[Paper Review] Uncertainty Quantification for Transport in Porous media using Parameterized Physics Informed neural Networks

Cédric Fraces Gasmi, Hamdi A. Tchelepi|arXiv (Cornell University)|May 19, 2022

Reservoir Engineering and Simulation Methods4 citations

TL;DR

This paper introduces a Parameterized Physics-Informed Neural Network (P-PINN) framework to quantify uncertainty in two-phase flow through heterogeneous porous media, treating reservoir properties (porosity, permeability) as random variables. By embedding stochastic input parameters into the PINN architecture, the model learns to predict full probability distributions of saturation and flow dynamics, achieving high accuracy comparable to Monte Carlo simulations with orders-of-magnitude faster inference.

ABSTRACT

We present a Parametrization of the Physics Informed Neural Network (P-PINN) approach to tackle the problem of uncertainty quantification in reservoir engineering problems. We demonstrate the approach with the immiscible two phase flow displacement (Buckley-Leverett problem) in heterogeneous porous medium. The reservoir properties (porosity, permeability) are treated as random variables. The distribution of these properties can affect dynamic properties such as the fluids saturation, front propagation speed or breakthrough time. We explore and use to our advantage the ability of networks to interpolate complex high dimensional functions. We observe that the additional dimensions resulting from a stochastic treatment of the partial differential equations tend to produce smoother solutions on quantities of interest (distributions parameters) which is shown to improve the performance of PINNS. We show that provided a proper parameterization of the uncertainty space, PINN can produce solutions that match closely both the ensemble realizations and the stochastic moments. We demonstrate applications for both homogeneous and heterogeneous fields of properties. We are able to solve problems that can be challenging for classical methods. This approach gives rise to trained models that are both more robust to variations in the input space and can compete in performance with traditional stochastic sampling methods.

Motivation & Objective

• To address uncertainty in reservoir engineering by modeling stochastic variations in porosity and permeability as random fields.
• To extend Physics-Informed Neural Networks (PINNs) to handle ensemble solutions of hyperbolic PDEs under uncertainty.
• To demonstrate that P-PINNs can accurately predict statistical moments (mean and variance) of saturation and front propagation in the Buckley-Leverett problem.
• To show that the interpolation capability of neural networks improves solution smoothness and convergence in stochastic hyperbolic problems.
• To compare P-PINN performance against traditional Monte Carlo simulations in terms of computational efficiency and accuracy.

Proposed method

• The P-PINN framework parameterizes input uncertainty by treating porosity and permeability as random variables with specified probability distributions.
• The model is trained to solve the stochastic Buckley-Leverett equation, a hyperbolic conservation law with non-convex fractional flow, by minimizing residual loss over space, time, and stochastic parameters.
• The method uses a Gaussian process to generate random velocity fields with exponential correlation structure, enabling stochastic field realization sampling.
• The network architecture is designed to predict full probability distributions of saturation (S_w) across space and time, conditioned on input parameters like mean and correlation length.
• The loss function incorporates both the PDE residual and boundary/initial conditions, with stochastic parameters explicitly included as network inputs.
• Training is performed on a 256-grid-block spatial domain with 100 time steps using a Godunov scheme for reference data, and inference is evaluated on statistical moments and realizations.

Experimental results

Research questions

• RQ1 Can P-PINNs accurately predict the statistical moments (mean and variance) of saturation in a stochastic Buckley-Leverett problem with heterogeneous permeability fields?
• RQ2 How does the inclusion of stochastic input parameters (e.g., velocity field mean and correlation length) affect the generalization and accuracy of the PINN model?
• RQ3 Does the interpolation capability of neural networks improve convergence and smoothness in hyperbolic PDEs with shocks under uncertainty?
• RQ4 How does the performance of P-PINN compare to Monte Carlo simulation in terms of computational cost and accuracy for uncertainty quantification?
• RQ5 Can P-PINNs recover full ensemble realizations of saturation dynamics when provided with explicit parametric representations of the stochastic field?

Key findings

• The P-PINN model achieves high accuracy in predicting both the mean and variance of saturation profiles across five time steps, closely matching reference results from Monte Carlo simulations using finite volume methods.
• For a heterogeneous random field with correlation length s=1.5, the P-PINN model reduced total computation time from 86 minutes (Monte Carlo) to under 16 minutes for 1,000 simulations, with inference time under 10 seconds per realization.
• The method demonstrated improved convergence and smoother solutions in hyperbolic problems due to the additional dimensions introduced by stochastic parameters, which regularize sharp gradients.
• Performance degradation was observed when decreasing the mean velocity or increasing the variance of the velocity field, likely due to limitations in first-order moment approximations.
• The P-PINN framework successfully captured shock front propagation and saturation dynamics in both homogeneous and heterogeneous porous media, even under high uncertainty.
• The model showed robustness to input variations and maintained consistent performance across different stochastic realizations when properly parameterized.