Y. Li, X. Li, J. Shi, H. Wang, and L. Wu, China University of Petroleum; S. Teng, SINOPEC Corp.

SPE-169939-MS A Nano-Pore Scale Gas Flow Model for Shale Gas Reservoir Y. Li, X. Li, J. Shi, H. Wang, and L. Wu, China University of Petroleum; S. Teng, SINOPEC Corp. Copyright 2014, Society of Petroleum Engineers This paper was prepared for presentation at the SPE Biennial Energy Resources Conference held in Port of Spain, Trinidad, 09 11 June 2014. This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents of the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright. Abstract Many shale/tight gas reservoirs can have pore scale values in the range from one to hundreds of nanometer. And the flow in nano-scale deviate the Darcy s law. Knudsen diffusion and/or gas slippage effects usually have modeled to character the non-darcy flow mechanisms by many authors. In this paper, we investigate the non-darcy flow mechanisms in unconventional gas reservoirs, and classify these various mechanisms based on different pore scale and pressure. Then, based on the change of pore scale and pressure, the models of gas flow that consider the absorption, desorption, slip flow, transition flow, Knudsen diffusion and continuous flow in nano-pore have been proposed to evaluate the flow character. Then, the relationship between the absorbed layers and pressure or Langmuir coefficient has been built and the influences of absorption of gas molecule have been studied on the permeability change. Compared with experimental value, the model could agree with the experimental value very well. And, desorption of the absorbed layers make the pore diameter become larger. When the thickness of the absorbed layers and the pore diameter ratio is larger than 0.1, the effect of adsorbed layer becomes very significant. With this study, the change of permeability and the gas rate on entire long term production performance could be understood better and predicted, and it is very important for the optimization of production performance and adjustment. Introduction Shale gas reservoirs have tiny and complicated pore scale values. It was found that the diameters of nano-pores were around 1~200nm (Cipolla, 2009) and the main pore sizes were around 5~20nm according to the nuclear magnetic resonance, scanning electron microscope, high-pressure mercury and JW-DA technique (Sakhaee-Pour, 2011). Many studies have discovered that it was very important to study the non-darcy flow mechanisms and governing equations for the gas in the nano-pores of the organic matter in the shale gas. Jones & Owens (1980), Sampath & Keighin (1982) and Florence et al. (2007) obtained the expression of slip factor based on the Klinkenberg slip factor, which did not take the influence of gas diffusion into consideration. Javadpour F (2007, 2009) studied the apparent permeability taking Knudsen diffusion and gas slippage effects into consideration, but this model ignored transition flow and the effect of desorption of adsorbed gas on the flow. Li et al. (2012) studied the flow characteristics in the nano-pores using the solid deformation theory based on the capillary model. However, this model only

2 SPE-169939-MS considered the effects of the deformation of the pores based on the pressure and permeability, which ignored the effect of the adsorption and desorption. Sakhaee-Pour et al. (2011) considered the adsorption on permeability and showed that permeability was over estimated at high pressures, without taking into account of transport in adsorbed layer, but he didn t consider the desorption of adsorbed layer with the decrease of pressure. X. Xiong et al. (2012) studied adsorbed phase transport based on the Fick s law of diffusion, but haven t considered the adaptation of transition flow model. Zhang etal. (2013) studied the transport mechanism of gas moving through matrix pores, but he also haven t considered the effect of adsorbed layer and the influence of desorption of adsorbed layer. In this paper, we investigate the non-darcy flow Figure 1 Pore-size classification for mud rocks pores, Rouquerol et al. (1994) Loucks, et al. (2012). mechanisms in unconventional gas reservoirs, and classify these various mechanisms based on different pore scales and pressure. Then the effect of adsorbed layer and the influence of desorption of adsorbed layer have been studied. Finally, the models of gas flow considering the absorption, desorption, Knudsen diffusion and flow in nano-pore have been proposed to evaluate the flow character. Flow characteristic of shale gas reservoir in various pole scales and molecule scales The complicated pores of shale gas reservoirs could be classified based on pore scales by Rouquerol et al. (1994) and Loucks, et al. (2012), shown in Fig. 1. Fig. 2 shows the distribution of the pore sizes in the experiment of mercury intrusion of Barnet shale, which shows the range of the pores is around 2 50nm, especially the main pore size is around 2 20 nm. The relationship between pore sizes and flow characteristics could be built by using Knudsen number. Fig. 3 shows the rule of seepage and diffusion with various Knudsen number. Knudsen number is: (1) Where represents the mean free path of gas molecules, d pore represents the characteristic length of the channels. The following expression can be obtained by substituting the molecular free path into Eq. 1: (2) Where is viscosity, mpa.s; is the gas density, kg/m 3 ; T is the temperature of the gas, K; R is the universal gas constant, 8.314; M is molar mass, kg/mol; d pore is the diameter of channels, m. The flow characteristic differs with Knudsen number, and the flow regime can be classified as continuous flow, slip flow, transition flow and molecular free movement flow based on Knudsen number. Continuous flow regime As is shown in Fig. 3, the typical continuous flow exists when Kn 10 3, whose momentum transfer is controlled by gas viscosity. And collisions among molecules are feeblish, so the equation of motion can be expressed by Darcy law. Then, the mass flow of unit area of laminar flow in circular tubes can be deduced using Hagen-Poiseuille equation. As Kn is gradually increased with the reduce of collisions

SPE-169939-MS 3 among molecules, and molecular mass is intensified, the typical continuous flow model could be describe as follows: (3) Slip flow regime The slip flow becomes predominant, when Kn ranges from 10 3 to 10 1. Karniadakis et al. (2005) presented the boundary conditions of slippage effect according to the effects of collisions between the molecular layers. Gas permeability is: (4) Where is related to the geometric construction of the conduit, whose value is around 5. The slip flow regime can also be represented by the Dust gas model (DGM) (Mason, etal, 1983), which predicts Figure 2 Pore size distribution of a Barnett shale sample obtained from mercury intrusion capillary pressure, Sakhaee-Pour, et al. (2011). all flows by using a linear combination of the gas transmission mechanism based on empirical observation and theoretical explanation (Graham, 1976). The permeability of the gas phase can be represented as: (5) Where 1 represents the enhanced degree of permeability, whose value is 13.58 by using the Dust gas model (Sakhaee-Pour et al. 2011). When, the equation should become Klinkenberg slippage formula. Brown et al. (2004) and Javadpour F et al. (2009) introduced the dimensionless coefficient F to simulate the slippage velocity in nano-pores: (6) Where ranges from 0 to 1 theoretically, which relies on the smooth extent of the nano-pore wall, gas type, temperature and pressure. should be determined by experiment for special mudstone system. Javadpour F et al. (2009) obtained the mass flow of slip flow, that is: (7) Transition flow regime Gas flow is in the stage of transition flow regime when Kn ranges from 10 1 to 10 1. In this regime, the model should be modeled by Monte Carlo simulation (Karniadakis et al. 2005), which simulates the Navier-stokes equations under various shear stresses. It also reflects the slip boundary conditions by using high order gradient of velocities, and the corresponding mass flow of unit area is shown as follows: (8) Therefore, the value which the permeability of gas phase divides the permeability ignored the slip should be:

4 SPE-169939-MS Figure 3 Knudsen number regimes. Roy, et al. (2011). From Eqs. 8 and 9, we can conclude that the permeability is not the liner function of Kn, therefore, Klinkenberg slippage formula cannot be used in this regime. From Eq. 9, we know that when Kn ranges from 0.1 to 0.8, the tangent function is not positive, while when Kn ranges from 0.8 to 10 1, tangent function is not always positive, and Eq. 8 could not be used to model the flow. Zhang, et al. (2013) proposed the gas transport model in this regime by the weighted average of slip flow model and Knudsen diffusion model: (10) (9) Where, and this model can be extending to slip flow regime, continuum flow regime, and free-molecule flow regime. So, the mass flow of unit area is also the weighted average of slip flow model and Knudsen diffusion model: (11) Molecular free flow regime Gas flow is in the stage of molecular free movement when Kn is over 10 1. Knudsen. (1909) presented the gas flow model of this regime: (12) The following equation can be obtained by deforming Eq. 12: (13) Where, J d represents the diffusion mass flow; D Kn represents Knudsen diffusion coefficient, which can be obtained by experiment (Reinecke et al. 2002); n represents concentration gradient. Roy et al. (2003) presented the expression of Knudsen diffusion coefficient in a circular cross section long tube: (14) So, the mass flow of unit area of Molecular free flow regime is:

SPE-169939-MS 5 Figure 4 Gas molecules desorption schematic diagram in nano-pores (15) The influence of the desorption of adsorbed layer on permeability As is shown in Fig. 2, the pore size of shale reservoirs that is less than 10 nm in nano-scale occupies a large part of proportion. When local pressure is greater than desorbed pressure, the rock surface will adsorb gas molecules and form adsorbed layer. if the difference between thickness of adsorbed layer and pore size is not very large, the adsorbed layer will hinder the flow, making the effective pore size of flow reduce as well as the seepage ability. The adsorption and adsorption of shale gas could be characterized by the Langmuir isotherm adsorption law: (16) Where, V m is Langmuir volume, which is the biggest volume of adsorbed gas when solid surface is covered by molecular layer, m 3 /t. b 1/P L is the reciprocal of Langmuir pressure, P L is Langmuir pressure when reaching the maximum adsorption, MPa. The effect of adsorption of adsorbed layer is studied by parallel capillary bundle model, shown in Fig. 4. The nano-pore radius is r. Fig. 4 (1) shows nano-pore desorption diagram under the initial condition. Local pore pressure is larger than the saturation pressure (Langmuir pressure), all the solid surface is covered by molecular gas layer and the thickness of adsorbed layer on the nano-pore is d, the volume of adsorbed gas in the V m, and the effective nano-pore radius is r-d. When local pore pressure is less than the saturation pressure, the adsorbed layer will desorb, the corresponding relative molecular layer thickness is reduced, and the pore diameter that the molecules of gas could move freely will increase, as shown in Fig. 4 (2) (3). Supposing that after desorption and under the pressure P, the equivalent thickness of molecular layer is d, as shown in Fig. 4 (3). According to the theory of parallel capillary bundle, the adsorbed volume of gas molecules under the original pressure (P i P L ) is: V m 2An rld; When local pressure drops to P (P P L ), the adsorption volume of gas molecules is: V 2An rld ; According to the law of Langmuir adsorption, Substituting above two equations into Langmuir equation, the equivalent thickness of molecular layer with the change of pressure is:

6 SPE-169939-MS (17) The effective flow nano-pore radius considering adsorption and desorption should be: (18) As is shown in Eq. 18, the equivalent thickness of the adsorbed layer is mainly affected by the initial thickness, pressure and Langmuir pressure. Nano-pore multi-scale model The flow model that considering slip flow, transition flow, Knudsen diffusion and continuous flow in nano-pore The flow characteristics for different pore scales are different based on Knudsen number such as the molecular diffusion in the smallest scale, transition flow in larger scale, the slip flow in some larger scales, and the continual flow in the largest scale. The multi-scale pore size in shale reservoir as shown in Fig. 2, these flows could be exist at the same time, so only considering molecular diffusion form, or one of them is not very comprehensive. All stages should be considered in the flow, and the mass flow equation considering slip flow, transition flow, molecular diffusion flow and continuous flow is: (19) Submitting Eqs. 3, 7, 11 and 15 into Eq. 19, then simplifies: (20) So, the apparent permeability by considering slip flow, transition flow, the molecular diffusion and continuous flow is: (21) The Darcy s law could characterize the continuous flow, and its permeability is: of apparent permeability and Darcy permeability is:. So the ratio (22) Roy et al (2003). studied the nano flow by cylindrical linear nano-pore (200 nm in diameter) of homogeneous porous medium, the length is 60 m, the porosity is 0.2 ~ 0.3, gas viscosity is 2.2 e-5, Ns/m 2, the temperature is 300 K. The argon gas has been injected under different pressure gradient, and the mass flow rate is as shown in Fig. 5. The molecular weight of argon gas is M 39.948 g/mol. Knudsen number is 9.85 in this nano-pore, so the flow is in a stage of transition flow, and transition flow stage must be

SPE-169939-MS 7 Figure 5 Comparison prediction with measured values Figure 6 The proportion of mass flow of different flow regimes considered in the flow. Coefficient alpha is 0.84, the model presented in this paper, compared with the experiment results found that this model can fit experimental measurement data very well, and the error is 3%. Fig. 6 shows the proportion of mass flow for slip flow, transition flow, molecular diffusion flow and continuous flow. The results show that if the pore size is smaller than 10nm, the slip flow dominated the mass flow; if the pore size is 10~10000nm, the slip flow and the transition flow dominated the mass flow. The molecular diffusion flow may influence the flow, but the proportion is very small. Fig. 7 is the proportion of mass flow for different flow regimes of Javadpour s model, and the slip flow dominated the flow when the pore size is smaller than 10000nm.

8 SPE-169939-MS Figure 7 The proportion of mass flow of different flow regimes of Javadpour model The flow model that considering absorption, desorption, slip flow, transition flow, Knudsen diffusion and continuous flow in Nano-pore Eq. 18 is the effective flow nano-pore, so the effective Knudsen number is: (23) The flow model that considers the absorption, desorption, slip flow, transition flow, Knudsen diffusion and continuous flow in nano-pores is (24) Assume that the nano-pore size is 5 nm, methane molecular layer thickness is 0.7 nm, and Fig. 8 shows the permeability curves under different Langmuir coefficient b and different pressure. The apparent permeability decrease due to the adsorption, but the apparent permeability will increase with desorption of adsorbed layer. And the influence of desorption on the permeability decrease with large Langmuir coefficient b, which means the less the adsorption, the less sensitive the permeability is. Fig. 9 shows when P 30MPa, the change of permeability under different initial thickness of adsorbed layer to nano-pore size ratio. The effect of desorption on the permeability will increase with the increase of the initial thickness of adsorbed layer to nano-pore size ratio. What s more, when the ratio is greater than 0.1, the adsorbed layer effect on the permeability is very significant. Conclusion 1. This paper investigates the non-darcy flow mechanisms in unconventional gas reservoirs, and classifies these various mechanisms based on different pore scale and pressure. Then, based on the change of pore scales and pressure, the gas flow models that considering the absorption, desorption, slip flow, transition flow, Knudsen diffusion and continuous flow in Nano-pore have been proposed to evaluate the flow character.

SPE-169939-MS 9 Figure 8 Permeability curve under different Langmuir coefficient b 2. Desorption of adsorbed layer will increase the flow nano-pore size, leading to the increase of permeability. And based on the Langmuir isotherm adsorption law, the equivalent molecular layer thickness have been proposed for studying the influence of desorption of adsorbed layer, which is function about the initial thickness of absorbed layer, pressure and Langmuir coefficient b. The effect of adsorption and desorption on the permeability will increase with the increase of the initial thickness of adsorbed layer to nano-pore size ratio. What s more, when the ratio is greater than 0.1, the effect of adsorbed layer on the permeability is very significant. When the ratio is less than 0.1, the influence on permeability is less significant. Acknowledgment This research was supported by National Natural Science Foundation Project (U1262113) and Science Foundation of China University of Petroleum, Beijing (YJRC-2013 37). Nomenclature Mean free path of gas molecules, m; d pore Mean diameter of the pore, m; r d pore /2 Mean radius of the pore, m; Kn Knudsen number, dimensionless; M Molar mass (kg/mol) or (lb/lbmol) R Gas constant (J/mol/K) or (psi. ft 3 /lbmol/r) avg Average gas density, kg/m 3 ; k g Apparent gas permeability, m 2 ; k l Liquid permeability, m 2 ; J a Continuous mas flow rate of gas, kg/s/m 2 J Slip Slip mass flow rate of gas, kg/s/m 2 J T Transition mass flow rate of gas, kg/s/m 2 J d Molecular free flow rate of gas, kg/s/m 2 (k g ) Free Apparent gas permeability in slip flow regime, m 2 ; (k g ) Slip Apparent gas permeability in slip flow regime, m 2 ; (k g ) Trans Apparent gas permeability in transition flow regime, m 2 ; k D Apparent gas permeability in continuous flow regime, m 2 ; = Tangential momentum accommodation coefficient, fraction; b= Klinkenberg slippage coefficient, dimensionless; d eff The effective flow diameter of the pore, m; r eff The effective flow radius of the pore, m; Figure 9 The permeability deviation under different initial thickness of adsorbed layer

10 SPE-169939-MS d The initial thickness of adsorbed layerr, m; d= The equivalent layer thickness with pressure, m; T Temperature, K; P Pressure, Pa; Gas viscosity, Pa s; f Weight coefficient, dimensionless; D Kn Knudsen diffusivity coefficient, m 2 /s; V m Langmuir volume (m 3 /t); b 1/P L is the reciprocal of Langmuir pressure, 1/MPa; Pressure for gas adsorption amount reaching maximum adsorption, MPa. P L Reference Cipolla, C.L., Lolon, E.P. and Mayerhofer, M. J. 2009. Reservoir Modeling and Production Evaluation in Shale-Gas Reservoirs. Paper IPTC 13185 presented at the International Petroleum Technology Conference held in Doha, Qatar, 7 9 December. doi: 10.2523/13185-MS. Xiong, X., Devegowda, D., Villazon, M., German, G., Sigal, R. F., & Civan, F. (2012, January). A fully-coupled free and adsorptive phase transport model for shale gas reservoirs including non-darcy flow effects. In SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers. Zhang L, Li X, Wang T, et al. Diffusion and Flow Mechanisms of Shale Gas through Matrix Pores and Gas Production Forecasting[C]//SPE Unconventional Resources Conference Canada. Society of Petroleum Engineers, 2013. A. Sakhaee-Pour, and Steven L. Bryant. Gas Permeability of Shale his paper was prepared for presentation at the SPE Annual Technical Conference and Exhibition held in Denver, Colorado, USA, 30 October 2 November 2011. Jones, F.O. and Owens, W.W. 1980. A Laboratory Study of Low-Permeability Gas Sands. Journal of Petroleum Technology, 1631 1640. Paper SPE 7541-PA doi: 10.2118/7551-PA. Sampath, C.W., Keighin, K. 1982. Factors affecting gas slippage in tight sandstones. Paper SPE 9872. J. Petrol. Technol, 34, 2715 2720. doi: 10.2118/9872-PA Florence, F. A., Rushing, J. A., Newsham, K. E. and Blasingame, T. A. 2007. Improved Permeability Prediction Relations for Low-Permeability Sands. Paper SPE 107954 presented at SPE Rocky Mountain Oil & Gas Technology Symposium held at Denver, Colorado, 16 18 Apr. doi: 10.2118/107954-MS. Klinkenberg, L.J. 1941. The Permeability of Porous Media to Liquid and Gases. API Drilling and Production Practice p. 200 213. Javadpour, F., Fisher, D., and Unsworth, M. 2007. Nanoscale Gas Flow in Shale Gas Sediments. Journal of Canadian Petroleum Technology, 46 (10), 55 61. doi: 10.2118/10.2118/07 10-06. Javadpour, F. 2009. Nanopores and Apparent Permeability of Gas Flow in Mudrocks (Shales and Siltstone). Journal of Canadian Petroleum Technology, 48(8), 16 21. doi: 10.2118/09 08-16-DA. Li Z, Li Z. Dynamic characteristics of shale gas flow in nanoscale pores[j]. The natural gas industry, 2012, 32(4): 50 53. Najeeb Alharthy, Mohammed Al Kobaisi and, Mehmet A. Torcuk, et al. Physics and Modeling of Gas Flow in Shale Reservoirs. This paper was prepared for presentation at the Abu Dhabi International Petroleum Exhibition & Conference held in Abu Dhabi, UAE, 11 14 November 2012. Loucks, R., Reed, R., Ruppel, S., and Hammes, U. 2012. Spectrum of pore types and networks in mudrocks and a descriptive classification for matrix-related mudrocks pores. The American Association of Petroleum Geologists Bulletin, v. 96, No. 6, pp. 1071 1098. Choquette, P.W., and L.C. Pray, 1970. Geologic nomenclature and classification of porosity in sedimentary carbonates. American Association of Petroelum Geologists Bulletin, v. 54, pp. 207 244.

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