外文翻譯--頁(yè)巖氣藏生產(chǎn)率分析的新方法及改進(jìn)外文翻譯（英文）

1、New and Improved Methods for Performing Rate-Transient Analysis of Shale Gas ReservoirsMorteza Nobakht,* SPE, University of Calgary and Fekete Associates, C.R. Clarkson, SPE, and D. Kaviani,** SPE, University of CalgaryS

2、ummaryMultifractured horizontal wells are currently the most popular method for exploiting low-permeability tight and shale gas reser- voirs. Production data analysis is the most widely used tool for analyzing these rese

3、rvoirs for the purpose of reserves estimation, hydraulic fracture stimulation optimization, and development planning (Ambrose et al. 2011). However, as pointed out by Clark- son et al. (2012), a fundamental problem with

4、the application of conventional production data analysis to ultralow permeability res- ervoirs is that current methods were derived with the assumption that flow can be described with Darcy’s law. This assumption may not

5、 be valid for tight/shale gas reservoirs, as they contain a wide distribution of pore sizes, including in some cases nanopores (Loucks et al. 2009). Therefore, the mean-free path of gas mole- cules may be comparable to o

6、r larger than the average effective rock pore throat radius, causing the gas molecules to slip along pore surfaces. This results in slippage non-Darcy flow, which is not accounted for in conventional production data anal

7、ysis. Clarkson et al. (2012) modified the pseudovariables used for analyzing gas reservoirs in production data analysis to account for slippage. They demonstrated that if the effect of slippage is not con- sidered, it le

8、ads to noticeable errors in reservoir characterization. Clarkson et al. (2012) also mentioned that even after using the modified pseudovariables, the values for permeability and fracture half-length do not exactly match

9、the input data to simulation. In this paper, a methodology to properly analyze the production data from a fractured well in a tight/shale gas reservoir producing under a con- stant flowing pressure in the presence of des

10、orption and slippage is presented. This method uses a new pseudotime definition instead of the conventional pseudotime currently being used in production data analysis. The method is validated using a number of numeri- c

11、ally simulated cases. It is found that the newly developed analyti- cal method results in a more reliable estimate of fracture half-length or contacted matrix surface area, if permeability is known.IntroductionHorizontal

12、 wells (cased or open hole) with multiple fractures are commonly used to develop wells in most tight/shale gas plays. Because of massive hydraulic fractures in these wells, the domi- nant flow regime observed is linear f

13、low, which may continue for several years. It is documented in the literature that linear flow appears as a straight line on the square-root-of-time plot, which is a plot of normalized pressure vs. square root of time (W

14、atten- barger et al. 1998; El-Banbi and Wattenbarger 1998). Using the slope of this line overestimates the value of xf ffiffi ffi k p , where xf is frac- ture half-length and k is permeability, or Acm ffiffi ffi k p , wh

15、ere Acm is contacted matrix surface area, calculated from linear-flow analy- sis for constant-flowing-pressure production (Ibrahim and Watten-barger 2005, 2006; Nobakht et al. 2010; Nobakht and Clarkson 2011a). Ibrahim a

16、nd Wattenbarger (2005, 2006) proposed to mul- tiply xf ffiffi ffi k p , obtained using the slope of the square-root-of-time plot by an empirically obtained correction factor, fCP, under con- stant-flowing-pressure condit

17、ionfCP ¼ 1 ? 0:0852DD ? 0:0857D2 D; ð1Þwhere DD is the drawdown parameter and is related to pseudo- pressure at initial pressure, ppi, and pseudopressure at flowing pressure, ppwf, using Eq. 2:DD ¼ pp

18、i ? ppwf ppi : ð2ÞNobakht and Clarkson (2011a) studied the linear flow under constant-flowing-pressure production in detail. They ran a number of simulation cases for the reservoir geometry shown in Fig. 1 and

19、showed that the correction factor calculated from Eq. 1 is not correcting xf ffiffi ffi k p values obtained from the square-root-of-time plot to match input values to the numerical simulation. Nobakht and Clarkson (2011a

20、) explained that the overestimation of xf ffiffi ffi k p is because the square-root-of-time plot does not account for chang- ing gas viscosity and gas compressibility, which are incorporated into pseudotime, ta, (Fraim a

21、nd Wattenbarger 1987; Agarwal et al. 1999) asta ¼ ðlgctÞið t0dt? lg? ct : ð3ÞHere, ? lg and ? ct are gas viscosity and total compressibility at the average reservoir pressure. Nobakht and Cl

22、arkson (2011a) pro- posed to use corrected pseudotime in which the gas viscosity and gas compressibility in Eq. 3 are evaluated at the average pressure in the region of influence (Anderson and Mattar 2005). Finally, they

23、 developed an analytical method to correct overestimation of xf ffiffi ffi k p calculated from the square-root-of-time plot. The analytical method developed by Nobakht and Clarkson (2011a) does not account for desorption

24、 and gas slippage effects (non-Darcy flow). As noted by Clarkson et al. (2012), common assumptions used for the development of conventional production data analysis are not true for unconventional reservoirs with extreme

25、ly low perme- ability. One of these limitations is the existence of gas slippage (non-Darcy flow) in low-permeability reservoirs (Ozkan et al. 2010; Clarkson et al. 2012). In low-permeability reservoirs, the gas molecule

26、s may slip along the pore surfaces (i.e., the gas velocity at pore surfaces is not zero) and cause additional flux on top of the vis- cous flow expressed by Darcy’s law. Because of this additional flux, apparent gas perm

27、eability, ka, becomes higher than the liquid- equivalent permeability, k1, of the same porous medium. Clarkson et al. (2012) used the following pseudopressure (Eq. 4) and pseudo- time (Eq. 5) to include slippage effect i

28、nto production data analyis:p? pi ? p? pwf ¼ 2ð pipwfkr lgZ pdp; ð4Þt? a ¼ ðlgctÞið t0? krdt? lg? ct : ð5Þ. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .

29、 . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .** Now with ConocoPhilips CanadaCopyright V C 2012 Society of

30、 Petroleum EngineersThis paper (SPE 147869) was accepted for presentation at the SPE Asia Pacific Oil and Gas Conference and Exhibition, Jakarta, 20–22 September 2011, and revised for publication. Original manuscript rec

31、eived for review 26 July 2011. Revised manuscript received for review 5 February 2012. Paper peer approved 7 March 2012.* Now with Encana CorporationJune 2012 SPE Reservoir Evaluation ð15Þwhere kai is the appa

32、rent permeability at initial pressure. Apparent permeability at initial pressure, kai, is used in this equation as the pressure propagation is occurring against initial pressure, and therefore, permeability at initial pr

33、essure is used in Eq. 15 to cal- culate the distance of investigation. Using the definition of kai ðkai ¼ k1kriÞ; Eq. 15 becomesy ¼ 0:159ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffik1kritð

34、;/lgctÞis: ð16ÞHere, kri is the permeability ratio at initial pressure and k1 is the liquid-equivalent reservoir permeability. The contacted gas in place (i.e., gas in place in the region of influence), in

35、cluding adsorbed gas, isG ¼ Ah /Sgi Bgi þ 0:031214qBVLpi pL þ pi? ?: ð17ÞHere, A is the area of the region of influence; Sgi is the initial gas saturation; Bgi is the initial gas formation volume

36、 factor; qB is the shale bulk density; VL is the Langmuir volume; pL is the Langmuir pressure; and pi is the initial reservoir pressure. It is assumed that the gas content follows the Langmuir isotherm. Eq. 17 can be rep

37、- resented asG ¼ Ah/Sgi B? gi ; ð18Þwhere B? gi is the initial gas formation volume factor, adjusted to account for desorption effect, and is defined as (King 1993; Clark- son et al. 2007).1B? gi ¼ 1B

38、gi þ 0:031214 qB /SgiVLpi pL þ pi : ð19ÞUsing the definition of area of the region of influence, A ¼ 2xey ¼ 4xf y; and replacing y from Eq. 16 results inG ¼ 4 ? 0:159h/Sgixf ffiffiffiff

39、iffiffiffiffiffiffiffi k1kri pB? giffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ð/lgctÞiq ffiffi t p : ð20ÞThe unit of G in this equation is scf. The average pressure in the region of influenc

40、e, ? p, can be calculated using the following equa- tion (Moghadam et al. 2011):? p ? Z?? ¼ pi Z?? i 1 ? Gp G? ?: ð21ÞHere, ? Z?? and Z?? i are modified Z-factors [introduced by Mogha- dam et al. (2011)] a

41、t average pressure in the region of influence and initial reservoir pressure, respectively. Substituting Gp and G from Eq. 14 and Eq. 20, respectively, into Eq. 21 leads to? p ? Z?? ¼ pi Z?? i 1 ?2; 000B? giffiffiff

42、iffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ð/lgctÞiq4 ? 0:159mh/Sgixf ffiffiffiffiffiffiffiffiffiffiffi k1kri p2435: ð22ÞThis equation shows that the average pressure in the region of influence is n

43、ot time dependent. Nobakht and Clarkson (2011a) reported the same finding in the absence of slippage and desorp- tion. Because the average pressure in the region of influence is constant, using Eq. 5, the corrected pseud

44、otime, t? a, becomest? a ¼? krðlgctÞi ? lg? ct t: ð23ÞThis means that the corrected pseudotime has a linear relationship with time. Eq. 23 also shows that the slope of the 1=q vs. ffiffiffiffi t?

45、 a p plot, m0, and the slope of the 1=q vs. ffiffi t p plot, m, have the following relationship:m ¼ m0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ? krðlgctÞi ? lg? cts: ð24ÞIn order t

46、o obtain the correct value for xf ffiffi ffi k p when gas is being analyzed, the slope of the 1=q vs. ffiffiffiffi t? a p plot, m0, should be used in Eq. 12 (Nobakht and Clarkson 2011a). Therefore, using Eq. 24, the foll

47、owing equation can be used to calculate xf ffiffi ffi k p from the slope of the 1=q vs. ffiffi t p plot, m:xf ffiffiffiffiffiffi k1 p ¼ 315:4Th ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ð/lgctÞi

48、q ? 1p? pi ? p? pwf ? 1m ?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ? krðlgctÞi ? lg? cts:? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ð25ÞComparing Eqs. 12 and 25, the correction factor fCP th

49、at is used to improve the value of xf ffiffi ffi k p calculated from the slope of the 1=q vs. ffiffi t p plot becomesfCP ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi? krðlgctÞi ? lg? cts: ð

50、;26ÞSubstituting xf ffiffiffiffiffiffi k1 p from Eq. 25 and B? gi ¼ 0:0282Z? i T=pi into Eq. 22 results in? p ? Z?? ¼ pi Z?? i 1 ? 0:281ðZ?lgctÞiðp? pi ? p? pwf ÞSgipiffiffiffiffiffiffi

51、ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi? lg? ctkri ? krðlgctÞis “ #:? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ð27ÞHere, Z* is the gas compressibility factor, adjusted to account for desorption ef

52、fect (King 1993; Clarkson et al. 2007). Assuming oil, water, and formation compressibilities are negligible, water influx is negligible, and gas saturation ¼ 100%, Z* becomes (King 1993; Clarkson et al. 2007).Z?

53、88; Z1 þ 0:031214ZTpscVLqB ZscTscðpL þ pÞ/: ð28ÞIn this equation, psc, Zsc, and Tsc are pressure, gas compressibility factor, and temperature at standard conditions, respectively. Eq. 27 sho

54、ws that the average pressure in the region of influence depends on initial pressure, flowing pressure, reservoir tempera- ture, gas properties, gas saturation, and kr defined in Eq. 9. Eq. 27 can be solved to obtain aver

55、age pressure in the region of influence and then the correction factor, fCP, can be calculated using Eq. 26. To improve linear-flow analysis, xf ffiffiffiffiffiffi k1 p calculated from Eq. 12 can be multiplied by fCP. Th

56、is is similar to the Nobakht and Clark- son (2011a) procedure. Note that in the absence of slippage (i.e., kr ¼ 1) and desorption (i.e., Z* ¼ Z), Eq. 27 is identical to the deri- vation presented by Nobakht and

57、 Clarkson (2011a). If kr ¼ 1, Eq. 27 becomes independent of reservoir permeability, and therefore the correction factor that is calculated from Eq. 26 is independent of permeability. In the presence of slippage, the

58、 permeability ratio that is calculated from Eq. 9 depends on reservoir permeability (liquid equivalent) and as a result, average pressure in the region of influence calculated from Eq. 27 depends on the reservoir liquid-

59、equivalent permeability.ValidationTo validate the methodology proposed in this study to analyze the linear flow in tight/shale gas reservoirs, a number of test cases. . . . . . . . . . . . . . . . . . . . . . .. . . . .

60、. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . ..

61、. . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .June 2012 SPE Reservoir