Limiting the injection rate to restrict the pressure below a threshold at a critical location can be an important goal of simulations that model the subsurface pressure between injection and extraction wells. The pressure is approximated by the solution of Darcy’s partial differential equation for a given permeability field. The subsurface permeability is modeled as a random field since it is known only up to statistical properties. This induces uncertainty in the computed pressure. Solving the partial differential equation for an ensemble of random permeability simulations enables estimating a probability distribution for the pressure at the critical location. These simulations are computationally expensive, and practitioners often need rapid online guidance for real-time pressure management. An ensemble of numerical partial differential equation solutions is used to construct a Gaussian process regression model that can quickly predict the pressure at the critical location as a function of the extraction rate and permeability realization. The Gaussian process surrogate analyzes the ensemble of numerical pressure solutions at the critical location as noisy observations of the true pressure solution, enabling robust inference using the conditional Gaussian process distribution.

中文翻译：

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用于通过多孔介质的地数值解的计算效率和误差感知代理构造


限制注入速率以将关键位置的压力限制在阈值以下可能是模拟注入井和开采井之间地下压力的模拟的一个重要目标。该压力由给定磁导率场的达西偏微分方程的解近似。次表面渗透率被建模为随机场，因为它只知道统计属性。这会导致计算压力的不确定性。求解随机磁导率模拟系综的偏微分方程，可以估计临界位置压力的概率分布。这些模拟的计算成本很高，从业者通常需要快速的在线指导才能进行实时压力管理。使用数值偏微分方程解的集合来构建高斯过程回归模型，该模型可以快速预测临界位置的压力作为提取速率和磁导率实现的函数。高斯过程代理将关键位置的数值压力解集合分析为真实压力解的噪声观测值，从而能够使用条件高斯过程分布进行稳健推理。