The Pressure Equation

Pressure Spreads Like Heat

Substituting Darcy's law into the mass balance for a slightly compressible fluid gives the diffusivity equation $\dfrac{\partial P}{\partial t} = \eta,\nabla^2 P$, where the hydraulic diffusivity is $\eta = \dfrac{k}{\phi,\mu,c_t}$. Pressure spreads through the reservoir the way heat spreads through a solid.

The Radius of Investigation

A well's drawdown does not appear everywhere at once; it diffuses outward as an error-function front. The distance it has reached, the radius of investigation, grows with $\sqrt{\eta,t}$, so quadrupling the time only doubles the reach.

Reading Permeability from Pressure

Higher diffusivity, from higher permeability or lower compressibility, pushes the front out faster. A pressure-transient well test inverts exactly this relationship to infer permeability from how fast the drawdown spreads.