[论文解读] Modern theory of hydraulic fracture modeling with using explicit and implicit schemes

A. M. Linkov|arXiv (Cornell University)|May 8, 2019

Hydraulic Fracturing and Reservoir Analysis参考文献 51被引用 6

一句话总结

本文通过强调速度方程在水力压裂（HF）建模中的关键作用，推进了水力压裂建模的发展，表明显式时间积分在近前缘区域由于CFL条件限制更宽松，可优于隐式格式。本文引入了通用渐近包络概念，并表明前缘区域的波状传播特性使得迎风格式得以高效应用，从而使显式方法在水力压裂模拟中具有竞争力或优势，尤其在具有动态裂纹轮廓的三维模拟中表现更优。

ABSTRACT

The paper presents novel results, obtained on the basis of the modified theory of hydraulic fractures (HF). The theory underlines significance of the speed equation. When applied to numerical simulation of HF, the theory reveals three distinct issues: (i) modeling the central part of a HF; (ii) modeling the near-front zone; and (iii) tracing changes in the shape of a fracture contour. Modeling the central part leads to a stiff system of ODE in time, what strongly complicates its integration. For explicit schemes, it requires small time steps to meet the CFL condition. For implicit schemes, it requires proper preconditioners. The gains and flaws of the two strategies are discussed. It is noted that a rough spatial mesh may be used in the central part. Modeling the near-front zone reveals the vital role of the speed equation for HF modeling by any method. Its asymptotic analysis has resulted in the fundamental concept of the universal asymptotic umbrella. For the near-front zone, it is also established that a notable part of the zone adjacent to the front propagates virtually as a simple wave. This implies that the CFL condition of stability for this zone is much less restrictive than for the central part of the fracture. Of essence is also that the wave-like propagation of the near front zone makes preferable upwind schemes of time stepping. On whole, the analysis implies that explicit time stepping may be complementing, competitive and even superior over implicit integration. Tracing changes in the front shape appears merely in 3D problems when the contour changes its form. The problem, being essentially geometrical, it may be solved separately by various methods, including fast marching, level set and the simplest string/marker methods. In 1D cases, it does not arise at all. Quantitative estimations and numerical examples illustrate the theoretical conclusions.

研究动机与目标

• 为解决模拟水力压裂过程中的计算挑战，特别是中心区域的刚性问题和前缘传播的复杂性。
• 评估显式与隐式时间积分格式在水力压裂建模中的性能与稳定性。
• 为追踪三维中裂纹轮廓演化（尤其是形状变化时）建立稳健的框架。
• 基于渐近分析，建立在近前缘区域使用迎风格式的理论与数值基础。
• 证明显式格式在某些裂纹区域可与隐式格式竞争甚至更优，这与传统假设相反。

提出的方法

• 采用一种改进的水力压裂理论，强调速度方程在建模动力学中的核心作用。
• 对近前缘区域进行渐近分析，导出通用渐近包络概念。
• 识别出近前缘区域表现为简单波，从而导致CFL条件显著放宽。
• 由于波状传播特性，提出适用于近前缘区域的迎风时间推进格式。
• 对三维问题采用独立的几何追踪方法，如快速行进法、水平集法或标记/绳索法。
• 通过数值算例与定量估算，验证不同网格分辨率和格式下的理论结论。

实验结果

研究问题

• RQ1 在模拟水力压裂时，显式与隐式时间积分格式在稳定性与效率方面如何比较？
• RQ2 速度方程在决定近前缘区域水力压裂传播行为中起什么作用？
• RQ3 考虑到稳定性约束，显式格式是否能在水力压裂建模中与隐式格式竞争甚至更优？
• RQ4 前缘传播的波状特性如何影响数值格式选择与时间步长限制？
• RQ5 在三维水力压裂模拟中，追踪裂纹轮廓演化最有效的方法是什么？

主要发现

• 近前缘区域表现出波状传播特性，导致CFL条件相比裂纹中心部分显著放宽。
• 速度方程控制近前缘区域的动力学行为，其渐近分析导出通用渐近包络概念，统一了不同模型间的演化行为。
• 由于近前缘区域稳定性约束放宽，显式时间积分可与隐式格式竞争甚至更优。
• 由于波状传播特性，迎风格式特别适用于近前缘区域，可增强数值稳定性和计算效率。
• 在三维模拟中，裂纹轮廓随时间变化，需采用独立的几何追踪方法（如快速行进法或水平集法），而一维情况则无需此类追踪。
• 粗糙的空间网格足以满足裂纹中心区域的模拟需求，可在不损失精度的前提下降低计算成本。