Generalized Lattice Boltzmann Research Articles

A consistent generalized model-based lattice Boltzmann flux solver for incompressible porous flows

The recently developed lattice Boltzmann flux solver (PLBFS) for the incompressible porous flow is free from the limitations of coupled streaming time step and the mesh spacing, and the uniform meshes and the complex distribution function treatment at the boundary. However, the local flux reconstruction is inconsistent with the global governing equations in PLBFS. To overcome the drawback, a consistent generalized lattice Boltzmann flux solver for the incompressible porous flow is proposed based on the generalized lattice Boltzmann method (GLBM). The recovered macroscopic governing equations given by the Chapman–Enskog analysis of GLBM are globally resolved by the finite volume method. Specifically, the macroscopic variables are updated at cell centers using the three-step Runge–Kutta method, while the solution of the GLBM is locally applied for the fluxes reconstruction at cell interfaces. Unlike the PLBFS, the forcing term can be naturally incorporated into the interface fluxes reconstruction, which gives the present method a stronger physical basis and ensures global consistency. Moreover, different from the PLBFS, the streaming time step used at the cell interface is decoupled from the updating time step at the cell center in the present solver. Furthermore, a simplified flux reconstruction strategy is proposed to avoid complex calculations and save computing resources. Several numerical examples have been adopted to test the proposed method. The simulations of the nonlinear lid-driven cavity flow show that our method is more accurate than the original PLBFS. Results also demonstrate that the simplified method can reduce the computational time by 43%.

A Simplified Lattice Boltzmann Boundary Conditions for Gas Transport in Self-Affine Microchannels with an Inherent Roughness of in a Tight Reservoir

A simplified method of determining lattice Boltzmann boundary conditions based on self-affine microchannels with an inherent roughness in a tight reservoir is presented in this paper to address nonlinear efficiency problems in fluid simulation. This approach effectively combines the influence of rough surfaces in the simulation of the flow field, the description of L-fractal theory applied to rough surfaces, and a generalized lattice Boltzmann method with equivalent composite slip boundary conditions for inherent roughness. The numerical simulations of gas slippage in a two-dimensional plate model and rough surfaces to induce gas vortex reflux flow are also successfully carried out, and the results are in good agreement with the simulation results, which establishes the reliability and flexibility of the proposed simplified method of rough surfaces. The effects of relative average height and fractal dimensions of the rough surfaces under exact boundary conditions and equivalent coarsened ones are investigated from three perspectives, namely those of the average lattice velocity, the lattice velocity at average height position at the outlet, and the coefficient of variation for lattice velocity at average height position. It was found that the roughness effect on gas flow behavior was more obvious when it was associated with the enhanced rarefaction effect. In addition, the area of gas seepage was reduced, and the gas flow resistance was increased. When the fractal dimension of the wall was about 1.20, it has the greatest impact on the fluid flow law. In addition, excessive roughness of the wall surface tends to lead to vortex backflow of the gas in the region adjacent to the wall, which greatly reduces its flow velocity. For gas flow in the nanoscale seepage space, wall roughness hindered gas migration rate by 84.7%. For pores larger than 200 nm, the effects of wall roughness on gas flow are generally negligible.

Nano-modification effect on the pore characteristics and the water multiscale transport properties of concrete in winter construction of tunnel engineering

Seepage has become the dominant damage that affected tunnel stability. Although drainage and the waterproofing system are frequently employed in seepage tunnels, the poor performance of shotcrete is still an inducement for the waterproofing system failure. Further, the frost damage caused by seepage becomes more pronounced when tunnel construction locates in cold region environments. Therefore, the objective of the study is to explore the nano-modification effects on the water multiscale transport behaviors of concrete in winter construction environments. Firstly, the pore characteristics of the hydrated cement were experimentally determined by the mercury intrusion porosimetry (MIP) test. The meso-structures of concrete were obtained via the X-ray computed tomography (CT) test. Secondly, the fractal analysis was conducted to derive the permeability of the hydrated cement based on the capillary model with the interaction of solid and fluid molecules considered. Then, the permeability distribution of concrete was calculated by combining test results at the micro-scale and that at the meso-scale. Finally, the unified multiscale simulations based on the generalized Lattice Boltzmann method (GLBM) were conducted to obtain the permeability of all groups. The water transport process of concrete on the meso-scale was simulated by considering the influence of micro-structures. The permeability tests were conducted to verify the simulation results. Results indicated that the addition of nano-materials can markedly increase the complexity of the pore structure and decrease the porosity, thereby enhancing the impermeability of concrete.

Apparent Permeability Prediction of Coal Matrix with Generalized Lattice Boltzmann Model considering Non-Darcy Effect

Methane flow in coal is associated with the content of both organic matter (OM) and inorganic matter (IOM). Coal matrix contains nanopores ranging in size from a few to hundreds of nanometers, which leads to a non-Darcy effect where the measured permeability of a gas (apparent permeability) is higher than that of a liquid (intrinsic permeability). In this study, a generalized Lattice Boltzmann model (GLBM) is employed for gas flow through the reconstructed coal matrix consisting of OM, IOM, and fractures. The apparent permeability model is proposed to calculate the total flow flux accounting for multiple transport mechanisms including viscous flow, slip flow, transitional flow, and the Knudsen diffusion. The impact of effective pore radius and gas surface diffusion on permeability in the gas adsorption-desorption process is also considered in the model. What’s more, the weighting factors are adopted to adjust the contribution of the viscous flow and the Knudsen flow. The effect of total organic/inorganic content and the development of fractures on the apparent permeability of the reconstructed coal matrix is also studied. It is found that the apparent permeability is extremely low when a fracture is nonexistent, and varies almost linearly with the total organic/inorganic content. A fracture plays a significant role in determining apparent permeability and the velocity distribution of the coal matrix.

A generalized lattice Boltzmann model for solid–liquid phase change with variable density and thermophysical properties

In this paper, we modify the calculations of volume fraction of liquid and temperature, and propose a generalized two-relaxation-time (TRT) lattice Boltzmann model for the solid–liquid phase change problem with variable density and thermophysical properties. Two benchmark problems are used to validate the new model. The results show that this generalized model can handle the differences in both density and thermophysical properties.

A generalized lattice Boltzmann model for fluid flow system and its application in two-phase flows

In this paper, a generalized lattice Boltzmann (LB) model with a source term in the continuity equation is proposed to solve both incompressible and nearly incompressible Navier–Stokes (N–S) equations. This model can be used to deal with single-phase and two-phase flows problems with a source term in the continuity equation. From this generalized model, we can not only get some existing models, but also derive new models. Moreover, for the incompressible model derived, a modified pressure scheme is introduced to calculate the pressure, and then to ensure the accuracy of the model. In this work, we will focus on a two-phase flow system, and in the frame work of our generalized LB model, a new phase-field-based LB model is developed for incompressible and quasi-incompressible two-phase flows. A series of numerical simulations of some classic physical problems, including a spinodal decomposition, a static droplet, a layered Poiseuille flow, and a bubble rising flow under buoyancy, are performed to validate the developed model. Besides, some comparisons with previous quasi-incompressible and incompressible LB models are also carried out, and the results show that the present model is accurate in the study of two-phase flows. Finally, we also conduct a comparison between quasi-incompressible and incompressible LB models for two-phase flow problems, and find that in some cases, the proposed quasi-incompressible LB model performs better than incompressible LB models.

Generalized lattice Boltzmann model for frosting.

Frosting is a multiscale and multiphysics problem, which presents a significant challenge for numerical methods. In this study, a generalized lattice Boltzmann (LB) model is developed to simulate the frosting of humid air at representative elementary volume scale. In this model, three LB equations are introduced to describe the evolution of distribution functions for velocity, temperature, and humidity (i.e., mass fraction of water vapor in the humid air) fields, respectively. The frost layer is regarded as a porous medium, while the humid air is treated as a plain one. This unified LB model can be applied to describe the phase change and transport processes in these two subdomains seamlessly. Through the Chapman-Enskog analysis, the macroscopic equations for the frosting process can be recovered from the present LB model. Benchmark problems in conduction solidification, convection melting and frosting are simulated, and the numerical results match well with analytical or experimental solutions. Finally, this model is applied to simulate frost formation between two parallel plates, and the influences of air velocity, humidity, temperature, and cold wall temperature are evaluated.

The transport behaviors of oil in nanopores and nanoporous media of shale

Shale, known as the “tight” rock with abundant nanopores, exhibits extremely low permeability on the order of micro/nanodarcy. The classic Darcy law, being widely and successfully used in conventional porous media, becomes insufficient for the shale. In this work, on the basis of molecular dynamics (MD) simulations data available in the literature, a model for oil transport through a single nanopore is established by considering the boundary slip and the varying viscosity of the confined oil. Then, the established model for the single nanopore is integrated into the generalized lattice Boltzmann method (GLBM) to mimic the oil transport in the shale with nanopore networks, successfully scaled up to the nanoporous media. The results show that, to accurately predict the oil transport properties in inorganic and organic nanopores, the viscosity correction for the confined oil transport in the nanopores is necessary. The oil transport capability in organic nanopores is greatly enhanced compared with that predicted by the no-slip Poiseuille equation, significantly enhancing the flow capability in the scale of nanoporous media, while the small slip length in the inorganic matter (IOM) has neglected effect. The large liquid slip in organic matter (OM) also results in the apparent oil permeability of shale increases with the total organic content (TOC), although the fact that pore sizes within OM are universally smaller than that in IOM. This work provides a better understanding of the oil transport behaviors in a single nanopore and the nanoporous shale, and paves a feasible way to the exact numerical simulation for oil transport in nanoporous shale.

The effect of obstacles’ characteristics on heat transfer and fluid flow in a porous channel

The fluid flow and heat transfer around obstacles are an engineering and research interest. This paper dealt with this matter in a porous channel. It explained the features of the presence of hot solid obstacles in porous media. These obstacles were located at different positions inside the medium. The particularity of this work is coupling two complex phenomena: the heat transfer in porous media and the presence of hot solid obstacles. This is the first time that these phenomenona were studied. The diffusion–convection equation is adopted to calculate the temperature. The viscous heat dissipation and compression work due to the pressure were not taken into consideration. This choice and assumptions were based on our previous work. The numerical simulation was done using the generalized lattice Boltzmann method. To ensure that our numerical code is free of errors, we resorted to benchmark cases. Then, we were interested in the effect of triangular and rectangular obstacles on the heat transfer and fluid flow in a porous channel. The isotherms and the velocity contours were studied for several dynamic parameters. The fluid behavior was described by the streamlines and the velocity fields. The velocity profile was followed along the porous channel. These results allow concluding that the Reynolds number increment led to the increase in the heat transfer and the fluid velocity. The increment of the distance between the inlet and the obstacle generates the same conclusion.

Modelling the water transport behavior in organic-rich nanoporous shale with generalized lattice Boltzmann method

The hydraulic fracturing fluid could easily infiltrate the ultra-tight shale matrix due to the large slip of the liquid flow, showing higher-than-expected fluid-loss in varieties of shale gas development cases. One possible reason is that the water transport behaviors through the pores with nanoscale significantly deviate from that occurring at larger scales. The classic Darcy law, being widely and successfully used in conventional porous media becomes insufficient for the nanoporous shale. In this study, a generalized lattice Boltzmann method with liquid slip effect incorporated is established to understand the transport behavior of hydraulic fracturing fluid in nanopores dominated shale matrix and to demonstrate a new insight into the transport behaviors. The results show that the flow capability of fracturing fluid in the shale matrix with strong hydrophobic organic nanopores is significantly improved due to the huge wall-fluid interaction. And this would considerably change the flow field (magnitudes and preferred pathway) with and without the micro-fractures, contributing a lot to the huge hydraulic fracturing fluid loss reported from the practical fields. The huge fluid-loss emphasizes the importance for liquid slip effect in organic nanopores of shale matrix. Especially, in the organic-rich shale gas reservoir, the fracturing fluid can be infiltrated into the ultra-tight shale formation easier than commonly expected during the hydraulic fracturing operation.

Finite element lattice Boltzmann method for fluid flow through complex fractured media with permeable matrix

Abstract This study proposed an improved finite element based generalized lattice Boltzmann model (FE-GLBM) to simulate the fluid flow in extremely heterogeneous porous media, in which the generalized lattice Boltzmann equations proposed by Guo and Zhao (2002) are solved using the characteristic Galerkin finite element method (GFEM), and the FEM is applied to improve the flexibility of GLBM and to optimize the flow simulation in complex geometries. An image-based automatic meshing method is adopted to generate advanced unstructured mesh of the structure. The proposed numerical model is validated with analytical results, finite-difference results, and previously published data in three cases, including the generalized two-dimensional Poiseuille flow, the lid-driven cavity flow, and the circular cylinder flow, respectively. The sensitivity study of the fluid flow in three-dimensional fractured porous media with a disordered fracture network and permeable matrix is carried out as a practical application example, and the results indicate that the matrix permeability plays an important role in controlling the flow dynamics in fractured porous media.

Gas transport in self-affine rough microchannels of shale gas reservoir

The matrix of shale gas reservoirs is always characterized by rough surfaces and the roughness of the topography significantly influences the gas flow characteristics. However, most existing models are based on smooth surface. Moreover, the roughness effect leads to a complex velocity distribution and can influence the apparent permeability through its own transport mechanism. In this study, a generalized lattice Boltzmann method with nonequilibrium extrapolation boundary conditions was developed to simulate the rarefied shale gas flow through nanochannels with rough surface. Fractal geometry was introduced to model the rough surface, and the roughness was characterized by two parameters, namely, relative roughness height and fractal dimension. Triple-effect permeability estimation model was adopted to calculate the apparent permeability accounting for multiple roughness characteristics including surface tortuosity, hydraulic tortuosity, and local roughness. Effects of fractal dimension, relative roughness, and tortuosity on velocity distribution were investigated. The effect of pressure and roughness characteristic parameters on the apparent permeability at different Knudsen numbers was also studied. The results showed that the influence of fractal dimension and relative roughness on the permeability was affected by the rarefaction effect. The roughness effect on gas flow behavior became more evident when associated with the enhanced rarefaction effect. Moreover, relative roughness was found to play a significant role in determining the permeability in the transient flow regime.

A Generalized Lattice Boltzmann Model for Simulating Axisymmetric Convective Flow in Porous Media

Axisymmetric convective flow in porous media is frequently encountered in nature and industry applications. In recent years, Lattice Boltzmann method has been developed as a powerful tool for such kind of flow and heat transfer. Despite its success in many problems regarding to porous flow, the existing LB model for the axisymmetric thermal flow in porous media at the representative volume scale suffers from a serious drawback. That is, it can not handle the cases where the heat capacitance of porous media varies spatially obviously. In this paper, a generalized LB model for axisymmetric temperature field is proposed to remedy this shortcoming. Chapman-Enskog analysis demonstrates that the energy equation in the cylindrical coordinates system can be recovered by the proposed model. Natural convection in a vertical annulus filled with saturated porous media, natural convection in a vertical annulus without porous media, have been carried out, and the results predicted by the present LB model agree well with the existing numerical data. More, natural convection in a vertical annulus with spatially varying heat capacitance shows that the present model can address the problem where the heat capacitance varies spatially obviously successfully.

Field-wide flow simulation in fractured porous media within lattice Boltzmann framework

In this paper, a generalized lattice Boltzmann model for simulating fluid flow in porous media at the representative volume element scale is extended towards applications of hydraulically and naturally fractured reservoirs. The key element within the model is the development of boundary conditions for a vertical well and horizontal fracture with minimal node usage. In addition, the governing non-dimensional equations are derived and a new set of dimensionless numbers are presented for the simulation of a fractured reservoir system. Homogenous and heterogeneous vertical well and fracture systems are simulated and verified against commercial reservoir simulation suites. Results are in excellent agreement to analytical and finite difference solutions.

Apparent permeability prediction of organic shale with generalized lattice Boltzmann model considering surface diffusion effect

Gas flow in shale is associated with both organic matter (OM) and inorganic matter (IOM) which contain nano-pores ranging in size from a few to hundreds of nano-meters. In addition to the non-continuum effect which leads to an apparent permeability of gas higher than the intrinsic permeability, the surface diffusion of adsorbed gas in organic pores also can influence the apparent permeability through its own transport mechanism. In this study, a generalized lattice Boltzmann model (GLBM) is employed for gas flow through the reconstructed shale matrix consisting of OM and IOM. The Expectation–Maximization (EM) algorithm is used to assign the pore size distribution to each component, and the dusty gas model (DGM) and generalized Maxwell–Stefan model (GMS) are adopted to calculate the apparent permeability accounting for multiple transport mechanisms including viscous flow, Knudsen diffusion and surface diffusion. Effects of pore radius and pressure on permeability of both IOM and OM as well as effects of Langmuir parameters on OM are investigated. The effect of total organic content and distribution on the apparent permeability of the reconstructed shale matrix at different surface diffusivity is also studied. It is found that the influence of pore size and pressure on the apparent permeability of organic matter is affected by the surface diffusion of adsorbed gas. Moreover, surface diffusion plays a significant role in determining apparent permeability and the velocity distribution of shale matrix.

Finite-volume method with lattice Boltzmann flux scheme for incompressible porous media flow at the representative-elementary-volume scale.

Based on the Darcy-Brinkman-Forchheimer equation, a finite-volume computational model with lattice Boltzmann flux scheme is proposed for incompressible porous media flow in this paper. The fluxes across the cell interface are calculated by reconstructing the local solution of the generalized lattice Boltzmann equation for porous media flow. The time-scaled midpoint integration rule is adopted to discretize the governing equation, which makes the time step become limited by the Courant-Friedricks-Lewy condition. The force term which evaluates the effect of the porous medium is added to the discretized governing equation directly. The numerical simulations of the steady Poiseuille flow, the unsteady Womersley flow, the circular Couette flow, and the lid-driven flow are carried out to verify the present computational model. The obtained results show good agreement with the analytical, finite-difference, and/or previously published solutions.

Permeability prediction of shale matrix reconstructed using the elementary building block model

Representative elementary volume (REV)-scale structure of shale matrix is reconstructed based on elementary building block (EBB) model using a stochastic reconstruction method called Quartet Structure Generation Set. In the EBB model, various constituents with different pore morphologies in shale matrix including organic matter and inorganic minerals are considered as different EBBs, and in each EBB, specific structural parameters and transport properties are locally defined. A generalized lattice Boltzmann model for fluid flow through tight porous media with slippage is employed to simulate fluid flow through the reconstructed REV-scale structures. A four-EBB shale matrix including clay, calcite, pyrite and organic matter is studied and its permeability is predicted. Effects of organic content, grain size, interparticle pores on the permeability of the REV-scale matrix are investigated. It is found that smaller grain size and interparticle pores can increase the permeability. The influences of complex physical processes such as slippage and adsorption on the REV-scale permeability are also explored. Slippage effect increases as the pore size decreases. Adsorption has two opposite effects on the permeability, and which one dominates depend on pressure. The present study can help understand gas transport in shale matrix and improve reservoir scale studies.

Generalized lattice Boltzmann model for flow through tight porous media with Klinkenberg's effect.

Gas slippage occurs when the mean free path of the gas molecules is in the order of the characteristic pore size of a porous medium. This phenomenon leads to Klinkenberg's effect where the measured permeability of a gas (apparent permeability) is higher than that of the liquid (intrinsic permeability). A generalized lattice Boltzmann model is proposed for flow through porous media that includes Klinkenberg's effect, which is based on the model of Guo et al. [Phys. Rev. E 65, 046308 (2002)]. The second-order Beskok and Karniadakis-Civan's correlation [A. Beskok and G. Karniadakis, Microscale Thermophys. Eng. 3, 43 (1999) and F. Civan, Transp. Porous Med. 82, 375 (2010)] is adopted to calculate the apparent permeability based on intrinsic permeability and the Knudsen number. Fluid flow between two parallel plates filled with porous media is simulated to validate the model. Simulations performed in a heterogeneous porous medium with components of different porosity and permeability indicate that Klinkenberg's effect plays a significant role on fluid flow in low-permeability porous media, and it is more pronounced as the Knudsen number increases. Fluid flow in a shale matrix with and without fractures is also studied, and it is found that the fractures greatly enhance the fluid flow and Klinkenberg's effect leads to higher global permeability of the shale matrix.

Discrete adjoint sensitivity analysis for fluid flow topology optimization based on the generalized lattice Boltzmann method

A discrete adjoint sensitivity analysis for fluid flow topology optimization based on the lattice Boltzmann method (LBM) with multiple-relaxation-times (MRT) is developed. The lattice Boltzmann fluid solver is supplemented by a porosity model using a Darcy force. The continuous transition from fluid to solid facilitates a gradient based optimization process of the design topology of fluidic channels. The adjoint LBM equation, which is used to compute the gradient of the optimization objective with respect to the design variables, is derived in moment space and found to be as simple as the original LBM. The moment based spatial momentum derivatives used to express the discrete objective functional (cost function) have the advantage that the local stress tensor is a local quantity avoiding the numerical computation of gradients of the discrete velocity field. This is particularly useful if dissipation is a design criterion as demonstrated in this paper. The method is validated by a detailed comparison with results obtained by Borrvall et al. for Stokes flow. While their approach is only valid for Stokes flow (i.e. very low Reynolds numbers) our approach in its present form can in principle be applied for flows of different Reynolds numbers just like the Navier–Stokes equation based approaches. This point is demonstrated with a bending pipe example for various Reynolds numbers.

Particle Tracking in Large Eddy Simulated Turbulent Channel Flow Using Generalized Lattice Boltzmann Method

The deposition of spherical solid particles with 0.00192 ≤ τ+ ≤ 160.56 in three-dimensional turbulent vertical and horizontal channel flows with friction Reynolds number was studied. A generalized lattice Boltzmann equation (GLBE) was used for computation of instantaneous turbulent flow field for which large eddy simulation (LES) was employed. The subgrid scale (SGS) turbulence effects were simulated through a shear-improved Smagorinsky model (SISM), which can predict turbulent near wall region without any wall function. The Lagrangian particle tracking approach was used, and qualitative distribution of particles in a vertical channel center plane and in a near wall region was presented. It was shown that, in horizontal channels, the rate of deposition is higher than in vertical channels since gravitational sedimentation is the dominant mechanism of deposition on the lower wall. Combination of SISM in GLBE with particle tracking analysis in turbulent channel flow is the novelty of the present work. Since the accuracy of the flow field is essential to correctly predict particles' behavior, a reliable method should be used to simulate the flow field. In the proposed method, the flow field equations are solved locally. The collaboration of this method and particle tracking can be considered a good candidate for simulation of particulate flow.