A set of deterministic fractal-fracture networks at different iterations are generated as described in Roy et al. (2007). These are essentially a set of Sierpinski lattices that comprise self-similar line segments as shown in Fig. 1. Three parameters are used in generating these lattices: the scale factor, b, the iteration level, i, and the initial number of un-fractured blocks, n. As seen in Fig. 1, a sequence of hierarchically fractured patterns are generated by selecting the spatial locations of the fractured and un-fractured blocks in a deterministic manner. In our particular case, Fig. 1a is the initiator that divides the entire area into b2=22=4 blocks and Fig. 1b is the generator that shows how one such block (thus n=1), the upper left corner, is left un-fractured and scaled down versions of the initiator is used for populating the remaining three blocks. Since this is a deterministic fractal, at each iteration starting from i=2 (Fig. 1c), the un-fractured block is placed at the exact location (upper-left corner in this case) as in the generator.

All the models described here have the same set of values for b and n where, b=2, n=1, the theoretical box-counting fractal dimension, D=log⁡(b2-n)/log⁡b=log⁡3/log⁡2=1.585. Since D is independent of the iteration, i, all of the patterns have the exact same theoretical fractal dimension. The empirical box-counting values of this dimension as documented in Roy et al. (2007) stabilize at i=3 and closely approximates the theoretical D value as seen in Fig. 2.

From a visual inspection of Fig. 1 and especially, Fig. 3 it is apparent that as the number of iterations, i is increased from 1 to 2 and 4 to 6 respectively, the square domain which remains at the same size, is populated with more and more fractures. Hence, while the patterns still have the same fractal dimension, the fracture intensity increases with increase in iteration. This leads to a possible increase in the scale-dependent clustering attribute of the fracture networks as well as that in their topological connectivity and hence, the fluid recovery. These parameters are discussed in some detail in the next sections.

Lacunarity is a parameter that characterizes the distribution of spaces or gaps in a pattern as a function of scale and quantifies the degree of clustering at a given spatial resolution. In essence, lacunarity is a scale-dependent measure of heterogeneity in datasets (Plotnick et al., 1993). Quantifying lacunarity as a function of scale can be achieved by using the gliding-box algorithm (Allan and Cloitre, 1991; Plotnick et al., 1996). This algorithm slides a window or box of a given length, r, translated in increments of a chosen unit length across the pattern. The box-size, r, is generally a multiple of this assigned unit length. The interrogator box searches for occupied sites in the pattern at each step and counts them. The total number of steps, N(r), required to cover the entire pattern is given by: 1Nr=rt-r+1E Here, E is the Euclidean dimension of the pattern and rt is the total length of the set. The first and second moments of the distribution of the number of occupied sites at each step, Z1(r), and Z2(r) respectively, are given by (Plotnick et al., 1996): 1aZ1r=s(r)1bZ2r=ss2r+[s(r)]2 Here, s(r) and ss2(r) are the arithmetic mean and variance respectively. The lacunarity is then defined as a function of box-size, L(r), as (Allain and Cloitre, 1991): 2Lr=Z2r/[Z1r]2 A single value for lacunarity, 〈L〉 is computed from the values of lacunarity obtained at different scales, r by log transforming the parameters and applying the following Eq. (3) as described in Roy and Perfect (2014): 3〈L〉=∑[log⁡L(r)⋅log⁡r]∑log⁡r

The topological connectivity is a measure of intersections of the fracture segments in a fracture network. It can be computed through a combination of fracture intersections (X-nodes) and abutments or splays (Y-nodes). Barton and Hsieh (1989) invoked the use of ternary diagrams to characterize the connectivity, where the relative frequencies of all three node types: intersections (X-nodes), abutments (Y-nodes) and isolated line tips (I-nodes) present in a fracture network system are plotted as a point. Figure 4 illustrates the X, Y and I nodes present in a fracture network. The connectivity of a fracture network is a useful tool to forecast the fluid flow and transport characteristics in a fractured reservoir.

Manzocchi (2002) showed that the connectivity can also be expressed in terms of a single parameter, n, defined by Eq. (4): 4n=[41-PI/(1-PX)] Where, PI = Proportion of I (isolated) nodes, PX = Proportion of X (intersection) nodes, PY = Proportion of Y (abutment) nodes, PX+PY+PI=1.

Evaluating the flow response is key in understanding the importance of fractures in any given network, especially for hydrogeologists and petroleum engineers. The deterministic fractal-fracture networks are modeled as a grid of 500×500×1 cells. This is a fracture-continuum model similar to those described by Langevin (2003), Neuman (2005), Svensson (2001) and Tsang et al. (1996), by means of which discrete fractures are converted into permeability structures on a model grid. This is done in a manner such that each cell gets a characteristic porosity and permeability value depending on whether or not it is occupied by a fracture.

In our research, a cell occupied by a fracture is assigned porosity and permeability of 95 % and 106 md respectively, whereas all the others (“matrix” cells) have values 5 % and 102 md. These values are chosen so as to ensure that flow occurs only through the fractures. A pair of injection and production wells are placed at diagonally opposite corners of the grid i.e., at (1, 1) and (500, 500) such that there is a total areal swept of the fluids present in the modelled rock volume. These models are then flow simulated in Trace3D, a Darcy-based streamline simulator, at constant boundary conditions with reservoir pressure of 2480 psi and an injection rate of 500 bbl d-1 for a period of 1000 d. The overall fluid recovery values thus obtained from each deterministic fractal-fracture model is used for parametric characterization of the flow behavior.

The lacunarity parameter, 〈L〉, which is a measure of scale dependent clustering, is calculated for the set of eight deterministic fractal-fracture patterns (Table A1 in the Appendix) and plotted as shown in Fig. 5. Although all the fractal-fracture patterns have the same theoretical fractal dimension, the lacunarity value increases with increase in the number of iterations. The lacunarity, which is a measure of fracture clustering thus increases because in going from one iteration to the next higher one, more and more fractures are added in populating the same domain which increases the fracture intensity.

The connectivity values of the set of eight deterministic fractal-fracture patterns are computed (Table A1) and plotted as shown in Fig. 6. Similar to the lacunarity parameter, the connectivity also increases with the increase in the number of iterations and somewhat appears to “stabilize” at higher iterations. However, at least until i=8, the connectivity actually does not “stabilize”. This behaviour is different when compared with Fig. 2 in Sect. 2.1 where the empirical fractal dimension of the same patterns stabilizes at iteration i = 3. This points to the fact that while the empirical fractal dimension stabilizes around its theoretical value (all patterns have the same dimension), it is not so in the case of the connectivity. This is because it increases with every iteration as does the intensity of these fractal-fracture networks.

The differences in the connectivity of these fractal-fracture networks generated at different iterations can also be demonstrated in the form of a ternary diagram (Fig. 7) that plots the relative proportions of I, X and Y nodes of a network as a point representing the connectivity of that network. As seen in the figure, an increase in the iteration results in the formation of a higher relative number of X and Y nodes as compared to the number of I nodes.

Such an increase in the X and Y nodes relative to I, hence, the connectivity of fractal-fracture networks, at increasing iterations is possibly related to the fact that fracture intensity also increases as i goes from 1 to 8. It has been shown previously that fracture clustering (lacunarity) also increases with iteration as more and more fractures populate the domain whose size remains fixed.

Lacunarity can therefore, be implemented in investigating possible relationships between clustering and connectivity by cross-plotting the two parameters. Figure 8 shows an appreciable positive correlation between these two thus indicating that instead of the fractal dimension which remains unchanged at different iterations (Fig. 2), it is the lacunarity parameter which can be used as a proxy for connectivity. From a computational point of view, the prime advantage of using lacunarity over connectivity lies in the fact that it does not require maps to be presented in vector formats, in other words, a simple bitmap figure file (raster data) can be used for analysis.

The fluid recovery obtained by flow simulation of deterministic fractal-fracture models increases with increase in the number of iterations and is positively correlated to lacunarity as seen in Fig. 9. This can possibly be attributed to the fact that recovery is directly controlled by connectivity. The latter in turn, is related to fracture clustering which is quantified by lacunarity. As the number of iterations increase, more fractures are added that increases lacunarity which leads to better connectivity that enhances transport of fluid from the injection point to the production point of the fractured domain. This observation thus indicates that instead of the fractal dimension which remains unchanged at different iterations (Fig. 2), the lacunarity parameter is better suited to represent fluid recovery of deterministic fractal-fracture networks.