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Geothermal
modelling
A. Speranzaet al.
Geothermalsystems
The physical
problem
Mathematicalmodel
The modellingweek problem
Finalconsiderations
Modelling of geothermal reservoirs
Alessandro Speranza1 Iacopo Borsi2 Maurizio Ceseri2
Angiolo Farina2 Antonio Fasano2 Luca Meacci2
Mario Primicerio2 Fabio Rosso2
1Industrial Innovation Throught Technological Trasnfer, I2T32Dept. of Mathematics, University of Florence
Modelling week 2009, Madrid
A. Speranza et al. Geothermal modelling
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Geothermal
modelling
A. Speranzaet al.
Geothermalsystems
The physical
problem
Mathematicalmodel
The modellingweek problem
Finalconsiderations
Outline
1 Geothermal systems
A. Speranza et al. Geothermal modelling
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Geothermal
modelling
A. Speranzaet al.
Geothermalsystems
The physical
problem
Mathematicalmodel
The modellingweek problem
Finalconsiderations
Outline
1 Geothermal systems
2 The physical problem
A. Speranza et al. Geothermal modelling
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Geothermal
modelling
A. Speranzaet al.
Geothermalsystems
The physical
problem
Mathematicalmodel
The modellingweek problem
Finalconsiderations
Outline
1 Geothermal systems
2 The physical problem
3 Mathematical model
A. Speranza et al. Geothermal modelling
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Geothermal
modelling
A. Speranzaet al.
Geothermalsystems
The physical
problem
Mathematicalmodel
The modellingweek problem
Finalconsiderations
Outline
1 Geothermal systems
2 The physical problem
3 Mathematical model
4 The modelling week problem
A. Speranza et al. Geothermal modelling
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Geothermal
modelling
A. Speranzaet al.
Geothermalsystems
The physical
problem
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
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Geothermal
modelling
A. Speranzaet al.
Geothermalsystems
The physical
problem
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 the
reservoir
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Geothermal
modelling
A. Speranzaet al.
Geothermalsystems
The physical
problem
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
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Geothermal
modelling
A. Speranzaet al.
Geothermalsystems
The physical
problem
Mathematicalmodel
The modellingweek problem
Finalconsiderations
Geothermal areas in Europe
Geothermic potential is
widely spreadHowever, not all canbe exploited
High geothermalgradient in Toscany
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Geothermal
modelling
A. Speranzaet al.
Geothermalsystems
The physical
problem
Mathematicalmodel
The modellingweek problem
Finalconsiderations
High geothermal potential in Toscany
High geothermal gradient(>10 C) in Toscany
Larderello is the oldest
exploited reservoir (1905)
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Geothermal
modelling
A. Speranzaet al.
Geothermalsystems
The physical
problem
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 (> 100
bar) 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).
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Geothermal
modelling
A. Speranzaet al.
Geothermalsystems
The physical
problem
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, fluid
properties, pressure, temperature, boundaries etc., on thereservoir are often unknown or very uncertain.
ENEL provided the most of the data we will use.
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Geothermal
modelling
A. Speranzaet al.
Geothermalsystems
The physical
problem
Mathematicalmodel
The modellingweek problem
Finalconsiderations
Thermodynamics of the reservoir, water only
Water vapour pressure
P(T) 961 exp
17.27 (T 273)
T A. Speranza et al. Geothermal modelling
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Geothermal
modelling
A. Speranzaet al.
Geothermalsystems
The physical
problem
Mathematicalmodel
The modellingweek problem
Finalconsiderations
Mixture, in the real world
Polydispersity
Phase envelope changes with concentrations
Gas-liquid equilibrium, within a region of phase diagram
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Geothermal
modelling
A. Speranzaet al.
Geothermalsystems
The physical
problem
Mathematicalmodel
The modellingweek problem
Finalconsiderations
Mass/energy conservation law
Mass conservation
t(xi S
) + (xi Svi) =
= Mi
Mtot
1
Vextext + (x
i S)
where
xi is mass fraction of i-th component in phase S is saturation of phase
is porosityvi velocity of the i-th component in phase ext is total mass of extracted/injected fluid per time unitVext is total volume of the extraction/injection well mass exchanged per unit time, due to phase change
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Geothermal
modelling
A. Speranzaet al.
Geothermalsystems
The physical
problem
Mathematicalmodel
The modellingweek problem
Finalconsiderations
Momentum conservation
Assume Darcys law for fluid velocity
q =Sv = Kkr
(P + g) ,
Where kr is relative permeability and is dynamicviscosity of phase
Assume, e.g., isotropic absolute permeability
K= KId,
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Geothermal
modelling
A. Speranzaet al.
Geothermalsystems
The physical
problem
Mathematicalmodel
The modellingweek problem
Finalconsiderations
Energy conservation
Total energy conservation
t
(1 )rcrT+
Su
+
(hq) =
+ [mixT] ,
where
u is the internal energy density (per mass unit) of phase h is the henthalpy density of phase
andmix= (1 )r+
S
/r is the heat conductivity of phase /rock
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Geothermal
modelling
A. Speranzaet al.
Geothermalsystems
The physical
problem
Mathematicalmodel
The modellingweek problem
Finalconsiderations
Coupling with thermodynamics
Phase equilibrium conditions couple with the set of PDEsAt a given T, given a set ofparent densities,
(0)i =
=l,g
xi S,
Two phases are in equilibrium when
Li =Gi ,
where
i=
iF(i,T),
are the chemical potentialsAlso impose lever ruleand volume conservation
SGGi +SLLi =
(0)i S
G+SL = 1
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Geothermal
modelling
A. Speranzaet al.
Geothermalsystems
The physical
problem
Mathematicalmodel
The modellingweek problem
Finalconsiderations
Final considerations
Sum mass conservation equations over phases, to get ridof mass transfer due to phase change
Get a set ofn (mass equations) + 1 (energy equation) +n (chemical potentials equality) + n (lever rule) + 1(volume conservation) = 3n + 2 Equations.
In (0)i ,
i =xi , S
G, SL, T, i.e., 3n + 3 unknowns.
Pressures are given by EOS, P =P(i ,T)
Add extra constitutive equation over P
PG
=PL
in equilibriumPG =PL +Pc
in case of capillary pressure
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Geothermal
modelling
A. Speranzaet al.
Geothermalsystems
The physical
problem
Mathematicalmodel
The modellingweek problem
Finalconsiderations
Other considerations
Need to impose boundary conditions for (0)i and T (orP
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
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Geothermalmodelling
A. Speranzaet al.
Geothermalsystems
The physical
problem
Mathematicalmodel
The modellingweek problem
Finalconsiderations
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 x
Gas reservoir starting atx
=L
s=
1300Impose fixed pressure value P=Ps atx=Lsto simulate extraction well.Sharp (moving) interface s(t) betweengas and liqud.Assume saturated vapour pressure on s
Liquid between x=s(t) andx=Li= 3000Assume fixed pressure value at bottomP(x=Li) =PiAssume no bottom flux (isolated reservoir)
A. Speranza et al. Geothermal modelling
Fi l id i
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Geothermalmodelling
A. Speranzaet al.
Geothermalsystems
The physical
problem
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 Monteverdi
MarittimaPossible further step, go cylindrical symmetry and add avaporization front.
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Geothermalmodelling
A. Speranzaet al.
Geothermalsystems
The physical
problem
Mathematicalmodel
The modellingweek problem
Finalconsiderations
Good work
A. Speranza et al. Geothermal modelling
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