A fully coupled dynamic model for two-phase fluid 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.

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.

A model of solid-water-air coupling in triphasic mixtures is compared with solid-water coupling in biphasic mixtures with an application to partially saturated porous media. Based on thermodynamics, the mathematical framework governing the behavior of a partially saturated soil is derived using balance equations, and the numerical implementation and drainage tests of a soil column are carried out to validate the obtained formulations. The role of the air phase in the hydro-mechanical behavior of triphasic mixtures can be analyzed from the interactions among multiple phases for the constitutive behavior of a solid skeleton, and the triphasic mixture model can be applied in geotechnical engineering problems, such asCO2sequestration and air storage in aquifers.

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.

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.

This is the third and conclusive part of a three‐paper series and describes the application of a numerical model for saturated‐unsaturated flow in deformable porous media. In all, 10 illustrative examples are presented not only to demonstrate the validity of the method but also to highlight the fundamental unity that exists in the sic principles of the fields of hydrogeology, soil mechanics, and soil physics. The chosen examples involve such diverse phenomena as soil consolidation, infiltration, and drainage and generation of fluid pressures due to cyclic loading such as earthquakes.

An integrated finite difference algorithm is presented for numerically solving the governing equation of saturated‐unsaturated flow in deformable porous media. In recognition that stability of the explicit equation is a local phenomenon a mixed explicit‐implicit procedure is used for marching in the time domain. In this scheme the explicit changes in potential are first computed for all elements in the system, after which implicit corrections are made only for those elements for which the stable time step is less than the time step being used. Time step sizes are automatically controlled in order to optimize the number of iterations, to control maximum change in potential during a time step, and to obtain desired outputs. Time derivatives, estimated on the basis of system behavior during two previous time steps, are used to start the iteration process and to evaluate nonlinear coefficients. Boundary conditions and sources can vary with time or with the dependent variable. Input data are organized into convenient blocks. Accuracy of solutions can be affected by modeling errors, different types of truncation errors, and convergence errors. The algorithm constitutes an efficient tool for analyzing linear and nonlinear fluid flow problems in multidimensional heterogeneous porous media with complex geometry. An important limitation is that the model cannot conveniently handle arbitrary anisotropy and other general tensorial quantities.

Effect of the inertia term in one-dimensional fluid flow in deformable porous media

11R59. Computational Methods in Environmental Fluid Mechanics. - O Kolditz (Center for Appl Geosci, Univ of Tubingen, Sigwartstr 10, Tubingen, D-72076, Germany). Springer-Verlag, Berlin. 2002. 378 pp. ISBN 3-540-42895-X. $54.95.Reviewed by LA Glenn (Computational Phys Group, Geophys Div, MS L-200, LLNL, 7000 East Ave, Livermore CA 94550-9900).This is intended to be a graduate-level textbook for students in civil and environmental engineering. It is organized into four parts: Continuum Mechanics, Numerical Methods, Software Engineering, and Selected Topics. The first part considers the general balance equations of mass, momentum, and energy; averaging concepts for turbulence; a discussion of porous media; and the mathematical and physical classification of the partial differential equations (PDEs) governing fluid flow and related transport processes. The second part deals with basic concepts for solving PDEs; concepts of approximation theory; and a description of finite difference, finite element, and finite volume methods, with application to diffusion, advection, and transport processes. This material spans roughly half of the book and, while reasonably well organized, really covers no new ground that is not readily available in numerous other standard texts on fluid mechanics. In fact, the roughly 100 pages focusing on numerical methods affords only quite skimpy treatment of many important topics that would be required before a student could reasonably be expected to apply these methods to actual problems. In Parts 3 and 4, by contrast, this book breaks new ground. The author believes that object-oriented programing methods are important tools for modeling complex systems, and Part 3 is an introduction to these methods and their application to coupled processes in subsurface systems (geomechanics, single and multiphase flows, heat and mass transport, and chemical and biological processes). Unfortunately, Part 3 covers only 40 pages so that, here again, one gets only the most meager treatment of the subject. This reviewer found himself wishing that Parts 1 and 2 had been dispensed with and this part extended accordingly. The last part of the book is divided into four chapters, each of which is a fairly self-contained segment dealing with problems of particular interest to the author: nonlinear flow in fractured media, heat transport in fractured porous media, density dependent flow in porous media, and multiphase flow in deformable porous media. On each topic, there is a nice introduction to its relevance in environmental fluid mechanics, a description of the governing equations and approximations, an outline of the numerical scheme employed for solution, comparison of solution results with experimental data, and a bibliography giving relevant papers and background material. Computational Methods in Environmental Fluid Mechanics is well illustrated throughout, and the text and mathematical derivations are clear and relatively easy to follow. One complaint, a minor one to be sure, is that the index is arranged rather poorly so that some topics are hard to locate. Although it may serve as a useful reference, the use of this book as a graduate text in computational fluid mechanics is problematical since many important practical issues arising in the application of the numerical methods are either ignored or given only very skimpy treatment.

Flow in deformable porous media: Modelling and simulations of compression moulding processes

Numerical modeling of two-phase flow in deformable porous media: application to CO$$_2$$ injection analysis in the Otway Basin, Australia

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

In this research study, we conducted a coupled thermal-stress-fluid flow numerical model on the UTF Phase A SAGD project in order to investigate the fundamental geomechanical behaviour involved in the SAGD process, and to gain insight into the reservoir response to temperature and pore pressure changes. The numerical simulation is carried out by using a self-developed coupled finite element model which incorporates our proposed strain-induced permeability model. The obtained simulation results were compared with the measured data. Introduction Thermal recovery processes involve coupling between heat transfer, multiphase flow and stress/deformation, which has become an increasingly important subject in the petroleum field(1). Particularly, the coupling is crucial in problems such as borehole stability, hydraulic fracturing and injection/production induced deformation of the ground surface during the thermal recovery process in heavy oil or oil sand reservoirs. Numerical modelling of the coupled processes is historically carried out in the areas of geomechanics modelling and reservoir simulation. The former is to compute the stress-strain behaviour; therefore, the deformation. The latter is to essentially model the multiphase flow and heat transfer in porous media. Each of these disciplines simplifies the part of the problem that is not of primary interest. These approaches are unacceptable in situations where the coupling is strong and the changes of porosity and permeability cannot be accounted for by rock compressibility alone. Gutierrez and Lewis(2) extend Biot's theory to multiphase fluid flow in deformable porous media. Based on their formulation, they conclude that the coupling between the geomechanics and the multiphase flow occurs simultaneously. Thus, fully coupled system equations of deformations, multiphase flow and heat transfer should be solved simultaneously. Development of such kinds of fully coupled geomechanics-multiphase flow-heat transfer simulators needs tremendous effort, since the existing FEM geomechanics codes and FDM reservoir simulators cannot be used. Settari and Mourits(3) present an approach to couple the stress-strain behaviour to multiphase flow-heat transfer using porosity as a coupling parameter. The geomechanics module and the thermal reservoir simulator are used in a staggered manner. Pore pressure and temperature changes are calculated from the thermal reservoir simulator and transferred to the geomechanics module. The stress and the displacement changes are then calculated in the geomechanics simulation. An iterative algorithm is used to ensure that the porosity calculated from the geomechanics module is the same as that from the thermal reservoir simulator. The staggered technique employed to solve the coupled system equations allows for the use of the existing geomechanics codes in conjunction with a standard reservoir simulator. Currently, most of the commercial coupled geomechanics-multiphase flow-heat transfer simulators are developed in this way. The disadvantage of these kinds of coupled simulators is that the thermal reservoir module, usually developed using the finite difference method (FDM), cannot accommodate the full permeability tensor, since they adopt the standard discretization scheme, such as 5-spot for 2D problems and 7-spot for 3D problems. In this paper, the development of a coupled geomechanics-multiphase flow-heat transfer simulator using the finite element method (FEM) is described with the use of Galerkin's least squares (GLS) technique(4) to stabilize the saturation equation.

In this chapter we review the basic governing equations for fluid flow in permeable media. Our treatment is general and includes coupled deformation and fluid flow. Starting with the basic conservation laws we derive the pressure and transport equations for single phase and multiphase flow in porous media. We cover physical processes involving both miscible and immiscible displacements as well as mass transfer from compositional effects and present the equations describing these processes. We have included a fairly extensive discussion on the underlying principles behind elastic deformation and how the concepts carry over for modeling fluid flow in deformable porous media. We conclude the chapter with a discussion on poroelasticity and the associated equations for fluid flow in deformable porous matrix in the presence of one or more fluid phases. In summary, this chapter provides the foundation and notation for subsequent developments. However, in each of the subsequent chapters we will start with the relevant governing equations so the reader may comfortably skip this chapter and refer back to it as necessary.

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.