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.     

High order discontinuous Galerkin method for simulating miscible flooding in porous media

We present a high-order method for miscible displacement simulation in porous media. The method is based on discontinuous Galerkin discretization with weighted average stabilization technique and flux reconstruction post processing. The mathematical model is decoupled and solved sequentially. We apply domain decomposition and algebraic multigrid preconditioner for the linear system resulting from the high-order discretization. The accuracy and robustness of the method are demonstrated in the convergence study with analytical solutions and heterogeneous porous media, respectively. We also investigate the effect of grid orientation and anisotropic permeability using high-order discontinuous Galerkin method in contrast with cell-centered finite volume method. The study of the parallel implementation shows the scalability and efficiency of the method on parallel architecture. We also verify the simulation result on highly heterogeneous permeability field from the SPE10 model.     

Relative permeability of CBM reservoirs: Controls on curve shape

Relative permeability to gas and water for 2-phase flow coalbed methane (CBM) reservoirs has long been known to exhibit a strong control on (gas and water) production profile characteristics. Despite its important control on both primary and enhanced recovery of CBM for coal seams that have not been fully dewatered, relative permeability in coal has received little attention in the literature in the past decade. There are few published laboratory-derived curves; these studies and their resulting data represent a small subset of the commercial CBM reservoirs and do not allow for a systematic investigation of the physical controls on relative permeability curve shape. Other methods for estimation of relative permeability curves include derivation from simulation history-matching, and production data analysis. Both of these methods will yield pseudo-relative permeability curves whose shapes could be affected by several dynamic CBM reservoir and operating characteristics.The purpose of the current work is to perform a systematic investigation of the controls on CBM relative permeability curve shape, including non-static fracture permeability and porosity, multi-layer effects and transient flow. To derive the relative permeability curves, effective permeability to gas and water are obtained from flow equations, flow rates and pressure data. Simulated cases are analyzed so that derived and input curves may be compared allowing for investigation of CBM reservoir properties on curve shape. One set of relative permeability curves that were input into the simulator were obtained from pore-scale modeling. Field cases from two basins are also examined and controls on derived relative permeability curve shape inferred. The results of this work should be useful for future CBM development and greenhouse gas sequestration studies, and it is hoped that it will spark additional research of this critical CBM flow property.     

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.     

An Approach for Modeling Rock Discontinuous Mechanical Behavior Under Multiphase Fluid Flow Conditions

In this paper, the two computer codes TOUGH2 and RDCA (for “rock discontinuous cellular automaton”) are integrated for coupled hydromechanical analysis of multiphase fluid flow and discontinuous mechanical behavior in heterogeneous rock. TOUGH2 is a well-established code for geohydrological analysis involving multiphase, multicomponent fluid flow and heat transport; RDCA is a numerical model developed for simulating the nonlinear and discontinuous geomechanical behavior of rock. The RDCA incorporates the discontinuity of a fracture independently of the mesh, such that the fracture can be arbitrarily located within an element, while the fluid pressure calculated by TOUGH2 can be conveniently applied to fracture surfaces. We verify and demonstrate the coupled TOUGH–RDCA simulator by modeling a number of simulation examples related to coupled multiphase flow and geomechanical processes associated with the deep geological storage of carbon dioxide—including modeling of ground surface uplift, stress-dependent permeability, and the coupled multiphase flow and geomechanical behavior of fractures intersecting the caprock.     

Fast 3D Reservoir Simulation and Scale Up Using Streamtubes

This paper presents an implementation of a semianalytical method for oil recovery calculation in heterogeneous reservoirs that is both fast and accurate. The method defines streamline paths based on a conventional single-phase incompressible flow calculation. By calculating the time-of-flight for a particle along a streamline and assigning a volumetric flux to each streamline, the cumulative pore volume of a streamtube containing the streamline can be calculated. Subsequently, the streamtube geometries are kept constant and the effects of the time varying mobility distribution in two-phase flow are accounted for by varying the flow rate in each streamtube, based on fluid resistance changes along the streamtube. Oil recovery calculations are then done based on the 1D analytical Buckley–Leverett solution. This concept makes the method extremely fast and easy to implement, making it ideal to simulate large reservoirs generated by geostatiscal methods. The simulation results of a 3D heterogeneous reservoir are presented and compared with those of other simulators. The results shows that the new simulator is much faster than a traditional finite difference simulator, while having the same accuracy. The method also naturally handles the upscaling of absolute and relative permeability. We make use of these upscaling abilities to generate a coarse curvilinear grid that can be used in conventional simulators with a great advantage over conventional upscaled Cartesian grids. This paper also shows an upscaling example using this technique.     

A phase-field method for the direct simulation of two-phase flows in pore-scale media using a non-equilibrium wetting boundary condition

Advances in pore-scale imaging (e.g., μ-CT scanning), increasing availability of computational resources, and recent developments in numerical algorithms have started rendering direct pore-scale numerical simulations of multi-phase flow on pore structures feasible. Quasi-static methods, where the viscous and the capillary limit are iterated sequentially, fall short in rigorously capturing crucial flow phenomena at the pore scale. Direct simulation techniques are needed that account for the full coupling between capillary and viscous flow phenomena. Consequently, there is a strong demand for robust and effective numerical methods that can deliver high-accuracy, high-resolution solutions of pore-scale flow in a computationally efficient manner. Direct simulations of pore-scale flow on imaged volumes can yield important insights about physical phenomena taking place during multi-phase, multi-component displacements. Such simulations can be utilized for optimizing various enhanced oil recovery (EOR) schemes and permit the computation of effective properties for Darcy-scale multi-phase flows.We implement a phase-field model for the direct pore-scale simulation of incompressible flow of two immiscible fluids. The model naturally lends itself to the transport of fluids with large density and viscosity ratios. In the phase-field approach, the fluid-phase interfaces are expressed in terms of thin transition regions, the so-called diffuse interfaces, for increased computational efficiency. The conservation law of mass for binary mixtures leads to the advective Cahn–Hilliard equation and the condition that the velocity field is divergence free. Momentum balance, on the other hand, leads to the Navier–Stokes equations for Newtonian fluids modified for two-phase flow and coupled to the advective Cahn–Hilliard equation. Unlike the volume of fluid (VoF) and level-set methods, which rely on regularization techniques to describe the phase interfaces, the phase-field method facilitates a thermodynamic treatment of the phase interfaces, rendering it more physically consistent for the direct simulations of two-phase pore-scale flow. A novel geometric wetting (wall) boundary condition is implemented as part of the phase-field method for the simulation of two-fluid flows with moving contact lines. The geometric boundary condition accurately replicates the prescribed equilibrium contact angle and is extended to account for dynamic (non-equilibrium) effects. The coupled advective Cahn–Hilliard and modified Navier–Stokes (phase-field) system is solved by using a robust and accurate semi-implicit finite volume method. An extension of the momentum balance equations is also implemented for Herschel–Bulkley (non-Newtonian) fluids. Non-equilibrium-induced two-phase flow problems and dynamic two-phase flows in simple two-dimensional (2-D) and three-dimensional (3-D) geometries are investigated to validate the model and its numerical implementation. Quantitative comparisons are made for cases with analytical solutions. Two-phase flow in an idealized 2-D pore-scale conduit is simulated to demonstrate the viability of the proposed direct numerical simulation approach.     

Upscaling of Navier–Stokes equations in porous media: Theoretical, numerical and experimental approach

The accurate estimation of hydraulic conductivity is important for many geotechnical engineering applications, as the presence of fluids affects all aspects of soil behaviour, including its strength. Darcy’s law is the key experimental (or phenomenological) equation employed to model ground water flow. Yet, this phenomenological equation can be linked to a more fundamental microscale model of flow through the pore spaces of the porous material. This paper provides an experimental verification of the relationships between Darcy’s law (macroscale) and the Navier–Stokes equations (microscale) for actual complex pore geometries of a granular material. The pore geometries are experimentally obtained through state-of-the-art X-ray computer assisted micro-tomography. From the numerical modelling of the microscale flow based on actual pore geometries, it is possible to quantify and visualize the development of pore-scale fluid preferential flow-paths through the porous material, and to assess the importance of pore connectivity in soil transport properties.     

Permeability Tensors of Anisotropic Fracture Networks

Analytical models are presented to provide enhanced capabilities for modeling fluid flow through natural fractures nested in parallel plate type configurations. The modeled fractures may be arbitrarily positioned, but subgrouped according to the consistent parallel sequences. The derived analytical expressions for fracture permeability can be considered as an extension to those in which flow within fractures is uniform and isotropic. This modification offers a correction for the traditional permeability calculations whenever fractures are oblique to the flow orientation. For the fracture flow scenarios, the graphical solutions show the permeability envelope normal to any arbitrary planes within the calculated domain. Consideration of rock anisotropy may significantly improve the accuracy in determining the formation permeability in cases where natural fractures exhibit a dominant control in regional fluid flow.     

A parallel ensemble-based framework for reservoir history matching and uncertainty characterization

We present a parallel framework for history matching and uncertainty characterization based on the Kalman filter update equation for the application of reservoir simulation. The main advantages of ensemble-based data assimilation methods are that they can handle large-scale numerical models with a high degree of nonlinearity and large amount of data, making them perfectly suited for coupling with a reservoir simulator. However, the sequential implementation is computationally expensive as the methods require relatively high number of reservoir simulation runs. Therefore, the main focus of this work is to develop a parallel data assimilation framework with minimum changes into the reservoir simulator source code. In this framework, multiple concurrent realizations are computed on several partitions of a parallel machine. These realizations are further subdivided among different processors, and communication is performed at data assimilation times. Although this parallel framework is general and can be used for different ensemble techniques, we discuss the methodology and compare results of two algorithms, the ensemble Kalman filter (EnKF) and the ensemble smoother (ES). Computational results show that the absolute runtime is greatly reduced using a parallel implementation versus a serial one. In particular, a parallel efficiency of about 35 % is obtained for the EnKF, and an efficiency of more than 50 % is obtained for the ES.     

Numerical modeling of stress-permeability coupling in rough fractures

A numerical model is described for coupled flow and mechanical deformation in fractured rock. The mechanical response of rock joints to changes in hydraulic pressure is strongly influenced by the geometric characteristics of the joint surfaces. The concept of this work is to combine straightforward finite element solutions with complex and realistic fracture surface geometry in order to reproduce the non-linear stress-deformation-permeability coupling that is commonly observed in fractures. Building on the numerous studies that have expanded the understanding of the key parameters needed to describe natural rough-walled fractures, new methods have been developed to generate a finite element mesh representing discrete fractures with realistic rough surface geometries embedded in a rock matrix. The finite element code GeoSys/Rockflow was then used to simulate the coupled effects of hydraulic stress, mechanical stress, and surface geometry on the evolving permeability of a single discrete fracture. The modeling concept was experimentally verified against examples from the literature. Modeling results were also compared to a simple interpenetration model.     

Investigation of thermal-hydro-mechanical coupled fracture propagation considering rock damage

Thermal-hydro-mechanical (THM) coupled fracture propagation is common in underground engineering. Rock damage, as an inherent property of rock, significantly affects fracture propagation, but how it influences the THM coupled fracturing remains stubbornly unclear. A pore-scale THM coupling model is developed to study this problem, which combines the lattice Boltzmann method (LBM), the discrete element method (DEM), and rock damage development theory together for the first time. This model can more accurately calculate the exchanged THM information at the fluid-solid boundary and fluid conductivity dependent on fracture and rock damage. Based on the developed model, the synergistic effect of injected temperature difference (fluid temperature below rock temperature) and rock damage (characterized by the parameter “critical fracture energy”, abbreviated as “CFE”) on fracture propagation of shale are investigated particularly. It is found that: (1) the generation of branched cracks is closely related to the temperature response frontier, and the fracture process zone of single bond failure increases in higher CFE. (2) through the analysis of micro failure events, hydraulic fracturing is more pronounced in the low CFE, while thermal fracturing displays the opposite trend. The fluid conductivity of fractured rock increases with a higher injected temperature difference due to the more penetrated cracks and wider fracture aperture. However, this enhancement weakens when rock damage is significant. (3) in the multiple-layered rock with various CFEs, branched cracks propagating to adjacent layers are more difficult to form when the injection hole stays in the layer with significant rock damage than without rock damage.

     

Review: Mathematical expressions for estimating equivalent permeability of rock fracture networks

Fracture networks play a more significant role in conducting fluid flow and solute transport in fractured rock masses, comparing with that of the rock matrix. Accurate estimation of the permeability of fracture networks would help researchers and engineers better assess the performance of projects associated with fluid flow in fractured rock masses. This study provides a review of previous works that have focused on the estimation of equivalent permeability of two-dimensional (2-D) discrete fracture networks (DFNs) considering the influences of geometric properties of fractured rock masses. Mathematical expressions for the effects of nine important parameters that significantly impact on the equivalent permeability of DFNs are summarized, including (1) fracture-length distribution, (2) aperture distribution, (3) fracture surface roughness, (4) fracture dead-end, (5) number of intersections, (6) hydraulic gradient, (7) boundary stress, (8) anisotropy, and (9) scale. Recent developments of 3-D fracture networks are briefly reviewed to underline the importance of utilizing 3-D models in future research.     

Significance of shape factor on permeability anisotropy of sand: representative elementary volume study for pore-scale analysis

Acta Geotechnica - This paper is a study of determination of representative elementary volume (REV) size suitable for pore-scale flow simulation (PFS) and evaluation of permeability anisotropy for...     

一种基于地质指标的裂隙岩体渗透系数估算模型

裂隙岩体的渗透特性受控于裂隙的发育特征、连通特性和充填情况,并与岩体的地应力水平具有显著的相关性。基于裂隙岩体渗透性的影响因素,并考虑现有渗透系数估算模型的不足,利用钻孔压水试验和钻孔电视图像资料,建立考虑埋深(Z)、岩石质量指标(RQD)以及充填物指标(FSD)等3个指标的渗透系数估算ZRF模型,并应用到牙根二级水电站及其他工程区的渗透系数估算中。结果表明,与已有的渗透系数估算模型相比,ZRF模型较好地反映了岩体渗透性的影响因素,且模型参数物理意义明确,便于获取,对分析裂隙岩体渗透性具有一定的工程参考价值。     

Smooth particle hydrodynamics simulations of low Reynolds number flows through porous media

In this paper, a three‐dimensional smooth particle hydrodynamics (SPH) simulator for modeling grain scale fluid flow in porous media is presented. The versatility of the SPH method has driven its use in increasingly complex areas of flow analysis, including the characterization of flow through permeable rock for both groundwater and petroleum reservoir research. SPH provides the means to model complex multi‐phase flows through such media; however, acceptance of the methodology has been hampered by the apparent lack of actual verification within the literature, particulary in the three‐dimensional case. In this paper, the accuracy of SPH is addressed via a comparison to the previously recognized benchmarks of authors such as Sangani and Acrivos (Int. J. Multiphase Flow 1982; 8 (4): 343–360), Zick and Homsy (J. Fluid Mech. 1982; 115 :13–26) and Larson and Higdon (Phys. Fluids A 1989; 1 (1):38–46) for the well‐defined classical problems of flow through idealized two‐ and three‐dimensional porous media. The accuracy of results for such low Reynolds number flows is highly dependent on the implementation of no‐slip boundary conditions. A new, robust and numerically efficient, method for implementing such boundaries in SPH is presented. Simulation results for friction coefficient and permeability are shown to agree well with the available benchmarks. Copyright © 2010 John Wiley & Sons, Ltd.     

Hybrid three-scale model for evolving pore-scale geometries

We consider flow and upscaling of flow properties from pore scale to Darcy scale, when the pore-scale geometry is changing. The idea is to avoid having to solve for the pore evolution at the pore scale, because this results in unmanageable complexity. We propose to use stochastic modeling to parametrize plausible modifications of the pore geometry and to construct distributions of permeability parametrized by Darcy-scale variables. To localize the effects of, e.g., clogging, we introduce an intermediate scale of pore-network models. We use local Stokes solvers to calibrate the throat permeability.     

GSHR-Tree:一种基于动态空间槽和哈希表的网格环境下的空间索引树

为提高网格环境下海量空间数据管理与并行化处理效率,将网格环境下的分布并行处理技术与空间索引相融合,提出了一种空间索引框架(grid slot and hash R tree,GSHR-Tree).该索引树结构基于散列hash表和动态空间槽,结合R树结构的范围查询优势和哈希表结构的高效单key查询,分析改进了索引结构的组织和存储.构造了适合于大规模空间数据的网格并行空间计算的索引结构,该索引树算法根据空间数据划分策略,动态分割空间槽,并将它们映射到多个节点机上.每个节点机再将其对应空间槽中的空间对象组织成R树,以大节点R树方式在多个节点上分布索引数据.以空间范围查询并行处理的系统响应时间为性能评估指标,通过模拟实验证明,该GSHR-Tree索引满足了当前网格环境空间索引的需要,并具有设计合理、性能高效的特点.      

A level-set corrector to an adaptive multiscale permeability prediction

We consider the inverse problem of permeability estimation for two-phase porous-media flow. The novel approach is based on regularization by zonation, where the geometry and size of the regions are chosen adaptively during the optimization procedure. To achieve this, we have utilized level-set functions to represent the permeability. The available data are sparsely distributed in space; hence, it is reasonable to confine the estimation to coarse-scale structures. The level-set approach is able to alter the boundaries between regions of different permeability without strict restrictions on their shape; however, when the data are sparse, a reasonable initial guess for the permeability is required. For this task, we use adaptive multiscale permeability estimation, which has the potential of identifying main permeability variations. These are described by a piecewise constant function, where the constant values are attained on rectangular zones. In the current work, we develop a level-set corrector strategy, assuming adaptive multiscale permeability estimation as a predictor.     

Simulation of CO<Subscript>2</Subscript> hydrate formation in cold aquifers: nonequilibrium approach

In this paper, we focus on the geological storage of CO₂ in reservoirs with zones that are cold enough to facilitate CO₂ hydrate formation at local pressures. A 2D hydro-chemical mechanical model which has five layers (three layers with aquifers and two layers with cap rock in which we introduced two fractures) is created. We apply a reactive transport reservoir simulator, RetrasoCodeBright (RCB), in which hydrate is treated as a pseudo mineral. Following the recent modifications to account for hydrate dynamics in the code through a kinetic approach (Kvamme et al., Proceedings of the 7th International Conference on Gas Hydrates (ICGH 2011), 2011b), we have further improved the simulator to implement the nonequilibrium thermodynamic calculations. In the present study, we spot the light on the hydrate formation effects on porosity in different regions, as well as on the flow pattern. These simulations are based on classical relationships between porosity and permeability, but the outline of ongoing modifications is presented as well. A critical question in such systems is whether hydrate formation can contribute to stabilizing the storage, given that hydrates are pore filling and cannot be stable toward mineral surfaces. The implications of hydrate formation on the geo-mechanical properties of the model reservoir are other aspects addressed in this study.