P. Cheng and M. Karmarkar Department of Mechanical Engineering University of Hawaii Honolulu, Hawaii 96822

Transcription

1 THE APPLICATION OF TWO-PHASE CRITICAL FLOW MODELS FOR THE DETERMINATION OF. GEOTHERMAL WELLBORE DISCHARGE CHARACTERISTICS P. Cheng and M. Karmarkar Department of Mechanical Engineering University of Hawaii Honolulu, Hawaii INTRODUCTION f- At the Fourth Workshop on Geothermal Reservoir Engineering, we presented some preliminary results1 on the evaluation of James' empirical f~rmulae~,~ for the determination of geothermal wellbore discharge characterfstics, based on Fauske's two-phase critical flow the~ry.~ Since that time, we have performed further work on the comparison of James' formulae with other two-phase critical flow models, and have also investigated the effects of noncondensable gases in wellbore discharge. The following is a summary of the work we have done. COMPARISON OF JAMES' FORMULAE WITH OTHER TWO-PHASE CRITICAL FLOW MODELS There are three different two-phase critical flow models proposed respectively by Fa~ske,~ Moody,5 and Levy.6 These theoretical models differ from one another in essentially two aspects: (1) the governing equations used for analysis, and (2) the criterion for two-phase flow to occur. -The latter leads to different expressions for the velocity slip ratio. Moody's and Levy's models were recomputed for the range of parameters.suitable for geothermal well tests, and results are presented in an easy-to-use form in Figs. 1 and 2. A comparison of results based on the three theoretical models to those based on James' empirical formulae for a few specific cases is also shown in Table.1. Note that although the James' formulae were determined for pressure below 64 psia, it is shown that they are in agreement with theoretical results even above 64 psia. In fact, results based on James' method are within 8% deviation with those based on the three theoretical models for theentirerange of lip pressure considered. EFFECTS OF NONCONDENSABLE GASES IN WELLBORE DISCHARGE U The presence of noncondensable gases would affect the critical discharge pressure of steam due to the partial pressure of gases in the gaseous phase of the two-phase flow. To take into consideration the presence of C02 in the wellbore discharge, James8 proposed to correct -49-

2 -50- the measured lip pressure p by p' formula: according to the following empirical P' = p(1 - ~/.2) where y is the ratio of the weight of CO present in the steam phase to the weight of steam in the discharge. dth p' and the measured water flowrate, the stagnation enthalpy h and the total mass flowrate are then determined by the usual procedures. 0 To evaluate James' modified empirical method, we have extended Fauske's critical model to include the presence of C02 in gaseous phase of the two-phase flow. Figure shows the weir flowrate versus steam quality and stagnation enthalpy at selected steam pressures for y = 0.2 according to the modified Fauske's model. The effects of the presence of CO in presented in Table 2. It is shown thag as charge: the wellbore discharge are a result of CO in the dis- 2 (1) the total mass flowrate decreases lip pressure and weir flowrate; and for a given set of values of (2) the specific enthalpy of the vapor phase is lower due to the reduced vapor pressure and the reduction of steam content because of the presence of COP in the gaseous phase. A comparison of the results based on the modified James' method and the modified Fauske's model for two specific cases is presented in Table. It is shown that the results based on the empirical formula are in agreement with those based on the theoretical model. REFERENCES 1. Cheng;?'.,rand Karmarkar,M: "An Evaluationof James' Empirical Formulae for the Determination of Two-Phase Flow Characteristics in Geothermal Wells," Proc., Fourth Workshop on Geothermal Reservoir Engineering," Stanford University (1978). 2. James, R.: "Measurement of Steam-Water Mixtures Discharging at the Speed of Sound to the Atmosphere," New Zealand Engineering, (1966), James, R.: "Steam-Water Critical Flow Through Pipes," &., Inst. Mech. Engrs., (1962),- 176, Fauske, H.: "Contribution to the Theory of Two-Phase, One Component Critical Flow," Argonne National Laboratory Report No. ANL-66 (1962). 5. Moody, F.: ''Maximum Flow Rate of a Single Component Two-Phase Mixture," J. Heat Transfer (1965), 14-. c

3 -51- u 2 6. Levy, S.: "Prediction of Two-Phase Critical Flow Rate," ASME Paper 64-HT-8. c.~? 7. Levy, S. : "Steam-Slip Theoretical Prediction from Momentum Model," J. Heat Transfer (1960), 82, James, R.: "Factors Controlling the Borehole Performance," Geothermics (1970), 2, Part 2, 1502, t U

4 -52- TABLE 1: COMPARISON OF RESULTS OBTAINED BASED ON JAMES' TH EORET I CAL YODELS. Case# 9 (psi4? (+) U ft -set G METHOD AND OTHER --- X Method b t a James (J) 0.54 Fauske (F) 0.55 bloody (M) F M L F M L J F M t F -. i'. ;r F M L

5 -5- b# f TABLE 2: EFFECTS OF NONCONOENSABLE GASES ON THE WELLBORE DISCHARGE CHARACTER1 STICS 8 - Lip Pressure Steam Pressure

6 -54- TABLE : COMPARISON OF RESULTS OF MODIFIED JAMES' METHOD AND THE MODI FI ED FAUSKE I S MODEL w P. Y G Method? Modi f i ed James Method (MJ) Modi f i ed Faus ke ' s Model (MF) MJ I MF 8

7 z -55-

8 500 \ t m 600. so0 I w Steam ~uoii~y. X' SfaQnotion t ntholpy.ho (61u/lbrn) R * Fig. 2. Weir Flow Rate vs. Steam Quality (Left) and Weir Flow Rate vs. Stagnation Enthalpy (Right) at Selected Lip Pressure According to Levy's Model. 8 '*

9 c -57- is 0 c P 0 N 4 z I I I I I I 1 1 t 1 u