Thermomechanical modelling of non-isothermal porous materials with application to enhanced geothermal systems

Abstract

Geothermal heat is the energy stored in the Earth that can be considered inexhaustibly over any reasonable time frame. Moreover, The higher temperatures of greater depths can be used to generate electricity and district heating. To use the geothermal heat in an economical way, temperatures of at least 80 to 100 degrees Celsius are necessary. Depending on the geological conditions, the geothermal gradient, that is the change of temperature relative to the depth, has a large range from one up to hundreds degrees Celsius per hundred metres. For an enhanced geothermal system (EGS), the permeability of the low-porosity rock usually has to be increased. This is commonly done by hydraulic stimulations. To operate an EGS, at least two wells must be drilled. After stimulation and, consequently, creation of the reservoir, the cold water is pumped down through the injection well (IW) into the subsurface. It flows through the porous rock and is, thereby, heated at the contact surfaces with the hot rock. Subsequently, the heated water is delivered back to the surface through the production well (PW), where the heat energy of the water can be used directly for heating applications, or can be converted into electricity. Finally, returning the cooled water into the IW closes the loop. Although the system is supplied continuously from the lower layers with heat, the temperature of the reservoir eventually decreases and, thus, the productivity of the reservoir diminishes in the course of years. In order to make predictions about the development of a reservoir in advance, not only the properties of the water and the rock must be accurately described, but also the propagation of heat, in particular the transition of heat from the rock to the water, must be taken into account. The subsurface consists of two components, namely the fractured rock and the water in the cracks. Driven by a pressure gradient, the water flows through the cracks. There are different mechanisms how the heat propagates within the reservoir. Depending on the temperature difference between the components and according to the second law of thermodynamics, the heat exchange takes place at the contact surfaces with the rock. Furthermore, a heterogeneous temperature distribution in each component results in a heat flow by conduction along the temperature gradient in the rock as well as in the water. Moreover, the heat is transported by the flow of water, thus, convectively. The system is further supplied with heat from deeper layers via conduction. The model is capable to describe the mechanical behaviour of the components, which is determined by the deformation of the rock due to a mechanical load, the flow velocity of the water within the cracks and the mechanical interaction between the components. Moreover, and this is the main focus here, the temperatures of the components are different in general and are, therefore, considered individually. Thus, the thermo-mechanical coupling, the heat transfer based on conduction within the components, the convection because of the flow of water, and the heat exchange between the constituents are considered. It is the aim of this contribution to derive a biphasic, thermodynamically consistent porous-media model, where a viscous fluid, namely water, flows through the pores of the elastic and incompressible solid skeleton, namely the rock, while both constituents are under non-isothermal conditions. The continuum-mechanical model is embedded in the framework of the TPM, where the rock and the water are respectively represented by a porous solid and a pore fluid. To achieve this goal, in addition to the kinematic relationships, which directly result from the spatial configurations and their changes over time, the balances of mass, linear momentum and energy are axiomatically introduced. These balance relations build a system of partial differential equations (PDE), which must be closed by means of constitutive assumptions. The constitutive assumptions have to satisfy the constraints arising from the evaluation of the entropy inequality. It ensures the thermodynamically consistency of the formulation. Furthermore, the primary variables for the solution of the PDE system are considered to be the solid displacement, the pore-fluid pressure, and the fluid and solid temperatures. To describe the materially incompressible solid, an elastic material behaviour is assumed. Due to the fact that the deformations of the solid are relatively small, it is sufficient to consider linearised stress and strain tensors. Furthermore, the thermal expansion of the solid constituent is neglected. The viscous fluid is also considered materially incompressible, but its density changes dependent on the temperature. Moreover, quasi-static conditions are considered, because of negligible accelerations of the constituents and of the overall aggregate. The numerical implementation of the model is achieved via the finite-element method (FEM), using the FE-Tool PANDAS. In the context of geothermal systems, the heat transfer via convection is usually dominant. Different techniques were implemented and their performance was investigated to overcome the problem of oscillations due to the convection-dominant system. Finally, an initial-boundary-value problem is solved according to a circulation test in a geothermal plant in Soultz-sous-Forêts (France).