  Geological parameterization enables the representation of geomodels in terms
of a relatively small set of variables. Parameterization is therefore very
useful in the context of data assimilation and uncertainty quantification. In
this study, a deep-learning-based geological parameterization algorithm,
CNN-PCA, is developed for complex 3D geomodels. CNN-PCA entails the use of
convolutional neural networks as a post-processor for the low-dimensional
principal component analysis representation of a geomodel. The 3D treatments
presented here differ somewhat from those used in the 2D CNN-PCA procedure.
Specifically, we introduce a new supervised-learning-based reconstruction loss,
which is used in combination with style loss and hard data loss. The style loss
uses features extracted from a 3D CNN pretrained for video classification. The
3D CNN-PCA algorithm is applied for the generation of conditional 3D
realizations, defined on $60\times60\times40$ grids, for three geological
scenarios (binary and bimodal channelized systems, and a three-facies
channel-levee-mud system). CNN-PCA realizations are shown to exhibit geological
features that are visually consistent with reference models generated using
object-based methods. Statistics of flow responses ($\text{P}_{10}$,
$\text{P}_{50}$, $\text{P}_{90}$ percentile results) for test sets of 3D
CNN-PCA models are shown to be in consistent agreement with those from
reference geomodels. Lastly, CNN-PCA is successfully applied for history
matching with ESMDA for the bimodal channelized system.
