Darcy's Law, Revisited

Your supervisor pulls up the Darcy's Law simulation you first saw as a homepage demo and finally explains it properly: it's not just an animation, it's the physical law that will constrain everything you build in this mission.

Darcy's Law says groundwater flows from high head (water table elevation) to low head, at a rate set by the hydraulic gradient and the material's conductivity. Near a pumping well, head drops toward the well, creating a 'cone of depression' that gets shallower the further you travel from the well, but never reverses direction.

For our transect, that means head as a function of distance must change consistently as distance increases, it can flatten out far from the well, but it cannot rise-then-fall-then-rise. That single constraint eliminates the vast majority of curves that would otherwise fit our eight noisy points.

Think of the dip a mattress makes when someone sits on it. Right under them, it sinks the most. Moving away from them, it gradually flattens back to normal. It never bounces back up above normal, dips again, then rises. The water table around a pumping well behaves the same predictable way.

Darcy's Law in one dimension, and the physical constraint it implies for our transect:

Revisit the Darcy's Law simulation, note how head always changes in one consistent direction along the flow path.

Every physics-informed model starts by writing down a governing equation like this one and asking: what does it forbid? Hydrogeologists, climate scientists, and structural engineers all use this same move, turning a physical law into a hard constraint a model must respect, not just a suggestion.

In plain language, explain why a 'cone of depression' can't have a bump in the middle.