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Two-phase geothermal flows with conduction and the connection with Buckley-Leverett theory

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Abstract

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.

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Abbreviations

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

P(x,t):

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

U(x,t):

shock velocity [m s−1]

x:

spatial position (primary independent variable) [m]

X :

representative length

x, y :

flow plane coordinates

z :

depth variable (+z vertically downwards) [m]

δ P ,δ S :

remainder terms [Pa s−1], [s−1]

δ:

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]

г:

line envelope

λ =D K /D 0 :

diffusivity ratio

Φ :

porosity (constant)

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 :

rock matrix

s :

steam (vapour)

w :

water (liquid)

t :

total

av:

average

0:

without conduction

K :

with conduction

References

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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

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Key words

  • Geothermal
  • two-phase
  • saturation
  • convection
  • conduction
  • Buckley-Leverett