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Application of ensemble transform data assimilation methods for parameter estimation in reservoir modeling

机译：集成变换数据同化方法在油藏建模参数估计中的应用

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Over the years data assimilation methods have been developed to obtain estimations of uncertain model parameters by taking into account a few observations of a model state. The most reliable Markov chain Monte Carlo (MCMC) methods are computationally expensive. Sequential ensemble methods such as ensemble Kalman filters and particle filters provide a favorable alternative. However, ensemble Kalman filter has an assumption of Gaussianity. Ensemble transform particle filter does not have this assumption and has proven to be highly beneficial for an initial condition estimation and a small number of parameter estimations in chaotic dynamical systems with non-Gaussian distributions. In this paper we employ ensemble transform particle filter (ETPF) and ensemble transform Kalman filter (ETKF) for parameter estimation in nonlinear problems with 1, 5, and 2500 uncertain parameters and compare them to importance sampling (IS). The large number of uncertain parameters is of particular interest for subsurface reservoir modeling as it allows us to parameterize permeability on the grid. We prove that the updated parameters obtained by ETPF lie within the range of an initial ensemble, which is not the case for ETKF. We examine the performance of ETPF and ETKF in a twin experiment setup, where observations of pressure are synthetically created based on the known values of parameters. For a small number of uncertain parameters (one and five) ETPF performs comparably to ETKF in terms of the mean estimation. For a large number of uncertain parameters (2500) ETKF is robust with respect to the initial ensemble, while ETPF is sensitive due to sampling error. Moreover, for the high-dimensional test problem ETPF gives an increase in the root mean square error after data assimilation is performed. This is resolved by applying distance-based localization, which however deteriorates a posterior estimation of the leading mode by largely increasing the variance due to a combination of less varying localized weights, not keeping the imposed bounds on the modes via the Karhunen–Loeve expansion, and the main variability explained by the leading mode. A possible remedy is instead of applying localization to use only leading modes that are well estimated by ETPF, which demands knowledge of which mode to truncate.
机译：多年来，通过考虑对模型状态的一些观察，已经开发了数据同化方法来获得不确定模型参数的估计。最可靠的马尔可夫链蒙特卡洛（MCMC）方法在计算上昂贵。诸如集合卡尔曼滤波器和粒子滤波器之类的顺序集合方法提供了一种有利的选择。但是，集合卡尔曼滤波器具有高斯假设。集成变换粒子滤波器不具有此假设，并且已被证明对具有非高斯分布的混沌动力学系统中的初始条件估计和少量参数估计非常有帮助。在本文中，我们采用集合变换粒子滤波器（ETPF）和集合变换卡尔曼滤波器（ETKF）来估计具有1、5和2500个不确定参数的非线性问题的参数，并将它们与重要性抽样（IS）进行比较。大量不确定性参数对于地下储层建模尤为重要，因为它允许我们对网格的渗透率进行参数化。我们证明了由ETPF获得的更新参数在初始集合范围内，而ETKF则不是这样。我们在一个双实验装置中检查了ETPF和ETKF的性能，在该装置中，压力的观察是基于已知参数值综合创建的。对于少量不确定参数（一和五个），就平均估计而言，ETPF与ETKF的性能相当。对于大量不确定参数（2500个），ETKF相对于初始整体具有鲁棒性，而ETPF由于采样误差而敏感。此外，对于高维测试问题，ETPF在执行数据同化后会增加均方根误差。这可以通过应用基于距离的局部化来解决，但是由于结合了较少变化的局部权重，未通过Karhunen-Loeve展开保持模式的强制边界，从而大大增加了方差，从而恶化了主导模式的后验估计，领先模式解释了主要可变性。一种可能的解决方法是代替应用本地化以仅使用由ETPF很好估计的领先模式，这要求了解截断哪种模式。

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