Modeling natural convection in heterogeneous porous media using encoder-decoder convolutional neural networks

Natural convection is an important phenomenon in porous media problems. It is encountered in a variety of applications, including in enhanced oil recovery systems and geothermal reservoirs. Physics-based numerical models are widely used to simulate natural convection in porous media. Although these models are usually effective, they commonly suffer from high computational costs. This is notably problematic in repetitive runs at large time and space scales, as in uncertainty analysis, data assimilation, and sensitivity analysis. In recent years, at least four different methods have been proposed to overcome this challenge, including optimizing the numerical solution algorithm, parallel computing, cloud computing, and data-driven methods. In most cases, while data-driven models are capable of handling low-dimensional problems, they have not been very successful in dealing with high-dimensional problems, both accurately and time efficient. To overcome these challenges, we propose using the encoder-decoder convolutional neural networks (ED-CNNs) for heterogeneous porous media. We apply the ED-CNN in the context of ‘image-to-image’ regression in the following two use cases in the context of natural convection simulations: (1) as a meta-model to estimate the heat map from the Rayleigh number distribution, and (2) as an optimizer to estimate the Rayleigh number distribution from the heat map.

The proposed ED-CNN is employed to model the hypothetical example of a square porous enclosure filled with a saturated porous medium. The boundaries are impermeable, and temperatures at two opposite side walls are different, resulting in the formation of natural convection. Heterogeneity in the Rayleigh number across the problem domain is applied through zonation.A numerical modeling tool is used to generate steady-state heat maps based on a number of randomly selected Rayleigh numbers. The numerical model input-outputs are transformed into square-shaped jpg images of 64 × 64 resolution. Two ED-CNNs are trained, one as a meta-model and the other as an optimizer. Different numbers of training input-output images (including 1000, 2000, 4000, and 5000) generated from the numerical model are employed to evaluate the performance of proposed networks. Two evaluation criteria are used to assess the performance of the developed ED-CNN models: (1) the root mean squared error (RMSE), and (2) the coefficient of determination (R^2-score). The ED-CNNs have been developed using Keras and Tensorflow python libraries.

Results show that the ED-CNN accuracy, both as a meta-model and as an optimizer, is satisfactory. For the meta-model case (i.e. prediction of the temperature distribution from the Rayleigh map), the RMSE is mostly smaller than 0.15, and the R^2-score is around 0.92. In the case of ED-CNN as optimizer (i.e. estimation of the Rayleigh distribution from the heat map), RMSE is mostly in the interval [0.017-0.034], while the R^2-score is around 0.89. Acceptable results can be obtained using 2000 input-output image pairs and 150 epochs for the meta-model case, and 4000 image pairs and 200 epochs for the optimizer case. Analysis of the spatial distribution of errors shows that maximum errors occur in the middle of the problem domain where the heat map is least sensitive to the Rayleigh number. The ED-CNN model is also evaluated as an uncertainty analysis tool by comparing maps of mean and standard deviation based on the numerical model and ED-CNN predictions, showing a significant agreement with estimation error between them.

In this paper, we examine the performance of ED-CNNs, as a specialized architecture of deep neural networks, to solve the forward and inverse problems of natural convection in porous media. For this purpose, we frame the problem as one of image-to-image regression and show that the developed model is able to provide high accuracy approximations with limited training samples, effectively solving the curse of dimensionality problem associated with heterogeneous domains. In practice, the proposed methodology can be applied to image datasets obtained from not only numerical modeling, but also high-resolution imaging and non-destructive scanning techniques, to either estimate the temperature distribution due to natural convection, or to characterize the porous media based on the temperature distribution.