A finite volume / discontinuous Galerkin method for the advective Cahn–Hilliard equation with degenerate mobility on porous domains stemming from micro-CT imaging

A numerical method is formulated for the solution of the advective Cahn–Hilliard (CH) equation with constant and degenerate mobility in three-dimensional porous media with non-vanishing velocity on the exterior boundary. The CH equation describes phase separation of an immiscible binary mixture at constant temperature in the presence of a conservation constraint and dissipation of free energy. Porous media / pore-scale problems specifically entail images of rocks in which the solid matrix and pore spaces are fully resolved. The interior penalty discontinuous Galerkin method is used for the spatial discretization of the CH equation in mixed form, while a semi-implicit convex–concave splitting is utilized for temporal discretization. The spatial approximation order is arbitrary, while it reduces to a finite volume scheme for the choice of element-wise constants. The resulting nonlinear systems of equations are reduced using the Schur complement and solved via inexact Newton’s method. The numerical scheme is first validated using numerical convergence tests and then applied to a number of fundamental problems for validation and numerical experimentation purposes including the case of degenerate mobility. First-order physical applicability and robustness of the numerical method are shown in a breakthrough scenario on a voxel set obtained from a micro-CT scan of a real sandstone rock sample.     

Method of negative saturations for flow with variable number of phases in porous media: extension to three-phase multi-component case

Various EOR methods lead to the appearance of specific macroscopic surfaces called interfaces of phase transition (IPT) such that the number of phases on two sides of an IPT is different, and fluids separated by an IPT are in non-equilibrium. Therefore, the flow equations are also different on two sides of an IPT and cannot be deduced from each other by a continuous degeneration, which imposes difficulties in numerical modelling. To describe such systems, we developed a new conceptual mathematical method based on the replacement of the real single-phase fluid by an imaginary multi-phase multi-component continuum having fictitious properties. As the result, the fluid over all zones becomes multi-phase and can be described by uniform multi-phase hydro- and thermodynamic equations, which allows applying the direct numerical simulation. The equivalence principle determines the physical properties of the fictitious multi-phase fluid, as well as the structure of the uniform multi-phase equations. It also proves that the saturation of each phase becomes an extended function negative or higher than unity in non-equilibrium zones, which becomes the efficient method of tracking the interfaces, the number of phases at any point, and their degree of disequilibrium. The method was developed in [1, 2] for the two-phase case. In the present paper, the new version of the method is developed for the three-phase case with gravity, diffusion, and capillarity. We have obtained the new equivalent uniform multi-phase equations which contain additional non-classical terms responsible for the diffusion and gravity across an IPT. The comparison with classical method is presented. The presentation is illustrated by several examples of simulation by means of the code developed by the research group; their concern: EOR by miscible methods and CO ₂ bubble raising in aquifer.     

Disturbed state model for sand–geosynthetic interfaces and application to pull-out tests

Successful numerical simulation of geosynthetic-reinforced earth structures depends on selecting proper constitutive models for soils, geosynthetics and soil–geosynthetic interfaces. Many constitutive models are available for modelling soils and geosynthetics. However, constitutive models for soil–geosynthetic interfaces which can capture most of the important characteristics of interface response are not readily available. In this paper, an elasto-plastic constitutive model based on the disturbed state concept (DSC) for geosynthetic–soil interfaces has been presented. The proposed model is capable of capturing most of the important characteristics of interface response, such as dilation, hardening and softening. The behaviour of interfaces under the direct shear test has been predicted by the model. The present model has been implemented in the finite element procedure in association with the thin-layer element. Five pull-out tests with two different geogrids have been simulated numerically using FEM. For the calibration of the constitutive models used in FEM, the standard laboratory tests used are: (1) triaxial tests for the sand, (2) direct shear tests for the interfaces and (3) axial tension tests for the geogrids. The results of the finite element simulations of pull-out tests agree well with the test data. The proposed model can be used for the stress-deformation study of geosynthetic-reinforced embankments through numerical simulation. Copyright © 1999 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.     

Numerical modeling of two-phase hysteresis combined with an interface condition for heterogeneous porous media

This paper presents a numerical implementation of two-phase capillary hysteresis and its combination with a capillary interface condition for the treatment of heterogeneities. The hysteresis concepts chosen in this work are first implemented in a node-centered FV discretization scheme and subsequently combined with the interface condition that predicts sharp saturation discontinuities at material interfaces, based on a pressure equilibrium concept. This approach allows for the approximation of history-dependent, and at the same time discontinuous, saturations at material interfaces. The resulting model provides a well-defined evolution of the hysteretic capillary pressure–saturation relationships at material interfaces that is independent of the grid spacing. As demonstrated with a simple 1-D example, this concept therefore offers the advantage that the solution of a two-phase flow problem involving hysteresis does not relate to the grid resolution at the material interfaces.     

Numerical simulation of two-phase flow in conceptualized fractures

Two-phase flow in fractured rock is an important phenomenon related to a range of practical problems, including non-aqueous phase liquid contamination of groundwater. Although fractured rocks consist of fracture networks, the study of two-phase flow in a single fracture is a pre-requisite. This paper presents a conceptual and numerical model of two-phase flow in a variable fracture. The void space of the fracture is conceptualized as a system of independent channels with position-dependent apertures. Fundamental equations, governing two-phase displacement in each channel, are derived to represent the interface positions and fractional flows in the fracture. For lognormal aperture distributions, simple approximations to fractional flows are obtained in analytical form by assuming void occupancy based on a local capillary allowability criterion. The model is verified by analytical solutions including two-phase flow in a parallel-plate fracture, and used to study the impacts of aperture variation, mobility ratio and fracture orientation on properties of two-phase flow. Illustrative examples indicate that aperture variation may control the distribution of wetting and non-wetting fluids within the fracture plane and hence the ability of the fracture to transmit these fluids. The presence of wetting fluid does little to hinder non-wetting fluid flow in fractures with large aperture variations, whereas a small volume of non-wetting fluid present in the fracture can significantly reduce wetting fluid flow. Large mobility ratios and high fracture slope angles facilitates the migration of non-wetting fluid through fractures.     

Dynamic capillary effects in heterogeneous porous media

In standard multi-phase flow models on porous media, a capillary pressure saturation relationship developed under static conditions is assumed. Recent experiments have shown that this static relationship cannot explain dynamic effects as seen for example in outflow experiments. In this paper, we use a static capillary pressure model and a dynamic capillary pressure model based on the concept of Hassanizadeh and Gray and examine the behavior with respect to material interfaces. We introduce a new numerical scheme for the one-dimensional case using a Lagrange multiplier approach and develop a suitable interface condition. The behavior at the interface is discussed and verified by various numerical simulations.     

长江中游洞庭湖防洪系统水流模拟——Ⅰ．建模思路和基本算法

针对长江中游洞庭湖防洪系统规模庞大、水流复杂、资料短缺和预报时限紧迫的实际条件,提出了具有层次和模块结构特点、一维与二维水流模拟、水力学与水文学方法、理论模型与补充信息相结合的建模途径.所建模型的湖泊部分采用无结构网格二维非恒定流高性能有限体积格式,以适应湖区复杂的边界形状和保持水量平衡;河网部分采用一维非恒定流守恒型显格式,避免隐格式矩阵算法的复杂性,同时有利于与二维模型的耦合及与各种复杂连通关系的显式连接.这种一、二维混合非恒定流模型可用于长江干流、洞庭湖河网及湖泊、堤垸区的洪水演进和防洪调度的水流仿真.     

Node-centered finite volume discretizations for the numerical simulation of multiphase flow in heterogeneous porous media

Three node-centered finite volume discretizations for multiphase porous media flow are presented and compared. By combination of these methods two additional discretization methods are generated. The ability of these schemes to describe flows at textural interfaces of different geologic formations is investigated. It was found that models with nonzero-entry pressures for the capillary pressure-saturation relationship in conjunction with the Box discretization may give rise to spurious oscillations for flows around low permeable lenses. Furthermore, the applicability and sensitivity of the discretization methods with regard to the used computational grids is discussed. The schemes are used for the numerical study of two-phase flow in porous media with zones of different material properties. This revised version was published online in July 2006 with corrections to the Cover Date.     

Time‐scale analysis in unsaturated porous media under external wave loads

Extreme waves caused by tsunamis and storm surges can lead to soil failures in the near‐shore region, which may have severe impact on coastal environments and communities. Multiphase flows in deformable porous media involve several coupled processes and multiple time scales, which are challenging for numerical simulations. The objective of this study is to investigate the roles of the various processes and their interactions in multiphase flows in unsaturated soils under external wave loading, via theoretical time‐scale analysis and numerical simulations. A coupled geomechanics–multiphase flow model based on conservation laws is used. Theoretical analysis based on coupled and decoupled models demonstrates that transient and steady‐state responses are governed by pore pressure diffusion and saturation front propagation, respectively, and that the two processes are essentially decoupled. Numerical simulations suggest that the compressibility of the pore fluids and the deformation of the soil skeleton are important when the transient responses of the media are of concern, while the steady‐state responses are not sensitive to these factors. The responses obtained from the fully coupled numerical simulations are explained by a simplified time‐scale analysis based on coupled and decoupled models. Copyright © 2010 John Wiley & Sons, Ltd.     

Crossover from Nonlinearity Controlled to Heterogeneity Controlled Mixing in Two-Phase Porous Media Flows

We use high resolution Monte Carlo simulations to study the dispersive mixing in two-phase, immiscible, porous media flow that results from the interaction of the nonlinearities in the flow equations with geologic heterogeneity. Our numerical experiments show that distinct dispersive regimes occur depending on the relative strength of nonlinearity and heterogeneity. In particular, for a given degree of multiscale heterogeneity, controlled by the Hurst exponent which characterizes the underlying stochastic model for the heterogeneity, linear and nonlinear flows are essentially identical in their degree of dispersion, if the heterogeneity is strong enough. As the heterogeneity weakens, the dispersion rates cross over from those of linear heterogeneous flows to those typical of nonlinear homogeneous flows.     

Taxila LBM: a parallel,modular lattice Boltzmann framework for simulating pore-scale flow in porous media

The lattice Boltzmann method is a popular tool for pore-scale simulation of flow. This is likely due to the ease of including complex geometries such as porous media and representing multiphase and multifluid flows. Many advancements, including multiple relaxation times, increased isotropy, and others have improved the accuracy and physical fidelity of the method. Additionally, the lattice Bolzmann method is computationally very efficient, thanks to the explicit nature of the algorithm and relatively large amount of local work. The combination of many algorithmic options and efficiency means that a software framework enabling the usage and comparison of these advancements on computers from laptops to large clusters has much to offer. In this paper, we introduce Taxila LBM, an open-source software framework for lattice Boltzmann simulations. We discuss the design of the framework and lay out the features available, including both methods in the literature and a few new enhancements which generalize methods to complex geometries. We discuss the trade-off of accuracy and performance in various methods, noting how the Taxila LBM makes it easy to perform these comparisons for real problems. And finally, we demonstrate a few common applications in pore-scale simulation, including the characterization of permeability of a Berea sandstone and analysis of multifluid flow in heterogenous micromodels.     

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.     

Mortar coupling and upscaling of pore-scale models

Pore-scale models are becoming increasingly useful as predictive tools for modeling flow and transport in porous media. These models can accurately represent the 3D pore-structure of real media. Currently first-principles modeling methods are being employed for obtaining qualitative and quantitative behavior. Generally, artificial, simple boundary conditions are imposed on a model that is used as a stand-alone tool for extracting macroscopic parameters. However, realistic boundary conditions, reflecting flow and transport in surrounding media, may be necessary for behavior that occurs over larger length scales or including pore-scale models in a multiscale setting. Here, pore-scale network models are coupled to adjacent media (additional pore-scale or continuum-scale models) using mortars. Mortars are 2D finite-element spaces employed to couple independent subdomains by enforcing continuity of pressure and flux at shared boundary interfaces. While mortars have been used in the past to couple subdomains of different models, physics, and meshes, they are extended here for the first time to pore-scale models. The approach is demonstrated by modeling single-phase flow in coupled pore-scale models, but the methodology can be utilized to model dynamic processes and perform multiscale modeling in 3D continuum simulators for flow and transport.     

Impact of single inclined faults on the fluid flow and heat transport: results from 3-D finite element simulations

The impact of inclined faults on the hydrothermal field is assessed by adding simplified structural settings to synthetic models. This study is innovative in carrying out numerical simulations because it integrates the real 3-D nature of flow influenced by a fault in a porous medium, thereby providing a useful tool for complex geothermal modelling. The 3-D simulations for the coupled fluid flow and heat transport processes are based on the finite element method. In the model, one geological layer is dissected by a dipping fault. Sensitivity analyses are conducted to quantify the effects of the fault’s transmissivity on the fluid flow and thermal field. Different fault models are compared with a model where no fault is present to evaluate the effect of varying fault transmissivity. The results show that faults have a significant impact on the hydrothermal field. Varying either the fault zone width or the fault permeability will result in relevant differences in the pressure, velocity and temperature field. A linear relationship between fault zone width and fluid velocity is found, indicating that velocities increase with decreasing widths. The faults act as preferential pathways for advective heat transport in case of highly transmissive faults, whereas almost no fluid may be transported through poorly transmissive faults.