CALCULATION AND USE OF STEAM/WATER RELATIVE PERMEABILITIES IN GEOTHERMAL RESERVOIRS KIYOSHI SHINOHAR4. June 1978 A WSTER OF SCIENCE REPORT

Transcription

1 CACUATION AND USE OF STEAM/WATER REATIVE PERMEABIITIES IN GEOTHERMA RESERVOIRS i A WSTER OF SCIENCE REPORT BY KIYOSHI SHINOHAR4 i I June 1978

2 ABSTRACT A new method to alulate the steam/water relative permeabilities in geothermal reservoirs was developed and applied to field data from Wairakei in New Zealand. This method has the following harateristis ompared to previous methods. This method : (i) needs the prodution flow rate history and the wellhead tempera- ture alone, (ii) evaluates the features of eah well separately, and i (iii) dereases the satter of the points on the alulated relative permeability urves. Bottomhole values of the parameters are needed for a more aurate determination of relative permeability urves. There are two ways to evaluate bottomhole onditions. One is by alulation, and the other is by measurement. Nethods to alulate the pressure drop and the heat loss in the wellbore were demonstrated. In partiular, a new method to alu- late the heat loss in the wellbore for finite flow rate was developed. It was also determined whih parameters should be measured in future field experiments. Finally, the study showed how to use the resulting relative permea- bility urves as a basis for analysis of future well tests for geothermal reservoirs. ii

3 TABE OF CONTENTS I IV. V. VI. VII. VIII. IX. ABSTRACT.... ii IST OF FIGURES... iv INTRODUCTION PRZVIOUS PIETHODS OF CACXATING STEAM/TtJATER REATIVE PERPlEX3IITIES IN GEOTHE?XA XZSERVOIFG... 2 NEW METHOD TO CACUATE STEA?/WATER lxeative PER4EABIITIES... 6 DESIGN OF AN OPERATIONA TEST TO OBTAIN XORE ACCURATE ESTIMATES USE OF REATIVE PEFWZABIITY CURVES CONCUSIONS NO?,IENCATURE ACKNOWEDGEHENT... 4 REFERENCES iii

4 IST OF FIGURES I 1 Q V S h G W STEAY-WATER REATIVE PEW.EABIITIES FROM WAIRAKEI WE DATA PRODUCTION HISTORY OF WE PRODUCTION HISTORY OF \E PRODUCTION HISTORY OF WE PRODUCTION HISTORY OF WE i PRODUCTION HISTORY OF WE Q VS R/Qs OF WE REATIVE PERMEABIITY OF XE Q VS \/Q OF WE S REATIVE PEI"IEAB1ITY OF WE Q VS G/Q OF WE S REATIVE PEIQIEABIITY OF WE Q VS \/Q, OF :JE REATIVE PERMEABIITY OF WE Q VS R/Q. OF WE REATIVE PERMEABIITY OF WE REATIVE PERMEABIITY OF EVERY WE BOTTOMHOE PRESSURE FOR WE PROCEDUX TO CACUATE BOTTOPlHOE CONDITION iv

5 I. INTRODUCTION It was reported by Grant' and Home2 that the steamlwater relative permeabilities in geothermal reservoirs an be estimated from the history of prodution flow rates and the enthalpy of the fluid. However, in order to get the enthalpy of the fluid, we need to know the pressure and temperature, after whih the enthalpy is a funtion of prodution flow rate, pressure, and temperature. Therefore, if we know the pressure and temperature o the fluid, we are able to estimate relative permeabilities from the history of prodution flow rate only. In this researh work, a graphial method to obtain relative perm- eabilities from the prodution flow rate history was developed and applied to the field data at Wairakei in New Zealand. This method removed some 1 assumptions whih were neessary for the methods of Grant and Horne. In addition, further studies of wellbore pressure drop and heat loss i were performed. These onsiderations are neessary to extend this method to future geothermal well test experiments that might not have suh a long history. For the analysis of wellbore pressure drop, the Govier i et al. J method was applied to the field data, and the results were om- pared with the atual values. For the analysis of wellbore heat loss, a new method for finite flow rates was developed. 7 Finally, the relative permeability urves obtained an be used in the derivation of methods of well-test analysis for the steam/water geothermal systems. The foundation of this work was shown. -1-

6 1 11. PREVIOUS METHODS OF CACUATING STEAM/WATER REATIVE PERMEABIITIES IN GEOTHERMA RESERVOIRS 1 Grant and Horne2 introdued the method to obtain the steam/water relative pemeabilities in geothermal reservoirs. Grant made several assumptions, namely: (i) The temperature of the well is onstant and is the same for every well. (ii) The pressure gradient in the reservoir does not hange even if the enthalpy of fluid hanges. (iii) The produt of permeability times flowing area is onstant. (iv) Wellhead steam and water disharges are the same as bottomhole values, thus flashing of fluid in rhe wellbore is negleted. (v) Fluid flows aording to Dary's law. (vi) Flashing in the reservoir is negleted. Under these assumptions, Grant derived the following equations: QW = pw -.. k pw k - AD' W k Qs = P,.-. ks - Ap' where p' is the pressure gradient, and from an energy balane: -2-

7 -3- PW h = - Pwkw Psks f- 1J.W US Grant plotted Q vs h (see Fig. l), then, assuming that kap' z B was on- stant for eah well, and saling eah well by an estimated B so that all the points lay on the same "urve," he extrapolated the pure water point, Qo, from the graph. At the pure water point, F is unity, so that from Eq. 1: W Thus, for any point: and : %l-lw p.=k W + k s ( ) QO h ( p Pwkwhw + Pskshs pw ps As Grant assumed that temperature was onstant and the same for every well, the thermodynami variables, pw, ps, pw, k, hw, and hs, were also on- stant. Therefore, Eqs. 6 and 7 are two linear equations for the two unknowns kw and k-. The relative permeabilities, kw and ks, an be found by 3 solving these two equations simultaneously.

8 -4- io\.. \ FIG. 1 : Q vs h GXDH (GRANT') I " " I I 1 91 /I WATER SATURATION

9 -5-2 Horne suggested that the prodution flow rate should be taken at bottomhole onditions, and that the total prodution flow rate should be divided by the wellhead pressure in order to smooth the effet of pres- sure hange as the well ages. Horne also reommended use of the atual bottomhole temperature, density, visosity, and enthalpy. He applied his improved method under these onditions to the field data from Wairakei in New Zealand. He obtained a pair of relative permeability urves, shown in Fig. 2. The water saturation in this analysis is based on a mass ratio rather than the more usual volume ratio.

10 111. NEW METHOD TO CACUATE STE&Y/WATER REATIVE PERMEABIITIES As enthalpy is a funtion of prodution flow rate, pressure, and temperature, if one knows pressure and temperature, enthalpy is unneessary to obtain relative permeabilities. In other words, only prodution flow rate history, pressure, and temperature are needed to alulate steam/water relative permeabilities in this new method. This is onvenient for field data analysis. Also, we an remove some of the assumptions whih Grant' made, thus: (i) Temperature need not be the same for every well. (ii) The produt of pressure gradient, permeability, and flowing area an be alulated expliitly, and thus need not be estimated graphially in Grant's proedure. The removal of these assumptions is important, as it dereases the variability of the alulated results. That is, we an onsider eah well's ondition expliitly in this method. Assumptions used for a preliminary test of this new method are summarized here. (i) The pressure gradient is onstant for a short time for eah well. (Although unneessary for the data used in the preliminary test, we an remove this assumption. This will be disussed in Setion IV.) C (ii) The produt of permeability and flowing area is onstant for eah well. (iii) Wellhead steam and water disharges are the same as bottomhole values, thus negleting flashing of fluid in the wellbores. (The effet -6-

11 -7- of making this assumption is small in the ase onsidered here, and we an, if neessary, remove this assumption also. This will be disussed in Setion IV.) (iv) (v) Fluid flows following Dary's law. Flashing in the reservoir is negleted. Under these assumptions and from Dary's law: k * Ap' S then : R - = QS (?)( 2-) and : Q=(--q-+-q)kAp'=($)[:+(:)] OWkW Psks Up' (9) Sine kap' is onstant for eah well in the simplest ase, and sine Q vs QJQs is almost linear when QJQs is small (beause k is nearly onstant and equal to unity for low water saturations), we an find Up' from either the interept or the gradient of the line on the graph Q vs smoothed using the least squares method. Even if Q,/Qs is not small, and S %'QS Q vs Qw/Qs is not linear, we an find kap' from the interept of the graph

12 -8- Q Vs Qw/Qs by urve fitting. If the value of Q at the interept $/Q, = is Q*, then: Beause k = 1 at Qw =, then, substituting S Eq. 1 into Eqs. 1 and 2: and : QS k =- s Q* From Eqs. 11 and 12, relative permeabilities k k an be omputed if w' s we know Qw, Q,, for eah well. Q*, vw, and vs. Of ourse, Q*, vw, and v are different This method was applied to the field data from Wairakei in New Zealand. Figures 3-7 show the prodution flow rate histories for five typial wells. Only the data between 1968 and 197 were used in order to make the use of the assumption of onstant pressure gradient reasonable. S Figures 8-17 show the Q vs \/Qs graphs, inluding the least-square fitted line used to determine Q*, and inferred relative permeability urves for eah well. Figure 18 shows the relative permeability urves obtained when every point is plotted on the same graph. In this analysis, water saturation is based on the flowing mass ratio, beause the flowing volumetri ratio is small for the field data. Volumetri water saturation annot be obtained from the flowing volume ratio beause (1)gas and liquid veloities are unknown, and (2) the immobile water saturation is unknown. In the petroleum industry, reservoir fluid saturations are usually estimated from material balane onsiderations. In Fig. 18, the immobile

13 OUTPUT

14

15 i i I i

16

17

18 -14- I I I I I I I ". \I I k.l m > n a 2 VI VI 3 y.. o 3 H kl

19 -15- a 3 A + o +++ i- t + o. t I m rl (u I o d- * (5 D.. QI

20 f. I d 3 e (4) Q

21 o d- a i sy MT

22 -18- i i.

23 I -19- I l l I l l d." i nl 3 (D m d- m

24 -2- n 3

25 4: v) + A I- +

26 1-22-, m \ 3 D i.. \o rl

27 -23- i st + + u h i s ' M Y Y.. h

28 m o +f + +f + T d- o ;l G! W 3

29 water saturation is not onsidered in this illustration. The relative permeabilities obtained for eah well agree losely and form a single urve. The operational proedure for this method of alulating relative permeabilities is summarized below as Method A. METHOD A Well Data Required: Water Disharge Qw Steam Disharge Q S Temperature T at well head Well Data Inferred: Steam Visosity, Water Visosity, s W Proedure: graph paper. (2) Extrapolate the data to the Q,/Q =O axis to obtain Q*. (3) For eah of the data points, evaluate kw and k from Eqs. 11 and 12. (4) Combine the separate well results. S S

30 IV. DESIGN OF AN OPERATIONA TEST TO OBTAIN HORE ACCURATE ESTNATES If one an assume that wells are operated for a long time and that the reservoir has reahed steady-state or pseudo-steady state, then for a radial system: 1 - for steady state 2 a= - 3 for pseudo-steady state 4 Thus, if: = 2nkh r e -a Gn (y) W -26-

31 then : i and : R - = 'S (:)(?) Thus, we an find C from the interept of the graph Q/ (T-pBH) vs %/Qs. In order to use this method, reliable and pbh values are needed. Furthermore, for aurate alulation of relative permeabilities, as Horne suggested, we should use bottomhole values of Q and. However, to do so in ases where there are no available measured bottomhole values, we would have to alulate the pressure drop and heat loss in the wellbore W S onsidering flashing of fluid. Flashing, two-phase flow alulations are i not easy. Even if the flashing of fluid in the wellbore were negleted, the pressure alulation must evaluate a two-phase vertial pressure drop. 3 To investigate the possibility of doing this, the Govier et al. method was applied to the field data. Figure 19 shows the results of bottomhole pressure alulations in well 72. The alulations indiate that the flow regime is an annular mist flow. The effetive depth of the well was assumed to be 1, E t from the math point of the measured bottomhole pressure and alulated bottomhole pressure in 1967, whih is the only year for whih we have data for both measured bottomhole pressure and wellhead pressure. As a matter of fat, the asing program of the well shows asing of 7.2 inh diameter down to 92 ft, and 5.5 and 2.25 inh

32 ... x hl

33 -29- diameter down to 1,98 ft, so the estimate of the effetive depth is not unrealisti. As Fig. 19 shows, the alulated bottomhole pressures in well 72 agree with the atual values. However, we do not have measured bottomhole pressures after 1968, so we annot ompare exatly. For the alulation of heat loss from the wellbore, we ar, use Ramey's method. 576 However, this method is inaurate for small flow rates. As part of this work, a method to alulate the heat loss from the wellbore for finite flow rates was developed.' By solving the heat balane equations exatly, the following equation was derived: where: E = Wf (t) 2rk The heat loss from the wellbore an be obtained by this equation, and thus the bottomhole values of enthalpy and thene the steamlwater ratio an be alulated. However, if the well has been produed for a long time and is not too deep, the alulated heat loss from the wellbore beomes small, and the differene between wellhead and bottomhole steamlwater ratios is almost negligible. Thus, after an initial hek on the magnitude of the heat loss, this additional onsideration was not applied to the Wairakei field data used here, sine the downhole values of disharge differ from

34 -3- surfae values by less than 2%. However, it will probably be neessary to inlude alulation of the wellbore heat loss in the ase of newer wells. Having alulated pressure drop and heat loss in the wellbore, the bottomhole onditions may be omputed from the wellhead onditions by the proedure outlined in Fig. 2. The proedure in this ase is summarized below as Method B. PETHOD B Well Data Required: Water Disharge Q Steam Disharge Q Temperature, T Pressure, pbh - J Reservoir Pressure, p W S at well head Well Data Inferred: Visosities pw and Us Proedure: (1) For eah well, obtain bottomhole pressure, pbh using the two-phase pressure drop alulation. (2) Calulate the wellbore heat loss 7 and thus obtain the bottom- hole enthalpy, and thene bottomhole values of R and Qs. (3) Plot (R+Q~) /G-pBH) against QJQ, on normal Cartesian graph paper. (4) Extrapolate the data to the QJQs=O axis to obtain C i (Eq. 15). (5) 'Then evaluate k and k for eah point from Eqs. 16 and W S 17. I (6) Combine and ompare the separate well results.

35 -31- I FIG. 2: PROCEDURE TO CACUATE BOTTOMHOE CONDITION

36 -32- However, even though we an now alulate the pressure drop and heat loss in the wellbore more exatly, learly the best approah would be to obtain those data diretly. It would be useful also t o hek the re- liability of the heat and pressure loss alulation methods proposed in this report by using measured values. In suh experiments, the parameters that should be measured would be the bottomhole and wellhead pressures, the bottomhole and wellhead temperatures, the prodution flow rates, and the average reservoir pressure. Simultaneous measurement of these parameters would be neessary. In summary, when we do not have information other than the prodution flow rate and the wellhead temperature, we an apply Xethod A proposed in Setion I11 to obtain relative permeabilities in geothermal reservoirs. If we wish to find more aurate relative permeabilities, the bottomhole onditions an be estimated through the alulation proedure shown in Fig. 2, and these values an be used to obtain more aurate relative permeabilities as shown in?lethod B. However, the best proedure would be to perform the experiment and obtain bottomhole values, after whih the relative permeability urves an be estimated using Hethod B, starting at Step 3 of the proedure.

37 V. USE OF REATIVE PERMEABIITY CURVES Reliable relative permeability urves for geothermal reservoirs an be used as a basis for the development of new well-test analysis tehniques. This approah may be of partiular importane if the relative permeability urves are different for eah reservoir. In suh ases, obtaining the relative permeability urves would be essential to welltest analysis. Boiling, two-phase flow is generally non-isothermal. However, simplified equations an be written in terms of isothermal flow. From the material balane for an isothermal radial flow system: Assuming that k and 4 are onstant: Then : - P is given as: - %+Qs p = v +v W S -33-

38 -34- Then : k k W S /-. -\ If we an assume that the pressure gradient is negligible, then: k k W S (26) This equation is similar to the diffusivity equation of the liquid flow ase. If we an solve this equation, we an develop useful tehniques for well-test analysis of two-phase geothermal systems. In solving this equation, we would use the relative permeabilities obtained using the method proposed in this report. This should be a fruitful area for future researh.

39 VI. CONCUSIONS A new method has been developed to alulate steadwater relative permeabilities in geothermal reservoirs. This method has the following harateristis: (i) We an get the relative permeabilities from the prodution flow i rate history and wellhead temperature alone. (ii) This method avoids some of the simplifying assumptions whih 1 Grant made, and is an improvement on the method of Home2 in that the bottomhole harateristis an be expliitly onsidered. (iii) This method dereases the satter of the data by evaluating pre- isely Up' values for eah well. i Further, the method an be improved greatly if more omplete informa- tion on the bottomhole onditions of the well is available. In order to obtain bottomhole onditions, we must either alulate the pressure drop and heat loss in the wellbore, or must measure the bottomhole onditions diretly in the field. This study indiated how to alulate the pres- sure drop and heat loss in the wellbore, and has shown whih parameters we should measure in the field. The alulated relative permeability urves may be used as a basis for future well-test analysis of geothermal reservoirs. Thus we have designed an operational test proedure to obtain these urves in future appliations. With the limited data available to this study, steam/water -35-

40 -36- relative permeabilities have been obtained suessfully. With more om- plete information, the method developed herein should produe reliable results.

41 VII. NOMEXCATURF, 2 A = flowing area ( ) a = geothermal gradient (degree/) B = onstant (=k4p') = onstant (= -r:wkh ) Rn (--) -a e = water saturation based on a mass ratio H = depth of well () h S = enthalpy of StSam Ah = enthalpy loss M K = thermal ondutivity of earth 2 k = permeability ( ) k = relative permeability of water w k = relative perneability of steam S -37- T--degree

42 p' = pressure gradient -38- (+) - p = average reservoir pressure T-- (+) Pwn = wellhead pressure T-* (k) = bottomhole pressure P~~ p = pressure drop M R = mass prodution flow rate of water (-) T M Qs = mass prodution flow rate of steam (-) T.M Qo = mass prodution flow rate at pure water point \--) T M Q* = mass prodution flow rate at pure steam point (-) T (Mi$ q = heat loss - r = external radius of reservoir () e r = wellbore radius () W t = prodution time (T) V = volumetri prodution flow rate of water (-) 3 W T V = volumetri prodution flow rate of steam (-) 3 S T w = mass prodution flow rate of fluid M (-) T x = wetness of fluid GREEK NOMENCATURE Q = onstant Q = thermal diffusivity of earth (-) 2 1 T = density of water M 3 P = density of steam (-1?I S 3

43 p = average density of fluid (7) M 9 = visosity of water (-) M W T'?is = visosity of steam (7) M w W = kineti visosity of l 7 m- water (-) T v s = kineti visosity of 2 steam (-) T $ = porosity..

44 VIII. ACI(IUOFTEDGEXE8T The enouragement and advie of Dr. R. X. Eiome during this re- searh is greatly appreiated. Finanial support for this study was provided by the Nippon Steel Corporation, Japan, by the Department of Energy Grant, B Contrat No , and by Stanford Gniversity. -4-

45 IX. REFERENCES Grant, M.A. : "Permeability Redution Fators at Wairakei," presented at the AIM-ASXE Heat Transfer Conferene, Salt ake City, Utah, Aug , Home, R.N., and Ramey, H. J., Jr. : "Steam/<ater Relative Permeabilities from Prodution Data," presented at the GRC Annual Eieeting, Hilo, Hawaii, July 25-27, Goviar, G.W., Aziz, K., and Fogarasi, M.: tfpressure Drop in Wells Produing Oil and Gas," J. of Can. Pet. Teh. (July-Sept. 1972), Govier, G.W., and Fogarasi, M.: "Pressure Drop in Wells Produing Gas and Condensate," J. of Can. Pet. Teh. (t.-de. 1975), Ramey, H.J., Jr.: "Wellbore Heat Transmission," J. Pet. Teh. (Apr. 1962), Ramey, H.J., Jr.: "How to Calulate Heat Transnission in Hot Fluid Injetion," Pet. Eng. (Nov. 1964), 11-12, Horne, R.N., and Shinohara, K.: "A Note on Wellbore Heat oss in Injetion and Prodution," Paper SPE 7153, submitted to J. Pet. Teh