A convergence study of monolithic simulations of flow and deformation in fractured poroelastic media

A consistent linearisation has been carried out for a monolithic solution procedure of a poroelastic medium with fluid‐transporting fractures, including a comprehensive assessment of the convergence behaviour. The fracture has been modelled using a sub‐grid scale model with a continuous pressure across the fracture. The contributions to the tangential stiffness matrix of the fracture have been investigated to assess their impact on convergence. Simulations have been carried out for different interpolation orders and for Non‐Uniform Rational B‐Splines as interpolants vs Lagrangian polynomials. To increase the generality of the results, Newtonian as well as non‐Newtonian (power‐law) fluids have been considered. Unsurprisingly, a consistent linearisation invariably yields a quadratic convergence, but comes at the expense of a loss of symmetry and recalculation of the contribution of the interface to the stiffness matrix at each iteration. When using a linear line search however, the inclusion of only those terms of the interface stiffness which result in a symmetric and constant tangential stiffness matrix is sufficient to obtain a stable and convergent iterative process

Convergence in non‐associated plasticity and fracture propagation for standard, rate‐dependent, and Cosserat continua

The use of pressure‐dependent plasticity models with a non‐associated flow rule causes a loss of the well‐posedness for sufficiently low hardening rates. Apart from a mesh dependence, this can result in poor convergence, or even divergence of the iterative procedure employed to find an equilibrium configuration. This can be aggravated when other nonlinear, dissipative mechanisms are introduced, for instance the propagation of cracks. This is demonstrated rigorously, as well as the regularizing effect of adding viscosity or employing a Cosserat continuum. In both cases the regularization is independent of the value of the internal length scale for a fairly wide range of parameters. The spatial discretization has been done using T‐splines, and the fracture is modeled using interface elements and propagated using mesh line insertions. The time integration has been done by an implicit Newmark scheme. The use of proper regularization techniques makes an implicit scheme feasible, resulting in a reduction in the number of time steps by an order of magnitude

Stick‐slip like behavior in shear fracture propagation including the effect of fluid flow

Shear‐based fracture propagation in fluid‐saturated porous materials is investigated using a displacement–pressure formulation that includes acceleration and inertial effects of the fluid. Pressure‐dependent plasticity with a nonassociated flow rule is adopted to realistically represent the stresses in the porous bulk material. The domain is discretized using unequal order T‐splines and cast into a finite element method using Bézier extraction. An implicit scheme is used for the temporal integration. The solid acceleration‐driven fluid flow reacts to stress waves, but it results in pressure oscillations. Adding fluid acceleration terms dampens these oscillations and increases the fluid pressure near the fracture tips. By simulating a typical shear fracture case, it is shown that stick‐slip like, or stepwise, fracture propagation occurs for a high permeability, also upon mesh refinement. The acceleration driven fluid flow results in a build‐up of pressure near the fracture tip. Once this pressure region encompasses the fracture tip, propagation arrests until the pressure has diffused away from the crack tip, after which propagation is resumed and the build‐up of pressure begins anew. This results in a stick‐slip like behavior, with large arrests in the fracture propagation. Stepwise propagation related to the initial conditions has also been observed, but disappears once the fracture length exceeds the size of the region influenced by the initial conditions

Strain relaxation in GaN grown on vicinal 4H-SiC(0001) substrates

The strain of GaN layers grown by Metal Organic Chemical Vapor Deposition (MOCVD) on three vicinal 4H-SiC substrates (0, 3.4 and 8 offcut from [0001] towards [11-20] axis) is investigated by X-ray Diffraction (XRD), Raman Scattering and Cathodoluminescence (CL). The strain relaxation mechanisms are analyzed for each miscut angle. At a microscopic scale, the GaN layer grown on on-axis substrate has a slight and homogeneous tensile in-plane stress due to a uniform distribution of threading dislocations over the whole surface. The GaN layers grown on miscut substrates presented cracks, separating areas which have a stronger tensile in-plane stress but a more elastic strain. The plastic relaxation mechanisms involved in these layers are attributed to the step flow growth on misoriented surfaces (dislocations and stacking faults) and to the macroscopical plastic release of additional thermoelastic stress upon cooling down (cracks)

Sub-grid models for multiphase fluid flow inside fractures in poroelastic media

Models have been developed for the simulation of multiphase fluid flow within fractured poroelastic media. They allow for the fluid phases to interact inside fractures, without requiring explicit simulations in the interior of the fractures. The models retain the ability to retrieve the fluid velocity profile in the fracture by post-processing. The models for flow within a fracture are combined with a formulation for multiphase flow within the poroelastic medium. They have been implemented using isogeometric analysis, cast into a traditional finite element format using Bézier extraction. Pressure oscillations around the fracture are prevented by using a lumped integration scheme for the pressure capacity. The effect of interactions between the fluid phases is first demonstrated for a single fracture through parameter studies. Next, two cases with a more practical orientation are simulated. They show, inter alia, that the inclusion of interactions between the fluid phases can result in fluid back-flow

Unequal order T-spline meshes for fracture in poroelastic media

Spline-based meshes allow for a higher inter-element continuity. For coupled problems, e.g. poroelasticity, different meshes with different orders of interpolation are normally used for the various fields in order to avoid spurious oscillations. When including discontinuities in these meshes, there exist several options for the discretisation. Herein we will discuss two options which use T-splines, one aiming at a minimum number of degrees of freedom around the crack tip, the other trying to maximise this number. Both meshes retain a higher-order continuity along the fracture, but the mesh which maximises the number of degrees of freedom mesh introduces two additional degrees of freedom around the crack tip to allow for a sharper crack. The two discretisations are used to simulate a pressurised fracture inside a poroelastic material and the results are compared to results obtained using a Non-Uniform Rational B-Spline (NURBS) mesh. A comparison between the two discretisations shows the effect of including additional degrees of freedom close to the crack tip. However, both meshes yield similar results further away from the crack tip. It is shown that both T-spline meshes capture a fully closed discontinuity at the fracture tip, whereas the NURBS mesh retains a small opening due to the discontinuity which exists for the cracked as well as the intact elements. A fully closed fracture aperture results in T-splines with a lower discontinuity pressure compared to NURBS, making T-splines more suitable for simulations in which the fracture propagation is limited by the fluid transport within the fracture

High proton conductivity in cyanide-bridged metal-organic frameworks: understanding the role of water

We investigate and discuss the proton conductivity properties of the cyanide-bridged metal–organic framework (MOF) [Nd(mpca)2Nd(H2O)6Mo(CN)8]·nH2O (where mpca is 5-methyl-2-pyrazinecarboxylate). This MOF is one of an exciting class of cyanide-bridged materials that can combine porosity with magnetism, luminescence, and proton conductivity. Specifically, we show that this material features highly hydrophilic open channels filled with water molecules. They enable a high proton conductivity, as much as 10−3 S cm−1. A rich hydrogen-bonding network, formed by the ligands' carboxylate groups with both coordinated and lattice water molecules, facilitates this high proton conductivity. Combined thermogravimetric studies, FTIR spectroscopy and PXRD analysis show that upon heating at 80 °C, the lattice water molecules are removed without any change in the framework. Further heating at 130 °C results in a partial removal of the coordinated water molecules, while still retaining the original framework. These activated MOFs shows an increasing conductivity from ∼10−9 S cm−1 to ∼10−3 S cm−1 when the relative humidity increases from 0% to 98%. Our studies show that the increase in proton conductivity is correlated with the re-hydration of the framework with lattice water molecules. The Arrhenius activation energy for the proton conductivity process is low (Ea = 37 kJ mol−1), indicating that the protons “hop” through the channels following the Grotthuss mechanism. The fact that this MOF is remarkably stable both under high humidity conditions and at relatively high temperatures (up to 130 °C) makes it a good candidate for real-life applications