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传播的脆性裂缝。在油气藏中故意制造高频，以改善开采井与储层其余部分之间的连通性。它们还被用来在采矿中对矿体进行预压裂；在地热储藏库中建立裂隙网络；以及处理废物。它们还将用于二氧化碳的封存。不幸的是，传播的HF可能会穿透为石油、废物或二氧化碳提供遏制的不渗透层，导致碳氢化合物的损失和严重的环境破坏。因此，拥有预测、监测和控制HF增长的分析工具是至关重要的。这项研究的目的是开发渐近解和数值算法，以准确和有效地模拟和监测高频传播。高频的数学模型通常由一组积分-偏微分方程组以及确定移动裂缝边界位置的边界条件和传播条件组成。在这个提案中，我概述了一个研究计划，该计划将建立在我们最近在平面高频的渐近分析和计算建模方面取得的关键进展的基础上。特别是，我将开发一个健壮的算法，它将能够模拟平面高频通过层状沉积岩的传播和后退。此外，我计划开发一项新的倡议，开发计算工具来模拟HF通过预先存在的裂缝网络的传播，这对于从页岩中的矿藏中提取石油非常重要。我还计划使用这些精确的计算模型来开发和校准计算要求较低的模型。然后，这些简化的模型将与卡尔曼滤波耦合，以便通过反演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.