Fracture scale fluid flow models for the simulation of poroelasticity

Fractures are a common occurrence in poroelastic materials: They are created to aid in underground resource recovery, or are unwanted during failure and collapse of materials. One of the main challenges for simulating these fractures is their small opening height compared to their length, making direct simulation of the interior of the fracture computationally expensive. In this thesis, models which reduce the two-dimensional fluid flow in the interior of the fracture to the in and outflow at a one-dimensional discontinuity are extended to include the complex fluid rheology of non-Newtonian power-law and Carreau fluids. One of the main advantages of the obtained sub-grid models is their ability to reconstruct the fluid behaviour through post processing the obtained results, allowing a detailed description of the fluid within the fracture to be re-obtained. In addition, these sub-grid models are also applied to multiphase flows, allowing the interactions between the fluid phases within the fracture to be included. Finally, a numerical two-scale model is presented, coupling numerically resolved velocity profiles within the fracture to the mass balance at the discontinuity. This allows for velocity profiles for which an analytic solution is not available to be included,
such as fluids displaying inertial effects.

These sub-grid models are implemented using finite element methods based on standard Lagrangian elements, non-uniform rational basis splines, and T-splines. While the Lagrangian elements are convenient and commonly used, it is shown that the increased inter-element continuity of Non-Uniform Rational B-Splines (NURBS) and T-splines is required to obtain continuous fracture outflows. It is furthermore shown that this increased continuity is beneficial for the convergence rate of the non-linear solver. The benefits of using lumped integration for the fracture inflow term are demonstrated, suppressing fluid velocity oscillations, and a special fracture tip integration scheme is presented which prevents non-physical fracture inflows for NURBS. Finally, a method to generate unequal order T-spline meshes is presented, allowing for interface elements to solely be inserted for fractured elements, and making mesh refinement near the discontinuity possible.

The fracture scale models and discretisation methods are used to investigate the interactions between the fluid and fracture propagation. It is shown that including a non-Newtonian fluid rheology can significantly alter the propagation velocity of the fracture, and the velocity of the fluid within the fracture. For multiphase flows, the fracture scale models show the importance of including inter-phase interactions within the fracture, providing significantly different results depending on the assumed flow model, either bubbly flow or separated flow. By comparing the fracture flow model to results obtained through direct simulation of the fracture flow the validity and accuracy of the fracture scale models are confirmed. Finally, simulations including inertial effects in the porous material show the interstitial fluid is capable of causing "stick-slip" like behaviour, and simulations using a numerical
two-scale approach give an indication of the possible pressure oscillations resulting from stepwise propagation.