Asymptotic and numerical analysis of hydraulic fractures

水力裂缝（HF）是由于注入粘性流体而在岩石中扩展的脆性裂缝。高频是在油气储层中故意产生的，以改善提取井眼与储层其余部分之间的连接。在采矿中也用于预破碎矿体；在地热储层中形成裂缝网络；并处理废物。它们还将被用于封存二氧化碳。不幸的是，传播的HF会击穿为石油、废物或二氧化碳提供密封的不透水层，导致碳氢化合物的损失和严重的环境破坏。因此，有分析工具来预测、监测和控制HF的生长是至关重要的。提出的研究目标是为高频传播的准确和有效建模和监测开发渐近解和数值算法。高频振动的数学模型通常由一组积分-偏微分方程以及边界和扩展条件组成，这些条件决定了移动裂缝边界的位置。在本提案中，我概述了一个研究计划，该计划将建立在我们最近在平面HF的渐近分析和计算建模方面取得的关键进展之上。特别是，我将开发一个强大的算法，将能够模拟平面HF通过层状沉积岩的传播和衰退。此外，我计划开发一项新的计算工具，以模拟HF在现有裂缝网络中的传播，这对于从页岩沉积物中提取石油很重要。我还计划使用这些精确的计算模型来开发和校准计算需求更少的模型。然后，这些简化的模型将与卡尔曼滤波器相结合，以便通过反演它们在岩石中引起的变形来监测HF的传播。

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.