What is the difference in using Laplace equation (say only in 1D) to simulate water flow in a porous media with that of using diffusion equation for same purpose?
The Laplace equation, (d^2 Ψ)/(dx^2 )+(d^2 Ψ)/(dy^2 )+(d^2 Ψ)/(dz^2 )=0, is just a steady state 3D flow equation. It's a black box conservation of hydraulic potential. Diffusion doesn't come into it. The Diffusion equation (assuming homogeneous isotropic conditions) is (∂^2 Ψ)/(∂x^2)+(∂^2Ψ)/(∂y^2)+(∂^2 Ψ)/(∂z^2)= S_s/K ∂h/∂t. This discretizes the time element, and is very much dependent upon the diffusive properties of the aquifer. Note that S_s/K is the reciprocal of the 'hydraulic diffusivity', hence the name of the equation.