Numerical Simulation of Drying of a Saturated Deformable Porous Media by the Lattice Boltzmann Method

A numerical study is reported here to investigate the drying of saturated deformable porous rectangular plate based on the Darcy–Brinkman extended model. All walls of the plate are maintained to a convective heat flux as well as the top and bottom faces are also subjected to a mass flux. The model for the energy transport is based on the local thermodynamic equilibrium between the fluid and the solid phases. The lattice Boltzmann method is used for solving the governing differential equations system. A comprehensive analysis of the influence of the Poisson’s coefficient, the Young’s modulus and the permeability on macroscopic fields is investigated throughout this work.     

Lattice Boltzmann Simulation of Flow-Induced Wall Shear Stress in Porous Media

The lattice Boltzmann method is increasingly utilized in the simulation of flow-induced wall shear stress needed in various applications. In image-based flow simulations, the simulation geometry is usually based on a three-dimensional reconstruction of the true structure of the pore space obtained, for example, by X-ray tomography. The geometry is then given in a voxel-based representation, which complicates an accurate determination of the surface-normal vectors that are necessary in the computation of the wall shear stress. To avoid this problem, we introduce here a method for the determination of surface-normal vectors directly from a greyscale image instead of its segmented binary image version. The proposed method is fast and automatic, and it can be used for an arbitrary pore space geometry provided in a greyscale form by any imaging modality. We show that this method can produce accurate surface-normal vectors even for binary images and that their accuracy is further increased when the original greyscale images are used instead. We compute wall shear stresses for generated benchmark geometries and then demonstrate the utility of the method for soil samples with ‘random’ pores imaged by X-ray tomography.     

Stochastic Modeling of the Permeability of Randomly Generated Porous Media via the Lattice Boltzmann Method and Probabilistic Collocation Method

Transport in Porous Media - The onset of convection in a porous layer saturated by a power-law fluid is here investigated. The walls are considered to be isothermal, isobaric and permeable in such...     

Percolation Thresholds in 2-Dimensional Prefractal Models of Porous Media*

Considerable effort has been directed towards the application of percolation theory and fractal modeling to porous media. We combine these areas of research to investigate percolation in prefractal porous media. We estimated percolation thresholds in the pore space of homogeneous random 2-dimensional prefractals as a function of the fractal scale invariance ratio b and iteration level i. The percolation thresholds for these simulations were found to increase beyond the 0.5927l... porosity expected in Bernoulli (uncorrelated) percolation networks. Percolation in prefractals occurs through large pores connected by small pores. The thresholds increase with both b (a finite size effect) and i. The results allow the prediction of the onset of percolation in models of prefractal porous media and can be used to bound modeling efforts. More fundamental applications are also possible. Only a limited range of parameters has been explored empirically but extrapolations allow the critical fractal dimension to be estimated for a large combination of b and i values. Extrapolation to infinite iterations suggests that there may be a critical fractal dimension of the solid at which the pore space percolates. The extrapolated value is close to 1.89 – the well-known fractal dimension of percolation clusters in 2-dimensional Bernoulli networks.     

Modeling Chemotactic Waves in Saturated Porous Media using Adaptive Mesh Refinement

Bacterial transport is heavily influenced by chemical gradients and interfaces that exist in the subsurface. The main aim of this article is to describe a method of simulating the propagation of a traveling bacterial wave in a contaminated region and the resulting degradation of the contaminant. The presence of the chemotactic term and the relatively small bacterial diffusion means that the wave contains a very sharp wavefront. We, therefore, use an upwind conservative numerical scheme to obtain accurate and numerically stable solutions. The accuracy of the method is verified by comparisons with an exact one-dimensional solution of a simplified problem to give the same wavespeed. The method is then used to simulate the propagation of a realistic chemotactic wave in one dimension. We then use adaptive mesh refinement (AMR) to compute the propagation of chemotactic waves in two dimensions using the simplified model calibrated to give the same wavespeed as the full model.     

An Orthorhombic Lattice Boltzmann Model for Pore-Scale Simulation of Fluid Flow in Porous Media

One application of the lattice Boltzmann equation (LBE) models is in combination with tomography to simulate pore-scale flow and transport processes in porous media. Most LBE models in the literature are based on cubic lattice, and if the voxels in a tomography image are not cubic or cannot be divided into cubes due to computational limitations, these models will lose most of their advantages. How to deal with such images is, hence, an interest in use of the LBE model to simulate pore-scale processes. In this paper, we present an orthorhombic LBE model based on the single-relaxation time approach with the relaxation parameter varying with lattice directions. The equilibrium distribution functions in the standard LBE model were modified to correct the anisotropy induced by the non-cubic lattice, and the calculations of the fluid density and momentum were also redefined in order to maintain the conservation of mass and momentum during the collision. We tested the model against analytical solution for fluid flow in a tube, and against the standard cubic-based LBE model for fluid flow in a duct with an island inside. The model was then applied to simulate fluid flow in a 3D image in attempts to analyse the errors if the voxels in the image are not cubic but are assumed to be cubic.     

Calculating the Anisotropic Permeability of Porous Media Using the Lattice Boltzmann Method and X-ray Computed Tomography

A lattice Boltzmann (LB) method is developed in this article in a combination with X-ray computed tomography to simulate fluid flow at pore scale in order to calculate the anisotropic permeability of porous media. The binary 3D structures of porous materials were acquired by X-ray computed tomography at a resolution of a few microns, and the reconstructed 3D porous structures were then combined with the LB model to calculate their permeability tensor based on the simulated velocity field at pore scale. The flow is driven by pressure gradients imposed in different directions. Two porous media, one gas diffusion porous layer used in fuel cells industry and glass beads, were simulated. For both media, we investigated the relationship between their anisotropic permeability and porosity. The results indicate that the LB model is efficient to simulate pore-scale flow in porous media, and capable of giving a good estimate of the anisotropic permeability for both media. The calculated permeability is in good agreement with the measured date; the relationship between the permeability and porosity for the two media is well described by the Kozeny–Carman equation. For the gas diffusion layer, the simulated results showed that its permeability in one direction could be one order of magnitude higher than those in other two directions. The simulation was based on the single-relaxation time LB model, and we showed that by properly choosing the relaxation time, it could give similar results to those obtained using the multiple-relaxation time (MRT) LB method, but with only one third of the computational costs of MRTLB model.     

Numerical Calculation of Effective Diffusion in Unsaturated Porous Media by the TRT Lattice Boltzmann Method

Numerical models that solve transport of pollutants at the macroscopic scale in unsaturated porous media need the effective diffusion dependence on saturation as an input. We conducted numerical computations at the pore scale in order to obtain the effective diffusion curve as a function of saturation for an academic sphere packing porous medium and for a real porous medium where pore structure knowledge was obtained through X-ray tomography. The computations were performed using a combination of lattice Boltzmann models based on two relaxation time (TRT) scheme. The first stage of the calculations consisted in recovering the water spatial distribution into the pore structure for several fixed saturations using a phase separation TRT lattice Boltzmann model. Then, we performed diffusion computation of a non-reactive solute in the connected water structure using a diffusion TRT lattice Boltzmann model. Finally, the effective diffusion for each selected saturation value was estimated through inversion of a macroscopic classical analytical solution.     

Statistical Scaling of Geometric Characteristics in Millimeter Scale Natural Porous Media

We analyze statistical scaling of structural attributes of two millimeter scale rock samples, Estaillades limestone and Bentheimer sandstone. The two samples have different connected porosities and pore structures. The pore-space geometry of each sample is reconstructed via X-ray micro-tomography at micrometer resolution. Directional distributions of porosity and specific surface area (SSA), which are key Minkowski functionals (geometric observables) employed to describe the pore-space structure, are calculated from the images, and scaling of associated order- $q$ sample structure functions of absolute incremental values is analyzed. Increments of porosity and SSA tend to be statistically dependent and persistent (tendency for large and small values to alternate mildly) in space. Structure functions scale as powers $\xi (q)$ of directional separation distance or lag, $s$ , over an intermediate range of $s$ , displaying breakdown in power law scaling at large and small lags. Powers $\xi \!\!\left( q \right) $ of porosity and SSA inferred from moment and extended self-similarity (ESS) analyses of limestone and sandstone data tend to be quasi-linear and nonlinear (concave) in $q$ , respectively. We observe an anisotropic behavior for $\xi (q)$ , which appears to be mild for the porosity of the sandstone sample while it is marked for both porosity and SSA of the limestone rock sample. The documented nonlinear scaling behavior is amenable to analysis by viewing the variables as samples from sub-Gaussian random fields subordinated to truncated fractional Brownian motion or fractional Gaussian noise.     

Acoustics of Rotating Deformable Saturated Porous Media

We investigate wave propagation in elastic porous media which are saturated by incompressible viscous Newtonian fluids when the porous media are in rotation with respect to a Galilean frame. The model is obtained by upscaling the flow at the pore scale. We use the method of multiple scale expansions which gives rigorously the macroscopic behaviour without any prerequisite on the form of the macroscopic equations. For Kibel numbers A A(1), the acoustic filtration law resembles a Darcys law, but with a conductivity which depends on the wave frequency and on the angular velocity. The bulk momentum balance shows new inertial terms which account for the convective and Coriolis accelerations. Three dispersive waves are pointed out. An investigation in the inertial flow regime shows that the two pseudo-dilatational waves have a cut-off frequency.     

Pore-Scale Simulation of Fluid Flow Through Deformable Porous Media Using Immersed Boundary Coupled Lattice Boltzmann Method

Transport in Porous Media - The interaction between the fluid flow and the deformable porous media is crucial in the applications of adsorption/absorption. The immersed boundary coupled lattice...     

Lattice Boltzmann Simulation of Immiscible Displacement in Porous Media: Viscous Fingering in a Shear-Thinning Fluid

Transport in Porous Media - In this work, we investigate immiscible displacement in porous media with the displaced fluid being shear-thinning. We focus on the influence the heterogeneous viscosity...     

Diffusion of Randon in Porous Media Saturated with Gels and Emulsions

The effective diffusion coefficient of radon was determined in polymer/silicate gels and clay suspension used as sealing materials in environmental protection. On the basis of the experimental findings, it was concluded that both materials drastically decrease the convective mass transport in porous media. Simultaneously, the effective diffusion coefficient was reduced significantly. Thus, the radon flux might be decreased by 5 to 6 orders of magnitude in porous systems originally having gas or low water saturation by injection of gel-forming materials or placement of clay suspensions. At high water saturation, however, the diffusion transport of radon can be slightly restricted in consolidated and unconsolidated porous media. The laboratory studies may firmly allow us to conclude that hydrogels and clay suspensions are prospective candidates in an integrated environmental technology to be used for restriction of radon migration in subsurface regions.     

Quasi-Static and Dynamic Behavior of Saturated Porous Media with Incompressible Constituents

In this paper the field equations governing the dynamic response of a fluid-saturated elastic porous medium are analyzed and built up for the study of quasi-static and dynamical problems like the consolidation problem and wave propagation. The two constituents are assumed to be incompressible. A numerical solution is derived by means of the standard Galerkin procedure and the finite element method.     

Ferrofluid Permeation into Three-Dimensional Random Porous Media: A Numerical Study Using the Lattice Boltzmann Method

Colloidal suspensions containing magnetic nanoparticles placed in appropriate carrier liquids present strong magnetic dipoles. These suspensions, in general, exhibit normal liquid behaviour coupled with super paramagnetic properties. This leads to the possibility of remotely controlling the flow of such liquids with a moderate-strength external magnetic field. In this study, we numerically investigate the viability of controlling and steering a base-fluid with magnetic-sensitive nanoparticles into randomly structured fibrous porous media. Three dimensional flow simulations are performed using the lattice Boltzmann method. The simulation results for the flow front are presented, and the effect of the magnetic field strength on the rate of ferrofluid penetration is discussed. It is shown that the porosity of the porous medium and the size of the fibres have a strong effect on the ferrofluid penetration rate.     

Wave Dynamics of Saturated Porous Media and Evolutionary Equations

Nonlinear wave dynamics of an elastically deformed saturated porous media is investigated following the Biot approach. Mathematical models under research are the Biot model and its generalization by consideration of viscous stresses inside liquids. Using two-scales and linear WKB methods, the classical Biot system is transformed to a first-order wave equation. To construct the solution of the other system, an asymptotic modified two-scales method is developed. Initial system of equations is transformed to a nonlinear generalized Korteweg–de Vries–Burgers equation for quick elastic wave. Distinctions of wave propagation in the context of the Biot model and its generalization are shown.     

Dispersion and Dispersivity Tensors in Saturated Porous Media with Uniaxial Symmetry

The coefficient of dispersion, D _ij , and the dispersivity, a _ijkl , appear in the expression for the flux of a solute in saturated flow through porous media. We present a detailed analysis of these tensors in an axially symmetric porous medium, e.g., a stratified porous medium, with alternating layers, and show that in such a medium, the dispersivity is governed by six independent moduli. We present also the constraints that have to be satisfied by these moduli. We also show that at least two independent experiments are required in order to obtain the values of these coefficients for any three-dimensional porous medium domain.     

Stress Generated During Drying of Saturated Porous Media

The article is a contribution for the modelling of heat and mass transfers coupled to strain–stress equations during drying of deformable two-phase media. Both unidirectional and bidirectional configurations are examined. In order to compare the results, one assumes the material of a convectively dried clay slab in two configurations. Numerical calculations of the temperature, drying curves variations and the spatio-temporal distributions of moisture, temperature and drying induced stresses are evaluated. A significant difference was observed between the results obtained for both configurations particularly in intensity of the shear stress that caused cracking.     

Modeling Geometric State for Fluids in Porous Media: Evolution of the Euler Characteristic

Multiphase flow in porous media is strongly influenced by the pore-scale arrangement of fluids. Reservoir-scale constitutive relationships capture these effects in a phenomenological way, relying only on fluid saturation to characterize the macroscopic behavior. Working toward a more rigorous framework, we make use of the fact that the momentary state of such a system is uniquely characterized by the geometry of the pore-scale fluid distribution. We consider how fluids evolve as they undergo topological changes induced by pore-scale displacement events. Changes to the topology of an object are fundamentally discrete events. We describe how discontinuities arise, characterize the possible topological transformations and analyze the associated source terms based on geometric evolution equations. Geometric evolution is shown to be hierarchical in nature, with a topological source term that constrains how a structure can evolve with time. The challenge associated with predicting topological changes is addressed by constructing a universal geometric state function that predicts the possible states based on a non-dimensional relationship with two degrees of freedom. The approach is validated using fluid configurations from both capillary and viscous regimes in ten different porous media with porosity between 0.10 and 0.38. We show that the non-dimensional relationship is independent of both the material type and flow regime. We demonstrate that the state function can be used to predict history-dependent behavior associated with the evolution of the Euler characteristic during two-fluid flow.

     

Resolution Effect: An Error Correction Model for Intrinsic Permeability of Porous Media Estimated from Lattice Boltzmann Method

Transport in Porous Media - In digital rock physics, the intrinsic permeability of a porous rock sample can be evaluated from its micro-computed tomography ($$\upmu$$-CT) image through lattice...