Pore-Scale Lattice Boltzmann Modeling and 4D X-ray Computed Microtomography Imaging of Fracture-Matrix Fluid Transfer

We present sequential X-ray computed microtomography (CMT) images of matrix drainage in a fractured, sintered glass-granule-pack. Sequential (4D) CMT imaging captured the capillary-dominated displacement of the oil-occupied matrix by the surfactant-brine-occupied fracture at the pore scale. The sintered glass-granule-pack was designed to have minimal pore space beyond the resolution of CMT imaging, ensuring that the pore space of the matrix connected to the fracture could be captured in its entirety. This provided an opportunity to validate the increasingly common lattice Boltzmann modeling technique against experimental images at the pore scale. Although the surfactant was found to alter the wettability of the originally weakly oil-wet glass to water-wet, the fracture-matrix fluid transfer is found to be a drainage process, showing minimal counter-current migration of the initial wetting phase (decane). The LB simulations were found to closely match experimental rates of fracture-matrix fluid transfer, and trends in the saturation profiles, but not the irreducible wetting-phase saturation behind the flooding front. The underestimation of the irreducible wetting phase saturation suggests that finer image and lattice resolutions than those reported here may be required for accurate prediction of some macroscale multiphase flow properties, at a sizable computational cost.     

Lattice Boltzmann modeling of pore‐scale fluid flow through idealized porous media

In order to capture the hydro‐mechanical impacts on the solid skeleton imposed by the fluid flowing through porous media at the pore‐scale, the flow in the pore space has to be modeled at a resolution finer than the pores, and the no‐slip condition needs to be enforced at the grain–fluid interface. In this paper, the lattice Boltzmann method (LBM), a mesoscopic Navier–Stokes solver, is shown to be an appropriate pore‐scale fluid flow model. The accuracy and lattice sensitivity of LBM as a fluid dynamics solver is demonstrated in the Poiseuille channel flow problem (2‐D) and duct flow problem (3‐D). Well‐studied problems of fluid creeping through idealized 2‐D and 3‐D porous media (J. Fluid Mech. 1959; 5 (2):317–328, J. Fluid Mech. 1982; 115 :13–26, Int. J. Multiphase Flow 1982; 8 (4):343–360, Phys. Fluids A 1989; 1 (1):38–46, Int. J. Numer. Anal. Meth. Geomech. 1999; 23 :881–904, Int. J. Numer. Anal. Meth. Geomech. 2010; DOI: 10.1002/nag.898, Int. J. Multiphase Flow 1982; 8 (3):193–206) are then simulated using LBM to measure the friction coefficient for various pore throats. The simulation results agree well with the data reported in the literature. The lattice sensitivity of the frictional coefficient is also investigated. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

Hybrid lattice Boltzmann finite‐difference simulation of axisymmetric swirling and rotating flows

The axisymmetric flows with swirl or rotation were solved by a hybrid scheme with lattice Boltzmann method for the axial and radial velocities and finite‐difference method for the azimuthal (or swirl) velocity and the temperature. An incompressible axisymmetric lattice Boltzmann D2Q9 model was proposed to solve the axial and radial velocities through inserting source terms into the two‐dimensional lattice Boltzmann equation. Present hybrid scheme was firstly validated by simulations of Taylor–Couette flows between two concentric cylinders. Then the benchmark problems of melt flow in Czochralski crystal growth were studied and accurate results were obtained. Numerical experiment demonstrated that present axisymmetric D2Q9 model is more stable than the previous axisymmetric D2Q9 model (J. Comp. Phys. 2003; 186 (1):295–307). Hence, compared with the previous model, present numerical method provides a significant advantage in simulation melt flow cases with high Reynolds number and high Grashof number. Copyright © 2006 John Wiley & Sons, Ltd.     

Lattice Boltzmann simulations for flow and heat/mass transfer problems in a three‐dimensional porous structure

The lattice Boltzmann method (LBM) for a binary miscible fluid mixture is applied to problems of transport phenomena in a three‐dimensional porous structure. Boundary conditions for the particle distribution function of a diffusing component are described in detail. Flow characteristics and concentration profiles of diffusing species at a pore scale in the structure are obtained at various Reynolds numbers. At high Reynolds numbers, the concentration profiles are highly affected by the flow convection and become completely different from those at low Reynolds numbers. The Sherwood numbers are calculated and compared in good agreement with available experimental data. The results indicate that the present method is useful for the investigation of transport phenomena in porous structures. Copyright © 2003 John Wiley & Sons, Ltd.     

Foreword of the Founding Editor of the Journal Transport in Porous Media,Jacob Bear

The intrinsic permeability is a crucial parameter to characterise and quantify fluid flow through porous media. However, this parameter is typically uncertain, even if the geometry of the pore structure is available. In this paper, we perform a comparative study of experimental, semi-analytical and numerical methods to calculate the permeability of a regular porous structure. In particular, we use the Kozeny–Carman relation, different homogenisation approaches (3D, 2D, very thin porous media and pseudo 2D/3D), pore-scale simulations (lattice Boltzmann method, Smoothed Particle Hydrodynamics and finite-element method) and pore-scale experiments (microfluidics). A conceptual design of a periodic porous structure with regularly positioned solid cylinders is set up as a benchmark problem and treated with all considered methods. The results are discussed with regard to the individual strengths and limitations of the used methods. The applicable homogenisation approaches as well as all considered pore-scale models prove their ability to predict the permeability of the benchmark problem. The underestimation obtained by the microfluidic experiments is analysed in detail using the lattice Boltzmann method, which makes it possible to quantify the influence of experimental setup restrictions.

     

Investigation on Heat and Mass Transfer in a Evaporator of a Capillary-Pumped Loop with the Lattice Boltzmann Method: Pore Scale Simulation

A two-dimensional transient numerical model based on the lattice Boltzmann method (LBM) for the global evaporator of a capillary-pumped loop (CPL) is proposed to describe heat and mass transfer with evaporation in the porous wick, heat conduction in the cover plate, and heat transfer in the vapor groove. To indicate the stochastic phase distribution characteristics of most porous wick, the quartet structure generation set (QSGS) is introduced for generating more realistic microstructures of porous media. By using the present lattice Boltzmann algorithm along with the porous structure, the heat and mass transfer of an evaporator on pore scale can be predicted without resorting to any empirical parameters determined case by case. The energy equations for entire evaporator are solved as a conjugate problem, which are solved by means of a spatially varying relaxation time in the lattice Boltzmann model and the liquid flow is driven via the interfacial mass flux. A convective boundary condition considering the latent heat during the evaporation on the interface is introduced into the lattice Boltzmann model based on the nonequilibrium extrapolation rule. Especially, the bounce-back rule and the equilibrium rule of the LBM are, respectively, introduced to deal with the momentum boundary conditions inside the porous wick and on the evaporation interface in order to ensure the stability and the efficiency of the LBM model. Numerical results corresponding to different working conditions and different working fluids are presented, which provide guidance for the evaporator design of a CPL system.     

Unstructured lattice Boltzmann method in three dimensions

Over the last decade, the lattice Boltzmann method (LBM) has evolved into a valuable alternative to continuum computational fluid dynamics (CFD) methods for the numerical simulation of several complex fluid‐dynamic problems. Recent advances in lattice Boltzmann research have considerably extended the capability of LBM to handle complex geometries. Among these, a particularly remarkable option is represented by cell‐vertex finite‐volume formulations which permit LBM to operate on fully unstructured grids. The two‐dimensional implementation of unstructured LBM, based on the use of triangular elements, has shown capability of tolerating significant grid distortions without suffering any appreciable numerical viscosity effects, to second‐order in the mesh size. In this work, we present the first three‐dimensional generalization of the unstructured lattice Boltzmann technique (ULBE as unstructured lattice Boltzmann equation), in which geometrical flexibility is achieved by coarse‐graining the lattice Boltzmann equation in differential form, using tetrahedrical grids. This 3D extension is demonstrated for the case of 3D pipe flow and moderate Reynolds numbers flow past a sphere. The results provide evidence that the ULBE has significant potential for the accurate calculation of flows in complex 3D geometries. Copyright © 2005 John Wiley & Sons, Ltd.     

Simulate natural convection in a 30 MeV cyclotron with effect of irradiation,lattice Boltzmann 3D compressible thermal model

The lattice Boltzmann equation method in three dimensions used to analyze compressible thermal flow in 30 MeV cyclotrons. Cyclotron produced radionuclides have diagnostic applications in nuclear medicine. This cyclotron has horizontal conically shaped and window air cooling in the front and water cooling of the body. Gas heated by irradiation. This scheme is a new type of simulation method for solving the time dependent Navier–Stokes equations in a compressible flow regime. The improved model is convenient to compromise the high accuracy and stability. The included dispersion term can effectively reduce the numerical oscillation at discontinuity. The lattice Boltzmann scheme with uniform mesh resolution is applied as a numerical research tool.     

Application of the lattice‐Boltzmann method to study flow and dispersion in channels with and without expansion and contraction geometry

The lattice‐Boltzmann (LB) method, derived from lattice gas automata, is a relatively new technique for studying transport problems. The LB method is investigated for its accuracy to study fluid dynamics and dispersion problems. Two problems of relevance to flow and dispersion in porous media are addressed: (i) Poiseuille flow between parallel plates (which is analogous to flow in pore throats in two‐dimensional porous networks), and (ii) flow through an expansion–contraction geometry (which is analogous to flow in pore bodies in two‐dimensional porous networks). The results obtained from the LB simulations are compared with analytical solutions when available, and with solutions obtained from a finite element code (FIDAP) when analytical results are not available. Excellent agreement is found between the LB results and the analytical/FIDAP solutions in most cases, indicating the utility of the lattice‐Boltzmann method for solving fluid dynamics and dispersion problems. Copyright © 1999 John Wiley & Sons, Ltd.     

Simulation of natural convection melting in a cavity with fin using lattice Boltzmann method

In this study, a numerical investigation of melting phenomenon with natural convection in a cavity with fin has been performed using enthalpy‐based lattice Boltzmann method. The lattice D2Q9 model was applied to determine the density and velocity fields, and the D2Q5 model for the temperature field. The effect of vertical position and length of the fin on the melting rate was studied. The simulations were carried out for Stefan number of 10, Rayleigh number of 10 ⁵ and relative thermal conductivity (k_fin∕k_fluid) ranging from 5 to 30. The obtained results show that the rate of melting increases when the relative thermal conductivity and the length of the fin become greater. We also found that the variation of vertical position of the fin from bottom to middle has an insignificant effect on melting while it causes the increase of full melting time when the fin is mounted on the top of the cavity. Copyright © 2011 John Wiley & Sons, Ltd.     

Pore Scale Modeling of Reactive Transport Involved in Geologic CO<Subscript>2</Subscript> Sequestration

We apply a multi-component reactive transport lattice Boltzmann model developed in previous studies for modeling the injection of a CO₂-saturated brine into various porous media structures at temperatures T = 25 and 80°C. In the various cases considered the porous medium consists initially of calcite with varying grain size and shape. A chemical system consisting of Na⁺, Ca²⁺, Mg²⁺, H⁺, CO₂^°(aq){{\rm CO}_2^{\circ}{\rm (aq)}}, and Cl⁻ is considered. Flow and transport by advection and diffusion of aqueous species, combined with homogeneous reactions occurring in the bulk fluid, as well as the dissolution of calcite and precipitation of dolomite are simulated at the pore scale. The effects of the structure of the porous media on reactive transport are investigated. The results are compared with a continuum-scale model and the discrepancies between the pore- and continuum-scale models are discussed. This study sheds some light on the fundamental physics occurring at the pore scale for reactive transport involved in geologic CO₂ sequestration.     

Lattice Boltzmann simulation of natural convection in nanofluid-filled 2D long enclosures at presence of magnetic field

In this paper, the effects of a magnetic field on natural convection flow in filled long enclosures with Cu/water nanofluid have been analyzed by lattice Boltzmann method. This study has been carried out for the pertinent parameters in the following ranges: the Rayleigh number of base fluid, Ra = 10³–10⁵, the volumetric fraction of nanoparticles between 0 and 6 %, the aspect ratio of the enclosure between A = 0.5 and 2. The Hartmann number has been varied from Ha = 0 to 90 with interval 30 while the magnetic field is considered at inclination angles of θ = 0°, 30°, 60° and 90°. Results show that the heat transfer decreases by the increment of Hartmann number for various Rayleigh numbers and the aspect ratios. Heat transfer decreases with the growth of the aspect ratio but this growth causes the effect of the nanoparticles to increase. The magnetic field augments the effect of the nanoparticles at high Rayleigh numbers (Ra = 10⁵). The effect of the nanoparticles rises for high Hartmann numbers when the aspect ratio increases. The rise in the magnetic field inclination improves heat transfer at aspect ratio of A = 0.5.     

High-Resolution 3D FIB-SEM Image Analysis and Validation of Numerical Simulations of Nanometre-Scale Porous Ceramic with Comparisons to Experimental Results

The development of focused ion beam-scanning electron microscopy (FIB-SEM) techniques has allowed high-resolution 3D imaging of nanometre-scale porous materials. These systems are of important interest to the oil and gas sector, as well as for the safe long-term storage of carbon and nuclear waste. This work focuses on validating the accurate representation of sample pore space in FIB-SEM-reconstructed volumes and the predicted permeability of these systems from subsequent single-phase flow simulations using a highly homogeneous nanometre-scale, mesoporous (2–50 nm) to macroporous (>50 nm), porous ceramic in initial developments for digital rock physics. The limited volume of investigation available from FIB-SEM has precluded direct quantitative validation of petrophysical parameters estimated from such studies on rock samples due to sample heterogeneity, large variations in recorded sample pore sizes and lack of pore connectivity. By using homogeneous synthetic ceramic samples we have shown that lattice-Boltzmann flow simulations using processed FIB-SEM images are capable of predicting the permeability of a homogeneous material dominated by 10–100 nanometre-scale pores (similar, albeit simpler, to those in natural samples) at the much larger scale where permeability measurements become practical. This result shows the LB flow simulations can be used with confidence in pores at this scale allowing future work to focus on sample preparation techniques for samples sensitive to drying and multiple FIB-SEM site selection for the population of larger-scale models for heterogeneous systems.     

Issue Information ‐ TOC

Numerical modeling of multiphase flow generally requires a special procedure at the solid wall in order to be consistent with Young's law for static contact angles. The standard approach in the lattice Boltzmann method, which consists of imposing fictive densities at the solid lattice sites, is shown to be deficient for this task. Indeed, fictive mass transfer along the boundary could happen and potentially spoil the numerical results. In particular, when the contact angle is less than 90 degrees, the deficiencies of the standard model are major. Various videos that demonstrate this behavior are provided (Supporting Information). A new approach is proposed and consists of directly imposing the contact angle at the boundaries in much the same way as Dirichlet boundary conditions are generally imposed. The proposed method is able to retrieve analytical solutions for static contact angles in the case of straight and curved boundaries even when variable density and viscosity ratios between the phases are considered. Although the proposed wetting boundary condition is shown to significantly improve the numerical results for one particular class of lattice Boltzmann model, it is believed that other lattice Boltzmann multiphase schemes could also benefit from the underlying ideas of the proposed method. The proposed algorithm is two‐dimensional, and the D2Q9 lattice is used. Copyright © 2016 John Wiley & Sons, Ltd.     

Permeability of Pore Networks Under Compaction

The characteristic pore length fixes the scale of permeability of a porous medium. For pore networks undergoing strong random compaction, this length becomes singular at transition porosities, revealing a change in the microstructure of the porespace controlling the transport. Nodal balances and lattice Boltzmann simulations of flow in pore networks under compaction show that the scaling between permeability and porosity changes near the transition porosities. Simulation results are compared with experimental permeability data from transparent two-dimensional micromodels of networks decorated with the same pore size distribution. Permeability?Cporosity data of media undergoing smooth compaction is well described by a single power law. Under strong compaction, however, the scaling between permeability and porosity is possible by traits only, the scaling exponent changes notably at given transition porosities. These behaviors are common to a wealth of permeability?Cporosity data thus far unexplained.     

Development of LBGK and incompressible LBGK‐based lattice Boltzmann flux solvers for simulation of incompressible flows

This paper presents lattice Boltzmann Bhatnagar–Gross–Krook (LBGK) model and incompressible LBGK model‐based lattice Boltzmann flux solvers (LBFS) for simulation of incompressible flows. LBFS applies the finite volume method to directly discretize the governing differential equations recovered by lattice Boltzmann equations. The fluxes of LBFS at each cell interface are evaluated by local reconstruction of lattice Boltzmann solution. Because LBFS is applied locally at each cell interface independently, it removes the major drawbacks of conventional lattice Boltzmann method such as lattice uniformity, coupling between mesh spacing, and time interval. With LBGK and incompressible LBGK models, LBFS are examined by simulating decaying vortex flow, polar cavity flow, plane Poiseuille flow, Womersley flow, and double shear flows. The obtained numerical results show that both the LBGK and incompressible LBGK‐based LBFS have the second order of accuracy and high computational efficiency on nonuniform grids. Furthermore, LBFS with both LBGK models are also stable for the double shear flows at a high Reynolds number of 10⁵. However, for the pressure‐driven plane Poiseuille flow, when the pressure gradient is increased, the relative error associated with LBGK model grows faster than that associated with incompressible LBGK model. It seems that the incompressible LBGK‐based LBFS is more suitable for simulating incompressible flows with large pressure gradients. Copyright © 2014 John Wiley & Sons, Ltd.     

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.