Representative elementary volume

Sections 1.2.1 to 1.2.3 present the basic equations for groundwater flow through a representative elementary volume of the porous medium (Bear, 1972) under certain restrictive assumptions, such as isothermal and isochemical subsurface conditions and absence of changes in tectonic stress. 

Darcy s law (Darcy, 1856) is a phenomenological law that is valid for the viscous flow of a single-phase fluid (e.g. groundwater flow) through porous media in any direction. This basic law of fluid flow is a macroscopic law providing averaged descriptions of the actual microscopic flow behaviour of the fluids over some representative elementary volume of the porous medium. For isothermal and isochemical subsurface conditions, the law can be written as (Hubbert, 1953)... 

Provided that the following assumptions are met, the time-rate of change of groundwater mass storage can be related to changes in the density of the groundwater, in the porosity of the porous medium and in the vertical dimensions of the representative elementary volume of the porous medium, i.e. 

There are no time-changes in stress externally imposed on the representative elementary volume of porous medium ... 

The groundwater pressure in a representative elementary volume of the porous medium fixed in space, changes directly with the hydraulic head, i.e. dp = p, gdh and Equation 1.24 can be written as... 

A general expression for the conservation of solute mass for a certain chemical component in a representative elementary volume of water-saturated porous medium fixed and rigid in space is (e.g. Garven, 1985 Garven and Freeze, 1984a)... 

Volume averaging is a technique in which the fundamental equations are spatially averaged over a representative elementary volume (REV) of porous media.f ° This approach has provided insight into the relationship between fundamental physics and larger-scale behavior but is rarely used for studying transport in specific media in a deterministic sense. 

Spatially periodic porous media are made up of structural elements whose arrangement in space is completely described by a single unit cell (similar to the representative elementary volume concept of Bear, 1969), that is then repeated ad infinitum (Adler, 1992). The structural elements can be discrete voids in a continuous solid phase or vice versa. The simplest spatially periodic models are comprised... 

Bear (1969, 1972) developed a dispersive flow model based upon the idea of building a continuum at the mesoscopic scale by statistically averaging microscopic quantities over a representative elementary volume, defined with respect to porosity. Phis geometric model is an assemblage of randomly interconnected tubes... 

Numerical models have also been applied to simulate macrodispersion in spatially variable ksat fields (e.g., Thompson Gelhar, 1990 Moissis Wheeler, 1990 Wheatcraft et al., 1991). This Darcian approach requires averaging of flow over some Representative Elementary Volume (REV). 

Berkowitz, B., and C. Braester. 1991. Dispersion in sub-representative elementary volume fracture networks Percolation theory and random walk approaches. Water Resour. Res 27 3159-3164. Berkowitz, B., and R.P. Ewing. 1998. Percolation theory and network modeling applications in soil physics. Surv. Geophys. 19 23-72. 

For dealing with variability of soil properties at the larger scale, a continuum approach is implemented. Thereby a representative elementary volume (REV) is considered to exist and material properties related to flow and transport are defined at the centre of this REV. Thermodynamic principles related to conservation of mass and momentum are further applied on the REV to obtain governing flow and transport equations. The... 

Other than the particle dimension d, the porous medium has a system dimension L, which is generally much larger than d. There are cases where L is of the order d such as thin porous layers coated on the heat transfer surfaces. These systems with Lid = 0(1) are treated by the examination of the fluid flow and heat transfer through a small number of particles, a treatment we call direct simulation of the transport. In these treatments, no assumption is made about the existence of the local thermal equilibrium between the finite volumes of the phases. On the other hand, when Lid 1 and when the variation of temperature (or concentration) across d is negligible compared to that across L for both the solid and fluid phases, then we can assume that within a distance d both phases are in thermal equilibrium (local thermal equilibrium). When the solid matrix structure cannot be fully described by the prescription of solid-phase distribution over a distance d, then a representative elementary volume with a linear dimension larger than d is needed. We also have to extend the requirement of a negligible temperature (or concentration) variation to that over the linear dimension of the representa-... 

For the analysis of the macroscopic heat flow through heterogeneous media, the local volume-averaged (or effective) properties such as the effective thermal conductivity k) = k, are used. These local effective properties such as the heat capacity (pcp), thermal conductivity (k), and radiation absorption and scattering coefficients (a ) and (a,) need to be arrived at from the application of the first principles to the volume over which these local properties are averaged, that is, the representative elementary volume. 

Local Thermal Equilibrium. In principle, determination of the thermal conductivity of saturated porous media involves application of the point conduction (energy) equation to a point in the representative elementary volume of the matrix and the integration over this volume. In doing so, we realize that at the pore level there will be a difference A7 between the temperature at a point in the solid and in the fluid. Similarly, across the representative elementary volume, we have a maximum temperature difference ATt. However, we assume that... 

With this assumed negligible local temperature difference between the phases, we assume that within the local representative elementary volume V= Vf+ V the solid and fluid phases are in local thermal equilibrium. This is stated using the phase (or intrinsic) and both-phases volume averaged temperatures as... 

This is a semiheuristic volume-averaged treatment of the flow field. The experimental observations of Dybbs and Edwards [27] show that the macroscopic viscous shear stress diffusion and the flow development (convection) are significant only over a length scale of i from the vorticity generating boundary and the entrance boundary, respectively. However, Eq. 9.22 predicts these effects to be confined to distances of the order oi Km and KuDN, respectively. We note that Km is smaller than d. Then Eq. 9.22 predicts a macroscopic boundary-layer thickness, which is not only smaller than the representative elementary volume i when i d, but even smaller than the particle size. However, Eq. 9.22 allows estimation of these macroscopic length scales and shows that for most practical cases, the Darcy law (or the Ergun extension) is sufficient. 

As before, a knowledge of b, the transformation vector, leads to the determination of 1C and Dd. The vector b, which is a function of position only, has a magnitude of the order of the representative elementary volume i and is determined from the differential equation and boundary conditions for T and its determination is discussed by Kaviany [9],... 

Inclusion of the pore-level (or particle-based) hydrodynamics along with the appropriate volume averaging allows for the inclusion of the local variation of D and D into the energy equations. In principle, these variations can only be included if the change from the bulk value to zero at the surface takes place over several representative elementary volumes. Otherwise it will not be in accord with the volume averaging. 

It is assumed that the particle size is much smaller than the linear size of the system. Then, the radiative properties are FIGURE 9.7 A schematic of the coordinate system. averaged over a representative elementary volume with a... 

Point scattering occurs, that is, the distance between the particles is large compared to their size. Thus, a representative elementary volume containing many particles can be found in which there is no multiple scattering and each particle scatters as if it were alone. Then this small volume can be treated as a single scattering volume. This leads to a limit on the porosity. 

For the transient behavior, it is assumed that the penetration depth (in the fluid and solid phases) is larger than the linear dimension of the representative elementary volume. This is required in order to volume-average over the representative elementary volume while sat-... 

As with the single-phase flows, the fractions of the representative elementary volume occupied by the liquid and gas phases are... 

Saturation. This is the extent to which the wetting phase occupies (averaged over the representative elementary volume) the pore space. At very low saturations the wetting phase becomes disconnected (or immobile). At very high saturations, the nonwetting phase becomes disconnected. 

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