Porous Flow - Submarine Hydrothermal Systems
Theoretical background
In the last lecture, we had made Navier-Stokes simulations on the pore-scale, which resolved the full dynamics of fluid flow. As we have already learned, in porous media studies flow is often approximated by Darcy’s, in which flow is proportional to the pressure gradient. The invovled constants are viscosity and permeability.
During this lecture we will study single-phase hydrothermal flow in submarine hydrothermal systems. The respective solver is named HydrothermalSinglePhaseDarcyFoam
. The hydrothermal fluid flow is governed by Darcy’s law (Eqn. equation (22)),
mass continuity (Eqn. equation (23)) and energy conservation (Eqn. equation (28)) equations shown below,
where the pressure equation equation (27) is derived from continuity equation equation (23) and Darcy’s law equation (22).
Implementation
The details of the OpenFoam implementation can be found in the HydrothermalFoam documentation. Here we only show a brief summary. Fig. 17 shows how the energy equation is solved within the OpenFoam framework.
Equation-of-state
The fluid properties like density, viscosity, specific heat are determined from the equation-of-state of pure water. Fig. 18 shows the phase diagram of pure water. At sub-critical conditions (P< 22 MPa), the boiling curve divides the regions of liquid water and water vapor. At super-critical conditions, there is a gradual transition from a liquid-like to a vapor-like fluid phase. HydrothermalFoam is a single phaes code and can only be used in regions, where a single fluid phase is present, i.e. under pure liquid water, water vapor, or supercritical conditions; boiling cannot be resolved. As we will find out later, the thermodynamic properties of water have first order control on flow dynamics and upflow temperatures in submarine hydrothermal systems.