darcy and continuity equations coupling
Hi everyone,
I am working on a finite volume modeling, the system is a porous media, composed of a solid phase and a fluid phase. unstead of the momentum equation, the velocity of the fluid is described by the darcy equation, where the velocity is directly related to the pressure gradient. my question is: should i apply the same methods (simple, simpler,piso..) algorithms for the coupling of darcy's law and the continuity equation? 
I think everything is much easier than general pressure/velocity coupling with NavierStokes. The face mass/volume fluxes arise from the face gradient of the pressurethat makes it a function of the two cells on either side of face (with, perhaps skew terms added in explicitly). Then substitute those face fluxes in the continuity equation. That gives you a Poisson(like) equation for pressure. Solve it, and explicitly update the velocities at the faces using the grad(p). There is your Darcy flow.
The SIMPLEx/PISO techniques are algorithms attempting to converge the effects of inertia and convective transport manifested in the momentum equations with the incompressibility constraint in the continuity equation. In Darcy flow, the momentum equation reduces so significantly that it no long includes the competing effects. Pressure dictates velocity directly. 
Thank you, dear Michael, for your detailed answer
So, there will be no checkboard pressure problem if I use the pressure values of the discretised poisson's equation (1D)(d²p/dx²= 0) to calculate the velocity values using the discretised form of the darcy equation? 
No, because you are not solving a linear system for velocity as you do with SIMPLEx/PISO applied to NS. Making the substitution I outlined leaves you with only one independent variablethe pressure. There is nothing left to get out of sync. The velocity field (at the faces) uniquely follows from the pressure field (on the cells).

Ok, Thank you so much. :)

I should point out a few things. There may be a gravity term in the Darcy law. That could give you something more complicated than a Laplace equation for pressure. The other issue is with the wildly varying permeability from cell to cell that can occur in real simulations. You will need to do something like harmonic averaging to get that formulated correctly. Good luck.

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