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Geothermal heat pipe stability: Solution selection by upstreaming and boundary conditions

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Abstract

In a geothermal reservoir, the heat pipe mechanism can transfer heat very efficiently, with vapor rising and liquid falling in comparable quantities, driven by gravity. For a given heat and mass flux that is not too large, there are two possible steady solutions with vapor-liquid counterflow, one liquid-dominated, and one vapor-dominated. Numerical solution of the equations for two-phase vertical counterflow displays intriguing stability behaviour. If pressure and saturation are fixed at depth, and heat and mass flux specified at the top, the vapor-dominated solution is almost always obtained. That is, for a variety of boundary values, the solution settles to the vapor-dominated steady-state, and only for very special values is it possible to obtain the liquid-dominated case. Similarly the liquid-dominated solution is almost always obtained if the boundary conditions are reversed, with pressure and saturation fixed at the top and heat and mass flux specified at depth.

This behaviour is here explained in two complementary ways. It is shown to be a consequence of upstream differencing of the flow terms in the numerical method. It is also shown to be expected behaviour for wavelike saturation solutions. Hence the observed behaviour is not only a direct consequence of the numerical method used, but is fundamental to geothermal heat pipes.

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Abbreviations

A :

a constant

c :

characteristic velocity

C :

a constant

d:

darcy, 10−12 m2, a unit for permeability

D :

diffusivity

E :

energy density

F :

mass flux density

h :

specific enthalpy

g :

gravitational constant

k :

permeability

k r :

relative permeability

M :

mass density

p :

pressure

s :

liquid saturation

U :

specific internal energy

z, t :

independent variables

δz :

distance between grid block centers

Q :

flux density

α :

a constant

Β :

a dummy subscript for phase

λ :

mobility

Μ :

dynamic viscosity

v :

kinematic viscosity

Φ :

porosity

σ :

≡1/gl

ρ :

density

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Authors and Affiliations

  1. Math Dept, Victoria University of Wellington, New Zealand

    M. J. McGuinness

  2. DES, University of Auckland, New Zealand

    M. Blakeley

  3. LBL, Berkeley, California, USA

    K. Pruess

  4. DES, University of Auckland, New Zealand

    M. J. O'sullivan

Authors
  1. M. J. McGuinness
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  2. M. Blakeley
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  3. K. Pruess
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  4. M. J. O'sullivan
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McGuinness, M.J., Blakeley, M., Pruess, K. et al. Geothermal heat pipe stability: Solution selection by upstreaming and boundary conditions. Transp Porous Med 11, 71–100 (1993). https://doi.org/10.1007/BF00614636

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  • DOI: https://doi.org/10.1007/BF00614636

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