Model equations similar to those for potential flow arise in flow in porous media, which has a wide variety of applications, e.g., in groundwater treatment, water-purity remediation, filtration, flow of a drug into a tissue such as a tumor, etc. The mathematical structure of the problem is presented here.

The model used is the Darcy equation, which states that the velocity is proportional to the pressure gradient,

\[\begin{equation*}v=-\frac{\mathcal{L}^{2}}{150 \mu} \frac{d P}{d z} \tag{15.84}\end{equation*}\]

where \(\mathcal{L}\) is a characteristic length parameter of the system.

Equation (15.84) has the form of the flux (velocity is a volumetric flux) as a function of a driving force (the pressure gradient) and is known as Darcy's law. Since all the parameters appearing on the RHS cannot be estimated exactly for porous media, they are combined into a constant \(\kappa\) defined as the Darcy permeability and the equation is written in a simple flux-driving force formula:

\[\begin{equation*}v_{z}=-\frac{\kappa}{\mu} \frac{d P}{d z} \tag{15.85}\end{equation*}\]

This equation is known as Darcy's law. The term \(\kappa\) is called the permeability of the porous medium. The equation may be generalized for three dimensions as

\[\begin{equation*}v=-\frac{\kappa}{\mu} abla P \tag{15.86}\end{equation*}\]

The vector \(v\) satisfies the continuity equation, i.e.,

\[abla \cdot \boldsymbol{v}=0\]

Hence, by applying the divergence operator on both sides of Eq. (15.86), verify that the pressure field is given by

\[\begin{equation*}abla^{2} P=0 \tag{15.87}\end{equation*}\]

Hence we find that the pressure field in a porous medium satisfies the Laplace equation. This equation has the same form as that for steady-state conduction, and hence similar solution methods can be used to compute the pressure field. Once the pressure field has been computed, the velocity field can be recovered by applying Eq. (15.86).