Apparent Permeability Model of Nanometer Pore in the Shale Reservoir Fengqiang Xu 1, a, Quan Zhang 1, b, Lingyan Niu 2, c, Xiaodong Wang 1, d 1 School of Energy Resources, China University of Geosciences, Beijing, 100083, China 2 Research Institute of Exploration and Development, Yumen Oilfield Company PetroChina, Jiuquan, 735019, China Abstract a 254545235@qq.com, b cugbszhangquan@163.com, c niuly@petrochina.com.cn, d wxd_cug@cugb.edu.cn The size of pore radius is primarily within the range of a few nanometers to a few tens of nanometers because of the very small pore structure, which can t satisfy the basic assumption of the classic Darcy law. The current research of the flow mathematical model for the shale gas reservoir focuses mainly on the correction of the viscous flow model and ignores the problem the transition flow between the viscous flow and molecular flow. Based on the pressure, temperature, pore structure and gas properties of the gas reservoir, this paper considered the effect of gas slippage and influence of Knudsen diffusion. Due to the above factors, in this paper, apparent permeability is considered as a function of the pore radius and gas pressure and different flow models of apparent permeability and slippage is calculated and analyzed. The law of the ratio of apparent permeability to true permeability and slippage with the gas pressure and porous configuration of different models is analyzed. The results show: the smaller the porosity of the shale reservoir is and the lower the gas pressure, the ratio of apparent permeability to true permeability is smaller. Keywords Shale Gas; Nanometer Pores; Apparent Permeability; Slippage; Knudsen Diffusion. 1. Introduction The pore size of conventional sandstone and carbonate roc reservoirs is between 1 and 100 micrometers, while the pore throat diameter of shale gas reservoir is between 5 and 1000 nanometers [1] or in a much smaller scale between 5 and 50 nanometers [2]. Because of the pore radius being at nanoscale sizes, the permeability of matrix is between micro-darcy and nano-darcy in the shale gas reservoir, which need fracturing reconstruction to get economic output. The flow characteristics of gas reservoir have many flow patterns from microcosmic flow in the nanometer pores to Darcy flow in the hydraulic fracture. Especially when the collisions between gas molecules in the matrix and pipe wall are much more frequent, the gas flow show the characteristic of molecular micro seepage pattern, but not follows the Darcy flow. The slippage and Knudsen diffusion can explain that the actual production of shale gas reservoir is higher than forecasting production. In view of various flow patterns such as viscous flow, slip flow, transition flow and molecular flow, based on the definition of apparent permeability, different models of apparent permeability of shale gas reservoir comparative analyzed and the ratio of apparent permeability to true permeability of different models and the law of the slip factor with gas pressure and pore radius is calculated [3]. In this paper, apparent permeability analytical models as a function of gas pressure and pore radius are discussed and analyzed. Meanwhile, the suitability of different analytical models of apparent permeability is assessed under different pressures and pore radiuses from theoretical formula and graphic method of analysis. 145
2. Knudsen diffusion and division of flow pattern The speed of fluid at pore wall is considered as zero in the fluid mechanics of continuous medium [4]. The pore radius is between 1 and 100 micrometers in the conventional reservoir, which can satisfy the condition of Darcy law. In the shale reservoir, the pore radius is between 1 and 200 nanometers [5] and the fluid mechanics of continuous medium should not be applied. Many molecules strie pore wall and tend to slide on the pore wall [6]. Jones and Owens[7] carried out an experiment to derive the formula of gas static slip factor and made a comparison with the result got by Klinenberg[8]; Sampath[9], Keighin and Florence [10,11] established an empirical formula similar to the equation derived by Jones and Owens. Ertein et al.[12] introduce dynamic slip factor as a function of pressure which was closely related to Knudsen diffusion coefficient and further developed the slip factor. Based on the wor of Beso and Karniadais, Michel et al.[13] developed slip factor by introducing Knudsen diffusion coefficient. Civan, Sahaee-Pour and Bryant [14,15,16] determined slip factor and derived apparent permeability as a function of Knudsen number. Javadpour [17] deduced apparent the permeability formula as a function of pressure, temperature too into account the influence of Knudsen diffusion and slip factor. Klinenberg was the first person that proposed gas slippage effect in petroleum industry. The apparent permeability formula that he raised is: b a (1 ) p Here, a is apparent permeability (md); is equivalent of fluid permeability (md); p avg is the average gas pressure under trial conditions; b is slip factor. The equation of slip factor is: 4c p b r Knudsen number is a vital parameter that decides the flow pattern (includes viscous flow, slip flow, transition flow and molecular flow). In math, the definition of Knudsen [18] number is: n r The mean free path of gas molecules [19] is: RT p 2M The flow pattern can be divided into four types. Different persons have different limits for the definition of the viscous flow area, which researchers generally believed that the critical value of Knudsen number of viscous flow is 0.001[20], while in other papers is 0.01[21,22]. In this paper, the critical value is 0.001. Assume that the pore radius is 6.5nm, the pressure is from 200psi to 7000psi and the average free path is from 0.6nm to 8nm, so the Knudsen number is from 0.09 to 1.23. The flow pattern of studied is in the area of slip flow and transition flow. Table.1 Division of flow pattern according to Knudsen number Knudsen number flow pattern Kn 0.001 avg viscous flow 0.001< Kn <0.1 Slip flow 0.1< Kn <10 transition flow Kn 10 molecular flow 146
3. Model of apparent permeability with slip factor and Knudsen diffusion 3.1 Model of absolute permeability with slip flow Jones and Owens [23] studied 100 samples of tight gas reservoir, raising an experimental correlation of slip factor and absolute permeability. Sampath and Keighin provided a calculating equator of slip factor for the core samples (net confining pressure is 34.5Mpa; the diameter is 0.025 meters; the length is 0.05 meters) at the temperature of 80 to 90. Florence et al put forward a relationship of nitrogen flow in porous media, which involve slippage effect in the formula of average free path and Knudsen number having nothing to do with pressure. In conclusion, the model of absolute permeability with slip factor can be seen in Tab.2. Table.2 Model of absolute permeability with slip factor time author slip factor significant factor 1980 Jones and Owens 1982 Sampath and Keighin b b 0.33 12.639( ) static slippage effect 0.53 13.581( ) static slippage effect 2007 Florence b 0.5 43.345( ) static slippage effect 3.2 Model of slip factor with Knudsen diffusion Ertein et al deduced the formula of Knudsen diffusion coefficient as shown in Tab.3. Michel et al established apparent permeability models referring to Beso and Karniadais. Michel et al too into consideration Knudsen diffusion, matrix porosity and tortuosity of porous media, average free path of gas molecules and pressure. The equation of equivalent matrix permeability could be seen in Tab.3 in SI. Table.3 Slip factor model with Knudsen diffusion time author Knudsen diffusion slip factor significant factor 1986 Ertein 2011 Michel D D 31.54 ( ) 0.67 M 2r 8RT ( ) 3 M 0.5 b b D c p g avg 1 3D m 16 m Knudsen diffusion and dynamic slippage effect Knudsen diffusion and dynamic slippage effect 3.3 Models of apparent permeability within Knudsen number Sahaee-Pour and Bryant studied flow characteristics of different numbers according to Beso and Karniadais s results and deduced flow equations of every region. They speculated that the relationship between apparent permeability and absolute permeability in transition flow is [24]: 2 (0.8453 5.4576 K 0.1633 K ) a n n Here, 0.1 K n 0.8. Civan studied the flow of shale reservoirs by Beso and Karniadais s way. The equation of apparent permeability based on Knudsen number is: f ( K ) Here, f (K n ) is the function of Knudsen number, tenuity factor and slip factor. 4Kn f ( Kn) (1 2Kn)(1 ) 1 bk The dimensionless tenuity factor α 2 is: a n n 147
B ( K n ) 2 0 B A K In the above equation, the down limit of α 2 is zero, which corresponds to slip flow; when K n tends to be infinite, the up limit of α 2 is α 0, which corresponds to free molecular flow and A and B are curve fitting coefficients. Civan fetch a specific set of data (α 0 =1.358, A=0.178, B=0.4348) for transition flow in tight sandstone. Civan came forward an approximate value of the above equation according to Florence s results. When Kn was less than or equal to 1, the dynamic slip factor is [25] (viscosity is a function of pressure): 2790 ( ) 0.5 b M 3.4 Models of apparent permeability within Knudsen diffusion and slip effect Javadpour et al. studied Knudsen diffusion and slip effect of nanopores in the shale reservoirs, and the formula of apparent permeability is: D M a FD RT Here, D is Knudsen diffusion coefficient; μ is gas viscosity; M is the molecular mass of gas molecules; R is gas constant; T is temperature; ρ is gas average density; F is slip factor; D is absolute permeability; The equation of slip factor is [26]: 8RT 2 F 1 1 M pr f In the above formula, pavg is average pressure; r is nanopore radius; f is the percentage of collision on the pore wall; the apparent permeability item that Javadpour introduced can revised to the absolute permeability as follows [27] : c D + F 4. Calculation and result analysis 0.5 a g D 4.1 Parameter value In this paper, apparent permeability along with pressure and pore radius is calculated according to the all above different researchers. Gas compression coefficient is calculated by the BWR relationship Matter raised. Gas viscosity is computed with Kobayashi and Burrow s method. The tortuosity value that wored out is 65. The pore radius is calculated from the below equation: m r 2.665 100 m Assume the matrix permeability is 12.5md and porosity is 0.15, the calculation result of pore radius is 6.5nm. Single-phase methane gas is selected in the calculation. Shale reservoir physical parameter and the formula constants are: M=0.018g/mol, R=8.314J/(mol K), f=0.8, A=0.178, B=0.4348, b=-1. 4.2 Ratio analysis of apparent and absolute permeability Ratio of apparent and absolute permeability of different models under different pressures is wored out as Fig.1 shown. Simulation results show: the lower the pressure is, the ratio is higher; when the pressure value is very high, the apparent permeability is approaching to absolute permeability. The ratio increases quicly with pressure decreasing when the pressure is less than 800psi. To a lesser degree, it can increase a few orders of magnitude; to a more degree, it can increase over twenty orders of magnitudes. n 0.45 148
100 K a /K m Florence(2007) Sampath-Keighin(1982) Jones-Owens(1980) Klinenberg Javadpour(2007) CivanA(2010) Fathi(2012) Michel(2011) 10 1 1 10 100 Pressure/MPa Fig.1 Diagram of the ratio of apparent permeability to Darcy permeability with pressure Value added of ratio is minimum in the Klinenberg and Jones-Owens s forecasting model. Their models assume a static slippage factor (based on study of tight gas samples), which might not hold very well for shale where matrix pore size is much smaller than tight gas reservoirs. Michel et al. too Knudsen diffusion into account, and the results fall in between those by Jones and Ertein. Ertein et al. who included a dynamic slippage factor which changes with pressure, predicts a greater change in apparent permeability with pressure as compared to static slippage models. Sahaee-Pour and Bryant used a relationship between apparent permeability and Knudsen number and the results were higher than the above models. Javadpour s model predicts much more permeability change than other models since it considers both Knudsen diffusion and slippage. The prediction using the Florence et al. theoretical model falls above those from other models-we believe the reason is the multiplying factor 43.345 which amplifies the effect. Also notable is Fathi et al. s prediction, which cuts across the other models. Figure 2 shows the variation in the ratio of apparent permeability to matrix Darcy permeability with respect to pore radius. As can be seen, the average pressure is 1000psi and the pore radius is in a more practical range, 1nm to 100nm. The smaller the pore radius is, the ratio of apparent permeability to Darcy permeability is higher. When the pore radius is 1nm, the ratio of every model reached a maximum. The ratio changed at 1.5 when the pore radius increased to 100nm, where apparent permeability was basically the same as Darcy permeability. 50 Javadpour(2007) 40 Florence(2007) Ertein(1986) Sahaee-pour(2012) 30 Fathi(2012) Jones-Owens(1980) 20 linenberg(1941) 10 0 1.0E-09 1.0E-08 1.0E-07 pore radius/m Fig.2 Diagram of the ratio of apparent permeability to Darcy permeability with pore radius 149
Slippage factor International Journal of Science Vol.3 No.12 2016 ISSN: 1813-4890 4.3 Slippage analysis of different models Figure 3 shows how the ratio of apparent permeability to matrix Darcy permeability varies with pressure. The value range of pressure is from 200psi to 8000psi and the pore radius is 6.5nm. As can be seen, models given by Jones-Owens, Sampat-Keighin and Florence predict the static slippage factor remains unchanged with pressure. As pressure increases, the slippage factor predicted by Ertein increase firstly and then decrease. Michel et al. too Knudsen diffusion into account and the results of slippage factor simulation continuously increases with pressure boosting. The slippage factor given by Klinenberg increases linearly with pressure boosting. 40 35 Slippage factor 30 25 20 15 Jones-Owens(1980) Florence et al(2007) Sampath-Keighin(1982) Ertein(1986) Michel(2011) linenberg(1941) 10 5 0 0 10 20 30 40 50 60 Pressure/MPa Fig.3 Diagram of slippage factor with pressure Figure 4 shows the variation in the ratio of apparent permeability to matrix Darcy permeability with respect to pore radius. As can be seen, the larger the pore radius is, the slippage factor is smaller; the smaller the pore radius is, the slippage factor is larger. The slippage factor got by Florence is highest and the factor by Jones-Owens is lowest for the pore radius in a range, 1nm to 1000nm. 100 Florence(2007) Jones-owens(1980) Michel-Klinenberg Ertein(1986) Sampath-Keighin(1982) 1 0 1.0E-09 1.0E-08 1.0E-07 1.0E-06 Pore radius/m Fig.4 Diagram of slippage factor with pore radius (0.1 to 1000nm) under the pressure 1000psi 150
5. Conclusion The ratio of apparent permeability to matrix Darcy permeability varies with pressure and maes important effects. The simulation results show: the higher the pressure is, the ratio is higher; for higher pressure, apparent permeability is closer to matrix Darcy permeability. Under the same pressure, the value by only taing static slippage into account is less than by Knudsen diffusion; the value by only considering Knudsen diffusion is less than by dynamic slippage effect; the value by dynamic slippage effect is less than by dynamic slippage effect and Knudsen diffusion. The pore radius has an important influence on the ratio of apparent permeability to matrix Darcy permeability. The smaller the pore radius is, the ratio is higher. When the pore radius is in a range, a few nanometers to a few tens of nanometers, the apparent permeability is several or dozens of times of Darcy permeability. The theoretical basis of Javadpour s model is reliable, which taes Knudsen diffusion and gas molecules slippage flow into account. The ratio of apparent permeability to Darcy permeability is the second largest from calculation analysis and the simulation result is conform to theory and accurate. The pore radius has an important influence on slippage factor. The slippage factor decreases with the pore radius becoming larger; the smaller the pore radius is, the slippage factor is higher. The Florence s simulation result is highest and the Jones-Owens s is lowest in a range of pore radius, 1nm to 10nm. References [1] Wang F.P. and Reed R.M. Pore Networs and Fluid Flow in Gas Shale[C].SPE124253, 2009. [2] Beso, A. and Karniadais, G.E. 1999. A Model for Flows in Channels, Pipes, and Ducts at Micro and Nano Scales. Microscale Thermophy. Eng. 3(1), 43 77. [3] Lan Ren, Liang Shu, Yongquang Hu and Jinzhou Zhao. Analysis of Gas Flow Behavior in Nano-scale Shale Gas Reservoir [J]. Journal of Southwest Petroleum University (Science & Technology Edition) 2014, 36(5):112-113 [4] Zhiping Li, Zhifeng Li. Dynamic Characteristics of Shale Gas Flow in Nanoscale Pores [J]. Natural Gas industry, 2012, 32(4): 50-53 [5] Heidemann, A., Jeje A.A and Mohtadi F. 2010. An Introduction to the Properties of Fluids and Solids. The University of Calgary Press, Calgary, Canada. [6] Yongming Li, Fengsheng Yao, Jinzhou Zhao and Xueping Zhang. Shale Gas Reservoir Nanometer-pore Microscopic Seepage Dynamic Research [J]. Science Technology and Engineering, 2013, 13(10): 2657-2662 [7] Jones, F.O. and Owens,W.W. A Laboratory Study of Low-Permeability Gas Sands. Journal of Petroleum Technology, SPE7541-PA, 1980:1631-1640. [8] Klinenberg L J. The Permeability of Porous Media to Liquids and Gases [J].API Drilling and Production Practice, 1941(2):200-213. [9] Sampath K,William Keighin C. Factors Affecting Gas Slippage in Tight Sandstones of Cretaceous Age in the Unita Basin [R].SPE 9872,1982:2715-2720. [10] Florence, F. A., Rushing, J. A., Newsham, K. E. and Blasingame, T. A. 2007. Improved Permeability Prediction Relations for Low-Permeability Sands.SPE 107954. [11] Ertein, T.King,G.R. and Schwerer, F.C. Dynamic Gas Slippage: A Unique Dual-Mechanism Approach to the Flow of Gas in Tight Formations. SPE 12045-PA, 1986:43 52. [12] Michel G.G., Sigal, R.F., Civan, F.et al. Parametric Investigation of Shale Gas Production Considering Nano-Scale Pore Size Distribution, Formation Factor, and Non-Darcy Flow Mechanisms. Paper SPE 147438, 2011:2-8. [13] Beso, A. and Karniadais, G.E. 1999. A model for flows in channels, pipes, and ducts at micro and nano scales. Microscale Thermophy. Eng. 3(1), 43 77. [14] Civan. Effective Correlation of Apparent Gas Permeability in Tight Porous Media. Transport in Porous Media, 2010, 82(2), 375-384. 151
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