A METHODOLOGY TO PREDICT THE GAS PERMEABILITY PARAMETERS OF TIGHT RESERVOIRS FROM NITROGEN SORPTION ISOTHERMS AND MERCURY POROSIMETRY CURVES

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1 SCA /1 A METHODOLOGY TO PREDICT THE GAS PERMEABILITY PARAMETERS OF TIGHT RESERVOIRS FROM NITROGEN SORPTION ISOTHERMS AND MERCURY POROSIMETRY CURVES Chito D. Takioglou 1, Adnan Al Hinai 2, Reza Rezaee 2 1 Foundation fo Reeach and Technology Hella Intitute of Chemical Engineeing Science, Stadiou t., Platani, 2654 Pata, Geece 2 Depatment of Petoleum Engineeing, Cutin Univeity, Peth, WA, Autalia Thi pape wa pepaed fo peentation at the Intenational Sympoium of the Society of Coe Analyt held in Tondheim, Noway, 27-3 Augut 218 ABSTRACT Fo the explicit computation of the abolute pemeability and Knuden diffuion coefficient of tight ock (hale fom poe tuctue popetie, a methodology i uggeted. The poe pace i epeented by a poe-and-thoat netwok quantified by bimodal poe and thoat ize ditibution, poe hape facto, and poe acceibility function. With the aid of pecolation theoy analytic equation ae developed to expe the N 2 adoption / deoption iothem and Hg intuion cuve a function of all petinent poe tuctue paamete. A multitep pocedue i adopted fo the ucceive etimation of each et of paamete by the invee modeling of N 2 adoption, N 2 deoption, and Hg intuion dataet. With the aid of citical path analyi of pecolation theoy, appoximate elationhip ae developed allowing the explicit calculation of the abolute pemeability and Knuden diffuion coefficient, fom etimated poe netwok popetie. Application of the methodology to the dataet of eveal hale ample enable u to evaluate the pedictability of the appoach. INTRODUCTION Tight and hale eevoi ytem poe a temendou potential eouce fo futue development, and tudy of thee ytem i poceeding apace. In paticula, hale ga eevoi poe many o-called unconventional featue and conideation on maco- and mico-cale of flow, and thei extemely low pemeability peent unuual challenge [4, 6]. Paticulaly, the tool of taditional poduction data analyi have not been ufficient fo detemining the eevoi matix pemeability in thee unconventional ytem. Depending on the poe ize ange, the flow of natual ga though poou conventional ock, tight fomation and hale ytem, i dominated by a vaiety of flow egime, including Knuden, tanition, lip and vicou Dacy flow [13]. Knowledge and undetanding of ock popetie, including poe geomety, pemeability, and fluid ditibution ae eential fo detemining hale hydocabon toage and ecovey. The micotuctual chaacteization of hale ample could be achieved by combining

2 SCA /1 mecuy injection capillay peue (MICP, with low-field nuclea magnetic eonance (NMR, nitogen adoption/deoption (N 2 and high eolution focued ion beam canning electon micocopy (FIB SEM image [1]. The ga pemeability of hale can be coelated with the capillay peue data fom mecuy injection meauement, by uing theoetical and empiical equation [2]. On the othe hand, the poe netwok appoach i a pactical way to explain the maco flow behavio of poou media fom a micocopic point of view, and the hale matix can be imulated by 3-dimenional poe netwok model that include typical bimodal poe ize ditibution, aniotopy and low connectivity [12]. Moeove, the digital ock with nanopoe could be contucted by 3D poe tuctue image obtained fom mico/nano CT and FIB-SEM image econtucted fom 2D SEM image of poe tuctue, and uing the Lattice Boltzmann method to calculate the intinic pemeability, pooity and totuoity; thee paamete ae ued to calculate the appaent pemeability unde conideation of diffeent combined ga tanpot mechanim [9]. In the peent wok, with the aid of pecolation theoy, analytic model of the N 2 adoption-deoption iothem and Hg intuion cuve a function of the poe and thoat adiu ditibution and poe netwok connectivity ae developed [1]. Then a tepwie methodology i adopted to etimate the poe netwok paamete with invee modeling of N 2 option and Hg intuion cuve and applied to dataet of eveal hale ample. The citical path analyi of pecolation theoy i ued to calculate the liquid pemeability and Knuden diffuion coefficient of poou media explicitly fom the afoementioned poe tuctue paamete and compae the pediction with expeimental meauement. METHODS AND MATERIALS Poe-cale model of dainage and imbibition The poe tuctue i epeented by a tochatic netwok of poe (ite-and-thoat (bond, the ize of which ae ampled fom bimodal ite ( f b f and bond adiu ( ditibution function. The thoat ae conideed a naow contiction without volume, while the poe ae aigned the entie poe volume accoding to the elation β ( V (1 whee β i a volume hape facto ( β. The co-ection of each poe i modeled by a high-ode egula polygon with numbe of ide (angle, aea, and peimete equal to n, A, and P, epectively. The poe hape facto, G, i given by [1] 2 G= A P (2 and the poe adiu,, i defined a the adiu of the incibed cicle in the egula polygon, given by = 2GP (3 Fom the polygon geomety, we obtain G= 1 4n tan( π n (4 The adii of poe follow a bimodal ditibution function compoed of log-nomal component ditibution function, f (;< 1 >,σ 1,< 2 >,σ 2,c, whee < i >, σ i, c ae the

3 SCA /1 mean adiu, tandad deviation, and contibution faction of component ditibution i to the oveall one. Repectively, the adii of thoat ( window connecting adjacent poe follow an analogou bimodal ditibution function f b (;< b1 >,σ b1,< b2 >,σ b2,c b. Model of N 2 adoption / deoption in poe-and-thoat netwok In geneal, the filling of poe with liquid nitogen i caied out though two paallel mechanim: (1 multilaye uface adoption of ga molecule on the flat poe-wall and (2 capillay condenation of liquid on the ough egion of the adobed laye. The thickne of the adobed laye, t c, i detemined by the genealized FHH equation [3] c ( 1 1 t = σ b ln x (5 whee x i the elative vapo peue, σ i the diamete of N 2 molecule (σ=3.54 A, b i a paamete depending on the popetie of the adobent and adoptive, and i an index with value anging fom 2 to 3. The cuvatue adiu of the liquid/vapo menici of the condenate, c, i given by Kelvin equation [3] = c ln 1 x c = V γ RT (6 c ( L Fo the ake of claity and phyical inpection of the poduced equation, the fluid atuation i fit calculated fo β = 2. and then the equation ae genealized to any β value. The atuation of a patially liquid N 2 -occupied poe conit of the adobed and condened phae and i given by { } ( σ( σ( Sf 2,c = 4FGc + 2 b c c b c c (7 whee F i the faction of the poe co-ection aea occupied by the wetting fluid (liquid N 2 and depending on it contact angle, θ (fo liquid N 2, θ = o, unle othewie tated. Auming that any change of fluid volume in a poe ae decibed by a elation analogou to Eq.(1, Eq.(7 can be ead a β 1 { } β ( σ( σ( β Sf 2,c = 4FGc + 2 b c c b c c (\8 The capillay condenation in a poe i analogou to wetting fluid (liquid N2 imbibition, and the citical cuvatue adiu of condenation, =, i given by { } ( 4 ( π θ ( 2 con = tc G F + n if F and θ π n (9a con = tc coθ if F < and θ > π n (9b Eq.(9, in conjunction with Eq.(5 and (6, yield the citical adiu =, given by 1 1 = 4G[ F + ( π nθ ] con + σ ( b c con if F and θ π n (1a 1 1 ( b c = 2 coθcon + σ con if F < and θ > π n (1b The liquid nitogen atuation of a poe netwok a a function of the elative vapo peue can be expeed by the elation c con

4 SCA /1 SLN 2( x = f ( V ( d + f ( V ( S f 2( d f ( V ( d (11 The capillay evapoation fom a poe i analogou to the wetting fluid (liquid N 2 dainage, and hence the citical cuvatue adiu of evapoation, evp, i given by evp = ( t c co θ ( t c co 4F 2 θ G 4F G if F and θ π n (12a evp = if F < and θ > π n (12b 2coθ which in conjunction with Eq.(5 and (6 eult in FGevp b = σ ( b c evp 2 coθ co θ 4FG if F and θ π n (13a 1 1 ( b c b = 2 coθevp + σ evp if F < and θ > π n (13b whee b i the citical adiu fo N 2 evapoating fom thoat with b. N 2 deob fom a poe if it adiu i le than the citical ize ( b o and ha acce eithe to the bulk vapo phae (pimay deoption o to iolated vapo pocket (econday deoption. At any tage of the deoption poce, the faction of bond (thoat q b and ite (poe q which ae allowable to the vapo phae ae given by q b = fb ( d and = f ( b q d (14 epectively. The faction of acceible ite (poe, Y i, to the vapo phae a a function of the faction of allowable bond (thoat, q b, i the poe acceibility function of the netwok, and accoding to pecolation theoy [1, 11], it i affected by the netwok topology, patial poe ize coelation and netwok ize. ( i q b inflection point of which, define the pecolation thehold atuation duing deoption i given by [1] S LN 2 ( x = f ( V ( q Yi d + q q i ( qb f f ( V ( Model of Hg Intuion in poe-and-thoat netwok i ( V ( Yi d + q d Y i a igmoid cuve, the q. Finally, the liquid N 2 bc ( qb f ( V ( S ( In geneal, the value of poe volume calculated fom N 2 option doe not coincide to the coeponding value calculated fom Hg intuion cuve. Although Hg i a non-wetting fluid fo the majoity of mateial, by convention the Hg contact angle, θ Hg, i meaued with epect to the wetting fluid (low peue ai and i conideed alway lowe than f 2 d (15

5 SCA /1 9o. In ode to etimate a unique poe volume value, Vp (cm3/g, the following method i adopted. At a given Hg intuion contact angle the maximum intuion capillay peue i conveted to an equivalent elative vapo peue. Thi value i intepolated in N2 deoption cuve to calculate the coeponding liquid nitogen volume. The um of the two volume yield the total poe volume. The pocedue i epeated fo vaiou value of θhg, and the poduced dataet (Vp v θhg ae fitted to a non-linea paametic function of the fom (θ θ w (16 V p = A1 + ( A2 A Hg { } to etimate the value of A1,A2,θ, w. The Hg intuion in a poe-and-thoat netwok i analogou to capillay evapoation fo qi= (pimay deoption and i contolled by thoat ize. The mecuy atuation in a poe netwok i given by [1, 11] S Hg (Pc = 1 f ( V ( d + q Y (qb Y (qb ( ( f V d + f ( V ( S f 1 ( d q q whee c = γ Ηg Pc ( f ( V ( d FHg = F θ = θ Hg ( b = ( 4FHg Gc co θ Hg co 2 θ Hg 4FHg G b = 2c co θ Ηg if (17 (18 if FHg > o θ Hg < π n FHg < o θ Hg π n (19a (19b Citical path analyi (CPA The fluid flow conduction though a poe ytem with a boad poe ize ditibution i govened by a citical poe conductance, gc, which i defined by the condition that all poe with conductance geate than gc ceate a netwok panning clute tanfeing mot of the flow [7]. Thi citical conductance gc depend on tuctual chaacteitic (e.g. poe hape, poe ize ditibution, connectivity, dimenionality, etc of the poe pace and CPA i ued to calculate the coeponding citical poe dimenion. Accoding to CPA [7], all conductance with value geate than gc ae et equal to gc, while all conductance malle than gc ae et equal to zeo. The tanpot popetie of the citical path may be dominated by univeal caling law pevailing in the vicinity of the pecolation thehold and decibing the vaiation of acceibility function with the faction of allowed thoat [11]. In mathematical tem, the vicou flow o Knuden flow poblem educe to the calculation of the citical poe dimenion that maximize the total conductance of the citical path [5, 11]. In geneal, the dependence of the local vicou flow conductance on poe ize may be diffeent fom that of the local Knuden diffuion conductance, o that the citical poe dimenion may depend on the tanpot mechanim [11]. The ga mola flux though a poou medium in the tanition egion between

6 SCA /1 Knuden diffuion and vicou flow can be decibed by the um of the individual contibution, namely k L 1 N = P+ Dkn P (2 µ g RT whee k L i the liquid pemeability, and D kn i the Knuden diffuion coefficient of the citical path. Accoding to the CPA [11], thee popetie ae appoximated by the elationhip 4 ϕ t kl = q 2 b ( qbc 8 (21 q 4 ϕ t 8 RT Dkn = ckn q 2 b ( qbc ckn = (22 q 3 2π M whee φ i the pooity, t i the univeal caling exponent of conductivity (t=2. fo 3-D ytem, and mean poe adiu, < q > can be expeed a function of thoat adiu,, by = f d (23 q ( The value of k L and D kn ae maximized fo = max,f and = max,d, epectively. Thee maximum value ae conideed appoximately equal to the pemeability and Knuden diffuion coefficient of the invetigated poou medium. Fo low pemeable poou media, the meaued ga pemeability i peue-dependent and i commonly expeed by the elationhip k = k 1+ b P (24 ( g L m whee b i the Klingenbeg facto, and P m i the mean ga peue [2]. Fo the teadytate ga flow though a poou medium, the integated fom of Dacy equation, including the Klingenbeg effect, take the fom uplµ PΔ P = k + bp (25 ( 2 2 g ( m L( 1 m whee u 2, P 2 ae the flow velocity and peue at the outlet, L i the ample length, µ g i the ga vicoity, and ΔP i the peue dop aco the ample. By combining Eq.(2 with Eq.(25, it i obtained ( D = bk µ (26 kn L g Expeimental dataet and multi-tep paamete etimation Fo the hale ample, the N 2 adoption/deoption iothem wee meaued with Micomeitic TiSta II 32, wheea the Hg intuion cuve wa meaued with Micometic Autopoe IV Pooimete [1, 2]. The ga pemeability and all petinent Klingenbeg paamete wee meaued with a modified teady-tate method, whee the ample i embedded in a ein dic and the ga flow ate i meaued at the outlet [2]. Fo the non-linea paamete etimation, numeical code in the envionment of Athena Viual Studio [8] wee developed, and a tep-wie pocedue wa adopted. Fit, the paamete n, β, and f (;< 1 >,σ 1,< 2 >,σ 2,c wee etimated with invee modeling of N 2 adoption iothem. Then, the foegoing paamete wee fixed, and 12

7 SCA /1 f b (;< b1 >,σ b1,< b2 >,σ b2,c b along with Y i (q b wee etimated with invee modeling of N 2 deoption iothem. Then, Y (q b wa etimated fom invee modeling of the Hg intuion cuve. Finally, the thoat ize ditibution, econday and pimay poe acceibility function, and Hg contact angle wee e-etimated, with the imultaneou invee modeling of N 2 adoption/deoption iothem and Hg intuion cuve. RESULTS AND DISCUSSION The methodology of the poe tuctue chaacteization fom N 2 adoption/deoption iothem and Hg intuion cuve wa applied to dataet of eveal hale ample (Al Hinai et al., 213 collected fom one well, at vaiou depth, of the Peth Bain (Autalia, and the eult ae hown in Table 1. Fo one ample (AC5, we wee unable to match peciely the Hg intuion cuve (Table 1, Fig.1c/S1, o that the lat tep of paamete etimation [f b (, Y i (q b, Y (q b, θ Hg ] wa epeated by changing the initial guee (Table 1, Fig.1c/S2. Not having impoed any limitation on the poe to the thoat ize atio, the etimated thoat adiu ditibution may extend to highe value, compaed to the poe adiu ditibution (Table 1, AC5, S1/S2. The phyical intepetation of uch a quantitative deciption i that the thoat ize ae compaable to the ize of inteconnecting poe, and pobably a imple model (e.g. a capillay tube netwok decibed by only one poe ize ditibution and one acceibility function might be moe adequate than the peent appoach fo the epeentation of the poe pace. The paamete value etimated fom the dataet of all ample wee ued to compute k L and D kn (Fig.2 and compae them with coeponding one fom expeimental meauement (Table 2. It eem that D kn i pedicted atifactoily fo almot all cae (Table 2, while the pemeability i almot pedictable only fo ample AC5, while it fail fo AC1 and AC8 (Table 2. The Knuden numbe i commonly ued to claify flow egime in fine poe whee deviation fom continuum flow i impotant [13]. It i defined a the atio of molecula mean fee path, λ (nm to the mean poe adiu, (nm, and i given by Kn = λ (27 If K n <.1, then vicou flow dominate, and the conventional fluid dynamic equation and Dacy equation ae applicable. If.1 < K n <.1, then lip flow egime occu and Dacy equation with Klinkenbeg o Knuden modification i applicable. If.1 < K n < 1, then the tanition flow egime pevail, whee both lip (continuum and diffuion flow can occu, and the taditional fluid dynamic equation tat to fail. The highe the Knuden numbe, the highe the chance of failue. Even though conventional equation could be applied (i.e., Dacy with Knuden coection, the validity of uch a fomulation i quetionable. It i afe to ue Knuden diffuion equation, epecially fo flow at highe Knuden numbe (cloe to 1. If K n > 1, then the Knuden (fee molecula flow egime pevail, whee the continuum fluid flow equation fully beak down, and diffuion-baed fomulation mut be applied. Evidently, fo the analyzed ample (Table 2, in the calculation of K n, the tanition (AC5 and Knuden flow egime (AC1, AC8 pevail. Theefoe, the Knuden flow i dominant ove all cae (AC1, AC5, AC8, but only in the one cae (AC5, vicou flow may be evident. Baed on the afoementioned

8 SCA /1 intepetation, failing to pedict the value of k L fo AC1 and AC8, and ucceeding to pedict atifactoily D kn fo all cae eem eaonable. CONCLUSIONS Analytical model of Hg intuion and N 2 adoption-deoption in poou mateial ae ued fo the invee modeling of expeimental dataet of hale ample to etimate a full et of paamete fo the poe pace, egaded a a poe-and-thoat netwok. Uing the citical path analyi of pecolation theoy, analytical elationhip ae developed fo the explicit computation of the Klinkenbeg paamete of the ga pemeability fom poe pace paamete. Application of the methodology to dataet of hale indicate that, baed on the etimated poe pace paamete, the liquid pemeability i pedictable at low K n value, ove which the vicou flow egime i valid, while the Knuden diffuion coefficient i pedictable at low and high K n value, ove which Knuden flow i feaible. REFERENCES 1. Al Hinai, A.A., Rezaee, R. Poe geomety in ga hale eevoi. In: Fundamental of Ga Shale Reevoi, R. Rezaee (ed, pp (215 John Wiley & Son, Inc. 2. Al Hinai, A.A., Rezaee, R., Saeedi, A., Lenomand, R. Pemeability pediction fom mecuy injection capillay peue: an example fom the Peth Bain, Weten Autalia. APPEA J. (213 53: Gegg, S.J., Sing, K.S.W. Adoption, Suface Aea and Pooity, Academic Pe, London ( Hill, D.G., Nelon, C.R. Ga poductive factued hale: an oveview and update. Ga TIPS (2 6(3: Hunt, A.G. Application of pecolation theoy to poou media with ditibuted local conductance. Adv.Wate Reou. (21 24: Javadpou, F., Fihe, D., Unwoth, M. Nanocale ga flow in hale ga ediment. J. Can. Pet. Technol. (27 46(1: Katz, A.J., Thompon, A.H., Quantitative pediction of pemeability in poou ock. Phy. Rev. B ( : Stewat, W.E., Caacotio, M. Compute-Aided Modeling of Reactive Sytem. John Wiley & Son, Hoboken, New Jeey ( Sun, H., Yao, J., Cao, Y-C, Fan, D-Y, Zhang, L. Chaacteization of ga tanpot behavio in hale ga and tight ga eevoi by digital ock analyi. Int. J. Heat Ma Tanfe (217 14: Takioglou, C.D., Bugano, V.N., Jacoben, J. Poe tuctue analyi by uing nitogen option and mecuy intuion data. AIChE J. (24 5: Takioglou, C.D., Ioannidi, M.A., Amithaaj, E., Vizika, O. A new appoach fo the chaacteization of the poe tuctue of dual pooity ock. Chem. Eng. Sci., (24 64: Zhang, P., Hu, L., Meegoda, J.N. Poe-Scale Simulation and Senitivity Analyi of Appaent Ga Pemeability in Shale Matix. Mateial (217 1: 14: Ziaani, A.S., Aguilea, R., Knuden Pemeability Coection fo Tight Poou Media. Tanp. Poou Media (212 91:

9 SCA /1 Table 1. Etimated paamete value fo hale ample AC1, AC5, AC8 Paamete AC1 AC5 (S1 AC5 (S2 AC8 n β < 1 > (nm σ 1 (nm < 1 > (nm σ 1 (nm c < > (nm σ (nm < b1 > (nm σ b1 (nm < b2 > (nm σ b2 (nm c b < b > (nm σ b (nm q bc 5.8x θ Hg 65 o 4.3 o 4.3 o 65 o LN2-occupied poe volume, V LN2 (cm 3 /g V p (cm 3 /g N2 adoption (expeiment N2 deoption (expeiment N2 adoption (S1 N2 deoption (S Relative Peue, P/P (a LN2-occupied poe volume, V LN2 (cm 3 /g N2 adoption (expeiment N2 deoption (expeiment N2 adoption (S2 N2 deoption (S Relative Peue, P/P Figue 1. Meaued (AC5 v pedicted (a N2 option iothem (imulated tuctue S1, (b N2 option iothem (imulated tuctue S2, (b Hg intuion cuve (imulated tuctue S1 & S2 k L (nd AC5 (S1 k L,max =36.8 nd AC5 (S2 k L,max =19.4 nd (μm (a Figue 2. (a Liquid pemeability, and (b Knuden diffuion coefficient of the citical path a a function of thoat adiu fo two imulated poe tuctue (S1,S2 of ample AC5 Table 2. Compaion of pedicted v expeimentally meaued flow popetie of hale Shale ample AC1 AC5 AC5 AC8 (k L exp (nd 3x (k L ped (nd (S (S2.6 (D kn exp =(bk L /µ g exp (m 2 / 2.26x x x x1-9 D kn (m 2 / 1.x1-8 8.x1-9 6.x1-9 4.x1-9 2.x1-9 (b Hg-occupied poe volume, V Hg (cm 3 /g.14 Hg intuion (expeim (c Hg intuion (S1.12 Hg intuion (S θ Hg =4 o V p =.1323 cm 3 /g x1-8 (b 1.4x1-8 AC5 (S1 D kn,max =6.7x1-9 m 2 / AC5 (S2 D 1.2x1-8 kn,max =1.4x1-8 m 2 / (μm Capillay peue, P c (kpa

10 SCA /1 (D kn ped (m 2 / 2.77x x1-9 (S1 1.4x1-8 (S2 1.6x1-9 λ/< >