SGP - TR - 30 SGP - TR - 30 CON-781222-26 PROCEEDINGS OURTH WORKSHOP GEOTHERMAL RESERVOIR ENGINEERING Paul Paul Krugerand and Henry.. Ramey, Ramey., r. r. Editrs December13-15, 13-15., 1978 DISTRIBUTION O THiS [;(~l~1e1h IS UNLlrl1lTED
AN EVALUATION O AMES' EMPIRICAL ORMULAE OR THE DETERMINATION O TWO-PHASE LOW CHARACTERISTICS IN GEOTHERMAL WELLS P. Cheng and M. Karmarkar University f Hawaii Hnlulu, Hawaii 96822 Intrductin One f the mst ecnmical and simple methds f determinatin f tw-phase flw parameters in gethermal well testing is the s-called ames' methd [1,2]. The methd cnsists f the measurements f lip pressure (p), and the flw rate f water (w) by a cnventinal weir. The stagnatin enthalpy (h ) is then determined frm a plt shwing versus w/po.96 which is gmpirically determined by ames [1,2]. The 0 mass flw rate is then determined frm the fllwing empirical frmula 0.96 G = 11,400 ~1.102 (1) where G is the ttal mass flw rate in 1b m /sec-ft 2, p is the lip pressure in psia, and h O is the specific enthalpy in BTU/lb m. The abve relatin is empirically determined fr discharge pressure up t 64 psia and pipe diamters up t 8". r pipe diamters smaller than 0.2", it has been suggested that the value f 11,400 be replaced by 12,800. In view f the widespread use f the ames' methd, it is imprtant t assess its accuracy and range f applicability. h Tw-Phase Critical lw Thery In this paper we shall cmpare the wellbre discharge characteristics btained frm ames' empirical frmulae t thse predicted by auske's tw-phase critical flw thery [3]. auske suggested that in tw-phase flw the maximum discharge rate may nt necessarily be accmplished by a shck frnt. He prpsed that at the critical flw cnditin the abslute value f the pressure gradient at a given lcatin is maximum but finite fr a given flw rate r quality, i.e., (dp/dz)g = maximum and finite,,x (2) where z is the crdinate alng the streamwise directin, and x the quality f the saturated mixture. Under the assumptins f (i) annular flw pattern, (ii) tw-phases -207-
in thermal equilibrium, (iii) negligible frictinal lss, and (iv) nedimensinal steady flw, auske (3] btained the fllwing analytical expressin fr the critical flw rate f a saturated mixture: (3) - 2xk 2 +k 2» dx/dp], g = 32.2 c 1b -ft m -=-----=-2 ' and 1b f -sec k v g and denting the specific vlume f the saturated vapr and liquid respectively. Thus, the critical flw rate can be calculated frm Eq.(3) if the steam quality and the lip pressure are knwn. The crrespnding stagnatin enthalpy can be determined frm 2 3 2 2 h f (l-x) + h x + (G /2g )«x v /Rg ) g c g 3 2 2 +(l-x) v f /(l-rg) )/, (4) where R is the gas vid fractin which is related t steam quality [4]. In cmpa~isn with experimental data, Levy [4] fund, hwever, that using Eq.(4) fr the cmputatin f h O wuld lead t under-predictin f the mass flw rate. r this reasn we shall cmpute the stagnatin enthalpy n the basis f a hmgeneus mdel, i.e., 2 2 h O = h f (l-x) + hgx + G v h /2g c, (5) where v h = v f (l-x) + vgx and = 778 ft-ibm/btu The weir flw rate is then determined frm w = G(l - x) (6) Results and Discussin r a given set f values f lip pressure and steam quality and with the data f saturated steam-water prperties [5], Eq.(3) can be used fr the cmputatin f ttal mass flw rate G. The stagnatin where Q [(l-x+kx) x (dv/dp) + (v g (1+2kx-2x) +v f (2xk-2k -208-
enthalpy and the weir flw rate are then determined frm Eqs.(5) and (6). The results f the cmputatins fr the lip pressure frm 14.7 psia t 200 psia fr gethermal well testing applicatins are pltted in igs. 1 and 2. When the lip pressure and the weir flw rate are measured in a gethermal well test. the stagnatin enthalpy f the reservir. the steam quality at the well head. and the ttal mass flw rate can easily be determined frm these plts. T assess the accuracy f the ames' methd. calculatins were carried ut fr five different sets f lip pressure and weir flw rate using ames' empirical frmulae and auske's theretical predictin (i.e igs. 1 and 2). The results fr ttal mass flw rate. the stagnatin enthalpy. and the steam quality are tabulated in Table 1 fr cmparisn. It is shwn that the results based n the tw methds differ within 8%. References L ames. R "Measurement f Steam Water Mixtures Discharging at the Speed f Sund t the Atmsphere. New Zealand Engineering. pp. 437-441 (1966). 2. ames. R. "Steam-Water Critical lw Thrugh Pipes." Prc. Inst. Mech. Engrs. v. 176. pp. 741-748 (1962). 3. auske. R "Cntributin t the Thery f Tw-Phase. One Cmpnent Critical lw." Argnne Natinal Labratry. Rept. N. ANL-6633 (1962). 4. Levy. S "Predictin f Tw-Phase Critical lw Rate." A.S.M.E. Paper 64-RT-8. 5. Keenan.. and Kays. Keys,. G., Steam Tables, hn Wiley and Sns. Inc New Yrk. N.Y. (1969). -209-
600 500, u."... 400 E -0, 0:: 300 ~ LL Lip Pressure (psia) 3= 200 100 400 500 600 700 800 900 1000 1100 1200 1300 Stagnatin Enthalpy, h (Btu/Ibm) ig. 1. Weir lw Rate vs. Stagnatin Enthalpy at Selected Values f Lip Pressure Accrding t auske's Thery --l-~,...---,----,----.---r---r-...--.--'--:~~~~~-- -210-
600 500 I U." '- 400 E...0 -. ō ly: 300 ~ LL Lip Pre ssure (psia) ::; 200 100 O-+---r-.---~---,---,------.-~..,.----,----r-.-.,...--,-----r=~-t- 0.2 0.3 0.4 0.5 06 0.7 Steam Quality, X ig. 2. Weir lw Rate vs. Steam Quality at Selected Values f Lip Pressure Accrding t auske1s Thery. -211-0.8 0.9 1.0
Table 1. COMPARISON O RESULTS BASED ON THE AMES' METHOD AND AUSKE'S ANALYTICAL MODEL Case p w h G x Methd (psia) (1 b m /sec-ft2) BTU71bm lb m /sec-ft 2 14.7 40 730.90 88.44.54 auske () 800.00 95.13.58 ames () 2 25.0 85.5 698.78 750.00 164.59 170.06.48.50 3 60.0 226.0 697.79 403.39 715.00 415.42.44.46 4 100.0 105.0 1004.52 419.84 985.00 476.60.75.78 5 150.0 53.0 1148.87 1130.00 523.41 590.00.90.90-212-