**Fluid-Driven Fracture in a Poroelastic Rock**

**Fluid-Driven Fracture in a Poroelastic Rock**

Yevhen Kovalyshen and Emmanuel Detournay

Additional information is available at the end of the chapter

Yevhen Kovalyshen and Emmanuel Detournay

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/56460

**1. Introduction**

10.5772/56460

Hydraulic fracturing is commonplace in the geo-industry, whether designed or unintended; e.g., stimulation of hydrocarbons reservoirs [1, 2], disposal of waste water [3], waterflooding operations [4], enhanced oil recovery by injection of CO2 [5], and preconditioning of rock mass in the mining industry [6, 7]. Nonetheless, modeling of hydraulic fracturing usually relies on oversimplified assumptions [1, 8]; in particular, fluid leak-off is often studied within Carter's model [9] that assumes that the transport of the filtrate and the porous fluid through the porous medium is one-dimensional. While this assumption is quite reasonable in the case of short treatments such as hydraulic fracturing of a hydrocarbons reservoir [2], it is unlikely to be applicable for injection operations over long periods of time.

This study is part of an ongoing effort to rigorously introduce large-scale 3D diffusion in a model of hydraulic fracture. The increase of pore pressure around the fracture caused by fluid leak-off from the fracture leads to an expansion of the porous medium. This expansion can be accounted for by the introduction of the so-called backstress [10, 11]. By definition, the backstress would be the stress induced across the fracture plane if the fracture were closed. Here we restrict our investigation to the toughness-dominated regime of propagation, for which the viscosity of the fluid is negligible. In other words we assume that the energy spent for hydraulic fracturing is mainly due to the rock damage rather than due to dissipation associated with viscous flow of the fracturing fluid. Setting the fracturing fluid viscosity to be equal to zero implies that the fluid pressure inside the fracture is uniform.

Previous works on the toughness-dominated regimes with leak-off include a detailed examination of the case of the Carter's leak-off model by means of scaling and asymptotic analyses [12] and an analysis of a "stationary" 3D leak-off under conditions of very slow fracture propagation, when the pore pressure around the fracture is always in equilibrium [13]. A model for the plane strain propagation of a natural fracture through a porous medium was proposed by [14], who introduced an efficient approach to calculate of the fluid exchange

of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Kovalyshen and Detournay; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

©2012 Kovalyshen and Detournay, licensee InTech. This is an open access chapter distributed under the terms

volume between the fracture and the medium. This approach relies on decomposing the evolving pressure loading inside the growing fracture into a series of pressure impulses and then on representing the actual fluid exchange volume by the superposition of fluid exchanges induced by a single impulse [11]. Despite algebraic errors in the main expression for the fluid exchange volume (equation (7) of [14]), the idea introduced by these authors is at the core of the approach summarized in this paper and described in more details in [15].

10.5772/56460

609

Fluid-Driven Fracture in a Poroelastic Rock http://dx.doi.org/10.5772/56460

**Figure 1.** sketch of the problem

is the fracture volume

*<sup>σ</sup><sup>b</sup>* (*r*, *<sup>t</sup>*). In (3), *<sup>E</sup>*′ <sup>≡</sup> *<sup>E</sup>*/

*G* (*ξ*,*s*) is given by

The elasticity integral equation [16, 17]

*<sup>w</sup>* (*r*, *<sup>t</sup>*) <sup>=</sup> <sup>8</sup>

*π R E*′ � <sup>1</sup>

�

*G* (*ξ*,*s*) =

Substitution of the elasticity equation (3) into (2) yields

*R*<sup>3</sup> (*t*) *E*′

3

volume *V*leak and the backstress *σ<sup>b</sup>* can be written as

*V*leak (*t*) =

*<sup>V</sup>*crack (*t*) <sup>=</sup> <sup>16</sup>

  1 *<sup>ξ</sup> F* �

1 *s F* �

with *F* (*φ*, *m*) denoting the incomplete elliptic integral of the first kind [18].

�

� *<sup>t</sup>* 0

*p* (*t*) − *σ*<sup>0</sup> + 3

As indicated earlier, we can represent the continuous evolution of the fluid pressure inside the crack by a sum of spatially uniform time impulses of pressure. Then, the leaked-off

The quantity *V*inject (*t*) = *Q*0*t* denotes the volume of injected fracturing fluid, while *V*crack (*t*)

� <sup>1</sup> 0

<sup>0</sup> {*<sup>p</sup>* (*t*) <sup>+</sup> *<sup>σ</sup><sup>b</sup>* [*<sup>R</sup>* (*t*)*s*, *<sup>t</sup>*] <sup>−</sup> *<sup>σ</sup>*0} *<sup>G</sup>*

links the fracture aperture *w* (*r*, *t*) to the fracturing fluid pressure *p* (*t*) and the backstress,

arcsin �1−*ξ*<sup>2</sup>

arcsin � <sup>1</sup>−*s*<sup>2</sup>

<sup>1</sup>−*s*<sup>2</sup> , *<sup>s</sup>*<sup>2</sup> *ξ*2 � , *ξ* > *s*

<sup>1</sup>−*ξ*<sup>2</sup> , *<sup>ξ</sup>*<sup>2</sup> *s*2 � , *ξ* < *s*

� <sup>1</sup> 0

*w* [*R* (*t*)*s*, *t*]*sds*. (2)

*sds*, (3)

, (4)

. (5)

� *r R* (*t*) ,*s* �

1 − *ν*2� denotes the plane strain modulus. The elasticity kernel

*σ<sup>b</sup>* [*R* (*t*)*s*, *t*]

�

*U* [*R* (*s*), *t* − *s*] [*p* (*s*) − *p*0] *ds*, (6)

<sup>1</sup> <sup>−</sup> *<sup>s</sup>*2*sds*�

*V*crack (*t*) = 2*πR*<sup>2</sup> (*t*)

In the above *R* (*t*) is the fracture radius, and *w* (*r*, *t*) is the fracture opening.

In this paper we build a general model of a penny-shaped hydraulic fracture driven by a zero viscosity fluid through a poroelastic medium. The model accounts for the backstress effect, which was not considered in earlier efforts [12–14]. This work makes use of the response of a poroelastic medium to an impulse of pore pressure applied to a penny-shaped domain; namely, *u* (*r*, *t*), the component of the fluid displacement, normal to the disc, and *Sb* (*r*, *t*), the normal stress component [15]. The main restrictive assumptions of this analysis is the absence of a low permeability cake build-up and the neglect of the poroelastic solid-to-fluid coupling. The later assumption was studied by [11], who have concluded that in the case of hydraulic boundary conditions when the pore pressure is prescribed, the fluid exchange between the fracture and the medium calculated via poroelastic theory is nearly identical to that computed by uncoupled diffusion equation. Throughout this work we intensively use scaling and asymptotic analyses; in particular, we show that the parametric space is a prism. In this parametric space, the case of the Carter's leak-off model [12] occupies one edge of this prism, whereas the pseudo steady-state model [13] covers one face.

The main objective of the analysis is to solve for the evolution of the fracture radius *R* (*t*), the fracturing fluid pressure *p*in (*t*), and the efficiency of the hydraulic fracturing E (*t*) ≡ *V*crack/*V*inject, where *V*crack is the volume of the fracture and *V*inject is the volume of the injected fracturing fluid.

#### **2. Mathematical model**

#### **2.1. Problem definition**

We consider a penny-shaped fracture driven by injection of an incompressible fluid, at a constant rate *Q*<sup>0</sup> (see Fig.1). The crack propagates through an infinite, homogeneous, brittle, poroelastic rock saturated by a fluid which has the same physical properties as the filtrate, i.e., these fluids are physically indistinguishable inside the porous medium. The medium is characterized by Young's modulus *E*, Poisson's ratio *ν*, fracture toughness *KIc*, intrinsic permeability *κ*, storage coefficient *S*, Biot coefficient *α*. Prior to the injection of fluid, the pore pressure field *p*<sup>0</sup> is uniform. Also there exists a far-field compressive stress *σ*0, perpendicular to the fracture plane.

#### **2.2. Dimensional formulation**

We start from the global fluid balance equation

$$V\_{\rm inject}\left(t\right) = V\_{\rm crack}\left(t\right) + V\_{\rm leak}\left(t\right). \tag{1}$$

**Figure 1.** sketch of the problem

2 Effective and Sustainable Hydraulic Fracturing

injected fracturing fluid.

**2. Mathematical model**

**2.1. Problem definition**

to the fracture plane.

**2.2. Dimensional formulation**

We start from the global fluid balance equation

volume between the fracture and the medium. This approach relies on decomposing the evolving pressure loading inside the growing fracture into a series of pressure impulses and then on representing the actual fluid exchange volume by the superposition of fluid exchanges induced by a single impulse [11]. Despite algebraic errors in the main expression for the fluid exchange volume (equation (7) of [14]), the idea introduced by these authors is at the core of the approach summarized in this paper and described in more details in [15]. In this paper we build a general model of a penny-shaped hydraulic fracture driven by a zero viscosity fluid through a poroelastic medium. The model accounts for the backstress effect, which was not considered in earlier efforts [12–14]. This work makes use of the response of a poroelastic medium to an impulse of pore pressure applied to a penny-shaped domain; namely, *u* (*r*, *t*), the component of the fluid displacement, normal to the disc, and *Sb* (*r*, *t*), the normal stress component [15]. The main restrictive assumptions of this analysis is the absence of a low permeability cake build-up and the neglect of the poroelastic solid-to-fluid coupling. The later assumption was studied by [11], who have concluded that in the case of hydraulic boundary conditions when the pore pressure is prescribed, the fluid exchange between the fracture and the medium calculated via poroelastic theory is nearly identical to that computed by uncoupled diffusion equation. Throughout this work we intensively use scaling and asymptotic analyses; in particular, we show that the parametric space is a prism. In this parametric space, the case of the Carter's leak-off model [12] occupies one edge of

this prism, whereas the pseudo steady-state model [13] covers one face.

The main objective of the analysis is to solve for the evolution of the fracture radius *R* (*t*), the fracturing fluid pressure *p*in (*t*), and the efficiency of the hydraulic fracturing E (*t*) ≡ *V*crack/*V*inject, where *V*crack is the volume of the fracture and *V*inject is the volume of the

We consider a penny-shaped fracture driven by injection of an incompressible fluid, at a constant rate *Q*<sup>0</sup> (see Fig.1). The crack propagates through an infinite, homogeneous, brittle, poroelastic rock saturated by a fluid which has the same physical properties as the filtrate, i.e., these fluids are physically indistinguishable inside the porous medium. The medium is characterized by Young's modulus *E*, Poisson's ratio *ν*, fracture toughness *KIc*, intrinsic permeability *κ*, storage coefficient *S*, Biot coefficient *α*. Prior to the injection of fluid, the pore pressure field *p*<sup>0</sup> is uniform. Also there exists a far-field compressive stress *σ*0, perpendicular

*V*inject (*t*) = *V*crack (*t*) + *V*leak (*t*). (1)

The quantity *V*inject (*t*) = *Q*0*t* denotes the volume of injected fracturing fluid, while *V*crack (*t*) is the fracture volume

$$V\_{\rm crack}\left(t\right) = 2\pi R^2\left(t\right) \int\_0^1 w\left[\mathcal{R}\left(t\right)s, t\right] s ds. \tag{2}$$

In the above *R* (*t*) is the fracture radius, and *w* (*r*, *t*) is the fracture opening.

The elasticity integral equation [16, 17]

$$w\left(r,t\right) = \frac{8}{\pi} \frac{R}{E'} \int\_0^1 \left\{ p\left(t\right) + \sigma\_b \left[ R\left(t\right)s, t\right] - \sigma\_0 \right\} G\left[\frac{r}{R\left(t\right)}, s\right] s ds,\tag{3}$$

links the fracture aperture *w* (*r*, *t*) to the fracturing fluid pressure *p* (*t*) and the backstress, *<sup>σ</sup><sup>b</sup>* (*r*, *<sup>t</sup>*). In (3), *<sup>E</sup>*′ <sup>≡</sup> *<sup>E</sup>*/ � 1 − *ν*2� denotes the plane strain modulus. The elasticity kernel *G* (*ξ*,*s*) is given by

$$G\left(\xi, s\right) = \begin{cases} \frac{1}{\xi} F\left(\arcsin\sqrt{\frac{1-\xi^2}{1-s^2}}, \frac{s^2}{\xi^2}\right), \xi > s\\ \frac{1}{s} F\left(\arcsin\sqrt{\frac{1-s^2}{1-\xi^2}}, \frac{\xi^2}{\xi^2}\right), \xi < s \end{cases} \tag{4}$$

with *F* (*φ*, *m*) denoting the incomplete elliptic integral of the first kind [18].

Substitution of the elasticity equation (3) into (2) yields

$$V\_{\rm crack}\left(t\right) = \frac{16}{3} \frac{R^3\left(t\right)}{E'} \left\{ p\left(t\right) - \sigma\_0 + 3 \int\_0^1 \sigma\_b\left[\mathcal{R}\left(t\right)s, t\right] \sqrt{1 - s^2} s ds \right\}. \tag{5}$$

As indicated earlier, we can represent the continuous evolution of the fluid pressure inside the crack by a sum of spatially uniform time impulses of pressure. Then, the leaked-off volume *V*leak and the backstress *σ<sup>b</sup>* can be written as

$$V\_{\text{leak}}\left(t\right) = \int\_0^t \mathcal{U}\left[\mathcal{R}\left(s\right), t - s\right] \left[p\left(s\right) - p\_0\right] ds,\tag{6}$$

$$
\sigma\_b \left( r, t \right) = \int\_0^t \mathcal{S}\_b \left[ R \left( s \right), r, t - s \right] \left[ p \left( s \right) - p\_0 \right] ds,\tag{7}
$$

611

• Backstress equation (7) after substitution of (8)

• Propagation criterion (9) combined with (11)

<sup>1</sup> <sup>=</sup> <sup>8</sup>

 <sup>1</sup> 0

 <sup>1</sup> 0

√*π*

*L* (*t*) *L* (*ts*)

and *Vbs* (*t*) is the change of the fracture volume due to the backstress

*<sup>ρ</sup>* (*ts*) *<sup>L</sup>* (*ts*)

G*<sup>σ</sup>* (*t*) +

<sup>4</sup>G*<sup>d</sup>* (*t*) *<sup>L</sup>*<sup>2</sup> (*t*) *L*<sup>2</sup> (*ts*)

Here *Kbs* (*t*) is the change of the stress intensity factor due to the backstress

G*<sup>σ</sup>* (*t*) +

1 *ρ* (*ts*)

> *L* (*t*) *L* (*ts*)

> *L* (*t*) *L* (*ts*)

*<sup>L</sup>* (*t*) *vbs <sup>L</sup>* (*t*) *<sup>ρ</sup>* (*t*)

 <sup>1</sup> 0

*L*<sup>2</sup> (*t*) *L*<sup>2</sup> (*ts*)

> ×

<sup>1</sup> <sup>=</sup> <sup>2</sup>

• Volume balance equation (1), where we substituted (5), (6), (8), and (12)

1 *ρ*<sup>2</sup> (*ts*) Ξ 

G*<sup>σ</sup>* (*t*) +

 *L* (*t*) *L* (*ts*)

*<sup>ξ</sup> <sup>L</sup>* (*t*) *<sup>ρ</sup>* (*t*) *L* (*ts*) *ρ* (*ts*)

> Π (*ts*)

<sup>3</sup> <sup>G</sup>*<sup>v</sup>* (*t*) *<sup>ρ</sup>*5/2 (*t*) [<sup>1</sup> <sup>+</sup> *Vbs* (*t*) <sup>−</sup> *Kbs* (*t*)] <sup>+</sup>

1 − *s ρ*<sup>2</sup> (*ts*)

*kbs <sup>L</sup>* (*t*) *<sup>ρ</sup>* (*t*) *L* (*ts*) *ρ* (*ts*)

> Π (*ts*)

*L* (*ts*) *ρ* (*ts*)

Π (*ts*) 

G*<sup>σ</sup>* (*t*) +

 *L* (*t*) *L* (*ts*)

, 4G*<sup>d</sup>* (*t*) *<sup>L</sup>*<sup>2</sup> (*t*)

, 4G*<sup>d</sup>* (*t*) *<sup>L</sup>*<sup>2</sup> (*t*)

*L*<sup>2</sup> (*ts*)

*L*<sup>2</sup> (*ts*)

, 4G*<sup>d</sup>* (*t*) *<sup>L</sup>*<sup>2</sup> (*t*)

<sup>√</sup>*<sup>π</sup> <sup>ρ</sup>*1/2 (*t*) <sup>Π</sup> (*t*) <sup>+</sup> *Kbs* (*t*), (12)

*L*<sup>2</sup> (*ts*)

1 − *s ρ*<sup>2</sup> (*ts*)

Fluid-Driven Fracture in a Poroelastic Rock http://dx.doi.org/10.5772/56460

*ds*, (11)

Π (*ts*) 

1 − *s ρ*<sup>2</sup> (*ts*)

*ds*, (14)

1 − *s ρ*<sup>2</sup> (*ts*)

*ds*, (15)

 ×

 ×

*ds*. (13)

 ×

Σ (*ξ*, *t*) = 4*η*G*<sup>d</sup>* (*t*)

+ 4G*<sup>c</sup>* (*t*)

 <sup>1</sup> 0

*Kbs* (*t*) <sup>=</sup> <sup>4</sup>*η*G*<sup>d</sup>* (*t*)

*Vbs* (*t*) <sup>=</sup> <sup>4</sup>*η*G*<sup>d</sup>* (*t*)

where

*ρ*5/2 (*t*)

*L* (*ts*) *<sup>L</sup>* (*t*) *<sup>ρ</sup>* (*ts*) <sup>Ψ</sup>

*ρ*1/2 (*t*)

where *U* (*R*, *t*) is the volume of the fracturing fluid that has escaped from a fracture of radius *R* at an elapsed time *t* after a uniform unit impulse of pressure has been applied, and *Sb* (*R*,*r*, *t*) is the generated backstress. In the following we refer to *U* (*R*, *t*) as the leak-off Green function, and to *Sb* (*R*,*r*, *t*) as the backstress Green function.

Simple scaling analysis reveals the following relations between these dimensional Green functions *<sup>U</sup>* (*R*, *<sup>t</sup>*), *Sb* (*R*,*r*, *<sup>t</sup>*) and the dimensionless ones <sup>Ψ</sup> (*τ*), <sup>Ξ</sup> (*ξ*, *<sup>τ</sup>*) [15]

$$\Delta U(R,t) = \frac{SR^3}{T\_R} \Psi \left(\frac{t}{T\_R}\right), \quad S\_b \left(R, r, t\right) = \frac{\eta}{T\_R} \Xi \left(\frac{r}{R}, \frac{t}{T\_R}\right), \quad T\_R = \frac{R^2}{4c},\tag{8}$$

where *c* = *κ*/*S* is the diffusion coefficient, *η* = *α* (1 − 2*ν*) /2 (1 − *ν*) is the poroelastic stress coefficient.

We close the formulation of the problem with the propagation criterion

$$K\_{\rm Ic} = \frac{2}{\sqrt{\pi}} \mathcal{R}^{1/2} \left( t \right) \int\_0^1 \frac{p \left[ s \mathcal{R} \left( t \right), t \right] + \sigma\_b \left[ s \mathcal{R} \left( t \right), t \right] - \sigma\_0}{\sqrt{1 - s^2}} s ds \,\tag{9}$$

The model has thus only two unknowns: the fracturing fluid pressure *p* (*t*) and the fracture radius *R* (*t*).

#### **2.3. Dimensionless formulation**

The problem depends on seven dimensional parameters: *KIc*, *<sup>E</sup>*′ , *Q*0, *c*, *S*, *σ*0, and *p*0, and one dimensionless parameter *η*. It is possible to reduce this set of parameters to five dimensionless quantities. Inspired by earlier works on hydraulic fracture [19], we introduce the scaling

$$r = \mathcal{R}\left(t\right)\sharp, \quad \mathcal{R}\left(t\right) = L\left(t\right)\rho\left(t\right),$$

$$p\left(t\right) - \sigma\_0 = \frac{K\_{\rm Ic}}{L^{1/2}\left(t\right)}\Pi\left(t\right), \quad \sigma\_b\left(r, t\right) = \frac{K\_{\rm Ic}}{L^{1/2}\left(t\right)}\Sigma\left(\xi\_r t\right). \tag{10}$$

where *ρ* (*t*) ∼ 1 is the dimensionless radius, Π (*t*) ∼ 1 is the dimensionless net pressure, Σ (*ξ*, *t*) is the dimensionless back stress, and *L* (*t*) ∼ *R* (*t*) is the characteristic size of the fracture. This scaling is thus time-dependent. Moreover we have not yet defined the parameter *L* (*t*). Below we show that the parameter *L* (*t*) can be defined for different propagation regimes in such a way that the dimensionless quantities *ρ*, Π, and Σ do not depend on time.

In the scaling (10) the governing equations transform as follows.

• Backstress equation (7) after substitution of (8)

4 Effective and Sustainable Hydraulic Fracturing

*<sup>U</sup>* (*R*, *<sup>t</sup>*) <sup>=</sup> *SR*<sup>3</sup>

coefficient.

radius *R* (*t*).

the scaling

depend on time.

*TR* Ψ *t TR* 

*KIc* <sup>=</sup> <sup>2</sup> √*π*

**2.3. Dimensionless formulation**

*σ<sup>b</sup>* (*r*, *t*) =

 *<sup>t</sup>* 0

Green function, and to *Sb* (*R*,*r*, *t*) as the backstress Green function.

functions *<sup>U</sup>* (*R*, *<sup>t</sup>*), *Sb* (*R*,*r*, *<sup>t</sup>*) and the dimensionless ones <sup>Ψ</sup> (*τ*), <sup>Ξ</sup> (*ξ*, *<sup>τ</sup>*) [15]

We close the formulation of the problem with the propagation criterion

 <sup>1</sup> 0

*R*1/2 (*t*)

The problem depends on seven dimensional parameters: *KIc*, *<sup>E</sup>*′

*L*1/2 (*t*)

In the scaling (10) the governing equations transform as follows.

*<sup>p</sup>* (*t*) <sup>−</sup> *<sup>σ</sup>*<sup>0</sup> <sup>=</sup> *KIc*

where *U* (*R*, *t*) is the volume of the fracturing fluid that has escaped from a fracture of radius *R* at an elapsed time *t* after a uniform unit impulse of pressure has been applied, and *Sb* (*R*,*r*, *t*) is the generated backstress. In the following we refer to *U* (*R*, *t*) as the leak-off

Simple scaling analysis reveals the following relations between these dimensional Green

, *Sb* (*R*,*r*, *<sup>t</sup>*) <sup>=</sup> *<sup>η</sup>*

where *c* = *κ*/*S* is the diffusion coefficient, *η* = *α* (1 − 2*ν*) /2 (1 − *ν*) is the poroelastic stress

The model has thus only two unknowns: the fracturing fluid pressure *p* (*t*) and the fracture

and one dimensionless parameter *η*. It is possible to reduce this set of parameters to five dimensionless quantities. Inspired by earlier works on hydraulic fracture [19], we introduce

*r* = *R* (*t*) *ξ*, *R* (*t*) = *L* (*t*) *ρ* (*t*),

where *ρ* (*t*) ∼ 1 is the dimensionless radius, Π (*t*) ∼ 1 is the dimensionless net pressure, Σ (*ξ*, *t*) is the dimensionless back stress, and *L* (*t*) ∼ *R* (*t*) is the characteristic size of the fracture. This scaling is thus time-dependent. Moreover we have not yet defined the parameter *L* (*t*). Below we show that the parameter *L* (*t*) can be defined for different propagation regimes in such a way that the dimensionless quantities *ρ*, Π, and Σ do not

<sup>Π</sup> (*t*), *<sup>σ</sup><sup>b</sup>* (*r*, *<sup>t</sup>*) <sup>=</sup> *KIc*

*L*1/2 (*t*)

*TR* Ξ *r R*, *t TR* 

*<sup>p</sup>* [*sR* (*t*), *<sup>t</sup>*] + *<sup>σ</sup><sup>b</sup>* [*sR* (*t*), *<sup>t</sup>*] − *<sup>σ</sup>*<sup>0</sup> √ 1 − *s*<sup>2</sup>

*Sb* [*<sup>R</sup>* (*s*),*r*, *<sup>t</sup>* − *<sup>s</sup>*] [*<sup>p</sup>* (*s*) − *<sup>p</sup>*0] *ds*, (7)

, *TR* <sup>=</sup> *<sup>R</sup>*<sup>2</sup>

4*c*

*sds*, (9)

, *Q*0, *c*, *S*, *σ*0, and *p*0,

Σ (*ξ*, *t*). (10)

, (8)

$$
\Sigma\left(\xi, t\right) = 4\eta \mathcal{G}\_d\left(t\right) \int\_0^1 \frac{L^2\left(t\right)}{L^2\left(ts\right)} \frac{1}{\rho^2\left(ts\right)} \Xi\left[\xi \frac{L\left(t\right)\rho\left(t\right)}{L\left(ts\right)\rho\left(ts\right)}, 4\mathcal{G}\_d\left(t\right) \frac{L^2\left(t\right)}{L^2\left(ts\right)} \frac{1-s}{\rho^2\left(ts\right)}\right] \times
$$

$$
\times \left[\mathcal{G}\_\sigma\left(t\right) + \sqrt{\frac{L\left(t\right)}{L\left(ts\right)}}\Pi\left(ts\right)\right] ds,\tag{11}
$$

• Propagation criterion (9) combined with (11)

$$1 = \frac{2}{\sqrt{\pi}} \rho^{1/2} \left( t \right) \Pi \left( t \right) + \mathcal{K}\_{\rm bs} \left( t \right) , \tag{12}$$

• Volume balance equation (1), where we substituted (5), (6), (8), and (12)

$$1 = \frac{8\sqrt{\pi}}{3} \mathcal{G}\_v\left(t\right) \rho^{5/2}\left(t\right) \left[1 + V\_{bs}\left(t\right) - K\_{bs}\left(t\right)\right] + \cdots$$

$$+4\mathcal{G}\_{\rm c}(t)\int\_{0}^{1}\frac{L\,\mathrm{(ts)}}{L\,(t)}\rho\,\mathrm{(ts)}\,\mathrm{\,\mathrm{\,}\mathrm{\,}^{2}}\left[4\mathcal{G}\_{\rm d}\,(t)\,\frac{L^{2}\,(t)}{L^{2}\,(\mathrm{ts})}\frac{1-s}{\rho^{2}\,(\mathrm{ts})}\right]\left[\mathcal{G}\_{\sigma}\,(t)+\sqrt{\frac{L\,\mathrm{(}t)}{L\,(\mathrm{ts})}}\,\Pi\,(\mathrm{ts})\right]ds.\tag{13}$$

Here *Kbs* (*t*) is the change of the stress intensity factor due to the backstress

$$K\_{bs}\left(t\right) = \frac{4\eta \mathcal{G}\_d\left(t\right)}{\rho^{1/2}\left(t\right)} \int\_0^1 \frac{L\left(t\right)}{L\left(ts\right)} \frac{1}{\rho\left(ts\right)} k\_{bs}\left[\frac{L\left(t\right)\rho\left(t\right)}{L\left(ts\right)\rho\left(ts\right)}, 4\mathcal{G}\_d\left(t\right) \frac{L^2\left(t\right)}{L^2\left(ts\right)} \frac{1-s}{\rho^2\left(ts\right)}\right] \times$$

$$\left[\mathcal{G}\_\sigma\left(t\right) + \sqrt{\frac{L\left(t\right)}{L\left(ts\right)}}\Pi\left(ts\right)\right]ds,\tag{14}$$

and *Vbs* (*t*) is the change of the fracture volume due to the backstress

$$
\mathcal{W}\_{\rm bs}(t) = \frac{4\eta \mathcal{G}\_{\rm d}(t)}{\rho^{5/2}(t)} \int\_0^1 \rho \left( \text{ts} \right) \frac{L\left(\text{ts} \right)}{L\left(t\right)} v\_{\rm bs} \left[ \frac{L\left(t\right)\rho\left(t\right)}{L\left(\text{ts}\right)\rho\left(\text{ts} \right)}, 4\mathcal{G}\_{\rm d}\left(t\right) \frac{L^2\left(t\right)}{L^2\left(\text{ts} \right)} \frac{1-\text{s}}{\rho^2\left(\text{ts} \right)} \right] \times
$$

$$
\left[ \mathcal{G}\_{\sigma}\left(t\right) + \sqrt{\frac{L\left(t\right)}{L\left(t\text{s} \right)}} \Pi\left(t\text{s} \right) \right] ds,\tag{15}
$$

where

$$k\_{bs} \left( \mathcal{R}, \tau \right) = \frac{2}{\sqrt{\pi}} \int\_0^{\mathcal{R}} \frac{\tilde{\xi} \Sigma \left( \tilde{\xi}, \tau \right)}{\sqrt{\mathcal{R}^2 - \tilde{\xi}^2}} d\tilde{\xi} \,\tag{16}$$

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• the *K*0-vertex represents the storage-dominated regime with 1D diffusion, during which

• the *K*˜ *<sup>κ</sup>*0-vertex is related to the leak-off-dominated regime with 1D diffusion, when the

• the *K*˜*σ*0-vertex is another leak-off-dominated regime with slow 1D diffusion, where from a diffusion point of view the fracturing fluid pressure *p* is approximately equal to the

• the *K*∞-vertex is the storage-dominated regime with pseudo steady-state 3D diffusion; • the *K*˜ *<sup>κ</sup>*∞-vertex is the pseudo steady-state (3D) diffusion version of the *K*˜ *<sup>κ</sup>*0-vertex; • the *K*˜*σ*∞-vertex is the pseudo steady-state (3D) diffusion version of the *K*˜*σ*0-vertex.

In the transition from one regime to another, the dominance of one physical process is displaced by the dominance of another one. For example the transition *K*∞*K*˜*κ*<sup>∞</sup> is governed by G*c*/G*v*, such that G*c*/G*<sup>v</sup>* = 0 for the *K*∞-vertex, and G*c*/G*<sup>v</sup>* = ∞ for the *K*˜*κ*∞-vertex. In another example, the transition from 1D to 3D diffusion is governed by G*d*, such that G*<sup>d</sup>* = <sup>0</sup>

Usually, each propagation regime is studied in an intrinsic time-dependent scaling, such that the propagation of a fracture in a given propagation regime does not depend on time in this scaling. In other words, each propagation regime is characterized by a self-similar solution. This intrinsic scaling is introduced in such a way that all dimensionless groups corresponding to the dominant physical processes are equal to 1, whereas all the other groups are small compared to 1. These small dimensionless groups are still time-dependent, therefore it is easy to estimate when a given propagation regime is valid. Also using these small time-dependent groups we can easily calculate the characteristic transition times between different propagation regimes. For example, in order to calculate the characteristic transition time *tAB* between the two propagation regimes *A* and *B*, one should follow the following procedure: first, introduce a scaling intrinsic to the propagation regime *A*; second, obtain in

most of the injected fluid is stored inside the fracture;

net fluid pressure *p* − *σ*<sup>0</sup> is large compared to *σ*<sup>0</sup> − *p*0;

confining stress *σ*0;

**Figure 2.** Parametric space

for 1D diffusion, and G*<sup>d</sup>* = <sup>∞</sup> for 3D diffusion.

$$v\_{\rm bs} \left( \mathcal{R}, \tau \right) = \frac{6}{\sqrt{\pi}} \int\_0^{\mathcal{R}} \xi \sqrt{\mathcal{R}^2 - \xi^2} \Xi \left( \xi, \tau \right) d\xi. \tag{17}$$

The above governing equations thus depend on four time-dependent dimensionless groups:

• Storage group

$$\mathcal{G}\_{\upsilon}\left(t\right) = \frac{K\_{Ic}}{Q\_0 E'} \frac{L^{5/2}\left(t\right)}{t} ,\tag{18}$$

which is proportional to the fraction of the injected fluid volume stored in the fracture;

• Leak-off group

$$\mathcal{G}\_{\mathcal{C}}\left(t\right) = \frac{c \mathcal{S} \mathcal{K}\_{\mathrm{Ic}}}{\mathcal{Q}\_{\mathrm{0}}} L^{1/2}\left(t\right),\tag{19}$$

which characterizes the amount of the fluid that has leaked into the formation;

• Diffusion group

$$\mathcal{G}\_d = \frac{ct}{L^2\left(t\right)'}\tag{20}$$

which is related to the diffusion process with √G*<sup>d</sup>* proportional to the ratio of the diffusion length scale to the fracture size. Thus this dimensionless group is small, G*<sup>d</sup>* ≪ 1, in the case of 1D diffusion and large in the case of 3D diffusion, G*<sup>d</sup>* ≫ 1;

• Pressure group

$$\mathcal{G}\_{\sigma} \left( t \right) = \frac{\sigma\_0 - p\_0}{K\_{lc}} L^{1/2} \left( t \right) \sim \frac{\sigma\_0 - p\_0}{p - \sigma\_0} \, ^\prime \tag{21}$$

which describes the effect of the material toughness on the net fluid pressure *p* − *σ*<sup>0</sup> compared to *σ*<sup>0</sup> − *p*0. Indeed, in the case of small toughness when *KIc* → 0 and G*<sup>σ</sup>* → ∞, the fracturing fluid pressure *p* can be assumed to be equal to the confining stress *σ*<sup>0</sup> from a diffusion point of view, i.e., *p* ≈ *σ*0. Whereas in the case of large toughness when *KIc* → ∞ and G*<sup>σ</sup>* → 0, the net fluid pressure *p* − *σ*<sup>0</sup> is large compared to *σ*<sup>0</sup> − *p*0.

The only unknown here are the fracture radius *ρ* (*τ*) and fracturing fluid pressure Π (*τ*).

#### **3. Propagation regimes**

The problem under study has six propagation regimes. Therefore it is convenient to represent the fracture propagation by a trajectory line lying inside the prismatic parametric space shown in Fig. 2. The vertices of this prism represent the different propagation regimes; namely,


**Figure 2.** Parametric space

6 Effective and Sustainable Hydraulic Fracturing

• Storage group

• Leak-off group

• Diffusion group

• Pressure group

**3. Propagation regimes**

namely,

*kbs* (R, *<sup>τ</sup>*) <sup>=</sup> <sup>2</sup>

√*π*

*vbs* (R, *<sup>τ</sup>*) <sup>=</sup> <sup>6</sup>

√*π*

 <sup>R</sup> 0 *ξ* 

<sup>G</sup>*<sup>v</sup>* (*t*) <sup>=</sup> *KIc*

<sup>G</sup>*<sup>c</sup>* (*t*) <sup>=</sup> *cSKIc*

which characterizes the amount of the fluid that has leaked into the formation;

case of 1D diffusion and large in the case of 3D diffusion, G*<sup>d</sup>* ≫ 1;

<sup>G</sup>*<sup>σ</sup>* (*t*) <sup>=</sup> *<sup>σ</sup>*<sup>0</sup> <sup>−</sup> *<sup>p</sup>*<sup>0</sup>

*KIc*

*KIc* → ∞ and G*<sup>σ</sup>* → 0, the net fluid pressure *p* − *σ*<sup>0</sup> is large compared to *σ*<sup>0</sup> − *p*0.

The only unknown here are the fracture radius *ρ* (*τ*) and fracturing fluid pressure Π (*τ*).

The problem under study has six propagation regimes. Therefore it is convenient to represent the fracture propagation by a trajectory line lying inside the prismatic parametric space shown in Fig. 2. The vertices of this prism represent the different propagation regimes;

The above governing equations thus depend on four time-dependent dimensionless groups:

*<sup>Q</sup>*0*E*′

*Q*<sup>0</sup>

<sup>G</sup>*<sup>d</sup>* <sup>=</sup> *ct L*<sup>2</sup> (*t*)

which is related to the diffusion process with √G*<sup>d</sup>* proportional to the ratio of the diffusion length scale to the fracture size. Thus this dimensionless group is small, G*<sup>d</sup>* ≪ 1, in the

which describes the effect of the material toughness on the net fluid pressure *p* − *σ*<sup>0</sup> compared to *σ*<sup>0</sup> − *p*0. Indeed, in the case of small toughness when *KIc* → 0 and G*<sup>σ</sup>* → ∞, the fracturing fluid pressure *p* can be assumed to be equal to the confining stress *σ*<sup>0</sup> from a diffusion point of view, i.e., *p* ≈ *σ*0. Whereas in the case of large toughness when

*<sup>L</sup>*1/2 (*t*) <sup>∼</sup> *<sup>σ</sup>*<sup>0</sup> <sup>−</sup> *<sup>p</sup>*<sup>0</sup>

*p* − *σ*<sup>0</sup>

which is proportional to the fraction of the injected fluid volume stored in the fracture;

 <sup>R</sup> 0

*ξ*Ξ (*ξ*, *τ*) R2 − *ξ*2

*L*5/2 (*t*)

*dξ*, (16)

R<sup>2</sup> − *ξ*2Ξ (*ξ*, *τ*) *dξ*. (17)

*<sup>t</sup>* , (18)

*L*1/2 (*t*), (19)

, (20)

, (21)

In the transition from one regime to another, the dominance of one physical process is displaced by the dominance of another one. For example the transition *K*∞*K*˜*κ*<sup>∞</sup> is governed by G*c*/G*v*, such that G*c*/G*<sup>v</sup>* = 0 for the *K*∞-vertex, and G*c*/G*<sup>v</sup>* = ∞ for the *K*˜*κ*∞-vertex. In another example, the transition from 1D to 3D diffusion is governed by G*d*, such that G*<sup>d</sup>* = <sup>0</sup> for 1D diffusion, and G*<sup>d</sup>* = <sup>∞</sup> for 3D diffusion.

Usually, each propagation regime is studied in an intrinsic time-dependent scaling, such that the propagation of a fracture in a given propagation regime does not depend on time in this scaling. In other words, each propagation regime is characterized by a self-similar solution. This intrinsic scaling is introduced in such a way that all dimensionless groups corresponding to the dominant physical processes are equal to 1, whereas all the other groups are small compared to 1. These small dimensionless groups are still time-dependent, therefore it is easy to estimate when a given propagation regime is valid. Also using these small time-dependent groups we can easily calculate the characteristic transition times between different propagation regimes. For example, in order to calculate the characteristic transition time *tAB* between the two propagation regimes *A* and *B*, one should follow the following procedure: first, introduce a scaling intrinsic to the propagation regime *A*; second, obtain in


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• *K*˜ *<sup>κ</sup>*0*K*˜ *<sup>κ</sup>*∞- edge

• *K*˜*σ*0*K*˜*σ*∞- edge

• *K*∞*K*˜ *<sup>κ</sup>*∞- edge

• *K*∞*K*˜*σ*∞- edge

of the *K*˜*κ*∞*K*˜*σ*∞-edge.

*K*∞*K*˜ *<sup>κ</sup>*∞*K*˜*σ*∞-face with *η* = 0.

**4. Methodology**

*tK*˜*κ*0*K*˜*κ*<sup>∞</sup> <sup>=</sup> *<sup>Q</sup>*<sup>4</sup>

*tK*˜*σ*0*K*˜*σ*<sup>∞</sup> <sup>=</sup> *<sup>Q</sup>*<sup>2</sup>

*tK*∞*K*˜*κ*<sup>∞</sup> <sup>=</sup> *<sup>Q</sup>*<sup>4</sup>

• *K*˜ *<sup>κ</sup>*∞*K*˜*σ*∞-edge is self-similar, i.e., the transition along this edge is impossible. Moreover all trajectory lines of the fracture propagation begin at the *K*0-vertex and end at some point

The case of the Carter's leak-off model studied in [12] corresponds to the *K*0*K*˜*σ*0-edge, whereas the pseudo steady-state model introduced in [13] corresponds to the

Interestingly the fracture radius *ρ* is the same in the two different storage-dominated regimes *K*<sup>0</sup> and *K*<sup>∞</sup> (see Table 1), while the fracturing fluid pressure is different [15]. Therefore the poroelastic effects split the storage dominated regime (denoted previously as the *K*-vertex [12, 17]) into two sub-regimes: the *K*0-vertex (1D diffusion) which is similar to the former *K*-vertex, and the *K*∞-vertex (pseudo steady-state diffusion) characterized by a higher pressure. The essence of the difference between these two regimes is illustrated in Fig. 3. Initially the fracture front propagates faster than the diffusion front, therefore the diffusion length scale is small compared to the size of the fracture and the diffusion is one dimensional. As time goes on, the diffusion front catches up and then passes the fracture front, making the diffusion length scale to be larger than the fracture size, and, as a result, switching the

propagation regime from the 1D diffusion to the pseudo steady-state (3D) diffusion.

Inclusion of diffusion and poroelastic effects into the model of penny-shaped hydraulic fracture model propagation in the toughness-dominated regime thus requires evaluating the convolution type integrals [see (11), (13)-(15)]. Indeed, calculation of the fracturing fluid volume which has leaked into the formation requires a "convolution" on Ψ (*τ*) [see (13)], whereas evaluation of the backstress <sup>Σ</sup> (*ξ*, *<sup>t</sup>*) and the related fracture volume *Vbs* (*t*) and stress intensity factor *Kbs* (*t*) changes requires "convolutions" on <sup>Ξ</sup> (*ξ*, *<sup>τ</sup>*). These "convolutions"

*tK*∞*K*˜*σ*<sup>∞</sup> =

0 *c*5*S*4*K*<sup>4</sup> *Ic* ;

0 *c*3*S*<sup>2</sup> (*σ*<sup>0</sup> − *p*0)

> 0 *c*5*S*5*E*′*K*<sup>4</sup> *Ic* ;

*Q*3 0*K*<sup>2</sup> *Ic <sup>c</sup>*5*S*5*E*′<sup>2</sup> (*σ*<sup>0</sup> <sup>−</sup> *<sup>p</sup>*0)

2 ;

5 .

**Table 1.** Propagation regimes and corresponding scalings

*d*

terms of this intrinsic scaling an expression for the dimensionless group G(*A*) *<sup>B</sup>* (*t*), which is dominant in the regime *B*, whereas it is small in the regime *A*; and third, solve the equation G(*A*) *<sup>B</sup>* (*tAB*) = 1 to obtain the characteristic transition time *tAB*.

Different scalings can be introduced by defining the length scale *L* (*t*), see (21)-(18). We define different propagation regimes in terms of the dimensionless groups (21)-(18) in Table 1, where we also introduce the scalings intrinsic to each of these propagation regimes. The transition times between different propagation regimes are given by

• *K*0*K*∞- edge

$$t\_{K\_0K\_\infty} = \frac{Q\_0^4 E'^4}{c^5 K\_{Ic}^4};$$

• *K*0*K*˜ *<sup>κ</sup>*0- edge

$$t\_{K\_0\mathbb{K}\_{\mathbb{X}0}} = \left[ c^{1/2} S \left( \frac{K\_{Ic}^2 E'^3}{Q\_0^2} \right)^{1/5} \left( 1 - \frac{4\eta}{E'S} \right) \right]^{-10} \geqslant$$

• *K*0*K*˜*σ*0- edge

$$t\_{K\_0\mathbb{K}\_{\mathcal{V}}} = \left[ c^{1/2} S \left( \sigma\_0 - p\_0 \right) \left( \frac{E'^4}{Q\_0 K\_{I\mathfrak{c}}^4} \right)^{1/5} \left( 1 - \frac{4\eta}{E'S} \right) \right]^{-10/3};$$

• *K*˜*κ*0*K*˜*σ*0- edge

$$t\_{\tilde{K}\ast\tilde{K}\circ 0} = \left[\frac{c^{1/2}S K\_{Ic}^4}{\left(\sigma\_0 - p\_0\right)^3 Q\_0} \left(1 - \frac{4\eta}{E'S}\right)\right]^2;$$

• *K*˜ *<sup>κ</sup>*0*K*˜ *<sup>κ</sup>*∞- edge

8 Effective and Sustainable Hydraulic Fracturing

*<sup>K</sup>*<sup>0</sup> <sup>≪</sup> 1 – ≫ G−1/2

Here <sup>G</sup>(1*D*) *<sup>c</sup>* <sup>≡</sup> <sup>G</sup>*<sup>c</sup>*

G(*A*)

• *K*0*K*∞- edge

• *K*0*K*˜ *<sup>κ</sup>*0- edge

• *K*0*K*˜*σ*0- edge

• *K*˜*κ*0*K*˜*σ*0- edge

*<sup>K</sup>*˜ *<sup>κ</sup>*<sup>0</sup> <sup>≪</sup> <sup>1</sup> <sup>≪</sup> <sup>1</sup> ≪ G−1/2

*<sup>K</sup>*˜*σ*<sup>0</sup> <sup>≪</sup> <sup>1</sup> <sup>≫</sup> <sup>1</sup> ≪ G−1/2

G1/2 *d* � <sup>1</sup> <sup>−</sup> <sup>4</sup>*<sup>η</sup> E*′*S* �

**Table 1.** Propagation regimes and corresponding scalings

*K*<sup>∞</sup> ≫ 1 – ≫ max [1, G*σ*] G*<sup>v</sup>* = 1

*K*˜ *<sup>κ</sup>*<sup>∞</sup> ≫ 1 ≪ 1 ≪ 1 G*<sup>c</sup>* = 1

*K*˜*σ*<sup>∞</sup> ≫ 1 ≫ 1 ≪ G*<sup>σ</sup>* G*c*G*<sup>σ</sup>* = 1 *<sup>Q</sup>*<sup>0</sup>

*<sup>B</sup>* (*tAB*) = 1 to obtain the characteristic transition time *tAB*.

transition times between different propagation regimes are given by

 *c*1/2*<sup>S</sup>*

*c*1/2*<sup>S</sup>* (*σ*<sup>0</sup> <sup>−</sup> *<sup>p</sup>*0)

*tK*˜ *<sup>κ</sup>*0*K*˜ *<sup>σ</sup>*<sup>0</sup> =

*tK*0*K*˜ *<sup>κ</sup>*<sup>0</sup> =

*tK*0*K*˜ *<sup>σ</sup>*<sup>0</sup> =

and I ≡ <sup>1</sup> <sup>−</sup> <sup>4</sup>*<sup>η</sup>*

terms of this intrinsic scaling an expression for the dimensionless group G(*A*)

**ver- definition scaling solution tex** G*<sup>d</sup>* G*<sup>σ</sup>* G*v*/G*<sup>c</sup>* definition *<sup>L</sup>* (*t*) *<sup>ρ</sup>*

*<sup>d</sup>* <sup>G</sup>(1*D*) *<sup>c</sup>* <sup>=</sup> <sup>1</sup>

*<sup>d</sup>* <sup>G</sup>*<sup>σ</sup>* <sup>G</sup>(1*D*) *<sup>c</sup>* <sup>G</sup>*<sup>σ</sup>* <sup>=</sup> <sup>1</sup>

*E*′*S*

dominant in the regime *B*, whereas it is small in the regime *A*; and third, solve the equation

Different scalings can be introduced by defining the length scale *L* (*t*), see (21)-(18). We define different propagation regimes in terms of the dimensionless groups (21)-(18) in Table 1, where we also introduce the scalings intrinsic to each of these propagation regimes. The

*tK*0*K*<sup>∞</sup> <sup>=</sup> *<sup>Q</sup>*<sup>4</sup>

� *E*′4 *Q*0*K*<sup>4</sup> *Ic*

� *c*1/2*SK*<sup>4</sup>

(*σ*<sup>0</sup> − *p*0)

*Ic*

<sup>3</sup> *<sup>Q</sup>*<sup>0</sup>

� <sup>1</sup> <sup>−</sup> <sup>4</sup>*<sup>η</sup> E*′*S*

� *K*2 *IcE*′<sup>3</sup> *Q*2 0

0*E*′<sup>4</sup> *c*5*K*<sup>4</sup> *Ic* ;

�1/5 �

<sup>1</sup> <sup>−</sup> <sup>4</sup>*<sup>η</sup> E*′*S*

�1/5 �

� 

<sup>1</sup> <sup>−</sup> <sup>4</sup>*<sup>η</sup> E*′*S*

> ��<sup>2</sup> ;

−10

� 

−10/3

;

;

� *<sup>Q</sup>*0*E*′ *KIc t*

� *<sup>Q</sup>*0I−<sup>1</sup>*<sup>t</sup>*

� *<sup>Q</sup>*0I−<sup>1</sup>*<sup>t</sup>*

� *<sup>Q</sup>*0*E*′ *KIc t*

> � *Q*<sup>0</sup> *cSKIc*

*cS*(*σ*0−*p*0)

1/2 *c*1/2*SKIc*

1/2 *c*1/2*S*(*σ*0−*p*0)

�2/5 � <sup>3</sup>

�2/5 � <sup>3</sup>

�<sup>2</sup> (1−*η*)

�1/2

�2/3 <sup>2</sup>2/3

8 √*π* �2/5

8 √*π* �2/5

*π*4/3

*π*−3/4

1−*η* 8

*<sup>B</sup>* (*t*), which is

2 16*π*

*<sup>d</sup>* max [1, <sup>G</sup>*σ*] <sup>G</sup>*<sup>v</sup>* <sup>=</sup> <sup>1</sup>

$$t\_{\mathcal{K}\kappa 0\mathcal{K}\kappa\infty} = \frac{Q\_0^4}{c^5 S^4 K\_{lc}^4}; \frac{1}{\cdot}$$

• *K*˜*σ*0*K*˜*σ*∞- edge

$$t\_{\mathbb{K}\sigma 0\mathbb{K}\sigma \infty} = \frac{Q\_0^2}{c^3 S^2 \left(\sigma\_0 - p\_0\right)^2};$$

• *K*∞*K*˜ *<sup>κ</sup>*∞- edge

$$t\_{K\_{\infty}\check{K}\kappa\infty} = \frac{Q\_0^4}{c^5 S^5 E' K\_{Ic}^4};$$

• *K*∞*K*˜*σ*∞- edge

$$t\_{K\_{\infty}\check{K}\sigma\infty} = \sqrt{\frac{Q\_0^3 K\_{I\varepsilon}^2}{c^5 S^5 E'^2 \left(\sigma\_0 - p\_0\right)^5}}.$$

• *K*˜ *<sup>κ</sup>*∞*K*˜*σ*∞-edge is self-similar, i.e., the transition along this edge is impossible. Moreover all trajectory lines of the fracture propagation begin at the *K*0-vertex and end at some point of the *K*˜*κ*∞*K*˜*σ*∞-edge.

The case of the Carter's leak-off model studied in [12] corresponds to the *K*0*K*˜*σ*0-edge, whereas the pseudo steady-state model introduced in [13] corresponds to the *K*∞*K*˜ *<sup>κ</sup>*∞*K*˜*σ*∞-face with *η* = 0.

Interestingly the fracture radius *ρ* is the same in the two different storage-dominated regimes *K*<sup>0</sup> and *K*<sup>∞</sup> (see Table 1), while the fracturing fluid pressure is different [15]. Therefore the poroelastic effects split the storage dominated regime (denoted previously as the *K*-vertex [12, 17]) into two sub-regimes: the *K*0-vertex (1D diffusion) which is similar to the former *K*-vertex, and the *K*∞-vertex (pseudo steady-state diffusion) characterized by a higher pressure. The essence of the difference between these two regimes is illustrated in Fig. 3. Initially the fracture front propagates faster than the diffusion front, therefore the diffusion length scale is small compared to the size of the fracture and the diffusion is one dimensional. As time goes on, the diffusion front catches up and then passes the fracture front, making the diffusion length scale to be larger than the fracture size, and, as a result, switching the propagation regime from the 1D diffusion to the pseudo steady-state (3D) diffusion.

#### **4. Methodology**

Inclusion of diffusion and poroelastic effects into the model of penny-shaped hydraulic fracture model propagation in the toughness-dominated regime thus requires evaluating the convolution type integrals [see (11), (13)-(15)]. Indeed, calculation of the fracturing fluid volume which has leaked into the formation requires a "convolution" on Ψ (*τ*) [see (13)], whereas evaluation of the backstress <sup>Σ</sup> (*ξ*, *<sup>t</sup>*) and the related fracture volume *Vbs* (*t*) and stress intensity factor *Kbs* (*t*) changes requires "convolutions" on <sup>Ξ</sup> (*ξ*, *<sup>τ</sup>*). These "convolutions"

**Figure 3.** Physical interpretation of the difference between *K*0- and *K*∞-vertices

involves both arguments of Ξ (*ξ*, *τ*) and are much more complicated than the "convolution" on the single argument Ψ (*τ*) [see (11), (14), and (15)].

To simplify the calculations of *Vbs* (*t*) and *Kbs* (*t*), we introduce two additional functions *vbs* (R, *<sup>τ</sup>*) and *kbs* (R, *<sup>τ</sup>*), such that "convolutions" on these functions yield *Vbs* (*t*) and *Kbs* (*t*) [see (14)-(17)]. Physically the function *vbs* (R, *<sup>τ</sup>*) can be interpreted as the volume change of a fracture of radius R > 1 at an elapsed time *τ* due to the backstress generated by a unit impulse of the pore pressure applied at time *τ* = 0 along the part of the fracture R located inside the unit circle *<sup>ξ</sup>* < 1. The function *kbs* (R, *<sup>τ</sup>*) is the corresponding change in the stress intensity factor. Note that there is a simple connection between *kbs* (R, *<sup>τ</sup>*) and *vbs* (R, *<sup>τ</sup>*) [see (16), (17)]

$$k\_{\rm bs} \left( \mathcal{R}, \tau \right) = \frac{2}{3} \frac{\partial v\_{\rm bs} \left( \mathcal{R}, \tau \right)}{\partial \mathcal{R}^2}. \tag{22}$$

10.5772/56460

617

�� , *<sup>x</sup>* ≤ 1,

Fluid-Driven Fracture in a Poroelastic Rock http://dx.doi.org/10.5772/56460

<sup>1</sup> <sup>−</sup> *<sup>ξ</sup>*<sup>2</sup> , *<sup>ξ</sup>* <sup>≤</sup> 1. (25)

*<sup>τ</sup>*<sup>−</sup>1/2. (26)

<sup>−</sup>3/2 , (27)

Therefore, *vbs* (R, *<sup>τ</sup>*) depends only on time [see (22)], and in order to define *vbs* (R, *<sup>τ</sup>*) we can evaluate it at any convenient point, e.g., R = 1. The expression for the stress distribution,

= (1 + *x*) *K*

*π*3/2*τ*1/2

� <sup>1</sup> 0

*vbs* (R, *<sup>τ</sup>*) <sup>=</sup> <sup>−</sup><sup>3</sup>

�

� *x*2 �

*E* � *ξ*2�

1 − *xt*<sup>2</sup> <sup>1</sup> <sup>−</sup> *<sup>t</sup>*<sup>2</sup> *dt*,

2

*δ* (*τ*) − 2 (*πτ*)

1, *ξ* ≤ 1

√ 1 *ξ*2−1 �

Substitution of this large-time asymptote Ξ∞ (*ξ*, *<sup>τ</sup>*) into the definitions of *kbs* (R, *<sup>τ</sup>*) and

−3/2�

− <sup>8</sup> <sup>3</sup> (*πτ*)

, *<sup>ξ</sup>* <sup>&</sup>gt; <sup>1</sup> .

, *x* ≤ 1,

given by (23), can be simplified by means of [20]

*K* �

= (1 + *x*)

4*x* (1 + *x*) 2 �

<sup>Ξ</sup><sup>0</sup> (*ξ*, *<sup>τ</sup>*) <sup>=</sup> <sup>−</sup> <sup>2</sup>

*E* (*x*) =

<sup>∞</sup> (*ξ*) �

> 

2 *<sup>π</sup>* arctan �

Now, using the integral representation of the elliptic integral

� 2*E* � *x*2 � − � 1 − *x*<sup>2</sup> � *K* � *x*2

4*x* (1 + *x*) 2 �

*E* �

one can calculate *vbs* (R, *<sup>τ</sup>*) [see (17)]

*Large-time asymptotes of kbs* (R, *<sup>τ</sup>*) *and vbs* (R, *<sup>τ</sup>*)

where *δ* (*τ*) is the Dirac delta function and

*vbs* (R, *<sup>τ</sup>*), given by (16) and (17), leads to

The large-time asymptote of Ξ (*ξ*, *τ*) is given by [15]

Ξ∞ (*ξ*, *<sup>τ</sup>*) = −Π˜ (0)

Π˜ (0) <sup>∞</sup> (*ξ*) =

Note that in the leading order we have separation of variables.

such that

*Small-time asymptotes of kbs* (R, *<sup>τ</sup>*) *and vbs* (R, *<sup>τ</sup>*)

The small-time asymptote of Ξ (*ξ*, *τ*) is given by [15]

$$\Xi\_0\left(\xi,\tau\right) = -\frac{1}{\pi^{3/2}\tau^{1/2}} \left\{ \left(1-\xi\right)^{-1} E\left[\frac{4\tilde{\xi}}{\left(1+\tilde{\xi}\right)^2}\right] + \left(1+\tilde{\xi}\right)^{-1} K \left[\frac{4\tilde{\xi}}{\left(1+\tilde{\xi}\right)^2}\right] \right\},\tag{23}$$

where *K* (*x*) and *E* (*x*) are the complete elliptic integrals of the first and second kinds respectively [18]. This asymptote has a strong singularity 1/ (1 − *ξ*), which causes significant problems in numerical simulations. Also one can observe a separation of variables, which simplifies the evaluation of *kbs* (R, *<sup>τ</sup>*) and *vbs* (R, *<sup>τ</sup>*) at small time.

Substitution of this small-time asymptote <sup>Ξ</sup><sup>0</sup> (*ξ*, *<sup>τ</sup>*) into the expression for *kbs* (R, *<sup>τ</sup>*) (16) yields

$$k\_{bs} \left( \mathcal{R}, \tau \right) = 0. \tag{24}$$

Therefore, *vbs* (R, *<sup>τ</sup>*) depends only on time [see (22)], and in order to define *vbs* (R, *<sup>τ</sup>*) we can evaluate it at any convenient point, e.g., R = 1. The expression for the stress distribution, given by (23), can be simplified by means of [20]

$$\begin{aligned} E\left[\frac{4\mathbf{x}}{\left(1+\mathbf{x}\right)^2}\right] &= \left(1+\mathbf{x}\right)\left[2E\left(\mathbf{x}^2\right) - \left(1-\mathbf{x}^2\right)K\left(\mathbf{x}^2\right)\right], \quad \mathbf{x} \le 1, \\\\ K\left[\frac{4\mathbf{x}}{\left(1+\mathbf{x}\right)^2}\right] &= \left(1+\mathbf{x}\right)K\left(\mathbf{x}^2\right), \quad \mathbf{x} \le 1, \end{aligned}$$

such that

10 Effective and Sustainable Hydraulic Fracturing

**Figure 3.** Physical interpretation of the difference between *K*0- and *K*∞-vertices

on the single argument Ψ (*τ*) [see (11), (14), and (15)].

*Small-time asymptotes of kbs* (R, *<sup>τ</sup>*) *and vbs* (R, *<sup>τ</sup>*)

<sup>Ξ</sup><sup>0</sup> (*ξ*, *<sup>τ</sup>*) <sup>=</sup> <sup>−</sup> <sup>1</sup>

The small-time asymptote of Ξ (*ξ*, *τ*) is given by [15]

*π*3/2*τ*1/2

simplifies the evaluation of *kbs* (R, *<sup>τ</sup>*) and *vbs* (R, *<sup>τ</sup>*) at small time.

(1 − *ξ*)

<sup>−</sup><sup>1</sup> *<sup>E</sup>* 

(16), (17)]

yields

involves both arguments of Ξ (*ξ*, *τ*) and are much more complicated than the "convolution"

To simplify the calculations of *Vbs* (*t*) and *Kbs* (*t*), we introduce two additional functions *vbs* (R, *<sup>τ</sup>*) and *kbs* (R, *<sup>τ</sup>*), such that "convolutions" on these functions yield *Vbs* (*t*) and *Kbs* (*t*) [see (14)-(17)]. Physically the function *vbs* (R, *<sup>τ</sup>*) can be interpreted as the volume change of a fracture of radius R > 1 at an elapsed time *τ* due to the backstress generated by a unit impulse of the pore pressure applied at time *τ* = 0 along the part of the fracture R located inside the unit circle *<sup>ξ</sup>* < 1. The function *kbs* (R, *<sup>τ</sup>*) is the corresponding change in the stress intensity factor. Note that there is a simple connection between *kbs* (R, *<sup>τ</sup>*) and *vbs* (R, *<sup>τ</sup>*) [see

3

where *K* (*x*) and *E* (*x*) are the complete elliptic integrals of the first and second kinds respectively [18]. This asymptote has a strong singularity 1/ (1 − *ξ*), which causes significant problems in numerical simulations. Also one can observe a separation of variables, which

Substitution of this small-time asymptote <sup>Ξ</sup><sup>0</sup> (*ξ*, *<sup>τ</sup>*) into the expression for *kbs* (R, *<sup>τ</sup>*) (16)

4*ξ* (1 + *ξ*) 2 

+ (1 + *ξ*)

<sup>−</sup><sup>1</sup> *<sup>K</sup>* 

*kbs* (R, *<sup>τ</sup>*) = 0. (24)

*<sup>∂</sup>vbs* (R, *<sup>τ</sup>*)

*<sup>∂</sup>*R<sup>2</sup> . (22)

4*ξ* (1 + *ξ*) 2 

, (23)

*kbs* (R, *<sup>τ</sup>*) <sup>=</sup> <sup>2</sup>

$$\Xi\_0\left(\xi,\tau\right) = -\frac{2}{\pi^{3/2}\pi^{1/2}}\frac{E\left(\xi^2\right)}{1-\xi^2}, \quad \xi \le 1. \tag{25}$$

Now, using the integral representation of the elliptic integral

$$E\left(\mathbf{x}\right) = \int\_0^1 \sqrt{\frac{1-\mathbf{x}t^2}{1-t^2}}dt\,\mathrm{s}$$

one can calculate *vbs* (R, *<sup>τ</sup>*) [see (17)]

$$
\sigma\_{bs} \left( \mathcal{R}, \tau \right) = -\frac{3}{2} \tau^{-1/2}. \tag{26}
$$

*Large-time asymptotes of kbs* (R, *<sup>τ</sup>*) *and vbs* (R, *<sup>τ</sup>*)

The large-time asymptote of Ξ (*ξ*, *τ*) is given by [15]

$$\Xi\_{\infty} \left( \mathfrak{f}, \tau \right) = -\tilde{\Pi}\_{\infty}^{(0)} \left( \mathfrak{f} \right) \left[ \delta \left( \tau \right) - 2 \left( \pi \tau \right)^{-3/2} \right] - \frac{8}{3} \left( \pi \tau \right)^{-3/2} \,, \tag{27}$$

where *δ* (*τ*) is the Dirac delta function and

$$\|\tilde{\Pi}^{(0)}\_{\infty}(\xi) = \begin{cases} 1, & \xi \le 1 \\ \frac{2}{\pi} \arctan\left(\frac{1}{\sqrt{\xi^2 - 1}}\right), \xi > 1 \end{cases}$$

Note that in the leading order we have separation of variables.

Substitution of this large-time asymptote Ξ∞ (*ξ*, *<sup>τ</sup>*) into the definitions of *kbs* (R, *<sup>τ</sup>*) and *vbs* (R, *<sup>τ</sup>*), given by (16) and (17), leads to

$$k\_{\rm bs} \left( \mathcal{R}\_{\prime} \,\mathrm{\,\tau} \right) = -\frac{2\delta \left( \tau \right)}{\sqrt{\pi}} + \mathcal{O} \left( \tau^{-3/2} \right), \tag{28}$$

619

;

;

<sup>+</sup> *<sup>η</sup>* <sup>1</sup> <sup>−</sup> *<sup>η</sup>* ,

*ρ*5/2 <sup>∞</sup> (*τ*) *<sup>τ</sup>* .

*Vbs<sup>κ</sup>* (*τ*) <sup>=</sup> <sup>−</sup><sup>3</sup>

<sup>2</sup> *<sup>ρ</sup>*−1/2

*ρσ* (*τ*) = <sup>2</sup>−1/2*π*−3/4*G*−1/2 *<sup>c</sup>*

*Vbs<sup>σ</sup>* (*τ*) <sup>=</sup> <sup>−</sup><sup>3</sup>

<sup>2</sup> *<sup>ρ</sup>*−1/2

*<sup>π</sup>* <sup>+</sup> <sup>2</sup> (<sup>1</sup> <sup>−</sup> *<sup>η</sup>*) /*Gc* <sup>−</sup> <sup>√</sup>*<sup>π</sup>* 4

4 *η*

<sup>√</sup>*<sup>π</sup> ηρ*1/2

<sup>Π</sup>*<sup>κ</sup>* (*τ*) <sup>=</sup> *<sup>π</sup>*1/2

<sup>Π</sup>*<sup>σ</sup>* (*τ*) <sup>=</sup> *<sup>π</sup>*1/2

*Vbs*<sup>∞</sup> (*τ*) <sup>=</sup> *Kbs*<sup>∞</sup> (*τ*) <sup>=</sup> <sup>−</sup> <sup>2</sup>

• *K*˜*σ*0-vertex

• *K*˜ *<sup>κ</sup>*∞*K*˜*σ*∞-edge

*ρ*<sup>∞</sup> (*τ*) =

**6. Transient solution**

(12)-(17) has been developed [15].

terminate on the *K*˜ *<sup>κ</sup>*∞*K*˜*σ*∞-edge.

2 *η* *π*1/2*τ*1/2 *ρ*1/2 *<sup>κ</sup>* (*τ*)

*<sup>κ</sup>* (*τ*), <sup>E</sup>*<sup>κ</sup>* (*τ*) <sup>=</sup> <sup>8</sup>

*πτ*1/2 *ρ*1/2 *<sup>σ</sup>* (*τ*)

*<sup>σ</sup>* (*τ*), <sup>E</sup>*<sup>σ</sup>* (*τ*) <sup>=</sup> <sup>8</sup>

<sup>1</sup> <sup>−</sup> *<sup>η</sup>Gv Gc*

> 1 +

2

To obtain a general trajectory of the system starting at the *K*0-vertex and ending at the *K*˜*κ*∞*K*˜*σ*∞-edge, an implicit numerical algorithm to solve the set of governing equations

The results of the numerical simulations for different values of the parameters *Gi* are presented in Figs 4-7. Depending on the values of the parameters *Gi*, the system can travel through different vertices, although the journey always has to start at the *K*0-vertex and

In some cases the propagation of the fracture terminates before it arrives to the *K*˜ *<sup>κ</sup>*∞*K*˜*σ*∞-edge (see Figs 6-7). In these cases, the system, going through a diffusion-dominated vertices, arrives to a point when the dilation of the poroelastic medium ∼ *Vbs* is very large, such that

Γ (5/3) <sup>Γ</sup> (13/6) <sup>+</sup>

> √*π* <sup>3</sup> *Gv*

<sup>−</sup>1/2

1 *ρ*1/2 *<sup>σ</sup>* (*τ*)

√*π* <sup>3</sup> *Gv*

, Π∞ (*τ*) <sup>=</sup> *<sup>π</sup>*1/2

<sup>∞</sup> (*τ*) [<sup>1</sup> <sup>+</sup> Π∞ (*τ*)] , <sup>E</sup><sup>∞</sup> (*τ*) <sup>=</sup> <sup>8</sup>

*π* 4*ρ*1/2 *<sup>κ</sup>* (*τ*)

*ρ*5/2

 ,

Fluid-Driven Fracture in a Poroelastic Rock http://dx.doi.org/10.5772/56460

*<sup>κ</sup>* (*τ*) [1 + *Vbs<sup>κ</sup>* (*τ*)] *τ*

*τ*1/4, *Kbs<sup>σ</sup>* (*τ*) = 0,

 ,

*<sup>σ</sup>* (*τ*) [1 + *Vbs<sup>σ</sup>* (*τ*)] *τ*

> *<sup>ρ</sup>*−1/2 <sup>∞</sup> (*τ*) 1 − *η*

> > √*π* <sup>3</sup> *Gv*

Γ (11/8) Γ (15/8)

2

*ρ*5/2

$$v\_{bs}\left(\mathcal{R},\tau\right) = -\frac{3\mathcal{R}^2 - 1}{\sqrt{\pi}}\delta\left(\tau\right) + \mathcal{O}\left(\tau^{-3/2}\right). \tag{29}$$

#### **5. Asymptotic solutions**

Details on the derivation of the asymptotic solutions, corresponding to each of the vertices of the parametric space, can be found in [15]. Here we simply list the solutions for the *K*0-, *K*˜ *<sup>κ</sup>*0-, and *K*˜*σ*0-vertices, as well as for the self-similar *K*˜ *<sup>κ</sup>*∞*K*˜*σ*∞-edge. These solutions are actually expressed in the same time-independent scaling that corresponds to G*<sup>σ</sup>* = 1. In other words, all the solutions are given in terms of the scaled radius *ρ*(*τ*) function of the dimensionless time *τ*

$$
\rho\left(\tau\right) = \frac{\mathcal{R}\left(t\right)}{L\_{\text{tr}}}, \qquad \tau = \frac{t}{T} \tag{30}
$$

where

$$L\_{\rm tr} = \left(\frac{K\_{\rm Ic}}{\sigma\_0 - p\_0}\right)^2, \qquad T = \frac{L\_{\rm tr}^2}{4c}. \tag{31}$$

Besides the asymptotic expressions for *<sup>ρ</sup>*(*τ*), *Kbs*(*τ*), *Vbs*(*τ*), <sup>Π</sup>(*τ*), we have also provided expressions for the hydraulic fracturing efficiency E, defined as E ≡ *<sup>V</sup>*crack/*V*inject.

• *K*0-vertex

$$\begin{aligned} \rho\_0(\tau) &= \left(\frac{3}{8\sqrt{\pi}G\_v}\right)^{2/5} \tau^{2/5}, & K\_{\rm bs0}(\tau) &= 0, \\\\ V\_{\rm bs0}\left(\tau\right) &= -\frac{3}{2}\eta \frac{\pi^{1/2}\tau^{1/2}}{\rho\_0^{1/2}\left(\tau\right)} \left[\frac{\Gamma\left(9/5\right)}{\Gamma\left(23/10\right)} + \frac{\pi^{1/2}}{2\rho\_0^{1/2}\left(\tau\right)} \frac{\Gamma\left(8/5\right)}{\Gamma\left(21/10\right)}\right], \\\\ \Pi\_0\left(\tau\right) &= \frac{\pi^{1/2}}{2}\rho\_0^{-1/2}\left(\tau\right), & \mathcal{E}\_0\left(\tau\right) &= \frac{8\sqrt{\pi}}{3}G\_v \frac{\rho\_0^{5/2}\left(\tau\right)\left[1 + V\_{\rm bs0}\left(\tau\right)\right]}{\tau}, \end{aligned}$$

• *K*˜ *<sup>κ</sup>*0-vertex

$$\rho\_{\mathbf{k}}\left(\tau\right) = \pi^{-4/3} G\_c^{-2/3} \left(1 - \frac{\eta G\_v}{G\_c}\right)^{-2/3} \tau^{1/3}, \qquad K\_{\rm b\mathbf{s}\mathbf{x}}\left(\tau\right) = 0,$$

$$\begin{aligned} V\_{\text{bşıx}}\left(\tau\right) &= -\frac{3}{2} \eta \frac{\pi^{1/2} \tau^{1/2}}{\rho\_{\text{x}}^{1/2}\left(\tau\right)} \left[ \frac{\Gamma\left(5/3\right)}{\Gamma\left(13/6\right)} + \frac{\pi}{4\rho\_{\text{x}}^{1/2}\left(\tau\right)} \right], \\\\ \Pi\_{\text{k}}\left(\tau\right) &= \frac{\pi^{1/2}}{2} \rho\_{\text{x}}^{-1/2}\left(\tau\right), \qquad \mathcal{E}\_{\text{k}}\left(\tau\right) = \frac{8\sqrt{\pi}}{3} G\_{\text{v}} \frac{\rho\_{\text{x}}^{5/2}\left(\tau\right) \left[1 + V\_{\text{bşıx}}\left(\tau\right)\right]}{\tau}; \end{aligned}$$

• *K*˜*σ*0-vertex

12 Effective and Sustainable Hydraulic Fracturing

**5. Asymptotic solutions**

time *τ*

where

• *K*0-vertex

• *K*˜ *<sup>κ</sup>*0-vertex

*kbs* (R, *<sup>τ</sup>*) <sup>=</sup> <sup>−</sup>2*<sup>δ</sup>* (*τ*)

*<sup>ρ</sup>* (*τ*) <sup>=</sup> *<sup>R</sup>* (*t*)

 *KIc σ*<sup>0</sup> − *p*<sup>0</sup>

expressions for the hydraulic fracturing efficiency E, defined as E ≡ *<sup>V</sup>*crack/*V*inject.

 3 8 <sup>√</sup>*πGv*

*π*1/2*τ*1/2 *ρ*1/2 <sup>0</sup> (*τ*)

*L*tr =

*ρ*<sup>0</sup> (*τ*) =

2 *η*

<sup>2</sup> *<sup>ρ</sup>*−1/2

*ρκ* (*τ*) <sup>=</sup> *<sup>π</sup>*−4/3*G*−2/3 *<sup>c</sup>*

*Vbs*<sup>0</sup> (*τ*) <sup>=</sup> <sup>−</sup><sup>3</sup>

<sup>Π</sup><sup>0</sup> (*τ*) <sup>=</sup> *<sup>π</sup>*1/2

*L*tr

2

Besides the asymptotic expressions for *<sup>ρ</sup>*(*τ*), *Kbs*(*τ*), *Vbs*(*τ*), <sup>Π</sup>(*τ*), we have also provided

2/5

Γ (9/5) <sup>Γ</sup> (23/10) <sup>+</sup>

<sup>0</sup> (*τ*), <sup>E</sup><sup>0</sup> (*τ*) <sup>=</sup> <sup>8</sup>

<sup>1</sup> <sup>−</sup> *<sup>η</sup>Gv Gc*

*vbs* (R, *<sup>τ</sup>*) <sup>=</sup> <sup>−</sup>3R<sup>2</sup> <sup>−</sup> <sup>1</sup>

√*π*

Details on the derivation of the asymptotic solutions, corresponding to each of the vertices of the parametric space, can be found in [15]. Here we simply list the solutions for the *K*0-, *K*˜ *<sup>κ</sup>*0-, and *K*˜*σ*0-vertices, as well as for the self-similar *K*˜ *<sup>κ</sup>*∞*K*˜*σ*∞-edge. These solutions are actually expressed in the same time-independent scaling that corresponds to G*<sup>σ</sup>* = 1. In other words, all the solutions are given in terms of the scaled radius *ρ*(*τ*) function of the dimensionless

√*π*

+ O *τ*−3/2

*δ* (*τ*) + O

, *<sup>τ</sup>* <sup>=</sup> *<sup>t</sup>*

, *<sup>T</sup>* <sup>=</sup> *<sup>L</sup>*<sup>2</sup>

tr

*τ*2/5, *Kbs*<sup>0</sup> (*τ*) = 0,

*π*1/2 2*ρ*1/2 <sup>0</sup> (*τ*)

*ρ*5/2

*τ*1/3, *Kbs<sup>κ</sup>* (*τ*) = 0,

√*π* <sup>3</sup> *Gv*

<sup>−</sup>2/3

Γ (8/5) Γ (21/10)

<sup>0</sup> (*τ*) [<sup>1</sup> <sup>+</sup> *Vbs*<sup>0</sup> (*τ*)] *τ*

 ,

;

 *τ*−3/2

, (28)

. (29)

*<sup>T</sup>* , (30)

<sup>4</sup>*<sup>c</sup>* . (31)

$$\rho\_{\sigma}\left(\tau\right) = 2^{-1/2}\pi^{-3/4}\mathcal{G}\_{\mathfrak{c}}^{-1/2}\left(1 - \frac{\eta\mathcal{G}\_{\mathfrak{v}}}{\mathcal{G}\_{\mathfrak{c}}}\right)^{-1/2}\tau^{1/4}, \qquad K\_{b\mathfrak{s}\sigma}\left(\tau\right) = 0,$$

$$\begin{aligned} V\_{\rm bs\sigma} \left( \tau \right) &= -\frac{3}{4} \eta \frac{\pi \tau^{1/2}}{\rho\_{\sigma}^{1/2} \left( \tau \right)} \left[ 1 + \frac{1}{\rho\_{\sigma}^{1/2} \left( \tau \right)} \frac{\Gamma \left( 11/8 \right)}{\Gamma \left( 15/8 \right)} \right], \\\\ \Pi\_{\sigma} \left( \tau \right) &= \frac{\pi^{1/2}}{2} \rho\_{\sigma}^{-1/2} \left( \tau \right), \qquad \mathcal{E}\_{\sigma} \left( \tau \right) = \frac{8 \sqrt{\pi}}{3} G\_{\nu} \frac{\rho\_{\sigma}^{5/2} \left( \tau \right) \left[ 1 + V\_{\rm bs\sigma} \left( \tau \right) \right]}{\tau}; \end{aligned}$$

• *K*˜ *<sup>κ</sup>*∞*K*˜*σ*∞-edge

$$\rho\_{\infty}(\tau) = \left[\frac{\sqrt{\pi + 2\left(1 - \eta\right)/\mathcal{G}\_c} - \sqrt{\pi}}{4}\right]^2, \qquad \Pi\_{\infty}(\tau) = \frac{\pi^{1/2}}{2} \frac{\rho\_{\infty}^{-1/2}(\tau)}{1 - \eta} + \frac{\eta}{1 - \eta}.$$

$$W\_{b\text{soc}}\left(\tau\right) = K\_{b\text{soc}}\left(\tau\right) = -\frac{2}{\sqrt{\pi}}\eta\rho\_{\text{ss}}^{1/2}\left(\tau\right)\left[1 + \Pi\_{\text{ss}}\left(\tau\right)\right], \quad \mathcal{E}\_{\text{ss}}\left(\tau\right) = \frac{8\sqrt{\pi}}{3}G\_{\upsilon}\frac{\rho\_{\text{ss}}^{5/2}\left(\tau\right)}{\tau}.$$

#### **6. Transient solution**

To obtain a general trajectory of the system starting at the *K*0-vertex and ending at the *K*˜*κ*∞*K*˜*σ*∞-edge, an implicit numerical algorithm to solve the set of governing equations (12)-(17) has been developed [15].

The results of the numerical simulations for different values of the parameters *Gi* are presented in Figs 4-7. Depending on the values of the parameters *Gi*, the system can travel through different vertices, although the journey always has to start at the *K*0-vertex and terminate on the *K*˜ *<sup>κ</sup>*∞*K*˜*σ*∞-edge.

In some cases the propagation of the fracture terminates before it arrives to the *K*˜ *<sup>κ</sup>*∞*K*˜*σ*∞-edge (see Figs 6-7). In these cases, the system, going through a diffusion-dominated vertices, arrives to a point when the dilation of the poroelastic medium ∼ *Vbs* is very large, such that the volume of the fracture becomes equal to zero. The fracture closure time can easily be estimated by substituting the above analytical expressions for *Vbs* and *Kbs* into *<sup>V</sup>*crack (*τ*) ∼ <sup>1</sup> + *Vbs* (*τ*) − *Kbs* (*τ*) = 0, and solving it with respect to time *<sup>τ</sup>*. Note that this estimate is based only on the total volume of fracture and does not address the issue as to when and where the two faces of the fracture first become into contact. In fact, the fracture will first evolve towards a viscosity-dominated propagation regime then close. In other words, a decrease of fracture opening leads to an increase of the pressure gradient of the viscous fracturing fluid, which in turn leads to an increase of the viscous dissipation and eventually to the violation of the zero viscosity assumption for the fracturing fluid. Moreover the fluid pressure profile inside the fracture becomes to be strongly nonuniform, and thus one cannot use the results of the auxiliary problem to model the poroelastic effects. More sophisticated models are needed to study this situation.

10.5772/56460

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Fluid-Driven Fracture in a Poroelastic Rock http://dx.doi.org/10.5772/56460

**(a)** Fracture radius *ρ* vs time *τ*

**(b)** Fracturing fluid pressure Π vs time *τ*

**(c)** Hydraulic fracturing efficiency E vs time *τ*

**Figure 4.** General case *Gv* = *Gc* = 1, *η* = 0.0, 0.25, 0.5
