Thermal modelling of the Larderello geothermal field (Tuscany, Italy)

Abstract

We present a 3-D thermal model of the Larderello geothermal field (Tuscany) to evaluate (1) the extent and contribution of the heat transfer mechanisms (conduction vs. convection) at the intermediate-upper crust levels, (2) the variability of the heat and mass fluxes entering from below and (3) the crucial role of the formation permeability. The model, composed by three main layers, considers the upper 10 km of the crust to better constrain the simulations with experimental data from borehole, fluid inclusion studies and hypocentral distributions. Several sets of simulations were carried out with different bottom boundary temperatures and different formation permeabilities for the two deeper layers. The results indicate that the present temperature (T) and pressure distributions in the Larderello field require deep reservoir rocks with higher permeability than the overlying capping units and underlying intermediate crust. Permeability values of 1 mDarcy for the reservoir rocks are enough to allow fluid convection, if the temperature at 10 km depth is as high as 500 ± 50°C. The presence of localized zones with formation permeability 50–100 times higher than the surrounding rocks strongly favours the migration of over-pressurized fluids, which episodically break through the overburden, feeding the presently exploited geothermal fields.

Appendix 1

The numerical modelling was performed using the commercial code SHEMAT 7.1 (Clauser 2003) considering various boundary conditions, inner geometries and formation permeabilities.

SHEMAT 7.1 code (Clauser 2003) was used to carry out the simulations of coupled heat and fluid fluxes. The solute transport was not simulated, although we assumed a purely speculative average density of the fluid of 1,100 kg m⁻³ (at room T). The 3-D regional conductive-convective model was realized by means of unsteady forward simulations, under the assumptions of impervious and isothermal top and bottom boundaries, lateral adiabatic faces and variable internal physical properties.

Solving the non stationary problem consists of finding the T, pressure and fluid velocity fields within the model domain, assuming appropriate initial and boundary conditions.

The code SHEMAT was used to simulate both the solid-state heat conduction and the conductive-convective flow of pore fluids. Non-stationary equations of non-isothermal hydrodynamics, accounting for phase transition, were used to carry out the three dimensional modelling of the convective flow. The most important assumptions were:

• flow conforms to Darcy’s law;

• effects of the capillary pressure are neglected;

• both phases (solid matrix, liquid) are considered to be in local thermal equilibrium;

• fluid flow does not affect the solid matrix.

The problem is described by a system of mass and energy conservation equations in Cartesian coordinates for convective two-phase flow through a porous medium (Faust and Mercer 1979):

$$ \nabla \cdot {\left( {\,\underline{{\lambda \,}} \nabla T - \rho _{f} \,c_{f} \,T\,\ifmmode\expandafter\vec\else\expandafter\vec\fi{v}\,} \right)} = \frac{{\partial T}} {{\partial t}}{\left[ {\,n\,\rho _{f} \,c_{f} + {\left( {1 - n} \right)}\;\rho _{m} \,c_{m} } \right]} $$

(1)

$$ {\underbrace {\rho _{f} \,g{\left( {\alpha + n\,\beta } \right)}}_{{S_{S} }}}\frac{{\partial h_{0} }} {{\partial t}} = \nabla \cdot {\left[ {{\underbrace {\frac{{\rho _{f} \,g\,\underline{k}}} {\mu }}_{\underline{K}}}{\left( {\nabla h_{0} + \rho _{r} \,\nabla z} \right)}} \right]} $$

(2)

where

α :

rock compressibility (Pa⁻¹);

β :

fluid compressibility (Pa⁻¹);

c :

specific heat capacity (J kg⁻¹ K⁻¹);

g :

gravitational acceleration (m s⁻²);

h :

hydraulic potential, head (m);

λ :

tensor of thermal conductivity (W m⁻¹ K⁻¹);

k :

tensor of permeability (m²);

K :

tensor of hydraulic conductivity (m s⁻¹);

n :

porosity (/);

ρ:

density (kg m⁻³);

S _S :

specific storage coefficient (m⁻¹);

T :

temperature (K);

t :

time (s);

$\ifmmode\expandafter\vec\else\expandafter\vec\fi{v}$ :

Darcy (filtration) velocity; specific discharge (m s⁻¹).

Subscripts

f :

liquid;

m:

matrix;

0:

reference condition.

SHEMAT assumes the properties of water as a function of fluid pressure and T for pressure greater than saturation pressure or T lower than critical T, limited to pressure and T below 100 MPa and 1,000°C, respectively (Meyer et al. 1979). SHEMAT accounts for the T dependence of rock thermal conductivity according to Zoth and Hänel (1988). Examples of the application of the SHEMAT code may be found in Clauser (2003).