What is the difference in using Laplace equation (say only in 1D) to simulate water flow in a porous media (aquifer) with that of using diffusion equation for same purpose?
Laplace equations simulates steady state flow (time independent, if there were not sources).
Diffusion equation simulates transient processes (time dependent). If boundary conditions do not change with time, then solution of diffusion equation for large time tends to solution of the Laplace equation with same boundary conditions.
Both the Laplace and Diffusion equations are used for simulation of water/fluid flow in porous media under steady state conditions .While under unsteady state the Diffusion equation represents the possible treatment when the boundary conditions are defined and after that the Laplace equation is used to obtain the ultimate simulation
I agreed with the submissions of Alexander. I also like to add further that, while the use of Laplace equation implies a non-anisotropy media with respect to hydraulic properties of the aquifers and therefore a steady state condition; the diffusion equation readily takes into consideration the anisitropy of the medium which of course could be overcome after certain period during which the flow becomes steady.
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