Theoretical Development of the Brooks-Corey Capillary Pressure Model from Fractal Modeling of Porous Media Kewen Li, SPE, Stanford University

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1 SPE Theoretial Development of the Brooks-Corey Capillary Pressure Model from Fratal Modeling of Porous Media Kewen Li, SPE, Stanford University Copyright 2004, Soiety of Petroleum Engineers In. This paper was prepared for presentation at the 2004 SPE/DOE Fourteenth Symposium on Improved Oil Reovery held in Tulsa, Oklahoma, U.S.A., April This paper was seleted for presentation by an SPE Program Committee following review of information ontained in a proposal submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Soiety of Petroleum Engineers and are subjet to orretion by the author(s). The material, as presented, does not neessarily reflet any position of the Soiety of Petroleum Engineers, its offiers, or members. Papers presented at SPE meetings are subjet to publiation review by Editorial Committees of the Soiety of Petroleum Engineers. Eletroni reprodution, distribution, or storage of any part of this paper for ommerial purposes without the written onsent of the Soiety of Petroleum Engineers is prohibited. Permission to reprodue in print is restrited to a proposal of not more than 300 words; illustrations may not be opied. The proposal must ontain onspiuous aknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O. Box , Rihardson, TX , U.S.A., fax Abstrat The apillary pressure model proposed empirially by Brooks and Corey has been used widely for several deades. However it is not lear why the Brooks-Corey apillary pressure model works so well. In this study, it has been found that the empirial Brooks-Corey apillary pressure model an be derived theoretially from fratal modeling of porous media. Also found was the orrelation between the pore size distribution index in the Brooks-Corey apillary pressure model and the fratal dimension. The pore size distribution index inreases with the derease in fratal dimension of the porous media. Capillary pressure urves of different types of rok samples were measured using a merury intrusion tehnique. The values of pore size distribution index and fratal dimension were alulated. The relationship between the two parameters obtained from the experimental data was onsistent with the relationship derived theoretially. This implies that the fratal dimension of porous media may be inferred diretly using the Brooks-Corey apillary pressure model instead of the fratal model. The theoretial development in this study demonstrates that the Brooks-Corey apillary pressure model, one onsidered as empirial, has a solid theoretial base. This may be why the Brooks-Corey apillary pressure model works satisfatorily in many ases. Introdution Capillary pressure plays an important role in many reovery proesses. It is essential to represent apillary pressure urves properly. The frequently-used model to express a apillary pressure urve mathematially is the Brooks-Corey apillary pressure model 1. Brooks and Corey 1 onduted analysis for the apillary pressure urves of a large number of onsolidated ore samples. The apillary pressure urves were measured using a desorption approah. Brooks and Corey 1 found that the relationship between the apillary pressure and the normalized or effetive wetting-phase saturation was a straight line on a log-log plot. The mathematial expression of this relationship was known as the Brooks-Corey apillary pressure model later. Residual wetting-phase saturation must be known or assumed to alulate the normalized or effetive wetting-phase saturation. In the analysis by Brooks and Corey 1, the residual wetting-phase saturation was hosen suh that the data fit as losely as possible to a straight line when plotted on log-log paper. The Brooks-Corey apillary pressure model works satisfatorily in many ases and has been utilized widely for several deades in petroleum and other industries 2-7. However it is not lear why the Brooks-Corey apillary pressure model works so well. Note that the apillary pressure model was proposed empirially by Brooks and Corey 1. Many researhers 8-20 have studied the fratal nature of reservoir roks and other porous media in the past two deades. It has been found that most natural porous media suh as reservoir rok are fratals and an be haraterized using a fratal model or a fratal urve whih represents the relationship between the number of pores and the radius of pores. Suh a fratal urve is a straight line on a log-log plot and the slope of the straight line is referred to as the fratal dimension of the porous media. The magnitude of fratal dimension is a representation of the heterogeneity of the porous medium. The greater the fratal dimension, the greater the heterogeneity of the porous media. Note that the pore size distribution index in the Brooks-Corey apillary pressure model is also a representation of the heterogeneity of porous media. The greater the pore size distribution index, the more homogeneous the porous medium. Attention has also been paid to the appliation of fratal modeling of porous media in reservoir engineering. The appliations inlude the development of relative permeability models, apillary pressure models, and the models to predit oil prodution rate, et. The author reviewed the literature in this area in a previous paper 18. The review shows that the fratal modeling of porous media is a powerful tool to haraterize the heterogeneity of porous media and to study fluid flow mehanisms. In this study, we onduted a theoretial development based on the fratal geometry to derive the Brooks-Corey apillary pressure model. Capillary pressure urves of Berea, halk, and reservoir sandstone were measured using a merury intrusion tehnique to infer the fratal dimension. The values of fratal dimension were alulated using the fratal model

2 2 SPE and the Brooks-Corey apillary pressure model respetively and the results were ompared. Methodology In this setion, the Brooks-Corey apillary pressure model is derived theoretially using the results from fratal modeling. Previously the author 18 obtained a relationship between the derivative of merury (nonwetting phase) saturation and the apillary pressure using the theory of fratal geometry. The relationship is expressed as follows: ds dp Hg ( 3 D = ap (1) where a is a onstant, S Hg is the merury (nonwetting phase) saturation, P is the apillary pressure, and D f is the fratal dimension of the porous media. As pointed out by the author 18, if a three-dimensional pore model, instead of a two-dimensional apillary tube model, were used to alulate the number of pores in porous media, the following equation an be obtained: ds dp Hg ( 4 D = ap (2) Eq. 2 was also derived by Friesen and Mikula 9 using a different approah. Eq. 2 an be represented as a more general form in terms of the nonwetting phase saturation: ds dp nw ( 4 D = ap (3) where S nw is the saturation of the nonwetting phase. Assuming that P approahes to p e when S nw = 0, integrate Eq. 3: 1 Sw P pe (4 D = 0 dsnw a P dp (4) where S w is the saturation of the wetting phase and p e is the entry apillary pressure. Aording to Eq. 4, one an obtain: ( 3 D (3 D e 1 S = b[ P p ] (5) w where b is another onstant. Assuming that P approahes infinity when S nw = 1- S wr, one an obtain aording to Eq. 5: (3 D ) 1 = f S wr bp e (6) where S wr is the residual saturation of the wetting phase. It is assumed that fratal dimension D f is less than 3 in deriving Eq. 6 from Eq. 5. Combining Eqs. 5 and 6: S 1 S P = 1 ( p 1 w (3 D wr Reduing Eq. 7, one an obtain: * 1 * = ( ) λ pe Sw e ) (7) P (8) here S w is the normalized saturation of the wetting phase and is expressed as follows: S * w Sw S = 1 S wr wr and λ = 3 - D f. Eq. 8 is the frequently-used Brooks-Corey apillary pressure model proposed empirially by Brooks and Corey 1 in One an see from the previous derivation of Eq. 8 that the Brooks-Corey apillary pressure model has a solid theoretial base. This may explain why this model has been found to be suitable for many types of rok, inluding reservoir and artifiial ore samples. The theoretial model showed that the pore size distribution index inreases with the derease in fratal dimension (λ = 3 - D. This is reasonable beause porous media with greater heterogeneity have smaller values of pore size distribution index. Note that the assumptions to derive Eq. 8 are: (1) fratal dimension D f is less than 3; (2) P approahes infinity when S nw = 1- S wr. The two assumptions are also the onstraints to use the Brooks-Corey apillary pressure model. Eq. 8 foresees that the relationship between the apillary pressure and the normalized saturation of the wetting phase is linear on a log-log plot. This is true in many ases. However the aurate estimation of the residual wetting-phase saturation (S wr ) is important to obtain suh a straight line from the apillary pressure data measured by a merury intrusion approah. Overestimation and underestimation of the residual wetting-phase saturation may hange the linear relationship on a log-log plot. This will be disussed later in more detail. Experimental Measurements Capillary pressure urves of different rok samples (Berea sandstone, halk, and sandstone from an oil field) were measured using a merury intrusion approah to obtain the fratal dimension. The porosity of the Berea sandstone sample was about 23.0% and the air permeability was about 804 md. The Berea sandstone sample used in this study was the same as that used by Li and Horne 17. The porosity of the halk sample was about (9)

3 SPE % and the air permeability was about 0.17 md. The porosity of the reservoir sandstone sample was about 27.1% and the air permeability was about 2131 md. The surfae tension of air/merury is 480 mn/m and the ontat angle through the merury phase is 140 o aording to the results reported by Purell 21. Results Fratal dimension of porous media an be inferred from the apillary pressure urves using two methods (Eqs. 3 and 8) respetively. The values of fratal dimension alulated using Eq. 3 should be equal to those alulated using Eq. 8. To verify this, both theoretial and experimental apillary pressure urves were used in this study. The results are presented and analyzed in this setion. Also disussed is the effet of the estimated residual wetting-phase saturation on the shape of the normalized apillary pressure urves. Theoretial apillary pressure urves were alulated using different values of pore size distribution index, λ, to demonstrate the relationship between the fratal dimension and λ. Fig. 1 shows the apillary pressure urves alulated using the Brooks-Corey model (Eq. 8) with different values of λ. For simpliity, the residual wetting-phase saturation was fixed at 20% and the entry apillary pressure was fixed at 0.4 atm for all of the apillary pressure urves shown in Fig. 1. The values of λ ranged from 0.3 to 1.9. Fratal dimension an be inferred from the values of λ (D f = 3 - λ). The normalized wetting-phase saturations were alulated aording to Eq. 9 and the relationships between apillary pressure and the normalized wetting-phase saturation were shown in Fig. 2. All of the relationships are straight lines as expeted. Aording to Eq. 3, the fratal dimension an be alulated one the relationship between apillary pressure and nonwetting-phase saturation gradient to the apillary pressure (ds nw /dp ) is known. The values of ds nw /dp were alulated using the apillary pressure data shown in Fig. 1. The urves representing the relationship between ds nw /dp and apillary pressure are shown in Fig. 3. All of the orrelations are linear as foreseen in Eq. 3. The values of fratal dimension were alulated using Eq. 3 and were also inferred from the values of λ (D f = 3 - λ). The results were ompared and are listed in Table 1. One an see from Table 1 that the values of the fratal dimension inferred from the relationship between ds nw /dp and apillary pressure (using Eq. 3) are very lose to those alulated from the value of λ (D f = 3 - λ). To further demonstrate the relationship between the fratal dimension and the pore size distribution index, experimental apillary pressure urves were used. Fig. 4 shows the apillary pressure urves measured by using a merury intrusion tehnique for different rok samples (Berea sandstone, halk, and reservoir sandstone). The apillary pressure urves are different beause the rok type and properties are different. Fig. 5 shows the orresponding normalized apillary pressure urves. Straight lines were obtained on a log-log plot. The values of the pore size distribution index were then alulated using Eq. 8 and the results are listed in Table 2. The values of fratal dimension were also determined from the values of λ (D f = 3 - λ). Fratal dimension an also be omputed using the fratal model (Eq. 3) from the relationship between ds nw /dp and apillary pressure. Fig. 6 shows the orrelations between ds nw /dp and apillary pressure of the three different rok samples. The values of the fratal dimension alulated using the data shown in Fig. 6 are listed in Table 2 and were ompared with those inferred from the values of λ. One an see from Table 2 that the values of the fratal dimension alulated using the two methods are almost the same for the ore samples studied. The previous alulations based on both the theoretial and experimental apillary pressure urves demonstrate that the fratal dimension may be inferred diretly using the Brooks- Corey apillary pressure model instead of using the relationship between ds nw /dp and apillary pressure in many ases. One advantage to determine the fratal dimension from the Brooks-Corey apillary pressure model is that there is no need to alulate ds nw /dp. As mentioned previously, auray in estimating the residual wetting-phase saturation is essential to obtain the linear relationship between the apillary pressure and the normalized wetting-phase saturation. Fig. 7 shows the effet of overestimation and underestimation of the residual wettingphase saturation on the normalized apillary pressure urves for λ=1.9. The true value of the residual wetting-phase saturation was 20% in Fig. 7. The values of the normalized wetting-phase saturation were alulated using Eq. 9 with different values of the residual wetting-phase saturation (see the numbers lose to eah urve in Fig. 7). One an see that the normalized apillary pressure urve is linear only when the estimated value of the residual wetting-phase saturation is equal to the true value (20%). The normalized apillary pressure urve is onave to the axis of the normalized wetting-phase saturation if the residual wetting-phase saturation is overestimated. Otherwise the normalized apillary pressure urve is onvex to the axis of the normalized wetting-phase saturation. One an see from Fig. 7 that the effet of the residual wetting-phase saturation on the shape of the normalized apillary pressure urves is signifiant. Fig. 8 also shows the effet of the estimated residual wetting-phase saturation on the shape of the normalized apillary pressure urves for λ=0.3. The phenomenon is similar to that shown in Fig. 7. However the effet of the estimated residual wetting-phase saturation on the normalized apillary pressure urves is more signifiant in the ase with smaller value of λ than in the ase with greater value of λ. Aording to the results shown in Figs. 7 and 8, one an see that it is important to estimate the residual wetting-phase saturation aurately. Usually the true value of the residual wetting-phase saturation is smaller than the wetting-phase saturation at the maximum intrusion pressure during the merury intrusion test. Brooks and Corey 1 used the trial and error tehnique to hoose the residual wetting-phase saturation. In this study, the solver funtion of Mirosoft Exel was used to estimate the residual wetting-phase saturation. Note that if the normalized apillary pressure urve is onave to the axis of the normalized wetting-phase saturation, it does not always imply that the residual wetting-

4 4 SPE phase saturation is overestimated. For example, Li and Horne 17 found that the apillary pressure urves of the rok from The Geysers geothermal field are onave to the axis of the normalized wetting-phase saturation. This is not brought about by the overestimation of the residual wetting-phase saturation beause the normalized apillary pressure urves are still onave to the axis of the normalized wetting-phase saturation even if the residual wetting-phase saturation is set to zero. Disussion Although a theoretial basis of the Brooks-Corey apillary pressure model has been found, it does not imply that the model an apply in all ases without onstraints. As mentioned previously, there are two known onstraints to use the Brooks-Corey apillary pressure model. One of the onstraints is that P approahes infinity when S nw = 1- S wr. This may not be true in the ase of imbibition apillary pressure urve. Sinnokrot 22 reported suh an example. Sinnokrot 22 measured the oil-water apillary pressures of different roks (limestones and sandstones) at different temperatures and found that the Brooks-Corey apillary pressure model ould model the drainage oil-water apillary pressure urves but not the imbibition ones. For this reason, Li and Horne 23 proposed an empirial apillary pressure model for the imbibition ase. Another onstraint to apply the Brooks-Corey apillary pressure model is that the fratal dimension D f is less than 3 (less than 2 if the apillary tube model is used in the fratal modeling of the porous media). However the fratal dimension of the porous media with great heterogeneity may be greater than this value. For example, Li and Horne 17 found that the apillary pressure urves of the rok (with high density mirofratures) from The Geysers geothermal field ould not be represented using the Brooks-Corey apillary pressure model (although it ould still be represented by a fratal model). Conlusions Based on the present work, the following onlusions may be drawn: 1. The Brooks-Corey apillary pressure model an be derived theoretially based on fratal modeling of porous media. This shows that the Brooks-Corey apillary pressure model, one onsidered as empirial, has a solid theoretial basis. 2. The pore size distribution index (λ) in the Brooks-Corey apillary pressure model inreases with the derease in fratal dimension (D of the porous medium. 3. The fratal dimension of a porous medium may be inferred diretly using the Brooks-Corey apillary pressure model instead of the fratal model in many ases. 4. Underestimation of the residual wetting-phase saturation may hange the shape of the normalized apillary pressure urve from linear to nonlinear (onvex to the axis of the normalized wetting-phase saturation) on a loglog plot. 5. Overestimation of the residual wetting-phase saturation may also hange the shape of the normalized apillary pressure urve from linear to nonlinear (onave to the axis of the normalized wetting-phase saturation) on a loglog plot. Aknowledgements This researh was onduted with finanial support from the US Department of Energy under grant DE-FG07-02ID14418, the ontribution of whih is gratefully aknowledged. Nomenlature a = onstant b = onstant D f = fratal dimension P = apillary pressure p e = entry apillary pressure S Hg = merury saturation S nw = nonwetting phase saturation S w = wetting phase saturation * S w = normalized wetting-phase saturation S wr = residual saturation of the wetting phase λ = pore size distribution index Referenes 1. Brooks, R.H. and Corey, A.T.: Hydrauli Properties of Porous Media, Colorado State University, Hydro paper No.5 (1964). 2. Erkal, A. and Numbere, D.T.: Relative Permeability Effets on the Migration of Steamflood Saturation Fronts, SPE 38299, presented at the 1997 SPE Western Regional Meeting, Long Beah, California, June 25-27, Siddiqui, F. I.: Defining Fluid Distribution and Fluid Contats for Dynamially Charged Reservoirs, SPE 56513, presented at the 1999 SPE Annual Tehnial Conferene and Exhibition held in Houston, Texas, Otober Mikes, D., Barzandji, O.H.M., Bruining, J., and Geel, C.R.: Upsaling of Flow Units for Reservoir Flow Inorporating Small-Sale Heterogeneities, SPE 68702, presented at the SPE Asia Paifi Oil and Gas Conferene and Exhibition held in Jakarta, Indonesia, April Kjosavik, A., Ringen, J.K., and Skjaeveland, S.M.: Relative Permeability Correlation for Mixed-Wet Reservoirs, SPEJ (Marh 2002), Li, K. and Horne, R.N.: Experimental Verifiation of Methods to Calulate Relative Permeability Using Capillary Pressure Data, SPE 76757, Proeedings of the 2002 SPE Western Region Meeting/AAPG Paifi Setion Joint Meeting held in Anhorage, Alaska, May 20-22, Papatzaos, P. and Skjæveland, S.M.: Relative Permeability from Capillary Pressure, SPE 77540, presented at the 2002 SPE Annual Tehnial Conferene and Exhibition, San Antonio, TX, USA, September 29 to Otober 02, Katz, A.J. and Thompson, A.H.: Fratal Sandstone Pores: Impliations for Condutivity and Pore Formation, Phys. Rev. Lett., 54, (1985).

5 SPE Friesen, W.I. and Mikula, R.J.: Fratal Dimensions of Coal Partiles, J. of Colloid and Interfae Siene, 120(1), (November 1987). 10. Hansen, J.P. and Skjeltorp, A.T.: Fratal Pore Spae and Rok Permeability Impliations, Phys. Rev. B, 38(4), (1988). 11. Krohn, C.E.: Fratal Measurements of Sandstones, Shales, and Carbonates, J. of Geophysial Researh, 93(B4), (1988). 12. Lenormand, R.: Gravity-Assisted Inert Gas Injetion: Miromodel Experiments and Model Based on Fratal Roughness, The European Oil and Gas Conferene, Altavilla Milia, Palermo, Siily, Otober 9-12, Angulo, R.F. and Gonzalez, H.: Fratal Dimensions from Merury Intrusion Capillary Tests, SPE 23695, presented at the 2 nd Latin Amerian Petroleum Engineering Conf. of SPE held in Caraas, Venezuela, Marh 8-11, Toledo, P.G., Novy, R.A., Davis, H.T., and Sriven, L.E.: Capillary Pressure, Water Relative Permeability, Eletrial Condutivity and Capillary Dispersion Coeffiient of Fratal Porous Media at Low Wetting Phase Saturations SPE Advaned Tehnology Series (1993), 2(1), Shen, P. and Li, K.: A New Method for Determining the Fratal Dimension of Pore Strutures and Its Appliation, Proeedings of the 10 th Offshore South East Asia Conferene, Singapore, Deember 6-9, Shen, P. and Li, K.: Quantitative Desription for the Heterogeneity of Pore Struture by Using Merury Capillary Pressure Curves, SPE 29996, Proeedings of the SPE International Meeting held in Beijing, China, November 14-17, Li, K. and Horne, R.N.: Fratal Charaterization of The Geysers Rok, presented at the Geothermal Resoure Counil 2003 annual meeting, Otober 12-15, 2003, Morelia, Mexio; GRC Trans. 27 (2003). 18. Li, K.: Charaterization of Rok Heterogeneity Using Fratal Geometry, SPE 86975, Proeedings of the 2004 SPE Western Region Meeting, Bakersfield, CA, USA, Marh 16-18, Abdassah, D., Permadi, P., and Sumantri, R.: Saturation Exponents Derived from Fratal Modeling of Thin Setions, SPE 36978, presented at the 1996 SPE Asia Paifi Oil and Gas Conferene held in Adelaide, Australia, Otober , Moulu, J-C., Vizika, O., Kalaydjian, F., and Duquerroix, J-P.: A New Model for Three-Phase Relative Permeabilities Based on a Fratal Representation of the Porous Media, SPE 38891, presented at the SPE Annual Tehnial Conferene and Exhibition, San Antonio, TX, USA, Otober 5-8, Purell, W.R.: "Capillary Pressures-Their Measurement Using Merury and the Calulation of Permeability", Trans. AIME, (1949), 186, Sinnokrot, A.A: The Effet of Temperature on Oil-Water Capillary Pressure Curves of Limestones and Sandstones, Ph.D. dissertation, Stanford University, Stanford, CA, USA (1969). 23. Li, K. and Horne, R.N.: An Experimental and Theoretial Study of Steam-Water Capillary Pressure, SPEREE (Deember 2001), p Table 1: Fratal dimension alulated using different methods. λ λ D f Eq Table 2: Fratal dimension and pore size distribution index alulated for different rok samples. rok Berea Chalk Reservoir rok λ D f 3-λ Eq E+07 1.E+06 1.E+05 1.E+04 λ =0.3 λ =0.5 λ =0.7 λ =1.1 λ = Wetting-Phase Saturation, fration Fig. 1: Capillary pressure urves with different values of pore size distribution index. 1.E+07 1.E+06 1.E+05 1.E+04 λ =0.3 λ =0.5 λ =0.7 λ =1.1 λ = Normalized Wetting-Phase Saturation, fration Fig. 2: Normalized theoretial apillary pressure urves with different values of pore size distribution index.

6 6 SPE ds nw /dp, 1/atm λ =0.3 λ =0.5 λ =0.7 λ =1.1 λ =1.9 ds nw /dp, 1/atm Berea Chalk Reservoir 1.E-03 1.E-04 Fig. 3: Relationship between ds nw /dp and apillary pressure for different values of pore size distribution index. 1.E Fig. 6: Relationship between ds nw /dp and apillary pressure for different roks. Berea Chalk Reservoir Wetting-Phase Saturation, fration Fig. 4: Capillary pressure urves of different roks (Berea sandstone, halk, and reservoir rok). Berea Chalk Reservoir Normalized Wetting-Phase Saturation, fration Fig. 5: Normalized apillary pressure urves of different roks (Berea sandstone, halk, and reservoir rok) Normalized Wetting-Phase Saturation, fration Fig. 7: Effet of residual wetting-phase saturation on the estimation of normalized apillary pressure urves (λ=1.9). 1.E+07 1.E+06 1.E+05 1.E Normalized Wetting-Phase Saturation, fration Fig. 8: Effet of residual wetting-phase saturation on the estimation of normalized apillary pressure urves (λ=0.3).