Tracing interfaces in porous media
Rafael Rangel1992, Physica A: Statistical Mechanics and its Applications
Sign up for access to the world's latest research
checkGet notified about relevant papers
checkSave papers to use in your research
checkJoin the discussion with peers
checkTrack your impact
Abstract
The geometry of the interface of clusters growing under both pressure gradients and capillary forces in porous media is mapped into a single value function by tracing the surface of the aggregate and recording the Y coordinate of the position of a walker moving along the perimeter of the clusters as a function of the arc length 1. We find a crossover behavior in the Hurst exponent of the self-afline function Y(I). For small scales, the Hurst exponent corresponds to invasion percolation with trapping (IPT) (0.73); for larger scales to diffusionlimited aggregation (DLA) (0.60). This is consistent with a previously found crossover length L, from IP to DLA (Phys. Rev. Lett. 67 (1991) 2958). p are respectively the velocity and the pressure. The incompressibility condition V-u = 0 then implies that the pressure satisfies Laplace's equation. The DLA model is therefore governed by viscous forces, and capillary and surfacetension effects are neglected. Laplace's equation is modelled by random walks in the defending (viscous) fluid, the pressure being constant in the invading fluid. The regime is called the open branching regime, due to the absence of encircled pockets of the defending fluid. 1
Related papers
Fragmentation of Invasion Percolation Cluster in Two-Dimensional Porous Media
Europhysics Letters (EPL), 1995
PACS. 47.55Mh -Flows through porous media. PACS. 47.55Kf -Multiphase and particle-laden flows.
Invasion percolation with viscous forces
Physical Review E, 1998
We study invasion percolation in the presence of viscous forces, as a model of the drainage of a wetting fluid from a porous medium. Using concepts from gradient percolation, we consider two different cases, depending on the magnitude of the mobility ratio M. When M is sufficiently small, the displacement can be modeled by a form of gradient percolation in a stabilizing gradient, involving a particular percolation probability profile. We develop the scaling of the front width and the saturation profile, in terms of the capillary number. In the opposite case, the displacement is described by gradient percolation in a destabilizing gradient and leads to capillary-viscous fingering. This regime is identified in the context of viscous displacements and in general differs from diffusion-limited aggregation, which also describes displacements at large M. Constraints for the validity of the two regimes are developed. Limited experimental and numerical results support the theory of stabilized displacement. The effect of heterogeneity is also discussed. ͓S1063-651X͑97͒08812-0͔
Migration and fragmentation of invasion percolation clusters in two-dimensional porous media
Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 1995
Experiments on and computer simulations of the migration of fractal, nonwetting Quid bubbles through a two-dimensional random porous medium saturated with wetting Quid are presented. A large invasion percolation bubble was initially formed by slow injection of a nonwetting Quid into a horizontal cell saturated with a denser wetting Quid. Slow, continuous tilting of the cell caused the bubbles to migrate through the medium. The interplay between local pinning forces and buoyancy led to fragmentation and coalescence of migrating bubbles. The process was simulated by a modified site-bond invasion percolation model. PACS number(s): 47.55.Mh, 05.40. +j, 47.55.Kf, 64.60.Ak
Geometry and dynamics of invasion percolation with correlated buoyancy
Physical Review E, 2000
We study an invasion percolation model for drainage where the disorder comes partly from capillary thresholds and partly from height differences in a rough self-affine landscape. As a function of the buoyancy, the geometry of the invaded clusters changes dramatically. Long-range correlations from the fracture topography induce a double cluster structure with strings and compact blobs. A characteristic length is introduced comparing the width of the capillary threshold distribution and gravity effects at the pore scale. We study electrical properties of percolating clusters. Current distributions along percolating clusters are shown to be multifractal and sensitive to the buoyancy.
Simulations of migration, fragmentation and coalescence of non-wetting fluids in porous media
Physica A: Statistical Mechanics and its Applications, 1993
Flow between two sites on a percolation cluster
PHYSICAL REVIEW E, 2000
We study the flow of fluid in porous media in dimensions dϭ2 and 3. The medium is modeled by bond percolation on a lattice of L d sites, while the flow front is modeled by tracer particles driven by a pressure difference between two fixed sites ͑''wells''͒ separated by Euclidean distance r. We investigate the distribution function of the shortest path connecting the two sites, and propose a scaling ansatz that accounts for the dependence of this distribution ͑i͒ on the size of the system L and ͑ii͒ on the bond occupancy probability p. We confirm by extensive simulations that the ansatz holds for dϭ2 and 3. Further, we study two dynamical quantities: ͑i͒ the minimal traveling time of a tracer particle between the wells when the total flux is constant and ͑ii͒ the minimal traveling time when the pressure difference is constant. A scaling ansatz for these dynamical quantities also includes the effect of finite system size L and off-critical bond occupation probability p. We find that the scaling form for the distribution functions for these dynamical quantities for dϭ2 and 3 is similar to that for the shortest path, but with different critical exponents. Our results include estimates for all parameters that characterize the scaling form for the shortest path and the minimal traveling time in two and three dimensions; these parameters are the fractal dimension, the power law exponent, and the constants and exponents that characterize the exponential cutoff functions.
Large-scale Invasion Percolation with Trapping for Upscaling Capillary-Controlled Darcy-scale Flow
Transport in Porous Media
We apply steady-state capillary-controlled upscaling in heterogeneous environments. A phase may fail to form a connected path across a given domain at capillary equilibrium. Moreover, even if a continuous saturation path exists, some regions of the domain may produce disconnected clusters that do not contribute to the overall connectivity of the system. In such cases, conventional upscaling processes might not be accurate since identification and removal of these isolated clusters are extremely important to the global connectivity of the system and the stability of the numerical solvers. In this study, we address the impact of percolation during capillary-controlled displacements in heterogeneous porous media and present a comprehensive investigation using random absolute permeability fields, for water-wet, oil-wet and mixed-wet systems, where J-function scaling is used to relate capillary pressure, porosity and absolute permeabilities in each grid cell. Important information is revealed about the average connectivity of the phases and trapping at the Darcy scale due to capillary forces. We show that in oil-wet and mixed-wet media, large-scale trapping of oil controlled by variations in local capillary pressure may be more significant than the local trapping, controlled by pore-scale displacement. Keywords Immiscible displacements • Capillary-driven flow • Large-scale percolation • Steady-state upscaling List of symbols A p Symmetric matrix used in percolation solver for a given phase A i j Cross-sectional area between two nodes (m 2) B Adjacent matrix of a simulation model B Hasan A. Nooruddin
Theory of boundary effects in invasion percolation
Journal of Physics A: Mathematical and General, 1998
We study the boundary effects in invasion percolation with and without trapping. We find that the presence of boundaries introduces a new set of surface critical exponents, as in the case of standard percolation. Numerical simulations show a fractal dimension, for the region of the percolating cluster near the boundary, remarkably different from the bulk one. In fact, on the surface we find a value of D sur = 1.65 ± 0.02 (for IP with trapping D sur tr = 1.59 ± 0.03), compared with the bulk value of D bul = 1.88 ± 0.02 (D bul tr = 1.85 ± 0.02). We find a logarithmic cross-over from surface to bulk fractal properties, as one would expect from the finite-size theory of critical systems. The distribution of the quenched variables on the growing interface near the boundary self-organises into an asymptotic shape characterized by a discontinuity at a value xc = 0.5, which coincides with the bulk critical threshold. The exponent τ sur of the boundary avalanche distribution for IP without trapping is τ sur = 1.56 ± 0.05; this value is very near to the bulk one. Then we conclude that only the geometrical properties (fractal dimension) of the model are affected by the presence of a boundary, while other statistical and dynamical properties are unchanged. Furthermore, we are able to present a theoretical computation of the relevant critical exponents near the boundary. This analysis combines two recently introduced theoretical tools, the Fixed Scale Transformation (FST) and the Run Time Statistics (RTS), which are particularly suited for the study of irreversible self-organised growth models with quenched disorder. Our theoretical results are in rather good agreement with numerical data.
Capillary displacement and percolation in porous media
Journal of Fluid Mechanics, 1982
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 a t 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. I n particular, we find a residual volume fraction of displaced phase which depends strongly on the sample size, but weakly or not a t all on the co-ordination number and microscopicsize distribution of the lattice elements.
Diffusion in sparsely connected pore spaces: Temporal and spatial scaling
Water Resources Research, 2002
[1] Diffusion both disperses and retards mass movement of contaminants through porous media. The retardation generally involves mass transfer of contaminants from faster to slower flow paths, making prediction more difficult and remediation slower. In ordinary (Fickian) diffusion, root-mean-square displacement of a population of diffusing molecules increases with the square root of the elapsed time, and diffusivity (D, L2T−1) of the porous medium is constant in time. The diffusivity can be measured using timedependent methods such as pulsed field gradient (PFG) magnetic resonance (MR), or steady state diffusion experiments. However, it is known that diffusion behaves anomalously in sparsely connected pore spaces at the percolation threshold. We conducted a series of Monte Carlo simulations using random walks, designed to mimic measurement of D using both time-dependent and steady state methods. In the time-dependent simulations, diffusivity scaled as D∼ T−0.48 at the percolation threshold. Above the percolation threshold, D decreased at early times, then became constant, with the time required for constant D being greater when the medium was less interconnected. In the steady state simulations we examined the effect of sample size on apparent diffusivity. Above the percolation threshold, diffusivity decreased, then became constant as sample size increased, but at the percolation threshold, diffusivity scaled as D ∼ L−1.83. The drop in diffusivity with sample size was caused by both a decrease in effective porosity and an increase in tortuosity, each of which also followed scaling laws. A single measurement will not suffice to determine whether a sample's pore space is at the percolation threshold, necessitating measurement of diffusivity at more than one scale in time or space. Using a scale-dependent measurement to make predictions at a different scale can result in serious overestimation or underestimation of diffusive mass transfer and consequent retardation.
Related topics
Related papers
Gradient stabilized and destabilized invasion percolation
Physica A: Statistical Mechanics and its Applications, 1992
The effects of gravity stabilization and destabilization on the slow displacement of a wetting fluid by a non-wetting fluid in two-dimensional and three-dimensional porous media have been investigated experimentally. The characteristic features of the resulting displacement patterns can be reproduced quite well by invasion percolation models with a spatial gradient added to the usual random threshold distribution. In the case of destabilized displacement the patterns can be described in terms of a string of blobs of size e that form a directed walk. The internal structure of the blobs is like that of an invasion percolation cluster (with trapping in the two-dimensional case). In the stabilized case the structure is like that of a fractal invasion percolation cluster on short length scales (lengths +[) and is uniform on longer lengths. The correlation length (5) also describes the maximum hole diameter. The invasion front is a self-similar fractal on length scales shorter than 5 and flat on longer length scales. In both the experiments and the simulations the correlation length .$ is related to the Bond number (B,, the ratio between buoyancy and capillary forces) by .$-1 BO[-y'(V+') where Y is the ordinary percolation correlation length exponent) in accord with the theoretical arguments of Wilkinson (Phys. Rev. A 30 (1984) 520; 34 (1986) 1380).
Cluster evolution in steady-state two-phase flow in porous media
Physical Review E, 2006
We report numerical studies of the cluster development of two-phase flow in a steady-state environment of porous media. This is done by including biperiodic boundary conditions in a twodimensional flow simulator. Initial transients of wetting and non-wetting phases that evolve before steady-state has occurred, undergo a cross-over where every initial patterns are broken up. For flow dominated by capillary effects with capillary numbers in order of 10 −5 , we find that around a critical saturation of non-wetting fluid the non-wetting clusters of size s have a power-law distribution ns ∼ s −τ with the exponent τ = 1.92 ± 0.04 for large clusters. This is a lower value than the result for ordinary percolation. We also present scaling relation and time evolution of the structure and global pressure.
The Dynamics of Viscous Penetration in Percolation Porous Media
We investigate the dynamics of viscous penetration in two-dimensional percolation networks at criticality for the case in which the ratio between the viscosities of displaced and injected fluids is very large. We report extensive numerical simulations that indicate that the scaling exponents for the breakthrough time distribution are the same as the previously reported values computed for the case of unit viscosity ratio. Our results are consistent with the possibility that viscous displacement through critical percolation networks constitutes a single universality class, independent of the viscosity ratio. We also find that the distributions of mass and breakthrough time of the invaded clusters have the same scaling form, but with different critical exponents.
Fragmentation and migration of invasion percolation clusters: Experiments and simulations
Physical Review E, 1997
Experimental studies of two fluid displacement processes in porous media involving extensive fragmentation of invasion percolationlike structures are described. In the first process, a two-dimensional porous cell saturated with a wetting fluid was slowly invaded by air. The air formed a fractal structure that fragmented when the pressure of the wetting fluid increased and the air was driven out of the system. In the second process, a fractal air structure migrated through a two-dimensional porous medium saturated with wetting fluid. The structure was driven by increasing buoyancy forces and fragmented. The fragments migrated, fragmented, and coalesced with other fragments. The processes were simulated using new site-bond invasion percolation models that captured the displacement mechanisms and reproduced the fragmentation events, and good agreement was found. In both processes, the fractal dimensionality of the fragments was equal to the dimensionality DϷ1.82 of the initial invasion percolationlike structures. The fragment size distributions measured in both processes and the dynamics of the migration process could be described by simple scaling forms.
Invasion percolation with long-range correlations: First-order phase transition and nonuniversal scaling properties
Physical Review E, 2000
We present the results of extensive Monte Carlo simulations of the invasion percolation model with trapping ͑TIP͒ with long-range correlations, a problem which is relevant to multiphase flow in field-scale porous media, such as oil reservoirs and groundwater aquifers, as well as flow in rock fractures. The correlations are generated by a fractional Brownian motion characterized by a Hurst exponent H. We employ a highly efficient algorithm for simulating TIP, and a novel method for identifying the backbone of TIP clusters. Both site and bond TIP are studied. Our study indicates that the backbone of bond TIP is loopless and completely different from that of site TIP. We obtain precise estimates for the fractal dimensions of the sample-spanning cluster ͑SSC͒, the minimal path, and the backbone of site and bond TIP, and analyze the size distribution of the trapped clusters, in order to identify all the possible universality classes of TIP with long-range correlations. For site TIP with HϾ1/2 the SSC and its backbone are compact, indicating a first-order phase transition at the percolation threshold, while the minimal paths are essentially straigth lines. For HϽ1/2 the SSC, its backbone, and the minimal paths are all fractal with fractal dimensions that depend on the Hurst exponent H. The fractal dimension of the loopless backbone for bond TIP is much less than that of site TIP for any H.
Dynamics of viscous penetration in percolation porous media
Physical Review E, 2001
We investigate the dynamics of viscous penetration in two-dimensional percolation networks at criticality for the case in which the ratio between the viscosities of displaced and injected fluids is very large. We report extensive numerical simulations that indicate that the scaling exponents for the breakthrough time distribution are the same as the previously reported values computed for the case of unit viscosity ratio. Our results are consistent with the possibility that viscous displacement through critical percolation networks constitutes a single universality class, independent of the viscosity ratio. We also find that the distributions of mass and breakthrough time of the invaded clusters have the same scaling form, but with different critical exponents.
Fragmentation Transition for Invasion Percolation in Hydraulic Gradients
Physical Review Letters, 1998
Air injected into a two-dimensional porous medium displaced a flowing defending fluid. At low flow rates the invading air formed chains of fractal clusters similar to those observed in gradient percolation. The defending fluid was excluded from the invading region and moved around the invading clusters. Above a critical flow rate the invaded region fragmented into a plumelike structure that permitted the defending fluid to flow through the invaded region. Invasion percolation simulations, modified to include fragmentation and pressure gradients due to flow, describe the observations well.
Surface effects in invasion percolation
Physical Review E, 1997
Boundary effects for invasion percolation are introduced and discussed here. The presence of boundaries determines a set of critical exponents characteristic of the boundary. In this paper we present numerical simulations showing a remarkably different fractal dimension for the region of the percolating cluster near the boundary. In fact, near the surface we find a value of D sur ϭ1.67Ϯ0.03, with respect to the bulk value of D bul ϭ1.87Ϯ0.01. Furthermore, we are able to present a theoretical computation of the fractal dimension near the boundary in fairly good agreement with numerical data.
Invasion percolation in a hydrostatic or permeability gradient: Experiments and simulations
Physical Review E, 1994
We present experiments and simulations of slow drainage in three-dimensional (3D) porous media, either homogeneous and in the presence of gravity or heterogeneous and in its absence. An acoustic technique allows for an accurate study of the 3D fronts and the crossover region. Our results suggest that both cases can be described by invasion percolation in a gradient. For the case of gravity, the front tail width scales with the Bond number as crFT-B. ', in agreement with the theory. For the case of permeability gradient a different scaling is found, in agreement with a modified theory of gradient percolation developed here.
Competition of gravity, capillary and viscous forces during drainage in a two-dimensional porous medium, a pore scale study
Energy, 2005
We have studied experimentally and numerically the displacement of a highly viscous wetting fluid by a non-wetting fluid with low viscosity in a random two-dimensional porous medium under stabilizing gravity. In situations where the magnitudes of the viscous-, capillary-and gravity forces are comparable, we observe a transition from a capillary fingering behavior to a viscous fingering behavior, when decreasing apparent gravity. In the former configuration, the vertical extension of the displacement front saturates; in the latter, thin branched fingers develop and rapidly reach breakthrough. From pressure measurements and picture analyzes, we experimentally determine the threshold for the instability, a value that we also predict using percolation theory. Percolation theory further allows us to predict that the vertical extension of the invasion fronts undergoing stable displacement scales as a power law of the generalized Bond number Bo à ¼ Bo À Ca, where Bo and Ca are the Bond and capillary numbers, respectively. Our experimental findings are compared to the results of a numerical modeling that takes local viscous forces into account. Theoretical, experimental and numerical approaches appear to be consistent. #
Loading Preview
Sorry, preview is currently unavailable. You can download the paper by clicking the button above.