Model errors are ubiquitous in practical history matching problems. A common approach in the literature to accounting for model errors is to treat them as random variables following certain presumed distributions. While such a treatment renders algorithmic convenience, its underpinning assumptions are often invalid. In this work, we adopt an alternative approach, and treat model-error characterization as a functional approximation problem, which can be solved using a generic machine learning method. We then integrate the proposed model-error characterization approach into an ensemble-based history matching framework, and show that, with very minor modifications, existing ensemble-based history matching algorithms can be readily deployed to solve the history matching problem in the presence of model errors.

To demonstrate the efficacy of the integrated history matching framework, we apply it to account for potential model errors of a rock physics model in 4D seismic history matching applied to the full Norne benchmark case. The numerical results indicate that the proposed model-error characterization approach helps improve the qualities of estimated reservoir models, and leads to more accurate forecasts of production data. This suggests that accounting for model errors from a perspective of machine learning serves as a viable way to deal with model imperfection in practical history matching problems.

在实际的历史匹配问题中普遍存在模型错误。文献中解决模型误差的一种常用方法是将它们视为遵循某些假定分布的随机变量。尽管这种处理为算法提供了便利，但其基础假设通常是无效的。在这项工作中，我们采用了一种替代方法，并将模型错误表征作为函数逼近问题，可以使用通用机器学习方法来解决。然后，我们将提出的模型错误特征描述方法集成到基于集成的历史匹配框架中，并表明，只需进行很小的修改，就可以轻松地部署现有的基于集成的历史匹配算法来解决模型存在时的历史匹配问题错误。

为了证明集成历史匹配框架的有效性，我们将其用于解决在应用于完整Norne基准案例的4D地震历史匹配中岩石物理模型的潜在模型错误。数值结果表明，所提出的模型误差特征描述方法有助于提高估计的储层模型的质量，并能更准确地预测生产数据。这表明从机器学习的角度考虑模型错误是解决实际历史匹配问题中模型不完善的一种可行方法。