A probability analysis of the problem of the breakthrough of a coarsely cellular foam in a porous medium

The problem of the carrying capacity of a system of film membranes (lamellae) in a porous medium, which block the free motion of a gas delivered from outside, is considered in a probability formulation. It is assumed that, in the initial state, the lamellae are only located in the pore throats. The system can be called a coarsely cellular foam since exactly one of its bubbles is actually enclosed in each pore. The lamellae are assumed to be immobile and are broken down if the pressure drop across them exceeds a certain critical value. The characteristics of the breakthrough cluster, which arises when a specified quantity of an ideal gas is injected into the medium, are investigated. The problem is formulated in terms of probability mechanics for the breakdown of discrete systems and is studied within the framework of lattice models. The pores (which, for simplicity, are of the same volume) are identified with lattice nodes and the lamellae are identified with links which are blocked in the initial state and which possess a random strength with a known probability distribution. The breakthrough process involves the successive rupture of overloaded lamellae and a corresponding enlargement of the domain of the pore space occupied by the injected gas. Analytic expressions for the probability of breakthrough to a specified depth are obtained for several forms of linear chains of lattice nodes and in the case when the structure of the system is a regular binary tree (a Cayley tree) Examples of calculations are presented Since the probability of the lamellae breaking down decreases as the breakthrough zone increases, the model considered is substantially different from traditional percolation models and, in particular, the breakthrough cluster is always bounded here (C) 2002 Elsevier Science Ltd. All rights reserved.