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Register a new userAn anisotropic viscoplasticity model for shale based on layered microstructure homogenization
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Viscoplastic deformation of shale is frequently observed in many subsurface applications. Many studies have suggested that this viscoplastic behavior is anisotropic---specifically, transversely isotropic---and closely linked to the layered composite structure at the microscale. In this work, we develop a two-scale constitutive model for shale in which anisotropic viscoplastic behavior naturally emerges from semi-analytical homogenization of a bi-layer microstructure. The microstructure is modeled as a composite of soft layers, representing a ductile matrix formed by clay and organics, and hard layers, corresponding to a brittle matrix composed of stiff minerals. This layered microstructure renders the macroscopic behavior anisotropic, even when the individual layers are modeled with isotropic constitutive laws. Using a common correlation between clay and organic content and magnitude of creep, we apply a viscoplastic Modified Cam-Clay plasticity model to the soft layers, while treating the hard layers as a linear elastic material to minimize the number of calibration parameters. We then describe the implementation of the proposed model in a standard material update subroutine. The model is validated with laboratory creep data on samples from three gas shale formations. We also demonstrate the computational behavior of the proposed model through simulation of time-dependent borehole closure in a shale formation with different bedding plane directions.
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The non-equilibrium transport of inhomogeneous and dense gases highly confined by surface is encountered in many engineering applications. For example, in the shale gas production process, methane is extracted from ultra-tight pores under high pressure so the gas is inhomogeneous and dense. Currently, the complex non-equilibrium transport of inhomogeneous and dense gases where gas surface interactions play a key role is commonly investigated by molecular dynamics or on a continuum-assumption basis. Here, a tractable kinetic model based on the generalized Enskog equation and the mean-field theory is employed to couple the effects of the volume exclusion and the long-range intermolecular attraction forces. The interactions between gas molecules and confined surface are modelled by a 10-4-3 Lennard-Jones potential, which can capture gas surface adsorption. The cross-sectional density profiles of methane under different confinements are in good agreement with the molecular dynamics results reported in the literature, and the transport behaviors are validated by the non-equilibrium molecular dynamics. The velocity of methane flow in shale matrix is plug-like due to its dense characteristics in nanopores. The influence of pressure, temperature, pore size and shale composition on density and velocity profiles is analyzed quantitatively. Our results show that the Klinkenberg correction is not applicable to model shale gas flow in the production process; the Navier-Stokes model using the second-order slip boundary condition cannot produce the proper velocity profiles, and consequently fails to predict the accurate flow rate in nanopores. This study sheds new light on understanding the physics of non-equilibrium dense gas flows in shale strata.
While the hydraulic fracturing technology, aka fracking (or fraccing, frac), has become highly developed and astonishingly successful, a consistent formulation of the associated fracture mechanics that would not conflict with some observations is still unavailable. It is attempted here. Classical fracture mechanics, as well as the current commercial softwares, predict vertical cracks to propagate without branching from the perforations of the horizontal well casing, which are typically spaced at 10 m or more. However, to explain the gas production rate at the wellhead, the crack spacing would have to be only about 0.1 m, which would increase the overall gas permeability of shale mass about 10,000$times$. This permeability increase has generally been attributed to a preexisting system of orthogonal natural cracks, whose spacing is about 0.1 m. But their average age is about 100 million years, and a recent analysis indicated that these cracks must have been completely closed by secondary creep of shale in less than a million years. Here it is considered that the tectonic events that produced the natural cracks in shale must have also created weak layers with nano- or micro-cracking damage. It is numerically demonstrated that a greatly enhanced permeability along the weak layers, with a greatly increased transverse Biot coefficient, must cause the fracking to engender lateral branching and the opening of hydraulic cracks along the weak layers, even if these cracks are initially almost closed. A finite element crack band model, based on recently developed anisotropic spherocylindrical microplane constitutive law, demonstrates these findings.
The analysis of the seismic vulnerability of urban centres has received a great attention in the last century. In order to estimate the seismic vulnerability of a densely populated urban area, it would in principle be necessary to develop in-depth analyses for predicting the dynamic behaviour of the individual buildings and their structural aggregation. Such analyses, however, are extremely cost-intensive, require great processing time and above all expertise judgement. It is therefore very useful to define simplified rules for estimating the seismic vulnerability of whole urban areas. In the last decades, the Self-Organized Criticality (SOC) scenario has gained increasing credibility as a mathematical framework for explaining a large number of naturally occurring extreme events, from avalanches to earthquakes dynamics, from bubbles and crises in financial markets to the extinction of species in the evolution or the behaviour of human brain activity. All these examples show the intrinsic tendency common to many phenomena to spontaneously organize into a dynamical critical state, whose signature is the presence of a power law behaviour in the frequency distribution of events. In this context, the Olami-Feder- Christensen (OFC) model, introduced in 1992, has played a key role in modelling earthquakes phenomenology. The aim of the present paper is proposing an agent-based model of earthquake dynamics, based on the OFC self- organized criticality framework, in order to evaluate the effects of a critical sequence of seismic events on a given large urban area during a given interval of time. The further integration of a GIS database within a software environment for agent-based simulations, will allow to perform a preliminary parametric study of these effects on real datasets. The model could be useful for defining planning strategies for seismic risk reduction
Recent failures of upstream-raised tailings storage facilities (TSF) raised con-cerns on the future use of these dams. While being cost-effective, they entail higher risks than conventional dams, as stability largely relies on the strength of tailings, which are loose and normally-consolidated materials that may exhibit strain-softening during un-drained loading. Current design practice involves limit equilibrium analyses adopting a fully-softened shear strength; while being conservative, this practice neglects the work input required to start the softening process that leads to progressive failure. This paper describes the calibration and application of the NGI-ADPSoft constitutive model to evaluate the potential of static liquefaction of an upstream-raised TSF and provides an indirect measure of resilience. The constitutive model incorporates undrained shear strength anisotropy and a mesh-independent anisotropic post-peak strain softening. The calibration is performed using laboratory testing, including anisotropically-consolidated triaxial compression tests and direct simple shear tests. The peak and residual undrained shear strengths are validated by statistical interpretation of the available CPTu data. It is shown that this numerical exercise is useful to verify the robustness of the TSF design.
We consider a variant of Gamows liquid drop model with an anisotropic surface energy. Under suitable regularity and ellipticity assumptions on the surface tension, Wulff shapes are minimizers in this problem if and only if the surface energy is isotropic. We show that for smooth anisotropies, in the small nonlocality regime, minimizers converge to the Wulff shape in $C^1$-norm and quantify the rate of convergence. We also obtain a quantitative extension of the energy of any minimizer around the energy of a Wulff shape yielding a geometric stability result. For certain crystalline surface tensions we can determine the global minimizer and obtain its exact energy expansion in terms of the nonlocality parameter.
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