**2. UFM model specifics**

A complex fracture network model, referred to as Unconventional Fracture Model (UFM), had recently been developed [22,23,24]. The model simulates the fracture propagation, rock deformation, and fluid flow in the complex fracture network created during a fracture treatment. The model solves the fully coupled problem of fluid flow in the fracture network and the elastic deformation of the fractures, which has similar assumptions and governing equations as conventional pseudo-3D fracture models. Transport equations are solved for each component of the fluids and proppants pumped. A key difference between UFM and the conventional planar fracture model is being able to simulate the interaction of hydraulic fractures with pre-existing natural fractures, i.e., determine whether a hydraulic fracture propagates through or is arrested by a natural fracture when they intersect and subsequently propagates along the natural fracture.

To properly simulate the propagation of multiple or complex fractures, the fracture model takes into account the interaction among adjacent hydraulic fracture branches, often referred to as "stress shadow" effect. It is well known that when a single planar hydraulic fracture is opened under a finite fluid net pressure, it exerts a stress field on the surrounding rock that is proportional to the net pressure. The details of stress shadow effect implemented in UFM are presented in [24].

The branching of the hydraulic fracture at the intersection with the natural fracture gives rise to the development of a complex fracture network. A crossing model that is extended from the Renshaw-Pollard [10] interface crossing criterion, applicable to any intersection angle, has been developed [14], validated against the experimental data [15], and was integrated in the UFM. The previous crossing model, showing good comparison with existing experimental data, does not account for the fluid impact on the crossing pattern.

The new crossing model (OpenT) which accounts for the fluid properties is presented in short below and is implemented in a new version of UFM.

### **3. New crossing model in UFM**

with high viscosity. Their results show that an increase in injection rate greatly increases the amount of tensile failure within the model leading potentially to creating more fractures, while a lower injection rate favours the creation of shear failure resulting mostly in activating

A new analytical model, called OpenT, for hydraulic fracture interaction with a pre-existing discontinuity has been developed to predict the fracture crossing or deflection at the encoun‐ tered interface [20, 21]. The new physically rigorous criterion of fracture re-initiation at the discontinuity has been implemented, which combines both stress criterion and energy release rate. It has been shown that the OpenT model adequately predicts the fracture crossing of noncohesive frictional interfaces observed in various laboratory experiments with different

The new crossing model predicts the dimensions of open and sliding zones created at cohesive and non-cohesive interfaces after the intersection with a fluid-driven fracture. Such informa‐ tion can be valuable, for example, in passive microseismic monitoring of fracture treatments in naturally fractured formations. By thoroughly examining the stress field generated by the hydraulic fracture and activated open and sliding zones at the discontinuity, it was shown that the new fracture initiation point is shifted along the inclined interface. The model predicts the offset of a secondary fracture as a function of the geometrical, loading, and mechanical parameters of the system, such as the fracture-interaction angle, in-situ stress components and

New OpenT crossing model incorporates the influence of rock properties (local horizontal stresses, rock tensile strength, toughness, pore pressure, Young's modulus, Poisson ratio), natural fracture properties (friction coefficient, toughness, cohesion, permeability), intersec‐ tion angle between hydraulic and natural fractures, fracturing fluid properties (viscosity, tip

This new OpenT model has been validated against laboratory experiments and against rigorous numerical models [3,20,21]. It was incorporated into the UFM model that simulates complex fracture network propagation in a formation with pre-existing natural fractures [22-24]. We present several UFM test cases showing the influence of injection rate and fluid viscosity on the generated hydraulic fracture footprint and production impact by comparing of two crossing criteria, the extended Renshaw & Pollard (hereafter referred as eRP) [14, 15]

A complex fracture network model, referred to as Unconventional Fracture Model (UFM), had recently been developed [22,23,24]. The model simulates the fracture propagation, rock deformation, and fluid flow in the complex fracture network created during a fracture treatment. The model solves the fully coupled problem of fluid flow in the fracture network and the elastic deformation of the fractures, which has similar assumptions and governing

(opening) pre-existing natural fractures.

186 Effective and Sustainable Hydraulic Fracturing

fracture toughness in rock.

and the OpenT.

**2. UFM model specifics**

interface orientations with respect to hydraulic fractures [21].

pressure), and injection rate to define crossing rules.

There are a few analytical criteria describing the mechanical HF-NF interaction devel‐ oped in the past [10, 11, 13, 14]. With their relative simplicity they do not take into account the influence of the fluid injection into the hydraulic fracture and the fluid infiltration into the natural fracture after contact. These criteria were designed to capture the effect of the fracture approach angle, the NF friction coefficient and the anisotropy of the in-situ stresses. To improve the description of HF-NF interaction a new analytical model that takes the mechanical influence of the HF opening and the hydraulic permeability of the NF into account has been developed.

The analytical model of the HF-NF interaction (OpenT) solves the problem of the elastic perturbation of the NF at the contact with the blunted HF tip, which is represented by a uniformly open slot (i.e. giving its name OpenT) [21]. The opening of the HF at the junction point *w*T (blunted tip) develops soon after contact, and approaches the value of the average opening of the hydraulic fracture *<sup>w</sup>*¯, defined by the injection rate *<sup>Q</sup>* and the fluid viscosity *μ*. In a viscosity-dominated regime, the average opening of the KGD fracture with half-length *L* and height *H* can be estimated as [25]

$$
\overline{w} = 2.53 \left[ \frac{\mathbb{Q} \mu L^2}{E'H} \right]^{1/4} \tag{1}
$$

positions in most situations correspond to the two opposite tips of the NF open zone (see Figure 2, right). In order to decide on the possibility of a secondary fracture (SF) re-initiation at these points, a criterion of fracture initiation which combines both stress criterion and energy release rate has been employed. The stress criterion requires that the maximum tensile hoop

the orientation of the NF must exceed the tensile strength of the rock *T0* along the distance *δ<sup>T</sup>*

<sup>0</sup> ( ,, ) *<sup>T</sup> j jr xr T*

<sup>1</sup> () , *inc <sup>C</sup> <sup>T</sup>* Á >Á <

d

 q

d

In addition, the energy criterion states that the elastic energy release rate due to the incremental initiation of a fracture of length *δl* must overcome the critical energy release rate for the given

> dd

The length of the fracture must not exceed the critical stress zone, *δT*. The mixed stress-energy

The model of HF-NF re-initiation has been validated against the results of various laboratory block tests [11, 12, 15]. The predictions of the analytical model for crossing and arresting behaviour agree with the experimental results for various fracture intersection angles, stress contrasts and fluid injection conditions used in different experimental groups. Figure 3 shows the comparison between different analytical models [13, 14, 21], and the experimental results

The experiments clearly show that the new model agrees with the experiments as well as other analytical models as it captures the first order crossing-arresting behavior. We note that the discrimination between the different models would require additional data points in the

Additionally, it should be noted that the injection rate and viscosity were not changed in this series of experiments, and so it was not possible to assess their effect on the fracture interaction outcome. In order to compensate for this lack of lab experiments, numerical experiments were conducted using MineHF2D code [4, 5, 8] to assess the sensitivity of the injection rate on fracture crossing. The results are demonstrated on Figure 4 and show that the OpenT model [20,21] agrees well with numerical computation results in the sense that it captures the

It should be mentioned that the OpenT incorporates the influence of rock properties (local horizontal stresses, rock tensile strength, toughness, pore pressure, Young's modulus, Poisson ratio), natural fracture properties (friction coefficient, toughness, cohesion, permeability), intersection angle between hydraulic and natural fractures, fracturing fluid properties (viscosity, tip pressure), and injection rate to define crossing rules [21]. The eRP criterion

having direction *θ<sup>j</sup>*

Effect of Flow Rate and Viscosity on Complex Fracture Development in UFM Model

<sup>&</sup>lt; £ - (3)

*l l* (4)

with respect to

189

http://dx.doi.org/10.5772/56406

stress *σθθ* in the vicinity of the stress concentration point *xj*

criterion has been verified experimentally [26].

transition zone, unfortunately not available here.

crossing-arresting transition.

rock

from [15].

qq

s

where *E* ′ =*E* / (1−*ν* <sup>2</sup> ), *E* is the Young modulus, *ν* is the Poisson coefficient. The OpenT model looks for the solution of the elastic problem for the NF perturbed by the HF, and outputs the profiles and boundaries of the opening and sliding zones as a result of the contact (*bo* and *bs* respectively shown in Figure 2, left).

The solution shows that the spatial extent of the open and sliding zones strongly depends on the fluid pressure inside the activated part of the NF. The larger the inner fluid pressure, the larger the open and sliding zones at the NF are. Consequently, it is expected that after the HF-NF contact, the injected fracturing fluid will gradually penetrate the NF with finite hydraulic permeability κ and thus enhance the inner fluid pressure within the NF, *pNF*.

**Figure 2.** Left –Schematic diagram of the HF-NF interaction. Right – result of the computed HF/NF interaction with the initiation of two secondary fractures and their subsequent propagation

The average pressure of the fracturing fluid penetrated the NF can be as approximated by the following function of the contact time *t*

$$p\_{NF}(t) = p\_f \tanh\left(\sqrt{\frac{2\kappa p\_f}{\mu b\_s^2}}t\right) \tag{2}$$

where *pf* is the fluid pressure at the contacting HF tip. As a result of the NF activation due to the fracturing, the fluid penetration becomes very active in highly permeable NFs or with low viscosity fracturing fluids. This could potentially prevent the HF from propagation across the weak interfaces.

The elasticity model of the fracture interaction enables the computation of the stress field in the vicinity of the activated NF. The analysis of the generated stress field gives the positions of sufficient tensile stress concentration where the new fractures can be nucleated. These positions in most situations correspond to the two opposite tips of the NF open zone (see Figure 2, right). In order to decide on the possibility of a secondary fracture (SF) re-initiation at these points, a criterion of fracture initiation which combines both stress criterion and energy release rate has been employed. The stress criterion requires that the maximum tensile hoop stress *σθθ* in the vicinity of the stress concentration point *xj* having direction *θ<sup>j</sup>* with respect to the orientation of the NF must exceed the tensile strength of the rock *T0* along the distance *δ<sup>T</sup>*

1/4 <sup>2</sup>

), *E* is the Young modulus, *ν* is the Poisson coefficient. The OpenT model

(1)

(2)

*E H* é ù m<sup>=</sup> ê ú ¢ ê ú ë û

looks for the solution of the elastic problem for the NF perturbed by the HF, and outputs the profiles and boundaries of the opening and sliding zones as a result of the contact (*bo* and *bs*

The solution shows that the spatial extent of the open and sliding zones strongly depends on the fluid pressure inside the activated part of the NF. The larger the inner fluid pressure, the larger the open and sliding zones at the NF are. Consequently, it is expected that after the HF-NF contact, the injected fracturing fluid will gradually penetrate the NF with finite hydraulic

**Figure 2.** Left –Schematic diagram of the HF-NF interaction. Right – result of the computed HF/NF interaction with the

The average pressure of the fracturing fluid penetrated the NF can be as approximated by the

( ) tanh *<sup>f</sup>*

ç ÷ <sup>=</sup> ç ÷

where *pf* is the fluid pressure at the contacting HF tip. As a result of the NF activation due to the fracturing, the fluid penetration becomes very active in highly permeable NFs or with low viscosity fracturing fluids. This could potentially prevent the HF from propagation across the

The elasticity model of the fracture interaction enables the computation of the stress field in the vicinity of the activated NF. The analysis of the generated stress field gives the positions of sufficient tensile stress concentration where the new fractures can be nucleated. These

*<sup>p</sup> ptp t <sup>b</sup>*

*NF f*

2 2

k

æ ö

m

è ø

*s*

2.53 *Q L <sup>w</sup>*

permeability κ and thus enhance the inner fluid pressure within the NF, *pNF*.

initiation of two secondary fractures and their subsequent propagation

following function of the contact time *t*

weak interfaces.

where *E* ′

=*E* / (1−*ν* <sup>2</sup>

188 Effective and Sustainable Hydraulic Fracturing

respectively shown in Figure 2, left).

$$\sigma\_{\partial\theta}(\mathbf{x}\_{\circ}, r, \theta\_{\circ})\_{r<\delta\_{\Gamma}} \le -T\_0 \tag{3}$$

In addition, the energy criterion states that the elastic energy release rate due to the incremental initiation of a fracture of length *δl* must overcome the critical energy release rate for the given rock

$$\mathfrak{T}\_{\rm inc}(\delta l) > \mathfrak{T}\_{1C'} \quad \delta l < \delta\_T \tag{4}$$

The length of the fracture must not exceed the critical stress zone, *δT*. The mixed stress-energy criterion has been verified experimentally [26].

The model of HF-NF re-initiation has been validated against the results of various laboratory block tests [11, 12, 15]. The predictions of the analytical model for crossing and arresting behaviour agree with the experimental results for various fracture intersection angles, stress contrasts and fluid injection conditions used in different experimental groups. Figure 3 shows the comparison between different analytical models [13, 14, 21], and the experimental results from [15].

The experiments clearly show that the new model agrees with the experiments as well as other analytical models as it captures the first order crossing-arresting behavior. We note that the discrimination between the different models would require additional data points in the transition zone, unfortunately not available here.

Additionally, it should be noted that the injection rate and viscosity were not changed in this series of experiments, and so it was not possible to assess their effect on the fracture interaction outcome. In order to compensate for this lack of lab experiments, numerical experiments were conducted using MineHF2D code [4, 5, 8] to assess the sensitivity of the injection rate on fracture crossing. The results are demonstrated on Figure 4 and show that the OpenT model [20,21] agrees well with numerical computation results in the sense that it captures the crossing-arresting transition.

It should be mentioned that the OpenT incorporates the influence of rock properties (local horizontal stresses, rock tensile strength, toughness, pore pressure, Young's modulus, Poisson ratio), natural fracture properties (friction coefficient, toughness, cohesion, permeability), intersection angle between hydraulic and natural fractures, fracturing fluid properties (viscosity, tip pressure), and injection rate to define crossing rules [21]. The eRP criterion [14,15] accounts for the local stress field, pore pressure, crossing angle, rock tensile strength and frictional properties of the natural fractures.

This new model has been implemented in UFM. Below we present comparison of UFM results with new and old crossing models and provide some analysis about the influence of fluid properties on the geometry of stimulated fracture network, and as a result on the production

Effect of Flow Rate and Viscosity on Complex Fracture Development in UFM Model

http://dx.doi.org/10.5772/56406

191

**4. Comparison of hydraulic fracturing simulations with OpenT versus eRP**

The comparison of results generated using two crossing criteria - eRP criterion [14,15] and new OpenT criterion [20, 21] - is presented in Figure 5 and Figure 6 for a simple example given in Table 1 (values shown in italic are used only in OpenT crossing criterion). The cohesion and

For the case of lower fluid viscosity (Figure 5a and Figure 6a) both criteria show similar hydraulic fracture patterns with no crossing of the natural fractures. For higher viscosity fluid OpenT crossing criterion shows that hydraulic fractures cross the NF#1 and NF#3 (Figure 6b), while with eRP the hydraulic fracture network (HFN) pattern does not change. The intersection angle between HF and NF#1 was 62.5 deg, between HF and NF#2 was 15 deg, and

toughness of natural fracture are considered to be negligible.

*Injection rate* 0.13 m3/s

Stress anisotropy 0.9 MPa

Poisson's ratio 0.2

Fluid Specific Gravity 1.0

Young's modulus 2.8 ×1010 Pa

*Fluid viscosity* 0.001-0.01 Pa-s

Fracture toughness 1.3 MPa-m0.5

Tensile strength 3.5 MPa

*NF permeability* 1 Darcy

NF friction Coefficient 0.5

the interaction angle between HF and NF#3 was 75 deg.

predictions.

**4.1. Influence of viscosity**

**Table 1.** Input data Example 1

**Figure 3.** Comparison between analytical models given in [13,14,15], OpenT [21], and the experimental results [15]

**Figure 4.** Comparison between numerical crossing-arresting HF-NF behavior using MineHF2D code [4,5,8]. The red crosses and squares respectively indicate crossing and arresting behavior from MineHF2D code, solid green curves cor‐ respond to analytical predictions using OpenT [21], dash yellow curve corresponds to Blanton criterion [13], and eRP criterion [14,15] is given by dash blue lines. The interaction is studied for various injection rate and relative stress dif‐ ference for two different HF-NF contact angles, β=90⁰ (left) and β=60⁰ (right).

This new model has been implemented in UFM. Below we present comparison of UFM results with new and old crossing models and provide some analysis about the influence of fluid properties on the geometry of stimulated fracture network, and as a result on the production predictions.
