Campus Theory / Science at the Edge Seminar

                      Friday, 02 May 2003


Speaker:   Bill Rossen,  University of Texas, Austin


Title:     Application of Percolation Concepts to
           Foam Flow Through Porous Media

Abstract:

Foams are injected into geological formations to redirect gas flow 
in enhanced-oil-recovery processes, to divert acid flow in oil-well 
stimulation, and optimize cleanup of contaminated groundwater.  In such 
applications, the complex geometry of the porespace meets the complex, 
changing rheology of the foam.  Percolation theory is useful in 
understanding the flow of Newtonian fluids through porous media, which
can be modeled as the flow of electricity through a network.  In foam flow, 
however, the fluid has nonlinear rheology, including a yield stress, and
the properties of the fluid can change abruptly as bubbles are created
or destroyed.  In these applications the bubbles in foam are as large as 
pores, so one should think not of shaving cream in a pipe but of 
sausage-shaped bubbles moving through an irregular, interconnected
porespace.

With foam, even at substantial pressure gradients, most gas in the 
porespace is trapped, and the foam that does flow moves along individual
paths called "bubble trains."  At the minimum pressure gradient for flow, 
the foam selects the path with the minimum resistance of all paths in the 
medium.  Calculations on a Bethe or Cayley tree network show that the path 
does not simply randomly sample the percolation-threshold fraction of 
lowest-resistance pores; rather, it minimizes overall resistance by 
occasionally choosing a high-resistance pore that gives access to a
large cluster of low-resistance pores.  The Bethe tree network has the
advantage that asymptotic behavior can be determined quickly for extremely
large networks.

Creation of foam in the porespace depends on mobilizing a small
population of soap films initially present so that they can multiply as 
they pass through pore junctions.  Mobilizing a film depends on the 
pressure drop across the film, and that in turn depends on the length of
the gas cluster blocked by the film.  Percolation theory allows one to 
relate the mobilization of soap films on the microscopic scale to the
size of clusters and then to the macroscopically observable parameters like 
injection rates of gas and liquid.  The process of foam generation is an
example of "catastrophe theory" in that there are multiple stable and 
unstable steady states, with jumps from one state to another at the limit 
of that state.  Percolation theory can help explain the jump from the 
"not-foam" state to the "foam" state.