Simplified mathematical modelling of geothermal reservoir
Problem raised by ENEL, Regione Toscana
Dr. Alessandro Speranza (Universitŕ degli Studi di Firenze)
Exposition of the problem:
The proposed problem concerns a simplified model of geothermal wells.
In general, a geothermal reservoir can be modeled as a multiphase fluid flowing in a porous medium. In fact, the medium is more like a fractured area of hard rocks, rather than a typical porous medium. However, the Darcy approximation is generally considered acceptable. The fluid generally consists of a mixture of brine (solution of salts in water) and a certain set of so called “non condensible gases”, such as CO2, N2, H2S and so on. The fluid can be found in one (gas or liquid) or two coexisting. In the first case, we have a “vapour or water dominated reservoir”. For instance the two main geothermal reservoirs in Toscany, Larderello and Amiata are, respectively known as vapour dominated and water dominated. However, a thermodynamic analysis shows that while Larderello could actually contain gas only, Amiata is in two-phase coexistence conditions, in terms of typical pressure and temperature.
The actual geometry of the reservoir is generally non known in details. However, this is a strongly three dimensional problem, in the sense that, generally, it is not easy to use any rotational or translational invariance to model effectively the fluid flow in the reservoir, unless one wants to limit the model to a small area in the nearby of the production well.
The proposed model, is limited to a simple one dimensional approximation of the geothermal reservoir. The geothermal fluid is assumed to be pure water H2O. Furthermore, temperature is assigned as varying linearly with depth. In spite of its drastic reduction, this simple model allows to draw some useful information on the real problem. Furthermore, in spite of the presence of a free boundary at the interface between the gas and liquid region of the well, the model allows both some analytical considerations as well as a numerical solution.
Scheme of the work to be done:
1) Introduction of the general problem and reduction to the simplified 1D model. Set up of the set of equations, and of boundary conditions. As a first step, one can assume the well to be isolated, and solve the dynamic problem, to find when the well reaches its exhaustion, in the sense that inside pressure has reached the outside value, and no more energy can be exctracted from it.
2) Introduction of a positive source term in the mass balance equation, for instance assuming a recharge of the extracted fluid at the boundary. In this case, one can look for the condition to impose at the boundary, in order to have an equilibrium solution
3) Numerical solution of the dynamic problem with a source term in the mass balance equation, in order to verify whether and when the problem actually reaches equilibrium, given the right choice for mass extraction from the top and fluid recharge at the lateral boundary.