Using fracture-based continuum modeling of coupled geomechanical-hydrological processes for numerical simulation of hydraulic fracturing

This paper describes numerical simulation of hydraulic fracturing using fracture-based continuum modeling(FBCM)of coupled geomechanical-hydrological processes to evaluate a technique for high-density fracturing and fracture caging.The simulations are innovative because of modeling discrete fractures explicitly in continuum analysis.A key advantage of FBCM is that fracture initiation and propagation are modeled explicitly without changing the domain grid(i.e.no re-meshing).Further,multiple realizations of a preexisting fracture distribution can be analyzed using the same domain grid.The simulated hydraulic fracturing technique consists of pressurizing multiple wells simultaneously:initially without permeating fluids into the rock,to seed fractures uniformly and at high density in the wall rock of the wells;followed by fluid injection to propagate the seeded fracture density hydraulically.FBCM combines the ease of continuum modeling with the potential accuracy of modeling discrete fractures and fracturing explicitly.Fractures are modeled as piecewise planar based on intersections with domain elements;fracture geometry stored as continuum properties is used to calculate parameters needed to model individual fractures;and rock behavior is modeled through tensorial aggregation of the behavior of discrete fractures and unfractured rock.Simulations are presented for previously unfractured rock and for rock with preexisting fractures of horizontal,shallow-dipping,steeply dipping,or vertical orientation.Simulations of a single-well model are used to determine the pattern and spacing for a multiple-well design.The results illustrate high-density fracturing and fracture caging through simultaneous fluid injection in multiple wells:for previously unfractured rock or rock with preexisting shallow-dipping or horizontal fractures,and in situ vertical compressive stress greater than horizontal.If preexisting fractures are steeply dipping or vertical,and considering the same in situ stress condition,well pressurization without fluid permeation appears to be the only practical way to induce new fractures and contain fracturing within the target domain.

Nonlinear wave dispersion in monoatomic chains with lumped and distributed masses:discrete and continuum models

The main objective of this paper is to investigate the influence of inertia of nonlinear springs on the dispersion behavior of discrete monoatomic chains with lumped and distributed masses.The developed model can represent the wave propagation problem in a non-homogeneous material consisting of heavy inclusions embedded in a matrix.The inclusions are idealized by lumped masses,and the matrix between adjacent inclusions is modeled by a nonlinear spring with distributed masses.Additionally,the model is capable of depicting the wave propagation in bi-material bars,wherein the first material is represented by a rigid particle and the second one is represented by a nonlinear spring with distributed masses.The discrete model of the nonlinear monoatomic chain with lumped and distributed masses is first considered,and a closed-form expression of the dispersion relation is obtained by the second-order Lindstedt-Poincare method(LPM).Next,a continuum model for the nonlinear monoatomic chain is derived directly from its discrete lattice model by a suitable continualization technique.The subsequent use of the second-order method of multiple scales(MMS)facilitates the derivation of the corresponding nonlinear dispersion relation in a closed form.The novelties of the present study consist of(i)considering the inertia of nonlinear springs on the dispersion behavior of the discrete mass-spring chains;(ii)developing the second-order LPM for the wave propagation in the discrete chains;and(iii)deriving a continuum model for the nonlinear monoatomic chains with lumped and distributed masses.Finally,a parametric study is conducted to examine the effects of the design parameters and the distributed spring mass on the nonlinear dispersion relations and phase velocities obtained from both the discrete and continuum models.These parameters include the ratio of the spring mass to the lumped mass,the nonlinear stiffness coefficient of the spring,and the wave amplitude.

Verification of a laboratory-based dilation model for in situ conditions using continuum models

With respect to constitutive models for continuum modeling applications, the post-yield domain remainsthe area of greatest uncertainty. Recent studies based on laboratory testing have led to thedevelopment of a number of models for brittle rock dilation, which account for both the plastic shearstrain and confining stress dependencies of this phenomenon. Although these models are useful inproviding an improved understanding of how dilatancy evolves during a compression test, there hasbeen relatively little work performed examining their validity for modeling brittle rock yield in situ. Inthis study, different constitutive models for rock dilation are reviewed and then tested, in the context of anumber of case studies, using a continuum finite-difference approach （FLAC）. The uncertainty associatedwith the modeling of brittle fracture localization is addressed, and the overall ability of mobilizeddilation models to replicate in situ deformation measurements and yield patterns is evaluated. 2014 Institute of Rock and Soil Mechanics, Chinese Academy of Sciences. Production and hosting byElsevier B.V. All rights reserved.

A CONTINUUM MODEL FOR AXIAL-STRAIN-INDUCED TORSION IN SINGLE-WALL CARBON NANOTUBES

Using the atomistic-based finite-deformation shell theory, we analytically investigate the coupling between the axial deformation and the torsion in single-wall carbon nanotubes. We find that the axial-strain-induced torsion（ASIT） response is limited only to chiral nanotubes. This response is affected by chiralities and radii of carbon nanotubes. Our results are similar to that of molecular dynamic simulations reported in the literatures.

Static analysis of ultra-thin beams based on a semi-continuum model

A linear semi-continuum model with discrete atomic layers in the thickness direction was developed to investigate the bending behaviors of ultra-thin beams with nanoscale thickness.The theoretical results show that the deflection of an ultra-thin beam may be enhanced or reduced due to different relaxation coefficients.If the relaxation coefficient is greater/less than one,the deflection of micro/nano-scale structures is enhanced/reduced in comparison with macro-scale structures.So,two opposite types of size-dependent behaviors are observed and they are mainly caused by the relaxation coefficients.Comparisons with the classical continuum model,exact nonlocal stress model and finite element model （FEM） verify the validity of the present semi-continuum model.In particular,an explanation is proposed in the debate whether the bending stiffness of a micro/nano-scale beam should be greater or weaker as compared with the macro-scale structures.The characteristics of bending stiffness are proved to be associated with the relaxation coefficients.

The density wave in a new anisotropic continuum model

In this paper the new continuum traffic flow model proposed by Jiang et al is developed based on an improved car-following model, in which the speed gradient term replaces the density gradient term in the equation of motion. It overcomes the wrong-way travel which exists in many high-order continuum models. Based on the continuum version of car-following model, the condition for stable traffic flow is derived. Nonlinear analysis shows that the density fluctuation in traffic flow induces a variety of density waves. Near the onset of instability, a small disturbance could lead to solitons determined by the Korteweg-de-Vries （KdV） equation, and the soliton solution is derived.

Numerical Simulation of Transport Phenomena in Solidification of Multicomponent Ingot Using a Continuum Model

A continuum model proposed for dendrite solidification of multicomponent alloys, with any partial solid back diffusion, was used to numerically simulate the macroscopic solidification transport phenomena and macrosegregations in an upwards directionally solidified plain carbon steel ingot. The computational results of each macroscopic field of the physical variables involved in the solidification process at a middle solidification stage were presented.

Non-classical solutions of a continuum model for rock descriptions

The strain-gradient and non-Euclidean continuum theories are employed for construction of non- classical solutions of continuum models. The linear approximation of both models＇ results in identical structures in terms of their kinematic and stress characteristics. The solutions obtained in this study exhibit a critical behaviour with respect to the external loading parameter. The conclusions are obtained based on an investigation of the solution for the scalar curvature in the non-Euclidean continuum theory, The proposed analysis enables us to use different theoretical approaches for description of rock critical behaviour under different loading conditions.

Motivation in Foreign Language Learning: Applying the Time Continuum Model of Motivation in FLL

The research is: by using Wdolkowski's Time Continuum Model throughout a lesson plan enables the teacher to increase students'motivation and help them move closer to success in a learning environment. This research supports the theory that instruction is a network of interactions between the teacher and learner that promotes a successful learning experience. It identifies a three-part learning sequence-a beginning, middle and an end. Each part has two of six key motivational factors that when applied correctly by the teacher will maximize the success and continued motivation of the learner.

Seepage simulation of high concrete-faced rockfill dams based on generalized equivalent continuum model

This research focused on the three-dimensional(3 D) seepage field simulation of a high concrete-faced rockfill dam(CFRD) under complex hydraulic conditions. A generalized equivalent continuum model of fractured rock mass was used for equivalent continuous seepage field analysis based on the improved node virtual flow method. Using a high CFRD as an example, the generalized equivalent continuum range was determined, and a finite element model was established based on the terrain and geological conditions, as well as structural face characteristics of the dam area. The equivalent seepage coefficients of different material zones or positions in the dam foundation were calculated with the Snow model or inverse analysis. Then, the 3 D seepage field in the dam area was calculated under the normal water storage conditions, and the corresponding water head distribution, seepage flow, seepage gradient, and seepage characteristics in the dam area were analyzed. The results show that the generalized equivalent continuum model can effectively simulate overall seepage patterns of the CFRD under complex hydraulic conditions and provide a reference for seepage analysis of similar CFRDs.

A review on the application of modified continuum models in modeling and simulation of nanostructures

Analysis of the mechanical behavior of nanos- tructures has been very challenging. Surface energy and non- local elasticity of materials have been incorporated into the traditional continuum analysis to create modified continuum mechanics models. This paper reviews recent advancements in the applications of such modified continuum models in nanostructures such as nanotubes, nanowires, nanobeams, graphenes, and nanoplates. A variety of models for these nanostructures under static and dynamic loadings are men- tioned and reviewed. Applications of surface energy and nonlocal elasticity in analysis of piezoelectric nanomateri- als are also mentioned. This paper provides a comprehensive introduction of the development of this area and inspires fur- ther applications of modified continuum models in modeling nanomaterials and nanostructures.

Analytical Polarizable Continuum Model for Wavelets on NURBS Patches

This article concerns the application of wavelet techniques on molecular surfaces constituted of four-sided patches. The Polarizable Continuum Model, which is governed by the Poisson-Boltzmann equation, is treated by means of boundary integral equations. The media inside and outside the molecular surface consist respectively of the solute and the solvent. For a given electrically charged molecule, the principal unknown is the electrostatic solvation energy when the permittivity is specified. The wavelet basis functions are constructed on the unit square which are subsequently mapped onto the patches that are assumed to be isotropically shaped and to admit similar surface areas. The initial transmission problem is recast as an integral equation in term of both the single and the double layers. Domain decomposition preconditioner serves as acceleration of the linear solver of the single layer which is badly conditioned.

Predicting effects of tunnel throttling of annular freeway vehicular flow by a continuum model

Fluid flow throttling is common in industrial and building services engineering.Similar tunnel throttling of vehicular flow is caused by the abrupt number reduction of roadway lane,as the tunnel has a lower lane number than in the roadway normal segment.To predict the effects of tunnel throttling of annular freeway vehicular flow,a three-lane continuum model is developed.LaneⅢof the tunnel is completely blocked due to the need of tunnel rehabilitation,etc.There exists mandatory net lane-changing rate from laneⅢto laneⅡjust upstream of the tunnel entrance,which is described by a model of random number generated through a golden section analysis.The net-changing rate between adjacent lanes is modeled using a lane-changing time expressed explicitly in algebraic form.This paper assumes that the annular freeway has a total length of 100 km,a two-lane tunnel of length 2 km with a speed limit of 80 km/h.The free flow speeds on lanesⅠ,ⅡandⅢare assumed to be 110,100 and 90 km/h respectively.Based on the three-lane continuum model,numerical simulations of vehicular flows on the annular freeway with such a tunnel are conducted with a reliable numerical method of 3rd-order accuracy.Numerical results reveal that the vehicular flow has a smaller threshold of traffic jam formation in comparison with the case without tunnel throttling.Vehicle fuel consumption can be estimated by interpolation with time averaged grid traffic speed and an assumed curve of vehicle performance.The vehicle fuel consumption is lane number dependent,distributes with initial density concavely,ranging from 5.56 to 8.00 L.Tunnel throttling leads to an earlier traffic jam formation in comparison with the case without tunnel throttling.

Continuum modeling of freeway traffic flows:State-of-the-art,challenges and future directions in the era of connected and automated vehicles

Connected and automated vehicles(CAVs)are expected to reshape traffic flow dynamics and present new challenges and opportunities for traffic flow modeling.While numerous studies have proposed optimal modeling and control strategies for CAVs with various objectives(e.g.,traffic efficiency and safety),there are uncertainties about the flow dynamics of CAVs in real-world traffic.The uncertainties are especially amplified for mixed traffic flows,consisting of CAVs and human-driven vehicles,where the implications can be significant from the continuum-modeling perspective,which aims to capture macroscopic traffic flow dynamics based on hyperbolic systems of partial differential equations.This paper aims to highlight and discuss some essential problems in continuum modeling of real-world freeway traffic flows in the era of CAVs.We first provide a select review of some existing continuum models for conventional human-driven traffic as well as the recent attempts for incorporating CAVs into the continuum-modeling framework.Wherever applicable,we provide new insights about the properties of existing models and revisit their implications for traffic flows of CAVs using recent empirical observations with CAVs and the previous discussions and debates in the literature.The paper then discusses some major problems inherent to continuum modeling of real-world(mixed)CAV traffic flows modeling by distinguishing between two major research directions:(a)modeling for explaining purposes,where making reproducible inferences about the physical aspects of macroscopic properties is of the primary interest,and(b)modeling for practical purposes,in which the focus is on the reliable predictions for operation and control.The paper proposes some potential solutions in each research direction and recommends some future research topics.

A model of continuum damage mechanics for high cycle fatigue of metallic materials

A non-linear continuum damage model was presented based on the irreversible thermodynamics framework developed by LEMAITRE and CHABOCHE. The proposed model was formulated by taking into account the influence of loading frequency on fatigue life. The parameters H and c are constants for frequency-independent materials, but functions of cyclic frequency for frequency-dependent materials. In addition, the expression of the model was discussed in detail at different stress ratios （R）. Fatigue test data of AlZnMgCu1.5 aluminium alloy and AMg6N alloy were used to verify the proposed model. The results showed that the model possesses a good ability of predicting fatigue life at different loading frequencies and stress ratios.

Numerical simulation for macrosegregation in direct-chill casting of 2024 aluminum alloy with an extended continuum mixture model

An extended continuum mixture model for macrosegregation is applied to predicting Cu and Mg segregation in large-size ingot of 2024 aluminum alloy during direct chill casting（DC）. A microsegregation model using the approximate phase diagram data was coupled with macroscopic transport equations for macrosegregation profiles. Then, the impacts of transport mechanisms on the formation of macrosegregation were discussed. It is found that copper and magnesium have a similar segregation configuration from the billet center to surface. Negative segregation is observed in the centerline and subsurface, whereas positive segregation is obtained in the surface and somewhat underestimated positive segregation in the middle radius. Further, the discrepancy between the predicted and experimental results was discussed in detail. The results show that the magnesium to some extent alleviates the copper segregation in ternary alloy, compared with that in binary alloy. The predicted results show good agreement with measured experimental data obtained from literatures.

Modes of grain growth and mechanism of dislocation reaction under applied biaxial strain:Atomistic and continuum modeling

The phase field crystal method and Continuum Modeling are applied to study the cooperative dislocation motion of the grain boundary(GB)migration,the manner of the nucleation of the grain and of the grain growth in two dimensions(2 D)under the deviatoric deformation at high temperature.Three types of the nucleation modes of new finding are observed by the phase field crystal simulation:The first mode of the nucleation is generated by the GB splitting into two sub-GBs;the second mode is of the reaction of the sub-GB dislocations,such as,the generation and annihilation of a pair of partial Frank sessile dislocation in 2 D.The process can be considered as the nucleation of dynamic recrystallization;the third mode is caused by two oncoming rows of the dislocations of these sub-GBs,crossing and passing each other to form new gap which is the nucleation place of the new deformed grain.The research is shown that due to the nucleation of different modes the mechanism of the grain growth by means of the sub-GB migration is different,and therefore,the grain growth rates are also different.Under the deviatoric deformation of the applied biaxial strain,the grain growth is faster than that of the grain growth without external applied stress.It is observed that the cooperative dislocation motion of the GB migration under the deviatoric deformation accompanies with local plastic flow and the state of the stress of the system changes sharply.When the system is in the process of recrystallized grain growth,the system energy is in an unstable state due to the release of the strain energy to cause that the reverse movement of the plastic flow occurs.The area growth of the deformed grain is approximately proportional to the strain square and also to the time square.The rule of the time square of the deformed grain growth can also be deduced by establishing the continuum dynamic equation of the biaxial strain-driven migration of the GB.The copper metal is taken as an example of the calculation,and the obtained result is a good agreement with that of the experiment.

Numerical Simulation of Macrosegregation During Steel Ingot Solidification Using Continuum Model

A continuum model is adopted to study the macrosegregation phenomena during solidification of large steel ingots.Evolution of temperature,melt velocity,and compositional concentration field during a 22 t steel ingot solidification are illustrated by using the finite volume method.Numerical results of temperature distribution are validated by experiments.The influence of local permeability relates to the friction that the melt experienced in mushy region is investigated.It is shown that the continuum model is able to predict the temperature field,and the variation of permeability obviously affects the melt flowing behavior and the final compositional distribution.

A simulation based on the cosserat continuum model of the vortex structure in granular materials

Displacement fluctuation is the difference between the real displacement and the affine displacement in deforming granular materials. The discrete element method （DEM） is widely used along with experimental approaches to investigate whether the displacement fluctuation represents the vortex structure. Current research suggests that the vortex structure is caused by the cooperative motion of particle groups on meso-scales, which results in strain localization in granular materials. In this brief article, we investigate the vortex structure using the finite element method （FEM） based on the Cosserat cor[tinuum model. The numerical example focuses on the relationship between the vortex structure and the shear bands under two conditions： （a） uniform granular materials; （b） granular materials with inclusions. When compared with distributions of the effective strain and the vortex structure, we find that the vortex structure coexists with the strain localization and originates from the stiffness cooperation of different locations in granular materials at the macro level.