Relationship between pore size and velocity probability distributions in stochastically generated porous media

Distribution of liquid flow in a pore network during evaporation

The variation of the distribution of the liquid flow in porous media during evaporation is still a puzzle. We resolve it with the pore network modeling approach. The distribution of the evaporation-induced liquid flow in a pore network composed of about 2.5 million pores is determined. The probability density function of the magnitude of the normalized liquid flow rate is obtained. For the low normalized liquid flow rate, the probability density function is power-lawlike. The power-law exponent depends on both the liquid saturation and the location of the moving meniscus in the main liquid cluster. The evaporation-induced liquid flow in the pores in the pore network can be correlated. Whether the liquid flow distributions in various zones in the pore network are similar or not relies significantly on the location of the moving meniscus in the main liquid cluster. The functions depicting the relation between the power-law exponent and the local liquid saturation for the zones adjacent to and away from the open side of the pore network are different. These findings from the pore scale studies provide insights into developing the accurate continuum model for evaporation in porous media.

Flow Path Resistance in Heterogeneous Porous Media Recast into a Graph-Theory Problem

Abstract

This work aims to describe the spatial distribution of flow from characteristics of the underlying pore structure in heterogeneous porous media. Thousands of two-dimensional samples of polydispersed granular media are used to (1) obtain the velocity field via direct numerical simulations, and (2) conceptualize the pore network as a graph in each sample. Analysis of the flow field allows us to distinguish preferential from stagnant flow regions and to quantify how channelized the flow is. Then, the graph’s edges are weighted by geometric attributes of their corresponding pores to find the path of minimum resistance of each sample. Overlap between the preferential flow paths and the predicted minimum resistance path determines the accuracy in individual samples. An evolutionary algorithm is employed to determine the “fittest” weighting scheme (here, the channel’s arc length to pore throat ratio) that maximizes accuracy across the entire dataset while minimizing over-parameterization. Finally, the structural similarity of neighboring edges is analyzed to explain the spatial arrangement of preferential flow within the pore network. We find that connected edges within the preferential flow subnetwork are highly similar, while those within the stagnant flow subnetwork are dissimilar. The contrast in similarity between these regions increases with flow channelization, explaining the structural constraints to local flow. The proposed framework may be used for fast characterization of porous media heterogeneity relative to computationally expensive direct numerical simulations.

Article Highlights

Microscale modeling of nondilute flow and transport in porous medium systems

Nondilute transport occurs routinely in porous medium systems. Experimental observations have revealed effects that seemingly depend upon density, viscosity, velocity, and chemical activity. Macroscale models based upon averaged behavior over many pores have been relied upon to describe such systems to date, which require parametrization of important physical phenomena in material coefficients. To advance fundamental understanding of these complex systems, we examine nondilute transport from a fundamental microscale, or pore-scale, continuum modeling perspective. We approximate the solution of a model based upon the variable-density Navier-Stokes equations and a nondilute species transport equation. Known dependencies of the densities, viscosities, chemical activity, and diffusion for a salt solution on chemical composition are included in the model. Microscale model solutions are averaged to the macroscale and compared with extant experimental observations. Investigation of the effects of various physical phenomena on the microscale velocity distribution and the observed macroscale dispersion are considered using dimensional analysis and constrained simulations. Simulation results are used to explain observed experimental results in light of underlying mechanisms. Conditions under which the various physicochemical effects investigated are important are revealed.

Pore-scale velocities in three-dimensional porous materials with trapped immiscible fluid

We study and document the influence of wetting and nonwetting trapped immiscible fluid on the probability distribution of pore-scale velocities of the flowing fluid phase. We focus on drainage and imbibition processes within a three-dimensional microcomputed tomographic image of a real rock sample. The probability distribution of velocity magnitude displays a heavier tail for trapped nonwetting than wetting fluid. This behavior is a signature of marked changes in the distribution and strength of preferential flow paths promoted by the wettability property of the trapped fluid. When the latter is wetting the host solid matrix, high-velocity areas initially present during single-phase flow conditions are mainly characterized by increased or decreased velocity magnitudes, and the velocity field remains correlated with its counterpart associated with the single-phase case. Otherwise, when the trapped fluid is nonwetting, features that are observed to prevail are appearance and disappearance of high-velocity areas and a velocity field that is less correlated to the one obtained under single-phase conditions.

Pore-scale statistics of flow and transport through porous media

Flow in porous media is known to be largely affected by pore morphology. In this work, we investigate the effects of pore geometry on the transport and spatial correlations of flow through porous media in two distinct pore structures arising from three-dimensional assemblies of overlapping and nonoverlapping spheres. Using high-resolution direct numerical simulations (DNS), we perform Eulerian and Lagrangian analysis of the flow and transport characteristics in porous media. We show that the Eulerian velocity distributions change from nearly exponential to Gaussian distributions as porosity increases. A stretched exponential distribution can be used to represent this behavior for a wide range of porosities. Evolution of Lagrangian velocities is studied for the uniform injection rule. Evaluation of tortuosity and trajectory length distributions of each porous medium shows that the model of overlapping spheres results in higher tortuosity and more skewed trajectory length distributions compared to the model of nonoverlapping spheres. Wider velocity distribution and higher tortuosity for overlapping spheres model give rise to non-Fickian transport while transport in nonoverlapping spheres model is found to be Fickian. Particularly, for overlapping spheres model our analysis of first-passage time distribution shows that the transport is very similar to those observed for sandstone. Finally, using three-dimensional (3D) velocity field obtained by DNS at the pore-scale, we quantitatively show that despite the randomness of pore-space, the spatially fluctuating velocity field and the 3D pore-space distribution are strongly correlated for a range of porous media from relatively homogeneous monodisperse sphere packs to Castlegate sandstone.

Pore-Scale Hydrodynamics in a Progressively Bioclogged Three-Dimensional Porous Medium: 3-D Particle Tracking Experiments and Stochastic Transport Modeling

Biofilms are ubiquitous bacterial communities that grow in various porous media including soils, trickling, and sand filters. In these environments, they play a central role in services ranging from degradation of pollutants to water purification. Biofilms dynamically change the pore structure of the medium through selective clogging of pores, a process known as bioclogging. This affects how solutes are transported and spread through the porous matrix, but the temporal changes to transport behavior during bioclogging are not well understood. To address this uncertainty, we experimentally study the hydrodynamic changes of a transparent 3-D porous medium as it experiences progressive bioclogging. Statistical analyses of the system's hydrodynamics at four time points of bioclogging (0, 24, 36, and 48 h in the exponential growth phase) reveal exponential increases in both average and variance of the flow velocity, as well as its correlation length. Measurements for spreading, as mean-squared displacements, are found to be non-Fickian and more intensely superdiffusive with progressive bioclogging, indicating the formation of preferential flow pathways and stagnation zones. A gamma distribution describes well the Lagrangian velocity distributions and provides parameters that quantify changes to the flow, which evolves from a parallel pore arrangement under unclogged conditions, toward a more serial arrangement with increasing clogging. Exponentially evolving hydrodynamic metrics agree with an exponential bacterial growth phase and are used to parameterize a correlated continuous time random walk model with a stochastic velocity relaxation. The model accurately reproduces transport observations and can be used to resolve transport behavior at intermediate time points within the exponential growth phase considered.

Solvable continuous-time random walk model of the motion of tracer particles through porous media

We consider the continuous-time random walk (CTRW) model of tracer motion in porous medium flows based on the experimentally determined distributions of pore velocity and pore size reported by Holzner et al. [M. Holzner et al., Phys. Rev. E 92, 013015 (2015)PLEEE81539-375510.1103/PhysRevE.92.013015]. The particle's passing through one channel is modeled as one step of the walk. The step (channel) length is random and the walker's velocity at consecutive steps of the walk is conserved with finite probability, mimicking that at the turning point there could be no abrupt change of velocity. We provide the Laplace transform of the characteristic function of the walker's position and reductions for different cases of independence of the CTRW's step duration τ, length l, and velocity v. We solve our model with independent l and v. The model incorporates different forms of the tail of the probability density of small velocities that vary with the model parameter α. Depending on that parameter, all types of anomalous diffusion can hold, from super- to subdiffusion. In a finite interval of α, ballistic behavior with logarithmic corrections holds, which was observed in a previously introduced CTRW model with independent l and τ. Universality of tracer diffusion in the porous medium is considered.

Statistics of highly heterogeneous flow fields confined to three-dimensional random porous media

We present a strong relationship between the microstructural characteristics of, and the fluid velocity fields confined to, three-dimensional random porous materials. The relationship is revealed through simultaneously extracting correlation functions R_{uu}(r) of the spatial (Eulerian) velocity fields and microstructural two-point correlation functions S_{2}(r) of the random porous heterogeneous materials. This demonstrates that the effective physical transport properties depend on the characteristics of complex pore structure owing to the relationship between R_{uu}(r) and S_{2}(r) revealed in this study. Further, the mean excess plot was used to investigate the right tail of the streamwise velocity component that was found to obey light-tail distributions. Based on the mean excess plot, a generalized Pareto distribution can be used to approximate the positive streamwise velocity distribution.

Topological analysis of non-granular, disordered porous media: determination of pore connectivity, pore coordination, and geometric tortuosity in physically reconstructed silica monoliths

Different approaches are applied and compared, which are universally applicable to quantify pore coordination, pore and pore-throat connectivity, and geometric tortuosity.

Power-exponential velocity distributions in disordered porous media

Velocity distribution functions link the micro- and macro-level theories of fluid flow through porous media. Here we study them for the fluid absolute velocity and its longitudinal and lateral components relative to the macroscopic flow direction in a model of a random porous medium. We claim that all distributions follow the power-exponential law controlled by an exponent γ and a shift parameter u_{0} and examine how these parameters depend on the porosity. We find that γ has a universal value 1/2 at the percolation threshold and grows with the porosity, but never exceeds 2.

Intermittent Lagrangian velocities and accelerations in three-dimensional porous medium flow

Intermittency of Lagrangian velocity and acceleration is a key to understanding transport in complex systems ranging from fluid turbulence to flow in porous media. High-resolution optical particle tracking in a three-dimensional (3D) porous medium provides detailed 3D information on Lagrangian velocities and accelerations. We find sharp transitions close to pore throats, and low flow variability in the pore bodies, which gives rise to stretched exponential Lagrangian velocity and acceleration distributions characterized by a sharp peak at low velocity, superlinear evolution of particle dispersion, and double-peak behavior in the propagators. The velocity distribution is quantified in terms of pore geometry and flow connectivity, which forms the basis for a continuous-time random-walk model that sheds light on the observed Lagrangian flow and transport behaviors.