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 (PDE) 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 PDE 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 PDE 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. Our first novel contribution is to identify a sampling methodology for the random environment and matching kernel technology for which fitting the Gaussian process regression model scales as O(n log n) instead of the typical O(n^3) rate in the number of samples n used to fit the surrogate. The surrogate model allows almost instantaneous predictions for the pressure at the critical location as a function of the extraction rate and permeability realization. Our second contribution is a novel algorithm to calibrate the uncertainty in the surrogate model to the discrepancy between the true pressure solution of Darcy's equation and the numerical solution. Although our method is derived for building a surrogate for the solution of Darcy's equation with a random permeability field, the framework broadly applies to solutions of other PDE with random coefficients.

翻译：限制注入速率以将关键位置的压力控制在阈值以下，是模拟注入井与生产井之间地下压力时的重要目标。压力通过达西偏微分方程在给定渗透率场下的解来近似。由于地下渗透率仅已知其统计特性，因此被建模为随机场，这给计算压力带来了不确定性。求解一组随机渗透率模拟的偏微分方程，可以估计关键位置压力的概率分布。这些模拟计算成本高昂，而从业者通常需要快速在线指导以实现实时压力管理。利用数值偏微分方程解构成的集合，建立高斯过程回归模型，该模型能够快速预测关键位置压力随开采速率和渗透率实现的变化。我们的第一项创新贡献是确定了一种随机环境采样方法及匹配的核函数技术，使得高斯过程回归模型的拟合复杂度从常规的O(n³)降低至O(n log n)（n用于拟合代理的样本数）。该代理模型几乎可即时预测关键位置压力随开采速率和渗透率实现的变化。第二项贡献是一种新算法，用于校准代理模型中的不确定性，以匹配达西方程真实解与数值解之间的差异。尽管该方法专为构建随机渗透率场下达西方程解的代理而设计，但其框架可广泛适用于其他具有随机系数的偏微分方程的解。