基于多松弛格子玻尔兹曼的弯道水流三维数值模拟

为研究弯道水流现象并且拓展格子玻尔兹曼(Lattice Boltzmann Method,LBM)在水利工程领域的应用,建立了弯道水流的三维多松弛LBM演进模型,其中包括水流自由表面模拟、弯曲固壁边界处理以及紊流模型耦合求解等关键技术。应用该模型模拟研究了不同流量下U型弯道的水流状态,得到了水位、最大水深平均流速以及紊动能等参数的分布规律。分析和实践表明,该模型能较好地模拟三维弯道水流现象。     

On the implicit immersed boundary method in coupled discrete element and lattice Boltzmann method

The coupled discrete element method and lattice Boltzmann method (DEMLBM) has increasingly drawn attention of researchers in geomechanics due to its mesoscopic nature since 2000. Immersed boundary method (IBM) and immersed moving boundary (IMB) are two popular schemes for coupling fluid particle in DEMLBM. This work aims at coupling DEM and LBM using the latest IBM algorithm and investigating its accuracy, computational efficiency, and applicability. Two benchmark tests, interstitial fluid flow in an ideal packing and single particle sedimentation in viscous fluid, are carried out to demonstrate the accuracy of IBM through semi-empirical Ergun equation, finite element method (FEM), and IMB. Then, simulations of particle migration with relatively large velocity in Poiseuille flow are utilized to address limitations of IBM in DEMLBM modeling. In addition, advantages and deficiencies of IBM are discussed and compared with IMB. It is found that the accuracy of IBM can be only guaranteed when sufficient boundary points are used and it is not suitable for geomechanical problems involving large fluid or particle velocity.     

多尺度多孔介质有效气体输运参数的分形特征

非常规油气资源的孔隙结构及其连通性非常复杂,其孔隙尺度从毫米到纳米跨越多个量级.多孔介质中气体的输运过程不仅依赖于介质的多尺度微观结构特征,还依赖于气体的相关属性.气体在多尺度多孔介质中的输运过程包括无滑流、滑脱流和过渡流,涉及分子扩散和努森扩散等多种机制,因此很难用唯一的连续介质理论来描述气体的输运特征.大量的数据表明真实多孔介质中的内部孔隙具有分形标度特征,因此采用分形几何表征多尺度多孔介质的孔隙结构,引入孔隙分形维数和迂曲度分形维数定量表征多孔介质的微结构和弯曲流道,建立多尺度多孔介质气体输运过程的细观模型;推导了多尺度多孔介质中气体的有效渗透率和有效扩散系数,并讨论了多尺度多孔介质微结构参数和气体属性对于气体等效输运特性的定量影响.该研究不仅可以丰富渗流理论,且有利于深入理解非常规油气藏的产出机制.      

用格子波耳兹曼方法模拟双重孔隙介质中的流体迁移

作者在本文中介绍了基于格子波耳兹曼模型的双重孔隙介质中流体运移的数值模拟计算方法。我们从格子波耳兹曼碰撞模型出发，利用格子波耳兹曼方程、Chapman-Enskog展开，以及多尺度技术，得到了描述双重孔隙介质中流体迁移的二维扩散方程。利用格子气自动机方法计算该扩散方程，实现了对双重孔隙介质中流体运移过程的数值模拟仿真。数值实验表明，我们所使用的方法正确、有效。     

Hydro-micromechanical modeling of wave propagation in saturated granular crystals

Biot theory predicts wave velocities in a saturated granular medium using the pore geometry, viscosity, densities, and elastic moduli of the solid skeleton and pore fluid, neglecting the interaction between constituent particles and local flow, which becomes essential as the wavelength decreases. Here, a hydro-micromechanical model, for direct numerical simulations of wave propagation in saturated granular media, is implemented by two-way coupling the lattice Boltzmann method (LBM) and the discrete element method (DEM), which resolve the pore-scale hydrodynamics and intergranular behavior, respectively. The coupling scheme is benchmarked with the terminal velocity of a single sphere settling in a fluid. In order to mimic a small amplitude pressure wave entering a saturated granular medium, an oscillating pressure boundary on the fluid is implemented and benchmarked with the one-dimensional wave equation. The effects of input waveforms and frequencies on the dispersion relations in 3D saturated poroelastic media are investigated with granular face-centered-cubic crystals. Finally, the pressure and shear wave velocities predicted by the numerical model at various effective confining pressures are found to be in excellent agreement with Biot analytical solutions, including his prediction for slow compressional waves.     

Visualizing complex pore structure and fluid flow in porous media using 3D printing technology and LBM simulation

Pore structures of porous media and properties of fluid flow are key factors for the study of non-Darcy groundwater flow. However, it is difficult to directly observe pore structures and flow properties, resulting in a “black box” problem of porous media. This problem has hindered the in-depth study of the groundwater flow mechanism at the pore scale. In recent years, 3D rapid prototyping technology has seen tremendous development. 3D printing provides digital models and printing models of porous media with clear internal structure. Thus, Lattice Boltzmann Method can be used to simulate the flow processes at the pore scale based on real pore structures. In this study, 3D printing cores and Lattice Boltzmann Method were coupled to conduct both laboratory and numerical experiments in spherical porous media with different sphere diameters and periodic arrays. The LBM simulation results show a good agreement with laboratory experimental results. With the advantages of LBM and 3D printing, this approach provides a visualization of the complex pore structure and fluid flow in pores, which is a promising method for studies of non-Darcy groundwater flow at the pore scale.     

A technique to test finite difference schemes to model some geophysical processes in a geological structure

A preliminary problem to solve in the set-up of a mathematical model simulating a geophysical process is the choice of a suitable discrete scheme to approximate the governing differential equations. This paper presents a simple technique to test finite difference schemes used in the modeling of geophysical processes occurring in a geological structure. This technique consists in generating analytical solutions similar to the ones characterizing a geophysical process, given general information on some relevant parameters. Useful information for the choice of the discrete scheme to employ in the mathematical model simulating the original geophysical process can be obtained from the comparison between the analytical solution and the approximated numerical solutions generated by means of different discrete schemes. Two classes of numerical examples approximating the differential equation that governs the steady state earth's heat flow have been treated using three different finite differences schemes. The first class of examples deals with media whose phenomenological parameters vary as continuous space functions; the second class, instead, deals with media whose phenomenological parameters vary as discontinuous space functions. The finite difference schemes that have been utilized are: Centered Finite Difference Scheme (CDS), Arithmetic Mean Scheme (AMS), and Harmonic Mean Scheme (HMS).The numerical simulations showed that: the CDS may yield physically inconsistent solutions if the lattice internodal distance is too large, but in case of phenomenological parameters varying as a continuous function, this pitfall can be avoided increasing the lattice node refinement. In case of phenomenological parameters varying as a discontinuous function, instead, the CDS may yield physically inconsistent solutions for any lattice-node refinement. The HMS produced good results for both classes of examples showing to be a scheme suitable to model situations like these.     

Multiscale mixed/mimetic methods on corner-point grids

Multiscale simulation is a promising approach to facilitate direct simulation of large and complex grid models for highly heterogeneous petroleum reservoirs. Unlike traditional simulation, approaches based on upscaling/downscaling, multiscale methods seek to solve the full flow problem by incorporating subscale heterogeneities into local discrete approximation spaces. We consider a multiscale formulation based on a hierarchical grid approach, where basis functions with subgrid resolution are computed numerically to correctly and accurately account for subscale variations from an underlying (fine-scale) geomodel when solving the global flow equations on a coarse grid. By using multiscale basis functions to discretise the global flow equations on a (moderately sized) coarse grid, one can retain the efficiency of an upscaling method and, at the same time, produce detailed and conservative velocity fields on the underlying fine grid. For pressure equations, the multiscale mixed finite-element method (MsMFEM) has been shown to be a particularly versatile approach. In this paper, we extend the method to corner-point grids, which is the industry standard for modelling complex reservoir geology. To implement MsMFEM, one needs a discretisation method for solving local flow problems on the underlying fine grids. In principle, any stable and conservative method can be used. Here, we use a mimetic discretisation, which is a generalisation of mixed finite elements that gives a discrete inner product, allows for polyhedral elements, and can (easily) be extended to curved grid faces. The coarse grid can, in principle, be any partition of the subgrid, where each coarse block is a connected collection of subgrid cells. However, we argue that, when generating coarse grids, one should follow certain simple guidelines to achieve improved accuracy. We discuss partitioning in both index space and physical space and suggest simple processing techniques. The versatility and accuracy of the new multiscale mixed methodology is demonstrated on two corner-point models: a small Y-shaped sector model and a complex model of a layered sedimentary bed. A variety of coarse grids, both violating and obeying the above mentioned guidelines, are employed. The MsMFEM solutions are compared with a reference solution obtained by direct simulation on the subgrid.     

A pore-scale numerical model for flow through porous media

A pore-scale numerical model based on Smoothed Particle Hydrodynamics (SPH) is described for modelling fluid flow phenomena in porous media. Originally developed for astrophysics applications, SPH is extended to model incompressible flows of low Reynolds number as encountered in groundwater flow systems. In this paper, an overview of SPH is provided and the required modifications for modelling flow through porous media are described, including treatment of viscosity, equation of state, and no-slip boundary conditions. The performance of the model is demonstrated for two-dimensional flow through idealized porous media composed of spatially periodic square and hexagonal arrays of cylinders. The results are in close agreement with solutions obtained using the finite element method and published solutions in the literature. Copyright © 1999 John Wiley & Sons, Ltd.     

Free-surface Simulations of Newtonian and Non-Newtonian Fluids with the Lattice Boltzmann Method

This paper describes the application of a three-dimensional lattice Boltzmann method (LBM) to Newtonian and non-Newtonian (Bingham fluid in this work) flows with free surfaces. A mass tracking algorithm was incorporated to capture the free surface, whereas Papanastasiou’s modified model was used for Bingham fluids. The lattice Boltzmann method was first validated using two benchmarks: Newtonian flow through a square cross-section tube and Bingham flow through a circular cross-section tube. Afterward, the dam-break problem for the Newtonian fluid and the slump test for Bingham fluid were simulated to validate the free-surface-capturing algorithm. The numerical results were in good agreement with analytical results, as well as other simulations, thereby proving the validity and correctness of the current method. The proposed method is a promising substitute for time-consuming and costly physical experiments to solve problems encountered in geotechnical and geological engineering, such as the surge and debris flow induced by a landslide or earthquake.     

Laboratory validation of lattice Boltzmann method for modeling pore-scale flow in granular materials

Characteristics of fluid flow through various engineering structures, such as granular filters and asphalt pavements, influence their design life. Numerical simulation of fluid flow is useful for evaluating the hydraulic characteristics of these materials. Among various techniques, the lattice Boltzmann (LB) method is widely accepted due to the ease of implementing boundary conditions and the numerical stability in a wide variety of flow conditions. It has proven to be extremely efficient in the simulation of fluid flow through the complex geometries of granular materials. In this study, two-dimensional and three-dimensional LB models were developed to represent pore-scale monophasic Newtonian incompressible fluid flow in granular materials. Three-dimensional geometries of compacted aggregates and asphalt specimens were generated from X-ray Computed Tomography technique and used as input for the LB model. The accuracy of the models was verified by comparing the results with analytical solutions of simple geometries and hydraulic conductivity measurements on the compacted aggregates and hot mix asphalt specimens. The results of LB simulations were in excellent agreement with those obtained from analytical calculations and laboratory measurements.     

Debris flows in the Swiss National Park: the influence of different flow models and varying DEM grid size on modeling results

In the Swiss National Park, debris flows are a frequent phenomenon and have repeatedly affected highways and hiking structures. In this study, we first investigated the main characteristics and dimensions of current debris flows by field work and empirical parameterization schemes. Additionally, we evaluated a topography-based flow-trajectory geographic information system model (MSF) and a flow-routing model (FLO-2D) in terms of debris flow-affected areas. Three generically different digital elevation models (DEM) with grid spacing of 25, 4, and 1 m were used in conjunction with the flow models. The evaluation of the DEM grid spacing shows that for both flow models the 25-m DEM can give an approximate estimation of the potential hazard zone. Four- and one-meter DEMs mostly confine the simulated debris flow to existing channels and are in accordance with observations of recent debris-flow events. The study shows that DEM quality and grid resolution are crucial for the resulting delineation of potentially affected areas and thus for hazard assessment and mapping.     

A comparison of multiscale methods for elliptic problems in porous media flow

We review and perform comparison studies for three recent multiscale methods for solving elliptic problems in porous media flow; the multiscale mixed finite-element method, the numerical subgrid upscaling method, and the multiscale finite-volume method. These methods are based on a hierarchical strategy, where the global flow equations are solved on a coarsened mesh only. However, for each method, the discrete formulation of the partial differential equations on the coarse mesh is designed in a particular fashion to account for the impact of heterogeneous subgrid structures of the porous medium. The three multiscale methods produce solutions that are mass conservative on the underlying fine mesh. The methods may therefore be viewed as efficient, approximate fine-scale solvers, i.e., as an inexpensive alternative to solving the elliptic problem on the fine mesh. In addition, the methods may be utilized as an alternative to upscaling, as they generate mass-conservative solutions on the coarse mesh. We therefore choose to also compare the multiscale methods with a state-of-the-art upscaling method – the adaptive local–global upscaling method, which may be viewed as a multiscale method when coupled with a mass-conservative downscaling procedure. We investigate the properties of all four methods through a series of numerical experiments designed to reveal differences with regard to accuracy and robustness. The numerical experiments reveal particular problems with some of the methods, and these will be discussed in detail along with possible solutions. Next, we comment on implementational aspects and perform a simple analysis and comparison of the computational costs associated with each of the methods. Finally, we apply the three multiscale methods to a dynamic two-phase flow case and demonstrate that high efficiency and accurate results can be obtained when the subgrid computations are made part of a preprocessing step and not updated, or updated infrequently, throughout the simulation. The research is funded by the Research Council of Norway under grant nos. 152732 and 158908.     

Effect of macropore tortuosity and morphology on preferential flow through saturated soil: A Lattice Boltzmann study

Unlike micropores where water moves upward or downward based on hydraulic gradient, in macropores, water flows predominantly downward due to the gravity. Therefore, models based on capillary flow are not capable of simulating macropore flow. There are attempts to model the macropore flow using two domains, one for capillary flow and another one for macropores. These models use Richard’s equation for capillary flow and Poiseuille’s law for macropores in which the macropore is approximated to be cylindrical or planar. This study quantifies the magnitudes of the errors induced by this assumption. Influence of macropore shapes and tortuosity was quantified by using a 3D Lattice Boltzmann model, which is capable of simulating fluid flow in micropores as well as macropores of cracked clays. Artificial macropores of constant sectional area and volume, but different shapes were generated in 3D and the influence of macropore shapes, shape related parameters, and tortuosity were systematically investigated. Macropore flow rate decreases with different shapes compared to cylindrical macropores and increase in aspect ratio of sectional shape leads to decrease in macropore flow rate. The maximum effect of bends/turnings along the tortuous macropore was about 25% on overall decrease of flow rate due to tortuosity. However, more detailed study is required on the influence of bends on macropore flow rate. The macropore flow rate reduces by about 70% for tortuosity of 1.41. A prediction equation is verified to predict the flow rate of different shapes and tortuous macropores based on straight cylindrical macropore using aspect ratio and tortuosity factor.     

Grid adaptation for the Dirichlet–Neumann representation method and the multiscale mixed finite-element method

A Dirichlet–Neumann representation method was recently proposed for upscaling and simulating flow in reservoirs. The DNR method expresses coarse fluxes as linear functions of multiple pressure values along the boundary and at the center of each coarse block. The number of flux and pressure values at the boundary can be adjusted to improve the accuracy of simulation results and, in particular, to resolve important fine-scale details. Improvement over existing approaches is substantial especially for reservoirs that contain high-permeability streaks or channels. As an alternative, the multiscale mixed finite-element (MsMFE) method was designed to obtain fine-scale fluxes at the cost of solving a coarsened problem, but can also be used as upscaling methods that are flexible with respect to geometry and topology of the coarsened grid. Both methods can be expressed in mixed-hybrid form, with local stiffness matrices obtained as “inner products” of numerically computed basis functions with fine-scale sub-resolution. These basis functions are determined by solving local flow problems with piecewise linear Dirichlet boundary conditions for the DNR method and piecewise constant Neumann conditions for MsMFE. Adding discrete pressure points in the DNR method corresponds to subdividing faces in the coarse grid and hence increasing the number of basis functions in the MsMFE method. The methods show similar accuracy for 2D Cartesian cases, but the MsMFE method is more straightforward to formulate in 3D and implement for general grids.     

Multiscale formulation for coupled flow-heat equations arising from single-phase flow in fractured geothermal reservoirs

Efficient heat exploitation strategies from geothermal systems demand for accurate and efficient simulation of coupled flow-heat equations on large-scale heterogeneous fractured formations. While the accuracy depends on honouring high-resolution discrete fractures and rock heterogeneities, specially avoiding excessive upscaled quantities, the efficiency can be maintained if scalable model-reduction computational frameworks are developed. Addressing both aspects, this work presents a multiscale formulation for geothermal reservoirs. To this end, the nonlinear time-dependent (transient) multiscale coarse-scale system is obtained, for both pressure and temperature unknowns, based on elliptic locally solved basis functions. These basis functions account for fine-scale heterogeneity and discrete fractures, leading to accurate and efficient simulation strategies. The flow-heat coupling is treated in a sequential implicit loop, where in each stage, the multiscale stage is complemented by an ILU(0) smoother stage to guarantee convergence to any desired accuracy. Numerical results are presented in 2D to systematically analyze the multiscale approximate solutions compared with the fine scale ones for many challenging cases, including the outcrop-based geological fractured field. These results show that the developed multiscale formulation casts a promising framework for the real-field enhanced geothermal formations.     

A new computational strategy for solving two‐phase flow in strongly heterogeneous poroelastic media of evolving scales

We develop a new computational methodology for solving two‐phase flow in highly heterogeneous porous media incorporating geomechanical coupling subject to uncertainty in the poromechanical parameters. Within the framework of a staggered‐in‐time coupling algorithm, the numerical method proposed herein relies on a Petrov–Galerkin postprocessing approach projected on the Raviart–Thomas space to compute the Darcy velocity of the mixture in conjunction with a locally conservative higher order finite volume discretization of the nonlinear transport equation for the saturation and an operator splitting procedure based on the difference in the time‐scales of transport and geomechanics to compute the effects of transient porosity upon saturation. Notable features of the numerical modeling proposed herein are the local conservation properties inherited by the discrete fluxes that are crucial to correctly capture the fingering patterns arising from the interaction between heterogeneity and nonlinear viscous coupling. Water flooding in a poroelastic formation subject to an overburden is simulated with the geology characterized by multiscale self‐similar permeability and Young modulus random fields with power‐law covariance structure. Statistical moments of the poromechanical unknowns are computed within the framework of a high‐resolution Monte Carlo method. Numerical results illustrate the necessity of adopting locally conservative schemes to obtain reliable predictions of secondary recovery and finger growth in strongly heterogeneous deformable reservoirs. Copyright © 2011 John Wiley & Sons, Ltd.     

Multiscale simulations of porous media flows in flow-based coordinate system

In this paper, we propose a multiscale technique for the simulation of porous media flows in a flow-based coordinate system. A flow-based coordinate system allows us to simplify the scale interaction and derive the upscaled equations for purely hyperbolic transport equations. We discuss the applications of the method to two-phase flows in heterogeneous porous media. For two-phase flow simulations, the use of a flow-based coordinate system requires limited global information, such as the solution of single-phase flow. Numerical results show that one can achieve accurate upscaling results using a flow-based coordinate system.     

三棱柱形网格下自由表面流的三维数值模拟

为了模拟自由表面流动问题,建立了基于三棱柱形网格的三维Navier-Stokes方程的半隐式动压离散模型。出流边界采用在动量方程右端附加线性衰减项的海绵层处理方式来消减波能。引入k-ε双方程紊流模型求解涡黏性系数,使方程达到封闭。用规则周期波通过淹没障碍物的传播变形和复合明渠流动算例,验证了当遇到短波高频问题时,所述模型模拟结果与测量值吻合良好,体现了模型的精确性和有效性。     

Pressure transient analysis of multi-fractured horizontal wells in tight oil reservoirs with consideration of stress sensitivity

Multi-fractured horizontal well (MFHW) is an effective technique to develop unconventional reservoirs. Complex fracture network around the well and hydraulic fractures is formed during the fracturing process. Fracture network and hydraulic fractures are the main seepage channel which is sensitivity to the effective stress. However, most of the existing models do not take the effect of stress sensitivity into account. In this study, a new analytical model was established for MFHW in tight oil reservoirs based on the trilinear flow model. Fractal porosity and permeability were employed to describe the heterogeneous distribution of the complex fracture network, and the stress sensitivity of fracture was considered in the model. The Pedrosa substitution and perturbation method were applied to eliminate the nonlinearity of the model. By using the Laplace transformation method, the analytical solutions in Laplace domain were obtained. Then, validations were performed to show that the model is valid. Finally, sensitivity analysis was discussed. The presented model provides a new approach to the estimation of fracturing effect and can be also utilized for recognizing formation properties of tight oil reservoirs.