Hydraulics apertures

Further improvements of the model may incorporate different aspects such as the influence of the formation of infilling material due to shear displacements between crack interfaces. Lee and Cho (2002) have investigated the variation of intrinsic permeability when fractured rocks (granite and marble) are subjected to normal and shear deformations. This work shows that mechanical and hydraulic apertures are not always in linear trend. For instance, deviation takes place for large shear strains due to formation of infilling materials. The overall effect is that permeability was bounded by a maximum value. This effect may justify the inclusion in the model of a maximum hydraulic aperture. 

For the hypothetical case presented in this analysis we assume bh = 0.85x by, where bi, is the hydraulic aperture defined from the parallel plate flow relationship... 

To test the significance of the calculated initial mechanical apertures, a constant initial aperture of 77 pm was used and the simulations repeated. The results produced different results to those for the calculated hydraulic apertures but the trend was similar (Table 3). Thus, the initial mechanical apertures emerge to have an impact on the resulting hydraulic apertures but, for the results presented, the significance appears to be less than the impact of the variation of the mechanical properties. From these results the importance of the mechanical properties and their spatial distribution in the rock mass to the estimation of hydraulic aperture appears to be strong. 

Figure 7. Cumulative distributions of hydraulic aperture for one DFN for different depth below ground level.

The effect of increasing stresses on the hydraulic aperture has been tested by simulating increasing depth of burial of the DFN for the median fracture frequency case. The calculated median hydraulic aperture was 23.7 pm for a burial depth of 50 m... 

Further the results indicate that changes to the distribution of the hydraulic apertures progressively decrease with increasing depth (Figure 7). There is almost no difference between the calculated aperture distributions at 750 m and at 1000 m depth a residual hydraulic aperture distribution is achieved at around 750 m depth. The anisotropy in the aperture distribution is removed between 250 m and 500 m depth. Above 250 m depth the anisotropy can have a substantial impact on the permeability and the principal direction, which is further discussed in detail in Blum et al. (2003). 

Finally, the sensitivity of fracture aperture to block size and fracture length was analysed. Fracture networks with domain sizes of 5 m x 5 m and 10 mx 10 m revealed differences in median hydraulic apertures of less than 0.5 pm. Even with a block size of 15mxl5m and effectively infinitely long fractures, the change in median hydraulic aperture remained less than I pm. It can be concluded that the size of the REV determined for flow only is also suitable for mechanical calculations, and that, for the assumed spatial distribution and orientation of fracturing, fracture length has only a minor impact on the hydraulic aperture distribution. 

Two models are presented and compared in this paper (1) a homogeneous hydraulic base case assuming constant hydraulic apertures throughout the model domain and (2) a non-homogeneous hydro-mechanical base case with hydraulic apertures determined from analysis of HM coupling... 

Figure 2. Hydraulic conductivity ellipses for the three rock formations (network size = 10 m X10 m, hydraulic aperture (a/,) = 50 jjm, X- and y-axis = k, in m/s).

JCS) values were used to represent the range of mechanical properties observed in all three formations (Formation 1 see Blum et al. 2(X)3 Formation 2 JRC (/- ) = 4.28, 5.98, 4.18, 2.29 JCS (1-4) = 39.3. 31.9, 90.9, 43.1 in MPa and the unchanged uniaxial compressive strength for all 4 cases UCS = 120.0 MPa Fault Zone JRC = 4.22, JCS = 105.9 MPa and the UCS = 128.4 MPa). In case of Formation 2 only four pairs were available, thus the entire data set is provided here. Stress conditions corresponding to five depths were also applied to the DFN. Table 1 summarizes the hydro-mechanical modelling results in terms of the median hydraulic apertures. Values range between 0.3 pm and 180.7 pm. 

For the hydraulic base case, constant hydraulic apertures and fracture densities were considered. Homogeneous hydraulic conductivity tensors and porosities were applied to each formation. The case with a constant hydraulic aperture of 10 im and medium fracture density for all formations is illustrated in Figure 5, where the streamlines and the particle travel times are shown. The time of travel between each marker is 1000 days. Owing to the low effective porosity values and the relatively high fracture density, particle travel times through the host rock are fast. The mean particle travel time from the repository to the seabed is only 123 years, when a constant hydraulic aperture of 10 pm is used. 

The results for the particle travel times for various constant hydraulic apertures and fracture densities from the repository to the seabed are presented graphically in Figure 4. For the low fracture density case, the hydraulic conductivity estimated for a block size of 25 m x 25 m has been used even though this size does not correspond to the REV, which is estimated to be greater than 100 m x 100 m. This block size does allow, however, a calculation of the mechanical closure of the fracture apertures for the HM case and therefore a comparison of the results between the two cases for low-density conditions. The results of the continuum model based on constant hydraulic apertures display very rapid mean particle travel times from the repository to the seabed. For example, for the low and high fracture density networks adopting a constant hydraulic aperture of 10 pm, particle travel times from the repository to the seabed are 580 years and 106 years respectively. A doubling of the aperture increases the conductivity by a factor of... 

For the HM base case the mean mechanical properties have been used to calculate the hydraulic aperture distributions over the depth of the model. The continuum model and the applied methodology for the HM coupling in fractured rock does not allow the modelling of a fully HM-coupled system, hence the HM-modified hydraulic conductivity tensors were calculated at the mid point values of several depth ranges (Table 1). The results were assigned uniformly to the formation within each depth range (25m=>0m-50 m, 75 m => 50 m - 100 m, 175 m => 100 m -250 m, 375 m => 250 m - 500 m and 750 m 500 m - 1000 m). The variation in the calculated aperture values decreases as depth increases, which allows for the larger depth bands at the base of the model. 

The result for the HM base case is shown in Figure 6. The streamlines reveal a quite different geometry to the H base case. The impact of the anisotropy of Formation 2 is apparent. The increase to 4,860 years for the particle travel time from the repository to the seabed from the H base case is predominantly caused by the tight upper Formation 2 with an average hydraulic aperture for the entire formation of around 1.0 pm. The low-conductivity upper formation significantly reduces the total flow through the whole system, which consequently lowers the fluid velocities through the host rock. 

Figure 5. Hypothetical host rock with streamlines and particle travel times for the H base case (medium-density DFN with constant hydraulic aperture of 10 fjmj.

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). 

Porosities The unstressed original porosities (due to mechanical and hydraulic apertures) attributed to unstressed fractures are given by ... 

The hydraulic aperture used for the real joint is supposed to be 6.5 10 m. We have considered a hydraulic aperture of lO" m for the fictitious joint. We have checked that the equivalent hydraulic permeability tensor computed considering a fictitious joint network is 5 orders of magnitude smaller than if we consider a real joint network. 

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). 

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... 

The obtained fracture normal stresses, a (MPa), were related to hydraulic apertures, b (pm), using an approximate empirical relationship based on the laboratory loading-unloading tests of core data in Equation 1. These hydraulic apertures, at 100 years of heating, were in turn related to fracture transmissivity, using the cubic law , and compared to the original, in-situ, fracture transmissivity. 

The evolving change in fracture aperture, under a constant effective stress of 3.5 MPa, is evaluated from the recorded flow-rates and differential pressures, as shown in Figure 2. The initial aperture of 12.3 pm at 20°C, falls to 11.4 pm with the initiation of the test. This closure is interpreted as the minor crushing of asperities and interstitial propping grains, as the fracture seats. Heating the sample to 80°C results in a sharp decline in effective hydraulic aperture that ultimately asymptotes to 8.3 pm at 75 hours. 

Figure 2. Change in fracture aperture with test duration. Hydraulic aperture evaluated from measured steady flux and axial pressure differential. Scan sequences, temperature transitions, brief test interruptions and aqueous...

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