Fluid flow and transport characteristics in fractured media are controlled by connectivity and conductivity of fracture networks. The effect of connectivity on flow behavior of fracture networks is well studied by many researchers (Robinson, 1983; Berkowitz and Balberg, 1993; Berkowitz, 1995; de Dreuzy et al., 2002; Sahu and Roy, 2020). However, when it comes to simulating flow in such networks, an average value of aperture is assigned in most cases (Min et al., 2004; Hardebol et al., 2015; Chen, 2020) even though it has been recognized that such assumption of constant fracture aperture is an oversimplification (Odling and Roden, 1997). Aperture is the void space between two surfaces of a fracture. They vary in size from micrometres to centimetres as they result not only from mechanical misfit of fracture walls but also from chemical dissolution, and overburden pressure (Bonnet et al., 2001). Hence, the characterization of aperture distribution is important as it controls the rock mass hydro-mechanical properties of fracture networks (Kaulatilake et al., 2006). Neretnieks et al. (1982) experimentally illustrated that flow channelling occurs in individual fractures and mostly through highly preferential pathways in fracture networks. Cacas et al. (1990a, b) showed that broad distribution of fracture conductivities (apertures) is the main cause of high degree of flow channelling. Berkowitz (2002) pointed out that even a well-connected fracture network can display sparse preferential flow paths if the distribution of fracture aperture is significantly wide. Klepikova et al. (2014) showed that inversion of borehole temperature profiles under predefined pumping conditions can be applied for the detection of main permeable fractures. Although this approach of using temperature tomography does not identify all flowing fractures, it allows the detection of the most transmissive ones. A study on fracture insertion and deletion based on DFN transdimensional inversion using reversible jump Markov Chain Monte Carlo (rjMCMC) algorithm was carried out by Somogyvari et al. (2017). One interesting conclusion of this research is that no breakthrough is observed in fractures with smaller apertures within the simulated time. However, there is limited information in the literature that help researchers in realistically assigning a distribution of apertures in fracture network models.

In many cases, such fracture aperture distribution is observed to follow a power law in naturally fractured reservoirs (Bour and Davy, 1997) and hence, the occurrence of smaller fractures with narrow aperture is more frequent than fractures of longer length with broader aperture (Somogyvari et al., 2017). The fluid flow behavior in each individual fracture element is controlled by the hydraulic aperture and length of the fractures (Bour and Davy, 1997; de Dreuzy et al., 2002; Darcel et al., 2003; Baghbanan and Jing, 2007). Therefore, the characterization of aperture distribution in fracture networks and its effect on flow behaviour is crucial in identifying the dominant fractures that serve as preferential pathways in fracture networks.

This research investigates how aperture distribution affects the connectivity and fluid flow in fractal-fracture networks and whether fractures with smaller apertures contribute significantly in terms of the overall flow recovery. One of the objectives of this study is to aid modellers in their decision-making process on how much of information (data) should be incorporated in building fracture models that are geologically realistic, i.e., the smallest apertures that can be eliminated from models without significantly affecting the flow while being computationally less expensive. In order to meet this objective, dominant sub-networks have been identified that are generated by the elimination of smaller fractures from a parent fractal-fracture network but nonetheless, have very similar fluid recovery to that of the latter. Fractures are assumed to have clean openings, that is, those that have the entire fracture width available for flow. In other words, we consider the hydraulic aperture (Snow, 1969; Neuzil and Tracy, 1981; Barton et al., 1985).

The connectivity and flow behavior of a set of synthetic fractal-fracture networks, both deterministic and random (Roy et al., 2007), and their dominant sub-networks, where apertures scale in accordance to their position in the hierarchy of a fractal model, are studied. A dominant sub-network is obtained by systemically eliminating fractures with smaller aperture from the parent network (Gong and Rossen, 2017). In order to simulate flow, a fracture continuum (FC) model is considered which is similar to those used by Langevin (2003), Neuman (2005), Svensson (2001a, b) and Tsang et al. (1996). Cells in the FC model are assigned porosity and permeability values depending on whether they represent fracture or matrix. A high porosity value is assigned to all the cells that represent fractures. Permeability values of “fracture cells” are computed based on the cubic law (aperture) and there is nearly no matrix porosity or permeability. Trace3D, a Darcy based streamline simulator (Datta-Gupta and King, 2007) is used for modelling fluid flow in the FC models. The concept thus developed, is then applied to a natural fracture map (Odling, 1997) and its dominant sub-network to demonstrate that in case of real data, fractures with smaller apertures may be excluded from the flow model without significantly affecting recovery. The dominant sub-network in this case has been generated from a lower resolution map of original fracture map which captured only the thick and longer fractures.

Fluid recovery can serve as a parametric estimation of flow characteristics in any fracture network. Figure 5 shows the recovery curves for the deterministic and random fractal-fracture networks and their respective dominant sub-networks. For the deterministic network (solid lines in Fig. 5) it can be seen that the dominant sub-networks are almost indistinguishable from their parent network in terms of recovery. A similar behavior is seen for the random network (dashed lines in Fig. 5), although some differences exist in the dominant sub-networks in this case. This indicates that even though the parent fractal-fracture networks and their dominant sub-networks have very different number of fractures (Fig. 2), they have almost similar fluid flow characteristics. It is also seen from the figure that the response from dominant sub-network-I of the deterministic pattern actually coincides with that of the parent network of the random pattern. As because the random and deterministic patterns have the same length distribution it may be conjectured that the parent fractal-fracture network is very similar to the sub-networks in terms of recovery.

The overall recovery values thus obtained for all fracture networks studied are shown in Table 3. Fluid recovery for dominant sub-network-I is about 96 % and 93 % to that of the parent network for the deterministic and random patterns respectively. In case of dominant sub-network-II, these values are about 85 % and 83 %. Hence, it may be hypothesized that while modelling such fracture networks, eliminating a large number fractures with smaller apertures will not significantly affect the fluid flow.

Time of flight (TOF) is an important parameter for visualizing the fluid velocity at a given location within the flow domain. The TOF can be defined as a spatial coordinate representing the distance along streamline to where the reservoir will be contacted at that instance of time. It is worth mentioning that along preferential flow paths, the TOFs show higher resolution as the streamline's tends to cluster along the high permeability streaks. The TOF's depicting fluid-front through the deterministic and random fractal-fracture network along with their dominant sub-networks at different time steps (500, 1500 and 3000) are shown in Fig. 6. It is observed that flow patterns are almost similar for the parent fractal-fracture network and their dominant sub-networks. The TOFs clearly show flow channelling through fractures with larger apertures of the dominant sub-networks in both deterministic and random fractal-fracture networks. It can be visually seen from Fig. 6 that these fractures serve as preferential pathways for conduit of major amount of fluid in the reservoir and is in agreement with the experimental analysis of Neretnieks et al. (1982). The time of flight for fractures with larger apertures are much smaller than those with smaller apertures hence, fluid movements are faster in the former. This results in almost similar recovery in the original fractal-fracture network and its dominant sub-networks which contain only the larger fractures.

A subset of natural fracture networks is chosen from a set of seven nested fractal-fracture maps gathered from the Devonian Sandstones of Hornelen basin, Norway (Odling, 1997). These maps have been studied by many researchers in relation to the characterization of various statistical and geometrical properties of fracture networks in the last three decades (Bonnet et al., 2001; Bour et al., 2002; Odling and Roden, 1997; Roy et al., 2007, 2010). Hence, they may serve as good outcrop analogs for fractured reservoirs. They cover areas from 18 to 720 m across at a range of resolutions forming a nested series. Each map represents a “window” or sub-sample of the fracture trace population with a lower bound controlled by the resolution and an upper bound controlled by the size of the region observed (Bour et al., 2002; Odling, 1997; Roy et al., 2007). The scale, resolution and the height of observation of these maps along with the mapping technique are described in Odling (1997). We chose map 1 and map 2 from the set such that the latter contains the former but, at a lower resolution as shown in Fig. 7. The aperture of the natural fracture map is scaled considering the trace-length distribution of individual maps which follows a power law (Odling, 1997).

A dominant sub-network of map-1 is generated by considering a smaller section from map-2 that represents map-1 (at a lower resolution) and rotating it to match its orientation with the latter. Thus, the dominant sub-network of map-1 retains only the longer fractures (with large apertures) as the shorter fractures (with smaller apertures) are “lost” due to decrease in resolution. The count of fractures in both the maps is estimated using FracPaQ and the dominant sub-network contains only about 15 % of the fracture segments from the original parent fracture network. The natural fracture network (map-1) and its sub-dominant network are shown in Fig. 8.

The proportion of X, Y and I nodes present in the natural fracture map and its dominant sub-network are estimated using FracPaQ and their connectivity values are calculated from Eq. (2). The connectivity of the dominant sub-network is observed to decrease by 15 % of that of its parent fracture map as shown in Table 2. The natural fracture map and its dominant sub-network are modeled as grids of 1042×1042×1 cells invoking the fracture continuum (FC) concept as before. Similar to our approach in Sect. 2.3, the permeability of cells that represent fractures are calculated from the cubic law, and a single high porosity of 95 % is assigned to all cells representing fractures. As with the models, matrix porosity and permeability values are assigned as 0.05 % and 0.98×10-16 m2 respectively, such that flow through the matrix is negligible. The other inputs used in the simulator are also kept the same as mentioned in Sect. 2.3. The fracture maps are flow simulated using Trace3D, with an injection and a production well placed at diagonally opposite corners of the model grid (i.e., 1×1 and 1042×1042). The recovery plots of the natural fracture map and its dominant sub-network are shown in Fig. 9. Their TOFs that represent fluid movement at different time steps (500, 1500 and 3000) are shown in Fig. 10.

The overall recovery values of each of the two maps are shown in Table 3. The fluid recovery value for the dominant sub-network is almost 88 % of that of the original map although the former retains only 15 % of the fractures from that of the latter. This clearly indicates that fractures with smaller aperture have a rather negligible contribution to the overall fluid flow. The TOFs for the natural fracture map and its dominant sub-network show almost no difference at least up to 1500 time steps. Some minor differences are observed between the two networks at 3000 time steps. This shows that the smaller fractures present in the original map (and those that are captured only in high resolution data) actually do not significantly contribute to the flow. From a modelling perspective, this observation is very helpful because geoscientists need not include data on smaller fractures in their models as they do not significantly affect the connectivity and overall fluid recovery.

The influence of aperture distribution on connectivity and fluid flow in fractal-fracture networks has been explored by studying a set of model networks, a natural map and their respective dominant sub-networks. Although the dominant sub-networks were generated by removing a large number of smaller aperture fractures from the parent networks, the connectivity was observed to decrease only by about 10 %–25 %. The networks were then flow simulated in Trace3D invoking the FC model. It was seen that in networks with a broad distribution of aperture, most of the fractures with smaller apertures can be eliminated without any significant reduction of overall fluid recovery. This is because dominant subnetworks containing only about 15 % of the total fracture segments from their respective parent networks yielded recovery values nearly 85 % of that of the latter.

The same strategy was applied to a natural fracture map where smaller fractures were removed by comparing it to its lower-resolution version such that only the long and wide fractures that appeared in the latter were retained. Here, the connectivity and fluid recovery of the dominant sub-network map was found to be around 84 % and 88 % of the parent network respectively. Thus, the results from this experiment proved to be very similar to what was obtained from the study of model fractal-fracture networks. It may therefore be concluded that even in a well-connected fracture network, fluid flow may be unequally distributed based on the aperture distribution and most of the network can be excluded, 85% in this particular case, without significantly affecting its fluid recovery.

In terms of decision making, this result will aid geoscientists in eliminating smaller fractures from the data that do not contribute significantly to the flow thus, building fracture models that are computationally less expensive. The implications of this finding for characterization of fractured rock mass can prove useful in primary as well as enhanced methods of fluid recovery, where the dominant sub-network fractures are the one that carry most of the produced and injected fluids as the larger aperture fractures provide preferential pathways for fluid flow.