WETTABILITY DETERMINATION OF GEOTHERMAL SYSTEMS. Kewen Li and Roland N. Horne

Transcription

1 PROCEEDINGS, Tety-Seveth Workshop o Geothermal Reservoir Egieerig Staford Uiversity, Staford, Califoria, Jauary 8-30, 00 SGP-TR-7 WETTABILITY DETERMINATION OF GEOTHERMAL SYSTEMS Kee Li ad Rolad N. Hore Staford Geothermal Program, Staford Uiversity Staford, CA keeli@staford.edu hore@staford.edu ABSTRACT A method has bee developed to ifer the ettability of steam-ater-rok systems based o the relatioship betee permeability ad apillary pressure by Purell. The Purell model as exteded to to-phase flo to verify hether the ettability is a futio of ater saturatio. The results alulated usig the experimetal data shoed that the ettability idex i the draiage proess as greater tha that i the imbibitio. The orrespodig otat agle i draiage as smaller tha that i imbibitio, hih is realisti ad osistet ith the existig experimetal results i apillary tubes. Both the reedig (draiage) ad advaig (imbibitio) otat agles i steam-ater-rok systems ere reasoably idepedet of ater saturatio ragig from about 5 to 85%. It as also foud that the ettability model ould be redued to the model reported by Slobod ad Blum (95) i speifi ases. The method developed i this study to estimate ettability of steam-ater-rok systems ould be applied diretly to other gas-liquid-rok ad liquidliquid-rok systems. INTRODUCTION Gas-liquid-rok systems are usually osidered to be strogly liquid-et, hih is true i most atural fluid-rok systems. It is also ofte assumed that the otat agle through the liquid phase is zero i gasliquid-rok systems (Slobod ad Blum, 95), hih may ot be true. Li ad Firoozabadi (000) disussed the pheomea of ozero otat agle i more detail. The ettability i differet gas-liquid-rok (or other porous media) systems may ot be the same. For example, the itrisi otat agle of ater agaist air o smooth PTFE (Teflo) measured by Morro ad MCaffery (978) as 08 o hile the otat agles i most atural gas-liquid-rok systems are muh smaller, as is already ko. Al- Siyabi et al. (997) measured the gas-oil otat agles of four biary mixtures (C /C 4, C /C 8, C /C 0,adC /C 4,) at reservoir oditios. Their results shoed that the gas-oil otat agles ere about 0 o. Li ad Hore (00a) foud sigifiat differees betee steam-ater ad air-ater apillary pressures, ad Hore et al. (000) foud sigifiat differees betee steam-ater ad air-ater relative permeabilities. We therefore speulated that there might be differees of ettability i differet gas-liquid-rok systems suh as steam-ater-rok ad air-ater-rok systems. Slobod ad Blum (95) developed a method to evaluate the ettability of reservoir roks from the threshold apillary pressures measured i oil-aterrok ad air-oil-rok systems respetively. Hoever the semiquatitative method as based o the assumptio that the otat agle through the liquid phase i gas-liquid-rok systems is zero. Hee this method ould ot be suitable for the ettability determiatio i may gas-liquid-rok systems. Purell (949) developed a model to orrelate the rok permeability ad the pore size distributio that ould be iferred from apillary pressure urves measured by the tehique of merury ijetio. Aordig to this relatioship, e developed a quatitative method to alulate the ettability idex or apparet otat agle i steam-ater-rok systems from the data of steam-ater apillary pressure. This method as also exteded to tophase flo to verify hether the ettability is a futio of fluid saturatio. I this ase, both apillary pressure ad relative permeability data are required to alulate the ettability idex at differet values of ater saturatio. Usig the experimetal data of apillary pressure ad relative permeability measured simultaeously i the same rok at the same temperature of about 0 o C, the values of the ettability idex i both the draiage ad the imbibitio proesses i steamater-berea systems ere alulated ad ompared. Some other published experimetal data of oil-ater flo ere also used to odut the alulatio ad the ompariso i order to further ofirm the pheomea observed i steam-ater systems.

2 METHOD The method to determie the ettability of gas-ater rok systems is derived i this setio. Usig Poiseuille's equatio ad Dary's La, Purell (949) derived a relatioship betee the rok permeability ad the apillary pressure urve as follos: ds k = Fφ ( σ osθ ) 0 () P here k ad φ are the absolute permeability ad porosity of the rok; F is the so-alled lithology fator. σ ad θ are the iterfaial tesio betee the to fluids ad the otat agle through the liquid phase; P ad S are the apillary pressure ad the saturatio of the ettig phase. To alulate the rok permeability usig Eq., it is oveiet to have a mathematial represetatio for the apillary pressure urve. Brooks ad Corey (966) suggested a futio to represet apillary pressure urves i porous media as follos: / λ = pe( S) P () here p e is the etry apillary pressure ad λ is the pore size distributio idex; S is the ormalized ettig phase saturatio. For the draiage proess, it is expressed as follos: S S S = S r r (3) here S r is the residual saturatio of the ettig phase, represetig the residual ater saturatio i this study. Although the apillary pressure futio (see Eq. ) suggested by Brooks ad Corey (966) as origially for the draiage ases istead of imbibitio ases, i this study it as used to alulate the ettability idex i the imbibitio by defiig the ormalized ettig phase saturatio as follos: S S Sr = S S r r (4) here S r is the residual saturatio of the oettig phase, represetig the residual steam saturatio i this study. Substitutig Eq. ito Eq. to obtai the rok permeability by usig the apillary pressure data: σ osθ λ k = Fφ( ) (5) λ + P e The ettability idex W i as defied as osθ ad from Eq. 5, it a be alulated as follos: λ + k Pe W i = osθ = ( )( ) (6) λ Fφ σ The oly uko parameter i Eq. 6 is the lithology fator, F, oe the apillary pressure urve is available. Purell (949) measured the values of the lithology fator for umerous rok samples by meas of omparig the air permeability to the permeability alulated usig Eq.. The apillary pressure urves ere measured by the tehique of merury ijetio. The lithology fator raged from 0.08 to 0.36 ad as foud to be smaller i loer permeability roks (Purell, 949). The otat agle through the liquid phase a also be alulated from Eq. 6 oe the apillary pressure urve ad the lithology fator are ko. Note that the otat agle alulated i suh a ay may be differet from that defied i a apillary tube or o a flat solid surfae. Atually, this value may represet the marosopi average otat agle of the fluidrok system. Suh a otat agle may be amed as a apparet otat agle for the sake of oveiee. Assumig that Eq. applies i to-phase flo, it is ritte as follos: S = ds k Fφ ( σ osθ) 0 (7) P here k ad θ are the effetive permeability of the ettig phase ad the otat agle through the ettig phase. Beause F is a parameter represetig lithology, it as assumed i this study that it does ot vary ith the saturatio of the ettig phase. Substitutig Eq. ito Eq. 7, the folloig equatio is obtaied: λ + k P W i = ( )( ) (8) λ FSφ σ here W i is the ettability idex, defied as osθ, at the ettig phase saturatio of S. The effetive permeability of the ettig phase ad the apillary pressure i Eq. 8 are a futio of the ettig phase saturatio. I terms of relative permeability, Eq. 8 a also be expressed as follos:

3 λ + k kr P W i = ( )( )( ) (9) λ Fφ S σ here k r is the relative permeability of the ettig phase. Aordig to Eq. 9, the ettability of gasliquid-rok systems at ay ater saturatio ould be determied oe the experimetal data of the apillary pressure ad the relative permeability are available. We a see from the proess of the derivatio of Eq. 9 that the method ould be used to determie the ettability i either gas-liquid-rok or liquid-liquidrok systems. This is beause there are o speial restritios or assumptios assiged to a speifi system durig the derivatio. It a be see from Eq. 9 that if the ettability of a fluid-rok system does ot hage ith the fluid saturatio, as it is usually assumed, the k ( S r ) P should be ostat. This may be verified by usig the data from the simultaeous measuremets of apillary pressure ad relative permeability urves. The experimetal ofirmatio ill be disussed later i more detail. Aother importat sigifiae of Eq. 9 is that it may be possible to determie the ettability of steamater-rok systems by usig the data from a simple spotaeous ater imbibitio experimet. Beause the relative permeability ad the apillary pressure at a speifi ater saturatio a be alulated simultaeously from oe sigle spotaeous ater imbibitio test aordig to the method developed by Li ad Hore (000a), the ettabiblity idex ould be obtaied usig Eq. 9. Therefore, e may obtai the ettability iformatio ithout measurig the hole apillary pressure urve if the ettability does ot vary ith the ater saturatio. The rok property fators, iludig the pore size distributio idex λ, the permeability k, the porosity φ, ad the lithology fator F, may be redued he Eq. 9 is used to ompare the ettability differees betee to differet fluid pairs suh as air-ater ad steam-ater (i the same rok). The ratio of the ettability idex of fluid pair to fluid pair at the same ater saturatio ad i the same rok a be obtaied from Eq. 9 as follos: i i r r W k P σ = (0) W k P σ i r here W, k, P, ad σ (=, ) are the ettability idex, the relative permeability of the ettig phase, the apillary pressure, ad the iterfaial tesio i fluid pair (=, ) respetively. It is oly eessary to obtai the apillary pressure ad the relative permeability data i order to ompare the ettability i to differet fluid systems. Eq. 0 ould be redued to the form of the Slobod ad Blum (95) model, hih is expressed i Eq., if the relative permeabilities of the ettig phase i the to fluid pairs are equal. i i W P σ W r = = () W P σ here W r is the ratio of the ettability idex of fluid pair to fluid pair. As stated previously, it as assumed that the otat agle through the liquid phase i the gas-liquid-rok system as zero i the W i Slobod ad Blum model (95). Hee (represetig ettability idex of the gas-liquid fluid pair) is equal to oe i this ase. Note that the relative permeabilities of the ettig phase measured usig to differet fluid pairs may ot be alays equal. EXPERIMENTS Experimetal data of relative permeability ad apillary pressure measured simultaeously i the same rok ad at the same temperature have bee rare i the literature, espeially for steam-ater flo. Mahiya (999) ad Li ad Hore (000b) reported suh a set of data for steam-ater flo i Berea sadstoe. These experimetal data of steam-ater relative permeability ad apillary pressure ere used i this study to determie the ettability of suh a geothermal system. The experimetal apparatus test proedures, ad the properties of the rok ad fluid samples, ere desribed by Mahiya (999) ad Li ad Hore (000b). For oveiee, a brief summary of the steam-ater flo tests is give here. Distilled ater asusedastheliquidphaseadtogeeratesteam; the speifi gravity ad visosity ere.0 ad.0 p at 0 o C. The steam properties at high temperatures ere alulated from the steam property softare from Tehare Egieerig Appliatios, I., based o the measured values of pressure ad temperature. The surfae tesio of vapor/ater at 0 o C as 7.75 dyes/m. A Berea sadstoe sample fired at a temperature 450 o C as used; its permeability ad porosity ere 400 md ad 4.8%; the legth ad diameter ere 43. m ad 5.04 m, respetively. RESULTS As stated previously, the method developed i this study to determie the ettability of gas-liquid-rok systems ould also be used i oil-ater-rok systems.

4 To verify the method e developed, the experimetal data of relative permeability ad apillary pressure of oil-ater flo i porous media from Kleppe ad Morse (974) ere used to alulate the ettability idies at differet ater saturatios. We ould also verify if the ettability varies ith the fluid satruatio. The imbibitio oil-ater apillary pressure data from Kleppe ad Morse (974) are sho i Fig. i log-log oordiates to obtai the value of the pore size distributio idex λ. Capillary Pressure (psi) Normalized Water Saturatio (%) Figure : Imbibitio oil-ater apillary pressure data from Kleppe ad Morse (974). The apillary pressure data sho i Fig. ere measured i a Berea sadstoe sample by ater ijetio. The porosity ad permeability of the ore sample ere.5% ad 90 md respetively. The oil phase as kerosee. We a see from Fig. that the relatioship betee the apillary pressure ad the ormalized ater phase saturatio is liear. So the relatioship ould be represeted by Eq. suggested by Brooks ad Corey (966). The value of the pore size distributio idex λ that e obtaied as about.06. This value is reasoable aordig to the study of Brooks ad Corey (966). Beause the value of the lithology fator is uko for this rok sample, both the miimum ad maximum values determied by Purell (949) ere used i the alulatio of the ettability idex. The alulated results may sho hether the effet of the lithology fator o the ettability idex is sigifiat or ot. The values of the lithology fator determied by Purell (949), as previously desribed, raged from 0.08 to 0.36 for the rok samples ith permeability ragig from 3 to 500 md. The permeability of the rok sample used by Kleppe ad Morse (974) as about 90 md hih as ithi this rage. The ettability idies ad the orrespodig otat agles i the oil-ater-rok system studied by Kleppe ad Morse (974) ere alulated usig the method developed i this ork (see Eq. 6) ad the results are listed i Table. Table : Results of Wettability Idex ad Cotat Agle Usig Differet Values of the Lithology Fator F. F (Lithology Fator) W i (Wettability Idex) θ (Cotat Agle) The alulated results i Table demostrate that the effet of the lithology fator o the otat agle is ot very sigifiat i this ase. Therefore the miimum value, 0.08, of the lithology fator determied by Purell (949) as used i the rest of the alulatios i this study. Note that the effet of the lithology fator o the otat agle may be sigifiat at some speifi ases. It a also be see from Table that the oil-aterrok (Berea sadstoe) system is ater-et, hih is osistet ith the atual ettability. The sigifiae of the value of the ettability idex i Table may ot be exatly the same as those of the ettability idex obtaied by the Amott (959) ad the USBM (Doaldso et al., 969) methods. Hoever, it is ertai that the loser the value of the ettability idex to.0, the stroger the ettability of the liquid phase. To alulate the ettability idex usig Eq. 9 ith to-phase flo data, it is eessary to have the orrespodig relative permeability data. The imbibitio oil-ater relative permeability data from Kleppe ad Morse (974) are sho i Fig. Relative Permeability k ro k r Water Saturatio (%) Figure : Imbibitio oil-ater relative permeability data from Kleppe ad Morse (974). Eve though it is usually assumed that the ettability does ot vary ith ater saturatio, there are fe experimetal data available to verify this assumptio due to the sarity of methods to evaluate the ettability at speifi values of ater saturatio. For this purpose, the values of the ettability idex ad the orrespodig otat agle at differet ater saturatios ere alulated usig Eq. 9 ith the

5 measured data of the oil-ater apillary pressure ad the relative permeability from Kleppe ad Morse (974). The alulated results are plotted i Fig. 3. Wettability Idex Model Data Test Data Normalized Water Saturatio (%) Figure 3: Wettability idex alulated usig the data of relative permeability ad apillary pressure from Kleppe ad Morse (974). The "model data" sho i Fig. 3 ere alulated usig the data of the apillary pressure ad the relative permeability from modelig. I this study, the apillary pressure data ere alulated usig Eq. ad the relative permeability data ere omputed usig the Purell model, hih is expressed as follos: +λ ( S ) λ k = () r The reaso e used the Purell model as that Li ad Hore (00b) foud it the best fit to the ettig phase relative permeability. It a be see from Fig. 3 that the ettability idex does ot vary sigifiatly ith the ater saturatio ad is very lose to the model data. The ettability idex alulated usig the modelig apillary pressure ad the relative permeability data is equal to that alulated usig Eq. 6 (i hih oly apillary pressure data ere used). The alulated results of the orrespodig otat agle are sho i Fig. 4. Aordigly, the otat agles alulated at differet ater saturatio are almost ostat ad lose to the model results. This implies that e may be able to determie the ettability idex or the apparet otat agle usig the relative permeability ad the apillary pressure at oly oe poit of speifi ater saturatio istead of over the hole rage of the urve. This is iterestig beause Li ad Hore (000a) reported that the relative permeability of the ettig phase ad the apillary pressure at speifi ater saturatio ould be alulated simultaeously from spotaeous ater imbibitio tests i gas-saturated roks. Cotat Agle Model Data Test Data Normalized Water Saturatio (%) Figure 4: Apparet otat agle alulated usig the data of relative permeability ad apillary pressure from Kleppe ad Morse (974). No e ill disuss the alulatios of the ettability idex i steam-ater-rok systems. The steam-ater apillary pressure urves from Li ad Hore (000b) ere used. Both the draiage ad the imbibitio apillary pressure urves are plotted i Fig. 5 i log-log oordiates to obtai the values of the pore size distributio idex λ for the alulatios usig Eq. 6 or Eq. 9. These data ere measured i a fired Berea sadstoe sample at a temperature of about 0 o C usig a steady-state flo method. The values of λ i the draiage ad the imbibitio ases ere ad 0.75 respetively. Capillary Pressure (psi) Draiage Imbibitio 0 00 Normalized Water Saturatio (%) Figure 5: Draiage ad imbibitio steam-ater apillary pressure data from Li ad Hore (000b). The experimetal data of the steam-ater relative permeability from Mahiya (999) are plotted i Fig. 6. These data ere measured simultaeously alog ith the apillary pressure sho i Fig. 5. The relative permeabilities of the ater phase i the draiage ad the imbibitio proesses are losely similar. The to solid lies i Fig. 6 are regressio urves for the draiage proess. Note that the steam relative permeability data sho i Fig. 6 have bee alibrated uder the osideratio of gas slip effet

6 (Klikeberg Effet) i to-phase flo measured by Li ad Hore (00). Relative Permeability Water Saturatio (%) k ro _draiage k r _draiage k ro _imbibitio k r _imbibitio Figure 6: Steam-ater relative permeability data from Mahiya (999). Fig. 7 shos the values of the ettability idex alulated usig Eq. 9 ith the model data of the steam-ater apillary pressure ad the relative permeability represetig the experimetal data from Li ad Hore (000b) ad Mahiya (999). The results alulated diretly usig the measured data are ot plotted i Fig. 7 beause several values of the ettability idex are greater tha.0, hih is ot reasoable. The reaso may be due to the satterig i the steam-ater apillary pressure data. It takes a log time to odut steady-state steam-ater flo tests i porous media ad it is very diffiulty to keep the steam-ater flo i ostat adiabati eviromet for suh a log time. Wettability Idex Draiage Imbibitio Normalized Water Saturatio (%) Figure 7: Wettability idies alulated usig the experimetal data from Li ad Hore (000b) ad Mahiya (999). We a see from Fig. 7 that the ettability idex i the imbibitio proess is less tha that i the draiage, hih is theoretially orret ad has bee prove experimetally by Morro ad MCaffery (978). The ratio of the ettability idex i the imbibitio proess to that i the draiage is about The values of the orrespodig apparet otat agle ere alulated usig the ettability idex data from Fig. 7 ad the results are sho i Fig. 8. Cotat Agle Draiage Imbibitio Normalized Water Saturatio (%) Figure 8: Apparet otat agle alulated usig the experimetal data from Li ad Hore (000b) ad Mahiya (999). The apparet otat agle related to the draiage proess is usually referred to as the reedig otat agle ad that related to the imbibitio proess as the advaig otat agle. As expeted, Fig. 8 shos that the reedig otat agle is smaller tha the advaig otat agle. Comparig the values of the ettability idex i Fig. 7 to those i Fig. 3, e foud that the ettability idies (W i =0.96 for draiage ad 0.56 for imbibitio) i steam-ater-rok systems are muh greater tha that (W i =0.) i oil-ater-rok systems as expeted. This observatio demostrates that the method developed i this study to determie the ettability of steam-ater-rok systems ould be useful. A importat feature of this method is that it is ot oly appropriate for liquid-liquid-rok systems but also for gas-liquid-rok systems. DISCUSSION Usig the method developed here, e demostrated that the steam-ater-rok system is muh more ater-et tha the oil-ater-rok system, as expeted. The apparet otat agle i the draiage proess is muh smaller tha that i imbibitio, hih is reasoable ad osistet ith the existig experimetal results. Oe fudametal oer i the study of geothermal fluid flo is the ettability differee betee the steam-ater-rok ad airater-rok systems. It is possible to determie the ettability differee betee the to systems oe the orrespodig experimetal data are available. These data ilude the apillary pressure of both the steam-ater ad the air-ater flo i the same rok

7 ad at the same temperature. The experimetal data of relative permeability i both steam-ater ad airater flo ould also be useful. Ufortuately suh data have bee very fe. The sigifiae of the method that e developed to determie the ettability is the ability to ompare the ettability differee betee ay to systems, either gas-liquid-rok or liquid-liquid-roks. CONCLUSIONS Based o the preset study, the folloig olusios may be dra:. A method as developed to determie the ettability of steam-ater-rok systems usig the data from experimetal measuremets of steam-ater apillary pressure ad relative permeability.. The method is appropriate to determie the ettability of the gas-liquid-rok systems as ell as the liquid-liquid-rok systems. 3. It is possible to determie the ettability at ay speifi ater saturatio by usig the method developed i this study although e verified that the ettability ould ot hage ith ater saturatio i the ases studied. 4. The ettability idex alulated usig the method i the imbibitio ase is smaller tha that i the draiage, hih has bee prove experimetally. 5. The values of the ettability idex i the gasliquid-rok systems are muh greater tha that i oil-ater-rok systems, hih is reasoable ad osistet ith existig observatios. ACKNOWLEDGEMENTS This researh as oduted ith fiaial support to the Staford Geothermal Program from the Geothermal ad Wid divisio of the US Departmet of Eergy uder grat DE-FG07-99ID3763, the otributio of hih is gratefully akoledged. NOMENCLATURE F = lithology fator k = absolute permeability k = effetive permeability of the ettig phase k r = relative permeability of the ettig phase k r = relative permeability of the ettig phase i fluid pair (=, ) P = apillary pressure P = apillary pressure i fluid pair (=, ) P e = etry apillary pressure S r = residual saturatio of the oettig phase S = ater saturatio S r = residual ater saturatio S = ormalized ater saturatio W i = ettability idex W i = ettability idex at the ettig phase saturatio of S W i = ettability idex i fluid pair (=, ) W r = ratio of the ettability idex of fluid pair to fluid pair φ = porosity λ = pore size distributio idex θ = apparet otat agle through liquid phase θ = apparet otat agle through ater phase at the ater saturatio of S σ = iterfaial tesio σ = iterfaial tesio i fluid pair (=, ) REFERENCES Al-Siyabi, Z.K., Daesh, A., Tohidi, B., ad Todd, A.C.: Measuremet of Gas-Oil Cotat Agle at Reservoir Coditios, Proeedigs of 997 Europea Symposium o Improved Oil Reovery, The Hague, The Netherlads, Ot. 0-. Amott, E.: "Observatios Relatig to the Wettability of Porous Rok," Tras. AIME, (959), 6, 56. Brooks, R. H. ad Corey, A. T.: "Properties of Porous Media Affetig Fluid Flo", J. Irrig. Drai. Div., (966), 6, 6. Doaldso, E.C., Thomas, R.D., ad Lorez, P.B.: "Wettability Determiatio ad Its Effet o Reovery Effiiey," SPEJ (Marh 969), 3-0. Hore, R.N., Satik, C., Mahiya, G., Li, K., Ambusso, W., Tovar, R., Wag, C., ad Nassori, H.: Steam- Water Relative Permeability, preseted at World Geothermal Cogress, Kyushu-Tohoku, Japa, May 8-Jue 0, 000. Kleppe, J. ad Morse, R. A.: "Oil Produtio from Fratured Reservoirs by Water Displaemet," SPE 5084, preseted at the 974 SPE Aual Tehial Coferee ad Exhibitio, Housto, TX, USA, Otober 6-9, 974. Li, K. ad Firoozabadi, A.: Experimetal Study of Wettability Alteratio to Preferetial Gas-Wetess i Porous Media ad its Effet, SPEREE (April 000), Li, K. ad Hore, R. N. (000a): Charaterizatio of Spotaeous Water Imbibitio ito Gas-Saturated Roks, SPE 655, preseted at the 000

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