Asymptotic and numerical analysis of hydraulic fractures

水力压裂（HF）是由于粘性流体的注入而在岩石中传播的脆性裂缝。在油气藏中故意产生HF，以改善开采钻孔与储层其余部分之间的连接。它们还用于在采矿中预先压裂矿体;在地热储层中建立裂缝网络;以及处理废物。它们还将用于封存二氧化碳。不幸的是，传播的HF会破坏为油、废物或CO2提供密封的不可渗透层，导致碳氢化合物的损失和严重的环境破坏。因此，至关重要的是要有分析工具来预测，监测和控制HF的增长。拟议的研究的目的是开发渐近解和数值算法的准确和有效的建模和监测的HF传播。HF的数学模型通常由积分-偏微分方程系统以及确定移动裂缝边界位置的边界和传播条件组成。在这个建议中，我概述了一个研究计划，将建立在我们最近取得的进展，在渐近分析和计算建模的平面HF的关键。特别是，我将开发一个强大的算法，将能够模拟通过层状沉积岩的平面HF的传播和衰退。此外，我计划一个新的举措，开发计算工具，通过预先存在的裂缝，这是重要的页岩中的石油矿床的提取网络HF的传播模型。我还计划使用这些精确的计算模型来开发和校准具有降低计算要求的模型。这些简化的模型将与卡尔曼滤波器耦合，以便通过反转它们在岩石中引起的变形来监测传播的HF。

Hydraulic fractures (HF) are brittle fractures that propagate in rock due to the injection of a viscous fluid. HF are deliberately created in oil and gas reservoirs to improve the connection between the extraction borehole and the rest of the reservoir. They are also used to pre-fracture ore-bodies in mining; to create fracture networks in geothermal reservoirs; and to dispose of waste. They will also be used in the sequestration of CO2. Unfortunately, propagating HF can perforate the impermeable layers that provide containment to the oil, waste, or CO2, leading to loss of hydrocarbons and severe environmental damage. It is thus crucial to have analysis tools to predict, monitor, and control the growth of HF. The objective of the proposed research is to develop asymptotic solutions and numerical algorithms for the accurate and efficient modeling and monitoring of HF propagation. The mathematical models for HF typically consist of a system of integro-partial differential equations together with boundary and propagation conditions that determine the location of the moving fracture boundaries. In this proposal, I outline a program of research that will build on the key developments that we have recently achieved in the asymptotic analysis and computational modeling of planar HF. In particular, I will develop a robust algorithm that will be able to model the propagation and recession of a planar HF through layered sedimentary rock. In addition, I plan a new initiative to develop computational tools to model the propagation of HF through networks of pre-existing fractures, which is important for the extraction of oil from deposits in shales. I also plan to use these accurate computational models to develop and calibrate models with reduced computational requirements. These reduced models will then be coupled with the Kalman Filter in order to monitor propagating HF by inverting the deformations that they induce in the rock.