# Velocity field to a permeability field using poisson pressure equation

I have a velocity field and I want to get a pressure field. In my experiment we're controlling the pressure at the inlet and the outlet. I have Dirichlet boundary conditions at the inlet and outlet and I'm applying Neuman boundary conditions at the vertical walls. To derive the hydraulic pressure map, I'm taking the divergence of the steady- state NaviereStokes equation (NSE) for an incompressible fluid. The poisson pressure equation reads

$$−\frac{1}\rho(\nabla^2p)=−\frac{1}{\rho}\left(\frac{∂^2p}{∂x^2}+\frac{∂^2p}{∂y^2}\right)=\left(\frac{∂u}{∂x}\right)^2+ 2\cdot \frac{∂u}{∂y}\frac{∂v}{∂x}+\left(\frac{∂v}{∂y}\right)^2$$

In my code I tried different velocity fields ( constante, linear, random etc...) but I'm always getting a linear pressure field which I find not logical. Do you have an idea what I'm doing wrong? because i think It's not normal to get always a linear pressure. Do you have another proposition to solve my problem?

In my case, I'm working on concrete so the velocity is in range of $10^{-6}\frac{m}{s}$ and the pressure on the bondary conditions are in range of $10^6$ Pa. When I'm searching the $(\nabla^2p)$, I'm getting values close to zero which is normal (since the range of the velocity is very small comparing to the pressure field) so at the end I'm getting a linear pressure field.

For info, I'm solving my problem using finite differences. What do you think?

• If you have the velocity field, why aren't you using Darcy's law to get the pressure field? Any why is the density in there, since this is a viscous-dominated flow? – Chet Miller Oct 16 '17 at 15:17
• Have you checked your Poisson solver by itself? Maybe on more basic forcing functions? – Darwin Oct 16 '17 at 15:19
• @ChesterMiller The aim of my study is to get the permeability field so I need to search the pressure field first. – M. Yehya Oct 17 '17 at 8:53
• @Darwin yes my poisson solver is working well. But I'm confused because the results are not similar to what I can get when using another software that from a permeability field I can get a pressure field and later a velocity field – M. Yehya Oct 17 '17 at 8:58
• From your description, I don't really know what is known and what is not known. As best I can tell, you know the permeability of the porous medium and the pressure boundary conditions, and you are trying to determine the interior pressure distribution. Is that correct? And why are you using the Navier Stokes equations (including inertial terms) if you know that the medium is porous, and the inertial terms are negligible? – Chet Miller Oct 17 '17 at 12:45

The equations that describe flow through a porous medium are $$\frac{k}{\mu}\vec{\nabla} P=\vec{v}$$ and $$\vec{\nabla} \centerdot \vec{v}=0$$So, $$\frac{\partial}{\partial x}\left(k\frac{\partial P}{\partial x}\right)+\frac{\partial}{\partial y}\left(k\frac{\partial P}{\partial y}\right)=0$$