[Hisham]

Dear Foamers,

We have a problem with numerical modelling of the Darcy porous flow at the
interface between layers of two different materials (i.e. different
permeabilities - with large permeability difference)


Motivation
============

This work is part of a study of the interface conditions at marine
granular filter base on seabed (gravel-sand interface under wave
action). To enhance the basic understanding, lab tests are carried
out with oscillatory "pressurized" flow through a closed tube with
gravel-sand layers set vertically (interface is
horizontal). Currently, we are working on reproducing the experiments
numerically.


Introduction
==============

Using the Finite Volume Method through OpenFOAM CFD solvers (no free
surface), we encounter a problem at the interface between two porous
materials.

First, the description of the simulated case configurations; two
layers of different porous materials on top of each other (as shown in
given sketch) for 1D flow in a circular container (modelled as one
quarter of the 3D geometry with symmetry BCs., we had some problems with axisymmetry)

------ U- zeroGradient (Neumann)
| f | & p- as given oscillatory value (Dirichlet)
| i | --> codedFixedValue as a sin wave
| l |
| t |
| e |
| r | k_1 = 3.8x10^(-4) , n_1 = 0.4
------ (the interface)
| s | k_2 = 2.5x10^(-3) , n_2 = 0.4
| e |
| a |
| b |
| e |
| d |
------ U- zeroGradient (Neumann) & p- as given
oscillatory value (Dirichlet) -->
codedFixedValue as a sin wave

A sketch of the case configuration

zeroGradient for velocity to allow flow in and out according to a
given dynamic harmonic pressure that is set to values from the linear
wave theory.

We have been simulating with the $u-p$ approximation of Biot equations
as well as full CFD (PISO algorithm), with the same problems.


The Problems
==============

Velocity
~~~~~~~~~~

There is a sudden change in velocity at the interface


Pressure
~~~~~~~~~~

The mean value for pressure oscillation at the interface is not zero
as it expected and thus there is a net flow, although according to the
given BCs, the flow is oscillatory with zero net flow.


Tested Solutions
==================

Case Configurations
~~~~~~~~~~~~~~~~~~~~~

Changing the case configurations (i.e. mesh refinement at the
interface does not seem to help, no effect --> as expected)


Modification of the Solution
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

To overcome the pressure gradient discontinuity (pressure gradient
calculated at cell centers instead of faces), we made a correction of
the Darcy permeability at the interface (holding continuity and
momentum balance at the interface) according to Eq. (1) (Szymkiewicz,
2012)



(delta_1 + delta_2) * k_1 * k_2
K_eq = ------------------------------------ Eq. (1)
delta_1 * k_2 + delta_2 * k_1


Assumptions: Mesh orthogonality

K_eq --> equivalent permeability at surface between two adjacent cells
delta_1 --> distance from 1st cell center to face center between cells
1 & 2
delta_2 --> distance from 2nd cell center to face center between
cells 1 & 2
k_1 --> permeability at 1st cell center
k_2 --> permeability at 2nd cell center

The properties multiplied by fluxes in the governing equations are now defined at
cell surfaces instead of center and values are interpolated (permeability is
calculated according to Eq. 1)


Nevertheless, we still have the problem. We would appreciate any suggestions for solving the problem. Thanks in advance.

Reference:

Szymkiewicz A. (2012) "Modelling Water Flow in Unsaturated Porous
Media"


On behalf of my colleagues, best regards,
Hisham

[cdegroot]

I have seen the same issue (in my own code) at clear fluid and porous media interfaces. I wonder if it is the same problem. I have developed some interface conditions to ensure balance of viscous and pressure forces. I have described them in the following article:

A Finite-Volume Model For Fluid Flow and Nonequilibrium Heat Transfer in Conjugate Fluid-Porous Domains Using General Unstructured Grids

http://dx.doi.org/10.1080/10407790.2011.601180