Robert Gracie

A review of extended/generalized finite element methods for material modeling

The extended and generalized finite element methods are reviewed with an emphasis on their applications to problems in material science: (1) fracture, (2) dislocations, (3) grain boundaries and (4) phases interfaces. These methods facilitate the modeling of complicated geometries and the evolution of such geometries, particularly when combined with level set methods, as for example in the simulation growing cracks or moving phase interfaces. The state of the art for these problems is described along with the history of developments.

Simulation of cracks in a Cosserat medium using the extended finite element method

Cosserat (micropolar, asymmetric) elasticity can better predict the mechanical behavior of the materials with a characteristic length scale than the classical theory of elasticity. However, the area of fracture modelling in a Cosserat medium is not widely presented in the literature. The simulation of cracks in a Cosserat medium using the eXtended Finite Element Method (XFEM) is presented in this paper. The proper crack tip enrichment of the translational and microrotational fields is important for the robustness and efficiency of the XFEM/Cosserat model. Using the example of an edge crack of the Mode I in this paper, we have shown that the values of the J-integral for the Cosserat solid are higher than those for the equivalent classical elastic solid. The difference becomes significant for the cases of materials with strong micropolar properties. The dependence of the fracture behavior of the crack of different sizes on the Cosserat elastic constants was found to be significant. Therefore, it is recommended that the special tip enrichment is considered for the microrotational field, and a careful analysis of the material parameters is performed in order to model fracture in a Cosserat medium.

Electromechanical simulations of dislocations

Improving the reliability of micro-electronic devices depends in part on developing a more in-depth understanding of dislocations because dislocations are barriers to charge carriers. To this end, the quasi-static simulation of discrete dislocations dynamics in materials under mechanical and electrical loads is presented. The simulations are based on the extended finite element method, where dislocations are modelled as internal discontinuities. The strong and weak forms of the boundary value problem for the coupled system are presented. The computation of the Peach–Koehler force using the J-integral is discussed. Examples to illustrate the accuracy of the simulations are presented. The motion of the network of the dislocations under different electrical and mechanical loads is simulated. It was shown that even in weak piezoelectric materials the effect of the electric field on plastic behaviour is significant.

A 3-Dimensional Eulerian Finite Element Model for Ice Scour

The main objective of this paper is to present a Finite Element (FE) numerical model of the ice scour process. The FE model is developed to study the soil deformation and transport process around the scouring ice and to investigate the effects of the ice scour on a pipeline buried or laid in a trench cut on the seabed. The focus of this paper is on the scours caused by ice ridges commonly observed in the Beaufort Sea. The developed FE model is a new application of the Arbitrary-Lagrangian-Eulerian (ALE) method to a soil mechanics problem involving very large deformations. Soil material, originally positioned in front of the ice ridge, is transported forward and sideways through the FE mesh and deposited in the berms formed on both sides of the scour. The soil material below the scour depth similarly moves across the mesh simulating the subscour effect. An inviscid CAP plasticity constitutive model is used to model the soil material. This paper focuses on the interaction between the ice ridge and the seabed. It describes soil transport process involved during the interaction. The soil deformation field obtained from the model is compared with the empirical deformation functions commonly used in current design methods. Future papers will report on the interaction between the ice ridge, the infill in the pipeline trench, and the pipeline; the influence of the soil properties of the trench and the seabed will also be studied.Copyright

A 3-Dimensional Continuum ALE Model for Ice Scour: Study of Trench Effects

A new Arbitrary Lagrangian Eulerian (ALE) Finite Element (FE) model of ice scour was recently developed by the authors. It is based on continuum representation of the soil. It was shown in recent papers that such a model can characterize the mechanics of the ice-soil-pipeline interaction without requiring any of the assumptions that the Winkler models depend on. The model utilizes soil properties obtained by conventional laboratory testing. In a recent paper, this model was used to show that the subscour deformations and ice-soil interaction forces are very sensitive to ice ridge geometries for shallow slope ice features. In this paper, the ALE FE ice scour model is utilized to study the effects of the pipeline trench on the scour process and the forces transmitted to the pipeline. Two different infill soil properties and two different ice ridge geometries are analyzed with a 36 inch diameter pipe buried to in a trench of 1.5 m cover. It is shown that the scour process near and in the trench is significantly different than in the ambient seabed soils and the recognition of this may present some potential advantages for the protection of the pipelines not recognized by the Winkler models. It is also shown that the pipeline loads generated during the scour process are cyclic. They build slowly as the ice moves over the trench and then reverse as the ice ridge moves away from the trench. This is in contrast to monotonic and rapidly growing loads predicted by the Winkler models. The paper shows that the loads transferred to the pipeline depend on the infill soil properties placed in the trench. It is shown that loads experienced by the pipeline are less for the softer infill than stiffer soils.Copyright

Unscented importance sampling for parameter calibration of carbon sequestration systems

A Bayesian importance sampling method is developed to efficiently and accurately calibrate the parameters of non-linear and non-Gaussian system models. The unscented importance sampling (UIS) consists of two stages. The first stage uses the latest monitoring data to generate a Gaussian approximation of the true posterior distribution of the uncertain parameters and utilizes the measurement update stage of the unscented Kalman filter (UKF) to approximate the posterior. The second stage of UIS uses a mixture of approximate posterior computed in the first stage and a heavy tailed distribution as the proposal distribution for Bayesian importance sampling. UIS is repeated whenever new monitoring data becomes available. Two case studies were developed to study the UIS method and to compare it UKF and importance sampling (IS) methods: a non-linear analytical system model and synthesized CO2 injection model using a numerical multi-phase flow simulator. In analytical case study, it is shown that UIS is more accurate than both UKF and traditional IS with static proposal and the relative accuracy of the UIS over traditional IS increases with dimensionality of the parameter space. The higher accuracy of UIS compared to UKF and traditional IS with static proposal is also shown in the CO2 injection case study. It is also shown that increasing number of samples and a defensive mixture distribution with a mixture ratio between 0.1 and 0.25 enhances the performance of UIS.

On Applications of XFEM to Dynamic Fracture and Dislocations

The application of the extended finite element method, which can model arbitrary discontinuities in finite elements to dynamic fracture and dislocations, is described. The dynamic fracture methodology is applied to a problem of crack branching and compared to element deletion and interelement crack methods. The dislocation method directly models the dislocation as a tangential discontinuity. This makes the method readily applicable to problems with interfaces and aniso-tropy.

Characterizing the Stimulated Reservoir Volume During Hydraulic Fracturing-Connecting the Pressure Fall-Off Phase to the Geomechanics of Fracturing

Microseismic imaging of the hydraulic fracturing operation in the naturally fractured rocks confirms the existence of a stimulated volume (SV) of enhanced permeability. The simulation and characterization of the SV evolution is uniquely challenging given the uncertainty in the nature of the rock mass fabrics as well as the complex fracturing behavior of shear and tensile nature, irreversible plastic deformation and damage. In this paper, the simulation of the SV evolution is achieved using a nonlocal poromechanical plasticity model. Effects of the natural fracture network are incorporated via a nonlocal plasticity characteristic length, ‘. A nonlocal Drucker–Prager failure model is implemented in the framework of Biot’s theory using a new implicit C finite element method. First, the behavior of the SV for a two-dimensional (2D) geomechanical injection problem is simulated and the resulting SV is assessed. It is shown that breakdown pressure and stable fracturing pressure are the natural outcomes of the model and both depend upon ‘. Next, the post-shut-in behavior of the SV is analyzed using the pressure and pressure derivative plots. A bilinear flow regime is observed and it is used to estimate the flow capacity of the SV. The results show that the flow capacity of the SV increases as ‘ decreases (i.e., as the SV behaves more like a single hydraulic fracture); however, for 0:1m ‘ 1m, the calculated flow capacity indicates that the conductivity of the SV is finite. Finally, it is observed that as ‘ tends to zero, the flow capacity of the SV tends to infinity and the SV behaves like a single infinitely conducting fracture. [DOI: 10.1115/1.4040479]

Petroleum Geomechanics: A Computational Perspective

Petroleum geomechanics is concerned with rock and fracture behavior in reservoir, drilling, completion, and production engineering. Typical problems in petroleum geomechanics include subsidence, borehole stability, and hydraulic fracturing. All are coupled problems that involve heat transfer, fluid flow, rock/fracture deformation, and/or solute transport. Numerical solutions through modeling are desired for such complicated systems. In this chapter, we present the mathematical descriptions of these typical problems in petroleum geomechanics, point out the challenges in solving these problems, and address those challenges by a variety of classical and emerging numerical techniques.