The basic equations of Darcy flow coupled to thermal transport are: $$$$\nabla \cdot \vec{u} = 0 \\ \vec{u} = -\frac{\mathbf{K}}{\mu} (\nabla p - \rho \vec{g}) \\ \sigma \frac{\partial T}{\partial t} + \vec{u}\cdot\nabla T - k\nabla^2 T = 0$$$$ where $$$\vec{u}$$$ is the fluid velocity, $$$\mathbf{K}$$$ is the permeability tensor, $$$\mu$$$ is fluid viscosity, $$$p$$$ is the pressure, $$$\rho$$$ is the fluid density, $$$\vec{g}$$$ is the gravity vector, $$$T$$$ is the temperature, $$$\sigma$$$ is the ratio of the heat capacity of the medium to the heat capacity of the fluid, and $$$k$$$ is the effective thermal diffusivity of the saturated medium.

We shall henceforth assume that $$$\vec{g} = \vec{0}$$$. If gravity is important, it is possible to include buoyancy effects in the fluid density by introducing Boussinesq's "slightly compressible" approximation, $$$\rho = \rho_0 (1 - \alpha(T - T_0))$$$. Using this assumption, taking the divergence of the second equation, and imposing the divergence-free condition leads to the following system of two equations in the unknowns $$$p$$$ and $$$T$$$: $$$$-\nabla \cdot \frac{\mathbf{K}}{\mu} \nabla p = 0 \\ \sigma \frac{\partial T}{\partial t} + \vec{u}\cdot\nabla T - k\nabla^2 T = 0$$$$ where $$$\vec{u} = -\frac{\mathbf{K}}{\mu} \nabla p$$$. That is, the pressure satisfies Laplace's equation (with possibly non-constant coefficients) and the temperature satisfies the linear convection-diffusion equation.