**Appendix**

Superposition of stress intensity factors for two radial cracks of length *a* emanating from an internally pressurized (*Pinj* - injection pressure in the borehole, *P frac*- pressure inside the fracture) circular hole of radius *r* in an infinite plate subjected to an isostatic far-field stress *Pm* as described by [2] and [3] :

$$K\_I\left(P\_m\right) = P\_m \sqrt{r} \, \, ^\*f\_{Pm}\left(a\_\prime r\right) \tag{7}$$

( )

case:

**Acknowledgements**

with geomecon GmbH, Potsdam.

Ruhr-University Bochum, Germany

1957;210:153–68.

\*Address all correspondence to: sebastian.brenne@rub.de

**Author details**

Sebastian Brenne\*

**References**

*<sup>a</sup> f ar*

p

1 <sup>2</sup> 2 1 <sup>1</sup> , 1 1 sin

*r a*

 æ ö æ ö

p

æ ö ç ÷ ç ÷ æ ö =+ - ç ÷ ç ÷ ç ÷ è ø èø ç ÷ ç ÷ <sup>+</sup> è ø è ø

Note: In equations 10 and 12 the borehole was excluded from the integration of stresses (cf. equation 4). The critical fracture propagation pressure at a given fracture length *a*, borehole radius *r* and mode I fracture toughness *KIC* for the unjacketed (*Pc*−*MF* ) and the jacketed (*Pc*−*SMF* )

\* *<sup>m</sup>*

*K*

*IC c SMF m P P*

The authors wish to thank the German Federal Ministry for the Environment, Nature Conser‐ vation and Nuclear Safety for financing our project (FKZ 0325279B). Many core specimens were prepared and analyzed by our student staff: T. Hoferichter, J. Braun, S. Hönig, K. Bartmann and A. Kraft. A great praise to the precision mechanics workshop guys for the construction of the fine working pressure intensifier system. We appreciate fruitful discussions

, Michael Molenda, Ferdinand Stöckhert and Michael Alber

[1] Hubbert M, Willis D. Mechanics of hydraulic fracturing. Petroleum Transactions.

*K <sup>P</sup> P f <sup>f</sup> <sup>r</sup>* -

*IC*

æ ö

æ ö

*m*

*r*

Hydraulic and Sleeve Fracturing Laboratory Experiments on 6 Rock Types

= + ç ÷ è ø (13)

= + ç ÷ è ø (14)

(12)

435

http://dx.doi.org/10.5772/56301

<sup>1</sup> *frac <sup>P</sup>*

1

*inj frac*

1

*inj*

*c MF m P P P*

*<sup>P</sup> P f f f <sup>r</sup>* -

$$K\_I\left(P\_{inj}\right) = P\_{inj}\sqrt{r} \triangleq f\_{P\_{inj}}\left(a, r\right) \tag{8}$$

$$K\_I \left( P\_{frac} \right) = P\_{frac} \sqrt{r} \, \, ^\* f\_{P\_{fuc}} \left( a, r \right) \tag{9}$$

$$f\_{Pm}(a,r) = 2\left(1+\frac{a}{r}\right)^2 \left[\frac{\left(1+\frac{a}{r}\right)^2 - 1}{\left(\pi\left(1+\frac{a}{r}\right)\right)^2}\right]^{\frac{1}{2}} + \left(\pi\left(1+\frac{a}{r}\right)\right)^{\frac{1}{2}} \left[1-\frac{2}{\pi}\sin^{-1}\left(\frac{1}{1+\frac{a}{r}}\right)\right] \tag{10}$$

$$f\_{P\_{\text{inj}}}\left(a,r\right) = \left| 1.3 \frac{\frac{a}{r}}{1 + \left(1 + \frac{a}{r}\right)^3} + \frac{7.8\left(\sin\left(\frac{2a}{r}\right)\right)}{2\left(1 + \frac{a}{r}\right)^5} \right| \tag{11}$$

Hydraulic and Sleeve Fracturing Laboratory Experiments on 6 Rock Types http://dx.doi.org/10.5772/56301 435

$$f\_{P\_{fuc}}\left(a,r\right) = \left(\pi \left(1 + \frac{a}{r}\right)\right)^{\frac{1}{2}} \left(1 - \frac{2}{\pi} \sin^{-1}\left(\frac{1}{1 + \frac{a}{r}}\right)\right) \tag{12}$$

Note: In equations 10 and 12 the borehole was excluded from the integration of stresses (cf. equation 4). The critical fracture propagation pressure at a given fracture length *a*, borehole radius *r* and mode I fracture toughness *KIC* for the unjacketed (*Pc*−*MF* ) and the jacketed (*Pc*−*SMF* ) case:

$$P\_{c-MF} = \frac{1}{f\_{P\_{inj}} \ast f\_{P\_{f\_{max}}}} \left(\frac{K\_{IC}}{\sqrt{r}} + P\_m f\_{P\_m}\right) \tag{13}$$

$$P\_{c-SMF} = \frac{1}{f\_{P\_{inj}}} \left(\frac{K\_{IC}}{\sqrt{r}} + P\_m f\_{P\_m}\right) \tag{14}$$

#### **Acknowledgements**

Due to high data scatter, the theoretical scale effect (critical injection pressure *Pc* is higher for smaller borehole radii) cannot be resolved by our data. However, tests with a larger

The simple fracture mechanics model is able to explain the higher *PAE* in SMF experiments. Equations 5 and 6 include the influence of fractures (with or without pressure inside), which is omitted in the classical approach (Equation 1). The high coefficient c in SMF test can only be

We excluded poroelastic effects in our analysis due to the use of initially dry rocks with low

Superposition of stress intensity factors for two radial cracks of length *a* emanating from an

fracture) circular hole of radius *r* in an infinite plate subjected to an isostatic far-field stress *Pm*

( ) \* , ( ) *inj*

( ) \* , ( ) *frac*

1 2 2

2 1 , 21 1 1 sin

7

( ) 3 5

*<sup>r</sup> <sup>r</sup> f ar*

1 1

*a*

æ ö

*a a <sup>r</sup> f ar*

p

, 1.3

*inj P*

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

p

ç ÷ æ ö æ ö æ ö ç ÷ + - ç ÷ æ ö ç ÷ ç ÷ æ ö æ ö è ø = + ç ÷ ++ - ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ è ø è ø èø ç ÷ ç ÷ ç ÷ æ ö æ ö ç ÷ ç ÷ <sup>+</sup> ç ÷ ç ÷ ç ÷ <sup>+</sup> è ø è ø è ø è ø è ø

*r r <sup>a</sup> <sup>a</sup>*

<sup>1</sup> <sup>1</sup>

2 2

*a a*

æ ö ç ÷ æ ö æ ö ç ÷ ç ÷ è ø è ø = +

1 1 2 1 1.7

*a a r r*

æö æö ++ + - ç÷ ç÷ è ø èø èø


( ) \* , ( ) *K P P r f ar I m m Pm* = (7)

*K P P r f ar I inj inj P* = (8)

*K P P r f ar I frac frac P* = (9)

p


(10)

(11)

1

*<sup>r</sup> <sup>r</sup>*

<sup>2</sup> 7.8 sin

(*r* =6.35 mm) borehole give some support to the notion.

explained by assuming high microcrack lengths (a0 ≈6 mm).

permeabilities.

**Appendix**

internally pressurized (*Pinj*

434 Effective and Sustainable Hydraulic Fracturing

as described by [2] and [3] :

( )

*Pm*

The authors wish to thank the German Federal Ministry for the Environment, Nature Conser‐ vation and Nuclear Safety for financing our project (FKZ 0325279B). Many core specimens were prepared and analyzed by our student staff: T. Hoferichter, J. Braun, S. Hönig, K. Bartmann and A. Kraft. A great praise to the precision mechanics workshop guys for the construction of the fine working pressure intensifier system. We appreciate fruitful discussions with geomecon GmbH, Potsdam.

#### **Author details**

Sebastian Brenne\* , Michael Molenda, Ferdinand Stöckhert and Michael Alber

\*Address all correspondence to: sebastian.brenne@rub.de

Ruhr-University Bochum, Germany

#### **References**

[1] Hubbert M, Willis D. Mechanics of hydraulic fracturing. Petroleum Transactions. 1957;210:153–68.

[2] Rummel F. Fracture Mechanics Approach to Hydraulic Fracturing Stress Measure‐ ments. In: Atkinson BK, editor. Fracture mechanics of rock. Academic Press geology series. London [.u.a.]: Academic Pr; 1987. p. 217–39.

**Section 6**

**Induced Seismicity and Slip**


**Induced Seismicity and Slip**

[2] Rummel F. Fracture Mechanics Approach to Hydraulic Fracturing Stress Measure‐ ments. In: Atkinson BK, editor. Fracture mechanics of rock. Academic Press geology

[3] Winter R. Bruchmechanische Gesteinsuntersuchungen mit dem Bezug zu hydrauli‐ schen Frac-Versuchen in Tiefbohrungen. Berichte des Instituts für Geophysik der

[4] Ito T, Hayashi K. Physical background to the breakdown pressure in hydraulic frac‐ turing tectonic stress measurements. International Journal of Rock Mechanics and

[5] Griffith AA. The Phenomena of Rupture and Flow in Solids. Philosophical Transac‐ tions of the Royal Society of London. Series A, Containing Papers of a Mathematical

[6] Irwin GR. Analysis of stresses and strains near the end of a crack traversing a plate.

[7] Sih GC. Handbook of stress-intensity factors: Stress-intensity factor solutions and formulars for reference. Bethlehem, Pa: Lehigh Univ., Inst. of Fracture and Solid Me‐

[8] Tada H, Paris PC, Irwin GR. The stress analysis of cracks handbook. 3rd ed. New

[9] Ulusay R, Hudson JA, editors. The complete ISRM suggested methods for rock char‐ acterization, testing and monitoring: 1974-2006. 2007th ed. Ankara: Commission on

[10] Mutschler T. Neufassung der Empfehlung Nr. 1 des Arbeitskreises "Versuchstechnik Fels" der Deutschen Gesellschaft für Geotechnik e. V.: Einaxiale Druckversuche an

[11] Selvadurai APS, Jenner L. Radial Flow Permeability Testing of an Argillaceous Lime‐

[12] ASTM E976. Standard guide for determining the reproducibility of acoustic emsis‐ sion sensor response. American Society for Testing and Materials. 1994;386-391.

Testing Methods, International Society of Rock Mechanics; 2007.

zylindrischen Gesteinsprüfkörpern. Bautechnik. 2004;81:825–34.

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Ruhr-Universität Bochum: Reihe A. Bochum; 1983.

or Physical Character. 1921;221:163–98.

York: ASME Press; 2000.Ulusaihutsen

stone. Ground Water. 2013;51:100–07.

chanics; 1973.

436 Effective and Sustainable Hydraulic Fracturing

Journal of Applied Mechanics. 1957;24:361–64.

Mining Sciences & Geomechanics Abstracts. 1991;28:285–93.

**Chapter 21**

**Microseismic Monitoring Developments in Hydraulic**

The last decade has seen a significantly increased interest in microseismic monitoring by the hydrocarbon industry due to the recent surge in unconventional resources such as shale-gas and heavy-oil plays. Both hydraulic fracturing and steam injection create changes in local pore pressures and in situ stresses and thereby brittle failure in intact rock plus additional slip/shearing in naturally fractured rock. Local rock failure or slip yields an acoustic emis‐ sion, which is also known as a microseismic event. The microseismic cloud represents thus a volumetric map of the extent of induced fracture shearing, opening and closing. Microseis‐ mic monitoring can provide pertinent information on in situ reservoir deformation due to fluid stimulation, thus ultimately facilitating reservoir drainage. This paper reviews some of the current key questions and research in microseismicity, ranging from acquisition, proc‐

Microseismic events are very small earthquakes of generally negative moment magni‐ tude1 that are often associated with hydraulic fracturing or fluid flow in reservoirs. Build‐ ing upon long-standing applications of microseismic methods, such as monitoring of stability in underground mines (e.g., Gibowicz and Kijko, 1994; Urbancic and Trifu, 2000)

1 Earthquake magnitude is measured on a logarithmic scale. Various roughly equivalent amplitude-based magnitude

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

© 2013 van der Baan et al.; 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,

**Fracture Stimulation**

Maurice Dusseault

essing to interpretation.

**1. Introduction**

**Abstract**

http://dx.doi.org/10.5772/56444

Mirko van der Baan, David Eaton and

Additional information is available at the end of the chapter

scales are in use, of which moment magnitude is the most general.
