Two-phase flows of boiling water and steam in geothermal reservoirs satisfy a pair of conservation equations for mass and energy which can be combined to yield a hyperbolic wave equation for liquid saturation changes. Recent work has established that in the absence of conduction, the geothermal saturation equation is, under certain conditions, asymptotically identical with the Buckley-Leverett equation of oil recovery theory. Here we summarise this work and show that it may be extended to include conduction. In addition we show that the geothermal saturation wave speed is under all conditions formally identical with the Buckley-Leverett wave speed when the latter is written as the saturation derivative of a volumetric flow.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
- C(P, S,q):
geothermal saturation wave speed [ms−1] (14)
- c t (P, S):
two-phase compressibility [Pa−1] (10)
- D(P, S):
diffusivity [m s−2] (8)
- E(P, S):
energy density accumulation [J m−3] (3)
- g :
gravitational acceleration (positive downwards) [ms−2]
- h w (P),h w (P):
specific enthalpies [J kg−1]
- J M (P, S,∇P):
mass flow [kg m−2 s−1] (5)
- J E (P, S,∇P):
energy flow [J m−2s−1] (5)
- k :
absolute permeability (constant) [m2]
- k w (S),k s (S):
relative permeabilities of liquid and vapour phases
- K :
formation thermal conductivity (constant) [Wm−1 K−1]
- L :
lower sheetC<0 in flow plane
- m, c :
gradient and intercept
- M(P, S):
mass density accumulation [kg m−3] (3)
- O :
flow plane origin
pressure (primary dependent variable) [Pa]
- q :
volume flow [ms−1] (6)
- S(x, t):
liquid saturation (primary dependent variable)
- S *(x,t):
normalised saturation (Appendix)
- t :
time (primary independent variable) [s]
- T :
temperature (degrees Kelvin) [K]
- T sat(P):
saturation line temperature [K]
- T′≡dT sat/dP :
saturation line temperature derivative [K Pa−1] (4)
- T c ,T D :
convective and diffusive time constants [s]
- u w (P),u s (P),u r (P):
specific internal energies [J kg−1]
- U :
upper sheetC > 0 in flow plane
shock velocity [m s−1]
spatial position (primary independent variable) [m]
- X :
- x, y :
flow plane coordinates
- z :
depth variable (+z vertically downwards) [m]
- δ P ,δ S :
remainder terms [Pa s−1], [s−1]
double-valued saturation region in the flow plane
- δh =h s −h w :
latent heat [J kg−1]
- δρ = δρ w −ρ s :
density difference [kg m−3]
- λ =D K /D 0 :
- Φ :
- Μ w (P),Μ s (P),Μ t (P, S):
dynamic viscosities [Pa s]
- v w (P),v s (P):
kinematic viscosities [m2s−1]
- v 0 =kδh/KT′ :
kinematic viscosity constant [m2 s−1]
- Μ 0 =v 0δρ :
dynamic viscosity constant [m2 s−1]
- ρ w (P),ρ s (P):
density [kg m−3]
- r :
- s :
- w :
- t :
- K :
Allen III, M. B., Behie, G. A. and Trangenstein, 1988,Multiphase Flow in Porous Media, Lecture Notes in Engineering 34, Springer-Verlag, Berlin.
Fayers, F. J., 1962, Some theoretical results concerning the displacement of a viscous oil by a hot fluid in a porous medium,J. Fluid Mech. 13, 65–76.
Grant, M. A., Donaldson, I. G., and Bixley, P. F., 1982,Geothermal Reservoir Engineering, Academic Press, New York.
Kissling, W., McGuinness, M., McNabb, A., Weir, G. J., White, S., and Young, R. M., 1992a, Analysis of one-dimensional horizontal two-phase flow in geothermal reservoirs,Transport in Porous Media 7, 223–253.
Kissling, W., McGuinness, M., Weir, G. J., White, S., and Young, R. M., 1992b, Vertical two-phase flow in porous media,Transport in Porous Media 8, 99–131.
Marle, C. M., 1981,Multiphase Flow in Porous Media, Editions Technip, Paris.
Pruess, K. 1988, SHAFT, MULKOM, TOUGH: A set of numerical simulators for multiphase fluid and heat flow,Geothermia, Revista Mexicana de Geoenergia 4, 185–202.
Pruess, K. and Narasimhan, T. M., 1982, On fluid reserves and the production of superheated steam from fractured, vapor-dominated geothermal reservoirs,J. Geophys. Res. 87(B11), 9329–9339.
Romero, L. A. and Nilson, R. H., 1981, Shock-like structure of phase-change flow in porous media,J. Fluid Mech. 104, 467–482.
Schubert, G. and Strauss, J. M., 1979, Steam-water counterflow in porous media,J. Geophys. Res. 84(B4), 1621–1628.
Sheu, J. P., Torrance, K. E. and Turcotte, D. L., 1979, On the structure of two-phase hydrothermal flows in permeable media,J. Geophys. Res. 84(B13), 7524–7532.
Trangenstein, J. A., 1988, Numerical analysis of reservoir fluid flow,in Allenet al.(see above).
Weir, G. J., 1991, Geometric properties of two-phase flow in geothermal reservoirs,Transport in Porous Media 6, 501–517.
Weir, G. J. and Young, R. M., 1991, Quasi-steady two-phase flows in porous media,Water Resour. Res. 27(6), 1207–1214.
Zazovskii, A. F., 1987, Displacing oil with hot water and steam,J. Appl. Mechanics Techn. Physics N6, 891–899.
About this article
Cite this article
Young, R. Two-phase geothermal flows with conduction and the connection with Buckley-Leverett theory. Transp Porous Med 12, 261–278 (1993). https://doi.org/10.1007/BF00624461