= Solution
{c}
Use a fixed representative volume $V$ containing many pores. Let $\varphi$ be the <porosity> and let $\mathbf q$ be the <Darcy velocity>, defined per unit total area. The <conservation of mass> is
$$
\frac{d}{dt}\int_V\varphi\rho\,dV
=-\int_{\partial V}\rho\mathbf q\cdot\mathbf n\,dS.
$$
The solid matrix is rigid and fixed, and the liquid is fully saturating and has constant <mass density>, so the stored mass on the left is constant. Microscopic internal liquid-solid boundaries contribute no through-flux by the <no-penetration boundary condition>. Applying the <divergence theorem> to the external flux gives
$$
\int_V\nabla\cdot(\rho\mathbf q)\,dV=0.
$$
Since the representative volume is arbitrary, the macroscopic <continuity equation> is
$$
\boxed{\nabla\cdot(\rho\mathbf q)=0,\qquad \nabla\cdot\mathbf q=0\quad\text{for constant }\rho.}
$$
There is no extra factor of $\varphi$ in the flux: that factor is already incorporated in the definition of the <Darcy velocity>. A compressible liquid or deformable matrix would require retaining the storage term.
Back to article page
