Fracture: Its been over three decades now that the statistical physicists started paying continual attention to failure and breakdown properties of disordered materials. Primarily its importance lie in the extreme nature of the statistics, translating into safety and integrity issues of large as well as small structures. It also has the fascinating aspect of universality in its response statistics that opened up new applications of concepts of phase transition and criticality in this field. One of the interesting questions in this regard is the forewarning of breakdown. Other than the nature of the acoustic signals emitted prior to a breakdown, even simply the duration of such signals is crucially important. It is well known that a higher disordered sample has more prior warnings of its imminent breakdown. A brittle material, on the other hand, will have an understandably shorter time window for such signals. The system size scaling, the probability distribution functions of the failure times of disorder materials have been an actively pursed question. I have extensively studied a disordered system, known as the fiber bundle model, in order to understand different models of failure that material goes through depending of the strength of disorder, size of the system, range of stress relaxation, temperature etc. We observe that for different values of above-mentioned parameters a failure can be spatially or temporally correlated or uncorrelated. The introduction of temperature makes the failure process rather interesting where it is very similar to the creep failure where the strain rate in the primary and secondary region shows time evolution like Omori and inverse Omori like behavior respectively making the model rather relevant to the study of seismic events.
Flow: The study of flow involves understanding the behavior of large scale processes (oil recovery, CO2 sequestration, groundwater collection, blood flow in capillary vessels etc.) by breaking down the dynamics of fluid in the pore scale. This is widely known as the upscaling process and has been studied extensively. In the pore scale, when two immiscible fluids flow in a porous media the flow does not obey linear Darcy law in the regime where the capillary forces are comparable to the viscous forces. The flow rate was observed both numerically and experimentally to scale in a quadratic manner with the pressure gradient. The disorder in capillary barriers at pores effectively creates a yield threshold in the porous medium, introducing an overall threshold pressure in the system making the fluids reminiscent of a Bingham viscoplastic fluid. The main motive of my work is to have a clear understanding of how this transition point and the threshold barrier behaves with system size and system disorder, where the later has two main origins: (i) fluctuation in pore sizes and (ii) interface configurations between the two liquids inside a pore. For this purpose, I have studied the dynamics of multiphase flow in a porous media (both 2d and 3d) through the pore network modeling where time evolution of the interfaces are observed and the local flowrate is determined from the Washburn equation. The flow criterion is determined from Kirchhoff's law and by solving the conjugate gradient equation for the future positions of the interfaces. A well-understood rheology can be obtained from the scaling of pressure gradient and flow rate (or the capillary number). I have started with a one-dimensional prototype of a porous media, known as the tube capillary bundle model that can be represented by parallel tubes carrying two immiscible fluids. The model is the only analytically solvable model from this context and can offer a detailed overview of the transition from non-linear to Darcy like liner rheology. We recently have explored a new theory for flow in porous media that is based on thermodynamic consideration. It is similar to the analysis based on the conservation of mass and rather in contrast to relative permeability equations that rely on specific physical assumptions. Such a thermodynamic study concerns the relation between velocities of the two phases, wetting, and non-wetting, during a flow. 4 of 5The theory also defines thermodynamic velocities corresponding to the phases, as derivatives (which is a homogenous Euler function) of the total volumetric flowrate. The relation between the superficial fluid velocity within pores (also known as the seepage velocity) and the thermodynamic velocities can be established through the co-moving velocity. The same has been repeated with a different formulation, which is based on the distribution of pore areas for wetting, non-wetting, and total fluid.