Biot’s model

To get better the understanding of the system, the mechanical behaviour of the clay has been taken into account. The initial conditions are null total stresses everywhere. So, in-situ mechanical stresses are not taken into account. The results of our THM calculation show only stresses induced by thermal-hydro-mechanical couplings. The contact between the EB and the canister is once again supposed to be perfect, so that no radial displacement of the clay is allowed at that boundary. Biot s poroelastic model is chosen to represent clay behaviour. It takes partial saturation into account via an equivalent pressure which includes capillary effects, involving both gas and liquid, Dangla (1998). Biot s model is added as fourth equation to the system. The associated main variable is total stress state. The couplings with thermal-hydraulics behaviour are introduced by... 

Although their boundary conditions are not totally satisfying, Ci , Ci and C calculations have been performed. A void of 1.7 cm is sufficient to allow free displacement of the EB during the whole saturation process. Comparing the five calculations and regarding the model, final values of total and effective stress are a linear function of radial displacement of the EB near the canister. This point is shown on figure 2. It is a consequence of the linearity of poroelastic Biot s model. 

Gassmann s model assumes no relative motion between the rock skeleton and the fluid (no pressure gradient) during the pass of a wave (low-frequency case). Biot s model (Biot, 1956a,b, 1962) considers a relative fluid motion of rock skeleton versus fluid. With this step combined with Gassmann s material parameters, fluid viscosity and hydraulic permeability k must be implemented. The implementation of viscous flow results in ... 

Geertsma and Smith (1961) derived an approximate solution for velocity equations in Biot s model and expressed deformation properties in terms of compressional moduli for practical purpose (see also Bourbie et al., 1987) Compressional wave velocity as a fimction of frequency / is... 

An extended overview of the various theoretical concepts and their applications is given by Mavko et al. (1998). The different types of fluid motion in the pore space are discussed and developed with particular emphasis. Gassmann s model considers no fluid flow (static case), whereas Biot s model assumes a global flow . Murphy (1982, 1984) and Mavko and Jizba (1991)... 

A chiral object and its mirror image are enantiomorphous, and they are each other s enantiomorphs. Louis Pasteur (Figure 2-37) was the first who suggested that molecules can be chiral. In his famous experiment in 1848, he recrystallized a salt of tartaric acid and obtained two kinds of small crystals which were mirror images of each other as seen by Pasteur s models in Figure 2-38 preserved at Institut Pasteur at Paris. Originally Pasteur may have been motivated to make these large-scale models because Jean Baptiste Biot, the discoverer of optical activity had very poor vision by the time of Pasteur s discovery [42], Pasteur demonstrated chirality to Biot, who was visibly affected... 

In this section first Biot s viscoelastic model is summarized which simulates high- and low-frequency wave propagation in water-saturated sediments by computing phase velocity and attenuation curves. Subsequently, analysis teehniques are introduced whieh derive P-wave veloeities and attenuation eoefficients from ultrasonie signals transmitted radially across sediment eores. Additional physieal properties like... 

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... 

Fig. 2.16 Comparison of P-wave attenuation and velocity dispersion data derived from ultrasonic transmission seismograms with theoretical curves based on Biot-Stoll s model for six traces of the turbidite layer of gravity core GeoB1510-2. Permeabilities vary in the model curves according to constant ratios K/a = 0.030, 0.010, 0.003 (K = permeability, a = pore size parameter). The resulting permeabilities are given in each diagram. Modified after Breitzke et al. (1996).

Abstract The Canadian Nuclear Safety Commission (CNSC) used the finite element code FRACON to perform blind predictions of the FEBEX heater experiment. The FRACON code numerically solves the extended equations of Biot s poro-elasticity. The rock was assumed to be linearly elastic, however, the poro-elastic coefficients of variably saturated bentonite were expressed as functions of net stress and void ratio using the state surface equation obtained from suction-controlled oedometer tests. In this paper, we will summarize our approach and predictive results for the Thermo-Hydro-Mechanical response of the bentonite. It is shown that the model correctly predicts drying of the bentonite near the heaters and re-saturation near the rock interface. The evolution of temperature and the heater thermal output were reasonably well predicted by the model. The trends in the total stresses developed in the bentonite were also correctly predicted, however the absolute values were underestimated probably due to the neglect of pore pressure build-up in the rock mass. 

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 ... 

In this analysis, the transient tunnelling process was simulated in a two-dimensional section across the FEBEX tunnel. A coupled HM analysis was conducted using a Biot (1941) model with Young s modulus E = 24.68 GPa, Poisson s ratio v = 0.37, Biot s coefficient b = 1 (Terzaghi assumption), and a Biot s modulus M equal to infinity (the storage phenomena is caused only by skeleton strain). The hydraulic permeability was set to 7xl0 m after model calibration against observed water inflow into the FEBEX tunnel. 

The numerical procedure is to impose a fluid pressure built up from P to P + AP within the model, assuming no mechanical deformation at the boundaries. In this case the Biot s equations (4 5) can be rewritten as follows ... 

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. 

Direct time integration is performed using the Newmark total equilibrium method (Carr 2007). MATLAB codes were developed for the time domain analysis incorporating both classical Rayleigh viscous and non-viscous damping models. In the case of non-viscous damping, a single exponential model called Biot s relaxation function is used as the Kernel function. Biot s relaxation function is of the form... 

To describe the spatially distributed strain of the PVME gel system, the THB equation of motion [5] can be used. Even if is an oversimplified version of the Biot s poroelastic model [7], it adequately describes the diffusion kinetics of gel matrices [3, 5-7] and reads ... 

S. Cros, C. Herve du Penhoat, N. Bouchemal, H. Ohassan, A. Imberty, and S. Perez, Int. ]. Biot. Macromol., 14,313 (1992). Solution Conformation of a Pectin Fragment Disaccharide Using Molecular Modelling and Nuclear Magnetic Resonance. 

R438 D. G. Lynn and S. C. Meredith, Review Model Peptides and the Physicochemical Approach to P-Amyloids , J. Struct. Biot, 2000, 130, 153... 

Ravi, G., Viswanathan, S., Gupta, S. K and Ray, M. B. (2003). Multi-objective optimization of venture scrubbers using a three-dimensional model for collection efficiency, J. Chem. Technol. Biot, 78, pp. 308-313. 

Variation of phase velocity over a range of solid-volume percentages are calculated for the above models for two types of particles, i.e., glass beads and kaolins (with acoustic impedances of 21.12 x 10s and 10.66 x 105 g/cm2-s, respectively). Calculated results are shown in Fig. 5-27 for glass beads and in Fig. 5-28 for kaolins. All models, except Biot-2, show decreasing phase velocity at lower volume fractions, then increasing phase velocity at higher volume fractions. 

In what follows the theoretical background of the most common physical properties and their measuring tools are described. Examples for the wet bulk density and porosity can be found in Section 2.2. For the acoustic and elastic parameters first the main aspects of Biot-Stoll s viscoelastic model which computes P- and S-wave velocities and attenuations for given sediment parameters (Biot 1956a, b, Stoll 1974, 1977, 1989) are summarized. Subsequently, analysis methods are described to derive these parameters from transmission seismograms recorded on sediment cores, to compute additional properties like elastic moduli and to derive the permeability as a related parameter by an inversion scheme (Sect. 2.4). 

Table 2.2 Physical properties of sediment grains, pore fluid and sediment frame used for the computation of attenuation and phase velocity curves according to Biot-Stoll s sediment model (Fig. 2.12).

Suppose that a current electrode is placed in a uniform conducting medium so that the distribution of currents possesses the spherical symmetry (Fig. 1.32a). It is then a simple matter to realize that the magnetic field is zero everywhere in the medium. This follows directly from Biot-Savart law and the symmetry of the model. In other words, one can always find two current elements which are located symmetrically with respect to the observation point and of which the magnetic field differ by sign only. Let us notice that Ampere s law does not apply here because the current lines are not closed. 

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