A Capillary Pressure Model for Geothermal Reservoirs

Transcription

1 Geothermal Resoures Counil Transations, Vol. 26, September 2002 A Capillary ressure Model for Geothermal Reservoirs Kewen Li and Roland N. Horne Stanford Geothermal rogram, Stanford University (kewenli@stanford.edu) Keywords Steam-water apillary pressure; Mathematial models; Steady-state flow in porous media. Abstrat Steam-water apillary pressure is often either ignored or determined artifiially in geothermal reservoir engineering. However steam-water apillary pressure plays an important role in geothermal reservoir performane and may not be substituted by air-water apillary pressure. To this end, preliminary mathematial models have been developed to alulate drainage and imbibition steam-water apillary pressure respetively. Using these models, steam-water apillary pressure in geothermal reservoirs an be omputed one porosity, permeability, and reservoir temperature are known. Experimental data of steam-water apillary pressure measured in Berea sandstone and inferred from adsorption tests in the rok from The Geysers were ompared and used to develop the steam-water apillary pressure models in both drainage and imbibition ases. Introdution In reent years, muh attention has been paid to the study of steam-water relative permeability (Sanhez and Shehter, 1990, Ambusso, 1996, Satik, 1998, Mahiya, 1999, Li and Horne, 1999, and Horne et al., 2000). However less attention has been paid to steam-water apillary pressure, even though apillary pressure is of equal signifiane to relative permeability and plays an important role in geothermal reservoir performane. As an example, Tsypkin and Calore (1999) investigated the vaporization proess and found that apillary pressure an play a stabilizing role for the vaporization front. On the other hand, Li and Horne (2001a) showed that steam-water apillary pressure was signifiantly different from air-water apillary pressure. Urmeneta et al. (1998) also studied the role of apillary fores in fratured geothermal reservoirs and found that apillary pressure tended to keep the vapor phase in the fratures and the liquid phase in the matrix. The numerial results from Urmeneta et al. (1998) showed that apillary fores ontrol the transfer of fluids between fratures and matrix, the stability of the liquiddominated two-phase zone, and the distribution of steam and water in geothermal reservoirs. Hene steam-water apillary pressure will influene the estimation of the energy reserves and prodution performane. Sta. Maria and ingol (1996) inferred values of apillary pressure from the adsorption data of Horne et al. (1995) for rok samples from The Geysers geothermal field. ersoff and Hulen (1996) inferred the vapor-water apillary pressure from adsorption data measured at room temperatures for The Geysers rok samples using different salt solutions to obtain a wide range of vapor pressures. Li and Horne (2001b) developed a method to alulate steam-water apillary 1

2 pressure in Berea sandstone using the experimental data from steady-state steam-water flow tests onduted by Mahiya (1999). The values of steam-water apillary pressure in both drainage and imbibition were obtained. It would be useful for reservoir engineers to have an approah to estimate the values of steamwater apillary pressure for geothermal roks with any porosity and permeability at any reservoir temperature. Until now, geothermal reservoir engineers have usually hypothesized the form of the steam-water apillary pressure urve used for numerial simulation, or ignored it entirely. In this paper, the steam-water apillary pressure data alulated by Li and Horne (2001b) in Berea sandstone were saled and ompared to the data from ersoff and Hulen (1996) using a J- funtion. Both drainage and imbibition steam-water apillary pressure models were developed based on these data for the appliation of geothermal reservoir engineering. Bakground Based on the Kelvin equation, Li and Horne (2001b) derived an equation to alulate steamwater apillary pressure from the experimental data of liquid phase pressure, temperature, and related parameters measured in a steady-state flow test. The equation is expressed as follows: = p v p w ρ wrt p0 = ln( ) M p w v (1) where is the apillary pressure. p 0 and p v are the vapor pressures when the vapor-liquid interfae is flat and urved respetively; R is the gas onstant, T the absolute temperature, M w the moleular weight of liquid, and ρ w the density of liquid. The units used in Equation 1 are listed here. p v, p w, p 0 : ka (absolute), ρ w : g/ml, R = 8310 (ka.ml)/(ºk.mole), T: ºK, and M w : g/mole. In steady-state steam-water flow experiments (Mahiya, 1999), p w and T an be measured at the same time and the same loation, while p 0 an be alulated aording to the measured saturation temperature. Therefore, p v, as the only unknown parameter in Equation 1, an be obtained by Newton iteration. The apillary pressure is then omputed. One the steam-water apillary pressure in the Berea sandstone was available, we ould infer the steam-water apillary pressure in geothermal roks. The proedure is desribed here. Capillary pressures in roks with different porosity and permeability may be orrelated using the J- funtion suggested by Leverett (1941) as follows: σ osθ = J ( k φ S w ) (2) where k, φ, S w, and J(S w ) are permeability, porosity, water saturation, and J-funtion respetively. Assuming that the J-funtion in both Berea and geothermal rok samples are the 2

3 same, we an alulate the steam-water apillary pressure in geothermal roks using the following equation: G kb σ G osθ φ G B B ( Sw) = ( Sw) σ B osθb kg φ G (3) G B Here ( S w ) and ( S w ) are the steam-water apillary pressures at a water saturation of S w in a geothermal rok with a permeability of k G and a porosity of φ G and in a Berea sandstone with a permeability of k B and a porosity of φ B respetively. Considering that the temperatures may be different in the two systems, σ B, the surfae tension in the steam-water-berea system, and σ G, the surfae tension in the steam-water-geothermal rok system, are introdued in Equation 3. Similarly, θ G and θ B are the ontat angles in steam-water-berea and steam-water-geothermal rok systems respetively. Equation 3 was derived by applying Equation 2 to eah type of rok: Berea and geothermal. Sine the ontat angle in steam-water-geothermal rok systems is not available, we assumed in this study that the ontat angles in both Berea and geothermal rok samples are the same. Furthermore, if we sale the experimental data to the same temperature, the surfae tension will be the same. Therefore, Equation 3 would be redued to: G kb φb B ( Sw) = ( Sw) kg φ G (4) Based on Equation 4, the steam-water apillary pressure in geothermal roks an be omputed one the steam-water apillary pressure in the Berea sandstone, and the permeability and porosity in both Berea and geothermal roks are known. We ompared the steam-water apillary pressure alulated using Equation 4 for a rok from The Geysers geothermal field with the steam-water apillary pressure measured in the same rok by ersoff and Hulen (1996) using an adsorption method. Beause the adsorption tests by ersoff and Hulen (1996) were onduted at a temperature of 28.5 o C and the steady-state flow tests were onduted at a temperature of 120 o C, it is neessary to sale up the apillary pressure measured by ersoff and Hulen (1996) to the same temperature, 120 o C. This was ahieved using the following equation: G, T2 σ T 2 G, T1 ( Sw ) = ( S σ T1 w ) (5) 3

4 G, T where 1 G, T2 ( Sw ) and ( Sw) are the apillary pressure for the same rok at the same water saturation of S w but at different temperatures of T 1 and T 2. σ T and σ 1 T are the surfae tensions at 2 temperatures T 1 and T 2. Results Steam-water apillary pressure in Berea sandstone. The drainage and imbibition steam-water apillary pressures measured by steady-state flow tests are shown in Figure 1 (Li and Horne, 2001b). The solid irles represent the drainage apillary pressure urve and the solid squares represent the imbibition apillary pressure urve. The urves represent the moving trend of the data points (the same for the rest figures exept those speified) Capillary ressure (Ma) Drainage Imbibition Water Saturation (%) Figure 1: Steam-water apillary pressure urve (drainage and imbibition) alulated from the data of steady-state flow of steam and water in a Berea sandstone sample. Steam-water apillary pressure in rok from The Geysers. As mentioned before, we an use Equation 4 to alulate the steam-water apillary pressure in geothermal roks one the steamwater apillary pressure in a Berea sandstone sample is available. The purpose is to ompare the results with those measured by ersoff and Hulen (1996) and hene evaluate the appropriateness of the assumptions of Equation 4. First of all, we need to know the porosity and permeability of the geothermal roks. We based these values on the ore sample with a porosity of 1.9% and a permeability of 1.3 nd in whih ersoff and Hulen (1996) measured the vapor-water apillary pressure at a temperature of 28.5 o C. The steam-water apillary pressure data in both drainage and imbibition were omputed using Equation 4 for this sample. Figure 2 shows the alulated apillary pressure urves. Note that the shape of the steam-water apillary pressure urves in Figure 2 is similar to that in Figure 1 but the values of steam-water apillary pressure are muh greater beause of the low permeability of The Geysers geothermal rok. 4

5 Capillary ressure (Ma) Drainage Imbibition Water Saturation (%) Figure 2: Steam-water apillary pressure urves alulated using a saling method for a ore sample from The Geysers field. Beause the reservoir temperature of The Geysers geothermal field is greater than the test temperature, the neessary alibration for the steam-water apillary pressure due to the differene in temperature ould be made using Equation 5. Comparison of drainage steam-water apillary pressure. The adsorption/desorption tests used to infer apillary pressure are stati proesses in whih there is no steam-water flow. In atual geothermal reservoirs, however, apillary pressure plays its important role while steam and water flow simultaneously through the rok. Hene the proess governing an adsorption test may not represent the mehanisms under atual fluid flow onditions in geothermal reservoirs. Consequently the apillary pressures alulated using adsorption test data may or may not be the same as those measured using a dynami method in whih steam and water flow simultaneously through the porous medium. It is known that apillary pressure is influened signifiantly by the ontat angle. The ontat angle in a stati state (no fluid flow) is often different from that in a dynami state (with fluid flow). Hene the apillary pressure is likely to be different under stati and dynami onditions. It is interesting to ompare the steam-water apillary pressure data obtained using different methods. Figure 3 shows the omparison of the drainage steam-water apillary pressure urve shown in Figure 2 with that measured in the drying proess by ersoff and Hulen (1996) using an adsorption method. Note that the drainage steam-water apillary pressure data by Li and Horne (2001b) were measured at about 120 o C and those by ersoff and Hulen (1996) were measured at about 28.5 o C. Therefore we saled the experimental values of steam-water apillary pressure from ersoff and Hulen (1996) to 120 o C using Equation 5. The surfae tension of steam/water at 120 o C is dynes/m. The water saturation dereases in the drying proess, whih is similar to the drainage proess. 5

6 Capillary ressure (Ma) SS flow (120 o C) Adsorption (120 o C) Water Saturation (%) Figure 3: Drainage steam-water apillary pressure urves by steady-state (SS) flow and adsorption methods for a rok sample from The Geysers field. We an see from Figure 3 that the values of drainage steam-water apillary pressure measured using different methods are onsistent for water saturation greater than about 60%. However the drainage steam-water apillary pressure measured using the desorption method are smaller than those measured using a steady-state flow tehnique for water saturation less than about 60%. Steam-water apillary pressure model for drainage. Usually the development of a geothermal reservoir before water injetion is a drainage proess if there is no bottom water or aquifer and is an imbibition proess otherwise. Beause both drainage and imbibition proesses may be involved in the development of a geothermal reservoir, steam-water apillary pressure models were developed for drainage and imbibition respetively. The reservoir rok in geothermal fields has different porosity and permeability and it may be impossible to measure the steam-water apillary pressure for every rok sample. Therefore we need to establish a orrelation between the steam-water apillary pressure of rok samples with different porosities and permeabilities. In order to onstitute suh a orrelation for geothermal roks, the drainage steam-water apillary pressure data from Li and Horne (2001b) were hosen. The reason is desribed as follows. As we pointed out previously, there is no steam-water flow in adsorption/desorption tests, whih is not a representation of the steam-water status in geothermal reservoirs in whih both steam and water may be mobile. Considering this and the inonsisteny between the drainage steam-water apillary pressures measured from the steady-state flow tests and those measured using adsorption approahes in some range of water saturation, we hose the data from the steady-state flow tests to onstitute the steam-water apillary pressure model. Beause the drainage steam-water apillary pressure data from Li and Horne (2001b) were measured in high permeability rok, the results were saled to the same porosity and the same 6

7 permeability as used by ersoff and Hulen (1996) using Equation 4. The drainage steam-water apillary pressure data were plotted vs. the normalized water saturation, as shown in Figure 4. Capillary ressure (Ma) Experimental Model Normalized Water Saturation (%) Figure 4: Normalized drainage steam-water apillary pressure for a rok from The Geysers field. The normalized water saturation is alulated using the following equation: S wd Sw S = 1 S wr wr (6) where S wr and Swd are the residual water saturation and normalized water saturation. The Brooks- Corey (1964) apillary pressure funtion is often used to model the apillary pressure urve, as given by: 1/ λ = pe ( Swd ) (7) where p e is the entry apillary pressure and λ is the pore size distribution index. We used the Brooks-Corey apillary pressure funtion to fit the data. Figure 4 shows a math to the data saled from Li and Horne (2001b). The values of the best-fit parameters are S wr = 0.20, p e = Ma and λ = Note that these values are only valid when the normalized water saturation is expressed as a fration rather than as a perentage. Beause the steam-water apillary pressures shown in Figure 4 were obtained from a rok sample with a permeability of about 1.3 nd and a porosity of 1.9% at a temperature of 120 o C, we would need to sale the data for roks with different porosity and permeability or for different temperatures. This an be done using Equation 2. Using this approah, we have reated a 7

8 drainage steam-water apillary pressure model based on the experimental data for geothermal roks as follows: σ = ( Swd ) k φ (8) where the units of, σ and k are Ma, dynes/m, and nd respetively; φ and S w are expressed as frations. The porosity and permeability of reservoir roks would need to be measured. The surfae tension an be alulated one the reservoir temperature is known. Therefore the steamwater apillary pressure urve for geothermal reservoir roks may be obtained using Equation 8. The model expressed in Equation 8 is suitable for drainage proess and is based on the assumptions: (1) ontat angle does not hange with permeability and temperature; (2) rok samples have the same J-funtions. Comparison of imbibition steam-water apillary pressure. The omparison of the imbibition steam-water apillary pressure urve measured from steady-state flow tests (see Figure 2) to that measured in the wetting proess by ersoff and Hulen (1996) using an adsorption method is shown in Figure Capillary ressure (Ma) SS flow (120 o C) Adsorption (120 o C) Water Saturation (%) Figure 5: Imbibition steam-water apillary pressure urves by steady-state (SS) flow and adsorption methods for a rok sample from The Geysers field. The water saturation inreases in the wetting proess, whih is similar to the imbibition proess in the steady-state flow tests. The porosity and permeability of the rok sample from The Geysers geothermal field that ersoff and Hulen (1996) used were 1.9% and 1.3 nd. The imbibition steam-water apillary pressure data from Li and Horne (2001b) were measured in Berea sandstone with a higher porosity and permeability, therefore the results were saled to the same porosity and the same permeability as used by ersoff and Hulen (1996) using Equation 4. 8

9 Similarly, the imbibition steam-water apillary pressure data from ersoff and Hulen (1996) were measured at a temperature of 28.5 o C, therefore the data were saled to the same temperature (120 o C) using Equation 5. We an see from Figure 5 that the imbibition steam-water apillary pressures obtained from the steady-state flow tests (Li and Horne, 2001b) are onsistent with those measured by ersoff and Hulen (1996) after the temperature alibration. Note that this is different from the drainage ase in whih the two urves were different in some range of water saturation (see Figure 4). The mehanisms assoiated with this observation are not lear. Steam-water apillary pressure model for imbibition. We hose the data from the steady-state flow tests to onstitute the steam-water apillary pressure model. To do so, the normalized water saturation in imbibition ase is defined as follows (Li and Horne, 2001b): S wimb Sw Swi = 1 S S wi sr (9) where S wi is the initial water saturation for imbibition; it is equal to the residual water saturation by drainage in this study. S sr is the residual steam saturation by imbibition and S wimb is the normalized water saturation for an imbibition apillary pressure model. The Brooks-Corey apillary pressure funtion (Equation 7) was originally developed for drainage proess. Sinnokrot (1969) measured the oil-water apillary pressures of different roks (limestones and sandstones) at different temperatures and found that the Brooks-Corey funtion ould model the drainage oil-water apillary pressure urves but not the imbibition ones. hysially, it may not be appropriate to model imbibition apillary pressure urves using Equation 7. For example, there should be no parameters suh as entry pressure in an imbibition apillary pressure model. Considering this, we hose the imbibition apillary pressure funtion that we proposed previously (Li and Horne, 2001b): = p (1 S ) m wimb m (10) here p m is the apillary pressure at S wi ; m is a fitting oeffiient for the imbibition apillary pressure funtion. The imbibition apillary pressure funtion expressed in Equation 10 was used to fit the experimental data from steady-state flow tests after saling to the porosity of 1.9% and the permeability of 1.3 nd. Figure 6 shows the math between the imbibition apillary pressure funtion and the experimental data, given that S sr is equal to 13% (Horne et al., 2000). The values of the best-fit parameters are S wi = 0.20, p m = Ma and m = Also note that these values are only valid when the normalized water saturation is expressed as a fration rather than as a perentage. 9

10 Capillary ressure (Ma) Experimental Model Normalized Water Saturation (%) Figure 6: Normalized imbibition steam-water apillary pressure for a rok from The Geysers field. Using Equation 2 and the values obtained from fitting, we reated an imbibition steam-water apillary pressure model for geothermal rok. The model is expressed as follows: = σ S (1 wimb k φ ) (11) where the units of, σ and k are Ma, dynes/m, and nd respetively; φ and S wimb are expressed as frations. The surfae tension an be alulated one the reservoir temperature is known. Therefore, the imbibition steam-water apillary pressure urve for geothermal reservoir roks may be obtained using Equation 11. The model expressed in Equation 11 is suitable for imbibition proesses and is based on the same assumptions as the drainage model. Conlusions Based on the present work, the following onlusions may be drawn: 1. Both drainage and imbibition steam-water apillary pressure models have been developed for geothermal roks based on experimental data. These models are given by Equations 8 and Steam-water apillary pressure in geothermal reservoirs with any porosity, permeability, and temperature an be alulated using the proposed mathematial models. 3. The saled imbibition steam-water apillary pressure from a steady-state flow tehnique is onsistent with that measured by an adsorption method. However the drainage results from the two approahes are not onsistent in some range of water saturation. 10

11 Aknowledgments This researh was onduted with finanial support to the Stanford Geothermal rogram from the Geothermal and Wind division of the US Department of Energy under grant DE-FG07-99ID13763, the ontribution of whih is gratefully aknowledged. Nomenlature J(S w ) = J-funtion k = permeability k B = permeability in a Berea ore sample k G = permeability in a geothermal rok sample m = fitting oeffiient in the imbibition model M w = moleular weight of liquid p 0 = vapor pressure on flat vapor-liquid interfae p = apillary pressure G,T p 1 = apillary pressure of a geothermal rok sample at a temperature of T 1 G,T p 2 = apillary pressure of a geothermal rok sample at a temperature of T 2 p e = entry apillary pressure p m = apillary pressure at initial water saturation in imbibition p v = vapor pressure on urved vapor-liquid interfae p w = pressure of liquid phase r = radius of a apillary tube R = gas onstant S sr = residual steam saturation S w = water saturation S = normalized water saturation in drainage wd S wimb = normalized water saturation in imbibition S wr = residual water saturation T = temperature φ = porosity φ G = porosity in a geothermal rok sample φ B = porosity in a Berea sandstone sample θ = ontat angle measured through the liquid phase θ B = ontat angle in a steam-water-berea system θ G = ontat angle in a steam-water-geothermal rok system ρ w = density of liquid water σ = surfae tension σ B = surfae tension in a steam-water-berea system σ G = surfae tension in a steam-water-geothermal system σ = surfae tension at a temperature of T 1 T 1 σ T 2 = surfae tension at a temperature of T 2 λ = pore size distribution index 11

12 Referenes Ambusso, W.J.: Experimental Determination of Steam-Water Relative ermeability Relations, MS report, Stanford University, Stanford, California (1996). Brooks, R.H. and Corey, A.T.: Hydrauli roperties of orous Media, Colorado State University, Hydro paper No.5 (1964). Horne, R.N., Ramey, H.J. Jr., Shang, S., Correa, A., and Hornbrook, J.: The Effets of Adsorption and Desorption on rodution and Reinjetion in Vapor-Dominated Geothermal fields, ro. of the World Geothermal Congress 1995, Florene, Italy, May, 1995, Horne, R.N., Satik, C., Mahiya, G., Li, K., Ambusso, W., Tovar, R., Wang, C., and Nassori, H.: Steam-Water Relative ermeability, presented at World Geothermal Congress, Japan, May 28-June 10, Leverett, M.C.: Capillary Behavior in orous Solids, Trans., AIME, 142, , Li, K., and Horne, R.N.: "Aurate Measurement of Steam Flow roperties," GRC Trans. 23 (1999). Li, K. and Horne, R.N. (2001a): An Experimental Method of Measuring Steam-Water and Air-Water Capillary ressures, roeedings of the etroleum Soiety s Canadian International etroleum Conferene 2001, Calgary, Alberta, Canada, June 12 14, Li, K. and Horne, R.N. (2001b): An Experimental and Theoretial Study of Steam-Water Capillary ressure, SEREE (Deember 2001), p Mahiya, G.F.: Experimental Measurement of Steam-Water Relative ermeability, MS report, Stanford University, Stanford, Calif., ersoff,. and Hulen, J.B.: Hydrologi Charaterization of Four Cores from the Geysers Coring rojet, ro. of 21 st Workshop on Geothermal Reservoir Engineering, Stanford, Calif., Sanhez, J.M. and Shehter, R.S.: "Comparison of Two-hase Flow of Steam/Water through an Unonsolidated ermeable Medium," SE Reservoir Engineering, Aug. (1990), pp Satik, C.: "A Measurement of Steam-Water Relative ermeability," ro. of 23 rd Reservoir Engineering, Stanford University, Stanford, CA (1998). Workshop on Geothermal Sinnokrot, A.A: The Effet of Temperature on Oil-Water Capillary ressure Curves of Limestones and Sandstones, h.d. dissertation, Stanford University, Stanford, CA, USA (1969). Sta. Maria, R.B. and ingol, A.S.: Simulating the Effets of Adsorption and Capillary Fores in Geothermal Reservoirs, ro. of 21 st Workshop on Geothermal Reservoir Engineering, Stanford, Calif., Tsypkin, G.G. and Calore, C.: Capillary ressure Influene on Water Vaporization in Geothermal Reservoirs, ro. of 24 th Workshop on Geothermal Reservoir Engineering, Stanford, Calif., Urmeneta, N.A., Fitzgerald, S., and Horne, R.N.: The Role of Capillary Fores in the Natural State of Fratured Geothermal Reservoirs, ro. of 23 rd Workshop on Geothermal Reservoir Engineering, Stanford, Calif.,