Numerical methods for hydraulic fracture propagation: a review of recent trends

The recently introduced phase-field approach for pressurized fractures in a porous medium offers various attractive computational features for numerical simulations of cracks such as joining, branching, and nonplanar propagation in possibly heterogeneous media. In this paper, the pressurized phase-field framework is extended to fluid-filled fractures in which the pressure is computed from a generalized parabolic diffraction problem. Here, the phase-field variable is used as an indicator function to combine reservoir and fracture pressure. The resulting three-field framework (elasticity, phase field, pressure) is a multiscale problem that is based on the Biot equations. The proposed numerical solution algorithm iteratively decouples the equations using a fixed-stress splitting. The framework is substantiated with several numerical benchmark tests in two and three dimensions.

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A phase-field method for propagating fluid-filled fractures coupled to a surrounding porous medium

Propulsion at the microscale requires unique strategies such as the undulating or rotating filaments that microorganisms have evolved to swim. These features however can be difficult to artificially replicate and control, limiting the ability to actuate and direct engineered microdevices to targeted locations within practical timeframes. An alternative propulsion strategy to swimming is rolling. Here we report that low-strength magnetic fields can reversibly assemble wheel-shaped devices in situ from individual colloidal building blocks and also drive, rotate and direct them along surfaces at velocities faster than most other microscale propulsion schemes. By varying spin frequency and angle relative to the surface, we demonstrate that microwheels can be directed rapidly and precisely along user-defined paths. Such in situ assembly of readily modified colloidal devices capable of targeted movements provides a practical transport and delivery tool for microscale applications, especially those in complex or tortuous geometries.

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Surface-enabled propulsion and control of colloidal microwheels

We consider the dynamics of an elastic sheet lubricated by the flow of a thin layer of fluid that separates it from a rigid wall. By considering long wavelength deformations of the sheet, we derive an evolution equation for its motion, accounting for the effects of elastic bending, viscous lubrication, and body forces. We then analyze various steady and unsteady problems for the sheet, such as peeling, healing, levitating, and bursting, using a combination of numerical simulation and dimensional analysis. On the macroscale, we corroborate our theory with a simple experiment, and, on the microscale, we analyze an oscillatory valve that can transform a continuous stream of fluid into a series of discrete pulses.

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Peeling, healing, and bursting in a lubricated elastic sheet.

In this paper, new nonlinear dynamic properties of electrostatically actuated microstructures [referred to as electrostatic microelectromechanical systems (MEMS)] observed under superharmonic excitations are presented using numerical simulations. Application of a large dc bias (close to the pull-in voltage of the device) is found to bring the device to a nonlinear state. This nonlinear state (referred to as "dc-symmetry breaking") can be clearly observed from the characteristic change in the phase-plot of the device. Once a steady nonlinear state is reached, application of an ac signal at the Mth superharmonic frequency with an amplitude around "ac-symmetry breaking" gives rise to M oscillations per period or M-cycles in the MEM device. "ac-symmetry breaking" can also be observed by a characteristic change in the phase-plot of the device. On further increasing the ac voltage, a period doubling sequence takes place resulting in the formation of 2/sup n/M-cycles in the system at the Mth superharmonic frequency. An interesting chaotic transition (banded chaos) is observed during the period doubling bifurcations. The nonlinear nature of the electrostatic force acting on the MEM device is found to be responsible for the reported observations. The significance of the mechanical and the fluidic nonlinearities is also studied.

K.J. Stout

Papers

Fluid film lubrication