abstract
New concepts are introduced to describe single-component two-phase flow under gravity. The phases can flow simultaneously in opposite directions (counterflow), but information travels either up or down, depending on the sign of the wavespeedC. Wavespeed, saturation and other quantities are defined on a two-sheeted surface over the mass-energy flow plane, the sheets overlapping in the counterflow region. A saturation shock is represented as an instantaneous displacement along a line of constant volume fluxJ Q in the flow plane. Most shocks are of the wetting type, that is, they leave the environment more saturated after their passage. When flow is horizontal all shocks are wetting, but it is a feature of vertical two-phase flow that for sufficiently small mass and energy flows there also exist drying shocks associated with lower final saturations.
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Abbreviations
- a, b, c, d :
-
constants
- C :
-
characteristic velocity (wavespeed)
- c :
-
specific heat capacity
- D :
-
diffusivity
- E :
-
energy density
- F :
-
two-phase mobility
- f :
-
saturation forcing function
- g :
-
acceleration due to gravity
- G M, GE, G:
-
gravitational flow terms
- h :
-
flowing enthalpy (specific enthalpy when subscripted)
- J M, JE :
-
mass and energy flows
- J l, Jv :
-
liquid and vapour flows
- J 1,J 2,J Q2,J * Q :
-
flow constants
- J Q :
-
volume flow
- k :
-
absolute permeability
- k l, kv :
-
liquid and vapour relative permeabilities (functions of saturation S)
- L :
-
block length
- M :
-
mass density
- P :
-
pressure
- P′≡∂P/∂ Z :
-
pressure gradient
- S :
-
(liquid) saturation
- \(\tilde S\) :
-
normalized saturation
- S *, S* :
-
critical and residual (liquid) saturations
- t :
-
time
- T CL, TDL :
-
convective and diffusive time scales
- T :
-
temperature
- u :
-
specific internal energy
- U :
-
shock velocity
- x≡k l :
-
liquid relative permeability
- z :
-
depth (+z vertically downwards)
- Z :
-
shock location
- a :
-
after (the arrival of the shock)
- b :
-
before (the arrival of the shock)
- l :
-
liquid (e.g. water)
- v :
-
vapour (e.g. steam)
- h vl=hv−hl :
-
latent heat
- m :
-
rock matrix
- 1, 2:
-
referring to, respectively, upper (C>0) and lower (C<0) branches of the two-sheeted wavespeed surface
- Q :
-
pertaining to volume flux
- α=(Μ l−Μv)/Μl :
- δ:
-
discriminant
- δρ=ρ l−ρv :
- δΜ=Μ l−Μv :
- δS=S *−S* :
- Μ :
-
dynamic viscosity
- v :
-
kinematic viscosity
- Φ :
-
porosity
- ρ :
-
density
- ℰ s :
-
involves saturation derivatives: see Equation (4b)
- ℰ p :
-
involves pressure derivatives: see Appendix
References
Burnell, J. G., McNabb, A., Weir, G. J., and Young, R. M., 1989a, Two phase boundary layer formation in a semi-infinite porous slab,Transport in Porous Media 4, 395–420.
Burnell, J. G., McNabb, A., Weir, G. J., and Young, R. M., 1989b, The formation of boundary layers in two phase regions,Proc 4th Australasian Conference on Heat and Mass Transfer, 543–550.
Burnell, J. G., McNabb, A., Weir, G. J., White, S., and Young, R. M., 1989c, Response of a porous medium to pressure changes: The example of a finite length, horizontal, two-phase column,Proc. 4th Australasian Conference on Heat and Mass Transfer, 543–550.
Burnell, J. G., Kissling, W., McNabb, A., Weir, G. J., White, S., and Young, R. M., 1989d, Two phase flow studies,Proc. 11th New Zealand Geothermal Workshop, 241–246.
Burnell, J. G., Weir, G. J., and Young, R. M., 1991, Self-similar radial two-phase flows,Transport in Porous Media 6, 359–390.
Faust, C. R. and Mercer, J. W., 1979, Geothermal reservoir simulation 1. Mathematical models for liquid- and vapour-dominated hydrothermal systems,Water Resour. Res. 15(1), 23–30.
Garg, S. K. and Pritchett, J. W., 1977, On pressure-work, viscous dissipation and the energy balance relation for geothermal reservoirs,Adv. Water Resour. 1, 41–47.
Grant, Malcolm A., Donaldson, Ian G., and Bixley, Paul F., 1982,Geothermal Reservoir Engineering, Academic Press, New York.
Kissling, W., McGuinness, M., McNabb, A., Weir, G. J., White, S., and Young, R. M., 1992, Analysis of one-dimensional horizontal two-phase flow in geothermal reservoirs.Transport in Porous Media 7, 223–253.
Lax, P. D., 1973, Hyperbolic systems of conservation laws and the mathematical theory of shock waves,SIAM Regional Conference Series in Applied Mathematics 11.
McGuinness, M., 1990, Heat pipe stability in geothermal reservoirs,Proc. 1990 Geothermal Symposium, Geothermal Resour. Council Trans. 14, Part II, 1301–1307.
Pruess, K., 1983, Development of the general purpose simulator MULKOM,Annual Report 1982, Earth Sciences Division Report LBL-15500, Lawrence Berkeley Laboratory.
Pruess, K., 1985, A quantitative model of geothermal reservoirs as heat pipes in fractured porous rock,Trans. of 1985 Symposium on Geotherm. Energy, Geotherm. Resour. Council 1985 Annual Meeting 9(II) 353–362.
Pruess, K. and Tsang, Y. W., 1989, On relative permeability of rough-walled fractures,Proc. of 14th Workshop on Geothermal Reservoir Engineering, Stanford University, January 24–26, 1989.
Weir, G. J., 1991, Geometric properties of two phase flow in geothermal reservoirs,Transport in Porous Media 6, 501–517.
Whitham, G. B., 1974,Linear and Nonlinear Waves, Wiley, New York.
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Kissling, W., McGuinness, M., Weir, G. et al. Vertical two-phase flow in porous media. Transp Porous Med 8, 99–131 (1992). https://doi.org/10.1007/BF00617113
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DOI: https://doi.org/10.1007/BF00617113