LA JUVENTUD SE HACE MÁS ALTA

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

129 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.

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Network Modeling at Waterloo (Waterloo Network Model)

WATNEMO is based on the work done by Chatzis (1976), (1980), the extensions given by Diaz (1984) and Kantzas (1985). A random network involves three key parameters: the average coordination number z, the pore body size distribution, and the pore throat size distribution. A 3-D random model is generated using the cubic lattice as a complete topological representation of an irregular network. The case of 2-D networks is not considered in this work since, as was pointed out by Chatzis (1976), 2-D networks cannot be used as physically sound models of real porous media. The main reason for this is that, in a 2-D network, it is impossible to have nodes and bonds fully occupied by one phase to form a continuum, while the other phase fully occupies nodes and bonds that a continuum as well (i.e., bicontinua cannot exist).

Using the fact that the real network of pores in a porous medium is in 3-D, one can create many different types of 3-D networks characterized by different values of the coordination number. It has been found that it is very convenient to work-with regular networks that have a cubic lattice arrangement and a coordination number of six. This simplification introduces the "physical' assumption that the pores of the medium are well connected, and facilitates straightforward computer algorithms. Taking the above fact into account, the other parameters to be defined are the size distributions of pore throats, assigned to the bonds, and of the pore bodies assigned to the nodes of the network.

The network models generated using bond correlated site percolation have been found to bemore suitable models of pore structure in sandstones (Chatzis (1980), Diaz (1984)). The principle of generating such networks is simple. The nodes of the network are assigned with indices 1,2 ....n at random. The nodes assigned the index 1 represent the largest pore bodies and the nodes assigned index n are the smallest pore bodies. A pseudo-random number generation technique can be used to provide a random setting of those indices in the sites of the network. The next step is to define the bonds by assigning an index equal to the larger index of the two nodes it connects. The accessibility properties of random network models using the bond correlated site percolation scheme have been investigated at Waterloo with special attention given to cubic networks (Chatzis (1980), Diaz (1984), Kantzas (1985)). Generalized number based accessibility functions have been obtained and applied to model capillary pressure saturation relationships for mercury-air systems and water-oil systems with entrapment. Having specified the number based pore body size distribution, the pore throat size distribution and pore geometry, there are algorithms available to convert the number based accessibility data to volume based capillary pressure curve data (Chatzis (1980), Chatzis and Dullien (1982), Diaz (1984), Kantzas (1985)).

In addition to the simulation of capillary pressure curves, the WATNEMO has been applied successfully to model the "dendritic" non-wetting phase saturation during primary drainage in mercury-air experiments (Chatzis and Dullien (1982)) and relative permeability characteristics in mercury- permeametry experiments. The work of Diaz (1984) expanded the capability of WATNEMO to model fluid distributions of the non-wetting phase as well as the wetting phase, as a function of saturation and saturation history, while Kantzas (1985) used the same principles for the simulation of two phase relative permeabilities as described below.

131 The conductivity properties of random network models of interest to us for the simulation of relative permeability behaviour in oil-water systems have not been explored in detail. Results presented by Kirkpatrick (1973) for the cubic networks are limited by the fact that all bonds in the network allowed to conduct flow had the same conductivity value. There are no analytical methods for the calculation of the conductivity properties of random networks with size distribution of conductivities or with variable coordination numbers.

Effective medium theory approximations (Kirkpatrick 1973, Koplik 1982) have been applied by Larson et al (1981 A), Heiba et al (1982) to two phase flow problems of relevance to relative permeability behaviour. Mohanty and Salter (1982) have also looked at the application of the conductivity properties of random networks to simulate relative permeability behaviour, but their simulation results were not in good agreement with experimental results. Moreover, the information provided in their paper is not sufficient to warrant a detailed criticism.

In all of the past studies involved with the simulation of relating permeability characteristics, several tacit assumptions are made. These include: 1) The nodes in the network simulation have no resistance to fluid flow; 2) Only the bonds in the network carry the information of the resistance to flow in pore networks; and 3) With the exception of Chatzis and Dullien (1982), a node in the network can be simultaneously part of the non-wetting phase network as well as part of the wetting phase network (e.g., Fatt (1956C), Heiha et al (1992), Winterfeld et al (1981)).

In addition to the above assumptions, several inconsistencies arise in transforming the relative conductivities of random networks into the form of relative permeability curves. For example, for the conductivity of a pore with diameter D, g(D) may be taken to be proportional to D cubed or D to the power of four, while the volume of such a pore V(D) is proportional to D to the power of 0.84 instead of V(D) proportional to D cubed or D squared as it should be (e.g., Heiba et al (1982), Soo and Slattery( 1983)).

Part of this work is devoted to clarify some of these inherent inconsistencies in past studies. To do so, however, the capability of WATNEMO to model relative permeability behavior had to be developed. This was the main objective of Kantzas (1985) - the development of software for the investigation of the conductivity properties of random networks of the bond correlated site percolation type. The following sections form the basis for understanding the development of this software for simulating immiscible two-phase flow problems in porous media with applications to predictions of relative permeability behavior.