng 2 (5): SPE-5796-PA. htt://dx.do.org/.28/5796-pa. Anderson, D.M. and Mattar, L. 25. An Imroved Pseudo-Tme for Gas Reservors wth Sgnfcant Transent Flow. Paer CIPC 25-4 resented at the Canadan Internatonal Petroleum Conference, Calgary, 7 9 June. htt://dx.do.org/.28/25-4. Anderson, D.M., Nobakht, M., Moghadam, S., and Mattar, L. 2. Analyss of Producton Data from Fractured Shale Gas Wells. Paer SPE 3787 resented at the SPE Unconventonal Gas Conference, Pttsburgh, Pennsylvana, USA, February. htt://dx.do.org/.28/ 3787-MS. Bello, R.O. and Wattenbarger, R.A. 28. Rate Transent Analyss n Naturally Fractured Shale Gas Reservors. Paer SPE 459 resented at the CIPC/SPE Gas Technology Symosum, Calgary, 6 9 June. htt://dx.do.org/.28/459-ms. Bello, R.O. and Wattenbarger, R.A. 2. Modellng and Analyss of Shale Gas Producton Wth a Skn Effect. J. Cdn. Pet. Tech. 49 (2): SPE PA. htt://dx.do.org/.28/43229-pa. June 22 SPE Reservor Evaluaton & Engneerng 383

15 Carlson, E.S. and Mercer, J.C Devonan Shale Gas Producton: Mechansms and Smle Models. J Pet Technol 43 (4): SPE- 93-PA. htt://dx.do.org/.28/93-pa. El-Banb, A.H. and Wattenbarger, R.A Analyss of Lnear Flow n Gas Flow Producton. Paer SPE resented at the SPE Gas Technology Symosum, Calgary, 5 8 March. htt://dx.do.org/.28/39972-ms. Fram, M.L. and Wattenbarger, R.A Gas Reservor Declne Curve Analyss Usng Tye Curves wth Real Gas Pseudoressure and Normalzed Tme. SPE Form Eval 2 (4): SPE-4238-PA. htt:// dx.do.org/.28/4238-pa. Ibrahm, M. and Wattenbarger, R.A. 25. Rate Deendence of Transent Lnear Flow n Tght Gas Wells. Paer CIPC resented at the Canadan Internatonal Petroleum Conference, Calgary, 7 9 June. htt://dx.do.org/.28/ Ibrahm, M. and Wattenbarger, R.A. 26. Analyss of Rate Deendence n Transent Lnear Flow n Tght Gas Wells. Paer SPE 836 resented at the Abu Dhab Internatonal Petroleum Exhbton and Conference, Abu Dhab, UAE, 5 8 November. htt://dx.do.org/.28/ 836-MS. Mayerhofer, M.J., Lolon, E.P., Youngblood, J.E., and Henze, J.R. 26. Integraton of Mcrosesmc Fracture Mang Results wth Numercal Fracture Network Producton Modelng n the Barnett Shale. Paer SPE 23 resented at the SPE Annual Techncal Conference and Exhbton, San Antono, Texas, USA, Setember. htt:// dx.do.org/.28/23-ms. Moghadam, S., Jeje, O., and Mattar, L. 2. Advanced Gas Materal Balance n Smlfed Format. J. Cdn. Pet. Tech. 5 (): SPE PA. htt://dx.do.org/.28/39428-pa. Nobakht, M. and Clarkson, C.R. 22. A New Analytcal Method for Analyzng Producton Data from Shale Gas Reservors Exhbtng Lnear Flow: Constant-Rate Boundary Condton. SPE Res Eval & Eng 5 (): SPE-4399-PA. htt://dx.do.org/.28/4399-pa. Nobakht, M. and Mattar, L. 22. Analyzng Producton Data from Unconventonal Gas Reservors wth Lnear Flow and Aarent Skn. J. Pet. Tech. 5 (): SPE PA. htt://dx.do.org/.28/ PA. Nobakht, M., Mattar, L., Moghadam, S., and Anderson, D.M. 2. Smlfed Yet Rgorous Forecastng of Tght/Shale Gas Producton n Lnear Flow. Paer SPE 3365 resented at the SPE Western Regonal Meetng, Anahem, Calforna, USA, May. htt://dx.do.org/.28/3365-ms. Wattenbarger, R.A., El-Banb, A.H., Vllegas, M.E., and Maggard, J.B Producton Analyss of Lnear Flow Into Fractured Tght Gas Wells. Paer SPE 3993 resented at the SPE Rocky Mountan Regonal/Low-Permeablty Reservors Symosum, Denver, 5 8 Arl. htt://dx.do.org/.28/3993-ms. Morteza Nobakht s a reservor engneer wth Encana n Calgary, where he s workng on exloraton and develoment of Montney lay. Before jonng Encana, he worked at Fekete Assocates for 4 years, where he was the lead techncal advsor for Fekete FAST RTA TM software. Nobakht secalzed n the analyss of roducton data from conventonal and unconventonal reservors, and he has authored more than 2 techncal aers. He holds BSc degrees n mechancal engneerng and etroleum engneerng from Sharf Unversty of Technology n Iran and an MASc degree n etroleum systems engneerng from the Unversty of Regna. Nobakht s also ursung a PhD degree n etroleum engneerng at the Unversty of Calgary. Chrstoher R. Clarkson s a rofessor and the Encana Char n Unconventonal Gas n the Deartment of Geoscence and an adjunct rofessor wth the Deartment of Chemcal and Petroleum Engneerng at the Unversty of Calgary. Hs work focus has been on exloraton for and develoment of coalbed methane, shale, tght gas sand, tght ol, and conventonal reservors. Clarkson has studed both rmary and enhanced coalbed methane recovery rocesses n the San Juan Basn of New Mexco/Colorado and s artcularly nterested n the hyscs of gas storage and transort n coalbed methane and shale reservors and the alcaton of roducton data analyss/reservor smulaton to otmze gas recovery and well erformance. He holds a PhD degree n geologcal engneerng from the Unversty of Brtsh Columba, Canada. The author of numerous artcles n eer-revewed scentfc and engneerng journals, Clarkson receved the Rosster W. Raymond Memoral Award from AIME and the Alfred Noble Prze from ASCE for hs aer Alcaton of a New Multcomonent Adsorton Model to Coal Gas Adsorton Systems ublshed n SPE Journal (Setember 23). He was selected as an SPE Dstngushed Lecturer for the 29 2 lecture season. 384 June 22 SPE Reservor Evaluaton & Engneerng