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1 Author s Accepted Manuscript Productivity analysis or a vertically ractured well under non-darcy low condition Wanjing Luo, Xiaodong Wang, Yin Feng, Changu Tang, Yingang Zhou PII: DOI: Reerence: To appear in: S (16) PETROL3536 Journal o Petroleum Science and Engineering Received date: 16 October 2015 Revised date: 6 January 2016 Accepted date: 5 July 2016 Cite this article as: Wanjing Luo, Xiaodong Wang, Yin Feng, Changu Tang and Yingang Zhou, Productivity analysis or a vertically ractured well under non Darcy low condition, Journal o Petroleum Science and Engineering This is a PDF ile o an unedited manuscript that has been accepted o publication. As a service to our customers we are providing this early version o the manuscript. The manuscript will undergo copyediting, typesetting, and review o the resulting galley proo beore it is published in its inal citable orm Please note that during the production process errors may be discovered which could aect the content, and all legal disclaimers that apply to the journal pertain

2 Productivity analysis or a vertically ractured well under non-darcy low condition Wanjing Luo a *, Xiaodong Wang a, Yin Feng b,changu Tang c,yingang Zhou d a. School o Energy Resources, China University o Geosciences (Beijing), Beijing, China, b. University o Louisiana at Laayette, Laayette, LA, USA c. Exploration Research Institute, Anhui Provincial Bureau o Coal Geology, Heei, Anhui d. School o engineering, university o Aberdeen, Aberdeen, the UK AB24 3DB * Corresponding author: luowanjing@cugb.edu.cn Abstract In this paper, a semi-analytical solution o the pseudosteady-state (PSS) productivity index under non-darcy low condition is proposed. Based on the model, a new method o optimization o the racture conductivity has been developed or non-darcy low in the racture. Meanwhile, the eects o the Reynolds number, the proppant number and the racture conductivity on the dimensionless productivity index have been discussed in detail. Results show that: (1) the classic UFD (Uniied Fracture Design) curves o the Darcy-low model underestimate the eect o the proppant number and are unsuitable or the non-darcy-low racture optimization. Our discretized model provides a new tool to obtain the optimal racture parameters or the case o non-darcy low condition. (2) or a given penetration ratio, the non-darcy low behavior exerts a strong inluence on the productivity index with the dimensionless racture conductivity C D = and a considerable productivity index drop can be observed. For the extremely low (C D <0.1) or high (C D >1000) dimensionless racture conductivity, the eect o non-darcy low becomes negligible. (3) the eect o the non-darcy low on the productivity index becomes pronounced at large value 1

3 o proppant number, N p. For a given proppant number, the optimal racture conductivity is slowly increasing as the Reynolds number increases. However, the magnitude o the eect on the productivity index is gradually declining. At the Reynolds number less than 5, the non-darcy low has a relatively strong impact on the productivity index and an apparent all o the maximum productivity index can be noticed, especially or a large proppant number. Beyond the value o 5, the declining trend o the maximum productivity index gradually slows down as the Reynolds number increases. Kewords: productivity index; non-darcy low; vertically ractured well; optimization o the racture parameters; rectangular reservoir Introduction As an eicient stimulation technique, hydraulic racturing has been widely used to increase productivity index (PI) by increasing production rate or decreasing pressure drawdown, especially or tight / shale oil/ gas reservoirs. Mathematical models have been widely used to calculate the productivity o a ractured well under Darcy low condition. Prats (1961) introduced the concept o equivalent wellbore radius to consider the eect o the racture. Reymond and Blinder (1967) provided ways o designing racture treatments and evaluating their results in a damaged ormation with a mathematical model relating stimulation ratio 2

4 to the relative conductivity o ractures. They showed that their model was in agreement with the McGuire-Sikora curves or the racture penetration ratios less than one-hal o the drainage radius. The eects o racture penetration ratio on reciprocal eective wellbore radius have been presented or the uniorm-lux and ininite-conductivity racture (Gringarten and Ramey, 1974; Raghvan et al., 1978). Cinco-Ley and Samaniego (1981) introduced a pseudo-skin unction to characterize the impact o a inite-conductivity racture on the perormance o a vertical well. Riley et al. (1991) provided a solution or the equivalent wellbore radius based on the assumption o elliptical inite-conductivity ractures. According to the deinition o pseudo-skin and pseudo-skin unction, dimensionless pseudo-steady state (PSS) productivity index can be expressed by three parameters, i.e., boundary radius, hal length o a racture and pseudo-skin unction. In calculations, it was convenient to use an explicit expression to replace the pseudo-skin unction (Economides et al., 2002). ValkÓ and Economides (1998) introduced the concept o proppant number and proposed the UFD (Uniied Fracture Design) method or conductivity optimization. For a ixed proppant number, the maximum productivity index can be achieved by racture conductivity optimization in a square drainage area. Diego J.Romero et al. (2003) extended ValkÓ-Economides method to a stimulated well with racture aces and choke skins. A.S. Demarchos et al. (2004) pushed the physical limits o racturing in a wide range o reservoirs with a series o parametric studies. Meyer and Jacot (2005) used the pseudosteady-state resistivity model to calculate the 3

5 productivity o a ractured well. Daal and Economides (2006) calculated the pseudosteady-state productivity index o a ractured well in a rectangular drainage area using the Direct Boundary method. Wang and Jia (2014) established a model to calculate the productivity o multi-ractured horizontal well. For a ractured gas well, the eect o non-darcy low on the productivity cannot be ignored (Holditch and Morse, 1976; Guppy et al., 1982a, b). Vincent et al. (1999) presented several cases o non-darcy low eects on productivity. For a speciied mass o proppant, a shorter and wider racture can be used to compensate or the non-darcy eects (Vincent et al., 1999 ; Hernandez, 2004 ; Kakar et al., 2004). Gil et al. (2003) discussed the design and analysis o ractured-gas-well tests or the case o non-darcy low within the racture. They used a rate-dependent skin to represent the additional pressure drop caused by the eect o non-darcy low. They pointed out that non-darcy low eect may be reduced to tolerate ranges by design considerations. An eective permeability was introduced to account or the eect o non-darcy low (Henry D. Lopez-Hernandez et al., 2004; Y. Wei and Economides, 2005). Based on the deinition o the eective permeability, the equation orm o non-darcy low can be transormed into Darcy low equation orm. Thus the ValkÓ-Economides optimization curves (UFD curves) and an iterative process started with a Reynolds number guess have been used to obtain the maximum productivity index and optimal racture conductivity. Zeng and Zhao (2010) also presented a dierent optimal method or a vertical ractured well under non-darcy low eects. Based on the assumption o the ininite homogeneous 4

6 reservoir, a constant value o 0.5 with the pressure derivatives was chosen as the pseudo-steady state characteristic or hydraulically ractured wells. They suggested that the racture geometry optimization could involve two stages: racture volume optimization and racture length optimization. As can be seen rom the literature review above, the racture can be handled by three methods or the productivity calculation. Firstly, the inite-conductivity racture can be taken as an equivalent parameter, or example, equivalent wellbore radius (Prats, 1961; Gringarten and Ramey, 1974; Raghvan et al., 1978; Riley et al., 1991), pseudo-skin unction (Cinco-Ley and Samaniego, 1981; Economides et al., 2002). This method is relatively simple and easy to calculate. However, it is diicult to extend. Secondly, the racture can be approximately replaced by multi-equally spaced point-source wells (Direct Boundary method) (ValkÓ and Economides, 1998; Diego J.Romero et al., 2003; A.S. Demarchos et al., 2004; Daal and Economides, 2006). Without the racture low model, it is hard to incorporate non-darcy low eects. Thus, an eective permeability model has been introduced to transorm non-darcy low model into the Darcy low model (Henry D. Lopez-Hernandez et al., 2004; Y. Wei and Economides, 2005). Lastly, the racture has been divided into n segments and each segment can be taken as a short racture with uniorm low rate. This is a general racture model which can be used or the Darcy and non-darcy low. However, the existing methods or the productivity calculation are based on the assumption o ininite-acting reservior (Cinco-Ley and Samaniego, 1981; Gil et al., 2003; Zeng and Zhao, 2010). 5

7 In this paper, a racture model incorporating into the eect o non-darcy low in a rectangular drainage area has been presented. Based on the model, we discussed the eect o the Reynolds number, the racture conductivity and the proppant number on the productivity index with 2-D and 3-D igures in detail. Then, a new procedure has been employed to obtain the optimal conductivity, maximum productivity index, racture length and width under non-darcy low condition. This method is dierent rom the eective permeability model (Henry D. Lopez-Hernandez et al., 2004; Y.Wei and Economides, 2005) and pseudosteady-state equivalent model (Zeng and Zhao, 2010). This paper is organized as ollows : First o all, we present a PSS mathematical model o luid low in a racture in a rectangular drainage area. Secondly, the eects o the parameters on the productivity index are discussed in detail. Then, an application o the optimization is presented to illustrate the advantage o the new model. Finally, some conclusions are made. 2 Mathematical models 2.1 Model descriptions Fig.1 shows the conceptual model used in this paper. The ollowing assumptions are made. (1) A vertical well is intercepted by a symmetric racture with a hal length, x, in a closed rectangular drainage area. And the racture is assumed to ully penetrate the reservoir. 6

8 (2) The reservoir is o uniorm thickness h, porosity φ, and permeability k. A single phase low in the reservoir is assumed to obey Darcy s law and the viscosity o the luid is μ. The total compressibility actor, c t is also uniorm. (3) One-dimensional non-darcy low occurs in the racture. The racture permeability k (x) and width w (x) are changing along the racture. The racture storage capacity is ignored. The vertically ractured well produces at a constant rate q (oil well) or q gsc (gas well) in the wellbore. 2.2 Dimensionless deinitions o the variables For the sake o simplicity, the ollowing dimensionless variables will be used. For an oil well, the dimensionless reservoir and racture pressure are p D 2 kh( pi p), qb p D 2 kh( pi p), qb p D 2 kh( pi p ) (1) qb For a gas well, the dimensionless reservoir and racture pressure are p D k hz T p p p T z q 2 2 g gsc gsc i gsc g gsc, p D k hz T p p p T z q 2 2 g gsc gsc i gsc g gsc, p D k hz T p p p T z q 2 2 g gsc gsc i gsc g gsc (2) For the racture low model, the dimensionless racture conductivity, C D, is C D k x w x kx (3) The penetration ratio in the x direction is deined as I x 2x x e (4) The proppant number (Daal and Economides, 2006) is N p 2k V k V prop res (5) 7

9 From the deinition o the penetration ratio and dimensionless racture conductivity, we obtain 2k V 4k x wh x N C I prop 2 e p D x k Vres kxe yeh ye (6) given as The dimensionless low rate o the i-th segment or the oil well and gas well are q Di ql i, q q gscdi ql i (7) q gsc For an oil well, the dimensionless productivity index in the wellbore is J D q B p p 2 kh w (8) For a gas well, the dimensionless productivity index in the wellbore is J D q p T z gsc gsc g 2 2 w g gsc gsc p p k hz T (9) Other dimensionless deinitions in the reservoir model are L Di L x i y ; D y yw ye, ywd, yed x x x x ; D x xw xe, xwd, xed (10) x x x As can be seen rom the dimensionless deinitions above, the same dimensionless variables can be obtained by dierent combinations o the variables or the oil well and gas well. For simpliication, we develop the mathematical models or the oil reservoir in Section 2 and Section 3. An example is presented to illustrate the application or a gas well in Section Fluid low in the reservoir The dimensionless pressure drop o a point located at (x Di, y Di ) caused by a 8

10 point source (x wdj, y wdj ) in a rectangular reservoir is presented by Ozkan (1988). p Dvi 2 2 yed 1 y y y Di Di wdj pd 2 ( ) 2 x 3 y 2y ed ed ed yed ydi y wdj yed ydi y wdj chk chk 1 x x x wdj ed x ed Di 2 cos k cos k k 1 k xed xed y ed shk xed (11) The dimensionless pressure drop o a point (x Di, y Di ) caused by a segment centered at (x wdj, y wdj ) with hal length L Di in the x direction can be obtained by integrating Eq. 11 rom x wdj -L Dj to x wdj +L Dj with respect to x wdj. x L wdj Dj 1 pdi pdvi xdi, ydi, u, ywdj, LDj, xed, yed du 2L Dj xwdj LDj 2 2 y ed 1 y ydi y Di wdj pd 2 2 x ed 3 yed 2y ed 2x m L ed F m Dj m x mx Di wdj sin cos cos 2 L Dj m1 m xed xed xed (12) where F m m yed ydi ywdj cosh yed ydi ywdj m cosh x ed x ed y ed sinh m x ed (13) We deine F ij as the inluence unction F p p ij Di D (14) In act, The racture can be divided into 2N segments and each segment is taken as a symmetrical racture located at (x wdj, y wdj ) with hal length L Dj in the x direction (Fig.2). Regarding the assumption o symmetric racture, only hal o the racture is considered in the ollowing section. According to the superposition principle, the 9

11 pressure solution o the point (x Di, y Di ) caused by the production o the racture (2N segments) can be expressed as 2 N Di D Dj ij Di, Di, wdj, wdj, Dj, ed, ed j1 p p q F x y x y L x y, i=1, 2 N (15) The sensibility o the discretized segments, 2N, is tested by increasing the number o the segments and comparing the error o the contiguous segment number or dierent dimensionless racture conductivity. It is ound that the error is decreasing with the increase o discretized segments. We choose the discretized segments, 2N, corresponding to the value o the error less than 1% as the optimal segments. Thus, N=100 and 800 are used or C D greater than 1 and less than 0.1, respectively. N=400 is used or the calculation while C D is between 0.1 and Fluid low in the racture Luo and Tang (2015) derived a general solution or the varying conductivity racture under non-darcy low condition. In this paper, we present the derivation in brie. To account or the high velocity low in the racture, the ollowing Forchheimer equation will be used (Guppy et al. 1982), p x v v k ( x) 2 (16) where v is low velocity in the racture, ρ is luid density, and the term β is called the β actor (Guppy et al.,1982). Eq.16 can be written in the orm o Darcy low 10

12 k ( x) p x, app qcb w ( x) h (17) The apparent racture permeability in Eq. 17 is deined as k, app k ( x) ( x) (18) 1 k ( x) v / In order to analyze the eect o non-darcy low in the racture, the Reynolds number was deined as N Re k q w h (19) Substituting Eq.19 into Eq.18 yields 1 k, app ( x) k ( x) 1 N q Re cd (20) Thus, Eq.17 can be written in the ollowing dimensionless orm C D 1 N ( x Re D ) q cd p x D D 2 q cd 0 (21) Combination o the inner and outer boundary conditions results in the discretized orm o the racture model ( Luo and Tang, 2015) p wd p Di N ( Di Di 1/ 2) Di q Dj 2 j1 2 Di ˆ i1 j C D Dj Di Dn j1 2 n1 2 q q Di Dj (22) where x D dxd D D( xd) Cˆ D (23) C ( x 1 0 D D) ˆ dxd C D 1/, x D [0,1 ] (24) C ( x ) 0 D D 11

13 2.5 Productivity index o a ractured well The dimensionless productivity index o the i-th segment can be deined as J Di q Di (25) p p D wd The corresponding total dimensionless productivity index o the oil well in the wellbore is J D q B q 2 kh p p p p Di N i1 2 2 N w D wd i1 J Di (26) The PSS equation (Eq.15) and the racture equation (Eq.22) o the i-th segment can be urther expressed as N Di D Dj ij i, jn j1 p p q ( F F ) (27a) p wd p Di N j1 q Dj A ij (27b) According to the continuity condition, or the pressure and the lux to be continuous along the racture surace, the ollowing conditions must hold along the racture segment p x, y p x, y,,, Di Di Di Di Di Di Substituting Eq.27b and Eq.28 into Eq.27a yields q x y q x y (28) Di Di Di Di Di Di N p p q ( F F ) q A D wd Dj ij i, jn Dj ij j1 j1 N,i=1, 2,, N (29) By substituting Eq.25 into Eq.29, we obtain N J Dj ( Fij Fi, jn Aij ) 1, i=1, 2,, N (30) j1 Eq.30 can be written in matrix orm as 12

14 ( F A) J 1 (31) Where J D1 J D2 J J DN, a11 A ai 1 an1 a a 1 j a ij Nj a a 1N F F F F F F F F F F F F F F F F F F F a in NN 11 1, N , N 2 1N 1, N N 21 2, N , N 2 2N 2, N N N1 N, N 1 N 2 N, N 2 NN N, N N (32) (33) The calculation o a ij is presented by Luo and Tang (2015) in detail. By solving Eq.31, the dimensionless productivity index J Di o each segment can be obtained. Then the total productivity index J D o the vertical racture in the wellbore can be calculated by Eq Results 3.1 Veriication o the model We compare our solutions with results presented by ValkÓ and Economides (1998). As shown in Table 1, our results are highly consistent with the results when the dimensionless racture conductivity is greater than 1. For the case o the low dimensionless racture conductivity, the variance can be observed which is caused by the discretized segments, dierent calculation methods and selection o the pressure point or calculation. ValkÓ and Economides introduced many point sources (vertical wells) to approximately represent the racture and the calculated pressure point is at (x wd, y wd + ε) or the point source instead o a line source (racture) 13

15 calculated at (x wd, y wd ) in this paper. 3.2 Eect o the racture penetration ratio and Reynolds number on the productivity index Fig.3 illustrates the eect o the racture penetration ratio (I x ) and Reynolds number (N Re ) on the dimensionless productivity index (J D ) in a square drainage area. It is shown that J D increases as the penetration ratio increases at the same racture conductivity. For a speciic penetration ratio, there is a slight rise o J D when C D is below 0.1. In the range o 0.1 to 1000 or C D, a relatively rapid increase o J D can be observed, especially or the high penetration ratio (I x =0.8). The curves latten out at large C D which delineates the ininite-conductivity racture perormance. Moreover, the existence o non-darcy low (N Re >0) in the racture will result in the reduction o J D compared with the Darcy low (N Re =0, black line). A stronger eect can be ound when the racture conductivity, C D, is in the range o with a large penetration ratio (I x =0. 8). 3.3 Dimensionless productivity index J D at N p less than 0.1 The eects o the Reynolds number, the racture conductivity and the proppant number on the productivity index are presented in Fig.4 through Fig.12 or the case o low proppant number (N p <0.1). Some 2-D and 3-D igures are used to elaborate the eects. Fig.4 shows the dimensionless productivity index as a unction o 14

16 dimensionless racture conductivity with the proppant number as a parameter or N p <0.1 under the Darcy low condition. This igure was irst drawn by ValkÓ and Economides (1998), who pointed out that or a given value o N p, the maximum productivity index is achieved at a well-deined value o the dimensionless racture conductivity located at the peak o the individual curve. At proppant numbers less than 0.1, the dimensionless racture conductivity corresponding to the maximum J D will always occur at C D =1.6. For non-darcy low in the racture, the Reynolds number exerts great inluence on the dimensionless productivity index. As shown in Fig.5, the optimal conductivities will not stay constant at C D =1.6 but luctuate around C D =3.06 or the case o N Re =10. Fig.6 urther reveals the eect o the Reynolds number on the productivity index at a proppant number o With the increase o the Reynolds number, the optimal conductivity becomes larger and a steady all in the productivity index can be noticed. The optimal conductivity is at C D =1.6 corresponding to the Darcy low while it is at the Reynolds number equal to 20. Meanwhile, the maximum productivity index reduces rom at N Re =0 to at N Re =20. In addition, some 3-D color igures and their top views are presented to illustrate the eects o both the racture conductivity and the Reynolds number on the productivity index or a given proppant number in Fig.7 through Fig.12. For an extremely low proppant number, such as N p =0.0003, the Reynolds number has a relatively weak impact on the productivity index. The peak o the J D curve (Fig.7) 15

17 drops steadily with the increase o the Reynolds number. The optimal racture conductivity changes rom 1.6 at N Re =0 to 3.2 at N Re =20 (Fig.8). On the contrary, or a relatively large proppant number, such as N p =0.03, a considerable all o the peak o the J D curve (Fig.11) can be noticed at a low Reynolds number. The scope o the optimal racture conductivity is between 1.6 and 4.47 (Fig.12). In general, the eect o non-darcy low on the productivity index becomes more and more signiicant with the increase o the proppant number. However, the magnitude o the eect is gradually declining as the Reynolds number increases or a given proppant number (Fig.7, Fig.9 and Fig.11). Comparing with Fig.8, Fig.10 and Fig.12, an approximate linear relationship on a semi-log plot between the Reynolds number and the optimal racture conductivity corresponding to the maximum productivity index can be observed when the Reynolds number is greater than 5. Moreover, the relatively stronger impact on the productivity index occurs at Reynolds number less than 5 (Fig.7 through Fig.12). 3.4 Dimensionless productivity index J D at N p greater than 0.1 Fig.13 and Fig.14 present the J D curves with the Darcy and non-darcy low eect at N p greater than 0.1, respectively. The maximum productivity index increases with increase in the proppant number. The eect o the non- Darcy low will not change the trend o the curves, but decrease the range o the optimal conductivity C D rom at N Re =0 to at N Re =10. Comparing Fig.6 with Fig.15, the same phenomena can be ound that the presence o non-darcy low 16

18 reduces the productivity index and enlarges the optimal racture conductivity. However, the eect o the non- Darcy low at a large proppant number is more obvious than that at a low proppant number. Compared to the Darcy low in the racture, the maximum productivity index J D at N Re =20 drops by 13.5% when N p = 0.01 (Fig.6) and by 42.89% when N p = 3 (Fig.15), respectively. In addition, or the ininite-conductivity racture (C D >300), the curves with dierent Reynolds number overlay each other, implying a negligible non-darcy eect or an extremely large racture conductivity (Fig.6 and Fig.15). The 3-D color igures and their top views are presented in Fig.16 through Fig.21 at N p =0.3, 3, and 30, respectively. For comparison, we use the same scales with C D = , N Re =0-20 and J D =0.2-2 to illustrate the curves o the productivity index. As seen, the maximum productivity index drops dramatically when N Re <5 or a large proppant number, such as N p =3 (Fig.18) and N p =30 (Fig.20). Beyond the value o 5 or the Reynolds number, the declination o the maximum productivity index gradually slows down (Fig.16, Fig.18 and Fig.20) and an approximate linear relationship on the semi-log plot between the maximum productivity and the racture conductivity can also be observed just like the cases with the low proppant number (N p <0.1) (Fig.8, Fig.10, Fig.12, Fig.17, Fig.19 and Fig.21). 3.5 Comparison o our method with the equivalent model The curves o the productivity index can be used to determinate the optimal 17

19 racture length and width or a given proppant number (ValkÓ and Economides, 1998; Economides et al., 2002; Diego J.Romero et al., 2003; Daal and Economides, 2006). The UFD (Uniied Fracture Design) method has been widely used or the racture design under the Darcy low condition (Economides et al., 2002). For non-darcy low in the racture, an approximately equivalent model which introduces the concept o the eective permeability to transorm non-darcy low into Darcy low was developed to obtain the optimal racture length and width by using the Darcy-low curves o the productivity index (Henry D. Lopez-Hernandez et al., 2004; Y. Wei and Economides, 2005). In this section, we discuss the adaptability o the equivalent model to deal with non-darcy low in the racture. The parameters o a circular drainage area presented by Henry D. Lopez-Hernandez et al. (2004) were used to calculate the productivity index (Table 2). Although our model is based on the assumption o the rectangular drainage area, it can also be used to calculate the perormance o the circular drainage area when x y r (Ozkan, 1988; Economides et al., 2002). e e e We calculate the proppant number. Reservoir volume (V res ) V = r h =6.810 t (34) res e The volume o proppant injected (V i-2w ) V i-2w 0.016( M ) (60,000) = =610 (1- ) SG (1-0.40) 2.62 p p-2w 3 P t (35) Volume o proppant reaching the pay (V p-2w ) is estimated rom the ratio o pay to the racture height: 18

20 h 39 Vp-2w= Vi -2w =610 =171 t h (36) The real proppant number (N p ) is obtained rom N p 2k V p2w 7 kg Vres (37) In the case o non-darcy low condition, Henry D. Lopez-Hernandez et al. (2004) introduced the concept o the eective permeability to calculate the proppant number. We name it as an equivalent proppant number in this paper. For a Reynolds number guess, such as N Re =9.82, the eective permeability k k e N Re md (38) And the equivalent proppant number N p 2k e Vp2w k V g res (39) The Darcy-low curve o the productivity index at N Re =0 and N p =0.31 which represents the non-darcy-low curve at N Re =9.82 and N p =3.383 was used to obtain the maximum productivity index (Henry D. Lopez-Hernandez et al., 2004). In order to reveal the dierence between the two methods, the equivalent proppant number under dierent Reynolds number is calculated (Table 3). Fig.22 illustrates the comparison o our model (red lines) with Henry D. Lopez-Hernandez et al. method (black lines). The dots and the blue lines denote the maximum productivity index on each curve. It is shown that a huge dierence can be observed and the dierence becomes larger as the Reynolds number increases (blue lines). The errors o the optimal racture conductivity o the two methods are 19

21 between 49.14% and 80.43%, while it is rom 15.75% to 29.66% o the errors o the maximum productivity index when N Re =2 to 20 (Table 3). As discussed, the equivalent model (Henry D. Lopez-Hernandez et al., 2004; Y. Wei and Economides, 2005) which underestimates the eect o the proppant number will lead to a large error or calculation o the maximum productivity index and optimal racture conductivity. Thus, it is not suitable or racture optimization under non-darcy low condition. 4 Application Based on the non-darcy-low model developed in this study, a method to determine the optimal racture parameters is proposed. As stated by Henry D. Lopez-Hernandez et al. (2004), an iterative process starts with a Reynolds number guess. A new Reynolds number is calculated at the end o the process. The iteration stops when error in step 6 is 0.1% or less. For comparison, the parameters presented by Henry D. Lopez-Hernandez et al. (2004) are used (Table 2). (1) Calculate optimal dimensionless racture conductivity (C Dopt ) and optimal dimensionless productivity index (J Dopt ) By setting an initial guess o the Reynolds number as 16.5, the curve o the productivity index with N p =3.383 and N Re =16.5 is presented in Fig.23. As can be seen, the maximum productivity index is corresponding to the dimensionless racture conductivity 9.942, i.e, C Dopt = and J Dopt = (2) Calculate optimal racture dimensions Optimal racture length (x ) 20

22 x 1 2 kv p1w C Doptkgh (40) The optimal propped width (w ) w 1 2 C DoptkgVp1w kh (41) (3) Calculate gas production (q gsc ) q gsc kgh p = 1,424 zt 2 2 ( - p ) res w g res Dopt ( ) J = (4) Calculate gas velocity within the racture Gas ormation volume actor (B g ) Msc / day (42) ztres Bg = = = rc p 1400 sc w (43) Gas velocity (v) 500Bq g gsc v= = = hw t / day (44) (5) Calculate the Reynolds number Molecular weight o the mixture (M g ) Density o the gas (ρ g ) M = M SG = =18.68 (45) g air g pm (46) = w g g zrtres R = psia t lbmole R (47) z-actor is calculated at wellbore lowing conditions. 21

23 Thereore, Beta actor (β) g = = lbm / t (48) 11 a = = b c k p (49) Reynolds number (N Re ) N Re = k v g g = = (50) (6) Error o Reynolds number NRe NRe new old Error= 100= 100=0.073% N 16.5 Reold (51) By comparing with Henry D. Lopez-Hernandez et al. results, it is shown that a greater productivity index can be achieved with a longer and narrower racture (Table 4). 5 Conclusions This paper ocuses on the PSS productivity index in a rectangular reservoir. According to the results and observations, some conclusions can be drawn as ollows: (1) Unlike the existing model, a line source has been used to describe the racture instead o the point source. The characteristic o the luid low o the line source is more close to the luid low in the racture than the point source. Based on the line-source model, a semi-analytical solution o the PSS productivity index under 22

24 non-darcy low condition is proposed. (2) It is diicult to consider the eect o non-darcy low in the racture or the existing point-source model. Thus, an equivalent model or non-darcy low (Henry D. Lopez-Hernandez et al., 2004; Y. Wei and Economides, 2005) has been used to approximately calculate the productivity index. It is shown that the equivalent model underestimates the eect o the proppant number and leads to large errors in calculation o the maximum productivity index. This equivalent model is unsuitable or the optimization or the racture conductivity under non-darcy low condition. A new and accurate method is presented or the racture optimization accounting or the eect o non-darcy low in the racture. (3) For a given penetration ratio, the presence o non-darcy low in the racture makes productivity index drop compared with the Darcy low. For a large penetration ratio, a stronger eect can be ound in the range o the racture conductivity C D = The eect o non-darcy low can be ignored or extremely large racture conductivity (C D >1000). (4) In general, the presence o non-darcy low reduces the productivity index and increases the optimal racture conductivity or a given proppant number. Moreover, the eect o non-darcy low on the productivity index becomes more signiicant with the increase o the proppant number. However, the magnitude o the eect gradually declines as the Reynolds number increases or a given proppant number. (5) When the Reynolds number is less than 5, it has a strong impact on the 23

25 productivity index and an apparent all in maximum productivity index can be noticed, especially or the large proppant number. Beyond the value o 5, the declining trend o the maximum productivity index gradually slows down. An approximately linear relationship on the semi-log plot between the Reynolds number and the optimal racture conductivity can be observed when the Reynolds number is greater than 5. Acknowledgement This work was supported by the National Natural Science Foundation o China No and the Fundamental Research Funds or the Central Universities. 24

26 Nomenclature B Volume actor, RB/STB C D dimensionless racture conductivity h net pay, t h racture height, t I x penetration ratio J D dimensionless productivity index k resrevoirreservoir permeability, md k racture permeability, md k -e eective permeability o non-darcy low in the racture, md L hal length o a discretization segment M air molecular weight o air, lb/lb mole M g molecular weight o gas mixture, lb/lb mole M p-2w injected proppant mass, lbm N Re Reynolds number N Re new Reynolds number calculated at the end o actual iteration N Re old Reynolds number calculated in the previous iteration N p proppant number p p pressure, psi average pressure, psi p i initial ormation pressure, psi p racture pressure, psi 25

27 p w wellbore pressure, psi q q ~ total oil rate in wellbore, STB /day low rate o per unit racture length rom ormation, i.e., low rate strength, STB /d/t q Di dimensionless low rate o the i-th segment q gsc gas rate production at standard conditions, Msc/day q cd dimensionless cross-sectional low rate within the racture R universal constant o gas law, psia.t 3 /(lb. mole. R ) r e drainage radius, t r w wellbore radius, t SG p SG g proppant speciic gravity gas speciic gravity T temperature, R v gas velocity, t/day V volume, t 3 V p-2w volume o proppant in the net pay, t 3 V p-1w volume o proppant in the net pay in one wing, t 3 V i-2w total volume o proppant to be injected, t 3 w width o the racture, t x coordinate in the x direction, t x w dimensionless wellbore coordinate in the x direction x e reservoir length, t 26

28 x racture hal length, t y coordinate in the y direction, t y e reservoir width, t y w dimensionless wellbore coordinate in the y direction z gas compressibility actor β non-darcy low actor, t -1 φ porosity, raction φ p proppant porosity, raction µ luid viscosity, cp ρ luid density, lbm/cu t ξ D dimensionless coordinate in the ξ direction Di dimensionless discretized step o the i-th segment in the ξ direction Special Subscripts: D g i opt res w gsc v Dimensionless racture property gas well initial or segment i Optimal Reservoir wellbore property standard conditions vertical well (point source) 27

29 Reerences A.S. Demarchos et al Pushing the Limits in Hydraulic Fracture Design. SPE SPE International Symposium and Exhibition on Formation Damage Control, February, Laayette, Louisiana Cinco-Ley, H. and Samaniego-V Transient Pressure Analysis: Finite Conductivity Fracture Case versus Damaged Fracture Case. SPE Daal and Economides Optimization o Hydraulically Fractured Wells in Irregularly Shaped Drainage Areas. SPE SPE International Symposium and Exhibition on Formation Damage Control, February, Laayette, Louisiana, USA Diego J.Romero et al Optimization o the Productivity Index and the Fracture Geometry o a Stimulated Well With Fracture Face and Choke Skins. SPE PA. 18(1):57-64 Dyes, A. B., Kemp, C. E. and Caudle, B. H Eect o Fractures on Sweep-Out Pattern. Trans., AIME (1958) 213, 245. Economides, M., Oligney, R., and ValkÓ, P Uniied Fracture Design, Orsa Press, Alvin, Texas. Gil et al Fractured-Well-Test Design and Analysis in the Presence o Non-Darcy Flow. SPE Reservoir Evaluation & Engineering. 6(03): Gringarten, A. C., Ramey, H. J., and Raghavan, R Unsteady-State Pressure Distributions Created by a Well with a Single Ininite- Conductivity Fracture, SPEJ, August 1974,

30 Guppy, K.H., Cinco-Ley, H., Ramey Jr., H. J. and Samaniego-V. F Non-Darcy low in wells with inite-conductivity vertical ractures. Soc. Pet. Eng. J. 22 (5), Guppy, K.H., Cinco-Ley, H., Ramey Jr., H. J Pressure buildup analysis o ractured wells producing at high low rates. J. Pet. Tech. 34(11), Henry D. Lopez-Hernandez et al Optimum Fracture Treatment Design Minimizes the Impact o Non-Darcy Flow Eects. SPE SPE Annual Technical Conerence and Exhibition, September, Houston, Hernandez, H. D. J. L Optimal Fracture Treatment Design or Dry Gas Wells maximizes Well perormance in the Presence o Non-Darcy Flow Eects. Master thesis. Texas A&M University. Texas. Holditch, S. A., Morse, R. A., The eects o non-darcy low on the behavior o hydraulically ractured gas wells. JPT 28 (10), SPE-5586-PA. John M. Tinsley et al Vertical Fracture Height--Its Eect on Steady-State Production Increase. SPE 1900, , May, 1969 Kakar, A. M. Zheng. S., Stewart. G Well test analysis o hydraulically ractured gas wells or non-darcy low eects. Presented at the SPE Regional Conerence in Pakistan, October 8-9. Luo and Tang A Semianalytical Solution o a Vertical Fractured Well With Varying Conductivity Under Non-Darcy-Flow Condition. SPE PA McGuire, W. J. and Sikora, V. J The Eect o Vertical Fracture on Well Productivity. Trans., AIME (1960)219,

31 Meyer, B. R., Jacot, R. H Pseudosteady-state analysis o inite-conductivity vertical ractures. Paper SPE presented at the 25th Annual Technical Conerence and Exhibition. Dallas, Texas, October Ozkan Perormance o horizontal wells. Ph.D. Dissertation, the University o Tulsa, Tulsa, OK. Prats, M Eect o Vertical Fractures on Reservoir Behavior-Incompressible Fluid Case, SPEJ June, 1961, Raghavan, R. and Hadinoto, Nico Analysis o Pressure Data or Fracture Wells: The Constant-Pressure Outer Boundary. SPEJ April, Raymond, L. R., and Binder, G. G Productivity o Wells in Vertically Fractured, Damaged Formations. JPT, Riley, M.F., Brigham, W.E., Horne, R.N Analytical solutions or elliptical inite-conductivity ractures. Paper SPE presented at the 66th Annual Technical Conerence and Exhibition, Dallas, TX, USA, 6 9 Oct 1991 ValkÓ and Economides Heavy Crude Production rom Shallow Formations: Long Horizontal Wells Versus Horizontal Fractures. SPE SPE International Conerence on Horizontal Well Technology, 1-4 November, Calgary, Alberta, Canada Vincent, M. C., Pearson, C. M., Kullman, J Non-Darcy and multiphase low in propped ractures: case studies illustrate the dramatic eect on well productivity. Paper SPE presented at the 1999 SPE Western Regional Meeting held in Anchorage, Alaska, May

32 Wahl, Harry A., Jr An Electrolytic Model Study o the Eect o Horizontal Fractures on Well Productivity, MA thesis, The U. o Oklahoma, Norman, Okla Wang and Jia, A general productivity model or optimization o multiple ractures with heterogeneous properties. Journal o Natural Gas Science and Engineering. 21 (2014) Y. Wei and Economides, Transverse Hydraulic Fractures From a Horizontal Well. SPE SPE Annual Technical Conerence and Exhibition, 9-12 October, Dallas, Texas Zeng, F. and Zhao, G The Optimal Hydraulic Fracture Geometry Under Non-Darcy Flow Eects. J. Pet. Sci. Eng. 72 (1 2): Fig.1 Schematic o a vertical racture in a rectangular reservoir 31

33 Fig.2 Fracture divided into 2N equal segments Fig.3 Fracture perormance as a unction o dimensionless racture conductivity, penetration ratio and the Reynolds number (square drainage area) 32

34 Fig.4 Fracture perormance at selected proppant number (N Re =0, Np<0.1) (square drainage area) Fig.5 Fracture perormance at selected proppant number (N Re =10, Np<0.1, square drainage area) 33

35 Fig.6 The eect o N Re on the productivity index (Np=0.01) (square drainage area) Fig.7 The 3-D color map and its top view o the productivity index, racture conductivity and the Reynolds number at Np= (square drainage area) 34

36 Fig.8 The 2-D color map o the productivity index, racture conductivity and the Reynolds number at Np= (square drainage area) Fig.9 The 3-D color map and its top view o the productivity index, racture conductivity and the Reynolds number at Np=0.003 (square drainage area) 35

37 Fig.10 The 2-D color map o the productivity index, racture conductivity and the Reynolds number at Np=0.003 (square drainage area) Fig.11 The 3-D color map and its top view o the productivity index, racture conductivity and the Reynolds number at Np=0.03 (square drainage area) 36

38 Fig.12 The 2-D color map o the productivity index, racture conductivity and the Reynolds number at Np=0.03 (square drainage area) Fig.13 Fracture perormance at selected proppant number (N Re =0, Np>0.1) (square drainage area) 37

39 Fig.14 Fracture perormance at selected proppant number (N Re =10, Np>0.1) (square drainage area) Fig.15 The eect o N Re on the productivity index (Np=3) (square drainage area) 38

40 Fig.16 The 3-D color map and its top view o the productivity index, racture conductivity and the Reynolds number at Np=0.3 (square drainage area) Fig.17 The 2-D color map o the productivity index, racture conductivity and the Reynolds number at Np=0.3 (square drainage area) 39

41 Fig.18 The 3-D color map and its top view o the productivity index, racture conductivity and the Reynolds number at Np=3 (square drainage area) Fig.19 The 2-D color map o the productivity index, racture conductivity and the Reynolds number at Np=3 (square drainage area) 40

42 Fig.20 The 3-D color map and its top view o the productivity index, racture conductivity and the Reynolds number at Np=30 (square drainage area) Fig.21 The 2-D color map o the productivity index, racture conductivity and the Reynolds number at Np=30 (square drainage area) 41

43 Fig.22 Comparisons o the productivity index with Lopez et al. method Fig.23 The optimization o the racture conductivity with Np=3.383 and N Re =16.5 (square drainage area) 42

44 Table 1 Comparison between ValkÓ-Economides result and our work (xed/yed=1) Ix=0.1 Ix=0.3 Ix=0.6 Ix=1.0 C D ValkÓ-Ec onomides (1998) This paper Error ValkÓ-Ec onomides (1998) This paper Error ValkÓ-Ec onomides (1998) This paper Error ValkÓ-Ec onomides (1998) This paper Error % % % % Ave rage Table 2 Basic parameters or calculation o the productivity index (Henry D. Lopez-Hernandez et al., 2004) Basic model parameters values Basic model parameters values Drainage radius, r e, t 745 Viscosity o gas, μ, mpa.s Fracture height, h, t 139 Gas compressibility actor, z Reservoir thickness, h, t 39 Injected proppant mass, M p-2w, lbm Reservoir permeability, k, md 0.2 Proppant speciic gravity, SGp 2.62 Fracture permeability, k, md Gas speciic gravity, SGg Reservoir porosity, φ, % 10 Molecular weight o air, Mair, lb/lb mole 29 Fracture porosity, φ p, % 40 Coeicient o Beta actor, a 3.7E11 Reservoir temperature, T res, R 680 Coeicient o Beta actor, b 1.35 Reservoir pressure, p res, psi 5254 Coeicient o Beta actor, c 0 Bottom pressure, p w, psi

45 Table 3 Comparisons o the optimal racture conductivity and maximum productivity index with Henry D. Lopez-Hernandez et al. method This paper Henry D. Lopez-Hernandez et al. (2004) Error,% N Re N p C Dopt J Dopt Equivalent N Re Equivalent N p C Dopt J Dopt C Dopt J Dopt Table 4 Basic parameters and the results or the racture optimization Parameters kg k k -e N p N Re C Dopt J Dopt x w Units md md md t t This paper Henry D. Lopez-Hernandez et al. (2004) Appendix A: Unit Conversion Factors SI Metric Conversion Factors bbl m 3 cp t Pa s m 44

46 t m 2 psi kpa Highlights: (1) Propose a semi-analytical solution o the pseudosteady-state (PSS) productivity index under the non-darcy low condition in a rectangular reservoir (2) Reveal the eect o the Reynolds number, the proppant number and the racture conductivity on the dimensionless productivity index (3) Discuss the adaptability o the Darcy-low UFD (Uniied Fracture Design) curves under the non-darcy low condition (4) Develop a new model or optimization o the racture parameters 45