Geothermal Reservoir Modelling

8/12/2019 Geothermal Reservoir Modelling

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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


Finalconsiderations

Outline

1 Geothermal systems



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Geothermal

modelling

A. Speranzaet al.

Geothermalsystems

The physical

problem

Mathematicalmodel


Finalconsiderations

Outline


2 The physical problem



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Geothermal

modelling

A. Speranzaet al.

Geothermalsystems

The physical

problem

Mathematicalmodel


Finalconsiderations

Outline



3 Mathematical model



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Geothermal

modelling

A. Speranzaet al.

Geothermalsystems

The physical

problem

Mathematicalmodel


Finalconsiderations

Outline




4 The modelling week problem



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Geothermal

modelling

A. Speranzaet al.

Geothermalsystems

The physical

problem

Mathematicalmodel


Finalconsiderations

Outline




4 The modelling week problem

5 Final considerations



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Geothermal

modelling

A. Speranzaet al.

Geothermalsystems

The physical

problem

Mathematicalmodel


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


Finalconsiderations

The geothermal system

Geothermal reservoirs con-sist of

A deep heat source(magma intrusion)

A fractured rock layer

A water reservoir



10/26

Geothermal

modelling

A. Speranzaet al.

Geothermalsystems

The physical

problem

Mathematicalmodel


Finalconsiderations

Geothermal areas in Europe

Geothermic potential is

widely spreadHowever, not all canbe exploited

High geothermalgradient in Toscany



11/26

Geothermal

modelling

A. Speranzaet al.

Geothermalsystems

The physical

problem

Mathematicalmodel


Finalconsiderations

High geothermal potential in Toscany

High geothermal gradient(>10 C) in Toscany

Larderello is the oldest

exploited reservoir (1905)



12/26

Geothermal

modelling

A. Speranzaet al.

Geothermalsystems

The physical

problem

Mathematicalmodel


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


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.



14/26

Geothermal

modelling

A. Speranzaet al.

Geothermalsystems

The physical

problem

Mathematicalmodel


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


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


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


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,



19/26

Geothermal

modelling

A. Speranzaet al.

Geothermalsystems

The physical

problem

Mathematicalmodel


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


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


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


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


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)


Fi l id i


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Geothermalmodelling

A. Speranzaet al.

Geothermalsystems

The physical

problem

Mathematicalmodel


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


Finalconsiderations

Good work


Geothermal Reservoir Modelling

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