Invasion Percolation Process Research Articles

MRI and PFG NMR studies of percolation effects in advanced melting during a cryoporometry characterisation of disordered mesoporous alumina

Cryoporometry (or thermoporometry) offers a way of pore structural characterisation for mesoporous materials that often needs little sample preparation, is relatively quick, and is statistically-representative for macroscopic samples. While it is well-known that freezing is controlled by pore-blocking, and is thus an invasion percolation process, the percolative nature of pore-to-pore co-operative advanced melting effects has been much less studied. In this work, PFG NMR studies, of diffusivity within the molten phase, have shown that the early melting process follows the scaling law, expected from percolation theory, below the percolation threshold. The percolation threshold thereby obtained was that for a 3D isotropic Poisson polyhedral lattice, consistent with the observation of patchwise macroscopic heterogeneities in the spatial distribution of local average pore size seen in MR relaxation time-weighted images. MRI has shown that once advanced melting effects kicked-in, around the percolation threshold, they occurred to different degrees in different slices along the length of the extrudate pellet. The macroscopic banding in pore-blocking, during freezing, and advanced melting effects, along the axis of the extrudate was consistent with anisotropic diffusional properties observed with MRI. Hence, it has been shown how the pore-pore co-operative effects can be utilised to improve structural characterisation of mesoporous solids.

Discrete fracture network model analysis of the effects of fluid transport on the morphology of a cluster of activated fractures.

Convective transport in low-permeability rocks can be enhanced by the injection of a pressurized fluid to activate preexisting weak planes (fractures). These fractures are initially closed, but fluid-pressure-induced slippage creates void space that allows for fluid flow. The Coulomb-Mohr criterion yields a critical pressure required to open each of the fractures. Due to the intrinsic porosity of the rock, the injected fluid can flow from the fractures' surfaces to the rock matrix through a process referred to as leakoff. Following this activation mechanism, the connectivity of the cluster of activated fractures is strongly dependent on the ratio F_{N} of the standard deviation of the critical pressures to the viscous pressure drop over a fracture's length. Recently, we proposed a continuum model to predict the effects of fluid transport on the morphology of the cluster of activated fractures formed by this process over a specific intermediate range of values of F_{N} [Alhashim and Koch, J. Fluid Mech. 847, 286 (2018)JFLSA70022-112010.1017/jfm.2018.313]. In this paper, the activation process of a discrete well-connected network of preexisting fractures embedded in a highly heterogeneous rock is modeled to analyze the effects of fluid transport on the resulting cluster's morphology for a wider range of F_{N} and show how the idealized averaged equation solution arises in a discrete system. We derive a length scale ξ_{ch}, which is a function of F_{N} above which the viscous pressure drop is important. This length scale, along with the radius of the cluster R and the average separation between the preexisting fractures ξ_{0}, can be used to define distinct growth regimes where different models can be used to describe the growth dynamics and predict the connectivity of the active network. When ξ_{0}∼ξ_{ch}≪R, the cluster is well connected and a linear pressure diffusion equation can be used to describe the cluster's growth. When ξ_{ch}≫R≫ξ_{0}, a fractal network is formed by an invasion percolation process. In an intermediate regime ξ_{0}≪ξ_{ch}≪R, percolation theory relates the porosity and permeability of the network to the local fluid pressure. For this regime, we validate the predictions of the continuum theory we recently developed to describe the cluster growth on length scales larger than ξ_{ch}.

The Brooks and Corey Capillary Pressure Model Revisited from Pore Network Simulations of Capillarity-Controlled Invasion Percolation Process

Relating the macroscopic properties of porous media such as capillary pressure with saturation is an on-going problem in many fields, but examining their correlations with microstructural traits of the porous medium is a challenging task due to the heterogeneity of the solid matrix and the limitations of laboratory instruments. Considering a capillarity-controlled invasion percolation process, we examined the macroscopic properties as functions of matrix saturation and pore structure by applying the throat and pore network model. We obtained a relationship of the capillary pressure with the effective saturation from systematic pore network simulations. Then, we revisited and identified the microstructure parameters in the Brooks and Corey capillary pressure model. The wetting phase residual saturation is related to the ratio of standard deviation to the mean radius, the ratio of pore radius to the throat length, and pore connectivity. The size distribution index in the Brooks and Corey capillary pressure model should be more reasonably considered as a meniscus size distribution index rather than a pore size distribution index, relating this parameter with the invasion process and the structural properties. The size distribution index is associated with pore connectivity and the ratio of standard deviation to mean radius (σ0/r¯), increasing with the decline of σ0/r¯ but the same for networks with same σ0/r¯. The identified parameters of the Brooks and Corey model might be further utilized for correlations with other transport properties such as permeability.

Dynamics of water injection in an oil-wet reservoir rock at subsurface conditions: Invasion patterns and pore-filling events.

We use fast synchrotron x-ray microtomography to investigate the pore-scale dynamics of water injection in an oil-wet carbonate reservoir rock at subsurface conditions. We measure, in situ, the geometric contact angles to confirm the oil-wet nature of the rock and define the displacement contact angles using an energy-balance-based approach. We observe that the displacement of oil by water is a drainagelike process, where water advances as a connected front displacing oil in the center of the pores, confining the oil to wetting layers. The displacement is an invasion percolation process, where throats, the restrictions between pores, fill in order of size, with the largest available throats filled first. In our heterogeneous carbonate rock, the displacement is predominantly size controlled; wettability has a smaller effect, due to the wide range of pore and throat sizes, as well as largely oil-wet surfaces. Wettability only has an impact early in the displacement, where the less oil-wet pores fill by water first. We observe drainage associated pore-filling dynamics including Haines jumps and snap-off events. Haines jumps occur on single- and/or multiple-pore levels accompanied by the rearrangement of water in the pore space to allow the rapid filling. Snap-off events are observed both locally and distally and the capillary pressure of the trapped water ganglia is shown to reach a new capillary equilibrium state. We measure the curvature of the oil-water interface. We find that the total curvature, the sum of the curvatures in orthogonal directions, is negative, giving a negative capillary pressure, consistent with oil-wet conditions, where displacement occurs as the water pressure exceeds that of the oil. However, the product of the principal curvatures, the Gaussian curvature, is generally negative, meaning that water bulges into oil in one direction, while oil bulges into water in the other. A negative Gaussian curvature provides a topological quantification of the good connectivity of the phases throughout the displacement.

Terrain Amplification with Implicit 3D Features

While three-dimensional landforms, such as arches and overhangs, occupy a relatively small proportion of most computer-generated landscapes, they are distinctive and dramatic and have an outsize visual impact. Unfortunately, the dominant heightfield representation of terrain precludes such features, and existing in-memory volumetric structures are too memory intensive to handle larger scenes. In this article, we present a novel memory-optimized paradigm for representing and generating volumetric terrain based on implicit surfaces. We encode feature shapes and terrain geology using construction trees that arrange and combine implicit primitives. The landform primitives themselves are positioned using Poisson sampling, built using open shape grammars guided by stratified erosion and invasion percolation processes, and, finally, queried during polygonization. Users can also interactively author landforms using high-level modeling tools to create or edit the underlying construction trees, with support for iterative cycles of editing and simulation. We demonstrate that our framework is capable of importing existing large-scale heightfield terrains and amplifying them with such diverse structures as slot canyons, sea arches, stratified cliffs, fields of hoodoos, and complex karst cave networks.

Drying of sapwood analyzed as an invasion percolation process

The sapwood fibers in green softwood are normally filled – or nearly filled – with free water. This water forms a continuous phase through the narrow bordered pits between fiber lumens. In a drying process water evaporates from the menisci formed at these narrow throats and initially located close to the surface of the wood. As the amount of free water decreases in this way, a retraction of the meniscus will happen in the widest throat, which can be located anywhere, not necessarily close to the surface. As the individual size of these narrow throats is a stochastic variable, this process will produce complicated water phase structures that gradually fragmentize. If this drying behavior is viewed as such an invasion percolation process it is obvious that the result differs from normal diffusion, which however most often is used as the basis for wood drying modeling. A computer based simulation model that utilizes the above principle has been developed. This model has been used for the simulation of a few situations of practical and theoretical importance. It has been found that this invasion percolation approach can reproduce some previously unexplained features seen in softwood drying processes. Some detailed examples are presented in this paper

Pore connectivity, electrical conductivity, and partial water saturation: Network simulations

AbstractThe electrical conductivity of brine‐saturated rock is predominantly dependent on the geometry and topology of the pore space. When a resistive second phase (e.g., air in the vadose zone and oil/gas in hydrocarbon reservoirs) displaces the brine, the geometry and topology of the pore space occupied by the electrically conductive phase are changed. We investigated the effect of these changes on the electrical conductivity of rock partially saturated with brine. We simulated drainage and imbibition as invasion and bond percolation processes, respectively, in pipe networks assumed to be perfectly water‐wet. The simulations included the formation of a water film in the pipes invaded by the nonwetting fluid. During simulated drainage/imbibition, we measured the changes in resistivity index as well as a number of relevant microstructural parameters describing the portion of the pore space saturated with water. Except Euler topological number, all quantities considered here showed a significant level of “universality,” i.e., insensitivity to the type of lattice used (simple cubic, body‐centered cubic, or face‐centered cubic). Hence, the coordination number of the pore network appears to be a more effective measure of connectivity than Euler number. In general, the simulated resistivity index did not obey Archie's simple power law. In log‐log scale, the resistivity index curves displayed a substantial downward or upward curvature depending on the presence or absence of a water film. Our network simulations compared relatively well with experimental data sets, which were obtained using experimental conditions and procedures consistent with the simulations. Finally, we verified that the connectivity/heterogeneity model proposed by Bernabé et al. (2011) could be extended to the partial brine saturation case when water films were not present.

Role of overpressurized fluid and fluid-driven fractures in forming fracture networks

A 2-D numerical study was carried out, using a fully coupled rock deformation and fluid flow hydraulic fracturing model on fracture network formation in a low-permeability naturally fractured rock, in which new fractures are allowed to nucleate and grow driven by overpressurized fluid. Fracture seeds either are defined as small flaws or are nucleated based on a stress and fracture energy criterion. Fracture intersection, fluid flow and frictional slip along fractures are also explicitly simulated in the model. In particular, we consider a few artificial fracture network geometries with sets of finite discrete or isolated fractures in order to study the roles of fluid viscosity, injection conditions, fracture intersections and offsetting at pre-existing or natural fractures in determining the paths which hydraulic fractures follow through the network. The newly fluid-created fracture segments, which are oriented locally normal to the least compressive stress, are the most conductive parts along the hydraulic fracture path. The resulting network will allow for long-term fluid flow to occur more easily provided that these segments retain their conductivity. However, the existence of intersections and offsets in the main fracture path act to impede fracture growth and fluid flow. A fracture nucleation event associated with higher fluid pressure may allow a fracture path to bypass these barriers, thus leading to a more planar fracture geometry. If fracture nucleation does not occur, fluid may enter other cross cutting natural fractures or enter along a barrier fracture to the tip of the barrier fracture, both processes that require a larger expenditure of energy reflected in an increasing fluid pressure. The results clearly demonstrate the importance of these stress and flow barriers in forming a preferential fracture and flow path. Fluid invasion into and fracture growth of natural fractures can still continue to occur at the high pressure region near the hydraulic fracture entry, even after a complete hydraulic fracture pathway has developed, showing strong similarity to the invasion percolation process. In contrast to fixed-aperture, connected fracture network models, the fracture growth-generated patterns also depend on the in situ stresses, which affect mechanical interaction between the main hydraulic fracture and natural fractures. The numerical results provide improved understanding of fluid-driven fracture growth into and through a network of pre-existing natural fractures.

Liquid water distribution in hydrophobic gas-diffusion layers with interconnect rib geometry: An invasion-percolation pore network analysis

Water distribution in gas diffusion layers (GDLs) of polymer electrolyte membrane fuel cells (PEMFCs) is determined by the pore morphology of the GDL as well as the flow conditions between the GDL and the gas flow field, where interconnect ribs and gas channels are placed side-by-side. The present study employs a steady state pore network model based on the invasion-percolation (IP) process to investigate the water transport in the under-rib region, in the under-channel region, and in between those regions inside the GDL. The interconnect rib partially blocks the outlet surface of the GDL, which forces water transport paths from the under-rib region to grow towards the gas channel through an extra IP process. The pore network model predicts spatially non-uniform water distributions inside the GDL due to the interconnect ribs, especially with an increased saturation level in the under-rib region. Parametric studies are also conducted to investigate the effects of several geometrical factors, such as width of the rib and the channel, thickness of the GDL, and water intruding condition at the inlet surface of the GDL.

Stress- and fluid-driven failure during fracture array growth: Implications for coupled deformation and fluid flow in the crust

Brittle experimental deformation on dolomite rocks shows for the fi rst time the differ- ence in growth of fracture networks by ordinary percolation and invasion percolation processes. Stress-driven fracture growth, in the absence of fl uid pressure, is an ordinary percolation process characterized by distributed nucleation and growth of microfractures, which coalesce with increasing strain to form a connected fracture network. Fluid pres- sure-driven fracture growth is more akin to an invasion percolation process characterized by preferential fracture growth occurring initially at the high fl uid pressure part of the rock. With progressive deformation, the network propagates rapidly through the sample and away from the high fl uid pressure reservoir. X-ray microtomography analysis suggests that the fracture network in three dimensions (3-D) is probably a fully connected network at peak stress conditions, whereas conventional 2-D analysis suggests that connectivity only occurs at shear failure. The development of 3-D fracture connectivity prior to shear failure has important implications for fl uid fl ow and fl uid pressure changes immediately prior to rupture nucleation in active fault zones, for fl uid migration in ore-producing hydrothermal systems, and for reservoir integrity in hydrocarbon systems.

A Non-isothermal Pore Network Drying Model with Gravity Effect

The concept of immiscible displacement as an invasion percolation (IP) process driven by heat and mass transfer is used in a pore network model for convective drying of capillary porous media. The coupling between heat and mass transfer occurs at the liquid–gas interface through temperature-dependent equilibrium vapor pressure and surface tension as well as the phase change enthalpy (in evaporation and condensation). The interfacial effects due to capillary forces and gravity are combined in an invasion potential; viscous forces are neglected. Simulation results show stabilized invasion patterns and finite drying front width by the influence of gravity.

INVASION PERCOLATION IN THE PRESENCE OF NANOPORES

Site invasion percolation (IP) processes are combined with bond percolation model, to study the effects of size restriction on low capillary number immiscible displacement in heterogeneous nanoporous media. Both cases of compressible (NTIP) and incompressible defender fluid (TIP) are considered. It is found that in site IP the value of mass uptake increases with the size of invader particles, if the latter is not greater than a critical value. This occurs when the accessible porosity of the medium decreases as the size of fluid particles increases. We also investigate the effect of nanopore's concentration on the mass and the anisotropy of sample spanning cluster as well as the critical exponent of trap numbers.

The shape of invasion percolation clusters in random and correlated media

The shape of two-dimensional invasion percolation clusters are studied numerically for bothnon-trapping (NTIP) and trapping (TIP) invasion percolation processes. Two differentanisotropy quantifiers, the anisotropy parameter and the asphericity, are used for probingthe degree of anisotropy of clusters. We observe that, in spite of the difference in scalingproperties of NTIP and TIP, there is no difference in the values of anisotropyquantifiers of these processes. Furthermore, we find that, in completely randommedia, the invasion percolation clusters are on average slightly less isotropic thanstandard percolation clusters. Introducing isotropic long-range correlations into themedia reduces the isotropy of the invasion percolation clusters. The effect is morepronounced for the case of persisting long-range correlations. The implicationof boundary conditions on the shape of clusters is another subject of interest.Compared to the case of free boundary conditions, IP clusters of conventionalrectangular geometry turn out to be more isotropic. Moreover, we see that inconventional rectangular geometry the NTIP clusters are more isotropic than the TIPclusters.

Structural and isotopic constraints on fluid flow regimes and fluid pathways during upper crustal deformation: An example from the Taemas area of the Lachlan Orogen, SE Australia

Structural and stable isotope studies of calcite veins and host rocks in a 1000 m thick, Devonian limestone sequence in the Taemas area of the Lachlan Orogen in SE Australia indicate that externally derived fluids migrated through the sequence over a protracted interval during folding and associated reverse faulting at temperatures of ∼200°C. The evolution of fluid pathways was governed by growth of fold‐related and fault‐related fracture networks at transiently supralithostatic fluid pressures. The internal structures of veins indicate growth during repeated cycles of permeability enhancement and sealing. Systematic increase inδ18O of vein calcite upward through the limestone sequence, and the presence of an O isotope alteration front in the vein system, is related to buffering of fluid compositions by fluid‐rock reaction during upward migration of overpressured, lowδ18O fluids. Repeated switching of fractures between high‐permeability and low‐permeability states promoted episodic flow and changes in flow paths during progressive deformation. A lack of significant migration of the isotopic front during the flow history is consistent with fluid‐rock interaction being associated with numerous, separate pulses of fluid ascending through the percolation network. Irregular variations in fluidδ18O with time during vein growth are interpreted to be driven by repeated changes in reactive path lengths, reaction rates, and flow rates in a discontinuous flow regime. Systematic upward changes in veinδ18O over an area of 20 km2requires that on a time‐averaged basis, most of the vein system was hydraulically well connected to the external fluid reservoir and that growth of the fracture network occurred by fluid‐driven, invasion percolation processes. Repeated failure and sealing events in the fracture system are interpreted to have been driven by episodic migration of fluid pressure waves up through the fracture network, possibly in response to deeper level fault rupture events repeatedly breaching an overpressured fluid reservoir.

Invasion percolation between two sites

We investigate the process of invasion percolation between two sites (injection and extraction sites) separated by a distance r in two-dimensional lattices of size L. Our results for the nontrapping invasion percolation model indicate that the statistics of the mass of invaded clusters is significantly dependent on the local occupation probability (pressure) Pe at the extraction site. For Pe = 0, we show that the mass distribution of invaded clusters P(M) follows a power-law P(M) approximately M(-alpha) for intermediate values of the mass M, with an exponent alpha = 1.39+/-0.03. When the local pressure is set to Pe = Pc, where Pc corresponds to the site percolation threshold of the lattice topology, the distribution P(M) still displays a scaling region, but with an exponent alpha = 1.02+/-0.03. This last behavior is consistent with previous results for the cluster statistics in standard percolation. In spite of these differences, the results of our simulations indicate that the fractal dimension of the invaded cluster does not depend significantly on the local pressure Pe and it is consistent with the fractal dimension values reported for standard invasion percolation. Finally, we perform extensive numerical simulations to determine the effect of the lattice borders on the statistics of the invaded clusters and also to characterize the self-organized critical behavior of the invasion percolation process.

Perimeter growth of a branched structure: application to crackle sounds in the lung.

We study an invasion percolation process on Cayley trees and find that the dynamics of perimeter growth is strongly dependent on the nature of the invasion process, as well as on the underlying tree structure. We apply this process to model the inflation of the lung in the airway tree, where crackling sounds are generated when airways open. We define the perimeter as the interface between the closed and opened regions of the lung. In this context we find that the distribution of time intervals between consecutive openings is a power law with an exponent beta approximately 2. We generalize the binary structure of the lung to a Cayley tree with a coordination number Z between 2 and 4. For Z=4, beta remains close to 2, while for a chain, Z=2 and beta=1, exactly. We also find a mean field solution of the model.

Invasion percolation: new algorithms and universality classes

Employing highly efficient algorithms for simulating invasion percolation (IP), whose execution time scales as O[Mlog(M)] or better for a cluster of M sites, and for determining the backbone of the cluster, we obtain precise estimates for the fractal dimensions of the sample-spanning cluster, the backbone, and the minimal path in order to identify the universality classes of four different IP processes (site and bond IP, with and without trapping). In two dimensions IP is characterized by two universality classes, one each for IP without trapping, and site and bond IP with trapping. In a three-dimensional site IP with and without trapping is in the universality class of random percolation, while bond IP with trapping is in a distinct universality class, which may be the same as that of optimal paths in strongly disordered media.

Isothermal drying on non-hygroscopic capillary-porous materials as an invasion percolation process

Numerical simulations of drainage and drying at low capillary number are performed. Drainage is simulated by means of the standard invasion percolation algorithm. Simulations of drying are based on an invasion algorithm combining elements of the invasion percolation algorithm with the computation of the vapour flux at each elementary liquid-gas interface. The simulations show that the invasion front is the very same fractal object in drainage and in drying. Specific features of drying are investigated. It is shown that the evaporation front should be clearly distinguished from the invasion front. In the presence of gravity forces, the disconnected cluster erosion mechanism is very effective and no disconnected cluster can survive outside the invasion front region. In the absence of gravity forces, the simulations indicate that drying of an initially saturated capillary-porous medium cannot be described according to the continuum approach to porous media.

Invasion percolation in a destabilizing gradient.

Extensive two- and three-dimensional computer simulations have been carried out to investigate the effect of a destabilizing external field (such as gravity) on invasion percolation processes. The displacement patterns found under these conditions are dominated by the growth of a single branch. This branch can be described in terms of a connected string of blobs of size ${\ensuremath{\xi}}_{\mathit{w}}$, which form a directed random walk along the direction of the field. On length scales smaller than ${\ensuremath{\xi}}_{\mathit{w}}$, the displacement figures have the structure of invasion percolation clusters without a destabilizing field. The dependence of the correlation length ${\ensuremath{\xi}}_{\mathit{w}}$ on the magnitude of the field gradient g is given by ${\ensuremath{\xi}}_{\mathit{w}}$\ensuremath{\sim}\ensuremath{\Vert}g${\mathrm{\ensuremath{\Vert}}}^{\mathrm{\ensuremath{-}}\ensuremath{\nu}/(\ensuremath{\nu}+1)}$ (where \ensuremath{\nu} is the ordinary percolation correlation length exponent) in accord with the theoretical arguments of Wilkinson [Phys. Rev. A 30, 520 (1984); 34, 1380 (1986)]. For continuous threshold distributions the exponent relating ${\ensuremath{\xi}}_{\mathit{w}}$ to g does not depend on the shape of the distribution. For discontinuous distributions the behavior is quite different.

The use of macroscopic percolation theory to construct large‐scale capillary pressure curves

This paper introduces a macroscopic invasion percolation process suitable for simulating the displacement of one immiscible fluid by another through porous media under conditions of capillary‐dominated flow. The theory is similar to classical percolation theory in that the structure of a real porous medium is represented as an ordered lattice, but differs in that each point of the lattice is assigned a local‐scale porosity, permeability, and capillary pressure‐saturation relationship, rather than microscale pore and pore throat dimensions. Unlike pore‐scale percolation theory, each node of the lattice may be occupied by either one or two phases, thereby allowing bicontinua of fluids in two dimensions. To illustrate an application of the theory, both wetting and non wetting fluids are percolated through a heterogeneous porous medium while accounting for fluid trapping such that a hysteretic, large‐scale capillary pressure‐saturation curve is constructed. The simulations are carried out in spatially correlated, random permeability fields assuming that the local‐scale capillary pressure‐saturation relationships are perfectly correlated with permeability. The resulting large‐scale capillary pressure curves are found to be influenced by the mean and variance of the assigned lognormal distribution of permeabilities. Very little sensitivity to the ratio of correlation lengths was observed. It is found that the threshold saturation giving rise to an initial percolating cluster of nonwetting fluid across the lattice corresponds to between 18.8% and 29.5% nonwetting saturation, depending on the statistics of the permeability field. Comparison of the percolation‐derived capillary pressure curves to those based on a direct arithmetic average demonstrates that an arithmetic average is only valid through the range of fluid saturations where no trapping occurs.