CALCULATION OF STEAM AND WATER RELATIVE PERMEABILITIES USING FIELD PRODUCTION DATA, WITH LABORATORY VERIFICATION

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1 CALCULATION OF STEAM AND WATER RELATIVE PERMEABILITIES USING FIELD PRODUCTION DATA, WITH LABORATORY VERIFICATION Jericho L. P. Reyes, Chih-Ying Chen, Keen Li and Roland N. Horne Stanford Geothermal Program, Department of Petroleum Engineering Stanford University, Stanford, CA, U.S.A. KEYWARDS Steam-ater relative permeability; production data; The Geysers; Salton Sea field; DOGGR database ABSTRACT The steam and ater relative permeabilities at The Geysers and Salton Sea geothermal reservoirs ere calculated from available production data. A method as used to estimate the relative permeability curves using Darcy s la from mass production rates of steam and ater that are available from the DOGGR database. A verification as also conducted using data measured in laboratory steam-ater flo experiments. The laboratory results sho good agreement ith the relative permeabilities calculated from a standard Darcy's La approach. The ater saturation estimated from the production data (i.e. the floing ater saturation) as found to be a significant underestimate compared to the in-place (static) saturation. From the laboratory experiments, the relationship beteen the floing ater saturation and the in-place ater saturation as developed. The relative permeability curves inferred from field production data, corrected to static saturation, sho a behavior that is very similar to that seen in laboratory experiments. INTRODUCTION There are to types of geothermal reservoirs: the vapor-dominated reservoir here steam is the principal recovery fluid and the liquid-dominated reservoir here liquid ater is the principal recovery fluid. In both cases, the interaction beteen these to different phases has been the subject of numerous studies. Many measurements have encountered experimental difficulty due to the phase changes during the flo. An alternative ay of determining ho these to phases interact hile in a state of flo ould be very useful in the prediction of the ultimate recovery of the resource. Quantifying this interaction, by calculating the relative permeability of each of the phases, is of particular importance. The objective of this study as to develop a method to calculate the relative permeabilities of steam and ater by using production data from active geothermal fields, and to verify and calibrate this method using data from laboratory experiment. Knoledge of the relative permeabilities of steam and ater ill provide better understanding of the fluid flo interactions in the geothermal reservoir, and this is valuable in estimating the performance of a geothermal field and its capacity for further exploitation.

2 BACKGROUND There have been numerous attempts to characterize the steam and ater relative permeability curves both experimentally and theoretically. The main difficulty of direct measurement has been the phase changes that occur during steam and ater multiphase flo. A number of experiments have been made in nonboiling flo in fractured media, such as in air-ater (Diomampo, 2) and ater-oil. Current research on steam-ater relative permeability in fractures (Chen et al. 22, 23) gives us a preliminary insight on the characteristics of the interaction of these to phases ith one another. The to frequently used functions for relative permeability are the linear model (X-curve) and the Corey-model (Corey, 954). These functions are dependent on phase saturation. The X- curve has a linear relationship ith saturation: k = () rl S l k = (2) rg S g here S l and S g are the liquid and gas saturation respectively. The Corey model is expressed as follos : *4 k rl = S (3) k rg S * * 2 *2 = ( S ) ( S ) (4) = ( S S ) /( S S ) (5) l rl rl rg Chen et al. (22) developed a method to compare steam- and air-ater transport through fractured media. The main finding as that steam-ater flo behavior in fractures is different from that of nitrogen-ater flo. Chen et al. (23) found less phase interference in steamater flo, and sa the behavior of the steam-ater relative permeabilities behave closer to the X-curve. The DOGGR Database has been made available publicly by the California Division of Oil, Gas and Geothermal Resources. The database contains production histories of, among others, the Geysers and Salton Sea geothermal ells. The data include temperature, pressure and steam and ater production rates, and these parameters ere used here in this study. The Geysers Geothermal Field, a vapor-dominated reservoir field, is located in Northern California about 3 km north of San Francisco. The Salton Sea Geothermal Field, a liquid-dominated reservoir field, is located in Imperial County in Southern California. METHOD Shinohara (978) described a method to estimate the steam and ater relative permeabilities in geothermal reservoirs, and applied this method to production data from the Wairakei geothermal field in Ne Zealand. This method is simple and useful, in that it only needs the production flo rate history and the temperature of the reservoir, as ell as the ability to evaluate each ell separately. Some of the assumptions of this method include: () The pressure gradient is constant for a short time in each ell. (2) The product of permeability and floing area is constant in each ell. (3) Fluid flo follos Darcy s La. 2

3 Under these assumptions and from Darcy s la: k Q = ρ Ap' µ (6) k Qs = ρ s s Ap' µ (7) s here Q is the mass flo rate, ρ is the density, µ is the dynamic viscosity, k r is the relative permeability, k is the absolute permeability of the geothermal rock, A is the cross sectional area of flo, and p is the pressure gradient. The subtitles and s refer to ater and steam respectively. Dividing Equation 6 by Equation7 gives us: Q υs = (8) Q υ k s rs here υ is the kinematic viscosity. Taking the sum of Equations 6 and 7 gives us: k k k Q r rs rs Q = Q + Qs = ρ + ρ s kap' = + kap' s s Q (9) µ µ υ s here Q is the total of mass production rate of steam and ater. If e assume kap' is constant in each ell, then Equation 9 shos that a plot of Q vs Q/Q s ould be almost linear hen Q /Q s is small, and e can find the value of kap' from either the intercept or the gradient of the line on the graph. This intercept, here Q/Q s =, becomes Q*, here: Q* = kap' () υ s Because k rs = at Q =, then, substituting Equation into Equation 6 and 7, υ Q = () υs Q * Qs s = (2) Q * Therefore knoing Q*, e can calculate k rs and k r by also knoing Q, Q s, υ s, and υ Unfortunately, the actual ater saturation cannot be obtained in actual geothermal reservoirs. To estimate ater saturation roughly using the production data only, the volumetric ratios can be used to infer the reservoir ater saturation in the absence of residual saturation and for homogeneous flo of both phases. This estimated ater saturation is called the floing ater saturation, and can be calculated from: ( x) ν S, f = (3) ( x) ν + xν here x is the mass fraction of steam and ν and ν s is the specific volume of ater and steam, respectively. This floing saturation is often referred to as the fractional flo. It must be 3

4 understood that the floing saturation is different from the actual (in-place) saturation in a geothermal reservoir. In the next section, e ill describe the application of Shinohara s method to the production data from The Geysers and Salton Sea geothermal fields. After that, e ill present a verification of Shinohara s method by applying it to laboratory data in hich the steam-ater relative permeabilities ere already knon. Comparison ith the laboratory data also reveals the relationship beteen the floing saturation and the actual (in-place) saturation. RESERVOIR APPLICATIONS The production data in the Geysers and Salton Sea geothermal fields include temperature, pressure and steam and ater production rates. In choosing the ells to be used in this study, a number of issues had to be addressed. First, for the vapor-dominated reservoir, e had to find data from ells that had both steam and ater production. Of the 53 ells made available to us from The Geysers, 25 of them produced ater. Nine ells ere ultimately used, as these ells had a sufficient number of readings for the calculation. Also, the first assumption of Shinohara s method tells us that it is necessary to choose a short time period over hich e can assume a constant pressure gradient. Since production data are usually intermittent in nature and often have periodic fluctuations, e had to find data sets that had significant stable periods. Of the 28 ells documented in the database that belong to the Salton Sea field operated by CalEnergy, e used six ells for our liquid-dominated case. Figures and 2 are examples of steam and ater production histories from Coleman 4-5, a Geysers ell, and IDD 9, a ell from the Salton Sea geothermal field. Well IDD-9 (Figure 2) from Salton Sea had zero production for much of its history. We chose an interval that e can assume to have a roughly constant pressure gradient. For this ork e chose an interval from mid-99 to late-992. Choosing the time interval for the vapor-dominated ell is much easier. We tried to omit extreme readings from our analysis, therefore the spike seen in as not chosen as part of the range. For this ork, e used a data interval from mid-987 to 989. Figures 3 and 4 sho the Q vs Q /Q s graphs for Coleman 5-5 and IDD 9, respectively. The value of Q* is inferred from the y-intercept value from the linear fit to the graph. Table shos the Q* inferred from all the ells used in the study Water Production Rate Steam Production Rate Florate ( kg/s) Figure : Steam and Water Production History of Coleman 4-5, The Geysers Geothermal Field. 4

5 Flo Rate ( kg/s) Year Water Production Rate Steam Production Rate Figure 2: Steam and Water Production History of IID - 9, Salton Sea Geothermal Field Q (Q + Qs) 2 5 y = x Q/Qs Figure 3: Q vs. Q/Qs to infer Q* for Coleman 4-5, The Geysers Geothermal Field Q (Q + Qs) y = 9.258x Q/Qs Figure 4: Q vs. Q/Qs to infer Q* for IID - 9, Salton Sea Geothermal Field. 5

6 Table : Inferred Q* values for the Geysers and Salton Sea Geothermal Field Wells. Geysers Wells Q* Salton Sea Wells Q* Coleman IID 9 Coleman Sinclair 2 75 Coleman Vonderahe Francisco Sinclair Coleman A Elmore 2 Thorne Sinclair Thorne Francisco CA E If e compare the Q* values beteen The Geysers ells and the Salton Sea ells, e can see that The Geysers Q* values are smaller than those in the Salton Sea. Also, The Geysers Q* values are close to each other. This is an extension of the second assumption made by Shinohara in developing his method. Not only is kap constant in a ell, ells that are near each other or belong to the same geothermal field also have similar kap values. Since the ells in a certain geothermal field mainly have the same k values, and to a certain extent, A and p, then our inferred values are consistent ith each other. The Salton Sea ells have a ider range of values of Q*, but are generally of the same magnitude and larger than those in The Geysers. To evaluate the kinematic viscosities and mass production rates of the steam and ater correctly, e must infer the bottomhole conditions, as these reflect the true floing conditions of the ell. We made temperature corrections based on the documented depths of the ells. We can no use Equations and 2 to calculate the relative permeabilities of steam and ater. Figures 5 and 6 shos us a plot of relative permeability ith ater saturation for The Geysers and Salton Sea geothermal ells, respectively. Note that these graphs are plotted against the floing saturation, S,f, as defined by Equation 3. The floing saturation excludes the immobile ater and steam fractions. Th ater saturation as estimated by using Equation 3 since the actual (in-place) ater saturation as not available. Figure 5 shos The Geysers relative permeability plot. Because The Geysers is a vapor-dominated reservoir, e expected the lo ater saturation values. Figure 6, the Salton Sea examples, shos us a larger range for floing ater saturation, ith a maximum at around.25. Even ith a vapor-dominated reservoir, e see that, volumetrically, the steam saturation values still dominate, even if, by mass, ater production is greater. We can see the general trend of the relative permeability curves by plotting both ell samples into Figure 7. From Figure 7, e see that the relative permeability values for the vapor-dominated and liquid-dominated samples are only partially consistent ith each other. For the relative permeability of steam, The Geysers calculation gives us a sharp drop in k rs at small values of S,f. We then see a plateau of values approaching S,f =. from the Salton Sea values. For the relative permeability of ater, e see a more constant and stable rise as the ater saturation increases. The steepness of the rise for both sets of ell samples is consistent. The ater saturation in the figures seem to be much smaller than the traditional behavior of relative permeability curves, because of the use of floing saturation based on Equation 3, rather than the true in-place saturation. A mapping beteen floing and static ater saturations based on laboratory experiment ill be address in the next section. 6

7 S,f - C 4-5 s - C C 5-5 s - C C 3-5 s C F 2-5 s - F 2-5 s C A-5 C A-5 - T s - T - CA56 s - CA56 - T 6 s - T 6 - F 5-5 s - F 5-5 Figure 5: Plot of relative permeability curves against floing ater saturation for The Geysers Geothermal Field S,f k - IID 9 ks - IID 9 k - V ks - V k - E ks - E k - S ks - S k - S 2 ks - S 2 Figure 6: Plot of relative permeability curves against ater saturation for the Salton Sea Geothermal Field k,salton Sea ks,salton Sea k,geysers ks,geysers S,f Figure 7: Plot of relative permeability curves against ater saturation for The Geysers and Salton Sea Geothermal Reservoir Fields. 7

8 To compare the estimated relative permeability values ith the to most commonly assumed models of relative permeabilities, namely Corey and X curves, e plot the computed k r and k rs values ith these model curves in Figure 8. For The Geysers samples, e see that the relative permeability follos the Corey-model. On the other hand, the Salton Sea values lie more in the region beteen the X-curve and Corey-curve. s Salton Sea data The Geysers data X curve Corey Figure 8: Plot of k r vs k rs for The Geysers and Salton Sea Geothermal Field, ith the Corey and X-curves. We plot the data from Figure 8 again, this time ith logarithmic axes, in Figure 9. We see from this graph that the calculated values lie beteen the X- and Corey-curves, for both The Geysers and Salton Sea ells.. s.. Salton Sea data The Geysers data X curve Corey Figure 9: Logarithmic Plot of k r vs k rs for The Geysers and Salton Sea Geothermal Field, ith the Corey and X-curves. LABORATORY VERIFICATION To confirm Shinohara s method and examine the relationship beteen the floing saturation and the actual in-place (static) saturation, data from laboratory experiments ere used. These data 8

9 ere obtained from steam-ater flo experiments conducted by Chen et al. (23). In these experiments, a FFRD (fractional flo ratio detector) device as used to sense both steam and ater production rates. Flo visualization and image processing techniques ere used to determine the ater saturation (static), and differential pressure transducers ere used to measure the pressure drop through the artificial reservoir (a single fracture). We used this "production data" from the laboratory to estimate relative permeabilities using Shinohara s method, and compared the results ith those from the standard porous media approach provided by Chen et al. (23). Since the pressure gradient in the laboratory scale experiment as not constant, e scaled the data to a constant pressure gradient prior to the calculations. Figure shos the Q vs. Q /Q s plot in the experiment. The value of Q* is.32 ml/sec in this case. The steam-ater relative permeabilities calculated from Shinohara s method (Equations and 2) ere compared ith those from the porous media approach (Equation 6 and 7). A close agreement of relative permeability values from these to methods is shon in Figure. The steam-phase and aterphase values sho less than 5% relative error beteen the to methods. The relationship beteen floing and static ater saturations as examined by comparing the actual (static) ater saturation measured in the experiment ith the floing ater saturation calculated from Equation 3. From Figure 2, it is evident that the floing ater saturation is significantly less than the actual ater saturation. The relationship beteen the to saturations can be expressed by a logarithmic trend as shon in Figure 3. Q (ml/sec) y =.37x R = Q/Qs Figure : Q vs. Q/Qs to infer Q* for the steam-ater experiment of Chen et al. (23). 9

10 .4.2 s,chen,chen s-shinohara -Shinohara s,chen,chen s,shinohar,shinohar a b S S Figure : Comparison of steam-ater relative permeabilities from porous media approach and Shinohara s method for the steam-ater experiments: (a) generalized from five experimental runs; (b) averaged values..4.2 s(s) (S) s(s,f) (S,f) s(s) (S) s(s,f) (S,f) a S or S,f b S or S,f Figure 2: Comparison of vs. S and vs. S,f from Shinohara s method for the steam-ater experiments: (a) generalized from five experimental runs; (b) averaged values as y =.946Ln(x) +.89 R 2 = S,f bs y =.52Ln(x) R 2 = Figure 3: The floing ater saturation versus actual (static) ater saturation: (a) generalized from five experimental runs; (b) averaged values. S,f

11 From Figures, 2 and 3, Shinohara s method can been demonstrated to obtain accurate relative permeabilities if the reservoir pressure gradient is close to constant. Hoever, the floing saturation values inferred from Equation 3 are a significant underestimate of the static (in-place) saturation. The difference beteen the to saturations is due to the velocity differences beteen steam and ater phases, to the effects of immobile phases, and to the phase transformation effects. Therefore, e cannot simply use the floing saturation to substitute for the real reservoir saturation. Figure 3b provides a mapping equation to relate the floing and the actual ater saturations for the laboratory scale measurements. The relationship beteen S,f and S can be expressed as: S.52 ln(, ) (4) = S f By applying Equation 4 to convert the (floing) ater saturation values estimated from both the Geysers and Salton Sea production data, Figure 7 can be replotted against the corrected (static) ater saturation, as shon in Figure 4. Comparing Figure 4 ith Figure 7, the underestimated ater saturation has been improved, and Figure 4 shos more conventional relative permeability behavior. The relative permeabilities previously spanning from to.23 (floing) ater saturation no range from to.7 (static) ater saturation after applying Equation 4. The solid curves in Figure 4 sho the approximate trends of the relative permeability values for The Geysers and Salton Sea data, hereas the dashed lines are the trends for the experiments of Chen et al. (23) shon earlier in Figure 3b. The estimates from the field production data still lie beteen the Corey and X-curves, and they are loer than the values measured by Chen et al. (23). This observation may imply more phase interference in the actual geothermal reservoirs, hich is reasonable. Greater phase interference may be attributed to several issues. First of all, the fracture and matrix interaction as not included in Chen et al. s single fracture apparatus. The surface morphology, the complexity of nature fracture netork, and the ettability difference of the materials ere not considered in the smooth-alled fracture model of Chen et al. (23). Moreover, the scale difference may be important too. Nonetheless, the similarities beteen the to sets of curves are striking S ks,all field data k,all field data s,chen,chen

12 Figure 4: Relative permeability vs. mapped static ater saturation from the field production data for The Geysers and Salton Sea Geothermal fields, comapred to experimental results from Chen et al. (23). CONCLUSIONS ) We can infer the steam and ater relative permeabilities from field measurements of the production flo rate history and bottomhole temperature. Comparison ith laboratory data demonstrated that this method can estimate the relative permeabilities accurately. 2) The estimated values of relative permeability in The Geysers and the Salton Sea geothermal fields lie beteen the X-curve and the Corey curve. 3) There is a sharp decline in the relative permeability of steam at small values of floing ater saturation, and this decline moderates as the saturation increases. 4) The relationship beteen the floing ater saturation and the actual (static) ater saturation is close to logarithmic. After applying this correction, The k r versus S curves from field data sho a more conventional appearance. 5) In comparison to laboratory measurements from Chen et al. (23), the estimated relative permeabilities from the field production data are loer, hich implies that more phase interference occurs in the actual geothermal reservoirs. REFERENCES Chen, C.-Y., Diomampo, G., Li, K. and Horne, R.N.: "Steam-Water Relative Permeability in Fractures," Geothermal Resources Council Transactions Vol.26, pp , 22. Chen, C.-Y., Li, K. and Horne, R.N.: "Difference Beteen Steam-Water and Air-Water Relative Permeabilities in Fractures," Geothermal Resources Council Transactions Vol.27, pp , Oct., 23. Corey, A.T., 954. "The Interrelations Beteen Gas and Oil Relative Permeabilities," Producers Monthly, Vol. 9, p Diomampo, G., "Relative Permeability through Fractures", MS report, Stanford University, Stanford, California (2). Shinohara, K., Calculation and Use of Steam/Water Relative Permeabilities in Geothermal Reservoirs, MS report, Stanford University, Stanford, California (978). 2