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Comparison of fully-coupled and sequential solution methodologies for enhanced geothermal systems
Computer Methods in Applied Mechanics and Engineering ( IF 7.2 ) Pub Date : 2021-01-01 , DOI: 10.1016/j.cma.2020.113554
Bruce Gee , Robert Gracie

Abstract The simulation of enhanced geothermal systems (EGS) requires finding the solution to a highly coupled nonlinear set of partial differential equations. Verification of EGS modelling has been difficult due to a lack of analytical or semi-analytical solutions and is typically limited to a subset of processes. A comparison of alternative solution schemes is one method to numerically verify the coupling between all processes simultaneously and provide confidence in the solutions produced by the different solution schemes. This article presents the first such comparison of monolithic and sequential solution schemes for the modelling of EGS wells. A custom thermo-hydro-mechanical finite element model for an EGS well connected by planar fractures is developed along with two monolithic schemes; one using the Newton–Raphson iterative method and another using the Newton–Raphson iterative method modified by Aitken’s Δ 2 relaxation method. The two monolithic schemes are compared against two sequential schemes and one loosely-coupled scheme in terms of accuracy and computational efficiency. The monolithic schemes are shown to have an optimal rate of convergence with respect to mesh and timestep refinement. The monolithic Newton–Raphson scheme is shown to require small timesteps that violate the minimum timestep size of the β -method of time integration, introducing spurious oscillations into the solution. Aitken’s Δ 2 relaxation method is shown to improve the ability of the Newton–Raphson scheme to find a converged solution at larger timesteps, avoiding spurious oscillations and reducing overall computation time. The sequential solution schemes are shown to provide optimal rates of convergence with respect to mesh and timestep refinement and converge to the same solution as the monolithic solution schemes, verifying the results of both methodologies and previous modelling efforts. It is demonstrated that the sequential solution schemes are computationally more expensive than the monolithic schemes when seeking well-converged solutions. The loosely-coupled scheme demonstrates an optimal rate of convergence with mesh and timestep refinement, but is also shown to be much less accurate than the other schemes. The recommended solution scheme for the efficient simulation of EGSs is thus the Newton–Raphson scheme with Aitken’s Δ 2 relaxation.

中文翻译:

增强地热系统的全耦合和顺序求解方法的比较

摘要 增强型地热系统 (EGS) 的模拟需要找到一组高度耦合的非线性偏微分方程的解。由于缺乏分析或半分析解决方案,EGS 建模的验证一直很困难,并且通常仅限于过程的子集。替代解决方案的比较是一种同时对所有过程之间的耦合进行数值验证的方法,并提供对不同解决方案产生的解决方案的信心。本文首次介绍了用于 EGS 井建模的整体和顺序解决方案方案的此类比较。与两个整体方案一起开发了用于由平面裂缝连接的 EGS 井的定制热-水-机械有限元模型;一种使用 Newton-Raphson 迭代法,另一种使用由 Aitken 的 Δ 2 松弛法修改的 Newton-Raphson 迭代法。在准确性和计算效率方面,将两种单片方案与两种顺序方案和一种松耦合方案进行比较。整体方案显示出在网格和时间步长细化方面具有最佳收敛速度。整体 Newton-Raphson 方案被证明需要小时间步长,这违反了时间积分 β 方法的最小时间步长大小,从而在解决方案中引入了虚假振荡。Aitken 的 Δ 2 弛豫方法被证明可以提高 Newton-Raphson 方案在更大的时间步长上找到收敛解的能力,避免虚假振荡并减少整体计算时间。顺序求解方案显示出在网格和时间步长细化方面提供最佳收敛速度,并收敛到与整体求解方案相同的解决方案,验证了两种方法和先前建模工作的结果。结果表明,在寻求良好收敛的解决方案时,顺序解决方案方案在计算上比整体方案更昂贵。松散耦合方案展示了网格和时间步细化的最佳收敛速度,但也显示出远不如其他方案准确。因此,用于有效模拟 EGS 的推荐解决方案是具有 Aitken Δ 2 弛豫的 Newton-Raphson 方案。
更新日期:2021-01-01
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