1. Introduction
With the increase in global energy utilization, oil and gas extraction has also shifted from conventional to unconventional. Unconventional energy can be defined as reservoir resources that cannot be economically effectively recovered without implementing special stimulation measures. Existing stimulation technologies include fracking, acid fracturing, supercritical CO
2 fracturing, and discharge shock waves [
1,
2,
3].
Discharge shock waves are a relatively new reservoir modification technology that releases a large amount of stored electrical energy in a short period of time, exerting pressure on the reservoir rocks along the water medium, causing micro-cracks and extending the original cracks in the reservoir rock near the wellbore area.
Repeated action of discharge shock wave will cause fatigue damage to the rock. When rocks are subjected to impact, the strain or stress accumulates until damage occurs, which manifests in the form of micro-cracks. With the continuous increase of the damage degree of the body, the irreversible deformation of the rock will continue to accumulate, and in this process, rock defects such as micro-cracks will also continue to generate and develop, eventually lead to the reduce of the rock strength and failure. Therefore, it can be considered that the development state of micro-cracks characterizes the damage and failure behavior of rocks under the repeated action of the shock wave. Therefore, to explore the fatigue characteristics of rock under the action of the discharge shock wave, it is particularly important to analyze the development state of micro-cracks in the rocks under load.
The fatigue damage of rock under repeated impact is mainly affected by factors such as shock wave load parameters [
4,
5,
6,
7,
8]. In recent years, many efforts have been made to investigate the fracture behaviors of rock under cyclic loading. For instance, Xiurun found that the upper-limit stress and amplitude are the main factors that influence the fatigue life of sandstone [
9]. Zhou studied the effect of frequency on the mechanical properties of sandstone subjected to cyclic loading, and found that as the frequency increases, the mechanical properties will increase at first and then decrease [
10]. Song studied the fatigue behavior of rock materials under multi-stage repeated impact loads and proved that the failure mode and crack size are directly related to the loading mode [
11,
12].
The mechanical behaviors and evolution characteristics of mineral particles at the mesoscale are internal factors leading to the macromechanical properties of rocks [
13,
14]. In recent years, the effects of various types of defects, such as cracks, beddings, and holes, on the strength, deformation characteristics, and fracture evolution laws of rocks or rock-like materials have been systematically investigated [
15,
16,
17].
In recent years, numerical simulations have been widely used to investigate the macro- and mesomechanical properties and failure mechanism of rocks. Budiansky analyzed the dispersed crack group in rock and established the corresponding rock fracture damage model [
18]. Grady considered that there are a large number of primary cracks obeying the two-parameter Weibull distribution in the rock [
19]. Taylor introduced the expressions of the effective bulk modulus and Poisson’s ratio of R.J.O, Connel, with the micro-crack density and the expression formula of the fragment size given by the TCK model [
20]. Then, many scholars have proposed a variety of damage constitutive models from the perspective of the dynamic fracture of rock [
21,
22].
Although the damage caused by impact load to rock is considered in the above model, the dynamic state damage caused by repeated action to the interior of intact rock is not considered. This is because, with the hange of impact load parameters, the micro-crack state generated in the rock body is dynamic. However, the theoretical circle has not yet proposed a dynamic damage constitutive model that can reflect its influence on rock mechanical properties and micro-crack changes. At the same time, previous experiments have also proved that the fatigue damage of rock is dynamically affected by discharge impact energy and impact times. In this study, is based on the theory of fracture mechanics and the classical rock dynamic damage constitutive model (TCK model). Firstly, the cohesive force model is introduced, then the weakening formula of rock mechanics is deduced by experimental means, and finally, the modified dynamic damage model of rock is established by combining the above formulas. It is verified by AUTODYN simulation software. The influence of different discharge energy and times on rock compressive strength and micro-crack formation can be quantitatively obtained by the modified model. The establishment of this model enables a more simplified and accurate application for the reformation of sandstone reservoirs under various reservoir conditions. It greatly advances the application of discharge impact technology in the field of oil and gas reservoirs.
2. Existing Rock Damage Constitutive Model
2.1. TCK Model
According to the existing damage models, when the impact load does not exceed the plastic yield limit of the rock, the fatigue damage of the rock is mainly manifested as an elastic-brittle failure caused by micro-crack propagation. Therefore, the evolution of rock fatigue damage is generally determined by the change in rock element micro-crack density.
The TCK model is a dynamic damage constitutive model of rock based on the microscopic damage mechanism. The basic assumption is that rock is the isotropic material, and under the action of external force, the rocks defects will initiate, expand and even connect, leading to its failure [
23].
where
σ is the stress in tensor form (MPa),
ε is the volume strain (dimensionless),
E is the elastic modulus (MPa),
D is the damage variable;
n is the number of discharges (dimensionless).
The damage coefficient
D caused by micro-cracks is defined by the bulk modulus of the medium as follows:
Budiamsky and O’Connel proposed an expression for the effective bulk modulus of cracked solids [
18].
where
is the effective bulk modulus (MPa),
K is the original bulk modulus (MPa),
is the effective Poisson’s ratio (dimensionless),
Cd is micro-crack density (quantity/cm
2).
The crack density is the ratio of the rock volume in the crack-affected zone to the total rock volume.
where
β is the coefficient (dimensionless),
β = 3,
a is the average radius of micro-cracks (mm),
N is the number of cracks activated by rock impact (quantity).
Under volume tension, the number of activated micro-cracks in the rock obeys the Weibull distribution [
19].
where
k,
m are rock parameters in the damage model (dimensionless),
k = 7.47 × 10
15,
m = 6.
Parameter m relates to the ultimate tensile stress and strain rate of rock materials. Grady believed that m is a constant.
The material parameter k is obtained by tensile fracture experiments at different strain rates.
Combined with the above formula, the damage coefficient is related to the crack density, and the damage coefficient is defined as follows:
2.2. Existing Problems
The above rock fatigue damage model treats the dynamic damage of rock under load as a continuous process and defines the damage variable as a function of crack density from the perspective of micromechanics, but the current theoretical model does not consider the dynamic hange of the average micro-crack radius parameter of crack density under the influence of different discharge energy and discharge times during the discharge impact process in practical application.
However, how to describe the effect of different discharge energy and discharge times on the weakening of rock mechanical properties and the formation of micro-cracks is the key and difficult problem in the current study of discharge impact dynamics.
To solve this problem, many scholars have put forward many different methods from different angles. For example, Kyoya defined the damage tensor of rock mass with a set of parallel cracks as follows [
24]:
where Ω is the damage tensor of jointed rock mass (MPa),
l is the average crack spacing (mm),
V is the sample volume (mm
3),
N is the number of cracks in the sample;
ak is the surface area of the
k-th crack in the sample (mm
2),
nk is the unit normal vector on the surface of the
k-th crack in the sample (dimensionless).
This is also one of the most commonly used damage tensor calculation methods in rock damage theory, but it also has certain defects. Beceuse this method only considers the effect of micro-crack length and does not consider the effect of crack strength properties, which is inconsistent with the rock mechanical properties that weaken the compressive strength of rocks under repeated discharge impact. Therefore, Kawamoto modified the above model by introducing crack pressure transfer and shear transfer coefficient to consider how the characteristics of those joints can transfer partial compressive stress and shear stress under compressive loads [
25].
It can be seen from the current research that for the fatigue damage constitutive model of rocks, it is generally believed that the joint effects of crack geometry and strength characteristics, such as compressive strength, should be considered simultaneously. However, the current research is still to first define the damage tensor according to the geometric characteristics of the cracks and then modify the above calculation results through the strength characteristics of the cracks, which not only causes inconvenience in the use of the model but also makes the theoretical calculation results difficult to apply to engineering practice due to the randomness of the empirical parameter values.
Based on the previous research on rock damage mechanics, this paper uses experimental methods to establish a calculation method that can obtain the crack geometry and mechanical properties through the hange of discharge energy and discharge parameters based on the action characteristics of the discharge shock wave and improves the correction formula of the TCK model.