submitted to manuel rocha medal, isrm · submitted to manuel rocha medal, isrm barcelona, december...

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


-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


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


Load Cell

LVDT


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)

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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


0

20

40

60

80

[K

Pa]

a) Net Normal Stress 30 KPa

0 1 2 3 4


-0.2

-0.15

-0.1

-0.05

0

0.05

Nor

mal

dis

plac

emen

ts [

mm

]

0 1 2 3


0

40

80

120

[K

Pa]

b) Net Normal Stress 60 KPa

0 1 2 3


-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


-0.2

-0.15

-0.1

-0.05

0

0.05

Nor

mal

dis

plac

emen

ts [

mm

]0 1 2 3 4


0

20

40

60

80

[K

Pa]


0 1 2 3 4


-0.2

-0.15

-0.1

-0.05

0

0.05

Nor

mal

dis

plac

emen

ts [

mm

]

0 1 2 3


0

40

80

120

[K

Pa]


0 1 2 3


-0.2

-0.15

-0.1

-0.05

0

0.05

Nor

mal

dis

plac

emen

ts [

mm

]


0

40

80

120

160

200

[K

Pa]

Suction [MPa]200 100 20



-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


0

40

80

120

160

[K

Pa]


0 2 4 6


-0.1

0

0.1

0.2

0.3

Nor

mal

dis

plac

emen

ts [

mm

]

0 2 4 6 8


0

50

100

150

200

250

[K

Pa]


0 2 4 6 8


0

0.2

0.4

Nor

mal

dis

plac

emen

ts [

mm

]

0 2 4 6 8


0

100

200

300

400

[K

Pa]

Suction [MPa]20010020


0 2 4 6 8


-0.1

0

0.1

0.2

Nor

mal

dis

plac

emen

ts [

mm

]

=20 MPa =100 MPa =200 MPa

-19-


0 1 2 3 4


0

100

200

300

400

[K

Pa]


0 1 2 3 4


0

0.2

0.4

0.6

0.8

Nor

mal

dis

plac

emen

ts [

mm

]

0 2 4 6 8


0

100

200

300

400

[K

Pa]

b) Net normal stress 60 KPa

0 2 4 6 8


0

0.2

0.4

0.6

0.8

1

Nor

mal

dis

plac

emen

ts [

mm

]

0 2 4 6 8


0

200

400

600

[K

Pa]

Suction [MPa]200

100 20


0 2 4 6 8


0

0.4

0.8

1.2

1.6

Nor

mal

dis

plac

emnt

s [m

m]

=20 MPa =100 MPa =200 MPa

-20-


0 1 2 3 4 5


0

100

200

300

400

[K

Pa]


0 1 2 3 4 5


0

0.2

0.4

0.6

0.8

Nor

mal

dis

plac

emen

ts [

mm

]

0 1 2 3 4 5


0

100

200

300

400

[K

Pa]


0 1 2 3 4 5


0

0.2

0.4

0.6

0.8

1

Nor

mal

dis

plac

emen

ts [

mm

]

0 1 2 3 4 5


0

200

400

600

[K

Pa]


0 1 2 3 4 5


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-


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

100

200

300

[KP

a]a) Net Normal Stress 30 KPa


0

0.2

0.4

0.6

0.8

1

Nor

mal

dis

plac

emen

ts [

mm

]

0 1 2 3 4


0

100

200

300

400

500

[K

Pa]

b) Net normal stress 60 KPa

0 1 2 3 4


-0.2

0

0.2

0.4

0.6

0.8

Nor

mal

dis

plac

emnt

s [m

m]

0 1 2 3 4


0

200

400

600

800

[K

Pa]

Suction [MPa]200

100 20


0 1 2 3 4


-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


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



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


0

200

400

600

800

Shea

r St

ress

[K

Pa]


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


0

200

400

600

800

She

ar S

tres

s [K

Pa]


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


0

200

400

600

800

She

ar S

tres

s [K

Pa]


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


0

200

400

600

800

She

ar S

tres

s [K

Pa]


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


0

20

40

60

80

[K

Pa]

Suction [MPa]200 Experimental200 Model100 Experimental100 Model20 Experimental

20 Model


0 1 2 3 4


-0.08

-0.04

0

0.04

Nor

mal

dis

plac

emen

ts [

mm

]

0 1 2 3


0

40

80

120

[K

Pa]


0 1 2 3


-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]


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


0

40

80

120

160

[KP

a]Suction [MPa]

200 Experimental200 Model100 Experimental100 Model20 Experimental20 Model


0 2 4 6


-0.1

0

0.1

0.2

0.3

0.4

Nor

mal

dis

plac

emen

ts [

mm

]

0 2 4 6 8


0

50

100

150

200

250

[K

Pa]


0 2 4 6 8


0

0.2

0.4N

orm

al d

ispl

acem

ents

[m

m]

0 2 4 6 8


0

100

200

300

400

[K

Pa]


0 2 4 6 8


-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


0

100

200

300

400

[KP

a]Suction [MPa]

200 Experiemental200 Model100 Experimental100 Model20 Experimental20 Model


0 1 2 3 4


0

0.2

0.4

0.6

0.8

Nor

mal

dis

plac

emen

ts [

mm

]

0 2 4 6 8


0

100

200

300

400

[K

Pa]


0 2 4 6 8


0

0.2

0.4

0.6

0.8

1N

orm

al d

ispl

acem

ents

[m

m]

0 2 4 6 8


0

200

400

600

[K

Pa]


0 2 4 6 8


0

0.4

0.8

1.2

1.6

Nor

mal

dis

plac

emnt

s [m

m]


-32-

0 1 2 3 4 5


0

100

200

300

400

[KP

a]Suction [MPa]



0 1 2 3 4 5


0

0.2

0.4

0.6

0.8

Nor

mal

dis

plac

emen

ts [

mm

]

0 1 2 3 4 5


0

100

200

300

400

[K

Pa]


0 1 2 3 4 5


0

0.2

0.4

0.6

0.8

1N

orm

al d

ispl

acem

ents

[m

m]

0 1 2 3 4 5


0

200

400

600

[K

Pa]


0 1 2 3 4 5


-0.2

0

0.2

0.4

0.6

0.8

Nor

mal

dis

plac

emnt

s [m

m]


-33-


0

100

200

300

[KP

a]Suction [MPa]




0

0.2

0.4

0.6

0.8

1

Nor

mal

dis

plac

emen

ts [

mm

]

0 1 2 3 4


0

100

200

300

400

500

[K

Pa]


0 1 2 3 4


-0.2

0

0.2

0.4

0.6

0.8N

orm

al d

ispl

acem

nts

[mm

]

0 1 2 3 4


0

200

400

600

800

[K

Pa]


0 1 2 3 4


-0.2

0

0.2

0.4

0.6

Nor

mal

dis

plac

emen

ts [

mm

]


-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.

submitted to manuel rocha medal, isrm · submitted to manuel rocha medal, isrm barcelona, december...

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