A comprehensive study of two-phase flow in deformable porous media

We propose a formulation for non-isothermal two-component two-phase flow through deformable porous media. The approach covers phase transitions among both phases, i.e. liquid phase components evaporate into the gas phase while gas phase components dissolve or condensate into the liquid phase. These phase transitions always take place in thermodynamic equilibrium. The set of model equations is thereby largely independent of the specific constitutive relations. Starting from general equilibrium equations, we show the evolution of the system of weak formulations of all governing equations, which are then discretised with Taylor-Hood elements in a standard finite element approach. The model equations and the construction of the constitutive equilibria are implemented in the open-source simulator OpenGeoSys, which can be freely used and modified. To verify the implementation, we have selected a number of complementary test cases covering a wide range of process couplings. The numerical model is compared with analytical and semi-analytical solutions of these problems as well as with experimental results. It is shown in the paper that by including thermodynamic effects, solid mechanics, and phase transition processes, the proposed numerical model covers many characteristic features of unsaturated geomaterials and can be employed for the description of a broad range of problems encountered in geotechnical engineering.Article highlightsAn open-source FEM tool for non-isothermal two-phase flow in deformable porous or fractured media is presented in detail.The model features phase transitions across both fluid phases based on simple equilibrium conditions.A variety of benchmark tests is presented and compared to other software results and to exact solutions.

Fully mass-conservative IMPES schemes for incompressible two-phase flow in porous media

Two-phase incompressible flow in porous media plays an important role in various fields including subsurface flow and oil reservoir engineering. Due to the interaction between two phases flowing through the pores, the fluid–fluid friction force may have a significant effect on each phase velocity. In this paper, we propose an energy stable (thermodynamically consistent) Maxwell–Stefan–Darcy model for two-phase flow in porous media, which accounts for the fluid–fluid friction. Different from the classical models of two-phase flow in porous media, the proposed model uses the free energy to characterize the capillarity effect. This allows us to employ the Maxwell–Stefan model to describe the relationships between the driving forces and the friction forces. The driving forces include the pressure gradient and chemical potential gradients, while both fluid–solid and fluid–fluid friction forces are taken into consideration. Thermodynamical consistency is the other interesting merit of the proposed model; that is, it satisfies an energy dissipation law and also obeys the famous Onsager's reciprocal principle. A linear semi-implicit numerical method is also developed to simulate the model. Numerical simulation results are provided to show that the fluid–fluid friction force can improve the oil recovery substantially during the oil displacement process.

A fully coupled dynamic model for two-phase fluid flow in deformable porous media

Homogenization of two-phase fluid flow in porous media via volume averaging

Purpose – The purpose of this paper is to present a novel sequential implicit discontinuous Galerkin (DG) method for two-phase incompressible flow in porous media. It is based on the wetting phase pressure-saturation formulation with Robin boundary condition (Klieber and Riviere, 2006) using H(div) velocity projection. Design/methodology/approach – The local mass conservation and continuity of normal component of velocity across elements interfaces are enforced by a simple H(div) velocity projection in lowest order Raviart-Thomas (RT0) space. As further improvements, the authors use the weighted averages and the scaled penalties in spatial DG discretization. Moreover, the Chavent-Jaffre slope limiter, as a consistent non-oscillatory limiter, is used for saturation values to avoid the spurious oscillations. Findings – The proposed model is verified by a pseudo 1D Buckley-Leverett problem in homogeneous media. Two homogeneous and heterogeneous quarter five-spot benchmark problems and a random permeable medium are used to show the accuracy of the method at capturing the sharp front and illustrate the impact of proposed improvements. Research limitations/implications – The work illustrates incompressible two-phase flow behavior and the capillary pressure heterogeneity between different geological layers is assumed to be negligible. Practical implications – The proposed model can efficiently be used for modeling of two-phase flow in secondary recovery of petroleum reservoirs and tracing the immiscible contamination in porous media. Originality/value – The authors present an efficient sequential DG method for immiscible incompressible two-phase flow in porous media with improved performance for detection of sharp frontal interfaces and discontinuities.

Coupled multiphase flow and poromechanics play a fundamental role in various Earth science applications, from subsurface energy extraction to induced seismicity. However, the inherent complexity of subsurface environments—characterized by fluid compressibility, capillary effects, and heterogeneous permeability—poses significant computational challenges, particularly in high-resolution three-dimensional simulations.To overcome these challenges, we develop a high-performance computational framework optimized for Graphics Processing Units (GPUs) to simulate two-phase flow in deformable porous media. Our approach introduces a novel formulation of the poro-visco-elasto-plastic equations, explicitly designed for GPU architectures. This framework accounts for compressible fluids with capillary pressure effects and employs a customized iterative solver that enhances computational efficiency. By leveraging modern GPU hardware, we enable large-scale simulations with unprecedented spatial resolution, facilitating faster computations and significantly larger grid sizes than previously achievable.Our results reveal that within shear bands, pressure drops occur similarly to single-phase fluid environments. However, in our two-phase flow model, pressure evolves differently due to the influence of strain localization on capillary pressure. This interaction between multiphase flow and mechanical deformation introduces new physical insights, suggesting that strain localization may play a critical role in modifying fluid distributions and capillary effects. These findings offer a deeper understanding of two-phase flow behavior in deforming porous media, with implications for geomechanics, fault stability, and fluid-driven deformation processes.

The global Riemann problem for a nonstrictly hyperbolic system of conservation laws modeling polymer flooding is solved. In particular, the system contains a term that models adsorption effects.

The focus of the underlying research work is on the macroscopic modeling of unstable multiphase fluid flow in deformable porous media, where a lower‐viscous fluid is displaced by a more viscous fluid. This process leads to the formation of channel‐like networks, called viscous fingering. The instability effect is involved in a wide range of different fields in engineering. Some of the most common applications include carbon sequestration to store carbon dioxide (CO2) in underground reservoirs, contaminant transport in geostructures, in industrial processes, such as filtration, catalytic reactions, and in the operation of polymer electrolyte membrane fuel cells with multiphase flow in the gas diffusion layers. In this work, ideal miscible water–glycerin fluids are considered. It is assumed that the interacting fluids in the deformable porous media are incompressible. In addition, a dispersion–diffusion law is applied to capture the fluid–fluid interactions. The presence of a deformable porous material adds additional effects to the problem of the multiphase flow in porous media. The stresses in the porous solid matrix lead to changes in the porosity and influence the flow velocity of the fluids. To couple the deformable porous media with the multiphase flow, a macroscopic approach is used that relies on the theory of porous media. For the porous solid matrix, a linear elastic material model within small strains assumption is applied. The influence of the deformation‐dependent porosity on the instability is studied for 2D simulations, such as a multilayered geometry with different elastic parameters. The presented coupled nonlinear system of differential equations is simulated with the finite element method. Furthermore, a stabilization technique based on the quasi‐compressibility method is used.

Numerical simulation of two-phase flow in deformable porous media: Application to carbon dioxide storage in the subsurface

We study the formation of viscous fingering and fracturing patterns that occur when air at constant overpressure invades a circular Hele-Shaw cell containing a liquid-saturated deformable porous medium -- i.e. during the flow of two non-miscible fluids in a confined granular medium at high enough rate to deform it. The resulting patterns are characterized in terms of growth rate, average finger thickness as function of radius and time, and fractal properties. Based on experiments with various injection pressures, we identify and compare typical pattern characteristics when there is no deformation, compaction, and/or decompaction of the porous medium. This is achieved by preparing monolayers of glass beads in cells with various boundary conditions, ranging from a rigid disordered porous medium to a deformable granular medium with either a semi-permeable or a free outer boundary. We show that the patterns formed have characteristic features depending on the boundary conditions. For example, the average finger thickness is found to be constant with radius in the non-deformable system, while in the deformable ones there is a larger initial thickness decreasing to the non-deformable value. Then, depending on whether the outer boundary is semi-permeable or free there is a further decrease or increase in the average finger thickness. When estimated from the flow patterns, the box-counting fractal dimensions are not found to change significantly with boundary conditions, but by using a method to locally estimate fractal dimensions, we see a transition in behavior with radius for patterns in deformable systems; In the deformable system with a free boundary, it seems to be a transition in universality class as the local fractal dimensions decrease towards the outer rim, where fingers are opening up like fractures in a paste.

This paper presents a numerical method to model the coupled thermo-hydro-mechanical (THM) processes in porous media saturated with two immiscible fluids. The basic equations of the system have been derived based on the averaging theory, considering skeleton deformation, two-phase fluid flow, and heat transport. As applying the standard Galerkin finite element method (GFEM) to solve this system of partial differential equations may lead to oscillatory results for saturation and temperature profiles, a hybrid numerical solution is proposed. In this frame, the GFEM is combined with a control volume based finite element (CVFE) approach, and a streamline upwind control volume finite element (SUCVFE) scheme, respectively for the mechanical, hydraulic and thermal part of the system. The CVFEM has been adopted to provide a smooth saturation profile by ensuring local mass conservation, while the streamline upwind scheme has been applied to remove the spurious temperature oscillation by adding stabilizing terms to the thermal part of the system. The CVFE and SUCVFE formulations have been derived using a similar approach as the standard FE practice in the context of weighted residual technique, but using different weighting functions. This will significantly facilitate the implementation of the proposed model in existing FE codes. Accuracy and efficiency of the proposed method have been justified using several numerical examples and comparing the results with available analytical or numerical solutions.

A sequential discontinuous Galerkin method for two-phase flow in deformable porous media

Relative permeability of two-phase flow in three-dimensional porous media using the lattice Boltzmann method

When neglecting capillarity, two-phase incompressible flow in porous media is modelled as a scalar nonlinear hyperbolic conservation law. A change in the rock type results in a change of the flux function. Discretizing in one-dimensional with a finite volume method, we investigate two numerical fluxes, an extension of the Godunov flux and the upstream mobility flux, the latter being widely used in hydrogeology and petroleum engineering. Then, in the case of a changing rock type, one can give examples when the upstream mobility flux does not give the right answer.