**2. Problem statement**

Let us consider a horizontal well of radius *a* drilled in the direction of the minimum horizontal stress *σ<sup>h</sup>* . In this ideal configuration, we are interested in the initiation and propagation of a

hydraulicfracturefromaradialaxisymmetricnotchoflengthl*<sup>o</sup>* transversetothewellbore(Figure. 1). As previously mentioned, we assume that the radial notch length is sufficient to favor a transverse fracture as compared to a longitudinal one (see [3] for discussion on the effect of the stress field on the competition between transverse and longitudinal fracture initiation).

Intheabsenceofanystressbarriersorotherheterogeneities,thisaxisymmetrictransversefracture of radius *R* will transition toward a penny-shaped geometry when its radius is much greater than the wellbore radius (Figure. 1). We will denote as *p <sup>f</sup>* the fluid pressure inside the fracture (andinthewellbore), *p* = *p <sup>f</sup>* - *σ<sup>h</sup>* thenetpressureopeningthefracturefaceand*w* thecorrespond‐ ing fracture width. The axial stress variation azimuthally around the wellbore is given by [9]

$$
\sigma\_a = \sigma\_h \text{ - } 2\nu \left( \sigma\_H \text{ - } \sigma\_v \right) \Big|\_{r^2}^{a^2} \cos 2\theta \tag{1}
$$

where *r* and *θ* are the polar coordinates centered at the wellbore, *σ<sup>H</sup>* is the maximum horizontal stress, *σv* is the vertical stress, *ν* is the poisson's ratio and *a* is the wellbore radius. The azimuthal average of the axial stress (i.e. integrating the above equation from *θ* =0 to *θ* =2*π*) reduces to *σ<sup>h</sup>* . We will neglect the azimuthal variation of the axial stress close to the wellbore wall as a first approximation. The axial stress therefore reduces to the minimum horizontal stress. Under such an approximation, the model is truly axi-symmetric, i.e. independent of *θ*.

**Figure 1.** Sketch of a transverse fracture propagating from a horizontal wellbore (top), axisymmetric model (bottom)

#### **2.1. Elasticity**

horizontal stress in order to create hydraulic fractures perpendicular to the wellbore and therefore maximize the drainage from the reservoir. During the pressurization of the wellbore, tangential tensile stress is generated which can result in the initiation of longitudinal fractures parallel to the wellbore axis [1,2]. In an isotropic elastic medium, depending upon the stress field, the initiation of fractures transverse to the wellbore axis is favored by creating an initial flaw, i.e. an axisymmetric notch, of sufficient depth [3] (see Figure. 1 for a sketch). We focus

In the initial stage of propagation, these transverse fractures can be idealized as axisymmetric (radial) fractures around the wellbore until they hit a stress barrier or other type of heteroge‐ neities. We investigate the initiation and propagation of such a fracture under constant injection of a Newtonian fluid, focusing on the case of "tight" rocks where leak-off is typically negligible. In particular, we are interested in clarifying when the effects associated with the near-wellbore region dissipate and no longer affect the hydraulic fracture propagation and

The initiation and the early stage of the propagation of such a hydraulic fracture is affected by two "near wellbore" effects: i) the finiteness of the wellbore and the initial flaw length and ii) a transient phenomenon associated with the release of the fluid stored by compressibility in the wellbore during the pressurization phase prior to the initiation of the fracture. This second effect is ultimately linked to the compressibility of the injection system (i.e. mostly the fluid volume stored by compressibility within the wellbore). These effects have been investigated for the case of plane-strain fractures in [4,5] and for the case of axisymmetric hydraulic fractures driven by an inviscid fluid (i.e. zero-viscosity) in [6]. In this paper, we extend these contribu‐

A detailed dimensional analysis is performed indicating various time scales, length scales and dimensionless numbers controlling the problem. A numerical solver, along the lines of previous contributions [7,8], is presented. A series of numerical simulations are performed in order to study the transition from the near-wellbore propagation regime to the case of a radial

Let us consider a horizontal well of radius *a* drilled in the direction of the minimum horizontal stress *σ<sup>h</sup>* . In this ideal configuration, we are interested in the initiation and propagation of a hydraulicfracturefromaradialaxisymmetricnotchoflengthl*<sup>o</sup>* transversetothewellbore(Figure. 1). As previously mentioned, we assume that the radial notch length is sufficient to favor a transverse fracture as compared to a longitudinal one (see [3] for discussion on the effect of the stress field on the competition between transverse and longitudinal fracture initiation).

Intheabsenceofanystressbarriersorotherheterogeneities,thisaxisymmetrictransversefracture of radius *R* will transition toward a penny-shaped geometry when its radius is much greater than the wellbore radius (Figure. 1). We will denote as *p <sup>f</sup>* the fluid pressure inside the fracture

solely on transverse hydraulic fractures in this contribution.

tions to the case of viscous flow in axi-symmetric fractures.

hydraulic fracture in an infinite medium under constant injection rate.

how this transition takes place.

402 Effective and Sustainable Hydraulic Fracturing

**2. Problem statement**

The relation between the fracture width and the net loading acting on the crack is ex‐ pressed by a hyper-singular boundary integral equation following the method of distribut‐ ed dislocations [10]:

$$p\left(r\right) = p\_f\left(r\right) - \sigma\_h = \frac{E}{2\pi} \int\_a^r f\left(r\_r \mid r\right) \frac{\partial w\left(r\right)}{\partial r} dr \,\tag{2}$$

where the singular kernel *J*(*r*, *r '*) is given by [11]. It provides the normal stress due to an axisymmetric dislocation around a wellbore of radius *a*. Details of this kernel are recalled in Appendix A for completeness.

#### **2.2. Fluid flow**

We neglect the fluid compressibility compared to the fracture compliance as is usually done in modeling hydraulic fractures. The balance of mass then reduces to a strict volume balance:

$$\frac{\partial w}{\partial t} + \frac{1}{r} \frac{\partial}{\partial r}(rq) = 0 \tag{3}$$

The fluid flux entering the fracture is equal to the total fluid injection rate (for example the injection rate at the wellhead) minus the fluid volume stored in the well due to its compressi‐ bility (essentially the compressibility of the fluid inside the wellbore). Therefore, from the fluid mass conservation in the wellbore (between the injection point and the fracture inlet), one can

Initiation and Breakdown of an Axisymmetric Hydraulic Fracture Transverse to a Horizontal Wellbore

∂ *pb*

where *U* (*barrels* / *psi*) is the injection system compressibility and *pb* denotes the wellbore pressure. Pressure continuity is ensured at the fracture in-let by the following condition

It is important to note that no friction pressure drop (e.g. perforation drop) is taken into account

Prior to any opening of the initial defect of length l*o*, the pressurization rate is uniform (*q* =0 in Equation (7)). We thus start the simulation only when the fluid pressure in the wellbore and in the notch has reached the minimum horizontal stress, which is the pressure at which this initial defect starts to open: we will denote this start-up time *to*. For modeling purposes, at *to*, the initial condition will be taken as a vanishingly small net pressure (i.e. a fluid pressure just slightly above the minimum horizontal stress). Due to the continuous fluid injection consid‐ ered here, the wellbore pressure keeps increasing. The fracture will initiate its growth once the stress intensity factor reaches its critical value *KIc*, i.e. the fracture toughness of the rock. Once fracture initiation has occurred, we assume that the fracture propagates under quasi-static equilibrium such that the stress intensity factor is always equal to its critical value. For a pure mode I fracture considered here, this condition can be expressed as an asymptote on the

*<sup>E</sup>* ' ((<sup>l</sup> <sup>+</sup> *<sup>a</sup>*) - *<sup>r</sup>*)1/2 <sup>1</sup> - *<sup>r</sup>*

Let us scale the variables involved in the problem in order to grasp the effects of various physical phenomena acting at various scales. We introduce a characteristic length scale *L* <sup>⋆</sup>,

<sup>∂</sup> *<sup>t</sup>* ), (7)

http://dx.doi.org/10.5772/56262

405

(<sup>l</sup> <sup>+</sup> *<sup>a</sup>*) <sup>≪</sup><sup>1</sup> . (9)

*p <sup>f</sup>* (*r* =*a*)= *pb*. (8)

write the following boundary condition at the inlet of the fracture:

in this injection boundary condition.

fracture opening near the tip:

= 32 / *πKIc*

where *K* '

**3. Scaling**

**2.4. Initiation and fracture propagation condition**

*<sup>w</sup>* <sup>~</sup> *<sup>K</sup>* '

*<sup>q</sup>* <sup>=</sup> <sup>1</sup>

<sup>2</sup>*π<sup>a</sup>* (*Qo* - *U*

Under the hypothesis of lubrication theory (low Reynolds number flow) for a Newtonian fluid, the fluid flux *q* is given by Poiseuille law

$$q = -\frac{w^3}{\mu} \frac{\partial \, p}{\partial r} \tag{4}$$

where *μ '* =12*μ* with *μ* the fluid viscosity.

#### **2.3. Boundary conditions**

During the propagation of a hydraulic fracture, a fluid lag may develop at the tip of the frac‐ ture [12,13]. Such a fluid lag is larger at early time during the propagation. In order to check whether the fluid lag should be taken into account, we can estimate the characteristic time‐ scale associated with the disappearance of the fluid lag, which is equal to *tom* <sup>=</sup> *<sup>E</sup> '*<sup>2</sup> *μ ' σo* <sup>3</sup> (see [14,7]), where *σo* denotes the far-field confining stress (here the minimum horizontal stress). This timescale is inversely proportional to *σ<sup>o</sup>* <sup>3</sup> which means that for higher values of the con‐ fining stress this timescale is very small (typically the case in deep formations). For illustra‐ tion, we use stress typical of an unconventional reservoir (e.g. Barnett shale) and typical rock parameters:

$$E = 5.4 \times 10^6 \text{ psi}, \quad \nu = 0.21, \quad \sigma\_o = \sigma\_h = 3390 \text{ psi (at the depth of } 5000 ft\text{)}, \quad K\_{1c} = 1500 \text{ psi/} \sqrt{lm}. \tag{5}$$

For the case of a slick water stimulation (viscosity of 1 cP), the characteristic time *tom* is equal to 0.0014 *sec*. For a gel treatment (tangent viscosity of approximately 100 cP), this characteristic time still remains small, *tom* =0.14 *sec*. The transient effect associated with the disappearance of the fluid lag can thus be ignored for the conditions typically encountered in slick water fracturing of deep horizontal wells (i.e. in the Barnett shale). In the remaining of this paper, we assume that the fluid front coincides with the fracture front.

The conditions at the tip of the fracture are thus:

$$w(r=l+a) = 0 \qquad \qquad w\left(r=l+a\right) = 0. \tag{6}$$

The fluid flux entering the fracture is equal to the total fluid injection rate (for example the injection rate at the wellhead) minus the fluid volume stored in the well due to its compressi‐ bility (essentially the compressibility of the fluid inside the wellbore). Therefore, from the fluid mass conservation in the wellbore (between the injection point and the fracture inlet), one can write the following boundary condition at the inlet of the fracture:

$$q = \frac{1}{2\pi a} \left( \mathbf{Q}\_o - \mathbf{U} \frac{\partial \cdot p\_b}{\partial t} \right) \tag{7}$$

where *U* (*barrels* / *psi*) is the injection system compressibility and *pb* denotes the wellbore pressure. Pressure continuity is ensured at the fracture in-let by the following condition

$$p\_f\left(r=a\right) = p\_b.\tag{8}$$

It is important to note that no friction pressure drop (e.g. perforation drop) is taken into account in this injection boundary condition.

#### **2.4. Initiation and fracture propagation condition**

Prior to any opening of the initial defect of length l*o*, the pressurization rate is uniform (*q* =0 in Equation (7)). We thus start the simulation only when the fluid pressure in the wellbore and in the notch has reached the minimum horizontal stress, which is the pressure at which this initial defect starts to open: we will denote this start-up time *to*. For modeling purposes, at *to*, the initial condition will be taken as a vanishingly small net pressure (i.e. a fluid pressure just slightly above the minimum horizontal stress). Due to the continuous fluid injection consid‐ ered here, the wellbore pressure keeps increasing. The fracture will initiate its growth once the stress intensity factor reaches its critical value *KIc*, i.e. the fracture toughness of the rock. Once fracture initiation has occurred, we assume that the fracture propagates under quasi-static equilibrium such that the stress intensity factor is always equal to its critical value. For a pure mode I fracture considered here, this condition can be expressed as an asymptote on the fracture opening near the tip:

$$w \sim \frac{K}{E} (\left(\mathbb{I} + a\right) \cdot r)^{1/2} \qquad \qquad 1 \cdot \frac{r}{\left(\mathbb{I} + a\right)} \ll 1 \,\,\,\,\,\tag{9}$$

where *K* ' = 32 / *πKIc*

#### **3. Scaling**

**2.2. Fluid flow**

404 Effective and Sustainable Hydraulic Fracturing

where *μ '*

**2.3. Boundary conditions**

rock parameters:

We neglect the fluid compressibility compared to the fracture compliance as is usually done in modeling hydraulic fractures. The balance of mass then reduces to a strict volume balance:

Under the hypothesis of lubrication theory (low Reynolds number flow) for a Newtonian fluid,

During the propagation of a hydraulic fracture, a fluid lag may develop at the tip of the frac‐ ture [12,13]. Such a fluid lag is larger at early time during the propagation. In order to check whether the fluid lag should be taken into account, we can estimate the characteristic time‐

[14,7]), where *σo* denotes the far-field confining stress (here the minimum horizontal stress).

fining stress this timescale is very small (typically the case in deep formations). For illustra‐ tion, we use stress typical of an unconventional reservoir (e.g. Barnett shale) and typical

*<sup>E</sup>* =5.4×10<sup>6</sup> *psi*, *<sup>ν</sup>* =0.21, *<sup>σ</sup><sup>o</sup>* <sup>=</sup>*σ<sup>h</sup>* =3390 *psi* (at the depth of <sup>5000</sup> *ft*) , *<sup>K</sup> Ic* =1500 *psi* / *in*. (5)

For the case of a slick water stimulation (viscosity of 1 cP), the characteristic time *tom* is equal to 0.0014 *sec*. For a gel treatment (tangent viscosity of approximately 100 cP), this characteristic time still remains small, *tom* =0.14 *sec*. The transient effect associated with the disappearance of the fluid lag can thus be ignored for the conditions typically encountered in slick water fracturing of deep horizontal wells (i.e. in the Barnett shale). In the remaining of this paper,

scale associated with the disappearance of the fluid lag, which is equal to *tom* <sup>=</sup> *<sup>E</sup> '*<sup>2</sup>

*<sup>q</sup>* <sup>=</sup> - *<sup>w</sup>* <sup>3</sup> *μ* ' ∂ *p*

<sup>∂</sup> *<sup>r</sup>* (*rq*)=0 (3)

<sup>∂</sup> *<sup>r</sup>* (4)

<sup>3</sup> which means that for higher values of the con‐

*q*(*r* =l + *a*) =0 *w*(*r* =l + *a*) =0. (6)

*μ ' σo* <sup>3</sup> (see

∂ *w* <sup>∂</sup> *<sup>t</sup>* + 1 *r* ∂

the fluid flux *q* is given by Poiseuille law

=12*μ* with *μ* the fluid viscosity.

This timescale is inversely proportional to *σ<sup>o</sup>*

we assume that the fluid front coincides with the fracture front.

The conditions at the tip of the fracture are thus:

Let us scale the variables involved in the problem in order to grasp the effects of various physical phenomena acting at various scales. We introduce a characteristic length scale *L* <sup>⋆</sup>, characteristic fracture width *w*⋆, characteristic pressure *p*⋆, characteristic fluid flux *q*⋆, and a characteristic time *t*⋆ and scale the variables as follow:

$$r = L\_{\ \star} \rho, \qquad l = L\_{\ \star} \gamma, \qquad a = L\_{\ \star} \mathcal{G}\_{\nu}, \qquad p = p\_{\star} \Pi, \qquad w = w\_{\star} \Omega, \qquad q = q\_{\star} \Psi, \qquad t = t\_{\star} \tau,\tag{10}$$

*ρ*, *γ*and *a* denotes respectively the dimensionless coordinate along the fracture, the dimen‐ sionless fracture length and the dimensionless wellbore radius. The dimensionless opening, net pressure and fluid flux are denoted as Ω, Π and Ψ respectively.

Using the above scaling, the governing equations are converted to dimensionless form where the different scales are yet to be defined:

**•** Elasticity operator

$$\Pi = \frac{g\_{\epsilon}}{2\pi \mathcal{G}\_{\mathbf{z}}} \left| \int\_{\cdot}^{\mathbf{1}^{+}} \mathbf{J}(\rho\_{\nu}, \rho^{\cdot}) \frac{\partial \Omega(\rho^{\cdot})}{\partial \rho^{\cdot}} d\rho^{\cdot} \right| \tag{11}$$

where *<sup>q</sup>* <sup>=</sup> *Qo*

where *<sup>k</sup>* <sup>=</sup> *<sup>K</sup> ' <sup>L</sup>* <sup>⋆</sup>

*E ' w*<sup>⋆</sup>

in the fracture in the absence of leak-off.

**•** The LEFM propagation condition reduces to

Ω~*<sup>k</sup>* ((*γ* + *a*) - *ρ*)1/2

*<sup>U</sup>* <sup>=</sup> *<sup>U</sup> <sup>p</sup>*<sup>⋆</sup>

*<sup>L</sup>* <sup>⋆</sup>*q*<sup>⋆</sup> is a dimensionless group associated with the effect of the injection rate, and

Initiation and Breakdown of an Axisymmetric Hydraulic Fracture Transverse to a Horizontal Wellbore

is a dimensionless group associated with the rock fracture toughness.

, for(*γ* + *a*) - *ρ*≪1 (15)

http://dx.doi.org/10.5772/56262

407

*<sup>L</sup>* <sup>⋆</sup>*q*⋆*t*⋆ is a dimensionless group associated with the injection system compressibility.

First, it is natural to set the dimensionless groups associated with the injection rate *<sup>q</sup>* (we inject fluid) and elasticity *<sup>e</sup>* to unity. In setting *<sup>e</sup>* to 1, we account for the fact that the crack opening is typically much smaller than the fracture length and that the net pressure is also much smaller than the rock elastic modulus. The dimensionless group associated with fluid conservation *r* is also set to unity to account for the fact that injected fluid volume remains

In the case of the propagation of a radial hydraulic fracture in an infinite medium, energy dissipation is attributed to two competing mechanisms i.e. viscous forces associated with fluid flow within the crack and the creation of new fracture surfaces (i.e. fracture toughness) [15]. At the beginning of the propagation, i.e. for small fracture radius, viscous forces are the dominant dissipative process, and fracture toughness can be neglected in such a viscosity dominated regime of propagation. A self-similar solution has been obtained in [15] for that case, and will be denoted as the *M* -vertex solution. As time increases, fracture energy slowly takes over viscous forces as the main dissipative mechanism. Ultimately, at large time, the fracture propagates in the so-called toughness dominated regime of propagation, where vis‐ cosity can be neglected. Here again, an analytical solution exists [16], and will be referred as the *K*-vertex solution. In an infinite medium, the radial hydraulic fracture therefore transi‐

This picture is modified when accounting for near-wellbore effects. These effects will eventual‐ ly dissipate for fracture length much larger than the wellbore radius. This transition from the near-wellbore to the infinite medium solution is of particular interest. The effect of the well‐ bore and of the system compressibility will affect the system response at early time, i.e. when the radius of the fracture is comparable to that of the wellbore and when the system compressi‐ bility still has an effect. At large time, the transient associated with the fracture breakdown and the release of the fluid stored by compressibility prior to the crack initiation will become insignificant: i.e. the fluid flux entering the crack will then be equal to the injected flow rate. The

It is therefore interesting to introduce two different scaling. The first scaling relates to the case where the system compressibility and toughness are important, i.e. at early time or for small fractures. We will denote such a scaling as the Compressibility-Toughness scaling and denote it as*UK*. This scaling is based on the characteristic time of transition from the compressibility

tion from the viscosity (*M* ) to the toughness (*K*) regime of propagation.

solution will thus behave as the infinite medium solution [15,16,17].

where *<sup>e</sup>* <sup>=</sup> *<sup>E</sup> ' w*<sup>⋆</sup> *<sup>p</sup>*⋆*<sup>L</sup>* <sup>⋆</sup> is a dimensionless group associated with the elasticity operator and *<sup>a</sup>* <sup>=</sup> *<sup>a</sup> L* <sup>⋆</sup> is a dimensionless group associated with the effect of the wellbore radius.

**•** Continuity equation

$$\frac{\partial \Omega}{\partial \tau} + \mathcal{G}\_r \frac{1}{\rho} \frac{\partial}{\partial \rho} (\rho \,\Psi) = 0 \tag{12}$$

where *<sup>r</sup>* <sup>=</sup> *<sup>t</sup>*⋆*q*<sup>⋆</sup> *<sup>w</sup>*⋆*<sup>L</sup>* <sup>⋆</sup> is a dimensionless group associated with fluid conservation within the fracture.

**•** Poiseuille law for lubrication

$$\mathbf{NP} = \mathbf{1} \cdot \frac{\Omega^{\beta}}{\mathcal{G}\_{m}} \frac{\partial \Pi\_{f}}{\partial \rho} \tag{13}$$

where *<sup>m</sup>* <sup>=</sup> <sup>μ</sup>' *q*⋆*L* <sup>⋆</sup> *w*<sup>⋆</sup> <sup>3</sup> *<sup>p</sup>*<sup>⋆</sup> is a dimensionless group associated with fluid viscosity.

**•** Inlet boundary condition

$$\Psi = \frac{1}{2\pi} \left( \mathcal{G}\_q \, \Psi\_o \, \text{--} \, \mathcal{G}\_{IJ} \, \frac{\partial \, \Pi\_b}{\partial \, \tau} \right) \tag{14}$$

where *<sup>q</sup>* <sup>=</sup> *Qo <sup>L</sup>* <sup>⋆</sup>*q*<sup>⋆</sup> is a dimensionless group associated with the effect of the injection rate, and *<sup>U</sup>* <sup>=</sup> *<sup>U</sup> <sup>p</sup>*<sup>⋆</sup> *<sup>L</sup>* <sup>⋆</sup>*q*⋆*t*⋆ is a dimensionless group associated with the injection system compressibility.

**•** The LEFM propagation condition reduces to

characteristic fracture width *w*⋆, characteristic pressure *p*⋆, characteristic fluid flux *q*⋆, and a

*r* = *L* <sup>⋆</sup>*ρ*, l= *L* <sup>⋆</sup>*γ*, *a* = *L* <sup>⋆</sup>*a*, *p* = *p*⋆Π, *w* =*w*⋆Ω, *q* =*q*⋆Ψ, *t* =*t*⋆*τ*, (10)

*ρ*, *γ*and *a* denotes respectively the dimensionless coordinate along the fracture, the dimen‐ sionless fracture length and the dimensionless wellbore radius. The dimensionless opening,

Using the above scaling, the governing equations are converted to dimensionless form where

) <sup>∂</sup> <sup>Ω</sup>(*<sup>ρ</sup>* ' ) <sup>∂</sup> *<sup>ρ</sup>* ' *<sup>d</sup><sup>ρ</sup>* '

*<sup>p</sup>*⋆*<sup>L</sup>* <sup>⋆</sup> is a dimensionless group associated with the elasticity operator and *<sup>a</sup>* <sup>=</sup> *<sup>a</sup>*

*<sup>w</sup>*⋆*<sup>L</sup>* <sup>⋆</sup> is a dimensionless group associated with fluid conservation within the

) (11)

<sup>∂</sup> *<sup>ρ</sup>* (*ρ*Ψ)=0 (12)

<sup>∂</sup> *<sup>ρ</sup>* (13)

<sup>∂</sup> *<sup>τ</sup>* ) (14)

*L* <sup>⋆</sup>

characteristic time *t*⋆ and scale the variables as follow:

the different scales are yet to be defined:

406 Effective and Sustainable Hydraulic Fracturing

**•** Elasticity operator

*w*<sup>⋆</sup>

**•** Poiseuille law for lubrication

*q*⋆*L* <sup>⋆</sup> *w*<sup>⋆</sup> <sup>3</sup> *<sup>p</sup>*<sup>⋆</sup>

**•** Inlet boundary condition

**•** Continuity equation

where *<sup>e</sup>* <sup>=</sup> *<sup>E</sup> '*

where *<sup>r</sup>* <sup>=</sup> *<sup>t</sup>*⋆*q*<sup>⋆</sup>

fracture.

where *<sup>m</sup>* <sup>=</sup> <sup>μ</sup>'

net pressure and fluid flux are denoted as Ω, Π and Ψ respectively.

Π= *<sup>e</sup>* 2*π<sup>a</sup>* ( *∫* 1

1+ *γ a* J(*ρ*, *ρ* '

is a dimensionless group associated with the effect of the wellbore radius.

1 *ρ* ∂

<sup>Ψ</sup><sup>=</sup> - <sup>Ω</sup><sup>3</sup> *m* ∂ Π *<sup>f</sup>*

<sup>Ψ</sup><sup>=</sup> <sup>1</sup>

is a dimensionless group associated with fluid viscosity.

∂ Π*<sup>b</sup>*

<sup>2</sup>*<sup>π</sup>* (*q*Ψ*<sup>o</sup>* - *<sup>U</sup>*

∂ Ω <sup>∂</sup> *<sup>τ</sup>* + *<sup>r</sup>*

$$\{\Omega \text{–} \mathcal{G}\_k(\{\gamma + \mathcal{G}\_a\} \text{-} \rho\}^{1/2}, \qquad \text{ for} \\ \{\gamma + \mathcal{G}\_a\} \text{-} \rho \ll 1 \tag{15}$$

where *<sup>k</sup>* <sup>=</sup> *<sup>K</sup> ' <sup>L</sup>* <sup>⋆</sup> *E ' w*<sup>⋆</sup> is a dimensionless group associated with the rock fracture toughness.

First, it is natural to set the dimensionless groups associated with the injection rate *<sup>q</sup>* (we inject fluid) and elasticity *<sup>e</sup>* to unity. In setting *<sup>e</sup>* to 1, we account for the fact that the crack opening is typically much smaller than the fracture length and that the net pressure is also much smaller than the rock elastic modulus. The dimensionless group associated with fluid conservation *r* is also set to unity to account for the fact that injected fluid volume remains in the fracture in the absence of leak-off.

In the case of the propagation of a radial hydraulic fracture in an infinite medium, energy dissipation is attributed to two competing mechanisms i.e. viscous forces associated with fluid flow within the crack and the creation of new fracture surfaces (i.e. fracture toughness) [15]. At the beginning of the propagation, i.e. for small fracture radius, viscous forces are the dominant dissipative process, and fracture toughness can be neglected in such a viscosity dominated regime of propagation. A self-similar solution has been obtained in [15] for that case, and will be denoted as the *M* -vertex solution. As time increases, fracture energy slowly takes over viscous forces as the main dissipative mechanism. Ultimately, at large time, the fracture propagates in the so-called toughness dominated regime of propagation, where vis‐ cosity can be neglected. Here again, an analytical solution exists [16], and will be referred as the *K*-vertex solution. In an infinite medium, the radial hydraulic fracture therefore transi‐ tion from the viscosity (*M* ) to the toughness (*K*) regime of propagation.

This picture is modified when accounting for near-wellbore effects. These effects will eventual‐ ly dissipate for fracture length much larger than the wellbore radius. This transition from the near-wellbore to the infinite medium solution is of particular interest. The effect of the well‐ bore and of the system compressibility will affect the system response at early time, i.e. when the radius of the fracture is comparable to that of the wellbore and when the system compressi‐ bility still has an effect. At large time, the transient associated with the fracture breakdown and the release of the fluid stored by compressibility prior to the crack initiation will become insignificant: i.e. the fluid flux entering the crack will then be equal to the injected flow rate. The solution will thus behave as the infinite medium solution [15,16,17].

It is therefore interesting to introduce two different scaling. The first scaling relates to the case where the system compressibility and toughness are important, i.e. at early time or for small fractures. We will denote such a scaling as the Compressibility-Toughness scaling and denote it as*UK*. This scaling is based on the characteristic time of transition from the compressibility effects (*U* ) to the infinite medium solution corresponding to toughness dissipation (*K*). The characteristic scales in that Compressibility-Toughness scaling are obtained as:

$$L\_{\rm uik} = \left(\underline{E}^{\prime} \underline{U}\right)^{1/3}, \quad p\_{\rm uk} = \frac{\stackrel{\cdot}{K}^{\prime}}{\stackrel{\cdot}{E}^{1/6} \stackrel{\cdot}{U}^{1/6}}, \quad \text{bz}\_{\rm uk} = \frac{\stackrel{\cdot}{K}^{\prime} \stackrel{\cdot}{U}^{1/6}}{\stackrel{\cdot}{E}^{5/6}}, \quad q\_{\rm uk} = \frac{Q\_o}{\stackrel{\cdot}{E}^{1/3} \stackrel{\cdot}{U}^{1/3}}, \quad \text{t}\_{\rm uk} = \frac{\stackrel{\cdot}{K}^{\prime} \stackrel{\cdot}{U}^{1/6}}{\stackrel{\cdot}{E}^{1/6} Q\_o} \tag{16}$$

Now introducing =*auk* <sup>=</sup> *<sup>a</sup>*

*L mk <sup>L</sup> uk* <sup>=</sup>*<sup>χ</sup>* 2/5

> *mmk muk*

ratio of the initial defect length to the wellbore radius l<sup>o</sup>

**3.1. Field and laboratory conditions: Scaling**

and

(*E '*

numbers for the two scalings previously presented can be written as:

*muk* <sup>=</sup>χ2/5

*<sup>U</sup> mk* <sup>=</sup>χ-6/5

*wuk* <sup>=</sup>*<sup>χ</sup>* 1/5

, *<sup>U</sup> mk <sup>U</sup> uk*

the key parameters to simulate the right physics in the laboratory [18].

field conditions and the laboratory sample for illustration.

, *wmk*

=*χ* -2/5

Correspondence between the two scalings can also be obtained as the function of *χ*:

, *pmk*

=*χ* -6/5

In addition to the above mentioned factors, the ratio of dimensionless initial defect length *γ<sup>o</sup>* to the dimensionless wellbore radius also effects the hydraulic fracture initiation. It can be concluded here that the problem of axisymmetric hydraulic fracture depends only on three dimensionless parameters: the timescale ratio *χ*, dimensionless wellbore radius and the

Laboratory experiments are typically performed in order to study one particular aspect of a problem independently. Results of these experiments are then complemented by numerical and theoretical studies and ultimately verified by field experiments. Conditions in the laboratory experiments should be controlled in such a manner that they represent as close as possible a scaled version of the field conditions to be investigated. The scaling and the dimensionless parameters, described in the previous section, are thus critical in identifying

We will compare the different scales (in the Compressibility-Toughness scaling *UK*) of the problem for "typical" laboratory and field conditions in order to illustrate their differences and the need to carefully design laboratory experiments. It is worthwhile to recall that these scaling are based on the assumption of an axisymmetric fracture initiating from a circular notch. Parameters representative from the Barnett shale (5) are again considered for both the

A typical wellbore diameter (2*a*) in the field is 8.75 *in* and a typical wellbore diameter in the laboratory is 1 *in*. The constant flow rate *Qo* in the field is about 20 *barrels* / *min* and in the

*puk* <sup>=</sup>*<sup>χ</sup>* -1/5

, *amk auk*

, *qmk*

=*χ* -2/5

*<sup>a</sup>* <sup>=</sup> <sup>γ</sup><sup>o</sup> .

*quk* <sup>=</sup>*<sup>χ</sup>* -2/5

*<sup>U</sup>* )1/3 , the ratio between the wellbore radius and a lengthscale

, *auk* = (21)

, *amk* <sup>=</sup>*<sup>χ</sup>* -2/5 (22)

, (23)

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409

. (24)

associated with the volume stored in the injection system by compressibility, the dimensionless

Initiation and Breakdown of an Axisymmetric Hydraulic Fracture Transverse to a Horizontal Wellbore

In particular, the dimensionless time in that scaling is*<sup>τ</sup>* <sup>=</sup> *<sup>t</sup> tuk* . The corresponding dimensionless numbers are:

$$\mathcal{G}\_{m\_{uk}} = \frac{e^{\frac{-1}{2}\zeta Q\_{ul}\mu^{-1}}}{K^{-4}\mathcal{U}^{-15}}, \quad \mathcal{G}\_{q\_{uk}} = 1, \quad \mathcal{G}\_{\mathcal{U}\_{uk}} = 1, \; \mathcal{G}\_{k\_{uk}} = 1, \; \mathcal{G}\_{a\_{uk}} = \frac{a}{(\mathcal{E}\cdot\mathcal{U})^{15}} \tag{17}$$

The second scaling of interest corresponds to the case where the transient effects associated with the wellbore and injection compressibility have vanished: i.e. when the model reduces to the case of radial fracture propagating in an infinite medium – we will call this scaling the Viscosity-Toughness scaling and denote it as *MK*. This scaling is based on the characteristic time of transition from the viscosity dissipation (*M* ) to the toughness dissipation (*K*). The corresponding characteristic scales are [15]:

$$L\_{\
u \text{mk}} = \frac{E^{\text{-}3} \text{Q}\_{\text{v}} \mu^{\text{-}}}{K^{\text{-}4}}, \quad p\_{\text{mk}} = \frac{K^{\text{-}3}}{E^{\text{-}3/2} \sqrt{Q\_{\text{v}} \mu^{\text{-}}}}, \text{ or }\\
\mu\_{\text{mk}} = \sqrt{\frac{E^{\text{-}} Q\_{\text{v}} \mu^{\text{-}}}{K^{\text{-}2}}}, \; q\_{\text{mk}} = \frac{K^{\text{-}4}}{E^{\text{-}3} \mu^{\text{-}}}, \; t\_{\text{mk}} = \frac{E^{\text{-}132} Q\_{\text{v}}^{\text{-}3/2} \mu^{\text{-}5/2}}{K^{\text{-}9}} \tag{18}$$

The dimensionless time is here defined as *τ*˜ <sup>=</sup> *<sup>t</sup> tmk* and the different dimensionless numbers in that scaling are given as

$$\mathcal{G}\_{mmk} = 1, \quad \mathcal{G}\_{q\_{mk}} = 1, \quad \mathcal{G}\_{\mathcal{U}\_{mk}} = \frac{K^{\circ 2} \mathcal{U}}{E^{\circ} \mathcal{Q}\_{o}^{\circ} \mu^{\circ 3}}, \quad \mathcal{G}\_{k \, mk} = 1, \; \mathcal{G}\_{amk} = \frac{aK^{\circ 4}}{E^{\circ} \mathcal{Q}\_{o} \mu} \tag{19}$$

In order to grasp the transition from the early time where the near-wellbore effects are important to the large-time solution of propagation in an infinite medium, we introduce *χ* as the ratio of the timescales associated with the previously defined scalings:

$$\chi\_{\mathcal{X}} = \frac{t\_{mk}}{t\_{nk}} = \frac{E^{-20/3} Q\_o^{5/2} \mu^{5/2}}{K^{-10} L^{-56}}.\tag{20}$$

Large values of *χ*corresponds to high viscosity *μ '* and/or low injection system compressibility *U* . In this case, the transition from the near wellbore solution to the infinite medium solution occurs prior to the transition from the viscosity (*M* ) to the toughness (*K*) dominated regime of propagation of an infinite radial hydraulic fracture. Smaller values of *χ* correspond to lower viscosity/higher injection system compressibility. In this case the transition from near wellbore solution to the infinite medium solution occurs in the *K* regime of the infinite medium solution. Now introducing =*auk* <sup>=</sup> *<sup>a</sup>* (*E ' <sup>U</sup>* )1/3 , the ratio between the wellbore radius and a lengthscale associated with the volume stored in the injection system by compressibility, the dimensionless numbers for the two scalings previously presented can be written as:

$$\mathcal{G}\_{muk} = \chi^{2/5}, \quad \mathcal{G}\_{auk} = \mathcal{\mathcal{A}} \tag{21}$$

and

effects (*U* ) to the infinite medium solution corresponding to toughness dissipation (*K*). The

*<sup>U</sup>* 1/3 , *quk* =1, *<sup>U</sup> uk* =1, *<sup>k</sup> uk* =1, *auk* <sup>=</sup> *<sup>a</sup>*

The second scaling of interest corresponds to the case where the transient effects associated with the wellbore and injection compressibility have vanished: i.e. when the model reduces to the case of radial fracture propagating in an infinite medium – we will call this scaling the Viscosity-Toughness scaling and denote it as *MK*. This scaling is based on the characteristic time of transition from the viscosity dissipation (*M* ) to the toughness dissipation (*K*). The

*Qoμ* '

*<sup>K</sup>* '2 , *qmk* <sup>=</sup> *<sup>K</sup>* '4

*E* '3

*<sup>μ</sup>* '3 , *<sup>k</sup> mk* =1, *amk* <sup>=</sup> *aK* '4

, *wmk* <sup>=</sup> *<sup>E</sup>* '

*E* '8 *Qo* 3

the ratio of the timescales associated with the previously defined scalings:

*<sup>χ</sup>* <sup>=</sup> *tmk*

*tuk* <sup>=</sup> *<sup>E</sup>* '20/3

In order to grasp the transition from the early time where the near-wellbore effects are important to the large-time solution of propagation in an infinite medium, we introduce *χ* as

> *Qo* 5/2 *μ* '5/2

*U* . In this case, the transition from the near wellbore solution to the infinite medium solution occurs prior to the transition from the viscosity (*M* ) to the toughness (*K*) dominated regime of propagation of an infinite radial hydraulic fracture. Smaller values of *χ* correspond to lower viscosity/higher injection system compressibility. In this case the transition from near wellbore solution to the infinite medium solution occurs in the *K* regime of the infinite medium solution.

*U* 1/6

*<sup>E</sup>* '5/6 , *quk* <sup>=</sup> *Qo*

*<sup>E</sup>* '1/3 *<sup>U</sup>* 1/3 , *tuk* <sup>=</sup> *<sup>K</sup>* '

(*E* '

*<sup>μ</sup>* ' , *tmk* <sup>=</sup> *<sup>E</sup>* '13/2

*Qo* 3/2 *μ* '5/2

*tmk* and the different dimensionless numbers in

*E* '3

*<sup>K</sup>* '10*<sup>U</sup>* 5/6 . (20)

and/or low injection system compressibility

*U* '5/6 *E* '1/6 *Qo*

*<sup>U</sup>* )1/3 (17)

*<sup>K</sup>* '9 (18)

*Qoμ* ' (19)

*tuk* . The corresponding dimensionless

(16)

characteristic scales in that Compressibility-Toughness scaling are obtained as:

*<sup>E</sup>* '1/6 *<sup>U</sup>* 1/6 , *wuk* <sup>=</sup> *<sup>K</sup>* '

*L uk* =(*E* '

*<sup>L</sup> mk* <sup>=</sup> *<sup>E</sup>* '3

that scaling are given as

numbers are:

*U* )1/3

408 Effective and Sustainable Hydraulic Fracturing

*muk* <sup>=</sup> *<sup>E</sup>* '8/3

corresponding characteristic scales are [15]:

*<sup>K</sup>* '4 , *pmk* <sup>=</sup> *<sup>K</sup>* '3

The dimensionless time is here defined as *τ*˜ <sup>=</sup> *<sup>t</sup>*

Large values of *χ*corresponds to high viscosity *μ '*

*<sup>E</sup>* '3/2 *Qoμ* '

*mmk* =1, *qmk* =1, *<sup>U</sup> mk* <sup>=</sup> *<sup>K</sup>* '12*<sup>U</sup>*

*Qoμ* '

, *puk* <sup>=</sup> *<sup>K</sup>* '

*Qoμ* ' *K* '4

In particular, the dimensionless time in that scaling is*<sup>τ</sup>* <sup>=</sup> *<sup>t</sup>*

$$\mathcal{G}\_{\rm LI\\_mk} = \chi^{-6/5}\prime\prime \mathcal{G}\_{a\_{mk}} = \mathcal{\mathcal{A}}\chi^{-2/5} \tag{22}$$

Correspondence between the two scalings can also be obtained as the function of *χ*:

$$\frac{L\_{\phantom{mk}}}{L\_{\phantom{mk}}} = \chi^{2/5}, \quad \frac{w\_{mk}}{w\_{uk}} = \chi^{1/5}, \quad \frac{p\_{mk}}{p\_{uk}} = \chi^{-1/5}, \quad \frac{q\_{mk}}{q\_{uk}} = \chi^{-2/5}, \tag{23}$$

$$\frac{\mathcal{G}\_{m\boldsymbol{m}}}{\mathcal{G}\_{m\boldsymbol{m}}} = \chi^{-2/5} \quad \frac{\mathcal{G}\_{\boldsymbol{U}\boldsymbol{m}}}{\mathcal{G}\_{\boldsymbol{U}\boldsymbol{m}}} = \chi^{-6/5} \quad \frac{\mathcal{G}\_{\boldsymbol{s}\boldsymbol{m}}}{\mathcal{G}\_{\boldsymbol{s}\boldsymbol{m}}} = \chi^{-2/5}.\tag{24}$$

In addition to the above mentioned factors, the ratio of dimensionless initial defect length *γ<sup>o</sup>* to the dimensionless wellbore radius also effects the hydraulic fracture initiation. It can be concluded here that the problem of axisymmetric hydraulic fracture depends only on three dimensionless parameters: the timescale ratio *χ*, dimensionless wellbore radius and the ratio of the initial defect length to the wellbore radius l<sup>o</sup> *<sup>a</sup>* <sup>=</sup> <sup>γ</sup><sup>o</sup> .

#### **3.1. Field and laboratory conditions: Scaling**

Laboratory experiments are typically performed in order to study one particular aspect of a problem independently. Results of these experiments are then complemented by numerical and theoretical studies and ultimately verified by field experiments. Conditions in the laboratory experiments should be controlled in such a manner that they represent as close as possible a scaled version of the field conditions to be investigated. The scaling and the dimensionless parameters, described in the previous section, are thus critical in identifying the key parameters to simulate the right physics in the laboratory [18].

We will compare the different scales (in the Compressibility-Toughness scaling *UK*) of the problem for "typical" laboratory and field conditions in order to illustrate their differences and the need to carefully design laboratory experiments. It is worthwhile to recall that these scaling are based on the assumption of an axisymmetric fracture initiating from a circular notch. Parameters representative from the Barnett shale (5) are again considered for both the field conditions and the laboratory sample for illustration.

A typical wellbore diameter (2*a*) in the field is 8.75 *in* and a typical wellbore diameter in the laboratory is 1 *in*. The constant flow rate *Qo* in the field is about 20 *barrels* / *min* and in the laboratory setting is around 5 *ml* / *min*. The pressurization rate *β* before breakdown in the field will be taken as 60 *psi* /*s* due to higher pumping rates in the field whereas in the laboratory it is about 1.2 *psi* /*s*. The wellbore compressibility *U* results from the compressibility of the fluid in the wellbore and the injection lines, as well as the compressibility of the wellbore and injection lines themselves. It is expressed as the ratio between the constant flow rate and the pressurization rate *U* =*Qo* / *β*. The typical fluid used in the field is slick water [19] with a viscosity of 1 *cP*. We consider glycerin as the fluid used in the laboratory with viscosity 1000 *cP*.

experimental campaign. For example, if the goal is to study hydraulic fracture propagation then the compressibility effects must be reduced in order to speed up the convergence to the infinite medium solution. These effects can be reduced by manipulating the material of the test block, using smaller injection lines and by using needle control valves as it was done for

Initiation and Breakdown of an Axisymmetric Hydraulic Fracture Transverse to a Horizontal Wellbore

The governing equations are solved in their dimensionless form described in Section 3. The elasticity equation is discretized by Displacement Discontinuity Method (DDM) using piecewise constant elements with the tip element correction [22] for better accuracy. The fluid

At time *tn*, the fracture length is given as *γn*, fracture opening as Ω*n* and net pressure Π*n*. The algorithm consists of two nested loops. In the outer loop (time-stepping loop), a fracture increment Δ*γ* =Δ*ρ* is specified and the time step Δτn, for which the tip asymptotic condition (15) is satisfied, is found iteratively. In the inner loop (i.e. the Reynolds solver), the coupled system ofelasticityandlubricationequationsaresolvedinordertofindthefractureopeningandpressure profilescorrespondingtoagivenfracturelengthand(trial)timestep.Detailsofthetimestepping

The problem of the initiation of an axisymmetric hydraulic fracture from a wellbore depends only on three dimensionless parameters i.e. the timescale ratio *χ*, dimensionless wellbore

space for these three dimensionless quantities is now explored. We aim to see their effect on the transition to the analytical solution of a penny-shaped hydraulic fracture propagating in an infinite medium [15,17] as well as on the breakdown pressure (i.e. the maximum pressure

In Figure. 2, the evolution of fracture length *γ*, inlet opening Ω(0) and wellbore pressure Π*wb*

is done in order to investigate the transition to the infinite medium solution. These results are presented in the Viscosity-Toughness (*MK*) scaling. The infinite medium solution goes from the viscosity dominated dissipation (*M* vertex solution) to the toughness dominated dissipation (*K* vertex solution). This transition is called the *MK* edge of the semi-infinite me‐ dium solution. It can be observed in Figure. 2 that fracture length, opening and wellbore pressure of our numerical solution accounting for the wellbore effects converges at large time to the *MK* edge solution in an infinite medium. The timescales ratio *χ* governs where the transient effects end along the MK edge of the infinite medium solution. For larger value

are plotted for different values of the timescale ratio *<sup>χ</sup>* for values of =0.1 and *γ<sup>o</sup>*

*<sup>a</sup>* <sup>=</sup> <sup>γ</sup><sup>o</sup>

. The parameter

http://dx.doi.org/10.5772/56262

411

=0.3. This

loop are given in Appendix B and the Reynolds Solver is described in Appendix C.

radius and the ratio of the initial defect length and the wellbore radius l<sup>o</sup>

example in [20,21].

**4. Numerical algorithm**

**5. Results and discussion**

recorded).

flow is discretized by the finite volume method.

The compressibility length scale *L uk* and time scale *tuk* corresponding to these parameters are displayed in Table 1. It is shown in the previous section that the dimensionless parameters depend upon the two dimensionless numbers i.e. and *χ*. These dimensionless numbers are also given in Table 1. While *L uk* and *tuk* give the length and the timescale associated with the compressibility effects, the value of *χ* determine the infinite medium propagation regime after the dissipation of compressibility effects. Large value of *χ* i.e. (*χ* >1) means that the fracture will propagate in the viscosity regime whereas, smaller values of *χ* i.e. (*χ* <1) means that the fracture will propagate in the toughness regime after the dissipation of compressibility effects.


**Table 1.** Characteristic length scales and dimensionless parameters for the field and the laboratory conditions.

It is obvious from Table 1, that in the field, the fracture propagates in the viscosity dominated regime (*χ* >1) whereas in the laboratory for the parameters chosen here, the fracture propagates in the toughness dominated regime (*χ* <1) after the dissipation of the early-time compressi‐ bility effects. For the field conditions, the compressibility length scale is equal to 56 *ft*. Which means that the propagation in the field is dominated by the compressibility effect until the fracture reaches about 56 *ft* (i.e. 150 times the wellbore radius) in to the formation. Even though this is a large value as compared to the wellbore radius, the length scale is still small as compared to the final fracture length which may be of the order of 800 to 1000 *ft* in the field. For laboratory conditions, the length scale is 2.4 *ft* (i.e. about 60 times the wellbore radius) which is smaller as compared to the field conditions, still the length-scale is large enough as compared to the specimen size that entire fracture propagation is dominated by the injection system compressibility.

The previous example has emphasized the differences with field conditions for a particular set of experimental parameters. However, these laboratory parameters can be appropriately adjusted in order to study a given regime of propagation. There can be different goals for an experimental campaign. For example, if the goal is to study hydraulic fracture propagation then the compressibility effects must be reduced in order to speed up the convergence to the infinite medium solution. These effects can be reduced by manipulating the material of the test block, using smaller injection lines and by using needle control valves as it was done for example in [20,21].
