Evaluación estadística de correlaciones de fracción volumétrica de vapor para la modelación numérica de flujo bifásico en pozos geotérmicos

Statistical evaluation of void fraction correlations for the numerical modeling of two–phase flow in geothermal wells

A. Álvarez del Castillo¹, E. Santoyo²*, O. García–Valladares² y P. Sánchez–Upton¹

¹ Centro de Investigación en Energía (UNAM), Posgrado en Ingeniería (Energía–Geotermia), Privada Xochicalco s/n, Centro, Temixco, Mor., 62580, México

² Centro de Investigación en Energía (UNAM), Sistemas Energéticos, Privada Xochicalco s/n, Centro, Temixco, Mor., 62580, Mexico. *Autor para la correspondencia. E–mail: esg@cie.unam.mx

Recibido 3 de Julio 2009.
Aceptado 5 de Marzo 2010.

Resumen

La predicción de los perfiles de presión y temperatura en pozos geotérmicos bifásicos es una tarea fundamental para estudiar sus mecanismos de producción de flujo. La fracción volumétrica de vapor es uno de los parámetros más importantes requeridos para la predicción realista de los perfiles de producción. En este trabajo se evaluaron ocho correlaciones empíricas (Bonnecaze, modelo de Dix, Duns–Ros, Krilov, Hasan, Rouhani, modelo Homogeneo y Orkiszewski) para estimar la fracción volumétrica de vapor y modelar sus implicaciones en el flujo bifásico de pozos geotérmicos productores. Estas correlaciones fueron evaluadas mediante la modelación del flujo bifásico usando los simuladores GEOPOZO y GEOWELLS en 4 pozos productores de campos geotérmicos de México: Los Azufres, Mich. (Az–18), Los Humeros, Pue. (H–1) y (Corro Prieto, B.C. (M–90 y M–201). Los perfiles de presión y temperatura obtenidos por simulación fueron estadísticamente comparados con datos medidos en los pozos. Se encontró sistemáticamente que los perfiles simulados con la correlación del modelo de Dix provee las aproximaciones más aceptables (< 10%) al compararse con los datos medidos en los pozos evaluados, sugiriendo su uso para simular el flujo bifásico en pozos geotérmicos, ante la ausencia de correlaciones más confiables para estimar la fracción volumétrica de vapor.

Palabras clave: fracción volumétrica de gas, fracción volumétrica de líquido, flujo vapor–líquido, flujo vertical–inclinado, perfiles de producción, energía geotérmica.

Abstract

Predicting flowing pressure and temperature profiles in geothermal wells is a fundamental task to study the in flow production mechanisms. The gas void fraction is one of the most important parameters required for the better prediction of production profiles. Eight empirical correlations (Bonnecaze, Dix model, Duns–Ros, Krilov, Hasan, Rouhani, Homogeneous Model and Orkiszewski) for the estimation of gas void fractions and to model their implications on the two–phase flow inside geothermal wells were evaluated. These correlations were assessed through the two–phase flow modeling (using the wellbore simulators GEOPOZO and GEOWELLS) in four producer wells from Mexican geothermal fields: Los Azufres, Mich. (Az–18), Los Humeros, Pue. (H–1), and Cerro Prieto, B.C. (M–90 and M–201). The simulated pressure and temperature profiles were statistically compared with actual field data. A n acceptable agreement (< 10%) between the simulated profiles, obtained wit h the Dix model correlation, and measured data was obtained. These results enabled the modeling of two–phase flow inside geothermal wells to be reliably performed, which constitute an advantage due to the limited number of available correlations to calcula te the gas void fraction in geothermal wells.

DESCARGAR ARTÍCULO EN FORMATO PDF

Referencias

Alkan, H. y Satman, A. (1990). A new lumped parameter model for geothermal reservoirs in the presence of carbon dioxide. Geothermics 19, 469–479.         [ Links ]

Ambastha, A.K. y Gudmundsson, J.S. (1986). Pressure profiles in two–phase geothermal wells: comparison of field data and model calculations. SGP–TR–93, June 21–23,         [ Links ]

Stanford, California, USA: 11th Workshop on Geothermal Reservoir Engineering.         [ Links ]

Aragón, A., García, A., Baca, A. y González, E. (1999). Comparison of measured and simulated pressure and temperature profiles in geothermal wells. Geofísica Internacional 38, 35–42.         [ Links ]

Barelli, A., Corsi, R., Del Pizzo, G. y Scali, C. (1982). A two–phase flow model for geothermal wells in the presence of non–condensable gas. Geothermics 11, 175–191.         [ Links ]

Barnett, V. y Lewis, T. (1994). Outliers in statistical data. John Wiley & Sons, Chichester, UK.         [ Links ]

Barragán, R.M., Nieva, D., Santoyo, E., González, E., Verma, M. y López–Mendiola, J.M. (1991). Geoquímica de fluidos del campo geotérmico de Los Humeros, Puebla (México). Geotermia, Revista Mexicana de Geoenergía 7, 23–48.         [ Links ]

Battistelli, A., Calore, C. y Pruess, K. (1997). The simulator TOUGH2/EWASG for modeling geothermal reservoirs with brines and non–condensible gas. Geothermics 26, 437–464.         [ Links ]

Bonnecaze, R.H., Erskine, W. y Greskovich, E.J. (1971). Holdup and pressure drop for two phase slug flow in inclined pipes. Journal ofAmerican Institute of Chemical Engineers 17, 1109–1113.         [ Links ]

Castañeda, M., Abril, A., Arellano, V. y McCoy, R.L. (1981). Well log analysis applied to Cerro Prieto geothermal field. SGP–TR–55, December 16–18. Stanford, California, USA: 7th Workshop Geothermal Reservoir Engineering.         [ Links ]

Chadha, P.K., Malin, M.R. y Palacio–Póerez, A. (1993). Modelling of two–phase flow inside geothermal wells. Applied Mathematical Modelling 17, 236–245.         [ Links ]

Dalkilic, A.S., Laohalertdecha, S. y Wongwises, S. (2009). Effect of void fraction models on the two–phase friction factor of R134a during condensation in vertical downward flow in a smooth tube. International Communications in Heat and Mass Transfer 35, 921–927.         [ Links ]

Dix, G.E. (1971). Vapor void fractions for forced convection with subcooled boiling at low flow rate. Tesis de Doctorado, University of California, USA.         [ Links ]

Duns, H. y Ros, N.C.J. (1963). Vertical flow of gas and liquid mixtures in wells. Paper 10132, June 19–26. Frankfurt, Germany: 6th World Petroleum Congress.         [ Links ]

Espinosa–Paredes, G. y Soria, A. (1998). Method of finite difference solutions to the transient bubbly air–water flows. International Journal for Numerical Methods in Fluids 26, 1155–1180.         [ Links ]

Espinosa–Paredes, G., Cazarez–Candia, O., García–Gutiérrez, A. y Martínez–Méndez, J. (2002). Void fraction propagation in a bubbly two–phase flow with expansion effects. Annals of Nuclear Energy 29, 12611298.         [ Links ]

Espinosa–Paredes, G. y García–Gutiérrez, A. (2004). Thermal behaviour of geothermal wells using mud and air–water mixtures as drilling fluids. Energy Conversion and Management 45, 1513–1527.         [ Links ]

Espinosa–Paredes, G., Cazarez–Candia, O. y Vazquez, A. (2004). Theoretical derivation of the interaction effects with expansion effects to bubbly two–phase flows. Annals ofNuclear Energy 31, 117–133.         [ Links ]

Espinosa–Paredes, G., Salazar–Mendoza, R. y Cazarez–Candia, O. (2007). Averaging model for cuttings transport in horizontal wellbores. Journal ofPetroleum Science and Engineering 55, 301–316.         [ Links ]

García, A., Ascencio, F., Espinosa, G., Santoyo, E., Gutiérrez, H. y Arellano, V. (1999). Numerical modeling of high–temperature deep wells in the Cerro Prieto geothermal field, Mexico. Geofísica Internacional 38, 251–260.         [ Links ]

García–Gutiérrez, A., Espinosa–Paredes, G. y Hernández–Ramírez, I. (2002a). Study on the flow production characteristics of deep geothermal wells. Geothermics 31, 141–167.         [ Links ]

García, A., Espinosa–Paredes, G. y Barragán, R.M. (2002b). Effect of non–condensable gases on the flow of water and steam in geothermal wells. Geofísica Internacional 41, 377–383.         [ Links ]

García–Valladares, O., Sánchez–Upton, P. y Santoyo, E. (2006). Numerical modeling of flow processes inside geothermal wells: An approach for predicting production characteristics with uncertainties. Energy Conversion and Management 47, 1621–1643.         [ Links ]

Garg, S.K., Pritchett, J.W. y Alexander, J.H. (2004). A new liquid holdup correlation for geothermal wells. Geothermics 33, 795–817.         [ Links ]

Gokcen, G. y Yildirim, N. (2008). Effect of non–condensable gases on geothermal power plant performance. Case study: Kizildere Geothermal Power Plant–Turkey. International Journal of Exergy 5, 684–695.         [ Links ]

Gunn, C.I.M., Freeston, D.H. y Hadgu, T. (1992). Principles for wellbore simulator validation and calibration using matching analysis – I. Analytical techniques. Geothermics 21, 341–361.         [ Links ]

Hasan, A.R. y Kabir, C.S. (1992). Two–phase flow in vertical and inclined annuli. International Journal of Multiphase Flow 18, 279–293.         [ Links ]

Hasan, A.R., Kabir, C.S. y Sayarpour, M. (2007). A basic approach to wellbore two–phase flow modeling. Paper 109868–MS, November 1114. Anaheim, California, USA: Society of Petroleum Engineers Annual Technical Conference and Exhibition.         [ Links ]

Herzig, C.T. (1990). Geochemistry of igneous rocks from the Cerro Prieto geothermal field, northern Baja California, Mexico. Journal of Volcanology and Geothermal Research 42, 261–271.         [ Links ]

Horner, D.R. (1951). Pressure build–up in wells. Paper 4135, May 28 – June 6. The Hague, The Netherlands: 3rd World Petroleum Congress.         [ Links ]

Hughmark, G.A. (1962). Holdup in gas–liquid flow. Chemical Engineering Progress 53, 6265.         [ Links ]

Hurtig, E., Grosswig, S. y Kiihn, K. (1997). Distributed fibre–optic temperature sensing: a new tool for long term temperature monitoring in boreholes. Energy Sources 19, 55–62.         [ Links ]

Jung, D.B., Wai, K.W. y Howard, W.T. (2001). Geothermal flow metering errors. Geothermal Resources Council Transactions 25, 23–25.         [ Links ]

Kelessidis, V.C., Karydakis, G.I. y Andritsos, N. (2007). Method for selecting casing diameters in wells producing low–enthalpy geothermal waters containing dissolved carbon dioxide. Geothermics 36, 243–264.         [ Links ]

Lu, X., Watson, A., Gorin, A.V. y Deans, J. (2006). Experimental investigation and numerical modelling of transient two–phase flow in a geysering geothermal well. Geothermics 3, 409–427.         [ Links ]

Nieva, D., Verma, M., Santoyo, E., Barragán, R.M., Portugal, E., Ortiz, J. y Quijano, L. (1987). Chemical and isotopic evidence of steam up–flow and partial condensation in Los Azufres reservoir. SGP–TR–109, January 20–22. Stanford, California, USA: 12th Workshop on Geothermal Resources Engineering.         [ Links ]

NIST/ASME Steam v2.2, (1996). Formulation for General and Scientific Use. NIST Standard References Database Number 10.         [ Links ]

Orkiszewski, J. (1967). Predicting two–phase pressure drop in vertical pipes. Journal of Petroleum Technology 19, 829–838.         [ Links ]

Ortiz–R, J. (1983). Two–Phase Flow in Geothermal Wells: Development and Uses of a Computer Code. Internal Report SGP–TR–66. June. Stanford Geothermal Program, Stanford University, California, USA.         [ Links ]

Rice, C.K. (1987). The effect of void fraction correlation and heat flux assumption on refrigerant charge inventory predictions. ASHRAE Transactions 93, 341–367.         [ Links ]

Rouhani, S.Z. y Axelsson, E. (1970). Calculation of void volume fraction in the sub cooled and quality boiling regions. International Journal ofHeat and Mass Transfer 13, 383393.         [ Links ]

Rybach, L. (2003). Geothermal energy: sustainability and the environment. Geothermics 32, 463–470.         [ Links ]

Santoyo, E., Verma, S.P., Nieva, D. y Portugal, E. (1991). Variability in the gas phase composition of fluids discharged from Los Azufres Geothermal Caldera (Móexico). Journal of Volcanology and Geothermal Research 47, 161–181.         [ Links ]

Satman, A. y Ugûr, Z. (2002). Flashing point compressibility of geothermal fluids with low CO2 content and its use in estimating reservoir volume. Geothermics 31, 29–44.         [ Links ]

Suárez, M.C., Tello, M.R. y Samaniego, F. (2000). Geochemical evolution of the Los Azufres, Mexico, geothermal reservoir. Part II: Non–condensable gases. Paper R0375, May 28 – June 10, Japan: World Geothermal Congress 2000.         [ Links ]

Szilas, A. P. y Patsch, F. (1975). Flow in geothermal hot water wells. Geothermics 4, 79–88.         [ Links ]

Tian, S. y Finger, J.T. (2000). Advanced geothermal wellbore hydraulics model. Journal of Energy Resources Technology 122, 142–146.         [ Links ]

Torres–Alvarado, I.S., Pandarinath, K., Verma, S.P., y Dulski, P. (2007). Mineralogical and geochemical effects due to hydrothermal alteration in the Los Azufres geothermal field, Mexico. Revista Mexicana de Ciencias Geológicas 24, 15–24.         [ Links ]

Verma, S.P. (2005). Estadística básica para el manejo de datos experimentales: Aplicación en la geoquímica (Geoquimiometría). (UNAM), México.         [ Links ]

Verma, M.P. (2009). Steam Tables: An approach of multiple variable sets. Computers & Geosciences 35, 2145–2150.         [ Links ]

Verma, S.P. (2009). Evaluation of polynomial regression models for the Student t and Fisher F critical values, the best interpolation equations from double and triple natural logarithm transformation of degrees of freedom up to 1000, and their applications to quality control in science and engineering. Revista Mexicana de Ciencias Geológicas 26, 79–92.         [ Links ]

Verma, S.P. y Andaverde, J. (1996). Temperature distribution from cooling of a magma chamber in Los Azufres geothermal field, Michoacón, Mexico. Geofísica Internacional 35, 105–113.         [ Links ]

Verma, S.P. y Andaverde, J. (2007). Coupling of thermal and chemical simulations in a 3–D integrated magma chamber–reservoir model: A new geothermal energy research frontier. En: Geothermal Energy Research Trends, pp. 149–188, Nova Science Publishers, New York, USA.         [ Links ]

Verma, S.P., Andaverde, J. y Santoyo, E. (2006a). Statistical evaluation of methods for the calculation of static formation temperatures in geothermal and oil wells using an extension of the error propagation theory. Journal of Geochemical Exploration 89, 398–404.         [ Links ]

Verma, S.P., Quiroz–Ruiz, A. y Díaz–González, L. (2008). Critical values for 33 discordancy test variants for outliers in normal samples up to sizes 1000, and applications in quality control in Earth Sciences. Revista Mexicana de Ciencias Geológicas 25, 82–96.         [ Links ]

Verma, S.P., Díaz–González, L., Sánchez–Upton, P. y Santoyo, E. (2006b). OYNYL: A new computer program for ordinary, York, and New York least–squares linear regressions. WSEAS Transactions Environment Development 2, 997–1002.         [ Links ]

Wallis, G.B. (1969). One–dimensional two–phase flow. Editorial McGraw–Hill, USA.         [ Links ]

Woldesemayat, M.A. y Ghajar, A.J. (2007). Comparison of void fraction correlations for different flow patterns in horizontal and upward inclined pipes. International Journal of Multiphase Flow 33, 347–370.         [ Links ]