3.2.1. Rock [32]
The primary mineral composition of the shale [
31] in the reservoir examined in this chapter consists of albite (69.37%), dolomite (9.78%), illite (8.27%), and calcite (6.89%). The fundamental physical and mechanical properties of the reservoir are as follows: density
= 2370 kg/m
3, uniaxial compressive strength
= 192.5 MPa, elastic modulus
= 112.02 GPa, Poisson’s ratio
= 0.29, and porosity
= 15.9% [
33]. The maximum tensile strength
, shear modulus
, and bulk modulus
can be calculated using Equations (3)–(5), respectively.
Obtain = 8.602 MPa, = 4.659 GPa, = 8.465 GPa
The strength model is described by the dimensionless equivalent stress
, which is formulated as:
where
represents the actual equivalent stress and
denotes the uniaxial compressive strength. Additionally,
can be defined in terms of damage, which may be expressed as:
where
represents the damage parameter, with values ranging from 0 to 1. Specifically,
= 1 signifies complete failure of the rock, while
= 0 indicates an undamaged state.
The parameters
,
, and
are associated with the yield surface of the HJC model. In the absence of damage (
= 0) and without considering strain rate effects, Equation (7) can be simplified as follows:
Based on the principles of plasticity theory and without accounting for damage evolution or strain rate effects, both the HJC constitutive model and the Mohr–Coulomb constitutive model intersect at the corresponding points of pure shear and uniaxial compression on the compressive meridian plane. Consequently, the relationship between the dimensionless viscous strength coefficient
, cohesion
, and uniaxial compressive strength
can be derived.
Based on the Mohr–Coulomb strength criterion, the cohesion
can be determined using the following equation:
In the formula, and represent the maximum and minimum principal stressors at the point of static compressive failure, respectively, while denotes the angle of internal friction.
The Hoek–Brown criterion is widely recognized for its utility in assessing the strength index of rock under triaxial static compression. In scenarios where static triaxial compression test data are unavailable, Xie Xianli et al. [
34] and Banadaki et al. [
35] employed this criterion to calibrate parameters for the RHT model and JH-2 model, respectively, achieving satisfactory results in numerical simulations. Consequently, this study adopted this method to obtain static compressive strength data of rocks under varying confining pressure conditions using the following Hoek–Brown empirical formula:
where
represents the static uniaxial compressive strength, with
;
,
, and
denote constants, which are typically 24, 1, and 0.5, respectively, for intact rock materials.
The data presented in
Table 2 were derived by applying various confining pressures
in conjunction with Equation (11). Through linear regression analysis based on Equation (10), as illustrated in
Figure 2, the following expression was obtained:
In conjunction with Equation (10), the values
= 45.585° and
= 48.266 MPa can be derived from Equation (13).
From Equation (9), we derive
= 0.251. Subsequently, Equation (8) can be transformed into
. The standardized equivalent stress
and the standardized hydrostatic pressure
can be determined using the following respective formulas:
By substituting the static compressed data presented in
Table 2 into Equations (14) and (15), seven sets of data listed in
Table 3 can be derived. Through fitting these seven data sets with the equation
, the parameters
= 2.11 and
= 0.894 are obtained, as illustrated in
Figure 3.
The maximum normalized intensity
signifies the critical threshold beyond which
ceases to increase with further increments in
. When taking the value, it is imperative that the condition
is satisfied. Based on the maximum value of the second invariant of the partial stress tensor for reservoir rock under hydrostatic pressure [
36], this study adopted
= 6.3.
When the loading rate is within the range of 10
−5/s to 10
−1/s, the rock chamber test is classified as a static test. Both the triaxial compression process and the confining pressure increase process are characterized as static processes [
37]. Therefore, the influence of strain rate on the test results is negligible, leading to the assumption that the strain rate coefficient
.
The damage parameter
can be determined using Equation (16), where
, resulting in
= 0.0473. The parameters
and
exert minimal influence on the simulation outcomes; therefore, following the literature [
38], the values adopted are
= 1 and
= 0.01.
When the rock attains its elastic limit, the crushing pressure
and the crushing volume strain
can be determined using Equations (17) and (18).
Obtain = 64.167 MPa, = 0.00758.
represents the volumetric strain at the compaction limit state, which can be determined using the following equations:
In these equations, represents the real-time density of the material under pressure; denotes the initial porosity of the rock, with = 15.9%.
By applying Equations (19) and (20), we obtain = 0.189.
For conventional rock materials, in the absence of Hugoniot experimental data, an empirical formula can be derived as follows:
In the formula,
represents the initial density of the rock, with a value of 2370 kg/m
3.
and
are empirical parameters. For the selection of these parameters, Meyers provides values for
and
for various materials in Table 5.1 of the literature [
39]. For albite, which constitutes the highest content in shale reservoirs, it primarily consists of SiO
2 (68.8%), Al
2O
3 (19.4%) and Na
2O (11.8%) [
40]. The
and
values for the mixture can be determined using the following formula:
where
represents the percentage of the element. From Equation (22), we determine that
= 4291.92 m/s and
= 1.273.
The pressure constants
,
, and
can be utilized to fit Equation (21) to a cubic polynomial as described in Equation (23).
The final fitting yielded a set of parameters, which are presented in
Figure 4. From this, the values obtained are
= 45 GPa,
= 58 GPa, and
= 55 GPa.
The value represents the longitudinal coordinate at the intersection of the fitting curves corresponding to the plastic and compact stages in the equation of state. Due to insufficient experimental data in the plastic stage, the relationship between the modified volumetric strain and the axial static pressure P under varying confining pressures is described by Equations (20) and (23). Consequently, we determine that = 1.452 GPa.
The failure type
is utilized to govern the removal of failed material elements and is defined as follows in the ANSYS/LS-DYNA User Manual [
41]:
When the material is in tension,
can be defined as:
- 2.
When , invaid.
- 3.
, failure when the equivalent plastic strain reaches .
Regarding the value of
, there is a greater inclination in the academic community to accept failure strain (pressure-induced failure mode) as its intrinsic failure mechanism. Sun et al. [
42] proposed several assumptions concerning the value of
. They posited that
represents the critical threshold of plastic strain in the HJC material model, which determines whether the material will fail under pressure
. This relationship can be mathematically expressed by Equation (26).
Furthermore, when the pressure on the material under compression increases exponentially and the corresponding bulk strain change rate is less than 1% [
42], the rock material attains its maximum density
. Based on this premise, Lin et al. [
43] proposed that the maximum density
is approximately 1% greater than the compaction density
, which can be simplified as
. Consequently, by substituting the calculated value of
from Equations (20) and (23) into Equation (26), the final factor
= 0.726.
The selection of the reference strain rate
has a minimal impact on the simulation outcomes. In this chapter, the final HJC constitutive parameters are determined using
, as detailed in
Table 4.
In order to cause cracks caused by removal of elements when modeling with LS-DYNA finite element software, in addition to defining the rock constitutive model with failure mode, cracks caused by failure of rock materials after blasting can be controlled by adding the *MAT_ADD_EROSION option.
Given that the tangential tensile stress induced by the propagation of the detonation wave into the fracture zone exceeds the dynamic tensile strength of the rock, the rock within this zone experiences tensile failure due to the stress wave. Therefore, to control the fracture propagation following the tensile failure calculation, the minimum hydrostatic pressure failure criterion is established. This criterion can be derived from Equation (27), yielding a result of −8.195 MPa.
Furthermore, shear failure is inevitable when rock is subjected to compression failure. Therefore, the maximum shear strain failure criterion (EPSSH = 500 MPa) has been incorporated into the numerical simulation.