Governing equations in geothermal energy

Fimbul currently supports pure H₂O as a single-component, two-phase fluid system. Water may be present in a liquid phase ($L$) and/or a vapor phase ($V$).

Conservation of mass

Mass conservation is written as

$${\partial }_t([1-\varphi ]{\rho }_R+\varphi [{\rho }_L{s}_L+{\rho }_V{s}_V])+\mathrm{\nabla }\cdot ({\rho }_L{\overrightarrow{v}}_L+{\rho }_V{\overrightarrow{v}}_V)=Q_M,$$

where $\varphi$ is porosity, ${\rho }_{\alpha }$ is density for the rock ($\alpha =R$) or fluid phase ($\alpha \in \{L,V\}$), and $s_{\alpha }$ is phase saturation (the fraction of pore volume occupied by phase $\alpha$, with $s_L+s_V=1$). The term $Q_M$ denotes mass sources and sinks, including wells and boundary fluxes. Phase velocities are given by Darcy's law,

$${\overrightarrow{v}}_{\alpha }=-{\lambda }_{\alpha }\mathbf{K}(\mathrm{\nabla }p-{\rho }_{\alpha }g\mathrm{\nabla }z),$$

where ${\lambda }_{\alpha }=k_{r,\alpha }/{\mu }_{\alpha }$ is phase mobility (ratio of relative permeability to viscosity), $\mathbf{K}$ is absolute permeability, $p$ is pressure, $g$ is gravitational acceleration, and $z$ is elevation.

NOTE: In practice, the interface between liquid and vapor phases in an H₂O system may exhibit non-negligible capillary pressure effects. This is currently not modeled in Fimbul; support is planned for a future release.

Conservation of thermal energy

Thermal energy conservation is expressed as

$${\partial }_t([1-\varphi ]{\rho }_R{u}_R+\varphi [{\rho }_L{s}_L{u}_L+{\rho }_V{s}_V{u}_V])+\mathrm{\nabla }\cdot ({\rho }_L{h}_L{\overrightarrow{v}}_L+{\rho }_V{h}_V{\overrightarrow{v}}_V)-\mathrm{\nabla }\cdot \left(\mathbf{\Lambda }\mathrm{\nabla }T\right)=Q_H,$$

where $u_{\alpha }$ is specific internal energy, $h_{\alpha }=u_{\alpha }+p/{\rho }_{\alpha }$ is specific enthalpy, $T$ is temperature, and $\mathbf{\Lambda }$ is the effective thermal conductivity tensor of the rock–fluid mixture. The second term on the left represents advective heat transport by the flowing phases; the third term is conductive heat transport via Fourier's law,

$${\overrightarrow{q}}_c=-\mathbf{\Lambda }\mathrm{\nabla }T.$$

The term $Q_H$ denotes thermal energy sources and sinks.

Fluid properties

The fluid properties of H₂O—such as density, viscosity, enthalpy, and phase state—depend on both pressure and enthalpy and are provided through precomputed lookup tables.

For the high-enthalpy (two-phase) regime, the steam tables are generated using the CoolProp library [1] via the FimbulCoolPropExt extension. They are stored as Artifacts that are loaded automatically when using the high-enthalpy formulation. The tables can also be loaded directly using the Artifacts API. To generate your own tables, see build_steam_tables_h2o in FimbulCoolPropExt.

For the low-enthalpy (single-phase liquid) regime, Fimbul uses PVT tables from JutulDarcy generated using the NIST database.

The figure below shows the temperature, density, and vapor saturation of H₂O as functions of pressure and specific enthalpy, covering the full range of both single-phase and two-phase states. The saturation line separates the single-phase liquid region (left), two-phase region (center), and single-phase vapor region (right). These phase diagram contours can be visualized using plot_phase_diagram_contours.

Single- and two-phase systems

Most Fimbul examples currently operate in the low-enthalpy regime, i.e., single-phase liquid conditions. The high-enthalpy examples use the two-phase steam-table formulation described above.

Multi-continuum and fractures

The formulation above assumes local thermal equilibrium between rock and fluid, i.e., a single shared temperature field $T$. This is appropriate for many porous-medium geothermal applications, but may not be adequate when flow is dominated by fractures or other high-contrast heterogeneities.

To model fractured systems, Fimbul can be combined with the discrete fracture modeling framework in JutulDarcy—see e.g., ftes.