submitted to manuel rocha medal, isrm · submitted to manuel rocha medal, isrm barcelona, december...
TRANSCRIPT
Summary of the Thesis
“THERMO-HYDRO-MECHANICAL ANALYSIS OF JOINTS
A THEORETICAL AND EXPERIMENTAL STUDY”
by Maria Teresa Zandarin Iragorre
Department of Geotechnical Engineering and Geosciences
Universitat Politècnica de Catalunya, Barcelona, Spain
Submitted to Manuel Rocha Medal, ISRM
Barcelona, December 2010
TABLE OF CONTENTS
1. INTRODUCTION .......................................................................................................... 2 2. A COUPLED THERMO-HYDRO-MECHANICAL FORMULATION OF JOINTS .. 4 3. DISCRETIZATION OF EQUATIONS OF STRESS EQUILIBRIUM, MASS AND ENERGY BALANCE ......................................................................................................... 5 4. NUMERICAL SIMULATION OF A HYDRAULIC SHEAR TEST ON ROUGH GRANITE FRACTURES ................................................................................................... 6
4.1 Geometry and material parameters of the model ................................................... 7 4.2 Numerical results against test data ......................................................................... 9
5. DIRECT SHEAR TESTS ON ROCK JOINTS WITH SUCTION CONTROL .......... 10 5.1 Direct shear cell apparatus with suction control by a vapour transfer technique . 10 5.2 Characterisation of the rock tested ....................................................................... 12 5.3 Direct shear test. Testing methodology ................................................................ 13
5.3.1 Sample preparation ........................................................................................ 14 5.3.2 Measurement of surface roughness ............................................................... 15 5.3.3 Equilibration with Relative Humidity environment ...................................... 15 5.3.4 Direct shear testing ........................................................................................ 15
5.4. Tests results and discussion ................................................................................. 16 5.4.1 Shear Strength ............................................................................................... 16 5.4.2 Normal displacements ................................................................................... 21 5.4.3 Rock joint surface damage ............................................................................ 22
6. INFLUENCE OF SUCTION AND ROUGHNESS ON YIELD SURFACE PARAMETERS ................................................................................................................. 24 7. NUMERICAL SIMULATION OF DIRECT SHEAR TESTS ON LILLA CLAYSTONE ................................................................................................................... 27
7.1 Geometry and parameters adopted in the model .................................................. 28 8. CONCLUSIONS .......................................................................................................... 34 APPENDIX. A COUPLED THERMO-HYDRO-MECHANICAL FORMULATION OF JOINTS .............................................................................................................................. 39
A1. Mechanical formulation ....................................................................................... 39 A2. Mass and energy balance equation ...................................................................... 40 A3. Constitutive Models ............................................................................................. 43
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THERMO-HYDRO-MECHANICAL ANALYSIS OF JOINTS
A THEORETICAL AND EXPERIMENTAL STUDY
Maria Teresa Zandarin
Department of Geotechnical Engineering and Geosciences
Universitat Politècnica de Catalunya, Barcelona, Spain
ABSTRACT
The thesis presents a thermo-hydro-mechanical (THM) model for joints in rock. It
describes also a pioneering set of experiments to investigate suction effects on the shear
behaviour of rock discontinuities.
A formulation for the coupled analysis of thermo- hydro- mechanical problems in joints is
first presented. The work involves the establishment of equilibrium and mass and energy
balance equations. Balance equations were formulated taking into account two phases:
water and air.
The joint element developed was implemented in a general purpose finite element
computer code for THM analysis of porous media (Code_Bright). The program was then
used to study a number of cases ranging from laboratory tests to large scale “in situ” tests.
A numerical simulation of coupled hydraulic shear tests of rough granite joints is first
presented. The tests as well as the model show the coupling between permeability and the
deformation of the joints. The formulation was also used to simulate the behaviour of
interfaces presents on a large scale test select to nuclear waste research. Vapour
diffusivity and gas flow were specific processes simulated in this case.
The experimental investigation focused on the effects of suction on the mechanical
behaviour of rock joints. Available experimental data on the effect of moisture on joint
behaviour was very scarce. Laboratory tests were performed in a direct shear cell
equipped with suction control. Suction was imposed using a vapour forced convection
circuit connected to the cell and controlled by an air pump. Artificial joints of Lilla
claystone were prepared. Joint roughness of varying intensity was created by carving the
surfaces in contact in such a manner that rock ridges of different tip angles were formed.
These angles ranged from 0º (smooth joint) to 45º (very rough joint profile). The
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geometric profiles of the two surfaces in contact were initially positioned in a “matching”
situation. Several tests were performed for different values of suction (200, 100 and 20
MPa) and for different values of vertical stress (30, 60 and 150 kPa). A constitutive model
including the effects of suction and joint roughness is proposed to simulate the
unsaturated behaviour of rock joints. The new constitutive law was incorporated in the
code and experimental results were numerically simulated.
1. INTRODUCTION
Discontinuities of the rock mass are the result of the origin of the rock and the subsequent
deformations imposed, in most cases by tectonic activity. According to Jennings (1969),
two sets of discontinuities could be typically defined as major and minor or secondary.
Major discontinuities include bedding planes, faults, contacts and dykes, while minor are
joints of limited length, i.e. cross joints in sedimentary rocks.
Taking into account their origin, joints can be classified as: bedding planes, which are
associated with sedimentary rocks and appear when there is a change in the characteristics
of the deposited material; stress relief joints, which form as a result of erosion of
weathered rock; tension joints, which are the result of cooling and crystallization of
igneous rock; and faults, which result in a plane of shear failure that exhibits obvious
signs of differential movement of the rock mass on each side of the central plane. Usually,
faults are linked to the movement of tectonic plates.
The characteristics of the planar surfaces constituting a joint depend on the geological
history of the rock mass. They are the results of mechanical, hydraulic, depositional,
chemical and other processes. The void structure of discontinuities has a dominant effect
on its hydro-mechanical behaviour.
Finite element formulations describing joint behaviour started in the pioneering
contribution of Goodman et al., (1968). Since then published formulations have steadily
improved the capabilities of the joint models. In particular, attention is given here to the
coupled hydraulic and mechanical behaviour of joints. Recent contributions were
published by Guidicci et al., (2002) and Segura (2008).
The hydro-mechanical behaviour under varying normal stress has been extensively
studied. The experimental results obtained by Hans et al. (2002) show that transmissivity
decreased as normal stress increased. This decrease is due to the reduction of the void
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space between the discontinuity walls, the increase of the contact area and the changes in
tortuosity. When the compression stress on the discontinuity is released, a non reversible
behaviour can be observed, i.e. the transmissivity at zero stress is lower than the initial
reference value.
When a shear stress is applied, before peak conditions are attained, transmissivity initially
decreases. However when peak conditions are met transmissivity increases substantially
(approximately two orders of magnitude). The increment of transmissivity is directly
related to joint dilatancy (Lee & Cho, 2002). Even if dilatancy increases continuously
with relative shear displacements, joint permeability reaches a constant value. This is a
consequence of the gouge material generated by the breakage of asperities. The roughness
degradation depends on the strength of asperities, the applied normal load and the shear
stiffness. Olsson & Barton (2001) described this behaviour and proposed a model to
consider these phenomena. Indraratna et al., (2003) reported an analytical and
experimental study of the two phase flow trough rock joints.
Taking these results as a starting point a finite element formulation for the coupled
thermo-hydro-mechanical behaviour of joint elements has been developed. It considers a
two phase (air and water) flow and vapour diffusivity through joints. A further motivation
for this work was related to the conditions found in nuclear waste disposal designs.
Bentonite barriers, initially unsaturated exhibit strong suctions at early phases. The heat
generation imposed by nuclear canisters results in a drying of the engineering barrier,
which is also subjected to inflow from the host rock. In the long term the gas generated in
the waste may escape through interfaces and rock joints, a phenomenon which depends on
gas generation rates. The set of conditions outlined imply that artificial joints (those
existing between engineering barriers and excavated rock surface, for instance) and
natural rock joints may be exposed to partially saturated conditions.
Finally joints above an existing water level or exposed to ambient conditions are involved
in slope stability and excavations. It was then natural to attempt a generalized formulation
of joint behaviour for partially saturated conditions. This is achieved by providing a
separate consideration to water and air transfer. In addition, since heat transfer is also
involved in some applications (notably nuclear waste disposal) an energy balance was
added to the field equations.
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The effect of suction on the mechanical behaviour of rock joints has not been reported in
the literature to the author’s knowledge. Since the prevailing suction has a very significant
effect on rock strength (Oldecop & Alonso, 2001) it was anticipated that rock joint
behaviour is also significantly affected. This was the motivation for the performance of a
laboratory testing programme concentrated on the mechanical behaviour of rock joints
subjected to direct shear under suction control. Suction was controlled by a vapour
equilibrium technique (Fredlund & Rahardjo, 1993; Romero, 2001). Artificially prepared
joints of Lilla claystone were tested. Joint roughness of varying intensity was created by
carving the surfaces in contact in such a manner that rock ridges of different tip angles
were formed. These angles varied between 0º (smooth joint) to 45º (very rough joint
profile). The geometric profiles of the two surfaces in contact were initially positioned in
a “matching” situation. Several tests were performed for different values of suction (200,
100 and 20 MPa) and for different values of vertical stresses (30, 60 and 150 kPa). From
the analysis of test results a constitutive law was proposed. It takes into account the effect
of suction on the strength parameters and the degradation of rock joints. The performance
of the model was checked against the recorded shear stress-relative displacements.
2. A COUPLED THERMO-HYDRO-MECHANICAL FORMULATION OF
JOINTS
The THM formulation of the joint is described in an Appendix to this Summary Report.
The equations of mass balance of water and air as well as the energy balance equations
were formulated for the joint Their solution requires a set of constitutive equations. The
Appendix includes the laws for longitudinal and transversal flow, the water retention
curve and its relationship with permeability and temperature, the relative permeability and
the heat conduction equations.
The mechanical model includes a strain softening law for the shear stress-relative
displacement relationship. A hyperbolic yield function is used. The formulation is done
within a viscoplastic framework. The model for the joint has been included into a general
Finite Element program (Code_Bright) with the purpose of solving a number of cases
described in the remaining of the Thesis.
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3. DISCRETIZATION OF EQUATIONS OF STRESS EQUILIBRIUM, MASS
AND ENERGY BALANCE
The discrete form of stress equilibrium relations can be directly established for the joint
element. Then, the integration to average the residuals provides:
4 44 4 2 2
4 4
10
2u T u u T pmp mp j mp mp j
lmp lmp
dl P dl
I I
N r D r N I I u N r m N I I bI I
(1)
where b is the vector of the external body forces.
In order to describe the numerical treatment of mass and energy balance equations, the
water mass balance equation is used as an example. Only terms describing water vapour
transfer will be considered. For the remaining mass and energy balance equations the
treatment is identical (see also Olivella et al., 1995).
The weighted residual method is applied to obtain the discrete form of balance equations.
The discrete forms of the terms of the equations are given as follows:
Storage changes of mass or energy at constant joint opening
2 2
2 2
4
1 1a a
2 2
a
4
w w w w w wl l g g l l g gp p
mp mp
lmp lmp
w w wl l g g
S S S Sdl dl
t t
S S l
t
I I
N NI I
I
(2)
Storage change induced by changes of joint opening
24 4
2
24 4
2
1
2
1
2
p T w w T ump l l g g mp
lmp
p T w w T ump l l g g mp
lmp
duS S dl
dt
uS S dl
t
IN m rN I I
I
IN m rN I I
I
(3)
Advective fluxes
The nodal liquid pressures Plj are differentiated to provide the mid-plane values of the
joint pressure drop:
2 2p
mp mp jp N l I I P
(4)
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The discretized expression for the transversal advective flux is:
2 22 2
2 2
wp T w p T prlt lmp l lt mp lt mp j
llmp lmp
kq dl k dl l
I I
N N N I I PI I
(5)
The liquid pressure at the mid-plane is calculated by averaging the nodal liquid pressures:
2 2
1
2p
mp mp jp N l I I P
(6)
The discretized expression for the longitudinal advective flux is:
2
2
2 22 2
2 2
1
2
1 1 1
2 2 2
p T pmp mpll rl l
ll l llmp
p T p p Tw wmp mp mprl l l rl l l l
ll j lll l l l llmp lmp
pk kdl
k kk dl l k dl
NIg
I
N N NI II I P g
I I
(7)
The discretized equations for the non-advective fluxes and heat conduction are analogous
to the equations for advective fluxes given above.
4. NUMERICAL SIMULATION OF A HYDRAULIC SHEAR TEST ON ROUGH
GRANITE FRACTURES
The hydraulic shear tests selected to check the capabilities of the model were performed
on granitic rock from Korea (Lee & Cho, 2002). A intact rock block was sawed to obtain
samples with a length of 160 mm and equal values of width and height (120 mm) (Figure
1a). The fracture surfaces were created by means of a tensile fracture exerted by a splitter.
The fracture opening was measured using a 3-D laser profilometer. The mean value of the
opening was 0.65mm.
Shear hydraulic tests were performed maintaining constant normal stresses of 1, 2 and 3
MPa. The tangential displacement was applied at a rate of 0.05-0.08mm/seg. The
hydraulic pressure applied to the joint varied from 4.91 kPa to 19.64 kPa. For each stage
of shear displacement of about 1mm, hydraulic pressure was kept constant. When the
fluid flow reached steady state, the mean flow rate was calculated recording the amount of
outflow measured for a period of 2 minutes. These measurements were also used to
calculate the permeability of the joint.
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120
110
160 mm
120
Normal Load
Shear Load
A B
C D
E F
G H
Pl
120 mm
550,
65
110
us us
Figure 1: a) Joint specimens of Hwangdeung granite. b) Discretized geometry and boundary conditions used for the hydro-mechanical simulation of the test performed by Lee et al., (2002).
The shear behaviour of the rock joint is shown in Figures 6a and b. The results obtained
are characterized by a peak shear strength and a pronounced dilation, that greatly affected
the hydraulic behaviour of the rough fractures. Dilatancy increases rapidly before shear
stress reaches its peak value. Then, dilatancy increases at a lower rate during the shear
stress drop to reach residual values.
The permeability changes, with respect to the increments of shear displacements, are
plotted in Figure 2c. The fracture permeability changes slightly during the initial stage of
shear loading. But, as dilation occurs close to peak strength, permeability increases
dramatically, about 2 orders of magnitude. When shear displacements reach 7 mm,
permeability become constant.
4.1 Geometry and material parameters of the model
The tests described above were modelled using the coupled hydro-mechanical
formulation described before implemented into Code_Bright. The model is 120mm high
and 110mm wide (Fig. 1 b). The rock matrix was discretized using 200 quadrilateral
continuum elements having 4 nodes and the joint was discretized by means of 10 joint
elements. The normal stress is applied at the AB boundary, while shear displacements are
applied at AC and BD boundaries. Boundaries EG and FH are horizontally fixed and GH
is vertically fixed. The water injection (Pl) on the joint was applied at CE boundary,
while at DF a drainage boundary condition was considered. The pressure at CE was
increased when the shear displacement increased 1mm, as done in the real test. The joint
is considered to be saturated (Sl= 1) during the test.
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Table 1.a: Hydro-mechanical parameters of granite matrix and joint used in the numerical model.
Hwangdeung granite parameters
Mechanical parameters Symbol Units Value
Young’s modulus E MPa 54100
Poisson’s ratio 0.29
Porosity no % 49.0
Hydraulic parameters
Intrinsic permeability k m2 1x10-16
Table 1.b: Hydro-mechanical parameters of the joint model used in calculations. Rock joint parameters
Mechanical parameters Symbol Units Value
Initial normal stiffness parameter m MPa 90
Tangential stiffness Ks MPa/m 1500
Initial cohesion c0 MPa 0.02
Initial friction angle 0 47º
Residual friction angle res 37º
Initial opening a0 mm 0.65
Minimum opening amin mm 0.065
Viscosity parameter s-1 1x10-4
Stress power N 2.0
Critical displacement for cohesion uc* mm 15.0
Critical displacement for tan u *tan mm 15.0
Uniaxial compressive strength qu MPa 151
Model parameter d 40
Joint Roughness Coefficient JRC 2.70
Hydraulic parameters
Hydraulic opening e mm 0.035
Longitudinal intrinsic permeability kl m2 1x10-8
Transversal intrinsic permeability kt m2 1x10-16
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A linear elastic constitutive law was used to simulate the mechanical behaviour of the
rock matrix and the intrinsic permeability was considered constant during the test. The
parameters adopted for the granite matrix are summarised in Table 1a.
The mechanical behaviour for the rock joint was modelled using the elasto-visco-plastic
constitutive laws described before. The longitudinal permeability changes during the test
according to the joint opening. The parameters for rock joint are indicated in Table 1 b.
0 2 4 6 8 10 12 14 16
Shear displacements [mm]
0
0.5
1
1.5
2
2.5
3
3.5
4
[M
Pa]
Net Normal Stresses3 MPa (Test)3 MPa (Model)2 MPa (Test)2 MPa (Model)1 MPa (Test)1 MPa (Model)
0 2 4 6 8 10 12 14 16
Shear displacements [mm]
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
Nor
mal
dis
plac
emen
ts [
mm
]
0 2 4 6 8 10 12 14 16
Shear displacements [mm]
1.00E-007
1.00E-006
1.00E-005
1.00E-004
1.00E-003
1.00E-002
Intr
insi
c P
erm
eabi
lity
[cm
2 ]
a) b)
c)
Figure 2: Comparison between experimental results obtained by Lee et al. (2002), and results from numerical simulation. a) Shear stress-shear displacement curve. b) Normal displacement vs. shear displacement and c) Intrinsic permeability vs. shear displacement.
4.2 Numerical results against test data
The results obtained from the simulation are compared with the tests results in Figures 2
a, b and c. The mechanical behaviour of the joint is closely reproduced by the model. The
numerical formulation is able to reproduce the increment of peak shear stress with normal
stresses. Also, it is possible to capture how the shear strength decreases with
displacements. The figure also compares the measured and calculated dilatancy of the
joint (Fig. 2 b).
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The evolution of the intrinsic permeability of the joints is also simulated. Even though
permeabilities in the model increase continuously with dilatancy, the permeability
measured in the test for different normal stresses became constant and independent of
normal stress. This is mainly caused by the gouge materials generated from the
degradation of asperities during shearing. This phenomenon is not considered in the
model (Fig. 2 c).
5. DIRECT SHEAR TESTS ON ROCK JOINTS WITH SUCTION CONTROL
In the following section the equipment used to perform direct shear test with suction
control on rock joints is first described. Then, the mechanical properties and geology of
Lilla claystone, the sample preparation and the procedure followed during the test are
presented. The section ends with an analysis of test results followed by an explanation of
the model proposed and a comparison of simulated model results and actual testing
measurements.
5.1 Direct shear cell apparatus with suction control by a vapour transfer technique
The direct shear cell apparatus is constituted by the following main parts: a mobile base
with a ceramic disc which slides on two metallic guides with bearings to reduced friction;
an electrical motor which moves the push rod with different displacement rates
(0.005mm/min to 2000mm/min); an air chamber with an inner diameter of 220mm which
encloses a shear box. The shear box proper is constituted by two parts; the lower part
(fixed to the mobile base by means of four screws) has a height of 10mm and an inner
hole, 50mm in diameter, where the sample is placed. The upper part has a height of 21mm
and slides over the lower part. This part covers the entire sample and it hosts a metallic
porous disc and the loading cap. The metallic porous disc has a diameter of 50mm and a
height of 10mm. The loading cap has a special design. It allows the free swing of the
loading bushing which centers the piston in order to apply a centered vertical load for any
relative shear displacement. Finally, the air pressure chamber is formed by an upper lid
with a valve connected to the air pressure and by a lower piece which holds the Bellofram
seal. The vertical load is applied by the diaphragm pressure acting on the piston. The
maximum pressure is limited to 1MPa. The air pressure is controlled by a throttle and
measured by a manometer. A scheme of the direct shear device is shown in Figure 3.
(Escario & Sáez, 1986)
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Air pump
Hermetic Recipient
SalineSolution
Vapour InletVapour Outlet
Sensor of temperature andRelative Humidity
Air Chamber
Porousceramic disc
Rock Sample Load Cell
LVDT
Data adquisition system
Vapour Inlet
Air Pressure
Figure 3: a) Scheme of the direct shear device. b) Photograph of testing arrangement.
The shear load is transmitted through a piston in contact with the shear box to a load cell.
The load cell has a nominal capacity of 500kg. The vertical displacements of the sample
are measured by a Linear Variable Differential Transformer (LVDT) with a range of
measurement of ± 5mm. The LVDT is fixed to a steel stem and it is located over the
horizontal extension of the vertical loading piston.
Some improvements were made to the device to perform shear tests with suction control
reported here. One of them consists in connecting the air chamber to the vapour
circulation system by two pipe connections with the objective of controlling the relative
humidity of the sample during the shear test. A further improvement was to incorporate a
sensor to measure the temperature and relative humidity of the air within the chamber.
This was made possible by building an airtight chamber which holds the transducer (Fig.
3). The sensor is able to measure relative humidity from 0% to 100% and temperatures
from 0º to 60ºC. The data acquisition system was also improved by incorporating a
Shear Cell Load Cell
LVDT
Humidity and Temperature Sensor
Air pump
Data Adquisition
Load Cell
LVDT
Humidity and Temperature Sensor
Load Cell
LVDT
Humidity and Temperature Sensor
Air pump
Data Adquisition
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multifunctional analogue to digital, digital to analogue and digital input/output board
using a USB6009 data acquisition device. A digital program was designed, using
LabVIEW programming language, for data acquisition. Data recorded was finally stored
in a PC operating in a Windows environment.
5.2 Characterisation of the rock tested
The characterisation of the rock was taken from previous works performed by García-
Castellanos et al., (2003), Berdugo (2007), Tarragó (2005) and Pineda (2010). The rock
tested (Lilla claystone), is a sulphated-bearing argillaceous rock located in the Lower
Ebro Basin, in northeast of Spain. These sulphated rocks formed during the Tertiary
Period range from Early Eocene to Late Miocene in age. Lilla claystones has two main
components; the host argillaceous matrix (composed by illite, paligorskite, dolomite and
quartz), and the sulphated crystalline fraction (composed mainly by anhydrite and
gypsum. The X-Ray diffraction analysis was applied to the crushed rock fraction having
particle size smaller than 20 μm in order to identify the mineral phases of the rock. The
main minerals were dolomite (32.31%), anhydrite (44.32%), illite (15.85%) and
paligorskite (8.51%).
The density of the rock varies from 2.56 to 2.58 g/cm3. The clay matrix has a low
plasticity. The porosity varies from 0.09 to 1.1. The Young modulus E0 varies from 26.5
to 28.5GPa and the shear stiffness G0 varies from 11 to 12.5 GPa.
The water retention curve for unweathered Lilla claystone is shown in Figure 8 (some
data from Pineda, 2010, is given in the figure). The methodology followed to measure this
curve consists in subjecting an intact sample, 15mm in diameter and a 10 mm in height, to
a wetting-drying cycle under unstressed condition. The initial state of the sample was in
equilibrium with a RH~50% and a temperature equal to 20º. The wetting and drying path
were applied using the vapour equilibrium technique. The wetting path was applied by
using a hermetic vessel with distilled water. Air drying was then used to induce increasing
suction until a relative humidity of RH=50% was reached. Suction was measured after a
24 hours equalization period, using a chilled-mirror dew-point psychrometer.
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Gravimetric water content, w [%]
1
10
100
1000
Tot
alsu
ctio
n[M
Pa]
Initial state of WRC(Pineda et al., 2010)
Initial state
Initial state
Wetting path
Drying path
Wetting path (Pineda et al., 2010)Drying path (Pineda et al., 2010)
ZandarinZandarin
1 2 3 4 5 6 7
Figure 4: Water retention curve for unweathered Lilla claystone and suctions vs. gravimetric water content used in the tests perform presented here.
In this study suction and gravimetric water content were also measured on six samples of
unweathered Lilla claystone. The initial state of all samples was a temperature of 20º and
a RH~50% (laboratory conditions). These samples, in pairs, were subjected to different
conditions of relative humidity using the vapour equilibrium technique. The conditions
mentioned before consist in a drying path placing the samples in a hermetic vessel in an
atmosphere controlled by lithium chloride (RH~20% at 20º); air drying maintaining the
samples at laboratory conditions and a wetting path placing the samples in a hermetic
vessel in equilibrium with distilled water (RH~98% at 20º). Suction was measured after
fifteen days, using a chilled-mirror dew-point psychometer and the gravimetric content
was determined after 24 hours of oven drying (Fig. 4).
5.3 Direct shear test. Testing methodology
The testing methodology consists in: (1) preparing the samples by carving joints with
different geometric angles; (2) measuring the profile of the joint wall surface; (3) applying
a wetting or a drying cycle on the samples using vapour equilibrium technique; (4)
performing the direct shear test with suction control and (5) measuring the profile of the
joints surface after the test.
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Figure 5: Samples construction. a) Drilling of borehole core and joints carving with the diamond drill. b) Rock joints having different geometric angles of 0º, 15º, 30º and 45 degrees respectively.
5.3.1 Sample preparation
The samples were extracted from a borehole core of Lilla claystone drilled from the floor
of Lilla tunnel. The core had 110mm in diameter and a length of one meter. The core was
cut into pieces with a nominal length of 50mm. Then, these pieces were drilled and cut in
a machine to obtain samples 50mm in diameter and 12 mm in height. Then, the joints
were carved with a diamond drill in order to create regular geometric asperities having
“opening” angles of 5º, 15º, 30º, and 45º degrees respectively (Fig. 5). The intention was
to test different asperity roughness.
a)
b)
-15-
5.3.2 Measurement of surface roughness
To obtain the topographical data of rock fracture surfaces, 2-D laser-scanning profiles of
both sides of joints were measured before and after the shear test. A laser and a LVDT
were used to obtain the X-Y profile. The Laser has a range of measurements which varies
from 25 to 30mm with a precision of 0.03%. The LVDT used has a measurement range of
+/-25mm and a sensitivity of ± 0.05% mV/mm. A computer performs data collection and
processing in real time.
5.3.3 Equilibration with Relative Humidity environment
Prior to shearing, each sample was equilibrated at the required suction. Samples were
placed in a desiccator with a solution, whose concentration is known, at a constant
temperature of 20ºC. Some of the samples were dried placing them in an atmosphere of
pure lithium chloride. The pure lithium chloride solution takes the humidity to a value
RH=20% (approximately equivalent to a total suction of 200MPa). Others samples were
wetted using distilled water. A quasi-saturation condition was obtained for a RH~86%
and a suction of 20MPa. Others samples were exposed to the laboratory room
environment (RH~50%, equivalent to a suction of 100MPa). The equilibrium was
considered complete when there was no measurable change in the weight of samples (no
changes in water content). Samples weights were measured and the total suction was
measured on small samples of rock using a dew-point psychrometer (WP4, Decagon
Device). A small sample of rock was placed in the desiccator together with the joints
samples and its suction was used as a standard average value. It was assumed that the
suction measured with the psychrometer is the suction of the joint. Samples reached the
equalization stage after a period of fifteen days.
5.3.4 Direct shear testing
The initial step was placing the sample in the shear cell, controlling that the joint was
aligned with the direction of shear displacements. Then the centering bushing was
positionated and the piston was carefully inserted into its axis. Once the shear cell was
assembled and the sensors were positioned, the vapour system was connected to the shear
cell. When the relative humidity measured by the sensor reached the constant value of
suction required, the test began by applying the vertical load. Three different values of air
pressure (net normal stress) were applied: 30, 60 and 150 kPa respectively. When the
vertical displacements induced by the vertical stress remained constant the shear
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displacements were applied at a rate of 0.05mm/min. The shear test ended when the shear
stress reached its residual value. The recorded shear stress versus shear displacements and
normal displacements against shear displacements are plotted and analyzed in the
following section.
5.4. Tests results and discussion
The results of the tests are plotted in Figures 6 to 10 for the different joint roughness (a =
0º, 5º, 15º, 30º and 45º) and for the different normal stresses a) 30, b) 60 and c) 150 kPa.
The left side of the figures shows the evolution of the shear stress and normal
displacements with shear displacement. Each plot includes the data recorded for the three
values of suction. The normal and shear stresses plotted are the mean average value acting
on the middle plane of the joint. Photographs of samples, taken after the test, are shown
on the right side of the figures.
5.4.1 Shear Strength
The recorded plots of shear stress versus shear displacement show that the shear strength
of joints depends on the three variables namely normal stress, suction and joint roughness
angle.
The effect of the normal stress is well known; the larger the normal stress the higher the
shear strength.
The value of suction imposed also affect the peak and residual shear strength. Increasing
suction results in higher values of peak shear strength. However, the effect of suction on
the residual strength is not seen as clearly as in the peak strength. Residual strength
depends not only on suction, but also on the degradation of the roughness of asperities.
Degradation of asperities is influenced not only by suction but also by the irregular
matching due to defects of joint construction and the heterogeneity of the rock. This
implies the existence of contact areas with higher or lower strength. These phenomena
results in some heterogeneity of results.
Increasing the asperity roughness is associated with higher strength. Furthermore the
roughness also affects the strength softening of the joint. In joints with higher roughness
the residual strength is reached for smaller displacements. For example, for joints having
a roughness of 45º the residual strength is reached for a displacement of approximately
-17-
2.5mm, while for a joint having a roughness of 15º the residual strength occurs for a
displacement of 6mm (see Fig. 10 and Fig. 8). The flat joint shows a ductile behaviour; in
these cases the softening effect is negligible (Fig. 6).
0 1 2 3 4
Shear displacements [mm]
0
20
40
60
80
[K
Pa]
a) Net Normal Stress 30 KPa
0 1 2 3 4
Shear displacements [mm]
-0.2
-0.15
-0.1
-0.05
0
0.05
Nor
mal
dis
plac
emen
ts [
mm
]
0 1 2 3
Shear displacements [mm]
0
40
80
120
[K
Pa]
b) Net Normal Stress 60 KPa
0 1 2 3
Shear displacements [mm]
-0.2
-0.15
-0.1
-0.05
0
0.05
Nor
mal
dis
plac
emen
ts [
mm
]
0 1 2 3 4Shear displacements [mm]
0
40
80
120
160
200
[K
Pa]
Suction [MPa]200 100 20
c) Net Normal Stress 150 KPa
0 1 2 3 4Shear displacements [mm]
-0.2
-0.15
-0.1
-0.05
0
0.05
Nor
mal
dis
plac
emen
ts [
mm
]0 1 2 3 4
Shear displacements [mm]
0
20
40
60
80
[K
Pa]
a) Net Normal Stress 30 KPa
0 1 2 3 4
Shear displacements [mm]
-0.2
-0.15
-0.1
-0.05
0
0.05
Nor
mal
dis
plac
emen
ts [
mm
]
0 1 2 3
Shear displacements [mm]
0
40
80
120
[K
Pa]
b) Net Normal Stress 60 KPa
0 1 2 3
Shear displacements [mm]
-0.2
-0.15
-0.1
-0.05
0
0.05
Nor
mal
dis
plac
emen
ts [
mm
]
0 1 2 3 4Shear displacements [mm]
0
40
80
120
160
200
[K
Pa]
Suction [MPa]200 100 20
c) Net Normal Stress 150 KPa
0 1 2 3 4Shear displacements [mm]
-0.2
-0.15
-0.1
-0.05
0
0.05
Nor
mal
dis
plac
emen
ts [
mm
]
Figure 6: Evolution of shear strength and normal displacements with shear displacements for a = 0º.
-18-
Figure 7: Evolution of shear strength and normal displacements with shear displacements for a = 5º (left). Photographs of samples taken after the test (right).
0 2 4 6
Shear displacements [mm]
0
40
80
120
160
[K
Pa]
a) Net Normal Stress 30 KPa
0 2 4 6
Shear displacements [mm]
-0.1
0
0.1
0.2
0.3
Nor
mal
dis
plac
emen
ts [
mm
]
0 2 4 6 8
Shear displacements [mm]
0
50
100
150
200
250
[K
Pa]
b) Net Normal Stress 60 KPa
0 2 4 6 8
Shear displacements [mm]
0
0.2
0.4
Nor
mal
dis
plac
emen
ts [
mm
]
0 2 4 6 8
Shear displacements [mm]
0
100
200
300
400
[K
Pa]
Suction [MPa]20010020
c) Net Normal Stress 150 KPa
0 2 4 6 8
Shear displacements [mm]
-0.1
0
0.1
0.2
Nor
mal
dis
plac
emen
ts [
mm
]
=20 MPa =100 MPa =200 MPa
-19-
Figure 8: Evolution of shear strength and normal displacements with shear displacements for a = 15º (left). Photographs of samples taken after the test (right).
0 1 2 3 4
Shear displacements [mm]
0
100
200
300
400
[K
Pa]
a) Net Normal Stress 30 KPa
0 1 2 3 4
Shear displacements [mm]
0
0.2
0.4
0.6
0.8
Nor
mal
dis
plac
emen
ts [
mm
]
0 2 4 6 8
Shear displacements [mm]
0
100
200
300
400
[K
Pa]
b) Net normal stress 60 KPa
0 2 4 6 8
Shear displacements [mm]
0
0.2
0.4
0.6
0.8
1
Nor
mal
dis
plac
emen
ts [
mm
]
0 2 4 6 8
Shear displacements [mm]
0
200
400
600
[K
Pa]
Suction [MPa]200
100 20
c) Net Normal Stress 150 KPa
0 2 4 6 8
Shear displacements [mm]
0
0.4
0.8
1.2
1.6
Nor
mal
dis
plac
emnt
s [m
m]
=20 MPa =100 MPa =200 MPa
-20-
Figure 9: Evolution of shear strength and normal displacements with shear displacements for a = 30º (left). Photographs of samples taken after the test (right).
0 1 2 3 4 5
Shear displacements [mm]
0
100
200
300
400
[K
Pa]
a) Net Normal Stress 30 KPa
0 1 2 3 4 5
Shear displacements [mm]
0
0.2
0.4
0.6
0.8
Nor
mal
dis
plac
emen
ts [
mm
]
0 1 2 3 4 5
Shear displacements [mm]
0
100
200
300
400
[K
Pa]
b) Net Normal Stress 60 KPa
0 1 2 3 4 5
Shear displacements [mm]
0
0.2
0.4
0.6
0.8
1
Nor
mal
dis
plac
emen
ts [
mm
]
0 1 2 3 4 5
Shear displacements [mm]
0
200
400
600
[K
Pa]
c) Net Normal Stress 150 KPa
0 1 2 3 4 5
Shear displacements [mm]
0
0.2
0.4
0.6
0.8
Nor
mal
dis
plac
emnt
s [m
m] Suction [MPa]
200 100
20
=20 MPa =100 MPa =200 MPa
-21-
Figure 10: Evolution of shear strength and normal displacements with shear displacements for a = 45º (left). Photographs of samples taken after the test (right).
5.4.2 Normal displacements
The normal displacements recorded for the flat joint were negative. These joints present a
contractive behaviour. The contraction increases with the increment of net normal stresses
(Fig. 6 a, b and c). In joints having a smooth angle of roughness (a=5º) positive normal
displacements were recorded. Normal displacements (dilatancy) increase for joints having
a= 15º, 30º and 45º increase. However, if the normal displacements for a= 15º, 30º and
45º are compared it is noticed that dilatancy decreases with a. This is explained because
0 1 2 3 4Shear displacements [mm]
0
100
200
300
[KP
a]a) Net Normal Stress 30 KPa
0 1 2 3 4Shear displacements [mm]
0
0.2
0.4
0.6
0.8
1
Nor
mal
dis
plac
emen
ts [
mm
]
0 1 2 3 4
Shear displacements [mm]
0
100
200
300
400
500
[K
Pa]
b) Net normal stress 60 KPa
0 1 2 3 4
Shear displacements [mm]
-0.2
0
0.2
0.4
0.6
0.8
Nor
mal
dis
plac
emnt
s [m
m]
0 1 2 3 4
Shear displacements [mm]
0
200
400
600
800
[K
Pa]
Suction [MPa]200
100 20
c) Net Normal Stress 150 KPa
0 1 2 3 4
Shear displacements [mm]
-0.2
0
0.2
0.4
0.6
Nor
mal
dis
plac
emen
ts [
mm
]
=20 MPa =100 MPa =200 MPa
-22-
rougher asperities present more degradation which extends laterally affecting the whole
surface of the joint.
Also it is generally observed that increasing normal stress results in lower dilatancy.
However this behaviour was not always recorded. This is the case for a= 15º and 30º (see
Fig. 8 and Fig. 9 a, b). It is believed that this anomalous behaviour is due to some
irregularity of matching and probably a consequence of the heterogeneity of the rock
which influences the degradation of asperities.
The influence of suction on dilatancy is also apparent in plots. Joints equilibrated at low
suction (=20MPa, RH=86%) exhibit the lowest dilatancy. Dilatancy increases with
suction. The higher the suction, the higher the strength. The effect is particularly intensive
in this rock. It seems that the sliding of the joint walls, one over the other, occurs without
breakage of asperities when suction is high. Even if breakage occurs at a given suction it
is likely that the gouge material equilibrated at high suctions is capable of rolling on the
joint surface. The gouge material at lower suctions is easier to crush without rolling.
These phenomena are capable of explaining the recorded effect of suction on dilatancy.
5.4.3 Rock joint surface damage
Since the global damage of a rough joint may be a consequence of the work spent in
shearing the joint, there was an interest in relating a measure of the joint damage and the
irreversible (plastic) work induced by external stress. Joint damage was defined by an
index relating the weight of the unweathered samples (Wg) to the damaged sample (Wi).
In Figures 11, 12 and 13 the damage ratio (Wg/Wi) is plotted in terms of shear work,
dilatancy work and total work, respectively. In all cases, a maximum shear displacement
of 2.5 mm was considered to calculate the work. Comparing Figures 12 and 13 it is
observed that the dilatancy work is one order of magnitude smaller than shear work.
Therefore the total work is essentially the shear work. In order to determine a relationship
between joint damage and the work applied to the joint during shearing, the ratio Wg/Wi
is plotted against the total work (shear plus volumetric) in Figure 14. The plot shows that
suction is also a controlling factor not fully accounted for by the Work. The trend lines
plotted in the figure (dash blue, green and red line) indicate that the degradation of joints
increase with the work exerted, in all cases. In addition, the higher the suction, for a given
value of total work, the lower the joint degradation.
-23-
0 0.4 0.8 1.2 1.6 2
Shear Work [J]
0
0.01
0.02
0.03
0.04
0.05
Wg/
Wi
= 20MPa0º5º15º30º45º
=100MPa0º5º15º30º45º
=200MPa0º5º15º30º45º
Figure 11: Wg/Wi against shearing work.
-0.2 -0.1 0 0.1
Dilatancy Work [J]
0
0.01
0.02
0.03
0.04
0.05
Wg/
Wi
= 20MPa0º5º15º30º45º
=100MPa0º5º15º30º45º
=200MPa0º5º15º30º45º
Figure 12: Wg/Wi against dilatancy work. Negative values correspond to dilatant behaviour and positive values to contractant behaviour.
-24-
0 0.4 0.8 1.2 1.6 2
Total Work [J]
0
0.01
0.02
0.03
0.04
0.05
Wg/
Wi
= 20MPa0º5º15º30º45ºTrendline
=100MPa0º5º15º30º45ºTrendline
=200MPa0º5º15º30º45ºTrendline
Figure 13: Relationship between Wg/Wi, total work and suction
6. INFLUENCE OF SUCTION AND ROUGHNESS ON YIELD SURFACE
PARAMETERS
The parameters c0’ and tan Ф0’, which define the yield surfaces (eq. 14) were obtained for
each test by plotting the maximum shear stress measured against net normal stress (Fig.
18).
Values of c0’ and tan Ф0’ were plotted against roughness angle (aa) and suction (Figs. 15
a and 15 b respectively). Figure 15 a (above) shows that minimum values of cohesion are
obtained for aa=0º. Cohesion increases up to aa=15º, but remains essentially constant for
others values of aa. Figure 15 a (below) shows a linear increment of c0’with suction.
A mathematical expression is proposed for c0’ taking into account the effect of suction
and asperity roughness angle:
2 a
a
b tan0( , ) 0 1 0 1' c c b b 1c e (8)
where c’0(,aa) is the effective initial cohesion; is the total suction; c0 is the cohesion
for =0 and aa = 0º; c1 is the slope of c’0 vs. suction line for aa = 0º; b0 is the average
-25-
value of c0 for aa = 15º-45º; b1 ia a parameter of the model that controls the increment of
cohesion with for a aa = 5º-45º; and b2 is a parameter of the model.
0 40 80 120 160
Net Normal Stress [KPa]
0
200
400
600
800
Shea
r St
ress
[K
Pa]
Parameters of yield surfacesc'0=296.00KPa tgF'0=2.40 F'0=67.38º
c'0=244.38KPa tgF'0=1.39 F'0=54.26º
c'0=168.00KPa tgF'0=0.90 F'0=41.98º
Suction [MPa]200 (Experimental)200 (Fit)100 (Experiemtal)100 (Fit)20 (Experimental)20 (Fit)
Asperity angle 45º
0 40 80 120 160
Net Normal Stress [KPa]
0
200
400
600
800
Shea
r St
ress
[K
Pa]
Parameters of yield surfacesc'0=292.85KPa tgF'0=1.79 F'0=60.81º
c'0=237.27KPa tgF'0=1.27 F'0=51.78º
c'0=159.28KPa tgF'0=0.91 F'0=42.30º
Asperity angle 30º
0 40 80 120 160
Net Normal Stress [KPa]
0
200
400
600
800
She
ar S
tres
s [K
Pa]
Parameters of yield surfacesc'0=307.60KPa tgF'0=1.81 F'0=61.07º
c'0=225.00KPa tgF'0=1.10 F'0=47.72º
c'0=184.00KPa tgF'0=0.90 F'0=41.98º
Asperity angle 15º
0 40 80 120 160
Net Normal Stress [KPa]
0
200
400
600
800
She
ar S
tres
s [K
Pa]
Parameters of yield surfacesc'0=128.69KPa tgF'0=1.30 F'0=52.43º
c'0= 75.00KPa tgF'0=1.02 F'0=45.56º
c'0= 56.00KPa tgF'0=0.90 F'0=41.98º
Asperity angle 5º
0 40 80 120 160
Net Normal Stress [KPa]
0
200
400
600
800
She
ar S
tres
s [K
Pa]
Parameters of yield surfacesc'0=58.69KPa tgF'0=0.69 F'0=34.61º
c'0=28.42KPa tgF'0=0.84 F'0=40.03º
c'0= 9.22KPa tgF'0=0.86 F'0=40.70º
Asperity angle 0º
Figure 14: Peak shear stresses vs. net normal stresses for different values of a. The associated parameters of the yield surface are also given.
Figures 15 b show that tan Ф0’ increases with aa and suction. The increment with respect
to aa is considered dependent on tanaa and the increment with respect to suction is made
linear. The equation proposed for tan Ф0’ is:
a0( , ) 0 1 0 1 atan ' t t d d tan (9)
where tan Ф’0(,aa) is the tangent of the effective initial angle of internal friction; is
the total suction; t0 is the value of tan Ф’0 for = 0 and aa = 0º; t1 is the slope of tan Ф0’
vs. suction line for aa = 0º; d0 and d1 are model parameters which control the increment
of tanФ0’ with suction for aa = 5º-45º; and tan aa is the geometric tangent of the asperity
roughness.
-26-
Figures 15 a and 15 b show the fitting of the experimental values of c’0(,aa) and
tanФ’0(, aa) with the equations previously proposed. Parameters are listed in Table 2.
Figure 15: a) Effective cohesion vs. a (above) and effective cohesion vs. suction (below). b) Effective tangent of internal friction angle vs. aa (above) and effective tangent of internal friction angle vs. suction (below).
Table 2: Parameters used to adjust the variation of c0’(aa,Ψ) and tg Ф0 (aa,Ψ) with the asperity roughness angle and suction.
Parameter Value c0 2.8 kPa c1 0.3 b0 170.0 kPa b1 0.3 b2 5.0 t0 0.7 t1 0.001 d0 0.2 d1 0.008
0 10 20 30 40 50a
0.5
1
1.5
2
2.5
3
tan
' 0
Suction [MPa]200 (Experimental)200 (Model)100 (Experimental)100 (Model) 20 (Experiemntal)20 (Model)
0 50 100 150 200 250
Suction [MPa]
0.5
1
1.5
2
2.5
3
tan
' 0
Asperity angle0º (Experimental)0º (Model)5º (Experimental)5º (Model)15º (Experiemental)15º (Model)30º (Experimental)30º (Model)45º (Experimental)45º (Model)
0 10 20 30 40 50a
0
100
200
300
400
c'0 [
KP
a]
Suction [MPa]200 (Experimental)200 (Model)100 (Experimental)100 (Model)20 (Experimental)20 (Model)
0 50 100 150 200 250
Suction [MPa]
0
100
200
300
400
c'0 [
KP
a]
Asperity angle0º (Experimental)0º (Model)5º (Experimental)5º (Model)15º (Experimental)15º (Model)30º (Experimental)30º (Model)45º (Experimental)45º (Model)
a) b)
-27-
7. NUMERICAL SIMULATION OF DIRECT SHEAR TESTS ON LILLA
CLAYSTONE
The numerical simulation of the shear stress tests was carried out with the help of
Code_Bright using the joint element developed and the new mechanical constitutive law
proposed before.
The values of c’0(,aa) and tan Ф’0(,aa) incorporated the strain-softening law. This
allows considering the effect of suction and roughness angle in the softening of the joint.
The parameters fdil and fcdil, which control the dilatant behaviour of the joint with shear
stresses, were modified through the following expressions:
a
1 'tan 1 expdil
d du u
fq q
(9)
a
a
,
0 ,
'
'dil
c
cf
c
(10)
where a is the asperity roughness angle; qu is the uniaxial compression strength; d and
d are model parameters and the term atan considers the influence of roughness on
dilatancy.
Then, the amount of dilatancy depends on the level of the normal stress, on the roughness
of the joint surfaces (eq.9) and on the degradation of the interface surface, which varies
with suction (eq.10).
50 mm
100,
1
21
us us
A
B
C DE F
G H
Figure 16: Finite element mesh geometry used to numerical simulates the experimental results.
-28-
7.1 Geometry and parameters adopted in the model
The geometry of model is shown in Figure 16. The rock was assumed to be an elastic
material and the joint was modelled as a viscoplastic joint element. The joint is
discretized using 10 elements (in red). The rate of displacements used in the test
(0.05mm/min) is applied on boundaries AC and BD. Boundaries EG and FH are
horizontally fixed and boundary GH is vertically fixed. The net normal stresses used in
the test (30, 60 and 150 kPa) are applied on boundary AB. The initial liquid pressures are
-20, -100 and -200MPa which are the values of the applied suction.
Table 3: Material parameters
Rock Matrix
Mechanical Properties Value Unit
Young’s modulus [E] 27000 MPa
Poisson’s ratio [] 0.29
Rock Joint
Mechanical Properties Value Unit
Initial normal stiffness parameter [m] 100 MPa
Tangential stiffness [Ks] 500 MPa/m
Initial friction angle [0] 35º
Residual friction angle [res] 8º
Initial opening [a0] 0.1 mm
Minimum opening [amin] 0.01 mm
Viscosity [] 1 × 10-2 s-1
Stress power [N] 2.0
Uniaxial compressive strength [qu] 20 MPa
Model parameter [d] 0.3
Model parameter [d] 100
All the simulations were performed with the same parameters, except that the critical
values of shear displacements u*c and u*Ф were changed according to the strength
-29-
softening of shear stress and dilatancy of the joints. The parameters are listed in Table 3
and 4. The predictions of the numerical analysis are plotted in dashed line alongside test
measurements in Figures 17 to 21. In general the mathematical results predict well the
experimental results. However, it is not possible to simulate the contractant behaviour of
the flat joint.
0 1 2 3 4
Shear displacements [mm]
0
20
40
60
80
[K
Pa]
Suction [MPa]200 Experimental200 Model100 Experimental100 Model20 Experimental
20 Model
a) Net Normal Stress 30 KPa
0 1 2 3 4
Shear displacements [mm]
-0.08
-0.04
0
0.04
Nor
mal
dis
plac
emen
ts [
mm
]
0 1 2 3
Shear displacements [mm]
0
40
80
120
[K
Pa]
b) Net Normal Stress 60 KPa
0 1 2 3
Shear displacements [mm]
-0.2
-0.15
-0.1
-0.05
0
0.05
Nor
mal
dis
plac
emen
ts [
mm
]
0 1 2 3 4 5Shear displacements [mm]
0
40
80
120
160
200
[K
Pa]
c) Net Normal Stress 150 KPa
0 1 2 3 4 5Shear displacements [mm]
-0.2
-0.15
-0.1
-0.05
0
0.05
Nor
mal
dis
plac
emen
ts [
mm
]
Figure 17: Comparison of shear stress vs. shear displacements and normal displacements vs. shear displacements for the experimental tests and simulation results (a = 0º).
-30-
0 2 4 6
Shear displacements [mm]
0
40
80
120
160
[KP
a]Suction [MPa]
200 Experimental200 Model100 Experimental100 Model20 Experimental20 Model
a) Net Normal Stress 30 KPa
0 2 4 6
Shear displacements [mm]
-0.1
0
0.1
0.2
0.3
0.4
Nor
mal
dis
plac
emen
ts [
mm
]
0 2 4 6 8
Shear displacements [mm]
0
50
100
150
200
250
[K
Pa]
b) Net Normal Stress 60 KPa
0 2 4 6 8
Shear displacements [mm]
0
0.2
0.4N
orm
al d
ispl
acem
ents
[m
m]
0 2 4 6 8
Shear displacements [mm]
0
100
200
300
400
[K
Pa]
c) Net Normal Stress 150 KPa
0 2 4 6 8
Shear displacements [mm]
-0.1
0
0.1
0.2
Nor
mal
dis
plac
emen
ts [
mm
]
Figure 18: Comparison of shear stress vs. shear displacements and normal displacements vs. shear displacements for experimental tests and simulation results (a = 5º).
Table 4: Parameters of the softening law used for different asperity roughness angles.
0º 5º 15º 30º 45º
u*c [m] 1.0 × 10-2 8.0 × 10-3 8.0 × 10-3 3.5 × 10-3 2.0 × 10-3
u*Ф [m] 1.5 × 10-2 8.5 × 10-3 8.5 × 10-2 4.0 × 10-3 2.5 × 10-3
-31-
0 1 2 3 4
Shear displacements [mm]
0
100
200
300
400
[KP
a]Suction [MPa]
200 Experiemental200 Model100 Experimental100 Model20 Experimental20 Model
a) Net Normal Stress 30 KPa
0 1 2 3 4
Shear displacements [mm]
0
0.2
0.4
0.6
0.8
Nor
mal
dis
plac
emen
ts [
mm
]
0 2 4 6 8
Shear displacements [mm]
0
100
200
300
400
[K
Pa]
b) Net Normal Stress 60 KPa
0 2 4 6 8
Shear displacements [mm]
0
0.2
0.4
0.6
0.8
1N
orm
al d
ispl
acem
ents
[m
m]
0 2 4 6 8
Shear displacements [mm]
0
200
400
600
[K
Pa]
c) Net Normal Stress 150 KPa
0 2 4 6 8
Shear displacements [mm]
0
0.4
0.8
1.2
1.6
Nor
mal
dis
plac
emnt
s [m
m]
Figure 19: Comparison of shear stress vs. shear displacements and normal displacements vs. shear displacements for experimental tests and simulation results (a = 15º).
-32-
0 1 2 3 4 5
Shear displacements [mm]
0
100
200
300
400
[KP
a]Suction [MPa]
200 Experimental200 Model100 Experimental100 Model20 Experimental20 Model
a) Net Normal Stress 30 KPa
0 1 2 3 4 5
Shear displacements [mm]
0
0.2
0.4
0.6
0.8
Nor
mal
dis
plac
emen
ts [
mm
]
0 1 2 3 4 5
Shear displacements [mm]
0
100
200
300
400
[K
Pa]
b) Net Normal Stress 60 KPa
0 1 2 3 4 5
Shear displacements [mm]
0
0.2
0.4
0.6
0.8
1N
orm
al d
ispl
acem
ents
[m
m]
0 1 2 3 4 5
Shear displacements [mm]
0
200
400
600
[K
Pa]
c) Net Normal Stress 150 KPa
0 1 2 3 4 5
Shear displacements [mm]
-0.2
0
0.2
0.4
0.6
0.8
Nor
mal
dis
plac
emnt
s [m
m]
Figure 20: Comparison of shear stress vs. shear displacements and normal displacements vs. shear displacements for experimental tests and simulation results (a = 30º).
-33-
0 1 2 3 4Shear displacements [mm]
0
100
200
300
[KP
a]Suction [MPa]
200 Experimental200 Model100 Experimental100 Model20 Experimental20 Model
a) Net Normal Stress 30 KPa
0 1 2 3 4Shear displacements [mm]
0
0.2
0.4
0.6
0.8
1
Nor
mal
dis
plac
emen
ts [
mm
]
0 1 2 3 4
Shear displacements [mm]
0
100
200
300
400
500
[K
Pa]
b) Net Normal Stress 60 KPa
0 1 2 3 4
Shear displacements [mm]
-0.2
0
0.2
0.4
0.6
0.8N
orm
al d
ispl
acem
nts
[mm
]
0 1 2 3 4
Shear displacements [mm]
0
200
400
600
800
[K
Pa]
c) Net Normal Stress 150 KPa
0 1 2 3 4
Shear displacements [mm]
-0.2
0
0.2
0.4
0.6
Nor
mal
dis
plac
emen
ts [
mm
]
Figure 21: Comparison of shear stress vs. shear displacements and normal displacements vs. shear displacements for experimental tests and simulation results (a = 45º).
-34-
8. CONCLUSIONS
A coupled thermo-hydro-mechanical formulation for a joint element was proposed and
implemented in the finite element program Code_Bright.
A mechanical constitutive law considering the elastic and plastic displacements of the
joint is adopted to describe the stress-displacement behaviour of the joint. In the elastic
law normal stiffness depends on the evolution of the joint element opening. Plastic
behavior is defined by a hyperbolic yield surface and softening is based on a slip
weakening model. The equations theoretically developed were transformed into a
viscoplastic formulation.
Darcy’s law was adopted for the longitudinal hydraulic constitutive law. However, the
transversal flux is calculated proportional to pressure drop between joint surfaces (Segura,
2008). A retention curve with an air pressure entry dependant on joint aperture (Olivella
& Alonso, 2008) is adopted to calculate the degree of saturation of the joint. The vapour
diffusivity is calculated by Fick’s law and the heat conduction through the joint is
obtained by Fourier’s law.
A numerical simulation of rough rock joints under shear stress subjected to forced flow
along the joint was carried out to validate the numerical tool. The comparison between
test and numerical results was positive and it was concluded that the formulation is able to
reproduce the main characteristic of coupled mechanical-flow joint behaviour. Shear
stress softening and dilatancy with shear displacements as well as the increments of
permeability with displacement was well captured.
The influence of suction on joint behaviour was also experimentally investigated. It is
believed that this is an important issue in applications. No reference of this effect, which
was found to be very significant in the rock tested, was found in the literature.
An available direct shear device was successfully modified to test rock joints under
controlled relative humidity of the specimens. Modifications included the addition of a
vapour circulation system and the improvement of the acquisition data incorporating an
analogue data acquisition device.
The carving process adopted to build different asperity angles allowed exploring the
roughness effects on the shear strength and on dilatancy of joints.
-35-
The shear test results showed the marked dependency of peak shear stress and dilatancy
on suction and roughness. The shear strength and dilatancy decrease when suction
decreases. However, the dependency of residual strength with suction was not so clear.
Comparing the shear strength and dilatancy recorded for different roughness, it was noted
that greater roughness implied greater shear strength as expected. It was also observed
that a rougher asperity results in smaller values of displacements to reach residual
strength. In other words rougher joints are more brittle. This brittle behaviour induces
higher damage on the joints surfaces, and this damage extends to the whole joint surfaces.
A consequence of this phenomenon is that rougher surfaces exhibit a lower dilatancy.
It was also obtained that the degradation of joints increases with the applied work in all
cases. However, the additional effect of suction should be considered an independent
contribution. The higher the suction, for a given value of total work, the lower the joint
degradation
New mathematical expressions for the strength parameters (initial effective cohesion
(c0’) and initial effective tangent of internal friction angle (tanФ’0) of the asymptote of
the hyperbolic yield surface are proposed. These expressions consider the effects of
suction and asperity roughness on strength parameters. Also, the dilatancy parameters
were modified taking into account suction and geometry of joints. Both modifications
were introduced in the constitutive law of the interface element implemented in
Code_Bright.
The numerical simulation performed reproduces in a satisfactory manner the experiments
run on rock joints of Lilla claystone.
REFERENCES
Barton N., Bandis S. & Bakhtar K. (1985) Strength, deformation and conductivity
coupling of rock joints. International Journal of Rock Mechanics and Mining
Science & Geomechanics Abstracts, 1985, 22(3):121–140,
Berdugo, I.R. (2007) Tunnelling in sulphate-bearing rocks expansive phenomena. PhD
Thesis. Department of Geotechnical Engineering and Geosciences, UPC.
Carol I., Prat P. & Lopez C.M. (1997) A normal/shear cracking model. Application to
discrete crack analysis. ASCE Journal of Engineering Mechanics, 123(8):765–773.
-36-
CODE_BRIGHT. DIT-UPC. (2000) A 3-D program for thermo-hydro-mechanical
analysis in geological media. User’s guide. Barcelona: Centro Internacional de
Métodos Numéricos en Ingeniería (CIMNE).
Escario V. & Saez J. (1986) The shear strength of partly saturated soils, Geotechnique
36(3): 453-456.
Fredlund D.G. and Rahardjo H. (1993) Soils mechanics for unsaturated soils. New Wiley
& Sons.
Garcia-Castellanos, D., Vergés, J. Gaspar-Escribano, J. and Cloetingh, S. (2003) Interplay
between tectonics, climate, and fluvial transport during the Cenozonic evolution of
the Ebro Basin (NE Iberia). J. Geophys. Res., 108 (B7): ETG 8-1-8-18
Gens A., Carol I. & Alonso E.E. (1990) A constitutive model for rock joints; formulation
and numerical implementation. Computers and Geotechnics, 9:3–20.
Goodman R.E., Taylor R.L. & Brekke T.L. (1968) A model for the mechanics of jointed
rock. ASCE Journal of the Soil Mechanics and Foundations Division,
94(SM3):637–659.
Guiducci C., Pellegrino A., Radu J.P., Collin F. & Charlier R. (2002) Numerical
modelling of hydro-mechanical fracture behaviour. In Pande & Petruszczak, editor,
Numerical Models in Geomechanis-NUMOG VIII, pages 293–299, Lisse, Swets &
Zeitlinger.
Hans J. (2002) Etude expérimental et modélisation numérique multiéchelle du
comportamente hydromécanique de répliques de joints rocheux. Thése de doctorant-
Université Joseph Fourier Grenoble.
Indraratna B., Ranjith P.G., Price J. R. and Gale W. (2003) Two-Phase (Air and Water)
Flow through Rock Joints: Analytical and Experimental Study. Journal of
Geotechnical and Geoenvironmental Engineering, Vol.129 No.10, October.
Jennings J.E. & Robertson A.MacG.(1969) The stability of slopes cut into natural rock.
Conf. on Soil Mechanics and Foundation Engineering, Mexico,Vol. II, pp. 585-590.
Lee H.S. & Cho T.F. (2002) Hydraulic Characteristics of Rough Fractures in Linear Flow
under Normal and Shear Load. Rock Mechanics and Rock Engineering, 35(4),229-
318.
-37-
López, C.M. (1999) Análisis microestructural de la fractura del hormigón utilizando
elementos finitos tipo junta. Aplicación a diferentes hormigones. PhD thesis,
ETSECCPB, UPC, Barcelona, España.
Oldecop, L. & Alonso E.E. (2001) A model of rockfill compressibility. Geotechnique 51
No. 2, 127–139.
Olivella, S. & Alonso, E. E. (2008) Gas flow through clay barrier. Geotechnique 58 No. 3,
157–176.
Olivella, S., Gens, A., Carrera, J. & Alonso, E. E. (1995) Numerical formulation for a
simulator (CODE_BRIGHT) for the coupled analysis of saline media. Engng.
Comput. 13, No.7, 87–112.
Olsson R. & Barton N. (2001) An improved model for hydromechanical coupling during
shearing of rock joints. International Journal of Rock Mechanics and Mining
Sciences, 2001, 38:317–329.
Palmer A.C., & Rice J.R. (1973) The growth of slip surfaces in the progressive failure of
over-consolidated clay. Proc. Roy. Soc. Lond. A 332, 527-548.
Perzyna P. (1963) The constitutive equations for rate sensitive materials, Quarterly of
Applied Mathematics. no. 20, Pags. 321-332.
Pineda, J.A., De Gracia, M. and Romero E., (2010) Degradation of partially saturated
argillaceous rocks: influence on the stability of geotechnical structures. 4th Asia-
Pacific Conference on Unsaturated Soils, Newcastle, Australia. Unsaturated soils-
Buzzi, Fityus & Sheng (eds.). Taylor & Francis Group.
Romero, E.E., (2001) Controlled suction techniques. Proc 4º Simposio Brasileiro de Sols
Nao Saturados. Gehling and Schnaid Edits. Porto Alegre, Brasil, 2001, pp 535-542.
Segura J. Ma. (2008) Coupled HM analysis using zero-thickness interface elements with
double nodes. Tesis de Doctorado-Universidad Politécnica de Catalunya, Barcelona.
Tarragó, D. (2005) Degradación mecánica de argilitas sulfatadas y su efecto sobre la
expansividad. BSc dissertation. Universitat Politécnica de Catalunya, Barcelona.
Van Genuchten, M. Th. (1980) A closed-form equation for predicting the hydraulic
conductivity of unsaturated soils. Soil Sci. Soc.Am. J. 1980, 49, No. 9, 892–898
-38-
Zienkiewicz O.C. & Cormeau I. (1974) Visco-Plasticity-Plasticity and Creeping Elastic
Solids-A Unified Numerical Solution Approach. International Journal for
Numerical Methods in Engineering, Vol 8, 821-845.
-39-
APPENDIX. A COUPLED THERMO-HYDRO-MECHANICAL FORMULATION
OF JOINTS
A1. Mechanical formulation
The mechanical formulation of the joint element is defined by the relationship between
stress and relative displacements of the joint element mid-plane (Figure A1). Then, the
mid-plane relative displacements are interpolated using the nodal displacements and the
shape functions.
4 4n u
mp mp js mp
u
u
w r N I I u (A1)
where un and us are the normal and tangential relative displacements of the element’s
mid-plane (see Figure A1 b), r is the rotation matrix that transforms the relative
displacements in the local orthogonal coordinate system into the global coordinate
system, Nmpu is a matrix of shape functions, I4 is a identity matrix of 4th order and uj is the
vector of nodal displacements.
The stress tensor of the mid-plane is calculated as:
''mp mp
mp
σ D w (A2)
where ’mp is the net effective stress at the mid-plane of the element and it is defined as
’mp = mp- max{Pgmp; Plmp} (where mp is total mean stress; Pgmp is the gas pressure and
Plmp is the liquid pressure in the mid-plane of the element); is the tangential stress at
mid-plane and D is the stiffness matrix, which relates relative displacements to stress state
(see Figures A1a and Ab).
Note that the mechanical response is defined in terms of a net stress (excess of total stress
over air pressure) when the joint is not saturated. Once saturated, the definition adopted
for effective stress results in Terzaghi’s principle.
-40-
Figure A1: Joint element with double nodes. a) Stress state at the mid-plane of the joint element. b) Relative displacement defined at mid plane.
A2. Mass and energy balance equation
The two phase flow through a single joint is analyzed by formulating the water, air and
energy balance equations at the mid-plane of the element. The fluxes at mid-plane are
calculated by interpolating the leak-off at the element boundaries (see Figure A2).
Water mass balance equation
The water mass balance equation for a differential volume of joint is:
aa j ' j '
w wl l g g w w w w w
l l g g l gmp mp
S S ddl S S dl f
t dt
(A3)
where wl is the mass of water in liquid phase, w
g is the mass of gas in liquid phase
(vapour), a is the opening of the joint element, dl is the discrete length of the joint
element, lS is the liquid degree of saturation, gS is the gas degree of saturation,
j 'wl mp is the liquid flux at mid-plane, j 'wg mp
is the vapour flux at the mid- plane and
wf is an external supply of water. The first term of Equation A3 considers the storage
change of mass at constant volume, the second term is the storage change caused by
changes of joint opening,
The fluxes at mid-plane are calculated by:
a dl
0 0
a dl
0 0
j ' q q a a
j' q q a a
w w w w wl l lt lt l ll llmp
w w w w wg g gt gt g gl glmp
dl i dl i
dl i dl i
(A4)
a) b)
1 2
3 4
mp1 mp2
ut
dl
a0 aun
us
-41-
where q lt , qgt , q ll , and qgl are the advective (liquid or gas) transversal and longitudinal
fluxes at the element boundaries respectively and wlti ,
wgti ,
wlli ,
wgli are the nonadvective
(liquid or gas) transversal and longitudinal fluxes at the element boundaries (Fig. A2 a).
The first term of the equation 4 corresponds to the transversal fluxes at mid-plane of the
joint (calculated by the pressure drop between surfaces pmp1=P3-P1 and pmp2=P4-P2) (Fig.
A2 b). And the second term corresponds to the longitudinal fluxes at mid-plane calculated
considering the average pressure in nodes ( mp1 3 11
p P +P2
and mp2 4 21
p P +P2
) (Fig. A2 c)
P1 P2
P 4P3
pmp2=P4-P2
pmp1=1(P3+P1) pmp2=1(P4+P2)2 2
[qll+ill][qll+ill] mp2
transversal fluxes
longitudinal fluxes
mp1
dl
a
P1 P2
P3 P4
dl
a
dl
aaccumulatedmass energy
[qlt+ilt] 0
[qlt+ilt] a
[qll+ill] 0 [qll+ill]dl
[qlt+ilt]mp2[qlt+ilt]mp1
pmp1=P3-P1
1 2
43
mp1 mp2
Figure A2: a) Schematic view of the mass balance of joint element. b) Transversal fluxes. c) Longitudinal fluxes.
Air mass balance equation
The air mass balance equation considers the dry air and the air dissolved in the water
phase. Its expression is:
aa j' j '
a al l g g a a a a a
l l g g l gmp mp
S S ddl S S dl f
t dt
(A5)
where al is the mass of air dissolved in liquid phase, a
g is the mass of gas phase (dry air),
j 'al mp is the air dissolved fluxes at mid plane, j 'ag mp
is the gas flux at mid plane, and
af is an external supply of air.
a)
b)
c)
b) b)
-42-
Internal energy balance for the element
The internal energy balance for the element is expressed by:
aa
i j j
l l l g g g
l l l g g g
Ec El E gmp mp mp
E S E S ddl E S E S dl
t dt
f
(A6)
where the energy of the liquid and gas phases is calculated by:
w w a a w w a al l l l l l l l l l lE E E E E
w w a a w w a ag g g g g g g g g g gE E E E E
(A7)
where wlE and
alE are the internal energy of water and/or air in liquid phase per unit mass
of water and/or air respectively, wl and
al are the mass of water and/or air in liquid
phase, wgE and
agE is the internal energy of water and/or air in gas phase per unit mass of
water and/or air respectively, wg and
ag are the mass of water and/or air in gas phase
The conduction of heat at mid-plane of joint is calculated by:
a dl
0 0i ac ct clmp
i dl i (A8)
where [ic]mp is the heat flux at the mid-plane of the joint element, ict is the transversal heat
flux, and icl is the longitudinal heat flux at the element boundaries.
The energy fluxes are calculated considering the advective fluxes:
j j' j 'w w a aEl l l l lmp mp mp
E E
j j ' j 'w w a aE g g g g gmp mp mp
E E
(A9)
The weighted residual method is applied to obtain the discrete form of equations. Finally,
Equations A2, A3, A5, A6 are solved simultaneously. The unknown’s vector for each
node includes the normal and shear relative displacements (un, us), the gas and liquid
pressure (Pg, Pl) and the temperature (T).
-43-
A3. Constitutive Models
The mechanical response of the joint was modelled by means of nonlinear elasto-
viscoplastic formulation. The viscoplastic approach provides numerical advantages (no
need to use return algorithm in particular).
Darcy´s law describes the advective flow for longitudinal directions. A flow proportional
to the pressure drop is used in transversal direction. The non advective fluxes (vapour
diffusivity) were modelled by Fick´s law. The longitudinal permeability and the air entry
pressure of the joint depend on its opening. Finally, the heat conduction through the joint
is calculated by Fourier’s law.
Mechanical model based on elasto-viscoplastic formulation
The elastic formulation proposed describes the elastic normal stiffness by means of a
nonlinear law which depends on the joint opening (Gens et al., 1990).
The viscoplastic formulation (Perzyna,1963; Zienkiewicz et al., 1974) allows the
treatment of a non-associated plasticity and a softening behaviour of joints subjected to
shear displacements.
Total displacements w are calculated by adding reversible elastic displacements, we, and
viscoplastic displacements wvp, which are zero when stresses are below a threshold value
(the yield surface):
e vp w w w (A10)
Normal and shear displacements are represented by a two-element vector ,n su u in the
two-dimensional case:
,Tn su uw (A11)
a) Elastic behaviour
The elastic behaviour of the joint relates the normal effective (’) and the tangential
stresses () to the normal (un) and the tangential (us) displacement of the joint element
through the normal (Kn) and tangential stiffness (Ks), respectively. Normal stiffness
depends on the opening of the joint, as indicated in Figure A3 and equation (A13):
-44-
1/ 0 '
0 1/n n
s s
u K
u K
(A12)
mina an
mK
(A13)
where m is a parameter of the model; a is the opening of the element and amin is the
minimum opening of the element (at this opening the element is closed).
'
amin a
Figure A3: Elastic constitutive law of the joint element. Normal stiffness depends on joint opening.
b) Visco-plastic behaviour
The visco-plastic behaviour of the joint was developed taking into account the
formulations proposed by Gens et al., (1990) and Carol et al., (1997) for rock joints.
According to these theories, it is necessary to define a yield surface, a plastic potential and
a softening law.
Visco-plastic displacements occur when the stress state of the joint reaches a failure
condition. This condition depends on a previously defined yield surface. In this study a
hyperbolic yield surface (Figure A4) based on Gens et al., (1990) was adopted:
22 ' ' tan 'F c (A14)
where is the shear stress; c’ is the effective cohesion; ’is the effective net normal stress
and tanФ’ is the tangent of internal friction effective angle. Note that cohesion and
friction angle are defined for the asymptote of the hyperbolic yield function.
Variation of these parameters results in a family of yield surfaces (Fig. A4 a).
-45-
c) Softening law
The strain-softening of the joint subjected shear stress is modelled by means of the
degradation of the strength parameters. The degradation of parameters c’ and tan’
depends linearly on viscoplastic shear displacements. This is based on the slip weakening
model introduced by Palmer & Rice (1973). In this way the cohesion decays from the
initial value c0’ to zero and the tangent of friction angle decays from the peak (intact
material) to the residual value as a function of a critical visco-plastic shear displacements
(u*). Two different values of u* are used to define the decrease of cohesion (u*c’) and
friction angle (u*tanФ’) (see Figs. A4 b and c). The mathematical expressions are:
0 *
u' ' 1
u
vps
c
c c
(A15)
where c’ is the effective cohesion which corresponds to the visco-plastic shear
displacement usvp; c’0 is the initial value of the effective cohesion; u*c is the critical value
of shear displacement for which the value of c’ is zero
0 0 *
utan ' tan ' tan ' tan '
u
vps
res
(A16)
where tanФ’ is the tangent of internal friction effective angle, which corresponds to visco-
plastic shear displacement usvp; tanФ’0 is the tangent of the peak friction angle; tanФ’res is
the tangent of the internal friction effective residual angle and u*Ф is the critical value of
shear displacement when the value of tanФ’ is equal tanФ’res.
0
170°
res
1
2
c tan
us
c0
uc*
tan0
tanres
vp usvputan*
11
2
2
Figure A4: a) Evolution of the failure surface due to softening of the strength parameters. b) Softening law of cohesion. c) Softening law of tanФ.
d) Visco-plastic displacements
-46-
If F < 0, the stress state of the joint element is inside the elastic region. If F 0, the
displacements of the joint element have a visco-plastic component. Viscoplastic
displacements are calculated by:
0
vpd F G
dt F
w
σ
(A17)
where is a viscosity parameter. In order to ensure that there is no viscoplastic flow
below the yield, the following consistency conditions should be met:
0
0
0 0
0
Fif F
F
FF if F
F
(A18)
where F0 can be any convenient value of F to render the above expressions non-
dimensional. In this study F0 = 1.
The visco-plastic displacement rate is given by a power of law:
vp Nn
Gu F
vp Ns
Gu F
(A19)
e) Plastic potential surface and dilatancy
The associativity rule allows the calculation of displacements directions. The derivative of
G with respect to stresses includes the parameters fdil and fcdil which take into account
the dilatant behaviour of the joint under shear stresses (Lopez, 1999):
2 tan ' ' ' tan ' , 2Tdil dil
c
Gc f f σ
(A20)
The parameter fdil accounts for the decrease of dilatancy with the level of the normal
stress acting on the joint. And fcdil considers the degradation of the joint surfaces due to
shear displacements. The following expressions describe these effects:
-47-
' '1 expdil
du u
fq q
(A21)
where uq is the compression strength of the material for which dilatancy vanishes and d
is a model parameter.
0
'
'dil
c
cf
c (A22)
where c’ is the cohesion value for the visco-plastic shear displacement usvp and c0’ is the
initial value of the cohesion.
Hydraulic model
The transversal advective flux flow through the joint is calculated by means a transversal
intrinsic permeability and the pressure drop between joint surfaces (Segura, 2008).
Furthermore, the longitudinal advective flow is calculated using a longitudinal intrinsic
permeability and a generalized Darcy’s law. Therefore, it is necessary to define the
longitudinal and transversal intrinsic permeabilities of the joint. Likewise, in the case of
joints under unsaturated conditions, the water retention curve should be specified.
a) Advective fluxes
The transversal flux is calculated as:
lt rltl t mp
l
k kq p
(A23)
where ltk is the transversal intrinsic permeability for liquid; rltk is the transversal relative
permeability for the liquid, l is the dynamic viscosity of the liquid and mpp
is the
pressure drop between the two surfaces of the joint element.
The generalized Darcy’s law for the longitudinal flow reads:
mpll rllll
l
pk kq
l
g
(A24)
-48-
where llk is the longitudinal intrinsic permeability for the liquid, rllk is the longitudinal
relative permeability for the liquid, l is the dynamic viscosity of the liquid and g is the
gravity vector.
b) Non-advective fluxes (vapour diffusivity)
The nonadvective flux (vapour diffusivity) is calculated by means of Fick’s law:
w w wg g g g gS D i I (A25)
where is the tortuosity, wgD is the molecular diffusion coefficient, which depends of
temperature and gas pressure, I is the identity matrix and wg is the mass fraction of vapour
in gas phase.
c) Intrinsic Permeability
The longitudinal fluid flow has been analyzed as a laminar flow between two smooth and
parallel plates separated a given hydraulic opening (e). Based on this hypothesis, the
longitudinal hydraulic conductivity of the joint is calculated by means of cubic law:
3
12l
g eK
(A26)
where r is the fluid density, g is the gravity and is the fluid viscosity.
Then the equation of intrinsic permeability is given by:
2
12ll
ek
(A27)
The hydraulic opening (e) of joints will be related to its geometrical aperture (a) and to
the roughness of joint surfaces (JRC) by means of the law proposed by Barton et al.,
(1985). Substituting in eq. A27 Barton´s expression, the longitudinal intrinsic
permeability can be expressed as:
22
2.5
a 1
12llkJRC
(28)
-49-
The transversal intrinsic permeability ltk is considered equal to the value for the
continuum media.
d) Water retention curve
The degree of saturation of joints is calculated using the standard retention curve
proposed by Van Genuchten (1980),
1
1-
1lSP
(A29)
where Sl is the liquid degree of saturation; lg PP is the current suction; is a model
parameter and P is the air pressure entry necessary to desaturate the joints.
The air pressure entry of a joint depends on the hydraulic opening as suggested by
Olivella & Alonso (2008):
1 2
1 1 2P
r r e
(A30)
P is obtained when (1/r1) = 0 and r2 = e/2. The wetting angle has been assumed equal to
zero. If equation A30 is combined with equation A27 the air pressure is obtained as:
l
l
k
kPP 0
0 (A31)
Also, P is scaled with surface tension if temperature effect are considered:
0
00
l
l
k
kPP (A32)
e) Relative permeability
The relative permeability is calculated as:
nlrl ASk (A33)
where A=1.0 and n=3.
f) Thermal model
The heat conduction is given by Fourier´s law:
-50-
ci T (A34)
where is the thermal conductivity and T is the temperature gradient.
The thermal conductivity is made dependant on the degree of saturation of the joint as:
sat l dry lS S1 (A35)
where sat is the thermal conductivity of the water saturated joint, dry is the thermal
conductivity of the dry joint and Sl is the degree of saturation.