1. Introduction

2. Fracability evaluation method

2.1. Energy brittleness index method

The energy brittleness index used in this paper is based on the full stress-strain curve obtained from uniaxial compression experiments, which is divided into pre-peak brittleness index and post-peak brittleness index with the breaking point as the dividing line. Since there are nonlinear segments in the full stress-strain curve, which is more complicated in the analysis process, the full stress-strain curve is linearly simplified, as shown in Figure 1.

2.1.1. Pre-peak fragility index

In conventional uniaxial compression experiments, the rock sample is first elastic deformed under the action of axial load, and the external energy is accumulated inside the sample in the form of elastic energy. In Figure 2, We represents the elastic strain energy accumulated inside the rock sample during the elastic deformation phase of the rock. At this stage if the external stress is withdrawn, the deformation of the rock will be fully recovered and this part of the elastic strain energy will be fully released accordingly, at which time the slope of the straight line is the elastic modulus. Under the same deformation conditions, the larger the elastic modulus, the larger the elastic energy that the rock can accumulate. Therefore, the elastic modulus is not only a measure of the object's ability to resist elastic deformation, but also reflects the ability of the rock to accumulate energy.

The above analysis shows that the pre-peak stage dissipation energy (dWd) has a significant effect on the rock brittleness, and the root reason is that the pre-peak dissipation energy is closely related to the "energy storage limit" of the rock itself. The rock's energy storage limit, i.e. the amount of energy accumulated at the peak of elastic strain energy, is used to characterize the rock's ability to accumulate elastic strain energy and is related to the nature of the rock itself and the stress state it is located in. When the elastic energy accumulated in the rock increases and reaches it's energy storage limit, the energy accumulation effect stops and turns to release to the outside and the rock will occur the overall fracture and rupture.

It is generally believed that the more energy consumed to make the rock break, the harder the rock is to break, the less brittle it is, and the more difficult it is to fracture hydraulically. Applying the theory to the hydraulic fracturing process, hydraulic fracturing for the occurrence of starting fracture, the rock occurs elastic strain, at this time the energy is concentrated in the rock, but this part of the energy is only stored in the rock, in the subsequent changes in this part of the elastic energy will also be released, so the greater the proportion of this part, which means the smaller the proportion of energy dissipated before the peak, the bigger the rock brittleness. When the rock is stressed to the yield point during hydraulic fracturing, micro cracks and plastic strains are created inside the rock, and part of the energy is consumed, which cannot be recovered after consumption, so the larger the proportion of this part is, the larger the proportion of dissipated energy before the peak, and the smaller the rock brittleness is.

Due to the energy change in the elastic section is all elastic energy, and there is no energy dissipation, so the energy distribution before the peak occurs mainly in the plastic section, part of the plastic section energy into the plastic section elastic energy, part into the plastic section dissipation energy, when the plastic section elastic section in the plastic section of the larger the proportion of the total energy, the bigger the rock brittleness, respectively, expressing the plastic section total energy and plastic section elastic energy:

Plastic section elastic energy:

(1)

Plastic section total energy:

(2)

The pre-peak brittleness index can be represented by the ratio of the plastic section elastic energy to the plastic section total energy:

(3)

The index shows that the closer the pre-peak brittleness index is to 1, the closer the total energy of the plastic section is to the elastic energy of the plastic section, and the smaller the plastic section dissipation energy is, i.e., the more brittle the pre-peak section is.

2.1.2. Post-peak brittleness index

After reaching the peak strength σ_B, the rock enters the fracture damage stage, where the microfractures inside the rock further expand and converge to form macroscopic fracture cracks, and the rock sample is completely damaged and loses a certain load-bearing capacity.Figure 3 shows the internal energy classification of the rock during the post-peak fracture phase.

Usually, the post-peak curve of the rock sample does not fall vertically, but decreases gradually at a certain rate. This is due to that after the peak strength is reached, the elastic energy stored inside the rock is not sufficient to sustain further fracture damage and additional energy (dW_a) is needed from outside to support this process, under mechanical experimental conditions, this energy is partially provided by the continued loading of the experimental machine. It can be seen that during the post-peak fracture phase, the following changes occurred within the rock: The elastic energy accumulated within the rock (dW_e(B)) and the additional energy provided by the testing machine (dW_a) together provide the energy for the fracture damage of the rock specimen. When the rock reaches the residual strength σ_C, due to its not completely lost load-bearing ability, there is still residual elastic energy inside the rock(dW_e(C)). The difference between dW_e(B) + dW_a and dW_e(C) is the fracture energy (dW_F) released by the rock fracture process. The fracture energy dW_F is the key energy that determines the brittle characteristics of rocks in the fracture damage stage. The smaller the fracture energy, the less extra energy the rock needs from the outside, the more violent the process of releasing energy from the rock, and the stronger the rock has brittleness. Bringing the above theory into hydraulic fracturing, it can be seen that when the reservoir rock is fractured, the elastic energy stored in the rock before the peak is released, and this part of energy is used to continue to damage the rock to produce fractures, but the elastic energy is not enough to fully support the fracture extension process, and hydraulic fracturing still needs to continue to apply pump pressure to maintain fracture extension. And the lower the maintained pump pressure the easier it is to complete fracturing, which means the stronger the rock brittleness. The fracture energy required for the fracture extension process of hydraulically fractured rocks can be divided into two parts, one part is the energy generated by additional pump pressure applied after the peak, and the other part is the elastic energy of the rock itself released after the peak. The brittleness of the rock after the peak can be expressed by the ratio of the post-peak release elastic energy to the post-peak fracture energy, and the post-peak fracture energy to the post-peak release elastic energy can be expressed separately as:

Post-peak fracture energy:

Release of elastic energy after the peak:

(5)

The post-peak Brittleness Index can be represented by the ratio of the post-peak release elastic energy to the post-peak fracture energy:

(6)

The closer the post-peak brittleness index is to 1, the closer the post-peak dissipation energy is to the post-peak rock's own elastic energy consumption, that is, the release of elastic energy accumulated in the pre-peak section can complete most of the post-peak rock destruction process, and complete fragmentation can be achieved without applying additional energy, which means the more brittle the post-peak section of the rock is.

2.1.3. Combined brittleness index

The combined brittleness index is derived from the combination of the pre-peak brittleness index and the post-peak brittleness index, as shown in the following equation:

(7)

The above Eq.shows that the pre-peak brittleness index and post-peak brittleness index are both 1 for completely brittle rocks, so the neutralizing brittleness index for completely brittle rocks should also be 1. The larger the value, the weaker the brittleness.

The comprehensive brittleness index is calculated by strain energy, but the practical application of calculating strain energy is difficult, so the comprehensive brittleness index is simplified. The comprehensive brittleness index is proposed based on the uniaxial all-stress-strain curve, which has been simplified in Figure x by linearizing the all-stress-strain curve and splitting the whole curve into three linear segments, namely the elastic segment, the plastic segment and the post-peak segment; The slope of these three linear segments is calculated, and the slope of the elastic segment is the elastic modulus (E), the slope of the plastic segment is the yield modulus (D), and the slope of the post-peak segment is the post-peak modulus (M), as shown in Figure 4.

The elastic modulus, yield modulus and post-peak modulus are relatively easier to calculate and more intuitive, so this project simplifies the combined brittleness index by the three modulus.

For the pre-peak index it is obtained that

(8)

(9)

For the post-peak index it is obtained that

(10)

(11)

Therefore, the combined fragility index is

$$B=\frac{d{W}_d^{*}}{d{W}_{e(B)}-d{W}_{e(A)}}\frac{d{W}_f}{d{W}_{e(B)}-d{W}_{e(C)}}=\frac{E}{D}\frac{M-E}{M}$$

(12)

2.1.4. Calculation of brittleness index from logging data

The integrated brittleness index Eq.12 shows that to obtain the brittleness index, only the elastic modulus, yield modulus and post-peak modulus need to be calculated. In this section, the correlation between the logging data and the three modulus quantities is established separately along the above lines, and then the brittleness index is calculated.

The elastic modulus can be divided into dynamic elastic modulus and static elastic modulus. Since the elastic modulus used in the energy method is obtained in the full stress-strain curve, the elastic modulus is the static elastic modulus, which can be obtained by the dynamic elastic modulus, and the dynamic elastic modulus can be obtained by the transverse and longitudinal wave acoustic time difference and density.

The dynamic elastic modulus is calculated using the following equation:

$$E_d=\frac{ZDEN\times (3\times DT{S}^2-4\times DT{C}^2)}{DT{S}^2(DT{S}^2-DT{C}^2)}\times 9.299\times {10}^7$$

(13)

where is the dynamic modulus of elasticity, MPa; ZDEN is the density, g/cm³; DTC is the longitudinal acoustic time difference, μs/ft; DTS is the transverse acoustic time difference, μs/ft.

The above formula is used to calculate the dynamic modulus of elasticity, and the static modulus of elasticity can be calculated by Eq.14:

$$E_s=0.18945E_d+5.66963$$

(14)

The plastic modulus is less used in practical engineering, so there is no mature conversion equations like the elastic modulus. Therefore, we solve them based on the relevant parameters of the well logging curve through the stress-strain curve. For the plastic modulus, we need to solve for the peak stress and its corresponding strain value; Correspondingly, for the weakened modulus, we need to solve for the yield stress and its corresponding strain value. The solution of these stress and strain values can be derived through logging parameters and related physical models.

Firstly, the rock mechanics parameters are calculated based on the logging curve, and the main solving parameters are as follows:

(1)

Velocity conversion of longitudinal and transverse sound waves

$$V_p=304.8\times \frac{1}{∆t}$$

(15)

where, $V_p$ is the longitudinal wave velocity, km/s; $∆t$ is measured acoustic time difference，$\mathsf{\mu }\mathrm{s}/\mathrm{f}\mathrm{t}$.

(2)

Effective stress coefficient (Biot coefficient)

$$\mathsf{\alpha }=1-\frac{\rho (3V_p^2-4V_s^2)}{{\rho }_m(3V_{mp}^2-4V_{ms}^2)}$$

(16)

where,$\rho$ Is the density value of the formation, g/cm³; ${\rho }_m$ is the density of the skeleton rock material, g/cm³, taken from dense sandstone${\rho }_m=2.65$, input from other lithology; V_mp is the longitudinal wave velocity of the skeleton material, km/s, and V_mp =5. 95 or dense sandstone 95, artificial input from other lithologies; Vms is the shear wave velocity of the skeleton material, km/s, and V_ms = 3.0 is taken for dense sandstone, with input from other lithologies; V_p is the longitudinal wave velocity of the formation, km/s; Vs is the shear wave velocity of the formation, in km/s.

(3)

Mud content

Using gamma data, the mud content is calculated by calculating the mud content calculation formula as:

(17)

(18)

where I_sh is the mud mass fraction; ∆GR is the natural gamma difference; GR_max and GR_min are the maximum and minimum values of natural gamma in the logging curve, GAPI, respectively; and G_CUR is the formation age correction factor, 3.7 for new formations and 2.0 for old formations.

(4)

Uniaxial tensile strength of rocks

The uniaxial compressive strength is:

$$S_c=0.033{\rho }^2{V}_p^4\left(\frac{1+{\mu }_d}{1-{\mu }_d}{)}^2\right(1-2{\mu }_d\left)\right(1+0.78I_{sh})$$

(19)

$$S_t=\frac{S_c}{K}$$

(20)

where, S_c is the uniaxial compressive strength, MPa; ρ Is the rock mass density, g/cm3;; μ _D is the dynamic Poisson's ratio, dimensionless; S_t is the uniaxial tensile strength, MPa; The commonly used range of K values is 8-25, with a temporary value of 12.

(5)

Formation pore pressure

$$P_P=DEPT*1.2/100$$

(21)

where, DEPT is the depth of the well, m.

(6)

Vertical stress and maximum and minimum horizontal principal stresses

The maximum horizontal ground stress and the minimum horizontal ground stress need to be calculated when calculating the ground stress difference coefficient. There are many methods to calculate horizontal ground stress, among which Huang's model is the most widely used. In this paper, Huang's model is used for calculation, and the specific formula is as follows:

$${\sigma }_H=[\frac{\nu }{1-\nu }+A]\left({\sigma }_v-V{P}_P\right)+V{P}_P$$

(22)

$${\sigma }_h=[\frac{\nu }{1-\nu }+B]\left({\sigma }_v-V{P}_P\right)+V{P}_P$$

(23)

$${\sigma }_v=g{\displaystyle {\int }_0^D{\rho }_b(h)dh}$$

(24)

$$\nu =\frac{DT{S}^2-2DT{C}^2}{2(DT{S}^2-DT{C}^2)}$$

(26)

where,$\mathsf{\sigma }$_v is vertical stress, MPa; $\mathsf{\rho }$_b is density, g/cm³;$\mathsf{\nu }$ is Poisson's ratio, which can be calculated by Eq.29; DEPT is well depth, m; P_P is pore pressure, MPa; V is effective stress coefficient, which is taken as 0.8 according to the data; A and B are tectonic coefficients, which are taken as 0.575 and 0.315 respectively in this block.

By solving the above parameters, we can further calculate the required peak stress and yield stress. The peak stress is the highest point that appears on the stress-strain curve, also known as peak strength. The peak strength is calculated using the mud content and dynamic modulus of elasticity by using the compressive strength formula, which is calculated as:

$${\sigma }_b=(0.0045+0.0035I_{sh})E_d$$

(27)

where, σ_b is the uniaxial compressive strength, MPa.

The yield stress is the strength value at which a rock ruptures, and at this point, the stress does not significantly change with strain. Here, we use fracture pressure to approximate yield stress, and the specific solution for formation fracture pressure is as follows:

$${\sigma }_c=P_f=3{\sigma }_h-{\sigma }_H-\alpha P_p+S_t$$

(28)

where, σ_c is the yield stress, MPa; P_f is the formation fracture pressure, MPa; σ_H is the maximum horizontal geostress, MPa; σ_h is the minimum horizontal geostress, MPa; P_p is the formation pore pressure, MPa; α is the effective stress coefficient, dimensionless; S_t is the uniaxial tensile strength, MPa.

For the solution of strain values, the constitutive equation is obtained according to Lemaitre's strain equivalence principle [28,29]:

$${\epsilon }_i=\frac{1}{E}[{\sigma }_i-\mu ({\sigma }_j+{\sigma }_k)$$

(29)

where, ${\epsilon }_i$ is the strain value in the i direction; E is the elastic modulus; μ is Poisson's ratio;${\sigma }_i$，${\sigma }_j$，${\sigma }_k$ is the stress in the i, j, and k directions, respectively.

Assuming that the peak stress and yield stress are in the k direction, under tensile/compressive stress, the maximum horizontal principal stress is perpendicular to the tensile/compressive direction, and the minimum horizontal principal stress is parallel to the tensile/compressive direction. We use the maximum horizontal principal stress and the minimum horizontal principal stress to replace the stress values in the i and j directions. Therefore, the calculation results of strain values corresponding to peak stress and yield stress are as follows:

$${\epsilon }_b=\frac{1}{E}[{\sigma }_b-\mu ({\sigma }_{\mathrm{h}}+{\sigma }_H)$$

(30)

$${\epsilon }_c=\frac{1}{E}[{\sigma }_c-\mu ({\sigma }_h+{\sigma }_H)$$

(31)

By combining the strain values, we can calculate the yield modulus D and post-peak modulus M at each point in the logging data:

$$\mathrm{D}=\frac{{\sigma }_b-E}{{\epsilon }_{i\_b}-1}$$

(32)

$$\mathrm{M}=\frac{{\sigma }_b-{\sigma }_c}{{\epsilon }_{i\_c}-1}$$

(33)

3. Results and Discussion

Figure 12. Contour plot of fracability coefficient in the Z-direction (Z= 2400m/2600m /2800m/3000m).

Figure 12. Contour plot of fracability coefficient in the Z-direction (Z= 2400m/2600m /2800m/3000m).

4. Conclusion

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