# Problems of Multiscale Brittleness Estimation for Hydrocarbon Reservoir Exploration and Development

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Existing Brittleness Indices

- Mineral-based brittleness indices (MBIs) calculated from the relative fractions of different minerals contained in the rock mass;
- Log-based brittleness indices (LBIs) calculated from empirical correlations between particular well logs data and brittleness;
- Elastic-based brittleness indices (EBIs) calculated from elastic moduli characterizing the mechanical behavior of the rocks.

#### 2.1. Mineral-Based Brittleness Indices (MBIs)

_{i}stands for the weight fraction of the i-th mineral; a

_{i}, a

_{i}

^{’}, b

_{i}, and b

_{i}

^{’}are empirically driven coefficients. The first pair, a

_{i}and a

_{i}

^{’}, represent mineral specific brittleness factors: these parameters show the positive and negative effects of the particular mineral on overall brittleness respectively. The second pair, b

_{i}and b

_{i}

^{’}, are associated with the specifics of mineral distribution in the homogeneous rock: these parameters are affected by the geometrical properties of the inner structure of the rock. Equation (1) makes it possible to take into account the effects of various minerals on brittleness of the whole composition with regard to mineral-specific effects. The majority of MBIs obtained for different fields and formations can be set using the form of Equation (1) with site-specific values of a

_{i}, a

_{i}

^{’}, b

_{i}, and b

_{i}

^{’}factors typical for the studied rock. There are not many examples of Equation (1) being incapable of predicting brittleness of heterogeneous rocks: a notable exception has been reported by Rybacki et al., 2016 [24], who suggested introducing porosity into consideration.

_{i}, a

_{i}

^{’}, b

_{i}, and b

_{i}

^{’}factors in Equation (1), satisfying the mineral weight fractions and providing the MBIs corresponding to real brittleness of the studied rock samples. This regression problem is usually ill-posed: depending on the number of studied minerals and number of rock samples in the studied collection, there can be either too few or too many experimental results for obtaining the single stable solution. In the first case, there will be not enough data to obtain a single solution for the inverse problem; in the second case, the system can become overdetermined and therefore inconsistent. It is necessary to apply certain methods to obtain the proper correlation parameters for the studied set of data.

_{i}, a

_{i}

^{’}, b

_{i}, and b

_{i}

^{’}, so in fact, MBIs do not provide a definition of brittleness—instead, such equations provide correlations between mineral composition and some sort of parameter that is claimed to be brittleness. Definition of brittleness is not given here—brittleness as physical property of rock can usually be defined from mechanical perspective, based on failure mechanics. Consequently, TBIs, dealing with real rock mechanical behavior under loading, provide some insight on brittleness nature.

- The inner structure of rock samples is studied in detail to obtain mineral composition and geometry;
- Rock samples are subjected to loading under laboratory condition. TBIs are obtained from the stress–strain curve (the details on this determination will be given below, in the corresponding section);
- Data on inner structure are used to construct a mathematical model of the studied rocks. This model is used to establish the relationship between model parameters, including mineral-specific brittleness factor a and mineral distribution factor b, and brittleness obtained from loading tests. Note that mathematical model mentioned here is not obliged to be a certain rock physics model—any mathematical model, e.g., correlation dependencies—can be used for solving the problem of brittleness evaluation;
- Upscaling procedure is established: it is expected that the established relationship between inner structure and composition of the sample and its brittleness remains valid for samples of other sizes, including bigger samples. Nevertheless, bigger samples can contain inhomogeneities of scales exceeding the typical size of the laboratory studied cores. As a result, it is necessary to include inhomogeneities of bigger sizes into the model to predict brittleness of the bigger samples;
- An intermediate step can be completed to verify the model: if loading tests are performed on samples of different sizes, the assumption of the model being valid at any scale can be directly checked.

#### 2.2. Log-Based Brittleness Indices (LBIs)

_{norm}and v

_{norm}are normalized Young’s modulus and Poisson ratio respectively

_{max}and E

_{min}are the maximum and minimum Young’s moduli in the formation, v

_{max}and v

_{min}are maximum and minimum Poisson ratios in the formation. Equation (4) was proposed for static elastic moduli, yet we still consider brittleness index LBI

_{2}a log-based, as it was obtained from initial data on dynamic elastic moduli through the correlation between static and dynamic elastic moduli. Although Equation (4) is widely used in practice for brittleness evaluation, the other LBIs have been proposed as well: Sharma and Chopra, 2012 and Sun et al., 2013 have incorporated rock bulk density ρ to suggest two more brittleness indices [25,26]

_{5}and LBI

_{5′}are both used in practice to evaluate brittleness; they differ by taking rock density ρ into account.

_{c}had been established for such fracture propagation, resulting in the existence of a relationship between Young’s modulus and a certain fracability. The following failure mechanics equation was used to evaluate critical energy release rate [27]

_{IC}is fracture toughness, and E

^{’}is equal to Young’s modulus E for plane stress state and E’ = E/(1 − v

^{2}) for plane strain (this difference emerges from Hooke’s law form for the cases of one zero principal stress—plane stress state and one zero principal strain—plane strain state). Note that dynamic elastic moduli were used in Equation (7) to predict critical energy release rate. Plane strain state was considered by Jin et al., 2014 [21] to analyze hydraulic fracture propagation. Fracture toughness was obtained from existing correlations, so well logs were used to estimate profiles of critical energy release rate. Consequently, Equation (4) was modified to incorporate fracture toughness and critical energy release rate. The following equations were proposed for brittleness evaluation [21]

_{3}, critical energy release rate, and fracture toughness are normalized as [21]

_{i}is weight factor specific for i-th elastic modulus E

_{i}obtained from well logs data interpretation. Dynamic moduli are used in majority of Equations (4)–(8): e.g., fracture toughness K

_{IC}used in Equation (7) was, in fact, obtained from a linear correlation between fracture toughness and dynamic Young’s modulus typical for Barnett shale studied by Jin et al., 2014 [21]. Acquisition of weight factors remains the task of regression problem in a way similar to the one discussed above: laboratory studies on core material remain the main source of data for finding the proper correlation coefficients. Respectively, all problems discussed in the previous section remain actual, including the problem of difference between scales of laboratory tests and well logs data.

_{2}coincide with zones of minimum MBIs, and vice versa. Moreover, this difference is not within the confidence interval: it is typical for the indices introduced in the mentioned papers to have an error of ±0.1 [21], which is lower than the differences between brittleness indices plotted in Figure 2.

#### 2.3. Test-Based Brittleness Indices (TBIs)

_{a}and radial effective stress σ

_{r}. Radial stress is maintained at a certain level, while axial stress is gradually increased until the failure of the sample. Differential stress Δσ = σ

_{a}− σ

_{r}and axial and radial strains of the sample ε

_{a}and ε

_{r}are the parameters of stress–strain state of the sample obtained at any point of time during the test. Experimentally obtained axial strain dependency on differential stress plotted as a solid black line in Figure 3 is generally used to evaluate brittleness. This line has certain specific zones: linear elastic sector OA, inelastic strain accumulation AB, failure point B, post-peak rupture BQ, and relaxation zone QK. Specifics of behavior of the rock in these regions are used in many studies to estimate test-based brittleness indices (TBIs).

_{c}has been used for LBI calculation. In fact, within the failure mechanics concepts [28], this energy is associated with the area of free surface emerging during failure process. This free surface is, in turn, closely related to brittleness. Consider two media: very brittle and very ductile samples. A brittle sample subjected to external loading will fail with many micro fractures or one big fracture. In both cases, the cumulative area of free surfaces of all parts of the sample after failure will be large. At the same time, failure of the ductile sample will be associated with plastic flow and dislocations emergence and movement. The cumulative area of free surface of the sample after failure will be less, compared to the brittle sample. As a result, total energy stored in a ductile sample at the failure point will be higher than energy accumulated in brittle sample, as a considerable part of it was released to form free surfaces. That means that definition of brittleness index TBI

_{2}is, in fact, backed up by failure mechanics and remains in agreement with energetic concept of failure process. Various modifications of energy-based TBI

_{2}have been developed to take post-peak behavior into account [29,30,31,32,33]: it was shown that post-peak rupture of the sample has an important effect on rock brittleness. There are still other methods to evaluate brittleness index from results of special tests [34,35,36,37,38,39], their comprehensive review was provided by Zhang et al., 2016 [14], but their consideration is not within the scope of the current study.

_{2}, but not to TBI

_{3}and TBI

_{4}proposed by Equation (12) to analyze post-peak behavior. The proper normalization procedure for these indices is yet to be established.

_{1}: Kidybinski, 1981 [40] proposed the following procedure to evaluate effective brittleness of the layered formation: if the formation being investigated is not uniform in structure, a geological log is made, and for each layer of significance, a sample is taken and brittleness is determined in the laboratory. An average brittleness index value for the whole vertical section is then calculated as follows [40]

_{5}was used to obtain brittleness from elastic moduli measured in the laboratory—it is a complicated task to properly incorporate energy concept into brittleness determination.

## 3. Methodology and Materials

#### 3.1. Rock-Physics Models and Methods of Effective Medium Theory

#### 3.2. Rock-Physics Model of Effective Elastic Properties Used for LBI Analysis

_{p}) and shear (V

_{s}), equal to 6.54 and 3.35 km/s and density 2.71 g/cm

^{3}[54]. The properties of formation water are V

_{p}is 1.6 km/s and density is 1.1 g/cm

^{3}.

**C**

^{*}relates the stress and strain fields averaged over the representative volume of the heterogeneous medium via the Hook’s law ($\langle {\sigma}_{ij}\rangle ={C}_{ijkl}^{*}\langle {\epsilon}_{kl}\rangle ,\hspace{0.17em}\hspace{0.17em}i,\hspace{0.17em}j,\hspace{0.17em}k,\hspace{0.17em}l=1,\hspace{0.17em}2,\hspace{0.17em}3$). According to the self-consistent method, the formula for the effective stiffness tensor the has the form (in the tensorial form) [43,44]

**x**is a point within a representative volume whose physical properties coincide with the properties of the rock;

**C**(

**x**) is the stiffness tensor of a heterogeneity at the point

**x**;

**I**is the unit tensor of the fourth rank having the components ${I}_{ijkl}=\frac{1}{2}\left({\delta}_{ik}{\delta}_{jl}+{\delta}_{il}{\delta}_{jk}\right),\hspace{0.17em}i,\hspace{0.17em}j,\hspace{0.17em}k,\hspace{0.17em}l=1,\hspace{0.17em}2,\hspace{0.17em}3$. The volume averaging in Equation (14) is performed over all components with the use of their volume concentrations. In this method the ellipsoidal shape is assumed for all heterogeneities. The tensor

**g**depends on the shape of ellipsoidal heterogeneities and the effective properties. This tensor has the form [44,55]

_{1}, a

_{2}, and a

_{3}are semi-axes of ellipsoids describing the shape of heterogeneities. Commonly, the shape of heterogeneities is modeled by ellipsoids of revolution with a

_{1}= a

_{2}= a and a ≥ a

_{3}(oblate spheroids). In this case the shape can be characterized by an aspect ratio that is equal to a

_{3}/a. As seen Equation (14) contains the term

**C**

^{*}in the both left- and right-hand sides. In this case, the solution is found by iterations.

_{p}) and shear (V

_{s}) wave velocities are calculated from the effective stiffness tensor C

^{*}(in Voigt matrix representation) and density of rock $\langle \rho \rangle $ as

## 4. Results

#### 4.1. Testing of Rock-Physics Model Selected for LBI Analysis on Experimental Data

#### 4.2. Effect of Microstructure Parameters on Brittleness Indices

^{3}in the porosity range from 0 to the limiting value that depend on the aspect ratio (as described in the previous section). As expected, the Young’s modulus decreases as the porosity increases for a fixed aspect ratio. If porosity is a constant value, the Young’s modulus increases with the increase in aspect ratio. For the Poisson ratio, the opposite behavior is observed. A rather wide domain for almost constant Poisson ratio (around 0.32–0.33) is seen.

_{1}behavior. Among the indices, the LBI

_{5}is more variable whereas the smallest variability is observed for LBI

_{3}. Similar behavior is observed for LBI

_{8}(Figure 8). Its variability is comparable to LBI

_{3}.

_{6}and LBI

_{7}, first, the fracture toughness and critical strain energy release rate should be estimated. To obtain the fracture toughness we apply the regression equation from the work [56] that has the form K

_{IC}= 0.3 + 0.027E, where the fracture toughness K

_{IC}is given in MPa ×·m

^{0.5}, and E is the Young’s modulus (in GPa). The critical strain energy release rate is calculated by Equation (7). The fracture toughness and the critical strain energy release rate versus the pore space characteristics are shown in Figure 8. Since the relation between the fracture toughness and Young’s modulus is linear, the behavior of fracture toughness is similar. The strain energy release also decreases with the porosity and increases with the void opening.

_{6}, LBI

_{7}, and LBI

_{8}. As seen, for indices LBI

_{6}, LBI

_{7}the behavior is different as compared to other ones. The indices increase with the porosity. For small opening of voids (not greater than 0.01) the indices decrease with the opening. For more open voids the behavior of indices becomes nonmonotonic. It could be explained by that the regression equation used for the fracture toughness estimation is applicable for rather thin voids. The index LBI

_{6}is more variable compared to LBI

_{7}.

## 5. Discussion

_{6}deserves particular discussion. Recall that LBI

_{6}was proposed with energy concept kept in mind: critical energy release rate was considered as a factor controlling brittleness. Energy concept has been used for introduction of TBIs as well, so it can be used to establish a bridge between TBIs and LBIs. This link can be highlighted after considering of inner structure influence on failure process. Figure 10 represents brittleness index LBI

_{6}as function of fractures density. Note that the fracture density ε is related to the number of fractures N

_{fr}by the linear equation N

_{fr}= Vε/a

^{3}, where V is the volume containing the fractures, and a is the fracture’s half-length.

_{1}or TBI

_{2}. At the same time, rock physics modeling has been carried out for calculating LBIs, so the tendency shown in Figure 10 is in fact visualization of the connection between energetic concept of brittleness, rock physics model, and log-based empirical correlation for brittleness index evaluation.

_{4}that is rather simple, linear dependent on the Young’s modulus, and does not require normalization. In this case, we can compare the indices calculated for different directions and distinctly conclude that the index in the symmetry plane is greater than that in the perpendicular plane. However, when considering LBI

_{5′}or LBI

_{3}that are widely used in practice or other indices the comparison meets some difficulties since it is not clear which minimum and maximum of moduli should be used for the normalization—over all directions or only for the particular direction. The same problem arises when dealing with TBIs—although it is natural to treat TBIs obtained from samples with known spatial orientations as anisotropic indices, construction of a single parameter standing for brittleness in all directions remains a challenge. There are attempts to construct a tensor characterizing brittleness anisotropy [42], but a strict definition of anisotropic brittleness of tensor nature is yet to be established. These problems should be analyzed in more detail and this is out of scope of this paper.

_{6}and LBI

_{7}indices) have reduced values compared to the other brittleness values shown in Figure 7 and Figure 9. This is in line with the successful drilling of the wells and high-quality log data.

## 6. Conclusions

_{3}, LBI

_{4}, LBI

_{5}, LBI

_{5′}, and LBI

_{8}that are calculated from the elastic moduli and density, decrease with porosity and increase with the relative opening of voids. However, the LBIs, that are dependent on the fracture toughness and critical strain energy release rate, increase with the number of fractures and their opening. The rock physics analysis of anisotropic rocks is not trivial and requires special attention.

_{6}and TBI

_{2}in particular). Accumulation of plastic strains in the sample under stress can be realized through one of two ways: it is either related to slow movement of dislocations or increase in free surface of the sample, i.e., increase in number and density of the fractures. Both modeled log-based index and discussed test-based indices increase with growth of the amount of fractures, so there is a positive correlation between these two parameters.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Brittleness evaluation from different correlations for four formations. Modified after Mews et al., 2019 [4].

**Figure 3.**Typical single-stage loading test result for brittleness index determination. Modified after Ezhov et al., 2021 [27].

**Figure 4.**(

**a**) Compressional and (

**b**) shear wave velocities for a model of calcite matrix with randomly oriented brine-saturated voids versus decimal logarithms of porosity (poros) and void’s aspect ratio (AR). Red signs show values for West Siberian limestones. Blue signs are used for limestones from layers within a shale formation [54].

**Figure 5.**Relative difference (Diff) between the theoretical and experimental values of compressional (orange signs) and shear (green signs) wave velocities. (

**a**) West Siberian limestones and (

**b**) limestones from layers within a shale formation [54]. The horizontal axis shows experimental velocity values.

**Figure 6.**(

**a**) Young’s modulus and (

**b**) Poisson ratio for a model of carbonate matrix with randomly oriented brine-saturated voids versus decimal logarithms of porosity and void’s aspect ratio.

**Figure 7.**Brittleness indices (

**a**) LBI

_{3}, (

**b**) LBI

_{4}, (

**c**) LBI

_{5}, and (

**d**) LBI

_{5′}for a model of carbonate matrix with randomly oriented brine-saturated voids versus decimal logarithms of porosity and void’s aspect ratio.

**Figure 8.**(

**a**) Fracture toughness and (

**b**) critical strain energy release G

_{C}versus decimal logarithms of porosity and void’s aspect ratio.

**Figure 9.**Brittleness indices (

**a**) LBI

_{6}, (

**b**) LBI

_{7}, and (

**c**) LBI

_{8}for a model of carbonate matrix with randomly oriented brine-saturated voids versus decimal logarithms of porosity and void’s aspect ratio.

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**MDPI and ACS Style**

Dubinya, N.; Bayuk, I.; Bakhmach, M.
Problems of Multiscale Brittleness Estimation for Hydrocarbon Reservoir Exploration and Development. *Appl. Sci.* **2022**, *12*, 1134.
https://doi.org/10.3390/app12031134

**AMA Style**

Dubinya N, Bayuk I, Bakhmach M.
Problems of Multiscale Brittleness Estimation for Hydrocarbon Reservoir Exploration and Development. *Applied Sciences*. 2022; 12(3):1134.
https://doi.org/10.3390/app12031134

**Chicago/Turabian Style**

Dubinya, Nikita, Irina Bayuk, and Milana Bakhmach.
2022. "Problems of Multiscale Brittleness Estimation for Hydrocarbon Reservoir Exploration and Development" *Applied Sciences* 12, no. 3: 1134.
https://doi.org/10.3390/app12031134