Bridging Deep Convolutional Autoencoders and Ensemble Smoothers for Improved Estimation of Channelized Reservoirs

Abstract

One of the main problems associated with applying data assimilation methods for facies models is the lack of geological plausibility in updates. This issue is even more acute for channelized reservoirs, knowing that, without a reliable parameterization, the geometrical structure of the channels can hardly be reproduced in the updated step of any data assimilation method. This paper presents a new methodology for estimation and uncertainty quantification of facies fields in channelized reservoirs, bridging a deep convolutional autoencoder with an ensemble-based method. The proposed methodology is suitable for any geological simulation model and does not use the resampling from the training image when using a multipoint geostatistical simulation model. Besides the channel estimation, the proposed methodology preserves the geological plausibility in the updated step of the history-matching method. The new methodology employs, inside the parameterization, a deep convolutional autoencoder to reconstruct the channel geometry. The convolutional autoencoder is used for image reconstruction purposes. The input of the training set of the autoencoder consists of images (facies fields) generated with a parameterization of the facies fields and perturbed with a Gaussian noise having spatial correlation. This procedure ensures the consistency of the method in the sense that the input fields have a similar structure with the facies fields obtained after the history matching. The methodology is tested for channelized reservoirs, with different levels of complexity, and also by comparison with previous methods that use or not resampling from the training image. The results show an improvement in the geological plausibility, estimation, and uncertainty quantification of the channel distributions while achieving a good data match.

Appendices

Appendix A: Statistical Measurements of the ES-MDA Updated Step

Let us consider the experiment of ES-MDA applied in four assimilation cycles with the global parameterization from Eq. (4). After performing the first assimilation cycle, the parameter fields suffer an adjustment due to the effect of observations. At the top of Fig. 29 is presented the parameter field $\theta _{1}$ of the first member in the prior ensemble ($\theta _{1}^{pr}$, at the left side), in the updated ensemble ($\theta _{1}^{up}$ in the center), and the field calculated as the difference between the first two (at the right side). By applying the truncation rule for both parameter fields, we obtained the prior and updated facies fields (Fig. 29 at the left side and center, respectively). The updated facies field has lost the geological plausibility. Let $\theta _{1}^{inv} = \theta _{1}^{up}-\theta _{1}^{pr}$, and its spatial distribution is shown in the left side of Fig. 29 (first row). Then, $\theta _{1}^{up} = \theta _{1}^{pr}+\theta _{1}^{inv}$, and from this equation, the updated field can be seen as the perturbation of the prior with $\theta _{1}^{inv}$.

Fig. 29

Top: parameter field in the prior ensemble (right), the updated ensemble (center), and the innovation (left). Bottom: facies field in the prior ensemble (left), the updated ensemble (center), and the difference between first two (right)

From a statistical perspective, the field $\theta _1^{inv}$ has the histogram from Fig. 30 which exhibits Gaussian characteristics; the mean of the field values is $mean(\theta _1^{inv})=-0.04$, and the standard deviation is $std(\theta _1^{inv})=0.6$.

Fig. 30

The histogram of field $\theta _1^{inv}$

Considering the entire ensemble of fields $\theta _i^{inv}$, where $i \in \overline{1, n_e}$ ($n_e$ is the number of ensemble members), the mean of the mean field $mean(mean(\theta ^{inv}))=-0.01$ and the mean of the standard deviations $mean(std(\theta ^{inv}))=0.62$ are calculated. Consequently, $\theta ^{inv}$ is a sample from a random field, marginally Gaussian with a 0 mean and 0.6 standard deviation. This information is further used for generation of the input of training and validation sets of the deep convolutional autoencoder.

Appendix B: Architecture of the Convolutional Autoencoders

See Tables 2 and 3.

Table 2 Convolutional autoencoder architecture—small image ($50 \times 50$)

Table 3 Convolutional autoencoder architecture—large image ($100 \times 100$)