Coupling Fluid Flow Model of Multiscale Fractures in Tight Reservoirs

Available online at www.sciencedirect.com

ScienceDirect Procedia Engineering 126 (2015) 353 – 357

7th International Conference on Fluid Mechanics, ICFM7

Coupling fluid flow model of multiscale fractures in tight reservoirs Yuliang Su*, Binglin Li, Chen Xu, Yongmao Hao School of Petroleum Engineering, China University of Petroleum, No.66 at the Western Yangtze River Road, Qingdao 266580,China

Abstract In this study, on the basis of the fact that tight reservoirs exhibit fractures of different scales after hydraulic fracturing and given their coupling patterns, an improved 2nd-order finite element mixed model is proposed for fluid flow in such reservoirs. This model takes into account the threshold pressure gradient as well as cross flow from the matrix to the natural fractures. The natural fractures are assumed to be continuous, while the hydraulic fractures are treated as discrete ones. All the media are coupled by the cross flow between them. Sensitivity analyses are performed to determine the effects of the threshold pressure gradient and the permeabilities of the various media. It is concluded that the actual bottom-hole pressure can be increased by increasing the permeability of only the natural fractures or by decreasing the threshold pressure gradient. Increasing the permeabilities of all the media causes the bottom-hole pressure to decrease rapidly initially and then slowly and increases the cross flow to the artificial fractures. Thus, increasing the permeability of artificial fractures is a suitable strategy only in the case of low-permeability matrices. © Published by by Elsevier Ltd.Ltd. This is an open access article under the CC BY-NC-ND license ©2015 2015The TheAuthors. Authors. Published Elsevier (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of The Chinese Society of Theoretical and Applied Mechanics (CSTAM). Peer-review under responsibility of The Chinese Society of Theoretical and Applied Mechanics (CSTAM) Keywords: tight reservoirs; fractures of different scales; coupling fluid flow model; finite element method

1. Introduction Given the developments related to petroleum exploitation in China, especially exploitation in reservoirs with complex boundaries as well as under nonhomogeneous conditions and threshold pressure gradients—these increase the complexity of the behavior of the underground fluid—the storage of the extracted petroleum in tight reservoirs with fractures has become a critical issue. However, most previous studies have ignored the effects of the threshold

*

Corresponding author. Tel.: +86-532-8698-3115; Fax: +86-532-8698-3115; E-mail: [email protected]

1877-7058 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of The Chinese Society of Theoretical and Applied Mechanics (CSTAM)

doi:10.1016/j.proeng.2015.11.209

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Yuliang Su et al. / Procedia Engineering 126 (2015) 353 – 357

pressure gradient and the channeling of fluid from the matrix to the natural fractures and the well bore [1,2]. Moreover, Morita [3] has suggested that, compared to the traditional Laplace-Stehfest numerical inversion technique [2,4] and the finite difference method [1,5-7], the finite element method [3,8-12] is more suitable for solving fluid flow problems related to tight reservoirs with high accuracy, flexibility, and low grid orientation. Therefore, based on the results of previous studies, a single-phase finite element model was developed in this study while taking into account the threshold pressure gradient and fluid channeling from the matrix to the natural fractures and the well bore for the case where artificial and nature fractures coexist. The natural fractures were considered a continuous medium, while the hydraulic fractures were treated as discrete fractures. 2. Development of model As mentioned above, in this study, a single-phase flow finite element model was developed for all media while taking into account the threshold pressure gradient as well as the channeling of fluid from the matrix to the natural fractures and the well bore for the case where artificial and nature fractures coexist. The following assumptions were made while developing the model: the fluid flow in the fractures conforms to Darcy’s law and there exists a threshold pressure gradient in the matrix. Further, both the fluid and the rock are compressible. In addition, the compressibility coefficient, temperature, and many other physical parameters are constant throughout the reservoir. Finally, the boundary pressure remains constant, the effect of gravity is ignored, and it is assumed that the artificial fractures cut through the whole reservoir. x Flow model for the matrix:

Om2 ( p G ) q1* G q*3 Ct

wp wt

(1)

wp wt

(2)

wp wt

(3)

x Flow model for the natural fractures:

Olf 2 pq1* G q*2 Ct x Flow model for the artificial fractures:

O f 2 p q*2 q*3 G 2 q Ct

λm = ratio of the permeability of the matrix to its viscosity, μm·g/s; λlf = ratio of the permeability of the natural fracture to its viscosity, μm·g/s; λf = ratio of the permeability of the artificial fracture to its viscosity, μm·g/s; q1*= ratio of the volume of fluid channeled from the matrix to the natural fractures and the element volume, 1/s; q2*= ratio of the volume of fluid channeled from the natural fractures to the artificial fractures and the element volume, 1/s; q3*= ratio of the volume of fluid channeled from the matrix to the artificial fractures and the element volume, 1/s; δ = delta function; δ =1 when fluid channeling occurs; otherwise δ = 0; G = threshold pressure gradient, MPa/m; Ct = total compressibility coefficient, 1/MPa; δ2 = delta function, δ2 = 1 when a well bore exists, otherwise δ2 = 0.

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x Coupling means of each media It was assumed that there existed cross flow between all the media. The cross flow rates were regarded as corresponding to a source and a sink and were assumed to be proportional to the pressure difference between the two media in contact with each other. Their values were identical while their characteristics varied depending on the medium. During cross flow, the threshold pressure gradient was not considered. 3. Model solution The discretization method was used for all the media. The processes involved can be described as follows. The natural fractures were separated into 2nd-order rectangular elements, while the hydraulic fractures were separated into 2nd-order linear elements. Subsequently, the characteristics of all the grids were analyzed. Then, after incorporating the boundary conditions and the cross flow equations, the stiffness matrix for solving the finite element model for the fluid flow was obtained. Based on the Galerkin method, we get

( Om (

C dq* B1 B2 dq* 2 ) t B) p n Cmlf 1 Cmf 3 2 X Y T dV dV

COmG

Ct Bp n 1 T

(4)

Further, the equation for the artificial pressure is as follows:

( Of

B3 Ct C dq* dq* dq A2 ) p n Cq 2C f ( 2 3 ) t A2 p n 1 2 X T dV dV dV T

(5)

where X, Y = the size of the elements; 4. Example calculation In this study, the calculations were performed using MATLAB (2014 version). The model used was of a quadrate reservoir with a producing well in the center. The basic parameters for the reservoir are listed in Table 1. The sensitivity analyses were performed on the basis of these parameters. Table 1

Basic parameter of reservoir

Parameter

Value

Number of elements

10 × 10 2

Size of the reservoir (m )

60 × 60

Initial pressure (MPa)

18

Threshold pressure gradient (MPa/m)

8 × 10-2

3

Well rate (m /d)

20

Viscosity (mPa·s)

2

Permeability of the artificial fractures (md)

300

Permeability of the natural fractures (md)

3

Permeability of the matrix (md)

0.3

Total compressibility coefficient (1/MPa)

10-4

Width of the artificial fracture (cm)

0.8

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4.1. Effect of threshold pressure gradient on pressure at bottom hole In order to study the effect of the threshold pressure gradient of the matrix on the pressure at the bottom hole, the bottom-hole pressure was plotted for different threshold pressure gradients, as shown in Fig. 1a. It can be seen from the figure that increasing the threshold pressure gradient did not lower the energy of formation. Therefore, the initial bottom-hole pressure decreased rapidly. Further, the actual bottom-hole pressure also decreased, since the flow, which was only via the matrix, also decreased. Consequently, in order to maintain the bottom-hole pressure, the threshold pressure gradient should be lowered.

Pressure (MPa)

Pressure (MPa)

a

Time (s)

b

Time (s)

Fig. 1 (a). Curves of the bottom-hole pressure for different threshold pressure gradients and (b) curves of the bottom-hole pressure for natural fractures with different permeabilities.

4.2. Effect of permeability on pressure at bottom hole

Cross flow rate (cm3/s)

Pressure (MPa)

a

Time (s)

b

Time (s)

Fig. 2 (a) Curves of the bottom-hole pressure for artificial fractures with different permeabilities and (b) changes with time in the rate of cross flow from the matrix to the artificial fractures with different permeabilities.

Next, the effect of the medium permeability and that of the channeling of fluid from the matrix to the artificial fractures on the bottom-hole pressure were studied. The results are shown in Fig. 1b and Fig. 2. As shown in Fig. 1b, the actual bottom-hole pressure increased with an increase in the permeability of the natural fractures; this was because the flow was only through the natural fractures. The lower the permeability of the medium is, the bottomhole pressure will be; this is because the energy supplied to the bottom hole will be lower. Thus, the formation energy would be maintained better by increasing the permeability of the natural fractures. Next, as shown in Fig. 2, increasing the permeabilities of all the media rapidly decreased the bottom-hole pressure initially. The reason for this is the fact that, when the flow resistance decreases, cross flow to the artificial


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fractures increases. Further, the flow from the boundary via only the artificial fractures decreased, and the production rate can be used for contribution to the flow supply and to the boundary via the artificial fracture. As a result, the bottom-hole pressure decreased slowly, since energy was supplied to the bottom hole by the cross flow of the fluid. Once the bottom-hole pressure plateaued, it was not affected by the permeability. However, its recovery rate was lowered.The effect of decreasing the permeability of only the artificial fractures was not as conspicuous as that of decreasing the permeabilities of all the media. Although the flow resistance can be increased by decreasing the permeability of only the artificial fractures, fluid supply from the boundary would still occur. The lower the permeabilities of the various media are, the more conspicuous the effects of decreasing the permeability of only the artificial fractures will be. This is because when the degree of cross flow is decreased, the bottom-hole pressure also decreases rapidly. Therefore, increasing the permeability of the artificial fractures is a suitable strategy only in the case of low-permeability matrices. 5. Conclusions (1) Increasing the permeabilities of all mediums will cause the bottom-hole pressure to decrease rapidly at first and then slowly. This is because the degree of cross flow to the artificial fractures will increase. However, the recovery rate will be lowered. (2) The actual bottom-hole pressure can be increased by only increasing the permeability of the natural fractures. This will also help in maintaining the formation energy. (3) The actual bottom-hole pressure can be lowered by increasing the threshold pressure gradient. In this case, the formation energy will not be maintained. (4) Increasing the permeability of the artificial fractures is a strategy suitable only in the case of low-permeability matrices. Acknowledgements This study was supported by the National Basic Research Program of China (2014CB239103); the Program for Changjiang Scholars and Innovative Research Team in University (PCSIRT) (IRT1294); and the Shandong Province Natural Science Foundation (ZR2014EL014). References [1]Zhang Dongli, Li Jianglong, Progress flow mathematical model of hole type reservoir fluid and its application Journal of Southwest Petroleum University. 2009,Vol.31(6): pp.66-70. in Chinese [2]Chen Fangfang, Jia Yonglu, Qiao Zhongming, Yan Yupeng, Wang Feng, Three heavy medium reservoir seepage flow model and the welltest curve Xinjiang Petroleum Geology. 2008,Vol.29 (3): pp.350-3. in Chinese [3]Morita, Nobuo, Transient finite element code: A versatile tool for well performance AnalysesSPEJ. 1993, Vol.1 (02): pp.1-10. [4] Al-Ghamdi, Pressure transient analysis of dually fractured reservoirs, SPEJ. 1996, Vol.1 (01): pp.93-100. [5] Lange A, Basquet R, Bourbiaux B, Hydraulic characterization of faults and fractures using a dual medium discrete fracture network simulator, SPEJ. 2004, Vol.8 (07): pp.1-10. [6] Zhang Dezhi, Yao Jun, Wang Zisheng, =KDQ ALting, Three heavy medium reservoir well test interpretation model and pressure characteristics, Xinjiang Petroleum Geology. 2008,Vol.29 (2): pp.222-6. in Chinese [7]Zhang Dongli, Li Jianglong, Wu Yushu, A similar gap model of hole type reservoir three heavy media numerical well testing, Journal of Southwest Petroleum University. 2010,Vol.32 (2): pp.82-8. in Chinese [8]Huang Z, Yao J, Wang Y, Tao K, Numerical study on two-phase flow through fractured porous media, Science China Technological Sciences. 2011,vol.54(9): pp.2412-20. [9] Zhang Yong, He Guoliang, Double media fractal reservoir deformation of unsteady seepage flow mathematical model of finite element method for solving the research, Journal of Yangtze University (Natural Science Edition). 2013,Vol.10 (22): pp.13-17. [10]Zhang Na, Yao Jun, Huang Zhaoqin, Wang Yueying, The local conservation of finite element analysis of two-phase flow in porous media in, Computational Physics. 2013,Vol.30 (5): pp.667-675. [11] Wang Haiyong, Yu Xiaojie, 1DQ