Conductivity tensor, total

In the two-medium treatment of the single-phase flow and heat transfer through porous media, no local thermal equilibrium is assumed between the fluid and solid phases, but it is assumed that each phase is continuous and represented with an appropriate effective total thermal conductivity. Then the thermal coupling between the phases is approached either by the examination of the microstructure (for simple geometries) or by empiricism. When empiricism is applied, simple two-equation (or two-medium) models that contain a modeling parameter hsf (called the interfacial convective heat transfer coefficient) are used. As is shown in the following sections, only those empirical treatments that contain not only As/but also the appropriate effective thermal conductivity tensors (for both phases) and the dispersion tensor (in the fluid-phase equation) are expected to give reasonably accurate predictions. 

For each block of 10 by 10 cells in the small scale model, a block conductivity tensor is determined by solving the flow equation (2) for a set of boundary conditions and then determining which is the best conductivity tensor that is able to match the total flow crossing the block at the small scale (Gomez-Hemandez and Joumel, 1994). The procedure is sketched in Figure 3. 

Van den Brule proposed a connection between the thermal conductivity tensor and the total stress tensor as follows ... 

Using an equation of equilibrium or motion, which determines the deformation of the solid skeleton, and (5.58), a system of differential equations for specifying the mean velocity v (i.e., the conventional consolidation problem) is achieved. Note that in (5.58)

total head excluding the velocity potential), k is the hydraulic conductivity tensor, p is the pore pressure of the fluid, g is the gravity constant, and is the datum potential. Thus by starting with the mass conservation laws for both fluid and solid phases, we can simultaneously obtain the diffusion equation and the seepage equation which includes a term that accounts for the volumetric deformation of the porous skeleton. 

S is time-independent because of the constancy of the total momentum. Under conditions of spherical sjmmetry, as in a fluid, the conductivity tensor reduces to a scalar, and... 

In the last formulae a — db + Ad is a conductivity distribution, for which we calculate the forward modeling operator variation 6d is the corresponding variation of the conductivity a, which is obviously equal to the variation of the anomalous conductivity, 6d = 6Ad. Tensors Gg jf are electric and magnetic Green s tensors calculated for the given conductivity a. Vector E in expressions (10.54) and (10.55) represents the total electric field, E = E -t-E for the given conductivity d. 

Strictly speaking, the shift given by (18.23) is the isotropic component of the total shift tensor. It applies directly to nuclei at crystalline sites of cubic symmetry. Otherwise, in general, the shift tensor possesses additional components, as discussed in section 1.2.2. For axially symmetric conduction electron distributions, the axial component of the Knight shift in simple metals is given by ... 

Here p is the fluid density, g is the gravitational acceleration, Cp is the heat capacity at constant pressure (per unit mass), and q is the heat flux (taken to be given by Fourier s law q ——XV where X is the thermal conductivity). It is convenient to split the (total) stress tensor n into two parts n=pS+x where p is an isotropic pressure, is the unit tensor, and t is the (extra) stress tensor, which is that part of the total stress tensor that vanishes when the fluid is at rest. 

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