ISTP-16, 005, PRAGUE 16 TH INTERNATIONAL SYMPOSIUM ON TRANSPORT PHENOMENA PERMEABILITY-POROSITY RELATIONSHIP ASSESSMENT BY -D NUMERICAL SIMULATIONS J. Pnela*, S. Kruz*, A. F. Mguel* +, A. H. Res*, M. Aydn** * Évora Geophyscs Centre, Rua Romão Ramalho, 59, 7000-671 Évora, Portugal ** Dept Mech. Engneerng, Istanbul Techncal Unversty, 449 Gumussuyu, Istanbul, Turkey + afm@uevora.pt phone: +51667457 fax: +516674594 Keywords: porous structure, creepng flow, permeablty, porosty Abstract An accurate evaluaton of the permeablty of a porous structure s crtcal for predctng flud flow rates. In ths study, we report the results of two-dmensonal smulatons of creepng flow through porous structures wth porostes between 0. and 0.99. Our numercal results demonstrate that n ths range, permeablty s an exponental functon of the porosty. Ths result s found to be n good agreement wth expermental data. In addton, we compare our smulaton results wth some permeablty-porosty relatonshp avalable n lterature. We found that Carman Kozeny s correlaton agrees very well wth our results for porostes up to 0.87 and Koponen et al. s relatonshp holds well for porostes hgher than 0.95. Based on the results obtaned n ths study, a new permeablty-porosty relatonshp s presented, for the model system used. 1 Introducton Flud transport n porous structures s a topc of nterest n several felds of scence and technology, rangng from geoscences, petroleum engneerng and cleansng to lvng structures [1-4]. In a broad range of applcatons, the most mportant transport property of porous structures s permeablty [1,]. For a statonary creepng flow, permeablty coeffcent (K) can be obtaned from the Darcy law [1-4] K p u = µ y (1) where u s the superfcal velocty, µ s the dynamc vscosty and p/ y s the pressure drop across the porous structure. The permeablty coeffcent quantfes the ablty of the structure to be crossed by flud and can be obtaned based both on experments and on a number of analytcal approaches avalable n lterature [1-4]. The exstng studes show that the permeablty depends manly on the porosty and dameter of the pores. One of the earlest models avalable s the one that consders the porous structure as a bundle of tubes wth constant dameter placed n parallel [5]. Most of the models suggest a power-law model dependency between the permeablty and porosty [6]. However, among the avalable models, the Carman Kozeny s approach [7-9] s the most wdely known. Accordng to the hydraulc radus theory of Carman Kozeny, the permeablty coeffcent of a bed of partcles s related to the porous meda porosty (φ) and a characterstc dmenson (d) of the beds through: φ K = 6θ ( 1 φ ) d () The so-called Kozeny parameter θ vares wth the changes n pore structure sgnfcantly [1,7]. For a porous structure composed by granular bed, Carman [8] suggested that θ vares between 4 and 6. Alternatvely, for a structure composed 1
J. Pnela, S. Kruz, A. F. Mguel, A. H. Res, M. Aydn by cylnders parallel to flud flow t s suggested that [10]: 0.89φ θ = ( 1 φ )[ 4( 1 φ ) ( 1 φ ) whle for transverse flow [10] ln( 1 φ ) () Numercal method and model system The geometrc model s shown n Fg.1. It conssts of a two-dmensonal porous structure formed by regularly placed overlappng sold crcles wth constant radus. These crcles wth radus varyng between 1.1 and 9.4 mm are used for decoratng the underlyng lattces for predctons of permeablty as a functon of porosty. 0.89φ [1 + ( 1 φ ) ] θ = φ + ( 1 φ ) + ( 1 φ ) (4) Despte wdely used t s known that very often the Carman Kozeny type correlatons do not ft the expermental data [11-14]. Some authors have tred to present new expressons for the parameter θ, whle others have tred to establsh new correlatons, usually of emprcal type or based on numercal studes [1]. Koponen et al. [14] reported results of lattce-boltzmann smulatons of creepng flow through random fber webs and suggested that for porostes close to one, the permeablty coeffcent can be obtaned from 1.9 K = exp[10.1( 1 φ )] 1 d (5) It s clear from the above dscusson that, despte many mportant studes avalable n lterature there s stll no agreement about the relatonshp that s more approprate to compute the permeablty of a porous meda. The objectve of our work s to gan some nsght about the valdty of the most wdely used equatons. For ths purpose, a numercal smulaton of creepng flow across a smple two-dmensonal porous structure s carred out. Based on that, we study the dependence of the permeablty on the porosty of the porous structure and we compare these results wth those prevous studes avalable n lterature. Fg. 1. Geometry of porous structure The governng equatons for two-dmensonal flow feld are the contnuty equaton: (6) u = 0 and the Naver Stokes equaton: u j u 1 p = ρ u + ν j (7) These equatons were solved numercally usng a commercal fnte-volume-based program, FLUENT [15]. Calculatons were performed for Reynolds lower than 1 (Reynolds number Re s calculated accordng to Re = ρ udcrcles µ where ρ s densty of flud, u s averaged velocty at the nlet, d crcles s the dameter of the sold crcles, and µ s dynamc vscosty of flud). The nonslp boundary condtons were set along the sold crcles. The developed velocty profle was
PERMEABILITY-POROSITY RELATIONSHIP ASSESSMENT BY -D NUMERICAL SIMULATIONS prescrbed at the nlet whle outflow boundary condtons were set at the ext. Fg. Contours of stream functon (Re=0.1; φ=0.9) Fg. Contours of stream functon (Re=0.1; φ=0.4) The cell sze of the grd where hgh gradents of veloctes are expected has been vared to ensure a grd ndependent soluton. Soluton for low Reynolds number converges rapdly and monotoncally. For example, for Re=0.001 t takes around 50 teratons. Typcal stream functon contours for porostes 0.4 and 0.9, are shown n Fgs. and. These contours are consstent wth other results avalable n lterature [16]. Results and dscusson The effect of the superfcal velocty on the pressure drop across the porous structure was studed for porous structures wth porostes between 0. and 0.99. Fgs. 4 and 5 llustrate ths dependence for two of dfferent porostes tested (.e, porostes 0.4 and 0.8). The plots show a lnear varaton between the velocty (u) and the pressure drop. Ths varaton corresponds to Darcy's law, and the permeablty s obtaned from the best straghtlne ft of ths data. The results are presented n Fg. 6, n terms of dmensonless permeablty (K*) defned as the rato between the permeablty coeffcent (K) and the square dameter of sold crcles (d crcles ). The varaton of K* versus porosty dsplays three dfferent regmes: an ntal slow ncrease of K* (0. φ 0.85), followed a transton regon 0.85<φ 0.9), and for very hgh porostes (0.9<φ 0.99) a regon of abrupt ncrease (.e, t seems that K* dverges when φ ). In Fg. 7 we compare our results generated by the numercal smulatons wth reported results from Carman Kozeny s correlaton (Eq. wth c=5), Kavany (Eqs. and 4) and Koponen et al. s correlaton (Eq. 5). Pressure drop (Pa) Pressure drop (Pa) 0.0 0.01 0 0.0E+00.5E-04 5.0E-04 1.E-05 5.E-06 Superfcal velocty (m/s) (a) 0.E+00 0.0E+00.5E-04 5.0E-04 Superfcal velocty (m/s) (b) Fg. 4 Superfcal velocty versus pressure drop (φ=0.0 (a); φ=0.99 (b))
J. Pnela, S. Kruz, A. F. Mguel, A. H. Res, M. Aydn The plot shows that, for porostes lower 0.87 (θ=5) our results are n rather good agreement wth the results predcted by the hydraulc radus theory of Carman Kozeny but do not hold for hgher porostes (Carman Kozeny curve wth θ gven by Eq. 4 s always above our results). On the other hand, for porostes between 0.95 and 0.99 the relatonshp presented by Koponen et al [14] s n good agreement wth our smulatons but overestmates the smulated ponts up to a factor of 4.5 for porostes lower 0.95. The above results allow us to conclude that the permeablty can be descrbed by Carman Kozeny s for porostes up to 0.87 and Koponen et al. [14] relatonshp for porostes over 0.95. Besdes, there s a porostes range between 0.87 and 0.95 where these approaches do not agree wth our smulatons. As found by Koponen et al. [14], Fg. (6) reveals that K* ncreases exponentally wth porosty. The data depcted n ths fgure are ftted an equaton of the form For each of the three dfferent regmes depcted n Fg 6 are descrbed by the followng parameters: a 1 =1x10-5 a =11.09 (r =0.994) for 0. φ 0.85 a 1 =5x10-8 a =17.6 (r =0.996) for 0.85 φ 0.9 a 1 =1x10-4 a =57.99 (r =0.970) for 0.9<φ 0.99 The very hgh correlaton between the smulated ponts and ftted curve, for each one of the three regons, means that the exponental dependence of permeablty of porosty descrbes the porous structure under consderaton. Ths result s n agreement wth the experments performed by Mguel [17] on thermal and shadow screens made of random and regular fbbers. K * 1.E+0 1.E+01 1.E-01 K*=a 1 exp (a φ) (6) 1.E-0 For 0. φ 0.99, the best ft gves a 1 =x10-6 and a =1.6 (r =0.950). 1.E-05 0.00 0.5 0.50 0.75 1.00 Porosty K * 1 8 4 0 0.00 0.5 0.50 0.75 1.00 Porosty Fg. 5 Dmensonless permeablty K* versus porosty φ Fg. 6 Dmensonless permeablty K* versus porosty φ ( numercal smulaton; Carman Kozeny correlaton wth θ=5, Carman Kozeny correlaton wth θ gven by Eq. 4; Koponen et al. s correlaton (Eq. 5)). 4 Concluson We numercally studed creepng flow through porous structures for the determnaton of the permeablty of structures wth dfferent porostes. We observed that, for porostes rangng from 0.0 to 0.99, the varaton on porosty of permeablty dsplays three dfferent regmes: an ntal slow ncrease of K* (0. φ 0.85), followed a transton regme 0.85<φ 0.9), and for very hgh porostes 4
PERMEABILITY-POROSITY RELATIONSHIP ASSESSMENT BY -D NUMERICAL SIMULATIONS (0.9<φ 0.99) a regme of sharp ncrease. Besdes, we found that permeablty ncreases exponentally wth porosty of the porous structure. Ths result s confrmed by other expermental [17] and numercal studes [14]. We compared our results wth some analytcal relatonshps avalable n lterature. Carman Kozeny s correlaton holds very well wth our results for porostes up to 0.87 whle Koponen et al. [14] relatonshp fts our data for porostes over 0.95. However, these models fal to descrbe accurately the permeablty of porous structures for porostes rangng from 0.87 to 0.95. Fnally, we found that an expresson of the type K*=a 1 exp(a φ) fts our smulaton results wth a very hgh correlaton coeffcent. We present the coeffcents a 1 and a for the varous regmes. Acknowledgments The authors would lke to thank the Évora Geophyscs Centre (CGE) for the support to the development of ths work. [9] Carman P C. Flow of Gases Through Porous Meda, Butterworths, London, 1956 [10] Kavany M. Prncples of heat transfer n porous meda. Sprnger, New York, 1991 [11] Macdonald M J, Chu C F, Gullot P P and Ng K M. A generalzed Blake Kozeny equaton for a multszed sphercal partcles. AIChE J. Vol 7, pp 158 1587, 1991 [1] Duckett K E, Can J, Krowck R S and Thbodeaux D P. Automatng the aerolometer: examnaton of the Kozeny equaton, Textle Res. J., Vol 61, pp 09 18, 1991 [1] Kamst G F, Brunsma O S L and de Graauw J. Permeablty of flter cakes of palm ol n relaton to mechancal expresson. AIChE J., Vol 4, pp 67 680, 1997 [14] Koponen A, Kandha D, Hellné E, Alava M, Hoekstra A, Kataja M, Nskanen K, Sloot P and Tmonen J. Permeablty of threedmensonal random fber web. Phys. Rev. Lett., Vol 80, pp 716 719, 1998 [15] FLUENT 6 User s Gude, Fluent Inc., 00 [16] Choukh R, Guzan A, Maflej M. Numercal study of lamnar natural convecton flow around an array of two horzontal sothermal cylnders, Int. Comm. Heat Mass Transfer, Vol. 6, No,, pp. 9-8, 1999 [17] Mguel A F. Arflow through porous screens: from theory to practcal consderatons. Energy and Buldngs, Vol 8, pp 6-69, 1998 References [1] Bejan A, Dncer I, Lorente S, Mguel A F and Res A H. Porous and Complex Flow Structures n Modern Technologes, Sprnger-Verlag, New York, 004 [] Neld D A and Bejan A. Convecton n Porous Meda, Sprnger-Verlag, New York, 1999 [] Mguel A F. Contrbuton to flow characterzaton through porous meda. Internatonal Journal of Heat and Mass Transfer, Vol 4, pp 67-7, 000 [4] Mguel A F and Res A H. Transent forced convecton n an sothermal flud-saturated porous medum layer: effectve permeablty and boundary layer thckness. Journal of Porous Meda, Vol 8, pp 1-10, 005 [5] Bear J. Dynamcs of Fluds n Porous Meda. Elsever, New York, 197 [6] Guéguen Y and Palcauskas V. Introducton to the Physcs of Rocks. Prnceton Unversty Press, Prnceton, 1994. [7] Carman P C. Flud flow through a granular bed, Trans. Inst. Chem. Eng. London, Vol 15, pp 150 156, 198 [8] Kozeny J. Uber Kapllare Letung des Wassers m Boden, Stzungsber. Akad. Wss. Wen, Vol 16, pp. 71 06, 197 5