Equivalent Continuum Research Articles

Numerical simulation of cold and hot water injection into naturally fractured porous media using the extended–FEM and an equivalent continuum model

AbstractIn this paper, a computational technique is presented for the isothermal and non‐isothermal water injection into naturally fractured oil reservoirs. A remarkable number of naturally fractured reservoirs contain relatively heavy oils that could not be extracted economically; hence, the thermal recovery methods are extensively used for such reservoirs. In this study, the effectiveness of hot water injection over cold (isothermal) water injection in oil production is quantified. The influence of long and short fractures and their alignments on oil recovery are discussed. To this end, a 2D model for two‐phase fluid flow and heat transfer is presented. The medium is assumed to be partially saturated with water as the wetting phase and oil as the non‐wetting phase. The governing equations consist of the energy balance equation together with the continuity equations of both fluid phases. To take the long (macro) fractures into account, the extended finite element method is employed by considering permeable fracture faces; accordingly, the mass and heat transfer between the fracture and the matrix are captured accurately. In addition, the network of short (micro) fractures is incorporated by an equivalent continuum model. Several numerical examples are presented to demonstrate the influence of injection temperature, initial reservoir temperature, and the long and short fractures on oil production. Furthermore, the simultaneous effect of the macro‐ and micro‐fractures along with the injection temperature are examined. The results show that the proposed computational model is a promising tool that can be used in the cold or hot water injection, and can be employed in the economic feasibility studies of recovery methods in the future.

Incorporating defects into model predictions of metal lattice-structured materials

Metal lattice structures are optimized for high specific strength properties and are now customizable using additive manufacturing methods. However, many of these methods, like selective laser melting, can introduce defects such as porosity, parasitic material, and disconnected struts into the structure, which can negatively affect mechanical behavior. While there are many computational models used to predict the mechanical response of lattice-structured materials, most use an idealized structure and are often not robust enough to account for defects. In this study, metal lattice structures are characterized for defects and mechanically tested in compression. The defect types and distributions are characterized using x-ray micro-tomography and the tomography analyses is used in two different model predictions. First, in an equivalent continuum model, where the defects are used to predict the variability in mechanical properties, and second, in a finite element analysis, where the predicted stress-strain response for both realistic and idealized structures are compared. Investigating this further, a finite element analysis of an octet lattice quantifies the reduction in strength associated with disconnected struts and captures a dependence on the strut's orientation relative to the loading direction. Overall, incorporating defect information gleaned from tomography data improves predictions of mechanical properties by capturing a more realistic deformation response for lattice-structured material.

Determination of representative elementary volume (REV) for jointed rock masses exhibiting scale-dependent behavior: a numerical investigation

Representative elementary volume (REV) is defined as the usual size of a rock mass structure beyond which its mechanical properties are homogenous and isotropic, and its behavior can be modeled using the equivalent continuum approach. Determination of REV is a complex problem in rock engineering due to its definition ambiguity and application area. This study is one of the first attempts to define a REV for jointed rock masses using the equivalent continuum approach. It is aimed to numerically search a ratio between the characteristic size of an engineering structure and pre-existing joint spacing, which are the two most important contributing elements in assessing REV. For this purpose, four hypothetical engineering cases were investigated using the RS2 (Phase2 v. 9.0) finite element (FE) analysis program. An underground circular opening with a constant diameter, an open-pit mine with varying bench heights, a single bench with a constant height, and an underground powerhouse cavern with a known dimension were executed for possible changes in the safety factor and total displacement measurements under several joint spacing values. Different cut-off REVs were calculated for FE models depending on the type of excavation and measurement method. An average REV size of 19.0, ranging between a minimum of 2 for tunnels and a maximum of 48 for slopes, was found in numerical analysis. The calculated sizes of REV were significantly larger than the range of values (5 to 10) commonly reported in the relevant geotechnical literature.

Numerical Assessment of the Hybrid Approach for Simulating Three-Dimensional Flow and Advective Transport in Fractured Rocks

This study presents a hybrid approach for simulating flow and advective transport dynamics in fractured rocks. The developed hybrid domain (HD) model uses the two-dimensional (2D) triangular mesh for fractures and tetrahedral mesh for the three-dimensional (3D) rock matrix in a simulation domain and allows the system of equations to be solved simultaneously. This study also illustrates the HD model with two numerical cases that focus on the flow and advective transport between the fractures and rock matrix. The quantitative assessments are conducted by comparing the HD results with those obtained from the discrete fracture network (DFN) and equivalent continuum porous medium (ECPM) models. Results show that the HD model reproduces the head solutions obtained from the ECPM model in the simulation domain and heads from the DFN model in the fractures in the first case. The particle tracking results show that the mean particle velocity in the HD model can be 7.62 times higher than that obtained from the ECPM mode. In addition, the developed HD model enables detailed calculations of the fluxes at intersections between fractures and cylinder objects in the case and obtains relatively accurate flux along the intersections. The solutions are the key factors to evaluate the sources of contaminant released from the disposal facility.

Equivalent fracture network model for steady seepage problems with free surfaces

Based on the back analysis of equivalent continuum theory in frcatued media by Snow (1969), an equivalent fracture network model is proposed to evaluate steady seepage problems with free surface in porous media. In contrast to the continuum-based approaches on seepage flow, the continuum is replaced by an artificial fracture network system formed by two sets of fractures. The relationships between the permeability coefficients of porous media and equivalent parameters of fracture dip, spacing and aperture are deduced. The corresponding governing equations and finite element procedure according to the variational inequality theory are developed. Through good agreements of free surface locations between the proposed method, numerical data and field measurement in four typical examples, the proposed equivalent fracture network model is validated in modelling free surface flow in porous media. The solutions on the isotropic and anisotropic rectangular dams show that the proposed model has weak sensitivity of the fracture dip and mesh size, and its equivalence to the continuum-based method is also proved. In addition, the proposed model can be successfully used for the fractured media as well as porous media, the fracture direction has a significant impact on the free surface flow behavior in the fractured media, in which the fracture directions are negative for the antidip slope and positive for the dip slope to drain ground water. Finally, the results from the zoned dam with clay core and sand shell and the left bank abutment demonstrate the feasibility of the proposed model in complex dam engineerings with isotropic and anisotropic heterogeneity media.

Survey on Equivalent Continuum Modeling for Truss Structures and Their Nonlinear Dynamics and Vibration Control

Survey on Equivalent Continuum Modeling for Truss Structures and Their Nonlinear Dynamics and Vibration Control

Investigating the effects of intersection flow localization in equivalent-continuum-based upscaling of flow in discrete fracture networks

Abstract. Predicting effective permeabilities of fractured rock masses is a crucial component of reservoir modeling. Its often realized with the discrete fracture network (DFN) method, whereby single-phase incompressible fluid flow is modeled in discrete representations of individual fractures in a network. Depending on the overall number of fractures, this can result in high computational costs. Equivalent continuum models (ECMs) provide an alternative approach by subdividing the fracture network into a grid of continuous medium cells, over which hydraulic properties are averaged for fluid flow simulations. While continuum methods have the advantage of lower computational costs and the possibility of including matrix properties, choosing the right cell size to discretize the fracture network into an ECM is crucial to provide accurate flow results and conserve anisotropic flow properties. Whereas several techniques exist to map a fracture network onto a grid of continuum cells, the complexity related to flow in fracture intersections is often ignored. Here, numerical simulations of Stokes flow in simple fracture intersections are utilized to analyze their effect on permeability. It is demonstrated that intersection lineaments oriented parallel to the principal direction of flow increase permeability in a process we term intersection flow localization (IFL). We propose a new method to generate ECMs that includes this effect with a directional pipe flow parameterization: the fracture-and-pipe model. Our approach is compared against an ECM method that does not take IFL into account by performing ECM-based upscaling with a massively parallelized Darcy flow solver capable of representing permeability anisotropy for individual grid cells. While IFL results in an increase in permeability at the local scale of the ECM cell (fracture scale), its effects on network-scale flow are minor. We investigated the effects of IFL for test cases with orthogonal fracture formations for various scales, fracture lengths, hydraulic apertures, and fracture densities. Only for global fracture porosities above 30 % does IFL start to increase the systems permeability. For lower fracture densities, the effects of IFL are smeared out in the upscaling process. However, we noticed a strong dependency of ECM-based upscaling on its grid resolution. Resolution tests suggests that, as long as the cell size is smaller than the minimal fracture length and larger than the maximal hydraulic aperture of the considered fracture network, the resulting effective permeabilities and anisotropies are resolution-independent. Within that range, ECMs are applicable to upscale flow in fracture networks.

Goaf and Slope Stability: A Case Study on the Yangla Copper Mine

Considering the wide distribution range of deposits, the complicated geological conditions, and the potential detriment to slope stability during the caving period, underground goaf and slope stability is a critical concern in the Yangla copper mine. In this paper, goaf and slope stability is studied by implementing empirical analysis and numerical simulations with an equivalent continuum model. An overall understanding of underground goaf stability is obtained by the Laubscher graph method, and an approach composed of the geological modeling technology GoCAD and the specialized meshing tool Griddle is proposed to manage the geological database and prepare the numerical model for stability analysis with FLAC 3D . The Hoek–Brown constitutive model is introduced to simulate the complicated mechanical behavior of rock mass in the process of mining, i.e., the brittleness and confinement effect of rock mass. The key orebodies KT2 and KT5 develop in reverse inclination from slope surface to the inner slope with a maximum depth of 900 m, which means the rock mass bears a high stress induced by mining. The mechanical parameters are calibrated carefully by performing mining simulations that are consistent with the field instability phenomenon and measurement results. In particular, rock-mass behavior is predicted by sensitivity modeling in consideration of the goaf span, mining sequence, and material filling, the aim of which is to optimize the mining design to maintain slope stability during the third mining phase. The calculation results indicate that large deformations at the roof of the goaf and pillar are a critical problem in the Yangla copper mine as they are detrimental to slope stability. Moreover, many goafs collapse during the second and third phases of underground mining, which is also detrimental to slope stability, and the span of the goaf with a deep buried depth needs to be adjusted from 36 to 24 m in the third mining phase. The slope is stable in the second mining phase, but, in the third phase, it enters the large-scale instability state due to the destruction of the rock-mass structure near the slope surface by underground caving. Accordingly, goafs with the potential for collapse should be filled with high-strength materials to maintain slope stability.

Equivalent Shell Model of Elastic Gridshells Including the Effect of the Geometric Curvature

In this work, an equivalent continuum of a barrel gridshell is introduced. Constitutive identification procedures based on periodic homogenization are provided in the literature for this purpose, based on a flat Representative Element Volume (REV), notwithstanding that the geometry of the structures concerned is curved. Therefore, the novelty of the present study is the selection of a curved REV to obtain the equivalent elastic constants. The numerical validation of the identification procedure is made comparing gridshell response to that of the equivalent shell under homogeneous load conditions. Finally, in order to highlight the effect of the curved geometry on the constitutive law of the continuum, the response of the proposed model is also compared to that of a continuum obtained from a flat REV.

The influence of carbon nanotube coated-carbon fibers on thermal residual stresses of Multi-Scale hybrid composites: Analytical approach

The effect of carbon nanotube (CNT) coated-carbon fibers on thermal residual stresses of multi-scale hybrid composites is assessed employing analytical approach. The model comprises carbon fiber, coating region and surrounding matrix, in which the coating region around core fiber encompasses CNTs and matrix. Considering three configurations of grown CNTs on the fiber surface including axially, radially and randomly oriented, the mechanical properties of various coating regions are acquired employing the Eshelby–Mori–Tanaka method in conjunction with an equivalent continuum approach. Utilizing the total energy minimization method, the closed-form solution of the thermal residual stresses of hybrid composite is obtained. The results disclose a noteworthy influence of CNT–coating on the reduction of interfacial stresses which precludes debonding at interface and attenuates the effect of thermal expansions mismatch between the carbon fiber and matrix. Furthermore, the results demonstrate that unlike radially oriented CNTs, the existence of axially and randomly oriented CNTs at the coating region has a remarkable diminishing effect on residual interfacial stresses. It is also shown that increasing the coating thickness leads to reduction of maximum interfacial stresses even at a constant CNT volume fraction. A close agreement exists between predicted outcomes by the proposed analytical approach and published data in the literature.

Equivalent continuum method for anisogrid composite lattice conical shells with equiangular, equidistant and geodesic spiral ribs

This paper is concerned with filament winding trajectories of helical ribs of anisogrid composite lattice conical shells. Filament winding can be designed to coincide with equiangular, equidistant and geodesic spiral trajectories. The unified geometrical formulas of the three types of conical spiral and circumferential curves are established. As a consequence, three types of lattice conical shells are built and the corresponding geometric parameters of lattice conical shells are obtained. Additionally, the smearing method, adopting straight-line and curved-line assumptions for helical ribs, respectively, is applied to calculate the elastic properties and the global stiffness properties along with the classical membrane theory. The finite element analysis verifies the analytical results and also shows a closer correspondence with numerical results for the curved-line assumption than the straight-line assumption. Further, the global stiffness properties of three types of lattice conical shells are compared using the single variable method. The results indicate that the flexural stiffness and the torsional stiffness of the lattice conical shell with equiangular helical ribs are the maximum and the axial stiffness of the shell with geodesic helical ribs is the maximum among them, while the values of all stiffness properties of the shell with equidistant helical ribs are the minimum among them.

A Novel Approach for Design and Simulation of Equivalent Continuum Model of Lattice Structures Used in Tunnels and Caverns

Developing a continuum model of a lattice structure means expressing a complicated structure into a simpler one, and this is done by determining equivalent stiffness of the original structure, which can further be used to transform into desired simpler geometry for multiple benefits. Objective of this paper is to find a reliable method for determination of equivalent stiffness of lattice girder, comparing all the techniques suggested by various researchers. In present work the actual lattice girder structures and their equivalents were analysed using Finite Element Method tool Strand7. The primary focus is on lattice girder, which is a primary rock support system employed in tunnels and caverns. The equivalent stiffness so obtained is used as input parameter in numerical analysis tool STRAND7 for analysis of equivalent lattice beam. The maximum deflection and bending stress at centre of equivalent lattice beam were calculated and compared. In an attempt to develop a more accurate equivalent beam model a novel approach which is modification of the Method-B has been developed by authors. This Noval approach is the most convenient approach to develop an equivalent section with an extremely uncompromised representation of stress development and deformation characteristic. In this approach, the focus was made on the physical resemblance of a proposed equivalent beam with the actual lattice girder, leading us to adopt a beam of I-cross-section. This approach yielded similar stress and deflections in the equivalent beam as that of the actual lattice girder.

Hydrogeological Models of Water Flow and Pollutant Transport in Karstic and Fractured Reservoirs

AbstractIn a literature review of the recent advancements in mathematical hydrologic models applied in fractured karstic formations, we highlight the necessary improvements in the fluid dynamic equations that are commonly applied to the flow in a discrete fracture network (DFN) via channel network models. Fluid flow and pollutant transport modeling in karst aquifers should consider the simultaneous occurrence of laminar, nonlaminar, and turbulent fluxes in the fractures rather than the laminar flow by the cubic law that has been widely applied in the scientific literature. Some simulations show overestimations up to 75% of the groundwater velocity when non‐laminar flows are neglected. Moreover, further model development is needed to address the issues of tortuosity of preferential saturated fluid flow in fractures suggesting adjustments of the size of the mean aperture in DFN models. During the past decade, DFN mathematical models have been significantly developed aimed at relating the three‐dimensional structure of interconnected fractures within rocky systems to the specific fracture properties measurable on the rock outcrops with the use of reliefs, tracer/pumping tests, and geotechnical field surveys. The capabilities and limitations of previous reported hydrological models together with specific research advancements and findings in modeling equations are described herein. New software is needed for creating three‐dimensional contour maps in fractured aquifers corresponding to the outputs of particle tracking simulations. Existing software based on the equivalent continuum or multiple‐interacting continua cannot delineate the spread of pollutant migrations affected by the tortuous preferential flow pathways that occur in DFNs.

Computational assessment of Stone-Wales defects on the elastic modulus and vibration response of graphene sheets

In this work we assess the influence of Stone-Wales defects on the elastic modulus and vibration responses of single-layered graphene sheets. To this end, the atomic position of the C-atoms within the defective layers are inserted as initial conditions into finite element calculations used to estimate the mechanical properties and natural frequencies of carbon monolayers. Moreover, we consider two sets of initial atomic positions: the harmonic solution predicted by discrete dislocation theory and its fully-relaxed configuration, which is a novelty in this type of study. Thus, the mechanical properties of the beam elements are depicted based on the equivalence between the interatomic potential energy of the atomic model and the strain energy of the equivalent continuum model. Furthermore, based on the results, we propose an extension of the classical beam model that accurately defines the mechanical properties of the elements in terms of the actual length of each atomic bond. The finite element model is implemented in the commercial software Abaqus/Standard. The results obtained are in reasonable agreement with previous data presented by other authors. In this work we have specifically ascertained the importance of using a non-linear model to predict the Young’s modulus and vibration responses of defective graphene sheets.

Enhanced dynamic homogenization of hexagonally packed granular materials with elastic interfaces

It is well known that the classical energetically consistent micropolar model has limits in simulating the frequency band structure of packed granular materials (see Merkel et al., 2011). It is here shown that if a standard continualization of the difference equation of motion of the discrete model is carried out, an equivalent micropolar continuum is obtained which is able to accurately simulate the optical branches of the discrete model. Nevertheless, this homogenized continuum presents non-positive defined elastic potential energy, a deficiency that limits the reliability of the model and implies instability phenomena (destabilizing effects) in the acoustic branches. This drawback is circumvented here through an high-frequency dynamic homogenization scheme which is based on an enhanced continualization of the discrete governing equations into pseudo-differential equations. Through a formal Taylor expansion of the pseudo-differential operators a higher order differential equation corresponding to the governing equation of a non-local continuum thermodynamically consistent are obtained. The resulting approach allows obtaining an equivalent micropolar continuum characterized by inertial non-locality. Moreover, higher order continua with non-local constitutive and inertial terms may be derived. The proposed continuum models are proved to be able to accurately describe both the static and dynamic behavior of the discrete granular model. Finally, the convergence to the response of the discrete system is shown when increasing the order of the higher order continuum.

Equivalent Continuum Coupling-Based Slope Stability Analysis of Zhouning Pumped Storage Power Station

A 3D equivalent continuum coupling analysis of the slope of the Zhouning pumped storage power station was proposed in this study. Firstly, the hydraulic properties of rock mass, groundwater, and deformation mechanism under high-frequency fluctuations of the water level are analyzed. Secondly, the three-dimensional Digital Terrain Model (DTM) with surface image information was established by digital photogrammetry technology for cataloging field fracture information. Third, stress-seepage coupling simulation using a hybrid grid approach and the FLAC3D equivalent continuum model was implemented. The results show that the permeability of rock mass is stress-dependent and anisotropic, and the direction of maximum principal permeability of both bank slopes is approximately parallel to the river. Slope deformations induced by impoundment during the operation period are the superposition results of two mechanisms, referred to as “mattress effect” and “swelling effect.” Attention should be paid to the tensile strength of the rock mass during operation.

A multifield continuum model for the description of the response of microporous/microcracked composite materials

The description of the mechanical behavior of composite materials is an open challenge in engineering. The focus of this work is on micro-cracked/micro-porous composites, made of stiff grains embedded in a cracked or porous deformable matrix. Grains interact between each other and with cracks or pores localized at the interfaces, which reduce the contact area between grains. Pores’ interaction is also taken into account. We apply a two-scale modeling strategy based on a discrete to scale-dependent (non-classical) continuum equivalence procedure. The constitutive relationships of the resulting microcontinuum, identified by requiring equivalence, in terms of virtual work, between discrete and continuum non-local descriptions of the reference material, take directly into account the material internal structure, and, in particular, material internal lengths. Field and boundary conditions of the equivalent continuum are also derived based on work equivalence between internal and external actions. The response of the obtained one-dimensional equivalent model and of a finite element model of a bar with pores are compared. A good match was obtained in a wide range of pore densities.

Numerical Study on Application Conditions of Equivalent Continuum Method for Modeling Heat Transfer in Fractured Geothermal Reservoirs

The equivalent continuum method an effective approach for modeling heat transfer in fractured geothermal reservoirs. However, presently there is a lack of systematical and profound study on application conditions of the equivalent porous media (EPM) method. In this study, we numerically investigated the application conditions of the EPM method based on geological data of Yangbajing geothermal field. The results indicate that when fracture spacing is within 3–25 m, the results of the EPM method are basically in the same levels as those of the MINC method. However, when the fracture spacing is within 25–300 m, differences of the EPM method from the MINC method increase with the fracture spacing, so when the fracture spacing is within 25–300 m, it is unreasonable to adopt the EPM method to simulate the fractured reservoirs. With the fracture spacing increasing within 25–300 m, the system production temperature and electric power will gradually decrease; the injection pressure, reservoir impedance and pump power will gradually increase; and the energy efficiency will gradually decrease.

Effects of Gouge Fill on Elastic Wave Propagation in Equivalent Continuum Jointed Rock Mass

The presence of gouge in rock joints significantly affects the physical and mechanical properties of the host rock mass. Wave-based exploration techniques have been widely used to investigate the effects of gouge fill on rock mass properties. Previous research on wave propagation in gouge-filled joints focused on analytical and theoretical methods. The lack of experimental methods for multiple rock joint systems, however, has limited the verification potential of the proposed models. In this study, the effects of gouge material and thickness on wave propagation in equivalent continuum jointed rocks are investigated using a quasi-static resonant column test. Gouge-filled rock specimens are simulated using stacked granite rock discs. Sand and clay gouge fills of 2 and 5 mm thicknesses are tested to investigate the effects of gouge material and thickness. Comprehensive analyses of the effects of gouge thickness are conducted using homogeneous isotropic acetal gouge fills of known thickness. The results show that gouge fill leads to changes in wave velocity, which depend on the characteristics of the gouge fill. The results also show that particulate soil gouge is susceptible to preloading effects that cause permanent changes in the soil fabric and contact geometry and that increased gouge thickness causes a more significant stiffness contribution of the gouge material properties to the overall stiffness of the equivalent continuum specimen. The normal and shear joint stiffnesses for different gouge fill conditions are calculated from the experimental results using the equivalent continuum model and suggested as input parameters for numerical analysis.

Thermodynamically consistent non-local continualization for masonry-like systems

Masonry-like systems composed by modular stiff units bonded by soft connections represent an efficient, versatile and ultimately successful strategy for natural and artificial macro-scale architectures. The static and dynamic behavior of masonry-like materials characterized by a running bond periodic pattern of rigid rectangular blocks and elastic interfaces is described by formulating equivalent nonlocal continuum models. The paper discusses the thermodynamic restrictions limiting the consistency of the standard continualization strategies. The inherent pathologies recognized in the macroscopic quasi-static response and/or in the dynamic dispersion properties of different continuum models motivate the original proposal of an enhanced continualization strategy. Based on the series expansion of the pseudo-differential functions accounting for shift operators and proper downscaling laws, the enhanced continualization scheme allows formulating homogeneous non-local continuum models of increasing orders, analytically featured by characteristic non-local constitutive and inertial terms. The enhanced continualization shows thermodynamic consistency in the definition of the overall elastic moduli, as well as qualitative agreement and convergent matching of the frequency dispersion functions. The theoretical findings are successfully verified though the solution of representative static and dynamic benchmark problems.