Subsurface storage of CO$_2$ is an important means to mitigate climate change, and to investigate the fate of CO$_2$ over several decades in vast reservoirs, numerical simulation based on realistic models is essential. Faults and other complex geological structures introduce modeling challenges as their effects on storage operations are uncertain due to limited data. In this work, we present a computational framework for forward propagation of uncertainty, including stochastic upscaling and copula representation of flow functions for a CO$_2$ storage site using the Vette fault zone in the Smeaheia formation in the North Sea as a test case. The upscaling method leads to a reduction of the number of stochastic dimensions and the cost of evaluating the reservoir model. A viable model that represents the upscaled data needs to capture dependencies between variables, and allow sampling. Copulas provide representation of dependent multidimensional random variables and a good fit to data, allow fast sampling, and coupling to the forward propagation method via independent uniform random variables. The non-stationary correlation within some of the upscaled flow function are accurately captured by a data-driven transformation model. The uncertainty in upscaled flow functions and other parameters are propagated to uncertain leakage estimates using numerical reservoir simulation of a two-phase system. The expectations of leakage are estimated by an adaptive stratified sampling technique, where samples are sequentially concentrated to regions of the parameter space to greedily maximize variance reduction. We demonstrate cost reduction compared to standard Monte Carlo of one or two orders of magnitude for simpler test cases with only fault and reservoir layer permeabilities assumed uncertain, and factors 2--8 cost reduction for stochastic multi-phase flow properties and more complex stochastic models.

翻译：摘要：CO$_2$地下封存是缓解气候变化的重要手段，为研究CO$_2$在大型储层中数十年的归趋，基于现实模型的数值模拟至关重要。断层及其他复杂地质结构因数据有限导致其对封存作业的影响存在不确定性，从而引入建模挑战。本文提出一种不确定性前向传播计算框架，包括随机尺度升级和流动函数的Copula表示，并以北海斯梅阿希亚组维特断层带为测试案例针对CO$_2$封存场地展开研究。尺度升级方法可减少随机维度数量并降低储层模型评估成本。能够表示尺度升级数据的可行模型需捕捉变量间依赖关系并支持采样。Copula提供相依多维随机变量的表示，兼具数据拟合度佳、快速采样及通过独立均匀随机变量与前向传播方法耦合的优势。针对某些非平稳相关性的尺度升级流动函数，可采用数据驱动变换模型精确捕捉。通过两相系统数值储层模拟，将尺度升级流动函数及其他参数的不确定性传播至泄漏估计的不确定性。采用自适应分层采样技术估算泄漏期望值，通过将样本依次集中至参数空间区域以贪婪最大化方差缩减。研究表明，在仅假设断层和储层渗透率不确定的简单测试案例中，本方法相比标准蒙特卡罗方法降低一至两个数量级成本；对于随机多相流动特性及更复杂随机模型，成本降低幅度为2-8倍。