Saturated flow through sediments is governed by Darcy's Law, which relates the pressure drop across the porous medium to the permeability of the medium to calculate the fluid flow rate. The fluid viscosity affects the relationship but is usually assumed to be constant in groundwater studies. That may or may not be the case for your coffee (e.g. if the temperature changes).
If you know the gauge pressure on the pump side of the coffee grounds and if the pressure on the downstream side is atmospheric, then you know the pressure drop. You need to know/estimate/calibrate the permeability of the media. You don't say what kind of coffee maker, but it will depend on the grind and tamping for an espresso machine. You need to decide if you can assume the value is constant. If it is not, you need to determine how it varies.
Groundwater studies use analytical or digital model solutions to determine how flow changes with differences in boundary conditions and variation in permeability. Your system sounds similar to Darcy's original experiments so you can easily calculate flow if you know the other parameters.
