Effect of Mean Network Coordination Number on Dispersivity Characteristics

In this study, we investigate the role of topology on the macroscopic (centimeter scale) dispersion characteristics derived from pore-network models. We consider 3D random porous networks extracted from a regular cubic lattice with coordination number distributed in accordance with real porous structures. We use physically consistent rules including ideal mixing in pore bodies, molecular diffusion, and Taylor dispersion in pore throats to simulate transport at the pore-scale level. Fundamental properties of porous networks are based on statistical distributions of basic parameters. Theoretical calculations demonstrate strong correspondence with data obtained from numerical experiments. For low coordination numbers, we observe normal transport in porous networks. Anomalous effects expressed by tailing in concentration evolution are seen for higher coordination numbers. We find that the mean network coordination number has significant influence on averaged characteristics of porous networks such as geometric and hydraulic dispersivity, while other topological properties are of less significance. We give an explicit formula that describes the decrease of geometric dispersivity with growing mean coordination number. The results demonstrate the importance of network topology for models for flow and transport in porous media.     

Coupled Processes in Charged Porous Media: From Theory to Applications

Charged porous media are pervasive, and modeling such systems is mathematically and computationally challenging due to the highly coupled hydrodynamic and electrochemical interactions caused by the presence of charged solid surfaces, ions in the fluid, and chemical reactions between the ions in the fluid and the solid surface. In addition to the microscopic physics, applied external potentials, such as hydrodynamic, electrical, and chemical potential gradients, control the macroscopic dynamics of the system. This paper aims to give fresh overview of modeling pore-scale and Darcy-scale coupled processes for different applications. At the microscale, fundamental microscopic concepts and corresponding mass and momentum balance equations for charged porous media are presented. Given the highly coupled nonlinear physiochemical processes in charged porous media as well as the huge discrepancy in length scales of these physiochemical phenomena versus the application, numerical simulation of these processes at the Darcy scale is even more challenging than the direct pore-scale simulation of multiphase flow in porous media. Thus, upscaling the microscopic processes up to the Darcy scale is essential and highly required for large-scale applications. Hence, we provide and discuss Darcy-scale porous medium theories obtained using the hybrid mixture theory and homogenization along with their corresponding assumptions. Then, application of these theoretical developments in clays, batteries, enhanced oil recovery, and biological systems is discussed.

     

Models for flow of non-Newtonian and complex fluids through porous media

This article first provides a brief and simple account of continuum models for transport in porous media, and of the role of length scales in passing from pore-scale phenomena to “Darcy” continuum scale representations using averaged variables. It then examines the influence of non-Newtonian rheology on the single- and multi-phase transport parameters, i.e. Darcy viscosity, dispersion lengths and relative permeabilities. The aim is to deduce functional forms and values for these parameters given the rheological properties of the fluid or fluids in question, and the porosity, permeability, dispersion lengths and relative permeabilities (based on Newtonian fluids and equivalent capillary pressures) of the porous medium. It is concluded that micro-models, typically composed of capillary networks, applied at a sub-Darcy-scale, parameterised using data for flows of a well-characterised set of non-Newtonian fluids, are likely to provide the most reliable means.     

Automaton Simulations of Dispersion in Porous Media

A thermodynamic lattice gas (automaton) model is used to simulate dispersion in porous media. Simulations are constructed at two distinctly different scales, the pore scale at which capillary models are constructed and large scale or Darcy scale at which probabilistic collision rules are introduced. Both models allow for macroscopic (pore scale) phase separation. The pore scale models clearly show the effect of pore structure on dispersion. The large scale (mega scale) simulations indicate that when the pressure difference between the displacing phase and displaced phase is properly chosen (representing the average pressure gradient between the phases). The simulation results are consistent with both theoretical predictions and experimental observations.     

Pore-Scale Flow in Surfactant Flooding

Recovery of oil from the blocks of an initially oil-wet, naturally fractured, reservoir as a result of counter-current flow following introduction of aqueous wettability-altering surfactant into the fracture system is considered, as an example of a practical process in which phenomena acting at the single pore-scale are vital to the economic displacement of oil at the macroscopic scale. A Darcy model for the process is set up, and solutions computed illustrating the recovery rate controlling role of the bulk diffusion of surfactant. A central ingredient of this model is the capillary pressure relation, linking the local values of the pressure difference between the oleic and aqueous phases, the aqueous saturation and the surfactant concentration. Using ideas from single capillary models of oil displacement from oil-wet tubes by wettability-altering surfactant, we speculate that the use of a capillary pressure function, with dependences as assumed, may not adequately represent the Darcy scale consequences of processes acting at the single pore-scale. Multi-scale simulation, resolving both sub-pore and multi-pore flow processes may be necessary to resolve this point. Some general comments are made concerning the issues faced when modelling complex displacement processes in porous media starting from the pore-scale and working upwards.     

Erosion and deposition of solid particles in porous media: Homogenization analysis of a formation damage

The aim of the paper is to model at a large scale, the formation damage in porous media by erosion and deposition of solid particles. We start from the equations governing the pore-scale processes of erosion, deposition, convection and diffusion. The macroscopic equivalent behaviour is investigated by using a homogenization method. Four characteristic models with different dominating phenomena at the pore scale are determined. The main results are twofold: first dispersion-deposition and dispersion-erosion phenomena are shown at the macroscopic scale for peculiar values of the dimensionless numbers; furthermore, and contrarily to phenomenological models, erosion and deposition generally occur in regions of intense and slow flow, respectively.     

Filtration in a Porous Granular Medium: 1. Simulation of Pore-Scale Particle Deposition and Clogging

This paper presents a numerical model for simulating the pore-scale transport and infiltration of dilute suspensions of particles in a granular porous medium under the action of hydrodynamic and gravitational forces. The formulation solves the Stokes’ flow equations for an incompressible fluid using a fixed grid, multigrid finite difference method and an embedded boundary technique for modeling particle–fluid coupling. The analyses simulate a constant flux of the fluid suspension through a cylindrical model pore. Randomly generated particles are collected within the model pore, initially through contact and attachment at the grain surface (pore wall) and later through mounding close to the pore inlet. Simple correlations have been derived from extensive numerical simulations in order to estimate the volume of filtered particles that accumulate in the pore and the differential pressure needed to maintain a constant flux through the pore. The results show that particle collection efficiency is correlated with the Stokes’ settling velocity and indirectly through the attachment probability with the particle–grain surface roughness. The differential pressure is correlated directly with the maximum mound height and indirectly with particle size and settling velocity that affect mound packing density. Simple modification factors are introduced to account for pore length and dip angle. These parameters are used to characterize pore-scale infiltration processes within larger scale network models of particle transport in granular porous media in a companion paper. Articlenote: Currently at GZA GeoEnvironmental Inc., 1 Edgewater Drive, Norwood, MA 02062, U.S.A.     

Steady-State Local Diffusive Fluxes in Porous Geo-Materials Obtained by Pore-Scale Simulations

Computer simulations of non-sorbing tracers diffusing in fluid-saturated porous sediment/rock were performed using pore-scale X-ray microtomographic images to reveal the following. (i) The histogram of the magnitude of the local diffusive flux vector obeys a unimodal log-normal distribution having a long positive tail. Simulations using model images were also performed to show that the flux broadening in large pores and the flux mixing at the pore network junctions are responsible for the log-normal shape. (ii) The simulation enabled us to directly visualize pore voxels with large and small fluxes, confirming the existence of transport pores and stagnant pores. Because of the unimodal nature, however, it was difficult to distinguish transport pores from stagnant pores using an objective threshold in the histogram. (iii) Another histogram of the flux vector component along the direction of the macroscopic concentration gradient was analyzed. A negative tail was found in the histogram, indicating that local counter diffusion exists in the porous geo-materials. However, the population and intensity of the counter diffusion fluxes are too small and weak to contribute to the overall diffusive transport across the porous media system. A long positive tail representing a large-flux diffusive pathway was also observed in the histogram. However, again, the population of the large-flux transport pores is small. As a result, the main conveyer of the tracer is the stagnant pores (not the transport pores), which have small positive flux values but a large population.     

Analysis of hydrodynamic and thermal dispersion in porous media by means of a local approach

A pore scale analysis is implemented in this numerical study to investigate the behavior of microscopic inertia and thermal dispersion in a porous medium with a periodic structure. The macroscopic characteristics of the transport phenomena are evaluated with an averaging technique of the controlling variables at a pore scale level in an elementary cell of the porous structure. The Darcy–Forchheimer model describes the fluid motion through the porous medium while the continuity and Navier–Stokes equations are applied within the unit cell. An average energy equation is employed for the thermal part of the porous medium. The macroscopic pressure loss is computed in order to evaluate the dominant microscopic inertial effects. Local fluctuations of velocity and temperature at the pore scale are instrumental in the quantification of the thermal dispersion through the total effective thermal diffusivity. The numerical results demonstrate that microscopic inertia contributes significantly to the magnitude of the macroscopic pressure loss, in some instances with as much as 70%. Depending on the nature of the porous medium, the thermal dispersion may have a marked bearing on the heat transfer, particularly in the streamwise direction for a highly conducting fluid and certain values of the Peclet number.     

Multiscale Modeling of Colloid and Fluid Dynamics in Porous Media Including an Evolving Microstructure

We consider colloidal dynamics and single-phase fluid flow within a saturated porous medium in two space dimensions. A new approach in modeling pore clogging and porosity changes on the macroscopic scale is presented. Starting from the pore scale, transport of colloids is modeled by the Nernst?CPlanck equations. Here, interaction with the porous matrix due to (non-)DLVO forces is included as an additional transport mechanism. Fluid flow is described by incompressible Stokes equations with interaction energy as forcing term. Attachment and detachment processes are modeled by a surface reaction rate. The evolution of the underlying microstructure is captured by a level set function. The crucial point in completing this model is to set up appropriate boundary conditions on the evolving solid?Cliquid interface. Their derivation is based on mass conservation. As a result of an averaging procedure by periodic homogenization in a level set framework, on the macroscale we obtain Darcy??s law and a modified averaged convection?Cdiffusion equation with effective coefficients due to the evolving microstructure. These equations are supplemented by microscopic cell problems. Time- and space-dependent averaged coefficient functions explicitly contain information of the underlying geometry and also information of the interaction potential. The theoretical results are complemented by numerical computations of the averaged coefficients and simulations of a heterogeneous multiscale scenario. Here, we consider a radially symmetric setting, i.e., in particular we assume a locally periodic geometry consisting of circular grains. We focus on the interplay between attachment and detachment reaction, colloidal interaction forces, and the evolving microstructure. Our model contributes to the understanding of the effects and processes leading to porosity changes and pore clogging from a theoretical point of view.     

Droplet Imbibition into Paper Coating Layer: Pore-Network Modeling Simulation

Liquid penetration into thin porous media such as paper is often simulated using continuum-scale single-phase Darcy’s law. The underlying assumption was that a sharp invasion front percolates through the layer. To explore this ambiguous assumption and to understand the controlling pore-scale mechanisms, we have developed a dynamic pore-network model to simulate imbibition of a wetting phase from a droplet into a paper coating layer. The realistic pore structures are obtained using the FIB-SEM imaging of the coating material with a minimum resolution of 3.5 nm. Pore network was extracted from FIB-SEM images using Avizo software. Data of extracted pore network are used for statistically generating pore network. Droplet sizes are chosen in the range of those applicable in inkjet printing. Our simulations show no sharp invasion front exists and there is the presence of residual non-wetting phase. In addition, penetration of different sizes of droplets of different material properties into the pore network with different pore body and pore throat sizes are performed. We have found an approximately linear decrease in droplet volume with time. This contradicts the expected \(\sqrt{t}\)-behavior in vertical imbibition that is obtained using macroscopic single-phase Darcy’s law. With increase in flow rate, transition of imbibition invasion front from percolation-like pattern to a more sharper one with less trapping of non-wetting phase is also reported. Our simulations suggest that the single-phase Darcy’s law does not adequately describe liquid penetration into materials such as paper coating layer. Instead Richards equation would be a better choice.     

Stochastic Flow Simulation and Particle Transport in a 2D Layer of Random Porous Medium

A stochastic numerical method is developed for simulation of flows and particle transport in a 2D layer of porous medium. The hydraulic conductivity is assumed to be a random field of a given statistical structure, the flow is modeled in the layer with prescribed boundary conditions. Numerical experiments are carried out by solving the Darcy equation for each sample of the hydraulic conductivity by a direct solver for sparse matrices, and tracking Lagrangian trajectories in the simulated flow. We present and analyze different Eulerian and Lagrangian statistical characteristics of the flow such as transverse and longitudinal velocity correlation functions, longitudinal dispersion coefficient, and the mean displacement of Lagrangian trajectories. We discuss the effect of long-range correlations of the longitudinal velocities which we have found in our numerical simulations. The related anomalous diffusion is also analyzed.     

The Peculiarities of Linear Wave Propagation in Double Porous Media

The features of propagation of longitudinal and transverse waves (LW and TW) in fractured porous medium (FPM) saturated with liquid are investigated by methods of multiphase mechanics. The mathematical model of FPM accounting for inequality of velocities and pressures of liquid in pores and fractures, liquid mass exchange and nonstationary interaction forces is developed. Processes of monochromatic wave propagation are studied. The dispersion relation is obtained and the effect of model parameters on wave propagation is analysed. It is established that one transverse and three longitudinal waves propagate in FPM saturated with liquid. The fastest LW is a deformational wave and the two others are filtrational. Filtrational waves attenuate much stronger than deformational and transverse waves. Distinction of velocities and pressures in liquid in various pore systems provides an explanation for the existence of the two filtrational waves in porous medium with two different characteristic sizes of pores.     

AC Diffusion: Transport in Porous Networks Subjected to Zero-Time-Average Advective Flow

Diffusion is a slow transport mechanism and advective transport tends to dominate in large-size systems. An alternative transport mechanism is explored herein, whereby zero time-average cyclic fluid flow is compounded with pore-scale mixing to render effective transport. Two one-dimensional cyclic flow cases are analyzed: a rigid porous network with two open boundaries subjected to cyclic flow through, and a compressible porous network with only one open boundary subjected to cyclic compression. The corresponding analytical models predict diffusion-like macroscale response and provide explicit expressions for the effective diffusion coefficients in terms of the microstructure of the porous medium and flow conditions. A parallel experimental study is conducted to corroborate analytical predictions. Results confirm the relevance of pore-scale mixing in cyclic flow as a transport mechanism in porous networks.     

Particle Flow Code Method-Based Meso-scale Identification for Seepage Failure of Soil Levee

The development and validation of a grid-based pore-scale numerical modelling methodology applied to five different commercial metal foam samples is described. The 3-D digital representation of the foam geometry was obtained by the use of X-ray microcomputer tomography scans, and macroscopic properties such as porosity, specific surface and pore size distribution are directly calculated from tomographic data. Pressure drop measurements were performed on all the samples under a wide range of flow velocities, with focus on the turbulent flow regime. Airflow pore-scale simulations were carried out solving the continuity and Navier–Stokes equations using a commercial finite volume code. The feasibility of using Reynolds-averaged Navier–Stokes models to account for the turbulence within the pore space was evaluated. Macroscopic transport quantities are calculated from the pore-scale simulations by averaging. Permeability and Forchheimer coefficient values are obtained from the pressure gradient data for both experiments and simulations and used for validation. Results have shown that viscous losses are practically negligible under the conditions investigated and pressure losses are dominated by inertial effects. Simulations performed on samples with varying thickness in the flow direction showed the pressure gradient to be affected by the sample thickness. However, as the thickness increased, the pressure gradient tended towards an asymptotic value.     

Inertial and Darcy’s Terms Ratio in Boundary Layer at Fluid–Porous Medium Interface

The study considers the forced boundary-layer flow overlying the Darcy–Brinkman porous medium and gives a quantitative analysis of the nonlinear inertial terms in the Brinkman filtration equation. The inertial terms are shown to be larger than the Darcy’s drag near the porous medium interface. The applicability range of boundary-layer approach is determined. It is suitable in high-permeable media with moderate velocities of an external flow. If it is slow enough, the inertial terms can be omitted in spite of interface effect. On the other hand, fast external flow produces the filtration with large pore-scale Reynolds number; therefore, the Forchheimer’s drag should be taken into account. It is shown the Brinkman term as well as inertial terms have a significant role in boundary-layer formation within the porous medium.     

Modelling of Advection-Dominated Transport in a Porous Column with a Time-Decaying Flow Field

This paper examines the problem of the advective-dispersive movement of a non-decaying, inert chemical dye solution through the pore space of a fluid saturated porous column. The objective of the paper is to present a complete study of the one-dimensional advective-dispersive transport problem by considering certain analytical solutions, experimental results and their comparisons with specific computational simulations. Dye concentrations obtained by means of an image processing method are used in conjunction with an analytical solution to identify the hydrodynamic dispersion coefficient that governs the advective-dispersive transport problem. The experimental results and identified parameters are also used to assess the computational estimates derived from several stabilized computational schemes available in the literature, for examining advection-dominated transport processes in porous media.     

Numerical Modeling of Momentum Dispersion in Porous Media Based on the Pore Scale Prevalence Hypothesis

Transport in Porous Media - A macroscopic model that accounts for the effect of momentum dispersion on flows in porous media is proposed. The model is based on the pore scale prevalence hypothesis...