Biot’s theory

Furthermore, in gels the elastic moduli K and p treated so far, those of the so-called skeletal frame in Biot s theory, are much smaller than the bulk moduli of fluid and polymer. Note that K accompanies no changes in the total volume of network + solvent, whereas K, p, and pfcl involve them and are much larger than K. From this fact, without changing the essential physics, we assume that the solvent and polymer have the same constant specific volume, so that... 

The coupling effects of various poromechanical processes on the response of a porous medium have been successfully addressed by Biot s theory of poroelasticity and its extensions [3,4,5,8,2], The chemical effects have also been addressed by considering interaction between the porous matrix and a pore fluid comprising of a solute and solvent [10, 7, 6], Comprehensive anisotropic poromechanics formulations and corresponding solutions for the inclined borehole problem have been presented [4—2], However, the coupled chemo-thermo-hydro-mechanical response of an anisotropic porous medium has not been addressed to date. 

The macroscopic behavior of a saturated porous material undergoing a dissolution of its linear elastic solid matrix is therefore described by the classical Biot s theory, where the poroelastic properties now depend on the morphological parameter . Formally, plays the role of a damage parameter accounting for the dissolution. 

To deseribe wave propagation in marine sediments mathematieally, various simple to eomplex models have been developed which approximate the sediment by a dilute suspension (Wood 1946) or an elastic, water-saturated frame (Gassmann 1951 Biot 1956a, b). The most common model whieh considers the microstructure of the sediment and simulates frequency-dependent wave propagation is based on Biot s theory (Biot 1956a, b). It includes Wood s suspension and Gassmann s elastic frame model as low-frequency approximations and combines acoustic and elastic parameters - P- and... 

Berryman J.G, 1980. Confirmation of Biot s theory. Applied Physics Letters 37 382-384... 

The material properties to be homogenized and upscaled are drawn from Biot s theory extended for non-isothermal consolidation (Guvanasen and Chan 2000). The modified crack tensor theory of Oda (1986) has been modified to include other transport and thermoelastic properties specified by Guvanasen and Chan (2000). 

Forest L, Gibiat V, Woignier T (1998) Biot s theory of acoustic propagation in porous media applied to aerogels and alcogels. J Non-Cryst Solids 225 287-292... 

De Leo R (2008) Long-term operational experience with the HERMES aerogel RICH detector. Nuclear Instruments Methods Phys Res A Accelerators, Spectrometers, Detectors, and Associated Equipment 595 19-22 Forest L, Gibiat V, Woignier T (1998) Biot s theory of acoustic propagation in porous media applied to aerogels and alcogels. J Non-Cryst SoUds 225 287-292... 

According to Sontheimer s theory [41], two typical shapes of breakthrough curves may exist. In the case of the porous diffusion predomination (carbon F - 100) curve of breakthrough is vertical and exhibits convex shape to X - axis or when layer diffusion predominated (carbon N) concave shape of elongated S letter (it is marked by BIOT (Bi) - number). It was stated, that elongated shape of breakthrough curve is connected with dilution of adsorption face, what makes the achievement of equili-birium state difficult. Thus, adsorption described by curve of vertical shape is most convenient. 

Based on Biot s fundamental work Stoll (e.g. 1974, 1977, 1989) reformulated the mathematical background of this theory with a simplified uniform nomenclature. Here, only the main physical principles and equations are summarized. For a detailed description please refer to one of Stoll s publications or Biot s original papers. 

Fig. 2.20 (a) S-wave velocities and (b) attenuation coefficients (at 400 kHz) derived from least square inversion based on Biot-Stoll s theory versus porosities for the four sediment cores 40KL, 47KL, GeoB2821-l, and PS2567-2. NFO and FNO as in Figure 2.19. Modified after Breitzke (2000). 

In order to predict the T-H-M response of the bentonite, a coupled T-H-M transient analysis was performed with the Finite Element Code FRACON. The governing equations incorporated in the FRACON code were derived from an extension of Biot s (1941) theory of poro-elasticity to include the T-H-M behaviour of the unsaturated FEBEX bentonite. The model formulation(Nguyen, Selvadurai and Armand, 2003) resulted in three governing equations where the primary unknowns are temperature, the displacement vector and the pore fluid pressure, as follows ... 

The permeability of the rock matrix is negligible in comparison with that of the fractures. 8) Contributions from various fractures and fracture sets can be superimposed. 9) Conversion between mechanical energy and thermal energy is negligible. 10) Fractures do not contribute to thermoelastic strain and the matrix does not contribute to hydroelastic strain. II) Biot s (1955) theory of coupled hydroelastic processes is valid. 12) Porosities are based on mechanical and hydraulic apertures. Other assumptions are provided in Oda (1986), and Guvanasen and Chan (2000). 

The AECL team used an in-house MOTIF finite-element code (Guvanasen and Chan 2000), which is based on an extension of the classical poroelastic theory of Biot (1941). This code has undergone extensive verification and validation (Chan et al. 2003). The CTH team employed the commercially available, general-purpose finite-element code ABAQUS/Standard 6.3 (ABAQUS manuals). This code adopts a macroscopic thermodynamic approach. The porous medium is considered as a multiphase material, and an effective stress principle is used to describe its behaviour. ABAQUS allows the value of bulk modulus of the mineral grains as an input parameter. In order to select an appropriate value for this low-permeability, low-porosity rock, the CTH team compared the ABACjus solution with Biot s (1941) analytical solution for ID consolidation in the form presented by Chan et al. 2003). 

ABSTRACT With the increase of mine exploitation depth and appliance widely of large-scale full-mechanized equipment, coal block gas emission has been one of the most gas effusion source. Base on unsteady diffusion theory and mass transmission fundamental, the mathematical and physical model of gas diffusion through coal particles with third type boundary condition was founded and its analytical solution was obtained by separate variableness method. The characteristics of gas through coal particles was analyzed according as mass transmission theory of porous material. The results show that the Biot s criterion of mass transmission can reflect the resistance characteristic of gas diffusion and the Fourier s criterion of mass transmission can represent the dynamic feature of diffusion field varying with time. 

Conservation of Linear Momentum, Effective Stress and Biot s Consolidation Theory... 

FIGURE 6.22 Ratio of Plateau level surface compression to midplane tension as function of Biot number. Curve 1, experimetal results 2, predictions by Indenbom s theory 3, predictions by Bartenev s theory. (Boguslavskii et al.,... 

To calculate the pore pressure response due to a volume source we use the Green s function based on the effective Biot theory. We write the coupled system of equations directly from the constitutive relations given by Biot (1962). These are the total stress of the isotropic porous medium, the stress in the porous fluid, the momentum balance equation for total stress, and the generalized Darcy s law. Following Parra (1991) and Boutin et al. (1987), the coupled system of differential equations in the... 

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