Efficient and generalizable nested Fourier-DeepONet for three-dimensional geological carbon sequestration

TL;DR

A nested Fourier-DeepONet is developed by combining the expressiveness of the FNO with the modularity of a deep operator network (DeepONet) that is twice as efficient as a nested FNO for training and has at least 80% lower GPU memory requirement due to its flexibility to treat temporal coordinates separately.

Abstract

Geological carbon sequestration (GCS) involves injecting CO$_2$ into subsurface geological formations for permanent storage. Numerical simulations could guide decisions in GCS projects by predicting CO$_2$ migration pathways and the pressure distribution in storage formation. However, these simulations are often computationally expensive due to highly coupled physics and large spatial-temporal simulation domains. Surrogate modeling with data-driven machine learning has become a promising alternative to accelerate physics-based simulations. Among these, the Fourier neural operator (FNO) has been applied to three-dimensional synthetic subsurface models. Here, to further improve performance, we have developed a nested Fourier-DeepONet by combining the expressiveness of the FNO with the modularity of a deep operator network (DeepONet). This new framework is twice as efficient as a nested FNO for training and has at least 80% lower GPU memory requirement due to its flexibility to treat temporal coordinates separately. These performance improvements are achieved without compromising prediction accuracy. In addition, the generalization and extrapolation ability of nested Fourier-DeepONet beyond the training range has been thoroughly evaluated. Nested Fourier-DeepONet outperformed the nested FNO for extrapolation in time with more than 50% reduced error. It also exhibited good extrapolation accuracy beyond the training range in terms of reservoir properties, number of wells, and injection rate.

Figures (7)

• Figure 1: Nested machine learning pipeline for predicting the pressure buildup and gas saturation for five levels with gradually increasing resolution. There are two groups of inputs which include reservoir condition (3D permeability field, temperature, 3D initial pressure) and injection scheme (injection rate, and injection location). Each level starting from level 1 uses outputs from the previous level (grey arrows) on top of the reservoir condition and injection scheme (black arrows).
• Figure 2: 2-Dimensional sample visualizations of a global grid and local refinements in the dataset. (A) $x$-$y$ view. (B) $y$-$z$ view. The boundaries of different computational domains are drawn in black, red, orange, blue, and green solid lines for levels 0, 1, 2, 3, and 4, respectively. $z'$-coordinate is the depth relative to the top of the simulation domain.
• Figure 3: Fourier-DeepONet architecture. The branch and trunk nets receive two groups of inputs, which are encoded and then decoded by the Fourier layers.
• Figure 4: Effect of time batch size for Fourier-DeepONet. (A) GPU memory usage. (B) Training time per epoch. (C) Error, $\delta^{P}_{\Omega_{j}}$, for pressure buildup model at level 0 ($\mathcal{N}^P_0$). Networks are trained for 20 epochs.
• Figure 5: Examples of nested Fourier-DeepONet pressure buildup prediction for three different reservoirs on the top surface of the reservoir. (A) 2 injection wells. (B) 3 injection wells. (C) 4 injection wells.