A new kind of percolation problem is described which differs from ordinary percolation theory in that it automatically finds the critical points of the system. The model is motivated by the problem of one fluid displacing another from a porous medium under the action of capillary forces, but in principle it may be applied to any kind of invasion process which proceeds along a path at least resistance. The name invasion percolation is proposed for this new process. Similarities to, and differences from, ordinary percolation theory are discussed.

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Invasion percolation: a new form of percolation theory

In this paper, theoretical and experimental approaches to flow, hydrodynamic dispersion, and miscible and immiscible displacement processes in reservoir rocks are reviewed and discussed. Both macroscopically homogeneous and heterogeneous rocks are considered. The latter are characterized by large-scale spatial variations and correlations in their effective properties and include rocks that may be characterized by several distinct degrees of porosity, a well-known example of which is a fractured rock with two degrees of porosity---those of the pores and of the fractures. First, the diagenetic processes that give rise to the present reservoir rocks are discussed and a few geometrical models of such processes are described. Then, measurement and characterization of important properties, such as pore-size distribution, pore-space topology, and pore surface roughness, and morphological properties of fracture networks are discussed. It is shown that fractal and percolation concepts play important roles in the characterization of rocks, from the smallest length scale at the pore level to the largest length scales at the fracture and fault scales. Next, various structural models of homogeneous and heterogeneous rock are discussed, and theoretical and computer simulation approaches to flow, dispersion, and displacement in such systems are reviewed. Two different modeling approaches to these phenomena are compared. The first approach is based on the classical equations of transport supplemented with constitutive equations describing the transport and other important coefficients and parameters. These are called the continuum models. The second approach is based on network models of pore space and fractured rocks; it models the phenomena at the smallest scale, a pore or fracture, and then employs large-scale simulation and modern concepts of the statistical physics of disordered systems, such as scaling and universality, to obtain the macroscopic properties of the system. The fundamental roles of the interconnectivity of the rock and its wetting properties in dispersion and two-phase flows, and those of microscopic and macroscopic heterogeneities in miscible displacements are emphasized. Two important conceptual advances for modeling fractured rocks and studying flow phenomena in porous media are also discussed. The first, based on cellular automata, can in principle be used for computing macroscopic properties of flow phenomena in any porous medium, regardless of the complexity of its structure. The second, simulated annealing, borrowed from optimization processes and the statistical mechanics of spin glasses, is used for finding the optimum structure of a fractured reservoir that honors a limited amount of experimental data.

Flow phenomena in rocks : from continuum models to fractals, percolation, cellular automata, and simulated annealing

The mechanisms of displacement of one fluid by another are investigated in an etched network.Experiments show that both fluids are simultaneously present in a duct, the wetting fluid remaining in the extreme corners of the cross-section. Calculation of displacement pressures are in good agreement with experiments for drainage, imbibition and removal of blobs. The results may be related to some flow behaviour exhibited in porous media.

Mechanisms of the displacement of one fluid by another in a network of capillary ducts

We define six kinetic growth models that have witnessed an explosion of recent activity: 

1
Cancer Growth (Eden)

2
Colloid Growth (Langer-Muller-Krumbhaar; Witten-Sander)

3
Breakdown (Sawada et al.)

4
Crystallization (Rikvold)

5
Invasion Percolation (Schlumberger group)

6
Addition Polymerization (Manneville-de Seze; Herrmann et al.)





We also describe briefly some of the approaches used to study these models, with emphasis on the re normalization group approach being developed by us.

Kinetics of Aggregation and Gelation

We present a predictive calculation of two-phase relative permeabilities in granular porous media formed from a dense random packing of equal spheres. The spatial coordinates of every sphere in the pack have been measured, enabling the microstructure of the medium to be completely determined. From these data we extract a network model that replicates the pore space. By compacting the packing or swelling individual spheres, we may generate model porous media of different porosities whose microstructure is also completely determined. We simulate both viscous- and capillary-dominated invasion of a nonwetting fluid into a wetting fluid contained in these media. During invasion we calculate the average hydraulic conductance of each phase to obtain the relative permeabilities as a function of fluid saturation. Because the microstructure is known, the calculations do not involve any adjustable parameters or supplementary measurements of pore structure. The computed relative permeabilities are successfully compared with experimental values previously measured on sand packs, bead packs, and a simple sandstone.

Prediction of relative permeability in simple porous media.

We consider capillary displacement of immiscible fluids in porous media in the limit of vanishing flow rate. The motion is represented as a stepwise Monte Carlo process on a finite two-dimensional random lattice, where at each step the fluid interface moves through the lattice link where the displacing force is largest. The displacement process exhibits considerable fingering and trapping of displaced phase at all length scales, leading to high residual retention of the displaced phase. Many features of our results are well described by percolation-theory concepts. In particular, we find a residual volume fraction of displaced phase which depends strongly on the sample size, but weakly or not at all on the co-ordination number and microscopic-size distribution of the lattice elements.

Richard Chandler

Papers

Capillary displacement and percolation in porous media