The novel neural networks show great potential in solving partial differential equations. For single-phase flow problems in subsurface porous media with high-contrast coefficients, the key is to develop neural operators with accurate reconstruction capability and strict adherence to physical laws. In this study, we proposed a hybrid two-stage framework that uses multiscale basis functions and physics-guided deep learning to solve the Darcy flow problem in high-contrast fractured porous media. In the first stage, a data-driven model is used to reconstruct the multiscale basis function based on the permeability field to achieve effective dimensionality reduction while preserving the necessary multiscale features. In the second stage, the physics-informed neural network, together with Transformer-based global information extractor is used to reconstruct the pressure field by integrating the physical constraints derived from the Darcy equation, ensuring consistency with the physical laws of the real world. The model was evaluated on datasets with different combinations of permeability and basis functions and performed well in terms of reconstruction accuracy. Specifically, the framework achieves R2 values above 0.9 in terms of basis function fitting and pressure reconstruction, and the residual indicator is on the order of 1 & times; 10-4. These results validate the ability of the proposed framework to achieve accurate reconstruction while maintaining physical consistency.

With the rapid development of engineering applications such as groundwater resource exploitation, oil and gas recovery, geothermal utilization, and geological carbon sequestration, the efficient and accurate simulation of subsurface fluid flow has become an important research topic in computational science and engineering. This review focuses on the application of deep learning techniques to subsurface fluid flow problems and systematically summarizes recent progress in solving multiscale equations such as Darcy flow. In particular, it covers the applications of classical deep learning methods, physics-informed neural networks, and neural operators to flow simulation, while also introducing various scalable hybrid algorithms for solving multiscale partial differential equations. For each of these directions, this review summarizes representative existing results and key advances, and finally discusses the major challenges and future research opportunities in this field.

In the field of flow field reconstruction, traditional deep learning models predominantly rely on standard convolutions, but their predictive accuracy remains limited. To address this issue, we explore the potential of E(2)-equivariant convolutions to enhance the predictive accuracy of deep learning models for fast flow field prediction. Unlike conventional convolutions, E(2)-equivariant convolutions offer a richer representation capability by better capturing geometric and structural information. Our neural network integrates an attention mechanism that leverages the signed distance function (SDF) to encode geometric details and an indicator matrix to incorporate boundary conditions. The model predicts velocity and pressure fields as outputs. We conducted experiments specifically targeting non-uniform steady laminar flows, and the results show a 16.1% reduction in overall error compared to models based on traditional convolutions while maintaining high efficiency. These findings indicate that E(2)-equivariant convolution, coupled with an attention mechanism, significantly improves flow field prediction by focusing on critical information and better representing complex geometries.

Although Deep Learning (DL) models have been introduced in various fields as effective prediction tools, they often do not care about uncertainty. This can be a barrier to their adoption in real-world applications. The current paper aims to apply and evaluate Monte Carlo (MC) dropout, a computationally efficient approach, to investigate the reliability of several skip connection-based Convolutional Neural Network (CNN) models while keeping their high accuracy. To do so, a high-dimensional regression problem is considered in the context of subterranean fluid flow modeling using 376,250 generated samples. The results demonstrate the effectiveness of MC dropout in terms of reliability with a Standard Deviation (SD) of 0.012–0.174, and of accuracy with a coefficient of determination (R2) of 0.7881–0.9584 and Mean Squared Error (MSE) of 0.0113–0.0508, respectively. The findings of this study may contribute to the distribution of pressure in the development of oil/gas fields.

Deep feed-forward networks, with high complexity, backpropagate the gradient of the loss function from final layers to earlier layers. As a consequence, the "gradient " may descend rapidly toward zero. This is known as the vanishing gradient phenomenon that prevents earlier layers from benefiting from further training. One of the most efficient techniques to solve this problem is using skip connection (shortcut) schemes that enable the gradient to be directly backpropagated to earlier layers. This paper investigates whether skip connections significantly affect the performance of deep neural networks of low complexity or whether their inclusion has little or no effect. The analysis was conducted using four Convolutional Neural Networks (CNNs) to predict four different multiscale basis functions for the mixed Generalized Multiscale Finite Element Method (GMsFEM). These models were applied to 249,375 samples. Three skip connection schemes were added to the base structure: Scheme 1 from the first convolutional block to the last, Scheme 2 from the middle to the last block, and Scheme 3 from the middle to the last and the second-to-last blocks. The results demonstrate that the third scheme is most effective, as it increases the coefficient of determination (R-2) value by 0.0224-0.044 and decreases the Mean Squared Error (MSE) value by 0.0027-0.0058 compared to the base structure. Hence, it is concluded that enriching the last convolutional blocks with the information hidden in neighboring blocks is more effective than enriching using earlier convolutional blocks near the input layer.

The accuracy of Deep Learning (DL) algorithms can be improved by combining several deep learners into an ensemble. This avoids the continuous endeavor required to adjust the architecture of individual networks or the nature of the propagation. This study investigates prediction improvements possible using Deep Ensemble Learning (DEL) to determine four distinct multiscale basis functions in the mixed Generalized Multiscale Finite Element Method (GMsFEM), involving the permeability field as the only input. 376,250 samples were initially generated, filtered down to 367,811 after data pre-processing. A standard Convolutional Neural Network (CNN) named SkiplessCNN and three skip connection-based CNNs named FirstSkipCNN, MidSkipCNN, and DualSkipCN N were developed for the base learners. For each basis function, these fou r CNNs were combined into an ensemble model using linear regression and ridge regression, separately, as part of the stacking technique. A comparison of the coefficient of determination (R2) and Mean Squared Error (MSE) confirms the effectiveness of a l l three skip connections in enhancing the performance of the standard CNN, with DualSkip being the most effective among them. Additionally, as evaluated on the testing subset, the combined models meaningfully outperform the individual models for a l l basis functions. The case that applies linear regression delivers R2 ranging from 0.8456 to 0.9191 and MSE ranging from 0.0092 to 0.0369. The ridge regression case achieves marginally better predictions with R2 ranging from 0.8539 to 0.922, and MSE ranging from 0.009 to 0.0349 because its solution involves more evenly distributed weights.

Machine Learning (ML) and/or Deep Learning (DL) methods can be used to predict fluid flow in porous media, as a suitable replacement for classical numerical approaches. Such data-driven approaches attempt to learn mappings between finite-dimensional Euclidean spaces. A novel neural framework, named Fourier Neural Operator (FNO), has been recently developed to act on infinite-dimensional spaces. A high proportion of the research available on the FNO has focused on problems with large-shape data. Furthermore, most published studies apply the FNO method to existing datasets. This paper applies and evaluates FNO to predict pressure distribution over a small, specified shape-data problem using 1700 Finite Element Method (FEM) generated samples, from heterogeneous permeability fields as the input. Considering FEM-calculated outputs as the true values, the configured FNO model provides superior prediction performance to that of a Convolutional Neural Network (CNN) in terms of statistical error assessment based on the coefficient of determination (R2) and Mean Squared Error (MSE). Sensitivity analysis considering a range of FNO configurations reveals that the most accurate model is obtained using modes=15 and width=100. Graphically, the FNO model precisely follows the observed trend in each porous medium evaluated. There is potential to further improve the FNO's performance by including physics constraints in its network configuration.

In multiscale modeling of subsurface fluid flow in heterogeneous porous media, standard polynomial basis functions are replaced by multiscale basis functions. For instance, to produce such functions in the mixed Generalized Multiscale Finite Element Method (mixed GMsFEM), a number of Partial Differential Equations (PDEs) must be solved, which requires a considerable overhead. Thus, it makes sense to replace PDE solvers with data-driven methods, given their great capabilities and general acceptance in the recent decades. Convolutional Neural Networks (CNNs) automatically perform feature engineering, and they also need fewer parameters via defining two-dimensional convolutional filters without reducing the quality of models. This is why four distinct CNN models were developed to predict four different multiscale basis functions for the mixed GMsFEM in the present study. These models were applied to 249,375 samples, with the permeability field as the only input. The statistical results indicate that the AMSGrad optimization algorithm with a coefficient of determination (R2) of 0.8434–0.9165 and Mean Squared Error (MSE) of 0.0078–0.0206 performs slightly better than Adam with an R2 of 0.8328–0.9049 and MSE of 0.0109–0.0261. Graphically, all models precisely follow the observed trend in each coarse block. This work could contribute to the distribution of pressure and velocity in the development of oil/gas fields. Looking at this work as an image (matrix)-to-image (matrix) regression problem, the constructed data-driven-based models may have applications beyond reservoir engineering, such as hydrogeology and rock mechanics.

A technique of local volume averaging is applied to obtain general equations that depict mass and momentum transport of incompressible two-phase flow in porous media. Starting from coupled Stokes-Cahn-Hilliard equations for incompressible two-phase fluid flow, the averaging is performed without oversimplifying either the porous media or the fluid mechanical relations. The resulting equations are Darcy's law for two-phase flow with medium parameters which could be evaluated by experiment. The Richards equation of the mixed form can be deduced from the resulting equations. The differences between the resulting equations and the empirical two-phase fluid flow model adopted in oil industry are discussed using several numerical examples.

A technique of local volume averaging is employed to obtain general equations which depict mass and momentum transport of incompressible two-phase flow in porous media. Starting from coupled Navier–Stokes–Cahn–Hilliard equations for incompressible two-phase fluid flow, the averaging is performed without oversimplifying either the porous media or the fluid mechanical relations. The resulting equations are Darcy’s law for two-phase flow with medium parameters which could be evaluated by experiment. The Richards’ equation of the mixed form can be deduced from the resulting equations.The differences between the resulting equations and the empirical two-phase fluid flow model adopted in oil industry are discussed by several numerical examples.

In this report, we study a partitioned time stepping algorithm for a dual-porosity-Stokes model, which consists of dual-porosity media and macrofractures/conduits in the coupled system. More specifically, the dual-porosity-Stokes model uses two pressures, the matrix pressure and the fracture pressure, to couple with the Stokes equations. There are four physically valid interface conditions to couple the two models on the interface, including a no-exchange condition, a mass balance condition, a force balance condition, and the Beavers–Joseph condition. To decouple the complex model into three simple sub-problems and solve them separately, we propose a partitioned time stepping method. It solves one, uncoupled matrix pressure equation, microfracture pressure equation and Stokes equation per time step. Under a modest time step restriction of the form \(\triangle t\le C\)(depending on physical parameters) , we prove the stability of the method. We also derive its optimal error estimates. Numerical tests verify the theoretical results.

In this paper, a multiple-time-step decoupled algorithm for a non-stationary Stokes-Darcy problem is proposed and investigated. Under a modest time step restriction of physical parameters and the time step proportion, we give the stability analysis and convergence analysis of the decoupled scheme with different time step in fluid and porous subregions. Finally, a series of numerical experiments are provided to illustrate the accuracy, efficiency, and stability of the presented method for the coupled problem with the Beavers-Joseph-Saffman-Jones interface conditions.

This article presents a new tuning approach for an adaptive internal-model-principle based signal identification algorithm whose computational costs are low enough to allow a realtime implementation. The algorithm allows an instantaneous Fourier decomposition of non-stationary signals that have a strongly predictable component. The algorithm is implemented as a feedback loop resulting in a closed loop system with a frequency response of a bandpass filter with notches at the frequencies of the Fourier decomposition. This is achieved through real time selection of the coefficients of the transfer functions in the feedback loop. Previously these coefficients were selected by solving a large set of coupled linear equations. Rules for explicitly solving for these parameters are given that only involve evaluating frequency responses at the frequencies of the instantaneous Fourier decomposition. This allows realtime implementation on a low cost lap top with sampling rates up to 10 kHz.

The Northern Shaanxi Area, Ordos Basin is an important continental shale gas exploration area in China. The Chang 7 Member of Upper Triassic Yanchang Formation is a main exploration interval for this area. In order to strengthen research on nanometer pore structures of Chang 7 Member shale reservoirs in the Northern Shaanxi Area, the microscopic pore structure characteristics are studied through the nitrogen adsorption experiment. The main controlling factors of pore development are discussed and the significance of pore structure for shale gas storage is analyzed. Results show that the pores of shale reservoirs of Chang 7 Member are dominated by nanoscale ones and the average pore size of shale samples is between 5.7 and 7.9 nm. According to IUPAC classification, shale pores of Chang 7 Member are dominated by mesopores, accounting for 90.8%-95.5% of the total pore volume. The nanometer pore structure is relatively uniform and the adsorption/desorption curves are dominated by Type B in the De Boer classification. The micropore volume and the macropore volume have similar proportions, and they account for 4.5%-9.2% of the total volume, averaging at 7.2%. Characteristics of Type E indicate that most pores are open type ones, and slit-like pores and 'ink bottle' shaped pores are irregularly configured. Total organic carbon (TOC) has a weak negative correlation with specific surface area of the shale, and has a weak positive correlation with total pore volume. Maturity (Ro) has a weak negative correlation with both specific surface area and total pore volume. The content of brittle minerals and content of clay minerals have no clear correlation with the specific surface area and the pore volume.

Shale transport properties are investigated by numerical simulations of a simple, physics-based continuum model. Shale samples from the Yanchang Triassic Formation in Ordos Basin are tested to get the pore volume distribution of shale. Due to the existence of nanopores in shale reservoirs, there are great differences in gas transport mechanisms compared to those in conventional gas reservoirs. These mechanisms include Knudsen diffusion and adsorption. A physics-based continuum model is utilized to simulate the gas transport in a channel with nanoscale radius, where these mechanisms are taken into consideration by imposing a general boundary condition with two parameters. The correction factor which is the ratio between apparent permeability and intrinsic permeability is investigated for a wide pressure range, which shows it is always larger than 1, illustrating that Knudsen diffusion always plays a role in shale gas transport.

苏里格气田随着逐年开发,低压低产井越来越多.此类井产能较差,部分存在严重积液,排水采气工艺已在该区块广泛应用,但是各类工艺的参数优化,以及不同类型、不同时期产水气井对各种工艺的适用性如何,仍是急待解决的现实问题.该区块应用较多的是泡沫排水采气和井下工具类排水采气(速度管柱、雾化和涡流工具),本文对其工艺原理、工艺应用、工艺优化与适用性分析和室内及现场实验研究现状进行了综述,并提出了目前该领域研究需要关注和待解决的相关问题.

In this paper, we present a hybrid method, which consists of a mixed-hybrid finite element method and a penalty discontinuous Galerkin method, for the approximation of a fractional flow formulation of a two-phase flow problem in heterogeneous media with discontinuous capillary pressure. The fractional flow formulation is comprised of a wetting phase pressure equation and a wetting phase saturation equation which are coupled through a total velocity and the saturation affected coefficients. For the wetting phase pressure equation, the continuous mixed-hybrid finite element method space can be utilized due to a fundamental property that the wetting phase pressure is continuous. While it can reduce the computational cost by using less degrees of freedom and avoiding the post-processing of velocity reconstruction, this method can also keep several good properties of the discontinuous Galerkin method, which are important to the fractional flow formulation, such as the local mass balance, continuous normal flux and capability of handling the discontinuous capillary pressure. For the wetting phase saturation equation, the penalty discontinuous Galerkin method is utilized due to its capability of handling the discontinuous jump of the wetting phase saturation. Furthermore, an adaptive algorithm for the hybrid method together with the centroidal Voronoi–Delaunay triangulation technique is proposed. Five numerical examples are presented to illustrate the features of proposed numerical method, such as the optimal convergence order, the accurate and efficient velocity approximation, and the applicability to the simulation of water flooding in oil field and the oil-trapping or barrier effect phenomena.

Shale pore structure has an important effect upon shale-gas adsorption. In order to quantify their contribution to the total amount of adsorbed gas, we investigate the pore volume distribution on shale, coal, isolated kerogen, and clay samples using low pressure nitrogen and carbon dioxide adsorption method. And a series of methane adsorption isotherms on the samples was measured by a volumetric approach at 20 °C, 30 °C and 40 °C under pressures up to 10 MPa. Results indicate that a strong correlation exists between TOC and micropore volume of shale, but to verify the correlations between BET surface area and TOC, or micropore volume and Ro are still difficult. Certain isotherms on shales differ from Type I IUPAC (International Union of Pure and Applied Chemistry) isotherm. The unusual shape may be attributed to shale pore structure or composition. Finally, the modified D-A-Langmuir model including pore distribution parameters can provide a precise representation of the methane adsorption data on shales. The findings of this study are applicable to the accurate resource assessments and recovery technologies for shale-gas.

The upwind scheme is very important in the numerical approximation of some problems such as the convection dominated problem, the two-phase flow problem, and so on. For the fractional flow formulation of the two-phase flow problem, the Penalty Discontinuous Galerkin (PDG) methods combined with the upwind scheme are usually used to solve the phase pressure equation. In this case, unless the upwind scheme is taken into consideration in the velocity reconstruction, the local mass balance cannot hold exactly. In this paper, we present a scheme of velocity reconstruction in some H(div) spaces with considering the upwind scheme totally. Furthermore, the different ways to calculate the nonlinear coefficients may have distinct and significant effects, which have been investigated by some authors. We propose a new algorithm to obtain a more effective and stable approximation of the coefficients under the consideration of the upwind scheme.

This paper reports a numerical study of coupling two-phase fluid flow in a free fluid region with two-phase Darcy flow in a homogeneous and anisotropic porous medium region. The model consists of coupled Cahn-Hilliard and Navier-Stokes equations in the free fluid region and the two-phase Darcy law in the anisotropic porous medium region. A Robin-Robin domain decomposition method is used for the coupled Navier-Stokes and Darcy system with the generalized Beavers-Joseph-Saffman condition on the interface between the free flow and the porous media regions. Obtained results have shown the anisotropic properties effect on the velocity and pressure of the two-phase flow.