Deformable Porous Media Research Articles

Numerical Investigation of the Deformable Porous Media Treated by the Intermittent Microwave

A 2D axi-symmetric theoretical model of dielectric porous media in intermittent microwave (IMW) thermal process was developed, and the electromagnetic energy, multiphase transport, phase change, large deformation, and glass transition were taken into consideration. From the simulation results, the mass was mainly carried by the liquid water, and the heat was mainly carried by liquid water and solid. The diffusion was the dominant mechanism of the mass transport during the whole process, whereas for the heat transport, the convection dominated the heat transport near the surface areas during the heating stage. The von Mises stress reached local maxima at different locations at different stages, and all were lower than the fracture stress. A material treated by a longer intermittent cycle length with the same pulse ratio (PR) tended to trigger the phenomena of overheat and fracture due to the more intense fluctuation of moisture content, temperature, deformation, and von Mises stress. The model can be extended to simulate the intermittent radio frequency (IRF) process on the basis of which one can select a suitable energy source for a specific process.

A locally conservative mixed finite element framework for coupled hydro-mechanical–chemical processes in heterogeneous porous media

This paper presents a mixed finite element framework for coupled hydro-mechanical–chemical processes in heterogeneous porous media. The framework combines two types of locally conservative discretization schemes: (1) an enriched Galerkin method for reactive flow, and (2) a three-field mixed finite element method for coupled fluid flow and solid deformation. This combination ensures local mass conservation, which is critical to flow and transport in heterogeneous porous media, with a relatively affordable computational cost. A particular class of the framework is constructed for calcite precipitation/dissolution reactions, incorporating their nonlinear effects on the fluid viscosity and solid deformation. Linearization schemes and algorithms for solving the nonlinear algebraic system are also presented. Through numerical examples of various complexity, we demonstrate that the proposed framework is a robust and efficient computational method for simulation of reactive flow and transport in deformable porous media, even when the material properties are strongly heterogeneous and anisotropic.

Numerical Solution of a Two-Dimensional Problem of Fluid Filtration in a Deformable Porous Medium

The paper considers a two-dimensional mathematical model of filtration of a viscous incompressible fluid in a deformable porous medium. The model is based on the equations of conservation of mass for liquid and solid phases, Darcy’s law, the rheological relationship for a porous medium, and the law of conservation of the balance of forces. In this article, the equation of the balance of forces is taken in full form, i.e. the viscous and elastic properties of the medium are taken into account. The aim of the work is a numerical study of a model initial-boundary value problem. Section 1 gives a statement of the problem and a brief review of the literature on works related to this topic. In item 2, the original system of equations is transformed. In the case of slow flows, when the convective term can be neglected, a system arises that consists of a second-order parabolic equation for the effective pressure of the medium and the first-order equation for porosity. Section 3 proposes an algorithm for the numerical solution of the resulting initial-boundary value problem. For the numerical implementation, a variable direction scheme for the heat equation with variable coefficients is used, as well as the Runge — Kutta scheme of the fourth order of approximation.

Modeling fluid–structure interactions between cerebro-spinal fluid and the spinal cord

The spinal cord (SC) is a tissue composed of white and gray matter that is permeated by capillary blood, venular blood and cerebrospinal fluid (CSF). It is surrounded by an annular cavity, called the subarachnoid space (SAS), within which the CSF exhibits a pulsatile laminar flow. Insights into the functioning of cerebrospinal fluid are expected to reveal the pathogenesis of severe neurological diseases – such as syringomyelia – involving the formation of fluid-filled cavities (syrinxes) in the spinal cord. A fully analytical fluid–structure interaction model, based on the Multiple Network Porous Elastic Theory, is here proposed. The spinal cord is regarded as a deformable porous medium permeated by different fluids, while a pulsatile flow in the subarachnoid space is taken into account. By virtue of the slender body approximation, the mathematical model allows to analytically solve the space–time structure of the CSF flow field, the displacements in the spinal cord, and the pressures of the fluids inside the cord. The results are consistent with data reported in literature and with some numerical simulations. A sensitivity analysis shows that the deviation from the physiological values of the Young modulus, the capillary pressures at the SC–SAS interface and the permeability of blood networks can lead to a great increase in the CSF fluxes across the spinal cord. These findings bring new insights into the factors that affect CSF exchange, and they may support hypotheses which attribute syrinx formation to an abnormal exchange of cerebrospinal fluid between the SAS and the spinal cord.

Porochemothermoelastic solutions considering fully coupled thermo-hydro-mechanical-chemical processes to analyze the stability of inclined boreholes in chemically active porous media

The present paper presents a fully coupled thermo-hydro-mechanical-chemical (THMC) model for chemically active deformable porous media (such as clay or clay-like shale) under local thermal and chemical equilibrium (LTE and LCE) conditions. The proposed model is developed by combining the phenomenological method and the new entropy production derived based on the non-equilibrium thermodynamics. Our proposed THMC model introduces the following new coupled terms: (1) ultrafiltration effects in the solute flux equation, (2) thermal filtration and Dufour effects into the heat flux expression, and (3) strain, pressure, and solute mass-fraction in the thermal diffusion equation. This model also introduces the influence of bound water that occupies the interlayer space of clay platelets and governs the chemical and mechanical responses as the solid part of porous media. We derive semi-analytical solutions for the specific engineering application of stability of an inclined wellbore in a chemically active clay-like shale formation during petroleum drilling, considering the fully coupled porochemothermoelastic effects under the plane strain assumption. The wellbore loading is decomposed into axisymmetric and deviatoric cases. The time-dependent field variables including displacements, pore pressures, stresses, solute fraction, and temperatures are obtained by performing the inversion of the Laplace transforms. Based on the newly developed semi-analytical porochemothermoelastic solutions, several analyses are conducted to evaluate the influence of the newly introduced terms in the THMC model on pore pressures and stresses around a horizontal wellbore in a shale formation; and on time-dependent failure behavior, including shear failure and tensile fracturing. Results show that the significance of the newly introduced ultrafiltration, thermal filtration, and Dufour effects on induced pore pressure not only depends on the permeability, thermo-osmotic, and thermal diffusion coefficients, but also on the in-situ pressure, temperature, and chemical gradients. We suggest that the drilling engineers consider the time-dependent and location-dependent characteristics of borehole failure that depend on the thermo-hydro-mechanical-chemical process and appropriately design the mud weight required to ensure wellbore stability.

Understanding Ground Rupture Due to Groundwater Overpumping by a Large Lab Experiment and Advanced Numerical Modeling

AbstractGround rupture due to groundwater pumping is a major environmental hazard accompanying land subsidence in some areas. Ground ruptures usually develop when a significant compaction affects a sedimentary sequence with peculiar geological conditions, for example, the presence of a shallow bedrock with buried ridges. A laboratory test was developed with the aim at improving our understanding of the mechanisms responsible for rupture generation in this particular hydrogeological setting, typical, for instance, of the Guangming village, China. A 0.8‐m high concrete prism, representing a rocky ridge, was placed in a 4.0‐m long, 1.8‐m wide, and 1.4‐m high box and buried by alluvial material. The box was saturated and then drained, with the formation of a main crack above the prism‐shaped ridge. The measured water content, vertical displacements, strains, rupture initiation, and growth are analyzed through an original coupled 3D‐numerical model, simulating the variably saturated groundwater flow in a deformable and fractured porous medium. To our knowledge, this is the first application of such a kind of model to a lab‐scale experiment. Despite the uncertainty on the material parameters, the numerical model allows satisfactorily reproducing the observed groundwater flow, land subsidence, and rupture features (depth and width). The rupture dynamics is captured in part as well, despite the employment of a simple Mohr–Coulomb failure criterion and although other forces (e.g., electrochemical) may likely play a role at the metric scale of the lab test. The modeling outcomes provide a clear view on how the rupture develops from the surface and propagates downward.

Pore-Scale Simulation of Fluid Flow Through Deformable Porous Media Using Immersed Boundary Coupled Lattice Boltzmann Method

The interaction between the fluid flow and the deformable porous media is crucial in the applications of adsorption/absorption. The immersed boundary coupled lattice Boltzmann method is used in this study. The spring constant k of porous skeleton is introduced to consider the deformation effects. Subsequently, the evolutions of deformation of porous skeleton and hydrodynamic characteristics of fluid with k and Reynolds number are elucidated. The trend of deformation of porous skeleton appears as “concave front and convex back.” Moreover, the porosity of deformable porous media decreases gradually with the deformation growing. Meanwhile, the resistance of fluid flow increases due to the growing deformation of the porous skeleton. The larger k and Re contribute to the smaller drag coefficient. The deformation of the porous skeleton accelerates the detachment of the fluid boundary layer, and vortex patterns transfer from 2S to 2P mode with Reynolds number increasing when k = 0.05. Larger deformation leads to the higher pressure drop. The pressure drop for k = 0.01 is 21.7% higher than k = 0.05.

Nonlinear and non-local analytical solution for Darcy–Forchheimer flow through a deformable porous inclusion within a semi-infinite elastic medium

Permeability is a nonlinear and non-local function of the intimate coupling between pore fluid flow and solid deformation in porous media. A class of related problems involves the effect of fluid injection or withdrawal on the transport properties of geomaterials. This paper presents an analytical solution for the nonlinear and non-local problem of fluid flow through a disk-shaped porous elastic inclusion below the free surface of an elastic half-space. The solution accounts for the following nonlinear mechanisms: (i) variations in the permeability coefficient due to the flow-induced deformation of the inclusion; (ii) the inertial losses of the pore fluid flow. The former and latter mechanisms are formulated using the Green's function for a dilatation centre in an elastic half-space and the Darcy–Forchheimer equation for fluid flow through porous media, respectively. An analytical perturbation solution to the considered problem is developed and validated against the numerical finite element solution to the same problem. The described nonlinear mechanisms are represented by two dimensionless parameter groups. The extreme values of these dimensionless groups govern the solution asymptotic behaviours mimicking the special-case solutions in which either mechanism is forced to vanish. The applied aspects of the solution are demonstrated through the wellbore performance index parameter that quantifies the subsurface rock ability to deliver the pore fluid toward or away from a wellbore in a porous reservoir. Unlike the linear models, the presented nonlinear solution captures the observed dependence of the performance index on the wellbore flow rate.

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

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

A method for measuring fluid pressure and solid deformation profiles in uniaxial porous media flows.

Flow through rigid, reactive, or deformable porous media has relevance to the dynamical behavior studied across a broad range of problems in science and engineering. Here, we describe an apparatus designed to simultaneously measure the pervadic pressure profile, fluid volume flux, and deformation in an evolving poroelastic medium. We demonstrate the apparatus in measurements of flow-induced compression of a soft latex foam. The approach can be used in both rigid and reactive porous media.

Modeling Multiphase Flow Within and Around Deformable Porous Materials: A Darcy‐Brinkman‐Biot Approach

AbstractWe present a new computational fluid dynamics approach for simulating two‐phase flow in hybrid systems containing solid‐free regions and deformable porous matrices. Our approach is based on the derivation of a unique set of volume‐averaged partial differential equations that asymptotically approach the Navier‐Stokes Volume‐of‐Fluid equations in solid‐free regions and multiphase Biot Theory in porous regions. The resulting equations extend our recently developed Darcy‐Brinkman‐Biot framework to multiphase flow. Through careful consideration of interfacial dynamics (relative permeability and capillary effects) and extensive benchmarking, we show that the resulting model accurately captures the strong two‐way coupling that is often exhibited between multiple fluids and deformable porous media. Thus, it can be used to represent flow‐induced material deformation (swelling, compression) and failure (cracking, fracturing). The model's open‐source numerical implementation, hybridBiotInterFoam, effectively marks the extension of computational fluid mechanics into modeling multiscale multiphase flow in deformable porous systems. The versatility of the solver is illustrated through applications related to material failure in poroelastic coastal barriers and surface deformation due to fluid injection in poro‐visco‐plastic systems.

Computational homogenization of fully coupled multiphase flow in deformable porous media

In this paper, a computational modeling tool is developed for fully coupled multiphase flow in deformable heterogeneous porous medium that consists of complex and non-uniform micro-structures using the dual continuum scales based on the computational homogenization approach. The first-order homogenization technique is employed to perform the multi-scale analysis. The governing equations of two-phase flow of immiscible fluids, including an equilibrium equation and two mass continuity equations, are considered based on the appropriate main variables. According to the well-known Hill–Mandel principle of macro-homogeneity, the proper energy types are defined instead of conventional stress power for linking micro- and macro-scales, which plays a significant role in determination of consistent microscopic fields. The finite element squared strategy is utilized to resolve the two scales simultaneously. The periodic and linear boundary conditions are exploited in the micro-scale analysis, and the macroscopic quantities such as stress tensor, inertial force vector, flux vectors and fluid contents are determined from the boundary information of microscopic domain. Moreover, a general approach is defined depending on the type of boundary condition in which the macroscopic tangent operators can be extracted directly from the converged microscopic Jacobian matrix. Finally, in order to illustrate the efficiency and accuracy of the proposed computational algorithm, several numerical examples are solved, and the effects of various parameters, such as boundary conditions, RVE types, RVE length scale, and volume fraction of heterogeneities are investigated.

Entropy generation to predict irreversibilities in poroelastic film with multiple forces: spectral study

The present investigation is useful in medical engineering treatments, in which biothermal therapy is one of the popular treatments. The investigation of transport phenomena of fluid flow and heat in those applications requires multi-physical models featuring heat transfer and deformable porous media. In view of this, it is anticipated to study the influence of thermal buoyancy force and nonlinear radiation on the irreversibility, flow field, solid deformation, and heat transfer characteristics in viscous radiated fluid flow in a vertical deformable porous film. The combined phenomenon of the flow field movement and solid deformation in the porous medium is considered. The flow governing equations are non-dimensionalized and solved numerically by employing Chebyshev spectral method. The impact of important parameters on the fluid velocity, solid displacement, and fluid temperature profiles is depicted graphically and interpreted at length. In the deformable porous layer, it is perceived that the fluid velocity, solid displacement, and fluid temperature profiles decrease with an increase in suction/injection parameter values. The present physical model finds applications in geomechanics (Coussy in Mechanics of porous continua, Wiley, New York, 1995; Biot in J Appl Phys 12(2):155–164, 1941) and biomedical engineering (Mow et al. in J Biomech 17(5):377–394, 1984; Barry et al. in J Appl Math Phys 42:633–648, 1991; Lai et al. in J Biomech Eng 131:245-258, 1991).

An immersed phase field fracture model for microporomechanics with Darcy–Stokes flow

This paper presents an immersed phase field model designed to predict the fracture-induced flow due to brittle fracture in vuggy porous media. Due to the multiscale nature of pores in the vuggy porous material, crack growth may connect previously isolated pores, which leads to flow conduits. This mechanism has important implications for many applications such as disposal of carbon dioxide and radioactive materials and hydraulic fracture and mining. To understand the detailed microporomechanics that causes the fracture-induced flow, we introduce a new phase field fracture framework where the phase field is not only used as an indicator function for damage of the solid skeleton but also used as an indicator of the pore space. By coupling the Stokes equation that governs the fluid transport in the voids, cavities, and cracks and Darcy’s flow in the deformable porous media, our proposed model enables us to capture the fluid–solid interaction of the pore fluid and solid constituents during crack growth. Numerical experiments are conducted to analyze how the presence of cavities affects the accuracy of predictions based on the homogenized effective medium during crack growth.

On the existence of global solution of the system of equations of liquid movement in porous medium

The initial-boundary value problem for the system of one-dimensional isothermal motion of viscous liquid in deformable viscous porous medium is considered. Local theorem of existence and uniqueness of problem is proved in case of compressible liquid. In case of incompressible liquid the theorem of global solvability in time is proved in Holder classes. 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 poroelastiс properties. The transition from Euler variables to Lagrangian variables is used in the proof of the theorems.

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.

On modeling of airflow in human lungs: constitutive relations to describe deformation of porous medium

Within the framework of a multilevel mathematical model to describe the evolution of functional disorders of the human organism under the influence of environment factors, a mathematical model of the "meso-level" of the human respiratory system is developed. The article is deals with the development of the meso-level model - the formulation of a constitutive model to describe the airflow in a porous lung medium. Human lungs filled with small airways and alveoli, with air contained in them, are modeled by an elastically deformable saturated porous medium enclosed in an internal chamber with varying volume (movable walls). Conceptual and mathematical statements are presented. Air movement in the deformable porous medium of lungs is described by ratios of the mechanics of deformable solid body and filtration theory. As an element of this sub-model an analytical solution is obtained for an auxiliary geometrically linear problem of the all-round compression of an elastic thin-walled hollow sphere filled with air to determine the rate of mean stress of the two-phase medium of the lungs, taking into account the interaction between the lung tissue and the air contained in the lungs. To confirm the hypothesis on the acceptability of a linear solution of an auxiliary problem for large deformations, a similar problem was numerically solved in a geometrically nonlinear formulation. The results show that the obtained analytical solution is in satisfactory agreement with the solution of a similar problem in a nonlinear formulation both for calm and deep breathing, which indicates the possibility of using the former in the construction of the considered sub-model.

PorePy: an open-source software for simulation of multiphysics processes in fractured porous media

Development of models and dedicated numerical methods for dynamics in fractured rocks is an active research field, with research moving towards increasingly advanced process couplings and complex fracture networks. The inclusion of coupled processes in simulation models is challenged by the high aspect ratio of the fractures, the complex geometry of fracture networks, and the crucial impact of processes that completely change characteristics on the fracture-rock interface. This paper provides a general discussion of design principles for introducing fractures in simulators, and defines a framework for integrated modeling, discretization, and computer implementation. The framework is implemented in the open-source simulation software PorePy, which can serve as a flexible prototyping tool for multiphysics problems in fractured rocks. Based on a representation of the fractures and their intersections as lower-dimensional objects, we discuss data structures for mixed-dimensional grids, formulation of multiphysics problems, and discretizations that utilize existing software. We further present a Python implementation of these concepts in the PorePy open-source software tool, which is aimed at coupled simulation of flow and transport in three-dimensional fractured reservoirs as well as deformation of fractures and the reservoir in general. We present validation by benchmarks for flow, poroelasticity, and fracture deformation in porous media. The flexibility of the framework is then illustrated by simulations of non-linearly coupled flow and transport and of injection-driven deformation of fractures. All results can be reproduced by openly available simulation scripts.

Transport analysis in deformable porous media through integral transforms

AbstractGeomechanical deformation can alter the flow field that impacts solute mass fluxes. Despite its importance, the effects of the coupling between geomechanical deformation and the flow field on solute transport behavior are not fully known. In this paper, we study the impact of this coupling on the solute concentration distribution. The concentration field is semianalytically derived by making use of the generalized integral transform technique. We apply the semianalytical solution to two uniaxial consolidation problems, the classical Terzaghi's problem with a constant load and the case of periodic loading of a porous deformable layer. Our results indicate that geomechanical parameters, such as the Skempton's coefficient and the soil compressibility, can affect the peak concentration as well as the spatial moments of solute plume. In case of periodic loading, we show that the frequency of loading also plays a key role in regulating the temporal dynamics of the concentration field.

On the peridynamic effective force state and multiphase constitutive correspondence principle

This article concerns modeling unsaturated deformable porous media as an equivalent single-phase and single-force state peridynamic material through the effective force state. The balance equations of linear momentum and mass of unsaturated porous media are presented by defining relevant peridynamic states. The energy balance of unsaturated porous media is utilized to derive the effective force state for the solid skeleton that is an energy conjugate to the nonlocal deformation state of the solid, and the suction force state. Through an energy equivalence, a multiphase constitutive correspondence principle is built between classical unsaturated poromechanics and peridynamic unsaturated poromechanics. The multiphase correspondence principle provides a means to incorporate advanced constitutive models in classical unsaturated porous theory directly into unsaturated peridynamic poromechanics. Numerical simulations of localized failure in unsaturated porous media under different matric suctions are presented to demonstrate the feasibility of modeling the mechanical behavior of such three-phase materials as an equivalent single-phase peridynamic material through the effective force state concept.