Skip to main content
SpringerLink
Log in

Numerical experiments on convective heat transfer in water-saturated porous media at near-critical conditions

  • Published:
Transport in Porous Media Aims and scope Submit manuscript

Abstract

Fluid and heat flow at temperatures approaching or exceeding that at the critical point (374 °C for pure water, higher for saline fluids) may be encountered in deep zones of geothermal systems and above cooling intrusives. In the vicinity of the critical point the density and internal energy of fluids show very strong variations for small temperature and pressure changes. This suggests that convective heat transfer from thermal buoyancy flow would be strongly enhanced at near-critical conditions. This has been confirmed in laboratory experiments. We have developed special numerical techniques for modeling porous flow at near-critical conditions, which can handle the extreme nonlinearities in water properties near the critical point. Our numerical simulations show strong enhancements of convective heat transfer at near-critical conditions; however, the heat transfer rates obtained in the simulations are considerably smaller than data reported from laboratory experiments by Dunn and Hardee. We discuss possible reasons for this discrepancy and develop suggestions for additional laboratory experiments.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

b :

Dispersion factor, dimensionless

C p :

Specific heat at constant pressure, kJ/kg °C

D :

Dispersion tensor, dimensionless

D p :

Pressure diffusivity, m2/s

d :

Characteristic pore (or grain) dimension, m

f :

Dispersive enhancement factor, dimensionless (Equation (17))

g :

Gravity acceleration, 9.8 m/s2

G :

Heat flux, W/m2

h :

Enthalpy, kJ/kg

H :

Height, m

k :

permeability, m2 (or Darcy ≈ 1 × 10-12 m2)

Nu:

Nusselt number, dimensionless

P :

Pressure, Pa (or bar = 105 Pa)

R :

Radius, m

Ra:

Rayleigh number, dimensionless

t :

Time, sec

T :

Temperature, °C

v :

Velocity, m/s

x :

Distance, m

α :

Coefficient of thermal expansion, °C-1

β :

Compressibility, /Pa-1

κ :

Thermal diffusivity, m2/s

λ :

Thermal conductivity, W/m °C

μ :

Dynamic viscosity, Pa · s

ϱ :

Density, kg/m3

v :

Kinematic viscosity, m2/s

φ :

Porosity, dimensionless

c :

Critical

cond:

Conductive

conv:

Convective

dis:

Dispersive

f :

Fluid

i :

Inner

m :

Medium

o :

Outer

p :

Pressure

r :

Radial

R :

Rock

t :

Temperature

tot:

Total

w :

Wire

z :

In z-direction

R :

length

(C p ϱ) m R 2 /λ m :

time

κ m /R :

velocity

ΔΔT :

temperature

ϱvκ m /k :

pressure

References

  • Bear, J., 1972, Dynamics of Fluids in Porous Media, Elsevier, Amsterdam.

    Google Scholar 

  • Bejan, A., 1984, Convection Heat Transfer, Wiley, New York, p. 398.

    Google Scholar 

  • Benenati, R. F. and Brosilow, C. B., 1962, Void fraction distribution in beds of spheres, AIChE J. 8, 359–361.

    Google Scholar 

  • Cappetti, G., Celati, R. Cigni, U., Squarci, P., Stefani, G., and Taffi, L., 1985, Development of deep exploration in the geothermal areas of Tuscany, Italy, 1985 International Symposium on Geothermal Energy, International Volume, Geothermal J. Resources Council, pp. 303–309.

  • Carslaw, H. S., and Jaeger, J. C., 1959, Conduction of Heat in Solids, 2nd edn, Oxford University Press.

  • Cathles, L. M., 1977, An analysis of the cooling of intrusives by groundwater convection which includes boiling, Econ. Geol. 72, 804–826.

    Google Scholar 

  • Dunn, J. C. and Hardee, H. C., 1981, Superconvecting geothermal zones, J. Volcanol. and Geotherm. Res. 11, 189–201.

    Google Scholar 

  • Facca, G., 1985, Geothermal activity in Italy, Geothemal Resources Council Bulletin, January 1985, pp. 10–14.

  • Fournier, R. O. and Potter, R. W., 1982, An equation correlating the solubility of quartz in water from 25 °C to 900 °C at Pressures up to 10,000 Bars, Geochimica et Cosmochimica Acta 46, 1969–1973.

    Google Scholar 

  • Haar, L., Gallagher, J. S., and Kell, G. S., 1984, NBSINRC Steam Tables, Hemisphere Publishing Corp.

  • Hadley, G. R., 1982, Natural convection of a near critical fluid through a porous medium, Sandia Report SAND82-1072J.

  • Hong, J. T. and Tien, C. L., 1987, Analysis of thermal dispersion effect on vertical-plate natural convection in porous media, Int. J. Heat Mass Transfer 30, 143–150.

    Google Scholar 

  • Hong, T. T., Yamada, Y., and Tien, C. L., 1987, Effects of non-darcian and non-uniform porosity on vertical-plate natural convection in porous media, J. Heat Transfer 109, 356–362.

    Google Scholar 

  • Katto, Y. and Masuoka, T., 1967, Criterion for the onset of convective flow in a fluid in a porous medium, Int. J. Heat Mass Transfer, 10, 297–309.

    Google Scholar 

  • Kimura, S. G., Schubert, G. and Straus, J. M., 1986, Route to chaos in porous-Medium thermal convection, J. Fluid Mech. 166, 305–324.

    Google Scholar 

  • Kvernvold, O. and Tyvand, P., 1980, Dispersion effects on thermal convection in porous media, J. Fluid Mech. 99, 673–686.

    Google Scholar 

  • Narasimhan, T. N. and Witherspoon, P. A., 1976, An integrated finite differences method for analyzing fluid flow in porous media, Water Resour. Res 12, 57–64.

    Google Scholar 

  • Norton, D. and Knight, J., 1977, Transport phenomena in hydrothermal systems and cooling plutons, Am. J. Sci 77, 937–981.

    Google Scholar 

  • Prasad, V., Kulacki, F., and Kulkarni, A. V., 1986, Free convection in a vertical, porous annulus with constant heat flux on the inner wall - experimental results, Int. J. Heat Mass Transfer 29, 713–723.

    Google Scholar 

  • Platzman, G. W., 1965, The spectral dynamics of laminar convection, J. Fluid Mech. 23, 481–510.

    Google Scholar 

  • Pruess, K., 1983, Development of the general purpose simulator MULKOM, Annual Report 1982, Earth Sciences Division, Report LBL-15500, Lawrence Berkeley Laboratory.

  • Pruess, K., 1988, SHAFT, MULKOM, TOUGH: A set of numerical simulators for multiphase fluid and heat flow, Geotermia, Rev. Mex. Geoenergia 4, 185–202.

    Google Scholar 

  • Reda, D. C., 1986, Natural convection experiments with a finite length, vertical cylindrical heat source in a water-saturated porous medium, Nucl. Chem. Waste Management 6, 3–14.

    Google Scholar 

  • Tokunaga, Tetsu K., 1988, Laboratory permeability errors from annular wall flow, Soil Sci. Am. J. 52, 24–27.

    Google Scholar 

  • Vafai, K. and Tien, C. L., 1981, Boundary and inertia effects on flow and heat transfer in porous media, Int. J. Heat Mass Transfer 24, 195–203.

    Google Scholar 

  • Verma A. and Pruess, K., 1988, Thermohydrological conditions and silica redistribution near high level nuclear wastes emplaced in saturated geological formations, J. Geophys. Res. 93, 1159–1173.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Earth Sciences Division, Lawrence Berkeley Laboratory, University of California, 94720, Berkeley, CA, USA

    B. L. Cox & K. Pruess

Authors
  1. B. L. Cox
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. Pruess
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cox, B.L., Pruess, K. Numerical experiments on convective heat transfer in water-saturated porous media at near-critical conditions. Transp Porous Med 5, 299–323 (1990). https://doi.org/10.1007/BF00140018

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00140018

Keywords

Search

Navigation