Equivalent permeability

After the application of the various stress boundary conditions, differential hydraulic heads are applied to opposite model boundaries to introduce fluid flow through the models, and the equivalent permeability of the model is calculated as the function of the applied stress boundary conditions. 

The two-dimensional distinct element code, UDEC (Itasca, 2000) is applied for the modeling of mechanical and hydraulic behaviours. UDEC simulates the response of a fractured rock mass represented as an assemblage of discrete deformable blocks, subjected to the mechanical stress and hydraulic pressure boundary condition. Numerical experiment consists of 1) generation of a Discrete Fracture Network (DFN) as a geometric model, 2) application of various stress conditions, and 3) application of fluid boundary condition and calculation of equivalent permeability. 

Figure 5. Application of stress boundary condition and calculation of equivalent permeability in x- and y- directions.

A series of numerical experiments are conducted to calculate the equivalent permeability of the model under various stress conditions. The flow through fractures is calculated by the Cubic law and blocks are treated as impermeable medium. Mass conservation is assured at each fracture intersection and in domains between them, through an iterative process. Lateral sides of the model are set impermeable and x- and y-directional permeability with the specified hydraulic pressure boundary conditions are calculated by the following equations. 

Abstract A new upscaling method has been developed using 3D numerical tools (RESOBLOK 3DEC). This method has been successfully compared with standard analytical approaches in the case of a simple fracture network. This method has been applied to determine the equivalent permeability, stiffness and Biot tensor of a real fracture rock-mass at different scales. The effects of the fracture network properties and of the state of stress on the result have been investigated. 

A new upscaling method has been proposed by INERIS to determine the equivalent hydro mechanical properties of a fractured rock-mass. This method is based on the 3D numerical simulations of the behaviour of a "sample" of fractured rock-mass submitted to different hydromechanical boundary conditions. The simulations are defined in order to determine the equivalent permeability, stiffness and Biot tensor of a fractured rock-mass. 

After each computation, the equivalent vectorial flowrate is averaged from the flowrate value at different locations of the fracture network. The different terms of the equivalent permeability tensor are then computed from equation (1). 

The objective of this section is to compute, with the numerical approach already presented and validated, the equivalent permeability tensor of the fracture network of the WP3 case. The set of computation presented here concerned the fracture network of formation 1 (computation has also been done for formation 2 and the fault zone). 

We have assumed that the hydraulic apertures were constant and equal to 6,5 10 m. The equivalent permeability tensor is given below at 2 m scale for the fracture network of formation 1 and considering a fracture length threshold of 0.5 m ... 

For HM computation, 3DEC consider a relation between the hydraulic aperture "a" and the mechanical aperture "u" that can be written a = ao + Au, where ao is the zero stress aperture. A maximum and residual aperture is considered for numerical stability reason. We have assumed that ama, = ao = 6.5 lO m and that a s = 1.8 lO m. Figure 6 represent the evolution of the diagonal terms of the equivalent permeability tensor with stress applied on the model boundaries (at 2 m scale). 

Figure 6 Equivalent permeability tensor variation with stress.

We can see a decreasing of the permeability when the applied stress increases. This decreasing is much faster if the normal joint stiffness is assumed to be smaller (4.43 10 Pa/m instead of 4.43 lO" Pa/m). It can be notice that there is a residual equivalent permeability (around 10 m/s) that has to be related to the residual hydraulic aperture. That residual permeability is reached for o > 20 MPa (= 800 m) if the joint normal stiffness... 

In this work we have assumed no relation between the joint stiffness and the stress or with the fracture length. This assumption could be revisited later on. We have also considered an isotropic state of stress. Others stress conditions could be applied and would probably involve other kinds of evolution of the equivalent permeability tensor. 

The equivalent permeability tensor K j has been computed at 2 m scale considering the real length of the joint (we have shown the importance to avoid artificial joint prolongations). The REV could not be estimated due to code limitation reasons. The relation between permeability and stress (considering an isotropic state of stress) has been determined and is highly dependent on the joint stiffness value. 

Two different definitions of water vapor permeability are used in the BS and ASTM standards, and this can cause confusion. BS 3177 and BS 2782 define the water vapor transmission rate as the mass transfer rate of water vapor per unit area (g m 24h) (see Section 1). but they call it "permeability," When the material is homogeneous and the transfer rate is inversely proportional to thickness. BS 3177 defines an "equivalent permeability" as the "permeability" multiplied by the thickness of the test piece in thousandths of an inch (g mil m 24h). 

For low wind speeds, the inclusion of information on the subjacent defects, due to either poor workmanship or to aging, that allow water to come into an apartment is currently under development. An equivalent permeability will be included in the program that will allow water leakage at low wind speeds. It is noteworthy though that water entering at low speeds is frequent in buildings in Florida so a consideration of this factor is important. 

Liu, Q.S. et al. 2011. Discrete element analysis of effect of stress on equivalent permeability of fractured rock mass. Chinese Journal of Rock Mechanics and Engineering 30(01) 176-183. 

The membrane thickness Aw may, of course, be incorporated into the pointwise diffusion coefficient D, or the equivalent permeability coefficient P to obtain the overall coefficient. 

With the equivalent permeability of the gap known, the flow in the gap can be calculated using the standard solution techniques for the RTM process, with the flow velocity now given by Darcy s law with K given by this equivalent permeability. For less computational effort, one can obtain fairly... 

Chen, R., Dong, C., Liang, Z., Zhang, C. and Wang, B., 2004. Flow modelling and simulation for vacuum assisted resin transfer molding process with the equivalent permeability method . Polymer Composites, 25, 146-164. 

While data from animal skin are abundant, some studies suggest that animal skin is more permeable than that of humans [93,94]. To date, there is no consensus on the circumstances producing equivalent permeability in animal and human skin, and animal data must await further validation of employed models. Human in vivo models are currently considered a gold standard for studying human dermal risk assessment. Human in vitro studies, compared to in vivo studies, may be advantageous in that multiple replicate experiments are more easily performed and toxic compounds may be safely studied. 

Similar upscaling techniques, motivated by the need to reduce grid block number, are important in practice. But the equivalent permeabilities within any reservoir will change if the reservoir is produced by different arrangements or patterns of wells, because the parallel and serial nature of the flow has changed. Upscaled quantities are not properties of the formation but are also related to the production method. However, several simulators compute fixed upscaled properties and use them in contrasting production scenarios. 

Figure 12.24 The effect of the residual strain (after unloading a high performance FRC with dense matrix) on the equivalent permeability to water (after Charron et al. [120]).

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