Spherical Darcy injection ($q_r=\Pi a\Delta p/(\mu r^2)$)

For a sphere of radius $a$ injecting an incompressible flow into an infinite homogeneous isotropic porous medium, Darcy's law and conservation of mass give $r^2{q}_r=C$. Integrating $p_r=-\mu C/(\Pi r^2)$ with $p(a)-p(\infty )=\Delta p$ gives $C=\Pi a\Delta p/\mu$. The hydraulic resistance is $\Delta p/Q=\mu /(4\pi \Pi a)$. A common background hydrostatic pressure cancels gravity if the imposed excess pressure is measured relative to that background.