Application of the fractal geometry theory on fracture network simulation
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Abstract
In this paper, we introduce an Lsystem based on the fractal geometry to study complex fracture networks. Comparing with other simulation models, the fractal fracture could not only maintain the bifurcation of the fracture geometry but also represent the multilevel feature of the complex fracture networks according to its fractal characteristics. Since the fractal geometry is always connected with several controlling parameters, the factors affecting on the fracture propagation can be quantized. With the fractal fractures, further studies on the fracture geometry and the multileveled branches are carried out, and the results can be concluded as: (1) when the complex fracture geometry is considered, the influence of the connectivity and complexity of the fracture network on the well performance is hard to ignore, and enhancing the connectivity and complexity of the fracture network will perform better than making a long but disconnected fracture on the shale gas development; (2) of a multileveled fracture network, the main fractures near the horizontal well contributes to the initial production, but the conductivity ratio (conductivity of the secondary fractures versus the conductivity of the main fractures) influences the decline rate, so a high production rate cannot be maintained without an efficient contributing area covered with conductive secondary fractures.
Keywords
Unconventional reservoirs Hydraulic fracture simulation Fractal fracture geometry Multilevel fracturesIntroduction
Duo to the low permeability and porosity, economic development of the shale gas resources always needs a multistage hydraulic fracturing. Different from the traditional biwing hydraulic fractures, complex fracture networks are generated near the horizontal wells in the shale gas reservoirs, as monitored by microseismic events (MSE) (Fisher et al. 2002; Maxwell et al. 2002; Daniels et al. 2007). Mineback experiments and some field observations (Huang and Kunsoo 1993; Mayerhofer 2006) suggest that hydrofractures do not propagate linearly; when the reservoir is rich in natural fractures (NFs), the hydraulic fractures (HFs) may be created multibranched. And it is mentioned that the complexity of the fracture networks is the main factor differing from the biwing fractures that contributes to the well production (Jang et al. 2015).
Considering the complex geometry, width of the branches of the fracture network is smaller than a single biwing fracture, the proppant might not be able to transport to the tip of the total fracture network (Xu et al. 2009), and this leads to a multilevel feature of the fracture networks which has a significant influence on the well performance. Analytical methods such as rate transient analysis (RTA) and well logging both show that the critical zone of the stimulated reservoir is smaller than the total area monitored by MSE (Friedrich 2013; Rahimi et al. 2014). The complexity and connectivity are two key parameters of the fracture networks, they relate directly with the well production, and it is obvious that the complexity and connectivity of the fracture network near the horizontal well are higher (Jones et al. 2013; Chen et al. 2016).
For better studying the performance of the fracture networks, analytical and numerical methods are applied. Olsen et al. studied the interactions between HFs and NFs and introduce a method to characterize the propagation of the complex fracture network considering the heterogeneity of the reservoir and the irregular distribution of natural fractures (Olson 2008; Olson and Arash 2009). Besides the analytical studies, numerical models are developed for further study and engineering simulation. The dualporosity model, wiremesh model and unconventional fracture model (UFM) are three typical numerical models that can take the main characteristics of the fractures into consideration when simulating a complex fracture network. Dualporosity model is first introduced by Warren and Root (Warren and Root 1963) to characterize the behavior of naturally fractured reservoirs and now widely applied in fracture modeling (Zimmerman et al. 1993; Du et al. 2010; Cipolla et al. 2010). The stimulated region is regarded as dual porosity or even multiporosity, and the properties of the grids can be assigned independently. The wiremesh model is consisted of two perpendicular sets of vertical planar fractures, and it quantizes the complicated geological and engineering factors to the parameters controlling the propagation of the wiremesh network (Xu et al. 2009; Xu et al. 2010; Meyer and Lucas 2011). The properties of the planar fractures and their spacing are related to different engineering parameters even MSE, and the mechanical interactions between the fracturing fluid and fracture walls are its main consideration (Xu et al. 2010; Weng et al. 2011). However, the models above cannot display the fracture geometry and that is the reason UFM is developed (Weng et al. 2011; Weng 2015). The UFM mainly studies the interactions between the HFs and NFs and details the propagation of the fracture network within the unstructured grids. It couples the fracture geometry with the factors influencing the propagation such as the orientation of the NFs and the rock deformation.
In fact, of either wiremesh model or the UFM, the main focus is the description on the fracture propagation; the conceptual models mentioned by Jones et al. (2013), Chen et al. 2016), studying the influence of the fracture complexity and connectivity on the typical production curves, fail to consider the fracture bifurcation. So in this paper, we would like to introduce a method based on the fractal geometry theory to characterize the fracture network and to analyze the well performance. The fractal geometry theory was put forward by Mandelbrot (1979) and has been applied to rock mechanics since 1982 (Xie 1993; Wang et al. 2015a, b) utilized the iterated function system (IFS) to study the bifurcation performance of the fracture network. According to the fractal geometry theory, the fracture network we generated can be both bifurcated and multileveled, the fracture geometry can be related to few fractal controlling parameters, and these lead to the main advantages of the model: (1) characterizing the bifurcation feature of the fracture geometry; (2) quantifying the connectivity and complexity of the fracture network for analyzing; and (3) classifying the fractures into different levels to differ the main fractures and secondary fractures.
Fractal fractures based on the Lsystem
Characterization of fractal fracture model
Lsystem is a rewriting system that defines a complex object by replacing parts of the initial object according to rewriting rules, and it simulates development rules and topological structure well (Lindenmayer 1968; Han 2007). The system has the feature of selfadjusting when something bifurcates, and this feature can describe the growth of trees. However, in this paper, we first introduce the Lsystem into fracture characterization because a fracture also has similar development rules and topological structure as trees and the interaction between HFs and NFs could be regarded as a type of ruleadjusting procession affecting the propagation of the fracture, which coincides with the basement of the Lsystem.
 1.The fractal distance (d) mainly controls the extending distance of the fractal fractions, it relates closely to the halflength of the fracture obtained by MS monitoring, and the influence of the fractal distance is shown in Fig. 1.
 2.The deviation angle (α) controls the orientation when the fracture deviates or generates a secondary branch; it relates to the area of the stimulated reservoir, and when cooperating with the fractal distance, the size of the fractal fracture network can be adjusted for matching. Figure 2 shows the influence of the deviation angle on the orientation of the branches.
 3.
The number of iterations (n) controls the growth of a fractal fracture network. It depends on the complexity of the fracture network or the density of the MS events. This parameter relates to the multilevel feature of the fractal branches: in each iteration, the fractal fractures propagate from the original nodes following the given generating rules to construct part of the network. It is now considered that during the actual stimulating procession, the secondary fractures extend on the basis of the main fractures, so with this similarity, levels of the fractal fractures are also distinguished based on the generating orders according to the iteration times.
 4.
There are rules for controlling the growth of the bifurcation. To account for an irregular propagation mode of a complex fracture network, the rules for controlling are always preset as more than three. In conjunction with the iteration times, the fractal fracture model could model numerous fracture geometries under different conditions. And by adjusting the combination of the growth rules, the value of the fractal distance and deviation angle, the fractal fracture model can generate the best fractal geometry matching the given fracture network.
Geometry calibration with Lsystem
Simulation workflow
Well performance analysis based on a case study
The fractal fracture model (FFM) could analyze the influence of either individual fractures or the integral fracture geometry on the well production. In this part, we would like to study the influence of the multileveled fractures, fracture geometry and conductivity ratio, which are rarely analyzed before, on the well performance based on the fractal fracture model.
Parameters for the base model for simulation
Input parameter  Value 

Thickness (m)  40 
Porosity (%)  5 
Matrix permeability (mD)  0.001 
Natural fracture permeability (mD)  0.001–30 
Initial reservoir pressure (MPa)  32 
Initial gas saturation (%)  1 
Initial water saturation (%)  0 
Reservoir temperature (K)  345 
Total compressibility (MPa^{−1})  1.42E−5 
Well length (m)  750 
Fracture conductivity (mD m)  6 
Fracture halflength obtained by MSE (m)  109 
Multilevel feature of fractal fracture network
According to the simulation results, daily production rate of level 1 soon goes down and its cumulative production is far smaller than that of the other two. The low complexity of level 1 cannot provide a high production, and it also demonstrates that the biwing fracture is hard to fit the fracture modeling in shale gas reservoirs. Daily production rate of level 2 is similar to that of level 3, but it also soon goes down, resulting in the difference on their cumulative production, demonstrating that when the complexity of the fracture network is large enough, it will not influence the initial production rate, but it still influences the decline rate of the production curves.
Influence of the fracture geometry on the well production
Comparing the fracture geometry and production data of patterns (a), (c) and (d), the final production drops with the reduction in the fracture connectivity, although the synthesized length of the fracture increases. So when the complex geometry of the fracture network is considered, the fracture connectivity performs a more critical influence on the well performance than the facture halflength. Combined with the conclusion we drawn before, the efficiency of the fracture network, related to the complexity and connectivity, is a determining factor to the well production, and only considering the increase on the fracture halflength in the shale gas development may contribute little to the final recovery. On the other hand, both pattern (a) and pattern (b) are low in connectivity and production, but the final production of pattern (a) is higher, showing that the synthesized halflength of the fracture network is still positive to the fracture performance.
Influence of the conductivity ratio on the well production
 1.
The initial production is almost the same. Since the pressure wave does not spread to the secondary branches, the main fractures of the network determine the early production.
 2.
With the ratio changes and the conductivity of secondary fractures turns down, the decline rate of the production curves increases. So the decline rate, which is a key factor to maintain the high production rate, is connected directly with the overall effect of fracturing.
 3.
The contribution of the secondary branches maintains a great proportion to the well production, so in the hydraulic fracture network simulation and in the actual fracturing design, only concerning the stimulated results of the main fractures without optimizing the secondary fractures may finally result an inefficient stimulated reservoir area and a low level of production.
Conclusions

In this paper, a novel fracture model based on the fractal geometry is introduced, we detail the fractal controlling parameters and their influence on the fracture geometry, and two simple matching cases present that the fractal fracture could be utilized to match either the rock fractures or the fracture network in the shale gas reservoirs.

The factors affecting on the fracture propagation can be quantized into several fractal controlling parameters, the fractal fracture can be divided into different levels according to the iteration times, and these are two advantages of the fractal fracture differing from other normal fracture models.

With the features of the fractal fractures, further studies on the influence of the fracture geometry and the conductivity ratio of the fracture network on the well performance are carried out, and we obtain the following conclusions: (1) considering the complex fracture geometry, the complexity and connectivity of the fracture network perform a obvious influence on the well production, and (2) the secondary fractures also contribute greatly to the fracture performance. The conductivity ratio mainly influences the decline rate, and it is not sensible to concern over the synthesized fracture halflength of the covering area without considering the contributing efficiency of the fracture network during the fracturing design.
Notes
Acknowledgments
This work was supported by the National Basic Research Program of China (2014CB239103) and Graduates’ Innovation Foundation of China University of Petroleum (East) (YCXJ2016020). The authors would like to acknowledge the technical support of ECLIPSE in this paper.
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