Graph Network Surrogate Model for Subsurface Flow Optimization

TL;DR

This work introduces a graph network surrogate model (GNSM) to accelerate well placement and control optimization in subsurface flow. By deploying separate PresGNN and SatGNN networks within an encoder–processor–decoder architecture and enriching inputs with a single-phase pressure feature, the model delivers accurate predictions of pressure, saturation, and near-well rates on 2D unstructured reservoirs. A multistage multistep training strategy mitigates error accumulation during rollout, enabling efficient optimization via differential evolution with GNSM-predicted states that closely matches simulation-based results while achieving roughly 36x runtime speedups. The method demonstrates extrapolation to related permeability fields and holds promise for robust optimization across multiple geological realizations, with potential extensions to 3D problems and broader subsurface applications such as geological CO2 storage.

Abstract

The optimization of well locations and controls is an important step in the design of subsurface flow operations such as oil production or geological CO2 storage. These optimization problems can be computationally expensive, however, as many potential candidate solutions must be evaluated. In this study, we propose a graph network surrogate model (GNSM) for optimizing well placement and controls. The GNSM transforms the flow model into a computational graph that involves an encoding-processing-decoding architecture. Separate networks are constructed to provide global predictions for the pressure and saturation state variables. Model performance is enhanced through the inclusion of the single-phase steady-state pressure solution as a feature. A multistage multistep strategy is used for training. The trained GNSM is applied to predict flow responses in a 2D unstructured model of a channelized reservoir. Results are presented for a large set of test cases, in which five injection wells and five production wells are placed randomly throughout the model, with a random control variable (bottom-hole pressure) assigned to each well. Median relative error in pressure and saturation for 300 such test cases is 1-2%. The ability of the trained GNSM to provide accurate predictions for a new (geologically similar) permeability realization is demonstrated. Finally, the trained GNSM is used to optimize well locations and controls with a differential evolution algorithm. GNSM-based optimization results are comparable to those from simulation-based optimization, with a runtime speedup of a factor of 36. Much larger speedups are expected if the method is used for robust optimization, in which each candidate solution is evaluated on multiple geological models.

Figures (23)

• Figure 1: Schematic of model architecture (encoder-processor-decoder). The encoder and decoder are MLPs. The processor is a message passing neural network containing $N_{msg}$ message passing graph networks (MPGNs).
• Figure 2: MPNN schematic for $3 \times 3$ reservoir model (left). In the computational graph shown on the right, Node 5 is the target node to be updated with $N_{msg}=2$. Node 1 (black) corresponds to a cell with an injection well, and Node 9 (red) to a cell with a production well.
• Figure 3: MPNN example for a $7 \times 7$ model, with Node 5 as the target node. Nodes containing injection and production wells shown in black and red. Gray and brown nodes are matrix nodes. Gray nodes do not contribute to the computational graph for Node 5 (with $N_{msg} = 2$).
• Figure 4: Three example well configurations. All training configurations contain five injection wells (black circles) and five production wells (red circles). The background shows the log-permeability field (in md) and the unstructured simulation grid.
• Figure 5: Water-oil relative permeability curves used in this study.