Application of Bayesian Generative Adversarial Networks to Geological Facies Modeling

Abstract

Geological facies modeling is a key component in exploration and characterization of subsurface reservoirs. While traditional geostatistical approaches are still commonly used nowadays, deep learning is gaining a lot of attention within geoscientific community for generating subsurface models, as a result of recent advance of computing powers and increasing availability of training data sets. This work presents a deep learning approach for geological facies modeling based on generative adversarial networks (GANs) combined with training-image-based simulation. In a typical application of learned networks, all neural network parameters are fixed after training, and the uncertainty in the trained model cannot be analyzed. To address this problem, a Bayesian GANs (BGANs) approach is proposed to create facies models. In this approach, a probability distribution is assigned to the neural parameters of the BGANs. Only neural parameters of the generator in BGANs are assigned with a probability function, and the ones in the discriminator are treated as fixed. Random samples are then drawn from the posterior distribution of neural parameters to simulate subsurface facies models. The proposed approach is applied to the two different geological depositional scenarios, fluvial channels and carbonate mounds, and the generated models reasonably capture the variability of the training/testing data. Meanwhile, the model uncertainty of learned networks is readily accessible. To fully sample the spatial distribution in the training image set, a large collection of samples of network parameters is required to be drawn from the posterior distribution, thus significantly increasing computational cost.

Appendix: Quality Evaluation Metrics

To assess the variability and similarity between the generated and testing data, the technique of multidimensional scaling (MDS) is used to reduce the high-dimensional problem to a low-dimensional data space while preserving similarity measure for comparison (Cox and Cox 2008). The MDS-transformed data points are unitless and can be plotted in the common Cartesian space. The main steps in the classical MDS algorithm are summarized as follows: (i) setup of the squared proximity matrix with Euclidean distance and double centering; (ii) determination of the first largest eigenvalues and corresponding eigenvectors of the double-centered matrix; (iii) calculation of the object coordinates in the new space (Cox and Cox 2008). The variability and similarity can be evaluated in the reduced dimension for the two model sets. For example, the projected data points can be sparsely distributed across the Cartesian space when diverse patterns exist in the original data or distributed in small clusters for multimodal distributions of patterns. The similarity of the data sets can be quantified by the range of these two clusters, and it should become indistinguishable when the two data sets are similar (Zhang et al. 2021).

For a quantitative measure of the variability within the generated models, the perception-based structural similarity index (SSIM) is introduced, on the basis of the definition given by Wang et al. (2004) and Sun (2018)

$$\mathrm{SSIM}\left(y,{y}^{*}\right)= \frac{(2{\mu }_{y}{\mu }_{{y}^{*}}+{c}_{1})(2{\sigma }_{y{y}^{*}}+{c}_{2})}{({\mu }_{y}^{2}+{\mu }_{{y}^{*}}^{2}+{c}_{1})({\sigma }_{y}^{2}+{\sigma }_{{y}^{*}}^{2}+{c}_{2})},$$

(A1)

where $y$ and ${y}^{*}$ represent two random images from the generated models, respectively; $\mu$ is the mean and ${\sigma }^{2}$ is the variance; ${c}_{1}$ and ${c}_{2}$ are constant values to stabilize the denominator, and are taking the values of ${0.01}^{2}$ and ${0.03}^{2}$, respectively, for the gray images. The value range of SSIM is between −1 and 1, where 1 means that the two images are identical, and −1 is reached when the two images are completely different (Sun 2018). In image recognition, SSIM quantifies image degradation as structural information change where pixels that are spatially close with each other should have strong inter-dependencies. The perceptual phenomena, including the masking terms of luminance and contrast, are also accounted for in the SSIM to ensure that image distortions are detected as well (Wang et al. 2004). In general, compared to the mean squared error, SSIM is more indicative of perceived dissimilarity/similarity in the models.