Three-Dimensional Crack Propagation Algorithms Based on Universal Meshes and their Application to Fracking

准确预测裂缝何时以及如何在岩层中扩展和传播是工程中一个重要但尚未解决的问题。裂缝扩展和预测模型可以指导油气储层的跟踪策略设计，以及将压裂液和废水注入地下，并有助于评估由此产生的地震活动性。裂缝扩展模型也被用于储层固碳稳定性评价和地热储层工程中。此外，它们在发展对地球物理现象的理解方面发挥着重要作用，例如火山爆发和冰川断裂。该奖项支持制定和实施数值算法和方法来模拟裂缝在结构和地下岩石中的传播的基础研究。该项目的结果将减少预测的不确定性，以了解被压裂岩石的物理和结构。它将减轻或消除与计算方法相关的不确定性。通过改进这种预测的方式，该项目有可能在未来几年影响能源生产和相关问题。此外，该项目将在一个急需专家的领域培养一名研究生，并涉及来自主要代表性不足的少数民族院校的本科生，培训他们并让他们接触工程研究。为了控制与模拟裂纹扩展相关的数值误差，本项目将结合三个新的发展。首先，提出了一种鲁棒的四面体背景网格变形算法，使其能够精确地网格化断裂在三维空间中传播时获得的每个裂纹几何形状。这样的网格被称为通用网格。第二，建立了一类能够以高阶精度计算裂缝前缘应力强度因子的数值方法。第三，建立了三维脆性裂缝和水力裂缝的收敛演化计算算法。通用网格算法和高阶方法共同工作，可以使计算非常精确的裂纹演化，而无需在裂纹前缘周围进行不断的重新网格划分。

Accurate prediction of when and how a crack will grow and propagate through a rock formation is an important yet unsolved problem in engineering. Crack growth and prediction models guide the design of strategies to frack oil and gas reservoirs and to inject fracking fluids and waste water back underground, and help assess the resulting seismicity. Crack growth models are also used in evaluating the stability of reservoirs for carbon sequestration, and in the engineering of geothermal reservoirs. Additionally, they play an important role in developing understanding of geophysical phenomena, such as volcanic eruptions and fracture of glaciers. This award supports fundamental research to formulate and implement numerical algorithms and methods to simulate the propagation of cracks through structures and underground rocks. The outcomes of this project will reduce the predictive uncertainty to the understanding of the physics and the structure of the rock being fractured. It will alleviate or remove uncertainties associated with the computational methods. By improving the way such predictions are made, the project has the potential to impact energy production and associated concerns for years to come. Additionally, this project will train a graduate student in an area in which experts are highly sought, as well as involve undergraduate students from predominantly underrepresented minority institutions, training and exposing them to research in engineering. To attain control of the numerical errors associated with simulating the propagation of cracks, this project will combine three novel developments. First, the formulation of a robust algorithm to deform a single background mesh of tetrahedra so that it exactly meshes every cracked geometry obtained as the fracture propagates in three dimensions. Such mesh is called a universal mesh. Second, the formulation of a class of numerical methods that enable the computation of stress intensity factors along a crack front with high-order of accuracy. Third, the creation of an algorithm to compute convergent evolution of brittle and hydraulic fractures in three dimensions. Working jointly, the universal mesh algorithm and the high-order method should enable the computation of very accurate crack evolutions without constant re-meshing around the crack front.