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So far, there are few studies concerning the effect of closed “fluid inclusions” on the macroscopic constitutive relation of deep rock. Fluid-matrix element (FME) is defined based on rock element in statistical damage model. The properties of FME are related to the size of inclusions, fluid properties, and pore pressure. Using FME, the equivalent elastic modulus of rock block containing fluid inclusions is obtained with Eshelby inclusion theory and the double M-T homogenization method. The new statistical damage model of rock is established on the equivalent elastic modulus. Besides, the porosity and confining pressure are important influencing factors of the model. The model reflects the initial damage (void and fluid inclusion) and the macroscopic deformation law of rock, which is an improvement of the traditional statistical damage model. Additionally, the model can not only be consistent with the rock damage experiment date and three-axis compression experiment date of rock containing pore water but also describe the locked-in stress experiment in rock-like material. It is a new fundamental study of the constitutive relation of locked-in stress in deep rock mass.

As the research of rock mechanics gradually develops to deep and complicated geological conditions, the traditional theory of rock mechanics has been continuously improved and perfected. The application of CT technology in rock mechanics helps people have a new understanding of microscopic pore structure of rock, thus establishing a microscopic system of rock [

Pores are classified into connected pores and closed pores. Actually, the theoretical and experimental research on the connected pores in rock is relatively mature [

In general, the research on the constitutive relations of rock fluid inclusions under deep high pressure is very few. Therefore, this study attempts to establish a concise mathematical and physical model that reflects the constitutive relationship between locked-in stress and rock block. The relationship between microcosmic characteristics and macroscopic rules has good applicability, which also lays a good foundation for the mechanical problems of deep rock mass containing closed fluid inclusions.

The statistical damage model is a good mathematical tool for establishing the constitutive relationship between the microscopic and macroscopic properties of rocks. “Rock element” is the basic element of the rock statistical damage model and the “rock element” satisfies the D-P yield criterion. Rock elements gradually yield under external stress, and their strength intensity conforms to Weibull distributions. Moreover, the process can well reflect the stress-strain law of the rock under external stress.

According to the Lemaitre strain equivalence hypothesis, the elastic modulus of rock after damage is as follows [

The stress-strain curve is consistent with the law of linear elasticity before material damage. The macroscopic constitutive relation considering cumulative damage of rock can be expressed as

The solution to

According to the study of Wengui Cao, the yield judgment of rock element is appropriate with Drucker-Prager criterion, and the strength intensity of rock elements conforms to Weibull distribution [

Assuming that the strength of rock element obeys the Weibull distribution, its probability density function is

According to (

However, the damage defined in the traditional statistical model essentially reflects the damage caused by the yielding of the nondestructive matrix due to the external stress. Besides, the initial damage of the “fluid inclusions,” cracks, and holes existing in the rock before the external loading are not reflected in the model. Obviously, these are important factors influencing the initial elastic modulus of the rock. To establish the relationship between fluid inclusions and macroscopic constitutive relations of rocks, the initial factors of these rocks must be clearly described.

To solve this problem, it is necessary to establish the “rock element” which can reflect the constitutive relation of the rock part representatively, so fluid inclusion-matrix element (FME) is defined [

FME is mixed up with a fluid inclusion and rock matrix around a finite range, as shown in Figure

FME (a) and rock element (b).

Figure

Pore fluid volume modulus of FME

The compression factor of rock matrix can be defined as

From the point of the whole point of FME, the change of fluid inclusion volume is equal to the volume of fluid. So,

Based on (

From (

From (

According to the theory of solid mechanics, the compression coefficient of the skeleton under the condition of elastic deformation can be expressed as

In engineering applications, the contents of water, oil, and gas in the fluid inclusions are often different. When there are many kinds of fluid, the effect of fluid components on the fluid inclusion modulus should be considered. In the study of fluid replacement process, Murphy obtained the calculation method of fluid bulk modulus in fluid inclusion under different saturation [

Bulk modulus inside fluid inclusions.

Serial number | Depth of measuring point/m | Rock species | fluid inclusion | Modulus of elasticity/GPa | source |
---|---|---|---|---|---|

1 | 1335–1338 | Sandstone | Gas, water | 1.6~3.5 | [ |

1430–1438 | Carbonate | Gas, water | 2~4 | [ | |

3 | — | Dolomite | Oil | 1.3 | [ |

4 | — | Dolomite | Water | 2.6 | [ |

5 | — | Sand mudstone | Oil | 1.38 | [ |

6 | — | Sand mudstone | Water | 2.25 | [ |

7 | 2355–2360 | Sand mudstone | Water | 2.5 | [ |

8 | 3206–3210 | Sand mudstone | 47% oil 53% water | 1.8~4 | [ |

9 | 3166–3170 | Sand mudstone | Oil | 0.9~2.0 | [ |

10 | — | — | Oil | 1.7~2.4 | [ |

11 | — | — | 80% oil 20% water | 1.75~2.3 | [ |

12 | — | — | 60% oil 40% water | 1.85~2.2 | [ |

According to Table

Table

Study on compression coefficient or bulk modulus of FME is important for the equivalent elastic modulus based on Eshelby equivalent inclusion theory. It makes the fluid inclusion have practical meaning and physical meaning of certain form, which is different from the traditional solid mechanic problems or traditional inclusion damage model from this point. In addition, it is also a very important reason why this model is suitable to analyze locked-in stress.

The constitutive relationship of the FME containing different pore types is described by a uniform “equivalent elastic modulus,” which is based on the Eshelby equivalent inclusion theory and M-T homogenization method. In 1957, Eshelby studied the law of stress around a phase in an infinite homogeneous matrix [

As shown in Figure

FME.

If it is a spherical inclusion in FME,

In fact, the pore fluid inclusions in the FME are heterogeneous with the rock matrix and are subjected to external stress, which can be found in Figure

Compared with homogeneous material, the perturbation strain tensor produced by the substrate on the heterogeneous inclusions under the action of external stress is

Confinement strain tensor of matrix to inclusion is

Common strain tensor is

FME under stress.

As is shown in Figure

Average stress tensor of matrix in FME is

With Eshelby inclusion theory, (

It can be obtained from (

Under the action of an external stress tensor

From (

From (

The stress-strain relationship of the “fluid inclusions-matrix part” and the “matrix part containing no initial damage” has been obtained. The composition of the rock block is shown in Figure

FME, rock element, “matrix part,” and “fluid inclusion-matrix part.”

The stress-strain relationship between the two parts can be unified by the M-T homogenization method [

According to (

In this way, the equivalent elastic modulus of FME and the constitutive equation of rock are combined, and the equivalent elastic modulus of FME is closely related to the fluid properties and pressure of fluid inclusions in FME. Then, the relationship between stress-strain curve of rock and fluid inclusions is established.

Numerous studies have demonstrated that rock porosity changes under different confining pressures should be considered. For example, in the study of rock pore seepage process, it was found that the pore porosity will change with the confining pressure. Additionally, the correlation study shows that the porosity changes with the different confining pressure change can be expressed as [

It can be obtained from (

According to (

(1) Confining pressure

Figure

Influence of porosity on the full stress-strain curve under confining pressure.

(2) The initial porosity is not mentioned in the reference. When confining pressure is 6.9 MPa, the porosity

Comparison with experimental data; confining pressure is 6.9 MPa.

(3) When confining pressure is 13.8 MPa, the porosity

Comparison with experimental data; confining pressure is 13.8 MPa.

Since the porosity in the original references was not considered or given, two sets of numerical values were selected for simulation. Through comparing Figures

Compared with the experimental data of the stress-strain curve, it is reasonable and effective in combining the Eshelby equivalent inclusion theory and the method of statistical mathematics to establish the model. The model has practicality and physical meaning. However, the figure also shows that there is a certain gap between the model curves beyond the yield limit and the development trend of the test curve and that the rock strain softening part is not well modeled. The theoretical establishment of this part is a direction that needs to be conducted in the future.

Meanwhile, on the basis of verification rationality, this model should not only verify the general rock compression experiment, but also validate the rock containing different types of fluid inclusions, especially for the constitutive model of rock mixed up with closed fluid inclusions or locked-in stress. Actually, experimental and verification work in this area is relatively few but significant.

Experimental verification includes two types of situations: one is the rock which contains fluid-connected pores and the other is to simulate the rock containing closed pores (locked-in stress) in rock-like materials.

The influence of the fluid inclusion on the total stress-strain curve of rock is rarely involved in the traditional statistical model of rock damage. In recent years, there are more studies about the influence of fluid inclusion on the overall characteristics of rock. Therefore, in this paper, the rationality of the model is validated by the three-axis experiment data of rock containing pore water. If the influence of pore pressure variation on the total stress and strain of rock can be obtained, the applicability of the model with the connected pore in FME will be validated.

Through comparing the model with the sample data of rock in [

Figures

Comparison between experimental date and model date; fluid inclusion pressure of FME is 4 MPa; confining pressure is 20 MPa.

Comparison between experimental date and model date; fluid inclusion pressure of FME is 12 MPa; confining pressure is 20 MPa.

If the fluid state in FME can be reasonably reflected, the model not only can be used to analyze the rock block containing connected pores, but also has good applicability to the rock containing locked-in stress. However, the experimental researches about locked-in stress are few, and the experimental simulation itself is also confronted with a lot of difficulties. The change of equivalent fluid inclusion modulus of FME has mature application in fluid replacement in oil and gas engineering, while the experimental date about the constitutive relation of rock in this process is few. Consequently, in this part, the locked-in stress in rock is validated by simulating the closed pores with different equivalent modulus of elasticity in FME.

The experiment is designed that nitrile rubber inside rock matrix (rock-like material) can produce different expansion pressure at different temperature control conditions to simulate the different locked-in stress in rock. The equivalent elastic modulus of different inclusions in FME can be achieved from the above analysis. Besides, the matrix material consists of Portland cement and fine quartz sand in a certain proportion. The parameters are as follows:

Transformation relationship.

Experimental situation | Proportion of rubber material in matrix: 2%/4% | Locked-in stress: 0.8 MPa/1.2 MPa |

Theoretical transformation value | porosity: 2%/4% | Equivalent elastic modulus of fluid inclusion: 0.2 GPa/0.5 Gpa |

Figure

Comparison between experimental date and model date and fluid inclusion pressure of FME is 0.8 MPa.

Comparison between experimental date and model date; fluid inclusion pressure of FME is 1.2 MPa.

The locked-in stress exerts significant influence on the experiment. The data in Figures

Therefore, from the experimental verification results of this part, it can be seen that the influence of locked-in stress in the deep rock mass on the macroscopic constitutive relationship should not be ignored. High-pressure and high-energy fluid inclusions in deep rock mass are the important reasons for such phenomena as rock burst and large deformation. Based on the model, we can analyze the fluid inclusions in the deep rock mass to predict or analyze some rock phenomenon. Additionally, the model provides a new idea for the solution of deep rock mechanics problems.

With the development of deep rock mechanics, more and more attention has been paid to the locked-in stress in rocks. High-pressure fluid inclusions are important parts of the locked-in stress in deep rock mass. Nevertheless, there are few studies on the mechanical properties of rock mixed up with closed pores. Based on RVE, this paper unifies the analysis methods of closed pore and connected pore, and FME is defined. Based on FME, the relationship between the microscopic fluid inclusions and the macroscopic mechanical properties of the rock is established using the statistical mathematical method.

The main works of this paper are as follows: on the basis of the traditional damage model, firstly, the constitutive relation of FME is built on Eshelby equivalent inclusions theory, and the stress-strain relation of FME is then determined. Secondly, the influence of FME on initial modulus of elasticity of rock mass is analyzed and the relationship between FME and the whole stress and strain of the matrix is established based on the Weibull distribution. The effect of fluid inclusions in FME on the stress-strain curve of rock is analyzed. Compared with the general rock compression experimental data, the model is proved to be reasonable. On this basis, the model is compared with the three-axis confining pressure experimental date and locked-in stress simulation experiment of rock mass. Moreover, the model results are in good agreement with experimental date.

The main conclusions are as follows:

(1) FME is the medium of establishing the fluid inclusion and macroscopic stress-strain curve of rock. Combined with the principle of Eshelby equivalent inclusion, the FME can reflect the fluid state of the fluid inclusion and other factors with the concept of equivalent elastic modulus. FME is a basis of rock statistical damage model. It is an improvement of the traditional statistical damage model which cannot effectively reflect the initial fluid inclusion, crack, and other damage.

(2) The stress of the same strain changes with the increasing elastic modulus of the FME, but it does not influence the variation trend before and after the peak. However, the increase of porosity affects the gradient of the curve behind the peak, and the stress-strain curve becomes slow with the increase of porosity. The assumption that porosity varies with confining pressure can reflect the law of post-peak variation, which is greatly consistent with the experimental data.

(3) Compared with the different experimental data, the model based on FME and the mathematical statistics model is reasonable. Based on FME, it can reflect the influence of initial porosity or initial damage in a wider sense on rock mechanical properties, whether it is connected pore or closed pore in FME. As long as the initial damage can be reflected in FME, the influence of initial damage can be obtained from the model.

The innovations of the model are as follows:

(1) Using the Eshelby equivalent inclusion theory and M-T uniform theory, the initial elastic modulus of rock is analyzed. Besides, it is an improvement of the traditional damage model, which cannot reflect the initial hole and damage of the rock, making the new statistical damage model more practical and reasonable.

(2) Based on FME, the relationship between rock microcosmic and macroscopic is established. The macroscopic constitutive relation of rock is analyzed by fluid inclusions in FME. Few studies have been conducted from this point of view.

(3) The model is employed to analyze the locked-in stress in deep rock mass, and the locked-in stress simulation experiment is carried out, which is groundbreaking for analyzing geological phenomena through the point of locked-in stress. Briefly, one important reason why we call it a new model is the effect of the locked-in stress on the rock. Therefore, the mathematical statistics model built on it is reasonable and effective.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The authors acknowledge the support of Jiangsu Natural Science Foundation of China (BK20141067).