**2. Interaction between hydraulic and natural fractures**

while numerical modeling yields more detailed results which can be used for further inves‐ tigations but it is computationally more expensive and time-consuming than artificial neural network approach. On the other hand, since artificial neural network approach is mainly da‐ ta-driven if just the input data is available (even while fracturing) the hydraulic fracture path (crossing/turning into natural fracture) can be predicted real-time and at the same time

Hydraulic fracture growth through naturally fractured reservoirs presents theoretical, design, and application challenges since hydraulic and natural fracture interaction can significantly affect hydraulic fracturing propagation. Although hydraulic fracturing has been used for decades for the stimulation of oil and gas reservoirs, a thorough understanding of the inter‐ action between induced hydraulic fractures and natural fractures is still lacking. This is a key challenge especially in unconventional reservoirs, because without natural fractures, it is not possible to recover hydrocarbons from these reservoirs. Meanwhile, natural fracture systems are important and should be considered for optimal stimulation. For naturally fractured formations under reservoir conditions, natural fractures are narrow apertures which are around 10-5 to 10-3 m wide and have high length/width ratios (>1000:1) [1].Typically natural fractures are partially or completely sealed but this does not mean that they can be ignored while designing well completion processes since they act as planes of weakness reactivated during hydraulic fracturing treatments that improves the efficiency of stimulation [2]. The problem of hydraulic and natural fracture interaction has been widely investigated both experimentally [3, 4, 5, 6, 7, 8] and numerically [9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. Many field experiments also demonstrated that a propagating hydraulic fracture encountering natural fractures may lead to arrest of fracture propagation, fluid flow into natural fracture, creation of multiple fractures and fracture offsets [19, 20, 21, 22] which will result in a reduced fracture width. This reduction in hydraulic fracture width may cause proppant bridging and conse‐ quent premature blocking of proppant transport (so-called screenout) [23, 24] and finally treatment failure. Although various authors have provided fracture interaction criteria [4, 5, 25] determining the induced fracture growth path due to interaction with pre-existing fracture and getting a viewpoint about variable or variables which have a decisive impact on hydraulic fracturing propagation in naturally fractured reservoirs is still unclear and highly controver‐ sial. However, experimental studies have suggested that horizontal differential stress, angle of approach and treatment pressure are the parameters affecting hydraulic and natural fracture interaction [4, 5, 6] but a comprehensive analysis of how different parameters influence the fracture behavior has not been fully investigated to date. In this way, in order to assess the outcome of hydraulic fracture stimulation in naturally fractured reservoirs the following

that fracturing is happening.

1040 Effective and Sustainable Hydraulic Fracturing

questions should be answered:

What is the direction of hydraulic fracture propagation?

How will the propagating hydraulic fracture interact with the natural fracture?

**1. Introduction**

The interaction between pre-existing natural fractures and the advancing hydraulic fracture is a key issue leading to complex fracture patterns. Large populations of natural fractures are sealed by precipitated cements (Figure 1) which are weakly bonded with mineralization that even if there is no porosity in the sealed fractures, they may still serve as weak paths for the growing hydraulic fractures [2].

**Figure 1.** A weakly bonded fracture cement in a shale sample [26].

In this way, experimental studies [4, 5, 6] suggested several possibilities that may occur during hydraulic and natural fractures interaction. Blanton [4] conducted some experiments on naturally fractured Devonian shale as well as blocks of hydrostone in which the angle of approach and horizontal differential stress were varied to analyze hydraulic and natural fracture interaction in various angles of approach and horizontal differential stresses. He concluded that any change in angle of approach and horizontal differential stress can affect hydraulic fracture propagation behavior when it encounters a natural fracture which will be referred to as opening, arresting and crossing. Warpinski and Teufel [5] investigated the effect of geologic discontinuities on hydraulic fracture propagation by conducting mineback experiments and laboratory studies on Coconino sandstone having pre-existing joints. They observed three modes of induced fracture propagation which were crossing, arrest by opening the joint and arrest by shear slippage of the joint with no dilation and fluid flow along the joint. In 2008 [6] some laboratory experiments were performed to investigate the interaction between hydraulic and natural fractures. They also observed three types of interactions between hydraulic and pre-existing fractures which were the same as Warpinski and Teufel's obser‐ vations. The above referenced experimental studies have investigated the initial interaction between the induced fracture and the natural fracture, however, in reality may be the hydraulic fracture is arrested by natural fracture temporarily but with continued pumping of the fluid, the hydraulic fracture may cross (Figure 2) or turn into the natural fracture (Figure 3).

Alternatively, in some cases the hydraulic fracture may get arrested if the natural fracture is

Investigation of Hydraulic and Natural Fracture Interaction: Numerical Modeling or Artificial Intelligence?

For fracture propagation through numerical modeling an energy based criterion has been considered which is energy release rate, *G*. The energy release rate, *G*, is related to the stress

ν is the Poisson's ratio). Energy release rate has been calculated by the *J* integral using the domain integral approach [28] whereas *J* integral is equivalent to the definition of the fracture energy release rate, *G*, for linear elastic medium. If the *G* is greater than a critical value, *Gc*, the fracture will propagate critically. The direction of hydraulic fracture propagation will be

æ ö ç ÷ æ ö

2

= ±+ ç ÷ ç ÷ è ø è ø

4

*<sup>c</sup> K K II II*


*K K I I*

During hydraulic and natural fracture interaction at the intersection point the hydraulic fracture has more than one path to follow which are crossing and turning into natural fracture. The most likely path is the one that has the maximum *G*. So, at the intersection point energy release rate is calculated for both crossing (*Gcross*) and turning into natural fracture (*Gturn*), and if (*Gturn* /*Gcross*)>1 hydraulic fracture turns into natural fracture while if (*Gturn* /*Gcross*)< 1 crossing takes place and hydraulic fracture crosses the natural fracture. To examine the proposed mechanism, eXtended Finite Element method (XFEM) was applied which was first introduced by Belytschko and Black [30] in order to avoid explicit modeling of discrete cracks by enhancing the basic finite element solution. In comparison to the classical finite element method, XFEM provides significant benefits in the numerical modeling of fracture propagation and it overcomes the difficulties of the conventional finite element method for fracture analysis, such as restriction in remeshing after fracture growth and being able to consider arbitrary varying geometry of fractures [12]. XFEM enhances the basic finite element solution through the use of enrichment functions which are the Heaviside function for elements that are completely cut by the crack and Westergaard-type asymptotic functions for elements containing crack-tips [27]. The displacement field for a point "x" inside the domain can be approximated based on

<sup>1</sup> 2 2 *G KK* ( ) *I II <sup>E</sup>* (1)

*)* for plane strain (where

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1043

(2)

**3. Numerical modeling: Extended Finite Element Method (XFEM)**

= + ¢

where *E'* = *E* for plane stress (*E* is Young's modulus) and *E'* = *E*/*(1- ν<sup>2</sup>*

q

the XFEM formulation as below [31]:

long enough and favorably oriented to accept and divert the fluid.

intensity factors through Eq. 1 [27]:

calculated by Eq. 2 [29]:

**Figure 2.** Propagating hydraulic fracture crosses the natural fracture and keep moving without any significant change in its path: left image is a schematic view of crossing and right image is the result of experimental study [4].

**Figure 3.** Hydraulic fracture turns into the natural fracture and propagates along it: left image is a schematic view and right image is the result of experimental study [4].

Alternatively, in some cases the hydraulic fracture may get arrested if the natural fracture is long enough and favorably oriented to accept and divert the fluid.

#### **3. Numerical modeling: Extended Finite Element Method (XFEM)**

experiments and laboratory studies on Coconino sandstone having pre-existing joints. They observed three modes of induced fracture propagation which were crossing, arrest by opening the joint and arrest by shear slippage of the joint with no dilation and fluid flow along the joint. In 2008 [6] some laboratory experiments were performed to investigate the interaction between hydraulic and natural fractures. They also observed three types of interactions between hydraulic and pre-existing fractures which were the same as Warpinski and Teufel's obser‐ vations. The above referenced experimental studies have investigated the initial interaction between the induced fracture and the natural fracture, however, in reality may be the hydraulic fracture is arrested by natural fracture temporarily but with continued pumping of the fluid,

the hydraulic fracture may cross (Figure 2) or turn into the natural fracture (Figure 3).

**Figure 2.** Propagating hydraulic fracture crosses the natural fracture and keep moving without any significant change

**Figure 3.** Hydraulic fracture turns into the natural fracture and propagates along it: left image is a schematic view and

right image is the result of experimental study [4].

1042 Effective and Sustainable Hydraulic Fracturing

in its path: left image is a schematic view of crossing and right image is the result of experimental study [4].

For fracture propagation through numerical modeling an energy based criterion has been considered which is energy release rate, *G*. The energy release rate, *G*, is related to the stress intensity factors through Eq. 1 [27]:

$$G = \frac{1}{E} (K\_I^2 + K\_{II}^2) \tag{1}$$

where *E'* = *E* for plane stress (*E* is Young's modulus) and *E'* = *E*/*(1- ν<sup>2</sup> )* for plane strain (where ν is the Poisson's ratio). Energy release rate has been calculated by the *J* integral using the domain integral approach [28] whereas *J* integral is equivalent to the definition of the fracture energy release rate, *G*, for linear elastic medium. If the *G* is greater than a critical value, *Gc*, the fracture will propagate critically. The direction of hydraulic fracture propagation will be calculated by Eq. 2 [29]:

$$\theta\_{\mathcal{C}} = 2 \tan^{-1} \frac{1}{4} \left( \frac{K\_I}{K\_{II}} \pm \sqrt{\left( \frac{K\_I}{K\_{II}} \right)^2 + 8} \right) \tag{2}$$

During hydraulic and natural fracture interaction at the intersection point the hydraulic fracture has more than one path to follow which are crossing and turning into natural fracture. The most likely path is the one that has the maximum *G*. So, at the intersection point energy release rate is calculated for both crossing (*Gcross*) and turning into natural fracture (*Gturn*), and if (*Gturn* /*Gcross*)>1 hydraulic fracture turns into natural fracture while if (*Gturn* /*Gcross*)< 1 crossing takes place and hydraulic fracture crosses the natural fracture. To examine the proposed mechanism, eXtended Finite Element method (XFEM) was applied which was first introduced by Belytschko and Black [30] in order to avoid explicit modeling of discrete cracks by enhancing the basic finite element solution. In comparison to the classical finite element method, XFEM provides significant benefits in the numerical modeling of fracture propagation and it overcomes the difficulties of the conventional finite element method for fracture analysis, such as restriction in remeshing after fracture growth and being able to consider arbitrary varying geometry of fractures [12]. XFEM enhances the basic finite element solution through the use of enrichment functions which are the Heaviside function for elements that are completely cut by the crack and Westergaard-type asymptotic functions for elements containing crack-tips [27]. The displacement field for a point "x" inside the domain can be approximated based on the XFEM formulation as below [31]:

$$\mathbf{u}^{h}(\mathbf{x}) = \sum\_{I \in N\_{uu}} N\_I(\mathbf{x}) \left( \mathbf{u}\_I + \underbrace{H(\mathbf{x}) \mathbf{a}\_I}\_{I \diamond N\_{uu}} + \underbrace{\sum\_{a=1}^{4} F\_a(\mathbf{x}) \mathbf{b}\_I^a}\_{I \diamond N\_{ab}} \right) \tag{3}$$

**4. Artificial intelligence: Artificial Neural Network (ANN)**

which multiplies by the related input.

**Figure 5.** Schematic structure of an ANN.

Artificial Neural Network (ANN) is considered as a different paradigm for computing and is being successfully applied across an extraordinary range of problem domains, in all areas of engineering. ANN is a non-linear mapping structure based on the function of the human brain that can solve complicated problems related to non-linear relations in various applications which makes it superior to conventional regression techniques [33, 34]. ANNs are capable of distinguishing complex patterns quickly with high accuracy without any assumptions about the nature and distribution of the data and they are not biased in their analysis. The most important aspect of ANNs is their capacity to realize the patterns in obscure and unknown data that are not perceptible to standard statistical methods. Statistical methods use ordinary models that need to add some terms to become flexible enough to satisfy experimental data, but ANNs are self-adaptable. The structure of the neural network is defined by the intercon‐ nection architecture between the neurons which are grouped into layers. A typical ANN mainly consists of an input layer, an output layer, and hidden layer(s) (Figure 5). As shown in Figure 5, each neuron of a layer is connected to each neuron of the next layer. Signals are passed between neurons over the connecting links. Each connecting link has an associated weight

Investigation of Hydraulic and Natural Fracture Interaction: Numerical Modeling or Artificial Intelligence?

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Where *NI* is the finite element shape function, *uI* is the nodal displacement vector associated with the continuous part of the finite element solution, *H(*x*)is* the Heaviside enrichment function where it takes the value +1 above the crack and –1 below the crack, a*<sup>I</sup>* is the nodal enriched degree of freedom vector associated with the Heaviside (discontinuous) function, Fα(x) is the near-tip enrichment function, *bI <sup>α</sup>* is the nodal enriched degree of freedom vector associated with the asymptotic crack-tip function, *Nu* is the set of all nodes in the domain, *N<sup>α</sup>* is the subset of nodes enriched with the Heaviside function and *Nb* is the subset of nodes enriched with the near tip functions. At the intersection point, instead of Heaviside enrichment function, Junction function will be applied as shown in Figure 4 [32]. By all means, XFEM is well-suited for modeling hydraulic fracture propagation and diversion in the presence of natural fracture.

**Figure 4.** Definition of Junction function at the intersection point [32].
