Session F35: Porous Media Flows: Pore-Scale I

Prediction of velocity distribution in simple 3D porous media

Fluid flow and particle transport through porous media are determined by the geometry of the host medium itself. Despite the fundamental importance of the velocity distribution in controlling early-time and late-time transport properties (e.g., early breakthrough and superdiffusive spreading), direct relations linking velocity distribution with the statistics of pore structure in 3D porous media have not been established yet. High velocities are controlled by the formation of channels, while low velocities are dominated by stagnation zones. Recent studies have proposed phenomenological models for the distribution of high velocities including stretched exponential and power-exponential distributions but without an underlying mechanistic or statistical physics theory. Here, we investigate the relationship between the structure of the host medium and the resulting fluid flow in random dense spherical packs. We simulate flow at low Reynolds numbers by solving the Stokes equations and imposing a no-slip boundary condition at the boundary of each sphere. We show that the distribution of low velocities in 3D porous media is described by a Gamma distribution, which is robust to variations in the geometry of the porous media.   

Fast parallel algorithms for flow in porous media

In porous media flows, the variation in permeability over a large range of length scales gives rise to fluctuations in velocity which in turn lead to phenomena of interest, such as anomalous diffusion. However, due to the complex 3D structure of the medium, numerical simulation of flows in porous media remains a challenge and there is a need to develop efficient parallel algorithms for fast computation. We model porous media with a set of arbitrary shapes that are homeomorphic to a sphere. The stokes equations are often valid for these flows, which we solve using the boundary integral method. This reduces the dimensionality and number of unknowns for the problem. We evaluate far-field kernel summations using a parallel fast multipole method. Since the separation between spheres can be arbitrarily small, it is a challenge to resolve the flow near the surface. We use special singular and near singular quadratures to tackle this challenge and achieve high accuracy close to the spheres. We study hydrodynamic properties of the flow, such as dispersion, using the correlated continuous time random walk model. Our solver is highly scalable, solving for velocity fields in arbitrary geometries of several thousand spheres in a few minutes.   

Effect of particle size distribution on permeability in the randomly packed porous media

An answer of how porous medium heterogeneity influences the medium permeability is still inconclusive, where both increase and decrease in the permeability value are reported. A numerical procedure is used to generate a randomly packed porous material consisting of spherical particles. Six different particle size distributions are used including \textit{mono}-, \textit{bi}- and \textit{three}-disperse particles, as well as uniform, normal and \textit{log}-normal particle size distribution with the maximum to minimum particle size ratio ranging from three to eight for different distributions. In all six cases, the average particle size is kept the same. For all media generated, the stochastic homogeneity is checked from distribution of three coordinates of particle centers, where uniform distribution of $x$-, $y$- and $z$- positions is found. The medium surface area remains essentially constant except for \textit{bi}-modal distribution in which medium area decreases, while no changes in the porosity are observed (around 0.36). The fluid flow is solved in such domain, and after checking for the pressure axial linearity, the permeability is calculated from the Darcy law. The permeability comparison reveals that the permeability of the \textit{mono}-disperse medium is smallest, and the permeability of all \textit{poly}-disperse samples is less than ten percent higher. For \textit{bi}-modal particles, the permeability is for a quarter higher compared to the other media which can be explained by volumetric contribution of larger particles and larger passages for fluid flow to take place.   

Pore size distribution effect on rarefied gas transport in porous media

Gas transport phenomena in porous media are known to strongly influence the performance of devices such as gas separation membranes and fuel cells. Knudsen diffusion is a dominant flow regime in these devices since they have nanoscale pores. Many experiments have shown that these porous media have complex structures and pore size distributions; thus, the diffusion coefficient in these media cannot be easily assessed. Previous studies have reported that the characteristic pore diameter of porous media can be defined in light of the pore size distribution; however, tortuosity factor, which is necessary for the evaluation of diffusion coefficient, is still unknown without gas transport measurements or simulations. Thus, the relation between pore size distributions and tortuosity factors is required to obtain the gas transport properties. We perform numerical simulations to prove the relation between them. Porous media are numerically constructed while satisfying given pore size distributions. Then, the mean-square displacement simulation is performed to obtain the tortuosity factors of the constructed porous media..   

Pressure drop for inertial flows in elastic porous media

The effect of the porosity and of the elastic properties of anisotropic solid skeletons saturated by a fluid is studied for flows displaying unsteady inertial effects. Insight is achieved by direct numerical simulations of the Navier-Stokes equations for model porous media, with inclusions which can oscillate with respect to their reference positions because of the presence of a restoring elastic force modeled by a spring. The numerical technique is based on the immersed boundary method, to easily allow for the displacement of pores of arbitrary shapes and dimensions. Solid contacts are anelastic. The parameters examined include the local Reynolds number, $Re_d$, based on the mean velocity through the reference unit cell and the characteristic size of the inclusions, the direction of the macroscopic forcing pressure gradient, the reduced frequency, $f^*$, ratio of the flow frequency to the natural frequency of the spring-mass system, and the reduced mass, $m^*$, ratio of the solid to the fluid density. Results demonstrate the effect of these parameters, and permit to determine the filtration laws useful for the subsequent macroscopic modeling of these flows through the volume averaged Navier-Stokes equations.   

Reduced-Order Direct Numerical Simulation of Solute Transport in Porous Media

Pore-scale models are an important tool for analyzing fluid dynamics in porous materials (e.g., rocks, soils, fuel cells). Current direct numerical simulation (DNS) techniques, while very accurate, are computationally prohibitive for sample sizes that are statistically representative of the porous structure. Reduced-order approaches such as pore-network models (PNM) aim to approximate the pore-space geometry and physics to remedy this problem. Predictions from current techniques, however, have not always been successful. This work focuses on single-phase transport of a passive solute under advection-dominated regimes and delineates the minimum set of approximations that consistently produce accurate PNM predictions. Novel network extraction (discretization) and particle simulation techniques are developed and compared to high-fidelity DNS simulations for a wide range of micromodel heterogeneities and a single sphere pack. Moreover, common modeling assumptions in the literature are analyzed and shown that they can lead to first-order errors under advection-dominated regimes. This work has implications for optimizing material design and operations in manufactured (electrodes) and natural (rocks) porous media pertaining to energy systems.   

The Mechanism behind Erosive Bursts in Porous Media

We implemented a new model based on the Lattice Boltzmann method to simulate erosion and deposition in suspension flows through porous media. Using this model we show that the cause of erosive bursts in filtration experiments is the re-opening of clogged pores when the pressure difference between two opposite sites of the pore surpasses a certain threshold. We perform numerical simulations and find excellent agreement to experimental results when comparing shape and size distribution of pressure loss jumps, which are the direct result of erosive bursts. Furthermore, we find that erosive bursts only occur for pressure gradient thresholds within the range of two critical values, independent on how the flow is driven. We believe that our findings provide a better understanding of sudden sand production in oil wells and breakthrough in filtration.   

Lattice Boltzmann simulations of immiscible displacement process with large viscosity ratios

Immiscible displacement is a key physical mechanism involved in enhanced oil recovery and carbon sequestration processes. This multiphase flow phenomenon involves a complex interplay of viscous, capillary, inertial and wettability effects. The lattice Boltzmann (LB) method is an accurate and efficient technique for modeling and simulating multiphase/multicomponent flows especially in complex flow configurations and media. In this presentation we present numerical simulation results of displacement process in thin long channels. The results are based on a new psuedo-potential multicomponent LB model with multiple relaxation time collision (MRT) model and explicit forcing scheme. We demonstrate that the proposed model is capable of accurately simulating the displacement process involving fluids with a wider range of viscosity ratios (>100) and which also leads to viscosity-independent interfacial tension and reduction of some important numerical artifacts.   

Analysis of Non-equilibrium Capillary Pressure-Saturation Relation using Direct Numerical Simulations with Volume-Of-Fluid (VOF) Method

In traditional two-phase flow models of porous media, capillary pressure (P$_{\mathrm{c}})$ and saturation (S$^{\mathrm{w}})$ are hysteretically related, i.e. different P$_{\mathrm{c}}$-S$^{\mathrm{w}}$ curves are obtained for drainage and imbibition. Extended two-phase flow theories hypothesize that inclusion of specific interfacial area (a$^{\mathrm{wn}})$ will result in a unique relation between capillary pressure, saturation and interfacial area (P$_{\mathrm{c}}$--S$^{\mathrm{w}}$--a$^{\mathrm{wn}})$. Several studies have confirmed the reduction of hysteresis in the P$_{\mathrm{c}}$--S$^{\mathrm{w}}$--a$^{\mathrm{wn}}$ relation under quasi-static conditions. However, the uniqueness of the P$_{\mathrm{c}}$--S$^{\mathrm{w}}$--a$^{\mathrm{wn}}$ relation under transient conditions is not clear. We investigate role of specific interfacial area under dynamic conditions using pore-scale direct numerical simulations (DNS) on two micromodels with volume-of-fluid (VOF) method. From the DNS data, Pc-S$^{\mathrm{w}}$ curves are estimated for drainage and imbibition; validity of different macroscopic capillary pressure definitions is evaluated. The quasi-static and dynamic P$_{\mathrm{c}}$--S$^{\mathrm{w}}$--a$^{\mathrm{wn}}$ relations are examined for uniqueness.