Rate Decline Analysis of Vertically Fractured Wells in Shale Gas Reservoirs
Abstract
:1. Introduction
2. Pseudo-Functions Approach
2.1. Derivation of the Pseudo-Functions
2.2. Behaviors of the Pseudo-Time Factor
3. Mathematical Model
3.1. Model Assumption
3.2. Solution for the Model
3.3. Model Validation
4. Parametric Study on Type Curves
4.1. Effects of Depletion-Driven Fluid Propertiesand Gas Desorption
4.2. Effect of Langmuir Volume
4.3. Effect of Langmuir Pressure
4.4. Example Calculation
5. Conclusions
- (1)
- In this work, the application of the pseudo-functions approach has been extended to solve the nonlinear flow problems of shale gas. This is accomplished by the definition of the pseudo-time factor accounting for both the viscosity-compressibility changes and desorption effect during reservoir depletion. The best advantage of this approach is that some partial differential equations can be effectively linearized, which contributes to the comprehensive investigation of the production performance of a fractured well in a shale gas reservoir.
- (2)
- The material balance equation with gas desorption is derived by the integration of the continuity equation with definite conditions, which can be used to obtain the analytical results of material balance equation in the application of well testing.
- (3)
- The modified formulation is validated and verified with the commercial software, and the successful analytical match demonstrates that the proposed model can effectively capture the production performance of gas reservoirs with significant desorption effect.
- (4)
- At a later production period, the production behaviors are significantly affected by the depletion-driven fluid properties and gas desorption in a shale gas reservoir. The shale gas reservoir can receive support from desorption effect in this period. A larger Langmuir volume or larger Langmuir pressure leads to a greater energy supply and less rate decline under a constant bottom-hole pressure condition.
Acknowledgments
Author Contributions
Conflicts of Interest
Nomenclature
Dimensionless Variables
vtDf | dimensionless time |
pD | dimensionless pseudo pressure |
qD | dimensionless flow rate |
cfD | dimensionless fracture conductivity |
xD | dimensionless coordinate in the x direction |
yD | dimensionless coordinate in the y direction |
xeD | dimensionless reservoir length |
yeD | dimensionless reservoir width |
wfD | dimensionless fracture width |
s | time variable in Laplace domain, dimensionless |
dimensionless pseudo pressure pD of finite-conductivity fracture in Laplace domain | |
dimensionless flow rate qD of finite-conductivity fracture in Laplace domain |
Field Variables
x, y | plane coordinates |
wf | fracture width, m |
yf | fracture half-length, m |
xe | lateral boundary of reservoir, m |
ye | vertical boundary of reservoir, m |
p | pressure, MPa |
pi | initial pressure, MPa |
pwf | bottom-hole producing pressure, MPa |
pf | fracture pressure, MPa |
pL | Langmuir pressure, MPa |
pP | pseudo pressure, MPa |
pavg | average pressure in reservoir, MPa |
Ti | temperature in reservoir, K |
qg | gas flow rate, 104 m3/d |
qgsc | standard gas flow rate, 104 m3/d |
kg | gas reservoir permeability, 10−3 μm2 |
kf | fracture permeability, 10−3 μm2 |
cfD | fracture conductivity, dimensionless |
h | reservoir thickness, m |
μg | gas viscosity, mPa·s |
Bg | Formation volume factor, m3/m3 |
φ | reservoir porosity, fraction |
t | duration, day |
cg | isothermal gas compressibility factor, 1/MPa |
Swi | irreducible water saturation, % |
γg | specific gravity, fraction |
ρg | free gas density, kg/m3 |
ρa | adsorbed gas density, kg/m3 |
ρb | bulk density of shale, kg/m3 |
vg | Darcy velocity of gas, m/s |
Vg | gas volume of adsorption, m3/kg |
VL | Langmuir volume, m3/kg |
Z | gas compressibility factor, fraction |
GP | cumulative gas production, 104 m3 |
Gsc | original gas in place, 104 m3 |
β | pseudo-time factor, dimensionless |
αt | coefficient, 3.6 × 24 × 10−3 |
αp | coefficient, 2π × 3.6 × 24 × 10−7 |
Special Subscripts:
D | dimensionless |
g | gas property |
i | initial condition |
f | fracture property |
sc | standard condition |
Appendix A. Derivation of the Model
References
- Jarvie, D.M. Shale resource systems for oil and gas: Part 1—Shale-gas resource systems. AAPG Mem. 2012, 97, 69–87. [Google Scholar]
- Zhang, L.; Pan, R. Major Accumulation Factors and Storage Reconstruction of Shale Gas Reservoir. China Pet. Explor. 2009, 14, 20–23. [Google Scholar]
- Hu, Z.; Du, W.; Peng, Y.; Zhao, J. Microscopic pore characteristics and the source-reservoir relationship of shale—A case study from the Wufeng and Longmaxi Formations in Southeast Sichuan Basin. Oil Gas Geol. 2015, 36, 1001–1008. [Google Scholar]
- Martin, J.P.; Hill, D.G.; Lombardi, T.E.; Nyahay, R. A Primer on New York’s Gas Shales. In Field Trip Guidebook for the 80th Annual Meeting of the New York State Geological Association; New York State Geological Association: New York, NY, USA, 2010; pp. A1–A32. [Google Scholar]
- Tinni, A.; Sondergeld, C.; Rai, C. New Perspectives on the Effects of Gas Adsorption on Storage and Production of Natural Gas from Shale Formations. In Proceedings of the SPWLA 58th Annual Logging Symposium, Oklahoma, OK, USA, 17–21 June 2017; Society of Petrophysicists and Well-Log Analysts: Houston, TX, USA, 2017. [Google Scholar]
- Zhang, K.; Wang, M.; Liu, Q.; Wu, K.; Yu, L.; Zhang, J.; Chen, S. Effects of Adsorption and Confinement on Shale Gas Production Behavior. In Proceedings of the SPE/IATMI Asia Pacific Oil & Gas Conference and Exhibition, Bali, Indonesia, 20–22 October 2015; Society of Petroleum Engineers: Richardson, TX, USA, 2015. [Google Scholar]
- Mengal, S.A.; Wattenbarger, R.A. Accounting for adsorbed gas in shale gas reservoirs. In Proceedings of the SPE Middle East Oil and Gas Show and Conference, Manama, Bahrain, 25–28 September 2011; SPE 141085. Society of Petroleum Engineers: Richardson, TX, USA, 2011; pp. 1–15. [Google Scholar]
- Thompson, J.M.; M’Angha, V.O.; Anderson, D.M. Advancements in shale gas production forecasting-a Marcellus case study. In Proceedings of the SPE Americas Unconventional Gas Conference and Exhibition, The Woodlands, TX, USA, 14–16 June 2011; SPE 144436. Society of Petroleum Engineers: Richardson, TX, USA, 2011. [Google Scholar]
- Arps, J.J. Analysis of decline curves. Trans. AIME 1945, 160, 228–247. [Google Scholar] [CrossRef]
- Solar, C.; Blanco, A.G.; Vallone, A.; Sapag, K. Adsorption of Methane in porous materials as the basis for the storage of natural gas. In Natural Gas; InTech: Rijeka, Croatia, 2010. [Google Scholar]
- Busch, A.; Alles, S.; Gensterblum, Y.; Prinz, D.; Dewhurst, D.N.; Raven, M.D.; Stanjek, H.; Krooss, B.M. Carbon dioxide storage potential of shales. Int. J. Greenh. Gas Control 2008, 2, 297–308. [Google Scholar] [CrossRef]
- Guo, W.; Xiong, W.; Gao, S.; Hu, Z. Isothermal adsorption/desorption characteristics of shale gas. J. Cent. South Univ. 2013, 44, 2836–2840. [Google Scholar]
- Clarkson, C.; Haghshenas, B. Modeling of Supercritical Fluid Adsorption on Organic-Rich Shales and Coal. In Proceedings of the SPE Unconventional Resources Conference-USA, The Woodlands, TX, USA, 10–12 April 2013; SPE 164532. Society of Petroleum Engineers: Richardson, TX, USA, 2013; pp. 1–24. [Google Scholar]
- Cipolla, C.L.; Lolon, E.P.; Erdle, J.C.; Rubin, B. Reservoir modeling in shale-gas reservoirs. SPE Res. Eval. Eng. 2010, 13, 638–653. [Google Scholar] [CrossRef]
- Das, M.; Jonk, R.; Schelble, R. Effect of multicomponent adsorption/desorption behavior on Gas-In-Place (GIP) calculations and estimation of free and adsorbed CH4 and CO2 in shale gas systems. In Proceedings of the Annual Technical Conference and Exhibition, San Antonio, TX, USA, 8–10 October 2012; SPE 159558. Society of Petroleum Engineers: Richardson, TX, USA, 2012. [Google Scholar]
- Coletti, K. Hydraulic Fracturing in the Marcellus Shale Region of the US. 1970. Available online: http://www.northeastern.edu/nuwriting/hydraulic-fracturing-in-the-marcellus-shale-region-of-the-u-s/ (accessed on 11 October 2017).
- Frohne, K.H.; Mercer, J.C. Fractured Shale Gas Reservoir Performance Study-An Offset WellInterference Field Test. J. Pet. Technol. 1984, 36, 291–300. [Google Scholar] [CrossRef]
- Chen, C.; Ozkan, E.; Raghavan, R. A Study of Fractured Wells in Bounded Reservoirs. In Proceedings of the SPE Annual Technical Conference and Exhibition, Dallas, TX, USA, 6–9 October 1991; SPE 22717. Society of Petroleum Engineers: Richardson, TX, USA, 1991; pp. 565–576. [Google Scholar]
- Zhang, X.; Du, C.; Deimbacher, F.; Crick, M.; Harikesavanallur, A. Sensitivity Studies of Horizontal Wells with Hydraulic Fractures in Shale Gas Reservoirs. In Proceedings of the International Petroleum Technology Conference, Doha, Qatar, 7–9 December 2009; pp. 1–9, IPTC 13338. [Google Scholar]
- Deng, J.; Zhu, W.; Ma, Q. A new seepage model for shale gas reservoir and productivity analysis of fractured well. Fuel 2014, 124, 232–240. [Google Scholar] [CrossRef]
- Zhang, D.; Zhang, L.; Guo, J.; Zhou, Y.; Zhao, Y. Research on the production performance of multistage fractured horizontal well in shale gas reservoir. J. Nat. Gas Sci. Eng. 2015, 26, 279–289. [Google Scholar] [CrossRef]
- Wanniarachchi, W.A.M.; Ranjith, P.G.; Perera, M.S.A.; Lashin, A.; Al Arifi, N.; Li, J.C. Current opinions on foam-based hydro-fracturing in deep geological reservoirs. Geomech. Geophys. Geo-Energy Geo-Resour. 2015, 1, 121–134. [Google Scholar] [CrossRef]
- Ma, T.; Chen, P.; Zhao, J. Overview on vertical and directional drilling technologies for the exploration and exploitation of deep petroleum resources. Geomech. Geophys. Geo-Energy Geo-Resour. 2016, 2, 365–395. [Google Scholar] [CrossRef]
- Wang, W.; Zheng, D.; Sheng, G.; Zhang, Q.; Su, Y. A review of stimulated reservoir volume characterization for multiple fractures horizontal well in unconventional reservoirs. Adv. Geo-Energy Res. 2017, 1, 54–63. [Google Scholar]
- Wattenbarger, R.A.; El-Banbi, A.H.; Vilegas, M.E.; Maggard, J.B. Production analysis of linear flow into fractured tight gas wells. In Proceedings of the SPE Rocky Mountain Regional/Low-Permeability Reservoirs Symposium, Denver, CO, USA, 5–8 April 1998; SPE 39931. Society of Petroleum Engineers: Richardson, TX, USA, 1998; pp. 1–12. [Google Scholar]
- Cinco, L.; Samaniego, V.; Dominguez, A. Transient pressure behavior for a well with a finite-conductivity vertical fracture. Soc. Pet. Eng. J. 1978, 18, 253–264. [Google Scholar] [CrossRef]
- Cinco, L.; Satnaniego, V. Transient pressure analysis for fractured wells. J. Pet. Technol. 1981, 33, 1749–1766. [Google Scholar] [CrossRef]
- Brown, M.; Ozkan, E.; Raghavan, R.; Kazemi, H. Practical solutions for pressure transient responses of fractured horizontal wells in unconventional reservoirs. SPE Reserv. Eval. Eng. 2009, 14, 663–676. [Google Scholar] [CrossRef]
- Lee, S.; Brockenbrough, J. A new approximate analytic solution for finite-conductivity vertical fractures. SPE Form. Eval. 1986, 1, 75–88. [Google Scholar] [CrossRef]
- Azari, M.; Wooden, W.; Coble, L. A complete set of Laplace transforms for finite-conductivity vertical fractures under bilinear and trilinear flows. In Proceedings of the SPE Annual Technical Conference and Exhibition, New Orleans, LA, USA, 23–26 September 1990; SPE 20556. Society of Petroleum Engineers: Richardson, TX, USA, 1990; pp. 251–266. [Google Scholar]
- Azari, M.; Wooden, W.; Coble, L. Further Investigation on the Analytic Solution for Finite-Conductivity Vertical Fractures; SPE 21402; Society of Petroleum Engineers: Richardson, TX, USA, 1991; pp. 559–576. [Google Scholar]
- Meyer, B.R.; Bazan, L.W.; Jacot, R.H.; Lattibeaudiere, M.G. Optimization of Multiple Transverse Hydraulic Fractures in Horizontal Wellbores; SPE 131733; Society of Petroleum Engineers: Richardson, TX, USA, 2010; pp. 1–37. [Google Scholar]
- Brohi, I.; Pooladi-Darvish, M.; Aguilera, R. Modeling Fractured Horizontal Wells as Dual Porosity Composite Reservoirs-Application to Tight Gas, Shale Gas and Tight Oil Cases; SPE 144057; Society of Petroleum Engineers: Richardson, TX, USA, 2011; pp. 1–22. [Google Scholar]
- Stalgorova, E.; Mattar, L. Analytical Model for History Matching and Forecasting Production in Multifrac Composite Systems; SPE 162516; Society of Petroleum Engineers: Richardson, TX, USA, 2012; pp. 1–17. [Google Scholar]
- Stalgorova, E.; Mattar, L. Practical Analytical Model to Simulate Production of Horizontal Wells with Branch Fractures; SPE 162515; Society of Petroleum Engineers: Richardson, TX, USA, 2012; pp. 1–17. [Google Scholar]
- Yao, S.; Wang, X.; Zeng, F.; Li, M.; Ju, N. A Composite Model for Multi-Stage Fractured Horizontal Wells in Heterogeneous Reservoirs; SPE 182016; Society of Petroleum Engineers: Richardson, TX, USA, 2016; pp. 1–32. [Google Scholar]
- Guo, J.; Zeng, J.; Wang, X.; Zeng, F. Analytical Model for Multifractured Horizontal Wells in Heterogeneous Shale Reservoirs; SPE 182422; Society of Petroleum Engineers: Richardson, TX, USA, 2016; pp. 1–41. [Google Scholar]
- Zhang, M.; Vardcharragosad, P.; Ayala, H. The similarity theory applied to early-transient gas flow analysis in unconventional reservoirs. J. Nat. Gas Sci. Eng. 2014, 21, 659–668. [Google Scholar] [CrossRef]
- Nobakht, M.; Clarkson, C. A New Analytical Method for Analyzing Linear Flow in Tight/Shale Gas Reservoirs: Constant-Flowing-Pressure Boundary Condition; SPE 143989; Society of Petroleum Engineers: Richardson, TX, USA, 2012; pp. 370–384. [Google Scholar]
- Qanbari, F.; Clarkson, C. A new method for production data analysis of tight and shale gas reservoirs during transient linear flow period. J. Nat. Gas Sci. Eng. 2013, 14, 55–65. [Google Scholar] [CrossRef]
- Liu, H.H.; Ranjith, P.G.; Georgi, D.T.; Lai, B.T. Some key technical issues in modelling of gas transport process in shales: A review. Geomech. Geophys. Geo-Energy Geo-Resour. 2016, 2, 231–243. [Google Scholar] [CrossRef]
- Lee, W.J.; Holditch, S.A. Application of pseudotime to buildup test analysis of low-permeability gas wells with long-duration wellbore storage distortion. J. Pet. Technol. 1982, 34, 2877–2887. [Google Scholar] [CrossRef]
- Blasingame, T.A.; Lee, W.J. The variable-rate reservoir limits testing of gas wells. In Proceedings of the SPE Gas Technology Symposium, Dallas, TX, USA, 13–15 June 1988; Society of Petroleum Engineers: Richardson, TX, USA, 1988. [Google Scholar]
- Stanislav, J.F.; Kabir, C.S. Pressure Transient Analysis; Prentice Hall: Englewood Cliffs, NJ, USA, 1990. [Google Scholar]
- Raghavan, R. Well Test Analysis; Prentice Hall: Englewood Cliffs, NJ, USA, 1993. [Google Scholar]
- Palacio, J.C.; Blasingame, T.A. Decline Curve Analysis Using Type Curves—Analysis of Gas Well Production Data; SPE 25909; Society of Petroleum Engineers: Richardson, TX, USA, 1993; pp. 12–14. [Google Scholar]
- Horne, R.N. Modern Well Test Analysis: A Computer-Aided Approach; Petroway: Palo Alto, CA, USA, 1995. [Google Scholar]
- Lee, J.; Rollins, J.B.; Spivey, J.P. Pressure Transient Testing; SPE Textbook Series vol. 9; Society of Petroleum Engineers: Richardson, TX, USA, 2003. [Google Scholar]
- Kamal, M.M. Transient Well Testing; Mongraph Series Vol. 23; Society of Petroleum Engineers: Richardson, TX, USA, 2009. [Google Scholar]
- Sing, K.S.W.; Everett, D.H.; Haul, R.A.W.; Moscou, L.; Pierotti, R.A.; Rouquerol, J.; Siemieniewska, T. Reporting Physisorption Data for Gas/Solid Systems with Special Reference to the Determination of Surface Area and Porosity (Recommendations 1984). Pure Appl. Chem. 1985, 57, 603–619. [Google Scholar] [CrossRef]
- Sheindorf, C.H.; Rebhun, M.; Sheintuch, M. A Freundlich-type multicomponent isotherm. J. Colloid Interface Sci. 1981, 79, 136–142. [Google Scholar] [CrossRef]
- Dubinin, M.M. Fundamentals of the theory of adsorption in micropores of carbon adsorbents: Characteristics of their adsorption properties and microporous structures. Carbon 1989, 27, 457–467. [Google Scholar] [CrossRef]
- Langmuir, I. The Adsorption of Gases on Plane Surfaces of Glass, Mica and Platinum. J. Am. Chem. Soc. 1918, 40, 1403–1461. [Google Scholar] [CrossRef]
- Patzek, T.; Male, F.; Marder, M. Gas Production in the Barnett Shale Obeys a Simple Scaling Theory. Proc. Natl. Acad. Sci. USA 2013, 110, 19731–19736. [Google Scholar] [CrossRef] [PubMed]
- Yu, W.; Sepehrnoori, K. Simulation of Gas Desorption and Geomechanics Effects for Unconventional Gas Reservoirs. Fuel 2014, 116, 455–464. [Google Scholar] [CrossRef]
- Russell, D.G.; Goodrich, J.H.; Perry, G.E.; Bruskotter, J.F. Methods of Predicting Gas Well Performance; Paper SPE; Society of Petroleum Engineers: Richardson, TX, USA, 1966; pp. 99–108. [Google Scholar]
- Ye, P.; Ayala, H.L.F. A density-diffusivity approach for the unsteady state analysis of natural gas reservoirs. J. Nat. Gas Sci. Eng. 2012, 7, 22–34. [Google Scholar] [CrossRef]
- Fraim, M.; Wattenbarger, R. Gas reservoir decline curve analysis using type curves with real gas pseudopressure and normalized time. SPE Form. Eval. 1987, 2, 671–682. [Google Scholar] [CrossRef]
- King, G.R. Material Balance Techniques for Coal Seam and Devonian Shale Gas Reservoirs; SPE 20730; Society of Petroleum Engineers: Richardson, TX, USA, 1990; pp. 182–192. [Google Scholar]
- Moghadam, S.; Jeje, O.; Mattar, L. Advanced gas material balance in simplified format. J. Can. Pet. Technol. 2009, 50, 90–98. [Google Scholar] [CrossRef]
- Stehfest, H. Numerical Inversion of Laplace Transforms. Commun. ACM 1970, 13, 47–49. [Google Scholar] [CrossRef]
- Van Everdingen, A.F.; Hurst, W. The Application of The Laplace Transformation to Flow Problems in Reservoirs; Society of Petroleum Engineers: Richardson, TX, USA, 1949. [Google Scholar]
- Molina, O.; Zeidouni, M. An Enhanced Nonlinear Analytical Model for Unconventional Multifractured Systems. In Proceedings of the SPE Europec Featured at 79th EAGE Conference and Exhibition, Paris, France, 12–15 June 2017; Society of Petroleum Engineers: Richardson, TX, USA, 2017. [Google Scholar]
- IHS Harmony. Analysis Method Theory, IHS Inc. 2016. Available online: https://www.ihs.com/btp/fekete.html (accessed on 11 October 2017).
- IHS Course Manual Software Training Course: IHS Harmony and IHS Decline Plus Training. 2017. Available online: https://www.ihs.com/products/declineplus-training.html (accessed on 11 October 2017).
Parameter | Value | Unit |
---|---|---|
kg | 0.0008 | 10−3 µm2 |
φ | 14 | % |
Swi | 10 | % |
h | 25 | m |
γg | 0.6 | Value |
yf | 50 | m |
pi | 34.5 | MPa |
Ti | 327.6 | K |
ρb | 2.63 × 103 | kg/m3 |
cfD | 1.5 | Value |
Parameter | Value | Unit |
---|---|---|
Initial pressure pi | 16.3 | MPa |
Initial temperature Ti | 338.15 | K |
Formation thickness h | 39.7 | m |
Porosity φ | 5 | % |
Water saturation Swi | 34.75 | % |
Bottom-hole pressure Pwf | 4.82 | MPa |
Langmuir volume VL | 3 | m3/t |
Langmuir pressure pL | 2.8 | MPa |
Initial gas compressibility cgi | 0.0592 | MPa−1 |
Designed fracture half-length yf | 45 | m |
© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Zhang, X.; Wang, X.; Hou, X.; Xu, W. Rate Decline Analysis of Vertically Fractured Wells in Shale Gas Reservoirs. Energies 2017, 10, 1602. https://doi.org/10.3390/en10101602
Zhang X, Wang X, Hou X, Xu W. Rate Decline Analysis of Vertically Fractured Wells in Shale Gas Reservoirs. Energies. 2017; 10(10):1602. https://doi.org/10.3390/en10101602
Chicago/Turabian StyleZhang, Xiaoyang, Xiaodong Wang, Xiaochun Hou, and Wenli Xu. 2017. "Rate Decline Analysis of Vertically Fractured Wells in Shale Gas Reservoirs" Energies 10, no. 10: 1602. https://doi.org/10.3390/en10101602