Hamdi A. Tchelepi

Joshua A. White, Nicola Castelletto, Sergey Klevtsov et al.

Many applications involving porous media--notably reservoir engineering and geologic applications--involve tight coupling between multiphase fluid flow, transport, and poromechanical deformation. While numerical models for these processes have become commonplace in research and industry, the poor scalability of existing solution algorithms has limited the size and resolution of models that may be practically solved. In this work, we propose a two-stage Newton-Krylov solution algorithm to address this shortfall. The proposed solver exhibits rapid convergence, good parallel scalability, and is robust in the presence of highly heterogeneous material properties. The key to success of the solver is a block-preconditioning strategy that breaks the fully-coupled system of mass and momentum balance equations into simpler sub-problems that may be readily addressed using targeted algebraic methods. Numerical results are presented to illustrate the performance of the solver on challenging benchmark problems.

Per Pettersson, Hamdi A. Tchelepi

The generalized polynomial chaos method is applied to the Buckley-Leverett equation. We consider a spatially homogeneous domain modeled as a random field. The problem is projected onto stochastic basis functions which yields an extended system of partial differential equations. Analysis and numerical methods leading to reduced computational cost are presented for the extended system of equations. The accurate representation of the evolution of a discontinuous stochastic solution over time requires a large number of stochastic basis functions. Adaptivity of the stochastic basis to reduce computational cost is challenging in the stochastic Galerkin setting since the change of basis affects the system matrix itself. To achieve adaptivity without adding overhead by rewriting the entire system of equations for every grid cell, we devise a basis reduction method that distinguishes between locally significant and insignificant modes without changing the actual system matrices. Results are presented for problems in one and two spatial dimensions, with varying number of stochastic dimensions. We show how to obtain stochastic velocity fields from realistic permeability fields and demonstrate the performance of the stochastic Galerkin method with local basis reduction. The system of conservation laws is discretized with a finite volume method and we demonstrate numerical convergence to the reference solution obtained through Monte Carlo sampling.

Xin Ju, François P. Hamon, Gege Wen et al.

Deep-learning-based surrogate models provide an efficient complement to numerical simulations for subsurface flow problems such as CO$_2$ geological storage. Accurately capturing the impact of faults on CO$_2$ plume migration remains a challenge for many existing deep learning surrogate models based on Convolutional Neural Networks (CNNs) or Neural Operators. We address this challenge with a graph-based neural model leveraging recent developments in the field of Graph Neural Networks (GNNs). Our model combines graph-based convolution Long-Short-Term-Memory (GConvLSTM) with a one-step GNN model, MeshGraphNet (MGN), to operate on complex unstructured meshes and limit temporal error accumulation. We demonstrate that our approach can accurately predict the temporal evolution of gas saturation and pore pressure in a synthetic reservoir with impermeable faults. Our results exhibit a better accuracy and a reduced temporal error accumulation compared to the standard MGN model. We also show the excellent generalizability of our algorithm to mesh configurations, boundary conditions, and heterogeneous permeability fields not included in the training set. This work highlights the potential of GNN-based methods to accurately and rapidly model subsurface flow with complex faults and fractures.

Chiyu Max Jiang, Soheil Esmaeilzadeh, Kamyar Azizzadenesheli et al.

We propose MeshfreeFlowNet, a novel deep learning-based super-resolution framework to generate continuous (grid-free) spatio-temporal solutions from the low-resolution inputs. While being computationally efficient, MeshfreeFlowNet accurately recovers the fine-scale quantities of interest. MeshfreeFlowNet allows for: (i) the output to be sampled at all spatio-temporal resolutions, (ii) a set of Partial Differential Equation (PDE) constraints to be imposed, and (iii) training on fixed-size inputs on arbitrarily sized spatio-temporal domains owing to its fully convolutional encoder. We empirically study the performance of MeshfreeFlowNet on the task of super-resolution of turbulent flows in the Rayleigh-Benard convection problem. Across a diverse set of evaluation metrics, we show that MeshfreeFlowNet significantly outperforms existing baselines. Furthermore, we provide a large scale implementation of MeshfreeFlowNet and show that it efficiently scales across large clusters, achieving 96.80% scaling efficiency on up to 128 GPUs and a training time of less than 4 minutes.