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

a

*NH F* (3)

is the nodal displacement vector associated

*<sup>α</sup>* is the nodal enriched degree of freedom vector

a

Î

*I N*

*ub*

aÎ

1

4

æ ö ç ÷

è ø

+ +

*II I I*

å å <sup>14243</sup> <sup>14243</sup>

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,

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

Î -

*I N I N*

u (x)= (x) u (x)a (x)b *nu ua*

*h*

1044 Effective and Sustainable Hydraulic Fracturing

Fα(x) is the near-tip enrichment function, *bI*

is the finite element shape function, *uI*

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

Where *NI*

natural fracture.

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 which multiplies by the related input.

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

Also, to diversify the various processing elements, a bias is added to the sum of weighted inputs called net input shown in Eq. (4)

$$\mathbf{m} = \left(\mathbf{w}\_{\mathrm{S},1}\mathbf{p}\_1 + \mathbf{w}\_{\mathrm{S},2}\mathbf{p}\_2 + \mathbf{w}\_{\mathrm{S},3}\mathbf{p}\_3 + \dots + \mathbf{w}\_{\mathrm{S},\mathbb{R}}\mathbf{p}\_{\mathbb{R}}\right) + \mathbf{b}\_{\mathrm{S}} \tag{4}$$

Where *n* is the net input, *w* is the weight, *p* is the input, *b* is the bias, *S* is the number of neurons in the current layer and *R* is the number of neurons in the previous layer.

Each neuron applies an activation function to its input to determine its output signal [35]. Neurons may use any differentiable activation function to generate their output based on problem requirement. The most useful activation functions are as follows:

$$a = p u r l \text{in} \{ n \} \; = \; n \tag{5}$$

$$a = \operatorname{tmsig}\left(n\right) = \left(2 \land \left(1 + \exp\left(-2n\right)\right)\right) - 1\tag{6}$$

$$a = radbus(n) = \exp(-n^2) \tag{7}$$

**Figure 7.** A one layer network architecture with "R" inputs and "S" neurons.

within the data. One iteration of this algorithm can be written as Eq. 8:

The feed-forward network with back-propagation (FFBP) is one of the most eminent and widespread ANNs in engineering applications [38]. In addition, it is easy to implement and solves many types of problems correctly [39]. Usually, FFBP uses tansig and purelin as activation functions in the hidden and output layers, respectively and the net input is calcu‐ lated the same as Eq. 4. FFBP operates in two steps. First, the phase in which the input information at the input nodes is propagated forward to compute the output information signal at the output layer. In other words, in this step the input data are presented to the input layer and the activation functions process the information through the layers until the network's response is generated at the output layer. Second, the phase in which adjustments to the connection strengths are made based on the differences between the computed and observed information signals at the output. In this step, the network's response is compared to the desired output and if it does not agree, an error is generated. The error signals are then transmitted back from the output layer to each node in the hidden layer(s) [40]. Then, based on the error signals received, connection weights between layer neurons and biases are updated. In this way, the network learns to reproduce outputs by learning patterns contained

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**4.1. Feed-forward network with back-propagation**

where *a* is the neuron layer output. Purelin is a linear activation function (Figure 6A) defined in Eq. (5). Tansig is hyperbolic tangent sigmoid activation function (Figure 6B) mathematically shown in Eq. (6). Radbas is Gaussian activation function (Figure 6C) shown in Eq. (7). In Figure. 7, a one layer network with *R* inputs and *S* neurons is shown [36]. The optimum number of hidden layers and the number of neurons in these layers are determined by trial and error during the training/learning process. The hidden layers in the network are used to develop the relationship between the variables. In general, multilayer networks are more powerful than single-layer networks [37].

**Figure 6.** Common Activation functions.

**Figure 7.** A one layer network architecture with "R" inputs and "S" neurons.

#### **4.1. Feed-forward network with back-propagation**

Also, to diversify the various processing elements, a bias is added to the sum of weighted

Where *n* is the net input, *w* is the weight, *p* is the input, *b* is the bias, *S* is the number of neurons

Each neuron applies an activation function to its input to determine its output signal [35]. Neurons may use any differentiable activation function to generate their output based on

where *a* is the neuron layer output. Purelin is a linear activation function (Figure 6A) defined in Eq. (5). Tansig is hyperbolic tangent sigmoid activation function (Figure 6B) mathematically shown in Eq. (6). Radbas is Gaussian activation function (Figure 6C) shown in Eq. (7). In Figure. 7, a one layer network with *R* inputs and *S* neurons is shown [36]. The optimum number of hidden layers and the number of neurons in these layers are determined by trial and error during the training/learning process. The hidden layers in the network are used to develop the relationship between the variables. In general, multilayer networks are more powerful than

in the current layer and *R* is the number of neurons in the previous layer.

problem requirement. The most useful activation functions are as follows:

n = w p + w p + w p + … + w p + b ( S,1 1 S,2 2 S,3 3 S,R R S ) (4)

*a purelin n n* = = ( ) (5)

= = - <sup>2</sup> *a radbas n n* ( ) exp( ) (7)

*a tansig n* 2/ 1 2 1 = = +- - ( ) ( ( *exp n* ( ))) (6)

inputs called net input shown in Eq. (4)

1046 Effective and Sustainable Hydraulic Fracturing

single-layer networks [37].

**Figure 6.** Common Activation functions.

The feed-forward network with back-propagation (FFBP) is one of the most eminent and widespread ANNs in engineering applications [38]. In addition, it is easy to implement and solves many types of problems correctly [39]. Usually, FFBP uses tansig and purelin as activation functions in the hidden and output layers, respectively and the net input is calcu‐ lated the same as Eq. 4. FFBP operates in two steps. First, the phase in which the input information at the input nodes is propagated forward to compute the output information signal at the output layer. In other words, in this step the input data are presented to the input layer and the activation functions process the information through the layers until the network's response is generated at the output layer. Second, the phase in which adjustments to the connection strengths are made based on the differences between the computed and observed information signals at the output. In this step, the network's response is compared to the desired output and if it does not agree, an error is generated. The error signals are then transmitted back from the output layer to each node in the hidden layer(s) [40]. Then, based on the error signals received, connection weights between layer neurons and biases are updated. In this way, the network learns to reproduce outputs by learning patterns contained within the data. One iteration of this algorithm can be written as Eq. 8:

$$X\_{k+1} = X\_k - \alpha\_k g\_k \tag{8}$$

**Figure 8.** Schematic of hydraulic fracture intersecting pre-existing natural fracture [24].

**min. horizontal stress (psi)**

**Table 1.** Comparison of XFEM code results with Warpinski and Teufel's [5] experiments

experiments.

**Angle of approach (θo)**

**Max. horizontal stress (psi)**

So, a 2D XFEM code has been developed to model hydraulic fracture propagation in naturally fractured reservoirs and interaction with natural fractures. For this purpose, firstly Warpinski and Teufel's [5] experiments have been modeled to see how much the results of the developed XFEM model for hydraulic and natural fracture interaction, are compatible with them. Table 1, presents the results of XFEM code which can be compared with Warpinski and Teufel's [5] experiments. As shown in Table 1, the results of XFEM code indicate that at high to medium angles of approach, crossing and turning into natural fracture both are observed depending on the differential stress while at low angles of approach with low to high differential stress, the predominant case during hydraulic and natural fracture interaction is hydraulic fracture diversion along natural fracture which are in good agreement with Warpinski and Teufel's [5]

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

**Horizontal differential stress (psi)**

 1000 500 500 Turn into 3.46 Turn into 1500 500 1000 Turn into 2.05 Turn into 2000 500 1500 Turn into 1.29 Turn into 1000 500 500 Turn into 1.948 Turn into 1500 500 1000 Turn into 1.201 Turn into 2000 500 1500 Crossing 0.785 Crossing 1000 500 500 Turn into 1.013 Turn into 1500 500 1000 Crossing 0.833 Crossing 2000 500 1500 Crossing 0.598 Crossing

**Experimental results [5]**

*Gturn/Gcross* **XFEM results**

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where *Xk* is a vector of current weights and biases, *gk* is the current gradient, and *α<sup>k</sup>* is the learning rate. Once the network is trained, it can then make predictions from a new set of inputs that was not used to train the network.

#### **4.2. ANN performance criteria**

There are several quantitative measures to assess ANN performance that the most usual one in a binary classification test is accuracy [41] (Fawcett 2006). To understand the meaning of accuracy, some definitions like true positive, false positive, true negative and false negative should be explained. Imagine a scenario where the occurrence of an event is considered. The test outcome can be positive (occurrence of the event) or negative (the event doesn't occur). According to this scenario:


According to above definitions:

$$\text{Accuracy} = \left(\text{TP} + \text{TN}\right) / \left(\text{TP} + \text{TN} + \text{FP} + \text{FN}\right) \tag{9}$$

In general, the accuracy of a system is a degree of closeness of the measured values to the actual (true) values [42].

#### **5. Results and discussions**

Physically, modeling hydraulic fracturing is a complicated phenomenon due to the heteroge‐ neity of the earth structure, in-situ stresses, rock behavior and the physical complexities of the problem, hence if natural fractures are added up to the problem it gets much more complex in both field operation and numerical aspects. For simplicity, it is assumed that rock is a homogeneous isotropic material and the fractures are propagating in an elastic medium under plane strain and quasi-static conditions driven by a constant and uniform net pressure throughout the hydraulic fracture system. Fracturing fluid pressure is included in the model by putting force tractions on the necessary degrees of freedom along the fracture. A schematic illustration for the problem has been presented in Figure 8 which shows that hydraulic fracture propagates toward the natural fracture and intersects with it at a specific angle of approach, *θ*, and in-situ horizontal differential stress, (σ1 -σ3).

**Figure 8.** Schematic of hydraulic fracture intersecting pre-existing natural fracture [24].

*XXg k k kk* <sup>+</sup><sup>1</sup> = a

that was not used to train the network.

**4.2. ANN performance criteria**

1048 Effective and Sustainable Hydraulic Fracturing

According to this scenario:

According to above definitions:

**5. Results and discussions**

*θ*, and in-situ horizontal differential stress, (σ1 -σ3).

(true) values [42].

where *Xk* is a vector of current weights and biases, *gk* is the current gradient, and *α<sup>k</sup>* is the learning rate. Once the network is trained, it can then make predictions from a new set of inputs

There are several quantitative measures to assess ANN performance that the most usual one in a binary classification test is accuracy [41] (Fawcett 2006). To understand the meaning of accuracy, some definitions like true positive, false positive, true negative and false negative should be explained. Imagine a scenario where the occurrence of an event is considered. The test outcome can be positive (occurrence of the event) or negative (the event doesn't occur).

**•** True Positive (TP): The event occurs and it is correctly diagnosed as it occurs;

**•** False Positive (FP): The event doesn't occur but it is incorrectly diagnosed as it occurs;

**•** True Negative (TN): The event doesn't occur and it is correctly diagnosed as it doesn't occur;

In general, the accuracy of a system is a degree of closeness of the measured values to the actual

Physically, modeling hydraulic fracturing is a complicated phenomenon due to the heteroge‐ neity of the earth structure, in-situ stresses, rock behavior and the physical complexities of the problem, hence if natural fractures are added up to the problem it gets much more complex in both field operation and numerical aspects. For simplicity, it is assumed that rock is a homogeneous isotropic material and the fractures are propagating in an elastic medium under plane strain and quasi-static conditions driven by a constant and uniform net pressure throughout the hydraulic fracture system. Fracturing fluid pressure is included in the model by putting force tractions on the necessary degrees of freedom along the fracture. A schematic illustration for the problem has been presented in Figure 8 which shows that hydraulic fracture propagates toward the natural fracture and intersects with it at a specific angle of approach,

Accuracy = TP + TN / TP + TN + FP + FN ( ) ( ) (9)

**•** False Negative (FN): The event occurs but it is incorrectly diagnosed as it doesn't occur.

(8)

So, a 2D XFEM code has been developed to model hydraulic fracture propagation in naturally fractured reservoirs and interaction with natural fractures. For this purpose, firstly Warpinski and Teufel's [5] experiments have been modeled to see how much the results of the developed XFEM model for hydraulic and natural fracture interaction, are compatible with them. Table 1, presents the results of XFEM code which can be compared with Warpinski and Teufel's [5] experiments. As shown in Table 1, the results of XFEM code indicate that at high to medium angles of approach, crossing and turning into natural fracture both are observed depending on the differential stress while at low angles of approach with low to high differential stress, the predominant case during hydraulic and natural fracture interaction is hydraulic fracture diversion along natural fracture which are in good agreement with Warpinski and Teufel's [5] experiments.


**Table 1.** Comparison of XFEM code results with Warpinski and Teufel's [5] experiments

Meanwhile debonding of natural fracture prior to hydraulic and natural fracture intersection could also be modeled which is a complicated and very interesting phenomena that has been rarely investigated. Figure 9, presents pre-existing fracture debonding before intersection with hydraulic fracture at approaching angles of 30o , 60o , 90o in Warpinski and Teufel's [5] experi‐ ments. As it is clearly observed in stress maps in Figure 9, a tensile stress is exerted ahead of hydraulic fracture tip for all of the approaching angles which makes the natural fracture debonded. In addition, the length and the position of the debonded zone vary depending on natural fracture orientation and horizontal differential stress.

**Figure 10.** Debonded zones (highlighted in red) of natural fracture at the intersecting point with hydraulic fracture at 30o (horizontal differential stress=1500 psi), 60o (horizontal differential stress=1000 psi), 90o (horizontal differential stress=1500 psi): the upper images show the coordinates of hydraulic and natural fracture relative to each other where the debonded zones are highlighted in red and the images below them are the numerical deformed configura‐

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**Figure 11.** The results of hydraulic and natural fracture interaction after intersection: the left image is a natural frac‐ ture with the orientation of 30o (horizontal differential stress=1500 psi), the middle image is a natural fracture with the orientation of 60o (horizontal differential stress=1000 psi) and the right image shows a natural fracture with the

In the second step, a FFBP neural network has been applied for predicting growing hydraulic fracturing path due to interaction with natural fracture in such a way that horizontal differ‐ ential stress, angle of approach, interfacial coefficient of friction, young's modulus of the rock and flow rate of fracturing fluid are the inputs and hydraulic fracturing path (crossing or turning into natural fracture) is the output whereas tansig is an activation function. The data

tions (magnified by 3).

orientation of 90o (horizontal differential stress=1500 psi).

**Figure 9.** Natural fracture debonding before intersecting with hydraulic fracture at 30o (horizontal differential stress=1500 psi), 60o (horizontal differential stress=1000 psi), 90o (horizontal differential stress=1500 psi): the upper images show the coordinates of hydraulic and natural fracture relative to each other where the debonded zones are highlighted in red, the middle images the are the numerical deformed configurations (magnified by 3) and the images below them are the stress maps (σxx) (magnified by 3).

Figure 10, shows the debonded zone at the intersecting point of hydraulic and natural fracture and Figure 11 presents the result of hydraulic and natural fracture interaction for approaching angles of 30o , 60o , 90o.

Investigation of Hydraulic and Natural Fracture Interaction: Numerical Modeling or Artificial Intelligence? http://dx.doi.org/10.5772/56382 1051

Meanwhile debonding of natural fracture prior to hydraulic and natural fracture intersection could also be modeled which is a complicated and very interesting phenomena that has been rarely investigated. Figure 9, presents pre-existing fracture debonding before intersection with

ments. As it is clearly observed in stress maps in Figure 9, a tensile stress is exerted ahead of hydraulic fracture tip for all of the approaching angles which makes the natural fracture debonded. In addition, the length and the position of the debonded zone vary depending on

**Figure 9.** Natural fracture debonding before intersecting with hydraulic fracture at 30o (horizontal differential stress=1500 psi), 60o (horizontal differential stress=1000 psi), 90o (horizontal differential stress=1500 psi): the upper images show the coordinates of hydraulic and natural fracture relative to each other where the debonded zones are highlighted in red, the middle images the are the numerical deformed configurations (magnified by 3) and the images

Figure 10, shows the debonded zone at the intersecting point of hydraulic and natural fracture and Figure 11 presents the result of hydraulic and natural fracture interaction for approaching

, 60o

, 90o in Warpinski and Teufel's [5] experi‐

hydraulic fracture at approaching angles of 30o

1050 Effective and Sustainable Hydraulic Fracturing

below them are the stress maps (σxx) (magnified by 3).

angles of 30o

, 60o , 90o.

natural fracture orientation and horizontal differential stress.

**Figure 10.** Debonded zones (highlighted in red) of natural fracture at the intersecting point with hydraulic fracture at 30o (horizontal differential stress=1500 psi), 60o (horizontal differential stress=1000 psi), 90o (horizontal differential stress=1500 psi): the upper images show the coordinates of hydraulic and natural fracture relative to each other where the debonded zones are highlighted in red and the images below them are the numerical deformed configura‐ tions (magnified by 3).

**Figure 11.** The results of hydraulic and natural fracture interaction after intersection: the left image is a natural frac‐ ture with the orientation of 30o (horizontal differential stress=1500 psi), the middle image is a natural fracture with the orientation of 60o (horizontal differential stress=1000 psi) and the right image shows a natural fracture with the orientation of 90o (horizontal differential stress=1500 psi).

In the second step, a FFBP neural network has been applied for predicting growing hydraulic fracturing path due to interaction with natural fracture in such a way that horizontal differ‐ ential stress, angle of approach, interfacial coefficient of friction, young's modulus of the rock and flow rate of fracturing fluid are the inputs and hydraulic fracturing path (crossing or turning into natural fracture) is the output whereas tansig is an activation function. The data set used in this model consists of around 100 data based on experimental studies [4, 5, 6]. Table 2, represents the range of the parameters used in the developed ANN.

HF turns into NF

HF crosses NF

agree with the actual case.

\* T= Turning into natural fracture \* C= Crossing natural fracture

**Angle of approach (deg)**

**Horizontal Differential Stress (psi)**

**Figure 12.** Comparison between actual case and FFBP prediction for testing subset.

**Coefficient of friction**

**Table 4.** Comparison between actual case, XFEM results and FFBP prediction

As shown in Figure 12, FFBP predictions are in prominent agreement with actual measure‐ ments which shows the high efficiency of the developed FFBP neural network approach for predicting hydraulic fracturing path due to interaction with natural fracture based on hori‐ zontal differential stress, angle of approach, interfacial coefficient of friction, young's modulus of the rock and flow rate of fracturing fluid. Finally, both XFEM and ANN approaches have been examined by a set of experimental study data [4, 6] and the results have been compared. The results of a comparison are presented in Table 4. As shown in Table 4 both of the proposed approaches yield quite promising results and in just one case ANN approach result doesn't

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

**E\*106 (psi)**

 60 0.6 1.45 8.2 e-7 T\* T T 45 0.6 1.45 8.2 e-7 T T T 30 0.68 1.3 1 e-7 T T T 90 0.68 1.3 1 e-7 T T T 60 0.38 1.218 4 e-9 T T T 30 0.38 1.218 4 e-9 T T T 913.5 60 0.38 1.218 4 e-9 T T C\* 60 0.6 1.45 8.2 e-7 C C C 1522.5 90 0.89 1.218 4 e-9 C C C 90 0.89 1.218 4 e-9 C C C

**Flow rate of fracturing fluid (m3/s)**

**Actual Case**

**XFEM Result**

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**FFBP Prediction**


**Table 2.** Range of the parameters used in the FFBP model.

The data in the database were randomly divided into two subsets. The training subset used 70% and the remaining 30%, was used for the testing subset. For standardizing the range of the input data and improves the training process the data used in network development were pre-processed by normalizing. Normalizing the data enhances the fairness of training by preventing an input with large values from swamping out another input that is equally important but with smaller values [43]. The optimal number of the neurons of a single hidden layer network for the developed ANN using trial and error method based on accuracy is 19 which is shown in Table 3. The developed FFBP neural network represented a high accuracy of 96.66% which was so promising. Also, according to the dataset, around 30 data were assigned for testing subset. Figure 12, show the results of the developed FFBP neural network predictions with actual measurements for testing subset.


**Table 3.** Developed FFBP model designing with different neurons in the hidden layer.

**Figure 12.** Comparison between actual case and FFBP prediction for testing subset.

As shown in Figure 12, FFBP predictions are in prominent agreement with actual measure‐ ments which shows the high efficiency of the developed FFBP neural network approach for predicting hydraulic fracturing path due to interaction with natural fracture based on hori‐ zontal differential stress, angle of approach, interfacial coefficient of friction, young's modulus of the rock and flow rate of fracturing fluid. Finally, both XFEM and ANN approaches have been examined by a set of experimental study data [4, 6] and the results have been compared. The results of a comparison are presented in Table 4. As shown in Table 4 both of the proposed approaches yield quite promising results and in just one case ANN approach result doesn't agree with the actual case.


\* T= Turning into natural fracture

\* C= Crossing natural fracture

set used in this model consists of around 100 data based on experimental studies [4, 5, 6]. Table

**Parameter Min Max**

The data in the database were randomly divided into two subsets. The training subset used 70% and the remaining 30%, was used for the testing subset. For standardizing the range of the input data and improves the training process the data used in network development were pre-processed by normalizing. Normalizing the data enhances the fairness of training by preventing an input with large values from swamping out another input that is equally important but with smaller values [43]. The optimal number of the neurons of a single hidden layer network for the developed ANN using trial and error method based on accuracy is 19 which is shown in Table 3. The developed FFBP neural network represented a high accuracy of 96.66% which was so promising. Also, according to the dataset, around 30 data were assigned for testing subset. Figure 12, show the results of the developed FFBP neural network

**Number of hidden neurons Accuracy (%)**

10 83.33 11 90 12 86.67 13 80 14 90 15 83.33 16 83.33 17 90 18 93.33 19 96.66 20 90

Horizontal differential stress (psi) 290 2175 Angle of approach (deg) 30 90 Interfacial coefficient of friction 0.38 1.21 Young's modulus of the rock (psi) 1.218\*106 1.45\*106 Flow rate of fracturing fluid (m3/s) 4.2\*10-9 8.2\*10-7

2, represents the range of the parameters used in the developed ANN.

**Table 2.** Range of the parameters used in the FFBP model.

1052 Effective and Sustainable Hydraulic Fracturing

predictions with actual measurements for testing subset.

**Table 3.** Developed FFBP model designing with different neurons in the hidden layer.

**Table 4.** Comparison between actual case, XFEM results and FFBP prediction

#### **6. Conclusions**

Two new numerical modeling and artificial intelligence methodologies were introduced and compared to account for hydraulic and natural fracture interaction. First a new approach has been proposed through XFEM model and an energy criterion has been applied to predict hydraulic fracture path due to interaction with natural fracture. To validate and show the efficiency of the developed XFEM code, firstly the results ob‐ tained from XFEM model have been compared with experimental studies which shows good agreement. It's been concluded that natural fracture most probably will divert hydraulic fracture at low angles of approach while at high horizontal differential stress and angles of approach of 60 or greater, the hydraulic fracture crosses the natural fracture. Meanwhile, the growing hydraulic fracture exerts large tensile stress ahead of its tip which leads to debonding of sealed natural fracture before intersecting with hydraulic fracture that is a key point to demonstrate hydraulic and natural fracture behaviors before and after intersection. Then, a FFBP neural network was developed based on horizontal differen‐ tial stress, angle of approach, interfacial coefficient of friction, young's modulus of the rock and flow rate of fracturing fluid and the ability and efficiency of the developed ANN approach to predict hydraulic fracturing path due to interaction with natural fracture was represented. The results indicate that the developed ANN is not only feasible but also yields quite accurate outcome. Finally, both of the approaches have been compared and both of them yield promising results. Numerical modeling yields more detailed results which can be used for further investigations and it can explain different observed behaviors of hydraulic fracturing in naturally fractured reservoirs as well as activation of natural fractures. Also, the potential conditions that may lead to hydraulic fracturing operation failure can be investigated through numerical modeling but it is computationally more expensive and time-consuming than artificial neural network approach. In another hand, since artificial neural network approach is mainly data-driven it can be of great use in realtime experimental studies and field hydraulic fracturing in naturally fractured reservoirs. So, as one may conclude easily, numerical modeling and artificial intelligence both have some positive and negative points; hence simultaneous use of these methods will lead to both technical and economical advantages in hydraulic fracturing operation especially in the presence of natural fractures.

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[2] Gale, J. F. W, Reed, R. M, & Holder, J. (2007). Natural fractures in the Barnett Shale and their importance for hydraulic fracture treatments, AAPG Bulletin, , 91, 603-622.

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[4] Blanton, T. L. (1982). An Experimental Study of Interaction Between Hydraulically Induced and Pre-Existing Fractures. SPE 10847. Presented at SPE/DOE unconven‐

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[7] Athavale, A. S, & Miskimins, J. L. (2008). Laboratory Hydraulic Fracturing Tests on Small Homogeneous and Laminated Blocks. 42nd US Rock Mechanics Symposium and 2nd U.S.-Canada Rock Mechanics Symposium, San Francisco, June July 2., 29.

[8] Zhou and Xue(2011). Experimental investigation of fracture interaction between nat‐ ural fractures and hydraulic fracture in naturally fractured reservoirs. SPE EU‐ ROPEC/EAGE Annual Conference and Exhibition, Vienna, Austin, May , 23-26.

[9] Akulich, A. V, & Zvyagin, A. V. (2008). Interaction between Hydraulic and Natural

[10] Jeffrey, R. G, & Zhang, X. (2009). Hydraulic Fracture Offsetting in Naturally Frac‐ tured Reservoirs: Quantifying a Long-Recognized Process. SPE Hydraulic Fracturing

[11] Rahman, M. M, Aghighi, A, & Rahman, S. S. (2009). Interaction between Induced Hy‐ draulic Fracture and Pre-Existing Natural Fracture in a Poro-elastic Environment: Ef‐ fect of Pore Pressure Change and the Orientation of Natural Fracture. SPE 122574. Presented at SPE Asia Pacific Oil and Gas Conference and Exhibition, Indonesia, Au‐

[12] Dahi TaleghaniA. and Olson, J.E. (2009). Numerical Modeling of Multi-Stranded Hy‐ draulic Fracture Propagation: Accounting for the Interaction between Induced and Natural Fractures. SPE 124884. Presented at SPE Annual Technical Conference and

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#### **Author details**

Reza Keshavarzi and Reza Jahanbakhshi

Young Researchers and Elites Club, Science and Research Branch, Islamic Azad University, Tehran, Iran
