A Novel Deformation Analytical Solution and Constitutive Model for Fractured Rock Masses

Abstract

In order to study the deformation and stability of a fractured rock mass, existing research suggests that fracture deformation, which is usually obtained by evaluating the equivalent deformation modulus, dominates the deformation of a fractured rock mass. However, this parameter is difficult to obtain in theory and practice, which limits the application of rock mass deformation analysis methods. In order to calculate the deformation of fractured rock masses, the mass of the rock is regarded as a sponge-like material, and it is assumed that the deformation of a water-saturated fractured rock mass under external force load is approximately equal to the net flow of fracture water. Based on this assumption, firstly, the relationship between rock mass deformation and fracture flow is studied through a single-fracture rock mass model. The hydraulic properties of the fracture are characterized by the permeability coefficient, and the fracture deformation in the rock mass is equivalent to the fracture flow. The fracture deformation calculation formula is derived from the fracture hydraulics calculation formula, and this is compared with the measured data. The rationality of the calculation formula was verified. On this basis, the calculation formula for rock mass deformation, including multiple groups of fracture surfaces, is proposed, and the stress-strain constitutive relationship of a complex rock mass is established. The correctness of the calculation method was verified by comparing it with other theoretical calculation results.

1. Introduction

Rock masses are formed through geological evolution and cutting fracture surfaces. They are complex geological bodies, and their deformation and stability play an important role in the field of rock engineering. The deformation of fractured rock mass is composed of rock deformation and fracture deformation. As far as the rock mass in practical engineering is concerned, fracture deformation is the predominant composition of the deformation of a fractured rock mass. The deformation of a fractured rock mass is commonly obtained by evaluating the equivalent deformation modulus. At present, a large number of studies have investigated the equivalent deformation modulus of a fractured rock mass by using experimental [1,2,3,4], empirical [5,6,7,8,9,10], numerical [11,12,13], and analytical methods [14,15,16,17,18,19].

The experimental method can be utilized to predict the real deformation behavior within some field sites, whereas the cost of the field test is high, and the period is long. The empirical method is frequently utilized in practical engineering design; it is used to predict the deformation modulus according to rock mass classification. For example, Cai et al. (2007) [6] derived the deformation modulus of the jointed rock mass based on GSI values. The deformation modulus that was predicted from the GSI system is in good agreement with field test data. Sitharam et al. (2001) [7] obtained the empirical relations for the deformation modulus of rock masses based on the statistical analysis of a large amount of experimental data, which were used for representing the jointed rock mass as an equivalent continuum. The numerical method, as represented by the discrete element method by Potyondy et al. (2004) [8] and the discontinuous deformation analysis method proposed by Shi et al. (1984) [9], can calculate the deformation of fractured rock masses by establishing a numerical model considering fracture surfaces. For example, Wang (2009) [15] proposed a mathematical model for rock mass with multi-sets of ubiquitous joints accounting for the anisotropy in deformation induced by the existence of joints. Yang et al. (2016) [17] and Han et al. (2011) [18] estimated the deformation moduli of nonpersistent fractured rock masses based on the normal and shear stiffnesses of joints. The empirical methods are commonly used in engineering design, but when estimating the deformation modulus of rock masses using empirical methods, it is assumed that the rock mass is isotropic, which has certain limitations for surrounding rock masses with significant anisotropy. Meanwhile, multiple fracture surfaces are randomly crossed in a rock mass, and the computational efficiency of the numerical method is relatively slow, which is not applicable in practical engineering.

The analytical method has the advantages of direct relationships and clear expression. For rock masses with relatively developed fracture surfaces, obtaining the deformation modulus of engineering rock masses using the analytical method shows its superiority. As early as the 1980s, Fossum et al. (1985) [14] studied the analytical formula for the deformation modulus of rock masses under arbitrary through-fracture surfaces using the averaging method. However, until now, the analytical formula for the deformation modulus still requires mechanical parameters, such as normal stiffness and tangential stiffness of cracks, and sometimes, even geometric parameters, such as average spacing and the density of fracture surfaces. However, currently, obtaining these two parameters currently poses significant difficulties in both theory and practice, and there are still significant limitations in the application of analytical methods for the deformation of rock masses. In order to overcome the limitations, the area of fracture hydraulics is introduced into the analysis of the deformation of the fractured rock mass, and a novel analytical solution for the deformation of fractured rock masses with single fractures and multiple fractures is derived. The stress-strain constitutive relation is proposed for rock masses containing multiple fractures. The analytical solution is verified by comparing it with the experiment result and other existing analytical methods. This is a heuristic method for evaluating the deformation of a fractured rock mass by regarding the rock mass as sponge-like material and adopting the fracture hydraulics theory.

It should be pointed out that the deformation calculation of a naturally fractured rock mass containing saturated or unsaturated fracture water that considers fluid-soild coupling is not within the scope of application regarding the method herein. It is assumed that the fractured rock mass is full of water, and the purpose of the assumption is to adopt fracture hydraulics to calculate the fracture flow. This fracture flow is a virtual quantity, which is assumed to be equivalent to the deformation of a rock mass, and it is not concerned with the water content of the fracture itself.

2. Fracture Hydraulics

As the permeability of the fractures in the rock mass is far greater than the intact rock, the seepage of the rock mass is dependent on the fracture hydraulic in the rock mass [20,21]. Experiments have already been conducted for a single parallel fracture in the rock mass [17,18]. It has been observed that if the fracture width is relatively narrow, the water flow through the fracture is typically in a laminar state. The flow velocity, V, within the fracture can be described by the following equation:

$$V=-\frac{g{e}^2}{12\gamma }J$$

(1)

where g is the gravitational acceleration, e is the fracture width, γ is the coefficient of viscosity of the water, and J is the hydraulic gradient.

Essentially, the structural surfaces within the rock mass are rough and uneven, and it is often clogged with filler substances. Due to this, Louis (1974) [22] amended Equation (1) as

$$V=-\frac{\beta g{e}^2}{12\gamma c}J$$

(2)

where β is the connected parameter, and it is equal to the ratio of the connected area in accordance with the total area within the fracture, and c is the modified parameter of the relative roughness for the fracture surface; it is calculated as

$$c=1+8.8{(\frac{\Delta }{e})}^{1.5}$$

(3)

where Δ is the convex degree of the fracture surface.

According to Darcy’s law, the flow velocity, V, can be calculated as

$$V=-k_{f}J$$

(4)

where k_f is the permeability parameter of the fracture. According to Equation (2), k_f in Equation (4) can be written as

$$k_f=\frac{\beta g{e}^2}{12\gamma c}$$

(5)

According to fracture hydraulics, the flow rate in a fracture within a unit length, q, is

$$q=Ve$$

(6)

From Equation (6), the relation between the flow rate, q, and the fracture width, e, can be obtained.

Submitting Equation (2) into Equation (6), one obtains

$$q=-\frac{\beta gJ}{12\gamma c}{e}^3$$

(7)

It is observed from Equation (7) that the flow rate in a fracture within the unit length q is directly proportionally to the cube of the fracture width, e.

3. Analytical Solution of Fracture Deformation with a Single-Fracture Rock Mass Model

Under the effect of a load, the fracture deformation within a rock mass is much greater than the rock deformation itself. Thus, the rock mass deformation is primarily composed of fracture deformation. It is essential to solve the fracture deformation, whereas, due to the complexity of fracture distribution, it is remarkably difficult to obtain such fracture deformation directly. Herein, the rock mass is assumed to be a sponge-like material. In a rock mass, water is stored in internal fractures; similarly, in a sponge, water is stored in internal cavities. Generally, when all the holes in a sponge are considered to be in the saturated state, the change in the volume of a sponge is approximately equal to the volume that is discharged if the sponge is compressed. Similarly, the volumetric deformation of the water-saturated fractured rock mass with the external load is regarded as equal to the water flow through the fractures. On this basis, the analysis attempts to obtain the flow rate in a fracture as an indirect method for calculating the fracture deformation. Figure 1 plots the single-fracture rock mass model for calculating fracture deformation.

In this model, the fracture length is l₁, the width is e, and the horizontal angle is α. The vertical and horizontal pressures are P₁ and P₂, respectively. It is assumed that the fracture is completely saturated with water and that the water within the fracture is incompressible. According to the deformation compatibility conditions, the amount of fracture deformation is equivalent to the net water flow through the fracture. The rock in the rock mass is assumed to be non-deformable. The fracture deformation is mainly induced by the fracture closure. Then, the fracture deformation can be characterized by the flow rate in a fracture, Q.

As the water inside the fracture is hypothetical, the water pressure is initially 0. In order to calculate the flow rate in a fracture, Q, the water pressure within the fracture can be determined by a stress analysis. When reaching the equilibrium state, the water pressure, P, at the bottom of the fracture surface of rock component, A, is

$$P=P_1{\cos }^2\alpha +P_2{\sin }^2\alpha$$

(8)

When filler substances are present within the fracture, the hydrostatic pressure of the water within the fracture, P₀, can be described as

$$P_0=nP$$

(9)

where n is the porosity of the filler substance, and n = 1 represents no filler substance.

Under the effect of such water pressure, if the fracture flow is calculated by Equation (7), the water pressure within the fracture should be initially converted into the hydraulic gradient. The water within the fracture is assumed to flow from the middle location to both ends of the fracture simultaneously. The water head difference between the middle location and one of the ends of the fracture, ∆P, is

$$\Delta P=\frac{P_0}{J_0}$$

(10)

where J₀ is the head pressure per unit, which denotes the head pressure within 1 m, and it is 0.01 MPa. The corresponding hydraulic gradient is the ratio of the water head difference to the distance from the middle location to one of the ends, i.e.,

$$J=\frac{2\Delta P}{l_1}$$

(11)

The relation between the flow rate in a fracture, Q, and the flow rate in a fracture within the unit length q is

$$Q=q{l}_1$$

(12)

By combining Equations (7)–(12), Q is obtained as

$$Q=-\frac{\beta gn\left[P_1{\left(\cos \alpha \right)}^2+P_1{\left(\sin \alpha \right)}^2\right]}{6\gamma c{J}_0}{e}^3$$

(13)

It should be noted that the flow rate of the fracture Q is not concerned with seepage calculation or time. Thus, Q denotes the volume of the water flowing from the fracture per unit of time.

For simplicity, it is assumed that the fracture width is constant. Then, fracture deformation, Δe, can be regarded as the variation in the fracture width, i.e.,

$$\Delta e=\frac{Q}{l_1}$$

(14)

Substituting Equation (14) into Equation (13), one obtains

$$\Delta e=-\frac{\beta gn\left[P_1{\left(\cos \alpha \right)}^2+P_1{\left(\sin \alpha \right)}^2\right]}{6\gamma c{J}_0{l}_1}{e}^3$$

(15)

According to similar methods, the fracture deformation used in the single-fracture rock mass model for the other two conditions, i.e., pure shear and compression-shear (see Figure 2), is deduced as

$$\Delta e_{\mathrm{shear}}=-\frac{\beta gn\tau \sin 2\alpha }{3\gamma c{J}_0{l}_1}{e}^3$$

(16)

$$\Delta e_{\mathrm{c}-\mathrm{s}}=-\frac{\beta gn\left[P_1{\left(\cos \alpha \right)}^2+P_1{\left(\sin \alpha \right)}^2+2\tau \sin 2\alpha \right]}{6\gamma c{J}_0{l}_1}{e}^3$$

(17)

where τ is the effective shear stress on the rock mass.

Among the above equations, a negative value of the flow rate denotes that the water of the fracture is discharged; in contrast, a positive value denotes that the water is inflowing. A negative value for the fracture width represents that the fracture is closed, and a positive value means the fracture is open.

For the proposed analytical solution, since the water flow within the fracture is merely a medium for calculating rock mass deformation, the impact of the stress on the permeable behavior of the fracture is ignored. The permeability coefficient, k_f, is utilized to show the hydraulic behavior of the fracture (see Equation (4)). The permeability coefficient, k_f, can be solved by using the connected parameter, β, and the modified parameter of the relative roughness, c; these reveal the geometrical and geo-mechanical properties of the fracture surface (see Equation (5)). In comparison, for the traditional solution of fracture deformation, the mechanical behavior of the fracture is denoted by using the stiffness of the fracture surface, i.e., the normal stiffness, k_n, and the tangential stiffness, k_s, which includes more uncertainty. The stiffnesses are more simplified and rough for describing the fracture surface. Therefore, the advantage of the proposed analytical solution is that the high uncertainty of stiffnesses of the fracture surface can be avoided.

4. Analytical Solution of the Deformation of a Fractured Rock Mass Using a Multiple-Fracture Rock Mass Model

In Section 3, fracture deformation in terms of one single fracture is investigated. The deformation of the rock mass is composed of rock deformation and fracture deformation. Meanwhile, most rock masses contain multiple fractures. Due to this, in this section, the whole deformation of a rock mass, including rock deformation and fracture deformation, for a multiple-fracture rock mass model is discussed.

4.1. Constitutive Model

It is postulated that the size of the representative zone of the rock mass is L_x × L_y. L_x and L_y are the width and height of the rock mass, respectively. The rock mass deformation of the representative zone is solved by superimposing elastic rock deformation and fracture deformation. The rock is assumed to be an isotropic, elastic material. Then, a generalized version of Hooke’s law is utilized to solve the elastic rock strain:

$${\epsilon }_{ij}=\frac{1+\mu }{E_0}{\sigma }_{ij}-\frac{\mu }{E_0}{\sigma }_{kk}{\delta }_{ij}$$

(18)

where E₀ is the elastic modulus of the rock, and μ is the Poisson’s ratio for the rock, respectively. ε_ij, σ_ij, and δ_ij are the strain tensor, stress tensor, and identity tensor, respectively, in terms of the components in an orthonormal frame, and the subscripts i and j mean the free index. σ_kk is the trace of the stress tensor, and the two k subscripts represent a dummy index.

As for the fracture strain induced by fracture deformation, initially, fracture deformation, Δe, is decomposed into x–y co-ordinates. It is postulated that the representative rock mass zone contains n fractures. The normal direction of the rth fracture is n^r, and the corresponding fracture deformation is Δe^r; then, the equivalent deformation of any fracture, Δu_i, can be expressed as

$$\Delta u_i={\displaystyle {\sum }_{r=1}^n\Delta e^r{n}_i^r}$$

(19)

where n^r_i is the cosine of the direction of the rth fracture.

Because the definition of strain is the ratio of the amount of deformation to the original length, the equivalent fracture strain that is produced by fracture deformation, ε^c_ij, is

$${\epsilon }_{xx}^c=\Delta u_x/L_x$$

(20)

$${\epsilon }_{yy}^c=\Delta u_y/L_y$$

(21)

$${\epsilon }_{xy}^c={\epsilon }_{yx}^c=\Delta u_x/L_y$$

(22)

where L_x and L_y are the width and height of the rock mass, respectively. ε_xx^c and ε_yy^c are the normal strain in the direction of x and y, respectively. ε_xy^c and ε_yx^c represent the shear strain caused by fracture deformation.

The function of the rock mass strain is derived by superposing the elastic rock strain (see Equation (18)) and the equivalent fracture strain (see Equation (20)~(22)), i.e.,

$${\epsilon }_{ij}=\frac{1+\mu }{E_0}{\sigma }_{ij}-\frac{\mu }{E_0}{\sigma }_{kk}{\delta }_{ij}+{\epsilon }_{ij}^c$$

(23)

where ε_ij^c is the strain tensor caused by fracture deformation in terms of the components in an orthonormal frame.

Equation (23) is also the constitutive relation for the representative zone of the rock mass.

When using Equation (19), the variation in the fracture width, Δe, should be less than the initial fracture width, e₀. If Δe is greater than e₀, the rock on both sides of the fracture has experienced mutual penetration. This is not consistent with the assumption in this analysis. Thus, instead, it is taken that Δe = e₀ once Δe exceeds e₀ in the calculation. Furthermore, based on the relationship between fracture deformation and rock mass deformation, the positive and negative values of the strain in the above equations represent that the rock mass is tensed and compressed, respectively.

4.2. Verification

Nolte (1989) et al. [23] carried out hydraulic experiments on several fractured rock quartz monzonite samples that were recovered from a Swedish nuclear waste test site. Based on the three fractured rock samples (E30, E32, and E35), the compression of the fracture width with different normal stresses, as well as the fracture flow with different fracture widths, were tested. The diameter and the height of the fractured rock samples were 52 mm and 77 mm, respectively. Each fractured rock sample contained a single-fracture surface, the direction of which was vertical to the sample axis.

The normal stiffness of the fracture, k_n, and the maximum fracture compression, e_max, are directly measured by the fracture condition by Nolte (1989) et al. [23]. Nolte (1989) et al. [23] defined e_max as the average fracture width when three contact points exist within the two fracture surfaces under no normal load. In this analysis, e_max is regarded as equal to the initial fracture width, e₀. The measured results are presented in

Table 1. Hydraulic experiment parameters for several fractured rock quartz monzonite samples.

Fracture Rock Samples	Normal Stiffness of Fracture: kn (MPa/m)	Maximum Fracture Compression: emax (μm)
E30	7 × 106	12.5
E32	15 × 106	6.6
E35	3 × 106	46.0

.

Figure 3 plots the maximum fracture compression for the three fracture rock samples with different normal stresses. It is observed that the compression increases nearly linearly with the increase in normal stress. With different fracture stiffnesses, the fracture compression changes greatly.

The modified parameter for relative roughness, c, and convex degree, Δ, for the samples E30, E32, and E35 can be solved by fitting the curves in Figure 3. Specifically, the linear relation between fracture compression and normal stress are firstly derived through the linear fitting method based on the data in Figure 2. By substituting the linear relations into Equation (12), the modified parameter of relative roughness, c, can be fitted (see

Table 2. Fitted parameters based on the experiment results.

Fracture Rock Samples	Modified Parameter of the Relative Roughness c	Convex Degree Δ (μm)
E30	73.44	50.96
E32	18.70	10.52
E35	989.3	1070.8

). By substituting c into Equation (3), convex degree, Δ, is obtained (see Table 2). Table 2 displays the fitted values of c and Δ based on the experimental results. It is found that Δ in Table 2 is consistent with e_max in Table 1. This is in agreement with reality because if the convex of the fracture surface increases, the initial width of the fracture becomes large. This verifies the analytical method for solving Δ and c to some extent.

Based on the parameters in Table 1 and Table 2, three typical fractured rock mass models are presented in Figure 4. The direction of the fracture surface is assumed to be vertical to that of the load. The rock mass deformation obtained by using the traditional, equivalent deformation modulus method for a fractured rock mass by Goodman (1981) [16] is compared with the rock mass deformation of the proposed solution. In Goodman’s study [16], the equivalent deformation modulus can be obtained by using the following equation:

$$E=\frac{1}{\frac{1}{E_0}+\frac{1}{k_n\cdot s}}$$

(24)

where E is the equivalent deformation modulus of the rock mass, k_n is the normal stiffness of the fracture surface, and s is the length of the fracture. The rock mass deformation in the three models can be obtained by dividing the normal stress by the deformation modulus in Equation (24).

In Figure 4, the size of the rock mass model is 55 mm × 77 mm. The normal stresses corresponding to the three fractured rock mass samples are 20 MPa, 40 MPa, and 80 MPa, respectively. The rocks inside the fractured rock mass are assumed to be the elastic material. The elastic modulus is postulated as being 60 GPa. The coefficient of continuity of the fracture is β = 1, the porosity is n = 1, and the coefficient of viscosity of water at 20 °C is γ = 1.31 × 10⁻⁶ m²/s.

According to the proposed solution and Goodman’s solution, the rock mass deformations for the three models are presented in

Table 3. Rock mass deformation and fracture deformation according to the proposed solution and Goodman’s solution.

Normal Stress (MPa)	Different Solutions	Rock Mass Deformation and Fracture Deformation (10−6 m)
		One Single Fractures (a)	Two Fractures (b)	Three Fractures (c)
20	Proposed solution	27.23 (1.56)	28.14 (2.47)	33.92 (8.25)
	Goodman’s solution	27.71 (2.04)	28.66 (2.99)	33.42 (7.76)
40	Proposed solution	54.46 (3.13)	56.27 (4.94)	67.84 (16.51)
	Goodman’s solution	55.42 (4.08)	57.32 (5.99)	66.84 (15.51)
80	Proposed solution	108.92 (6.26)	112.54 (9.87)	135.68 (33.02)
	Goodman’s solution	110.83 (8.16)	114.64 (11.97)	133.69 (31.92)

. Fracture deformation is listed in the parenthesis for each case. Table 3 shows that rock mass deformation and fracture deformation are basically equivalent in the two solutions. This verifies that the proposed solution is reasonable. The c and Δ in the proposed solution are predicted by the fitting method, as mentioned before. This is the main factor that leads to the discrepancy between the proposed solution and Goodman’s solution.

The verification of the proposed solution for rock mass deformation mainly focuses on a parallel set of fractures, and the direction of the fractures is normal to the loading stress. A rock mass model for the other fracture distributions will be constructed in the next step in the near future. Meanwhile, the results in Table 3 also show that there is a certain correlation between the hydraulic parameters of the fractures and the mechanical parameters, which is consistent with the conclusions of Esaki et al. (1992) [24], Detournay [25] (1980), and Derek (2010) [26].

5. Conclusions

(1)

The relationship between rock deformation and the fracture flow rate was studied based on a single-fracture rock mass model. The fracture deformation calculation formula was derived from the fracture hydraulic calculation formula, and the rationality of the calculation formula was verified by comparing it with measured data.

(2)

Based on the calculation formula for fracture deformation, a calculation formula for rock mass deformation containing multiple sets of fracture surfaces was derived, and the stress-strain constitutive relationship of complex rock masses was established. The correctness of the calculation method in this paper was verified by comparing it with other theoretical calculation results.

It should be pointed out that the deformation calculation of a naturally fractured rock mass containing saturated or unsaturated fracture water that considers fluid-soild coupling is not within the scope of application of the calculation method herein. It is assumed that the fractured rock mass is full of water, and the purpose of this assumption is to adopt fracture hydraulics to calculate the fracture flow. This fracture flow is a virtual quantity, which is assumed to be equivalent to the deformation of rock mass and has nothing to do with the water content of the fracture itself. Similarly, for a complex rock mass model with randomly distributed fractures, according to the proposed method, the fracture flow of each fracture is solely calculated, and rock mass deformation is obtained by superposing the deformations of multiple fractures over fracture flow. This implies that the proposed method is not affected by the seepage path of the fracture water and the permeable behavior of the fracture.

The verification cases mainly focus on a parallel set of fractures, and the joint direction is perpendicular to the direction of the loading force. Other fracture distribution modes are not discussed in this work due to the lack of the corresponding experiment results. Related experimental work and analytical solutions for non-uniform fracture distribution will be carried out in the near future.