Shuyu Sun

Jisheng Kou, Shuyu Sun, Xiuhua Wang

In this paper, for the first time we propose two linear, decoupled, energy-stable numerical schemes for multi-component two-phase compressible flow with a realistic equation of state (e.g. Peng-Robinson equation of state). The methods are constructed based on the scalar auxiliary variable (SAV) approaches for Helmholtz free energy and the intermediate velocities that are designed to decouple the tight relationship between velocity and molar densities. The intermediate velocities are also involved in the discrete momentum equation to ensure the consistency with the mass balance equations. Moreover, we propose a component-wise SAV approach for a multi-component fluid, which requires solving a sequence of linear, separate mass balance equations. We prove that the methods preserve the unconditional energy-dissipation feature. Numerical results are presented to verify the effectiveness of the proposed methods.

Jisheng Kou, Shuyu Sun, Xiuhua Wang

The Peng-Robinson equation of state (PR-EoS) has become one of the most extensively applied equations of state in chemical engineering and petroleum industry due to its excellent accuracy in predicting the thermodynamic properties of a wide variety of materials, especially hydrocarbons. Although great efforts have been made to construct efficient numerical methods for the diffuse interface models with PR-EoS, there is still not a linear numerical scheme that can be proved to preserve the original energy dissipation law. In order to pursue such a numerical scheme, we propose a novel energy factorization (EF) approach, which first factorizes an energy function into a product of several factors and then treats the factors using their properties to obtain the semi-implicit linear schemes. We apply the EF approach to deal with the Helmholtz free energy density determined by PR-EoS, and then propose a linear semi-implicit numerical scheme that inherits the original energy dissipation law. Moreover, the proposed scheme is proved to satisfy the maximum principle in both the time semi-discrete form and the cell-centered finite difference fully discrete form under certain conditions. Numerical results are presented to demonstrate the stability and efficiency of the proposed scheme.

Jisheng Kou, Shuyu Sun

A general diffuse interface model with a realistic equation of state (e.g. Peng-Robinson equation of state) is proposed to describe the multi-component two-phase fluid flow based on the principles of the NVT-based framework which is a latest alternative over the NPT-based framework to model the realistic fluids. The proposed model uses the Helmholtz free energy rather than Gibbs free energy in the NPT-based framework. Different from the classical routines, we combine the first law of thermodynamics and related thermodynamical relations to derive the entropy balance equation, and then we derive a transport equation of the Helmholtz free energy density. Furthermore, by using the second law of thermodynamics, we derive a set of unified equations for both interfaces and bulk phases that can describe the partial miscibility of two fluids. A relation between the pressure gradient and chemical potential gradients is established, and this relation leads to a new formulation of the momentum balance equation, which demonstrates that chemical potential gradients become the primary driving force of fluid motion. Moreover, we prove that the proposed model satisfies the total (free) energy dissipation with time. For numerical simulation of the proposed model, the key difficulties result from the strong nonlinearity of Helmholtz free energy density and tight coupling relations between molar densities and velocity. To resolve these problems, we propose a novel convex-concave splitting of Helmholtz free energy density and deal well with the coupling relations between molar densities and velocity through very careful physical observations with a mathematical rigor. We prove that the proposed numerical scheme can preserve the discrete (free) energy dissipation. Numerical tests are carried out to verify the effectiveness of the proposed method.

Jisheng Kou, Shuyu Sun

In this paper, we consider a diffuse-interface gas-liquid two-phase flow model with inhomogeneous temperatures, in which we employ the Peng-Robinson equation of state and the temperature-dependent influence parameter instead of the van der Waals equation of state and the constant influence parameter used in the existing models. As a result, our model can characterize accurately the physical behaviors of numerous realistic gas-liquid fluids, especially hydrocarbons. Furthermore, we prove a relation associating the pressure gradient with the gradients of temperature and chemical potential, and thereby derive a new formulation of the momentum balance equation, which shows that gradients of the chemical potential and temperature become the primary driving force of the fluid motion. It is rigorously proved that the new formulations of the model obey the first and second laws of thermodynamics. To design efficient numerical methods, we prove that Helmholtz free energy density is a concave function with respect to the temperature under certain physical conditions. Based on the proposed modeling formulations and the convex-concave splitting of Helmholtz free energy density, we propose a novel thermodynamically stable numerical scheme. We rigorously prove that the proposed method satisfies the first and second laws of thermodynamics. Finally, numerical tests are carried out to verify the effectiveness of the proposed simulation method.

Jisheng Kou, Shuyu Sun

In this paper, we consider mathematical modeling and numerical simulation of non-isothermal compressible multi-component diffuse-interface two-phase flows with realistic equations of state. A general model with general reference velocity is derived rigorously through thermodynamical laws and Onsager's reciprocal principle, and it is capable of characterizing compressibility and partial miscibility between multiple fluids. We prove a novel relation among the pressure, temperature and chemical potentials, which results in a new formulation of the momentum conservation equation indicating that the gradients of chemical potentials and temperature become the primary driving force of the fluid motion except for the external forces. A key challenge in numerical simulation is to develop entropy stable numerical schemes preserving the laws of thermodynamics. Based on the convex-concave splitting of Helmholtz free energy density with respect to molar densities and temperature, we propose an entropy stable numerical method, which solves the total energy balance equation directly, and thus, naturally satisfies the first law of thermodynamics. Unconditional entropy stability (the second law of thermodynamics) of the proposed method is proved by estimating the variations of Helmholtz free energy and kinetic energy with time steps. Numerical results validate the proposed method.

Zhaoyang Ma, Xupeng He, Hyung Kwak et al.

In digital rock physics, analysing microstructures from CT and SEM scans is crucial for estimating properties like porosity and pore connectivity. Traditional segmentation methods like thresholding and CNNs often fall short in accurately detailing rock microstructures and are prone to noise. U-Net improved segmentation accuracy but required many expert-annotated samples, a laborious and error-prone process due to complex pore shapes. Our study employed an advanced generative AI model, the diffusion model, to overcome these limitations. This model generated a vast dataset of CT/SEM and binary segmentation pairs from a small initial dataset. We assessed the efficacy of three neural networks: U-Net, Attention-U-net, and TransUNet, for segmenting these enhanced images. The diffusion model proved to be an effective data augmentation technique, improving the generalization and robustness of deep learning models. TransU-Net, incorporating Transformer structures, demonstrated superior segmentation accuracy and IoU metrics, outperforming both U-Net and Attention-U-net. Our research advances rock image segmentation by combining the diffusion model with cutting-edge neural networks, reducing dependency on extensive expert data and boosting segmentation accuracy and robustness. TransU-Net sets a new standard in digital rock physics, paving the way for future geoscience and engineering breakthroughs.

Zhaoyang Ma, Xupeng He, Shuyu Sun et al.

Accurate image segmentation is crucial in reservoir modelling and material characterization, enhancing oil and gas extraction efficiency through detailed reservoir models. This precision offers insights into rock properties, advancing digital rock physics understanding. However, creating pixel-level annotations for complex CT and SEM rock images is challenging due to their size and low contrast, lengthening analysis time. This has spurred interest in advanced semi-supervised and unsupervised segmentation techniques in digital rock image analysis, promising more efficient, accurate, and less labour-intensive methods. Meta AI's Segment Anything Model (SAM) revolutionized image segmentation in 2023, offering interactive and automated segmentation with zero-shot capabilities, essential for digital rock physics with limited training data and complex image features. Despite its advanced features, SAM struggles with rock CT/SEM images due to their absence in its training set and the low-contrast nature of grayscale images. Our research fine-tunes SAM for rock CT/SEM image segmentation, optimizing parameters and handling large-scale images to improve accuracy. Experiments on rock CT and SEM images show that fine-tuning significantly enhances SAM's performance, enabling high-quality mask generation in digital rock image analysis. Our results demonstrate the feasibility and effectiveness of the fine-tuned SAM model (RockSAM) for rock images, offering segmentation without extensive training or complex labelling.

Jie Liu, Tao Zhang, Shuyu Sun

Pore-network models (PNMs) have become an important tool in the study of fluid flow in porous media over the last few decades, and the accuracy of their results highly depends on the extraction of pore networks. Traditional methods of pore-network extraction are based on pixels and require images with high quality. Here, a pixel-free method called the flashlight search medial axis (FSMA) algorithm is proposed for pore-network extraction in a continuous space. The search domain in a two-dimensional space is a line, whereas a surface domain is searched in a three-dimensional scenario. Thus, the FSMA algorithm follows the dimensionality reduction idea; the medial axis can be identified using only a few points instead of calculating every point in the void space. In this way, computational complexity of this method is greatly reduced compared to that of traditional pixel-based extraction methods, thus enabling large-scale pore-network extraction. Based on cases featuring two- and three-dimensional porous media, the FSMA algorithm performs well regardless of the topological structure of the pore network or the positions of the pore and throat centers. This algorithm can also be used to examine both closed- and open-boundary cases. Finally, the FSMA algorithm can search dead-end pores, which is of great significance in the study of multiphase flow in porous media.

Dengfei Zeng, Lijian Jiang, Shuyu Sun et al.

A likelihood-free transport filtering method is proposed based on the couplings between state and observation variables. By exploiting a block-triangular structure in the transport map, the analysis step of filtering is reformulated as the minimization of the maximum mean discrepancy (MMD) between the true joint measure and its transport-based approximation. To circumvent the non-convexity in the MMD optimization, we introduce a training-free transport filter method via gradient flows, which leads to an analytic computation for the transport map that implies the steepest descent direction of the MMD. The proposed approach accurately approximates non-Gaussian filtering posteriors and avoids particle collapse. We provide a convergence analysis for the expectation of the MMD between the approximated posterior and the truth posterior. Finally, we extend the method to high-dimensional problems through domain localization. Numerical examples demonstrate the superior performance of our approach over conventional filtering methods in nonlinear, non-Gaussian scenarios.

Yiteng Li, Tao Zhang, Shuyu Sun

Phase equilibrium calculation, also known as flash calculation, has been extensively applied in petroleum engineering, not only as a standalone application for separation process but also an integral component of compositional reservoir simulation. It is of vital importance to accelerate flash calculation without much compromise in accuracy and reliability, turning it into an active research topic in the last two decades. In this study, we establish a deep neural network model to approximate the iterative NVT-flash calculation. A dynamic model designed for NVT flash problems is iteratively solved to produce data for training the neural network. In order to test the model's capacity to handle complex fluid mixtures, three real reservoir fluids are investigated, including one Bakken oil and two Eagle Ford oils. Compared to previous studies that follow the conventional flash framework in which stability testing precedes phase splitting calculation, we incorporate stability test and phase split calculation together and accomplish both two steps by a single deep learning model. The trained model is able to identify the single vapor, single liquid and vapor-liquid state under the subcritical region of the investigated fluids. A number of examples are presented to show the accuracy and efficiency of the proposed deep neural network. It is found that the trained model makes predictions at most 244 times faster than the iterative flash calculation under the given cases. Even though training a multi-level network model does take a large amount of time that is comparable to the computational time of flash calculations, the one-time offline training process gives the deep learning model great potential to speed up compositional reservoir simulation.