Fundamentals of Fluid Flow in Porous Media
Two Phase Relative Permeability Literature Survey: Percolation Theory
There are many physical phenomena in which a fluid is “spreading” randomly in a medium. The terms “spreading”, “fluid” and “medium” are not necessarily used in their strict sense. Except for the spreading itself, external causes such as gravity forces, for example, may control the process and affect the random mechanism. The random mechanism may be ascribed to either the medium or the fluid, depending on the nature of the particular problem. Broadbent and Hammersley (1957) introduced the term “Percolation Process” for the process that ascribes the random mechanism to the medium. This term came to distinguish the above mathematical analysis from the ones that are confined to the random mechanism of a process generally ascribed to the fluid which are labeled as “diffusion processes”. Percolation theory has been a very important tool in the theoretical development of the conductivity of random mixtures of conducting and non-conducting materials (Kirkpatrick 1973, Shante and Kirkpatrick 1971).
In percolation theory a new terminology is introduced. The “medium” is defined to be an infinite set of abstract objects called “atoms”, or “nodes”, or “sites”. A fluid flows from the source site along paths connecting different sites. These paths are called “bonds”, and can be oriented or unoriented. A bond is defined as oriented when it permits the flow only in a specified direction. The fluid that flows along a bond will wet its two end points. The latter is used in porous media simulations. The coordination number, z, of a network of sites connected by bonds is defined to be the weighted average number of bonds leaving a node in a network (Chatzis 1980).
The random mechanism can be assigned to the medium in two distinct ways, leading to two different percolation problems. The first is the bond percolation problem in which each bond has a constant probability of transmitting fluid and the bonds “transmitting” fluid are assigned at random everywhere in the network. This probability is independent of the existence of other bonds at the level of a site. The second is the site percolation problem in which a site. A, has a certain probability of allowing, fluid reaching A to flow on, along bonds leaving A. In this case^every bond between two open sites is to flow and the random mechanism is assigned to the sites. When a site is missing, all bonds connecting it to its neighbours are missing too. There are quite a few works that are based on either problem (e.g., Chatzis (1976), (1980)). In the field of porous media the conventional methods treated drainage as a bond problem and imbibition as a site problem.
Chatzis (1980) used the site problem for drainage by taking the bonds into account. This is known as the bond correlated site problem in which the sites are assigned at random and the event of a bond being open is correlated with the event of two adjacent sites being open. Network models of pore structure with pore body sizes randomly distributed over the sites and the pore throat sizes-assigned according to a correlation scheme have been found to be sound models of simulating capillary pressure curves in sandstones (Chatzis and Dullien (1982), Diaz (1984)). Network models of pore structure obeying the bond percolation problem have been found to be unrealistic in simulating pore structure and flow behaviour (Chatzis (1980)). The most important finding of percolation theory is the critical percolation threshold. This is defined as the minimum fraction of bonds (bond percolation threshold, Pbc)’ or tne minimum fraction of sites (site percolation threshold, Pgc), that must be present in the network so that the “medium” is conducting to flow. A background to the approach taken in applying percolation theory to random network models of pore structure at the University of Waterloo is given next.