Two-phase Flow In Porous Media Research Articles

Editorial: Multiphase flow behavior in the deep-stratum and deep-water wellbores

state, heat/mass transfer between gas and liquid, and fluid hydrodynamics in the wellbore depend highly on the temperature and pressure conditions, the multiphase flow behaviors in deep-stratum and deep-water wellbores become much more complicated. Different from the conventional oil and gas exploitation, the production of oil and gas from the and deep stratum deep-water regions involves multiple complex physical phenomena, such as phase transition (gas-liquid-solid-supercritical), particle deposition (wax, hydrate, water scale, asphaltene, etc.), changes of heat/mass transfer induced by phase change, and so on. It is, therefore, of high importance to discuss the multiphase flow behavior in the deep-stratum and deep-water wellbores. This Research Topic aims to explore recent developments in this area and focused on the following two aspects: 1) Numerical simulations of gas-liquid or liquid-solid two-phase flow in porous media, 2) Experiments and simulations of multiphase flow in the pipelines and wellbores. There are a total of 12 articles under this Research Topic.

A physics-informed convolutional neural network for the simulation and prediction of two-phase Darcy flows in heterogeneous porous media

The physics-informed neural network (PINN) is a general deep learning framework for simulating flows with limited or no labeled data. In the current study, we develop a physics-informed convolutional neural network (PICNN) for simulating transient two-phase Darcy flows in heterogeneous reservoir models with source/sink terms in the absence of labeled data, where the finite volume method (FVM) is adopted to approximate the PDE residual in the loss function such that flux continuity between neighboring cells of different properties is defined rigorously, and a well model is adopted to approximate the high pressure gradient near sources or sinks. The implicit-pressure explicit-saturation (IMPES) scheme is employed such that only a single CNN needs to be trained per time step. Dirichlet boundary conditions are not a mandatory requirement for PICNN-based implicit solver but act as labeled data that can help enhance accuracy. The proposed approach is validated in homogeneous and heterogeneous reservoirs and aspects including efficiency and accuracy are discussed. In addition, we demonstrate that the CNN structure can be trained as a data-driven surrogate for two-phase Darcy flows given sufficient labeled samples.

Physics-informed machine learning for solving partial differential equations in porous media

Physical phenomenon in nature is generally simulated by partial differential equations. Among different sorts of partial differential equations, the problem of two-phase flow in porous media has been paid intense attention. As a promising direction, physics-informed neural networks shed new light on the solution of partial differential equations. However, current physics-informed neural networks’ ability to learn partial differential equations relies on adding artificial diffusion or using prior knowledge to increase the number of training points along the shock trajectory, or adaptive activation functions. To address these issues, this study proposes a physics-informed neural network with long short-term memory and attention mechanism, an ingenious method to solve the Buckley-Leverett partial differential equations representing two-phase flow in porous media. The designed network structure overcomes the dependency on artificial diffusion terms and enhances the importance of shallow features. The experimental results show that the proposed method is in good agreement with analytical solutions. Accurate approximations are shown even when encountering shock points in saturated fields of porous media. Furthermore, experiments show our innovative method outperforms existing traditional physics-informed machine learning approaches. Cited as: Shan, L., Liu, C., Liu, Y., Tu, Y., Dong, L., Hei, X. Physics-informed machine learning for solving partial differential equations in porous media. Advances in Geo-Energy Research, 2023, 8(1): 37-44. https://doi.org/10.46690/ager.2023.04.04

Experimental evaluation of fluid connectivity in two-phase flow in porous media

In this work, we provide a physically-consistent modeling approach for two-phase porous media flow, by including percolating interfacial area and saturation as state variables. For this purpose, we combine two continuum theories for two-phase flow which have been individually proven to be conditionally valid. This means the potential use of the connected-to-the-flow interfacial area as a state variable is tested utilizing time-resolved microfluidic experiments, for various flux boundary conditions. Moreover, we observe and study a linear relation between the percolating saturation and interfacial area, which is persistent for the tested boundary conditions. In our microfluidic experiments, we employ optical microscopy to perform cyclic immiscible displacement experiments. Our results show that a continuum model, where capillary pressure, saturation , and specific interfacial area of the clusters connected to the flow are considered, is closer to a universal description of two-phase flow than the common approaches, where the only state variable is saturation.

Machine Learning Prediction of Nanoparticle Transport with Two-Phase Flow in Porous Media

Reservoir simulation is a time-consuming procedure that requires a deep understanding of complex fluid flow processes as well as the numerical solution of nonlinear partial differential equations. Machine learning algorithms have made significant progress in modeling flow problems in reservoir engineering. This study employs machine learning methods such as random forest, decision trees, gradient boosting regression, and artificial neural networks to forecast nanoparticle transport with the two-phase flow in porous media. Due to the shortage of data on nanoparticle transport in porous media, this work creates artificial datasets using a mathematical model. It predicts nanoparticle transport behavior using machine learning techniques, including gradient boosting regression, decision trees, random forests, and artificial neural networks. Utilizing the scikit-learn toolkit, strategies for data preprocessing, correlation, and feature importance are addressed. Furthermore, the GridSearchCV algorithm is used to optimize hyperparameter tuning. The mean absolute error, R-squared correlation, mean squared error, and root means square error are used to assess the models. The ANN model has the best performance in forecasting the transport of nanoparticles in porous media, according to the results.

NUMERICAL ANALYSIS OF TWO IMMISCIBLE FLUIDS IN POROUS MEDIA BY THE MULTISCALE AND HOMOGENIZATION METHODS

This paper proposes a numerical model for simulating an immiscible compressible two-phase flow in a periodic heterogeneous porous media with application to a coal-bed methane extraction system. Methane flow in a coal sample submerged in water is simulated using the multiscale homogenization method and the averaged macroscopic approach. A weak formulation is used to discretize the governing equations coupled with the boundary conditions using the finite element method and solved using the open-source code Freefem++. It was observed that the general behavior of both methods was in good agreement with actual gas saturation evolution. However, the multiscale approach provided more information regarding the global pressure evolution behavior of the mitigating gas.

Uncertainty quantification of two-phase flow in porous media via the Coupled-TgNN surrogate model

The uncertainty quantification (UQ) of subsurface two-phase flow usually requires numerous executions of forward simulations under varying conditions. In this work, a novel coupled theory-guided neural network (TgNN) based surrogate model is built to facilitate computation efficiency under the premise of satisfactory accuracy. The core notion of this proposed method is to bridge two separate blocks on top of an overall network. They underlie the TgNN model in a coupled form, which reflects the coupling nature of pressure and water saturation in the two-phase flow equation. The TgNN model not only relies on labeled data, but also incorporates underlying scientific theory and experiential rules (e.g., governing equations, stochastic parameter fields, boundary and initial conditions, well conditions, and expert knowledge) as additional components into the loss function. The performance of the TgNN-based surrogate model for two-phase flow problems is tested by different numbers of labeled data and collocation points, as well as the existence of data noise. The proposed TgNN-based surrogate model offers an effective way to solve the coupled nonlinear two-phase flow problem, and shows good accuracy and strong robustness when compared with the purely data-driven surrogate model. By combining the accurate TgNN-based surrogate model with the Monte Carlo method, UQ tasks can be performed at a minimum cost to evaluate statistical quantities. Since the heterogeneity of the random fields strongly impacts the results of the surrogate model, corresponding variance and correlation length are added to the input of the neural network to maintain its predictive capacity. In addition, several more complicated scenarios are also considered, including dynamically changing well conditions and dynamically changing variance of random fields. The results show that the TgNN-based surrogate model exhibits satisfactory accuracy, stability, and efficiency in the UQ problem of subsurface two-phase flow.

A relaxation model for the non-isothermal Navier-Stokes-Korteweg equations in confined domains

The Navier-Stokes-Korteweg (NSK) system is a classical diffuse interface model which is based on van der Waals' theory of capillarity. Diffuse interface methods have gained much interest to model two-phase flow in porous media. However, for the numerical solution of the NSK equations two major challenges have to be faced. First, an extended numerical stencil is required due to a third-order term in the linear momentum and the total energy equations. In addition, the dispersive contribution in the linear momentum equations prevents the straightforward use of contact angle boundary conditions. Secondly, any real gas equation of state is based on a non-convex Helmholtz free energy potential which may cause the eigenvalues of the Jacobian of the first-order fluxes to become imaginary numbers inside the spinodal region.In this work, a thermodynamically consistent relaxation model is presented which is used to approximate the NSK equations. The model is complimented by thermodynamically consistent non-equilibrium boundary conditions which take contact angle effects into account. Due to the relaxation approach, the contribution of the Korteweg tensor in the linear momentum and total energy equations can be reduced to first-order terms which enables a straightforward implementation of contact angle boundary conditions in a numerical scheme. Moreover, the definition of a modified pressure function enables to formulate first-order fluxes which remain strictly hyperbolic in the entire spinodal region. The present work is a generalization of a previously presented parabolic relaxation model for the isothermal NSK equations.A high-order discontinuous Galerkin spectral element method which supports curved elements and hanging nodes is employed to discretize the system. The relaxation model and its corresponding boundary conditions are validated using solutions of the original NSK model and analytical results for one-, two- and three-dimensional test cases. The simulation of a spinodal decomposition in a three-dimensional porous structure underlines the capability of the presented approach.

A lattice Boltzmann exploration of two-phase displacement in 2D porous media under various pressure boundary conditions

While experimental designs developed in recent decades have contributed to research on dynamic nonequilibrium effects in transient two-phase flow in porous media, this problem has been seldom investigated using direct numerical simulation (DNS). Only a few studies have sought to numerically solve Navier–Stokes equations with level-set (LS) or volume-of-fluid (VoF) methods, each of which has constraints in terms of meniscus dynamics for various flow velocities in the control volume (CV) domain. The Shan–Chen multiphase multicomponent lattice Boltzmann method (SC-LBM) has a fundamental mechanism to separate immiscible fluid phases in the density domain without these limitations. Therefore, this study applied it to explore two-phase displacement in a single representative elementary volume (REV) of two-dimensional (2D) porous media. As a continuation of a previous investigation into one-step inflow/outflow in 2D porous media, this work seeks to identify dynamic nonequilibrium effects on capillary pressure–saturation relationship ( P c – S ) for quasi-steady-state flow and multistep inflow/outflow under various pressure boundary conditions. The simulation outcomes show that P c , S and specific interfacial area ( a nw ) had multistep-wise dynamic effects corresponding to the multistep-wise pressure boundary conditions. With finer adjustments to the increase in pressure over more steps, dynamic nonequilibrium effects were significantly alleviated and even finally disappeared to achieve quasi-steady-state inflow/outflow conditions. Furthermore, triangular wave-formed pressure boundary conditions were applied in different periods to investigate dynamic nonequilibrium effects for hysteretical P c – S . The results showed overshoot and undershoot of P c to S in loops of the nonequilibrium hysteresis. In addition, the flow regimes of multistep-wise dynamic effects were analyzed in terms of Reynolds and capillary numbers ( Re and Ca ). The analysis of REV-scale flow regimes showed higher Re (1 < Re < 10) for more significant dynamic nonequilibrium effects. This indicates that inertia is critical for transient two-phase flow in porous media under dynamic nonequilibrium conditions.

Generative Adversarial Neural Networks for the Heuristic Modelling of a Two-Phase Flow in Porous Media

Generative Adversarial Neural Networks for the Heuristic Modelling of a Two-Phase Flow in Porous Media

FRACTAL EVIDENCE OF REACTIVE TRANSPORT IN GEOLOGIC SYSTEMS BASED ON MULTIPHASE FLOW HYDRAULIC MODELS PART II: THE BIOGEOCHEMICAL ASPECT

In the petroleum and soil science literature, fractal models of porous systems abound. Consequently, any process, be it physicochemical or geomechanical, that imparts pore structure will equally impact the fractal nature of the system. Several studies have reported the effect of physicochemical processes, such as the water–rock interactions, on the fractal dimension of the porous system using different approaches involving sophisticated instrumentation. Others have also reported the effect of geomechanical processes such as compaction on fractal dimension using similar approaches. To the best of our knowledge, none or little has been reported regarding the use of multiphase flow hydraulic models for revealing the effect of water–rock interactions on the fractal dimension and pore size distribution of the porous system. Consequently, to fill the knowledge gap in this regard, data on two-phase flow of acid gas and brine in porous media have been used to show that the fractal dimension and pore size distribution of the porous system change in response to water–rock interactions. In view of the injection of acid gas, such as anthropogenic carbon dioxide, into saline aquifers being a feasible means for carbon immobilization by mineral carbonation and biogeochemical means, the effect of change in the fractal properties of the porous system on the microbial transport and distribution in porous system has also been reviewed in relationship to the principal findings of this paper. Most importantly, the results of this study show that data on two-phase flow in porous media involving acid gas and brine can be used to prove changes in the fractal nature of the porous system, and this change can also impact subsurface distribution of methanogens that have the potential to trap carbon by methanogenesis.

Numerical simulation of bubble rising in porous media using lattice Boltzmann method

Rising bubble systems in porous media exist in a variety of industrial processes. However, the flow characteristics of the issue are not well understood. In this work, the rising of bubble/bubbles through two types of porous structures, namely, in-line structured pore and staggered structured pore, are studied using a large density ratio lattice Boltzmann model. The effects of Eötvös number, pore shape, viscosity ratio, initial bubble number, and arrangement manner of the initial bubbles on the bubble deformation, bubble rising velocity, residual bubble mass, bubble perimeter, and the number of bubble breakups are investigated. It is found that as the Eötvös number increases, the bubbles are more easily broken during the process of passing through the porous media, the shapes of the sub-bubbles deviate from the original ones more and more, the bubble perimeter increases, and the difference between the bubble dynamics obtained by the in-line and staggered porous media decreases. Compared to the results of circular and rectangular pores, the bubble rising through the diamondoid pore has a more considerable deformation, which causes a slower rising speed. Furthermore, in the case that two bubbles are originally placed under the porous medium, the bubble deformation is greater and the bubble fracture times increase if the initial bubbles are aligned vertically. The findings of this work can contribute to the understanding of gas–liquid two-phase flow in porous media.

Effect of Pore Space Stagnant Zones on Interphase Mass Transfer in Porous Media, for Two-Phase Flow Conditions

Interphase mass transfer is an important solute transport process in two-phase flow in porous media. During two-phase flow, hydrodynamically stagnant and flowing zones are formed, with the stagnant ones being adjacent to the interfaces through which the interphase mass transfer happens. Due to the existence of these stagnant zones in the vicinity of the interface, the mass transfer coefficient decreases to a certain extent. There seems to be a phenomenological correlation between the mass transfer coefficient and the extent of the stagnant zone which, however, is not yet fully understood. In this study, the phase-field method-based continuous species transfer model is applied to simulate the interphase mass transfer of a dissolved species from the immobile, residual, non-aqueous phase liquid (NAPL) to the flowing aqueous phase. Both scenarios, this of a simple cavity and this of a porous medium, are investigated. The effects of flow rates on the mass transfer coefficient are significantly reduced when the stagnant zone and the diffusion length are larger. It is found that the stagnant zone saturation can be a proxy of the overall diffusion length of the terminal menisci in the porous medium system. The early-stage mass transfer coefficient continuously decreases due to the depletion of the solute in the small NAPL clusters that are in direct contact with the flowing water. The long-term mass transfer mainly happens on the interfaces associated with large NAPL clusters with larger diffusion lengths, and the mass transfer coefficient is mainly determined by the stagnant zone saturation.

A statistical mechanics framework for immiscible and incompressible two-phase flow in porous media

We construct a statistical mechanics for immiscible and incompressible two-phase flow in porous media under local steady-state conditions based on the Jaynes maximum entropy principle. A cluster entropy is assigned to our lack of knowledge of, and control over, the fluid and flow configurations in the pore space. As a consequence, two new variables describing the flow emerge: The agiture, which describes the level of agitation of the two fluids, and the flow derivative, which is conjugate to the saturation. Agiture and flow derivative are the analogs of temperature and chemical potential in standard (thermal) statistical mechanics. The associated thermodynamics-like formalism reveals a number of hitherto unknown relations between the variables that describe the flow, including fluctuations. The formalism opens for new approaches to characterize porous media with respect to multi-phase flow for practical applications, replacing the simplistic relative permeability theory while still keeping the number of variables tractable.

Fluid-Fluid Interfacial Area and Its Impact on Relative Permeability: A Pore Network Modeling Study

SummaryRelative permeability (kr) is commonly modeled as an empirical function of phase saturation. Although current empirical models can provide a good match of one or two measured relative permeabilities using saturation alone, they are unable to predict relative permeabilities well when there is hysteresis or when physical properties such as wettability change. Further, current models often result in relative permeability discontinuities that can cause convergence and accuracy problems in simulation. To overcome these problems, recent research has modeled relative permeability as a state function of both saturation (S) and phase connectivity (X). Pore network modeling (PNM) data, however, show small differences in relative permeability for the same S-X value when approached from a different flow direction. This paper examines the impact of one additional Minkowski parameter (Mecke and Arns 2005), the fluid-fluid interfacial area, on relative permeability to identify if that satisfactorily explains this discrepancy.We calculate the total fluid-fluid interfacial areas (IA) during two-phase (oil/water) flow in porous media using PNM. The area is calculated from PNM simulations using the areas associated with corners and throats in pore elements of different shapes. The pore network is modeled after a Bentheimer sandstone, using square, triangular prism, and circular pore shapes. Simulations were conducted for numerous primary drainage (PD) and imbibition cycles at a constant contact angle of 0° for the wetting phase. Simultaneous measurements of capillary pressure, relative permeability, saturation, and phase connectivity are made for each displacement. The fluid-fluid IA is calculated from the PNM capillary pressure, the fluid location in the pore elements, and the pore element dimensional data.The results show that differences in the relative permeability at the same (S, X) point are explained well by differences in the fluid-fluid interfacial area (IA). That is, for a larger change in IA at these intersection points, the permeability difference is greater. That difference in relative permeability approaches zero as the difference in IA approaches zero. This confirms that relative permeability can be modeled better as a unique function of S, X, and IA. The results also show that an increase in IA restricts flow decreasing the nonwetting (oil) phase permeability. This decrease is caused by an increase in the throat area fraction compared to the corner area as the total area IA increases. The wetting phase relative permeability, however, shows the inverse trend in that its relative permeability is greater when IA becomes larger owing to a greater fraction of the total area associated with the corners. The area IA, however, impacts the nonwetting phase relative permeability more than the wetting phase relative permeability. Corner flow improves the wetting phase relative permeability because the wetting phase is continuous there. Finally, a sensitivity analysis shows that relative permeability is more sensitive to change in S than it is for IA for the case studied, implying that if only two parameters are used to model relative permeability, it is better to choose S and X.

Thermodynamically consistent modeling of immiscible gas–liquid flow in porous media

Modeling of immiscible gas–liquid two-phase flow with gas compressibility in porous media plays an important role in shale gas production, geological sequestration of carbon dioxide, and underground gas storage. The second law of thermodynamics is universally recognized as an essential principle any promising model should obey. The existing models have no proper concept of free energies for such a problem, thereby failing to obey this law. In this paper, we first introduce free energies to account for the liquid–gas capillarity effect and gas compressibility, and then using the second law of thermodynamics, we rigorously derive a thermodynamically consistent model for immiscible gas–liquid two-phase flow in porous media. The proposed free energy that describes the capillarity effect is verified by the laboratory data. For gas flow, we use molar density rather than pressure as the primary variable and take the Helmholtz free energy density determined by a realistic equation of state to characterize the gas compressibility. Numerical simulation results are also presented to demonstrate the thermodynamical consistency of the model and the applicability to simulate the liquid and gas displacement processes.

Configurable Open-source Data Structure for Distributed Conforming Unstructured Homogeneous Meshes with GPU Support

A general multi-purpose data structure for an efficient representation of conforming unstructured homogeneous meshes for scientific computations on CPU and GPU-based systems is presented. The data structure is provided as open-source software as part of the TNL library (https://tnl-project.org/). The abstract representation supports almost any cell shape and common 2D quadrilateral, 3D hexahedron and arbitrarily dimensional simplex shapes are currently built into the library. The implementation is highly configurable via templates of the C++ language, which allows avoiding the storage of unnecessary dynamic data. The internal memory layout is based on state-of-the-art sparse matrix storage formats, which are optimized for different hardware architectures in order to provide high-performance computations. The proposed data structure is also suitable for meshes decomposed into several subdomains and distributed computing using the Message Passing Interface (MPI). The efficiency of the implemented data structure on CPU and GPU hardware architectures is demonstrated on several benchmark problems and a comparison with another library. Its applicability to advanced numerical methods is demonstrated with an example problem of two-phase flow in porous media using a numerical scheme based on the mixed-hybrid finite element method (MHFEM). We show GPU speed-ups that rise above 20 in 2D and 50 in 3D when compared to sequential CPU computations, and above 2 in 2D and 9 in 3D when compared to 12-threaded CPU computations.

Machine Learning Techniques Applied to Identify the Two-Phase Flow Pattern in Porous Media Based on Signal Analysis

The development of flow pattern identification technology for gas–liquid two-phase flow in porous media is of great significance to engineering research and production. In this paper, a high accuracy identification method for two-phase flow pattern in porous media is proposed with Machine learning techniques. The gas–liquid two-phase flow patterns and corresponding differential pressure signals in porous beds with particle diameters of 1.5 mm, 3 mm, and 6 mm are obtained through visual experiments. Three time domain characteristic parameters (Mean, Standard deviation, and Range) are calculated by a statistical method, while the EMD energy spectrum of the signal is obtained by empirical mode decomposition. Based on these parameters, machine learning technology, including support vector machine (SVM) and BP neural network, are employed to identify the flow pattern. Four flow pattern identification models are trained based on SVM and BP neural network, with accuracies of 94.77%, 93.4%, 96.08%, and 91.5%. Furthermore, the three models with good performance are integrated by integrated learning technology. An integrated identification model of gas–liquid two-phase flow pattern in porous media with an overall accuracy of 98.04% is finally obtained.

Taichi-LBM3D: A Single-Phase and Multiphase Lattice Boltzmann Solver on Cross-Platform Multicore CPU/GPUs

The success of the lattice Boltzmann method requires efficient parallel programming and computing power. Here, we present a new lattice Boltzmann solver implemented in Taichi programming language, named Taichi-LBM3D. It can be employed on cross-platform shared-memory many-core CPUs or massively parallel GPUs (OpenGL and CUDA). Taichi-LBM3D includes the single- and two-phase porous medium flow simulation with a D3Q19 lattice model, Multi-Relaxation-Time (MRT) collision scheme and sparse data storage. It is open source, intuitive to understand, and easily extensible for scientists and researchers.

Determination of dynamic capillary effect on two-phase flow in porous media: A perspective from various methods

The relationship between capillary pressure and saturation plays a critical role in the characterization of two-phase flow and transport in aquifers and oil reservoirs. This relationship is usually determined under the static condition, where capillary pressure is the only function of saturation. However, considerable experiments have suggested that the dependence of capillary pressure on desaturation rate is under the dynamic condition. Thus, a more general description of capillary pressure that includes dynamic capillary effect has been approved widely. A comprehensive understanding of the dynamic capillary effect is needed for the investigation of the two-phase flow in porous media by various methods. In general, dynamic capillary effect in porous media can be studied through the laboratory experiment, pore- to macro-scale modeling, and artificial neural network. Here, main principle and research procedures of each method are reviewed in detail. Then, research progress, disadvantages and advantages are discussed, respectively. In addition, upscaling study from pore- to macro-scale are introduced, which explains the difference between laboratory experiment and pore-scale modeling. At last, several future perspectives and recommendations for optimal solution of dynamic capillary effect are presented.