# Constitutive Model of Stress-Dependent Seepage in Columnar Jointed Rock Mass

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{4}square kilometers. Therefore, the study of CJRM mechanics and seepage characteristics is essential for engineering design and safe construction.

## 2. Establishment of Seepage Model for Stress-Dependent Fractured Rock Mass

#### 2.1. Seepage Model

_{f}is the hydraulic gradient parallel to the structural planes, b is the aperture, and μ is dynamic viscosity of fluid. Correspondingly, the permeability coefficient of a smooth joint can be expressed as K

_{f}= gb

^{2}/12μ.

_{h}is needed in the calculation. Lomize [24] and Louis [25] established the following expressions using the fracture surface roughness correction coefficient method:

_{Lomize}, C

_{Louis}are the correction coefficient of the fracture surface roughness and e is the absolute height of the protrusion of the fracture surface.

_{max}

^{2}is the maximum fracture displacement, and σ

_{n’}is the effective normal stress. K

_{n0}is initial normal stiffness when normal stress is zero.

#### 2.2. Stress-Dependent Seepage Model

_{1}, but also by lateral stress σ

_{2}and σ

_{3}in two directions parallel to the fracture when calculating the permeability coefficient; it is also subject to water pressure P, as shown in Figure 2.

_{0}, the normal stiffness of the crack is k

_{n}, the deformation of the crack under the normal stress can be expressed as follows:

_{f}can be expressed as:

_{f}of the jointed rock mass are only the effective normal stress σ

_{n’}and JRC, so the fracture permeability coefficient can be expressed as a function of the mechanical aperture, assuming that the fracture parameters are constant. When studying the seepage characteristics of CJRM, since the joint is in the form of a joint group, the results obtained by the superposition method of Formula (4) contains a large error, and the calculation of the permeability coefficient of Formula (4) is completely based on the physical test experience, which cannot accurately react to the permeability characteristics of columnar jointed rock masses. Therefore, it is necessary to propose a method for accurately calculating the analytical solution of the joint opening. In this article, which is based on the theory of composite mechanics and Goodman’s joint superposition principle, the constitutive model of joints of CJRM is derived on the basis of the Quadrangular prism, the Pentagonal prism and the Hexagonal prism model. Through these three models, the permeability characteristics of the columnar jointed rock mass can be reflected.

## 3. Establishment of Seepage Model for Stress-Dependent Fractured Rock Mass

#### 3.1. Stress-Dependent Seepage Model

_{s}} produced by the jointed rock mass comprises joint strain {ε

_{j}} and intact rock mass strain {ε

_{r}}:

_{s}] is the joint rock mass flexibility matrix, and {σ} is the stress matrix. Correspondingly, according to the theory of composite materials, the joint rock mass flexibility matrix minus the intact rock mass flexibility matrix [S

_{r}] can be used to obtain the joint flexibility matrix [S

_{j}], as follows:

_{r}, G

_{r}and v

_{r}, which represent the elastic modulus, shear modulus and Poisson’s ratio of the rock. The intact rock is an isotropic material, but the CJRM is a composite material with anisotropic characteristics, the flexibility matrix of which can be expressed by engineering constant as:

_{j}and joint connectivity rate A

_{j}[15], as shown in Figure 4. From the jointed rock mass unit, it can be seen that the joint is through in one direction (3-axis direction) and discontinuous in the other direction (2-axis direction).

_{n}and the shear stiffness k

_{s}commonly used in engineering are used to describe the elastic characteristics of the crack, and the Poisson effect of the crack is neglected, and the crack surface is regarded as anisotropic and in an elastic stage. The results obtained by solving the equivalent elastic parameters of the single set of jointed rock mass in the (1-2) plane are as follows:

_{j}is the mechanical aperture of the joint.

#### 3.2. Constitutive Equation of Joint in the 1-2 Plane

_{ij}are the composite engineering coefficients. In order to estimate the composite engineering constant of DJRM on the 1-2 plane, the Quadrilateral, Hexagonal and Pentagonal prism models are divided into jointed rock mass and intact rock mass. The flexibility matrix can be solved as shown in Figure 5.

_{0}corresponds to the flexibility matrix of the joint group whose spindle is parallel to the 1-2 axis. [K(φ)] is the plane coordinate transformation matrix, and its expression is as follows:

#### 3.3. Establishment of Three-Dimensional Joint Flexibility Matrix

_{nn}and b

_{sn}are the concentration factors of normal stress and shear stress caused by joint offset, respectively, Fs is the vertical offset between adjacent hidden joints, and S

_{s}and S

_{n}indicate the joint spacing between parallel and vertical column axis directions, respectively.

_{XYZ}and [S]

_{123}represent the flexibility matrix in the geodetic coordinate system x-y-z and the material spindle coordinate system 1-2-3, respectively; [T(α,β,γ)] is the coordinate transformation matrix, α is the deflection azimuth angle, β is the dip angle, and γ is the steering angle.

#### 3.4. Establishment of Permeability Coefficient K_{ZZ} of the Column-Axis Equation

_{1}, ε

_{2}, ε

_{3}, γ

_{1}, γ

_{2}, γ

_{3}} of the CJRM fissures can be obtained. For convenience of calculation, the joint arrangement of the Hexagonal prism and the Pentagonal prism model is simplified into a quadrilateral. This arrangement, applying the simplified approach proposed by Que [21], simplifies the geometric elements of the three models into a joint length S

_{2}parallel to the X axis and a joint width S

_{1}parallel to the Y axis; the cross-sectional shape of the Pentagonal prism and the Hexagonal prism model is simplified into a quadrilateral. This simplified method will lead to some errors when the variable angle θ ≠ 0 for the Pentagonal prism model, but the study by Que proved that this error is extremely small and can be ignored, and when θ = 0, the pentamorphic shape will become rectangular. The simplified cross-section is as shown in Figure 6:

_{ZX}and K

_{ZY}corresponding in the X and Y directions can be expressed as follows:

_{hx}and b

_{hy}are the equivalent hydraulic aperture of the X-axis and Y-axis, b

_{x0}and b

_{y0}are the maximum aperture at zero normal stress of the X-axis and Y-axis, respectively. According to Premise (4), the permeability coefficient K

_{ZZ}in the direction of the column axis can be solved as follows:

_{xi}and B

_{xi}are joint areas in the X-axis direction and the Y-axis direction, respectively. m and n are the number of joint groups of the X-axis and the Y-axis, respectively. By means of the above method, a formula is established for solving the permeability coefficient of CJRM in three-dimensional space.

## 4. Verification and Comparison of CJRM Seepage Constitutive Models

#### 4.1. Comparison and Analysis of Three Constitutive Models and Numerical Simulation Results

^{2}β) [1].

^{2}β

^{33-2}carried out in the dam site area of Baihetan Hydropower Station, the constitutive equation of the CJRM anisotropic seepage is verified. To study the influence of the water level variation on the safety of the dam foundation, Xiong [18] used the improved Voronoi algorithm to construct a three-dimensional columnar model of the columnar jointed rock mass of the Baihetan dam foundation, the analysis and calculation of the seepage tensor of the Baihetan dam foundation were carried out by 3DEC discrete element software. The numerical model is shown in Figure 7.

_{j}, s, s

_{3}), and then employed the engineering parameters of the Baihetan (kn, ks, Er, Gr) to perform numerical calculations. The above parameter values are shown in Table 2. The head difference between the upstream and downstream of the Baihetan arch dam is 200 m, the pore water pressure of 2 MPa is added to the corresponding surface of the model, and the remaining surfaces are set as impervious surfaces. The outer boundary of the model is subject to displacement constraints. By changing the inclination angle β of the cylinder, different inclination angle coefficients of column axial permeability K

_{ZZ}can be obtained. To verify the applicability of the constitutive model proposed in this paper, the parameters used in the constitutive equation are the same as those used in the numerical simulation. The permeability coefficients of columnar jointed rock masses at different deflection angles were calculated by using the Quadrangular prism, Pentagonal prism, and Hexagonal prism models, and the differences between the calculation results of the three seepage constitutive models and the numerical simulation results obtained at the same inclination were compared and analyzed.

_{0}= 0.8698 rad by inversion calculation. In the calculation process, a calculation method with equal circumference is adopted. Therefore, the contour of the method plan was selected to ensure that the different models had the same joint area. The polar coordinate diagram of the permeability coefficient value of the column axis direction changed with the deflection angle of the prism, as shown in Figure 8, and the calculation results were compared and analyzed.

#### 4.2. The Variation Law of Permeability Coefficient of Pentagonal Prism Model with Confining Pressure

_{ZZ}= a∙K

_{0}exp(−b∙σ). Both a and b parameters in the formula are related to the joint feature. Due to the introduction of three-phase stress, the three-phase stress will have an effect on the parameter b. The multivariate regression fitting curve and the Louis, Peuga and other scholars have the same form for the formula obtained from the normal stress, which can be expressed as a negative exponential function of pressure, as shown in Formula (5). Therefore, the constitutive model proposed in this paper expands the previous empirical formula, and is better able to describe the permeability characteristics of CJRM.

## 5. Conclusions

- The three models applied to the Baihetan project were able to reflect the basic seepage characteristics. Compared with the Quadrangular prism and the Hexagonal prism model, the calculation results of the Pentagonal prism model were most consistent with the numerical simulation results. This is because the Pentagonal prism model not only has a completely penetrating joint, but also a mutual bite between the prisms, which is able to truly reflect the seepage characteristics of the columnar jointed rock mass.
- The permeability coefficients calculated by the seepage constitutive model of the three columnar jointed rock masses reached their minimum when the deflection angle of the prism was β = n∙90°(n = 1,3), and reached their maximum when the deflection angle of the cylinder was β = (n − 1)∙90°(n = 1,3). This is because at β = n∙90°(n = 1,3), the confining pressure of the column is completely converted into normal stress, and the joint strain is high. As the deflection angle gradually decreases to β = (n − 1)∙90°(n = 1,3), the partial confining pressure causes the joint shear slip, and the joint strain becomes smaller, leading to a larger permeability coefficient of the joint.
- The law of permeability coefficient with the change of confining pressure calculated by the seepage constitutive models proposed in this paper can be expressed as a negative exponential function, which conforms to the general law of seepage of jointed rock mass, and expands the solution method for the permeability coefficient of columnar jointed rock mass under stress field.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Typical columnar joint, pictured (

**a**) parallel to the column axis direction; (

**b**) perpendicular to the column axis direction.

**Figure 3.**Typical columnar joint: (

**a**) Quadrangular prism model; (

**b**) Hexagonal prism model; (

**c**) Pentagonal prism model.

**Figure 5.**Composite solution method for joint flexibility matrix of CJRM: (

**a**) calculation method of Quadrangular flexibility matrix; (

**b**) calculation method of Hexagonal flexibility matrix; (

**c**) calculation method of Pentagonal flexibility matrix. s, and n are parallel and perpendicular to the direction of joint extension, respectively.

**Figure 6.**Simplified cross-section of Hexagonal and Pentagonal models: (

**a**) simplified Hexagonal prism model; (

**b**) simplified Pentagonal prism model.

**Figure 7.**Columnar joints with different joint deflection angles from Xiong: (

**a**) deflection angle of 0°; (

**b**) deflection angle of 30°; (

**c**) deflection angle of 60°; (

**d**) deflection angle of 90°.

**Figure 8.**Comparison of the permeability coefficient theoretical curve and numerical simulation curve.

**Figure 9.**Comparison of the permeability coefficient theoretical curve and fitted curve: (

**a**) deflection angle of 0°; (

**b**) deflection angle of 18°; (

**c**) deflection angle of 36°; (

**d**) deflection angle of 54°; (

**e**) deflection angle of 72°; (

**f**) deflection angle of 90°.

**Table 1.**Statistical table of sectional shape of columnar jointed basalt body column in typical areas.

Location and Country | Shape of Column Section | ||
---|---|---|---|

Quadrilateral | Pentagon | Hexagon | |

Craters of Moon (USA) (CAN) | 28% | 56% | 16% |

Dunsmuir (CAN) | 14.5% | 46% | 33.5% |

Lewiston (USA) | 7.5% | 45% | 41.5% |

Devils Tower (USA) | 17% | 42% | 35% |

BaihetanP_{2}β_{3}^{3} (CHN) | 49% | 46% | 5% |

k_{n} (GPa/m) | k_{s} (GPa/m) | K_{f}_{0} (cm/s) | t_{j} (m) | E_{r} (GPa) | G_{r} (GPa) | v_{r} | l (m) | s_{3} (m) | s (m) |
---|---|---|---|---|---|---|---|---|---|

100 | 50 | 8.599 | 1 × 10^{−3} | 65.1 | 25 | 0.23 | 0.2 | 2 | 0.05 |

_{n}, k

_{s}normal stiffness and shear stiffness, respectively. K

_{f}

_{0}is the initial permeability coefficient. E

_{r}, G

_{r}, v

_{r}are the elastic modulus, shear modulus and Poisson’s ratio, respectively, of the intact rock. l is the side length of CJRM, s

_{3}is the average column length, and s is the vertical staggered distance between the adjacent hidden joints.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Niu, Z.; Zhu, Z.; Que, X.
Constitutive Model of Stress-Dependent Seepage in Columnar Jointed Rock Mass. *Symmetry* **2020**, *12*, 160.
https://doi.org/10.3390/sym12010160

**AMA Style**

Niu Z, Zhu Z, Que X.
Constitutive Model of Stress-Dependent Seepage in Columnar Jointed Rock Mass. *Symmetry*. 2020; 12(1):160.
https://doi.org/10.3390/sym12010160

**Chicago/Turabian Style**

Niu, Zihao, Zhende Zhu, and Xiangcheng Que.
2020. "Constitutive Model of Stress-Dependent Seepage in Columnar Jointed Rock Mass" *Symmetry* 12, no. 1: 160.
https://doi.org/10.3390/sym12010160