Analysis of Flow in Spatially Correlated Systems by Applying the Percolation Theory

Abstract

Abstract The percolation theory provides universal laws which are helpful to describe geometrical and physical characteristics of a system. The present work verifies the influence of spatial correlation on percolation parameters and, as a consequence, on fluids flow properties. Monte Carlo simulation of stochastic processes was applied in association with the percolation theory of finitesized lattices, and with the invasion percolation model. In the first part, the sensibility of percolation parameters (as the percolation threshold and the connectivity function) to autocorrelation function in Gaussian process, and to attraction parameter in Markovian processes is observed. In the second part, the invasion percolation model is applied to inspect the behavior of fluids flow in pore-grain systems. The results show that spatial correlation plays an important role in the description of porous stems, strongly affecting the flow properties, the values of fluids recovery and residual saturations. Introduction The percolation theory, as termed by Broadbent and Hammersley in their classical work of 1957, was conceived to describe the morphology and nature of fluids transport in disordered media. It has undergone great progress, in several fields of science (physics, biology, chemistry), and has the distinguished quality to provide universal laws that describe the geometry and general properties of the tem. An important aspect concerning the percolation processes is the fact that its dynamics is ruled by the medium and not by the fluid, having outstanding importance the geometric and statistic characteristics of this medium. Assuming initially a square grid in which the cells are conductor of energy or insulate, let us suppose that a fraction p of them is conductor and randomly located at the grid, that is, each cell has a probability p of being conductor, independent of any other cell. Supposing that the energy propagation occurs only through conductor cells having a common face, the energy imposed to the top line may be transmitted to the bottom line or not; the passage of energy between opposite bounds is called percolation. ft can be noted that the greater the proportion of conductor cells, the greater the probability of occurring percolation; in addition, percolation only occurs above a certain proportion, called critical concentration or percolation threshold - pc. This model, in which the lattice is formed of connected cells, is denominated site percolation. A similar ease, in which all the cells are active, but the connections between them, called passages or bonds, may be active or not, is denominated bond percolation. These models are used to represent binary systems as permeable rock and shale, pore-grain systems, wetting fluid and non-wetting fluid, positive and negative spins, etc.