Characterization of Gas Flow Ability and Contribution of Diffusion to Total Mass Flux in the Shale

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1 Reseach Jounal of Applied Sciences, Engineeing and Technology 6(9): , 13 ISSN: ; e-issn: Maxwell Scientific Oganization, 13 Submitted: Januay 19, 13 Accepted: Febuay, 13 Published: July 15, 13 Chaacteization of Gas Flow Ability and Contibution of iffusion to Total Mass Flux in the Shale 1 Rui Wang, 1, Ningsheng Zhang, Xiaojuan Liu, Xinmin Wu and Jian Yan 1 College of Petoleum Engineeing, China Univesity of Petoleum-Beijing, Beijing 149, China College of Petoleum Engineeing, Xi an Univesity of Petoleum, Xi an 765, China Abstact: The aim of this study is to seach a paamete which chaacteize the flow ability and analyze the contibution of diffusion to total mass flux of gas flow in poe of shale whose size is as low as nanoscale. The diffusion coefficient of the flow egion which was detemined by Kundsen numbe was taken as the diffusion coefficient of system, then it was substituted into the equation which descibes gas diffusive and flow in nanopoous media, the appaent pemeability and mass flux wee calculated and the impacts of the poe adius and gas type wee analyzed finally. The esult showed that the diffusion of gas in shale was mainly in the tansition diffusion o Fick diffusion egion; The atio of the appaent pemeability of consideing the diffusion and slippage effect to acy pemeability and the atio of diffusion mass flux to total mass flux inceased with the deceasing of the poe adius; The diffusion ability of the methane was stonge than ethane s. The pesent study implied that the calculated esults of the diffusion coefficient which consides thee kind of diffusion was less than that one consideing udsen diffusion only; Consideing diffusion and slippage effect, the appaent pemeability of nanoscale poe of shale was 1-6 μm level, not 1-9 μm level in geneal tempeatue and pessue of shale esevoi. Keywods: Appaent pemeability, diffusion coefficient, diffusion flux, diffusion type, shale gas INTROUCTION The minimum diamete of poes in shale esevoi is nanoscale, which is close to the diamete of the methane molecule (Javadpou et al., 7; Javadpou, 9). It was found that the gas flow in shale esevois is in slippage and tansition flow egion by calculating Kundsen numbe, so consideing the slippage and diffusion effects wee needed (Sondegeld et al., 1b). Roy et al. (3) established a mathematical model which descibes gas diffuse in nanopoes, it was based on gas mass tansfe expeiment that A, N and O flow in alumina filtation membane whose apetue is nm. Javadpou (9) and Roy et al. (3) model to descibe gas diffuse and flow in shale (Javadpou et al., 7). Sigal and Qin (8) analyzed the impotance of self-diffusion in pocess of gas mass tansfe in shale by intoducing an effective tansfe coefficient. Javadpou (9) established an equation which consides Kundsen diffusion and slipping and intoduced a concept of appaent pemeability (Javadpou, 9). Sigal and Qin (8) built a finite diffeence model about that. These models didn t conside the type of diffusion. Javadpou (9); Javadpou et al. (7) and Shabo et al. (9) used udsen diffusion model; Javadpou (9) and Sigal and Qin (8) used Fick diffusion model only. We detemined the flow egion by Kundsen numbe and took the diffusion coefficient of the flow egion as the diffusion coefficient of system fistly, then calculated the appaent pemeability and mass flux, afte that calculated the atio of the appaent pemeability of consideing the diffusion and slippage effect to acy pemeability and the atio of the diffusion mass flux to total mass flux, finally analyzed the impact of the poe adius and gas type. METHOOLOGY Motion equations of the mass tansfe pocess of gas in nanopoes: Motion equations of the mass tansfe pocess of gas in nanopoes (Javadpou, 9): J = J + J a J = The total mass flux in kg/s/m = Mass flux of gas diffusion in kg/s/m J a = Mass flux of gas flow in kg/s / m (1) Roy et al. (3) ignoed viscous effect and intoduced a constant diffusion coefficient to establish a mathematical model which descibe gas diffuse in Coesponding Autho: Rui Wang, College of Petoleum Engineeing, China Univesity of Petoleum-Beijing, Beijing 149, China 1663

2 Res. J. Appl. Sci. Eng. Technol., 6(9): , 13 nanopoes by expeiments that A, N and O mass tansfeed in what-man alumina filtation membane (Anodisc13, poe diamete is nm, length is 6μm) (Javadpou et al., 7; Roy et al., 3): qρ p = φ AM RTL () q = Gas volume flow in m 3 /s ρρ = Gas density in kg/m 3 = Poosity in % A = The exposed aea of alumina filtation membane in m M = Mola mass of gas in kg/mol = iffusion coefficient (Javadpou, 9) used udsen diffusion coefficient, we assumed to be the geneal case)in m /s R = The univesal gas constant (8.314) in J/mol/K T = Tempeatue in K p = The pessue dop acoss the filtation membane in Pa L = The thickness of filtation membane in m So, mass flux of gas diffusion is: J M p = RT L (3) The mass flux of gas flow can be deived fom Hagen-Poiseuille s equation: Table 1: Classification of the flow egime by Numbe numbe Flow egime >1 udsen diffusion.1<<1 tansition diffusion.1<<.1 Fick diffusion.1<<.1 slippage flow <.1 acy flow T P = Tempeatue in K = Pessue in Pa In fact, the analysis of udsen flow was initially fo the pupose of eseach to flow of gas at low pessue, Matin udsen explained theoetically and demonstated expeimentally the flow of gas at low pessue by the tem of molecula flow fo the fist time in 199 (Steckelmache, 1999). Thee was a classification standad of gas flow egion accoding to the udsen numbe (Chen and Pfende, 1983). Civan et al. (11) used this classification standad and showed the coesponding elationship among the shale poe type, flow type, dominant paticle motion and flow egime. The standad that we used is descibed in Table 1: Getting diffusion coefficient of fluid diffusive in poous solid medium equied detemining the flow egion by numbe fistly, then took the diffusion coefficient of the flow egion as the diffusion coefficient of system. Additionally, thee was a method of taking diffusion coefficient and pemeability as a function of numbe (Civan et al., 11). The equation of udsen diffusion coefficient expessed as: p Ja = ρ 8µ L (4) The calculation of diffusion coefficient: The diffusion of gas o liquid diffusive in poous solid medium can be divided into the udsen diffusion (>1), Fick diffusion (<.1), tansition diffusion (.1<<1) and suface diffusion accoding to the udsen numbe in the chemical field. udsen numbe is defined as: (6) knudsen = udsen diffusion coefficient in m /s = Radius of poe in m u = Aveage velocity of gas molecules in m R = The univesal gas constant (8.314) in J/mol/K T = Tempeatue in K Fick diffusion coefficient is calculated by Stokesλ kt b =, λ = (5) Einstein equation. Stokes-Einstein equation was used to Λ πδ p calculate the diffusion coefficient of molecule in liquids, descibe the spheical paticles o molecules diffusive in a dilute solution especially. Fluid Fick λ = Gas mean-fee-path (calculated using the Eq. diffusive in solid poe medium is mainly a collision (5) usually, o Loeb method (Civan et al., 11) among fluid molecules athe than collision between in m; fluid molecules and wall suface of poe, as same as that gas diffusive in liquids. So we should use the Λ = iamete of poe in m viscosity of fluid which satuated in poous medium k b = The Boltzmann constant (1.385 ) in J/K, δ athe than the diffusive gas (Sigal and Qin, 8) is the collision diamete of the gas molecule in m while we calculated the Fick diffusion coefficient: 1664 knusden 8RT = u = ( ) 3 3 π M.5

3 Res. J. Appl. Sci. Eng. Technol., 6(9): , 13 fick kt b = 6πµ B A (7) The calculation model of diffusion coefficient what was selected by the value of : fick = Fick diffusion coefficient in m /s A = Radius of gas molecula in m μμ B = Viscosity of fluid in the poe of shale in Pa/s k b = Boltzmann constant 1.38 in J/K T = Absolute tempeatue in K fick, kn <.1 = tansition,.1 < kn < 1 knusden, kn > 1 (1) So, the mass flux of gas diffusion and the mass flux of gas flow of consideing slippage effect wee: In ou eseach, we assumed the fluid satuated in shale is N. The tansition diffusion coefficient calculated by udsen diffusion coefficient and Fick diffusion coefficient, is Bosanquit equation: (8) It was found that by the fomula (8), the tansition diffusion coefficient is smalle than not only udsen diffusion coefficient but also Fick diffusion coefficient. The calculation of appaent pemeability of consideing the diffusion and slippage effect: The flow velocity was equal to mass flux divided by density of fluid and then the pemeability can be obtained by contasting with acy equation. The acy pemeability which didn t conside the slippage effect was obtained by Hagen-Poiseuille equation: (9) (1) Thee is slippage effect in gas flow in nanopoes that can expess by intoducing a theoetical dimensionless coefficient to coect the Hagen- Poiseuille equation (Javadpou, 9): (11) k s = Coect coefficient of slippage effect in dimension p avg = Aveage pessue in Mpa α = Tangential momentum accommodation coefficient depending on pessue, tempeatue, gas type and smoothness of wall suface, the value of it is ~1 in dimension, we took.5 = ( + ) tansition fick knudsen Ja = ρavg 8 k dacy = 8 p µ L p Ja = ks ρavg 8µ L 8π RT.5 µ p = 1 + ( ) ( 1) ρavg M pavg α 8 µ L = Radius of poe in m Mµ 8π RT.5 µ ρavg 1 p J = J + Ja = ( ) ( 1) RT M pavg α 8 µ L (13) Using the concept which was poposed by Javadpou (9), the appaent pemeability is: k app M µ 8π.5 1 ( RT µ = + + ) ( 1) RT ρavg M pavg α 8 (14) is appaent pemeability of consideing diffusion in m. The appaent pemeability can chaacteize the flow ability of gas flow in poe of shale which as low as nanoscale. The atio of appaent pemeability to acy pemeability is: kapp Mµ 8 8π RT.5 µ = + 1 ( ) ( 1) + kdacy RT ρavg M pavg α RESULTS AN ISCUSSION (15) The paametes which involved in the calculations and thei ange of value: The system of poductive shale gas included fou poous Medias, inoganic matte, oganic matte, natual factues and atificial factues. Accoding the size of poe, thee wee two types of poe in shale matix, micopoe and nanopoe. Javadpou (9) pesented the images of nanopoes obtained by AFM fistly. Sondegeld et al. (1a) discoveed the size of poes in shale wee 3-8 nm in diamete by SEM image. Zou et al. (11) discoveed nano-poes in the eseach of unconventional tight sandstone and shale gas esevoi in China based on the SEM and the nano CT econfiguation technique and thei diamete was 5~3 nanometes and mainly was 8~ nanometes. Fo the size of the poe thoat, Sondegeld et al. (1a) obtained incemental cuves of Banett gas shale samples and showed that maximum injection pessue was 6, psi which equated to a poe thoat size of 1.8 nm. About natual factues, Gale and Holde (8) discoveed natual factues wee commonly naow (<.5 mm) in the Mississippian Banett shale. The size of natual factues can be consideed to micosize scale. The size of atificial factues was consideed to millisize scale. 1665

4 Res. J. Appl. Sci. Eng. Technol., 6(9): , 13 Table : The basic paametes of the gases Gas Molecula fomula Relative molecula mass Molecula diamete (nm) methane CH ethane C H Nitogen N Table 3: Calculation methods, units and ange of values of the paametes Paamete Calculation methods and ange of values M (kg/mol) Mola mass of gas Tempeatue,3~45 B (m) Radius of poe, ~ μ A,μ B (Pa/s) Viscosity of N and CH 4 (C H 6 ),calculated by lee-gonzalez-eakin method (Lee et al., 1966) ρ avg (kg/m 3 ) ensity of CH 4 (C H 6 ),Calculated by anchuk-puvis-robinson method p avg (MPa) Aveage pessue,.1~ p (Pa) Pessue diffeence,1 So, in ode to analyze the impact of poe size to diffusion and flow, the poe diamete that we assumed is.1~1 6 nm, including factue. The Banett shale has gas pessue of seveal thousand psi, tempeatue aound 35 K (Sigal and Qin, 8).We assumed poe pessue in shale gas esevoi is.1~ MPa, tempeatue is 3~4 K and nomal pessue is 5 MPa, tempeatue is 35 K. Thee ae a little expeimental data of composition of gas in shale. Zhan Jinchuan found that methane content of shale gas was 4.43% and ethane content was.4% by chomatogaphic analysis of gas samples which came fom well Yuye 1, Penshui county of Chongqing in China (Zhang et al., 1). We assumed methane o ethane content is %. The paametes involved in calculation wee as Table and 3: Calculations and analysis s of and diffusion coefficient: That the numbes of methane flowed in shale with diffeent diamete of poe (.1~1 6 nm) and pessue (.1~ MPa) at 35 K wee calculated by fomula (5). Figue 1 shows deceased with the inceasing of pessue and inceased with deceasing of poe diamete. The value of anged fom 6 to.3 while the pessue anged fom.1 to MPa and the diamete of poe was 1 nm. So the flow was in Fick diffusion o tansition diffusion egion. The values of diffusion coefficient of methane diffuse in shale with diffeent adius (.1~1 6 nm) at 3 K and 5 MPa wee calculated by fomula (6), (7), (8), (1). Figue shows Fick diffusion coefficient was independent of the change of poe adius, udsen diffusion coefficient inceased with the inceasing of poe adius. The Fick diffusion coefficient was equal to the udsen diffusion coefficient if poe adius is. nm. The tansition diffusion coefficient was smalle than udsen diffusion coefficient if poe adius <1 nm and the Fick diffusion coefficient was less than udsen diffusion coefficient too if poe adius >1 nm. So the diffusion coefficient of system consideing nm 1 nm nm 1 μm 1 μm μm 1 mm p/mpa Fig. 1: The tends of numbe with the diffeent poe diamete and pessue (35 K) s -1 /m fick 1 - knudsen B /m Fig. : The methane diffusion coefficient in diffeent poe adius (3K, 5MPa) udsen diffusion was bigge than the one which consideed thee kinds diffusion. The effect of poe adius to the atios of the appaent pemeability of consideing the diffusion and slippage effect to acy pemeability and the diffusion mass flux to total mass flux: The pemeability of methane flowed in shale with diffeent

5 k/ μm 1 1 kapp B /m Res. J. Appl. Sci. Eng. Technol., 6(9): , 13 k dacy Fig. 3: The methane appaent pemeability with the diffeent poe diamete (35 K, 5MPa) /k day /k dacy (J F) /k dacy /J(J F) /J /m Fig. 4: The atio of the methane appaent pemeability to acy pemeability and diffusion flux to total mass flux with the diffeent poe adius (35 K, 5MPa) /k dacy B /m k dacy (ethane) k dacy (methane) /J(ethane) /J(methane) Fig. 5: The atios of the methane and ethane appaent pemeability to the acy pemeability and diffusion flux to total mass flux with the diffeent poe adius (35 K, 5 MPa) ( /J)/% ( /J)/% 1667 adius (.1~1 6 nm) at 3 K and 5 MPa wee calculated by fomula (1) (14). Figue 3 shows appaent pemeability was bigge than acy pemeability if poe adius < nm and appaent pemeability was equal to acy pemeability if poe adius > nm. The acy pemeability was 1-9 μm if poe adius was.1 nm, howeve appaent pemeability was 1-7 μm unde same condition. This shows that acy pemeability of shale was eally as low as expessed in geneal idea what poe minimum size of shale was nanoscale and pemeability of it was nanodacy, but pemeability was 1-7 μm level baely if consideing diffusion effect. So, excepting slippage, the diffusion must be consideed to eseach the flow mechanism of shale gas and excepting pemeability, the diffusion coefficient must be consideed to chaacteize the flow ability of shale exactly. Besides the measuing of diffusion coefficient was easie than measuing of pemeability, since the pemeability of shale is too low. The atios of the appaent pemeability of consideing the diffusion and slippage effect to acy pemeability and the diffusion mass flux to total mass flux with diffeent adius (.1~1 6 nm) at 3 K and 5 MPa wee calculated by fomula (13) (15). The esults wee as Fig. 4, the cue maked J F wee calculated by Javadpou (9) F s model. Figue 4 shows the atio of appaent pemeability to acy pemeability calculated by fomula (15) was less than the one calculated by Javadpou F s model. The easons maybe wee that Javadpou F used the squae of the density and only consideed one kind of diffusion. The geneal laws wee the atio of appaent pemeability to acy pemeability deceased with the inceasing of poe adius, the deceasing was outstanding if poe adius <1 nm, fom at.1 nm to 1. at 1 nm, then closed to 1.. The atio of diffusion flux to total mass flux deceased with the inceasing of poe adius too, the deceasing was outstanding if poe adius <1 nm, fom 98.48% at.1 nm to 1.6% at 1 nm. And the atio diffusion flux to total mass flux was 5% if poe adius was 1~ nm, means that the contibution of diffusion and flow to total mass flux is same as half. The effect of gas type to the atios of the appaent pemeability of consideing the diffusion and slippage effect to acy pemeability and the diffusion mass flux to total mass flux: Javadpou (9) analyzed the effect of gas mola mass to diffusion. But the gas chaacteistic paametes of effecting diffusion wee not only molecula diamete, density, viscosity, expect fo mola mass, but also thee wee inteactions among them. So in ode to analyze compehensively, the type of gas instead of mola mass should be analyzed. The methane and ethane wee analyzed as follow. The atio of the methane and ethane appaent pemeability to acy pemeability and diffusion flux

6 to total mass flux at 3 K and 5 MPa with diffeent poe adius wee calculated. Figue 5 shows the atio of the methane appaent pemeability to acy pemeability and diffusion flux to total mass flux is geate than ethane s unde same condition. It means the diffusion ability of methane is stonge than ethane s. So the effect of using methane was moe obvious than using ethane in expeiment. But the diffusion and adsoption of methane is too difficult to distinguish, so whethe thee ae non-hydocabon gas should be seek. ANALYSIS AN ISCUSSION The mathematical model of gas diffusion in nanopoes was established by expeiments that A, N and O mass tansfeed in what-man alumina filtation membane and the diffusion was in udsen diffusion egion. Whethe this model can be used unde the condition consideing thee kind diffusions is unknown. Besides, the diffeence of fluid satuated in shale when calculated Fick diffusion coefficient and the coections of poosity and totuous of diffusion path to diffusion coefficient wee not consideed, besides the influents of pessue and tempeatue to total pocess ween t analyzed too. CONCLUSION Getting diffusion coefficient of fluid diffusion in the shale equied detemining the flow egion by numbe fistly, then took the diffusion coefficient of the flow egion as the diffusion coefficient of system. Gas flow in shale was in Fick diffusion o tansition diffusion egion. The diffusion coefficient of shale which consideed thee kinds of diffusion was less than the one consideed udsen diffusion only. The appaent pemeability of gas flow in shale was 1-7 μm instead of 1-9 μm level if consideing diffusion and slipping effect unde the condition of geneal pessue tempeatue of shale esevoi. The atio of appaent pemeability to acy pemeability and the diffusion flux to total mass flux inceased with the deceasing of poe adius, the inceasing was outstanding if poe adius <1 nm. In ode to analyze compehensively, the type of gas instead of mola mass should be analyzed. The diffusion ability of methane is stonge than ethane s. REFERENCES Chen, X. and E. Pfende, Effect of the udsen numbe on heat tansfe to a paticle immesed into a themal plasma. Plasma Chem. Plasma Pocess., 3(1): Res. J. Appl. Sci. Eng. Technol., 6(9): , Civan, F., C.S. Rai and C.H. Sondegeld, 11. Shalegas pemeability and diffusivity infeed by impoved fomulation of elevant etention and tanspot mechanisms. Tans. Poous Med., 86(3): Gale, J.F.W. and J. Holde, 8. Natual factues in the Banett Shale: Constaints on spatial oganization and tensile stength with implications fo hydaulic factue teatment in shale-gas esevois. Poceeding of the 4nd U.S. Rock Mechanics Symposium (USRMS), San Fancisco, CA: Ameican Rock Mechanics Association. Javadpou, F., 9. Nanopoes and appaent pemeability of gas flow in mudocks (Shales and Siltstone). J. Canadian Petol. Technol., 48(8): Javadpou, F.,. Fishe and M. Unswoth, 7. Nanoscale gas flow in shale gas sediments. J. Canadian Petol. Technol., 46(1): Lee, A., M. González and B. Eakin, The viscosity of natual gases. J. Petol. Technol., 18(8): Roy, S., R. Raju, H.F. Chuang, B.A. Cuden and M. Meyyappan, 3.Modeling gas flow though micochannels and nanopoes. J. Appl. Phys., 93(8): Shabo, V., F. Javadpou and C. Toes-Vedín, 9. A genealized Finite-iffeence iffusive- Advective (FA) model fo gas flow in micoand nano-poous media. Wold J. Eng., 6(3): Sigal, R.F. and B.F. Qin, 8. Examination of the impotance of self diffusion in the tanspotation of gas in shale gas esevois. Peto Phys., 49(3): Sondegeld, C.H., R.J. Ambose, C.S. Rai and J. Moncieff, 1a. Mico-stuctual studies of gas shales. Poceeding of the SPE Unconventional Gas Confeence, Pittsbugh, Pennsylvania, Society of Petoleum Enginees, USA. Sondegeld, C.H., K.E. Newsham, J.T. Comisky, M.C. Rice and C.S. Rai, 1b. Petophysical consideations in evaluating and poducing shale gasesouces, Poceeding of the SPE Unconventional Gas Confeence. Pittsbugh, Pennsylvania, Society of Petoleum Enginees, USA. Steckelmache, W., udsen flow 75 yeas on: The cuent state of the at fo flow of aefied gases in tubes and systems. Repots Pog. Phys., 49(1): 183. Zhang, J., Y. Li, H. Nie, P. Long, Y. Tang, X. Tang and W. Jiang, 1. Geologic setting and dilling effect of the shale coed well Yuye-1, Penshui County of Chongqing. Nat. Gas Ind., 3(1): Zou, C., R. Zhu, B. Bai, Z. Yang, S. Wu, L. Su,. ong and X. LI, 11. Fist discovey of nano-poe thoat in oil and gas esevoi in China and its scientific value. Acta Petol. Sinica, 7(6):