Rock Mech Rock Eng DOI 10.1007/s00603-014-0628-3

ORIGINAL PAPER

Water Pressure Effects on Strength and Deformability of Fractured Rocks Under Low Confining Pressures Majid Noorian Bidgoli • Lanru Jing

Received: 26 February 2014 / Accepted: 15 June 2014 Ó Springer-Verlag Wien 2014

Abstract The effect of groundwater on strength and deformation behavior of fractured crystalline rocks is one of the important issues for design, performance and safety assessments of surface and subsurface rock engineering problems. However, practical difficulties make the direct in situ and laboratory measurements of these properties of fractured rocks impossible at present, since effects of complex fracture system hidden inside the rock masses cannot be accurately estimated. Therefore, numerical modeling needs to be applied. The overall objective of this paper is to deepen our understanding on the validity of the effective stress concept, and to evaluate the effects of water pressure on strength and deformation parameters. The approach adopted uses discrete element methods to simulate the coupled stressdeformation-flow processes in a fractured rock mass with model dimensions at a representative elementary volume (REV) size and realistic representation of fracture system geometry. The obtained numerical results demonstrate that water pressure has significant influence on the strength, but with minor effects on elastic deformation parameters, compared with significant influence by the lateral confining pressure. Also, the classical effective stress concept to fractured rock can be quite different with that applied in soil mechanics. Therefore, one should be cautious when applying the classical effective stress concept to fractured rock media. Keywords Coupled hydro-mechanical  Effective stress  Discrete element methods (DEM-DFN)  UDEC  Failure criteria  Fractured crystalline rocks

M. Noorian Bidgoli (&)  L. Jing Department of Land and Water Resources Engineering, Engineering Geology and Geophysics Research Group, Royal Institute of Technology (KTH), Stockholm, Sweden e-mail: [email protected]

1 Introduction Nowadays, most slopes, foundations, and underground excavations are constructed in fractured rocks. A crystalline rock mass is a fractured medium, composed of intact rock matrix and rock fractures, containing fluids such as water, oil, natural gases and air, under complex in situ conditions of stresses, temperature and fluid pressures (Jing 2003). In crystalline rock masses, the fracture networks usually serve as the main flow channels or conduits for the movement of water through rock masses, especially below groundwater level at greater depths in the subsurface of the Earth’s crust. To enhance safety and performance of rock engineering projects, it is necessary to evaluate the strength and deformation parameters of fractured rocks while water pressure effects are taken into account through coupled stress-flow analyses. The subject is especially important for rock engineering projects of importance for energy resources and environment protection issues, such as underground nuclear waste repositories, gas/oil/water storage caverns and geothermal reservoirs. In the early 1960s, the coupling between hydraulic and mechanic processes in fractured rocks started to receive wide attention (Rutqvist and Stephansson 2003). Experimental and numerical studies on rocks without fractures (Gutierrez et al. 2000; Odedra et al. 2001; Tang et al. 2004; Wang 2006; Talesnick and Shehadeh 2007; Ba¨ckstro¨m et al. 2008; Wang et al. 2013) show that generally the presence of water reduced the strength of the rocks and affected the deformation behavior of the rocks. Many attempts have been made for fundamental studies on strength and deformation behavior of fractured rocks without considering the effect of water (Ramamurthy and Arora 1994; Hoek 1983; Liu et al. 2009; Chong et al.

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2013). Some attempts considered the effect of water (Goodman and Ohnishi 1973; Noorishad et al. 1982; Barton et al. 1985; Oda 1986; Bruno and Nakagawa 1991; Vlastos et al. 2006; Yuan and Harrison 2006; Gercek 2007; Zhang et al. 2007), but most of their models contained simple or regular fractured systems. In reality, more realistic representations of complex fracture system geometry of fractured rocks are needed in such studies. When working on problems of porous soils or fractured rocks where water pressure is involved, the concept of effective stress is used. The effective stress is a function of the total or applied stress and the pressure of the fluid in the pores of the material, known as the pore pressure or porewater pressure (Brady and Brown 2004). Terzaghi (1923) defined the effective stress concept to describe the deformation behavior of water saturated soil as the total stress minus the pore-water pressure, based on results of experiments on the strength and deformation of soils. The concept has been widely accepted for soil mechanics, but its validity for fractured rock masses has been questioned by many. Some research have been performed to investigate validity and limitation of effective stress concept in soils and rocks (Brace and Martin 1968; Nur and Byerlee 1971; Robin 1973; Carroll 1979; Walsh 1981; Bernabe 1986; Boitnott and Scholz 1990; Bluhm and Boer 1996; Oka 1996), with conclusions that the Terzaghi effective stress concept may not hold true universally, especially for fractured rock masses. However, the issue of validity of this concept for fractured crystalline rocks has not been adequately discussed in literature, and has almost never been thoroughly tested and verified in laboratory or field experiments. Therefore, further work is needed to establish a more systematic methodology for studying the effects of water pressure on strength and deformability of fractured rocks, which is the main aim of this research and which has not been adequately investigated in the past. The research presents a continued development from the previous research, as reported in (Noorian Bidgoli et al. 2013; Noorian Bidgoli and Jing 2014), as a systematic and comprehensive study about strength and deformability of fractured rocks. Intact rock samples of small sizes cannot contain complex fracture systems with many fractures of varying sizes, orientations and locations. Therefore, the ordinary rock mechanics laboratory tests cannot be used to obtain realistic results for studying problems involving water pressure and stress coupling problems of fractured rocks. On the other hand, large-scale in situ tests of fractured rock masses are usually very difficult to be conducted with ensured control of the initial and boundary (loading) conditions, besides challenges of time requirement and high costs, so that it becomes realistically impractical at present.

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Therefore, numerical methods need to be chosen as the available tool of investigation, due to their flexibility for representing the different hydro-mechanical and geometric features of fractures and the intact rock matrices. Among the current numerical methods, discrete element methods (DEM) are a suitable numerical tool for coupled stress-flow analysis of fractured rocks, with their advantages of explicit representations of fractures and their constitutive behaviors with the discrete fracture network (DFN) method, as a branch of DEM methods, for creating the realization of complex fracture system networks. In this research, the universal distinct element code (UDEC) (Itasca UDEC 2004) was used to perform numerical stressflow analyses of fractured rock models containing a large number of fractures of varying sizes and orientations, using realistic fracture system information from field mapping, with the assumption that intact rock matrix is impermeable and water flows through connected fractures only. The numerical results obtained were used for estimating the elastic modulus, Poisson’s ratio of fractured rocks and strength parameters of two of the most commonly accepted failure criteria, namely the Mohr-Coulomb (M-C) and Hoek-Brown (H-B) failure criteria, based on the effective principal stresses.

2 Numerical Simulation Methodology Clearly, the coupled hydro-mechanical behavior of a fractured rock is a very complex phenomenon and a very important subject in rock mechanics to improve our understanding of strength and deformability of fractured rocks with or without considering water pressure effects. Therefore, numerical modeling is required to simulate this complex interaction that happens in reality under a variety of boundary and initial conditions in many rock engineering problems, mainly because of the difficulty or impossibility of using the other available methods. In this study, systematic investigations were conducted using numerical experiments of combined compression tests under mechanical and hydraulic loading conditions, to generically estimate the water pressure effects on compressive strength and deformation behavior of fractured crystalline rocks. Figure 1 shows a flowchart of the conceptual approach for the numerical experiments that have been conducted in this research. 2.1 Numerical Experiment Setup The computer code used for this research is the universal distinct element code (UDEC), a powerful numerical tool based on the DEM method that has the capability to simulate water flow through the network fractures systems of

Water Pressure Effects on Strength and Deformability

Discrete Fracture Network Model Stress-Deformation analysis Numerical

Regularization of DFN Model (By UDEC)

Modeling

A few cycles to solution

Experiments

Stress-Flow analysis

Discrete Element Model

Examination of the model responses

(DFN)

(DEM) Fig. 1 Flowchart for numerical, coupled stress-flow processes in a fractured rock

fractured rocks. A few numerical experiments have been performed on the DFN model containing complex and irregular fracture network systems, subjected to varying mechanical and hydraulic loading conditions. During loading on a numerical model of the fractured rock concerned, both rock matrix and fractures were deformed or displaced, governed by the equations of motion of the rock blocks and constitutive models and material parameters for rock matrix and fractures, and the initial and boundary conditions. 2.1.1 General Assumptions The numerical modeling work presented in this research was carried out subjected to the following assumptions about model dimensions, rock matrix, fractures and water. 1. 2.

3. 4.

5. 6. 7.

The numerical model was defined in a two-dimensional (2D) space for a generic study. Simulations were performed under quasi-static plain strain loading conditions for stress-flow analysis, without considering effects of gravity. Rock matrix was a linear, isotropic, homogeneous, elastic, and impermeable material. The fractures follow an ideal elasto-plastic behavior of a Mohr–Coulomb model in the shear direction and a hyperbolic stress-displacement behavior (Bandis et al. 1985) in the normal direction. The initial aperture of fractures (without stress) was a constant. Water flow was allowed only in connected fractures and water was incompressible. The Cubic Law for fluid flow in smooth parallel plates was assumed for water flow in fractures.

These assumptions are necessary for a generic study, since our aim is to establish a numerical platform for predicting effects of coupled hydro-mechanical processes

on the strength and deformability of fractured rocks, not application for site-specific case studies. It should be noted that this research has been conducted for a granite rock matrix with low porosity, permeability, and water flow velocity. Therefore, the effects of water pressure on deformation and motion of rock matrix was mainly caused by water in fractures that was properly handled in UDEC models. Also, dead-ends and isolated and dangling fractures do not contribute to water flow in fracture systems, since they have no connection (isolated) or not enough connections (dead-ends) to the water conducting fracture systems, and therefore have no effects on water pressure. 2.1.2 DFN-DEM Model Size and Geometry As one can see in Fig. 1, the procedure of numerical simulations starts with generating a DFN model, containing a number of fractures of varying lengths and orientations, to represent the fractured rock masses as complex fracture networks. In this research, the square-shaped DFN model was extracted from the center of an original parent model of the fracture system, based on the same fracture system model data as was used in Min and Jing (2003) and (Noorian Bidgoli et al. 2013), with the geometric parameters of fracture systems based on the field mapping results of a site characterization at the Sellafield area, undertaken by the United Kingdom (Nirex 1997). The size of the square-shaped DFN model was chosen as 5 9 5 m, because of the fact that it was equal to the representative elementary volume (REV) size as defined in Min and Jing (2003), which is the minimum model size beyond which the elastic mechanical properties of the models remain basically constant. The resultant DFN model was used to generate a DEM model with internal discretization of finite difference elements, for coupled stress-flow simulations. Before performing the numerical simulations, the DFN models were

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Fig. 2 DEM model. a DEM model before regularization and position of monitoring points in the model; b DEM model after regularization and adding artificial fractures on two horizontal sides of the model Table 1 Material properties of intact rock and fractures Mechanical properties

Value

Intact rock Density

2,500 kg/m3

Young’s modulus (E)

84.6 GPa

Poisson’s ratio (m)

0.24

Uniaxial compressive strength (UCS)

157 MPa

Initial normal stiffness (Kn)

434 GPa/m

Fracture Shear stiffness (Ks)

434 GPa/m

Friction angle (u)

24.98

Dilation angle (ud)

58

Cohesion (C) Aperture for zero normal stress (maximum)

0 MPa 65 lm

Residual aperture at high stress (minimum)

1 lm

Shear displacement for zero dilation

3 mm

the monitoring points for stress, displacement, velocity, and water pressure in this study. Also, as shown in Fig. 2b, for prevention of large deformation and failing of corner blocks, and for having uniform water flow through fractured rock, a number of parallel artificial fractures with zero friction angle and cohesion were added on two horizontal sides of the DEM model. 2.1.3 Rock and Fracture Mechanical Properties The material parameters of the intact rock (granite) and fractures are shown in Table 1. This information was based on the laboratory test results reported in the Sellafield site investigation results (Min and Jing 2003; Noorian Bidgoli et al. 2013).

3 Coupled Stress-Flow Numerical Analysis regularized by deleting the isolated fractures and deadends, so that the resultant fractures were all connected and fractures that were isolated or were dead-ends were removed from the model during the meshing and the generation of the blocks, since they did not contribute to water flow. Figure 2 shows the DEM geometry model before (Fig. 2a) and after (Fig. 2b) the fracture system regulation. As shown in Fig. 2a, a grid of 36 monitoring points was defined at intersections of the six parallel horizontal and six vertical lines, with the same spacing. These points plus one point at the center of the DEM model were

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Two loading tasks were performed for this study, concerning purely mechanical process and coupled mechanical and water flow processes, respectively (Fig. 3). The first task was to simulate the stress–strain behavior of the DEM model during pure mechanical loading conditions without water flow. The objective was to generate the results for evaluating strength and deformability parameters without water pressure effects, so that comparison with results with the water pressure effects, as obtained from the second task, could be performed (Fig. 3).

Water Pressure Effects on Strength and Deformability Fig. 3 Flowchart for a numerical stress-deformationflow analysis processes in a DEM model

Running more time-step

Stress-Deformation analysis

Checking: 1. Horizontal and vertical velocity 2. Unbalanced forces versus time

Reaching to steady state flow

Reaching to equilibrium state

Checking: Equality of Inflow and outflow of boundaries of the DEM model

Water flow analysis

Running more time-step

The second task (Fig. 3) simulates the coupled water flow and stress processes with additional water pressures applied at the two vertical lateral boundaries, with the top and bottom boundaries sealed hydraulically. The hydraulic loading remained constant during each mechanical loading stage (by changing confining pressure), until the DEM model reaches steady state flow and mechanical equilibrium. The state of water flow was checked by checking the in-flow and out-flow rates at the ends of the model until a steady state flow was obtained.

3.1 Mechanical Boundary (Loading) Conditions As shown in Fig. 3, at the first stage, the stress-deformation analysis was performed step-by-step following the modeling procedure displayed in Fig. 3, by applying boundary conditionsofuniaxialandbiaxialcompressiontests,inawaysimilar to testing small intact rock samples in laboratory tests, on the

DEM model to generate the stress and strain data from the deformed DEM model. To minimize the influence of inertial effects on the response of the model, the uniaxial and biaxial loadings were performed with servo-control loading procedures by using a new FISH program in UDEC for selecting a propercyclicloadingrateinareasonablerangeofmaximumand minimum unbalanced forces, to prevent sudden failure of the DEM models. Figure 4a shows the mechanical loading conditions. A constant and very small axial load increment (Dry), equal to 0.05 MPa, was applied on the top of the DEM model in the vertical direction, while the bottom of the DEM model was fixed in the y-direction for simulating the biaxial compression test. Then, varying confining pressure (rx), equal to 0.5, 1, 1.5, 2, 2.5, and 3 MPa, respectively, was applied laterally on the two vertical boundary surfaces of the model, as in the laboratory biaxial compression tests. For uniaxial compression tests, the two vertical sides of the DEM model were kept as free surfaces.

σy + Δσy

Fig. 4 Typical boundary conditions for numerical experiments. a Mechanical boundary conditions, b hydraulic boundary conditions

Impermeable boundary

σx

σx

P2

Water flow through model

P1

Impermeable boundary

(a)

(b)

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The DEM model was loaded sequentially with a constant stress increment of 0.05 MPa, until a quasi-static equilibrium state of the model was reached. Equilibrium state of the DEM model was controlled by a velocity monitoring scheme during loading process. The velocities (in the both x and y directions) at a number of monitoring points (Fig. 2a) were checked to ensure that they become zero or very close to zero at the end of every loading step, and the resultant deformed DEM models at the end of loading cycling process generated stress–strain behavior of the fractured rock concerned, represented by the DEM model.

step at the end of each loading stage, following the abovementioned procedure, with different confining pressures at each stage. The normal stresses and strains in the x and y directions were calculated at the end of each loading stage, for all of the stages and from all monitoring points. The stress and strains were computed by taking the average values from the monitoring points by using the FISH function, the programming language embedded within UDEC. 4.1 Effect of Water Pressure on Deformation Behavior of the Fractured Rocks

3.2 Hydraulic Boundary Conditions At this stage, water flow through the deformed DEM models under mechanical loading was simulated under the specified hydraulic boundary conditions, representing a horizontal hydraulic gradient as illustrated in Fig. 4b. Water flow is governed by the Cubic Law with a pressure gradient between the two lateral boundaries of the DEM model, which is given by: Dp ¼ P2  P1

ð1Þ

Therefore, when a pressure difference (Dp) exists between two vertical sides of the DEM model, water flow will take place (Fig. 4b). This hydraulic condition allows water to flow horizontally, from the right side to the left side of the model, at all coupled-hydro-mechanical simulation stages with different confining pressures. In this research, a constant water pressure gradient equal to 0.001 MPa, with P1 = 1.0 MPa and P2 = 0.999 MPa, respectively, was used for evaluating the influence of water pressure on strength and deformability of fracture rocks. The choice of such a small hydraulic pressure gradient is the requirement for a basically uniform water pressure field of 1.0 MPa over the whole DEM model for strength and parameter evaluations. The values of confining pressure on the two lateral boundaries were lower than the water pressure during the two early loading steps of the numerical experiments, in order to check if the effective stresses in x-direction were still generally compressive. Also, this relatively low water pressure gradient is within the range of the water pressure gradient for in situ conditions in the shallow depth of the Earth’s crust, such conditions existing in low and intermediate radioactive waste repository considerations in a number of European countries.

4 Results of Water Pressure Effects on Strength and Deformability The effects of water pressure on the strength and deformation behavior of fractured rocks were evaluated step-by-

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Water pressure effects on stress–strain behavior of the DEM model were shown by plotting the numerical test results as the axial stress versus axial strain curves. Figure 5 shows a comparison for the axial stress–strain behaviors of the DEM model representing the fractured rock concerned, between pure mechanical stress–strain curves (solid lines) and stress–strain curves with combined stress and water pressure boundary conditions (dashed lines), with a horizontal hydraulic pressure (from right to left). These curves were used for evaluating equivalent strength and deformation parameters of the fractured rock concerned. As one can see in Fig. 5, in the absence of water (solid line curves), the DEM model deformed linearly and elastically before the axial stresses approached the elastic limit, but the slopes of the stress–strain curves within these ranges also changed slightly with different confining pressures. Further compression led to inelastic deformation up to the peak plastic strength. With increasing confining pressure, the axial stresses of the DEM models increased, and the stress–strain curves followed an elastic–plastic behavior with slight strain hardening. Also, in the present of water (dashed line curves), the DEM models had the same nonlinear stress–strain curves of elastic–plastic behavior, but with a significant reduction of axial stress and peak strength on the stress–strain curves, and the curves representing pure mechanical and coupled stress-flow behaviors were generally parallel in trend. It should be noted that a sharp increase of the axial stress-axial strain curve of the DEM model with a confining pressure of 1.0 MPa and the horizontal hydraulic pressure (dash blue line curve) occurred. The reason for this sharp change was local interlocking of DEM model blocks, as examined below. Figure 6 shows a comparison between major principal stress contours of the DEM model under 1.0 MPa confining pressure without water (dry condition, Fig. 6a, b) and with water (wet condition, Fig. 6c, d) at two specified points, representing the starting point (point 1) and finishing point (point 2) defining the sharp change interval, when

Water Pressure Effects on Strength and Deformability Fig. 5 Comparison of axial stress versus axial strain curves between mechanical analysis (solid lines) and hydromechanical analysis (dashed lines), with different confining pressure conditions and under horizontal hydraulic pressure (from right to left)

considering water pressure. It can be seen from Fig. 6 that while stress distributions were approximately the same at the two corresponding specified points of the curve without water (Fig. 6a, b), stress distribution is quite similar in both contour shape and magnitudes. However, between point 1 (Fig. 6c) and point 2 (Fig. 6d) for the wet condition, both shape and magnitude of the stress distribution changed with higher magnitude at point 2, with higher stress concentration around a few number of fracture intersections caused by interlocking of DEM model blocks, and caused by water pressure. This effect indicates that in reality, water pressure may cause local stress distribution patterns as the combined contributions of local fracture geometry and water pressure, and one may need to be concerned when evaluating large-scale in situ tests of stress and water flow behavior of fractured rocks.

condition order and Case 2 represents the hydraulicmechanical boundary condition order.. Figure 7 compares the obtained stress–strain curves using the Case 1 (solid line) and Case 2 (dashed line) loading condition orders with different confining pressures. Numerical results show insignificant differences between the obtained stress–strain curves. Therefore, applying hydraulic boundary conditions before or after applying axial compressive stress loading condition has no significant effect on the stress–strain behaviors of fractured rock. Choosing the order actually depends on the physical experiment requirements, such as sealing of water during the mechanical loading.

4.1.1 Effect of Hydraulic Conditions on Deformation Behaviors of Fractured Rock

Figure 8 shows the variation of the equivalent directional elastic modulus as a function of confining pressures and water pressure, in the axial loading direction (y-directional). The elastic modulus was calculated as the averaged slopes of the stress–strain curves of the DEM model during the stages of elastic deformation. As shown in Fig. 8, the elastic modulus decreased slightly when a water pressure was present, compared with that without water pressure. The discrepancy was moderate when confining pressure was smaller than 1.5 MPa, but insignificant afterwards. Slightly larger overall horizontal displacements or larger strain rates of the DEM model caused a slight decrease of the slopes of the stress–strain curves, confining pressures

As mentioned in Sect. 2, hydraulic boundary conditions were applied after applying axial compressive stress loading condition during the numerical experiments. In order to check the reliability of the numerical modeling developed and also to test changes in the sequence of applying mechanical and hydraulic boundary conditions, some numerical experiments were conducted with application of hydraulic conditions before applying axial compressive stress. We called these two choices Case 1 and Case 2. Case 1 represents the mechanical-hydraulic boundary

4.2 Effect of Water Pressure on Elastic Deformability Parameters

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(a)

(b)

(c)

(d)

Fig. 6 Comparison between major principal stress contours (Pa) of the DEM model under 1.0 MPa confining pressure. a, b With dry condition, and c, d with wet condition for two specified points; namely, before interlocking (point 1) and after interlocking (point 2)

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Water Pressure Effects on Strength and Deformability Fig. 7 The comparison between the axial stresses versus axial strain curves obtained during hydromechanical analysis of Case 1 (solid lines) and Case 2 (dashed lines), with different confining pressure conditions and under horizontal hydraulic pressure (from right to left)

Fig. 8 Estimated equivalent values of directional elastic modulus for DEM model without and with water pressure, under different confining pressures (the values of elastic modulus at zero confining pressure are 43 and 11 MPa for DEM model without and with water pressure, respectively, and cannot be more clearly presented in this figure due to their very small magnitudes)

were below 1.5 MPa, and water flow with a pressure of 1.0 MPa was involved. Generally, the magnitude of the directional elastic modulus for fractured rock is much less than the Young’s modulus of intact rock (84.6 MPa, Table 1). Figure 9 shows variation of the Poisson’s ratio as a function of confining pressure and effects of water pressure. The Poisson’s ratio was calculated as the ratio of the mean transverse strain to the mean axial strain at all monitoring points of the DEM models under the applied

Fig. 9 Estimated equivalent values of Poisson’s ratio for DEM model without and with water pressure under different confining pressures

axial loading with lateral mechanical confining pressure. The values of Poisson’s ratio decrease gradually with increase of confining pressure for both cases with and without water. The Poisson’s ratio increases slightly with water pressure. The reason for this difference can be the effect of parallel directions of the horizontal water flow direction and confining pressure. Generally, the magnitude of Poisson’s ratio for fractured rocks is much larger than that for intact rock. The obtained results of larger Poisson’s ratio indicated that water flow affected the deformability of the fractured rock in some confining pressure levels.

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M. Noorian Bidgoli, L. Jing 0.07

0.06 Numerical Results (Dry) M-C (Dry) Numerical Results (Wet) M-C (Wet)

0.05

0.04

0.03

0.02

0.00 0.000

0.005

0.010

0.015

0.02

0.020

0.00 0.000

0.005

0.010

0.015

0.020

Normalized Confining Pressure σ3 /σC

(a)

(b) The curve fitting results, as normalized strength versus normalized confining pressure, with M-C and H-B failure criteria were shown in Fig. 10. Table 2 shows the regressed strength parameters of both criteria and the correlation coefficients of the regression. Generally, based on correlation coefficient (R) obtained for two failure envelopes, the quality of fitting to the M-C and H-B strength envelopes were acceptable, with the M-C criterion showing a slightly better correlation. The results clearly show a general reduction of DEM models’ strength parameters of both M-C and H-B criteria, except parameter s for H-B criterion. The material constant parameters of Hoek-Brown failure criteria, namely m and s, seem to be more sensitive to the presence of water flow.

Mohr-Coulomb (M-C) and Hoek-Brown (H-B), as the two most popular failure criteria for rocks, were used for evaluating the equivalent strength of the DEM model, with and without considering water pressure effects. For this purpose, the major and minor effective principal stress pairs (r1, r3) obtained from the numerical experiments were used to fit both failure criteria. The Mohr-Coulomb (M-C) criterion was expressed by the following equation: 2c cos u 1 þ sin u þ r3 1  sin u 1  sin u

ð2Þ

where r1 is the major effective principal stress at failure or elastic strength, and r3 is the minor effective principal stress or confining pressure. The Hoek-Brown (H-B) criterion was expressed with the following equation (Hoek et al. 2002):  0:5 r3 r1 ¼ r3 þ rci m þs ð3Þ rci

5 Discussions on Effective Stress Effect In this study, we conducted numerical investigations of water pressure effects on strength and elastic deformability of fractured rocks, using an explicit DEM approach. The numerical results indicate significant differences of some strength parameters, but moderate effects on elastic deformation parameters of fractured rock models, when considering water pressure conditions. The main reason is

where rci is the uniaxial compressive strength of the intact rock, and m and s are the two parameters defining the H-B failure criterion.

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0.03

Normalized Confining Pressure σ3 /σC

4.3 Effect of Water Pressure on Strength of Fractured Rocks

Table 2 Equivalent material parameters of M-C and H-B failure criteria

0.04

0.01

0.01

r1 ¼

Numerical Results (Dry) H-B (Dry) Numerical Results (Wet) H-B (Wet)

0.05

Normalized Strength σ1 /σC

0.06

Normalized Strength σ1 /σC

Fig. 10 Strength curves for DEM models in the normalized effective stress space without (solid red line) and with (dashed blue line) water pressure condition. a M-C criterion, b HB criterion, respectively

Water pressure (Mpa)

Mohr-Coulomb

Hoek-Brown

Cohesion

Friction Angle

Correlation Coefficient

Parameters of Hoek-Brown

Correlation coefficient

C (Mpa)

u (°)

R

m

s

R

Dp = 0

0.1727

28.3

0.9950

0.0591

2.08E-06

0.9916

Dp = 0.001

0.0616

22.9

0.9853

0.0233

8.64E-06

0.9616

Water Pressure Effects on Strength and Deformability

-3.600E+06 -3.400E+06 -3.200E+06 -3.000E+06 -2.800E+06 -2.600E+06 -2.400E+06 -2.200E+06 -2.000E+06 -1.800E+06 -1.600E+06 -1.400E+06 -1.200E+06 -1.000E+06 -8.000E+05 -6.000E+05 -4.000E+05

-1.200E+06 -1.000E+06 -8.000E+05 -6.000E+05 -4.000E+05 -2.000E+05

(a) Confining pressure=0.5 MPa

-1.600E+07 -1.500E+07 -1.400E+07 -1.300E+07 -1.200E+07 -1.100E+07 -1.000E+07 -9.000E+06 -8.000E+06 -7.000E+06 -6.000E+06 -5.000E+06 -4.000E+06 -3.000E+06 -2.000E+06 -1.000E+06

-9.000E+06 -8.000E+06 -7.000E+06 -6.000E+06 -5.000E+06 -4.000E+06 -3.000E+06 -2.000E+06 -1.000E+06

(b) Confining pressure=1.5 MPa

-1.600E+07 -1.500E+07 -1.400E+07 -1.300E+07 -1.200E+07 -1.100E+07 -1.000E+07 -9.000E+06 -8.000E+06 -7.000E+06 -6.000E+06 -5.000E+06 -4.000E+06 -3.000E+06 -2.000E+06 -1.000E+06

-1.900E+07 -1.800E+07 -1.700E+07 -1.600E+07 -1.500E+07 -1.400E+07 -1.300E+07 -1.200E+07 -1.100E+07 -1.000E+07 -9.000E+06 -8.000E+06 -7.000E+06 -6.000E+06 -5.000E+06 -4.000E+06 -3.000E+06 -2.000E+06

(c) Confining pressure=2.5 MPa Fig. 11 Distribution of major principle stress contours (Pa) of the DEM model without (left column) and with (right column) water pressure condition under different confining pressures

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M. Noorian Bidgoli, L. Jing

Fig. 12 Normalized flow rates in each fracture intersecting the left vertical boundary of the DEM mode under different confining pressures

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Water Pressure Effects on Strength and Deformability

the effective stress induced by both direction of water flow and magnitude of water pressure, represented by the specified hydraulic boundary conditions due to the existence of the complex fracture system. Figure 11 shows the distribution of major principle stress contours of the DEM model without (left column) and with (right column) water pressure condition, as effective stresses under different confining pressures, respectively. Significant variation of stress distribution occurred, due to applying a constant water pressure and lateral confining pressures. The amounts of major principle stress decrease with water pressure. From Fig. 11a, one can see that no significant tensile stress was observed in the model when confining pressure is 0.5 MPa. Figure 12 shows the normalized flow rates in each fracture intersecting at the left vertical boundary (outlet flow side) of the DEM model with increasing confining pressure, illustrating the flow pattern changes with respect to the stress changes. Flow rates were normalized with respect to the mean flow rates (total flow rate divided by the number of fractures) at the boundary of the DEM model. One can see from these figures that the changes in flow-rate patterns in the lower confining pressure conditions, e.g., with confining pressure equal to 0.5 and 1.0 MPa, are more significant than that with confining pressures equal to or higher than the water pressure. Tables 3 and 4 show the calculated elastic limit and peak plastic strength values versus confining pressure in dry and wet conditions and differences between them, calculated using results in elastic and plastic deformation ranges, respectively. The results show that a water pressure of 1.0 MPa caused a reduction in strength values of 1.15–2.24 MPa, much larger than the water pressure applied at the lateral boundaries, with a larger mean difference for peak plastic strength. Therefore, the effective stress behavior is quite different from its classical definitions in Terzaghi and Biot’s poroelasticity theory (since the Poisson’s ratio is generally large than 0.5), due to the influence of fracture system complexity, direction of water pressure conditions applied and complex stress distribution in the tested volumes, especially at intersections of fractures. It should be noted that for fractured rock with irregular fracture systems, elastic deformation parameters cannot be calculated analytically. Several tests (numerical or physical) with linearly independent boundary conditions are needed, based on the general elasticity theory, as shown in Min and Jing (2003), using the equivalent elastic compliance tensor approach, which requires evaluation of a number of general compliance components and inversion of the compliance tensor. Since we considered only water pressure effects, not constitutive models of the rock mass concerned, the directional elastic modulus under axial

Table 3 Calculated elastic limit values versus confining pressures in dry and wet conditions and differences between them, with obtained results in the elastic deformation range Confined pressure (MPa)

Elastic limit (MPa)

Difference between dry and wet conditions (MPa)

Dry condition

Wet condition

0.5

1.23

0.486

1.0

2.5

0.974

1.526

1.5

3.73

2.18

1.55

2.0

5.05

3.45

1.60

2.5

6.34

4.73

1.61

3.0

7.6

5.96

1.64

0.744

Table 4 Calculated peak plastic strength values versus confining pressure in dry and wet conditions and differences between them, with obtained results in the plastic deformation range Confined pressure (MPa)

Peak plastic strength (MPa)

Difference between dry and wet conditions (MPa)

Dry condition

Wet condition

0.5

1.53

0.968

0.562

1.0

3.33

2.18

1.15

1.5

5.37

3.13

2.24

2.0

6.15

4.70

1.45

2.5

7.39

5.96

1.43

3.0

8.59

7.37

1.22

loading, not the strictly defined Young’s modulus, was adopted. The same reasoning was applied to the word ‘Poisson’s ratio’ here as well.

6 Concluding Remarks The water pressure effects on strength and deformability of fractured crystalline rocks is a subject of importance, and these were generically investigated through numerical modeling of coupled stress-flow processes of DEM models of realistic fracture system geometry. The novelty of the research is that the influences of water pressure on deformation and strength of the fractured rocks were systematically investigated for the first time, even though as a generic study, since a physical experiment on such a complex fracture system and with such a systematic approach is not yet available. This study provided several interesting insights that may lead to a better understanding of the effects of water pressure on strength and deformability of the fractured rocks, as summarized below: 1.

The numerical results demonstrated that the strength of fractured rocks is dependent on hydraulic boundary

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M. Noorian Bidgoli, L. Jing

2.

3. 4.

5.

conditions and mechanical confining pressures when testing in laboratory or in situ conditions. Unlike the confining pressure that generally causes strength increase of fractured rocks, water pressure generally causes strength decrease of fractured rock, due to the effective stress phenomenon, but the values of stress and strength reduction may or may not equal to the magnitude of water pressure, due to influence of fracture system complexity. The calculated results show that water pressure and confining pressure play a significant role in deformability parameters of fractured rocks. The trends of the elastic modulus differ more significantly than those of Poisson’s ratio with differences between water pressure and mechanical confining pressures. Both the M-C and H-B criteria give a fair estimate of the peak compressive strength of fractured rock. The results show that in general, the concept of effective stress may still be applicable for fractured crystalline rocks, but may differ quite significantly with that usually applied in soil mechanics or theories of poroelasticity or poroplasticity, with a larger difference between total and effective stress than water pressure. The theories of classical poroelasticity or poroplasticity with small deformation assumptions are not applicable if the equivalent Poisson’s ratio is larger than 0.5. The reason is non-uniform stress distribution in the tested volumes, especially at intersections of fractures, even when an approximate REV volume can be obtained and there are larger displacement distributions due to much weaker fracture shear strength. The hydraulic and mechanical boundary conditions also play an important role. Therefore, one should be cautious when applying the classical effective stress concept to fractured rock media and when the finite deformation assumption may need to be considered. This study is useful for the design of large-scale in situ experiments on coupled stress-flow processes of fractured rocks. In the current situation, many such in situ experiments were conducted in underground tunnels or caverns with disturbances to the local stress field (usually with stress releasing in the direction of excavated free face) and significant water flow situation changes (usually significant drainage), so that maintaining proper initial and boundary conditions of the tested volume become important. Therefore, comprehensive numerical modeling needs to be integrated with physical experiments from the start to the end of the project.

The methodology presented in this research provides a systematic platform for evaluating strength and deformability parameters of saturated fractured rock masses as a first hand

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evaluation approach. The current study focuses on hydromechanical effects on the strength and deformability of fractured rocks in 2D space. However, large-scale laboratory or in situ experiments are needed to verify the validity of the modelingapproachesandresults,andmoreinvestigationsneed to be conducted in 3D for more realistic simulations. Further work is under way to improve our knowledge about the uncertainty of the predicting of strength and deformability of fractured rock using stochastic analysis on multi-fracture system realizations.

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