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# General Information

## Oak Ridges Moraine Groundwater Program

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