Two-scale Model Research Articles

Prediction of mechanical properties in statistically inhomogeneous core–shell materials by second-order two-scale method

This paper develops a novel second-order two-scale (SOTS) method to predict the mechanical performance of statistically inhomogeneous core–shell materials. As for these kinds of materials, the sophisticated microscopic information of inclusions, including their shape, size, orientation, spatial distribution, volume fraction, and so on, leads to changes of the macroscopic mechanical properties. Firstly, the microscopic configuration for the core–shell structure with random distribution is briefly characterized. Secondly, the SOTS formulae for solving the mechanical problems are proposed, and the importance of the two-scale approximate solutions in pointwise sense is given. Then, the associated prediction algorithm based on the two-scale model is brought forward in detail. Finally, some numerical results for the effective properties of the materials with varying probability distribution models are calculated and compared with the data by analytical and experimental results. The comparison shows that the statistical second-order two-scale model is validated for determination of the mechanical properties of inhomogeneous core–shell materials and demonstrates its potential applications in actual engineering applications.

Effect of Natural Fractures on Wormhole-Propagation Behavior

Summary Natural fractures have significant influence on flow fields, thus affecting wormhole pattern in acidizing. This paper summarizes our research on wormholing behavior in naturally fractured carbonates. First, statistical natural-fracture models are established using the Monte Carlo method. Second, a two-scale continuum wormhole model is established to simulate wormhole propagation with natural fractures. Finally, extensive numerical simulation is conducted to investigate wormhole behavior and the effect of the natural-fracture parameters on wormhole pattern. In addition, possible wormhole-penetration distance is discussed. This study provides a theoretical basis for matrix-acidizing designs in naturally fractured carbonates.

On the lifetime prediction of rolling lobe air springs

Abstract In this paper, the fatigue life of rolling lobe air springs made from cord-rubber composites is investigated with particular focus on the crack nucleation approach. Commonly used test specimens like the simple tension test specimen or the dumbbell specimen fail in the fatigue analysis of rolling lobe air springs because the fatigue characteristic of the cord-rubber interface is represented insufficiently. Therefore a new cord-rubber specimen is developed. The fatigue characteristic of the new specimen is discussed and Wohler curves for two different predictors based on Cauchy stress and configurational stress are determined. Using FEA with a two-scale model of the air spring, the cycles-to-failure with respect to piston diameter and different inner pressures are determined and compared to test results taken from the literature. The results from the literature were derived at elevated temperature. Therefore, the Arrhenius approach is used to take into account different temperatures. In addition the crack growth approach using the cracking energy density is applied. The results show an improvement in the prediction of rolling lobe air springs fatigue life with respect to the crack initiation plane orientation and the cycles-to-failure using the new specimen. Compared to the crack nucleation approach, the crack growth approach seems to slightly overpredict the fatigue lives.

Two-Scale Geomechanics of Carbonates

The geomechanical characterization of a carbonate reservoir is required for formation stimulation and hydrocarbon recovery. The pertinent core- or block-scale (large-scale) characterizations are time consuming and expensive, and more importantly, cannot be used for drill cuttings. The present study proposes a two-scale model based on microscale (small-scale) measurements to predict the geomechanical properties of a carbonate formation at the core scale. At the small scale, we develop a physically representative element by accounting for the effective stiffness of a constitutive mineral and of voids. At the large scale, we account for the volume fraction of each mineral, the porosity, and the pore structure of the void space. The elastic deformation of a large-scale model is simulated using a finite element method (FEM), whose results are tested against independent lab measurements. The proposed two-scale model has applications for geomechanical characterization of a formation at the core scale from drill cuttings.

Features of X-Band Radar Backscattering Simulation Based on the Ocean Environmental Parameters in China Offshore Seas.

The X-band marine radar has been employed as a remote sensing tool for sea state monitoring. However, there are few literatures about sea spectra considering both the wave parameters and short wind-wave spectra in China Offshore Seas, which are of theoretical and practical significance. Based on the wave parameters acquired from the European Centre for Medium-Range Weather Forecasts reanalysis data (ERA-Interim reanalysis data) during 36 months from 2015 to 2017, a finite depth sea spectrum considering both wind speeds and ocean environmental parameters is established in this study. The wave spectrum is then built into a modified two-scale model, which can be related to the ocean environmental parameters (wind speeds and wave parameters). The final results are the mean backscattering coefficients over the variety of sea states at a given wind speed. As the model predicts, the monthly maximum backscattering coefficients in different seas change slowly (within 4 dB). In addition, the differences of the backscattering coefficients in different seas are quite small during azimuthal angles of 0° to 90° and 270° to 360° with a relative error within 1.5 dB at low wind speed (5 m/s) and 2 dB at high wind speed (10 m/s). With the method in the paper, a corrected result from the experiment can be achieved based on the relative error analysis in different conditions.

Hydromechanical model for hydraulic fractures using XFEM

In this study, a hydromechanical model for fluid flow in fractured porous media is presented. We assume viscous fluids and the coupling equations are derived from the mass and momentum balance equations for saturated porous media. The fluid flow through discrete cracks will be modelled by the extended finite element method and an implicit time integration scheme. We also present a consistent linearization of the underlying non-linear discrete equations. They are solved by the Newton-Raphson iteration procedure in combination with a line search. Furthermore, the model is extended to includes crack propagation. Finally, examples are presented to demonstrate the versatility and efficiency of this two-scale hydromechanical model. The results suggest that the presence of the fracture in a deforming, porous media has great impact on the fluid flow and deformation patterns.

Assessment of Wind Speed Estimation From C-Band Sentinel-1 Images Using Empirical and Electromagnetic Models

Surface wind speed estimation from synthetic aperture radar (SAR) data is principally based on empirical (EP) approaches, e.g., CMOD functions. However, it is necessary and significant to compare radar backscattering modeling based on EP and electromagnetic (EM) approaches for enhancing the understanding of the physical processes between radar signal and sea surface, which is important for the design of radar sensors (e.g., cyclone global navigation satellite system). Indeed, through comparisons, it is worth noticing that the scattering of wave breaking is not taken into account in the physical modeling of radar backscattering. Surface wind speed is selected here as a reference parameter for investigating the difference between EP and EM models, due to its important role in radar backscattering modeling. In addition, wind speed estimates can be easily compared to in situ measurements. For EP approach, CMOD5.N and Komarov’s model are selected for wind speed estimation from Sentinel-1 images. The CMOD5.N can offer wind speed estimates up to 25–35 m/s, while wind speed estimation based on Komarov’s model does not require wind direction input. For EM approach, the asymptotic models, i.e., composite two-scale model, small-slope approximation (SSA), and resonant curvature approximation (RCA), are investigated for wind speed retrieval. They are studied with two models of surface roughness spectrum: semi-EP spectrum and EP model. In general, normalized radar cross section (NRCS) calculated by CMOD5.N and SSA/RCA is quite similar for incidence angles below 40° in vertical polarized and below 30° in horizontal polarized. For larger ones, significant NRCS deviations between two approaches are demonstrated, due to the lack of wave breaking scattering in EM models. As a result, wind speed estimates by CMOD5.N and SSA/RCA are very close for low and moderate incidence angles, while SSA-/RCA-based wind speeds are overestimated for larger ones.

Stress relaxation of dense spongy-particle systems

Spongy particles have a permeable structure that allows them to undergo rate-dependent volume changes as their elastic network takes up or expels the viscous suspending solvent. Their ability to be jammed well above random close-packing makes them particularly attractive in applications where tailoring the overall properties is a requirement such as pharmaceuticals and foods. In this work, we independently vary the particle modulus and the particle permeability to study their effect on the stress-relaxation behavior of jammed permeable-particle suspensions. The dynamic two-scale model developed by Hutter et al. [Faraday Discuss. 158, 407–424 (2012)], which explicitly accounts for the particle size dynamics, is used for this purpose. We perform flow-cessation simulations of dense permeable-particle systems subjected to different preshear deformations. The stress relaxation occurs on shorter time scales in the case of permeable particles compared to impermeable particles. In terms of particle dynamics, stress relaxation is found to be promoted primarily by the motion of the particles within the cages formed by the surrounding particles, rather than by cage escape. The stress-relaxation process is accelerated by the permeability of spongy particles, namely, due to the sustained volume change that was induced during preshear, which renders their cages less effective.Spongy particles have a permeable structure that allows them to undergo rate-dependent volume changes as their elastic network takes up or expels the viscous suspending solvent. Their ability to be jammed well above random close-packing makes them particularly attractive in applications where tailoring the overall properties is a requirement such as pharmaceuticals and foods. In this work, we independently vary the particle modulus and the particle permeability to study their effect on the stress-relaxation behavior of jammed permeable-particle suspensions. The dynamic two-scale model developed by Hutter et al. [Faraday Discuss. 158, 407–424 (2012)], which explicitly accounts for the particle size dynamics, is used for this purpose. We perform flow-cessation simulations of dense permeable-particle systems subjected to different preshear deformations. The stress relaxation occurs on shorter time scales in the case of permeable particles compared to impermeable particles. In terms of particle dynamics, stre...

Damköhler number-based research on dividing dissolution patterns in carbonate acidizing

In general, the main dissolution patterns in carbonate acidizing include face dissolution, wormhole dissolution and uniform dissolution. Wormholes can also be divided into single, dominant wormhole, ramified wormhole and so on. The wormholes can establish advantageous channels for oil and gas in the reservoirs, increase the permeability of formations and achieve the ultimate goal of carbonate matrix acidizing. Considering the factors that affect carbonate acidizing, the flow rate of acid plays the key role in determining dissolution patterns. Many scholars have carried out the researches by means of numerical simulation or physical simulation experiment individually, but the reports which combine the two methods and define the flow rate boundary between different dissolution patterns in carbonate acidizing are rare. In this paper, the flow rate of acid was dimensionless on the basis of Damköhler number (Da), and the boundary of acid injection rate was confirmed by combining numerical simulation with physical simulation experiments. Among the various types of dissolution models, the two-scale continuum model comprehensively considers the convection, diffusion and acid-rock interfacial reaction and is capable of dynamically monitoring the acidizing region at different time steps. This paper, using the finite difference method, solved the typical two-scale continuum model. In order to extend the calculation of the range of Damköhler number, Steffensen's acceleration method was applied to the porosity variation equation. The previous conclusion that at higher injection rate the breakthrough pore volume of acid is in direct proportion to the one-third power of the average pore velocity was validated. Furthermore, the typical pressure fall-off curve was analyzed. Results show that the typical pressure drop curve changes from the power-law curve to the straight line, as the injection rate decreases. Based on the analysis of porosity field and concentration field, combined with physical simulation experiment, the change law of dissolution pattern with Da was analyzed, which clarify the injection rates boundary of dissolution pattern. The results show when Da is between 50 and 200, the core appears wormholes, when Da is between 1 and 50, the dissolution pattern is the transition stage from wormhole to uniform dissolution and when Da is between 200 and 500, the core dissolution pattern is the transition stage from wormhole to face dissolution. The Damköhler number and injection rates of different dissolution patterns can be used to guide the design of acidizing construction.

Two-Scale Geomechanics of Shale

Summary The geomechanical properties of reservoirs, which are important for formation stimulation, are often determined from triaxial tests on large-scale samples such as core plugs or blocks. It is difficult to recover large samples from shale formations because they are mechanically unstable and usually break down into pieces. The present study develops a two-scale model that uses drill cuttings to estimate the static elastic properties of shales at the core scale. We first propose a physically representative element to capture the elastic deformation of a solid grain with a known minerology by accounting for the grain size and its elastic properties using the structural-mechanics approach (a small-scale model). We then develop a core-scale model dependent on the volume fractions of the minerals, which are obtained from X-ray diffraction (XRD), for different realizations of the spatial distribution of the solid grains (large-scale model). The sensitivity of the large-scale model to the number of the elements is tested. The proposed model shows promising results for four shale formations (New Albany, Rocky Mountain Siliceous, Lower Bakken, and Barnett) and has major applications for the geomechanical characterization of a formation from drill cuttings.

Contribution of austenite-martensite transformation to deformability of advanced high strength steels: From atomistic mechanisms to microstructural response

Steels combining austenite (fcc) with lath martensite (bcc) in nanolaminate microstructures are tough, resistant to hydrogen-embrittlement, and inexpensive, making them attractive for many technological applications. Austenite provides plastic deformation while martensite provides strength, but the nanoscale processes that control plasticity in the austenite layers are not fully established. Recent atomistic simulations and crystallographic theory reveal a unified understanding of the structure and motion of the fcc austenite-bcc (lath) martensite interface in steels, with transformation strains up to ∼90% in Fe-C alloys. In this paper, the atomistic behaviour is connected to the ductility of nanolaminate microstructures. First, the mechanical response of the atomistic fcc/bcc interface under shear loading is analyzed. The interface motion follows a Schmid-type law for resolved shear stresses in the transformation direction. Furthermore, the forward fcc-to-bcc transformation is spontaneous while the reverse bcc-to-fcc transformation requires high stress. The asymmetry correlates well with the Peierls stresses for fcc and bcc screw dislocations, respectively. Second, the atomistic results guide the formulation of a two-scale continuum model for the phase transformation. The multi-scale strategy adopted here accounts for the relevant nano-scale mechanisms and enables modeling the mechanical response of real martensite microstructures, up to the scale of tens of micrometers - which would be untractable with direct atomistic simulations. Multi-scale simulations show that forward transformation contributes significantly to the apparent plasticity in lath martensite. This reinforces recent work highlighting the importance of such nanoscale austenite films for achieving ductility and toughness in lath martensite. Overall, the present work demonstrates how atomistic insights can directly inform continuum models of microstructural deformation, and points toward directions for material control and optimization to achieve enhanced mechanical performance in these steels.

Effects of Environment Knowledge in Evacuation Scenarios Involving Fire and Smoke: A Multiscale Modelling and Simulation Approach

We study the evacuation dynamics of a crowd evacuating from a complex geometry in the presence of a fire as well as of a slowly spreading smoke curtain. The crowd is composed of two kinds of individuals: those who know the layout of the building, and those who do not and rely exclusively on potentially informed neighbors to identify a path towards the exit. We aim to capture the effect the knowledge of the environment has on the interaction between evacuees and their residence time in the presence of fire and evolving smoke. Our approach is genuinely multiscale—we employ a two-scale model that is able to distinguish between compressible and incompressible pedestrian flow regimes and allows for micro and macro pedestrian dynamics. Simulations illustrate the expected qualitative behavior of the model. We finish with observations on how mixing evacuees with different levels of knowledge impacts important evacuation aspects.

On a robust FE2 model for delamination analysis in composite structures

The present paper deals with a robust two-scale delamination model based on a FE2 formulation, whereby a finite element model on the macro- and meso-scale are solved simultaneously. For a correct modeling of the delamination kinematics, the composite structure on macro-scale is discretized by a combined element. Only one single integration point over the full height of the composite structure contains the information about shell and interfacial kinematics, allowing a straight forward enhancement to the FE2 model. On the meso-scale, the combined kinematics are applied to a coupled representative volume element (cRVE), which describes the entire stacking sequence including an interfacial layer. The coupling between the shell and the interfacial kinematics results in a more robust delamination model. Numerical examples demonstrate the applicability and the improvements by reducing the oscillations of the load displacement curve during the softening process and the influence of the macroscopic mesh discretization under single and mixed mode loading.

Thermally insulating, fire-retardant, smokeless and flexible polyvinylidene fluoride nanofibers filled with silica aerogels

The thermally insulating, fire-retardant, smokeless and flexible polymer nanofiber is an incredibly important material for mankind. This polymer nanofiber is expected to breed technological innovations in applications such as thermally insulated clothing and building materials. Here, we propose an electrospinning process to fabricate silica aerogel-filled polymer nanofibers by modifying the typical sol–gel chemistry process for the synthesis of silica aerogel. Silica aerogel-filled polyvinylidene fluoride (PVdF) nanofiber webs, fabricated using the proposed process, were fire-retardant, smokeless and flexible; they also exhibited a low thermal conductivity of 0.028 W·m−1·K−1, which was 26.3% lower than that of pure PVdF nanofiber webs. To study the unique thermal characteristics of nanofibers theoretically, a micromechanics-based, two-scale multiphase model was proposed, with consideration of fiber structure including nano and micro pores. These developments make an important contribution to bringing silica aerogel products to everyday life.

A Robust Model To Simulate Dolomite-Matrix Acidizing

Summary The two-scale model for simulating carbonate acidizing has gained substantial attention recently. Five studies dealt with matching experimental data studying regular acid. Four studies considered limestone samples, while the fifth examined one dolomite core with face dissolution. The previous work only considered the pore volume (PV) to breakthrough (PVBT) to match experimental results. Researchers assumed linear kinetics for hydrochloric acid (HCl) carbonate reaction and relied on changing Carman-Kozeny exponents to match experimental data. Unlike previous studies, experiments were performed on 6-in.-long and 1.5-in.-diameter vuggy-dolomite cores at two sets of temperatures (150 and 200°F) and acid concentrations (15 and 20 wt% HCl). Computed tomography (CT) was used to scan the cores when dry, wet, and after acidizing. Porosity distribution calculated from the dry and wet scans was used to build a rectangular model with the cylindrical core inscribed inside. Nonlinear reaction kinetics were applied. The acid-reaction rate and diffusion coefficient were modified on the basis of X-ray-fluorescence (XRF) results and effluent chemical analysis. Wormhole 3D shape and experimental PVBT were used to assess the quality of model results. The tuned model was used to simulate a hypothetical 18-in. core as well as large-scale radial experiments to assess its prediction capabilities, and finally the model was used to predict the dolomite-acidizing performance under field conditions. The simulation runs emphasize that the exclusion of the wormhole shape and branching from the matching process results in an unrealistic match. It is important to simulate the cylindrical shape of the core using the actual porosity distribution to capture the wormhole growth, which is increasingly important when the wormhole propagates near the core perimeter. The present study highlights that matching parameters using experimental data yields a trustworthy model that matches both PVBT and wormhole spatial propagation. Accordingly, there is no need for excessively changing the Carman-Kozeny correlation exponents to match the dolomite-acidizing experiments. The current model accurately matches the wormhole propagation inside the core along with the PVBT. This model can be tuned using a few acidizing experiments and then can be used to generate an acid-efficiency curve with a high degree of confidence, thus avoiding the extra experimental cost. The current model was able to match two sets of experiments and follow the experimental trend of longer cores and large-scale radial experiments. It was used to predict acid performance under field conditions. The results show that the optimal PVBT under field conditions is always lower than the one predicted under laboratory conditions; the acid depth of penetration has a significant effect on the acid-efficiency curves; and the vertical flow of acid should be considered in acid-job design.

A multiscale model for reinforced concrete with macroscopic variation of reinforcement slip

A single-scale model for reinforced concrete, comprising the plain concrete continuum, reinforcement bars and the bond between them, is used as a basis for deriving a two-scale model. The large-scale problem, representing the “effective” reinforced concrete solid, is enriched by an effective reinforcement slip variable. The subscale problem on a Representative Volume Element (RVE) is defined by Dirichlet boundary conditions. The response of the RVEs of different sizes was investigated by means of pull-out tests. The resulting two-scale formulation was used in an FE^2 analysis of a deep beam. Load–deflection relations, crack widths, and strain fields were compared to those obtained from a single-scale analysis. Incorporating the independent macroscopic reinforcement slip variable resulted in a more pronounced localisation of the effective strain field. This produced a more accurate estimation of the crack widths than the two-scale formulation neglecting the effective reinforcement slip variable.

Sea Surface Radar Scattering at L-Band Based on Numerical Solution of Maxwell’s Equations in 3-D (NMM3D)

Radar scattering from ocean surfaces is investigated by 3-D numerical solution of Maxwell’s equations [numerical Maxwell’s model in 3-D (NMM3D)] using the ocean surface profiles stochastically generated from a 3-D Durden–Vesecky ocean spectrum. The surface integral equations (SIEs) are formulated for dielectric surfaces using Green’s functions of the air and the ocean permittivities with the surface tangential electric and magnetic fields as the unknowns. In solving the SIEs using the method of moment, a fast matrix solver of the sparse matrix canonical grid is used in conjunction with Rao–Wilton–Glisson basis functions. The computation has been implemented on a high-performance parallel computing cluster for problems with up to six million surface unknowns. Unlike the two-scale model (TSM) approximation, NMM3D does not require division of the surface spectrum into large- and small-scale ocean waves. The results of backscattering simulations are compared to Aquarius satellite radar measurements for wind speeds of 5, 8, and 10 m/s and for incidence angles of 29°, 39°, and 46°. The results show that NMM3D ocean backscattering solutions at L-band are in good agreement with Aquarius satellite radar data for co-polarized VV, HH, and cross-polarized VH returns as well as for the VV/HH ratio. The azimuthal dependence of L-band backscatter is also assessed. Finally, NMM3D results are compared to TSM solutions and are shown to lie close to Aquarius data in observed VV/HH ratio, and their azimuthal dependencies.

Numerical simulation and field application of diverting acid acidizing in the Lower Cambrian Longwangmiao Fm gas reservoirs in the Sichuan Basin

The Lower Cambrian Longwangmiao dolomite gas reservoirs in the Sichuan Basin are characterized by well-developed natural micro-fractures and dissolved pores and cavities. Due to the strong heterogeneity of reservoirs and the serious damage of drilling and completion fluids, acid placement is difficult, and especially the acidizing stimulation of long-interval highly deviated wells or horizontal wells is more difficult. In this paper, the diverting mechanism and rheological behavior of viscoelastic surfactant (VES) based diverting acid was firstly investigated, and the diverting acid with good diversion performance and low secondary damage was selected as the main acid. Then, based on the experimental results of its rheological behaviors, an empirical model of effective viscosity was fitted and a two-scale wormhole propagation model was coupled. And accordingly, a mathematical model for the acidizing of self-diverting acid was established to simulate the pH value, Ca2+ concentration, effective viscosity and wormhole shape under the effect of diverting acid in long-interval highly deviated wells that are non-uniformly damaged. Finally, gelled acid and 5% VES diverting acid were compared in terms of their etched wormhole shapes, flow rate distribution and acid imbibition profiles. It is shown that the diverting acid can obviously improve the acid imbibition profile of strong-heterogeneity reservoirs to intensify low-permeability reservoir stimulation. In view of the strong heterogeneity of Longwangmiao dolomite reservoirs and the complexities of drilling and completion fluid damage in the Sichuan Basin, a placement technology was developed for variable VES concentration diverting acid in horizontal wells and long-interval highly deviated wells completed with slotted liners. This acid placement technology has been practically applied in 8 wells and their cumulative gas production rate tested at the wellhead is 1233.46 × 104 m3/d. The average production stimulation ratio per well is up to 1.95. It provides a support for the efficient development of the Longwangmiao giant gas reservoir.

Modeling the shape dynamics of suspensions of permeable ellipsoidal particles

A dynamic two-scale model is developed for describing the mechanical behavior of suspensions of permeable ellipsoidal particles. The particle dynamics in the proposed model is described in terms of particle positions as well as conformation tensors that capture their size, shape, and orientation. Using non-equilibrium thermodynamics, the macroscopic fluid-dynamics and the particle dynamics on the microstructural level are mutually coupled in a consistent manner. So doing, the link between the macroscopic behavior, e.g. stresses, and the dynamics of the microstructure, e.g. particle shape and size, is established. Finally, the model is cast into a form in which the shape tensor is split into its volumetric and isochoric shape contributions, making it possible to model particles with both shape-preserving size-changes (e.g. swellable particles) and volume-preserving shape-changes (e.g. incompressible yet deformable particles). The size-shape model distinguishes itself in unifying prior knowledge of purely-shape models with that of purely-size models by appropriate choices of the Helmholtz free energy and the generalized mobility.

An Improved Facet-Based TSM for Electromagnetic Scattering From Ocean Surface

An improved facet-based model derived from a two-scale model (TSM) is proposed to evaluate the scattering strength of an electrically large ocean surface. In this model, a slope cutoff probability distribution function is introduced into geometrical optics to deal with the specular scattering of an individual facet. This modification allows us to approximate the rough surface with a large facet size without disrupting the accuracy of scattering evaluation. In addition, considering the sensitivity of the classical TSM to the cutoff wavenumber, the cutoff-invariant TSM is used into the facet-based model. Finally, our numerical results show that the facet-based model can obtain the comparable accuracy with the statistical models regardless of facet size and cutoff wavenumber. Due to the advantages of simplicity and efficiency, this model is of high applicable value for the synthetic aperture radar imaging.