Fluid-filled Fracture Propagation with a Phase Field Approach in Subsurface by Employing Nonlinear Strain Limiting Models and Enriched Galerkin Methods

该项目旨在研究加压和充满流体的裂缝或裂缝在地下材料中传播的方式。在诸如土壤和岩石之类的多孔材料中，流体通过材料孔隙的流动可以迫使固体多孔介质发生明显的变形（如裂缝和断裂）。这些孔隙力学相互作用对许多重要问题至关重要，如隧道施工、沉降、大坝或堤坝破坏以及二氧化碳封存。控制这些变形扩展的经典数学模型是通过将线性弹性或孔隙弹性与变形系统耦合而形成的。然而，经典线弹性模型的一个主要缺点是应变值与应力值成线性比例。因此，它与模型的假设相矛盾，并且可能无法准确预测现实情况。本课题的重点是建立非线性应变极限模型这一类新的理论模型。与经典线性化模型相比，非线性应变极限模型的优点是，即使应力趋于无穷大，应变仍保持有界，这对于充满流体的裂缝是至关重要的。新模型将被扩展到考虑孔隙弹性。接下来，将孔隙弹性模型与相场方法相结合，实现准静态充液裂缝扩展。此外，该项目将采用新颖的富伽辽金（EG）有限元近似来解决数值离散化的几个关键问题。众所周知，经典的伽辽金有限元方法通常不能保证局部质量守恒，这可能导致非物理振荡。为了克服这些挑战，我们将研究EG方法，并分析它们在孔隙弹性系统中的稳定性和收敛性。研究结果将用于开发预测工具，以预测准静态裂缝扩展路径，并用于评估和验证这些新模型的性能。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The project aims to investigate the way in which pressurized and fluid-filled cracks or fractures spread through subsurface materials. In porous materials such as soils and rocks, the flow of the fluids through the material's pores can force significant deformations (such as cracks and fractures) to occur in the solid porous media. These poromechanical interactions are crucial to many important problems such as tunnel construction, subsidence, dam or levee failure, and CO2 sequestration. The classic mathematical model governing the spread of these deformations is formulated by coupling linear elasticity or poroelasticity with deformation systems. However, one of the major disadvantages of classical linear elasticity models is that strain values are linearly proportional to stress values. Thus, it contradicts the assumptions of the model, and it may not accurately predict realistic scenarios. This project focuses on establishing the nonlinear strain limiting model, a new class of theoretical model. The advantage of the nonlinear strain limiting models over classical linearized models is that strain remains bounded even if the stress tends to infinity, which is critical for fluid-filled fractures. The new model will be extended to consider poroelasticity. Next, the poroelasticity model will be coupled with a phase field approach to implement quasi-static fluid-filled fracture propagation. Moreover, the novel enriched Galerkin (EG) finite element approximations will be employed in the project to address several crucial issues for numerical discretization. It is well known that classical Galerkin finite element methods generally do not guarantee local mass conservation, which could lead to non-physical oscillation. EG methods will be investigated to overcome these challenges, and their stability and convergence for the poroelasticity system will be analyzed. The findings will then be used to develop a forecasting tool to predict the path of quasi-static fracture propagation and will be utilized to evaluate and validate the performance of these new models.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.