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Engineering Conferences Interntionl ECI Digitl Archives 5th Interntionl Conference on Porous Medi nd Their Applictions in Science, Engineering nd Industry Refereed Proceedings Summer 6-4-4 Numericl

1 Engineering Conferences Interntionl ECI Digitl Archives 5th Interntionl Conference on Porous Medi nd Their Applictions in Science, Engineering nd Industry Refereed Proceedings Summer Numericl simultion of pressure pulse decy experiment on crushed low permebility rocks considering Klinkenberg effect nd gs bsorption/ desorption Bo Zhou Tsinghu University Rui-N Xu Tsinghu University Pei-Xue Jing Tsinghu University Follow this nd dditionl works t: Prt of the Mterils Science nd Engineering Commons Recommended Cittion Bo Zhou, Rui-N Xu, nd Pei-Xue Jing, "Numericl simultion of pressure pulse decy experiment on crushed low permebility rocks considering Klinkenberg effect nd gs bsorption/desorption" in "5th Interntionl Conference on Porous Medi nd Their Applictions in Science, Engineering nd Industry", Prof. Kmbiz Vfi, University of Cliforni, Riverside; Prof. Adrin Bejn, Duke University; Prof. Akir Nkym, Shizuok University; Prof. Oronzio Mnc, Second Università degli Studi Npoli Eds, ECI Symposium Series, (4). This Conference Proceeding is brought to you for free nd open ccess by the Refereed Proceedings t ECI Digitl Archives. It hs been ccepted for inclusion in 5th Interntionl Conference on Porous Medi nd Their Applictions in Science, Engineering nd Industry by n uthorized dministrtor of ECI Digitl Archives. For more informtion, plese contct frnco@bepress.com.

2 Proceedings of the 5th Interntionl Conference on Porous Medi nd its Applictions in Science nd Engineering ICPM5 June -7, 4, Kon, Hwii NUMERICAL SIMULATION OF PRESSURE PULSE DECAY EXPERIMENT ON CRUSHED LOW PERMEABILITY SHALE CONSIDERING GAS ADSORPTION/DESORPTION AND THE KLINKENBERG EFFECT Bo Zhou, Rui-N Xu, Pei-Xue Jing Therml Science nd Power Engineering Key Lbortory of Ministry of Eduction, Beijing Key Lbortory of CO Utiliztion nd Reduction Technology, Deprtment of Therml Engineering, Tsinghu University, Beijing, Chin ABSTRACT Pressure pulse decy method is widely used for permebility tests for low permebility rock plug smples. This method cn be used for crushed grin smples by removing the downstrem chmber in stndrd pulse decy tests. Processes in pulse decy tests for low permebility crushed shle re investigted using numericl simultion. Both the Klinkenberg slip effect for gs flows in low permebility rock nd the gs bsorption/desorption in the porous mtrix re considered. The complete mthemticl model is set up to include the two effects. Devition of the numericl pulse decy curve from the nlyticl one with n ssumption tht the pressure keeps constnt in the porous smple is investigted. The reltive importnce of gs bsorption/desorption nd gs compressibility is lso investigted quntittively. According to the present investigtion, gs compressibility nd dsorption both mke negtive contributions to the permeting process. A potentil two-curve method is proposed to decide bsolute permebility nd the Klinkenberg coefficient when these two prmeters cnnot be distinguished using one pulse decy curve during the inverse fitting procedure. These two prmeters cn be determined t the sme time only if the experiment is conducted under big initil pressure difference nd the Klinkenberg coefficient hs t lest the sme order of mgnitude s the pressure. INTRODUCTION Gs-bering shle is kind of porous medi with ultrlow permebility, whose informtion such s porosity nd permebility re very importnt in shle gs production industry. The pore size for shle mtrix rnges in nm nd nm [], which cuses non-drcy flows in nturl gs exploiting processes nd lbortory permebility tests [4,7]. When gs flowing through porous medium with nno-scle pore size, the velocity slip t the gs-solid interfce becomes significnt, which result in the Klinkenberg effect for mrco-scle porous medium flow [7, ]. The Klinkenberg reltion trets the effective permebility s function of locl gs pressure: bk k = k + p where k is the bsolute permebility nd b K is the Klinkenberg slip constnt. Florence hs pointed out the inner reltion between this formul nd rrefied gs flows in nno-scle chnnels [7]. The two prmeters k nd b K re usully decided by experiments. Another effect must be considered when modelling flows in shle is dsorption/desorption [, 3]. Kerogen is n importnt component of gs-shle, which provides very high internl surfce re to reserve nturl gs. The mount of gs dsorbed t the kerogen surfce chnges dynmiclly s the pore pressure chnges. A Lngmuir type isotherml eqution is usully used to describe gs dsorption in shle: q ql p = p + p L where q (std m 3 /kg) is the stndrd volume of gs dsorbed per solid mss, q L (std m 3 /kg) is the Lngmuir gs volume, p L is the Lngmuir pressure (P). Pressure pulse decy method is s stndrd technique to mesure the permebility for porous medi with ultrlow permebility, typiclly in the rnge of -9 D [, 3]. The conventionl stedy stte method which mesures the pressure drop nd the flow rte when test fluid flow through porous medium smple does not work for ultr-low porous medi becuse the flow rte re too low to mesure ccurtely. In pressure pulse decy experiment, only pressure (or pressure difference) decy () ()

3 curve is to be recorded while the test time is reltively short [9, 4]. The stndrd smple in rock nlysis is cylinder shpe plug smple. Gs plug smples cn be viewed s one-dimensionl, i.e. long the xis direction in pressure pulse decy experiments. Substitute smples re crushed qusi-sphericl smples, which re esier to cquired nd the corresponding pprtus is lso simpler thn tht for plug smples [5, 3]. The pprtus for crushed porous smples re shown in figure. The system contins two vessels, buffer vessel nd n experimentl vessel which contins sphericl smples (ll with rdius r ). First, let the gs fill the buffer vessel nd mintin reltively high pressure. Second, let the high pressure gs crushed into the experimentl vessel which is t pressure p i nd then turn off the vlve ( prehet process my be needed if the Joule-Thompson effect is significnt). At this instnt, the buffer vessel nd the void volume V of the experimentl vessel both hve pressure p i when equilibrium is reched. Then the gs in the volume V begins to permete in the low permebility smples long the rdius direction slowly until the pressure in the void volume nd the pore spce becomes equilibrium gin. The pressure difference guge records the process of pressure decy in the experimentl vessel nd n nlysis on the curve will provide informtion bout the porous medium. b K = the Klinkenberg coefficient c g = isotherml compressible coefficient e = error to the estimted prmeter k/ε F = dsorption-modified geometry constnt f = geometry constnt for the pprtus k = effective permebility k = bsolute permebility M g = gs mss (kg) per kmol p = pressure p i = initil pressure in void volume V p i = initil pressure p f = stedy stte pressure p L = the Lngmuir pressure q = dsorbte density per unit smple volume q = the stndrd volume of gs dsorbed per solid mss q L = the Lngmuir gs volume R = 834kJ/(kmol K)/M g r = spce coordinte r = smple rdius S s = totl smple surfce re T = temperture t = time coordinte t = chrcteristic time u = the Drcy velocity V = experiment vessel void volume V s = totl smple volume V std = idel gs volume per kmol in stndrd stte z = gs compressibility fctor x = ε /ε Greek Symbols ε = porosity ε = effective porosity cused by dsorption μ = gs viscosity ρ = gs density ρ s = solid density φ = root of the chrcteristic eqution Figure Apprtus for pressure pulse decy experiment for crushed sphericl porous medi smples with ultrlow permebility. In the present rticle, we will set up the complete mthemticl description of the pressure pulse decy experiment introduced bove. We will compre the pproximted nlyticl solution nd the ccurte numericl solution to show the importnce of the gs compressibility nd dsorption. Finlly, n investigtion on the inverse problem, i.e. the experiment dt processing method, will be shown. We provide potentil solution to decide both the bsolute permebility k nd the Klinkenberg slip constnt b K using two pressure decy curves. NOMENCLATURE A = the fitting constnt for dt processing Subscripts = vlue in the reference stte for gs properties dsor = dsorption comp = compressibility ref = reference ~ = non-dimensionl coordinte Governing equtions for pressure pulse decy experiments with sphericl smples Since the gs used in the pressure pulse decy permetes in ll the sphericl smples long the rdius direction, the governing eqution is one-dimensionl in spce. Considering the dsorption effect, the continuity eqution is:

4 ρ q ε + ( ε ) = ρ t t r r ( r u) (3) t r εµ c g = () k where ρ is gs density, q is dsorbte density per unit smple volume, ε is porous medium porosity, u is Drcy velocity. Apply the Drcy eqution, the Lngmuir isotherml nd the eqution of stte: k p u = µ r ρsmgq ρsmg ql p q = = Vstd Vstd pl + p ρ p = zρrt, cg = ρ p T where μ is gs viscosity, c g is isotherml compressible coefficient. Then (3) cn be converted to prbolic eqution of pressure [, 3] : p r ρk p = t cgρ( ε + ε ) r r µ r (4) (5) (6) (7) where ε is pressure dependent effective porosity cused by dsorption: ρ M ε = ( ε ) V pq ( p + p) s g L L std L c ρ As the gs permete into the smples, the gs mss in the void volume of the experimentl vessel decreses. Thus the boundry condition t the surfces of spheres is: ρ ρk p V = Ss (9) t r= r, t µ r r= r, t Another two boundry conditions re the symmetry condition t the sphere center nd step function s initil condition: g (8) p = () t r=, t pr ( r, t= ) = p, pr ( = r, t= ) = p () i i The solution of eqution (7) subject to boundry conditions (9), () nd () will provide the pressure dt of the smples nd the pressure p t the smple surfces cn be mesured by sensor. A fitting to curve p (t) with the permebility s prmeter provides method to determine permebility relted informtion, for exmple, k nd b K. Now set the stte of which the pressure is p i nd the temperture is T s reference stte. The corresponding gs properties re denoted with subscript. A nondimensionl time t cn be defined s: Neglecting the dsorption temporlly, stedy stte pressure cn be reched when the test time is sufficiently long. This pressure, denoting s p f, cn be clculted s: εvsp i + Vpi fp i + 3pi p f = = εvs + V f + 3 (3) ε rs f = s V (4) is n pprtus relted geometry constnt. The pressure p decys from p i to p f during the experiment (ssuming no dsorption). Using t nd r s non-dimensionlize prmeter for time nd spce, ssuming the gs is idel furtherly (z=, c g =/p, μ=μ ), eqution (7) becomes: pi p bk p = r + p t + ε / ε r p r nd the boundry condition (9) now reds: i p p p = f p t r r= r= (5) (6) Eqution (5) involves three effects explicitly: compressibility, dsorption nd the Klinkenberg slip. If we neglect the compressibility effect, i.e. ssuming the pressure in the smples keeps round constnt (p i ) ll the time, the non-linerity disppers nd n nlyticl solution cn be derived using the Lplce trnsformtion method [8]. Since the pressure in the smples rnges in p i nd p i, the nlyticl solution is resonble only if the reltive pressure difference (p i p i )/p i is smll. The pproximted nlyticl solution of eqution (5) subjected to corresponding boundry conditions re: 3pi + Fp i p(,) rt 3 F = + + bk / pi sin( φnr )exp( φnt )( pi p i) + x r [ φ cos φ + ( + F) sin φ ] n= n n n (7) where φ n is the nth root for chrcteristic eqution: Fφ tnφ = φ + F (8) F = f( + ε ( p ) / ε ) f( + x) (9) i Solution (7) cn be truncted to reserve only the first term in the summtion when the non-dimensionl time is sufficiently lrge. Figure shows tht when the nondimensionl time is lrger thn., the first root φ 3

5 domintes the decy. This behvior provides fst experiment dt processing method, which is clled lte-time technique in the literture [3]. This method is only resonble if the gs compressibility cn be neglected. Figure Anlyticl solutions clculted using the first roots or the first root of the chrcteristic eqution, where x=, b K =, f=, p i = 5 P, p i =8 4 P Impcts of compressibility nd dsorption In cses where the pressure drop in the smples is so lrge tht the gs compressibility must be tken into ccount, eqution (5) hs no nlyticl solution. The coefficient (+x) - contributed by the dsorption effect lso vries s the pressure distribution chnges in such cses. However, the qusi-liner eqution (5) cn be solved numericlly using the finite difference method. Figure 3 shows numericl results with severl modes. The gs used in the simultion is nitrogen. Prmeters involved in the dsorption effect re p L =7.5 6 P, q L =.m 3 /kg, ρ s =5kg/m 3, V std =.44std m 3 /kmol. The trnsport eqution (5) cn be written s: p p+ b K p = r t + ε / ε r pi r () Since p i is set s the reference stte in the nondimensionlize procedure, which is the highest pressure in the trnsport process, the non-dimensionl trnsport coefficient p/p i is lwys less thn when b K is bsent. Thus the compressibility lwys slows down the trnsport process, s tht is shown in figure 3. The contribution of dsorption to the trnsport coefficient is lso negtive, so the evlution towrds the equilibrium stte becomes slower when dsorption is considered. Adsorption lso influences the finl stedy stte pressure since this effect contributes n effective porosity to the porous medium. Further investigtions on compressibility nd dsorption respectively re shown in figure 4 nd figure 5. If the lte-time technique, i.e. the nlyticl solution, is still used to process n experiment pressure decy curve where the compressibility must be considered, n error to the prmeter k to be determined will occur. The Figure 3 Numericl solutions with or without dsorption nd compressibility, where b K =, f=, p i = 5 P, p i =5 4 P dimensionl form exponent in the nlyticl solution (the first term) gives: k Ar µ c = ε + ε φ g () where A(s - ) is fitting prmeter (ssuming n exct vlue here). When dsorption is bsent, the reltive devition of the porous medium trnsport prmeter k/ε is: e comp At = () φ Figure 4 shows contour of e comp s function of geometry prmeter f nd the reltive pressure difference (p i p i )/p i. As discussed bove, ignoring the compressibility will underestimte the trnsport coefficient. The error cused by ignoring compressibility increses s the non-dimensionl pressure difference increses. The geometry prmeter f lso influences this error becuse the root φ is function of f. Figure 4 Error of the porous medium trnsport coefficient k/ε cused by ignoring compressibility 4

6 Now we investigte on the dsorption effect if the gs cn be viewed s incompressible during the experiment. The model () is ccurte in such cse, the dsorption ffects the chrcteristic root φ only. The error of the trnsport coefficient k/(ε+ε ) is: e dsor φ ( F) = (3) φ ( f ) Figure 5 shows contour of e dsor s function of geometry prmeter f nd the prmeter x defined in (9). As x increses (the dsorption enhnces), the error cused by ignoring dsorption increses. The geometry prmeter f lso influences the error. ( ) M R( k, b ) = p ( k, b ) p (4) K, fit, n K, ref, n n= where p,fit,n is the nth dt point of the curve clculted using predicting prmeters k nd b K, p,ref,n is the nth dt point of the smple set. Finlly, minimize the function R subjected to positive k nd b K. The lgorithm using here is the Sequentil Lest SQures Progrmming (SLSQP) Constrined minimiztion method []. Figure 6 nd figure 7 show fitting result for cses k,ref = -8 m, b K,ref =5 4 P nd k,ref = -8 m, b K,ref = 5 P respectively. Other relted prmeters re p i = 5 P, f=, x=. The initil vlues (shown s hollow squres) for the optimiztion procedures re rndomly selected in the rnge n order of mgnitude more or less thn the reference prmeters. Two smple dt re evluted using different initil pressure p i for ech cse. Figure 5 Error of the porous medium trnsport coefficient k/(ε+ε ) cused by ignoring dsorption Determine the bsolute permebility nd the Klinkenberg coefficient by solving the inverse problem The impct of the Klinkenberg effect to the pressure decy curve p (t) is reltively cler. The positive constnt b K will enhnce the term (p+b K )/p i nd hence the effective permebility. Noticing tht the bsolute permebility k is involved in the non-dimensionl time, we conclude tht in the following two cses k nd b K cnnot be distinguished by pplying the inverse problem fitting on pressure decy curves: ) Incompressible cses. If the pressure in the smple remins closely to p i then the term (p+b K )/p i is lso nerly constnt. Only the combintion (p+b K )k /p i (which is constnt) cn be got in the inverse problem. ) Cses where b K p i. In such cses it is hrd to cpture b K numericlly since its contribution to the pressure decy curve is covered up by the bsolute permebility. The inverse problem is solved s the following steps. First evlute pressure decy curve using k,ref nd b K,ref, which re fitting prmeters. Second choose M discrete dt points s smple nd define n optimiztion function: Figure 6 Inverse fitting results for cse b K,ref =5 4 P Figure 7 Inverse fitting results for cse b K,ref = 5 P For ech cse the trget vlue of fitting is (,) in figure 6 nd figure 7. Fitting results with different initil pressure p i nd hence different pressure difference (p i p i )/p i give similr curves. Prmeters (k / k,ref, b K / 5

7 b K,ref ) on these curves cn ll mke clculted pressure decy curves nd the smple dt mtch. However, points stisfy both decy curves with different pressure difference concentrted ner the point (,). When b K is comprble to p i in order of mgnitude, n intersection zone of the fitting result curves grdully forms, which lso covers the desired point (,). This observtion verifies the prior nlysis in this section tht we cn hope to distinguish k nd b K only if b K is comprble to p i nd compressibility is considered. Furthermore, two sets of smple dt, i.e. two experiment pressure decy curves (with different initil pressure difference) re required t lest. A substitute wy to clculte k nd b K is to clculte the effective permebility t vrious pressure nd then fit the dt using the Klinkenberg reltion. This method is dopted in the conventionl stedy stte permebility test, which lso works here. More experiment pressure decy curves re needed to get ccurte fitting results. CONCLUSIONS A complete mthemticl model for the pressure pulse decy experiment for crushed shle mtrix smples with ultr-low permebility is built in the present rticle. For shle mtrix, both the Klinkenberg effect nd dsorption/desorption must be tken into ccount. After the non-dimensionlize procedure, we find tht compressibility, the Klinkenberg effect nd dsorption ll contributes to the effective permebility. The Klinkenberg effect mkes positive contribution while the other two effects mke negtive contributions. Key criterions on whether these effects re importnt re (p i p i )/p i for compressibility, x for dsorption nd b K /p i for the Klinkenberg effect. The geometry nondimensionl fctor f lso influences these effects. The investigtion on the inverse fitting procedure for k nd b K shows these two prmeters cnnot be distinguished using only one set of smple dt. For cses where incompressibility is ssumed or b K is too smll, the effort to distinguish them will lso fil. A prctice shows tht we cn hope to determine these two prmeters using two sets of smple dt produced by lrge initil pressure difference experiment if the prmeter b K is comprble to p i. ACKNOWLEDGEMENT The uthors would like to cknowledge support from the Ntionl Nturl Science Foundtion of Chin(No ) nd the Ntionl Science Fund for Cretive Reserch Groups of Chin(No. 53) REFERENCES [] Americn Petroleum Institute. Recommended Prctices for Core Anlysis, RP 4, th edition [R]. Wshington: API, 998: 6(37)-6(39) [] Civn F, Ri CS, Sondergeld CH () Shle-gs permebility nd diffusivity inferred by improved formultion of relevnt retention nd trnsport mechnisms. Trnsport in Porous Medi, 86(3): [3] Cui X, Bustin AMM, Bustin RM (9) Mesurements of gs permebility nd diffusivity of tight reservoir rocks: different pproches nd their pplictions. Geofluids, 9(3): 8-3. [4] Drbi H, Ettehd A, Jvdpour F, et l () Gs flow in ultr-tight shle strt. Journl of Fluid Mechnics, 7: [5] Egermnn P, Lenormnd R, Longeron D, Zrcone C (5) A fst nd direct method of permebility mesurements on drill cuttings. Society of Petroleum Engineers Reservoir Evlution nd Engineering, 4, [6] Florence, Frncois Andre, et l Improved permebility prediction reltions for low permebility snds. Rocky Mountin Oil & Gs Technology Symposium. Society of Petroleum Engineers, 7. [7] Freemn CM, Moridis GJ, Blsingme TA () A numericl study of microscle flow behvior in tight gs nd shle gs reservoir systems. Trnsport in porous medi, 9(): [8] Hsieh PA, Trcy JV, Neuzil CE, Bredehoeft JD, Sillimn SE (98) A trnsient lbortory method for determining the hydrulic properties of tight rocks: I. Theory. Interntionl Journl of Rock Mechnics nd Mining Sciences, 8, [9] Jones SC (997) A technique for fster pulse-decy permebility mesurements in tight rocks. SPE formtion evlution, (): 9-6. [] Klinkenberg LJ (94) The permebility of porous medi to liquids nd gses. Drilling nd production prctice. [] Krft, D (998) A softwre pckge for sequentil qudrtic progrmming. Tech. Rep. DFVLR-FB 88-8, DLR Germn Aerospce Center Institute for Flight Mechnics, Koln, Germny. [] Loucks RG, et l (9) Morphology, genesis, nd distribution of nnometer-scle pores in siliceous mudstones of the Mississippin Brnett Shle. Journl of Sedimentry Reserch, 79(): [3] Luffel DL, Hopkins CW, Schettler Jr PD. Mtrix permebility mesurement of gs productive shles. SPE Annul Technicl Conference nd Exhibition. Society of Petroleum Engineers, 993. [4] Ymd SE, Jones AH (98) A review of pulse technique for permebility mesurements. Society of Petroleum Engineers Journl, (5):

A Case Study of Fluid Transport in Shale Crushed Samples: Experiment and Interpretation

A Case Study of Fluid Transport in Shale Crushed Samples: Experiment and Interpretation SCA2014-071 1/12 A Cse Study of Fluid Trnsport in Shle Crushed Smples: Experiment nd Interprettion Tinho Wu, Zheng Jing, Dongxio Zhng Energy nd Resources Engineering, College of Engineering Peking University,