SPE Improved Permeability Prediction Relations for Low Permeability Sands

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1 SPE Imroved Permebility Prediction Reltions for Low Permebility Snds Frncois-Andre Florence, Texs A&M University T.A. Blsingme, Texs A&M University Dertment of Petroleum Engineering Texs A&M University College Sttion, TX Slide /2

2 Concet: Gs Slige Gs Slige Phenomenon: "Gs slige" occurs when the men free th () of the gs molecules is "not negligible" comred to effective ore throt rdius: the gs molecules "sli" on the surfces of the orous medi (i.e., the velocity v of the gs molecules t the wll is non-zero) : Gs Flow v y v x ( y 0) v 0, with v 0 dv c dy y0 Gs Slige /2 v x From: Kundt, A. nd Wrburg, E.: "Über Reibung und Wärmeleitung verdünnter Gse, " Poggendorfs Annlen der Physi und Chemie (875), 55, 337. Ignoring the gs slige effect led to n overestimtion of the gs flowrte nd the core ermebility (Drcy's lw). Slide 2/2

3 Concet: Gs Slige Gs Slige Phenomenon: The "gs slige" effect for orous medi ws first documented by Klinenberg. The gs ermebility tends to limiting vlue t n infinite men ressure which is referred to s equivlent liquid ermebility or Klinenberg-corrected ermebility. b K m Gs Slige 2/2 Where is the Klinenbergcorrected ermebility (identified s the bsolute ermebility), nd b K is the gs slige fctor. From: Klinenberg, L.J.: "The Permebility of Porous Medi to Liquids nd Gses," API Drilling nd Production Prctice (94). Slide 3/2

4 Concet: Correltions Correltions between b K nd the etrohysicl dt: The objective of this wor is to ccurtely correlte the b K -term with the Klinenberg-corrected ermebility nd other etrohysicl dt with the form: b K f (,...) This form is then substituted in the Klinenberg eqution: For given ir of vlues of gs ermebility nd men ressure The Klinenberg-corrected ermebility is then comuted by solving: Two tye of correltions re reorted in the literture: b K vs. b K vs. / f (,...) m f (,...) m 0 (Heid et l, Jones nd Owens) (Smth nd Keighin, squre-root (this wor)) Slide 4/2

5 Correltions: Heid et l. b K vs. /2 Discussion: Heid et l. Correltion is "ccetble" for vlues of ermebility greter thn md. Dt sctter must not be neglected Slide 5/2

6 Correltions: Jones-Owens nd Heid et l b K vs. 2/2 Discussion: Jones-Owens Although the correltion ers ccurte, the correltion errors re most significnt for low ermebility dt. Slide 6/2

Correltions: b K vs. / Bsed on 0 tight snd core smles, the Smth nd Keighin correltion reltes b K to /: b K 3.85 0.53 b K vs. /3 From: Smth, K., nd Keighin, C.W.

7 Correltions: b K vs. / Bsed on 0 tight snd core smles, the Smth nd Keighin correltion reltes b K to /: b K b K vs. /3 From: Smth, K., nd Keighin, C.W.: Fctors Affecting Gs Slige in Tight Sndstones, er SPE 9872 resented t the 98 SPE/DOE Low Permebility Symosium, Denver, CO., My (98) Discussion: Smth-Keighin The correltion is ccetble. Reltively smll dtbse. Slide 7/2

8 Correltions: b K vs. / The lter formultion is interesting s it is close to the theoreticl definition of cillry rdius r s function of the squre-root of ( /): r Klinenberg defines b K by: b K m 4 r m It is ossible to derive rigorously correltion between b K nd the squre-root of ( /) using the revious definitions nd the clssicl definition of the men free th of the gs molecules: This "squre-root" correltion hs the following form: / RT (, T ) / 2 [ (, T )] M 0.5 Where is rmeter (intercet) deending on the tye of gs used for the core b K Where m is the men free th of the gs molecules t the men ressure m flow exeriment usully nitrogen, for which = b K vs. 2/3 Slide 8/2

9 bk vs. 3/3 Correltions: Smth-Keighin, squre-root The squre-root model ers to give better results. The Smth-Keighin Model mtches minly their dt. Slide 9/2 Discussion: Smth-Keighin

10 Correltions: Comrison on Exmle Cse Errors re generlly less thn 30 ercent. Discussion: Lower Cotton Vlley Formtion No. The Jones-Owens nd Smth-Keighin overestimte the ermebility; the squre-root model underestimtes. Slide 0/2

11 Correltions: Comrison on Exmle Cse Errors re generlly less thn 25 ercent, lthough some outliers exhibit more thn 50 ercent error. Discussion: Lower Cotton Vlley Formtion No. 2 The Jones-Owens/Smth-Keighin correltions for > 0.00 md. The squre-root model gives good results for low ermebilities. Slide /2

12 Microflow model: Concet: The gs slige henomenon occurs s subset of lrger re of study nown s the theory of rrefied gses this theory exerienced substntil growth in the lst 50 yers (eronutics, erosce, MEMS ) The Klinenberg eqution is bsed on the results of this theory vilble in 94: b 4 K c m, with c m r The theory of rrefied gses describes the flow of gs t very low ressures, or in smll micro-chnnels. The flow regime for gs flowing in micro-chnnel is tyiclly determined by the vlue of the Knudsen number (Kn) : Kn l chr Where is the men free th of the gs molecules (i.e., the verge distnce (length) between 2 consecutive moleculr interctions), nd l chr is the chrcteristic length of the flow geometry (e.g., chnnel height, ie rdius). Slide 2/2

13 Microflow model: Concets The clssicl definition of the men free th from thermodynmics is: RT (, T ) / 2, where (, T ) M Limits of the different flow regimes, s function of the length chrcteristic of the geometry, l chr, nd the recirocl men free th normlized t tmosheric conditions nd 300 K. The lines defining the vrious Knudsen number regimes re bsed on ir t isotherml conditions (Modified from Krnidis nd Beso, 2002). Slide 3/2

14 Microflow model: Unified Flow Model: In 2002, Krnidis nd Beso develoed "unified flow model" for gs flows in micro-ies, vlid over the entire rnge of flow regimes: (for micro-ie of rdius r nd length L) r 4 4Kn 28 - q ( Kn) Kn, where( Kn) tn 4Kn 2 8 Kn L From: Krnidis, G.E. nd Beso, A.: Micro-flows, Fundmentls nd Simultion, Sringer-Verlg, New-Yor (2002). By nlogy with Klinenberg's derivtion using Poiseuille eqution (corrected for slige), we derive microflow model: Kn tn 4Kn Kn 2 5 Kn Where [+(Kn) Kn] is defined s the rrefction coefficient, ccounting for the chnge of gs density. Slide 4/2

15 Microflow model: Correltion of the Knudsen Number (Kn): The Knudsen number cnnot be mesured by direct lbortory mesurement we need to define "seudo"-knudsen number which is function of the mesurble rmeters ( m,, ) Assuming the isotherml flow of n idel gs, Kn is inversely roortionl to m nd l chr (tyiclly, the ore throt rdius, or function of nd ) For smle for which nd re nown, the "seudo"- Knudsen number is comuted by solving the following eqution: tn - 4Kn 0.4 4Kn Kn 0 Kn Slide 5/2

16 Microflow model: Aliction using Field Dt: To demonstrte the liction of the "seudo"-knudsen number (Kn ), we only use the Lower Cotton Vlley dt (No. 2): the seudo-knudsen numbers (Kn ) cn be relted to the vilble etrohysicl dt (,, nd ), yielding two reltions: Reltion : Kn 0.4 m Reltion 2: Kn 2.62 These two reltions cn be lied to the microflow model, yielding to n exlicit eqution (with Reltion ) or n imlicit eqution (with Reltion 2) tht cn be solved for : m tn - 4Kn 0.4 4Kn Kn 0 Kn Slide 6/2

17 Microflow model: Field Dt Results (2/2) Discussion: Imlicit nd Exlicit models The two models chieve firly good correltion the imlicit model seems to be more ccurte. Slide 7/2

18 "Fully imlicit" formultion : The Klinenberg-corrected ermebility my not be vilble in rctice. The rocess of solving simultneously for Kn nd requires n imlicit formultion with the form: The coefficients 0, nd 2 hve to be determined simultneously with the estimtion of. This cn be hndled (with sufficient quntity of dt) by using non-liner regression methods on the following eqution: The coefficients ( i ) comuted re not "universl" nd must be comuted for ech dtsets. Microflow model: Field Dt Results 2 0 m Kn / 4 4 tn m m m Slide 8/2

19 Microflow model: Field Dt Results (2/2) Discussion: "Fully imlicit" formultion The fully imlicit formultion chieves good mtch for this dt set. Slide 9/2

20 Summry, Issues, nd Chllenges: Summry: Primry objective: Jones-Owens, Smth-Keighin nd squre-root correltions cn be used Smth-Keighin nd squre-root correltions would be referred s they offer more consistency with the theory. Secondry objective: new theoreticl model (microflow model) for the estimtion of ermebility from gs flow mesurement hs been develoed nd needs to be further vlidted. Issues nd Chllenges: Continued vlidtion of the new microflow model requires lrge quntity of relible dt. Slide 20/2

21 SPE Imroved Permebility Prediction Reltions for Low Permebility Snds End of Presenttion Frncois-Andre Florence, Texs A&M University T.A. Blsingme, Texs A&M University Dertment of Petroleum Engineering Texs A&M University College Sttion, TX Slide 2/2

IPTC November 2005 IPTC (Doha, Qatar)

IPTC November 2005 IPTC (Doha, Qatar) November 005 IPTC (Doh, Qtr A Moifie Purcell/Burine Moel for Estimting Absolute Permebility from Mercury-Injection Cillry Pressure Dt CC Huet, Texs A&M University, JA Rushing, Anrko Petroleum Cor, KE Newshm,