TEAM-WATER RELATIVE PERMEABILITY BY THE CAPILLARY PREURE METHOD Keen Li and Roland N. Horne Department of Petroleum Engineering, tanford University Abstract Various capillary pressure techniques such as the Purcell, Burdine, Corey, and Broos- Corey methods ere utilized to calculate steam-ater relative permeabilities using the measured steam-ater capillary pressure data in both drainage and imbibition processes. The calculated results ere compared to the experimental data of steam-ater relative permeability measured in Berea sandstone. The steam-ater relative permeability and capillary pressure ere measured simultaneously. The differences beteen the Purcell model and the measured values ere almost negligible for ater phase relative permeability in both drainage and imbibition but not for the steam phase. The insignificance of the effect of tortuosity factor on the etting phase as revealed in this case. The steam phase relative permeabilities calculated by other models ere very close to the experimental values for drainage but very different for imbibition as expected. The same calculation as made for the nitrogen-ater flo to confirm the observation in the steam-ater flo. The results in this study shoed that it ould be possible and useful to calculate steam-ater relative permeability using the capillary pressure method, especially for the drainage case. Introduction team-ater relative permeability plays an important role in controlling reservoir performance for steam injection into oil reservoirs and ater injection into geothermal reservoirs here steam-ater flo exists. Hoever, it is difficult to measure steam-ater relative permeability because of the phase transformation and the significant mass transfer beteen the to phases as pressure changes. On the other hand, Li and Horne found significant differences beteen steam-ater and air-ater capillary pressures, and Horne et al. found significant differences beteen steam-ater and air-ater relative permeabilities. Therefore, steam-ater flo properties may not be simply replaced by air (or nitrogen-ater flo properties. It ould be helpful for reservoir engineers to be able to calculate steam-ater relative permeability once steam-ater capillary pressure is available. There are a lot of papers 3-3 related to the capillary pressure method for the calculation of oil-gas relative permeabilities. Honarpour et al. 4 revieed the literature in this field. The published literature and experimental data for relative permeability and capillary pressure ere not sufficient to conclude hich method should be the standard one. Unlie for oil-gas flo properties, there are fe studies for the calculation of steam-ater relative permeabilities by the capillary pressure technique. Historically, the capillary
pressure techniques ere developed for drainage situations and ere useful to obtain gasliquid (oil or ater relative permeability hen fluid flo tests ere not practical. As stated previously, it is difficult to measure steam-ater relative permeability. Therefore, e calculated the steam-ater relative permeability by different capillary pressure techniques and compared to the measured data in the same core sample. The steam-ater capillary pressure data used to compute steam-ater relative permeability ere measured by Li and Horne 5 at a temperature of about 0 o C in Berea sandstone. The experimental data of steam-ater relative permeability used to compare ith the calculated data ere measured by Mahiya 6 and Horne et al. in the same core sample and at the same temperature. Bacground We chose four representative models developed by various authors 3,7,8,0 to calculate steam-ater relative permeabilities using the capillary pressure techniques. The mathematical expressions of the four models are described briefly in this section. Purcell Model. Purcell 3 developed an equation to compute roc permeability by using capillary pressure data. This equation can be readily extended to the calculation of multiphase relative permeability. In to-phase flo, the relative permeability of the etting phase can be calculated as follos: r ³ 0 d /( Pc ³ 0 d /( Pc ( here r and are the relative permeability and saturation of the etting phase; P c is the capillary pressure as a function of. imilarly, the relative permeability of the nonetting phase can be calculated as follos: rn ³ d /( Pc ³ 0d /( Pc ( here rn is the relative permeability of the nonetting phase. It can be seen from Eqs. and that the sum of the etting and nonetting phase relative permeability at a specific saturation is equal to one. This may not be true in most porous media. In the next section, the relative permeabilities calculated using this method are compared to the experimental data. The comparison shos that Eq. is close to experimental values of the etting phase relative permeability but Eq. is far from the experimental results. Burdine Model. Burdine 7 developed equations similar to Purcell s method by introducing a tortuosity factor as a function of etting phase saturation in the calculation of relative permeability by the capillary pressure method. The relative permeability of the etting phase can be computed as follos 7 :
r ³ 0 d /( Pc ( Or ³ 0d /( Pc (3 here O r is the tortuosity ratio of the etting phase. According to Burdine 7, O r could be calculated as follos: O r W W (.0 ( m m (4 here m is the minimum etting phase saturation from the capillary pressure curve; W (.0 and W ( are the tortuosities of the etting phase hen the etting phase saturation is equal to 00% and respectively. In the same ay, relative permeabilities of the nonetting phase can be calculated by introducing a nonetting phase tortuosity ratio. The equation can be expressed as follos 7 : rn ³ d /( Pc ( Orn ³ 0d /( Pc (5 here rn is the relative permeability of the nonetting phase; O rn is the tortuosity ratio of the nonetting phase, hich can be calculated as follos 7 : O rn W W n n (.0 ( m e e (6 here e is the equilibrium saturation of the nonetting phase; W n is the tortuosity of the nonetting phase. Honarpour et al. 4 pointed out that the expression for the etting phase relative permeability (Eq. 3 fits the experimental data much better than the expression for the nonetting phase (Eq. 5. Corey Model. According to the Purcel and Burdine Models, an analytical expression for the etting and nonetting phase relative permeabilities may be obtained if capillary pressure curves can be represented by a simple mathematical function. Corey 8 found that oil-gas capillary pressure curves could be expressed approximately using the folloing linear relation: 3
* / P c C (7 here C is a constant and follos: * is the normalized etting phase saturation expressed as * r r (8a here r is the residual saturation of the etting phase or ater phase in steam-ater flo. In Corey s case, r is the residual oil saturation. Although the Corey Model as not originally developed for imbibition case, in this study it as used to calculate the imbibition steam-ater relative permeabilities by defining the normalized etting phase saturation as follos: * r r nr (8b here nr is the residual saturation of the nonetting phase, representing the residual steam saturation in this study. ubstituting Eq. 7 into Eqs. 3 and 5 ith the assumption that e =0 and m = r, Corey 8 obtained the folloing equations to calculate the etting (oil and nonetting (gas phase relative permeabilities for drainage cases: r ( * 4 (9 rn ( * * [ ( ] (0 A constraint to the use of Corey s Model (Eqs. 9 and 0 is that the capillary pressure curve can be represented by Eq. 7. Broos-Corey Model. Because of the limitation of Corey s Model, Broos and Corey 0 modified the representation of capillary pressure function to a more general form as follos: P c p ( e * / O ( here p e is the entry capillary pressure and O is the pore size distribution index. 4
ubstituting Eq. into the Burdine Model (Eqs. 3 and 5 ith the assumption that e =0, Broos and Corey 0 derived equations to calculate the etting and nonetting phase relative permeabilities as follos: r ( 3O * O (a rn ( O * * [ ( O ] (b When O is equal to, the Broos-Corey Model is reduced to the Corey Model. Results The data of both drainage and imbibitition steam-ater capillary pressure from Li and Horne 5 ere used to calculate the corresponding steam-ater relative permeability. Note that the capillary pressure data ere represented using Eq. in all the calculations by the Purcell Model. The calculated results ere compared to the experimental data of steam-ater relative permeability, 6. During the process of the fluid flooding tests, the ater saturation in the core sample as first decreased from 00% to the remaining ater saturation, about 8%, representing a drainage process. The ater saturation as then increased, representing an imbibition. The calculations and the comparisons are presented in this section. Fig. shos the experimental data of the steam-ater relative permeability and capillary pressure in drainage, 5-6. All these data ere measured at a temperature of about 0 o C in the same Berea core sample, 5-6. The roc and fluid properties ere described in References 5-7. Because the relative permeability and the capillary pressure ere measured simultaneously, the to curves had the same residual ater saturations. This feature is important and ill be discussed in more detail later. Note that the steam relative permeability data shon in Fig. have been calibrated under the consideration of gas slip effect (Klinenberg Effect in to-phase flo by Li and Horne 7. The drainage steam-ater relative permeabilities ere calculated using the experimental data of the drainage steam-ater capillary pressure shon in Fig. and plotted versus the normalized ater saturation that is defined in Eq. 8a. The calculated results and the comparison to the corresponding experimental data are shon in Fig.. The ater relative permeabilities calculated using the Purcell Model are the best fit to the experimental data. This implies that it may not be necessary to adjust the calculation of the etting phase relative permeabilities by introducing the concept of the tortuosity factor in such a case. The ater phase relative permeabilities calculated by all the other models are less than the experimental values. It can be seen from Fig. that the steam phase (nonetting phase relative permeabilities calculated by all the models but the Purcell Model are almost the same and consistent ith the experimental data for the 5
drainage case. The steam phase relative permeabilities calculated by the Purcell Model are not shon in Fig. and all the figures folloing in this section because the curve is concave to the axis of the normalized ater saturation on the Cartesian plot, hich is unexpected and far from the experimental values. The experimental data of the imbibition steam-ater relative permeability and the imbibition capillary pressure are shon in Fig. 3. These data ere also measured simultaneously in the same Berea core sample, 5-6 at a temperature of about 0 o C. The steam relative permeability data shon in Fig. 3 have also been calibrated under the consideration of gas slip effect in to-phase flo 7. The imbibition steam-ater relative permeabilities ere then calculated using the measured data of the imbibition steam-ater capillary pressure shon in Fig. 3 and also plotted versus the normalized ater saturation. Fig. 4 shos the calculated results and the comparison to the experimental values. The ater relative permeabilities from the Purcell Model are still the best fit to the experimental data. The results from the Corey Model are a good fit too. The ater phase relative permeabilities calculated by the Burdine and the Broos-Corey models are less than the experimental values. Actually the results calculated using the to models are the same if the capillary pressure data in the Burdine Model are represented using Eq.. The steam phase relative permeabilities calculated by all the models except the Purcell Model are not significantly different from each other but are much less than the experimental data for the imbibition case. In the folloing section, e ill discuss the calculated results and the comparison in nitrogen-ater systems. Li and Horne 7 measured the nitrogen-ater relative permeabilities in a fired Berea core sample similar to that used in the measurement of steam-ater relative permeabilities by Mahiya 6. In this study, e drilled a plug from another part of the same fired Berea sandstone that as used by Li and Horne 7. The length and diameter of the plug sample ere 5.09 cm and.559 cm respectively; the porosity as 4.37%. The drainage nitrogen-ater capillary pressure of the plug as measured by using the semipermeable porous-plate method. The measured data of the drainage nitrogen-ater capillary pressure along ith the relative permeabilities from Li and Horne 7 are plotted in Fig. 5. Although the capillary pressure and relative permeability curves ere not measured simultaneously, the residual ater saturations ere the same for both. The results calculated using the capillary pressure models for the nitrogen-ater flo (drainage and the comparison to the experimental data are shon in Fig. 6. The experimental data of ater relative permeability are located beteen the Purcell Model and the Corey Model. The to models provide a good approximation to the experimental data in this case. The features of gas phase relative permeability curve calculated by these models are similar to those of steam-ater flo (see Fig. 4 except that the calculated results are greater than the measured data. We made the same calculation and comparison using the data of oil-ater relative permeability and capillary pressure measured by Kleppe and Morse 8. We also observed 6
that the best fit to the etting phase relative permeability as from the Purcell Model. Hoever, e did not observe the same phenomenon for the data from Gates and Leitz 4. In summarizing all the calculations that e have made, the Purcell Model as the best fit to the etting phase relative permeability if the measured capillary pressure curve had the same residual etting saturation as the relative permeability curve. Discussion The technique of using capillary pressure to calculate relative permeability as developed in the late forties and as not idely utilized. Burdine 7 pointed out that the calculated relative permeabilities are more consistent and probably contain less maximum error than the measured data because the error in measurement is unnon. This may be true in some cases. Hoever, the differences beteen different capillary pressure models are obvious, especially for the etting phase. Therefore, one of the questions is hich model is most appropriate for practical use. The calculations in this study shoed that the Purcell Model as the best fit to the etting phase relative permeability. This seems surprising because the concept of the tortuosity factor as a function of etting phase saturation is not necessarily introduced for the calculation of the etting phase relative permeability in such a case. Burdine 7 obtained an empirical expression for the effective tortuosity factor as a function of etting phase saturation (see Eq. 4. O r is actually the ratio of the tortuosity at 00% etting phase saturation to the tortuosity at a etting phase saturation of. The tortuosity of etting phase is infinite at the minimum etting phase saturation according to Eq. 4. This may not be true because the etting phase may exist on the roc surface in the form of continuous film. In this case, W ( m may be close to W (.0, hich demonstrates the insignificant effect of the etting phase saturation on the tortuosity of the etting phase. imilarly, based on Eq. 6, the tortuosity of the nonetting phase is infinite hen the etting phase saturation is equal to - e. This may be true because the nonetting phase may exist in the form of discontinuous droplets. It can be seen from the analysis here that the tortuosity of etting and nonetting phases ould behave differently as a function of etting phase saturation. This may be hy it is necessary to introduce the tortuosity for the nonetting phase but not for the etting phase. Conclusions The folloing conclusions may be dran from the present study:. In steam-ater flo, the calculated results indicate that the Purcell model may be the best fit to the experimental data of the ater phase relative permeability for both drainage and imbibition processes but is not a good fit for the steam phase.. The Corey Model could also provide good approximation to the measured data of the etting phase relative permeability in some cases. 3. Except for the Purcell Model, the results of the steam phase relative permeability calculated using the models for the drainage case ere almost the same and very close to the experimental values. Hoever, those for the imbibition case ere smaller than the measured data. 4. Because of the difficulty of measuring steam-ater relative permeability, the capillary pressure technique ould be valuable. 7
Acnoledgments This research as conducted ith financial support to the tanford Geothermal Program from the Geothermal and Wind division of the U Department of Energy under grant DE-FG07-99ID3763, the contribution of hich is gratefully acnoledged. Nomenclature C =constant rn =relative permeability of nonetting phase r =relative permeability of etting phase P c =capillary pressure p e =entry capillary pressure e = equilibrium saturation of etting phase m = minimum etting phase saturation = etting phase saturation * = normalized etting phase saturation nr = residual saturation of nonetting phase r = residual etting phase saturation O = pore size distribution index O r = tortuosity ratio of etting phase O rn = tortuosity ratio of nonetting phase W = tortuosity of etting phase References. Li, K. and Horne, R.N.: An Experimental Method of Measuring team-water and Air-Water Capillary Pressures, paper 00-84, presented at the Petroleum ociety s Canadian International Petroleum Conference 00, Calgary, Alberta, Canada, June 4, 00.. Horne, R.N., ati, C., Mahiya, G., Li, K., Ambusso, W., Tovar, R., Wang, C., and Nassori, H.: team-water Relative Permeability, presented at World Geothermal Congress, Kyushu-Tohou, Japan, May 8-June 0, 000; GRC Trans. (000, 4. 3. Purcell, W.R.: "Capillary Pressures-Their Measurement Using Mercury and the Calculation of Permeability", Trans. AIME, (949, 86, 39. 4. Gates, J. I. and Leitz, W. J.: "Relative Permeabilities of California Cores by the Capillary Pressure Method", paper presented at the API meeting, Los Angeles, California, May, 950, 86. 5. Rapoport, L. A. and Leas, W. J.: "Relative Permeability to Liquid in Liquid - Gas ystem", Trans. AIME, (95, 9, 83. 6. Fatt, I. and Dystra, H.: "Relative Permeability tudies", Trans. AIME, (95, 9, 49. 7. Burdine, N. T.: "Relative Permeability Calculations from Pore ize Distribution Data", Trans. AIME, (953, 98, 7. 8. Corey, A. T.: "The Interrelation beteen Gas and Oil Relative Permeabilities", Prod. Mon., (954, 9, 38. 8
9. Wyllie, M. R. and Gardner, G. H. F.: "The Generalized Kozeny-Carmen Equation, Its Application to Problems of Multi-Phase Flo in Porous Media", World Oil, (958, 46,. 0. Broos, R. H. and Corey, A. T.: "Properties of Porous Media Affecting Fluid Flo", J. Irrig. Drain. Div., (966, 6, 6.. Land, C..: "Calculation of Imbibition Relative Permeability for To- and Three- Phase Flo from Roc Properties", PEJ, (June 968, 49.. Land, C..: "Comparison of Calculated ith Experimental Imbibition Relative Permeability", Trans. AIME, (97, 5, 49. 3. Huang, D. D., Honarpour, M. M., Al-Hussainy, R.: "An Improved Model for Relative Permeability and Capillary Pressure Incorporating Wettability", CA 978, proceedings of International ymposium of the ociety of Core Analysts, Calgary, Canada, eptember 7-0, 997. 4. Honarpour, M. M., Koederitz, L., and Harvey, A. H.: Relative Permeability of Petroleum Reservoirs, CRC press, Boca Raton, Florida, UA, 986, IBN 0-8493- 5739-X, 9. 5. Li, K. and Horne, R.N.: team-water Capillary Pressure, PE 634, presented at the 000 PE Annual Technical Conference and Exhibition, Dallas, TX, UA, October -4, 000. 6. Mahiya, G.F.: Experimental Measurement of team-water Relative Permeability, M report, tanford University, tanford, Calif., 999. 7. Li, K. and Horne, R.N.: Gas lippage in To-Phase Flo and the Effect of Temperature, PE 68778, presented at the 00 PE Western Region Meeting, Baersfield, CA, UA, March 6-30, 00. 8. Kleppe, J. and Morse, R. A.: "Oil Production from Fractured Reservoirs by Water Displacement," PE 5084, presented at the 974 PE Annual Technical Conference and Exhibition, Houston, TX, UA, October 6-9, 974. 9
Relative Permeability.0 0.8 0.6 0.4 0. ro, Exp. r, Exp. P c, Exp. 0.0 0 0 0 40 60 80 00 Water aturation (% Fig. Experimental data of drainage steam-ater relative permeability and capillary pressure 5-6. 40 30 0 0 Capillary Pressure (psi.0 Relative Permeability 0.8 0.6 0.4 0. 0.0 0 0 40 60 80 00 Normalized Water aturation (% rs, Exp r, Exp. r, Purcell r, Burdine rs, Burdine r, Corey rs, Corey r, Broos-Corey rs, Broos- Corey Fig. Calculated steam-ater relative permeability and the comparison to the experimental data in drainage case. 0
.0 Relative Permeability 0.8 0.6 0.4 0. ro, Exp. r, Exp. P c, Exp. 0 8 6 4 Capillary Pressure (psi 0.0 0 0 0 40 60 80 00 Water aturation (% Fig. 3 Experimental data of imbibition steam-ater relative permeability and capillary pressure 5-6..0 Relative Permeability 0.8 0.6 0.4 0. 0.0 0 0 40 60 80 00 Normalized Water aturation (% rs, Exp. r, Exp. r, Purcell r, Burdine rs, Burdine r, Corey rs, Corey r, Broos-Corey rs, Broos- Corey Fig. 4 Calculated steam-ater relative permeability and the comparison to the experimental data in imbibition case.
Relative Permeability.0 0.8 0.6 0.4 0. ro, Exp. r, Exp. P c, Exp. 80 60 40 0 Caillary Pressure (psi 0.0 0 0 0 40 60 80 00 Water aturation (% Fig. 5 Experimental data of drainage nitrogen-ater relative permeability and capillary pressure..0 Relative Permeability 0.8 0.6 0.4 0. 0.0 0 0 40 60 80 00 Normalized Water aturation (% rs, Exp r, Exp. r, Purcell r, Burdine rs, Burdine r, Corey rs, Corey r, Broos-Corey rs, Broos- Corey Fig. 6 Calculated nitrogen-ater relative permeability and the comparison to the experimental data in drainage.