Comparison of methods to calculate relative permeability from capillary pressure in consolidated water-wet porous media

Transcription

1 WATER RESOURCES RESEARCH, VOL. 42, W645, doi:1.129/25wr4482, 26 Comparison of methods to calculate relative permeability from capillary pressure in consolidated water-wet porous media Kewen Li 1 and Roland N. Horne 1 Received 3 August 25; revised 13 December 25; accepted 1 February 26; published 14 June 26. [1] The Brooks and Corey relative permeability model has been accepted widely as a way to calculate relative permeability using capillary pressure data. However, the Purcell model was found to be the best fit to the experimental data of the wetting-phase relative permeability in the cases studied here, as long as the measured capillary pressure curve had the same residual saturation as the relative permeability curve. The differences between the experimental data of relative permeability and the data calculated using the Purcell relative permeability model for the wetting phase were almost negligible. A physical model was developed to explain the insignificance of the effect of tortuosity on the calculation of the wetting-phase relative permeability. For the nonwetting-phase, the relative permeabilities calculated using the models were very close to the experimental values in drainage except for the Purcell model. However, in the case of imbibition, the relative permeabilities calculated using the models were different from the experimental data. This study showed that relative permeability could be calculated satisfactorily by choosing a suitable model, especially in drainage processes. In the reverse procedure, capillary pressure could also be computed once relative permeability data are available. Citation: Li, K., and R. N. Horne (26), Comparison of methods to calculate relative permeability from capillary pressure in consolidated water-wet porous media, Water Resour. Res., 42, W645, doi:1.129/25wr Introduction [2] Relative permeability is of central importance to soil science, petroleum engineering, and many other industries but may be difficult to measure in some cases. Such cases include extremely low permeability rocks and special fluid systems in which there are phase transformation and mass transfer between the two phases as pressure changes. Several mathematical models have been proposed to infer relative permeability because of the difficulty in making direct experimental measurements. Demond and Roberts [1993] conducted a comparison of experimental measurements with estimates generated with five common methods and their results showed that these methods are limited in their predictive capabilities. Another approach to estimating relative permeability is pore-scale network modeling. For example, Rajaram et al. [1997] used pore-scale network models to investigate the influence of correlations on the capillary pressure-saturation-relative permeability relationships for unconsolidated soils. The predicted relative permeabilities were compared to the measured values and predictions using the traditional van Genuchten [198] relationships. Rajaram et al. [1997] showed that the porescale model could fit the capillary-pressure saturation curves and predict the saturation-relative permeability curves with a degree of accuracy comparable to the van Genuchten [198] relationships. [3] Vapor-water flow in soil or rock is an example of a special fluid system in which there are phase transformation 1 Department of Petroleum Engineering, Stanford University, Stanford, California, USA. Copyright 26 by the American Geophysical Union /6/25WR4482 W645 and mass transfer between the two phases. Vapor-water relative permeability plays an important role in controlling fluid flow performance. Li and Horne [24a] found significant differences between vapor (steam)-water and air-water capillary pressures, and Horne et al. [2] found differences between vapor (steam)-water and air-water relative permeabilities. According to these studies, vaporwater flow properties may not be replaced simply by air-water or nitrogen-water flow properties. It would be helpful for engineers and scientists to be able to calculate steam-water relative permeability once steam-water capillary pressure data are available. [4] Capillary pressure and relative permeability are coupled. This feature is useful in many cases. For example, Parker et al. [1987] developed a parametric model to describe relative permeability relationships in two- or three-phase fluid flow from the scaled saturation-capillary pressure function. Wu and Pan [23] derived a class of analytical solutions for the transient flow into unsaturated rock matrix using specially correlated, physically meaningful relative permeability and capillary functions. [5] There are many papers related to techniques for the calculation of relative permeabilities from capillary pressure data. Purcell [1949] developed a method to calculate the permeability using pore size distribution derived from mercury-injection capillary pressure curves. This method has been used to calculate multiphase relative permeabilities, as reported by Gates and Leitz [195]. Later, Burdine [1953] introduced a tortuosity factor in the model. Corey [1954] and Brooks and Corey [1966] summarized the previous work and modified the method by representing capillary pressure curve as a power law function of the wetting-phase saturation. Later the modified model was known as the Brooks and Corey relative permeability 1of9

2 W645 LI AND HORNE: METHODS TO CALCULATE RELATIVE PERMEABILITY W645 model. This model has been used in many fields. These include vadose zone studies, subsurface remediation of nonaqueous phase liquids [Parker et al., 1987], and oilwater flow in reservoir rocks [Honarpour et al., 1986]. [6] Honarpour et al. [1986] reviewed the literature in this field. The published literature and experimental data for relative permeability and capillary pressure were not sufficient to conclude which method should be used in a specific case. [7] Historically, the Brooks and Corey [1966] capillary pressure technique was developed for drainage situations and has been useful to obtain gas-liquid relative permeability when fluid flow tests were not practical. [8] In this study, we calculated the gas-liquid and oilwater relative permeabilities using experimental data of capillary pressure by different methods. The calculated results were compared to the relative permeability data measured in the same core sample. The purpose of this study was to verify which capillary pressure model would achieve the best fit to the experimental data of relative permeability. We clarify that this study was limited to consolidated water-wet porous media, and did not consider unconsolidated oil-wet or mixed-wet porous media. 2. Mathematical Background [9] There are three main approaches to calculate relative permeability from capillary pressure data. One is the Purcell [1949] approach in which a tortuosity factor is not considered, and another is the Burdine [1953] approach in which a tortuosity factor is included. The third approach is the Mualem model [Mualem, 1976]. In this study, only the first two models were used. Different relative permeability models such as the Corey model and the Brooks and Corey model can be derived if different capillary pressure functions are chosen. The mathematical expressions of the models used in this article are described briefly in this section Purcell Approach [1] Purcell [1949] developed an equation to compute rock permeability by using capillary pressure data. This equation can be extended readily to the calculation of multiphase relative permeability. In two-phase flow, the relative permeability of the wetting phase can be calculated as follows: k rw ¼ Z Sw ds w = ðp c Þ 2 ds w = ðp c Þ 2 ð1þ where k rw and S w are the relative permeability and saturation of the wetting phase; P c is the capillary pressure as a function of S w. [11] Similarly, the relative permeability of the nonwetting phase can be calculated as follows: k rnw ¼ ds w = ðp c Þ 2 S w ds w = ðp c Þ 2 ð2þ where k rnw is the relative permeability of the nonwetting phase. It can be seen from equations (1) and (2) that the sum of the wetting and nonwetting-phase relative permeabilities at a specific saturation is equal to one. This is not true in most porous media. In the next section, the relative permeabilities calculated using this method are compared to the experimental data. The comparison shows that equation (1) is close to experimental values of the wetting-phase relative permeability but equation (2) for the nonwetting phase is far from the experimental results Burdine Approach [12] Burdine [1953] developed equations similar to Purcell s method by introducing a tortuosity factor as a function of wetting-phase saturation. The relative permeability of the wetting phase can be computed as follows: k rw ¼ ðl rw Þ 2 Z Sw ds w = ðp c Þ 2 ds w = ðp c Þ 2 ð3þ where l rw is the tortuosity ratio of the wetting phase. According to Burdine [1953], l rw could be calculated as follows: l rw ¼ t wð1:þ t w ðs w Þ ¼ S w S m ð4þ 1 S m where S m is the minimum wetting-phase saturation from the capillary pressure curve; t w (1.) and t w (S w ) are the tortuosities of the wetting phase when the wetting-phase saturation is equal to 1% and S w respectively. [13] In the same way, relative permeabilities of the nonwetting phase can be calculated by introducing a nonwetting-phase tortuosity ratio. The equation can be expressed as follows: k rnw ¼ ðl rnw Þ 2 ds w = ðp c Þ 2 S w ds w = ðp c Þ 2 ð5þ where l rnw is the tortuosity ratio of the nonwetting phase, which can be calculated as follows: l rnw ¼ t nwð1:þ t nw ðs w Þ ¼ 1 S w S e ð6þ 1 S m S e Here S e is the equilibrium saturation of the nonwetting phase; t nw is the tortuosity of the nonwetting phase. [14] Honarpour et al. [1986] pointed out that the expression for the wetting-phase relative permeability (equation (3)) fits the experimental data much better than the expression for the nonwetting phase (equation (5)) Corey Relative Permeability Model [15] According to the Purcell and Burdine models, an analytical expression for the wetting and nonwetting-phase relative permeabilities can be obtained if capillary pressure curves can be represented by a simple mathematical function. Corey [1954] found that oil-gas capillary pressure 2of9

3 W645 LI AND HORNE: METHODS TO CALCULATE RELATIVE PERMEABILITY W645 curves could be expressed approximately using the following linear relation: 1=P 2 c ¼ CS w* ð7þ where C is a constant and S w * is the normalized wettingphase saturation, which could be expressed as follows for the drainage case: S w * ¼ S w S wr 1 S wr ð8þ where S wr is the residual saturation of the wetting phase. In Corey s case, S wr is the residual oil saturation. [16] Although originally the Corey model was not developed for the imbibition case, in this study it was used to calculate the imbibition relative permeabilities by defining the normalized wetting-phase saturation as follows: S w * ¼ S w S wr 1 S wr S nwr ð9þ where S nwr is the residual saturation of the nonwetting phase. [17] Substituting equation (7) into equations (3) and (5) with the assumption that S e = and S m = S wr, Corey [1954] obtained the following equations to calculate the wetting (liquid) and nonwetting (gas) phase relative permeabilities for drainage cases: k rw ¼ ðs w * Þ 4 ð1þ h i k rnw ¼ ð1 S w * Þ 2 1 ðs w * Þ 2 ð11þ Equations (1) and (11) are referred to as the Corey relative permeability model for simplicity even though they are based on the Burdine approach (equations (3) and (5)) by using the Corey capillary pressure model (equation (7)). A constraint to the use of Corey s model (equations (1) and (11)) is that the capillary pressure curve should be represented by equation (7) Brooks-Corey Relative Permeability Model [18] Because of the limitation of Corey s model, Brooks and Corey [1966] modified the representation of capillary pressure function to a more general form as follows: P c ¼ p e ðs w * Þ 1=l ð12þ where p e is the entry capillary pressure and l is the pore size distribution index. [19] Substituting equation (12) into equations (3) and (5) with the assumption that S e =,Brooks and Corey [1966] derived equations to calculate the wetting and nonwettingphase relative permeabilities as follows: k rw ¼ Sw 2þ3l l h k rnw ¼ ð1 S w * Þ 2 1 ðs w * Þ 2þl l i ð13þ ð14þ Figure 1. Experimental data of drainage steam-water relative permeability from Mahiya [1999] and capillary pressure from Li and Horne [21]. Equations (13) and (14) are referred to as the Brooks-Corey relative permeability model. When l is equal to 2, the Brooks-Corey model reduces to the Corey model Purcell Relative Permeability Model [2] Substituting equation (12) into equations (1) and (2) with the assumption that S e =, one can obtain: k rw ¼ ðs w * Þ 2þl l h k rnw ¼ 1 Sw 2þl l i ð15þ ð16þ Equations (15) and (16) are referred to as the Purcell Relative Permeability Model. 3. Results [21] The experimental data of capillary pressure from our previous study [Li and Horne, 21] and from the literature were used to compare to the results calculated using three models. These models include: (1) the Purcell relative permeability model (equations (15) and (16)); (2) the Corey relative permeability model (equations (1) and (11)); and (3) the Brooks-Corey relative permeability model (equations (13) and (14)). The calculation and comparison in steam-water, nitrogen-water, oil-water, and oil-gas flow are presented and discussed in this section Vapor (Steam)-Water Flow [22] The data of both drainage and imbibition steamwater capillary pressure from Li and Horne [21] were used to calculate the corresponding steam-water relative permeability. The calculated results were compared to the experimental data of steam-water relative permeability measured by Mahiya [1999]. During the process of the fluid flooding tests, the water saturation in the core sample was first decreased from 1% to the residual water saturation, about 28%, representing a drainage process. The water saturation was then increased, representing an imbibition. [23] Figure 1 shows the experimental data of the steamwater relative permeability [Mahiya, 1999] and capillary pressure [Li and Horne, 21] in drainage. The symbols represent the experimental data and the solid lines are drawn 3of9

4 W645 LI AND HORNE: METHODS TO CALCULATE RELATIVE PERMEABILITY W645 Table 1. Properties of Rock and Fluids Core f, % k, md d, cm L, cm T, C Fluids a mn/m IFT, Rock S-W 55. Berea N-W 72.6 Berea N-W 72.6 Berea O(kerosene)-W Berea O(kerosene)-H Berea O(kerosene)-A Pyrex a S, steam; W, water; N, nitrogen; O, oil; H, helium; A, air. only for visualization purpose (the same for all the figures of capillary pressure and relative permeability). All these data were measured using a steady-state method at a temperature of about 12 C in the same Berea core sample. The permeability and porosity of this core were 14 md and 24.8%; the length and diameter were 43.2 cm and 5.4 cm, respectively (core number 1 in Table 1). Because the relative permeability and the capillary pressure were measured simultaneously, the two curves had the same residual water saturations. This feature is important and will be discussed later in more detail. Note that the steam relative permeability data shown in Figure 1 have been calibrated under the consideration of gas slip effect [Klinkenberg, 1941] in two-phase flow by Li and Horne [24b]. [24] The drainage steam-water relative permeabilities were calculated using the experimental data of the drainage steamwater capillary pressure shown in Figure 1 and plotted versus the normalized water saturation that is defined in equation (8). The calculated results and the comparison to the corresponding experimental data are shown in Figure 2. The relative permeabilities in Figure 2 were normalized to conduct the comparison. The method to do this is to divide the experimental relative permeabilities by the corresponding end-point values. The same normalization has been applied to the experimental relative permeabilities shown in the figures used to compare results in the remainder of this paper. [25] We can see from Figure 2 that the water relative permeabilities calculated using the Purcell relative permeability model (equations (15) and (16)) are the best fit to the experimental data. This implies that it may not be necessary to adjust the calculation of the wetting-phase relative permeabilities by introducing the concept of the tortuosity factor in such a case. The water phase relative permeabilities calculated by all the other models are less than the experimental values. It can be seen from Figure 2 that the steam phase (nonwetting phase) relative permeabilities calculated by the Corey model and the Brooks-Corey model (except the Purcell model) are almost the same and consistent with the experimental data for the drainage case. The steam phase relative permeabilities calculated by the Purcell model are not shown in Figure 2 and all the following figures because the curve is concave downwards, which is unexpected and far from the experimental values. [26] The experimental data of the imbibition steam-water relative permeability from Mahiya [1999] and the imbibition capillary pressure from Li and Horne [21] are shown in Figure 3. These data were also measured simultaneously in the same Berea core sample at a temperature of about 12 C. The steam relative permeability data shown in Figure 3 have also been calibrated under the consideration of gas slip effect in two-phase flow [Li and Horne, 24b]. [27] The imbibition steam-water relative permeabilities were then calculated using the measured data of the imbibition steam-water capillary pressure shown in Figure 3 and also plotted versus the normalized water saturation. Figure 4 shows the calculated results and the comparison to the experimental values. The water relative permeabilities from the Purcell relative permeability model are also the best fit to the experimental data, the same as in drainage. The results from the Corey relative permeability model are a good fit too. The water phase relative permeabilities calculated by the Brooks-Corey relative permeability models are less than the experimental values. The steam phase relative permeabilities calculated by the Corey model and the Brooks-Corey model(except the Purcell model) are not significantly different from each other but are less than the experimental data in the imbibition case. Figure 2. Calculated steam-water relative permeability and the comparison to the experimental data from Mahiya [1999] in drainage. 4of9 Figure 3. Experimental data of imbibition steam-water relative permeability [Mahiya, 1999] and capillary pressure [Li and Horne, 21].

5 W645 LI AND HORNE: METHODS TO CALCULATE RELATIVE PERMEABILITY W645 Figure 4. Calculated steam-water relative permeability and the comparison to the experimental data [Mahiya, 1999] in imbibition Nitrogen-Water Flow [28] In the following section, we will discuss the calculated results and the comparison in nitrogen-water systems to further confirm the phenomena that we observed. Li and Horne [24b] measured the nitrogen-water relative permeabilities using a steady-state method in a fired Berea core sample similar to that used in the measurement of steamwater relative permeabilities by Mahiya [1999]. The properties of the rock and fluids are listed in Table 1 (core number 2). In this study, we drilled a plug from another part of the same fired Berea sandstone that was used by Li and Horne [Li and Horne, 24b]. The length and diameter of the plug sample were 5.3cm and 2.56cm respectively; the porosity was 24.37% (core number 3 in Table 1). The drainage nitrogen-water capillary pressure of the plug was measured by using the semipermeable porous-plate method. The measured data of the drainage nitrogen-water capillary pressure along with the relative permeabilities from Li and Horne [24b] are plotted in Figure 5. Although the nitrogen-water capillary pressure and relative permeability Figure 6. Calculated nitrogen-water relative permeability and the comparison to the experimental data in drainage. curves were not measured simultaneously, the residual water saturations were almost the same for both. [29] The results calculated using the relative permeability models for the nitrogen-water flow (drainage) and the comparison to the experimental data are shown in Figure 6. The experimental data of water relative permeability are located between the Purcell and the Corey relative permeability models. The two 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 steamwater flow (see Figure 4) except that the calculated results are greater than the measured data Organic Liquid (Oil)-Water Flow [3] Organic liquid (oil)-water flow exists in the study of contamination in soils as well as in oil reservoirs. It may also be helpful to look at the case of oil-water flow. Kleppe and Morse [1974] reported the experimental data of imbibition oil-water relative permeability and capillary pressure in Berea sandstone with a permeability of 29 md and a porosity of 22.5% (core number 4 in Table 1). The three curves are shown in Figure 7. The calculated results of oil Figure 5. Experimental data of drainage nitrogen-water Figure 7. Imbibition oil-water relative permeability and relative permeability and capillary pressure. capillary pressure from Kleppe and Morse [1974]. 5of9

6 W645 LI AND HORNE: METHODS TO CALCULATE RELATIVE PERMEABILITY W645 Figure 8. Calculated oil-water relative permeability and the comparison to the experimental data from Kleppe and Morse [1974]. and water relative permeability and the comparison to the experimental data are plotted in Figure 8. In oil-water flow, the best fit to the wetting-phase (water phase in this case) relative permeability is also from the Purcell relative permeability model. The water phase relative permeabilities calculated using other models are not notably different from each other but are much less than the experimental data in this case. For the nonwetting-phase (oil phase in this case) relative permeability, the Corey model and the Brooks- Corey model (except the Purcell model) give good fits to the experimental data. [31] Beckner et al. [1988] reported imbibition oil-water relative permeability and capillary pressure data which were representative of actual field data (see Figure 9). The capillary pressure data were also used to calculate oil-water relative permeability with various methods. The results and the comparison are shown in Figure 1. The Purcell model produced the best fit to the water phase relative permeability, as observed previously. The water phase relative permeabilities calculated using other models are less than the data from Beckner et al. [1988]. Figure 1. Calculated oil-water relative permeability and the comparison to the data from Beckner et al. [1988] Organic Liquid (Oil)-Gas Flow [32] Organic liquid (oil)-gas flow also exists in the study of contamination in soils as well as in oil reservoirs. We made the same calculation and comparison using the data of oil-gas relative permeability and capillary pressure measured in Berea sandstone by Richardson et al. [1952]. The permeability and porosity of this core were 17 md and 17.7%; the length and diameter were 3.7 cm and 6.85 cm, respectively (core number 5 in Table 1). The oil phase was kerosene and the gas phase was helium. The experimental data of the drainage oil-gas relative permeability and the capillary pressure are shown in Figure 11. The calculated results of relative permeability and the comparison to the experimental values are demonstrated in Figure 12. We also observed that the best fit to the wetting-phase relative permeability in oil-gas flow was from the Purcell model. [33] All the relative permeability and capillary pressure curves we used in the previous sections have a common feature: the residual saturation from the capillary pressure curve is equal to that from the relative permeability curve. Gates and Leitz [195] reported oil-gas relative permeability and capillary pressure curves without such a feature. The Figure 9. Imbibition oil-water relative permeability and Figure 11. Drainage oil-gas relative permeability and capillary pressure from Beckner et al. [1988]. capillary pressure from Richardson et al. [1952]. 6of9

7 W645 LI AND HORNE: METHODS TO CALCULATE RELATIVE PERMEABILITY W645 Figure 12. Calculated oil-gas relative permeability and the comparison to the experimental data from Richardson et al. [1952]. Figure 14. Calculated oil-gas relative permeability and the comparison to the experimental data from Gates and Leitz [195]. experimental data of drainage oil-gas relative permeability and capillary pressure, taken from Figure 4 in the paper by Gates and Leitz [195], were used in this study and are depicted in Figure 13. These data were measured in a Pyrex core with a permeability of 137 md and a porosity of 37.4% (core number 6 in Table 1). The oil phase was kerosene and the gas phase was air. The residual oil saturation was about 3% according to the oil phase relative permeability curve but was about 12% according to the capillary pressure and the gas phase relative permeability curves (see Figure 13). The reason might be the evaporation of oil caused by continuous gas injection even after the residual oil saturation by displacement was reached. [34] The oil and gas relative permeabilities calculated using various capillary pressure techniques were compared to the experimental data measured by Gates and Leitz [195] and the results are shown in Figure 14. We observed that the Corey model and the Brooks-Corey model (except the Purcell model) yielded good fits to both the wetting and nonwetting-phase relative permeabilities. [35] In summarizing all the calculations that we have made, including some not presented here, the Purcell model Figure 13. Drainage oil-gas relative permeability and capillary pressure from Gates and Leitz [195]. 7of9 was the best fit to the wetting-phase (liquid) relative permeability if the measured capillary pressure curve had the same residual saturation as the relative permeability curve Calculation of Capillary Pressure Using Relative Permeability Data [36] In some cases, relative permeability data are available but capillary pressure data are not. A method to calculate capillary pressure function using relative permeability is proposed in this section. As observed previously, the Purcell model may be the best fit to the experimental data of the wetting-phase relative permeability. Therefore we can fit the experimental data of the wetting-phase relative permeability using equation (15) to obtain the value of the pore size distribution index l. According to equation (12), the corresponding capillary pressure function can be determined once the value of the pore size distribution index l is available. The entry capillary pressure may be measured readily or can be evaluated using other methods. 4. Physical Model and Discussion [37] The techniques using capillary pressure to calculate relative permeability were developed in the late 194s. Burdine [1953] 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 unknown. This may be true in some cases. However, the differences between different relative permeability models are obvious, especially for the wetting phase. Therefore, one important question is which model is most appropriate for practical use. The calculations in this study showed that the Purcell model was the best fit to the wetting-phase relative permeability. This seems surprising because the concept of the tortuosity factor as a function of wetting-phase saturation is not introduced for the calculation of the wetting-phase relative permeability in such a case. A physical model was developed to demonstrate the insignificant effect of the tortuosity factor on the wetting phase, as shown in Figure 15. L is the direct distance

8 W645 LI AND HORNE: METHODS TO CALCULATE RELATIVE PERMEABILITY W645 Figure 15. Tortuosity in a single capillary tube. between the ends of a single capillary tube and L a is the length of the tortuous capillary tube. [38] Burdine [1953] obtained an empirical expression of the effective tortuosity factor as a function of wetting-phase saturation (see equation (4)). l rw is actually the ratio of the tortuosity at 1% wetting-phase saturation to the tortuosity at a wetting-phase saturation of S w. According to equation (4), the tortuosity of the wetting phase is infinite at the minimum wetting-phase saturation, that is equal to residual water saturation S wr here. This may not be true for the wetting phase because the wetting phase may exist on the rock surface in the form of a continuous film, as shown in Figure 15b. In this case, t w (S m = S wr ) may be close to t w (1.) (see Figure 15a), which demonstrates that there is little effect of the wetting-phase saturation on the tortuosity of the wetting phase. Similarly, based on equation (6), the tortuosity of the nonwetting phase is infinite when the wetting-phase saturation is equal to 1-S e. This may be true because the nonwetting phase may exist in the form of discontinuous droplets (see Figure 15c). In this case, S e is equal to S gr. [39] It can be seen from the analysis here that the tortuosity of wetting and nonwetting phases would behave differently as a function of wetting-phase saturation. This may be why it is necessary to introduce the tortuosity for the nonwetting phase but not for the wetting phase. [4] As stated previously, capillary pressure techniques were developed originally in cases in which it is difficult to measure relative permeability. Actually these techniques may also be useful even in cases in which both relative permeability and capillary pressure data are available. In these cases, we can still calculate relative permeability using the appropriate models with the capillary pressure data and compare the results to the experimental values. If the calculated results are consistent with the experimental data, we may have more confidence on the experimental measurements. This idea may also be applied to numerical simulation. 5. Conclusions [41] Based on the present study, the following conclusions may be drawn: [42] 1. The calculated results indicate that the Purcell relative permeability model is the best fit to the experimental data of the wetting-phase relative permeability, which is independent of the fluids systems (either gas-liquid or liquid-liquid systems) and the saturation history (either drainage or imbibition) in the cases studied. However the Purcell relative permeability model is not a good fit for the nonwetting phase. [43] 2. For a consolidated water-wet porous medium, it is not necessary to introduce the tortuosity factor in calculating the wetting-phase relative permeability as long as the measured capillary pressure curve had the same residual saturation as the relative permeability curve. [44] 3. Except for the Purcell relative permeability model, the results of the nonwetting-phase relative permeability calculated using the Corey model and the Brooks-Corey model for the drainage case were almost the same and very close to the experimental values. However, those for the imbibition cases were different from the measured data. [45] 4. A physical model was proposed to explain the insignificant effect of tortuosity on the wetting-phase relative permeability in a consolidated water-wet porous medium. [46] 5. Capillary pressure function may also be calculated from relative permeability data. Notation C constant. k rnw relative permeability of nonwetting phase. k rw relative permeability of wetting phase. L direct distance between the ends of a single capillary tube. L a length of the tortuous capillary tube. P c capillary pressure. p e entry capillary pressure. S e equilibrium saturation of wetting phase. S m minimum wetting phase saturation. S w wetting phase saturation. S w * normalized wetting phase saturation. S nwr residual saturation of nonwetting phase. S wr residual wetting phase saturation. l pore size distribution index. l rw tortuosity ratio of wetting phase. l rnw tortuosity ratio of nonwetting phase. tortuosity of wetting phase. t w [47] Acknowledgments. This research was conducted with financial support to the Stanford Geothermal Program from the Geothermal and Wind division of the U.S. Department of Energy under grant DE-FG7-99ID13763, the contribution of which is gratefully acknowledged. References Beckner, B. L., A. Firoozabadi, and K. Aziz (1988), Modeling transverse imbibition in double-porosity simulators, paper presented at SPE California Regional Meeting, Long Beach, Calif., March. Brooks, R. H., and A. T. Corey (1966), Properties of porous media affecting fluid flow, J. Irrig. Drain. Div., 6, 61. Burdine, N. T. (1953), Relative permeability calculations from pore size distribution data, Trans. AIME, 198, 71. Corey, A. T. (1954), The interrelation between gas and oil relative permeabilities, Prod. Mon., 19, 38. Demond, A. H., and P. V. Roberts (1993), Estimation of 2-phase relative permeability relationships for organic liquid contaminants, Water Resour. Res., 29(4), Gates, J. I., and W. J. Leitz (195), Relative permeabilities of California cores by the capillary pressure method, paper presented at API meeting, Los Angeles, Calif., 11 May. Honarpour, M. M., L. Koederitz, and A. H. Harvey (1986), Relative Permeability of Petroleum Reservoirs, CRC Press, Boca Raton, Fla. Horne, R. N., C. Satik, G. Mahiya, K. Li, W. Ambusso, R. Tovar, C. Wang, and H. Nassori (2), Steam-water relative permeability, paper presented at World Geothermal Congress, Kyushu-Tohoku, Japan, 28 May to 1 June. 8of9

9 W645 LI AND HORNE: METHODS TO CALCULATE RELATIVE PERMEABILITY W645 Kleppe, J., and R. A. Morse (1974), Oil production from fractured reservoirs by water displacement, paper presented at SPE Annual Technical Conference and Exhibition, Houston, Tex., 6 9 Oct. Klinkenberg, L. J. (1941), The permeability of porous media to liquids and gases, in API Drilling and Production Practice, pp , Am. Pet. Inst., Washington, D. C. Li, K., and R. N. Horne (21), An experimental and analytical study of steam/water capillary pressure, SPEREE, Li, K., and R. N. Horne (24a), Steam-water and air-water capillary pressures: Measurement and comparison, J. Can. Pet. Technol., 43(7), Li, K., and R. N. Horne (24b), Experimental study of gas slippage in two-phase flow, SPEREE, Mahiya, G. F. (1999), Experimental measurement of steam-water relative permeability, M.S. report, Stanford Univ., Stanford, Calif. Mualem, Y. (1976), A new model for predicting the hydraulic conductivity of unsaturated porous media, Water Resour. Res., 12(3), Parker, J. C., R. J. Lenhard, and T. Kuppusamy (1987), A parametric model for constitutive properties governing multiphase flow in porous media, Water Resour. Res., 23(4), Purcell, W. R. (1949), Capillary pressures Their measurement using mercury and the calculation of permeability, Trans. AIME, 186, 39. Rajaram, H., L. A. Ferrand, and M. A. Celia (1997), Prediction of relative permeabilities for unconsolidated soils using pore-scale network models, Water Resour. Res., 33(1), Richardson, J. G., J. K. Kerver, J. A. Hafford, and J. S. Osoba (1952), Laboratory determination of relative permeability, Trans. AIME, 195, 187. van Genuchten, M. T. (198), A closed form equation for predicting the hydraulic conductivity of unsaturated soils, Soil Sci. Soc. Am. J., 44, Wu, Y. S., and L. H. Pan (23), Special relative permeability functions with analytical solutions for transient flow into unsaturated rock matrix, Water Resour. Res., 39(4), 114, doi:1.129/22wr1495. R. N. Horne and K. Li, Department of Petroleum Engineering, Stanford University, Stanford, CA 9443, USA. (kewenli@stanford.edu) 9of9