For most history matching problems, the posterior probability density function (PDF) may have multiple local maxima, and it is extremely challenging to quantify uncertainty of model parameters and production forecasts by conditioning to production data. In this paper, a novel method is proposed to improve the accuracy of Gaussian mixture model (GMM) approximation of the complex posterior PDF by adding more Gaussian components. Simulation results of all reservoir models generated during the history matching process, e.g., using the distributed Gauss-Newton (DGN) optimizer, are used as training data points for GMM fitting. The distance between the GMM approximation and the actual posterior PDF is estimated by summing up the errors calculated at all training data points. The distance is an analytical function of unknown GMM parameters such as covariance matrix and weighting factor for each Gaussian component. These unknown GMM parameters are determined by minimizing the distance function. A GMM is accepted if the distance is reasonably small. Otherwise, new Gaussian components will be added iteratively to further reduce the distance until convergence. Finally, high-quality conditional realizations are generated by sampling from each Gaussian component in the mixture, with the appropriate relative probability. The proposed method is first validated using nonlinear toy problems and then applied to a history-matching example. GMM generates better samples with a computational cost comparable to or less than other methods we tested. GMM samples yield production forecasts that match production data reasonably well in the history-matching period and are consistent with production data observed in the blind test period.

中文翻译：

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高斯混合模型拟合方法，通过对生产数据的条件进行不确定性量化

对于大多数历史匹配问题，后验概率密度函数（PDF）可能具有多个局部最大值，并且通过对生产数据进行条件化来量化模型参数和生产预测的不确定性极具挑战性。本文提出了一种新的方法，通过添加更多的高斯分量来提高复杂后验PDF的高斯混合模型（GMM）逼近的准确性。在历史匹配过程中生成的所有储层模型的模拟结果（例如，使用分布式高斯-牛顿（DGN）优化器）的模拟结果均用作GMM拟合的训练数据点。通过对所有训练数据点上计算出的误差求和，可以估算出GMM近似值与实际后PDF的距离。该距离是未知GMM参数（例如每个高斯分量的协方差矩阵和加权因子）的解析函数。这些未知的GMM参数是通过最小化距离函数来确定的。如果距离相当小，则接受GMM。否则，将迭代添加新的高斯分量，以进一步减小距离直至收敛。最后，通过以适当的相对概率从混合物中的每个高斯分量采样来生成高质量的条件实现。首先使用非线性玩具问题对提出的方法进行了验证，然后将其应用于历史匹配示例。GMM产生的更好样本的计算成本可与我们测试的其他方法相比，甚至更低。