text: The porous medium equation name originates from its use in describing the flow of an ideal gas in a homogeneous porous medium. We require three equations to completely specify the mediums density , flow velocity field , and pressure : the continuity equation for conservation of mass; Darcys law for flow in a porous medium; and the ideal gas equation of state.  These equations are summarized below:<math display="block">\begin{aligned}
  \varepsilon {\partial \rho\over{\partial t}} &= -\nabla \cdot (\rho {\bf v}) & (\text{Conservation of mass}) \\
  {\bf v} &= -{k\over{\mu}}\nabla p & (\text{Darcy's law}) \\
  p &= p_{0}\rho^{\gamma} & (\text{Equation of state})
  \end{aligned} is the porosity,  is the permeability of the medium,  is the dynamic viscosity, and  is the polytropic exponent (equal to the heat capacity ratio for isentropic processes). Assuming constant porosity, permeability, and dynamic viscosity, the partial differential equation for the density is:where  and .