## Summary

The objective of hydraulic fracturing is to design/execute a fracture treatment that achieves the desired fracture characteristics (length & conductivity) to maximize a wells production rate/reserves, creating a high-permeability flow channel towards the wellbore in the formation.

Hydraulic fracturing in rocks takes place when the fluid pressure within the rock exceeds the smallest principal stress plus the tensile strength of the rock, resulting in the tensile failure of the rock in the direction of the least resistance. Away from the borehole the fracture will propagate in the direction perpendicular to the smallest principal stress (Fjaer et al., 2008).

In this post I review the fundamental concepts of rock physics and its applications in the hydraulic fracturing, and review the equations that describe the rock failure. For most of the section I assume that rocks are homogeneous and isotropic.

## What is conventional? and what is UNCONVENTIONAL?

In a geological sense, in a conventional reservoir the hydrocarbon generated by a kerogen-rich rock migrates naturally and is stored by buoyant forces into the porous space of a reservoir rock, and subsequently is trapped by an impermeable seal. This geological definition of a petroleum system differentiates three rock types: source, reservoir and seal.

On the other hand, an unconventional reservoir is one where one single rock combines the previous rock characteristics, and the hydrocarbon storage in the rock pores (typically natural gas) does not flow naturally due to the low (> 0.1 mD) rock permeability. Many of these low-permeability rocks are shale and tight sandstone, but currently significant amounts of gas are also produced from low-permeability carbonates and coal bed methane.

Figure II-1 shows a map (EIA, 2012) representing that proliferation of the exploration activity into new shale plays has increased the shale gas resources in the U.S. from 1 TCF in 2006 to 336 TCF in August 2011. In this research I will focus on the Barnett Shale, located in the Fort Worth Basin (Texas).

Recent use of horizontal drilling in conjunction with sequenced multistage hydraulic fracturing has greatly expanded the ability of producers to profitably recover natural gas and oil from low-permeability geologic plays, particularly shale plays. An effective stimulation creates new fractures and reopens or connects preexisting natural fractures necessary to extract the gas from the rock matrix and achieve production. In some cases, natural fractures form sweet spots that become the target since they provide a conduit for the natural gas. In other cases, fracture swarms need to be avoided because they may be connected to an adjacent aquifer. Figure II-2 from the low permeability oil-saturated Woodford Shale compares fracture size and length with highway speed limits, indicating that the larger the fracture aperture (the wider the road), the greater the permeability of the rock (or speed limit).

**Figure II-1.**

**Figure II-2.** Photo of a Woodford Shale (Ardmore, Oklahoma) outcrop indicating several fracture sets that would give rise to differences in permeability.

## Brittleness definitions

The term brittleness is used differently by different authors, and is not a precise definition or concept. The measurement of brittleness has not been standardized. However, the principle of reversible strain – energy is generally used. Hucka and Das (1974) summarize different formulation of brittleness such as:

• Determination of brittleness from percentage of reversible strain:

where ϵ_{el} is the reversible elastic strain and the ϵ_{total} is the total strain.

• Determination of brittleness from the percentage of reversible work:

where W_{el} is the reversible elastic work and the W_{total} is the total work.

• Determination of brittleness from tensile and compressive strengths:

where σ_{c} is the compressive strength and the σ_{t} is the tensile strength.

• Determination of brittleness from Mohr’s envelope:

where φ is the angle of internal friction, determined from Mohr’s envelope at σ_{n}=0. Notice that σ_{n} is the normal stress on the plane of failure.

• Determination of brittleness from measurement of oblique shear:

This method is similar to determining brittleness from Mohr’s envelope; only here the practical test is different. The relation between the angle of oblique shear plane and the angle of internal friction is given by the Mohr’s stress circle.

Hence,

## Discussion

Based on the variety of definitions of brittleness, the values of brittleness are not identical. Each definition has its use in science and technology depending on practical utility. Intuitively a material is brittle if, when subjected to stress, it breaks without significant deformation (strain). Brittle materials absorb relatively little energy prior to fracture, even those of high strength (Figure II-3). If the rock has a large region of elastic behavior but only a small region of ductile behavior the rock is considered brittle. In contrast, if the material under stress has a small region of elastic behavior and a large region of ductile behavior, absorbing much energy before failure, it is considered ductile (opposite of brittle).

In this series I use the well-known and accepted brittleness definition proposed by Rickman et al. (2008). Using a crossplot of Young’s modulus and Poisson’s ratio he defined that the ductile points will fall to

where *E* is Young’s modulus, and *E*_{min} and *E*_{max} are the minimum and maximum Young’s modulus measured in the logged formation, and

where *ν* is Poisson’s ratio, and *v*_{max} and *v*_{min} are the maximum and minimum values of Poisson’s ratio logged in the formation.

Finally, they define a brittleness average, *BA*, as

**Figure II-3:** Strain vs. stress diagram comparing brittle (red) and ductile (blue) curves, where the area under the curve for ductile rocks (A_{ductile}) is larger than the area under the curve for the brittle rocks (A_{brittle}).

Rickman et al. (2008) derived the static Young’s Modulus and Poisson’s Ratio using the methodology described in Mullen et al. (2007). However, the modulus important to fracturing is the static linear elastic rock property, and must be measured by stress-strain testing from core measurements. Rickman et al. (2008) based his brittleness definition using acoustic log derived Young’s modulus. However, the modulus determined in this manner represents a dynamic value, and nearly always differs greatly from static lab measurement. In fact, variations between static and dynamic modulus of a factor of two are common, and even larger variations have been reported. Also, in general the dynamic modulus is always greater than the static modulus, and since modulus directly controls the fracture geometry, significant errors in these predictions result from utilizing dynamic modulus from logs. In general, for a specific formation, it is best to use lab tests to determine a specific static-dynamic relation before sonic logs can be used quantitatively. Finally, note that while special (long spaced, dipole) sonic logs must be used to measure shear and compression velocity to calculate Poisson’s ration, a usable value for dynamic Young’s modulus can be found from a simple sonic (along with a density) log.

## REFERENCES

Fjaer, E., R. M. Holt, P. Horsrud, A. M. Raan, and R. Risnes, 2008, Petroleum related rock mechanics: Elsevier.

Hucka, V. and B. Das, 1974, Brittleness determination of rocks by different methods: International Journal of Rock Mechanics and Mining Sciences & Geomechanics, 11, 389 – 392.

Mullen, M., R. Roundtree, R. Barree, and G. Turk, 2007. A composite determination of mechanical rock properties for stimulation design (What to do when you don’t have a sonic log): Paper SPE 108139 presented at the SPE Rocky Mountain Oil and Gas Technology Symposium, Denver, CO, 16-18 April.

Perez, R., 2013, Brittleness Estimation from Seismic Measurements in Unconventional Reservoirs: Application to the Barnett Shale: The University of Oklahoma, Ph.D. dissertation.

Rickman, R., M. Mullen, E. Petre, B. Grieser, and D. Kundert, 2008, A practical use of shale petrophysics for stimulation design optimization: all shale plays are not clones of the Barnett Shale: Presented at the 2008 Annual Technical Conference and Exhibition, SPE, SPE 115258.

## ADDITIONAL INFORMATION

For additional information and material visit: www.roderickperezaltamar.com and www.rockphysicsapplications.com.

## Your Turn

**What do you think? Leave a comment below.**

#### Roderick Perez

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