Percolation invasion

Lenormand R, Zarcone C (1985) Invasion percolation in an etched network Measurement of fractal dimension. Phys Rev Lett 54 2226-2229... 

Wilkinson D (1986) Percolation effects in immiscible displacement. Phys Rev A 34 1380-1391 Wilkinson D, WiUemsen JE (1983) Invasion percolation A new form of percolation theory. J Phys A 34 1380-1391... 

Since water must be forced into a dry GDM regardless of PTFE-content, water is effectively the non-wetting phase. This process has the characteristics of primary drainage, for which invasion percolation30 is an adequate model.6,31 Invasion percolation describes... 

The first one is the Katz and Thompson s model (1986) which interprets transports within pore solids in terms of these percolation ideas [2]. From that, the authors introduced a fractal percolation model to predict the permeability of a disordered porous media. In invasion percolation, a non-wetting fluid can have access to the first connection from one face of the sample to the other only when the driving pressure is sufficient to penetrate the smallest pore-throat of radius rc in the most efficient conducting pathway. So, the permeability of rocks saturated with a single liquid phase is given from the following relationship ... 

Drainage (i.e., displacement of a wetting fluid by a non-wetting fluid) was modelled by an invasion-percolation process (Wilkinson and Willemsen, 1983). This implies that viscous forces may be ignored (i.e., pressure drops in both fluids may be ignored) and that capillary forces (pressure difference AF between the two fluids) control the process ... 

Berkowitz and Ewing (1998) also describe distinctions between three types of percolation ordinary, invasion, and invasion with trapping. In ordinary percolation, sites contain tracer simply on the basis of their connection with the source. Figures 3-21 and 3-22 are examples of ordinary percolation. In invasion percolation, sites fill with tracer in the order determined by the size of pore necks. Invasion percolation and invasion percolation with trapping apply primarily to multiphase conditions (e.g., infiltration of water into soil containing air, or water/oil systems) and are not considered further here. 

Zhou, D., and E.H. Stenby. 1993. Interpretation of capillary pressure curves using invasion percolation theory. Transp. Por. Media 11 17-31. 

Optimal paths in the presence of strong disorder is found to belong to the universality class of the shortest paths on self-organised critical clusters in invasion percolation with and without trapping [38]. The length L of the shortest paths on percolation clusters scales with the end-to-end distance R as... 

The above algorithm is commonly referred to as invasion percolation as defined by Wilkinson and Willemsetf and has been employed by Bazylak et al ... 

When the assumption of negligible viscous forces is not made, researchers have modified the invasion percolation algorithm to incorporate a flow-induced pressure drop within the invading cluster in a variety of ways typically following a dynamic invasion algorithm, such as the following ... 

Ceballos, L., and Prat, M., 2010, Invasion percolation with inlet multiple injections and the water management problem in proton exchange membrane fuel cells , J. Power Sources, 195 (3) pp. 825. 

Kang, J. H., Lee, K.-J., Nam, J. H. et al. 2010. Visuahzation of invasion-percolation drainage process in porous media using density-matched immiscible fluids and refractive-index-matched solid structures. Journal of Power Sources 195 2608-2612. 

Visualizations of the density distributions in the porous materials reveal insights into the mechanisms of intrusion and extrusion in the systems. We find that intrusion resembles an invasion percolation process with a front of liquid dually moving into frie porous material from the external si ace. However vapor transport and liquid condensation ahead of this fiomt may also contributral to the mechanism. An important aspect of extrusion is Ihe fingmentation of the liquid. This means that the transport processes in extrusion involve vapor transport. This provides a mechanism for m cury entrapment. We have made dynamics studies of these effects in our recent work [21,23],... 

For invasion percolation, the growth is started from a seed on a lattice with a random distribution of site probabilities, and at each time unit growth occurs at the perimeter site with the lowest probability. Figure 2a shows a cluster of mass M = 4000. In the infinite mass limit there exists a critical probability pc at which the normalized distribution of probabilities AT(p) along the perimeter jumps from zero to the constant value 1/(1 — Pc)- In this limit the average probability becomes... 

Fig. 2. (a) Invasion percolation cluster of mass M = 4000. The surviving paths are shown as heavy lines. The rest is a disconnected set of extinct branches, (b) The size distribution 1 (5) of extinct branches, (c) The distribution D t) of lifetimes t. The slopes of the straight lines are -1.50 and -0.92, respectively. The distributions are based on 40 clusters of mass M = 4000. The circles represent binned data, (d) DLA of mass M = 4000. The heavy lines are surviving paths. The rest is extinct branches. 

In both cases, a convenient model is percolation, and theoretical models have been elaborated to describe the dynamics of these fronts. The dynamics of diffusion fronts was first studied [3] in the case of length (or surface) fluctuations while a model for the temporal development of invasion percolation was published at the same time [4]. Dynamics of invasion percolation, directly derived from the theory of the dynamics of diffusion fronts was also studied when a gravity field is present [5]. Two different expressions were found in [4] and [5] for the number of events of a given size, per unit time. Veiy surprisingly these expressions give the same result in d = 2 but not in d = 3. 

Independently, a two dimensional approach of the interface motion through a disorder media, led Martys, Robbins and Cieplak [6] to an expression of the number of events similar to the one obtained in [5] (their study includes invasion percolation and compact growth). 

The expression obtained by Roux and Guyon in their study of temporal development of invasion percolation [4], to determine the bursts distribution is only valid in d=2, and the expression (1), correct in d = 1 and in d = 2 and in d > 6 is expected to be general (its proof does not involve any restriction on the dimensionality). A precise numerical confumation remains necessary in d = 3. In d = 2, numerical calculations have been performed in the bond percolation case [4,6,8], and in the site percolation case [3,5], leading to results compatible with equation (1). This supposes that for site percolation, when removal a site, the contribution due to many daughter clusters remains marginal [8]. 

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