Viscoelastic Porous Medium Research Articles

Моделирование захоронения углекислого газа в вязкоупругой пористой среде

Предложена математическая модель фильтрации газа в среде с переменной пористостью. В двумерном случае определяющая система уравнений в предположении малости скорости твердой фазы сведена к параболическому уравнению для эффективного давления среды и уравнению первого порядка для пористости. Проведено численное исследование полученной начально-краевой задачи. Исследовано несколько вариантов параметров нагнетания углекислого газа в пласт с малой начальной пористостью. В ходе численных расчетов определены оптимальные варианты нагнетания газа для его хранения в геологической среде в долгосрочной перспективе. Purpose. This paper presents a model that describes gas filtration in a poroelastic medium. The initial-boundary problem describing carbon dioxide injection into a geological environment is formulated. The purpose of the research is numerical solution of the initial-boundary value problem and the analysis of results of the presented numerical simulation. Methodology. The assumption that the solid phase velocity is small allows reducing the original system of constitutive two-dimensional equations to two equations for the effective pressure and porosity. For numerical implementation, the scheme of alternating directions and the Runge - Kutta method of the fourth order of accuracy are used. A parabolic equation for the effective pressure was solved using the stabilizing correction scheme and the explicit scheme for the heat equation. Grid convergence is tested by a series of computational experiments on sequences of refined grids. Findings. Several options for gas injection are considered. Optimal parameters for storing carbon dioxide in geological formations in the long term have been found. Originality/value. Taking into account the poroelastic properties and the assumption of the deformability of the solid phase of the medium allows making new predictions of the distribution of carbon dioxide in geological media. The research results can be useful for improving existing carbon dioxide storage models and for developing new ones.

Analysis of the viscoelasticity in coal based on the fractal theory

Coal is a complex viscoelastic porous medium with fractal characteristics at different scales. To model the macroscale structure of coal, a fractal viscoelastic model is established, and the P-wave velocity dispersion and attenuation characteristics are discussed based on the complex modulus derived from this model. The numerical simulation results indicate that the fractional order [Formula: see text] and relaxation time [Formula: see text] greatly affect the P-wave velocity dispersion and attenuation. The fractal viscoelastic model indicates a full-band velocity dispersion between 1 Hz and 104 Hz. Meanwhile, the P-wave velocity has a weaker dispersion with the fractal viscoelastic model than with the Kelvin-Voigt model and Zener model between 1 Hz and 104 Hz for the same relaxation time and elastic modulus, but the velocity at 1 Hz based on the fractal viscoelastic model is higher with the Kelvin-Voigt model and Zener model. Simultaneously, the velocities of five coal samples are tested, and the attenuation factor is calculated using a low-frequency system. The experimental results indicate a strong dispersion in coal in the range of 10–250 Hz. The classic Kelvin-Voigt model and Zener model cannot describe the dispersion characteristics of coal, but the fractal viscoelastic model can describe them well by using the appropriate fractional order and relaxation time.

Vibrational analysis of Love waves in a viscoelastic composite multilayered structure

A unified procedure is presented to investigate the dispersion and damping behavior of Love waves in a double-layered structure. The double-layered structure is comprised of a viscoelastic fiber-reinforced medium (Medium I) embedded between a viscoelastic sandy layer (Medium II) and a viscoelastic porous semi-infinite medium (Medium III). The displacement vectors for the respective medium have been obtained using analytical techniques. Using the separation of variables method and admissible boundary conditions at the free surface and the interfaces, the complex frequency equation is obtained in closed form. The substantial impact of all material parameters involved in the considered model on the phase velocity as well as damped velocity has been observed numerically and shown by graphical illustrations. The outcomes from this analysis may serve as a dynamic tool in various fields including geophysics, geotechnical, and earthquake engineering.

The problem of determining the 2D-kernel in a system of integro-differential equations of a viscoelastic porous medium

Under consideration is the system of integro-differential equations of a viscoelastic porous medium. The direct problem is to define the $$y$$ -component of the displacement vectors of the elastic porous body and the liquid from the initial boundary value problem for these equations. We assume that the kernel of the integral term of the first equation depends on time and one of the spatial variables. To determine the kernel, some additional condition is given on the solution of the direct problem for $$z=0 $$ . The inverse problem is replaced by an equivalent system of integro-differential equations for the unknown functions. We apply the method of scales of the Banach spaces of analytic functions. The local solvability of the inverse problem is proved in the class of the functions analytic in $$x$$ and continuous in $$t $$ .

The Problem of Determining the 2D Kernel in a System of Integro-Differential Equations of a Viscoelastic Porous Medium

The Problem of Determining the 2D Kernel in a System of Integro-Differential Equations of a Viscoelastic Porous Medium

Coefficient determination problem in the system of integro-differential equation for visco-elastic porous medium

Coefficient determination problem in the system of integro-differential equation for visco-elastic porous medium

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.

Asymptotic Stability of Solutions to the Porous Media System with Hysteresis

The long time behavior of solutions to the autonomous PDE system describing fluid diffusion in a viscoelastic porous medium with capillary hysteresis is studied with homogeneous Dirichlet conditions for the displacement of the solid and homogeneous Neumann boundary conditions for the capillary pressure. Although the set of possible equilibria is very large, a detailed investigation of the hysteresis memory dynamics shows that all global solution trajectories converge to an equilibrium in the state space of all admissible memory configurations as time tends to infinity.

Inverse Problem for A System of Integro-Differential Equations for SH Waves in A Visco-Elastic Porous Medium: Global Solvability

We consider a system of hyperbolic integro-differential equations for SH waves in a visco-elastic porous medium. The inverse problem is to recover a kernel (memory) in the integral term of this system. We reduce this problem to solving a system of integral equations for the unknown functions. We apply the principle of contraction mappings to this system in the space of continuous functions with a weight norm. We prove the global unique solvability of the inverse problem and obtain a stability estimate of a solution of the inverse problem.

Multiscale modelling of sound propagation through the lung parenchyma

In this paper we develop and study numerically a model to describe some aspects of sound propagation in the human lung, considered as a deformable and viscoelastic porous medium (the parenchyma) with millions of alveoli filled with air. Transmission of sound through the lung above 1 kHz is known to be highly frequency-dependent. We pursue the key idea that the viscoelastic parenchyma structure is highly heterogeneous on the small scale e and use two-scale homogenization techniques to derive effective acoustic equations for asymptotically small e. This process turns out to introduce new memory effects. The effective material parameters are determined from the solution of frequency-dependent micro-structure cell problems. We propose a numerical approach to investigate the sound propagation in the homogenized parenchyma using a Discontinuous Galerkin formulation. Numerical examples are presented.

Transient Dynamic Analysis of Gradient-Saturated Viscoelastic Porous Media

In the present work, by considering the inhomogeneity, solid skeleton viscosity, and compressibility of solid particles and fluid, the calculation formulations of reverberation ray matrix method (RRMM) is established for transient response of viscoelastic porous medium. The derivation process of the formulas and calculation method is described in detail herein. Feasibility and efficiency of the present approach are validated by comparing the solutions with those obtained from other methods. As numerical examples, by assuming that the material properties of the saturated poroviscoelastic media have an exponential law distribution along the thickness coordinate, the transient response, in terms of displacement of solid skeleton, total stress, and pore fluid pressure, are examined by using numerical inverse Laplace transform. The effects of inhomogeneity on transient response of gradient saturated poroviscoelastic media are demonstrated and discussed. The results show that there are significant effects of inhomogeneity on compressive wave propagation and that RRMM is an effective way for solving the partial differential equations with variable coefficients in a mathematic model of saturated porous media with gradient-variation properties.

Mixed and Galerkin finite element approximation of flow in a linear viscoelastic porous medium

We propose two fully discrete mixed and Galerkin finite element approximations to a system of equations describing the slow flow of a slightly compressible single phase fluid in a viscoelastic porous medium. One of our schemes is the natural one for the backward Euler time discretization but, due to the viscoelasticity, seems to be stable only for small enough time steps. The other scheme contains a lagged term in the viscous stress and pressure evolution equations and this is enough to prove unconditional stability. For this lagged scheme we prove an optimal order a priori error estimate under ideal regularity assumptions and demonstrate the convergence rates by using a model problem with a manufactured solution. The model and numerical scheme that we present are a natural extension to ‘poroviscoelasticity’ of the poroelasticity equations and scheme studied by Philips and Wheeler in (for example) [Philip Joseph Philips, Mary F.Wheeler, Comput. Geosci. 11 (2007) 145–158] although — importantly — their algorithms and codes would need only minor modifications in order to include the viscous effects. The equations and algorithms presented here have application to oil reservoir simulations and also to the condition of hydrocephalus — ‘water on the brain’. An illustrative example is given demonstrating that even small viscoelastic effects can produce noticeable differences in long-time response. To the best of our knowledge this is the first time a mixed and Galerkin scheme has been analysed and implemented for viscoelastic porous media.

ON the linear theory of viscoelastic porous media

This paper is concerned with a reciprocity theorem and a variational theorem in the linear theory of viscoelastic porous media in t h e quasi-static case.

Energy velocity and quality factor of plane harmonic inhomogeneous waves in anisotropic poro-viscoelastic media

SUMMARY Energy flux vector and related wavefield quantities are studied for the propagation of harmonic plane waves in an anisotropic viscoelastic porous medium saturated with viscous fluid. In this dissipative medium, Biot's theory is used to define the propagation of four attenuating waves through different complex-valued slowness vectors. The attenuating waves are considered to be propagating as general inhomogeneous waves. The inhomogeneity strength of an attenuating wave is defined by a non-dimensional parameter. A fixed value of this inhomogeneity parameter is used to calculate the complex slowness vector of any attenuated wave in the medium for an arbitrarily chosen propagation direction. The complex slowness vector is then used to calculate the velocity and direction of energy flux for each of the four waves in the medium. Dissipation of energy is explained in terms of quality factors of attenuation such that the contributions from pore-fluid viscosity and anelastic porous-frame are separated. Deviations of the ray (or group velocity) direction from the propagation direction, the attenuation direction and the propagation–attenuation plane are also calculated. A numerical example is studied to analyse the variations in the energy-flux characteristics with propagation direction and inhomogeneity parameter.

Creep and relaxation functions of a heterogeneous viscoelastic porous medium using the Mori-Tanaka homogenization scheme and a discrete microscopic retardation spectrum

In this paper the macroscopic creep and relaxation functions of a heterogeneous viscoelastic porous medium are derived by using Mori-Tanaka homogenization scheme. Analytical and semi-analytical solutions can then be determined with a parametric number of heterogeneous phases embedded in a viscoelastic matrix whose behavior is described with a parametric number of analogical units. Under some simplifying assumptions, a solution strategy is presented in order to make explicit how the microscopic retardation and relaxation times of the viscoelastic matrix control the distribution of the retardation and relaxation times of the homogenized medium.

Numerical modeling of seismic wavefield in anisotropic viscoelastic porous medium with the pseudo-spectral method

In order to describe the reservoirs more accurately, the characteristics of the earth medium must be considered comprehensively, such as the anisotropy, viscoelasticity and pore, etc. Consequently, an anisotropic viscoelastic porous medium model is presented in this paper, and the elastic wave equation in this medium has been derived. The seismic wavefield numerical modeling was simulated by using the pseudo-spectral method. Synthetic seismograms are used to show the characteristics of the seismic wave in complex medium. The research work is useful to understanding seismic wave propagation in earth medium.

Wave propagation and the frequency domain Green’s functions in viscoelastic Biot/squirt (BISQ) media

In this paper, we examine the characteristics of elastic wave propagation in viscoelastic porous media, which contain simultaneously both the Biot-flow and the squirt-flow mechanisms (BISQ). The frequency-domain Green’s functions for viscoelastic BISQ media are then derived based on the classic potential function methods. Our numerical results show that S-waves are only affected by viscoelasticity, but not by squirt-flows. However, the phase velocity and attenuation of fast P-waves are seriously influenced by both viscoelasticity and squirt-flows; and there exist two peaks in the attenuation-frequency variations of fast P-waves. In the low-frequency range, the squirt-flow characteristic length, not viscoelasticity, affects the phase velocity of slow P-waves, whereas it is opposite in the high-frequency range. As to the contribution of potential functions of two types of compressional waves to the Green’s function, the squirt-flow length has a small effect, and the effects of viscoelastic parameter are mainly in the higher frequency range.

Analysis of lung parenchyma as a parametric porous medium

The dynamic behavior of the lung in health and disease depends on its viscoelastic properties. To better understand these properties, several mathematical models have been utilized by many investigators. In the present work, we present a new approach that characterizes the dynamics of gas flow into a viscoelastic porous medium that models the lung structure. This problem is considered in terms of the lung input impedance on a macro level and parenchymal tissue impedance on the level of an alveolar wall. We start from a basic theoretical analysis in which macroscopic tissue deformations are represented in accordance with the linearized Navier-Stokes equations. This approach has strong theoretical underpinnings in other situations but has not been applied to analyze the impedance of the inflated lung. Our analysis provides a theoretical basis for analyzing the interaction between flow into the lungs as a biophysical diffusion process and parenchymal viscoelasticity described phenomenologically, within the frameworks of standard viscoelasticity and structural damping. This lung impedance incorporates parameters of porosity, permeability, and viscoelasticity on micro and macro levels of parenchymal tissue. The analysis shows the theoretical basis of the transformation from the impedance of alveolar walls or isolated tissue strips to that of the intact parenchyma. We also show how the loading impedance at the lung boundary may have a significant impact on the dynamic behavior of whole lung viscoelasticity. Our analysis may be useful in directing specific tests of different models and for analyzing experimental measurements of viscoelastic parameters of lung material under normal and pathological conditions.

Modelling effective rheologies for viscoelastic porous media with application to silt, and medium and coarse sand

A modification of Biot's poroelastic differential equations is made to include matrix–fluid interaction mechanisms which assume a solid–fluid relaxation function coupling coefficient. Values of physical properties of sediments are incorporated into equations which define phase velocity and attenuation for porous media which are dependent on the composite densities of various media (silt, and medium and coarse sand). The results enable us to compare the attenuation and velocities of waves in these media. We observed that the density of coarse sand is greater than that of medium sand and this in turn is greater than that of silt—the same holds for the velocities of P-waves in these media but the situation is converse for shear waves in the same given media. As the densities of the media increase, their attenuation decreases as it was found that the attenuation of silt is the highest and that of coarse sand lowest for the media considered.

Biot/Squirt Model in Viscoelastic Porous Media

Viscoelastic wave equations are derived for both theBiot flow and squirt-flow involved. The relationship between phase velocityor attenuation of viscoelastic wave and macro parameters of the reservoiris clarified in a porous extensive dilatancy anisotropy medium.Numerical models clearly demonstrate that the viscoelasticproperty of rocks, not the squirt-flow, causes the dispersion andattenuation in the low-frequency range. The attenuation of the quasiSV-wave, SH-wave and the quasi P-wave depend strongly on the directionsof permeabilities. Furthermore, the trends of attenuation of the quasiP-wave fast and the quasi SV-wave are inversed in both the high-frequencyand low-frequency ranges. Our numerical result is in agreementwith the experimental result that was carried out for theBiot/Squirt model by Jorge [Geophys. 65(2000) 202]