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On vapour flow in a hot porous layer

Published online by Cambridge University Press:  26 April 2006

Shaun D. Fitzgerald
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics and Department of Earth Sciences, University of Cambridge CB3 9EW, UK Present address: Department of Petroleum Engineering, Stanford University, Standford, CA 94305, USA.
Andrew W. Woods
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics and Department of Earth Sciences, University of Cambridge CB3 9EW, UK

Abstract

The motion of isothermal vapour in a permeable rock is governed by a nonlinear diffusion equation for the vapour pressure. We analyse vapour flow described by this equation in both bounded and unbounded domains. We then apply these solutions to describe the controls on the rate of vaporization of liquid invading a hot permeable rock. In an unbounded domain, we determine asymptotic similarity solutions describing the motion of vapour when it is either supplied to or removed from the reservoir. Owing to the compressibility, these solutions have the property that vapour surfaces migrate towards the isobar on which the vapour has the maximum speed.

In contrast, if vapour is supplied to or removed from a closed bounded system sufficiently slowly then the vapour density and pressure rapidly become approximately uniform. As more vapour is added, the mean pressure gradually increases and vapour surfaces become compressed. If liquid slowly invades a hot bounded porous layer and vaporizes, the vapour pressure becomes nearly uniform. As more liquid is added, the reservoir gradually becomes vapour saturated and the vaporization ceases.

In an open bounded system, with a constant rate of vapour injection, the flux of vapour across the reservoir becomes uniform. If liquid is injected slowly and vaporizes then again the vapour flux becomes spatially uniform. However, the vapour flux now increases slowly as the liquid invades further into the rock, as a result of the decreased resistance to vapour flow from the interface to the far boundary.

Type
Research Article
Copyright
© 1995 Cambridge University Press

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References

Ames, W. F. 1977 Numerical Methods for Partial Differential Equations. Nelson.
Axelsson, G. & Bodvarsson, G. 1987 Analysis of production data from fractured liquid dominated geothermal reservoirs in Iceland. Trans. Geotherm. Res. Counc. 11, 573580.Google Scholar
Bear, J. 1972 Dynamics of Fluids in Porous Media. Dover.
Bodvarsson, G. 1972 Thermal problems in the siting of reinjection wells. Geothermics 1, 6366.Google Scholar
Cathles, L. M. 1977 An analysis of the cooling of intrusives by ground-water convection which includes boiling. Econ. Geol. 72, 804826.Google Scholar
Cline, J. S., Bodnar, R. J. & Rimstidt, J. D. 1992 Numerical simulation of fluid flow and silica transport and deposition in boiling hydrothermal solutions: application to epithermal gold deposits. J. Geophys. Res. 97 (B6), 90859103.CrossRefGoogle Scholar
D'amore, F., Celati, R., Ferrara, G. C. & Panichi, C. 1977 Secondary changes in the chemical and isotopic composition of the geothermal fluids in Larderello field. Geothermics 5, 153163.Google Scholar
Donaldson, I. G. 1962 Temperature gradients in the upper layers of the earth's crust due to convective water flows. J. Geophys. Res. 67, 34493459.Google Scholar
Donaldson, I. G. 1968 The flow of steam water mixtures through permeable beds: a simple simulation of a natural undisturbed hydrothermal region. N.Z. J. Sci. 11, 323.Google Scholar
Duchi, V., Minissale, A. & Manganelli, M. 1992 Chemical composition of natural deep and shallow hydrothermal fluids in the Larderello geothermal field. J. Volcanol. Geotherm. Res. 49, 313328.Google Scholar
Dullen, F. A. L. 1992 Porous Media Fluid Transport and Pore Structure. Academic.
Dunn, J. C. & Hardee, H. C. 1981 Superconvecting geothermal zones. J. Volcanol. Geotherm. Res. 11, 189201.Google Scholar
Elder, J. 1981 Geothermal Systems. Academic.
Enedy, K. L. 1989 The role of decline curve analysis at The Geysers. Trans. Geotherm. Res. Counc. 13, 383391.Google Scholar
Fitzgerald, S. D. & Woods, A. W. 1994 The instability of a vaporisation front in hot porous rock. Nature 367, 450453.Google Scholar
Fradkin, L. J., Sorey, M. J. & McNabb, A. 1981 On identification and validation of some geothermal models. Water Resources Res. 17, 929936.Google Scholar
Goyal, K. P. & Box, W. T. 1990 Reservoir response to production: Castle Rock Springs Area, East Geysers, California, USA. Proc. Stanford Geotherm. Workshop 15, 103112.Google Scholar
Grant, M. A., Donaldson, I. G. & Bixley, P. F. 1982 Geothermal Reservoir Engineering. Academic.
Haywood, R. W. 1972 Thermodynamic Tables in SI (Metric) Units. Cambridge University Press.
Hurst, A. W. & Dibble, R. R. 1981 Bathymetry, heat output and convection in Ruapehu crater lake, New Zealand. J. Volcan. Geotherm. Res. 9, 215236.Google Scholar
Kerr, R. A. 1991 Geothermal tragedy of the commons. Science 253, 134135.Google Scholar
Martini, M., Giannini, L., Buccianti, A., Prati, F., Cellini Legittimo, P., Iozzelli, P. & Capacciono, B. 1991 1980-1990: Ten years of geochemical investigation at Phlegrean Fields (Italy). J. Volcanol. Geotherm. Res. 48, 161171.Google Scholar
Norton, D. & Knight, J. 1977 Transport phenomena in hydrothermal systems: cooling plutons. Am. J. Sci. 277, 937981.Google Scholar
Parmentier, E. M. & Schedl, A. 1981 Thermal aureoles of igneous intrusions: some possible indications of hydrothermal convective cooling. J. Geol. 89, 122.CrossRefGoogle Scholar
Phillips, O. M. 1991 Flow and Reactions in Permeable Rocks. Cambridge University Press.
Pruess, K., Calore, C., Celati, R. & Wu, Y. S. 1987 An analytical solution for heat transfer at a boiling front moving through a porous medium. Intl J. Heat Mass Transfer 30, 25952602.Google Scholar
Rubin, A. & Schweitzer, S. 1972 Heat transfer in porous media with phase change. Intl J. Heat Mass Transfer 15, 4360.Google Scholar
Schroeder, R. C., O'Sullivan, M. J., Pruess, K., Celati, R. & Ruffilli, C. 1982 Reinjection studies of vapour-dominated systems. Geothermics 11, 93119.Google Scholar
Truesdell, A. H. & White, D. E. 1973 Production of superheated steam from vapour-dominated geothermal reservoirs. Geothermics 2, 154173.Google Scholar
Wohletz, K. & Heiken, G. 1992 Volcanology and Geothermal Energy. California.
Woods, A. W. & Fitzgerald, S. D. 1993 The vaporization of a liquid front moving through a hot porous rock. J. Fluid Mech. 251, 563579.Google Scholar