Collaborative Research: Adaptive Mixed-Dimensional Modeling and Simulation of Porous Media

本研究项目的总体目标是开发、分析和实现基于混维模型的裂隙多孔介质渗流模拟的稳定数值方法。裂隙多孔介质渗流是多物理、多尺度的，开发适用于这类系统的稳健有效的数值工具是计算数学中的一类重要挑战。例如，在许多实际应用中，骨折或其他特征，如大脑中的毛细血管，是低维的，并通过封装它们的三维域以复杂的方式相互作用。裂隙多孔介质的重要应用包括水力压裂、废物沉积和生物力学模型。该项目旨在提供新的计算范例，促进混合维度建模在这些领域和相关领域的使用，并缓解目前计算机模拟的限制。这些技术将在开放源码软件包中实施，并将向科学界提供。在本项目中，将对混合维度建模和仿真的三个重要方面进行研究。第一个研究目标是设计改进的稳定离散格式，将线弹性力学的稳定格式与达西流动的保结构离散格式相结合。稳定性和质量守恒将使用最少的自由度来实现。第二个目标是混合维度框架中的自适应近似。将开发和实施时空自适应，以提高精度，具有坚实的理论基础。第三个目标是研究所得到的离散线性系统的稳健线性求解器。基于混合维模型及其数值离散的物理和数学特性，将发展新的块预条件和整体多重网格方法。关于物理参数和离散化参数的稳健性将从理论和数值上得到证明。该项目的主要应用是裂隙多孔介质中的线性孔力学，但也将研究对非线性公式的可能概括，包括在物理和工程中的几个应用。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The overall goal of this research project is to develop, analyze, and implement stable numerical methods for the simulation of flow in fractured porous media, based on mixed-dimensional modeling. Fractured porous media flow is multi-physics and multi-scale, and the development of robust and effective numerical tools for such systems represents a class of important challenges in computational mathematics. For example, in many practical applications fractures or other features, such as capillaries in the brain, are lower-dimensional and interact in a complex way through the three-dimensional domains encapsulating them. Important applications of fractured porous media include hydraulic fracturing, waste deposition, and models from biomechanics. The project aims to provide new computational paradigms, promote the usage of mixed-dimensional modeling in these and related fields, and alleviate current limitations in computer simulations. The techniques will be implemented in an open-source software package and will be made available to the scientific community. In this project, three important aspects of mixed-dimensional modeling and simulation will be investigated. The first research objective is to design advanced stable discretizations, which couple stabilized schemes for linear elasticity with structure-preserving discretizations for Darcy flow in a mixed-dimensional setting. Stability and mass conservation will be achieved using a minimal number of degrees of freedom. The second objective is adaptive approximations in the mixed-dimensional framework. Space-time adaptivity will be developed and implemented to improve accuracy with solid theoretical foundations. The third objective is to study robust linear solvers for the resulting discrete linear systems. Based on physical and mathematical properties of the mixed-dimensional models and their numerical discretizations, new block preconditioners and monolithic multigrid methods will be developed. Robustness with respect to physical and discretization parameters will be justified both theoretically and numerically. The main application of this project is linear poromechanics in fractured porous media, but possible generalizations to nonlinear formulations will also be investigated, including several applications in physics and engineering.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.