**7. Discussion**

Let us consider the results from an application point of view. Table 2 list the parameters for a re-injection of production water [21]. The values for *S* and *c* were estimated on the assumption that *<sup>K</sup>*f/*<sup>E</sup>* ≪ 1 [22].

To characterize the propagation of a fracture, the transition times were calculated using the expressions found in Section 3, see Table 3. In this example, all time scales are well separated. As a result, the fracture follows the edges with the shortest transition time starting at the *K*0-vertex, passing through the *K*˜*σ*0-vertex, and ending up at the *K*˜*σ*∞-vertex. Moreover the transition time from the *K*0-vertex into the *K*˜*σ*∞-vertex is very small compared to the treatment time. This means that the treatment design can be based on a constant radius model. This analysis relies only on general results of the scaling analysis and does not involve any explicit asymptotic solutions.


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

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

14 Effective and Sustainable Hydraulic Fracturing

models are needed to study this situation.

involve any explicit asymptotic solutions.

assumption that *<sup>K</sup>*f/*<sup>E</sup>* ≪ 1 [22].

**7. Discussion**

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

Let us consider the results from an application point of view. Table 2 list the parameters for a re-injection of production water [21]. The values for *S* and *c* were estimated on the

To characterize the propagation of a fracture, the transition times were calculated using the expressions found in Section 3, see Table 3. In this example, all time scales are well separated. As a result, the fracture follows the edges with the shortest transition time starting at the *K*0-vertex, passing through the *K*˜*σ*0-vertex, and ending up at the *K*˜*σ*∞-vertex. Moreover the transition time from the *K*0-vertex into the *K*˜*σ*∞-vertex is very small compared to the treatment time. This means that the treatment design can be based on a constant radius model. This analysis relies only on general results of the scaling analysis and does not

> porosity *φ* (%) 10 20 permeability *k* (md) 10 100 Young's modulus *E* (GPa) 30 15 Poisson's ratio *ν* 0.2 0.25

diffusion coefficient *c* (m2/s) 0.212 1.04 storage coefficient *<sup>S</sup>* (Pa<sup>−</sup>1) 4.65 × <sup>10</sup>−<sup>11</sup> 9.49 × <sup>10</sup>−<sup>11</sup> poroelastic stress modulus *η* 0.225 0.2 reservoir thickness *H* (m) 50 5

rock toughness *KIc* (MPa · m1/2) 1.0 water bulk modulus *K*<sup>f</sup> (GPa) 2.2 water viscosity *µ* (mPa · s) 1.0 Biot coefficient *α* 0.6

confining stress *σ*<sup>0</sup> (MPa) 55 initial pore pressure *p*<sup>0</sup> (MPa) 30 injection rate *Q*<sup>0</sup> (m3/day) 750 treatment time *T* (days) 100

**Table 2.** Characteristic parameters during production water re-injection [21]

**low porosity mean porosity reservoir (LPR) reservoir (MPR)**

623

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 6.** Case *Gv* = <sup>10</sup>−15, *Gc* = <sup>3</sup> × <sup>10</sup>−11, *<sup>η</sup>* = 0.0, 0.5. Here the fracture goes through the *<sup>K</sup>*˜ *<sup>σ</sup>*0-vertex

**Figure 5.** Case *Gv* = <sup>10</sup>−5, *Gc* = <sup>10</sup>, *<sup>η</sup>* = 0.0, 0.25, 0.5. Here the fracture goes through the *<sup>K</sup>*˜ *<sup>κ</sup>*0-vertex

16 Effective and Sustainable Hydraulic Fracturing

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

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

**(c)** Hydraulic fracturing efficiency E vs time *τ* **Figure 5.** Case *Gv* = <sup>10</sup>−5, *Gc* = <sup>10</sup>, *<sup>η</sup>* = 0.0, 0.25, 0.5. Here the fracture goes through the *<sup>K</sup>*˜ *<sup>κ</sup>*0-vertex

**Figure 6.** Case *Gv* = <sup>10</sup>−15, *Gc* = <sup>3</sup> × <sup>10</sup>−11, *<sup>η</sup>* = 0.0, 0.5. Here the fracture goes through the *<sup>K</sup>*˜ *<sup>σ</sup>*0-vertex

625

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

**possible transitions current vertex transition time vertex vertex LPR**, **sec MPR**, **sec choice**

*K*˜ *<sup>κ</sup>*<sup>0</sup> 4.6 × 1016 8.3 × 109

*K*˜*σ*<sup>0</sup> *K*˜*σ*<sup>∞</sup> 5.8 × 103 11.9 *K*˜*σ*<sup>∞</sup>

At the *K*˜*σ*∞-vertex the fracture radius is equal to *R* ≈ 3.4 m and the net pressure is *p* − *σ*<sup>0</sup> ≈ 7.4 MPa in the LPR case, whereas *R* ≈ 0.35 m and *p* − *σ*<sup>0</sup> ≈ 7.2 MPa for the MPR case. If one does not take into account the poroelastic effect, the fracture would grow to *R* ≈ 4.4 m and *p* − *σ*<sup>0</sup> ≈ 0.071 MPa in the LPR case, while *R* ≈ 0.44 m and *p* − *σ*<sup>0</sup> ≈ 0.71 MPa in the MPR case. Thus the ultimate fracture radius decrease due to the poroelastic effects is not so significant. On the other hand, the net pressure increase is significant (about 100- and 10-fold

In the above analysis we have assumed that the fracture propagates in the toughness-dominated regime (fracturing fluid of zero viscosity). To check this assumption, the value of the following dimensionless group (known as the dimensionless viscosity [17])

*<sup>Q</sup>*0*E*′<sup>3</sup>

, (32)

*K*4 *IcL* (*t*)

where *L* (*t*) is the characteristic fracture size. If the fracture propagates in a viscosity-dominated regime, then G*<sup>m</sup>* (*t*) ≫ 1. In the toughness-dominated regime, G*<sup>m</sup>* (*t*) ≪ 1. Using the above data one can find that at the *K*˜*σ*∞-vertex this dimensionless group is equal to G*<sup>m</sup>* ≈ 90 for the LPR and G*<sup>m</sup>* ≈ 121 for the MPR. Thus fracturing fluid viscosity should be taken into account. Nevertheless, the above example illustrates the importance of the poroelastic effects. In fact a rigorous analysis of the viscosity-dominated regimes predicts

The numerical simulations reported in Figs 4-7 sweep huge time ranges. This is a consequence of the small-time asymptote (*K*0-vertex) as the initial condition combined with the need to start from a physically correct initial condition to construct accurate numerical solutions. In practice, however, a correct assessment of the relevant part of the parametric space can dramatically simplify the situation. Knowing this information one can use the analytical vertex asymptotes for preliminary estimation, and then optimize a numerical algorithm. For example Figs 5-7 illustrate that one can use the asymptotic solution of an intermediate vertex as the initial condition provided that the transition time from the *K*0-vertex into this vertex is small compared to the treatment time. In the above example of production water reinjection, we have shown that the fracture propagation arrests within a very short period of time compared to the characteristic treatment time. Thus from a practical point of view in this case one can simply use the analytical large-time asymptote to design

<sup>G</sup>*<sup>m</sup>* (*t*) <sup>=</sup> *<sup>µ</sup>*′

*<sup>K</sup>*˜*σ*<sup>0</sup> 0.087 1.7 <sup>×</sup> <sup>10</sup>−<sup>3</sup> *<sup>K</sup>*˜*σ*<sup>0</sup>

*K*<sup>0</sup> *K*<sup>∞</sup> 1.2 × 1013 3.1 × 108

**Table 3.** Crack propagation

has to be assessed

the treatment.

increase in the LPR and MPR case, respectively).

similar values for the fracture size and the net pressure [15].

**Figure 7.** Case *Gc* = <sup>10</sup>−10, *<sup>η</sup>* = 0.0, 0.5. Here the fracture goes through the *<sup>K</sup>*˜ *<sup>κ</sup>*0- and *<sup>K</sup>*˜ *<sup>σ</sup>*0-vertices


**Table 3.** Crack propagation

18 Effective and Sustainable Hydraulic Fracturing

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

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

**(c)** Hydraulic fracturing efficiency E vs time *τ* **Figure 7.** Case *Gc* = <sup>10</sup>−10, *<sup>η</sup>* = 0.0, 0.5. Here the fracture goes through the *<sup>K</sup>*˜ *<sup>κ</sup>*0- and *<sup>K</sup>*˜ *<sup>σ</sup>*0-vertices At the *K*˜*σ*∞-vertex the fracture radius is equal to *R* ≈ 3.4 m and the net pressure is *p* − *σ*<sup>0</sup> ≈ 7.4 MPa in the LPR case, whereas *R* ≈ 0.35 m and *p* − *σ*<sup>0</sup> ≈ 7.2 MPa for the MPR case. If one does not take into account the poroelastic effect, the fracture would grow to *R* ≈ 4.4 m and *p* − *σ*<sup>0</sup> ≈ 0.071 MPa in the LPR case, while *R* ≈ 0.44 m and *p* − *σ*<sup>0</sup> ≈ 0.71 MPa in the MPR case. Thus the ultimate fracture radius decrease due to the poroelastic effects is not so significant. On the other hand, the net pressure increase is significant (about 100- and 10-fold increase in the LPR and MPR case, respectively).

In the above analysis we have assumed that the fracture propagates in the toughness-dominated regime (fracturing fluid of zero viscosity). To check this assumption, the value of the following dimensionless group (known as the dimensionless viscosity [17]) has to be assessed

$$\mathcal{G}\_m\left(t\right) = \frac{\mu' Q\_0 E'^3}{K\_{Ic}^4 L\left(t\right)},\tag{32}$$

where *L* (*t*) is the characteristic fracture size. If the fracture propagates in a viscosity-dominated regime, then G*<sup>m</sup>* (*t*) ≫ 1. In the toughness-dominated regime, G*<sup>m</sup>* (*t*) ≪ 1. Using the above data one can find that at the *K*˜*σ*∞-vertex this dimensionless group is equal to G*<sup>m</sup>* ≈ 90 for the LPR and G*<sup>m</sup>* ≈ 121 for the MPR. Thus fracturing fluid viscosity should be taken into account. Nevertheless, the above example illustrates the importance of the poroelastic effects. In fact a rigorous analysis of the viscosity-dominated regimes predicts similar values for the fracture size and the net pressure [15].

The numerical simulations reported in Figs 4-7 sweep huge time ranges. This is a consequence of the small-time asymptote (*K*0-vertex) as the initial condition combined with the need to start from a physically correct initial condition to construct accurate numerical solutions. In practice, however, a correct assessment of the relevant part of the parametric space can dramatically simplify the situation. Knowing this information one can use the analytical vertex asymptotes for preliminary estimation, and then optimize a numerical algorithm. For example Figs 5-7 illustrate that one can use the asymptotic solution of an intermediate vertex as the initial condition provided that the transition time from the *K*0-vertex into this vertex is small compared to the treatment time. In the above example of production water reinjection, we have shown that the fracture propagation arrests within a very short period of time compared to the characteristic treatment time. Thus from a practical point of view in this case one can simply use the analytical large-time asymptote to design the treatment.

#### **8. Conclusions**

This paper has described a detailed study of a penny-shaped fracture driven by a zero viscosity fluid through a poroelastic medium. The main contribution of this study is the consideration of large scale 3D diffusion and the related poroelastic effect (backstress). We have shown that the problem under consideration has six self-similar propagation regimes (see Section 3). In particular we have demonstrated the existence of a regime (*K*˜ *<sup>κ</sup>*∞*K*˜*σ*∞-edge) where the fracture stops propagating. In this regime the fracturing fluid injection is balanced by the 3D fluid leak-off. This stationary solution in the case of zero backstress, *η* = 0, was originally obtained by [13].

10.5772/56460

627

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

[6] A. van As and R.G. Jeffrey. Caving induced by hydraulic fracturing at Northparkes

[7] R.G. Jeffrey and K.W. Mills. Hydraulic fracturing applied to inducing longwall coal mine goaf falls. In *Pacific Rocks 2000*, pages 423–430, Rotterdam, 2000. Balkema.

[8] J. Adachi, E . Siebrits, A. P. Peirce, and J. Desroches. Computer simulation of hydraulic fractures. *Int. J. Rock Mech. Min. Sci.*, 44(5):739–757, International Journal of Rock

[9] E.D. Carter. Optimum fluid characteristics for fracture extension. In G.C. Howard and C.R. Fast, editors, *Drilling and Production Practices*, pages 261–270. American Petroleum

[10] M. P. Cleary. Analysis of mechanisms and procedures for producing favourable shapes of hydraulic fractures. In *Proc. 55th Annual Fall Technical Conference and Exhibition of the Society of Petroleum Engineers of AIME*, volume SPE 9260, pages 1–16, September 1980.

[11] E. Detournay and A.H-D. Cheng. Plane strain analysis of a stationary hydraulic fracture

[12] A.P. Bunger, E. Detournay, and D.I. Garagash. Toughness-dominated hydraulic fracture

[13] S. A. Mathias and M. Reeuwijk. Hydraulic fracture propagation with 3-d leak-off.

[14] I. Berchenko, E. Detournay, and N. Chandler. Propagation of natural hydraulic fractures.

[15] Y. Kovalyshen. *Fluid-Driven Fracture in Poroelastic Medium*. PhD thesis, University of

[16] D.T. Barr. *Leading-Edge Analysis for Correct Simulation of Interface Separation and Hydraulic Fracturing*. PhD thesis, Massachusetts Institute of Technology, Cambridge MA, 1991.

[17] A.A. Savitski and E. Detournay. Propagation of a fluid-driven penny-shaped fracture in an impermeable rock: Asymptotic solutions. *Int. J. Solids Structures*, 39(26):6311–6337,

[18] M. Abramowitz and I.A. Stegun, editors. *Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables*. Dover Publications Inc., New York NY, 1972.

[19] E. Detournay. Propagation regimes of fluid-driven fractures in impermeable rocks. *Int.*

[20] I.S. Gradshteyn and I.M. Ryzhik. *Table of Integrals, Series and Products*. Academic Press,

in a poroelastic medium. *Int. J. Solids Structures*, 27(13):1645–1662, 1991.

with leak-off. *Int. J. Fracture*, 134(2):175–190, 2005.

mines. In *Pacific Rocks 2000*, pages 353–360, Rotterdam, 2000. Balkema.

Mechanics and Mining Sciences 2007.

Institute, Tulsa OK, 1957.

*Transport in Porous Media*, 2009.

*J. Geomechanics*, 4(1):1–11, 2004.

San Diego CA, 5th edition, 1994.

2002.

*Int. J. Rock Mech. Min. Sci.*, 34(3-4), 1997.

Minnesota, Minneapolis, February 2010.

Numerical simulations illustrate that poroelastic effects could have a significant influence on the propagation of a hydraulic fracture. Namely in the case of 3D diffusion, the backstress effect leads to a decrease of the fracture radius (see Figs 4a and 5a) accompanied by an increase of the fracturing fluid pressure (see Figs 4b and 5b). Moreover, the poroelastic effects can lead to premature closure of a fracture propagating in a leak-off dominating regime with 1D diffusion.

The technique developed in this paper could be also applied to the problem of in situ stress determination by hydraulic fracture [23]. In this application the *in situ* stress determination relies on the interpretation of the fracture breakdown and reopening fluid pressure as well as of the fracture closure pressure during the shut-in phase of experiment. It is obvious that the poroelastic effects could lead to a significant corrections into the stress measurements.

#### **Author details**

Yevhen Kovalyshen1,<sup>⋆</sup> and Emmanuel Detournay1,2


### **References**


[6] A. van As and R.G. Jeffrey. Caving induced by hydraulic fracturing at Northparkes mines. In *Pacific Rocks 2000*, pages 353–360, Rotterdam, 2000. Balkema.

20 Effective and Sustainable Hydraulic Fracturing

This paper has described a detailed study of a penny-shaped fracture driven by a zero viscosity fluid through a poroelastic medium. The main contribution of this study is the consideration of large scale 3D diffusion and the related poroelastic effect (backstress). We have shown that the problem under consideration has six self-similar propagation regimes (see Section 3). In particular we have demonstrated the existence of a regime (*K*˜ *<sup>κ</sup>*∞*K*˜*σ*∞-edge) where the fracture stops propagating. In this regime the fracturing fluid injection is balanced by the 3D fluid leak-off. This stationary solution in the case of zero backstress, *η* = 0, was

Numerical simulations illustrate that poroelastic effects could have a significant influence on the propagation of a hydraulic fracture. Namely in the case of 3D diffusion, the backstress effect leads to a decrease of the fracture radius (see Figs 4a and 5a) accompanied by an increase of the fracturing fluid pressure (see Figs 4b and 5b). Moreover, the poroelastic effects can lead to premature closure of a fracture propagating in a leak-off dominating regime with

The technique developed in this paper could be also applied to the problem of in situ stress determination by hydraulic fracture [23]. In this application the *in situ* stress determination relies on the interpretation of the fracture breakdown and reopening fluid pressure as well as of the fracture closure pressure during the shut-in phase of experiment. It is obvious that the poroelastic effects could lead to a significant corrections into the stress measurements.

1 Drilling Mechanics Group, CSIRO Earth Science and Resource Engineering, Australia

[1] D.A. Mendelsohn. A review of hydraulic fracture modeling - part i: General concepts,

[2] M.J. Economides and K.G. Nolte, editors. *Reservoir Stimulation*. John Wiley & Sons,

[3] A.S. Abou-Sayed. Safe injection pressures for disposing of liquid wastes: a case study for deep well injection (SPE/ISRM-28236). In Balkema, editor, *Proceedings of the Second*

[4] J. Hagoort, B. D. Weatherill, and A. Settari. Modeling the propagation of waterflood-induced hydraulic fractures. *Soc. Pet. Eng. J.*, pages 293–301, August 1980.

[5] M. Blunt, F.J. Fayers, and F.M. Orr. Carbon dioxide in enhanced oil recovery. *Energy*

*SPE/ISRM Rock Mechanics in Petroleum Engineering*, pages 769–776, 1994.

**8. Conclusions**

originally obtained by [13].

1D diffusion.

**Author details**

**References**

Yevhen Kovalyshen1,<sup>⋆</sup> and Emmanuel Detournay1,2

Chichester UK, 3rd edition, 2000.

*Convers. Mgmt*, 34(9-11):1197–1204, 1993.

<sup>⋆</sup> Address all correspondence to: Yevhen.Kovalyshen@csiro.au

2 Department of Civil Engineering, University of Minnesota, USA

2d models, motivation for 3d modeling. *jert*, 106:369–376, 1984.


[21] P. Longuemare, J-L. Detienne, P. Lemonnier, M. Bouteca, and A. Onaisi. Numerical modeling of fracture propagation induced by water injection/re-injection. *SPE European Formation Damage, The Hague, The Netherlands*, May 2001. (SPE 68974).

**Chapter 30**

**Provisional chapter**

**Pressurization of a PKN Fracture in a Permeable Rock**

The aim of the present work is to investigate injection of a low-viscosity fluid into a pre-existing fracture within a linear elastic, permeable rock, as may occur in waterflooding and supercritical CO2 injection. In conventional hydraulic fracturing, high viscosity and cake building properties of injected fluid limit diffusion to a 1-D boundary layer incasing the crack. In the case of injection of low viscosity fluid into a fracture, diffusion will take place over wider range of scales, from 1-D to 2-D, thus, necessitating a new approach. In addition, the dissipation of energy associated with fracturing of the rock dominates the energy expended to flow a low viscosity fluid into the crack channel. As a result, the rock fracture toughness is an important parameter in evaluating the propagation driven by a low-viscosity fluid. We consider a pre-existing, un-propped, stationary Perkins, Kern and Nordgren's (PKN) fracture into which a low viscosity fluid is injected under a constant flow rate. The fundamental solution to the auxiliary problem of a step pressure increase in a fracture [1] is used to formulate and solve the convolution integral equation governing the transient crack pressurization under the assumption of negligible viscous dissipation. The propagation criterion for a PKN crack [2] is then used to evaluate the onset of propagation. The obtained solution for transient pressurization of a stationary crack provides initial

The problem of injection of a low-viscosity fluid into a pre-existing fracture may arise in several rock engineering areas, such as, injection of liquid waste (e.g., supercritical CO2) into deep geological formations for storage [3,4,5], waterflooding process to increase recovery from an oil reservoir [6], and control of possible leaks from pre-existing fractures around

and reproduction in any medium, provided the original work is properly cited.

©2012 Sarvaramini and Garagash, licensee InTech. This is an open access chapter 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 Sarvaramini and Garagash; 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,

**Pressurization of a PKN Fracture in a Permeable**

**Rock During Injection of a Low Viscosity Fluid**

**During Injection of a Low Viscosity Fluid**

Erfan Sarvaramini and Dmitry I. Garagash

Additional information is available at the end of the chapter

conditions to the fracture propagation problem.

Erfan Sarvaramini and Dmitry I. Garagash

http://dx.doi.org/10.5772/56474

10.5772/56474

**Abstract**

**1. Introduction**

Additional information is available at the end of the chapter

