Geothermal modelling A. Speranza et al. Geothermal systems The physical problem Mathematical model The modelling week problem Final considerations Modelling of geothermal reservoirs Alessandro Speranza 1 Iacopo Borsi 2 Maurizio Ceseri 2 Angiolo Farina 2 Antonio Fasano 2 Luca Meacci 2 Mario Primicerio 2 Fabio Rosso 2 1 Industrial Innovation Throught Technological Trasnfer, I2T3 2 Dept. of Mathematics, University of Florence Modelling week 2009, Madrid A. Speranza et al. Geothermal modelling
1Industrial Innovation Throught Technological Trasnfer, I2T32Dept. of Mathematics, University of Florence
Modelling week 2009, Madrid
A. Speranza et al. Geothermal modelling
Geothermalmodelling
A. Speranzaet al.
Geothermalsystems
The physicalproblem
Mathematicalmodel
The modellingweek problem
Finalconsiderations
Dip. Di Matematica “U. Dini”
MAC-GEO Project
A. Speranza et al. Geothermal modelling
Geothermalmodelling
A. Speranzaet al.
Geothermalsystems
The physicalproblem
Mathematicalmodel
The modellingweek problem
Finalconsiderations
Outline
1 Geothermal systems
2 The physical problem
3 Mathematical model
4 The modelling week problem
5 Final considerations
A. Speranza et al. Geothermal modelling
Geothermalmodelling
A. Speranzaet al.
Geothermalsystems
The physicalproblem
Mathematicalmodel
The modellingweek problem
Finalconsiderations
Outline
1 Geothermal systems
2 The physical problem
3 Mathematical model
4 The modelling week problem
5 Final considerations
A. Speranza et al. Geothermal modelling
Geothermalmodelling
A. Speranzaet al.
Geothermalsystems
The physicalproblem
Mathematicalmodel
The modellingweek problem
Finalconsiderations
Outline
1 Geothermal systems
2 The physical problem
3 Mathematical model
4 The modelling week problem
5 Final considerations
A. Speranza et al. Geothermal modelling
Geothermalmodelling
A. Speranzaet al.
Geothermalsystems
The physicalproblem
Mathematicalmodel
The modellingweek problem
Finalconsiderations
Outline
1 Geothermal systems
2 The physical problem
3 Mathematical model
4 The modelling week problem
5 Final considerations
A. Speranza et al. Geothermal modelling
Geothermalmodelling
A. Speranzaet al.
Geothermalsystems
The physicalproblem
Mathematicalmodel
The modellingweek problem
Finalconsiderations
Outline
1 Geothermal systems
2 The physical problem
3 Mathematical model
4 The modelling week problem
5 Final considerations
A. Speranza et al. Geothermal modelling
Geothermalmodelling
A. Speranzaet al.
Geothermalsystems
The physicalproblem
Mathematicalmodel
The modellingweek problem
Finalconsiderations
Geothermal energy
The geothermal energy is dueto the heat deep under theground
Need contemporary presenceof water and a heat source.
Only a fractured soil canmake “productive” thereservoir
A. Speranza et al. Geothermal modelling
Geothermalmodelling
A. Speranzaet al.
Geothermalsystems
The physicalproblem
Mathematicalmodel
The modellingweek problem
Finalconsiderations
The geothermal system
Geothermal reservoirs con-sist of
A deep heat source(magma intrusion)
A fractured rock layer
A water reservoir
A. Speranza et al. Geothermal modelling
Geothermalmodelling
A. Speranzaet al.
Geothermalsystems
The physicalproblem
Mathematicalmodel
The modellingweek problem
Finalconsiderations
Geothermal areas in Europe
Geothermic potential iswidely spread
However, not all canbe exploited
High geothermalgradient in Toscany
A. Speranza et al. Geothermal modelling
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A. Speranzaet al.
Geothermalsystems
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High geothermal potential in Toscany
High geothermal gradient(> 10◦ C) in Toscany
Larderello is the oldestexploited reservoir (1905)
A. Speranza et al. Geothermal modelling
Geothermalmodelling
A. Speranzaet al.
Geothermalsystems
The physicalproblem
Mathematicalmodel
The modellingweek problem
Finalconsiderations
Main types of geothermal reservoirs
Geothermal reservoirs are typically
Water dominated: water is mostlty found in liquid phase,e.g., Amiata. Characterized by very high pressure (> 100bar) and temperature (> 300◦ C).
Vapour dominated: water is mostly found in gas phase,e.g., Larderello. Characterized by fairly low pressure (∼ 70bar) and high temperature (> 300◦ C).
In some vapour dominated reservoirs, the fluid could befound in a mixture of liquid and gas phases (e.g.,Monteverdi Marittima).
A. Speranza et al. Geothermal modelling
Geothermalmodelling
A. Speranzaet al.
Geothermalsystems
The physicalproblem
Mathematicalmodel
The modellingweek problem
Finalconsiderations
The physical model
Need to express in mathematical terms, the complexphysics of a geothermal reservoir.
The aspects to consider involve
Thermodynamics of mixtures of water, gases (NCGs) andsaltsFluid motion in porous (fractured) mediumHeat conduction/convection
Numerical data, such as, petrophysical properties, fluidproperties, pressure, temperature, boundaries etc., on thereservoir are often unknown or very uncertain.
ENEL provided the most of the data we will use.
A. Speranza et al. Geothermal modelling
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The physicalproblem
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Thermodynamics of the reservoir, water only
Water vapour pressure
P?(T ) ' 961 exp
{17.27 (T − 273)
T
}A. Speranza et al. Geothermal modelling
Geothermalmodelling
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Mixture, in the real world
Polydispersity
Phase envelope changes with concentrations
Gas-liquid equilibrium, within a region of phase diagram
A. Speranza et al. Geothermal modelling
Geothermalmodelling
A. Speranzaet al.
Geothermalsystems
The physicalproblem
Mathematicalmodel
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Finalconsiderations
The physical model
Assume general 3D geometry
Assume Darcy’s law is valid in fractured medium(equivalent porosity/permeability)
xαi is mass fraction of i-th component in phase αSα is saturation of phase αφ is porosityvαi velocity of the i-th component in phase αΨext is total mass of extracted/injected fluid per time unitVext is total volume of the extraction/injection wellΓα mass exchanged per unit time, due to phase change
A. Speranza et al. Geothermal modelling
Geothermalmodelling
A. Speranzaet al.
Geothermalsystems
The physicalproblem
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Momentum conservation
Assume Darcy’s law for fluid velocity
qα = φSαvα = −Kkrα
µα(∇Pα + ραg) ,
Where krα is relative permeability and µα is dynamicviscosity of phase α
Assume, e.g., isotropic absolute permeability
K = K Id,
A. Speranza et al. Geothermal modelling
Geothermalmodelling
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Geothermalsystems
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Energy conservation
Total energy conservation
∂
∂t
[(1− φ)ρrcrT + φ
∑α
ραSαuα
]+
∑α
∇ · (hαqα) =
+ ∇ · [λmix∇T ] ,
where
uα is the internal energy density (per mass unit) of phase αhα is the henthalpy density of phase αand
λmix = (1− φ)λr + φ∑α
λαSα
λα/r is the heat conductivity of phase α/rock
A. Speranza et al. Geothermal modelling
Geothermalmodelling
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Geothermalsystems
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Coupling with thermodynamics
Phase equilibrium conditions couple with the set of PDEsAt a given T , given a set of parent densities,
ρ(0)i =
∑α=l ,g
ραxαi Sα,
Two phases are in equilibrium when
µLi = µG
i ,
where
µi =∂
∂ρiF (ρi ,T ),
are the chemical potentialsAlso impose lever rule and volume conservation
SGρGi + SLρL
i = ρ(0)i SG +SL = 1
A. Speranza et al. Geothermal modelling
Geothermalmodelling
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Final considerations
Sum mass conservation equations over phases, to get ridof mass transfer due to phase change
Get a set of n (mass equations) + 1 (energy equation) +n (chemical potentials equality) + n (lever rule) + 1(volume conservation) = 3n + 2 Equations.
In ρ(0)i , ραi = ραxαi , SG , SL, T , i.e., 3n + 3 unknowns.
Pressures are given by EOS, Pα = P(ραi ,T )
Add extra constitutive equation over Pα
PG = PL
in equilibriumPG = PL + Pc
in case of capillary pressure
A. Speranza et al. Geothermal modelling
Geothermalmodelling
A. Speranzaet al.
Geothermalsystems
The physicalproblem
Mathematicalmodel
The modellingweek problem
Finalconsiderations
Other considerations
Need to impose boundary conditions for ρ(0)i and T (or P
and x(0)i )
Need to set appropriate initial values
All the data above are usually unknown
Petrophysical properties can be only guessed
Coupling of PDEs and thermodynamics is not an easy task
A. Speranza et al. Geothermal modelling
Geothermalmodelling
A. Speranzaet al.
Geothermalsystems
The physicalproblem
Mathematicalmodel
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Finalconsiderations
Possible simplifying assumptions
Model only well region
Cylindrical symmetry could be reduced to 1D
Assume water only, thus
Liquid density is constantGas density is given by Ideal Gas EOSPhase coexistence only on vapour pressure curve
Assume temperature, varying linearly with depth andconstant in t (no energy conservation)
Assume no extraction/injection; just set a lower value of Pat the top boundary
Can assume (we will) natural recharge
A. Speranza et al. Geothermal modelling
Geothermalmodelling
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Geothermalsystems
The physicalproblem
Mathematicalmodel
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Free boundary problem
In case of gas/liquid phase separationBecomes a 1D free boundary problem
Impermeable rocks at the top (x=0)Assume constant (in time) temperatureT = T (x), linear in xGas reservoir starting at x = Ls = −1300Impose fixed pressure value P = Ps atx = Ls to simulate extraction well.Sharp (moving) interface s(t) betweengas and liqud.Assume saturated vapour pressure on sLiquid between x = s(t) andx = Li = −3000Assume fixed pressure value at bottomP(x = Li ) = Pi
Assume no bottom flux (isolated reservoir)
A. Speranza et al. Geothermal modelling
Geothermalmodelling
A. Speranzaet al.
Geothermalsystems
The physicalproblem
Mathematicalmodel
The modellingweek problem
Finalconsiderations
Final considerations
Full model is very complex
No analysis can be made, only full 3D simulations.
Several commercial codes simulate such systems ofequations, with some simplifications on thermodynamics(e.g., TOUGH2)
However, simple 1D can help to understand how thingsgo, e.g., how a vapor/liquid reservoir could evolve into avapor dominated one, such as in the case of MonteverdiMarittima
Possible further step, go cylindrical symmetry and add avaporization front.
