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General Information

Oak Ridges Moraine Groundwater Program

Geothermal transport modelling



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