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Characteristics of non-Darcy flow in low-permeability reservoirs

Characteristics of non-Darcy flow in low-permeability reservoirs parameters of low-permeability reservoirs and effectively analyzing the deliverability. Well test models must comply with the particular characteristics of flow in low-permeability reservoirs in order to obtain reasonable well test interpretation. At present, non-Darcy fl ow in low-permeability reservoirs is attracting much attention. In this study, displacement tests were conducted on typical cores taken from low-permeability reservoirs. Two dimensionless variables were introduced to analyze the collected experimental data. The results of the dimensionless analysis show whether non-Darcy fl ow happens or not depends on the properties of fl uid and porous media and the pressure differential. The combination of the above three parameters was named as dimensionless criteria coeffi cient (DCC). When the value of the DCC was lower than a critical Reynolds number (CRN), the fl ow could not be well described by Darcy’s law (so-called non-Darcy fl ow), when the DCC was higher than CRN, the fl ow obeyed Darcy’s law. Finally, this paper establishes a transient mathematical model considering Darcy flow and non- Darcy fl ow in low-permeability reservoirs, and proposes a methodology to solve the model. The solution technique, which is based on the Boltzmann transformation, is well suited for solving the flow model of low-permeability reservoirs. Based on the typical curves analysis, it was found that the pressure and its derivative curves were determined by such parameters as non-Darcy flow index and the flow characteristics. The results can be used for well test analysis of low-permeability reservoirs. Low-permeability, non-Darcy fl ow, dimensionless analysis, transient fl ow, well testing Key words: (Yang, 2007). The threshold pressure gradient and other 1 Introduction parameters mentioned in these models were only determined Low-permeability reservoirs form one of the important by specifi c core tests. This makes the model diffi cult to apply resources in China according to the statistics of Jiang et to general cases. al (2004). Low-permeability reservoirs have different Ren et al (2009) proposed a new Non-Darcy flow characteristics from the conventional ones. Non-Darcy fl ow equation for single-phase flow (oil or water) in low- occurs in the low-permeability reservoirs (Yan et al, 1990). permeability porous media. This equation took into account When the pressure gradient ∆p is relatively low, the fluid the pore structure of reservoir rocks, fl uid properties, and the flow in low-permeability reservoirs does not obey Darcy’s pressure gradient. However, this equation is not applicable law, i.e. the fl uid velocity v is not proportional to the pressure to low-velocity fluid flow in porous media. For example, gradient; when the pressure gradient is relatively high, the the fl ow rate calculated from this equation is not zero when fluid velocity increases linearly with the pressure gradient the pressure gradient is zero, which contradicts the flow (Huang, 1998). Extrapolating the straight part of the ∆p–v characteristics in actual reservoir formations. Based on the curve to the pressure gradient axis, the pseudo threshold core flow tests, Yao and Ge (2001) concluded that the flow pressure gradient is obtained. The equation including the within the low-permeability cores was non-linear when the threshold pressure gradient was fi rstly used to describe fl uid dimensionless criteria coeffi cient (DCC) was lower than the -5 fl ow in low-permeability reservoirs. To increase the prediction critical Reynolds number (CRN, 8.5×10 ); and the fl ow was accuracy of the fl ow equation, a three-parameter model was linear when it exceeded CRN. The experimental results were presented (Deng and Liu, 2001; Deng et al, 2007; 2009). also in agreement with microscopic boundary layer theory. Later, the model was simplified to a two-parameter model In order to achieve more satisfactory results, the resistance coeffi cient of the above model was modifi ed by Liu and Liu (2003). However, the new resistance coefficient is not easy *Corresponding author. email: yaoyuedong@163.com to apply to general cases due to varying threshold pressure Received January 15, 2010 56 Pet.Sci.(2011)8:55-62 gradient in different cores. Using the method proposed by of parameters such as the distribution of flow pattern and Yao and Ge in 2001, Li et al (2005) drew the same conclusion seepage index, and then analyzed the transient pressure and except for a slightly difference in the critical Reynolds its infl uencing factors. −5 number (Re =8.95×10 ). 2 Flow tests on low-permeability cores For non-Darcy transient flow in low-permeability reservoirs, it is convenient to choose the Izbash equation 2.1 Cores and fl owing medium (Bordier and Zimmer, 2000) to describe non-Darcy flow. The Izbash equation indicates that the hydraulic gradient Flow tests were performed on 28 cylindrical core samples is a power function of the flow rate. Izbash’s law has been taken from low-permeability reservoirs. The core plugs were preferred to derive drainage equations because it is in 8 cm long and 2.5 cm in diameter. The gas permeability and –3 2 continuity with Darcy’s law and facilitates the development of porosity of these samples ranged from 0.017 to 50×10 μm an analytical solution. Because of its nonlinearity, the Izbash and 0.095 to 0.218, respectively. There were no structural equation is usually simplifi ed by the linear approximation, and defects like surface cracks, bedding planes, and corrosion then an approximate solution is obtained (Ikoku and Ramey, holes etc. All samples were consistent with the characteristics 1979). The above approximation is applicable only for fl uid of a sedimentary environment. The fluids used in core flow near the borehole or in large time, and the radial flow flow tests were 8% KCl solution, kerosene, and simulated rate calculated is only accurate in the wellbore. Assuming that oils A and B. Simulated oils A and B were prepared by the fl ow rate is constant at any time and any space, the longer blending degassed oil with kerosene in a ratio of 1:1 and 1:3, the distance from the wellbore and the earlier the time is, the respectively. The physical properties of these fluids were greater the error is. The same approximate linear method was listed in Table 1. applied to study non-Newton fluid flow (Ikoku and Ramey, 1982; Tong and Shi, 2004; Tong and Wang, 2004; Li et al, Table 1 Physical properties of fl uids used in fl ow tests 2007; Zhang and Yue, 2007). Other researchers studied the Fluid Relative density Viscosity, mPa·s transient fl ow in low-permeability reservoirs using the non- Darcy flow model with the additional threshold pressure 8% KCl solution 1.01 0.89 gradient (Liu, 1982; Cheng et al, 1996; Song and Liu,1999; Kerosene 0.80 0.90 Hou and Tong, 2009; Xiong et al, 2009), while the above method is restricted to set the exact value of the threshold Simulated oi l A 0.84 5.46 pressure gradient just as mentioned above. By using the Simulated oil B 0.82 2.13 power function of the Izbash equation (Wen et al, 2008), a single power function was used in the equation of the whole 2.2 Experimental equipment and method flow regime. The above method should divide into regimes individually according to the distribution of fl ow pattern. The The experimental equipment was mainly composed dimensionless bottom-hole pressure solution of above model of a flowing system, a pressure control system, and a data is lower than Darcy’s fl ow, it is obviously unreasonable. acquisition system, as shown in Fig. 1. Core flooding In this study, we conducted a number of core tests on low- tests were conducted in a conventional manner at room permeability reservoir rocks to determine the flow equation temperature (20 °C). The core was saturated with the fl uid of and critical Reynolds number for non-Darcy flow in low- known density and viscosity. permeability reservoirs. According to the characteristics The pump connected to the experimental fl uid container of fluid flow in low-permeability reservoirs, we developed maintained the desired fl ow rate through the core. The outlet a complex model for transient flow in low-permeability of the pump was connected to the core. The effl uents from the reservoirs, solved the flow model considering the variation core outlet were collected in a fraction collector. A differential Pressure Pressure transducer Core holder transducer Quizix pump Computer Experimental fluid Fluid receiver Hand pump Fig. 1 A schematic diagram of the fl ow test apparatus Pet.Sci.(2011)8:55-62 57 58 Pet.Sci.(2011)8:55-62 60 Pet.Sci.(2011)8:55-62 Pet.Sci.(2011)8:55-62 61 62 62 Pet.Sci.(2011)8:55-62 Development. 1996. 23(4): 50-53 (in Chinese) 2003. 22(4): 556-561 (in Chinese) Den g Y E, Huang R Q and Liu C Q. Nonlinear flow law and Ren X J, Zhang G H and Liao F F. Criterion of starting pressure gradient consolidation in unsaturated low-permeability clays. Journal of existence of non-Darcy fl owing in low permeability porous media. Hydrodynamics, Serial A. 2009. 24 (1): 99-105 (in Chinese) Journal of Liaoning Technical University (Natural Science). 2009. Den g Y E and Liu C Q. A mathematical model of nonlinear flow law 28(Suppl): 273-276 (in Chinese) in low permeability porous media and its application. Acta Petrolei Son g F Q and Liu C Q. Analysis of two-phase fluid flow in low Sinica. 2001. 22(4): 72-76 (in Chinese) permeability reservoirs with the threshold pressure gradient. Journal Den g Y E, Xie H P, Huang R Q, et al. Law of nonlinear fl ow in saturated of the University of Petroleum, China. 1999. 23(3): 47-50, 56 (in clays and radial consolidation. Applied Mathematics and Mechanics. Chinese) 2007. 28 (11): 1427-1436 Ton g D K and Shi L N. The generalized fl ow analysis of non-Newtonian Hou Y M and Tong D K. Nonsteady fl ow of non-Newtonian power-law visco-elastic fl uid fl ows in porous media. Journal of Hydrodynamics, fl uids of low permeability with moving-boundary in double porous Serial A. 2004. 19(6): 695-701 (in Chinese) media and fractal reservoir. Engineering Mechanics. 2009. 26(8): Ton g D K and Wang R H. Analysis of non-Newtonian visco-elastic fl uid 245-250 (in Chinese) fl ows in fractal reservoirs. Science in China, Series G. 2004. (01): Hua ng Y Z. Fluid Mechanics in Low-permeability Reservoir. Beijing: 102-109 (in Chinese) Petroleum Industry Press. 1998. 131-135 (in Chinese) Wen Z, Huang G H and Zhan H B. An analytical solution for non- Iko ku C U and Ramey Jr H J. Transient fl ow of non-Newtonian power Darcian flow in a confined aquifer using the power law function. law fl uids in porous media. SPE Journal. 1979. 44 (3): 164-174 Advances in Water Resources. 2008. (31): 44-55 Iko ku C U and Ramey H J. Pressure behavior during polymer fl ow in Xio ng W, Shen R and Gao S S. Non-linear flow theory in low petroleum reservoirs. Journal of Energy Resources Technology. 1982. permeability reservoir and its preliminary application. Journal of 104: 149-156 Liaoning Technical University (Natural Science). 2009. 28(Suppl): Jia ng L Z, Gu J Y and Guo B C. Characteristics and mechanism of low 58-60 (in Chinese) permeability clastic reservoirs in Chinese petroliferous basin. Acta Yan Q L, He Q X, Wei L G, et al. A laboratory study of percolation Sedimentologica Sinica. 2004. 22(1): 13-18 (in Chinese) characteristics of single phase fl ow in low-permeability reservoirs. Li M, Diao N R and Fang Z H. Analysis of seepage fl ow in a confi ned Journal of Xi’an Shiyou University. 1990. 5(2): 1-6 (in Chinese) aquifer with a standing column well. Journal of Hydrodynamics, Yan g Q L. Nonlinear flow theory in ultra-low permeability reservoirs Serial B. 2007. 19(1): 84-91 and its application. Ph.D Thesis. Institute of Porous Flow and Fluid Li Z F, He S L and Men C Q. Study on the non-Darcy percolation rules Mechanics of CAS. 2007 (in Chinese) in the low permeable oilfield. Well Testing. 2005. 14(3): 14-17 (in Yao Y D and Ge J L. Study of the fluid flow in low permeability Chinese) reservoirs. Petroleum Exploration and Development. 2001. 28(4): Liu C Q. Approximation solution of seepage with threshold pressure 73-75 (in Chinese) gradient. Chinese Journal of Geotechnical Engineering. 1982. 4(3): Zha ng L J and Yue X A. Mechanism for viscoelastic polymer solution 107-109 (in Chinese) percolating through porous media. Journal of Hydrodynamics, Serial Liu J J and Liu X G. Study of nonlinear seepage of rock of low B. 2007. 19 (2): 241-248 permeability. Chinese Journal of Rock Mechanics and Engineering. (Edited by Sun Yanhua) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Petroleum Science Springer Journals

Characteristics of non-Darcy flow in low-permeability reservoirs

Yao, Yuedong; Ge, Jiali
Petroleum Science , Volume 8 (1) – Feb 15, 2011
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Publisher
Springer Journals
Copyright
Copyright © 2011 by China University of Petroleum (Beijing) and Springer-Verlag Berlin Heidelberg
Subject
Earth Sciences; Mineral Resources; Industrial Chemistry/Chemical Engineering; Industrial and Production Engineering; Energy Economics
ISSN
1672-5107
eISSN
1995-8226
DOI
10.1007/s12182-011-0115-3
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Abstract

parameters of low-permeability reservoirs and effectively analyzing the deliverability. Well test models must comply with the particular characteristics of flow in low-permeability reservoirs in order to obtain reasonable well test interpretation. At present, non-Darcy fl ow in low-permeability reservoirs is attracting much attention. In this study, displacement tests were conducted on typical cores taken from low-permeability reservoirs. Two dimensionless variables were introduced to analyze the collected experimental data. The results of the dimensionless analysis show whether non-Darcy fl ow happens or not depends on the properties of fl uid and porous media and the pressure differential. The combination of the above three parameters was named as dimensionless criteria coeffi cient (DCC). When the value of the DCC was lower than a critical Reynolds number (CRN), the fl ow could not be well described by Darcy’s law (so-called non-Darcy fl ow), when the DCC was higher than CRN, the fl ow obeyed Darcy’s law. Finally, this paper establishes a transient mathematical model considering Darcy flow and non- Darcy fl ow in low-permeability reservoirs, and proposes a methodology to solve the model. The solution technique, which is based on the Boltzmann transformation, is well suited for solving the flow model of low-permeability reservoirs. Based on the typical curves analysis, it was found that the pressure and its derivative curves were determined by such parameters as non-Darcy flow index and the flow characteristics. The results can be used for well test analysis of low-permeability reservoirs. Low-permeability, non-Darcy fl ow, dimensionless analysis, transient fl ow, well testing Key words: (Yang, 2007). The threshold pressure gradient and other 1 Introduction parameters mentioned in these models were only determined Low-permeability reservoirs form one of the important by specifi c core tests. This makes the model diffi cult to apply resources in China according to the statistics of Jiang et to general cases. al (2004). Low-permeability reservoirs have different Ren et al (2009) proposed a new Non-Darcy flow characteristics from the conventional ones. Non-Darcy fl ow equation for single-phase flow (oil or water) in low- occurs in the low-permeability reservoirs (Yan et al, 1990). permeability porous media. This equation took into account When the pressure gradient ∆p is relatively low, the fluid the pore structure of reservoir rocks, fl uid properties, and the flow in low-permeability reservoirs does not obey Darcy’s pressure gradient. However, this equation is not applicable law, i.e. the fl uid velocity v is not proportional to the pressure to low-velocity fluid flow in porous media. For example, gradient; when the pressure gradient is relatively high, the the fl ow rate calculated from this equation is not zero when fluid velocity increases linearly with the pressure gradient the pressure gradient is zero, which contradicts the flow (Huang, 1998). Extrapolating the straight part of the ∆p–v characteristics in actual reservoir formations. Based on the curve to the pressure gradient axis, the pseudo threshold core flow tests, Yao and Ge (2001) concluded that the flow pressure gradient is obtained. The equation including the within the low-permeability cores was non-linear when the threshold pressure gradient was fi rstly used to describe fl uid dimensionless criteria coeffi cient (DCC) was lower than the -5 fl ow in low-permeability reservoirs. To increase the prediction critical Reynolds number (CRN, 8.5×10 ); and the fl ow was accuracy of the fl ow equation, a three-parameter model was linear when it exceeded CRN. The experimental results were presented (Deng and Liu, 2001; Deng et al, 2007; 2009). also in agreement with microscopic boundary layer theory. Later, the model was simplified to a two-parameter model In order to achieve more satisfactory results, the resistance coeffi cient of the above model was modifi ed by Liu and Liu (2003). However, the new resistance coefficient is not easy *Corresponding author. email: yaoyuedong@163.com to apply to general cases due to varying threshold pressure Received January 15, 2010 56 Pet.Sci.(2011)8:55-62 gradient in different cores. Using the method proposed by of parameters such as the distribution of flow pattern and Yao and Ge in 2001, Li et al (2005) drew the same conclusion seepage index, and then analyzed the transient pressure and except for a slightly difference in the critical Reynolds its infl uencing factors. −5 number (Re =8.95×10 ). 2 Flow tests on low-permeability cores For non-Darcy transient flow in low-permeability reservoirs, it is convenient to choose the Izbash equation 2.1 Cores and fl owing medium (Bordier and Zimmer, 2000) to describe non-Darcy flow. The Izbash equation indicates that the hydraulic gradient Flow tests were performed on 28 cylindrical core samples is a power function of the flow rate. Izbash’s law has been taken from low-permeability reservoirs. The core plugs were preferred to derive drainage equations because it is in 8 cm long and 2.5 cm in diameter. The gas permeability and –3 2 continuity with Darcy’s law and facilitates the development of porosity of these samples ranged from 0.017 to 50×10 μm an analytical solution. Because of its nonlinearity, the Izbash and 0.095 to 0.218, respectively. There were no structural equation is usually simplifi ed by the linear approximation, and defects like surface cracks, bedding planes, and corrosion then an approximate solution is obtained (Ikoku and Ramey, holes etc. All samples were consistent with the characteristics 1979). The above approximation is applicable only for fl uid of a sedimentary environment. The fluids used in core flow near the borehole or in large time, and the radial flow flow tests were 8% KCl solution, kerosene, and simulated rate calculated is only accurate in the wellbore. Assuming that oils A and B. Simulated oils A and B were prepared by the fl ow rate is constant at any time and any space, the longer blending degassed oil with kerosene in a ratio of 1:1 and 1:3, the distance from the wellbore and the earlier the time is, the respectively. The physical properties of these fluids were greater the error is. The same approximate linear method was listed in Table 1. applied to study non-Newton fluid flow (Ikoku and Ramey, 1982; Tong and Shi, 2004; Tong and Wang, 2004; Li et al, Table 1 Physical properties of fl uids used in fl ow tests 2007; Zhang and Yue, 2007). Other researchers studied the Fluid Relative density Viscosity, mPa·s transient fl ow in low-permeability reservoirs using the non- Darcy flow model with the additional threshold pressure 8% KCl solution 1.01 0.89 gradient (Liu, 1982; Cheng et al, 1996; Song and Liu,1999; Kerosene 0.80 0.90 Hou and Tong, 2009; Xiong et al, 2009), while the above method is restricted to set the exact value of the threshold Simulated oi l A 0.84 5.46 pressure gradient just as mentioned above. By using the Simulated oil B 0.82 2.13 power function of the Izbash equation (Wen et al, 2008), a single power function was used in the equation of the whole 2.2 Experimental equipment and method flow regime. The above method should divide into regimes individually according to the distribution of fl ow pattern. The The experimental equipment was mainly composed dimensionless bottom-hole pressure solution of above model of a flowing system, a pressure control system, and a data is lower than Darcy’s fl ow, it is obviously unreasonable. acquisition system, as shown in Fig. 1. Core flooding In this study, we conducted a number of core tests on low- tests were conducted in a conventional manner at room permeability reservoir rocks to determine the flow equation temperature (20 °C). The core was saturated with the fl uid of and critical Reynolds number for non-Darcy flow in low- known density and viscosity. permeability reservoirs. According to the characteristics The pump connected to the experimental fl uid container of fluid flow in low-permeability reservoirs, we developed maintained the desired fl ow rate through the core. The outlet a complex model for transient flow in low-permeability of the pump was connected to the core. The effl uents from the reservoirs, solved the flow model considering the variation core outlet were collected in a fraction collector. A differential Pressure Pressure transducer Core holder transducer Quizix pump Computer Experimental fluid Fluid receiver Hand pump Fig. 1 A schematic diagram of the fl ow test apparatus Pet.Sci.(2011)8:55-62 57 58 Pet.Sci.(2011)8:55-62 60 Pet.Sci.(2011)8:55-62 Pet.Sci.(2011)8:55-62 61 62 62 Pet.Sci.(2011)8:55-62 Development. 1996. 23(4): 50-53 (in Chinese) 2003. 22(4): 556-561 (in Chinese) Den g Y E, Huang R Q and Liu C Q. Nonlinear flow law and Ren X J, Zhang G H and Liao F F. Criterion of starting pressure gradient consolidation in unsaturated low-permeability clays. Journal of existence of non-Darcy fl owing in low permeability porous media. Hydrodynamics, Serial A. 2009. 24 (1): 99-105 (in Chinese) Journal of Liaoning Technical University (Natural Science). 2009. Den g Y E and Liu C Q. A mathematical model of nonlinear flow law 28(Suppl): 273-276 (in Chinese) in low permeability porous media and its application. Acta Petrolei Son g F Q and Liu C Q. Analysis of two-phase fluid flow in low Sinica. 2001. 22(4): 72-76 (in Chinese) permeability reservoirs with the threshold pressure gradient. Journal Den g Y E, Xie H P, Huang R Q, et al. Law of nonlinear fl ow in saturated of the University of Petroleum, China. 1999. 23(3): 47-50, 56 (in clays and radial consolidation. Applied Mathematics and Mechanics. Chinese) 2007. 28 (11): 1427-1436 Ton g D K and Shi L N. The generalized fl ow analysis of non-Newtonian Hou Y M and Tong D K. Nonsteady fl ow of non-Newtonian power-law visco-elastic fl uid fl ows in porous media. Journal of Hydrodynamics, fl uids of low permeability with moving-boundary in double porous Serial A. 2004. 19(6): 695-701 (in Chinese) media and fractal reservoir. Engineering Mechanics. 2009. 26(8): Ton g D K and Wang R H. Analysis of non-Newtonian visco-elastic fl uid 245-250 (in Chinese) fl ows in fractal reservoirs. Science in China, Series G. 2004. (01): Hua ng Y Z. Fluid Mechanics in Low-permeability Reservoir. Beijing: 102-109 (in Chinese) Petroleum Industry Press. 1998. 131-135 (in Chinese) Wen Z, Huang G H and Zhan H B. An analytical solution for non- Iko ku C U and Ramey Jr H J. Transient fl ow of non-Newtonian power Darcian flow in a confined aquifer using the power law function. law fl uids in porous media. SPE Journal. 1979. 44 (3): 164-174 Advances in Water Resources. 2008. (31): 44-55 Iko ku C U and Ramey H J. Pressure behavior during polymer fl ow in Xio ng W, Shen R and Gao S S. Non-linear flow theory in low petroleum reservoirs. Journal of Energy Resources Technology. 1982. permeability reservoir and its preliminary application. Journal of 104: 149-156 Liaoning Technical University (Natural Science). 2009. 28(Suppl): Jia ng L Z, Gu J Y and Guo B C. Characteristics and mechanism of low 58-60 (in Chinese) permeability clastic reservoirs in Chinese petroliferous basin. Acta Yan Q L, He Q X, Wei L G, et al. A laboratory study of percolation Sedimentologica Sinica. 2004. 22(1): 13-18 (in Chinese) characteristics of single phase fl ow in low-permeability reservoirs. Li M, Diao N R and Fang Z H. Analysis of seepage fl ow in a confi ned Journal of Xi’an Shiyou University. 1990. 5(2): 1-6 (in Chinese) aquifer with a standing column well. Journal of Hydrodynamics, Yan g Q L. Nonlinear flow theory in ultra-low permeability reservoirs Serial B. 2007. 19(1): 84-91 and its application. Ph.D Thesis. Institute of Porous Flow and Fluid Li Z F, He S L and Men C Q. Study on the non-Darcy percolation rules Mechanics of CAS. 2007 (in Chinese) in the low permeable oilfield. Well Testing. 2005. 14(3): 14-17 (in Yao Y D and Ge J L. Study of the fluid flow in low permeability Chinese) reservoirs. Petroleum Exploration and Development. 2001. 28(4): Liu C Q. Approximation solution of seepage with threshold pressure 73-75 (in Chinese) gradient. Chinese Journal of Geotechnical Engineering. 1982. 4(3): Zha ng L J and Yue X A. Mechanism for viscoelastic polymer solution 107-109 (in Chinese) percolating through porous media. Journal of Hydrodynamics, Serial Liu J J and Liu X G. Study of nonlinear seepage of rock of low B. 2007. 19 (2): 241-248 permeability. Chinese Journal of Rock Mechanics and Engineering. (Edited by Sun Yanhua)

Journal

Petroleum Science – Springer Journals

Published: Feb 15, 2011

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