Fluid Flows in Porous Media

Svetlana Tlupova
MSRI Summer School on Mathematical Graphics at Reed College


Fluid flows though porous media have many practical applications, which include filtration, permeation of water or oil within the matrix of a porous rock and so on. In particular, filtration of blood in the kidneys is the first step in urine formation, where very small capillaries filter minerals, wastes and water but retain red cells, proteins and large molecules. The following applet (Henry's favourite) is a simple demonstration of the process. Red cells and proteins cannot cross the membrane, but small minerals can.

So what do we solve for a fluid flow near a porous boundary?

Consider a viscous two-dimensional incompressible flow in a channel where the bottom boundary is adjacent to a porous medium. A uniform pressure gradient is maintained in the longitudinal direction, the velocity is zero on the top boundary and the normal component of the inflow and outflow velocity is zero. For the porous interface, we will use the slip boundary condition derived by Beavers and Joseph in 1967. Here p is the pressure and (u,v) is the velocity of the fluid.

The following applet shows a flow that solves the above problem. You'll see two colors: the yellow particles move as if the lower boundary weren't porous at all, so they have zero velocity on both walls of the channel. The red particles satisfy the slip condition, so their velocity on the lower boundary is not zero.


Modeling flow through a porous medium.
We've considered a flow of free fluid near a porous boundary. What happens if the region is porous? A porous medium can be modeled as a collection of point obstacles in the fluid. In this case the problem becomes one where we are solving forced Stokes equations:

The fundamental solution, i.e. the velocity due to a point force, is called a Stokeslet, and the flow due to a distribution of point forces can be computed using the superposition of these. To avoid the singularity which is proportional to log(r) in 2D we will be solving the above equations where the force is

Here the function

is an example of a smooth approximation of the delta function, also called a cutoff function. The exact solution of the problem is

where

If there are several forces, the velocity can be computed as the sum of all the Stokeslets:

In our case, the velocities of fixed obstacles have to equal to zero, so a system of equations can be set up to compute the forces that satisfy those conditions. Once the forces are known, the velocity at any point in the fluid can be computed using the Stokeslet formula.

My last applet shows fluid flow around an arbitrary number of obstacles. The particles are given a velocity (1,0). There are no boundaries in this case, so adding point obstacles will possibly make the fluid go out of the shown region. The top picture will scale the color down to the velocity of the points, so regions will velocity close to zero become nearly black. So add as many obstacles as you wish and see what happens! New particles can be added at any time.



Many thanks to the organizers of the workshop - this has been a great experience! Thank you for all your help during the development of this project.


References
1. R. Cortez, The method of regularized Stokeslets, SIAM J. Sci. Comput., v. 23 (2201), pp. 1204-1225.
2. G.S. Beavers and D.D. Joseph, Boundary conditions at a naturally permeable wall, J. Fluid Mech., 30 (1967), pp. 197-207.
3. W.Jager and A.Mikelic, On the interface boundary conditions by Beavers, Joseph and Saffman. SIAM J. Appl. Math., v. 60 (2000), pp. 1111-1127.