Physics-Informed Spatio-Temporal Graph Neural Network for Waterflood Management
Authors: Darabi Hamed , Kianinejad Amir , Salehi Amir 
Published: October 2022

Abstract
Conventional approaches such as operational spreadsheets and reservoir simulation are ineffective for actively managing waterflooding; either too simplistic or challenging to (re)calibrate in a short timeframe for operational decision-making. This paper presents a framework that optimally blends physics-based and data-driven approaches for fast and reliable subsurface modeling. The formulation is based on a graph neural network (GNN), capable of capturing spatial and temporal patterns, and leverages critical physics to improve model accuracy and generalization.
We represent the reservoir by a heterogeneous, dynamic, directed graph with injector and producer nodes where directed edges connect nearby nodes. We selected GNN for modeling the waterflood network since other deep-learning approaches (CNN and RNN) are generally suitable for regular Euclidean data (2D grids and 1D sequences). We use message passing with attention to leverage the graph's topology and reduce the number of trainable parameters. Production in each producer is equal to the weighted summation of signals received by nearby injector/aquifer nodes, where the connection's strength (well allocation factor) and efficiency (oil-cut function) represent weights. Strength quantifies the hydraulic communication between the node pairs and is estimated by solving single-phase pressure and tracer equations on an adaptive 3D unstructured PEBI grid. Efficiency indicates the percentage of total strength that contributes to oil production and is characterized by a sigmoid-like function with trainable parameters estimated by minimizing a regression loss using gradient-descent-based optimization. Also, a Markov-Chain Monte-Carlo (MCMC)-based uncertainty quantification framework is developed to quantify the model parameters' uncertainty.
The proposed methodology was successfully applied to many reservoirs across the world. This paper shows the results for a carbonate field with more than 150 wells, 60 years of history, and a 50% water cut. The asset team's objective was to increase oil production while maintaining water production. The GNN model was trained with test-set (last 12 months of production history was held out a test set) accuracy of 90% and then used to optimize the waterflooding strategy for the next six months. After applying a non-linear constrained pattern search method, the optimized strategy resulted in a 26,100 STB/D increase in oil production without a drastic change in water production level. This outcome was achieved only by adjusting the injection rates and producers' operational conditions without drilling or major workovers.
The presented modeling approach has many benefits for actively managing waterflooding: a) more than 90% speed-up for model building and (re)training compared to conventional workflows, b) super-fast simulations with GNN, c) improved model accuracy/generalization leveraging a physics-informed machine learning, d) more robust decision making through uncertainty quantification, and e) significantly shorter decision cycles in waterflood operations for well control optimization to increase oil recovery and/or reduce water production.