  This study investigates the integration of machine learning (ML) and data
assimilation (DA) techniques, focusing on implementing surrogate models for
Geological Carbon Storage (GCS) projects while maintaining high fidelity
physical results in posterior states. Initially, we evaluate the surrogate
modeling capability of two distinct machine learning models, Fourier Neural
Operators (FNOs) and Transformer UNet (T-UNet), in the context of CO$_2$
injection simulations within channelized reservoirs. We introduce the
Surrogate-based hybrid ESMDA (SH-ESMDA), an adaptation of the traditional
Ensemble Smoother with Multiple Data Assimilation (ESMDA). This method uses
FNOs and T-UNet as surrogate models and has the potential to make the standard
ESMDA process at least 50% faster or more, depending on the number of
assimilation steps. Additionally, we introduce Surrogate-based Hybrid RML
(SH-RML), a variational data assimilation approach that relies on the
randomized maximum likelihood (RML) where both the FNO and the T-UNet enable
the computation of gradients for the optimization of the objective function,
and a high-fidelity model is employed for the computation of the posterior
states. Our comparative analyses show that SH-RML offers better uncertainty
quantification compared to conventional ESMDA for the case study.
