Modelling of geothermal reservoirs

ling The ling 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 ling

ling Dip. Di Matematica U. Dini The ling MAC-GEO Project ling

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energy ling The ling The geothermal energy is due to the heat deep under the ground Need contemporary presence of water and a heat source. Only a fractured soil can make productive the reservoir ling

The geothermal system ling The ling reservoirs consist of A deep heat source (magma intrusion) A fractured rock layer A water reservoir ling

areas in Europe ling The ling Geothermic potential is widely spread However, not all can be exploited High geothermal gradient in Toscany ling

High geothermal potential in Toscany ling High geothermal gradient (> 10 C) in Toscany Larderello is the oldest exploited reservoir (1905) The ling ling

Main types of geothermal reservoirs ling The ling 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 ( 70 bar) and high temperature (> 300 C). In some vapour dominated reservoirs, the fluid could be found in a mixture of liquid and gas phases (e.g., Monteverdi Marittima). ling

ling The ling Need to express in mathematical terms, the complex physics of a geothermal reservoir. The aspects to consider involve Thermodynamics of mixtures of water, gases (NCGs) and salts Fluid motion in porous (fractured) medium Heat conduction/convection Numerical data, such as, petrophysical properties, fluid properties, pressure, temperature, boundaries etc., on the reservoir are often unknown or very uncertain. ENEL provided the most of the data we will use. ling

Thermodynamics of the reservoir, water only ling The ling Water vapour pressure { } 17.27 (T 273) P (T ) 961 exp T ling

Mixture, in the real world ling The ling Polydispersity Phase envelope changes with concentrations Gas-liquid equilibrium, within a region of phase diagram ling

ling The ling Assume general 3D geometry Assume Darcy s law is valid in fractured medium (equivalent porosity/permeability) General mixture of n components Assume gas-liquid phase coexistence; phase equlibrium (!) Conservation laws (mass and energy) Set suitable boundary conditions ling

Mass/energy conservation law ling The ling Mass conservation t (ρα x α i S α φ) + (ρ α x α i S α φv α i ) = = Mα i M tot 1 V ext Ψ ext + (ρ α x α i S α φ) Γ α where xi α is mass fraction of i-th component in phase α S α is saturation of phase α φ is porosity vi α velocity of the i-th component in phase α Ψ ext is total mass of extracted/injected fluid per time unit V ext is total volume of the extraction/injection well Γ α mass exchanged per unit time, due to phase change ling

Momentum conservation ling The ling Assume Darcy s law for fluid velocity q α = φs α v α = K k rα µ α ( P α + ρ α g), Where k rα is relative permeability and µ α is dynamic viscosity of phase α Assume, e.g., isotropic absolute permeability K = K Id, ling

Energy conservation ling The ling Total energy conservation [ (1 φ)ρ r c r T + φ ] ρ α S α u α t α + (h α q α ) = α + [λ mix T ], where u α is the internal energy density (per mass unit) of phase α h α is the henthalpy density of phase α and λ mix = (1 φ)λ r + φ λ α S α α λ α/r is the heat conductivity of phase α/rock ling

Coupling with thermodynamics ling The ling Phase equilibrium conditions couple with the set of PDEs At a given T, given a set of parent densities, = ρ α xi α S α, ρ (0) i α=l,g Two phases are in equilibrium when µ L i = µ G i, where µ i = F (ρ i, T ), ρ i are the chemical potentials Also impose lever rule and volume conservation S G ρ G i + S L ρ L i = ρ (0) i S G +S L = 1 ling

ling The ling Sum mass conservation equations over phases, to get rid of 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, ρ α i = ρ α x α i, S G, S L, T, i.e., 3n + 3 unknowns. Pressures are given by EOS, P α = P(ρ α i, T ) Add extra constitutive equation over P α P G = P L in equilibrium in case of capillary pressure P G = P L + P c ling

Other ling The ling Need to impose boundary conditions for ρ (0) i and x (0) i ) Need to set appropriate initial values All the data above are usually unknown Petrophysical properties can be only guessed and T (or P Coupling of PDEs and thermodynamics is not an easy task ling

Possible simplifying assumptions ling The ling Model only well region Cylindrical symmetry could be reduced to 1D Assume water only, thus Liquid density is constant Gas density is given by Ideal Gas EOS Phase coexistence only on vapour pressure curve Assume temperature, varying linearly with depth and constant in t (no energy conservation) Assume no extraction/injection; just set a lower value of P at the top boundary Can assume (we will) natural recharge ling

Free boundary ling The ling In case of gas/liquid phase separation Becomes a 1D free boundary Impermeable rocks at the top (x=0) Assume constant (in time) temperature T = T (x), linear in x Gas reservoir starting at x = L s = 1300 Impose fixed pressure value P = P s at x = L s to simulate extraction well. Sharp (moving) interface s(t) between gas and liqud. Assume saturated vapour pressure on s Liquid between x = s(t) and x = L i = 3000 Assume fixed pressure value at bottom P(x = L i ) = P i Assume no bottom flux (isolated reservoir) ling

ling The ling Full is very complex No analysis can be made, only full 3D simulations. Several commercial codes simulate such of equations, with some simplifications on thermodynamics (e.g., TOUGH2) However, simple 1D can help to understand how things go, e.g., how a vapor/liquid reservoir could evolve into a vapor dominated one, such as in the case of Monteverdi Marittima Possible further step, go cylindrical symmetry and add a vaporization front. ling

ling The ling Good work ling