# 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

## 1 Answer

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, $$\frac{\partial}{\partial x}\left(k\frac{\partial P}{\partial x}\right)+\frac{\partial}{\partial y}\left(k\frac{\partial P}{\partial y}\right)=0$$

You parameterize k as a function of x and y, then you solve the PDE subject to the imposed pressure conditions. Then you calculate the velocities at the measurement points, determine the least square error, and minimize the error with respect to the parameterization parameters of k.

• We tried to do that but my code was not converging. because I dont know how to minimize the error with respect to the parameterization of K. I did that but I'm not able to find my K espacially when I'm having too many nodes and elements. Do you have some references or some examples of people who did that? Thanks – M. Yehya Oct 18 '17 at 8:28
• Before I would give further advice on something like this, I would have to know more details of the geometry, the locations and methods of measuring velocity, and the locations of the pressure boundary conditions. – Chet Miller Oct 18 '17 at 10:48
• I'm testing saturated concrete by injecting high pressure. So in my experiment I'm controlling the inlet pressure and the outlet pressure is the atmospheric pressure. By image analysis, I'm getting a velocity field and I want to go to a permeability field that's why I was using poisson pressure equation since there's a direct relation between the velocity V and the pressure P. I tried to do the inverse analysis of darcy's law using finite elements but the problem is too complex. The velocity is measured by following the water front in the radiographies (2d images). – M. Yehya Oct 18 '17 at 12:11
• The sample is cylindrical? The flow is basically 1D? The inlet area is equal to the outlet area? You know the volumetric flow rate? Why do you want to know the permeability field? – Chet Miller Oct 18 '17 at 12:29
• The sample is cylindrical and the flow is 1D and the inlet area is equal to the oulet area. I dont know the volumetric flow rate. I want to search a permeability field in my concrete to see how the permeability is changing in the sample when I have a crack for example or a bar etc.... – M. Yehya Oct 18 '17 at 13:54