### The Delicate Matter of Rock Brittleness; Part II: The Role of Elastic Properties

**Find the Complete PDF version of this series here: The Delicate Matter of Rock Brittleness**

**There’s something brittle in me that will break before it bends. **(Mark Lawrence, Prince of Thorns)

Considering the definition of rock brittleness discussed in the previous post of this series, it is clear that the dynamic elastic properties measured by sonic or seismic waves and even the static ones measured in laboratories are not the most appropriate parameters for determining this property of rock as these parameters only describe the rock deformation before it reaches near failure or fracturing. This is truer for dynamic elastic properties as they only agitate the rock with very small deformations (see my other post for more on this). In fact, rock’s response during cracking or failure needs to be defined by fracturing or failure criteria. However, there must be some justifiable reasons for using dynamic elastic properties so frequently for identifying brittle rocks. Of course, the main reason is the abundant availability of these data from wireline logs and seismic surveys but, in addition to data availability, there are some technical evidence to justify using elastic properties as proxies for brittleness.

In a general sense, more brittle rocks (e.g., granite or over-consolidated cemented sandstones) are usually expected to show less deformation within their elastic limits and before yielding (or failure) in comparison to more ductile rocks. Although this general pattern may work for some rocks, in reality, brittleness/ductility is not equal to stiffness/softness. I will talk about the inconsistencies later in this post but let’s see how the idea of ‘less deformability=more brittleness’ is exploited to determine brittleness index using elastic properties. In elasticity, less axial deformation means higher Young’s modulus (*E*) and less lateral deformation means less Poisson’s ratio (*v*). Therefore, according to the discussed hypothesis, these two parameters seem to be good candidates for estimation of rock brittleness. This is the basis for the frequently used form of elastic brittleness index in petroleum industry as proposed by Rickman et al. (2009) who gave the same weight to the effects of Young’s modulus and Poisson’s ratio by simple arithmetic averaging:

*max* and *min* subscripts in this equation denote the maximum and minimum values of the elastic parameters for the formation(s) of interest. Figure 1 shows an example of using this equation where Young’s modulus and Poisson’s ratio are acquired from sonic and density logs. Figure 2 shows the cross-section of a shale formation showing the brittleness index variation. The dynamic elastic parameters in this case were derived from both sonic and seismic data.

*Figure 1. An example of brittleness calculated from sonic and density log data. The direction of arrow shows increasing in brittleness index and the data points are color-coded by this parameter. (Modified after Varga et al., 2013).*

*Figure 2. An example of brittleness index profile derived from seismic and sonic data. Yellower colours stand for higher brittleness and bluer colours indicate less brittle rock (Source: Varga et al., 2013).*

**Reasons for Doubt**

The idea of identifying brittle rocks using abundantly available log- or seismic-based dynamic elastic properties sounds very appealing but there are reasons that make it hard to accept the credibility of this method inclusively. I discussed one of these reasons before but there are more. One obvious reason is the hesitation in accepting the relation between rock brittleness and elastic properties. As it will be discussed later in this series, brittleness and fraccability are functions of more than just elastic properties even in materials that follow the rules of linear elastic fracture mechanics (LEFM). In LEFM, other independent parameters such as fracture toughness or fracture energy are the major factors that govern fracture propagation or blunting. This becomes even more complex in inelastic or plastic materials such as rocks.

On the other hand, during shear failure experiments, there are occasions that rocks with less elastic deformibality (i.e., higher Young’s modulus and lower Poisson’s ratio) show less brittle behaviour compared to the ones with more elastic deformability. Another contradictory observation is the change in rock behaviour by increase in confining stress during triaxial tests as discussed in the previous post of this series. This behaviour might not be the case for all the rocks but there are several types of rocks that follow this rule. The example given in Figure 3 shows that with increase in confining pressure on samples in triaxial test, the elastic deformability of rock is not significantly affected but it becomes more ductile in a shear failure mode. In other words, by increasing the confining stress, rocks become more ductile and less brittle while their Young’s moduli do not change; something that is not aligned with the assumptions of elastic brittleness index given above.

*Figure 3. This figure shows how the mechanical behaviour of a limestone in a shear mode changes with increase in confining stress in triaxial test. Apparently, Young’s modulus does not change with confining pressure while rock’s behaviour becomes more ductile (source: www.higgs-palmer.com).*

Different contradictory examples discussed here show that the idea of ‘lower *v* and higher *E* = higher brittleness’ may not work as expected all the time. In addition, there are others reasons that undermine credibility of this hypothesis. One is the effect of high pore pressure in hydrocarbon plays on dynamic elastic properties. High pore pressure is known to make the rock less consolidated and more deformable but the question is whether ‘dynamic Young’s modulus is always lower and dynamic Poisson’s ratio is always higher’ for rocks with higher pore pressure or not. The answer seems to be YES for Young’s modulus and NO for Poisson’s ratio as depicted in Figure 4. This figure shows Poisson’s ratio is only higher for higher pressures if the occupying fluid is incompressible enough (e.g., brine) but for more compressible fluids such as light oil or gas (as it is the case for many unconventional plays), increase in pore pressure leads to decrease in Poisson’s ratio. In the case of more compressible fluids, higher pore pressure still leads to less stiff and more deformable rocks but this does not result in higher dynamic Poisson’s ratio as is the case for brine-filled rocks.

One other thing about the brittleness index defined in the given equation is the fact that it is hard to agree with Young’s modulus and Poisson’s ratio having exactly the same share of influence on rock brittleness as imposed by arithmetic averaging in the equation. Also, we must remember that there are cases where a rock might have a very low elastic deformability but, at the same time, it can also have a very high resistance against fracturing due to its higher yield and failure strength. In these cases, for a certain amount of stress or fluid pressure, this rock does not fracture while rocks with higher elastic deformability (less brittleness index) and less resistance may fracture. Everything said here shows that the presented equation for brittleness might not be always trusted as a measure of rock brittleness but we cannot completely deny that there are cases where this index can work as a proxy for this property of rock.

*Figure 4. These graphs show how pore pressure increase affects elastic properties such as (a) compressional wave velocity and (b) compressional wave impedance and Poisson’s ratio for different types of filling fluids including brine, oil, and gas (Source: Dvorkin).*

**Elastic Parameters and Hydraulic Fractures’ ****Characteristics**

Now, let’s take a close look at some theories that relate elastic properties to fracture characteristics. It is important to remember that what we will discuss in the following is not directly related to brittleness or criteria for fracturing and it merely explains the effect of elastic properties on the fractures without considering their resistance against propagation. In other words, these fractures are considered to propagate as ‘knife cuts through butter’[i].

In practice, it is assumed that the volume of injected fracturing fluid is either lost to the formation through leakoff or helped in creating the volume of fracture. This forms a volume balance equation that is completed by knowing the fracture geometry. The necessary equations for finding fracture geometry are usually derived using the principals of LEFM that assumes elastic behaviour for the rock. Therefore, elastic parameters such as Young’s modulus and Poisson’s ratio have a direct role in determining the characteristics of hydraulic fractures. Depending on the complexity of modeling, different solutions have been suggested and used in industry. In here, without getting into detail, we will look at one of the simplest (although still practical) solutions called Perkins-Kern (PKN) that is based on assuming plane-strain geometry for the fracture. This model assumes an ellipsoidal shape for the fracture with a half-length of *x _{f }*, maximum width of

*w*at the borehole wall, and constant depth of

_{f}*h*(Figure 5). If we inject a Newtonian fracturing fluid with an injection rate of

_{f}*q*and viscosity of

*m*for a time period of

*t*, it is possible to find fracture geometry parameters using the following equations with assumption of no fluid loss to the formation:

The net pressure (i.e., the difference between fracturing fluid pressure and in-situ stress required for fracture propagation) at the initiating point of the fracture at the wellbore wall can be found using the following equation:

In these equations, *E _{ps}* is an elastic parameter called plane-strain Young’s modulus that can be written as a function of basic elastic properties:

*G* in this equation is rock’s shear modulus. Now, let’s examine and see how changes in *E* and *v* will affect the geometry and net pressure of the fracture for a given injection rate for a certain period of time. Figure 6a to 6c show the effects of these changes on *x _{f}*,

*w*, and

_{f}*p*. According to these figures, increase in both

_{n}*E*and

*v*leads to longer and narrower fractures that require higher net pressure for propagation. Nevertheless, unlike the discussed brittleness index, significance of Poisson’s ratio in determining fracture characteristics is much less than Young’s modulus. These results show that the dependency of fracture characteristics to elastic properties is more complex than ‘high is good, low is bad’.

Depending on hydraulic fracturing design priorities, different values of elastic parameters may be preferred but usually wider and longer fractures that can propagate with less pressure of fracturing fluid are favoured.

*Figure 5. Geometry of Perkin-Kern (PKN) model for hydraulic fractures (source: modified after Economides and Valkó, 1996)*

*Figure 6. Variation of different fracture characteristics in the PKN model with Young’s modulus and Poisson’s ratio: (a) fracture’s half-length (x _{f},), (b) fracture’s maximum width (w_{f}), and (c) fracture’s net pressure (p_{n}).*

**Elastic Parameters and In-situ Stresses**

Hydraulic fracturing engineers, geophysicists, and petrophysicists frequently use elastic properties to calculate horizontal in-situ stresses in rocks (See Figure 7 for an example). These calculations are usually performed by assuming poroelastic and uniaxial vertical deformation during sedimentary deposition of rocks. Using this approach, horizontal stresses are calculated simply by using vertical stress (*S _{v}*), pore pressure (

*P*), and elastic properties. Considering tectonic strains (

_{p}*e*and

_{Hmax }*e*) or stresses in these calculations, it is possible to account for the effect of tectonics and stress anisotropy in the rock. For instance, minimum and maximum horizontal stresses (

_{Hmin}*S*and

_{hmin }*S*respectively) for a homogeneous isotropic rock are calculated using the following equations:

_{hmax},*a* in this equation is Biot’s coefficient. In relaxed basins with low tectonic effects (i.e., *e _{Hmax }*and

*e*~0),

_{Hmin}*S*and

_{hmin}*S*become equal. We must remember that these are not the favorite equations of many geomechanics experts as they believe the assumptions are over-simplistic and imprecise.

_{hmax}A hydraulic fracture propagates only if the pressure of the injected fracturing fluid exceeds both *S _{hmin}* and fracture resistance. Therefore, higher

*S*means less potential for fracture propagation with a certain injection pressure. Now, let’s see according to the uniaxial poroelastic theory, how this critical parameter is affected by variation in elastic properties. Figure 8 shows an example of how

_{hmin }*S*in the given equation changes with variation in Young’s modulus (

_{hmin}*E*) and Poisson’s ratio (

*v*). According to this figure, increase in both

*E*and

*v*leads to higher

*S*or, in other words, fractures will have a harder time to propagate. This might indirectly contradict Rickman’s assumption that higher elastic Young’s modulus leads to more fraccability.

_{hmin}**Wrap Up**

It seems that the simple assumption of using high or low elastic properties (or any similar assumption) for brittleness evaluation does not seem to have extensive theoretical and experimental support. Elastic properties are not satisfactory enough to characterize rock’s brittleness and more parameters are required for this purpose. On the other hand, considering their abundant availability from seismic surveys and sonic logs, it is not clever to completely rule them out as with proper and precise treatment they might be able to act as proxies for brittleness. If we want to use these parameters properly, probably the right option is using them besides other means such as laboratory measurements of brittleness and correlating them with observed fractures in the field from cores, image logs, and sonic scanners, or microseismic surveys. We can also compare them with other types of brittleness such as mineralogical brittleness to ensure their repressiveness (as will be discussed in the next post of this series). We also need to remember that instead of classifying the rocks based on high or low values of elastic parameters, we need to find specific ranges of them that suit rock fraccability.

*Figure 7. This example shows minimum horizontal in-situ stress (S _{hmin}) calculated using elastic equations for the some selected formations (source: Gray et al., 2012)*

*Figure 8. An example graph showing how minimum horizontal stress (S _{hmin}) changes with variation in Young’s modulus and Poisson’s ratio according to the given equation in the text. In this example, vertical stress is 20 MPa, pore pressure is 10 MPa, Biot’s coefficient is 1.0, maximum tectonic strain is 0.0001, and minimum tectonic strain is assumed to be zero.*

**Endnote**

[i] I burrowed this term from ‘Hydraulic Fracture Mechanics by Economides and Valkó (1996). The equations for the PKN model have also been adopted from this book.