Geothermal transport modelling
Theory
Assumptions:
- incompressible flow
- fully saturated
- isothermal
- background/regional temperature $\approx 10^oC$
Saturated flow:
\[\nabla\mathbf{q} + S_s\frac{\partial h}{\partial t} + Q=0\]where from Darcy’s law:
\[\mathbf{q}=-\mathbf{K}\nabla h\]Thermal transport in saturated porous media:
\[\frac{\partial \rho_b c_b T}{\partial t} = -\nabla\left[\mathbf{q} \rho_w c_w T -(k_b+\rho_bc_b\mathbf{D})\nabla T\right] + Q_T\] \[\frac{\partial \rho_b c_b T}{\partial t} = -\nabla\left[\mathbf{q} \rho_w c_w T -\lambda\nabla T\right] + Q_T\] \[\lambda=k_b+\rho_bc_b\mathbf{D}\]where $Q$ is the sink/source term (i.e., well pumping/injection) at temperature $Q_T$, $\mathbf{K}$ is the saturated hydraulic conductivity tensor, the dispersion tensor:
\[\mathbf{D}=f\left(\alpha_l, \alpha_{th}, \alpha_{tv}, D^*\right),\]bulk density:
\[\rho_b = (1-\phi)\rho_s+\phi \rho_w,\]and bulk heat capacity:
\[c_b = (1-\phi)c_s+\phi c_w.\]| Parameter | Description | Assigned value |
|---|---|---|
| $\alpha_l$ | longitudinal dispersivity | 1.0 |
| $\alpha_{th}$ | transverse horizontal dispersivity | 0.1 |
| $\alpha_{tv}$ | transverse vertical dispersivity | 0.1 |
| $D^*$ | effective molecular diffusion coefficient | 1.4e-7 |
| $k_b$ | bulk thermal conductivity | 0.6 |