Pressure transient analysis of multiple fractured horizontal wells in naturally fractured unconventional reservoirs based on fractal theory and fractional calculus

Abstract

Currently, most models for multiple fractured horizontal wells (MFHWs) in naturally fractured unconventional reservoirs (NFURs) are based on classical Euclidean models which implicitly assume a uniform distribution of natural fractures and that all fractures are homogeneous. While fractal theory provides a powerful method to describe the disorder, heterogeneity, uncertainty and complexity of the NFURs. In this paper, a fractally fractional diffusion model (FFDM) for MFHWs in NFURs is established based on fractal theory and fractional calculus. Particularly, fractal theory is used to describe the heterogeneous, complex fracture network, with consideration of anomalous behavior of diffusion process in NFURs by employing fractional calculus. The Laplace transformation, line source function, dispersion method, and superposition principle are used to solve this new model. The pressure responses in the real time domain are obtained with Stehfest numerical inversion algorithms. The type curves of MFHW with three different outer boundaries are plotted. Sensitivity analysis of some related parameters are discussed as well. This new model provides the relatively more accurate and appropriate evaluation results for pressure transient analysis for MFHWs in NFURs, which could be applied to accurately interpret the real pressure data of an MFHW in field.

Keywords

Fractal theory / Anomalous diffusion / Fractional calculus / Naturally fractured unconventional reservoirs / Multiple fractured horizontal well / Pressure responses

Acknowledgements

The Authors would like to acknowledge the financial support provided by the China Joint foundation for Petrochemical Industry (A) (No.U1562102).

References

[1] E. Ozkan, M. Brown, R. Raghavan, et al., Comparison of fractured horizontal well performance in conventional and unconventional reservoirs,in:SPE Paper 121290-MS Presented at SPE Western Regional Meeting, San Jose, California, USA, 2009.

[2] E. Ozkan, M. Brown, R. Raghavan, H. Kazemi, Comparison of fractured horizontal-well performance in tight sand and shale reservoirs, SPE Reserv. Eval. Eng. 14 (2) (2011) 248-259.

[3] M. Brown, E. Ozkan, R. Raghavan, et al., Practical solutions for pressure transient responses of fractured horizontal wells in unconventional reservoirs, SPE Reserv. Eval. Eng. 14 (6) (2011) 663-676.

[4] I. Brohi, M. Pooladi-Darvish, R. Aguilera, Modeling fractured horizontal wells as dual porosity composite reservoirs-application to tight gas, shale gas and tight oil cases[C], in: Paper SPE 144057 Presented at the SPE Western North American Region Meeting, Anchorage, Alaska, USA, 2011.

[5] C.M. Freeman, G.J. Moridis, T.A. Blasingame, A numerical study of microscaleflow behavior in tight gas and shale gas reservoir systems, Transp. Porous Med. 90 (1) (2011) 253-268.

[6] L. Tian, C. Xiao, M. Liu, D. Gu, G. Song, H. Cao, et al., Well testing model for multi-fractured horizontal well for shale gas reservoirs with consideration of dual diffusion in matrix, J. Nat. Gas Sci. Eng. 21 (21) (2014) 283-295.

[7] S.S. Yao, F.H. Zeng, H. Liu, et al., A semi-analytical model for multi-stage fractured horizontal wells[J], J. Hydrol. 507 (2013) 201-212.

[8] Y.L. Zhao, L.H. Zhang, F. Wu, et al., Analysis of horizontal well pressure behavior in fractured low permeability reservoirs with consideration of threshold pressure gradient, J. Geophys. Eng. 10 (2013) 1-10.

[9] Y.L. Zhao, L.H. Zhang, J.X. Luo, B.N. Zhang, Performance of fractured horizontal well with stimulated reservoir volume in unconventional gas reservoir, J. Hydrol. 512 (2014) 447-456.

[10] D. Yang, F. Zhang, J.A. Styles, J. Gao, Performance evaluation of a horizontal well with multiple fractures by use of a slab-source function, SPE J. 20 (3) (2015) 652-662.

[11] H.T. Wang, Performance of MFHWs in shale gas reservoirs with consideration of multiple mechanisms, J. Hydrol. 510 (2014) 299-312.

[12] M. Liu, C. Xiao, Y. Wang, et al., Sensitivity analysis of geometry for multistage fractured horizontal wells with consideration of finite-conductivity fractures in shale gas reservoirs, J. Nat. Gas Sci. Eng. 22 (2015) 182-195.

[13] T. Samardzioska, V. Popov, Numerical comparison of the equivalent continuum, non-homogeneous and dual porosity models for flow and transport in fractured porous media, Adv. water Resour. 28 (3) (2005) 235-255.

[14] A. Peratta, V. Popov, A new scheme for numerical modelling of flow and transport processes in 3D fractured porous media[J], Adv. Water Resour. 29 (1) (2006) 42-61.

[15] J. Erhel, J.R. De Dreuzy, B. Poirriez, Flow simulation in three-dimensional discrete fracture networks[J], SIAM J. Sci. Comput. 31 (4) (2009) 2688-2705.

[16] J.R. Dreuzy, Y. Méheust, G. Pichot, Influence of fracture scale heterogeneity on the flow properties of three-dimensional discrete fracture networks (DFN)[J], J. Geophys. Res. Solid Earth 2012 (1978-2012), 117(B11).

[17] J. Chang, Y.C. Yortsos, Pressure-transient analysis of fractal reservoirs, SPE Form. Eval. 5 (1990) 31-38.

[18] J.A. Acuna, Y.C. Yortsos, Application of fractal geometry to the study of networks of fractures and their pressure transient, Water Resour. Res. 31 (1995) 527-540.

[19] R.A. Beier, Pressure-transient Model for a Vertically Fractured Well in a Fractal Reservoir, SPEFE, June 1994, p. 122.

[20] J. Olarewaju, Modeling fractured reservoirs with stochastic fractals, in: Paper Presented at Abu Dhabi International Petroleum Exhibition and Conference. Abu Dhabi, United Arab Emirates, 1996. 36207-MS.

[21] F. Flamenco-Lopez, R. Camacho-Velazquez,Fractal transient pressure behavior of naturally fractured reservoirs, in:Paper 71591 Presented at the Annual Technical Conference and Exhibition, Soc. Pet. Eng., New Orleans, LA, 2001.

[22] F. Flamenco-Lopez, R. Camacho-Velazquez, Determination of fractal parameters of fracture networks using pressure-transient data, SPE Reserve Eval. Eng. 6 (2003) 39-47.

[23] Y. Zhao, L. Zhang, Solution and type curve analysis of fluid flow model for fractal reservoir, World J. Mech. 1 (2011) 209-216.

[24] M. Cossio Santizo, A Semi-analytic Solution for Flow in Finite-conductivity Vertical Fractures Using Fractal Theory[D], Texas A&M University, 2012.

[25] M. Cossio, G. Moridis, T.A. Blasingame, A semianalytic solution for flow in finite-conductivity vertical fractures by use of fractal theory[J], SPE J. 18 (01) (2013) 83-96.

[26] W. Wang, et al., A semi-analytical fractal model for production from tight oil reservoirs with hydraulically fractured horizontal wells, Fuel 158 (2015a) 612-618.

[27] W. Wang, Y. Su, G. Sheng, et al., A mathematical model considering complex fractures and fractal flow for pressure transient analysis of fractured horizontal wells in unconventional reservoirs, J. Nat. Gas Sci. Eng. 23 (2015b) 139-147.

[28] G. Sheng, Y. Su, W. Wang, Q. Zhang, J. Liu, Flow behavior analysis and application of complex-flow model in explosive fracturing well using fractal theory, in: Paper SPE 172761-MS Presented at SPE Middle East Oil & Gas Show and Conference, Manama, Bahrain, 2015.

[29] Q. Zhang, Y. Su, W. Wang, et al., A new semi-analytical model for simulating the effectively stimulated volume of fractured wells in tight reservoirs, J. Nat. Gas Sci. Eng. 27 (2015) 1834-1845.

[30] X.-H. Tan, X.-P. Li, Transient flow model and pressure dynamic features of tree-shaped fractal reservoirs, J. Hydrodyn., Ser. B 26 (4) (2014) 654-663.

[31] X.-H. Tan, X.-P. Li, J.-Y. Liu, G.-D. Zhang, L.-H. Zhang, Analysis of permeability for transient two-phase flow in fractal porous media, J. Appl. Phys. 115 (11) (2014) 113502.

[32] X.-H. Tan, X.-P. Li, J.-Y. Liu, L.-H. Zhang, Z. Fan, Study of the effects of stress sensitivity on the permeability and porosity of fractal porous media, Phys. Lett. A 379 (39) (2015a) 2458-2465.

[33] X.-H. Tan, J.-Y. Liu, X.-P. Li, L.-H. Zhang, J. Cai, A simulation method for permeability of porous media based on multiple fractal model, Int. J. Eng. Sci. 95 (2015b) 76-84.

[34] F. Zhang, D. Yang, Determination of minimum permeability plateau and characteristic length in porous media with non-darcy flow behaviour, J. Pet. Sci. Eng. 119 (7) (2014a) 8-16.

[35] F. Zhang, D. Yang, Determination of fracture conductivity in tight formations with non-darcy flow behaviour, SPE J. 19 (1) (2014b) 34-44.

[36] R. Metzler, W.G. Glockle, T.F. Nonnenmacher, Fractional model equation for anomalous diffusion, Phys. A 211 (1994) 13-24.

[37] R. Metzler, T.F. Nonnenmacher,Fractional diffusion: exact representation of spectral functions, J. Phys. A Math. Gen. 30 (1997) 1089-1093.

[38] R. Metzler, T.F. Nonnenmacher, Fractional diffusion, waiting-time distributions, and Cattaneo-type equations, Phys. Rev. E 57 (1998), 6409-6914.

[39] I. Pudlubny, Fractional Differential Equations, Academic Press, London, U.K, 1999.

[40] H.W. Park, J. Choe, J.M. Kang, Pressure behavior of transport in fractal porous media using a fractional calculus approach, energy sources, Part A, Recovery, Util. Environ. Eff. 22 (10) (2000) 881-890.

[41] J. Tian, D. Tong, The flow analysis of fluids in fractal reservoir with the fractional derivative, J. Hydrodyn. 18 (3) (2006) 287-293.

[42] R. Raghavan, Fractional diffusion: performance of fractured wells, J. Pet. Sci. Eng. 92-93 (2012a) 167-173.

[43] R. Raghavan, Fractional derivatives: application to transient flow, J. Pet. Sci. Eng. 80 (2012b) 7-13.

[44] R. Raghavan, C. Chen, Fractured-well performance under anomalous diffusion, SPE Reserv. Eval. Eng. 16 (3) (2013a) 237-245.

[45] R. Raghavan, C. Chen, Fractional diffusion in rocks produced by horizontal wells with multiple, transverse hydraulic fractures of finite conductivity, J. Pet. Sci. Eng. 109 (2013b) 133-143.

[46] R. Raghavan, C. Chen, Anomalous subdiffusion to a horizontal well by a subordinator, Transp. Porous Med. 107 (2) (2015) 387-401.

[47] O. Ozcan, Fractional Diffusion in Naturally Fractured Unconventional Reservoirs. Master of Science Thesis, Colorado School of Mines, Golden, CO, 2014.

[48] O. Ozcan, H. Sarak, E. Ozkan, et al., A trilinear flow model for a fractured horizontal well in a fractal unconventional reservoir,in:Paper SPE-170971-MS Presented at the SPE Annual Technical Conference and Exhibition, Amsterdam, Netherlands, 27e29 October, 2014, 2014.

[49] J. Ren, P. Guo, Anomalous diffusion performance of MFHWs in shale gas reservoirs, J. Nat. Gas Sci. Eng. 26 (2015) 642-651.

[50] R. Camacho-Velázquez, G. Fuentes-Cruz, M. Vásquez-Cruz, Decline-curve analysis of fractured reservoirs with fractal geometry, SPE Reserve Eval. Eng. 11 (2008) 606-619.

[51] K. Razminia, A. Razminia, J.A. Tenreiro Machado, Analysis of diffusion process in fractured reservoirs using fractional derivative approach, Commun. Nonlinear Sci. Numer. Simul. 19 (2014) 3161-3170.

[52] K. Razminia, A. Razminia, J.J. Trujillo, Analysis of radial composite systems based on fractal theory and fractional calculus, Signal Process. 107 (2015) 378-388.

[53] K. Razminia, A. Razminia, J.A.T. Machado, Analytical solution of fractional order diffusivity equation with wellbore storage and skin effects[J], J. Comput. Nonlinear Dyn. 11 (1) (2016) 011006.

[54] K. Razminia, A. Razminia, D. Torres, Pressure responses of a vertically hydraulic fractured well in a reservoir with fractal structure, Appl. Math. Comput. 257 (2015) 374-380.

[55] Y. Wei, D. He, J. Wang, et al., A coupled model for fractured shale reservoirs with characteristics of continuum media and fractal geometry[C]// SPE Asia pacific unconventional resources conference and exhibition, 9-11 November, Brisbane, Australia, Soc. Petrol. Eng. (2015). SP--176843-MS.

[56] J. Wang, Y. Wei, Y. Qi, Semi-analytical modeling of flow behavior in fractured media with fractal geometry[J], Transp. Porous Media (2016) 1-30.

[57] A.F. Van Everdingen, W. Hurst, The application of the Laplace transformation to flow problems in reservoirs, JPT J. Pet. Technol. 1 (1949) 305-324.

[58] H. Stehfest, Numerical inversion of Laplace transforms, Commun. ACM 13 (1) (1970) 47-49.