Fractures are added and removed from the fracture networks before further investigation is carried out on the various parameters of the network. The reason for doing this is explained in this section. The processes of including fractures from beyond the observation domain and the removal of isolated fractures have the effect of changing the fracture density. The notation i is introduced to indicate the initial fracture density before any manipulation of the
fracture network, whereas denotes the actual fracture density of the network being investigated. The MATLAB script for truncating fractures beyond the observation domain is recorded in Appendix B.
Since the parameter is an important parameter in the subsequent investigations, some work was carried out to address the relationship between i and . In this section the domain
size L is the main variable, and the effect that L has on for a given value of i is
investigated. As a starting point, networks with uniform fracture lengths are investigated. Fracture networks are generated stochastically, with i = 20 m-2 and fracture lengths l = 1 m,
similar to the distribution plotted in Figure 2.3a. The domain lengths L are varied between 2 to 22 m, in 2 m increments. Ten realisations were generated for each set of parameters.
A fracture from beyond the observation domain is included if part of it lies within the observation domain. For the case of uniform fracture lengths, no fractures with centres more than l/2 away from the boundary can reach the observation domain, meaning that a fracture seeding domain size of l + L is sufficient. After all the relevant fractures are included, isolated fractures can then be removed from the fracture networks using the methods described in previous sections. The resultant for each fracture network is obtained, and the relationship between and i (note that i is a constant) is plotted against L in Figure 2.4.
The main conclusion that can be drawn from Figure 2.4 is that the spread of increase as the size of L decrease, which is consistent with the concept of representative elementary volume (REV), in that once L exceeds a certain size, the spread will become negligible. The value of seems to converge to some value asymptotically as L tends to a large value. There will be further discussion regarding the REV in later chapters.
39 Figure 2.4. Fractional difference between and i, for different domain sizes L. For each L,
ten realisations are carried out. Each fracture has of length 1 m.
Now consider the two steps of manipulating the network (including all the relevant fractures from fractures with centres that lie outside of the simulation domain; then removing of isolated fractures from the fracture network) separately. It seems likely that those two processes will have different effects on , as a function of L.
The change of due to adding and removing of fractures, along with the total change, is plotted in Figure 2.5a, where the ensemble averages of the ten realisations are plotted. The figure either shows that each component is converging on a certain value as L increases, or is converging very slowly to 0. It is expected that as L tends to infinity boundary effects, such as adding relevant fractures from beyond the simulation domain, should become irrelevant. The data points are plotted differently in the next Figure 2.5b to verify this point.
0% 10% 20% 30% 40% 50% 60% 70% 0 2 4 6 8 10 12 14 16 18 20 22 24 % change from i (20 m -2) L, m
40 (a)
(b)
Figure 2.5. Change in relative to i due to adding fractures from beyond the domain,
removing of isolated fractures, and the sum of those changes, for a network of uniform-length fractures. This is plotted against (a) domain length and (b) length ratio between fracture length and domain length.
-30% -10% 10% 30% 50% 70% 0 2 4 6 8 10 12 14 16 18 20 22 24 % change from i (20 m -2) L, m Adding Fracture Removing Fractures Total Change -30% -20% -10% 0% 10% 20% 30% 40% 50% 60% 70% 0 0.1 0.2 0.3 0.4 0.5 0.6 % change from i (20 m -2) l/L Add Fractures Remove Fractures Total Change
41 In Figure 2.5b, the change from i is plotted against the length ratio l/L instead. All the
lines are now straight lines through the origin, showing clearly that as L tends to infinity (as l/L tends to 0), the boundary effect of adding fractures become insignificant. It is interesting to note that the change due to removing isolated fractures also become insignificant for large L. It was initially thought that the number of isolated fracture in a fracture network is independent of boundary effect, and is purely a geometrical property of the fracture network. However, as seen in Figure 2.3a, the fracture networks currently chosen here are very well connected, and in the absence of boundary effect seem to be entirely connected (i.e. no isolated fractures). With such a high fracture density, it is unlikely for a 1 m long fracture not to hit another fracture. In this scenario, isolated fractures arise when fractures, either from within or without the domain, are truncated by the boundary. The gradients of the three lines, in descending order, are 1.31, 0.89, and -0.42 (dimensionless).
This exercise is repeated with a different set of fracture networks. All the other parameters are the same as above, but instead of uniform fracture lengths of 1 m, fracture lengths here are distributed by a lognormal distribution with mean length of 1 m and lognormal standard deviation ln = 1, as in Figure 2.3b. A domain size must be defined for fracture seeding since the fracture lengths are no longer constant. Using the justification given in the previous section, the fracture seeding domain length is set to 40 m.
As before, the change in due to including fractures from beyond the observation domain is directly proportional to l/L, and become insignificant for large L. Here, l refers to the average length. The change in due to removing isolated fractures also varies linearly with l/L, but even when L is infinite, 10% of the fractures are still removed (shown by where the line representing fracture removal intercepts with the axis in Figure 2.6). This residual 10% is due to the shorter unconnected fractures present, because of the length distribution of fractures. The gradients of the three lines, in descending order, are 1.27, 0.84, and -0.42 (dimensionless).
It can be concluded that the current method of modifying the fracture networks changes
, and three kinds of change are identified: the inclusion of fractures with centres beyond the simulation domain, isolated fractures created by boundary effects, and isolated fractures due to the fracture distribution itself. The first two effects are dependent on the domain size, and disappear when the domain become large, whereas the last effect is independent of the domain size.
42
Figure 2.6. Change in relative to i due to adding fractures from beyond the domain,
removing of isolated fractures, and the sum of those changes, for a fracture network with a lognormal length distribution, as a function of the ratio between fracture length and domain length.