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A diffuse interface model and fully decoupled, energy-stable scheme for the two-phase ferrofluid flows in porous media

For all societies that rely or have relied fully or partly on nuclear energy the long-term management of radioactive waste remains a complex challenge for generations to come. Issues are the multidisciplinary scientific understanding of key processes often coupled over time and space scales, the search of appropriate and acceptable disposal sites, the development of engineering solutions and the organization of transparency for public engagement. Deep geological repositories (DGRs) are industrial megaprojects, internationally recognised as the most robust solution, designed to isolate hazardous materials from the biosphere for more than tens of thousands of years [1]. First repositories have been licenced recently.The European Joint Programme on Radioactive Waste Management (EURAD) provides a collaborative roadmap for this work, consolidating research, development, knowledge management and demonstration (RD&D) to strengthen the scientific basis for geological disposal [2]. A cornerstone of EURAD is the systematic production of Stateof-the-Art (SoTA) reports in various research fields. These documents synthesise and review current understanding, identify uncertainties, and guide future research directions.This editorial introduces the SoTA reports published in this Research Topic and produced within EURAD-1, highlighting their contributions to the international knowledge base for geological disposal.Demonstrating the ongoing innovations in radioactive waste disposal material choices, the workpackage ConCorD report on container corrosion under disposal conditions (Muñoz et al., 2024) describe the long-term integrity of sealed containers. It covers a range of materials used to provide the long-term primary containment barrier, typical for high-level waste and spent fuel disposal containers. It surveys the current understanding of degradation processes across a range of environments, including the effects of irradiation, chemical transients, microbial activity, and extreme environmental fluctuations. It also explores the potential of novel container materials and highlights approaches to integrate experimental data into predictive models of container lifetime. Its inclusion affirms the importance of rigorous, coupled analysis of container behaviour as a keystone in safety case development.The chemical environment of a disposal cell in deep geological formations is shaped by the interactions between wasteforms, engineered barriers, and surrounding host rock. Two companion SoTA reports address this complexity.• Part I (Neeft et al.) examines processes at interfaces and the overall chemical evolution at the disposal cell scale, focusing on safety, performance, and optimisation.• Part II (Deissmann et al.) describes the experimental and modelling tools that underpin our understanding and support the construction of coherent narratives of disposal cell evolution.Together, these reports provide a comprehensive framework for evaluating chemical evolution in disposal systems and its significance for long-term safety. They cover disposal cells and material interfaces common to cemented intermediate level waste disposal concepts and vitrified high level waste in carbon steel overpacks with bentonite or concrete buffer materials.Clay-based materials are integral to disposal concepts, both as engineered barriers and as host rock formations. Their thermo-hydro-mechanical (THM) behaviour at repositoryrelevant conditions is the focus of two SoTA reports.• Villar et al. present the state of the art on clay buffers, with particular attention to the effects of elevated temperatures. Together, these studies improve understanding of how clay barriers evolve under thermal, hydraulic, and mechanical stresses, reinforcing confidence in their role towards contributing to the long-term containment of radioactive waste in DGRs.The long-term safety of geological disposal relies on the ability of engineered and natural barriers to retain radionuclides and limit their transport. The SoTA report by Maes et al. reviews current knowledge of radionuclide retention and migration in both clay and crystalline host rocks. This synthesis clarifies the key processes that govern radionuclide mobility and highlights areas where further research is needed.Repository performance assessments increasingly depend on advanced modelling of complex, coupled processes at various time and space scales. The SoTA report by Claret et al. reviews recent developments in numerical tools and high-performance computing for reactive transport, two-phase flow, and THM modelling in porous and fractured media. These advances enable more robust sensitivity analyses and uncertainty quantification, which are essential for reliable safety assessments.The SoTA reports in this collection demonstrate the scientific depth and collaborative effort underpinning the geological disposal of radioactive waste. By consolidating existing knowledge, identifying gaps, and providing direction for future research, they strengthen confidence in geological disposal as a safe and responsible long-term solution.Through these contributions, EURAD continues to build a shared, internationally recognised evidence base that will support informed decision-making on radioactive waste management for decades to come.

In this work, we propose an improved discretization, in terms of stability and accuracy, for the incompressible two-phase Darcy flows in a heterogeneous porous medium with discontinuous capillary forces. For this purpose, the total velocity formulation of the model is used. The coupled system is composed of a degenerate parabolic equation for the non-wetting phase and a pressure equation for the total velocity. We combine a positive Vertex Approximation Gradient (VAG) type scheme for the gradient fluxes with a hybrid upwinding of the mobilities. This approach entails a maximum principle on the saturations, which remain in their physical ranges. Energy estimates are obtained by selecting key approximations of the fluxes. These stability results allow to prove the existence of discrete solutions. Numerical experiments on complex test-cases show the robustness of the new approach in terms of the accuracy as well as the nonlinear convergence. Comparison to the usual phase potential upwinding approach and to a previous hybrid upwinding scheme are also provided.

ABSTRACT The mechanisms of CO2 Miscible Front Migration is a key factor for enhanced oil recovery in CO2 flooding, it is difficult to be described due to complex interactions between fluids, therefore, the CO2 Miscible Front Migration characteristics are unclear. In this paper, we established an injection-production pair model for coupled Gas-Oil Two-Phase Flow in Miscible Flooding based on the theory of Fluid Interactions in Miscible Flooding and gas-oil two-phase fluid flow in porous media, considering the heterogeneity, reservoir dip angle and miscibility degree. The results show that with the increase in gas viscosity, dip angle and decrease of intralayer heterogeneity, the gas breakthrough time is later and the distance of miscible front migration is longer; With the increase of interlayer heterogeneity and reservoir pressure, the gas breakthrough time is earlier and the distance of miscible front migration is shorter. At the same time, due to the negligence effect between CO2 and oil, and well pattern in the analytical model, an ideal model based on reservoir parameters from Block G66 × 1 fault-block reservoir in Jidong Oilfield is established. The model identified the CO2-crude oil miscible zone by using oil saturation and CO2 molar fraction in oil and characterized the migration patterns of the CO2 miscible front through bottom-hole pressure, gas-oil ratio, and dynamic front-position monitoring. Simulation results show when areal heterogeneity increases from 1 to 9, CO2 breakthrough time is earlier, the contribution of the pure oil flow zone to recovery factor decreased from 72.62% to 26.31%, the contribution of CO2 mass transfer zone to recovery factor increased from 14.06% to 47.11%. When the formation dip angle increased from 0° to 35°, CO2 breakthrough time is earlier, the contribution of the pure oil flow zone to recovery factor increased from 59.32% to 65.62%, the contribution of CO2 mass transfer zone to recovery factor decreased from 17.54% to 16.32%. With initial water saturation increasing from 50% to 90%, displacement efficiency continued decreasing, CO2 flooding breakthrough time is later. The contribution of the pure oil flow zone to recovery factor increased from 55.68% to 82.58%, and the contribution of CO2 mass transfer zone to recovery factor decreased from 27.42% to 10.20%. The results provide the basis for front prediction and control in CO2 miscible flooding development.

Abstract Carbon capture and storage (CCS) technology harnesses the unique properties of supercritical carbon dioxide (scCO 2 ) under reservoir conditions to achieve efficient geological sequestration and is widely recognized as a vital strategy for reducing CO 2 emissions and addressing climate change. However, the flow behavior of CO 2 within reservoirs is strongly influenced by the heterogeneity of the porous media in reservoir rocks, which serves as a critical factor in determining both the efficiency and safety of sequestration. In this study, permeability heterogeneity models were constructed using random functions, and the Volume of Fluid method was employed to simulate scCO 2 displacement of water and investigate the effects of porous media heterogeneity on the two-phase flow behavior. The results reveal that heterogeneity significantly impacts displacement patterns, dominant flow paths, and displacement efficiency during scCO 2 invasion. At low injection capillary number ( Ca ), capillary forces dominate, resulting in capillary fingering, while at high Ca , viscous forces prevail, leading to viscous fingering. In models with weaker heterogeneity, a transition zone from capillary fingering to viscous fingering is observed at intermediate Ca . Conversely, in models with stronger heterogeneity, no transition zone is detected due to the formation of fixed dominant flow paths. Additionally, the final displacement efficiency increases with Ca but decreases with increasing heterogeneity. Greater heterogeneity intensifies fingering effects, leading to higher residual water saturation and reduced invasion efficiency. These findings provide valuable theoretical insights for optimizing CO 2 geological storage operations, highlighting the importance of tailoring injection strategies to reservoir heterogeneity. Properly adjusting injection parameters can suppress fingering effects and improve sequestration efficiency, thereby enhancing the overall effectiveness of CO 2 storage.

This work develops and systematically evaluates a physics-informed neural network (PINN) solver for the fully coupled, time-dependent Muskat–Leverett system with capillarity modeled in the pressure equation. A single shallow–wide multilayer perceptron jointly predicts wetting pressure and water saturation; physical capillary pressure regularizes the saturation front, while a small numerical diffusion term in the saturation residual acts as a training stabilizer rather than a shock-capturing device. To guarantee admissible states in stiff regimes, we introduce a saturation soft-clamping head enforcing 0<Sw<1 and activate it selectively for stiff mobility ratios. Using IMPES solutions as reference, we perform a sensitivity study over network depth and width, interior collocation and boundary data density, mobility ratio, and injection pressure. Shallow-wide networks (10 layers × 50 neurons) consistently outperform deeper architectures, and increasing interior collocation points from 5000 to 50,000 reduces mean saturation error by about half, whereas additional boundary data have a much weaker effect. Accuracy is highest at an intermediate mobility ratio and improves monotonically with higher injection pressure, which sharpens yet better conditions the front. Across all regimes, pressure trains easily while saturation determines model selection, and the PINN serves as a physics-consistent surrogate for what-if studies in two-phase porous-media flow.

Immiscible two-phase flow in porous media occurs in many processes, such as enhanced oil recovery (EOR), as well as oil spill and soil remediation. These processes involve a fluid displacing another immiscible fluid within the confines of a heterogeneous porous structure. The invasion pattern generally remains the same under constant conditions but can also evolve over time in the presence of surfactants, which alter the interfacial tension (IFT) and surface wettability. The dynamics under such conditions extend beyond the usual way in which such immiscible displacement is modeled. Here, we develop a time-dependent pore network model (PNM) to simulate the effects of surfactant-induced IFT reduction on immiscible displacement driven by constant inlet pressure, with pressure drops across the network calculated using a random resistor network and mass conservation equations. Node-specific flux and velocity are derived using the Hagen-Poiseuille equation, and surfactant adsorption is modeled using the Langmuir isotherm, capturing its impact on fluid-fluid and solid-fluid interfaces within the invaded path. Since the evolution of the invasion pattern comprises the cooperative mechanisms of surfactant mass transfer to the interfaces and the resulting changes in capillary and Laplace pressures, we employ two strategies to quantify this complex feedback behavior: mass transfer based, introducing a mass transfer timescale, and Laplace pressure based, scaling with the inlet pressure. Results reveal that heavy-tailed pore throat distribution accelerates the onset of secondary invasions, which enhances the dominance of Laplace pressure. As the distribution becomes more symmetric or Gaussian, mass transfer becomes the dominant mechanism. This interplay highlights the intricate balance between mass transfer and capillary effects in governing the spatiotemporal evolution of immiscible fluid invasion.

Two-phase flow in porous media underpins a wide range of natural and industrial processes, but its transient dynamics remain challenging to capture at the spatiotemporal resolution required to resolve pore-scale phenomena. We present a method for rapid one-dimensional (1D) magnetic resonance imaging (MRI) profiling that simultaneously acquires spin-echo signal intensity and phase angle profiles with 98μm spatial and 20ms temporal resolution. The technique enables real-time observation of both fluid saturation and velocity fluctuations across a porous medium. We demonstrate its capabilities through three benchmark experiments: (1) controlled drainage and filling of a cylindrical tank, (2) buoyancy-driven rise of oil droplets in water, and (3) drainage and imbibition of a model porous medium. The results reveal dynamic interfacial behavior, velocity fluctuations linked to Haines jumps, and flow-dependent signal attenuation effects. We further analyze the relationship between flow velocity and signal attenuation in porous media using stop-motion dual-echo experiments. Our findings show that rapid magnetic resonance imaging provides a sensitive tool for probing two-phase flow dynamics, with implications for understanding complex fluid behavior in porous materials.

The effective permeability is a core macroscopic parameter describing two-phase flow in porous media, and it plays a crucial role in fields such as oil and gas exploitation and CO2 geological sequestration. However, the mechanism by which it is affected by the capillary number (Ca) remains unclear. In this study, with the pore-scale two-phase displacement pattern as a bridge and combined with the numerical model constructed based on real pore structures, immiscible two-phase flow simulations were conducted over a wide Ca range of 10−6 to 10−3. The results demonstrate that significant variations in pore-scale interfacial dynamics under different Ca conditions give rise to a series of distinct displacement patterns, which are the key factors responsible for the differences in effective permeability. Specifically, the invading phase fluid consistently flows within its “unique” spatial domain at different displacement stages, and the effective permeability is largely governed by the spatial distribution characteristics of this phase. Further investigations reveal that both inactive zones and active zones coexist during the seepage process in porous media. The evolutionary characteristics of effective permeability strongly depend on the saturation properties of the active zones; in other words, the effective permeability under different displacement patterns corresponds to the intrinsic permeability associated with the active flow paths of the invading phase under that specific pattern. Finally, the aforementioned conclusions were verified by employing an effective permeability calculation method based on fluid distribution images.

Gas–water two-phase flow in porous media is vital in groundwater management and hydrocarbon development, yet most experiments use small cores (5–10 cm) or etched micro-models. These studies often overlook the quantitative characterization of residual gas, long-distance gas–water flow behavior, and effects of gas–water flow on pore structure. This study presents a series of 3 m-long artificial unconsolidated sandstone models with permeabilities of 5, 10, 30, 50, and 100 mD, fabricated via rock–electric testing techniques to simulate edge-water invasion in gas reservoirs. The results indicate that (1) by adjusting clay content, cementing agents, grain size, and sand mix, artificial cores achieve permeability, porosity, cementation strength, sensitivity, and pore structure similar to natural cores; this approach addresses the sampling challenge from unconsolidated sandstone. (2) During long-distance gas–water flow, pressure drops rapidly in the gas–water transition zone. As permeability increases, the zone shifts downstream and becomes narrower. (3) The flow of gas–water causes a large number of particles to gather near the gas–water interface and block the throat, and effective stress on unconsolidated sandstone intensifies this blockage effect. (4) Residual gas exists in the forms of dead-end trapped gas, bypass trapped gas, and snap-off trapped gas. The residual gas volume is mainly controlled by gas saturation and pressure, but the largest amount of residual gas accumulates near the gas–water interface. This study addresses the research gap in understanding long-distance gas–water flow and presents a novel experimental method for unconsolidated porous media.

High-order bound-preserving finite difference methods for incompressible two-phase flow in porous media

This article presents ArcNum, a framework built on the Arcane platform, designed to easily and efficiently develop and maintain the numerical core in finite-volume and finite-element applications. This framework first enables the automatic generation of code needed to instantiate and handle complex physical models from a textual description, through its component called GUMP. ArcNum also offers software components to create, register and evaluate a set of physical laws, requiring only the specification of their inputs, outputs, and corresponding mathematical formulations. Finally, the framework includes a component named Contribution, which combines law evaluation with automatic differentiation to assemble linear systems efficiently. The framework ArcNum has been used to develop an open source Arcane-based porous media flow simulation proxy-app, named ShArc. Single and two-phase porous media flow simulations performed with ShArc are presented to complete the framework description. In order to illustrate the ability to use ArcNum for High Performance Computing, massively parallel simulations conducted with ShArc are finally presented.

We have developed a new theory relating partial molar volumes of binary mixtures to the intrinsic (Voronoi) volumes. The theory gives detailed insights into the physical meaning of partial molar volumes in terms of the actual volumes occupied by the molecules. The partial molar quantities are defined through the use of the Euler theorem for homogeneous functions. These properties have been in use for a long time, despite the fact that they do not give an intuitive picture of the properties they are to represent. For instance, the partial molar volume of a given component in a mixture is the change in the total volume with a change in composition; hence, it represents the derivative of a volume. The molar volume is a measurable property in the laboratory, and as such, a body of thermodynamics but the derived partial molar volume is not a direct measure of the physical volume occupied by the added component. On the other hand, the physical volume can be computed using, e.g., molecular dynamics simulations by Voronoi tessellation. To bridge the partial molar volume and the intrinsic volume so determined, we define a single new thermodynamic variable-the co-molar volume-thus bringing the latter into thermodynamics. We demonstrate this bridge through molecular dynamics simulations. The co-molar volume is closely related to the co-moving velocity defined in immiscible two-phase flow in porous media.

Coupled pressure and saturation prediction for two-phase flow in porous media using physics-informed neural networks (PINNs)

This study investigates hydrodynamic parameters governing pollutant degradation in a low-temperature plasma (LTP) reactor utilizing porous alumina ceramic media. A validated 3D Volume of Fluid (VOF) model simulated air–water two-phase flow to resolve film thickness, wettability, wetting area, residence time, and liquid hold-up across varying flow rates. Experimental measurements confirmed the CFD predictions and showed that increasing the flow rate led to a sharp decline in degradation efficiency due to reduced residence time and increased film thickness. Notably, maximum degradation (~33.4 mg/L) occurred at intermediate flow conditions (Q ≈ 7.0 L/h, RT ≈ 6.1 s), whereas degradation stagnated at higher flow rates due to shortened treatment time and possible side reactions indicated by conductivity shifts. Kinetic analysis of the experimental data confirmed a zero-order degradation mechanism, with a strong linear correlation between residence time and indigo carmine removal (R² = 0.997). Regression and desirability-based optimisation identified 10.97 L/h as the ideal flowrate, balancing surface wetting and residence time for effective degradation (~31.2 mg/L). Sensitivity analysis confirmed that film thickness and residence time were the most influential factors. The study offers a quantitative framework for optimising LTP reactors by integrating CFD, experiments, and multi-criteria optimisation. • CFD–VOF model predicts flow and saturation in porous ceramics with high accuracy • Film thickness and residence time control plasma-assisted degradation efficiency • Maximum degradation of 33.4 mg/L observed at ~7 L/h and RT ≈ 6.1 s • Degradation follows zero-order kinetics with R² = 0.997 across flow conditions • Multi-criteria optimisation identifies 10.97 L/h as the optimal flowrate for LTP

Impact of wetting films on the stability of two-phase flow in porous media: A pore-doublet perspective

Existence of weak solutions for nonisothermal immiscible compressible two-phase flow in porous media

Effects of discontinuous fluid interfaces on pressure drops and relative permeabilities for gas-liquid two-phase flow in porous media

Gaining pore-scale insights into relative permeability curves of two-phase flows in porous media: a modified LBM study