Deformable Porous Media Research Articles

Hydro-mechanical simulation of the saturated and semi-saturated porous soil–rock mixtures using the numerical manifold method

Dynamics of the widespread saturated and semi-saturated soil–rock mixtures (SRMs) are of importance in practical engineering. In this paper, a numerical manifold model is presented for hydro-mechanical simulation of the saturated and semi-saturated SRMs. In the case of the semi-saturated SRMs, the model is based on the generalized Biot theory involving immiscible two-phase wetting and non-wetting fluids in deformable porous media. The wetting, non-wetting fluid pressure and skeleton displacement are chosen to be primary variables which are related to the wetting and non-wetting saturation and permeability by experimental relationships. For the mechanical problem, the material interfaces between soil and rock are deemed discontinuous by imposing a stick–slip contact constraint using an augmented Lagrange multiplier approach. For the hydraulic problem, the material interfaces are continuous for the fluid pressure and flux fields. Within the framework of the numerical manifold method (NMM), the discretized model including the interfacial discontinuities always can be established using the triangular mesh and the lumped mass representation is always available to increase computational efficiency. Besides the benchmark problems of saturated and semi-saturated porous media, two examples of soil–rock foundation and slope are performed to demonstrate the versatility and robustness of the model and to investigate the hydro-mechanical responses of the porous SRMs.

Wettability control on deformation: Coupled multiphase fluid and granular systems.

Understanding of multiphase flow in porous media is important for a wide range of applications such as soil science, environmental remediation, energy resources, and CO_{2} sequestration. This phenomenon depends on the complex interplay between the fluid and solid forces such as gravitational, capillary, and viscous forces, as well as wettability of the solid phase. Such interactions along with the geometry of the medium give rise to a variety of complex flow regimes. Although much research has been done in the area of wettability, its mechanical effect is not well understood, and it continues to challenge our understanding of the phenomena on macroscopic and microscopic scales. In this paper, therefore, the effect of wettability on the deformation of porous media and fluid-fluid patterns is studied through a series of three-dimensional (3D) simulations. To this end, the discrete element method (DEM) and volume of fluid (VOF) are coupled to accurately model free-surface flow interaction in a cylindrical pack of spheres. The fluid-particle interactions are modeled by exchanging information between DEM and VOF, while the effect of wettability is considered to study how it controls fluid displacement. The results indicate that the drag force and deformation in the pack vary with the change in wettability and capillary number. To demonstrate the effect of both wettability and capillary number, a series of numerical experiments were conducted with two capillary numbers and three wettability conditions. Our results show that the drag force was greatest for near extreme wettability conditions, which resulted in a larger deformation.

Pore-scale simulation of heat and mass transfer in deformable porous media

Most porous media are susceptible to deformation during the heat and mass transfer process. In this paper, the bi-dispersed porous media model, which was generated through digitized image reconstruction, was used to effectively and conveniently simulate the real porous media. The thermal-hydro-mechanics model, which was established based on the Navier-Stokes equations and stress differential equations, was solved according to the finite element method. Meanwhile, the arbitrary Lagrange-Euler theory was applied to deal with the fluid-solid boundary. The results suggested favorable agreement between the numerical results and the experimental data. Moreover, the detailed distributions of velocity, temperature and water content in the porous media were obtained, which varied depending on the structural properties of porous media. Further, the stress and deformation were comparatively analyzed in the open and closed pore regions, and it was found that the wall of closed pore was prone to stress concentration. The deformation of the bi-dispersed porous sample resulted in an increase in pore area fractal dimension and a decrease in tortuosity fractal dimension. In addition, the effective transport properties and volume shrinkage of the deformable porous media at the representative elementary volume (REV) scale were also computed based on the obtained pore scale results, which confirmed that it was necessary to consider the effects of deformation on the real flexible porous media.

Experimental investigation on foam formation through deformable porous media

AbstractFoam formation in porous media is a topic of growing scientific and industrial interest due to its range of applications, from daily life consumer products to oil recovery. Despite the work done so far on foams flowing through complex structures, such as rigid porous media, this subject still needs to be fully elucidated. An additional complexity to the problem arises when the porous medium is deformable, a situation which has only been faced, to our knowledge, from a modelling point of view. In this work, the investigation of foam formation in deformable porous media is carried out by using commercial sponges as a deformable porous media system, with special emphasis on the effect of confinement on foam bubble size distribution. Foam is formed by wetting the sponge with an aqueous surfactant solution and then squeezing the sponge either between two glass cover slides or between a plastic net and a cover slide. Our experimental data reveal that the latter system allows the formation of drier foams (ie, with lower liquid fraction, fL < 0.3), more similar to the ones obtained in dish‐washing applications. Moreover, the effect of sponge type, in terms of material and microstructure, on final foam is presented. Our results are of potential interest for the optimization of foams in complex structures, such as in deformable porous media.

Elastohydrodynamics of a deformable porous packing in a channel competing under shear and pressure gradient

Elastohydrodynamics of a deformable porous medium sandwiched between two parallel plates is investigated under the influence of an externally applied pressure gradient as well as an induced shear due to the movement of the upper plate. Biphasic mixture theory is used to describe the macroscopic governing equations for the fluid velocity and the solid displacement, assuming the deformable porous medium as a continuum space. The corresponding reduced mathematical model is a coupled system of elliptic partial differential equations. It is assumed that the fluid at the lower plate experiences slip due to the surface roughness of the plate. The exact solution for unidirectional fluid velocity and solid deformation resembling plain Poiseuille–Couette flow are presented for steady and unsteady states. Asymptotic analysis of the biphasic mixture in the case of low and high Darcy numbers is performed to validate the obtained solution using Prandtl’s matching technique. It is observed that the Womersley number dictates whether the fluid is trapped inside the channel or escapes the channel. The competition between the shear and the pressure gradient is analyzed, and a critical criterion is established that dictates the dominant factor. A mathematical analysis of the current problem is invaluable in understanding the mechanical behavior of biomass under pressure-driven flow in applications such as tissue engineering or shear driven flow inside endothelial glycocalyx layers, which are discussed in brief. In this context, our analysis on the extent of tissue deformation in response to frequency variations is expected to give useful insights to identify the right diagnosis.

A staggered finite element procedure for the coupled Stokes-Biot system with fluid entry resistance

We develop a staggered finite element procedure for the coupling of a free viscous flow with a deformable porous medium, in which interface phenomena related to the skin effect can be incorporated. The basis of the developed simulation procedure is the coupled Stokes-Biot model, which is supplemented with interface conditions to mimic interface-related phenomena. Specifically, the fluid entry resistance parameter is used to relate the fluid flux through the interface to the pressure jump across the interface. The attainable jump in pressure over the interface provides an effective way of modeling sharp pressure gradients associated with the possibly reduced permeability of the interface on account of pore clogging. In addition to the fluid entry resistance parameter, the developed simulation strategy also includes the possibility of modeling fluid slip over the porous medium. Sensitivity studies are presented for both the fluid entry resistance parameter and the slip coefficient, and representative two- and three-dimensional test cases are presented to demonstrate the applicability of the developed simulation technique.

Modeling wavefields in saturated elastic porous media based on thermodynamically compatible system theory for two-phase solid-fluid mixtures

A two-phase model and its application to wavefields numerical simulation are discussed in the context of modeling of compressible fluid flows in elastic porous media. The derivation of the model is based on a theory of thermodynamically compatible systems and on a model of nonlinear elastoplasticity combined with a two-phase compressible fluid flow model. The governing equations of the model include phase mass conservation laws, a total momentum conservation law, an equation for the relative velocities of the phases, an equation for mixture distortion, and a balance equation for porosity. They form a hyperbolic system of conservation equations that satisfy the fundamental laws of thermodynamics. Two types of phase interaction are introduced in the model: phase pressure relaxation to a common value and interfacial friction. Inelastic deformations also can be accounted for by source terms in the equation for distortion. The thus formulated model can be used for studying general compressible fluid flows in a deformable elastoplastic porous medium, and for modeling wave propagation in a saturated porous medium. Governing equations for small-amplitude wave propagation in a uniform porous medium saturated with a single fluid are derived. They form a first-order hyperbolic PDE system written in terms of stress and velocities and, like in Biot’s model, predict three type of waves existing in real fluid-saturated porous media: fast and slow longitudinal waves and shear waves. For the numerical solution of these equations, an efficient numerical method based on a staggered-grid finite difference scheme is used. The results of solving some numerical test problems are presented and discussed.

Study on Viscous Fluid Flow in Disordered-Deformable Porous Media Using Hydro-mechanically Coupled Pore-Network Modeling

We investigate viscous fluid flows and concurrent fluid-driven deformations in porous media. The hydro-mechanically (H-M) coupled pore-network model (PNM) is developed, which combines the two-dimensional square-lattice PNM and block-spring model. The single-/two-phase flows into saturated deformable porous media are simulated through iterative two-way coupling method in H-M coupled PNM. A comparison between simulations and laboratory observations on flow patterns, solid deformation behaviors, and pressure responses ensures the validity of our H-M coupled PNM in both single-/two-phase flows. Parametric studies using the validated model examine the effects of mechanical coupling, stiffness of solid particles, the viscosity of invading fluids, injection flow rate, and degree of disorder during immiscible viscous fluid injection. The viscous fluid-driven deformation increases the pore throat size and hence reduces the injection pressure. In particular, the viscosity of invading fluid significantly alters the patterns of fluid propagation and solid deformation, along with a transition from the viscous fingering to the stable displacement with increasing viscosity. Moreover, the structural disorder in porous networks magnifies the irregular flow pattern, the pressure fluctuation associated with Haines jumps, and the poromechanical deformation. The particle-level force analysis delineates two distinct regimes: fluid invasion with no deformation and drag-driven deformation, which depends on the balance between the seepage drag force and the skeletal force. The presented results contribute to a better understanding of the H-M coupled fluid flows during the injection of viscous fluids into disordered-deformable porous media.

Impact of matrix deformations on drying of granular materials

Drying of deformable porous media is encountered in soil, paints, coatings, food, and building materials. During drying, capillary forces can displace solid particles and modify invasion thresholds, opening preferential paths for air to invade. Since drying rates are crucially dependent on liquid connectivity through the porous medium, they are affected by the drying patterns. We study the interplay between solid matrix deformation and drying patterns and rates, and the impact of pore-size heterogeneity and the confining stress in a granular material. We couple a pore-scale model of drying with a particle-scale mechanical model, to capture the two-way coupling between the evolving drying pattern and solid particle displacements. Our simulations show that for a low pore-size disorder, and low confining stress, matrix deformations are favorable and lead to preferential drying patterns which maintain high drying rates. This effect is observed throughout the drying process when the disorder is low, but disappears after breakthrough for low confining stress, indicating a different mechanism by which deformation can impact drying. These results could be significant for various industrial applications, including fabrication of advanced nano-materials, where patterned drying can be used to obtain the desired structure.

Numerical Solotion of a Problem of Fluid Filtration in a Viscoelastic Porous Medium

The paper considers a model for filtering a viscous incompressible fluid in a deformable porous medium. The filtration process can be described by a system consisting of mass conservation equations for liquid and solid phases, Darcy's law, rheological relation for a porous medium, and the law of conservation of balance of forces. This paper assumes that the poroelastic medium has both viscous and elastic properties. In the one-dimensional case, the transition to Lagrange variables allows us to reduce the initial system of governing equations to a system of two equations for effective pressure and porosity, respectively. The aim of the work is a numerical study of the emerging initial-boundary value problem. Paragraph 1 gives the statement of the problem and a brief review of the literature on works close to this topic. In paragraph 2, the initial system of equations is transformed, as a result of which a second-order equation for effective pressure and the first-order equation for porosity arise. Paragraph 3 proposes an algorithm to solve the initial-boundary value problem numerically. A difference scheme for the heat equation with the righthand side and a Runge–Kutta second-order approximation scheme are used for numerical implementation.

On Global Solvability of a Problem of a Viscous Liquid Motion in a Deformable Viscous Porous Medium

The initial-boundary value problem for the system of one-dimensional motion of viscous liquid in a deformable viscous porous medium is considered. The introduction presents the relevance of a theoretical study of this problem, scientific novelty, theoretical and practical significance, methodology and research methods, a review of publications on this topic. The first paragraph shows the conclusion of the model and the statement of the problem. In paragraph 2, we consider the case of motion of a viscous compressible fluid in a poroelastic medium and prove the local theorem on the existence and uniqueness of the problem. In the case of an incompressible fluid, the global solvability theorem is proved in the Holder classes in paragraph 3. In paragraph 4, an algorithm for the numerical solution of the problem is given. Mathematical models of fluid filtration in a porous medium apply to a broad range of practical problems. The examples include but are not limited to filtration near river dams, irrigation, and drainage of agricultural fields, oil and gas production, in particular, the dynamics of hydraulic fractures, problems of degassing coal and shale deposits in order to extract methane; magma movement in the earth's crust, geotectonics in the study of subsidence of the earth's crust, processes occurring in sedimentary basins, etc. A feature of the model of fluid filtration in a porous medium considered in this paper is the inclusion of the mobility of the solid skeleton and its poroelastic properties.

A comparative study on simulating flow-induced fracture deformation in subsurface media by means of extended FEM and FVM

Accurate and efficient simulation on the fluid flow and deformation in porous media is of increasing importance in a diverse range of engineering fields. At present, there are only several methods can be used to simulate the deformation of fractured porous media. It is very important to know their application scopes, advantages, and disadvantages for solving the practical problems correctly. Therefore, in this paper, we compared two numerical simulation methods for flow-induced fracture deformation in porous media. One is the Extended Finite Element Method (XFEM), which is based on the classical finite element method and can simulate strong or weak discontinuous problems. The other is developed within the finite-volume framework, termed Extended Finite Volume Method (XFVM). We designed three test cases, including single fracture, cross fractures and eight discrete fractures, to investigate the accuracy and efficiency of XFEM and XFVM. The reference solutions were provided by the commercial software, COMSOL, where the standard finite element method is implemented. The research findings showed that the accuracy of the XFEM was slightly higher than that of the XFVM, but the latter was more efficient. These results are likely to be useful in decision making regarding choice of solving methods for the multi-field coupling problem in fractured porous media.

Mathematical Model of Multiphase Nonisothermal Filtration in Deformable Porous Media with a Simultaneous Chemical Reaction

The authors have developed a mathematical model of multiphase nonisothermal filtration with chemical reactions and phase transitions in deformable porous media. A modified Biot approach was used to describe filtration through a deformable porous medium. Equations for the stress–strain state have been formulated in terms of displacements, and the elastic rheology of a porous skeleton in nonisothermal form was used in the model. The dependence of the permeability on porosity has been written in the form of a Kozeny–Carman equation. A test problem of combustion in a low-permeability sample of a bituminous reservoir has been solved. Results of the work can be used to numerically substantiate experiments on the technology of interbedding combustion in low-permeability beds.

Imaging fluid injections into soft biological tissue to extract permeability model parameters

One of the most common health care procedures is injecting fluids, in the form of drugs and vaccines, into our bodies, and hollow microneedles are emerging medical devices that deliver such fluids into the skin. Fluid injection into the skin through microneedles is advantageous because of improved patient compliance and the dose sparing effect for vaccines. Since skin tissue is a deformable porous medium, injecting fluid into the skin involves a coupled interaction between the injected fluid flow and the deformation of the soft porous matrix of skin tissue. Here, we introduce a semiempirical model that describes the fluid transport through skin tissue based on experimental data and constitutive equations of flow through biological tissue. Our model assumes that fluid flows radially outward and tissue deformation varies spherically from the microneedle tip. The permeability of tissue, assumed to be initially homogeneous, varies as a function of volumetric strain in the tissue based on a two-parameter exponential relationship. The model is optimized to extract two macroscopic parameters, k0 and m, for each of the seven experiments on excised porcine skin, using a radial form of Darcy’s law, the two-parameter exponential dependence of permeability on strain, and the experimental data on fluid flow recorded by a flow sensor and tissue deformation captured in real time using optical coherence tomography. The fluid flow estimated by the permeability model with optimized macroscopic parameters matches closely with the recorded flow rate, thus validating our semiempirical model.

Fluid-driven mechanical responses of deformable porous media during two-phase flows: Hele-Shaw experiments and hydro-mechanically coupled pore network modeling

Injecting fluid into a porous material can cause deformation of the pore structure. This hydro-mechanically coupled (i.e., poromechanical) phenomenon plays an essential role in many geological and biological operations across a wide range of scales, from geologic carbon storage, enhanced oil recovery and hydraulic fracturing to the transport of fluids through living cells and tissues, and to fuel cells. In this study, we conducted an experimental and numerical investigation of the hydro-mechanical coupling during fluid flows in porous media at the fundamental pore-scale. First, experimental demonstrations were undertaken to ascertain the effect of the hydro-mechanical coupling for two-phase fluid flows in either deformable or non-deformable porous media. Next, a hydro-mechanically coupled pore network model (HM-PNM) was employed to test a various range of influential parameters. The HM-PNM results were consistent with the experimental observations, including the advancing patterns of fluids and the development of the poroelastic deformation, when the viscous drop was incorporated. The hydro-mechanical coupling was observed to reduce the inlet pressure required to maintain a constant flow rate, whereas its effect on the pattern of fluid flow was minimal. The interfacial tension alteration also changed the pressure and deformation. The viscosity of invading fluid showed significant effects on both the patterns of fluid displacement and mechanical deformation.

Reactivation of natural fractures network by hydraulic fracturing

Modelling the process of induced fracture initiation, propagation and interaction with natural fractures is a very challenged task. Significant progress has been made in recent years in the development of complex fracture models to address the needs for more suitable design tools than the conventional planar fracture models. However, some aspects of this complex process are not still fully understood in terms of their impact or importance to the overall fracture geometry, or the complexity of simulating them is beyond the current modelling capabilities or requires computation effort that is not practical for engineering purpose. A technique that has been developed to represent the process of fracture propagation is the Continuous Approximation of Strong Discontinuities, which introduces a special kinematics, capable of representing the process of degradation of the material. One way to implement this approximation is to introduce the effects of a very narrow band of localized deformations within the existing finite elements. In this paper was used a finite element procedure that performs numerical analysis of fluid flow in a deformable porous media in a fully coupled scheme. In this analysis the propagation of the hydraulic fracture occurs specially along the pathway of the natural fractures, due to their lower tensile strength and greater permeability.

A Scalable Multigrid Reduction Framework for Multiphase Poromechanics of Heterogeneous Media

Simulation of multiphase poromechanics involves solving a multi-physics problem in which multiphase flow and transport are tightly coupled with the porous medium deformation. To capture this dynamic interplay, fully implicit methods, also known as monolithic approaches, are usually preferred. The main bottleneck of a monolithic approach is that it requires solution of large linear systems that result from the discretization and linearization of the governing balance equations. Because such systems are non-symmetric, indefinite, and highly ill-conditioned, preconditioning is critical for fast convergence. Recently, most efforts in designing efficient preconditioners for multiphase poromechanics have been dominated by physics-based strategies. Current state-of-the-art "black-box" solvers such as algebraic multigrid (AMG) are ineffective because they cannot effectively capture the strong coupling between the mechanics and the flow sub-problems, as well as the coupling inherent in the multiphase flow and transport process. In this work, we develop an algebraic framework based on multigrid reduction (MGR) that is suited for tightly coupled systems of PDEs. Using this framework, the decoupling between the equations is done algebraically through defining appropriate interpolation and restriction operators. One can then employ existing solvers for each of the decoupled blocks or design a new solver based on knowledge of the physics. We demonstrate the applicability of our framework when used as a "black-box" solver for multiphase poromechanics. We show that the framework is flexible to accommodate a wide range of scenarios, as well as efficient and scalable for large problems.

Periodic solutions of a hysteresis model for breathing

We propose to model the lungs as a viscoelastic deformable porous medium with a hysteretic pressure–volume relationship described by the Preisach operator. Breathing is represented as an isothermal time-periodic process with gas exchange between the interior and exterior of the body. The main result consists in proving the existence of a periodic solution under an arbitrary periodic forcing in suitable function spaces.

Entropy Generation in Couette Flow Through a Deformable Porous Channel

Abstract The present study examines the entropy generation on Couette flow of a viscous fluid in parallel plates filled with deformable porous medium. The fluid is injected into the porous channel perpendicular to the lower wall with a constant velocity and is sucked out of the upper wall with same velocity .The coupled phenomenon of the fluid flow and solid deformation in the porous medium is taken in to consideration. The exact expressions for the velocity of fluid, solid displacement and temperature distribution are found analytically. The effect of pertinent parameters on the fluid velocity, solid displacement and temperature profiles are discussed in detail. In the deformable porous layer, it is noticed that the velocity of fluid, solid displacement and temperature distribution are decreases with increasing the suction/injection velocity parameter. The results obtained for the present flow characteristic reveal several interesting behaviors that warrant further study on the deformable porous media. Furthermore, the significance of drag and the volume fraction on entropy generation number and Bejan number are discussed with the help of graphs.

A generalized poroelastic model using FEniCS with insights into the Noordbergum effect

Understanding the coupled behavior of fluid flow and solid deformation in porous media is often a critical aspect of geoscientific investigations, including studies of injection-induced seismicity, geological carbon sequestration, or surface deformation from pumping operations. In 1941, Biot first outlined the standard poroelastic formulation that accounted for this coupling within an isotropic and elastic porous medium. This fully coupled system of partial differential equations typically requires the use of numerical modeling for any practical purposes, necessitating either lengthy code development or the use of a complex, prebuilt model that often lacks flexibility. However, due to recent advances in automated approaches to solving systems of differential equations, problems of poroelasticity can now be easily handled in a streamlined yet flexible manner. Here, we present a poroelastic model built within the framework of FEniCS – an open source, general purpose finite element method software – which solves the system monolithically and can produce a continuous and mass-conserving solution for specific discharge. We present both a linear model and a generalized, nonlinear model. The nonlinear model allows for variable fluid density and employs a fluid pressure and volumetric strain dependent porosity relationship. The behavior of the linear model is verified with common benchmark problems, whereas results from the nonlinear model are compared to assess the impact of including its generalizations. Finally, the model is applied to a two-dimensional problem of fluid extraction meant to replicate the well-known Noordbergum effect (in which the fluid pressure in layers adjacent to the pumped layer temporarily increases during pumping). This is often referred to as ”reverse water fluctuation”. This showcases not only the flexibility of the model but also its ability to simulate numerically challenging scenarios. The results from this simulation suggest a novel explanation of the physical mechanism generating the Noordbergum effect: strain gradients.