Research Areas

Modeling material failure poses significant challenges for Lagrangian FE codes. A testament to their difficulty are the results published by Sandia National Laboratories in their blind predictions for fracture. They observed that despite all the recent advances in finite element technology, the most robust methodology for modeling material failure was the so-called "element-death" approach. In this approach, those elements that are considered to have failed based on a particular failure criterion are entirely removed from calculations. While robust, these methodologies leave a lot to be desired from an accuracy standpoint.

Fracture problems can be broadly considered to fall under a class of evolving interface problems. As the failure surface evolves, the background finite element mesh has to be adapted to accommodate the changing topology. One of the most widely used methods for this class of problems is the cohesive element approach. These methods assume that every finite element facet is a potential failure surface. Once the stress on any given facet exceeds the tensile strength of the material, the facet is allowed to fail and a fracture surface is generated. The method has been successful for a wide variety of applications including dynamic fragmentation. The method is also fully scalable and well-suited for massively parallel codes.

However, the method is not without its limitations. One of the well-known challenges for the method involves its ability to represent arbitrary crack paths when the crack trajectory is not known apriori. This is not that surprising considering we are only allowing failure to propagate along certain fixed directions. Consequently, severe mesh-dependence of results is commonly observed in these methods. While remeshing in the direction of the propagating fracture is a perfectly valid solution, it gets cumbersome rather quickly.

The eXtended Finite Element Method (X-FEM) and related technologies seek to alleviate these meshing constraints by modifying the underlying discretization spaces such that the evolving failure surface is represented independently of the background grid. These methods were developed simultaneously at Northwestern University and University of Texas, Austin in late 90s and have largely lived up to their early promise and have begun to appear in commercial software packages.

However, subsurface is often characterized by complex natural fracture networks. Treatment of such complex fracture patterns and their evolution still remains a challenge within X-FEM frameworks. A combination of the above techniques that represent cracks as discrete entities along with continuum damage mechanics, or phase-field methodologies that represent cracks as diffuse entities are now gaining traction. These are open areas of research and our research group is actively engaged in investigating such problems.

For material systems with an inherent microstructure and heterogeneity such as rocks, embedded boundary methods can be very powerful. Once again, the primary advantage comes from decoupling the numerical approximation with the underlying geometry. Such a decoupling facilitates fast computations on structured grids for relatively complex microstructures.

From a modeling standpoint, the presence of material interfaces results in weakly discontinuous problems. In such problems, the primary field variable (such as displacements) as well as the fluxes (such as surface tractions) remain continuous across the interface. Only the gradient of the field variable (for example, strains) is discontinuous. By contrast, a fracture surface results in a strong discontinuity, namely, the primary field variable exhibits a discontinuity across the fracture surface. The utility of enrichment methodologies described above is obvious for this class of strongly discontinuous problems. However, X-FEM methods are also very useful in modeling weakly discontinuous problems

In some ways, weakly discontinuous problems are more difficult to address within the X-FEM framework. Either specialized enrichment functions need to be constructed, or continuity requirements need to be enforced at the material interface. Moreover, the dissipation mechanisms such as interfacial slip introduce additional numerical challenges. Our group focuses on developing numerical algorithms to treat such problems accurately and efficiently.

In the past decade, the application of hydraulic fracturing technology has resulted in a dramatic increase in production of shale oil and natural gas. Extracting energy from previously inaccessible reservoirs has fundamentally altered the global energy landscape while also reducing the world's dependence on coal. However, several questions remain both with respect to improving reservoir efficiency as well as environmental impacts of recovery operations.

Computer models and physics-based simulations are important allies in providing us with some answers to the above questions. Leveraging high performance computing, we can provide a platform to drill several "virtual wells" and guide/optimize real-world scenarios.

The goal is to develop first principle equations for the main physical processes involved: (a) poroelastic deformation of the rock medium, (b) fluid-flow through fractured medium, and (c) fluid flow through the porous rock medium. The coupled interactions between the three processes are also honored in the models. Finally, we provide realistic data on the Elastic material constants, pore pressure, and in-situ stresses where available. Solution of these equations in three-dimensional space gives us a perspective on how the real-world hydraulic fracturing operations would behave. Our group will focus on advancing the numerical methodologies to model hydraulic fracturing as a multi-physics process.