Shell Oil Company. Houston, Texas, USA

Transcription

1 P z SHUT-IN ANI) FLOWING BOTTOM HOLE PRESSURE CALCULATION FOR GEOTHERMAL STEAM WELLS Mihael J. Eonomi des Shell Oil Company Houston, Texas, USA ABSTRACT t n z r The suess of pressure transient analysis often depends on the aurate measurement or estimation of the bottom hol e pressure. Measurement an be aomplished by a desending probe. Estimation may be realized via a alulation method. Although a pressure survey may be a more desirable mode, it is nevertheless time onsuming and ostly. In the ase of a geothermal well, two additional shortomjngs are obvious high temperature and frequent presene of highly orrosive nonondensable gases. The latter may render expensive pressure bombs obsolete. A alulation method for prediting bottom hole pressures based on easily obtainable well head parameters is therefore not only desirable, but neessary. Several orrelations are presently available, This paper presents four alulation proedures for the estimation of bottom hole pressures. Two of the methods are for stati pressure, suitable for buildup analysis, while the remaining two are for flowing wells. In both ases, the first proedure is the established, lassi tehnique followed by a novel orrelation, taflored to suit wells that operate at either saturated or slightly superheated onditions. These orrelations are partiularly appliable to the reservoirs in The Geysers area, where the properties of the geothermal fluid losely math the assumptions in this paper. INTRODUCTION Reservoir engineering priniples long established in oil and gas fields have proven valid in geothermal reservoirs. In spite of minor idiosynraies, saturated or superheated steam reservoirs behave like gas ondensate or pure gas reservoirs. In this vein, the lassi methods for alulattng bottom hole pressures for gas wells an be readily extended to steam wells. Perhaps the best known are: (a) the method for stati and flowing gas olymns outlined in the State of Texas Railroad Commission Bak-Pressure and (b) the Stati and Flowing Gas Column Method, by Cullender and The latter method is based upon a mehanial energy balane. The alulation proedure for the flowing bottom hole pressure that is presented in this report is also based on a mehanial energy balane. Fowler3 and Sukkar and Cornel14 presented a general orrelation in whih they utilized an integral form of the gas law deviation fator, Z, and they assumed a -139-

2 -140- onstant average temperature. This is not a shortoming of the method, sine the length of the wellbore an be divided into several setions. The Sukkar and Cornel1 method.requires, however graphial interpolations, a somewhat mhbersome proedure. Messer et al.5 presented a method for the alulation of the bottom hole pressures for "deep, hot, sour gas wells," whih inadvertently gave rise to the method for the flowing ase presented in this report. DISCUSSION LJ 1 1 P The lassi approah to the shut-in bottom hole pressure alulation originates from the pressure gradient in a gas olumn (desription and units of all vartables appear in the nomenlature setion). The gas law an be expressed as: dp,l dh 144 PV = ZNRT () from whih an expression for density, p, an be extrated: where M is the moleular weight. an equation of the form: From Eqs. 1 and 3, we an easily develop - = t~ dp P dh (4) The onstant is a result of introduing the value of the moleular weight of steam [18), the universal gas onstant,r (10.73 psi ft3/lb mole OR), and onverting psf to psi. Equation 4 an be formally integrated over the range of the wellbore, yielding: H/= pws = Ptse (5) where 7 and T are onstant average values. Equation 5 an be easily solved in the ase of saturated steam via trial and error. One may assume a bottom hole pressure, P, whih will in turn furnish a bottom hole temperature (from the steam tak!?es). Armed with the pressures and temperatures of both wellhead and bottom hole, the averages an be alulated whkh an then provide a value for the gas deviation fator,. P an tfien be alulated via Eq. 5 and ompared with the assumed value. 'The proedure an be repeated until a desirable agreement is attained. An example alulation using this approah an be found in Appendix C. The method just desribed is of a general utility. It an be applied in the ase of geothermal wells that operate anywhere in the domain of saturated steam or superheated steam. In the ase of reservoirs suh as the ones at The Geysers, where the produt is either saturated or slightly superheated, the following analysis is proposed. f -

3 -141- U By simple observation (see Fig. l), one may reah the seemingly unexpeted onlusion that vapor density is a linear funtion of pressure Pn the usual range of a geothermal steam well. Appendix A ontains some theoretial justifiation for this observation. The funtion is of the form: p=a+f3p (6) Introdution of Eq. 6 in Eq. 1, separation of variables, and integration yields : i a i The onstants a and 6 an be obtained with a least squares fit of readily available steam table data. Following suh a fit of values between 50 psia and 500 psia, the alulated onstants are: a = =.001 Equation 7 an be then manipulated into the following form: = -6 + (6+Pts)e, H pws The major and obvious advantage of Eq. 8 is that it an afford diret alulation of Pws without trial and error. One needs only the wellhead pressure, P, and the depth, H. P values for a geothermal well at The Geysers alhated by the two metho% desribed above are the same (to the nearest psi),as shown in Appendix C. Both methods desribed above presume a wellbore that is truly stati. Lingerlng transient effets may influene the auray of the methods at early shut-in times due to inertia, totally disregarded in the original assumptions. The phenomenon has not esaped the attention of reservoir engineers., An osillating front may be isolated and tested for the momentum ausing fore; The latter is related to the driving fore in the reservoir, whih in turn an be analyzed for the estimation of reservoir parameters. However, for the purposes of this report, very early transient analysis beomes a moot point sine the intent of bottom hole pressure alulation is to assfst well testing tehniques. In order to avoid both storage and inertia effets, all analyzable points are 1-1/ log yles of time removed from the essation of the harateristi 45 line (in a log-log plot of, PIP versus time). Wholesale redene should not be attributed to alulated bottom hole pressures at very early times. The results would be useful, though, in observing the effets of the osillating front and its duration. (8)

4 -14- DYNAMIC OR ROWING BOTTOM HOLE PRESSURE The so-alled exponential form of the flowing 'bottom hole pressure alulation is a modifiation of the stati olum method whih utilizes the Moody frition fator and an average Z and T.g The most familfar form of the equation is: -ii t i. A BPtf + AZ - Pp.0375 L/E (9 1 I where 6 = H/L - 5 and A = 667 fmq T /d Pp Equation 9 is generally appliable to any gas olumn. an be modified to apply in a geothermal well : The same equation where A = + A = (BPtf+A)e.033H/n Bpwf fmt - T W C d3 The proedure ditated by this orrelation is again trial and error. One assumes P obtains Twf, f, 7, and 7. Equations 10 and 10a an then be used to yfulate P and ompare it to the assumed value. An example alulation fllu;tr#fing this approah is provided in Appendix C. a Equation 10 an be applied to a stati gas olumn as well. If there is no flow, A = 0; by simply taking the square root of both sides of the equation, we an extrat Eq. 5. The alulation method to be desribed uses the same original basis as Cullender and Smith, Sukkar and Cornell, and Messer et a1. It will digress, though, from the somewhat umbersome proedures that the fi rst two methods require, thanks to the apparent linearity of the gas deviation fator, Z, with respet to pressure over the domain of interest of a geothermal well. All of the above-mentioned methods start from a mehanial energy bal ane : UdU VdP + dh dwf = -dws ( where VdP = pressure - volume potential energy dh = potential energy due to position -= UdU kineti energy dwf = frition loss dws = shaft work % 4 i L

5 -143- L, h Negleting kineti energy and shaft work, introduing the value of the frition loss and representing veloity by U = WV/A, the following equation emerges : H fmwvdt VdP t 7 dl +'- = 0 L g,da' Equation 1 an be solved for dl, yielding: dl = V dp fmw ' H gda +s (1 3) The value of the speifi volume, V, an be alulated using the gas law and, in the ase of steam: P (I V = TZ/P (14) Equations 13 and 14 an be then ombined and manipulated into:.01165(h/l) dl = -ZdP/ P f ~10:~ fmwt(z/p) d5(h/l). (1 5) B 'F (d ' Introduing a parameter A: Then Eq. 15 beomes: 4.84~1. A = 5 d P (H/t),.01165(H/L)dL T Oe4 fmwt -(Z/P) dp l+a(z/p) P. Remembering that PIP, = P, and that dp/p dp&, then Eq. 16 beomes: Z- dpr.01165(h/l)dl E Pr T 1 +A (Z/P,) (1 6) Again, observing a plot of the gas deviation fator as a funtion of pressure, with temperature as a parameter, one onludes that for the domain of interest of a geothermal well, the relationship is linear and of the form: Z = 1 + mpr (18) Figure is a plot that an be desribed by Eq. 18. The interept is equal to unity, while the slope, m, is a harateristi of the flowing average temperature. Equation 18 an be ombined with Eq. 17 to yield: (1 7)

6 -144- The left side of Eq. 19 an be integrated readily, whereas the right side an be formally integrated wwi some diffiulty. Appendix B ontains the. result of the integration. Although the equation thus obtained is far from being linear, it an Be easily programmed in a handheld alulator to obtain diret readings for the bottom hole pressure. NOMENCLATURE A = ross-setional area of pipe, ft d = diameter of pipe, inhes fm = Moody frition fator 9, = 3.174, onversion fator, (ft-lb mass/( ft-lb fore))(ft/se) H = vertial distane, ft y = gas gravity (to air) L = distane along tubing, ft n = number of moles P P = pressure, psia = ritial pressure, psia b Pr = redued pressure Pt = tubing head pressure, psia = Pw bottom hole pressure, psia q = flowrate at 50 F and psia, MMsf/D R = gas law onstant T = average temperature, OR U = veloity, ft/se VI = total volume, ft3 V, = molar volume, sf/lb-mol (see Eq. A-8) V = speifi volume of flowing fluid, u ft/lb mass W = mass flowrate, lb mass/hr Wf = energy loss due to frition, ft-lb fore/lb mass = shaft work done by flowing fluid Ids Z = gas law deviation fator p = density X = molal heat of vaporization REFERENCES 1. Anon.: "Bak-Pressure Test for Natural Gas Wells," State of Texas, Railroad Commission, Oil & Gas Division.. Cullender, M.H., and Smith, R.V.: "Pratial Solution of Gas-Flow Equa- Li tions for \jells and Pipe1 ines with Large Temperature Gradients," Trans. AIME (1956), 07, 't

7 -145- k h '. 6. Fowler, F.C.: "Calulations of Bottom-Hole Pressures," Petrol. Engr.. (1947), 19, NO. 3, Sukkar, Y.K., and Cornel'l, D.: "Diret Calulation of Bottom-Hole Pressures l'n Natural Gas Mells," Trans - ' AIME (1955),- 04, Messer, P.H., Raghavan, R., and Ramey, H.J., 3r. : "Calulation o f Bottom Hole Pressures for Deep, Hot,-Sour Gas Wells," J. Pet. Teh. (Jan. 1974), 85. Katf; D.K., and-coats,? K.H.: Underground Storage of Fluids. (1968),.C h -7 i

8 ~ APPENDIX A The apparent linearity between vapor density and pressure at the saturation ondition an be aptly demonstrated using the Clausius/Clapeyron equation and the gas equation. The Clausius-Clapeyron equation is: -- dp - A dt ~to- Where P = Vapor pressure h Molal heat of vadorization Vg, V1 = Speifi volume bf gas and 1 <quid respetively The gas law states that P = ZpRT M Differentiating (A-) with respet to pressure yields: (A-) (A-3) L The first term inside the parenthesis an be negleted sine it I has a very small value. Combining (A-1) and (A-3) and rearranging one an solve for 3: - M dp ZRT x (A-4) It an be easily onluded that the right hand side of the equation is roughly onstant for the range of geothermal interest.

9 -147- e, APPENDIX B 4' 5 The rr right hand side of equation (19) is.-.- f l+mp, Pt, 'Py) and it an be separated into: C i The first integral an be evaluated and it yields: -_ 0 I The seond integral is: 3 It t

10 -148- APPENDIX C: SAMPLE CALCULATIONS Stati Bottom Hole Pressure For the sample alulation of the stati bottom hole pressure, the following field data will fie given: pressure history of a buildup test and the depth of the well. The produt is either saturated or slightly superheated steam. H = 6615 ft t [se) t (min) Equation 5 an be utilized:.01165h/'rr pws = Ptse P (psig) (et.) At time 5 min, the wellhead pressure, P, is 7 psig or 85 psia. (Atmospheri pressure is 13. psia). Fromt%he steam tables, the wellhead temperature, T, is 41 F. At fprst trial, assume a bottom hole pressure of 30 psis, wkth furnishes a bottom hole temperature of 43OF. Average pressure, P, and temperature (T) are 30.5 psia and 877.5OR, respetively. The gas deviation fator,, is.905. Using Eq. 5, the P alulated is 314 psia, whih varies from the assumed value. A seond!pial with Pws assumed = 315 psia is suessful, with Pws alulated = 315 psia. Alternately, Eq. 8 avoids the trial and error approah: h = -6 pws + (6+Pts)e For t = 5 min, Pts = 85 psia and H = 6,615 ft Flowing Bottom Hole Pressure Pws = 315 psia For the flowing bottom hole pressure alulation, the following field data are given: pressure history, depth (7,500 ft), asing diameter (9-5/8"), H/L = 1, and flowrate.(100,000 lb/hr). Geothermal fluid is either saturated or superheated vapor. Equations 10 and loa an be used to alulate the flowing bottom hole pressure for this well. Let the wellhead pressure be 400 psia. The first task is to get a value for the Moody frition fator (fm) that appears in Eq. loa. A ursory look at the Moody grition fator hart would instrut that at highly turbulent flow (Re > 10 ), the frition fator depends only on the

11 -149- relative roughness of the onduit. Suh highly turbulent flow is mostly the rule i'n geothermal steam wells. The Reynolds number in the example an be alulated by the equation: Re = 6.3 W dv where w = flowrate (lb/hr) d = diameter (in] 1.1 = v.lsosity (p) 6.3 x 100,000 = 3.9x10 6 Re 9.65 x The roughness of the asing is E =.0015 in., and the relative roughness E/d = (d = 9.65 in.). The Moody frition fator is then fm =.0135 (from Fig. 3). The same value of fm is obtained for any Reynolds number larger than 106. It is neessary to note here that the roughndss of a wellbore will inrease with time as saling ours. The degree of saling varies signifiantly among geothermal reservoirs, and it is a funtion of the geohemistry of eah region. At a wellhead pressure of 400 psia, the saturation temperature is 444.6OF (from the steam tables). Assuming a bottom hole pressure of 450 sia, the assoiated saturation temperature is 456.8"F. Therefore, = 45OOF (910 R), P = 45 psia. The Z fator is equal to.855. We an then alulate parameter A in Eq. loa: A = 1.719x104 f$'f\.i' d5 A = 1:719~10-~ x.0135 x (,855) x (910)' (9.65)5 5 = 1.701~10 x (100,000)' From Eq. 10: pwf = (ptfta)e The value of Pwf alula A seond trial is-obviously in order. Assume P = 493 psia and Twf = 465.5OF. Therefore, T = 458.1OF (918OR), and = 4vg.6 psia. The 7 fator is equal to.851.

12 Then, from Eq. loa: and from Eq. 10: A = 1.715~10 L t J pwf = 493 psi'a (good agreement with assumed value) - Alternately, using the proedure developed in $his report and using T = 45OoF, the value of onstant A = 6.373~10'. The slope in Eq. 18 (from Fig. ) is m = -1.l. By solving equations in Appendix B, we an.obtain a value for Pwf = 49 psia. 4 I 1.0 J f J 8.a.t T, O F t FIG. 1: VAPOR DENSITY OF SATURATED STEAM

13 t z + P 1 - I I I I 1 I I I I I lS -1:. REDUCED PRESSUflE PI FIG. : COMPRESSIBILITY FACTOR FOR SATURATED AND SUPERHEATED STEAM I 6 L f W FIG. 3: FRICTION RELATIVE FACTOR AS A FUNCTION OF REYNOLDS NUMBER WITH ROUGHNESS AS A PARAMETER