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Reservoir History Matching of the Norne field with generative exotic priors and a coupled Mixture of Experts -- Physics Informed Neural Operator Forward Model

Clement Etienam, Yang Juntao, Oleg Ovcharenko, Issam Said

TL;DR

This work tackles reservoir history matching as an ill-posed Bayesian inverse problem by coupling a physics-informed neural operator forward model (PINO) with a cluster-classify-regress mixture of experts (CCR), enabling fast, uncertainty-aware inversion. The forward model uses a Fourier Neural Operator (FNO) trained with data and PDE residuals, producing pressure and saturations that feed a Peaceman-well surrogate within the CCR framework. Generative exotic priors are constructed with a Variational Convolutional Autoencoder (VCAE) and Denoising Diffusion Implicit Models (DDIM), while sparse Gaussian process experts provide localized, probabilistic regressions; aREKI with covariance localization guides ensemble updates to recover permeability, porosity, and fault multipliers. Implemented in NVIDIA Modulus, the approach yields forward-consistent surrogates and an inverse workflow that accelerates history matching by roughly 6000× on the Norne field, with training times around 5 hours for 100 samples and ensemble runs on GPUs. The results demonstrate accurate field-scale reconstruction and rapid posterior exploration, making the method well-suited for ensemble-based uncertainty quantification in challenging reservoir settings.

Abstract

We developed a novel reservoir characterization workflow that addresses reservoir history matching by coupling a physics-informed neural operator (PINO) forward model with a mixture of experts' approach, termed cluster classify regress (CCR). The inverse modelling is achieved via an adaptive Regularized Ensemble Kalman inversion (aREKI) method, ideal for rapid inverse uncertainty quantification during history matching. We parametrize unknown permeability and porosity fields for non-Gaussian posterior measures using a variational convolution autoencoder and a denoising diffusion implicit model (DDIM) exotic priors. The CCR works as a supervised model with the PINO surrogate to replicate nonlinear Peaceman well equations. The CCR's flexibility allows any independent machine-learning algorithm for each stage. The PINO reservoir surrogate's loss function is derived from supervised data loss and losses from the initial conditions and residual of the governing black oil PDE. The PINO-CCR surrogate outputs pressure, water, and gas saturations, along with oil, water, and gas production rates. The methodology was compared to a standard numerical black oil simulator for a waterflooding case on the Norne field, showing similar outputs. This PINO-CCR surrogate was then used in the aREKI history matching workflow, successfully recovering the unknown permeability, porosity and fault multiplier, with simulations up to 6000 times faster than conventional methods. Training the PINO-CCR surrogate on an NVIDIA H100 with 80G memory takes about 5 hours for 100 samples of the Norne field. This workflow is suitable for ensemble-based approaches, where posterior density sampling, given an expensive likelihood evaluation, is desirable for uncertainty quantification.

Reservoir History Matching of the Norne field with generative exotic priors and a coupled Mixture of Experts -- Physics Informed Neural Operator Forward Model

TL;DR

This work tackles reservoir history matching as an ill-posed Bayesian inverse problem by coupling a physics-informed neural operator forward model (PINO) with a cluster-classify-regress mixture of experts (CCR), enabling fast, uncertainty-aware inversion. The forward model uses a Fourier Neural Operator (FNO) trained with data and PDE residuals, producing pressure and saturations that feed a Peaceman-well surrogate within the CCR framework. Generative exotic priors are constructed with a Variational Convolutional Autoencoder (VCAE) and Denoising Diffusion Implicit Models (DDIM), while sparse Gaussian process experts provide localized, probabilistic regressions; aREKI with covariance localization guides ensemble updates to recover permeability, porosity, and fault multipliers. Implemented in NVIDIA Modulus, the approach yields forward-consistent surrogates and an inverse workflow that accelerates history matching by roughly 6000× on the Norne field, with training times around 5 hours for 100 samples and ensemble runs on GPUs. The results demonstrate accurate field-scale reconstruction and rapid posterior exploration, making the method well-suited for ensemble-based uncertainty quantification in challenging reservoir settings.

Abstract

We developed a novel reservoir characterization workflow that addresses reservoir history matching by coupling a physics-informed neural operator (PINO) forward model with a mixture of experts' approach, termed cluster classify regress (CCR). The inverse modelling is achieved via an adaptive Regularized Ensemble Kalman inversion (aREKI) method, ideal for rapid inverse uncertainty quantification during history matching. We parametrize unknown permeability and porosity fields for non-Gaussian posterior measures using a variational convolution autoencoder and a denoising diffusion implicit model (DDIM) exotic priors. The CCR works as a supervised model with the PINO surrogate to replicate nonlinear Peaceman well equations. The CCR's flexibility allows any independent machine-learning algorithm for each stage. The PINO reservoir surrogate's loss function is derived from supervised data loss and losses from the initial conditions and residual of the governing black oil PDE. The PINO-CCR surrogate outputs pressure, water, and gas saturations, along with oil, water, and gas production rates. The methodology was compared to a standard numerical black oil simulator for a waterflooding case on the Norne field, showing similar outputs. This PINO-CCR surrogate was then used in the aREKI history matching workflow, successfully recovering the unknown permeability, porosity and fault multiplier, with simulations up to 6000 times faster than conventional methods. Training the PINO-CCR surrogate on an NVIDIA H100 with 80G memory takes about 5 hours for 100 samples of the Norne field. This workflow is suitable for ensemble-based approaches, where posterior density sampling, given an expensive likelihood evaluation, is desirable for uncertainty quantification.
Paper Structure (27 sections, 75 equations, 12 figures, 1 table, 7 algorithms)

This paper contains 27 sections, 75 equations, 12 figures, 1 table, 7 algorithms.

Table of Contents

  1. Introduction.
  2. Overview
  3. Forward problem
  4. Surrogate forward modeling with a Fourier Neural operator (FNO) - Contribution
  5. Inverse Problem
  6. Mixture of experts.
  7. Cluster
  8. Classify
  9. Regress
  10. Bayesian Formulation
  11. Sparse Gaussian process experts
  12. Generative priors and Tree based learning
  13. Variational Convolutional Autoencoder (VCAE)
  14. Encoder
  15. Decoder
  16. ...and 12 more sections

Figures (12)

  • Figure 1: A standard Fourier neural operator (FNO) as adopted from li2021fourier
  • Figure 2: Forwarding of the Norne Field.$N_x=46, N_y=112, N_z=22$. Peaceman well model surrogate with CCR. The 66 tile indicates each of the 66 outputs for the wopr, wwpr, wgpr for the 22 wells. The accuracies are computed on the unseen validation text set.
  • Figure 3: Forwarding of the Norne Field.$N_x=46, N_y=112, N_z=22$. At Time = 8 days. Dynamic properties comparison between the pressure, water saturation, oil saturation and gas saturation field between Nvidia Modulus's PINO surrogate (left column), Numerical solver reservoir simulator (middle column) and the difference between both approaches (last column). They are 22 oil/water/gas producers (green), 9 water injectors (blue) and 4 gas injectors) red. We can see good concordance between the surrogate's prediction and the numerical reservoir simulator (Numerical solver).
  • Figure 4: Forwarding of the Norne Field.$N_x=46, N_y=112, N_z=22$. At Time = 968 days. Dynamic properties comparison between the pressure, water saturation, oil saturation and gas saturation field between Nvidia Modulus's PINO surrogate (left column), Numerical solver reservoir simulator (middle column) and the difference between both approaches (last column). They are 22 oil/water/gas producers (green), 9 water injectors (blue) and 4 gas injectors) red. We can see good concordance between the surrogate's prediction and the numerical reservoir simulator (Numerical solver).
  • Figure 5: Forwarding of the Norne Field.$N_x=46, N_y=112, N_z=22$. At Time = 2104 days. Dynamic properties comparison between the pressure, water saturation, oil saturation and gas saturation field between Nvidia Modulus's PINO surrogate (left column), Numerical solver reservoir simulator (middle column) and the difference between both approaches (last column). They are 22 oil/water/gas producers (green), 9 water injectors (blue) and 4 gas injectors) red. We can see good concordance between the surrogate's prediction and the numerical reservoir simulator (Numerical solver).
  • ...and 7 more figures