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In this work we present an enriched Petrov-Galerkin (EPG) method for the simulation of the Darcy flow in porous media. The new method enriches the approximation trial space of the conforming continuous Galerkin (CG) method with bubble functions and enriches the approximation test space of the CG method with piecewise constant functions, and it does not require any penalty term in the weak formulation. Moreover, we propose a framework for constructing the bubble functions and consider a decoupled algorithm for the EPG method based on this framework, which enables the process of solving pressure to be decoupled into two steps. The first step is to solve the pressure by the standard CG method, and the second step is a post-processing correction of the first step. Compared with the CG method, the proposed EPG method is locally mass-conservative, while keeping fewer degrees of freedom than the discontinuous Galerkin (DG) method. In addition, this method is more concise in the error analysis than the enriched Galerkin (EG) method. The coupled flow and transport in porous media is considered to illustrate the advantages of locally mass-conservative properties of the EPG method. We establish the optimal convergence of numerical solutions and present several numerical examples to illustrate the performance of the proposed method.

The success of geological H2 storage relies significantly on rock–H2–brine interactions and wettability. Experimentally assessing the H2 wettability of storage/caprocks as a function of thermos-physical conditions is arduous because of high H2 reactivity and embrittlement damages. Data-driven machine learning (ML) modeling predictions of rock–H2–brine wettability are less strenuous and more precise. They can be conducted at geo-storage conditions that are impossible or hazardous to attain in the laboratory. Thus, ML models were utilized in this research to accurately model the wettability behavior of a ternary system consisting of H2, rock minerals (quartz and mica), and brine at different operating geological conditions. The results revealed that the ML models accurately captured the wettability behavior at different geo-storage conditions by yielding less than 5% mean absolute percent error and above 0.95 coefficient of determination values. The partial dependency or sensitivity plots were generated to evaluate the impact of individual features on the trained models. These plots revealed that the models accurately captured the physics behind the problem. Furthermore, a mathematical equation is derived from the trained ML model to predict the wettability behavior without using any ML software. The accuracy of the predictions of the ML model can be beneficial for exactly predicting the H2 geo-storage capacities and assessing of H2 containment security of storage and caprocks for large-scale geo-storage projects.

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.

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.

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.

We propose an unconditionally energy-stable, orthonormality-preserving, component-wise splitting iterative scheme for the Kohn-Sham gradient flow based model in the electronic structure calculation. We first study the scheme discretized in time but still continuous in space. The component-wise splitting iterative scheme changes one wave function at a time, similar to the Gauss-Seidel iteration for solving a linear equation system. Rigorous mathematical derivations are presented to show our proposed scheme indeed satisfies the desired properties. We then study the fully-discretized scheme, where the space is further approximated by a conforming finite element subspace. For the fully-discretized scheme, not only the preservation of orthogonality and normalization (together we called orthonormalization) can be quickly shown using the same idea as for the semi-discretized scheme, but also the highlight property of the scheme, i.e., the unconditional energy stability can be rigorously proven. The scheme allows us to use large time step sizes and deal with small systems involving only a single wave function during each iteration step. Several numerical experiments are performed to verify the theoretical analysis, where the number of iterations is indeed greatly reduced as compared to similar examples solved by the Kohn-Sham gradient flow based model in the literature.

Hydrocarbons in subsurface nanoporous media, such as shale, are promising energy resources to compensate for the shortage of conventional reservoirs. Pore-network modeling serves as a valuable tool for simulating microscale fluid transport and elucidating flow physics in porous media. However, traditional pore-network models have failed to capture features of spatial structure and fluid flow in unconventional shale rock. This work presents a critical review of pore-network modeling of single-phase and two-phase flow in shale rock. Pore-network modeling advances of shale are reviewed based on three major parts: network morphology and geometries, flow principles in nanocapillaries, and pore-network computational algorithms. First, based on key geological features of shale rock, we analyze network topology, multiscale network, pore geometries, and network representativeness of shale pore-network models. Then, we discuss four important aspects that may influence flow principles of fluids in nanocapillaries: gas and liquid slippage, sorption and diffusion behavior, hydrocarbon thermodynamics, and the presence of water. Finally, we present pore-network modeling methods used for flow simulations in shale rock, including quasi-static and dynamic algorithms. We hope that this review could shed light on fundamentals of pore-network modeling of shale rock.

Fractured karst reservoirs play a significant role in hydrocarbon reserves and groundwater storage. They often exhibit complex multiscale heterogeneities involving pores, fractures and caves, whose length scales range from microns to tens or hundreds of meters. Thus, the study of fractured karst reservoirs is faced with a significant unresolved challenge in quantitatively characterizing the geometrical connectivity and hydraulic conductivity as well as their interrelationships in such strongly heterogeneous, multicomponent systems. In this paper, we propose an analytical formulation to characterize the connectivity of discrete fracture-cave networks building upon the excluded area concept of the percolation theory. By implementing a state-of-the-art computational model solving coupled Navier-Stokes (free flow) and Darcy (porous media flow) equations, we numerically derive the permeability of a fractured and karstified porous media, such that the relationship between the connectivity and permeability is further explored. The high-fidelity numerical model also permits us to elucidate the process of flow channelization within the fracture-cave network. The results show that the fracture-cave network connectivity correlates to the permeability via a power law scaling for connected systems. Significant flow channeling occurs around the percolation threshold where the flow is dominated by a limited number of preferential pathways. Caves play a crucial role in the flow due to the fact that caves could globally enhance the network connectivity and locally serve as hotspots for high fluid velocity. Finally, a semi-analytical permeability model for fractured karst reservoirs is developed. The results of our research and insights obtained have important implications for understanding the subsurface fluid flow in fractured karst reservoirs.

In this paper, we propose a physics-preserving multiscale method to solve an immiscible two-phase flow problem, which is modeled as a coupling system consisting of Darcy's law and mass conservation equations. We use a new Physics-preserving IMplicit Pressure Explicit Saturation (P-IMPES) scheme in order to maintain the local conservation of mass for both phases. Besides, this scheme is unbiased and if the time step is smaller than a certain value, the saturation of both phases are bounds-preserving. When updating velocity, MGMsFEM serves as an efficient solver by computing the unknowns on a coarse grid. We follow the operation splitting techinque to deal with the two-phase flow. In particular, we use an upwind strategy to iterate the saturation explicitly and the MGMsFEM is utilized to compute velocity with a decoupled system on a coarse mesh. To show the efficiency and robustness of the proposed method, we design a set of interesting experiments. A rigorous analysis is also included to serve as a theoretical base of the method, which is well verified by the numerical results. Both simulations and analysis indicate that the method attains a good balance between accuracy and computation cost., Comment: 38 pages

Molecular dynamics simulations were conducted to study the interfacial behavior of the CO2 + H2O and hexane + CO2 + H2O systems in the presence of hydrophilic silica at geological conditions. Simulation results for the CO2 + H2O and hexane + CO2 + H2O systems are in reasonable agreement with the theoretical predictions based on the density functional theory. In general, the interfacial tension (IFT) of the CO2 + H2O system exponentially (linearly) decreased with increasing pressure (temperature). The IFTs of the hexane + CO2 + H2O (two-phase) system decreased with the increasing mole fraction of CO2 in the hexane/CO2-rich phase x(CO2). Here, the negative surface excesses of hexane lead to a general increase in the IFTs with increasing pressure. The effect of pressure on these IFTs decreased with increasing x(CO2) due to the positive surface excesses of carbon dioxide. The simulated water contact angles of the CO2 + H2O + silica system fall in the range from 43.8 degrees to 76.0 degrees , which is in reasonable agreement with the experimental results. These contact angles increased with pressure and decreased with temperature. Here, the adhesion tensions are influenced by the variations in fluid-fluid IFT and contact angle. The simulated water contact angles of the hexane + H2O + silica system fall in the range from 58.0 degrees to 77.0 degrees and are not much affected by the addition of CO2. These contact angles increased with pressure, and the pressure effect was less pronounced at lower temperatures. Here, the adhesion tensions are mostly influenced by variations in the fluid-fluid IFTs. In all studied cases, CO2 molecules could penetrate into the interfacial region between the water droplet and the silica surface. (c) 2022 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

In this paper, a strongly mass conservative and stabilizer free scheme is designed and analyzed for the coupled Brinkman-Darcy flow and transport. The flow equations are discretized by using a strongly mass conservative scheme in mixed formulation with a suitable incorporation of the interface conditions. In particular, the interface conditions can be incorporated into the discrete formulation naturally without introducing additional variables. Moreover, the proposed scheme behaves uniformly robust for various values of viscosity. A novel upwinding staggered DG scheme in mixed form is exploited to solve the transport equation, where the boundary correction terms are added to improve the stability. A rigorous convergence analysis is carried out for the approximation of the flow equations. The velocity error is shown to be independent of the pressure and thus confirms the pressure-robustness. Stability and a priori error estimates are also obtained for the approximation of the transport equation; moreover, we are able to achieve a sharp stability and convergence error estimates thanks to the strong mass conservation preserved by our scheme. In particular, the stability estimate depends only on the true velocity on the inflow boundary rather than on the approximated velocity. Several numerical experiments are presented to verify the theoretical findings and demonstrate the performances of the method.

Phase equilibrium calculation, also known as flash calculation, plays significant roles in various aspects of petroleum and chemical industries. Since Michelsen proposed his milestone studies in 1982, through several decades of development, the current research interest on flash calculation has been shifted from accuracy to efficiency, but the ultimate goal remains the same focusing on estimation of the equilibrium phase amounts and phase compositions under the given variable specification. However, finding the transition route and its related saddle points are very often helpful to study the evolution of phase change and partition. Motivated by this, in this study we apply the string method to find the minimum energy paths and saddle points information of a single-component VT flash model with the Peng-Robinson equation of state. As the system has strong stiffness, common ordinary differential equation solvers have their limitations. To overcome these issues, a Rosenbrock-type exponential time differencing scheme is employed to reduce the computational difficulty caused by the high stiffness of the investigated system. In comparison with the published results and experimental data, the proposed numerical algorithm not only shows good feasibility and accuracy on phase equilibrium calculation, but also successfully calculates the minimum energy path and and saddle point of the single-component VT flash model with strong stiffness.

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. © 2018 Elsevier B.V.

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. © 2018 Elsevier B.V.

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. © 2018 Elsevier B.V.

In this work we consider a new efficient IMplicit Pressure Explicit Saturation (IMPES) scheme for the simulation of incompressible and immiscible two-phase flow in heterogeneous porous media with capillary pressure. Compared with the conventional IMPES schemes, the new IMPES scheme is inherently physics-preserving, namely, the new algorithm is locally mass conservative for both phases and it also enjoys another appealing feature that the total velocity is continuous in the normal direction. Moreover, the new scheme is unbiased with regard to the two phases and the saturations of both phases are bounds-preserving if the time step size is smaller than a certain value. The key ideas in the new scheme include that the Darcy flows for both phases are rewritten in the formulation based on the total velocity and an auxiliary velocity referring to as the capillary potential gradient, and the total discretized conservation equation is obtained by the summation of the discretized conservation equation for each phase. The upwind strategy is applied to update the saturations explicitly, and the upwind mixed finite element methods are used to solve the pressure-velocity systems which can be decoupled. We also present some interesting examples to demonstrate the efficiency and robustness of the new algorithm.

The subsurface flow is usually subject to uncertain porous media structures. In most cases, however, we only have partial knowledge about the porous media properties. A common approach is to model the uncertain parameters of porous media as random fields, then the statistical moments (e.g. expectation) of the Quantity of Interest(QoI) can be evaluated by the Monte Carlo method. In this study, we develop a full multigrid-multilevel Monte Carlo (FMG-MLMC) method to speed up the evaluation of random parameters effects on single-phase porous flows. In general, MLMC method applies a series of discretization with increasing resolution and computes the QoI on each of them. The effective variance reduction is the success of the method. We exploit the similar hierarchies of MLMC and multigrid methods and obtain the solution on coarse mesh $Q^c_l$ as a byproduct of the full multigrid solution on fine mesh $Q^f_l$ on each level $l$. In the cases considered in this work, the computational saving due to the coarse mesh samples saving is $20\%$ asymptotically. Besides, a comparison of Monte Carlo and Quasi-Monte Carlo (QMC) methods reveals a smaller estimator variance and a faster convergence rate of the latter approach in this study.

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.

In the past two decades, researchers have made remarkable progress in accelerating flash calculation, which is very useful in a variety of engineering processes. In this paper, general phase splitting problem statements and flash calculation procedures using the Successive Substitution Method are reviewed, while the main shortages are pointed out. Two acceleration methods, Newton's method and the Sparse Grids Method are presented afterwards as a comparison with the deep learning model proposed in this paper. A detailed introduction from artificial neural networks to deep learning methods is provided here with the authors' own remarks. Factors in the deep learning model are investigated to show their effect on the final result. A selected model based on that has been used in a flash calculation predictor with comparison with other methods mentioned above. It is shown that results from the optimized deep learning model meet the experimental data well with the shortest CPU time. More comparison with experimental data has been conducted to show the robustness of our model.

A review on the recent advances of the flow and transport phenomena in tight and shale formations is presented in this work. Exploration of oil and gas in resources that were once considered inaccessible opened the door to highlight interesting phenomena that require attention and understanding. The length scales associated with transport phenomena in tight and shale formations are rich. From nanoscale phenomena to field-scale applications, a unified frame that is able to encounter the varieties of phenomena associated with each scale may not be possible. Each scale has its own tools and limitations that may not, probably, be suitable at other scales. Multiscale algorithms that effectively couple simulations among various scales of porous media are therefore important. In this article, a review of the different length scales and the tools associated with each scale is introduced. Highlights on the different phenomena pertinent to each scale are summarized. Furthermore, the governing equations describing flow and transport phenomena at different scales are investigated. In addition, methods to solve these equations using numerical techniques are introduced. Cross-scale analysis and derivation of linear and nonlinear Darcy's scale laws from pore-scale governing equations are described. Phenomena occurring at molecular scales and their thermodynamics are discussed. Flow slippage at the nanosize pores and its upscaling to Darcy's scale are highlighted. Pore network models are discussed as a viable tool to estimate macroscopic parameters that are otherwise difficult to measure. Then, the environmental aspects associated with the different technologies used in stimulating the gas stored in tight and shale formations are briefly discussed.

A review on the recent advances of the flow and transport phenomena in tight and shale formations is presented in this work. Exploration of oil and gas in resources that were once considered inaccessible opened the door to highlight interesting phenomena that require attention and understanding. The length scales associated with transport phenomena in tight and shale formations are rich. From nanoscale phenomena to field-scale applications, a unified frame that is able to encounter the varieties of phenomena associated with each scale may not be possible. Each scale has its own tools and limitations that may not, probably, be suitable at other scales. Multiscale algorithms that effectively couple simulations among various scales of porous media are therefore important. In this article, a review of the different length scales and the tools associated with each scale is introduced. Highlights on the different phenomena pertinent to each scale are summarized. Furthermore, the governing equations describing flow and transport phenomena at different scales are investigated. In addition, methods to solve these equations using numerical techniques are introduced. Cross-scale analysis and derivation of linear and nonlinear Darcy's scale laws from pore-scale governing equations are described. Phenomena occurring at molecular scales and their thermodynamics are discussed. Flow slippage at the nanosize pores and its upscaling to Darcy's scale are highlighted. Pore network models are discussed as a viable tool to estimate macroscopic parameters that are otherwise difficult to measure. Then, the environmental aspects associated with the different technologies used in stimulating the gas stored in tight and shale formations are briefly discussed.

A review on the recent advances of the flow and transport phenomena in tight and shale formations is presented in this work. Exploration of oil and gas in resources that were once considered inaccessible opened the door to highlight interesting phenomena that require attention and understanding. The length scales associated with transport phenomena in tight and shale formations are rich. From nanoscale phenomena to field-scale applications, a unified frame that is able to encounter the varieties of phenomena associated with each scale may not be possible. Each scale has its own tools and limitations that may not, probably, be suitable at other scales. Multiscale algorithms that effectively couple simulations among various scales of porous media are therefore important. In this article, a review of the different length scales and the tools associated with each scale is introduced. Highlights on the different phenomena pertinent to each scale are summarized. Furthermore, the governing equations describing flow and transport phenomena at different scales are investigated. In addition, methods to solve these equations using numerical techniques are introduced. Cross-scale analysis and derivation of linear and nonlinear Darcy's scale laws from pore-scale governing equations are described. Phenomena occurring at molecular scales and their thermodynamics are discussed. Flow slippage at the nanosize pores and its upscaling to Darcy's scale are highlighted. Pore network models are discussed as a viable tool to estimate macroscopic parameters that are otherwise difficult to measure. Then, the environmental aspects associated with the different technologies used in stimulating the gas stored in tight and shale formations are briefly discussed.

A review on the recent advances of the flow and transport phenomena in tight and shale formations is presented in this work. Exploration of oil and gas in resources that were once considered inaccessible opened the door to highlight interesting phenomena that require attention and understanding. The length scales associated with transport phenomena in tight and shale formations are rich. From nanoscale phenomena to field-scale applications, a unified frame that is able to encounter the varieties of phenomena associated with each scale may not be possible. Each scale has its own tools and limitations that may not, probably, be suitable at other scales. Multiscale algorithms that effectively couple simulations among various scales of porous media are therefore important. In this article, a review of the different length scales and the tools associated with each scale is introduced. Highlights on the different phenomena pertinent to each scale are summarized. Furthermore, the governing equations describing flow and transport phenomena at different scales are investigated. In addition, methods to solve these equations using numerical techniques are introduced. Cross-scale analysis and derivation of linear and nonlinear Darcy's scale laws from pore-scale governing equations are described. Phenomena occurring at molecular scales and their thermodynamics are discussed. Flow slippage at the nanosize pores and its upscaling to Darcy's scale are highlighted. Pore network models are discussed as a viable tool to estimate macroscopic parameters that are otherwise difficult to measure. Then, the environmental aspects associated with the different technologies used in stimulating the gas stored in tight and shale formations are briefly discussed.

A review on the recent advances of the flow and transport phenomena in tight and shale formations is presented in this work. Exploration of oil and gas in resources that were once considered inaccessible opened the door to highlight interesting phenomena that require attention and understanding. The length scales associated with transport phenomena in tight and shale formations are rich. From nanoscale phenomena to field-scale applications, a unified frame that is able to encounter the varieties of phenomena associated with each scale may not be possible. Each scale has its own tools and limitations that may not, probably, be suitable at other scales. Multiscale algorithms that effectively couple simulations among various scales of porous media are therefore important. In this article, a review of the different length scales and the tools associated with each scale is introduced. Highlights on the different phenomena pertinent to each scale are summarized. Furthermore, the governing equations describing flow and transport phenomena at different scales are investigated. In addition, methods to solve these equations using numerical techniques are introduced. Cross-scale analysis and derivation of linear and nonlinear Darcy's scale laws from pore-scale governing equations are described. Phenomena occurring at molecular scales and their thermodynamics are discussed. Flow slippage at the nanosize pores and its upscaling to Darcy's scale are highlighted. Pore network models are discussed as a viable tool to estimate macroscopic parameters that are otherwise difficult to measure. Then, the environmental aspects associated with the different technologies used in stimulating the gas stored in tight and shale formations are briefly discussed.

A review on the recent advances of the flow and transport phenomena in tight and shale formations is presented in this work. Exploration of oil and gas in resources that were once considered inaccessible opened the door to highlight interesting phenomena that require attention and understanding. The length scales associated with transport phenomena in tight and shale formations are rich. From nanoscale phenomena to field-scale applications, a unified frame that is able to encounter the varieties of phenomena associated with each scale may not be possible. Each scale has its own tools and limitations that may not, probably, be suitable at other scales. Multiscale algorithms that effectively couple simulations among various scales of porous media are therefore important. In this article, a review of the different length scales and the tools associated with each scale is introduced. Highlights on the different phenomena pertinent to each scale are summarized. Furthermore, the governing equations describing flow and transport phenomena at different scales are investigated. In addition, methods to solve these equations using numerical techniques are introduced. Cross-scale analysis and derivation of linear and nonlinear Darcy's scale laws from pore-scale governing equations are described. Phenomena occurring at molecular scales and their thermodynamics are discussed. Flow slippage at the nanosize pores and its upscaling to Darcy's scale are highlighted. Pore network models are discussed as a viable tool to estimate macroscopic parameters that are otherwise difficult to measure. Then, the environmental aspects associated with the different technologies used in stimulating the gas stored in tight and shale formations are briefly discussed.

We study numerically the three-dimensional droplets spreading on physically flat chemically patterned surfaces with periodic squares separated by channels. Our model consists of the Navier-Stokes-Cahn-Hilliard equations with the generalized Navier boundary conditions. Stick-slip behavior and con-tact angle hysteresis are observed. Moreover, we also study the relationship between the effective advancing/receding angle and the two intrinsic angles of the surface patterns. By increasing the volume of droplet gradually, we find that the advancing contact line tends gradually to an equiangular octagon with the length ratio of the two adjacent sides equal to a fixed value that depends on the geometry of the pattern.

We study numerically the three-dimensional droplets spreading on physically flat chemically patterned surfaces with periodic squares separated by channels. Our model consists of the Navier-Stokes-Cahn-Hilliard equations with the generalized Navier boundary conditions. Stick-slip behavior and con-tact angle hysteresis are observed. Moreover, we also study the relationship between the effective advancing/receding angle and the two intrinsic angles of the surface patterns. By increasing the volume of droplet gradually, we find that the advancing contact line tends gradually to an equiangular octagon with the length ratio of the two adjacent sides equal to a fixed value that depends on the geometry of the pattern.

We study numerically the three-dimensional droplets spreading on physically flat chemically patterned surfaces with periodic squares separated by channels. Our model consists of the Navier-Stokes-Cahn-Hilliard equations with the generalized Navier boundary conditions. Stick-slip behavior and con-tact angle hysteresis are observed. Moreover, we also study the relationship between the effective advancing/receding angle and the two intrinsic angles of the surface patterns. By increasing the volume of droplet gradually, we find that the advancing contact line tends gradually to an equiangular octagon with the length ratio of the two adjacent sides equal to a fixed value that depends on the geometry of the pattern.

We investigate the two-phase flow in a horizontally placed Y-shaped tube with different Young's angle and width in each branch. By using a quasistatic approach, we can determine the specific contact position and the equilibrium contact angle of fluid in each branch based on the minimization problem of the free energy of the system. The wettability condition and the width of the two branches play important roles in the distribution of fluid in each branch. We also consider the effect of gravity. Some fluid in the upper branch will be pulled down due to the competition of the surface energy and the gravitational energy. The result provides some insights on the theory of two-phase flow in porous media. In particular, it highlights that the inhomogeneous wettability distribution affects the direction of the fluid penetrating a given porous medium domain. It also sheds light on the current debate whether relative permeability may be considered as a full tensor rather than a scalar. (C) 2014 Elsevier Ltd. All rights reserved.

In this article, we study two-phase fluid flow in coupled free flow and porous media regions. The model consists of coupled Cahn-Hilliard and Navier-Stokes equations in the free fluid region and the two-phase Darcy law in the porous medium region. We propose a Robin-Robin domain decomposition method 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. Numerical examples are presented to illustrate the effectiveness of this method. (C) 2014 Elsevier Inc. All rights reserved.

In this article, we study two-phase fluid flow in coupled free flow and porous media regions. The model consists of coupled Cahn-Hilliard and Navier-Stokes equations in the free fluid region and the two-phase Darcy law in the porous medium region. We propose a Robin-Robin domain decomposition method 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. Numerical examples are presented to illustrate the effectiveness of this method. (C) 2014 Elsevier Inc. All rights reserved.

We investigate the two-phase flow in a horizontally placed Y-shaped tube with different Young's angle and width in each branch. By using a quasistatic approach, we can determine the specific contact position and the equilibrium contact angle of fluid in each branch based on the minimization problem of the free energy of the system. The wettability condition and the width of the two branches play important roles in the distribution of fluid in each branch. We also consider the effect of gravity. Some fluid in the upper branch will be pulled down due to the competition of the surface energy and the gravitational energy. The result provides some insights on the theory of two-phase flow in porous media. In particular, it highlights that the inhomogeneous wettability distribution affects the direction of the fluid penetrating a given porous medium domain. It also sheds light on the current debate whether relative permeability may be considered as a full tensor rather than a scalar. (C) 2014 Elsevier Ltd. All rights reserved.

In this article, we study two-phase fluid flow in coupled free flow and porous media regions. The model consists of coupled Cahn-Hilliard and Navier-Stokes equations in the free fluid region and the two-phase Darcy law in the porous medium region. We propose a Robin-Robin domain decomposition method 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. Numerical examples are presented to illustrate the effectiveness of this method. (C) 2014 Elsevier Inc. All rights reserved.

In this paper, a semi-implicit finite element method is presented for the coupled Cahn-Hilliard and Navier-Stokes equations with the generalized Navier boundary condition for the moving contact line problems. In our method, the system is solved in a decoupled way. For the Cahn-Hilliard equations, a convex splitting scheme is used along with a P1-P1 finite element discretization. The scheme is unconditionally stable. A linearized semi-implicit P2-P0 mixed finite element method is employed to solve the Navier-Stokes equations. With our method, the generalized Navier boundary condition is extended to handle the moving contact line problems with complex boundary in a very natural way. The efficiency and capacity of the present method are well demonstrated with several numerical examples. (C) 2012 Elsevier Inc. All rights reserved.

In this paper, a semi-implicit finite element method is presented for the coupled Cahn-Hilliard and Navier-Stokes equations with the generalized Navier boundary condition for the moving contact line problems. In our method, the system is solved in a decoupled way. For the Cahn-Hilliard equations, a convex splitting scheme is used along with a P1-P1 finite element discretization. The scheme is unconditionally stable. A linearized semi-implicit P2-P0 mixed finite element method is employed to solve the Navier-Stokes equations. With our method, the generalized Navier boundary condition is extended to handle the moving contact line problems with complex boundary in a very natural way. The efficiency and capacity of the present method are well demonstrated with several numerical examples. (C) 2012 Elsevier Inc. All rights reserved.

In this paper, a semi-implicit finite element method is presented for the coupled Cahn-Hilliard and Navier-Stokes equations with the generalized Navier boundary condition for the moving contact line problems. In our method, the system is solved in a decoupled way. For the Cahn-Hilliard equations, a convex splitting scheme is used along with a P1-P1 finite element discretization. The scheme is unconditionally stable. A linearized semi-implicit P2-P0 mixed finite element method is employed to solve the Navier-Stokes equations. With our method, the generalized Navier boundary condition is extended to handle the moving contact line problems with complex boundary in a very natural way. The efficiency and capacity of the present method are well demonstrated with several numerical examples. (C) 2012 Elsevier Inc. All rights reserved.