## 1. Introduction

[2] Hydrofracking is a technology that utilizes highly pressurized fluid to create fracture networks in rock layers with low permeabilities [*Cosad*, 1992; *Brady et al.*, 1992]. A fracking fluid is injected into a cased wellbore, and the parts of the reservoir to be fractured are accessed by perforating the casing at the correct locations (Figure 1b). The injection well can be vertical or horizontal and several injection points may be active during the hydraulic stimulation of the system (Figures 1a, 1b). The highly pressurized fluid that flows into the reservoir increases the geomechanical stress around the injection points and causes the rock to fracture. Ideally, the hydrofracking creates long, distributed cracks that connect a large area of the reservoir to the well. If several injection points are active during the hydrofracking process, the final crack distribution strongly depends on the spacing of the points and the interaction between them during the injection process [*Cosad*, 1992; *Brady et al.*, 1992]. A comprehensive understanding of this interaction and its implications for the fracturing process is therefore crucial to optimize the functionality of the reservoir.

[3] The complexity of the fracture networks and the multiple scales involved represent challenges in the mathematical and numerical modeling of both the fluid flow and the fracturing processes. The different modeling approaches of the complex fluid flow can be classified into three main categories: discrete fracture-matrix (DFM) models, effective continuum models (also known as equivalent continuum models), and multiple-continuum models.

[4] DFM models explicitly resolves the fractures, which is feasible in situations where only a limited number of fractures dominate the fluid flow. The approach requires gridding of the fractures, either with an equidimensional approach where the fractures have the same dimension as the matrix, or with a hybrid approach where the fractures are modeled as planes in three-dimensional domains and lines in two-dimensional domains. The DFM model requires a detailed knowledge of the geometric properties of the fracture network, and also requires unstructured grids to account for the given geometry, but provides an accurate description of the flow when this information is available.

[5] The effective continuum model (see, e.g., *Bear et al.* [1993]) represents the fractures and the rock matrix as a single effective continuum. The method has been extensively used to model fluid flow in fractured porous media because of its computational efficiency and the limited data that is required. However, in general, the method is only feasible when considering transport processes under steady state conditions. The advantage of the effective continuum model is that it is simple, and it also provides a sufficient model for the flow in the case where the equivalent characteristics of the fractured porous media may easily be estimated, e.g., in the case of fractures parallel to the fluid flow [*Samardizioska and Popov*, 2005].

[6] Multiple-continuum approaches are an extension of the effective continuum model, developed to handle the different characteristics of the flow in the fractures and the matrix. The fractures and the matrix are considered as different continuums over the same volume, interacting through mass transfer coefficients, and hence representing the fractured porous medium. The approaches include the double-porosity model [*Warren and Root*, 1963], the dual-permeability concept, the dual-continuum generalization [*Pruess and Narasimhan*, 1985], and the multicontinuum model [*Wu and Pruess*, 1988]. The method offers more information than the effective continuum approach on averaged properties of the flow and transport processes in the matrix and the fractures, without the need of detailed and complex information of the geometry of the problem.

[7] The failure of homogeneous materials is well understood. A fracture is created when the strain energy-release rate exceeds a critical value, and it propagates along the direction that maximizes the energy release [*Knott*, 1973]. The failure of heterogeneous materials, in contrast, depends on many microscopic mechanisms and is challenging to model [*Hansen*, 2005]. To understand the fracturing process of a heterogeneous material on the microscale, statistical models such as the fuse and the beam models have been particularly useful [*Hansen*, 2005; *Herrmann and Roux*, 1990]. In these models, the heterogeneity of the system is taken into account by defining a local, position-dependent material strength that is drawn from a statistical distribution. The fracturing process is therefore not entirely controlled by the energy-release rate, but also depends on the disorder of the system and the distribution of the local strengths. Depending on the degree of disorder, the statistical models have shown that heterogeneous systems exhibit fracturing regimes that are distinct from the single-crack development observed in homogeneous systems [*Hansen*, 2005].

[8] Several models for hydrofracking have been developed [*Adachi et al.*, 2007], but a weakness of most of these models is that they do not account for the heterogeneities of the rock. However, some research based on extensions of the statistical beam model has been applied to the study of hydrofracking of heterogeneous systems [*Tzschichholtz et al.*, 1994; *Tzschichholtz and Herrmann*, 1995; *Tzschichholtz and Wangen*, 1998; *Flekkøy et al.*, 2002]. Recently, these studies have been further developed to provide a better description of the rock matrix. In the work of *Wangen* [2011] and *Wang et al.* [2009], the dynamics of the rock matrix are modeled using poroelastic theory, and the heterogeneity is treated by distributing the local material strength according to a probability distribution. *Wangen* [2011] applies an effective continuum approach to model the fluid flow, which captures the coupled dynamics of the fluid pressure and the fractured rock matrix.

[9] The creation of a fracture changes the geomechanical strain energy of the system. If the reservoir contains several fractures, the modification of the strain energy mediates an effective interaction between the cracks. The interactions between fractures have been studied theoretically in several works using both direct numerical simulations [*Akulich and Zvyagin*, 2008; *Aghighi and Rahman*, 2010] and statistical methods [*Masihi and King*, 2007; *Shekhar and Gibson*, 2011]. A geomechanical stress interaction also exists between two injection points during hydrofracking. However, a thorough investigation of this interaction and its consequences for the fracturing process is lacking. The main focus of the present paper is to investigate how this interaction in combination with disorder influences the hydrofracking process. To this end, we develop an effective continuum model of the hydrofracking of heterogeneous, poroelastic media. Our formalism is based on the previous work by *Wangen* [2011], which is extended to take into account the anisotropic fluid flow and the shear failure of the material. Our numerical simulations demonstrate, for the first time, how the fracturing process around each injection point is correlated with the neighboring point, and how this correlation depends on the degree of disorder in the system. We find that for two injection points that are separated by a distance that is smaller than a critical correlation length, the fracturing process around each injection point is strongly correlated with the position of the neighboring point. The critical correlation length at which this strongly correlated regime occurs depends on the degree of heterogeneity, with correlation lengths of ∼20 m for highly disordered systems and 45 m for weakly disordered systems. In the correlated regime, we predict a novel *effective fracture force*that attracts the fracture toward the neighboring injection point. Our results give important insights for optimizing the hydraulic stimulation of reservoirs. For well perforations separated by a distance that is less than the critical correlation length, the results imply a reduced effect of the stimulation because the fractures are attracted toward neighboring injection points. Knowing the correlation length of the system is therefore crucial for creating an effective and long-ranging fracture network.

[10] This paper is organized in the following manner: In section 2, we present the theory and governing equations of our hydrofracking model. The section concludes with a pseudo code of the algorithm. Section 3 describes the numerical solution strategy. The next two sections apply the formalism to the study of the interaction of two injection points during hydrofracking. Specifically, section 4 provides a description of the model system that we consider and section 5 presents our findings. We conclude and summarize our results in section 6.