Elastic Porous Media Research Articles

A mechanical picture of fractional-order Darcy equation

In this paper the authors show that fractional-order force-flux relations are obtained considering the flux of a viscous fluid across an elastic porous media. Indeed the one-dimensional fluid mass transport in an unbounded porous media with power-law variation of geometrical and physical properties yields a fractional-order relation among the ingoing flux and the applied pressure to the control section. As a power-law decay of the physical properties from the control section is considered, then the flux is related to a Caputo fractional derivative of the pressure of order 0⩽β≤1. If, instead, the physical properties of the media show a power-law increase from the control section, then flux is related to a fractional-order integral of order 0⩽β≤1. These two different behaviors may be related to different states of the mass flow across the porous media.

A Runge-Kutta/WENO method for solving equations for small-amplitude wave propagation in a saturated elastic porous medium

A high-accuracy Runge-Kutta/WENO method of up to fourth order with respect to time and fifth order with respect to space is developed for the numerical modeling of small-amplitude wave propagation in a steady fluid-saturated elastic porous medium. A system of governing equations is derived from a general thermodynamically consistent model of a compressible fluid flow through a saturated elastic porous medium, which is described by a hyperbolic system of conservation laws with allowance for finite deformations of the medium. The results of numerical solution of one- and two-dimensional wave fields demonstrate the efficiency of the method.

Flow of Stokesian fluid through a cellular medium and thermal effects

Abstract The thermal effects of a stationary Stokesian flow through an elastic micro-porous medium are compared with the entropy produced by Darcy’s flow. A micro-cellular elastic medium is considered as an approximation of the elastic porous medium. It is shown that after asymptotic two-scale analysis these two approaches, one analytical, starting from Stoke’s equation and the second phenomenological, starting from Darcy’s law give the same result. The incompressible and linearly compressible fluids are considered, and it is shown that in micro-porous systems the seepage of both types of fluids is described by the same equations.

A residue-based mode selection and sorting procedure for efficient poroelastic modeling in acoustic finite element applications.

Analysis of three-dimensional sound propagation in porous elastic media with the Finite Element (FE) method is, in general, computationally costly. Although it is the most commonly used predictive tool in complex noise control applications, efficient FE solution strategies for large-size industrial problems are still lacking. In this work, an original procedure is proposed for the sorting and selection of the modes in the solution for the sound field in homogeneous porous domains. This procedure, validated on several 2D and 3D problems, enables to reduce the modal basis in the porous medium to its most physically significant components. It is shown that the size of the numerical problem can be reduced, together with matrix sparsity improvements, which lead to the reduction in computational time and enhancements in the efficacy of the acoustic response computation. The potential of this method for other industrial-based noise control problems is also discussed.

Elastic Waves in Swelling Porous Media

Time harmonic waves in a swelling porous elastic medium of infinite extent and consisting of solid, liquid and gas phases have been studied. Employing Eringen’s theory of swelling porous media, it has been shown that there exist three dilatational and two shear waves propagating with distinct velocities. The velocities of these waves are found to be frequency dependent and complex valued, showing that the waves are attenuating in nature. Here, the appearance of an additional shear wave is new and arises due to swelling phenomena of the medium, which disappears in the absence of swelling. The reflection phenomenon of an incident dilatational wave from a stress-free plane boundary of a porous elastic half-space has been investigated for two types of boundary surfaces: (i) surface having open pores and (ii) surface having sealed pores. Using appropriate boundary conditions for these boundary surfaces, the equations giving the reflection coefficients corresponding to various reflected waves are presented. Numerical computations are performed for a specific model consisting of sandstone, water and carbon dioxide as solid, liquid and gas phases, respectively, of the porous medium. The variations of phase speeds and their corresponding attenuation coefficients are depicted against frequency parameter for all the existing waves. The variations of reflection coefficients and corresponding energy ratios against the angle of incidence are also computed and depicted graphically. It has been shown that in a limiting case, Eringen’s theory of swelling porous media reduces to Tuncay and Corapcioglu theory of porous media containing two immiscible fluids. The various numerical results under these two theories have been compared graphically.

Three-dimensional shear and compressional wave propagation of multiple point sources in fluid-saturated elastic porous media

A multiple point source model is developed in this research for studying both shear and compressional spherical wave propagation in a non-viscous fluid-saturated elastic porous medium. Relative displacement between the fluid and solid of the medium is quantified by the spherical wave governing equations, such that the waves described are more representative to that in engineering practices. The shear wave has shown significant influences on the characteristics of superposed shear and compressional waves generated by multiple point sources. Utilization of multiple point sources shows higher efficiency and effectiveness in generating desired waves, in comparing with that of single source. Specifically, the multiple sources model is more energy effective in comparing with the single source model by always producing larger magnitudes of relative displacement than a single source with the same energy, which becomes more significant when the distance between the source and the considered geological particles increases. Multiple point sources also show advantages on duration, direction control and magnitude adjustment for the waves generated. Numerical analyses are performed for comparing different shear, compressional and the superposed wave responses under single and multiple sources.

Field Detection of Microcracks to Define the Nucleation Stage of Earthquake Occurrence

Main shocks of natural earthquakes are known to be accompanied by preshocks which evolve following the modified Ohmori’s law in average over many samples. Individual preshock activity, however, is far less systematic for predictive purposes. On the other hand, the microcracks in laboratory rock experiments are always preceded to final rupture. And, previous investigations of field acoustic emissions showed that the activity increases prominently before and after the main shock. But there is no detection of any phenomena to identify the nucleation stage. Here we show that a special underground electric field measurement could detect microcracks. Pulse-like variations were classified into three groups (A, B, C) by frequency. The B-type is suggested to define the nucleation period: activity increases sharply following the modified Omori’s law before the main shock and there is no activity afterward. The B-type is subgrouped into three types possibly corresponding to crack-rupture modes. The variations are supposed to be induced by crack occurrence through electrokinetic effects in the elastic-porous medium. The detection distance is suggested to be several orders larger than that of the acoustic emission due to the effective smallness of dissipation rate, and the waveform can be used to infer the rupture mode.

Fundamental and plane wave solution in swelling porous medium

In the present paper propagation of plane waves in swelling porous medium (SP) is studied. The phase velocity and attenuation coefficients of these waves are computed numerically and presented graphically. The results so obtained have been compared to without swelling porous elastic medium (EL). The fundamental solution of the system of differential equations in swelling porous medium in case of steady oscillations in terms of elementary functions has been constructed. Some basic properties are established and particular case of interest is also deduced.

A Modified Stress-Displacement Solution for a Pressure Tunnel with a Permeable Liner in an Elastic Porous Medium Based on a New Model

Referring to previous literature, the elastic problem for a deep pressure tunnel with a permeable liner in a saturated elastic porous medium that obeys Terzaghi’s effective stress principle is further investigated. In the analysis, the influences of the changes of the pore water pressure around the opening, the construction sequence of the tunnel, and the interaction between the liner and the surrounding geomaterial on the mechanical response of the tunnel are appropriately considered. Also, the influences of the relative liner thickness and rigidity and the relative distance of the point under investigation from the tunnel axis on the stress-displacement fields for various combinations of the mechanical and geometric parameters are numerically evaluated and discussed. The results indicate that both the support pressure and displacement increase monotonically with increasing liner rigidity and thickness. The influence of the tunnel excavation upon the displacement field is significantly larger than upon the stress field. Either too high or too low values for the relative rigidity and thickness of the liner are unfavorable for structural stability.

Influence of fluid saturation on Rayleigh wave propagation

Based on the existing three-phase poroelastic theory, an analytical solution for Rayleigh wave propagation along the traction free, permeable surface of an elastic porous medium, saturated by two immiscible, viscous, compressible fluids, is derived. The analytical expression of the dispersion equation of Rayleigh wave is found to be a cubic polynomial. Through solving the dispersion equation, it is concluded that three different types of Rayleigh wave exist in such medium to be denoted as R1, R2 and R3, respectively. The influences of water saturation on these Rayleigh waves are numerically studied with varying frequencies from 1 Hz to 1 MHz. The results show that water content has significant effects on the velocities and attenuation of the three waves. The amplitudes of the three waves decay in both x - and z - directions, which are parallel to and perpendicular to the propagation direction of the waves, respectively. The R1 wave travels fastest, and the R3 and R2 waves have the second greatest and smallest phase velocities, respectively. The R1 wave is least attenuated, followed by the R2 and R3 waves. The attenuation coefficients of the R2 and R3 waves versus water saturation and excitation frequency have the similar variation trend.

NUMERICAL MODELING OF ELECTROACOUSTIC LOGGING INCLUDING JOULE HEATING

It is well known that electromagnetic field excites acoustic wave in a porous elastic medium saturated with fluid electrolyte due to electrokinetic conversion effect. Pride's equations describing this process are written in isothermal approximation. Update of these equations, which allows to take influence of Joule heating on acoustic waves propagation into account, is proposed here. This update includes terms describing the initiation of additional acoustic waves excited by thermoelastic stresses and the heat conduction equation with right side defined by Joule heating. Results of numerical modeling of several problems of propagation of acoustic waves excited by an electric field source with and without consideration of Joule heating effect in their statements are presented. From these results, it follows that influence of Joule heating should be taken into account at the numerical simulation of electroacoustic logging and at the interpretation of its log data.

Modeling seismic stimulation: Enhanced non-aqueous fluid extraction from saturated porous media under pore-pressure pulsing at low frequencies

Abstract Seismic stimulation, the application of low-frequency stress-pulsing to the boundary of a porous medium containing water and a non-aqueous fluid to enhance the removal of the latter, shows great promise for both contaminated groundwater remediation and enhanced oil recovery, but theory to elucidate the underlying mechanisms lag significantly behind the progress achieved in experimental research. We address this conceptual lacuna by formulating a boundary-value problem to describe pore-pressure pulsing at seismic frequencies that is based on the continuum theory of poroelasticity for an elastic porous medium permeated by two immiscible fluids. An exact analytical solution is presented that is applied numerically using elasticity parameters and hydraulic data relevant to recent proof-of-principle laboratory experiments investigating the stimulation-induced mobilization of trichloroethene (TCE) in water flowing through a compressed sand core. The numerical results indicated that significant stimulation-induced increases of the TCE concentration in effluent can be expected from pore-pressure pulsing in the frequency range of 25–100 Hz, which is in good agreement with what was observed in the laboratory experiments. Sensitivity analysis of our numerical results revealed that the TCE concentration in the effluent increases with the porous medium framework compressibility and the pulsing pressure. Increasing compressibility also leads to an optimal stimulation response at lower frequencies, whereas changing the pulsing pressure does not affect the optimal stimulation frequency. Within the context of our model, the dominant physical cause for enhancement of non-aqueous fluid mobility by seismic stimulation is the dilatory motion of the porous medium in which the solid and fluid phases undergo opposite displacements, resulting in stress-induced changes of the pore volume.

Dynamics of uncharged colloidal inclusions in polyelectrolyte hydrogels

We calculate the dynamics of an uncharged colloidal sphere embedded in a quenched polyelectrolyte hydrogel to (i) an oscillatory (optical and magnetic) force, as adopted in classical micro-rheology, and (ii) an oscillatory electric field, as adopted in electrical micro-rheology and electro-acoustics. The hydrogel is modelled as a linearly elastic porous medium with the charge fixed to the skeleton and saturated with a Newtonian electrolyte; and the colloidal inclusion is modelled as a rigid, impenetrable sphere. The dynamic micro-rheological susceptibility, defined as the ratio of the particle displacement to the strength of an applied oscillatory force, depends on the fixed-charge density and ionic strength and is bounded by the limits for incompressible and uncharged, compressible skeletons. Nevertheless, the influences of fixed charge and ionic strength vanish at frequencies above the reciprocal draining time, where the polymer and the electrolyte hydrodynamically couple as a single incompressible phase. Generally, the effects of fixed charge and ionic strength are small compared with, for example, the influences of polymer slip at the particle surface. The electrical susceptibility, defined as the ratio of the particle displacement to the strength of an applied oscillatory electric field, is directly influenced by charge at all frequencies, irrespective of skeleton compressibility. At low frequencies, polymer charge modulates the driving (electro-osmotic) and restoring (electrostatically enhanced elastic) forces, whereas charge has no influence on the restoring force at high frequencies where dilational strain is suppressed by hydrodynamic coupling with the electrolyte. In striking contrast to charged inclusions in uncharged hydrogels (Wang & Hill, J. Fluid Mech., vol. 640, 2009, pp. 357–400), the electrical susceptibility at high frequencies is independent of electrolyte concentration. Rather, the dynamics primarily reflect the elastic modulus, charge and hydrodynamic permeability, with a relatively weak dependence on particle size. Interestingly, the dynamic mobility in the zero-momentum reference frame, which is central to the electro-acoustic response, is qualitatively different from the dynamic mobility in the skeleton-fixed reference frame. Finally, we propose a phenomenological harmonic-oscillator model to address – in an approximate manner – the dynamics of charged particles in charged hydrogels. This shows that particle dynamics at low frequencies are dominated by particle charge, whereas high-frequency dynamics are dominated by hydrogel charge.

Efficient block preconditioners for the coupled equations of pressure and deformation in highly discontinuous media

SUMMARYLarge‐scale simulations of flow in deformable porous media require efficient iterative methods for solving the involved systems of linear algebraic equations. Construction of efficient iterative methods is particularly challenging in problems with large jumps in material properties, which is often the case in geological applications, such as basin evolution at regional scales. The success of iterative methods for this type of problems depends strongly on finding effective preconditioners. This paper investigates how the block‐structured matrix system arising from single‐phase flow in elastic porous media should be preconditioned, in particular for highly discontinuous permeability and significant jumps in elastic properties. The most promising preconditioner combines algebraic multigrid with a Schur complement‐based exact block decomposition. The paper compares numerous block preconditioners with the aim of providing guidelines on how to formulate efficient preconditioners. Copyright © 2010 John Wiley & Sons, Ltd.

Stress-induced crystal preferred orientation in the poromechanics of in-pore crystallization

A non-hydrostatic stress field affects the orientation of crystals growing in the pore network of an elastic porous medium. The hypothesis of a hydrostatic state of stress within the crystal has been implicitly made in the recent extension of poromechanics to in-pore crystalization ( Coussy, 2006). This underlying hypothesis is revisited on a small-scale conceptual model based on Eshelby's problem and shows that chemo-mechanical equilibrium requires that the crystal adapts its shape and orientation to the far-field stress, therefore resulting at equilibrium in a hydrostatic state of stress within the crystal. The optimum crystal shape as a function of the far-field stress is consistently investigated, highlighting limiting cases. The small scale model allows to understand the macroscopic effects associated with deviatoric stresses in the poromechanics of in-pore crystallization. Moreover, it provides the building block for an up-scaling of the macroscopic tangent poroelastic properties, which depend on both the current crystal saturation and the state of stress. A dilute micromechanical scheme illustrates the variation of the macroscopic Biot's coefficient tensor as a function of deviatoric stresses. A simple configuration akin to a potential laboratory experiment finally illustrates the strong induced anisotropy of the crystallization induced macroscopic strain when deviatoric stresses are applied to the material prior to crystallization.

Spatiotemporal BOLD dynamics from a poroelastic hemodynamic model

A quantitative theory is developed for the relationship between stimulus and the resulting blood oxygen level dependent (BOLD) functional magnetic resonance imaging (fMRI) signal, including both spatial and temporal dynamics for the first time. The brain tissue is modeled as a porous elastic medium, whose interconnected pores represent the vasculature. The model explicitly incorporates conservation of blood mass, interconversion of oxygenated and deoxygenated hemoglobin, force balance within the blood and of blood pressure with vessel walls, and blood flow modulation due to neuronal activity. In appropriate limits it is shown to reproduce prior Balloon models of hemodynamic response, which do not include spatial variations. The regime of validity of such models is thereby clarified by elucidating their assumptions, and when these break down, for example when voxel sizes become small.

Microstructural effects on the overall poroelastic properties of saturated porous media

At the macroscopic scale, the quasi-static deformation of an elastic porous medium saturated by an incompressible Newtonian fluid is described by the well-known Biot's model, which involves four effective parameters. In this work, the three effective poroelastic properties and the permeability of two periodic microstructures of saturated cohesive granular media, i.e. simple cubic (SC) and body-centered cubic (BCC) arrays of overlapping spheres, are computed by solving, over the representative elementary volume, boundary-value problems arising from the homogenization process. The influence of microstructure properties, i.e. solid volume fraction, arrangement of spheres, number of contacts as well as the intrinsic properties of the solid phase on the overall properties, is highlighted. Numerical results are then compared with rigorous bounds, self-consistent estimations, exact expansions and experimental results on ceramics and metals available in the literature. Finally, the capability of the obtained results on such periodic microstructures to describe the poroelastic properties of real porous media is discussed.

Evolution of the Pore-Pressure Field around a Moving Conical Penetrometer of Finite Size

A solution is developed for the evolution of buildup, steady, and postarrest dissipative pore-fluid pressure fields that develop around a finite-radius conical penetrometer advanced in a saturated linearly elastic porous medium. The analog with cone penetrometer testing is direct and is used to enable continuous distributions of permeability and diffusivity to be determined with depth. This analysis reveals the direct dependence of penetration rate on the induced fluid pressure field magnitudes, and predicts that a penetration rate threshold limit exists with respect to pore-pressure generation. This represents the essence of a partially drained system. The developed pore-pressure field is determined to be a function of the dissipation rate of the material, the penetration rate, and the storage effects of the advecting medium. Analysis of the pore-pressure field under start-up conditions reveals that the time required to reach steady state is strongly influenced by the penetration rate and the pressure-dissipation properties of the material. Analysis of the developed stable pressure fields illustrates the inversely proportional relationship that exists between penetration rate and pore-pressure magnitudes at the cone surface; representing the influence of storage in the medium on stable pore-pressure magnitudes. Stable pressure fields below the penetration threshold limit, UD10 1 , form a spherical response around the cone tip transitioning to an elongated radial response for penetration rates above this limit. Postarrest analysis indicates that the prearrest penetration rate strongly influences the dissipation rate and pattern of dissipation. The developed analysis can be correlated with CPTu-recovered data to independently evaluate permeability magnitudes during steady penetration.

Effects of Pressure Oscillations on Drainage in an Elastic Porous Medium

The effects of seismic stimulation on the flow of two immiscible fluids in an elastic synthetic porous medium is experimentally investigated. A wetting fluid is slowly evacuated from the medium, while a pressure oscillation is applied on the injected non- wetting fluid. The amplitude and frequency of the pressure oscillations as well as the evacuation speed are kept constant throughout an experiment. The resulting morphology of the invading structure is found to be strongly dependent on the interplay between the amplitude and the frequency of the applied pressure oscillations and the elasticity of the porous medium. Different combinations of these properties yield morphologically similar structures, allowing a classification of structures that is found to depend on a proposed dimensionless number.

Thermomechanical response of a poroelastic half-space soil medium subjected to time harmonic loads

The thermomechanical responses of a porous elastic medium subjected to time harmonic loads (normal force and thermal source) are investigated analytically in the context of generalised thermoelastic theory with one relaxation time. The material of the foundation, obeying Biot’s dynamic poroelastic theory, is idealised as a uniform, fully saturated poroelastic half-space stratum. The coupled governing equations are established based on Biot’s dynamic poroelastic theory and on generalised thermoelastic theory. Assuming the disturbances to be harmonically time dependent, the general solutions of stress, displacement, temperature distribution and excess pore water pressure are deduced using the Fourier transform, and the transformed solutions are numerically inverted. The differences among the coupled thermo-hydro-mechanical dynamic model (THMD), the hydro-mechanical dynamic model (HMD) and the thermo-elastic dynamic model (TMD) are discussed. In addition, the effects of the thermal loading frequency on the displacement, stress, temperature distribution and excess pore water pressure components are analysed in the numerical results.