Deformable Porous Media Research Articles

A Comparative Study of Different Drying Process for a Deformable Saturated Porous Medium

Drying of a deformable saturated porous medium based on convective tempering is a novel method that can enhance energy efficiency and the quality of the dried product itself. In this experimental investigation, the performances of this specific technique are compared with those of a standard stationary drying process in terms of deformation, drying kinetics, moisture redistribution, and energy consumption. In particular, the response of a deformable saturated porous medium (Kaolin) is considered. The results are critically discussed pointing out advantages and drawbacks.

Construction of a numerical model and algorithm for solving two-dimensional problems of filtration of multicomponent liquids, taking into account the moving “oil-water” interface

The paper considers a two-dimensional mathematical model of the filtration of a viscous, incompressible fluid in a deformable porous medium. The article describes a mathematical model of the problem at the “oil-water” interface with a system of parabolic differential equations. The problem posed is solved numerically using the differential-difference method based on the longitudinal-transverse scheme and the differential sweep method to determine the unknown boundary. To obtain a differential-difference problem, an algorithmic representation of a hidden scheme of alternating directions (longitudinal-transverse scheme) is used. The resulting system of differential-difference equations with initial conditions is known for each time layer along straight lines along the x axis, and then along each y axis it is solved by a differential sweep along a straight line, where the values just found correspond to time, i.e., the layer is taken as initial conditions. By approximating the differential equations in time, the position of the interface is determined for each time layer.

Pore-scale experimental study on fluid injection into two-dimensional deformable porous media

Fluid injection into a deformable (or soft) porous medium often results in the deformation and/or displacement of hosting solids, which in return, influences the transport of injected fluid. This study presents a summary of recent pore-scale experimental efforts that attempted to advance our understanding on the hydro-mechanical coupling at the pore scale. Our experimental study examines the effects of various fluid flow conditions, including miscibility, viscosity, injection rate, as well as mechanical stiffness/compression on the fluid-driven deformation and concurrent fluid transport more comprehensively. We also introduce four particle-level pressure terms, the yield (Py), fracture (Pf), capillary entry (Pc), and viscous drag (Pv) pressures, as primary indicators to properly capture the observed regimes of fluid-driven deformation and fluid transport, in addition to the traditional capillary and mobility numbers. The collected test results reveal that the single-phase fluid flow causes the fluid-driven solid deformation only when Pv/max(Py, Pf) ≥ 10−2, and its prevalent shape is volumetric deformation around the injection port. The two-phase fluid flow renders a volumetric deformation pattern when 10−5 < Pv/max(Py, Pf) < 10−2 and 10−2 < Pc/max(Py, Pf). Whereas, it causes fracture-like solid deformation when Pv/max(Py, Pf) ≥ 10−2 and Pc/max(Py, Pf) ≥ 10−2. A complex transition from the volumetric to fracture-like deformation, and then to pore invasion (without any noticeable solid deformation) may occur in a single porous medium if the injected fluid is low viscous and the local effective stress is relatively small. Furthermore, the results reveal a good correlation between the maximum inlet pressure and the magnitude of fluid-driven deformation. Lastly, a comparison of fluid injection energy is provided for those different regimes of fluid-driven deformation and fluid transport. This study provides unique pore-scale experiment results on flow-induced deformation and suggests criteria based on dimensionless parameters with the particle-level pressures, which advances our understanding and prediction capability on hydro-mechanical behavior associated with fluid injection. The observations from this study have implications for many geological and biological systems that are associated with porous materials and fluid injection, including geologic carbon sequestration and convection-enhanced drug delivery.

Numerical investigation of heating and cooling-induced damage and brine migration in geologic rock salt : Insights from coupled THM modeling of a controlled block scale experiment

Controlling brine flow is important for the safe disposal of radioactive waste in salt rocks. Thermal and mechanical processes can have a significant impact on brine flow, although this has never been thoroughly investigated. In this study, we conduct fully coupled THM modeling for analyzing brine flow in rock salt, considering non-isothermal two-phase flow through deformable porous media. To rigorously represent rock salt behavior, we incorporate suitable phenomenological models for creep and shear and tensile-induced dilatancy/damage, and their effect on the flow properties. To validate such a complex model, we use it to analyze an experiment on a meter-scale salt block subjected to multistage heating and cooling under controlled laboratory conditions. The experimental data and our model predictions of temperature and brine inflow show good agreement. Our modeling shows that it is important to consider the coupling between the heating- and cooling-induced damage and the flow properties for rigorously estimating brine inflow. Based on our modeling, a cooling-induced brine inflow spike is caused by a permeability increase due to tensile dilatancy. Thus, our modeling will be useful for a proper design of a cooling phase in the salt repositories to avoid or minimize damage and the resulting brine inflow.

Modeling gas breakthrough and flow phenomena through engineered barrier systems using a discrete fracture approach

Compacted clays are being considered to build engineered barrier systems (EBS) intended for the safe isolation of high-level nuclear waste (HLW) and spent nuclear fuel (SNF). The corrosion of the metallic canister containing the HLW/SNF will lead to the generation and buildup of the gas pressure in the more internal part of the clay buffer. This phenomenon would eventually trigger the formation and propagation of fractures in the clay barrier, jeopardizing its safety functions. In this work we propose to use the fragmentation technique (MFT) to model evolving fractures in clays triggered by gas pressurization. The MFT has been successfully used to model the formation of fractures in concrete, drying cracks in soil, hydraulic and thermo-fractures in rocks. In this work, we extend the MFT to deal with multiphase fluid flow in deformable porous media, and we upgraded a fully coupled computer finite element code using the extended technique. The proposed approach is first verified against analytical solutions and is then applied to model gas breakthrough experiments in clays. A very satisfactory performance of the method is observed in all the analyses, showing the potential of the MFT to tackle multiphase flow problems in deformable porous media with evolving discontinuities.

Coupling Chemotaxis and Growth Poromechanics for the Modelling of Feather Primordia Patterning

In this paper we propose a new mathematical model for describing the complex interplay between skin cell populations with fibroblast growth factor and bone morphogenetic protein, occurring within deformable porous media describing feather primordia patterning. Tissue growth, in turn, modifies the transport of morphogens (described by reaction-diffusion equations) through diverse mechanisms such as advection from the solid velocity generated by mechanical stress, and mass supply. By performing an asymptotic linear stability analysis on the coupled poromechanical-chemotaxis system (assuming rheological properties of the skin cell aggregates that reside in the regime of infinitesimal strains and where the porous structure is fully saturated with interstitial fluid and encoding the coupling mechanisms through active stress) we obtain the conditions on the parameters—especially those encoding coupling mechanisms—under which the system will give rise to spatially heterogeneous solutions. We also extend the mechanical model to the case of incompressible poro-hyperelasticity and include the mechanisms of anisotropic solid growth and feedback by means of standard Lee decompositions of the tensor gradient of deformation. Because the model in question involves the coupling of several nonlinear PDEs, we cannot straightforwardly obtain closed-form solutions. We therefore design a suitable numerical method that employs backward Euler time discretisation, linearisation of the semidiscrete problem through Newton–Raphson’s method, a seven-field finite element formulation for the spatial discretisation, and we also advocate the construction and efficient implementation of tailored robust solvers. We present a few illustrative computational examples in 2D and 3D, briefly discussing different spatio-temporal patterns of growth factors as well as the associated solid response scenario depending on the specific poromechanical regime. Our findings confirm the theoretically predicted behaviour of spatio-temporal patterns, and the produced results reveal a qualitative agreement with respect to the expected experimental behaviour. We stress that the present study provides insight on several biomechanical properties of primordia patterning.

Updated Lagrangian unsaturated periporomechanics for extreme large deformation in unsaturated porous media

Unsaturated periporomechanics is a strong nonlocal poromechanics based on peridynamic state and effective force state concept. In the previous periporomechnics the total Lagrangian formulation is adopted for the solid skeleton of porous media. In this article as a new contribution we formulate and implement an updated Lagrangian unsaturated periporomechanics framework for modeling extreme large deformation in unsaturated soils under drained conditions. In this new framework the so-called bond-associated sub-horizon concept is utilized to enhance the stability and accuracy at extreme large deformation of the solid skeleton. The stabilized nonlocal velocity gradient in the deformed configuration is used to update the effective force state from a critical state based visco-plastic constitutive model for unsaturated soils. The updated Lagrangian periporomechanics paradigm is numerically implemented through an explicit Newmark scheme for high-performance computing. Numerical examples are presented to demonstrate the stability of the computational updated Lagrangian periporomechanics paradigm and its efficacy and robustness in modeling extreme large deformation in porous media under drained conditions.

Variational geometric approach to the thermodynamics of porous media

AbstractMany applications of porous media research involves high pressures and, correspondingly, exchange of thermal energy between the fluid and the matrix. While the system is relatively well understood for the case of nonmoving porous media, the situation when the elastic matrix can move and deform, is much more complex. In this paper, we derive the equations of motion for the dynamics of deformable porous media, which includes the effects of friction forces, stresses, and heat exchanges between the media, by using the new methodology of variational approach to thermodynamics. This theory extends the recently developed variational derivation of the mechanics of deformable porous media to include thermodynamic processes and can easily include incompressibility constraints. The model for the combined fluid‐matrix system, written in the spatial frame, is developed by introducing mechanical and additional variables describing the thermal energy part of the system, writing the action principle for the system, and using a nonlinear, nonholonomic constraint on the system deduced from the second law of thermodynamics. The resulting equations give us the general version of possible friction forces incorporating thermodynamics, Darcy‐like forces, and friction forces similar to those used in the Navier–Stokes equations. The equations of motion are valid for arbitrary dependence of the kinetic and potential energies on the state variables. The results of our work are relevant for geophysical applications, industrial applications involving high pressures and temperatures, food processing industry, and other situations when both thermodynamical and mechanical considerations are important.

Cavitation in a soft porous material.

We study the collapse and expansion of a cavitation bubble in a deformable porous medium. We develop a continuum-scale model that couples compressible fluid flow in the pore network with the elastic response of a solid skeleton. Under the assumption of spherical symmetry, our model can be reduced to an ordinary differential equationthat extends the Rayleigh-Plesset equationto bubbles in soft porous media. The extended Rayleigh-Plesset equationreveals that finite-size effects lead to the breakdown of the universal scaling relation between bubble radius and time that holds in the infinite-size limit. Our data indicate that the deformability of the porous medium slows down the collapse and expansion processes, a result with important consequences for wide-ranging phenomena, from drug delivery to spore dispersion.

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

We formulate a numerical method for solving the two-phase flow poroelasticity equations. The scheme employs the interior penalty discontinuous Galerkin method and a sequential time-stepping method. The unknowns are the phase pressures and the displacement. Existence of the solution is proved. Three-dimensional numerical results show the accuracy and robustness of the proposed method.

Parameter-robust methods for the Biot–Stokes interfacial coupling without Lagrange multipliers

In this paper we advance the analysis of discretizations for a fluid-structure interaction model of the monolithic coupling between the free flow of a viscous Newtonian fluid and a deformable porous medium separated by an interface. A five-field mixed-primal finite element scheme is proposed solving for Stokes velocity-pressure and Biot displacement-total pressure-fluid pressure. Adequate inf-sup conditions are derived, and one of the distinctive features of the formulation is that its stability is established robustly in all material parameters. We propose robust preconditioners for this perturbed saddle-point problem using appropriately weighted operators in fractional Sobolev and metric spaces at the interface. The performance is corroborated by several test cases, including the application to interfacial flow in the brain.

Numerical simulation and analysis of multiscale interface coupling between a poroelastic medium and a lumped hydraulic circuit: Comparison between functional iteration and operator splitting methods

We consider a multiscale problem modeling the flow of a fluid through a deformable porous medium, described by a system of partial differential equations (PDEs), connected with a lumped hydraulic circuit, described by a system of ordinary differential equations (ODEs). This PDE/ODE coupled problem includes interface conditions enforcing the continuity of mass and the balance of stresses across models at different scales. In the present article, we address questions related to the solution methods of the PDE/ODE coupled problem via staggered algorithms, focusing on a detailed comparison between functional iterations and an energy-based operator splitting method and how they handle the interface conditions. We provide sufficient conditions for the convergence of functional iterations and prove that the energy-based operator splitting method is unconditionally stable with respect to the size of the time discretization step.

Physics-informed neural network simulation of multiphase poroelasticity using stress-split sequential training

Physics-informed neural networks (PINNs) have received significant attention as a unified framework for forward, inverse, and surrogate modeling of problems governed by partial differential equations (PDEs). Training PINNs for forward problems, however, pose significant challenges, mainly because of the complex non-convex and multi-objective loss function. In this work, we present a PINN approach to solving the equations of coupled flow and deformation in porous media for both single-phase and multiphase flow. To this end, we construct the solution space using multi-layer neural networks. Due to the dynamics of the problem, we find that incorporating multiple differential relations into the loss function results in an unstable optimization problem, meaning that sometimes it converges to the trivial null solution, other times it moves very far from the expected solution. We report a dimensionless form of the coupled governing equations that we find most favorable to the optimizer. Additionally, we propose a sequential training approach based on the stress-split algorithms of poromechanics. Notably, we find that sequential training based on stress-split performs well for different problems, while the classical strain-split algorithm shows an unstable behavior similar to what is reported in the context of finite element solvers. We use the approach to solve benchmark problems of poroelasticity, including Mandel’s consolidation problem, Barry–Mercer’s injection-production problem, and a reference two-phase drainage problem. The Python-SciANN codes reproducing the results reported in this manuscript will be made publicly available at https://github.com/sciann/sciann-applications.

Effect of bio-tissue deformation behavior due to intratumoral injection on magnetic hyperthermia

Thermal damage of malignant tissue is generally determined not only by the characteristics of bio-tissues and nanoparticles but also the nanofluid concentration distributions due to different injection methods during magnetic hyperthermia. The latter has more advantages in improving the therapeutic effect with respect to the former since it is a determining factor for the uniformity of nanofluid concentration distribution inside the tumor region. This study investigates the effect of bio-tissue deformation due to intratumoral injection on the thermal damage behavior and treatment temperature distribution during magnetic hyperthermia, in which both the bio-tissue deformation due to nanofluid injection and the mass diffusion after injection behavior are taken into consideration. The nanofluid flow behavior is illustrated by two different theoretical models in this study, which are Navier–Stokes equation inside syringe needle and modified Darcy’s law inside bio-tissue. The diffusion behavior after nanofluid injection is expressed by a modified convection–diffusion equation. A proposed three-dimensional liver model based on the angiographic data is set to be the research object in this study, in which all bio-tissues are assumed to be deformable porous media. Simulation results demonstrate that the injection point for syringe needle can generally achieve the maximum value in the tissue pressure, deformation degree, and interstitial flow velocity during the injection process, all of which then drop sharply with the distance away from the injection center. In addition to the bio-tissue deformation due to injection behavior, the treatment temperature is also highly relevant to determine both the diffusion duration and blood perfusion rate due to the thermal damage during the therapy.

Non-isothermal two-phase flow in deformable porous media: systematic open-source implementation and verification procedure

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.

A Fully Coupled Hydro-Mechanical-Gas Model Based on Mixture Coupling Theory

The interactions of gas migration, water transport and mechanical deformation of rocks are significant for geoenergy industry (e.g. Carbon Capture and Storage, radioactive waste disposal); however, the hydro-mechanical-gas coupled model remains a challenge due to the gap between multiple disciplines (e.g. Geomechanics and Geoenergy). This work presents a novel hydro-mechanical framework model of fully coupled two-phase fluid transport in a deformable porous media through extending mixture coupling theory which is based on non-equilibrium thermodynamics. The main difference between the mixture coupling theory approach and other approaches (ex., mechanic's approach) is that the mixture coupling theory uses energy and entropy analysis by utilizing the unbalanced thermodynamics, while the mechanic's approach analyses the stress–strain tensors. The gas free energy has been included in the Helmholtz free energy balance equation. Three main governing equations have been obtained for solid, liquid and gas phases. Benchmark experiments and modelling based on classical continuum mechanics approaches are used to validate the model by comparing the measured data to the simulation results. The results have a good agreement with experimental data, demonstrating that gas migration has a great influence on water transport and deformation of the solids. The novelty of this study is that it is providing a new approach to study the multiphase flow coupling in porous media rather than the classic mechanic’s approach.Article HighlightsA Hydro-Mechanical-Gas (HMG) model has been developed using the mixture coupling theory approach.The hydro-mechanical framework equations were established by using non-equilibrium thermodynamic and Darcy law.The model has been validated using published experimental data and the results of other researchers with different approaches.

Fluid flow modeling through pressure-dependent porous media: An analytical solution and a computational fluid dynamics approach

Fluid flow modeling in porous media requires solving relevant governing equations, which are inherently nonlinear. However, some simplifications are usually made to make them linear. One of those common assumptions is invariant rock properties with respect to reservoir pressure. However, there are some situations, including pressure-dependent porous media, where this assumption would not be applicable. Hence, the solution of the nonlinear diffusivity equation shall be sought for such cases. The main goal of this study is to present a new analytical solution to the diffusivity equation and a numerical scheme considering exponential decreases in the petrophysical properties with reservoir pressure. The diffusivity equation was solved using the assumed exponential functions, and analytical solutions are provided in this paper. Besides, Computational Fluid Dynamics methods were utilized to verify the accuracy of the derived equations. The developed analytical equations and numerical scheme were then applied over several pressure-dependent porous media cases, and our comparison study showed that neglecting the pressure-dependency of the petrophysical properties will lead to significant errors (approximately 40%). Based on the introduced solutions, we also proposed a new set of type-curves that can be used in well-testing analyses of the deformable porous media.

Numerical analysis of geomechanical behavior of fractures and faults in a deformable porous medium

In this study, geomechanical behavior of fractures and faults in rocks as a saturated deformable discrete fracture-porous medium has been evaluated using coupled fluid flow-geomechanics numerical modeling. The purpose of this paper is to observe and evaluate the effects of fractures and faults on the pore pressures during fluid flow through reservoirs. This issue involves solving the equations that have been derived from the Biot consolidation theory such as fluid mass balance equation, Darcy law and momentum balance equations. Govern coupled equations were solved using the standard Galerkin finite element method for continuous porous medium. Elements called "zero-thickness elements” were also used to discretize the fault as a discontinuous part of the porous medium. In compared with the previous and similar methods, the method introduced in this paper, made modifications in either the choice of the element and the method of solving the governing equations. The main advantage of this paper is providing clear precise formulations of the double node zero-thickness element in hydro-mechanical modeling of fractures and faults. Verification of the proposed process and models presented in this paper were done by providing three index problems which their analytical and numerical solutions are available. The results of our model provide a good agreement to these reference solutions which indicates the accuracy of the method presented in this paper.

Analysis of a tumor model as a multicomponent deformable porous medium

We propose a diffuse interface model to describe a tumor as a multicomponent deformable porous medium. We include mechanical effects in the model by coupling the mass balance equations for the tumor species and the nutrient dynamics to a mechanical equilibrium equation with phase-dependent elasticity coefficients. The resulting PDE system couples two Cahn–Hilliard type equations for the tumor phase and the healthy phase with a PDE linking the evolution of the interstitial fluid to the pressure of the system, a reaction-diffusion type equation for the nutrient proportion, and a quasistatic momentum balance.We prove here that the corresponding initial-boundary value problem has a solution in appropriate function spaces.

Modeling fracture propagation in a rock‐water‐air system with the assumed enhanced strain method

AbstractThis article presents a fully coupled finite element method for the single cohesive fracture propagation in a deformable porous medium (rock) interacting with two immiscible pore fluids, that is, water and air. The cohesive fracture is represented with the assumed enhanced strain method which can be cast into a classical plasticity framework. The governing equations for the solid skeleton deformation, pore water flow, and pore air flow are derived based on the mixture theory. The coupled nonlinear global system is then solved by the standard Galerkin finite element method. The numerical implementation is verified against the numerical and analytical solutions previously reported in the literature. The proposed model is capable of capturing fluid transfer between the fracture and the host rock due to the pressure difference. The numerical examples demonstrate that the behavior of fracture propagation can be significantly influenced by the capillary pressure inside fracture and vice versa. Since the fracture is represented at the level of Gauss points, the proposed numerical model can be easily implemented on any standard finite element code base.