# A Gradient-based Deep Neural Network Model for Simulating Multiphase Flow in Porous Media

arXiv: https://arxiv.org/abs/2105.02652 · HTML: https://mlsanity.com/paper/2105.02652/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 53 references that are in the map.
- First posted on arXiv 30 Apr 2021.

## Builds on (most cited first, 7 of 7)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Physics-guided Neural Networks (PGNN): An Application in Lake Temperature Modeling](https://mlsanity.com/paper/1710.11431/) (2017)
- [A deep-learning-based surrogate model for data assimilation in dynamic subsurface flow problems](https://mlsanity.com/paper/1908.05823/) (2019)
- [Deep Learning of Subsurface Flow via Theory-guided Neural Network](https://mlsanity.com/paper/1911.00103/) (2019)
- [Generalized Multiscale Finite Element Methods (GMsFEM)](https://mlsanity.com/paper/1301.2866/) (2013)

## Built on by (most cited first, 8 of 8)

- [Physics-Informed Graph Neural Network for Spatial-temporal Production Forecasting](https://mlsanity.com/paper/2209.11885/) (2022)
- [Physics-informed machine learning with differentiable programming for heterogeneous underground reservoir pressure management](https://mlsanity.com/paper/2206.10718/) (2022)
- [A Critical Review of Physics-Informed Machine Learning Applications in Subsurface Energy Systems](https://mlsanity.com/paper/2308.04457/) (2023)
- [Transfer learning-based physics-informed convolutional neural network for simulating flow in porous media with time-varying controls](https://mlsanity.com/paper/2310.06319/) (2023)
- [Physics-Informed DeepONet Coupled with FEM for Convective Transport in Porous Media with Sharp Gaussian Sources](https://mlsanity.com/paper/2508.19847/) (2025)
- [Graph Convolutional Networks for Simulating Multi-phase Flow and Transport in Porous Media](https://mlsanity.com/paper/2307.04449/) (2023)
- [Differentiable multiphase flow model for physics-informed machine learning in reservoir pressure management](https://mlsanity.com/paper/2508.19419/) (2025)
- [Estimating Parameter Fields in Multi-Physics PDEs from Scarce Measurements](https://mlsanity.com/paper/2509.00203/) (2025)

## Abstract (arXiv)

Simulation of multiphase flow in porous media is crucial for the effective management of subsurface energy and environment related activities. The numerical simulators used for modeling such processes rely on spatial and temporal discretization of the governing partial-differential equations (PDEs) into algebraic systems via numerical methods. These simulators usually require dedicated software development and maintenance, and suffer low efficiency from a runtime and memory standpoint. Therefore, developing cost-effective, data-driven models can become a practical choice since deep learning approaches are considered to be universal approximations. In this paper, we describe a gradient-based deep neural network (GDNN) constrained by the physics related to multiphase flow in porous media. We tackle the nonlinearity of flow in porous media induced by rock heterogeneity, fluid properties and fluid-rock interactions by decomposing the nonlinear PDEs into a dictionary of elementary differential operators. We use a combination of operators to handle rock spatial heterogeneity and fluid flow by advection. Since the augmented differential operators are inherently related to the physics of fluid flow, we treat them as first principles prior knowledge to regularize the GDNN training. We use the example of pressure management at geologic CO2 storage sites, where CO2 is injected in saline aquifers and brine is produced, and apply GDNN to construct a predictive model that is trained from physics-based simulation data and emulates the physics process. We demonstrate that GDNN can effectively predict the nonlinear patterns of subsurface responses including the temporal-spatial evolution of the pressure and saturation plumes. GDNN has great potential to tackle challenging problems that are governed by highly nonlinear physics and enables development of data-driven models with higher fidelity.