Charlez Rock Mechanics V1

101

Chapter 5. Plane theory of elasticity

5.6.3

Expression of stresses and displacements

The expressions of the stresses are obtained directly by deriving Airy's potential twice 3 [Eqs (5.11)]. Taking account of (5.73) and (5.79), one obtains

+ 17yy == 2 [
17xx

(l7 yy -

(5.81)

with


(5.82)

We may note finally that in polar coordinates Eqs (5.81) and (5.82) can be written

2G(u p

+ iue)

= e- i9

[(3 - 4v)
zcp(z) -1j;(z)]

+ 1790 = 2 [
I7pp

(5.83)

The solution of an elastic boundary problem is therefore based upon the determination of two complex functions, which have to be determined from the boundary conditions specific to each problem. The form of the complex potential can be better specified in the case of finite or infinite connected regions i.e. bounded by several simple contours L1, L2 ... Lm, Lm+I (Fig. 5.5) where Lm+l is supposed to contain all the other contours. For infinite regions, the contour Lm+I has entirely moved to infinity. Furthermore one assumes that these contours do not intersect themselves (L, n LJ =0 \;12 and VJ).

G Fig. 5.5. Multiply connected regions. (after Muscnelishvili. 1954).

3Indeed 8",(z)

I

8;;

== ",'(z).

102

Part II. Mechanism of material strain

In these conditions and remembering the uniqueness of the elastic solution both in terms of stresses and displacements, one can show (see Muskhelishvili pp. 121 to 126) that the complex potentials are analytic functions of the complex variable z i.e can be expanded in the form of a Laurent series. This fundamental property (in the case of multiply connected regions) will have an important practical repercussion on complex potential determination.

5.7

CONFORMAL MAPPING. TRANSFORMATION OF THE BASIC FORMULA

Complex representation is especially useful when the boundaries of the related domain are simple. In particular, they are perfectly adapted to circular boundaries. On the other hand, when the boundaries are more complicated (elliptic, square ... ), the boundary conditions can no longer be expressed in polar form. One of the great advantage of conformal mapping is to allow the transformation of a region limited by a curvilinear cavity into a region limited by a circle. The resolution of a curvilinear problem can thus be achieved in polar coordinates. A transformation between a real plane (compJex variable z) and an image plane (complex variable () is called conformal if the application of z in ( is univocal, reciprocal and maintains the angles. Let us consider (Fig. 5.6) in the real plane a point z = x + iy and its image ( such that ( = X + .Y = pe,e. The transformation will be written z

= x + iy = wee) = w(pe,e)

(5.84) y

y

o

x

--~------~--~--+-~....~

Fig. 5.6. Transformation of a curvilinear cavity inlo a cil"cular cavity (conformal mapping).

X

103

Chapter 5. Plane theory of elasticity

It is a univocal and plane. If the direction

COl'ref!J)o,naen<:e between the real plane and the of intensity Idz I according to of the image plane will undergo to direction OJ A'. ,-,H"'H}','"" in coordinates will therefore be such that "''-o""HV''-oU.'

dz

(5.85)

0: being the of the vectors OA and Ox. From (5.84) one can easily deduce an equation between 0: and B. Indeed

e,,:t

(5.86) ""'''"H''''~

Similarly, the complex

<1>(z) and \[1(z) can be written in the image plane <1> [wee)] =

\[1(z)

\[1 [wee)]

«) 1li 1

(5.87)

«)

and,

= Taking account of (5.86) and (5.87) and formed formulas for stresses

one obtains the trans-

+ \[1I()] (5.88)

+ Ilil«)] (()] Similarly, it is easily found that

+ ius) = p

5.8

EXPRESSION OF IN THE IMAGE

(5.89)

CONDITIONS

Equation (5.88) makes it possible therefore to solve an elastic

<1>1 and \[11 have been determined. This will be made via the

UUlIlIUa.lY

problem once conditions.

104

Part II. Mechanism of material strain

Let us consider 5.7) for this purpose the boundary AB and the surface force applied on a element ds. The boundary conditions can be written that is

Fn

cr xx

cos a

cryy

sin Q'

+ crxy sin + xy cos (J'

Q'

(5.90) Q'

y

dx

Bt..----

A

Fig. 5.7. Boundary conditions.

If one travels along AB an:C}-(:lo,:IiVI'I wards the negative x) while y increases is convex) dy cosO'

As

dx the

domain sin Q'

=

dx

(5.91)

(5.11) ay2

From (5.90) (5.91) (5.92) one obtains

Fi{ that is after

=

on AB

au +t.au)] ay

-

AB -

1

AB

+ iFi{)ds

(5.93)

105

Chapter 5. Plane theory of elasticity

or again taking account of (5.73) and (5.79) i {

~B

(F~ + iF~) ds = [rp(z) + zrp'(z) + 1j;(Z)]

AB

(5.94)

in which z belongs to the real (curvilinear) boundary. Eq. (5.94) can be expressed in the image plane by introducing the variable ( such that z = w(() which leads finally to w(() - - _ _

h+ih

+ w'(()

i {

(F~ + zF~) ds

with

=

h + ih

rp~(()+1fl(()

rpl(()

JAB

(5.95)

In the specific case for which the region is mapped on to a circle one introduces in Eq. (5.95) the notation (J = poe ifJ where Po is the radius of the mapped circle that is (the index 1 of complex potentials are not useful anymore)

rp((J)

5.9

+

w((J) rp'((J) + 1f((J) w'((J) (J = poe iB

= f((J)

(5.96)

DETERMINATION OF COMPLEX POTENTIAL BY BOUNDARY INTEGRALS

The objective is now to determine the two complex potentials rp(() and 1j;(() so as to satisfy the boundary conditions. (5.96) can be rewritten in the integral form (after multiplying each member by 1 / (J - ()

1 -y

rp((J) d(J (J - (

+

1 -y

W((J) rp'((J)~ w' ((J ) (J - (

+ l1j;((J)

d(J (J - ( f ((J)d(J = -y (J-(

I -y

(5.97)

where, is the unit circle. The interest of the boundary integration (5.97) is to reveal integrals of the Cauchy type that is in a general case g(t) dt (5.98)

1 -y

t- (

Indeed, the Cauchy integral has some mathematical properties of which one can take advantage for the determination of the complex potentials rp(() and 1j;((). Provided g(() is analytic (which is the case for multiply connected regions) at every point outside " (including infinity), one can show (for demonstration see Goodier and Timoshenko, 1970, pp. 207 to 209) that

1 -y

g(t)dt == -21rig(() t- (

and

1 -y

g(t)dt t- (

=0

(5.99)

106

5.10

Part II. Mechanism of material strain

APPLICATION TO THE CASE OF AN INFINITE PLATE CONTAINING AN ELLIPTICAL CAVITY

Let us consider an infinite elastic plate (Fig. 5.8) containing an elliptical cavity of axes a and b. That is the conformal transformation (5.100)

(J y

y

'--+ ~

x

x

......... ~---~ 2a For m= I the ellipse decays into a crac}(

or

length 2a

(J

Fig. '5.8. Plate containing an elliptical hole.

To the ellipse of the real plane corresponds a circle of unit radius in the image plane. Indeed, by identifying the real and imaginary parts of (5.100) one obtains for p = 1 (that is for ( = u = e,9)

x=R(1+m)cosO

y=R(1-m)sinO

(5.101)

which are the parametric equations of an ellipse in the real plane whose semi-axes are such that (5.102) b = R(1- m) a = R(l + m) Solving (5.102) with respect to Rand m , one obtains m= a-b

a+b

R= a+b 2

(5.103)

107

Chapter 5. Plane theory of elasticity

a and b, it is therefore easy to calculate Rand m which are the parameters of the transformation. We may note that this latter possesses two extreme cases: if m 0, the ellipse becomes a on the contrary, if m I, it is transformed into a line (in other words a crack of 2a see To solve the problem one has now to determine the '"v... v,,, .... nnr.pn'r.l In this particulate case,the boundary condition is written its expression 100) that is

dO'

--+ 0'-(

(5.104)

In (5.104) both second and third terms of the first member vanish 4 while the first one is such that r.p(O')dO'

1 'Y

nel·elCH'e.

0'-(

105)

the first complex potential is such that [replacing (5.105) into (5.104)]

r.p«)

=

1

i

0'-(

The second '"'v,,, .... ,,,.... potential 1/;(() can be determined by conjugating is after on l'

(5.106) that

Among the three Cauchy integrals of (5.107), the first one vanishes while the two others are such that

(5.108)

The second

",.."."..",1"" l'UI'''''''.l''''l

is then such that

1/;(() = 4FoT demonstration, see Goodier and Timoshenko, 1970, p. 212.

(5.109)

108

Part II. lvfecllanism of material strain

cavity

with a uniform pressure

In this case, the boundary conditions on an element ds can be written [Eq. (5.90)] -pcosa

FYn

The global boundary condition is obtained count of (5.91)] that is taking account of

h + ih = z

(5.1

-psin a

ac-

[Eq. (5.95)

1,

11)

the conjugate of which is (5.112) these values in (5.106) and (5.1 09), after integration one is led to

ip«() =

pRm

(5.113)

(

replacing (5.113) in (5.88), one obtains the OVlrw<,eelr>n of the stresses and in the expression of the displacements. These results are useful in the case of an infinitely flattened ellipse (crack) for which m L In this case, one obtains for the stress and fields after substitutions

(5.114) ---.~~~==~~==~-

2Gp pRp

--2G-'-r~~~~~

with 3 - 411 (plane state of strain) In the crack extension (that is for B a pp a XT and aeo = a yy )

0),

2p

a xx

== -.-, -1 p-

a yy

114) become (since in this direction

(5.115)

109

Chapter 5. Plane theory of elasticity

In other words, at the fracture tip (that is for p = 1), the elastic solution leads to a singular stress field. This observation, as we shall see, is of prime importance in the study of fissuring.

Infinite plate subjected to a far stress field (Fig.5.8) This classic problem has been variously solved by numerous authors (Muskhelishvili, 1953, Goodier, 1968, Sih and Leibowitz, 1968, Goodier and Timoshenko, 1970, Jaeger and Cook, 1979). The calculations are very similar to the previous problem. Calculation shows that the initial elliptical cavity strains but remains an ellipse whose new semi-axes are such that

a'

= a [ 1- 1 - E2v

2

(J'

]

(5.116)

The solution of the stress field is of the form (J'xx

+ (J'yy (5.117)

On the symmetry axis of the ellipse (J'xy ::::: O. (5.117) only takes account of the disturbed solution and to obtain the global solution, one has to add the trivial solution of the plate without a cavity subjected to the same loading. We shall see that Eqs (5.117) are the starting point for the study of fissured materials. It is also useful to calculate the volumic elastic strain energy associated with the elliptical cavity. This calculation can be easily performed from the stress and displacement fields but one can also evaluate it directly using complex potentials. For a biaxial loading (J', k(J'(k < 1) it is equal to (per unit length perpendicularly to the considered plane)

Wi ::::: 7r(J'2(1 - v

2

[(1 _ k?(a

)

4E

+ b)2 + 2(1 _

k 2 )(a 2

_

b2 ) (5.118)

+(1 + k)2(a 2 + b2 )]

5.11

CONCLUSION

While very condensed, this chapter provides readers with the mam bases for solving a number of plane elastic problems. Although insufficient to solve a concrete problem in rock mechanics, linear elasticity will enable us to introduce very intuitively the concept of a porous medium to which a large part of what follows will now be devoted.

110

Part IT. Mechanism of material strain

BIBLIOGRAPHY J.N., 1968, Mathematical theory of equilibrium cracks, in Mathematical fundamentals" (Vol. II), Liebowitz Ed. Academic Press New York, San GU"","V. London. GOURSAT, E., 1956, COUTS mathimatique, Vol. Gauthier-Villars, Paris. and COOK, N.G.W., 1979, Fundamentals of Rock Mechanics; 3rd ed. Chapman and Hall, London. LANDAU, L., and Thiorie de l'iiasticiti, Moscow. MUSKHELISHVILI, N.1., 1954, Some basic problems of the mathematical theory of elasticity, Noordhoff International Publishing. PARODI, M., 1965, Mathimatiques appliquies a l'aM de l'ingenieur, VoL III, "Fonction de variable complexe. symbolique", SEDES, Paris. PARTON, and 1981, Methodes de la tMorie matMmatique de l'elasticiU, Vol. I, II, Mir, Moscow. V., and P., 1977, Equations intigrales de la tMorie de Mir, Moscow. REKATCH, V., 1977, ProbJemes de La theorie de l'elasticiti, Mir, Moscow. SALEH, 1985, Determination de l'etat de contrainie et des pToprietis ",nCh"",,, d 'un massif rocheux par inversion des donnees ricolties lors d 'un essai de fracturation pressiometrique, Thesis Paris. and H., 1968, Mathematical theory brittle fracture, in "FracG ture, Mathematical fundamentals" (VoL ,Liebowitz Ed. Academic Press New San London. 1970, Theory of elasticity, Mac Graw Hill. and GOODIER, J

CHAPTER

6

Behaviour of a material containing cavities

In a fairly general way, a rock can be ranked with a continuous material containing a more or less substantial proportion of vacuums, often random in shape and geometrically complicated. In this chapter we shall study how the presence of vacuums can influence the macroscopic behaviour of an elastic material.

6.1

PHENOMENOLOGICAL ASPECT

The stress strain curve of the majority of sedimentary rocks (specially sandstones and limestones) offers in compression (uniaxial, triaxial or hydrostatic) a strongly nonlinear behaviour for low stresses (Fig.6.1). Under increasing loading, the modulus of the material (Young's modulus or hydrostatic bulk modulus) gradually increases until a certain value for which the curve becomes pseudo-linear. Furthermore, the rocks often display hysteresis after unloading. These two phenomena, namely non linearity and hysteresis are the dominant characteristics differentiating dry rocks (i.e without interstitial fluid for the moment) from continuous materials. We shall see that they are attributed to the presence of specific vacuums: microcracks.

6.2

STRAIN ENERGY ASSOCIATED WITH A CAVITY DEFINITION OF EFFECTIVE BULK MODULUS

In the preceding chapter we described the effect of a circular and then of an elliptical disturbances on stress and strain fields. From these redistributions, results a modification of the elastic strain energy: generally speaking, the presence of a cavity tends to increase the elastic strain energy of the same identically loaded solid but

112

Part II. Mechanísm

ol mateTÍal straín

Chapter 6. BeJum

without a cavity. Thereíore, the elastic strain energy oí a cavity IVc is the difference between the energy W oí the solid containing the cavity under a given loading system and that Wo oí the same solid identically loaded but without a cavity, that is

(6.1)

700

600

500

~

e

'.." ''"" ..'" e,

400

;:l

300

200

Replacing (<<

100

Fig. 6.1. Classical hydrostatic

behaviour

compression

of a porous

(Indonesian

rock under

Sandstone).

Let us consider a volume oí material (a cube oí side A) containing a cavity, and hydrostatically loaded (Fig. 6.2). In accordance with Clapeyron's theorem, the elastic strain energy accumulated in the material is equal to half the work carried out by the externalloads through the displacements; under hydrostatic loading one has thereíore

W

1

= 20"~V

(6.2)

in which ~ V is the volume variation oí the material between zero and absence oí a cavity, the stress field in the material is uniform so that 1

Wo

0".

In the

1 (j2

= 20"é

kk

V

=2K

V

(6.3)

in which K is the bulk modulus oí the solid material (known as matrix) and V the total volume oí the cube (V A3).

=

KeJf is can bulk modulus (l a cavity) knowi real material (iJ that Wc still hE modulus is less increases the 01

6.3

SPEC

PORl!

Equation (fi linear behaviou necessary to ex For instance tbl mmor axes a al

Chapter 6. Behaviour

of a material containing

•• the difference Ioading system • that is

113

cavities

a A

(6.1)

-

,-------'-------('.-

- - - - - - - - - - --

~aA

L--

-.-

---'

_

a Fig. 6.2. Cavity in a material a hydrostatic

under

loading.

Replacing (6_2) and (6.3) in (6.1), one obtains

~V

V

1 I<

2 Wc V (72

-=-+--=-(7

a cavity, and , the elastic out by the has therefore (6.2) In the

(6.3)

1 I
(6.4)

I
6.3

SPECIFIC TYPES OF CAVITIES: PORES AND MICROCRACKS

Equation (6.4) defines the effective modulus but do es not account for the nonlinear behaviour of the material such as is observed in Fig. 6_1. For this purpose it is necessary to expound Wc which can only be envisaged for simple cavity geometries. For instance the calculation is possible for an elliptical cavity of respective· major and minor axes a and b, under any type of loading (the calculation is performed on the

114

Part Il. Mechanism

of material strain

basis of the complex potentials determined in the previous chapter). In the case of hydrostatic loading the result becomes very simple [Eq. (5.118), with k 1]

=

a..pccr

6.4

'<1

6. B' .

EVOLU WITHl

(6.5) E and v being respectively Young's modulus and Poisson's ratio of the matrix. Replacing (6.5) in (6.4) one is led to _1_ Keff

=..!.. K

+

21l'"(1- v2) A( 2 EV a

b2)

+

Let us coasidI: der increasin~" disappear one a8I remain open, tha

(6.6)

The variation in total volume is equal to the volume variation of the matrix plus the volume variation of the cavity which for a value u of the loading will be equal to

(0'2

has been negl In that case I!

1

(6.7)

Ke, Initially, the volume of the cavity was such that Vo = 1rabA

(6.8)

The volume of the cavity for a value u of the loading will therefore be such that

R; being the día¡ The non-línea gradual closure (J tion of the pares range so that one

(6.9) or by introducing a

= bJa,

the shape factor of the cavity, (6.10)

Under compressive loading (u < O), the cavity volume will gradually be reduced and will be equal to zero for u e such that (6.11) The smaller the shape factor of the cavity, the lower the pressure required to close the cavity. One can therefore differentiate two categories of vacuums with respect to their closure pressure: (a) Those whose shape factor is small with respect to 1. These are sharply flattened ellipses closing under a low stress field. They are known as microcracks. (b) Those whose shape factor is large. They are circular vacuums (spherical or cylindrical). They can practically never close (indeed, a pressure of the order of magnitude of the matrix Young's modulus would be required). They are known as pores. By extension, a medium containing cavities is known as a porous medium.

is a constant. It I material whose b4 will become appa KB is always les! the microcracks a of the curve u - . this straight line the microcracks ( one can define tb

in which

with [Eq. (6.10)]

The crack poi open under u.

Chapter 6. Behaviour

In the case of i= 1]

6.4

115

oi a material containing cavities

EVOLUTION OF THE EFFECTIVE MODULUS WITH LOADING

(6.5) Let us consider a material containing initially np pores and nJ microcracks. Under increasing loading, only the microcracks will close and, their contributions will disappear one after the other. Therefore under loading 17, only n(17) microcracks will remain open, those whose shape factor is such that

of the matrix.

(6.6)

217(1 - 1I2) a> -

die matrix plus "will be equal to

(6.12)

E

(a has been neglected with respect to 1) 2

In that case Eq. (6.6) becomes 1

(6.7)

[{elJ

1

=

J{

n. 4(1 - 1I2) 7rR; A

+~

E

n(q)

2(1 - 1I2) 7ra; A

+~

V

E

V

(6.13)

R; being the diameter of a pore and

ai the length of any microcrack. The non-linear behaviour of rocks at low stress is explained therefore by the gradual closure of the microcracks during loading. On the contrary, the contribution of the pores in expression (6.13) remains quasi constant in a reasonable loading range so that one can consider that

(6.8) be such that

I (6.10) be reduced

(6.11)

2. ~

_1__

(6.9)

I

I

]{B

-

K

+ L..J

(6.14)

EV

.:1

is a constant. It represents the hydrostatic bulk modulus of an imaginary continuous material whose behaviour is equivalent to that of the real material. For reasons which will become apparent in the following chapters J{B is known as drained bulk modulus. J{B is always less than J{. The effective bulk modulus being equal to [{B when all the microcracks are closed, J{B is therefore represented by the slope of the linear part of the curve 17 - Ekk (in which En is equal to ~ VIV). The abscissa at the origin of this straight line represents the variation in relative volume due to the closing of all the microcracks (Fig. 6.3): it is a known as "initial crack porosity" T/a. Byextension, one can define the crack porosity under loading 17, T/( (7) such that T/(17)

I

=

¡

in which . sharply flat_ .icrocracks. (spherical or ofthe order )_ Theyare "- bown as a

4(1-1I2)1l"R;A

VP(17) V

(6.15)

n(q)

VF(17) =

¿ Vi(17)

(6.16)

.:1

with [Eq. (6.10)] Vi(17) = 7I"a;aA

+ 271"(1;

2 1I ) Aar17

The crack porosity is therefore representative open under 17.

a

«:

1

(6.17)

of the cracked volume remaining

116

Part Il. Mechanism

al material strain

a

Chapter 6. Behay¡

Taking acco

in which the sw a shape factor (

The shape ( a higher level a loading ranges whose shape faA

O~~~--~rr--

~"~ Now, d7](a) and a + da, in

Fig. 6.3. Drained bulk modulus (after Morlier. 1971 ).

and crack

porosity

ai

Figure 6.3 shows that '::kk(U)

=

u /{B

+ r¡a -

r¡(u)

(6.18)

in which d 2:, a and a + da. The sum exí

that is by deriving with respect to u

a.:: k k ou

1

=

or¡(u)

/{B-~

(6.19)

that is by repla

which can also be written

1 1 or¡(u) ----tc.¡ f J{B oU which is another form of the effective modulus evolution.

6.5

(6.20)

DETERMINATION OF THE CRACKING SPECTRUM USING MORLIER'S METHOD

The curve 7]( u) can be used to determine the distribution of cracks in accordance with their shape factor. Let us call dr¡( a) the porosity associated with all the crack s whose shape factor is comprised between a and a + da and given h( a) the density probability function such that d7](a) h(a)da (6.21)

=

or again taking

Replacing (~

Let us deriw

•• material

strain

Ghapter 6. Behaviour

of a material

117

containing cavities

Taking account of (6.15), (6.16) and (6.17), one has

=

dr¡(u) da

2

211"(1- v EV

A" 2 L..,¿ a,

)

O'

«1

(6.22)

• in which the sum is extended to all the cracks still open under u, in other words with a shape factor O' such that [Eq. (6.12)] 2u(1 - v2) E

a> -

0'«1

(6.23)

The shape of the compressibility curve leads one to think that O' is bounded by a higher level O'M. In other words for O' > O'M all the microcracks are closed in the loading ranges envisaged. The sum (6.22) is therefore only extended to the cracks whose shape factor is such that (u < O) 2
and

< O' < --

E

(6.24)

O'M

Now, dr¡(O') is the porosity of cracks whose shape factor is comprised between O' + da, in other words

d r¡( 0')

= --::-V-::-=---

O'

(6.25)

a;

(6.18)

is only relative to the cracks whose shape factor is comprised between in which d-¿. O' and O'+dO'. The sum extended to the interval (6.24) can then be calculated. Indeed (6.26)

(6.19)

that is by replacing (6.25) in (6.26)

L ·- j _ a2

(6.20)

aM

-

2<7(1-,,2)

,

E

V --d O' 11" Aa r¡( )

(6.27)

or again taking account of (6.21)

¡:a¡ - ja 2 _

V

1I"A

,

M

_2<7(1_,,2) E

h(O') --;;-dO'

(6.28)

Replacing (6.28) in (6.22), one obtains dr¡(u) do

= 2(1 - v2) E

ja

M

h(O') da

_2<7(1_,,2) E

O'

(6.29)

Let us derive Eq. (6.29) a second time with respect to u, that is d2r¡(u) da? (6.21)

=

2(1 - v2) E

[h(O') 00'] O'

o s«

OU _

2<7(1-,,2) E

(6.30)

118

Part II. Mechanism

of material strain

Chapter 6. BehaVÍOl

now, h(o:) being

0.182

or again by writn

•

0.181

¡

0.18

~...

0.179

o

'"

•..

0.178

Po

'"

0.177

E¡¡:;

0.176

O

0.175

¡::

o:: O

P...

a

Let us observe in fact, according

0.174

where

0.173

110

is the 1«

It is then neee

0.172

An example ( case O:M ~ 4.10-:

0.171 100

O

200

300

'100

PRESSURE(bar)

6.6

450 400

CLOSU A CON

350

Knowledge 01 closure under an Given a defec stress fieId (T1, (T2 (Fig. 6.5). The shape C~ a constant direct will be such that

300

c:I

~ .c

250 200 150 100 50

0.001

0.002

a Fig. 6.4. Cracking by Morlier's

spectrum

melhod

b+De ns ty probability í

of Vosges Sandstone

(tifter

a+Cornpre ss ibi Ií ty e urve

0.004

delermined

Under the effi and (6.32), the el

Segal,1989).

versus

versus

0.003

pressure.

sh ape coefficient.

Three cases e its orientation the contrary the be open if its din

su

crlmaterialstraín

Chapter 6. Behaviour

119

ol a material contaíning cavitíes

now, h(a) being a distribution h [_

function, h(aM)

=

20-(1-I/Z)]

E or again by writing

[C]-l

= O hence

2)]-1

[2(1_1/

E

=

2(1 E

2 d 'f}(0-)

o- do-z

1/2)

= Co-d2'f}(0-) 2

h(-C-10-)

(6.31)

(6.32)

(6.33)

d0-

Let us observe that (6.33) is not a true probability density since it is not normed; in fact, according to (6.21), one has

l

cxM

l

cxM

h(a)da =

d'f}(a) = 'f}a

(6.34)

where TIa is the total crack porosity. It is then necessary to divide h( a) by TIa to obtain a probability density. An example (for Vosge Sandstone) is taken from Fig. 6.4 and shows that in this case aM ~ 4.10-3 (h(a) in that example is divided by 'f}o).

6.6

CLOSURE OF A CRACK POPULATION UNDER A COMPRESSIVE STRESS FIELD

Knowledge of crack opening distribution h(a) makes it possible to follow crack closure under an increasing compressive stress field. Given a defect of length a, shape coefficient a submitted to a biaxial compressive stress field 0-1, 0-2 such that 0-1 < 0-2 and whose major axis makes an angle (3 with 0-2 (Fig. 6.5). The shape coefficient being small (a ~ 1) one can suppose that the normal ñ has a constant direction. The normal and tangential components applied to the defect will be such that

{:

(6.35)

~ (o-z - o-d sin 2(3

Under the effect of 0-, the defect initially open begins to close. According to (6.12) and (6.32), the critical value or o- for the crack to close is o-er

= -aC

(6.36)

Three cases can be considered: if O-er < 0-1, the defect will remain open whatever its orientation since the minor component is less compressive than O-er; if O-er > 0-2 on the contrary the defect cannot remain open; finally, if 0-1 < O-er < O-i, the defect will be open if its direction (3 is such that 7r

(3er

< (3 < "2

with o-«(3er) = o-er

120

Part Tl, Mechanism af material strain

Chapter 6. Behavi •••

36 34 32

y

30 28 26

E Z <, E-o Z

x

,,

"

\~" \.

24 22 20 18 16

a \

14 12 10

8 20t Fig. 6.5. Open crack

under

a b axi al loading. í

F.

01'

f3cr

=

. JlJ'l+O:C arcsm IJ'1 -

IJ'2

(o

One can easily find from (6.37) the two first cases: if IJ'1 = -o:C, f3cr = O (the -o:C, f3cr 'Ir/2 (the crack is closed whatever crack is open whatever 13is) and if IJ'2 13 is). Let us consider now N defects and given h(o:) the probability density on 0:. Among these N defects, a certain number d N¿ have a shape coefficient comprised between o: and o: + do: that is

=

01

(6.37)

=

(6.38)

ADDITl THE el

but, among these dN¿ defects, those whose argument is comprised between f3cr (1J'1, IJ'2, 0:) and 'Ir /2 will remain open. If we suppose furthermore a homogeneous distribution of 13, the number of open cracks wiU be then

Under uniaxial pressure shows thl

st«; = Nh(o:)do:

'Ir

"2 dNT = d N¿

f3cr(lJ'l, IJ'2, 0:) 'Ir

(6.39)

2 Under a compressive stress field IJ'1, IJ'2, the percentage of open cracks will be then such that NT (CXM 1 N = J ;h(o:) ['Ir- 2f3cr(1J'1, IJ'2, 0:)] do: (6.40) o

The numerical integration of (6.40) is represented on Fig. 6.6 for Vosge Sandstone and clearly shows that even for a smaU confining pressure IJ'2, only a few percent of defects remain open. Nevertheless we will see afterwards that these open cracks have a great importance as regards rupture.

6.7

(a) The micro Increases, 1 (b) The closun sil" microe tween lips] important non-lineari (e) In the san modulus .E

aJaterialstrain

Chapter 6. Behaviour

oi a material containing

121

cavities

36 J4 32 30 28 26

E

24

z )..

22

z,

20 lB

16 14 12

lO 8 200

300

400

lall Fig. 6.6. Percentage of confining

(6.37)

pressure

600

(bar)

of open cracks

for different

values

(Vosges sandstone)

(after Ctuirlez et al. 1991).

6.7 (6.38)

ADDITIONAL OBSERVATIONS CONCERNING THE CLOSURE OF THE MICROCRACKS

Under uniaxial compression, direct observation pressure shows that:

(6.39)

, (6.40)

500

of the microcracks closure under

(a) The microcracks perpendicular to the direction of the stress close as the stress increases, while those parallel to the latter have a tendency to open. (b) The closure of the microcracks is nearly always incomplete as far as the "fossil" microcracks are con cerned (non-superimposable lips, residual material between lips). The morphological characteristics of the microcracks playa very important part in their closure. This phenomenon alone explains why the non-linearity is due to the microcracks and not to the pores. (c) In the same way as we have defined J{E, one can define a drained Young's modulus EB and drained Poisson ratio VE such that

(6.41)

122

6.8

Part II. Mechanism

of material strain

CONCLUSION. CONCEPT OF POROSITY CHAPTER

The preceding paragraphs clearly show that a porous medium can only be envisaged as an equivalent material whose macroscopic behaviour is finally identical to that of the actual material. With this average material wiU be associated equivalent characteristic parameters which will only be representative of a sufficient quantity of material. Thus, independently of their shape, the vacuums will be characterized by an average parameter: "porosity" 0 which is by definition the volume ratio between the vacuum and the total volume, that is

7

Thern satura

(6.42) This concept, in the same way as EB, l/B and f{B, is a global parameter (in other words associated with a macroscopic quantity of material) and not a localquantity.

BIBLIOGRAPHY BRACE, W.F., WALSH, J.B. and FRANGOS, W.T., under high pressure, JGR, Vol. 73. BRACE, W.F., 1965, Some new mesurements Vol. 70, No 2.

1968, Permeabi/ity

of linear compressibi/ity

of granite

of rocks, JGR,

CHARLEZ, P., SEGAL, A., PERRIER, F. and DESPAX, D., 1991, Microstatistica/ behaviour of brittle rocks, submitted to lnt. Jour. Rock Mech. and Mining Sciences. GHIASSI, H., 1985, Détermination de l'état de contrainte microfissuration des roches. Thesis EC Paris.

in situ par analyse de la

JAEGER, J.C., 1966, Brittle fracture of rocks, in "Failure and breakage VIlIth Symposium on rock mechanics, City Press, Baltimore. MAVKO, M., and NUR, A., 1978, The Effect of non-elliptical ibility of rocks, Jour. Geoph. Research, Vol. 83.

of rocks",

cracks on the compress-

MORLIER, P., 1971, Description de l'état de fissuration d'une roche non destructifs simples, Rock mechanics, Vol. 3, pp, 125-128.

a

partir d'essais

SEGAL, A., 1989, Elaboration d'un modele microstaiistique linéaire de la rupture fragile et application a la stabilité des forages profonds. Thesis EC Paris. WALSH, J .B., 1965, The effect of cracks on Poisson's ratio -' (The effect of cracks on ihe uniazial elastic compression of rocks); - (The effect of cracks on the compressibility of rocks}, JGR, Vol. 70.

"

Porous media taining vacuums ~ beyond a certain could then be Ial associated a globs The chapters de-v pothesis: either o implicitly that th4 the presence of Yo continuous media tally filled by 0114 "interstitial presss If the connect saturated, each al certain conditions rated. Its behavi deals only with p4 Historically sp was supplemented (1975), who for th Mac Tigue (1985) equations the infll Lastly, quite 11 thermomechanics His achievemei and opens the wa¡

CHAPTER

7

Thermodynamics of saturated porous media

.!

(6.42)

..•

rupture

Porous media were envisaged in the previous chapteras continuous media containing vacuums of different forms (pores or microcracks). It was also observed that beyond a certain loading value, the microcracks completely close and the material could then be ranked with an equivalent continuous medium with which could be associated a global parameter characterizing the proportion of vacuums: porosity 0. The chapters devoted to porous media will be made in the scope of the following hypothesis: either one considers that the medium is not microcracked, or one assumes implicitly that the loading is sufficient for all the microcracks to be closed. However, the presence of vacuums is not the only dominant characteristic that differentiates continuous media and porous media. The connected porosity is always partly or totally filled by one or several fluids under pressure p, known as "pore pressure" or "interstitial pressure" . If the connected porosity is entirely filled with fluid, the medium is said to be saturated, each of the fluids participating in the saturation. On the contrary under certain conditions (particularly at low depth) the medium is only imperfectly saturated. Its behaviour becomes then very difficult to model. The following chapters deals only with porous media saturated by a single fluido Historically speaking, Biot was the first to deal with solid fluid coupling. His work was supplemented in the seventies first by Geertsma (1969), then by Rice and Cleary (1975), who for the first time provided a comprehensive formulation of poroelasticity. Mac Tigue (1985) completed the work of Rice and Cleary by adding to the poroelastic equations the influence of temperature. Lastly, quite recently Coussy (1989 a) (1989 b) developed a general theory of the thermomechanics of porous media by writing the major conservative laws. . His achievement makes it possible to go beyond the stage of thermoporoelasticity and opens the way to thermoporoplasticity.

124

7.1

Part II. Mechanism ol material strain

BASIC HYPOTHESIS OF THERMOPOROMECHANICS

7.3

The relevancy of the description of a porous medium is based on the following hypotheses. First of all (and this is the main difference between continuous and porous media) a porous medium is thermodynamically open: it exchanges Huid with the exterior. Secondly, although the medium is profoundly biphasic there will not be any differentiation between Huid and skeleton: the macroscopic description of phenomena will be considered as continuous. At least, the kinematic quantities associated with the medium (displacements, velocities, strains) are described with respect to the skeleton.

7.2

Chapter

•

Jiíodao

iífda

In the presen other hand, the I to (since the mal

where p and (2) present configura mass increment) be equal to m represents the strained configw The total ma

We saw in Chapter 1 that during a finite transformation, volume or area increments are modified and that elements in strained state (that is dV and da) could be determined from elements in the non-strained state (that is dVo and dao) provided the transformation gradient f and its determinant J (also known as Jacobian of the transformation) are known. These convective transports were expressed by the equations [Eqs (1.15) and (1.19)] JdVo

MASS'

Given a non-s Ro, and given 8)

THE IMPORTANCE OF THE LAGRANGIAN DESCRIPTION FOR WRITING CONSERVATIVE LAWS

dV

7. Thermo

(7.1)

Given W ·ñ(l normal Ti (conves is then an exit 11 Mass conserv

't

To give to (7.

with

that

f= o~ oX

x

J

= det If I

(7.2)

where is the coordinate of any point of the medium in the present configuration and X its initial position in the reference configuration (Lagrangian variable). In the case of continuous media, the conservative laws are often described neglecting the convective transports. This can be justified by the fact that the medium is thermodynamically closed. The open character of porous media necessitates writing the conservative laws in finite transformations for fear of committing serious errors particularly with regard to mass balance.

where dao is the to this surface E depends only on From (7.1) al

By inserting the expression ol

Chapter

7.3

7. Thermodynamics

oEsaturated

125

porous media

MASS CONSERVATION

Given a non-strained

volume element dVo with respect to an initial reference frame + solid) such that

Ro, and given 8Mo the total mass of this element (fluid 8Mo

= modVo

(7.3)

In the present strained state, the mass of solid remains unchanged while, on the other hand, the mass of fluid contained in the element of present volume dV is equal to (since the material is saturated) by taking account of (7.1)

= p0dV = J p0dVo

8M!

(7.4)

where p and 0 are respectively the fluid density and the medium porosity in the present configuration. In this way, the total mass increment .ó.M (equal to the fluid mass increment) between the initial configuration and the present configuration will be equal to J p0 dVo - po0o dVo mdVo .ó.M (7.5)

=

« area incre. c..d da) could _d dao) proas Jacobian i!l!!llllIeSsed by the

=

m represents therefore the mass increment between the initial configuration strained configuration divided by the initial volume. The total mass 8¡\.1 in the strained state will therefore be written 8M

= 8Mo

+.ó.M

= (m + mo)dVo

(7.6)

Given Mi· ñ da the mass of fluid flowing by time unit across the present area da of normal ñ (conventionally taken towards the exterior of the volume element). Mi· ñ da is then an exit flow. Mass conservation implies

f ~ (mo + m )dVo + t. f Mi· ñda = O i;

(7.1)

(7.7)

vt

To give to (7.7) a Lagrangian

form, let us introduce

that (7.2) c:onfiguration • le). t.:m»ed neglect• medium is .ates writing senous errors

and the

M

·ñodao

the Lagrangian

= Mi ·ñda

vector

M such (7.8)

where dao is the surface element in the non-strained state and ño the outside normal to this surface elemento In fact, M has no physical meaning, but M, unlike Mi, depends only on X and t. From (7.1) and (7.8) one can extract furthermore (7:9) By inserting (7.8) into (7.7) and by applying the expression of mass conservation

m=-V'·M

the divergence

theorem one is led to

(7.10)

126

Part Il. Mechanísm ol material straín

where m is a partial derivative with respect to time. The divergence operator is defined in (7.10) with respect to the reference configuration Ro. The usefulness of the Lagrangian description introduced by Biot (1977) in writing the conservative laws is especially apparent when one derives Eq.(7.5). The state variable which appears naturally is not p 0 (which is generally admitted) but J p 0. In the hypothesis of small deformations, the J acobian being such that J

=

€kk

Ghapter 7. Thermo

Taking accow gence theorem

where t = 1/2[v the external foro

+1

the variation in mass is therefore equal to (7.11)

7.5

FIRST

where ti is the velocity, and not

m= as is generally admitted

7.4

!

(p0)

(7.12)

when convective transfers are not taken into account.

This principk of the system au external forces ir

Given udV, t volume dV and variation in inte equal to

CONSERVATION OF LINEAR MOMENTUM AND MECHANICAL ENERGY BALANCE

If one neglects inertia and body forces, it is reduced to a static equilibrium (see Chapter 2). Writing directly the equations in the present configuration it will be written

the second term variables (7.20) I

(7.13) where T is the surface force applied on the present surface of the body. The boundary condition implies (7.14) where !?: is the Cauchy stress tensor also defined in the present configuration. It is not then necessary in the linear momentum balance to differentiate stresses in solid and skeleton. !?: and T are to be understood as average quantities. Substituting (7.14) in (7.13) and applying the divergence theorem, one is led to the classical equilibrium equation

The power o paragraph and o respect to the sil If Vr is the ~ surfacic force exl will be such that

(7.15) The mechanical energy balance is a consequence of linear momentum conservation. If v is the velocity of any point of the skeleton (with respect to which, we may recall, the kinematic quantities are defined), one can easily deduce from (7.15)

l

ti(\7 . !?:)dV= l

\7. io. !?:)dV- l!?::

(\7 0 v)dV = O

(7.16)

This equatioi

1For convenienc reference ís made • densíty of the medi

Chapter

7. Thermodynamics

oE saturated

127

porous medía

Taking account of (7.14), Eq. (7.16) can also be written, after applying the divergence theorem

(7.17)

w

a where ~ = 1/2[\7 <'9v + t(\7 <'9v)] is the strain velocity tensor and e the power of the external forces as if the Huid and the skeleton had the same velocity that is

w

a

e

=

v,

¡

(7.18)

v·Tda

(7.11)

7.5

FIRST PRINCIPLE OF THERMODYNAMICS1

This principle expresses the energy balance, that is the variation in internal energy of the system augmented by the rate in kinetic energy is equal to the power of the external forces increased by the heat rate:

(7.12)

(7.19) Given udV, the internal energy of the Huid and the solid contained in the present volume dV and given Um, the internal energy of the Huid per unit of mass. The variation in internal energy of the system during the increment of time dt will be equal to

r

u. = 8ta J v udV + l

A Um

a u. = 7i t

¡

uodVo

+

Vo

(7.20)

W . iida

the second term being due to the open character variables (7.20) can be expressed

(7.13)

..

j-

of the medium.

In Lagrangian

1-

(7.21)

umM . iiodao

Ao

The power of the external forces is the sum of W: expressed in the previous paragraph and of the power of the forces in the relative motion of the Huid with respect to the skeleton. If is the relative velocity of the Huid with respect to the skeleton and TF the surfacic force exerted on the Huid portion of the total area (that is 0da), this power will be such that in the present configuration

W;

v,.

(7.15)

W; =

¡

(7.22)

TF . VrOda

This equation can be written by introducing

the surfacic flowrate

W

such that

(7.23) (7.16)

lFor convenience, the state functíons (u, s, h, 1/J) are related to the unit volume except when reference is made to the Huid (specific state functíon with index m). This is the reason why the densíty of the medium does not appear in the equatíons as it was the case in Chapter 3.

128

Part II. Mechanism al material strain

and the pore pressure p such that

Chapter

7. Therm~

by temperature. 1 paragraphs, one CAl

(7.24) where the minus sign indicates a compression (ñ is towards the exterior). (7.23) and (7.24) into (7.22) one obtains

. ¡-

w; = -

Replacing

In (7.31),

p-

W . iida

(7.25)

AP

So

is

entropy of the flu written

or in Lagrangian variables taking account of (7.8)

. =w;

j

P-

Ao

-M P

·ñodao

(7.26)

Let us extract

Let us introduce finally the heat rate Q such that

Q=

- { q. ñda = - {

s.. Q.

lA

TS¡ ñodao

(7.27)

or after developme

{~

where q and Q are respectively the "Eulerian" and "Lagrangian" heat flows received (the minus sign compensates for the fact that the normal ñ points towards the exterior). The formula of convective transport of surfaces also makes it possible to express

Jq= F·

Q

(7.28)

Neglecting the kinetic energy rate k, substituting (7.21), (7.26), (7.27) in (7.19) one obtains by introducing, hm Um + p] p, the fluid mass enthalpy 2

=

(7.29) By applying the divergence theorem to the surface integral and substituting'' value of ea extracted from (7.17), one is led finally to

vv

Let us introdu thalpy gm of the :f

one is led to the

ti

the One will suppc

(7.30)

7.6

SECOND PRINCIPLE OF THERMODYNAMICS INEQUALITY OF CLAUSIUS-DUHEM

The second principle expresses that the variation in entropy of the system (again volume term plus convective term) is greater than or equal to the heat rate divided 2This hypothesis is not necessary. If k: is not nil, it appears in (7.18) and is eliminated between (7.18) and (7.19). See paragraph 3.11.2 and 3.11.3. 3Equation (7.17) can be written indifferently in Lagrangian or Eulerian configurations (expression of energy).

(7.10),

cll1 is known as

7.7

CHOICl

The inequality of the system (the

or material

Chapter

strain

7. Thermodynamics

129

of saturated porous media

by temperature. The approach being quite similar to that developed in the previous paragraphs, one can write the equation directly in Lagrangian variables that is (7.24) - r). Replacing

(7.31)

(7.25)

In (7.31), So is the volume entropy of the skeleton fluid whole and Sm the mass entropy of the fluido Locally, by applying the divergence theorem, (7.31) can be written

-

-

Tso + T'l· (smM) ~ -'l.

Q

Q + T . ('lT)

(7.32)

Let us extract -'1 . Q from (7.30) and introduce it into inequality (7.32),

(7.26)

(7.33) (7.27)

or after development

{Q': t - (uo

M}

- Tso - 1'so) - 1'so - (hm - Tsm)'l·

(7.34)

+ { - ~ srr - M . ['1 hm

-

T'l

sml } ~

O

Let us introduce into (7.34) the volumic free energy tPo and the specific free enthalpy gm of the fluid such that tPo = Uo - Tso (7.29)

(7.35)

one is led to the final expression of the inequality of Clausius Duhem (7.36) One will suppose the decoupling of the two dissipations that is, taking account of (7.10),

(7.30)

l

t - sor + gm m - ~o ~ o Q - T . ('lT) - M . ('1gm +

Q' :

Sm

<1>1 is known as inirinsic

,

7.7

dissipaiion,

(7.37)

'lT) ~

<1>2 as thermohydraulic

o

dissipation.

CHOICE OF STATE VARIABLES (intrinsic dissipation)

ted between (expression

The inequality of Clausius-Duhem which defines the thermodynamic admissibility of the system (the dissipation must always be positive) reveals three observable state

130

Part Il. Mechanism ol material strain

variables: the total strain ~, the temperature T and the mass variation of the system m (we should remember that it is only a Huid mass variation). This latter state variable characterizes simultaneously the presence of vacuums (through 0), the Huid compressibility (through p) and the medium compressibility (through tu). Taking into account dissipative phenomena such as plasticity necessitates the introduction of additional variables known as internal variables. These variables are of two types: plastic strain and hardening variables that characterize the memory of the material (see paragr. 3.12.3). For a porous medium, plastic strain results, from on the one hand, irreversible strain of the matrix ~~ (which is not an observable variable) and on the other from that of the interconnected porous space. In fact these variables are not independent. First of all, the hypothesis of small deformations induces,

Chapter 7. Therm~

that is replacing

-

(J'-~ (

7.9

<1

~

CASE O LAWS(

(7.38) where ~e and ~p are respectively the elastic and the plastic total strain. Secondly, the total plastic strain can be divided into a matrix plastic strain (on the portion 1-00 of the total volume) and a plastic porosity 0P such that (7.39) Physically, the plastic porosity represents the irreversible change of porosity after unloading. As a summary, taking account of (7.38) and (7.39), after elimination of ~e and ~~ the state variables can be chosen as ~' ~P, 0P, T, m and Vk in which Vk represents the hardening variables that characterize the possible memory of the material.

In case of reves leads to

In the same w; lastici ty. If 1/Jo is ¡ largely devoted ~

7.10

CASE

In the case of i of Clausius Duhen

7.8

CONSTITUTIVE STATE LAW AND THERMODYNAMIC POTENTIAL

In the same way as for continuous media we assume the existence of a scalar function 1/Jo (known as thermodynamíc potential) of the state variables. \Ve shall assume that this is free energy 1/Jo such that (7.40)

(7.44) defines thes

7.11

DIFFU THERJ

1/Jo being a state function one can calculate its total differential in order to eliminate

~o from the inequality of Clausius Duhem (7.37). By following the same reasoning as in [Eq. (3.57)], one is led, taking account of the strain partitioning rule, to

,i. _ {No .. e o/o - fJc_ e . f

+

{No T fJT

{No.

+ fJm m +

{No 0' fJ0P

P

(No

+ fJVík

Vi k

(7.41)

Intrinsic dissip Indeed, therrnopo pressure and temj tion results from 1

Chapter 7. Thermodynamícs

o( saturated

131

porous media

that is replacing (7.41) in the first Eq. (7.37),

(

-

(J" -

o'if;o )

o§.

--

e

:

.e

é -

+-

(J" :

.P

é -

_ o'if;o

o0P

7.9

+ ( gm -

o'if;o). m - ( So -Bm

el _ o'if;o Vi

Q'if;o) + -oT

T'

>o

OVk k -

(7.42)

.

CASE OF REVERSIBLE BEHAVIOUR LAWS OF THERMOPOROELASTICITY

(7.38) In case of reversible behaviour leads to

(tP = izl = Vk o'if;o gm=--

om

= O) (7.42) So

be comes equality which

= ---o'if;o oT

(7.43)

In the same way as for continuous media (7.43) define the laws of thermoporoelasticity. If 'if;o is a quadratic form, the behaviour becomes linear. Chapter 8 will be largely devoted to this.

(7.39)

P and é_M Vi represents material. OfE;e _

7.10

CASE OF IRREVERSIBLE BEHAVIOUR

In the case of irreversible behaviour, taking account of Eqs (7.42), the inequality of Clausius Duhem becomes

, of a scalar _ We shall

(7.40)

(7.41 )

(7.44) (7.44) defines thermoporoplasticity.

7.11

Chapter 10 will be devoted to this subject.

DIFFUSION LAWS OF THERMOPOROMECHANICS

Intrinsic dissipation describes only partially the behaviour of a porous medium. Indeed, thermoporomechanical coupling prescribes knowledge at every moment of pressure and temperature distribution in the medium. We shall see that this calculation results from two diffusion laws: Darcy's law and Fourier's law.

132

7.11.1

Part II. Mechanism of material strain

First diffusion law: hydraulic or Darcy's law

Chapter

diffusion law

Replacing the

This law expresses that the fluid velocity V through a porous medium is proportional to the interstitial pressure gradient 'Vp. The proportionality coefficient depends on the fluid (by means of its dynamic viscosity f-L) and the rock, by means of a tensor ~ characterizing the medium's percolation quality and known as permeability iensor, that is k v=_:::'.'Vp (7.45 )

M

-

M

= pil,

or after developrnei

p

k

with f{= :::.

-

-

in which 1f is known as the hydraulic diffusion

7.11.2

one obtains

.

= -l(. 'Vp

T'V

The first term o flow, the second, tI local increase in he member characteri irreversibilities or 1

(7.46)

f-L

tensor.

Second diffusion law: heat diffusion law or Fourier's law BIBLIOGRAl

This is similar to the previous one and expresses that the heat flow per unit of surface Q is proportional to the temperature gradient. The proportionality constant, which is also tensorial, characterizes the ability of the medium to diffuse heat, and is known as thermal conductibility tensor x, Fourier's law will be written

Q= 7.11.3

SI

one obtains

f-L

or by introducing the surface flow rate

7. Tbermodyi

Hydraulic

and thermal

(7.4 7)

-!S. 'VT

diffusivity laws

The drawback of the diffusion laws is that they depend on the flow rate M and the heat flow Q. Therefore diffusivity laws, (combining diffusion and conservation laws) are preferred. The hydraulic diffusivity law can be obtained by coupling Darcy's law and the mass conservation that is m=-'V·M (7.48) which Ieads after elimination of

M

to

m-

'V(pIJ. 'Vp) =

o

(7.49)

Similarly, the thermaI diffusivity law can be obtained by coupling Darcy's law and the two principles of thermodynamics [Eqs (7.30) and (7.37)]

úo

+ 'V.

(hmM)

M., 1941, ( 12. - 1972, Variation. chanics of porous 579-597. - 1977, The theor¡ 27,597-620. BIOT,

= q::

4. -

'V.

Q (7.50)

r

CHARLEZ, P., 1981 du Groupe TMP, 1

COUSSY, 0.,1989 du Groupe III du I - 1989, Thermod, J ournal of Mechan - 1989, A general March 1989. - 1988, Personal (

133

Cbapter 7. TbermodynanlÍcs of saturated porous media

Replacing the second of these equations in the first and taking into account that

'lj;o um is proporcient depends s of a tensor bility iensor,

uo - T So - soT (7.51)

m

1

(7.45)

one obtains

(7.52) or after development

(7.53) The first term of (7.53) represents the input of convective entropy due to the fluid flow, the second, the variation in entropy of the material over time and the third, the local in crease in heat due to the therrnal conductivity of the material. The right hand member characterizes the energy dissipated in heat, either mechanically in plastic irreversibilities or by the viscosity forces in the moving fluido

(7.46)

BIBLIOGRAPHY

tlow per unit of ity constant, heat, and is

BIOT,

M., 1941, General theory of ihree dimensional

consolidation,

J. Appl. Phys.,

12. - 1972, Variational Lagrangian thermodynamics of non-isothermal finite strain mechanics of porous solids and thermonuclear diffusion, Int. J. Solids Structures 13, 579-597. - 1977, The theory of finite deformations of porous solids, Indiana Univ. Math. J., 27,597-620.

(7.47)

¡ (7.48)

1

(7.49) Darcy's law and

r

(7.50)

J

CHARLEZ, P., 1989, Thermomécanique des milieux poreux saiurés, rapport de recherche du Groupe TMP, March 1989, unpublished.

COUSSY, 0.,1989, R61e du fluide interstitiel sur le comportement des roches, rapport du Groupe III du GS Mécanique des roches profondes. of saturaied porous solids in finite deformation, European - 1989, Thermodynamics Journal of Mechanics, A/Solids, Vol. 8, No 1. Transport in porous media, - 1989, A general theory of thermoporoelastoplasticity, March 1989. - 1988, Personal Communications, TOTAL CFP, unpublished.

CHAPTER

8

Infinitesimal thermoporoelasticity

In the case ofreversible (i.e. elastic) behaviour, the inequality ofClausius-Duhem becomes an equality which leads to the general equations of thermoporoelasticity! {)1/J s=--

{)T

(8.1)

To obtain a linear theory, it is sufficient to choose a quadratic form for the thermodynamic potential. The most physical approach is to find the thermodynamic potential by integrating the constitutive laws rather than to find the constitutive laws by deriving the thermodynamic potential. It is proposed first of all to determine ![, gm and s.

8.1

HOOKE'S LAW IN THERMOPOROELASTICITY CONCEPT OF ELASTIC EFFECTIVE STRESS

One can rank dry rocks with an equivalent continuous medium the mechanical behaviour of which is identical to that of the actual material. The laws of thermoporoelasticity are built on the following assumptions. First of all, one assumes that the equivalent dry material (i.e. without any fluid in the porous space) is linear, elastic and isotropic with elastic constants EB and v». Secondly one admits that the matrix (solid + unconnected porosity) is continuous isotropic and linearly elastic with elastic constants EM and VM. lln the previous chapter, index zero referred to a Lagrangian quantity. In the case of small perturbations (which is the scope of this chapter), Lagrangian and Eulerian configtirations being ident.ical, the index has been abolished.

136

Part Il, Mechanism

oi material strain

The second hypothesis as we shall see later is not necessary to build a general theory.

8.1.1

-p

•..•..

-

=

where KM EM/3{J Component 11 oí stress field r!: + [p 3D

Decomposition of the state of stress Hooke's law of a porous medium

The previous considerations enable one to determine easily the constitutive law of the equivalent material by decomposing the actual state of stress into two elementary components according to Fig. 8.1: the porous medium is subjected to a state of stress r!: and to a pore pressure P which are decomposed into a purely hydrostatic component 1 and a deviatoric component 1I. We shall mention that stresses are defined with respect to an initial state of stress r!:o different from zero (because of the hypothesis on microcracks closure). The pressure is also defined with respect to a reference value Po·

-

Chapter 8. Infinitesimal

The ~verall straÜI

in which KB = EBI Eq. (8.5) represents I

8.1.2

Biot's co

Let us write

®

~

az

@ Component

Fig. 8.1. Decomposition

with

~ +

1

o Component

of the state

After pooling Eq. ~-

a2+p

In other words, 11 as that of a continuo

II

(a) The stresses e (b) The elastic el EB, VB·

of stress

(cift er Carnet.1976).

Let us now apply Hooke's law to these two elementary states of stress [Eq. (4.30)] that is

~ v = 2G - ElJkk8'J

€'J

(8.2

)

Component 1 corresponds to the hydrostatic loading of the matrix (with pressure

p) and leads to a deformation state

€{J I

€'J

P = -3KM

8

'J

(8.3)

The stresses < O"ij coefficient. It is there the strain of a poron stress is linked to t

Chapter

137

8. Infinitesimal thermoporoelasticity

=

where KM EM /3(1 - 211M) is the bulk modulus of the matrix. Component 11 corresponds to the loading of the dry equivalent material by a stress field q;+ lP and leads to a deformation state c{f II

c~J

titutive law of two elementary a state of stress ic component . are defined with the hypothesis a reference value

.r

1+ IIB

= ---¡¡;;;-((f~J + p8~J)-

IIB

EB ((fa

+ 3p)8~J

(8.4)

The ~verall strain is obtained by adding (8.3) and (8.4) which leads to

(8.5)

=

in which KB EB/3(1 - 211B) is the bulk modulus of the equivalent dry material. Eq. (8.5) represents Hooke's law of poroelastic materials.

8.1.2

Biot's coefficient and elastic effective stress

Let us write with a

= 1- KB

KM

(8.6)

< (fa >= (fa + 3ap After pooling Eq. (8.5) becomes

(8.7) In other words, Hooke's law of a poroelastic medium is written in the same way as that of a continuous material provided one replaces: (a) The stresses (f'J by < (fij >. (b) The elastic characteristics by those of the equivalent continuous dry material

EB,

(8.2)

(8.3)

ve-

The stresses < (fij > (8.6) are known as effective stresses and a is known as Biot's coefficient. It is therefore the effective stresses and not the total stresses that govern the strain of a porous elastic material. We may note that this concept of effective stress is linked to the constitutive law. It is not a static concepto

138

Part II. Mechanism oEmaterial strain

Chapter 8. Infinit.

(8.11) and (1

8.2

VOLUME VARIATIONS ACCOMPANYING THE DEFORMATION OF A SATURATED POROUS MEDIUM Since dO'

8.2.1

Bulk volume variations

The bulk volume variations (matrix + pores) can easily be calculated from Hooke's law (8.5). Indeed, the volume strain f.u is such that f.u

=

6.VB VB

1 (Uu KB 3

=

+P

) -

P KM

Variation in pore volume

It is exclusively the variations in normal components that generate porous volume variations. Let us decompose the normal total stress into a hydrostatic part and a deviatoric part such that with /7

=

By eliminati

(8.8)

in which VB is the bulk volume.

8.2.2

=-

Uu

3

(8.9)

The final ex¡ jected to an ine pressure, is obu the two transfor in which the me remains constan pore pressure v¡ volume de crease increase in the Il dO' referring reciprocity theoi

Let us analyze separately the variations in pore volume due: (a) To the variations in mean stresses and interstitial pressure on the one hand. (b) To the variations in deviatoric stresses on the other.

By replacing

Let us assume that Vp (pore volume) and VB (bulk volume) are state functions of p and /7. One can therefore calculate their total differentials. The relative variations in pore volume will therefore be such that or taking accour dVp Vp

= _1

Vp

(OVp) 8U

p

dO'+ ~ (OVp) Vp op

dp

To eliminate/ (oVpfop)q from (8.10), let us consider the stress path -dp Equations (8.3) and (8.8) lead to dp --dVB = --= VB KM Since dVp

= dVB -

1 f.k/c

= --dVM VM

dO'

(8.10)

77

= -dp

= dO'. (8.11)

The same res pression

dVM taking account of (8.11) one obtains dVp

dp = --(VB KM

- VM)

dp = --,,-Vp AM

2The theory being linear, the partial derivatives are independent

The final exJ

(8.12) of the loading path.

Formula (8.1:

•• material strain

Cbapter 8. Innnitesimal tbermoporoelasticity

139

(8.11) and (8.12) lead to the identity

dVp Vp

dVM -VM

dVB -VB

dp KM

M=-dp

(8.13)

Since M = -dp, Eq. (8.10) will be written by taking account of (8.13)

By eliminating

{}(¡

p -

Vp

ap

(8.14)

7i

(aVp/ap)u between (8.10) and (8.14), one is led to dVp Vp

(8.9)

1 (avp)

1 (avp)

1

KM = Vp

~ (avp) {}(¡ Vp

= __ l_dp+ KM

(M+ dp) p

(8.15)

The final expression of the relative variation in pore volume for a material subjected to an increment M of mean total stress and to a variation dp of interstitial pressure, is obtained by applying Betty's reciprocity theorem (see paragr. 4.10) to the two transformations represented in Fig.8.2: on the one hand the transformation in which the mean total stress varies between 7f and 7f + M whereas the pore pressure remains constant and, on the other, that in which 7f remains constant whereas the pore pressure varies from p to p + dp. In the first case, the increment M induces a volume decrease (M negative when increasing compression) while dp induces a volume increase in the second case. M referring to the external surface (Fig.8.2) and dp to the pore surface, Betty's reciprocity theorem will be written

dVp,

X

dp + dVB

1

X

O = dVP2 X 0+ dVB

2

X

M

By replacing the differentials by their values (Fig. 8.2), one obtains

p (av{}(¡)

functions of ive variations

p

=

(aVB)

o»

(8.16)

u

or taking account of (8.8) (8.10) JIIdh

-dp

= M.

(avp) {}(¡

p

= VB

(

1 1)

(8.17)

KB - KM

The final expression of the relative variation will therefore be written (8.18)

(8.11) The same reasoning with the deviatoric components pression (8.12) path.

dVp Vp

=

1 ( 1 1)

0

KB

-

KM

of Eq, (8.9) leads to the ex-

(dUiAJ - M)

Formula (8.18) remains valid in the general case.

=O

(8.19)

140

Part Il, Mechanism

ol material strain

Chapter 8. Inñni

8.3

av

dV: =( --p) Pl

au

MAS! ACC( OFA

dCT p

The pore

VI

in which p is ti

-p

@

I

T

that is, after l.iJ

The differen (assumed to be

dV:B?=(

~

-aaVBP )-u dp which can also

Fig. 8.2. Applicatian af the variatian

8.2.3

of Betty's thearem

af the paraus

Dividing (8. after linearizati

far calculatian

:=(

valume.

in which m = 1 of per unit of t. 7Y and p, ar. pore pressure p

Relative porosity variation

By differentiating

I

porosity 0= Vp / VB one obtains

r i

8.4 d0 _ dVp._ dVB Vp VB

o -

(8.20)

UNDJ SKElV AND

that is by replacing (8.18) and (8.8) in (8.20) d0

o

(8.21 )

The name " do not induce t

· _material

strain

Chapter 8. InfinitesimaI thermoporoelasticity

8.3 iJvp

_ dO'

--)

iJO'

141

MASS VARIATIONS ACCOMPANYING THE DEFORMATION OF A SATURATED POROUS MEDIUM

P

1

it contains a mass of fluid M equal to

The pore volume being saturated,

M

= pVp

(8.22)

in which pis the fluid density. The differentiation of this expression leads to dM

= pdVp + Vpdp

(8.23)

1

that is, after linearization,

1

The difference in density can be expressed as a function of the fluid compressibility (assumed to be constant in the range of pressures used). Indeed [see Eq. (3.31)].

~M

= Po~Vp + (p -

_1 __ K¡ -

Po)Vp

(8.24)

.L (OVp) Vp

op

T

which can also be written after linearization

te¡ = ~

(8.25)

p- Po

Dividing (8.24) by PoVp and taking into account of (8.18) and (8.25), one obtains after linearization (8.26) in which m = ~M/VB is identical to that defined in Chapter 7 (variation fluid mass of per unit of total initial volume). (j' and p, are defined with respect to a reference state of stress [o and a reference pore pressure Po.

J (8.20)

I

(8.21 )

1

8.4

UNDRAINED BEHAVIOUR SKEMPTON'S COEFFICIENT AND UNDRAINED ELASTIC CONSTANTS

The name "undrained test" is given to a test during which the loading variations do not induce fluid mass variations. By taking m O, Eq. (8.26) becomes

=

p =-B(¡

(8.27)

Part Ir. Mechanism ol material strain

142

Chapter 8. Infiai

Identifying

in which 1

B =

1

J(B ¡{M o [¡~j - ¡}M] + [¡;B - ¡}M] -----

(8.28)

is known as "Skempton's coefficient" . Equation (8.27) shows that under undrained conditions, pressure in creases linearly with mean stress with a slope equal to the Skempton's coefficient (Fig.8.3). Taking account of (8.28) and (8.6), (8.26) is written in the general case

J(Bm = _0"+-

--

expo

p

which leads fin

(8.29)

B

p

which can also

Hooke's la1 elastic constan

B

~

~ •• ~a

or taking aCCO Fig. 8.3. Skempton's

Under undrained of (8.27)

coefficient

(undrained

conditíons).

conditions, Hooke's Eq. (8.5) is then written, by taking account

(8.30) or , by introducing

where

r¡ is a constant

Biot's coefficient ex one is led to (8.31 )

One defines then the undrained elastic constants

Vu

and E¿ such that

that is taking , (8.32)

Chapter 8. Infinitesimal

Identifying

thermoporoelasticity

143

(8.31) and (8.32), one obtains 1 + V« Eu

1 + VB EB

_

------===>

(8.28)

3VB -3v" = __ E"

G _ G

,,-

B

(8.33)

(1- 2vB)aB + 00....-_-=-'--_

EB

EB

which leads finally to the expressions Vu

+ (1 - 2VB )aB = 3VB 3 - (1 - 2vB)aB

(8.29) E¿

=

(8.34)

3EB 3 - (1 - 2vB)aB

K¿

=

(8.35)

/{B

(8.36)

1-aB which can also be written

taking account of the first of the Eqs (8.33) (8.37)

.

Hooke's law can be written in its general form as a function of the undrained elastic constants K¿ and E". By solving with respect to the stresses, (8.5) becomes

l

(8.38) or taking account of (8.36)

(8.39) where (8.40) (8.30)

r¡ is a constant

and depends on

/{u,

B and a. Indeed by combining (8.29) and (8.41)

(8.31 )

one is led to

--/{Bm = poa

rr

1\B€kk

+

(1-

aB) B

P

(8.42)

that is taking account of (8.36) (8.32)

(8.43)

144

Part II. Mechanism

of material stl'ain

Chapter 8. ImiII

(8.44)

The consti constants by i then written

By eliminating p between (8.40) and (8.43) one obtains r¡=--

BKu O'

Taking account of (8.44), Eq. (8.40) is finally rewritten p

= r¡ ( -O'éa + :)

(8.45)

Equation (8.45) enables one to interpret physically constant r¡: T}/ Po evaluates the excess of pressure that needs to be exerted with respect to a reference configuration to in crease the fluid content by a unit of mass per unit of total volume for an isochoric (éa O) and isothermal stress path since thermal phenomena have not been taken into account.

=

8.5

The last te

(p = Po), non-

taking accoun from an incres

THERMAL EFFECTS

In the gen

The constitutive Eqs (8.38), (8.39) and (8.45) established in the preceding paragraph consider implicitly that the temperature of the medium remains constant during the transformation. The main temperature effect on a medium (solid, liquid or gas) is to cause an in crease (or a diminution) in volume: this is the phenomenon of thermal expansion. The increase in incremental volume resulting from an increment in temperature dT is such that dV - = O'fdT (8.46)

8.6

ENT ATl

Vo

where O' J is the thermal expansion coefficient of the material. If one wishes to maintain the volume of material constant during transformation, one has thus to apply to this latter a compressive mean stress such that

(8.4 7) Let us now consider the case of a saturated porous medium and let us carry out an isochoric (ea O), undrained (m O) test with temperature variations. This causes the appearance of a thermal stress of the type (8.47) with K¿ (undrained bulk modulus) and O'u (volumic thermal expansion coefficient of the medium under undrained conditions) as parameters. In the general case, after linearization of Eq. (8.47) the constitutive equation will be therefore''

=

=

(Ku - 2~u ) éa5¡]

+ 2Gue.]

Entropy .cí obvious reaso and T rather After differen1

d

or dividing b3 d

(8.48) (ni (:)

To compm

S and the fre

Y

5,] - O'uJ(u(T - To)5,]

3We write now the equations taking into account the initial state.

Each part tests.

_ material strain

(8.44)

Chapter

8. InfinÍtesÍmal

145

thermoporoelastÍcity

The constitutive equation can also be expressed as a function of the drained elastic constants by introducing the drained thermal expansion coefficient «e- Eq. (8.38) is then written 2GB) ( J(B - -3-

+ 2GBC

1)

(8.49)

a(p - Po)8,) - O:BJ(B(T - To)8'J

(8.45) ", evaluates the configuration b an isochoric aot been taken

Ckk81)

The last term of (8.49) is therefore, characteristic of an isochoric (ckA: = O), drained (p = po), non-isothermal (T#To) transformation. By comparing (8.48) and (8.49) and Gu, one obtains the pressure variation resulting taking account of the fact that G B from an in crease in temperature in an isochoric undrained transformation, that is

=

p-

aBJ(B (T

_ auJ(u -

Po -

a

'T')

(8.50)

-.Lo

In the general case, Eq. (8.45) will be written

p - Po

8.6

= r¡ [m] -aCkk

+-

Po

+

«;«; - aBJ(B (T - To) o:

(8.51 )

ENTROPY VARIATION ACCOMPANYING A TRANSFORMATION

(8.46) To compute the thermodynamic

(8.47) carry out an _ This causes bulk mod_der undrained _ Eq. (8.47) the lIS

potential

1/J, one has to write the global entropy

S and the free enthalpy of the Huid gm. Entropy S being a state function, one can calculate its total differential. For obvious reasons, we will choose as state variables VE (bulk volume), M (total mass) and T rather than Ckk, m and T, the first two already being incremental After differentiation, one obtains

dS = ( oS ) aVB

dVB

+

M ,T

(OS) aT

vB,M

ar ; (

OS) aM

vB,T

quantities.

dM

(8.52)

or dividing by VB OS) ( aVB

(8.48) Each partial tests.

m,T

Ckk

+

1 VB dT

(OS) aT Ekk,m

+

(

os )

aM

m

.

(8.53)

Ekk,T

derivative can be expressed explicitly by assuming three different

146

8.6.1

Part II. Mechanism oE material strain

(m = O) isothermal

Undrained

(T

= To)

test

where s~ is ti one increases fluid, one has that is

In this case one has

(a~~k) 3

m,T

(::r)

(8.54) m,T

Now, it has been shown in [Eqs (3.31) and (3.37)] that

(a~:') 3

(avaT ) B

m,T

zs»: m,

where L is the therefore the _ the temperail latter transfos

= ll'uVB

3

The total, (8.55)

aO';k)

(

Chapter 8. lD&

aVB m,T

that is

Replacing (8.55) in (8.54), one obtains

(8.56)

8.6.2

Undrained

(m

= O) isochoric

Replacing of reference ".

(ckk = O) test

or, after lineal In this case, one has to introduce isochoric specific heat in paragraph 3.9.1 for Cv. The incremental increase in temperature dT (at m and supply of heat dQ to the system

C€kk

in the same way as

€kk constant)

induces a

(8.57) where Mo and To are the initial mass (invariable since the test is undrained) reference temperature. The second partial derivative is such that

and the

8.7

VAR

DUR

The free el (8.58)

By differes

8.6.3

Isochoric

(ckk = O) isothermal

(T = To) test

By introducing an additional mass offluid into the porous medium, one introduces an additional quantity of entropy such that (8.59)

where Vi is ti! medium is sat

ot material strain

Chapter 8. Infinitesimal thermoporoelasticity

147

where s~ is the specific entropy of the Huid at temperature To. But by injecting Huid, one in creases its pressure. Since compression tends to in crease the temperature of the Huid, one has to extract heat from the system to maintain its temperature constant that is dQ= -LdM (8.60) (8.54)

where Lis the latent heat per unit ofmass of fluid supplied. The latent heat represents therefore the quantity of heat to be removed per unit of mass of Huid for maintaining the temperature constant in an isochoric, isothermal test. There results from this latter transformation a decrease of entropy

dS2

=-

dQ To

= -~dM

(8.61)

To

The total entropy variation during the test is therefore (8.55)

dS

L

= dS + dS = s~dM 1

- -dM To

2

(8.62)

that is

(:!) (8.56)

€kk,T

= s~

- ~

(8.63)

Replacing (8.56), (8.58) and (8.63) in (8.53), and introducing of reference volume VB, one obtains finally

entropy s per unit

(8.64) or, after linearization, (8.65)

(8.57)

8.7

VARIATION IN FLUID FREE ENTHALPY DURING A TRANSFORMATION

ed) and the The free enthalpy of the Huid is such that (8.58) By differentiating

(8.59)

- TS,

(8.66)

(8.66) one is led to (see Chapter 3) dG,

one introduces

= H,

G,

= V,dp -

S,dT

(8.67)

where V¡ is the Huid volume contained in the porous medium (equal to Vp since the medium is saturated)

(OG,) op

= T

V '

(OG,) oT

= p

-S,

(8.68)

148

Part Il. Mechanism al material strain

V¡ and S¡ being state functions,

one can calculate (av¡) ap

dV¡

dS¡

+

dp T

(as¡) ap

+

dp T

their total differentials

(av¡) aT

dT p

gm

(as¡) aT

- 1 [l+a¡(T-To)--.,~

P

o

dT p

p - po] A¡

T-To

+ Cp--

Sm

where a¡ and K¡ are respectively

(8.70)

a¡

the thermal

Po

expansion

coefficient

static bulk modulus of the fluid, and Cp its specific heat at constant Replacing

(8.70) in (8.68) and after integration, _

gm -

o

(

9m

-

o T - To)sm

a¡ +-(T Po

8.8

p + --

- To)(p - Po) -

THERMODYNAMIC

and the hydropressure.

one obtains

Po

Po

-

Cp

2T,

o

(p - PO)2

(8.71)

one obtains afl

., 2poA.¡

POTENTIAL

tensor AV. such that

(b) The porous tensor (c) The thermaI tensor The constitutive

= (Kv. -

17

2A22 =-

+ G•.•(b'kbJl + b,ebJk)

E = a1J[.

-6 = a•.•Kv.1.

equations

-

2~ •.•) b'Jbkl

one obtains aft

The last co Now in a Ji¡

'" A~kl

The coeffici and by identifj

(T - To)2

The constitutive Eqs (8.48), (8.51), (8.65) and (8.71) define the thermomechanical behaviour of a poroelastic material. These equations can be written in a matricial form by introducing: (a) The elastic undrained

Contrary ti in T - To and is a quadratic while the term In these ea

-(p - Po)

-

To

= 9~-

(8.69)

which can also be written by taking account ofEqs (3.31) and (3.41), after linearization and division by mass (the index m indicates specific quantities) 1

Chapter 8. InfinI

= - + A'"'" 0 17

p = Po - R: €

-

or, taking acco

can be written,

~

:

e - (m) -

m

R -

+ 1J+ Po

Po

_

o

9m -gm

A (T - To) -

a •.•K •.•- aBKB (T - To) a

(8.72) Deriving (8 (8.73)

Chapter 8. Innnitesimal

149

thermoporoelasticity

mo(T - To) O To +Smm-

S=SO+{l.:f+GEkk gm

O

= gm -

O

(T - TO)sm

P-PO

+ --

(8.69)

PO

-

Gp

(T-To)2 2

Lm To

a¡

+ -(T

T.

- To)(p - PO)

Po

O

(8.74)

(8.75)

(p - PO)2 2pOK¡ linearization

Contrary to (8.72), (8.73) and (8.74), (8.75) is not linear but of the second order in T - To and P - Po. In a linear theory (i.e. for which the thermodynamic potential is a quadratic form), only the three first terms of (8.75) are taken into consideration while the terms of the second order are neglected. In these conditions, the thermodynamic potential can be written

(8.70) '1/;

A2(T

=

+ f : 411 : f + A22(T - TO)2 + A33m2 +;:h : f +

'1/;0

- To)

+ A3m +;:h2 : f (T - To) +;:h3 : f m + A23(T - To)m

(8.76)

The coefficients can be estimated by taking account ofthe partial derivatives (8-1) and by identifying subsequently with (8.72,8.73,8.74,8.75). Thus by calculating 0'1/; u=-

-

of

(8.71) one obtains

after identification

with (8.72) dt3

= -~

R

(8.77)

0'1/;

s=-one obtains

after identification

with (8.74)

mo

= -G

2An

oT

Ekk-

(8.78)

To

The last coefficients can be calculated Now in a linear theory gm is written O gm = gm - (T -

using the second partial

rro) .1. o

o

Sm

deri vati ve (8 - 1).

P+ --

Po Po

(8.79)

or, taking account of (8.73), gm

= gmo -

1J

: f Po

+

TI

2" m

Po

+ (T - To)

[ -Smo

a" te; - o s + -----poa

KB]

(8.80)

(8.72) Deriving

(8.76) with respect

to m, one obtains

0'1/;

(8.73)

gm = om = 2A33m

+ A3 + dt3

: f

+ A23

(T - To)

(8.81 )

150

Part

Ir.

Mechanism

ol material strain

Chapter 8. InñDi&

which leads after identification with (8.80) to

= -2r¡Po

O

A3

= gm

The expression of _ V; - V;o +[0: §, -

V;

+

A

2

A33

O

23

= -Sm +

Q'u

Ku - Q'B KB poQ'

(8.82) Substituting one finally obta

is therefore such that e :

mO

~u : é

so(T - To) + g!m o

-Smm

2

- C~kk 2To (T - To)

2

(

(,6 : §,)(T Lm

T - To)-+ -(T To

r¡

+ 2P6m

- To) -

2

3 : §,

(8.83)

(:)

8.10

- To)

Finally, identification of A23 in (8.78) and (8.82) leads to a relation between latent heat and thermal expansion coefficients L= To [Q'uKu -Q'BKB] Q' Po

8.9

(8.84)

RELATION BETWEEN THERMAL EXPANSION COEFFICIENTS

I~Vfl

= ~VÍ

= sv¡ -

(8.86)

~VB

I~Vfl = VfCtf(T - To) - VBCtB(T

Extracting I time, one obtaa

(8.85)

- ~Vp

+ ~VM - To) + VMCtM(T

Up to now Fourrier's law ti into the constin law is written ~

Mass conser

During a temperature variation, in drained conditions and at constant mean stress, fluid is expelled from the porous medium to maintain a constant pressure. In fact, the volume of fluid really expelled during the process is such that I~Vfl

EQU

Replacing (1

(8.87)

- To)

where Ctf and CtM are respectively the fluid and matrix expansion coefficients. IntroVf), one ducing the porosity and taking account that the medium is saturated (Vp is led to

=

I~Vfl

= [0oCtf

- CtB + (1- 00)Q'M]VB(T - To)

(8.88)

8.11

PAR:

(8.89)

Let us consa and (8.96) are 1

or again dividing the two members by VB m

- = -[00Ctf - CtB + (1- 00)CtM](T - To) Po

the minus sign indicating that the fluid is expelled from the porous space. On the other hand, in drained conditions (that is at constant pore pressure) and at constant mean stress, Eqs (8.49) and (8.51) are written

(8.90)

Gl material strain

Chapter 8. Infinitesimal

151

thermoporoelasticity

(8.91)

(8.82) Substituting (8.89) and (8.90) in (8.91) and taking account one finally obtains

of (8.36) and (8.44),

(8.92) (8.83)

8.10 between latent

(8.84)

EQUATION

OF HYDRAULIC

DIFFUSIVITY

Up to now we have not used the diffusion equations (Darcy's law for flow and Fourrier's law for heat) defined in the previous chapter. They have to be introduced into the constitutive laws in order to complete the formalismo As regards flow, Darcy's law is written [Chapter 7, Eq. (7.46)] (l{ is supposed to be a scalar)

!Vi

-

= -l{V'p

with l{

Po Mass conservation

= -/-tk

(8.93)

implies [Eq. (7.10)]

m=-V'·!Vi

(8.85) (8.86)

(8.94)

Extracting m from (8.73), taking account of (8.84) and deriving time, one obtains . Po {)p PoB ()€ P5L {)T m =- - + : - - 1]

Replacing

(8.87)

{)t

{)t

1]

1]To

with respect

{)t

to

(8.95)

(8.95) in (8.94) and taking into account of (8.93), one is led to

!{)p + a O.::kk_

Lpo {)T

1] {)t

T01]

PARTICULAR

{)t

(8.88)

8.11

(8.89)

Let us consider first an isothermal and (8.96) are written

= ~V'2p

{)t

/-t

(8.96)

CASES test at constant

mean stress.

In that case, (8.8)

(8.97)

(8.90)

(8.98)

oi materíal straín

Chapter 8. Infiair¡¡

By replacing (8.97) in (8.98) and by taking account of (8.36) and (8.44) one is led

Taking a.caJI

Part II. Mechanísm

152

to ~

= ~J1\12p

8p

Bl\.B 8t

(8.99)

or taking account of (8.28)

mass conservati 8p

(8.100)

8t

Eq. (8.95) and] 1

e¡

where

K¡

~[_l __1] __1 00

KB

KM

(8.101)

KM

are respectively the fluid and the pore volume compressibilities [see Eq. (8.18)] at constant mean stress. (8.100) is the classic equation used in reservoir engineering. Another interesting case is that of a rock with an incompressible matrix (KM -+ 00, O:M -+ O). For such a material, the thermoporoelastic coefficients become [see Eqs (8.6), (8.28), (8.36), (8.44), (8.84) and (8.92)]

=

Kf

K¡

(8.103)

+ 00KB K¡

00

To = -af1\¡

In Eq. (8.11 fluid flow is of theory. Taking

PoL

(8.104)

7]=-

L

since in a lineal

(8.102)

a=l

B

(8.108) can :

T.7

(8.105)

11

(8.113) is ti

Po

By replacing (8.102), (8.103), (8.104) and (8.105) in (8.96), the hydraulic diffusivity equation becomes _1_ 8p J{¡

8t

+ ~

fJckk

_ O:¡ 8T

00 8t

at

= .s: \12p 00J1

(8.106)

8.13

RES~

BOU

ANI:

which only depends on the fluid properties (that is J{¡ and a¡) and the porosity.

A general tl

8.12

EQUATION

OF THERMAL

DIFFUSIVITY

This derives directly from Eq, (7.53), assuming on the one hand 1>1 = O (no mechanical dissipation in an elastic material) and ignoring, on the other, the viscous dissipation in the fluido In this case, around the reference temperature To, one obtains (8.107)

(a) The tot (b) The su (c) The dis (d) The int (e) The teJ In the sam equations and form: the Belti

Chapter 8. Innnitesimal

153

thermoporoelasticity

Taking account respectively of Eq. (8.74)

as

(8.99)

0<;'

mo

aT

Lm

o.

at =-6 : 8i+CCkkTo 7it+smm-

(8.108)

To

mass conservation (8.109)

(8.100) Eq. (8.95) and Fourier's law

Q=

-/\,VT

(8.108) can be written (replacing matrices

(8.101)

r

(a,J'uTo

- LpQa)

(8.110)

-6

and

E

by their values)

aekk ( P6L2) er Tt + CCkkmO + r¡To 7it (8.111) _poL ap r¡

at

+ T.oMVs m

= /\,V2T

since in a linearized theory [Eqs (8.68) and (8.71)]

agm

Sm

= - aT =

o

(8.112)

sm

(8.102) In Eq. (8.111), the convective term ToMVsm representing the heat supply due to fluid fiow is of the second order and consequently can be neglected in a linearized theory. Taking account of (8.84), one obtains

(8.103)

PoL ap - ry at

(8.104)

+

( CEkkmO

+

P6L2) r¡To

er

.r

aekk

7it + aBJ'BT0Tt

=

2 /\'V

(8.113) is the linearized diffusivity equation of thermoporoelastic

(8.105)

(8.113)

T materials.

IIItkallllic diffusi-

8.13 (8.106)

RESOLUTION OF A THERMOPOROELASTIC BOUNDARY PROBLEM. BELTRAMI-MITCHELL AND CONSOLIDATION EQUATIONS

L

A general thermoporoelastic (a) (b) (c) (d) (e)

(8.107)

The The The The The

problem contents 17 unknowns that is:

total state of stress that is six unknowns. state of strain that is six unknowns. displacement field that is three unknowns. interstitial pressure. temperature.

In the same way as for elastic media, the Hooke's equations, the compatibility equations and the equilibrium equations can be expressed together into a differential form: the Beltrami-Mitchell equations.

Chapter 8. InlUútc

Part Il. Mechanism ol material strain

154

For this purpose, let us express the Hooke's equations in drained conditions with respect to the deformations. Taking account of Eq. (8.37) this is written [see Eq. (8.5)]

=

2GBt:'J

(O"J - O'?J) - -1 /lB + /lB (O'kk - O'Zk) Ó'J

3(/1" - /lB) + B(1 + /1,,)(1 + /lB) (p - PO)Ó'J

+ fJ2t:zz

Writing (3

=

= 2 {Pt:yz

oy2

Two further After summing

4

(T - To)ó'J

Consider now one of the compatibility equations, for instance [see Eq. (1.82)] {Pt:yy oz2

j

(8.114)

O:BEB

+ 3(1 + ve)

or by replacing

(8.115)

oyÓ Z

3(/1" - /lB) B(l + /1,,)(1 + /lB)

which is the Bell the hydraulic di the consolidatia

(8.116)

(8.114) in (8.115)

one obtains after substituting 020'yy

v»

020'kk

-----¡¡;2 - 1 + /lB -----¡¡;2 +

(302p OZ2

+

"Y

o2T Oz2

Taking into substituting in ~

(8.117)

where The equilibrium equations on the other hand are Adding Eqs 00' xx

00' xy

00' x z

00' yx

00' yy

00' yz

O

---¡¡;-+Ty+Tz

OO'zy

(8.118)

O

---¡¡;-+Ty+Tz OO'zx

e

2Jj

OO'zz

---¡¡;-+Ty+Tz

=

+-

O

Let us derive these equations respectively with respect to x, y and z. After summation of the two last and subtraction of the first, one is led to 02 2~=-~-~+ oyoz

that is, after substitution

1 ---

1 + v»

02

02 oy2

02

O'xx ox2

oz2

(8.119)

••

where CT is cal] Writing

allows one to ea

of (8.119) in (8.117) [020'kk 020'kk] ---+-oy2 oz2

+r

['V2T

"2-

v

O'

_ 02T] ox2

x

x+ (3

=O

["2 v

p

p--022 ]

ox

Equation (8.

(8.120)

of material strain

-

155

Chapter 8. Innnitesimal thermoporoelasticity

conditions with [see Eq. (8.5)]

f3 and

or by replacing

'V

2 [U

xx(1

+

i by their values

VB) - ukkl

(8.114)

_ aBEB 3

[PUkk

3(vu B(l

+ ---¡¡;;2 ('V2T

-

+

VB) V ) u

(2'V p -

[Pp) OX2

(8.121 )

=O

_ 02T)

Ox2

Two further expressions (with respect to y and z) can be derived in the same way. After summing of the three equations, one finally obtains 2 [

(8.115)

'V

Ukk + B(l

6(vu - VB) + v•.)(l _ VB/

2aBEB] VB) T

+ 3(1 _

=O

(8.122)

which is the Beltrami Mitchell equation of a thermoporoelastic medium. By combining the hydraulic diffusivity equation with the Beltrami Mitchell equation, one obtains the consolidation equation. Consider the diffusivity equation

(8.116)

2-

om at

Po

= K'V2p

(8.'i23)

Taking into account (8.37) and (8.48), deriving (8.51) with respect to time and substituting in (8.123), one obtains

(8.117)

2KGBB(1+vB)(1+vu)'V2 3(V,,-VB) \

where

1

C

P-

-!.- [ ot

(]H+

~P

~T]

B +

C

KB

= Ba

(8.124)

(8.125)

(aB - au)

Adding Eqs (8.122) and (8.124), one obtains 2 [

CT'V

(8.118)

6(vu

2KGBB(1 + vB)(l + 3CT(Vu-VB)

-.dz.

After sum-

~ (8.119)

VB)

vu)

] _

2aBEB 3(1-vB)T

+

!.- [

l

(8.126) lT]

P-atUkk+BP+C

where CT is called "isothermal consolidation coefficient" . Writing 6(vu - VB) 2KGBB(1 + vB)(l + vu) + B(l + vu)(l - VB) 3CT(Vu - VB)

3

=-B

(8.127)

allows one to compute CT, that is CT

.,]

-

0'1:.1:+B(l+vu)(l-vB)P+

=

2KGBB2(1 + vu)2(1- VB) 9(vu - VB )(1 - vu)

(8.128)

Equation (8.126) can then be written

(8.120) (8.129)

156

Part

Ir.

Mechanism

oE material strain

Chapter 8. InnnitesúJI

or by writing

- 1989, A general March 1989.

(8.130)

GEERTSMA, J.A., elasticity o/ satur.l - 1957, The effect Petroleum transact

(8.131) called "general consolidation equation". Coupled with the diffusivity Eqs (8.96) and (8.113) it allows one to calculate (TU, p and T. In the particulate case of an isothermal transformation, one obtains

CT'12 called isothermal

consolidation

[(TU +

!p]

=

%t [(TU +

!p]

NUR, A., and By¡ mation o/ rocks tf1Íj MACTIGUE, D.F., 9533-9542.

(8.132)

e

RICE, J.R., and saturaied elastic 1M No2.

equation.

SCHEIDEGGER, A.: New York.

8.14

CONCLUSION

The constitutive such as

equations ofthermoporoelasticity

encloses independent

constants

(8.133) These constants do not necessitate taking into account the hypothesis of homogeneity and isotropy of the matrix. This hypothesis is then not necessary to the establishment of the constitutive laws. The equations deriving from this hypothesis [expression of the Biot's coefficient (8.6), of the Skempton's coefficient (8.28) and relation (8.92)] have to be used carefully. It is better to measure directly fundamental constants like, J(u, a and B and to verify subsequently this hypothesis.

BIBLIOGRAPHY BEREST, P., 1988, Phénoménes canique des roches", BRGM.

thermiques

en géotechnique,

BOURBIE, T., COUSSY, O., and ZINNER, B., 1987, Acoustics nip, Paris. CORNET, F.R., 1976, Etude du comportement par un liquide, Annales du CF11R.

in "La Thermoméo/ porous media, Tech-

élastique et [raqile des Taches saturées

CORNET, F.R., and FAIRHURST, C., 1974, lnfiuence o/ pore pressure on the dejormation behaviour o/ saturaied rocks, in Proceeding of the 3rd Congress of ISRM, Vol. 1, part B, pp. 638-644, National Academy of Sciences, Washington. COUSSY, O., 1988, Thermoporoelastic

response o/ a Borehole, unpublished.

)-

Chapter 8. Innnitesimal

(8.130) (8.131)

(8.132)

thermoporoelasticity

157

- 1989, A general theory of thermoporoelastoplasticity, Transport in porous media, March 1989. GEERTSMA, J .A., 1957, Remarks on the analogy between thermoelasticity and the elasiicity of saturated porous media, J. Mech. Phys. Solids, 13-16 - 1957, The effeci of fluid pressure decline on volumetric changes of porous rocks, Petroleum transaction AIME, Vol. 210. NUR, A., and BVERLEE, J .D., 1971, An exact eJjective stress law [or elastic deformation of rocks with fluíd, JGR 76, 6414-6419. MAcTIGUE, D.F., 1986, Thermoelastíc response of a fluíd saturated porous rack, JGR 9533-9542. RICE, J .R., and CLEARV, M.P., 1976, Some basic stress diJjusíon solutíons [or fluíd saturated elastic porous media, Review of Geophysics and Space Physics, Vol. 14, No2. SCHEIDEGGER, A.E., 1960, The physícs New York.

(8.133)

media, Tech-

on the defor-

oí ISRM, Vol.

of flow through porous media, Mac Millan,

CHAPTER

9

The triaxial test and the measurement of thermoporoelastic properties

The triaxial cell is the most suitable tool for determining the thermoporoelastic properties of a geomaterial. This test derives in fact from soil mechanics where it has long been used (since the 1930s), except for its adaptation to high-pressure conditions. For a more detailed study therefore, readers are referred to the work of Bishop and Henkel which contains complete information on the subject.

9.1

DESCRIPTION OF THE TEST AND OF THE EXPERIMENTAL SYSTEM

In a triaxial test one applies to a cylindrical sample a vertical load ov and a radial confining pressure Pc. In a radial plane, the stress field is therefore isotropic (and equal to Pc). Furthermore, an interstitial pressure p is imposed in the interconnected porosity via a third pressurization circuit. The latter can moreover function (Fig.9.1) under drained (valve V1 open) or undrained conditions, (valve V1 closed). The Total Compagnie Francoise des Peiroles (Total CFP) experimental system is of entirely new design. It is schematized in Fig.9.2. Its originality lies in the fact that instead of conventionally regulating a vertical load and a confining pressure, one directly imposes a hydrostatic stress and a deviator. This is accomplished thanks to direct communication (through conduit AB) between the confining pressure and the chamber above the piston (chamber C).

160

Part II. l\1echanism of material strain

Chapter

9. The tria

In this way, tI The deviator is a load O"y is (Fig. 9 fluid inlet

VI

~

__

~I"""

-~

Fig. 9.1. Schematic

description

of the triaxial

test.

clcvia t.oric s trc ss ...., :;:; '"

g

.
~

Pv is the pre of the piston (eq

?'

'" ~

3

'Fl

:..

:..

in

~ ?'

0~

'Fl

'"

'" c, ?'

0c,

9.2

Fig. 9.2. TOTAL-CFP triaxial (Nicolas

patent).

cell

DRAll

The interstiti recovered) volun ment of the piS (Fig. 9.4). A set of two 1 in the table belo

161

Chapter 9. The triaxial test

of material strain

In this way, the confining pressure is in fact equivalent to a hydrostatic pressure u. The deviator is applied via a second circuit (chamber D) so that the actual vertical load o'V is (Fig. 9.3) "

..

(9.1)

a

,,..

a 2r

2R

Fig. 9.3. Delail

st.ress

.'

view or the piston.

Po is the pressure in chamber D, Rand r respectively of the piston (equal to the sample radius) (Fig.9.3).

9.2

the large and small radii

DRAINAGE CIRCUITS

The interstitial pressure is imposed via a servo-controlled jack. The injected (or recovered) volume can be evaluated with maximum precision by measuring the movement of the piston with the help of a displacement transducer linked to the rod (Fig. 9.4). A set of two valves (1 and 2) enables one to establish three types of drainage listed in the table below

162

Part II. Mechanism

Regime Undrained Drained at 2 ends Drained at 1 end

Vl

V2

Closed

Closed or open

Open

Open

Open

Closed

injection

VI

of material strain

j ac k

fluid inlet

Chapter

9. The trial

The displacement to the strain of tI cell base, in othes unwelcome strain The strain gal strain of the cell a (i.e. horizontal ta of the sample (sa; is not integrated reliable results tl dependent on the displacement trar to regulation exo

C2 regulation Cl }-

loop

....J

9.4

FRICTJ

Friction probl of two orders.

V2

9.4.1

Fig. 9.4. Diagram

,

of the circuit.

If Vl is closed, one is working under undrained conditions and one can no longer control the interstitial pressure since the sample is isolated from the injection jack. It is therefore essential for obvious reasons of regulation to schedule two measurements of pressure (transducers Cl and C2) so that the transducer on which the cylinder is regulated (C1 in our case) is not isolated from the jack. Under drained conditions, on the other hand, a single transducer would be sufficient (Cl and C2 do in fact provide an identical measurement). Valve V2 makes it possible to carry out drainage at both ends if it is open, at one end only if it is closed. Drainage at both ends speeds up the test quite notably, which is extremely useful when the sample has a low permeability.

! .

9.3

STRAINS MEASUREMENT

The measurement of strains is conventionally made either using displacement transducers (external or internal) or with strain gauges stuck to the sample via a layer of resin to prevent direct contact between the glue and the interstitial fluid.

Frictil

This relates tA ment of the plane manner the latteJ of limiting this E teflon, on the oth be homogeneous. correct.

of material strain

jack

lion loop

163

Chapter 9. The triaxial test

The displacement transducer can be placed outside the cell. In that case, in addition to the strain of the rock, one measures that of the cell. It can also be placed on the cell base, in other words in the confining chamber. In this case, one eliminates the unwelcome strain of the central block of the cell. The strain gauges have the advantage, on the one hand, of not integrating any strain of the cell and on the other of allowing possible measurement of the radial strain (i.e. horizontal tangential strain). However, their use requires meticulous preparation of the sample (sticking, wiring) and the measurement remains localized (the strain is not integrated over the whole sample). Generally, strain gauges give much more reliable results than displacement transducers the measurements of which are very dependent on the quality of sample faces parallelism. The author recommends that displacement transducers should not be used for strain measurements but restricted to regulation except when the material is highly deformable.

9.4

FRICTION PROBLEMS

Friction problems are the most crucial in experimental rock mechanics. They are of two orders.

9.4.1

Friction of the piston

This relates to the friction between rock and piston which prevents free displacement of the plane faces of the sample. Instead of becoming strained in a homogeneous manner the latter tends to assume the form of a barrel (Fig.9.5). There are two ways of limiting this edge effect: on the one hand friction can be limited by grease or teflon, on the other, one can increase the length of the sample for the central zone to be homogeneous. Ratio 2 between length L and diameter is generally admitted to be correct.

would be sufe V2 makes it if it is closed. cnremely useful .c


o .c

displacement sample via a . iderstitial fluid.

Fig. 9.5. Fric ti on effect (b orrcl

shape).

on the strain

of the sample

164

9.4.2

Part II. Mechanism

of material strain

Friction of moving piston

Friction of the moving piston is very troublesome when one inverses loading. thermore this friction generally increases very sharply with confining pressure, the loading-unloading loop increases (Fig.9.6). A very effective technical trick enables one to avoid this by making the piston slide over a teflon ring (Fig.9.7). o-ring located behind the ring ensures the tightness of the system. Thanks to method, it is possible to minimize to a large extent these disturbing effects.

Furand over An this

displacement Fig. 9.6. Typical stress strain curve with hysteresis due to friction during

9.5

unloading.

PREPARATION AND INSTALLATION OF THE SAMPLE

Chapter

9. The trial

damage minerals I However, it is espt a problem, much

After cleaning avoid such unwelc better to use a no distilled water is Once it has be porous plates an ensure better dist

9.6

COMP]

After 48 houn To be convinced hydrostatic stress connected porosit in the hydrostati increases slightly An undrained tion. Contrary tc

I

A classical dimension of the samples is 4 em diameter by 8 ern height. So as not to increase the heterogeneity of the stress field (already disturbed by friction problems), it is advised that one accurately monitors the parallelism of the faces and their perpendicularity to the symmetry axis of the core. A second essential point concerns saturation. Very often, the cores have not been protected and they require cleaning with solvents (alcohol, trichloroethylene, benzene ... ). These cleaning procedures are often disputed. Indeed the fluids used can

165

Chapter 9. The triaxial test

loading. Furpressure, and .cal trick over (Fig. 9.7). An Thanks to this 'effects.

damage minerals contained in the interconnected porosity (particularly clay minerals). However, it is especially at the physico-chemical level (wettability) that they may pose a problem, much more than at the purely mechanical level.

/

teflon

ring

\ o-ring

I piston

Fig. 9.7. Anlifriclion a tenon

ring

system

with

(Nicolas Patent).

After cleaning, the samples are presaturated in a vacuum system for 48 hours. To avoid such unwelcome phenomena as clay swelling (frequent in sandstones) it is often better to use a non-polar fluid such as a low-viscosity refined oil. Among polar fluids, distilled water is to be avoided. Once it has been presaturated the sample is introduced into a rubber sleeve. Two porous plates are placed between the pistons and the faces of the sample. These ensure better distribution of the fluid into the sample. .

9.6

COMPLETE

SATURATION

OF THE SAMPLE

After 48 hours in a vacuum system, the sample is generally very poorly saturated. To be convinced of this fact, it is sufficient to close the drainage and to increase the hydrostatic stress (f. If the sample is poorly saturated (presence of air in the interconnected porosity), the interstitial pressure will respond poorly to any incrementing in the hydrostatic load (f: either the interstitial pressure remains unchanged or it increases slightly with respect to (f and above all with a certain delay (Fig. 9.8). An undrained test is therefore a very reliable way of checking the sample saturation. Contrary to what is generally admitted, it is not a flow but a consolidation

166

Chapter 9. The tria:

Part II. Mechanism of material strain

that will permit the drainage and lower than the tc volume injected iJ The consolidatioi (time), the inject tion time, t100) i uniform interstiti

I

undrained

pressure

Ir

CALCl

9.7 saturated (instantaneous

sample

THEe

response)

unsaturated

Aside from 1>4 rect method of D For this puq initial and bourn transformation, ; isothermal conso

sample time

Fig. 9.8. Interstitial during

pressure

an undrained

response

test.

with

volume )

The boundar

• log(time)

Fig. 9.9. Consolidation evaluation

of 'tOO'

curve

with

where Pi is the conditions (9.4), Carslaw pp. 99-1

p(z in which Tv is a

almaterialstrain

167

Chapter 9. The triaxial test

'

•

that will permit complete saturation of the specimen. For this purpose, one opens the drainage and one imposes a certain pressure Pc in the injection jack (naturally lower than the total stress u reached in the previous phase). One then follows the volume injected into the interconnected porosity by maintaining the loading constant. The consolidation curve has the appearance given in Fig. 9.9: as a function of the log (time), the injected volume increases, then after a certain time (known as consolidation time, tl00) it stabilizes. At this stage, one may assume that the sample has a uniform interstitial pressure and that consequently it is saturated.

9.7

.', (

CALCULATION OF PERMEABILITY THE CONSOLIDATION TIME

FROM

Aside from being a very effective saturation process, consolidation is also an indirect method of measuring permeability via the isothermal consolidation coefficient. For this purpose one has to solve the consolidation equation for the imposed initial and boundary conditions imposed [Eq. (8.131)]. In the case of an isothermal transformation, a constant load and a linear flow (according to axis x of the core) the isothermal consolidation equation is written (9.2)

•

with

CT = 2kGBB2(1

•

9J1.(vu

-

+ vu)2(1

- VB)

vB)(l

Vu)

-

(9.3)

The boundary conditions (in the case of drainage at both ends) are such that

• ~ ••

p(x, t = 0) = 0

= -h, t) p(x = h, t)

p(x

Vx (9.4)

Pi } Vt Pi

where Pi is the consolidation pressure and h the half-length of the sample. With conditions (9.4), (9.2) admits the well-known solution (see for example Jaeger and Carslaw pp. 99-100)

P()z , t

= Pi -

4Pi ~

(_l)n

-;- L.J (2n

+ 1) e

(2n

_(2n+1)2,,2T. 4

cos

+ 1)1I'x 2h

(9.5)

n=O

~'

in which Tv is a dimensionless variable such that (9.6)

168

Part II. Mechanism

To evaluate the consolidation time, one introduces known as "degree of consolidation" such that

U

=

of material strain

a dimensionless

9. The triaJI

variable U, p(bar) 50

f~hP(x,t)dx

(9.7)

2hpi substituting

Chapter

(9.5) in (9.7), one obtains after integration, 8

U=l--

2:

00

11"2n=O

1

(2n

+ 1)2

e

_ (2n+l)'".'Tv 4

(9.8) 30

It will easily be verified that for

Tv=O

U=O

Tv=oo

U=l

t

20

,

The sample is therefore consolidated after an infinite time. It is possible to calculate the consolidation time from approximate expressions. That of Terzaghi for example provides a very accurate approximation up to a 90 % degree of consolidation

10

o

(9.9) In a first approximation, if one admits this equation valid up to 100 % of consolidation, one can easily evaluate the consolidation coefficient as a function of consolidation time tlOO that is

50

100


1I"h2

CT

= -4tlOO

(9.10)

Substituting (9.10) in (9.3), one obtains a relationship between the permeability and the poroelastic properties. Consolidation is therefore an indirect permeability evaluation.

300

200

9.8

UNDRAINED HYDROSTATIC MEASUREMENT OF BAND

COMPRESSION

x;

100

The second phase of the test consists in gradually increasing 0', in undrained conditions, up to the mean total in situ stress O'g. If complete saturation has been reached (it is generally the case) after the first consolidation, an increment of hydrostatic stress /:).0', induces an instantaneous interstitial pressure increase /:).p /:).p = -B/:).O'

(9.11)

where B is the Skempton's coefficient. However, the response is not always ideal as is shown by a few typical cases in Fig. 9.10: total destruction of the interconnected

o

20

40

of material strain

169

Chapter 9. The triaxial test

ess variable U, p(bar )

p(bar) 50

25

(9.7) 20 _1 -.,' •••.

(9.8)

.-:,,:, 30

~~-':....--•••.•..•••••..•.•• -""''''''''!'''

15

t

20

a

.

b

10

i

is possible to calof Terzaghi for of consolidation

10

5

1000(bar) o

(9.9)

50

100

150

200

250

o

300

50

100

150

200

250

p(bar) 150

(9.10) the permeability _.rect permeability

/

i

.'

:

/

300

./

100

i

i

.!

1~:

/

200

ION

..

'

.'

.'

/ d

c

50

100

•• undrained condihas been reached t of hydrostatic Lip

lal(bar)

Ev (*10000)

o

20

60

80

o

100

Fig. 9.10. Typical behaviour a-total

(9.11) •• always ideal as lite interconnected

destruction

b-unsaturated

of porous

under

100 undrained

space

sample

c-influence

of fissuring

on stress

strain

d-influence

of fissuring

on stress

pressure

(after

Charlez.1987)

curve curve

200

300

conditions

170

Part II. Mechanism of material

strain

Chapter 9. The trY

porosity (a) (at the end of the test, the porosity is no longer in communication with the pressure transducer) or quite simply (b) poor initial saturation. Contrary to what is generally accepted in soil mechanics, Skempton's coefficient is often much less than 1. Type of rock

Ruler sandstone* Tennessee marble* Charcoal granite* Berea sandstone* Westerly granite* Vosges sandstone Weher sandstone* Venezuela sandstone

B

0

0.88 0.51 0.55 0.62 0.85 0.41 0.73 0.30

0.02 0.02 0.02 0.19 0.01 0.18 0.15 0.05

* According to Rice and Cleary (1976). Under undrained conditions, the loading rate should have no effect on the response of the material which in poroelasticity is necessarily instantaneous. However, the rate of the test plays a very important part for two main reasons.

9.8.1

The measuring circuit of pore pressure

The pressure transducer measuring the interstitial pressure is connected to the porous medium by a steel tube via a porous stone (Fig. 9.11). Therefore, the various links in the measuring chain (rock - porous stone - steel tube) will not react in the same way to the variations in total stress and in pore pressure. Bishop (1962) analyzes the effect of the measurement circuit by introducing two additional parameters ml and m2. If ml is the compressibility of the porous disc, during a total stress increment .D.7f and a pore pressure increment Llp, the volume rejected LlVl will be (9.12) On the other hand, the tubing (whose compressibility is m2) is only sensitive to pore pressure variation .D.p. Its volume variation will be (9.13) If loading is applied quickly, these volumes must equalize and, consequently, the change in interstitial pressure PI observed immediately after an increment Ll7f will be Llp

1 Ll7f = -kLl7f _m_2-1

= PI = -=--mi

(9.14)

The instanta that of the mat coefficient). The rock not can take a certa that is Pt, can b the expression (4

where

an is the roo Po is the initi PI is the initi Pt is the pres poo is the eqt J1.

=

(7r /2) [h.

inverse of the oe

171

Chapter 9. The triaxial test

nnication with . Contrary to is often much less

porous

stone

(compressibility factor m L)

tube for pressure (compressibility Fig. 9.11. Pore pressure

I

me asurerne nt factor

m2)

measurement

system

(after Bishop and Henkel. 1962)

on the response :Dowever, the rate

The instantaneous response is therefore that of the measuring circuit Pl and not that of the material Po (which is equal to -Bl:!."ff in which B is the Skempton's coefficient) . The rock not being necessarily very permeable, equalization between Pl and Po can take a certain time. The pressure evolution in the measuring circuit at time t, that is Pt, can be evaluated by integrating the consolidation equation. One is led to the expression (Gibson, 1954) 1-~ Poo

(9.12)

(9.13)

(9.15)

where 0, an is the root of the equation an cot an Po is the initial pressure in the sample, Pl is the initial pressure in the measuring system, Pt is the pressure measured at instant t, Poo is the equilibrium pressure, f1 (7r/2)[hD2mv/(ml + m2)] in which D is the sample diameter and mv the inverse of the oedometric modulus

=

=

(9.14)

m,,=

1- v - 2v2 E(I-v)

172

Part II. Mechanism

In (9.15), Poo is the equilibrium pore pressure (theoretically infinite time, i.e. for Tv -+ 00) such that

of material strain

obtained

after an

1- PI } Poo

= Po { 1-

1 +~

(9.16)

When equalization is reached between the sample and the measuring system, one evaluates an apparent Skempton coefficient Bmes such that

= -BmesL:l.7i

Poo

(9.17)

The measured value ofSkempton's coefficient must be corrected; one easily obtains by replacing (9.14) and (9.17) in (9.16)

B

= Bmes

(1

+ ~) -

t

(9.18)

Generally, especially for high pressure systems, the tubing is much more compressible than the porous disc (ml > m2) so that k is closer to 1. As J1. is always greater than 100 (except for very incompressible samples), this correction modifies very slighty the results. An example of measurement of Skempton's coefficient is presented in Fig.9.12. 40 30

Chapter

9.8.2

9. The tI

The

The heterog consequence is become uniforn of the interstiti unwelcome phe

Aside from makes it possil phase of (Fig.! linearity in the observed whose ticularly specta instantaneouslj

9.9

snco

When the h; pressure (whid actual value ~ a final consolid permeability hi consolidation is the loading rate geostatic stress

36

•.. oj

.D

34 32

~ :..

30

;::l

25

OJ OJ

~ •.. c,

9.10

PAR

26 24

"@

~ ...,

22

OJ

20

...,~

.S

MEA

15 16 14 12 10 100

120

140

160

mean Fig. 9. 1.2-~lcasurement on Lavoux limestone.

stress

150

200

(bar)

of Skempton's

coefficient

220

A test unde carried out in s throughout the too fast the loe modulus will 1>4 The ideal sc: but this proced coefficient to ei excessive local] Let us imag this test is earn will be equal to

of material s~rain

obtained

after an

(9.16)

._ri·ng system, one

(9.17) 0Ile

easily obtains

(9.18) lIluch more com-

L As jJ is always tion modifies 's coefficient is

173

Chapter 9. The triaxial test

9.8.2

The heterogeneity of the stress field

The heterogeneity of the stress field within the sample is due to end effects. The consequence is a heterogeneity of the instantaneous pore pressure which tends to become uniform in the same way as during the previous process. The distribution of the interstitial pressure prior to the equilibrium is difficult to evaluate, and this unwelcome phenomenon often remains difficult to quantify. Aside from Skempton's coefficient, the undrained hydrostatic compressibility test makes it possible to obtain the undrained bulk modulus. In fact there is firstly a phase of (Fig.9.10-c) microcracks closure which is reflected in a very marked nonlinearity in the stress strain curve. During this phase, microruptures are occasionally observed whose effect is to increase the connected porosity; this phenomenon is particularly spectacular on the curves in Fig. 9.l0-d in which the interstitial pressure falls instantaneously by nearly 100 bars. This effect will be studied in the third part.

9.9

SECOND PHASE OF CONSOLIDATION

When the hydrostatic pressure (j reaches the mean geostatic stress, the interstitial pressure (which was not monitored during the previous phase) is readjusted to its actual value PR (corresponding generally to that of the reservoir) by carrying out a final consolidation. Generally speaking, during the preceding hydrostatic phase, permeability has decreased so that the consolidation time corresponding to this final consolidation is greater than the previous one. Measuring it enables one to readjust the loading rate of the subsequent phases which will always take place above the mean geostatic stress.

9.10

MEASUREMENT OF DRAINED ELASTIC PARAMETERS

A test under drained conditions, on a previously consolidated sample, must be carried out in such a way that the interstitial pressure remains constant and uniform throughout the sample. The loading rate plays therefore an essential part: if it is too fast the local excess pore pressure does not have time to dissipate, and Young's modulus will be closed to that observed in undrained conditions (i.e. Eu). The ideal solution would be to consolidate the sample at each loading increment but this procedure would be too long. Therefore, it is better to use the consolidation coefficient to evaluate a sufficiently slow loading rate that will allow dissipation of excessive local pore pressure. Let us imagine an increase in the total deviatoric stress (0"1 - 0"2) till rupture. If this test is carried out under undrained conditions the interstitial pressure at rupture will be equal to

_ P

Pu -

R-

B0"1-

3

0"2

(9.19)

174

Part II. Mechanism of material strain

On the contrary, if drainage had have changed during the deviatoric Under imperfect drainage conditions, of rupture) is equal (on average) to D such that D

Chapter

9. The

u

been perfect, the interstitial pressure would not phase, and would have remained equal to PR. the pressure dissipates slowly and, at tf (instant Pf(> PR). One defines the degree of dissipation P 1- Pf - R (9.20) Pu -PR

=

It varies between the two limits Pf

= PR

=1

perfectly drained test ==> D

(9.21)

Pi = Pu undrained test ==> D = 0 Once again, the dissipation degree can be evaluated by integrating the consolidation equation. However, Gibson and Henkel (1954) have found an empirical formula similar to that of Terzaghi and such that D

= 1-

h2 --

(9.22)

nCTtf

in which n is a constant depending on drainage conditions: it is equal to 3 for a sample drained at both ends, 0.75 for a sample drained at one end. If one supposes that a dissipation degree of 95 % is sufficient (order of magnitude verified by experimentation), the time t i required to reach rupture will be such that (in the case n 3)

=

20hZ tf =

3CT

(9.23)

and will ensure proper drainage of the sample. If R; represents the compression strength of the material under the considered confining pressure, a loading rate such that Rc (9.24) Ve=tf

will therefore ensure proper drainage of the sample. Once of the drained properties is easy (Fig.9.13):

Vc

is known measurement

(a) By increasing the deviatoric stress and maintaining the hydrostatic stant one measures EB. (b) By increasing the hydrostatic load and maintaining the deviatoric stant, one measures J{B. (c) By increasing the deviatoric load and maintaining the hydrostatic stant, one also has access to liB (provided EB is known).

9.11

MEASUREMENT PROPERTIES

OF UNDRAINED

load conload con-

9.12

MEA AND

load conLet us take; case of a hydro

ELASTIC

Measurement of undrained elastic properties (Ku, lIu, Eu) is carried out using the same loading paths, but the loading rate can be higher.

Two cases c

1. If one aI the hydJ

of material strain

Chapter 9. The triaxial test

would not equal to PRo d, at tf (instant of dissipation

250

(9.20)

230

175

.l!Ssure

240

220 210

(9.21)

200

the consolida~irical formula

(9.22)

~ .D

190

';:;'

t

0 ..,

180


'I

';: <1>

'"

170 160

'

..

equal to 3 for a H one supposes . ed by experithe case n 3)

150

I

140 130

=

120

(9.23)

110 -2

0

2 deformation

Fig. 9.13. Measurement

(9.24)

4

6

(10--4)

of elastic

properties

on Lavoux limestone.

tic load conric load con-

9.12

MEASUREMENT OF BlOT'S COEFFICIENT AND MATRIX BULK MODULUS

tic load conLet us take again the expression of the variation in total volume [Eq. (8.8)]; in the case of a hydrostatic loading 7f, it is written 1:!.£kk

1 = -}.' (6.7f + a6.p) '\..B

(9.25)

Two cases can be envisaged:

=

•• ......,;0.0. out using the

1. If one assumes an isochoric test (6.€kk 0) by regulating the jack acting on the hydrostatic pressure, one measures directly Biot's coefficient since 6.7f

= -a6.p

(9.26)

Part II. Mechanism

176 2. If the interstitial increment (~O' (9.25) becomes

of material strain

pressure increment remains equal to the hydrostatic pressure one measures directly the matrix bulk modulus since

~Ekk

= -}'iM

(9.27)

An example is presented in Fig. 9.14 for Lavoux limestone.

210

t*

l

190 180 170

(fJ

~ '".. '"~

....,

160

ro

150

~

s

:j:++'

140

t* t*

110

Th4

l

l

When the ts form

Generally tl perature. In th such that

where ar is the The volume tion will be sue

+1

/

/ l

++

t

or again

However, tl the order of 10' therefore relati

l

t

130 120

9.13.1

(-If

200

8

9. The CI

= -~p),

~(f

e

Chapter

/

/

+* +*

0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 axial def'orm ati on (10--4) Fig. 9.14. Meastli'ement

of grains

bulk modulus

of Lavoux limestone.

9.13

MEASUREMENT OF THE COEFFICIENTS OF THERMAL EXPANSION

The thermal expansion coefficients are probably the most difficult therrnoporoelastic constants to measure. In fact, there are four different coefficients (au, oe, au and aj), but only two of them are independent.

9.13.2

M.

In theory, I sample in a tri associated stra The heatin

1. Either· rock. 2. Or by c

In the first rogeneous due the confining ( of the rock an4

177

Chapter 9. The triaxial test

atic pressure modulus since

(9.27)

9.13.1

Thermal expansion coefficient of the fluid

When the temperature form

variations remain moderate, fluid obeys an equation of the (9.28)

Generally the fluid is in a solid receptacle which also undergoes changes of temperature. In this way, the receptacle which initially occupied the volume Vo expands such that (9.29) Vr Vo [1 + O'r(T - To)]

=

where O'r is the thermal expansion coefficient of the receptacle. The volume variation of fluid Va is therefore only apparent, and the actual variation will be such that (9.30) or again (9.31) However, the values of the thermal expansion coefficient of fluids are generally of the order of 10-3 while those of solids are rather 10-6 The correction remains therefore relatively small. 0'/ =O'a+O'r

rC

rC.

Thermal expansion coefficient of some common fluids at atmospheric pressure Fluid

0'/(10-3;oC)

Acetone ....... Benzene ....... Oil ............ Water ......... Mercury ......

1.32 1.16

0.9 0.5 0.18 .'

9.13.2

Measurement

of au and o»

In theory, their determination seems to be easy since it is sufficient to heat the sample in a triaxial cell under drained or undrained conditions and to evaluate the associated strains (or displacements). The heating can be carried out in two separate ways: 1. Either using an electric resistance placed between the rubber sleeve and the rock. 2. Or by directly heating the confining oil. thermoporoe-

ts (au, O'B, aM

In the first case, only the rock is heated, but the temperature distribution is heterogeneous due to heat losses by conduction in the piston steel and by convection in the confining oil. The temperature field depends greatly on the thermal conductivity of the rock and on the thermal convection coefficient in the oil.

178

of material strain

Chapter 9. Tbe CI

Rather than heat just the rock, it is better in order to obtain a homogeneous temperature field, to heat the whole of the cell via the confining oil and to measure the thermal strain with the aid of a strain gauge stuck on the sample so as to avoid the expansion of the various parts of the cell (or of the transducer itself if it is situated inside the cell). The disadvantage of this technique is that it restricts measurement to temperature fields below 1300 (problem of thermal instability of the gauge at high temperature). An example is presented in Fig. 9.15 for Lavoux limestone (undrained test ).

In SI units I The thermal but it remains 4

Part II. Mechanism

..•.

~

I

u

(

vert

'"

~-1

;..

...,;:J

\

•


~ c,

1-2 \

a

..., 40 (J)

hor

.,'

"t;j

35

-3

30 -4 vol

25

e

-5~~ 20

1 lime

3 (thousands

__ -L__~ __L-~ __-L

"~

o time

of s)

Fig. 9.15. Measurement

of thermal

expansion

(thousands

of s)

coefficient

on Lavoux Limestone.

9.14

THERMAL CONDUCTIVITY

Thermal conductivity measures the ability of the medium to let heat flow through it. This is expressed by Fourier's law which relates the heat flow Q to the temperature gradient Q -K'VT (9.32)

=

of material strain

Chapter 9. The triaxial test

a homogeneous and to measure so as to avoid if it is situated measurement the gauge at high ne (undrained

179

In SI units K. is expressed in Wjm.oC. The thermal conductivity of rocks is generally greater than that of gases and water, but it remains considerably less than that of metals (by approximately a factor 100). Thermal conductivity of different materials Materials

K.

Gas at 1 atm . ......... . Coal Wood Rocks and soils Building materials ...... Insulating materials ..... Non-metallic liquids ..... Metallic liquids ......... Pure metals ............. •••••••••••

00

••••••••••••

k

i \.

t:

0.006-0.180 0.167-0.377 0.100-0.400 0.100-8 0.628-2.930 0.025-0.25 0.1-1 8.5 20-400

•••••••

0.0

••

••••

0.0

•••

in W j m.oC

0

vert f

•

TABlE 1 Thcrmal

hor

o

conductivities

of common

2.5

rocks

at 20 C

5

7.5 --..

sedimentary

rocks

quartzite dolomite sandstone

•

vol

limestone

o

wet clay shales argillaceous

sand salt

anhydrite metamorphic

rocks gneiss

marble volcanic

rocks basalt g abbro

---

diabase granite

(9.32)

K(w/m.·C)

180

Part II. Mechanism of material strain

Chapter 9. The tr

Contrary to electrical conductivity or magnetic susceptibility, thermal conductivity changes are relatively small from one rock to another. Low for clays, medium for carbonates, it is relatively high for rock-salt.

The measur fluid at a given being known ill temperature of specific heat of

TABLE 2 of common

Specific heat 600

BOO

700

900

rocks

1000

at 20 C

1100

1200

1300

BIBLIOGB dolomite ---------

sandstone

-----------

BISHOP,

limestone

--

CHARLEZ,

argillaceous ----salt -----

-----

Ph., abilite d 'une fIl - 1987, The tr. mation, Augus

shales

metamorphic

A.W.

triaxial test, Ec

wet clay sand

anhydrite rocks

COSTE,

J.,

gneiss marble

GIBSON,

R.E.

rate of strain,

volcanic rocks

JAEGER, basalt

J

.C.:

Oxford.

--gabbro

REUSCHLE,

diabase

f--------------------

am

: "Plasticite et

T.

un calcaire, un

granitc

SKEMPTON,

A

pore pressure, TERZAGHI,

K.

- 1948, Soil AI

9.15

V Vol. I & II, 'I'I VUTUKURI,

SPECIFIC HEAT

WEAHLS,

For solids and fluids, it is not useful to distinguish specific heat at constant pressure and volume. One merely defines it at constant strain, that is C'kk. Specific heat was defined in Chapter 3. It represents the capacity of the material to store heat. It is defined by the quantity of heat required to increase by one degree the temperature of one gramme of material, that is (9.33)

Cekk is expressed as J /(kg. DC). Specific heat varies very little from one rock to Its order of magnitude is 800 J /kg.oC.

another.

H.}

the soil mecha

., material strain

181

Chapter 9. The triaxial test

The measurement of specific heat can be carried out in a calorimeter filled with fluid at a given temperature (water for example). The specific heat of this fluid being known as well as the masses of fluid m f and of rock mr, if T; is the initial temperature of the rock and TF the final equilibrium temperature of the mixture, the specific heat of the rock will be written

T!

c
-

C

mf T/ -TF f mr TF- Tl' i

(9.34)

BIBLIOGRAPHY I

i I

I ~

l (

BISHOP, A.W., and HENKEL, D.J., 1962, The measurement triaxial test, Edward Arnold Publisher.

of soil properties

in the

CHARLEZ, Ph., 1981, Etude critique des differentes methodes de mesure de la permeabilite d 'une roche au laboratoire, in French, unpublished. - 1987, The triaxial test. Its application to petroleum engineering, Petroleum Information, August. COSTE, J., and SANGLERAT, G., 1975, Cours pratique de mecanique : "Plasticite et calcul des tassernents", Dunod Technique.

des sols, Vol. I

GIBSON, R.E., and HENKEL, D.J., 1954, Influence of duration of tests at constant rate of strain on measured drained strength, Geotechnique 4: 6-15. JAEGER, J.C., Oxford.

and CARLSAW, 1959, Conduction

of heat in solids, Clarendon

REUSCHLE, T., and HEUGAS, 0., 1989, Mesures ihermoporoelasiiques un calcaire, unpublished (in French).

Press,

sur un gres et

SKEMPTON, A.W., 1960, Effective stress in soils, concrete and rocks, Proc. Conf. on pore pressure and suction in soils, 4-16, London, Butterworth. TERZAGHI, K., 1943, Theoretical Soil Mechanics, John Wiley. - 1948, Soil Mechanics in Engineering Practice, John Wiley. VUTUKURI, V.S., LAMA, R.D., and SALUJA, S.S., Mechanical Vol. I & II, Trans Tech Publications. WEAHLS, H.E., 1962, Analysis of primary the soil mechanics division ASCE, Dec.

(9.33)

tiom one rock to

f t

t

and secondary

properties

consolidation,

of rocks, Journal

of

CHAPTER

"':-'.-

10

Thermoporoelastoplasticity General theory and application

In the case of irreversible behaviour, it is necessary to introduce into the formalism the concepts of plastic strain f.P, plastic porosity 0P and hardening variable. These additional parameters will clarify the concept of "effective stress".

A.GENERALCONCEPTS 10.1

CONSTITUTIVE LAWS IN IDEAL THERMOPOROELASTOPLASTICITY

Let us consider the simplest case of ideal plasticity, ignoring therefore any hardening phenomenon. We propose in this scope to write the constitutive laws.

10.1.1

Variations in pressure associated transformation

=

with a TPEP

=

Let us imagine an isothermal (T To) and isochoric (ckk 0) transformation. Let us gradually inject a mass of fluid m from the reference pressure Po (Fig. 10.1). As long as p is less than a limit value v', in conformity with Eq. (8.73), the pressure p increases linearly, with a slope TJ/ Po. When the ideal plasticity threshold is reached, although

184

Part II. Mechanism

of material strain

=

fluid continues to be injected, the pressure remains constant (p pe). Finally, during a possible unloading, only the mass of fluid me will be recovered. The mass mP is the irreversible increase in fluid content that is

Chapter 10. Tben

or

(10.1)

10.1.3

Var trm

p

Let us imag of fluid minto j plasticity thresl variation is eqw

e P

'YJ

Po Po

~m me

~

P

111

Fig. J 0.1. Pressure

vari at.ions for an

tsot.hcrm al isoehorie

t ransrorm

at ion.

where 0P is the plastic porosity (irreversible variation or porosity when the material is unloaded). During the transformation, since only the increment of elastic mass me can generate an increase in pressure the constitutive equation of ideal plasticity for pressure is therefore such that p

= Po + !L(m -

p00P)

(10.2)

Po

The same reasoning for an undrained-isothermal tion p

= Po

-G:TJ(ckk

-

transformation

leads to the equa-

c1k)

(10.3)

in which cu and c1k are respectively total strain and plastic strain. For a general transformation, the overall variation in pressure will therefore be written (10.4)

10.1.2

Constitutive

law in TPEP

By the same reasoning one is led to

2 = 20 + 1;:B: (fo - foP) - al(p - Po) - G:B/{Bl(T - To)

(10.5)

The final te extracted from phase, one folka the material be yond point A d extract heat fro Entropy ine to AB' as woul point A. In this

.EmaUrial

strain

Finally, during mass mP is the

185

Chapter 10. Thermoporoelastoplasticity

or (7'

--

=

(7'0

+ AU

""

:

(€ --

€P)

R

-

(m -

-Po

0P)

-

A (T - To) -

(10.6)

(10.1)

10.1.3

Variation in entropy associated with a TPEP transformation

Let us imagine an isochoric isothermal transformation and let us inject a mass of fluid m into the connected porosity (Fig. 10.2). As long as one remains below the plasticity threshold (m < me), the behaviour being thermoporoelastic, the entropy variation is equal to [Eq. (8.74)] s=s

o

m---

m

Lm To

(10.7)

s

M

B

O=-~"-----'---L----

(10.2)

Fig. 10.2. Entropy variations an isochoric

isothermal

~

associated

m

with

transformation.

(10.3)

(lOA)

(10.5)

The final term of the right-hand member represents the quantity of heat to be extracted from the system to maintain the temperature constant. During this first phase, one follows path OA of Fig. 10.2. When the ideal plasticity threshold is reached, the material becomes strained at constant pressure. The additional fluid supply beyond point A does not create any increase in pressure; it is no longer necessary to extract heat from the system to maintain its temperature constant. Entropy increases therefore according to AB (parallel to OM) and not according to AB' as would have been the case if the elasticity limit had not been reached at point A. In this way, the variation in entropy is therefore such that s = s~m - ~ (m - P00P)

(10.8)

186

Part II. Mechanism of material strain

By repeating the same reasoning for an undrained obtains therefore for any transformation

isothermal transformation

Chapter 10. Thermo

one

(10.9)

-so(T - To)-

10.1.4

,

Variation in fluid free enthalpy

-(T-To

According to Eq. (8.71) the free enthalpy of the fluid in a linearized theory is such that (10.10)

10.2 or by taking into account that g! = 1,b!

INEQl

CONe

+ pol Po (10.11)

In the case of i is reduced to

that is by taking account of (10.4)

gm

1,b!+ -s!(T

10.1.5

in which Vk are tJ Taking accoun

:0 [PO+17[-o:(c;kk-c;~k)+(:-0P)]+P;~(T-To)] - To)

Thermodynamic

(10.12) The pore pres that the dissipatk

potential in TPEP

The thermodynamic potential can be obtained by integrating the constitutive via the equations of thermoporoelasticity

a1,b s=-aT

laws

(10.13)

However one must take care when integrating on variable m since (10.12) combines energy flows (1,b! and s!) and a pressure term. By the same reasoning as in the previous paragraph, it is clear that the integration of gm leads to (10.14)

since once plasticity appears the pressure remains constant. Taking account of (10.4) and of the two other partial derivatives of (10.11) one is led after integration to

where

is the thermodyni If one consides

(10.18) is writ

In the case of ciated with fP is ~ This effective stre As already menta It depends on the

Chapter

187

10. Thermoporoelastoplasticity

ormation one 'I/J= 'l/Jo+ 2:0 : (f. - f.P)

- 0P)

+ 'I/J~m + Po (:

(10.9) -so(T-To)+

,

-(T

[Cf.-f.p) :~u:

(f. - ~P)- s~m(T

- ToM:

+~

theory is such

(f.-f.p)]

- To)

[m _ 0

-0P)

/2- (: + L(m

8: (f.-e) T-To

(10.15)

- p00P)----r;;-

C€kkmO (T _ TO)2 2To

P] 2 _

2 Po

(10.10)

10.2

(10.11)

INEQUALITY OF CLAUSIUS-DUHEM AND CONCEPT OF PLASTIC EFFECTIVE STRESSES

In the case of irreversible behaviour [Eq. (7044)], the inequality of Clausius-Duhem is reduced to a : iP - 8'I/J 6P _ 8'I/J irk > 0

(T - To)]

-

80P

8Vk

(10.16)

-

in which Vk are the hardening variables. Taking account of (10.15) and (lOA), one obtains 8'I/J

(10.12)

p

(10.17)

= - 80P

The pore pressure is therefore the thermodynamic that the dissipation function can be written

force associated

with 0P so

(10.18) where

(10.13)

Ak = _ 8'I/J 8Vk force associated with Vk.

(10.19)

is the thermodynamic If one considers a plastically incompressible matrix that is

(10.20) (lD.18) is written (10.21) (10.14)

In the case of a plastically incompressible matrix, the thermodynamic force associated with P is £"+ [p, the effective stress conventionally used in soil mechanics. This effective stress is different from that defined in poroelasticity (equal to £"+alp). As already mentioned, the effective stress cannot be considered as a static concept. It depends on the constitutive law; it is a rheological concept.

t

i=

188

Part II. Mechanism of material strain

Chapter 10. Thermop

Secondly, we have seen (paragr. 3.19) that in the case of dissipative phenomena the normality concept was not necessarily verified and one has in the general case to differentiate the yield locus from the plastic potential F. Under the hypothesis of matrix incompressibility, the plastic How rules will be written (10.22) where). is the plastic multiplier. (10.22) clearly shows that the idea of plastic effective stress is connected to the concept of plastic flow and not to the concept of yield locus (Coussy, 1989). In the general case, even under the hypothesis of plastically incompressible matrix, one has

f(q;,p)#f(rz+l

p)

(10.23)

The concept of effective plastic stress on the contrary is very interesting for an associated plastic flow rule for which f == F. In that case, the poroplastic problem can be considered as a continuous plastic problem provided that one replaces total stress q;by effective plastic stress in the plastic formalism, exclusively. On the other hand, it makes no sense to write static conditions with effective (elastic or plastic) stresses. For example boundary conditions have always to be written in terms of total stress and pore pressure separately.

t

10.3

PHYSICAL CONCEPT OF HARDENING CALCULATION OF HARDENING MODULUS AND OF PLASTIC MULTIPLIER

At the end of Chapter 3, we have introduced without any supplementary tions the hardening modulus H such that [Eqs (3.90) and (3.91)]

~= ~

H

(0-:- Of) orz

First of all, let us observe that the sign of H is that of 0- : or nil so that (Fig. 10.3) H H

>0 ====?

hardening

<0 ====?

softening

====?

. of

u:7J> -

====?

loading

0

====?

unloading

(J'

. of

u:7J< -

informa-

of / 8q; since ~ is positive

====?

The plastic HOl

As for the elasti

the incremental ela

(10.24)

0

I

(J'

The physical meaning of H can be easily understood if one considers the simple a. case of an uniaxial stress path and an associated constitutive law such that, f

=

The hardening I and in this simple local stiffness). The hardening pends on several he In the stress 51 "deform" the yield The hardening time. They are the The evolution (] both in size (isotro across the stress sp scalar, tensorial or This is so for e hardening consists dening consists in a variable).

189

Chapter 10. Thermoporoelastoplasticity

a hardening

(10.22) softening

(10.23) !!jIIIftSLin,g for an •••• ti",c problem

replaces total _ On the other - or plastic) . _ tenus of total

OL--L------

~~

E Fig. 10.3. Physical concept

of hardening

modulus .

The plastic flow rule (10.24) is then written 'p

1. =-rr H

'e

1. = ~rr E

€

As for the elastic part, one has €

the incremental elastoplastic law is written

"'_uy informa(10.24)

The hardening modulus has then in plasticity (from an incremental point of view and in this simple case) the same meaning as Young's modulus in elasticity (i.e. a local stiffness). The hardening modulus describes the memory of the material and generally depends on several hardening variables. In the stress space these hardening variables appear as parameters which can "deform" the yield locus. The hardening variables describe therefore the evolution of the yield locus over time. They are the image of the past. The evolution of the yield locus can take several forms (Fig. 1004): it can change both in size (isotropic hardening), in shape (anisotropic hardening) or simply move across the stress space (kinematic hardening). The hardening variables can thus be scalar, tensorial or both. This is so for example in Fig. IDA in which the yield locus is a circle: isotropic hardening consists in an increase of radius R (scalar variable) while kinematic hardening consists in a displacement of the circle whose radius remains constant (tensorial variable).

190

Part II. Mechanism

of material strain

Chapter

10. Therm

the consistency e c

and the plastic fI

1_--+-4 __

~

By replacing

Fig. 10.4. Different

types

of hardening.

a-isotropic b-anisotropic c-kinematic

'-r ,

Given f-L and a respectively the scalar and tensorial hardening modynamic potential 'Ij;can be written!

The hardening variables are associated with the thermodynamic such that [see Eq. (3.65)]

variable, the ther-

forces Rand

Similarly, the by identifying (1

JS

INCH OF Aj

lOA (10.25)

If one supposes for 'Ij;a decoupling rule such that

If one asSUID dition will be WI and if one derives (10.25), with respect to time, one obtains (10.26)

The partitios

In the general case (no partitioning rule between 9" and f-L), one has to introduce crossed terms in the expression of'lj; (see Lemaitre and Chaboche, 1988, p.198). Writing {)2'1j; I< = {)2'1j; and L=-

with

(10.26) becomes

where A is the eI

{)f-L2

R=I
-

and

K

and

{)!J2

=f

.9"

(10.27)

==

As by definil

The yield locus being such that

I1j; is a volumic quantity.

2 One will write suppose furtheJ'IDIII

191

Chapter 10. Thermoporoelastoplasticity

the consistency condition

f

= 0 is written (10.28)

and the plastic flow rule with respect to the hardening variables [Eq. (3.95)] (10.29) By replacing (10.26) and (10.29) in (10.28) one finally obtains

of

.

~:

0"

uO"

-

(10.30)

Similarly, the hardening modulus H can be calculated from the plastic multiplier, by identifying (10.24) and (10.30) -

le, the therH=

10.4

of

(10.31)

o~

INCREMENTAL LAW IN THE CASE OF AN ASSOCIATED PLASTIC FLOW RULE

(10.25)

If one assumes that J1 is the only scalar hardening variable, the consistency condition will be written '

of . (/

00-' - .(10.26)

The partition

of. - 0

(10.32)

+ 0 p. J1-

rule is written (10.33)

with

i/=A:i:e

- ~-and (10.34) where A is the elastic tensor. (10.27)

=:

As by definition

e'

20ne will write the equations with respect to the plastic effective stress = £ +lp. One will suppose furthermore c<= 1 (elastic effective stress identical to plastic effective stress).

192

Part II. Mechanism

of material strain

or

.x = -~ of it

(10.35)

n o;

Taking account of (10.33) (10.34) and (10.35), (10.32) will be written

(€- -.x 8q;'8f)

of : A:

8q!

'"

- tt»

=0

(10.36)

~f : A: €

.

da'

8f

A= H

'" -

+ 8q! : 4:

of

(10.37)

8q;'

Taking account of (10.33) and (10.34), the incremental law can be written

8f)

U'=A:€-.x(A:

or by introducing

the elastoplastic

'" -

matrix

10. TherIlMlt

(b) If the stres contrary

The hyperplao of the incremental one side and pure! We shall thus 4 the incremental e independent of tit In a tensorial two tensorial ZOnE an arbitrary choic multiplier X and tl

(10.38)

'" oq;'

4ep

Chapter

10.6 and taking account of (10.37)

LAWS ZONE!

(10.39) with (10.40)

Identically one easily shows €

-

10.5

= (AeP)-l '"

: ;,

(10.41)

-

GENERALIZATION OF ELASTOPLASTICITY: CONCEPT OF TENSORIAL ZONE

To define the plastic flow rule (Chapter 3, paragr. 3.19), one assumes the existence of a constitutive matrix varying with the incremental stress direction: (a) If the increment dq; points outside the yield locus tutive relationship is such that

(8 f / 8q; : £' > 0) the consti-

In the case of the entire increme the elastic tensor] loading zone and; than two tensorial increment, sever~ law (four tensoria raise serious proh However, mod ample) is facilitat model, paragr. U

10.7

LAWS TENS4

If the numbs corresponds to a no longer disconl very difficult expe dissipative mecha

of material strain

(10.35)

(10.36)

(10.37)

lie written

193

Chapter 10. Thermoporoelastoplasticity

(b) If the stress increment contrary

i!

points inside the yield locus (8 f / 8![ : i!

< 0) on the

The hyperplan of equation 8 f / 8![ : f! is therefore a boundary between two regions of the incremental stress space for which the constitutive law is different, plastic on one side and purely elastic on the other. We shall thus call tensorial zone any domain of incremental stress space for which the incremental equation is linear, i.e. for which the behaviour of the material is independent of the stress increment. In a tensorial zone, the constitutive matrix is a constant. On the boundary of two tensorial zones, there must be continuity of the strain increment. This prohibits an arbitrary choice of constitutive matrix and leads to the introduction of the plastic multiplier X and the plastic potential F.

(10.38)

10.6

"'(10.37)

LAWS WITH MORE THAN TWO TENSORIAL ZONES: THEORY OF MULTIMECHANISMS

(10.39)

(10AO)

(lOA 1)

In the case of (linear or non-linear) elasticity a single tensorial zone exists: in the entire incremental stress space, the constitutive matrix is constant (and equal to the elastic tensor). Classical elastoplasticity possesses two tensorial zones: a plastic loading zone and an elastic unloading zone. When the constitutive law contains more than two tensorial zones, one speaks about multimechanisms: according to the loading increment, several types of plasticity can exist. It is the case of the octolinear Darve's law (four tensorial zones, Fig. 10.5). These extremely complex models occasionally raise serious problems of continuity for the transitions between tensorial zones. However, modelling of certain geomaterials of complex behaviour (chalk, for example) is facilitated by the introduction of multiple sources of plasticity (see Lade's model, paragr. 10.30).

10.7

LAWS WITH AN INFINITY OF TENSORIAL ZONES

If the number of tensorial zones becomes infinite, (each loading increment corresponds to a different constitutive matrix) the matrix varies continuously and no longer discontinuously as in the case of the multimechanisms, for it is always very difficult experimentally speaking to define the "boundaries between the different dissipative mechanisms" .

194

Part II. Mechanism

Chapter 10. Thermop

of material strain

transformation

will

where )

At least, instead void ratio which is

Fig 10.5. Octolinear

incremental

in the space of incremental The four tensorial

zones

law with four tensorial

zones

sollicitations.

are defined

by the dashed

in which 0 is the II One can dedua differentiating (10.'

lines.

(after Darve.1987)

if one neglects the (dVB == dVp). Fim

B. THE CAMBRIDGE

MODEL

Generally in books on rock mechanics, few pages are devoted to clay materials. Indeed, clays are very often solely listed under the word "soil", relating to un compacted materials existing at shallow depth. In fact, numerous oil drillings show compact rocks with a very high clay content at great depth. For this reason we describe in the following paragraph the classical Cam-Clay model recently generalized to thermoporous materials by Coussy (1989, 1991).

10.8

SPACE OF PARAMETERS

Let us consider a volume element submitted to a state of stress £" a pore pressure p and undergoing a total strain increment df. The strain energy associated with this

10.9

PHEN(

NORM HYDW Elastoplasticity memory of the grt:A The consolidati applying to a SaIIlJl Consolidation take Let us consider a completely unload!

195

Chapter 10. Thermoporoelastoplasticity

transformation

will be such that

+ pdv §.: d£ + Pdv

dW

![: df.

(10.42)

rdc+ Pdv where s = a-

-

-

P =

r

=

1 -fT""I

3

1

-

+P

-fTkk

3

(10.43)

J~(~:~)

At least, instead of porosity, as regards clay materials it is preferable to talk about. void rat.io which is t.he ratio between porous and solid volumes that is e

=

Vp Vs ==> e

=

in which 0 is the porosity. One can deduce an important relationship differentiating (10.44) one is led to de = dVp Vs

-2 dVs

Vs if one neglects the compressibility (dV8 ~ dVp). Finally one obtains

clay content the classical Coussy (1989,

(10.44)

from (10.43) and (10.44); indeed, by

Vp ~ dVp Vs

(10.45)

of the matrix with respect to that of the pores

dv~--

10.9

0 1_ 0

de l+e

(10.46)

PHENOMENOLOGICAL STUDY: NORMALLY CONSOLIDATED CLAY UNDER HYDROSTATIC COMPRESSION

Elastoplasticity of clays is based on the following concept: the material has the memory of the greatest consolidation stress undergone in the course of its history. The consolidation concept was described in details in Chapter 9: it consists in applying to a sample a constant mean total stress and a uniform interstitial pressure. Consolidation takes a certain time having regard for the permeability of the material. Let us consider a previously consolidated clay sample (at a value P pJ) then completely unloaded, and given eo the void ratio of the sample after removal of the

=

196

Chapter

Part II. Mechanism of material strain

load. Let us load the sample again under a hydrostatic compression P (Fig. 10.6) and under drained conditions. One assumes that loading is sufficiently slow for the interstitial pressure to remain uniform at all times.

10. TbenDIII

with

To introdnc a state function 01

e

For an isothen

while, for a transfC strain will increase

where Cl:B is the de during this transli 0) one can calcula to determine the Ii

In(-P)

Fig. 10.6. Behaviour hydrostatic

p

of a clay under

compression.

10.9.2 Initially the material displays a non-linear elastic behaviour such that the void ratio decreases linearly versus logarithm of mean effective stress P with a slope equal to K known as "swelling coefficient". At pJ, irreversible strains appear. The material hardens and the maximum value reached by the loading (PJ in Fig. 10.6) is memorized as a new consolidation stress. The maximum consolidation stress appears therefore as an elastic limit but also as a hardening variable, so that an unloading according to path Be will be purely elastic and during a subsequent reloading, plasticity will appear in B and no longer in A. During hardening, the material exhibits again a linear behaviour of void ratio versus In (-P), but with a deeper slope >., known as "compressibility coefficient".

When the me; irreversibilities apj (known as compre for strains, the COI e= or taking account,

that is

10.9.1

Behaviour in the elastic domain

If the mean effective stress is less than the current consolidation stress, the non linear constitutive law is written (Fig. 10.6? eO

= eo

- K In( - P)

for the fact that P is negative since it is compressive.

POI.

(10.47)

or, with respect to the mean pressure P and the volumetric elastic strain [see Eq. (10.46)) 3The minus sign compensates

where e~ is the pi

I

It is necessary tition rule) is not since, infinitely so ism. Although ope hypothesis issued

197

Chapter 10. ThermoporoeIastoplasticity

P

P (Fig. 10.6) slow for the

= Poexp

with

ko

{ko(ve

-

(10.48)

va)}

= _1 + eo K;

To introduce in the state law (10.7) thermal phenomena, let us consider P as a state function of u" and T that is

dP = For an isothermal path (dT (~;)

(o~) ov

dve

+

T

(OP)

dT

oT

(10.49)

u.

= 0), (10.48) remains valid so that

= Poko exp {ko(v

T

e

-

(10.50)

va)}

while, for a transformation at constant mean effective stress (dP strain will increase in such a way that

ve

Va = aB(T

-

= 0) the

volumetric

- To)

(10.51)

where aB is the drained expansion coefficient (the interstitial pressure being constant during this transformation). Replacing (10.50) and (10.51) in (10.49) (with dP 0) one can calculate the second partial derivative which allows one after integration to determine the general non isothermal state law

=

P = Po [1 + exp{ko(ve

10.9.2 that the void a slope equal The material ~ is memorized Illlealrs therefore according plasticity will

-

va)} - exp{koaB(T

- To)}]

(10.52)

Behaviour in the plastic domain

When the mean effective stress P reaches the consolidation stress Pal, plastic irreversibilities appear: the void ratio evolves linearly versus In( - P) with a slope >. (known as compressibility coefficient) greater than K;. Assuming the partition rule for strains, the constitutive law will be written (Fig. 10.6) e

= e"

+ eP

= eo + >.In( -POI)

- K; In( -POI) - Aln( -P)

or taking account of (10.47) eP that is

= -(A

- K;)[ln(-P) -In(-PoI)]

P = POI exp {keeP - eb)} 1

(10.53)

k=->._K; where eb is the plastic void ratio associated with the initial consolidation

pressure

POI' (10.4 7) •• (see Eq. (10.46)]

It is necessary to point out that the small perturbation hypothesis (i.e. the partition rule) is not strictly in accordance with a non linear elastic constitutive law since, infinitely small of power more than one are taken into account in the formalism. Although open to criticism from a theoretical point of view, we will adopt this hypothesis issued from the Cambridge School.

198

10.10

Part II. Mechanism

of material strain

Chapter

10. Thermop

however that, cone speaking refer to J: distinct structures..

BEHAVIOUR OF A CLAY UNDER DEVIATORIC STRESS CRITICAL STATE CONCEPT

=

To simplify, let us consider the case of a triaxial test (iI, (i2 (i3). After consolidation under P = POI (Fig.lD.7), let us maintain the confining pressure constant and increase the vertical load (iI, under drained conditions (that is at constant interstitial pressure p). In diagram, -Pr, the loading representative point follows path AB (Fig.lD.7) with a slope 3 since 4 dP = d(il

3

and

dr

= =dtr,

260 220

a e•..

180 140 100

60 20 0

elastic

f

limi t

(

IL- __ ~

__

~~_P

-JL-~

Pcr Fig. 10.7. Concept

of critical

state

The clay behaviour for such a loading path is displayed in Fig.lD.8. It can be divided into three phases: initially, the material is strained under growing deviatoric stress and undergoes a plastic volume decrease (contractant behaviour). The density of the sample increases, the material hardens. The second phase shows the appearance of an ideal plasticity stage: the material is then strained at constant plastic volume. This second type of behaviour has a limit, and gradually there is a strain localization in a shear band: this is the phenomenon known as bifurcation for which the behaviour becomes dilatant (dVB > D). Only the first two types of behaviour will be studied in this chapter; bifurcation will be dealt with in the third Part. We may mention 4Do not forget that

dlT~

is negative (compression).

The transition the change from a is associated with which the originaJi 1. Critical sta; P constant The critiea slope M . .Ii and of the i -Pr into • 'f)

= =rl P,

2. This first a density) de

flEJBaterial

strain

199

Chapter 10. Thermoporoelastoplasticity

however that, concerning the appearance of a shear band, one can no longer strictly speaking refer to rheological behaviour given the separation of the sample into two distinct structures.

-s)•

positive

Z50

confining pres(that is at tative point

hardening

/

phase

ideal pi aslic behaviour (cri t.ic al slale)

,...--r-~~~'

ZZO

00 bar

180

~ .0 J::'

140 00 bar

100 00 bar 50 100 bar 20 0

Z

5

·1

8

10

12

14

Ev(l~) Fig. 10.8. Stress (remoulded

strain

curve under

deviatoric

loading

deep clay from France).

The transition phase (which can be ranked with perfect plasticity), characterizes the change from contract ant to dilatant behaviour and is known as critical state. It is associated with two fundamental properties, corroborated by experiment and on which the originality of the Cambridge model is built: 1. Critical state appears for a ratio between deviatoric stress r and mean stress P constant. The critical state is reflected therefore in diagram -Pr, by a straight line slope M. M, being a material constant independent of the loading parameters and ofthe initial void ratio (Fig. 10.7). The critical state line divides the plane -Pr into a contracting zone and a dilating zone. Introducing the parameter 1] = -r/ P one has

1]

1]

< M ==? dvP < 0

hardening

1]

= M ==? dvP = 0

critical state

> M ==? dVB > 0 ==?

shear band

2. This first condition is not sufficient: in the critical state, the void ratio (or the density) depends only on the mean effective stress, which is expressed by e

+ A In( -P) = Const. = r

(10.54)

200

Part II. Jl,fechanism of material

strain

r, is a characteristic of the material in the critical state only. In another, plastically admissible state, Eq. (10.54) will remain valid but the constant will be different from r (see paragr. 10.15).

10.11

EXPRESSION

dvP (10.55)

=T}-M

One can then calculate plastic strain work dWP such that [see Eq. (10.42)] dWP

one is led after iDIIi

1

The first condition and its three associated zones can be synthesized by writing that the plastic volume strain, plastic deviatoric strain ratio is such that dc;p

10. TheJDllllll

The constant CI for instance the •• the yield locus eqIII!

OF THE PLASTIC WORK

-

Chapter

= Pdd' + rde"

(10.56)

= -111Pde"

(10.57)

which is the eq~ hardening variable becomes greater; II Hardening caa consolidation pns point P -Pert r

=

that is, taking account of (10.55) dWP

so that the yield ••

Equation (10.57) used in the original model has been modified by Burland so as to obtain better forecasts for low ratios -r / P (10.58) which leads by taking account of (10.56) and introducing the variable dc;P dvp

10.12

DETERMINATION

10.13

to

2T} T}2-

(10.59)

M2

HARI

To determine d (10.54). Indeed, iIi associated void rail

OF THE YIELD LOCUS

An additional hypothesis has to be introduced plastic flow rule. In this case the flow rule is writteu''

iP

T}

now: the associativeness

= ~.of or

of the

(10.60)

Let us unload" line whose ordinal!

Finally, a new that is

4

Taking account of (10.59)

of

or

of -

r

27] T}2-M2

T}

=-P

(10.61)

oP 5A

star has been put after the plastic multiplier to avoid a confusion with the compressibility coefficient.

Eliminating eo ; of (10.47) to

Per increases d the critical state. ']

201

Chapter 10. Thermoporoelastoplasticity

one is led after integration

to r2

P

+ M2 P = Const.

(10.62)

The constant can be calculated by choosing a particular "plastic" loading point, for instance the hydrostatic consolidation point r 0, P POI' One obtains finally the yield locus equation

=

=

(10.63) .lSilred by wri ting

(10.55) Eq. (10.42)]

which is the equation of ellipses in diagram -Pro In Eq. (10.63), POI appears as a hardening variable: if the sample is consolidated at a higher value, the elastic domain becomes greater; the material hardens. Hardening can also be characterized by the critical pressure Per instead of the consolidation pressure Po. Indeed, the ellipse (10.63) intercepts the critical line at point P = -Per, r = M Per such that

(10.56)

(10.57)

p __ POI er 2

Per>

0

(10.64)

so that the yield locus can also be written

(10.65) (10.58)

10.13 (10.59)

HARDENING

LAW

To determine the hardening law (i.e. how Per evolves), one has to use the condition

(10.54). Indeed, if Per is the current mean stress [positive according to (10.64)], the associated void ratio is such that (Fig. 10.9) Cer

- iveness of the

= Cer

Finally, a new elastic reloading (1 that is C

(10.61 )

r - ).In Per

(10.66)

Let us unload elastically the material from point A of Fig. 10.9. It follows a swelling line whose ordinate at origin Co will be Co

(10.60)

=

Eliminating of (10.47) to

Co

=

+ K In Per

(10.67)

< -P < Per) will follow the same swelling line Co

-

(10.68)

Kln(-P)

and Cer from (10.66), (10.67) and (10.68), one is led taking account Per = exp{ k( Co

+

cP

-

r)}

(10.69)

Per increases therefore during the hardening contractant phase and is constant in the critical state. The latter corresponds as indicated previously to perfect plasticity.

202

Part II. Mechanism of material strain

Chapter 10. Th_

with [see Eq. (10.' e

To calculate II (10.70); taking aCl

eo eer

t---==------.-

K A

I-------~--_I

•• ~

To calculate II sider the three pa

In(-P}

Per Fig. 10.9. Hardening

law for the Carn-Clay.

-ovof = (1+~ P

Substitution of (10.69) in (10.65) gives a more general form to the yield locus, that is taking account of (10.47) r2 + M2 P2] [ -2M2P

f(P,r,e)=e+Kln(-P)+(A-K)ln

-r=o

dvP )

( dc.:P

where -UP is such •

that is by replacil

Equation (10.1 in the critical sta! calculated

PLASTIC FLOW RULE AND HARDENING MODULUS

For the Cam-Clay, the relationship parameters will be such that

3

(10.70)

In the generalized space of the Cambridge parameters (P, r, e), the Cam-Clay can be ranked with ideal plasticity. The yield surface (10.70) is fixed since the only parameters figuring in it are material constants (A, K, M and I'). The representative points localized outside the surface (10.70) are therefore inaccessible to experimentation. In the space of the loading parameters P and r on the other hand, Eq. (10.70) shows that e appears as a hardening variable. To complete the formalism, it is necessary now to calculate the plastic flow rule and the hardening modulus.

10.14

In (10.73)," will be written6

between plastic strain increments and loading

= (N')-l ==

( dP ) dr

(10.71)

6The prime inda: stress. 7 Remember tluI& i

_material

strain

Chapter

203

10. Thermoporoelastoplasticit.r

with [see Eq. (10.41)) 1 of

of

(AP)-l = HoP oP '" [ 1 of of HoP Or

.2. of

of HoP or 1 of of --H or or

1 (10.72)

To calculate the partial derivatives and the hardening modulus, let us differentiate

(10.70); taking account of (10.46) and (10.47), one obtains df

= (1 + e)dvP

21]('\ -

K)

- P(1]2 + M2) dr (10.73)

(M2 _1]2)(,\ - K) dP + P(1]2 + M2)

To calculate the elastoplastic matrix and the hardening modulus, one has to consider the three partial derivatives

.lID

the yield locus,

(10.70) tile Carn-Clay can the only param-

of ~=(l+e) P uv

of or

of oP

(M2 _1]2)(,\ - K) P(1]2 + M2)

In (10.73), vP appears as a new hardening variable so that the consistency condition will be written" _ 0 of . " of. (10.75) !:l ,.0+zuvPp V U£" where

iJP

is such that [see Eq. (10.24) and (10.60))

(10.76)

tative points lIIperiJ" mentation.

lland, Eq. (10.70) law"ism, it is nee-

(10.74)

that is by replacing (10.76) in (10.75) and taking account of (10.74)

(1 + e)(M2 _1]2)(,\ - K) P(1]2 + M2)

(10.77)

Equation (10.77) shows that before the critical state (71 < M), H is positive" while 0 (ideal plasticity). The plastic matrix (10.72) can now be in the critical state H calculated

=

(10.78) 18 and loading

(10.71)

6The prime index is introduced in (10.75) and (10.76) to indicate that £'is an effective state of stress. 7Remember that in (10.77) P is negative.

204

Part II. Mechanism of material strain

Chapter 10. ThermopGI

for the plastic matrix and _ >.. - K (dP 1+e P

dtJ'

>.. -

K (

1+ e

for the incremental

10.15

+

=>

2TJdTJ ) TJ2 M2

+

2TJ

)

(2TJdTJ TJ2 M2

+

M2 - TJ2

dv

= dtf'

_ ~

(10.79)

dP)

+P

law.

APPLICATION OF THE CAMBRIDGE TO SOME SPECIFIC STRESS PATHS

Equations (10.79) can be integrated onto some particularly

10.15.1

dP

1+e P

MODEL

interesting stress paths.

Isotropic consolidation

In this case, r

= TJ = 0 which

leads by replacing in (10.70) to

e

+ >.. In( -P)

10.15.3

= .6..

This is charactee

with

.6.. =

r + (>" -

(10.43), K) In 2

(10.80)

The elastoplastic law has thus the same form, as for the critical state but the constant is no longer r but .6...

10.15.2

Oedos

Anisotropic

In this case, TJ = Const.

consolidation => r e

= -TJP

which leads by replacing in (10.70) to

+ >..In( -

P)

Let us seek a all For this purpose let· constant and equal • After substitutial

= e'l

TJo is therefore III

with (10.81 ) which is indeed a constant since TJ is constant in this type of test. Again the elastoplastic law is identical, but the constant is different. It will easily be verified that the three constants are such that r < e'l < .6..(Fig. 10.10). We may observe that in the specific case in which TJ = M (critical state) one has e'l = r. Lastly, for TJ = a (isotropic consolidation) one obtains e'l = .6...

which is a constant K,>" and M). Rather than 'lo. i

•

of material

strain

205

Chapter 10. Thermoporoelastoplasticity

e

isotropic

consolidation

(10.79) anisotropic

consolidation

critical

state

MODEL

stress paths.

OL-

~

Fig. 10.10. Admissible

10.15.3

in the e-P

diagram.

Oedometric consolidation

This is characterized (10.43),

by a zero radial strain that is de3 = O. Taking account of

(10.80) state but the

states

-p

de

2 = --dv 3

(10.82)

Let us seek a constant value of 7] (and equal to 7]0) for which (10.82) is verified. For this purpose let us replace in (10.82) Eq. (10.79) with d7] = 0 (since 7] is assumed constant and equal to 7]0 in the course of loading). After substitutions, one obtains 7]2

+ 3A7]

-

M2 = 0

in (10.70) to A=I-}" 7]0

(10.83)

K

is therefore the root of Eq. (10.83) that is 7]0

(10.81 ) which is a constant

= ~ [V9A2 +4M2

-

(it depends only on the material

3A]

(10.84)

parameters

A and M, that is,

K,>.. and M). Rather than

7]0,

it is often better to use the oedometric T/

_

no -

0";,- _ 0"1

327]0

7]0

+3

ratio Ko such that (10.85)

206

Part II. Mechanism

or, after substitution

of (10.84) in (10.85) J(

10.15.4

of material strain

-- 1 [

One can also c phase (~0'2 = 0). 9

-1]

3(1-A)+v'9A2+4M2

0-2

Chapter 10. ThenDllll

(10.86)

Undrained triaxial test

that is by elirmna.

Let us consider a clay sample previously consolidated isotropically at a value Po (Fig. 10.11). According to Eq. (10.80) the corresponding initial void ratio eo is such that eo + Aln(-Po) - r - (A - K)ln2 = 0 (10.87)

or by replacing

(It

M 240

10.16

200

DIFFl

THEi

1;1 160 .0

';:;' 120

In the Cam-OI C1k see FAI-I equation of fluid (

[0P

80

=

40

Fig. 10.11. Typical undrained behaviour of clay for different pressur-es (r'eruoul ded deep clay from Francc). consolidation

Let us now increase 0'1 alone under undrained conditions. Given the matrix incompressibility, this condition is obtained by assuming a constant void ratio during loading so that (10.70) can be written (introducing 'fJ instead of r). eo

+ Aln(-P)

- (A - K)ln

p

2 2

'fJ

-

r=0

(10.88) p

1+ M2 A

Equalling (10.87) 1 - KIA)

which only de~ therefore similar' I To determine • equation. One ~ The first deriw i at constant mean;

and (10.88) one finally obtains

(introducing

the

constant

=

(10.89)

By substitutisq led to

(

_.aterial strain

207

Chapter 10. Thermoporoelastoplasticity

One can also calculate interstitial 0). During the latter phase (Ll0"2

=

pressure increase resulting from the deviatoric

(10.86) LlO"l

P that is by eliminating

= Po + -3- + Llp

LlO"l

===} Llp = P - Po

at a value Po -.io eo is such

+ -r

3

or by replacing (10.89) (10.87) (10.90)

10.16

DIFFUSIVITY EQUATIONS ASSOCIATED WITH THE CAM-CLAY

In the Cam-Clay, one assumes elastic and plastic incompressibility

of the matrix

[0P = c1k see Eq. (10.20)]. Taking account of Eqs (8.102) to (8.105), the constitutive equation of fluid (10.4) becomes

(10.91)

the matrix inratio during

_d

(10.88)

,".

the

constant (10.89)

which only depends on fluid properties. The hydraulic diffusivity equation will be therefore similar to Eq. (8.106). To determine the thermal diffusivity equation, it is necessary to know the entropy equation. One will proceed as in Chapter 8, paragraph 8.6. The first derivative of Eq. (8.55) consists in a non isothermal undrained elastic test at constant mean total stress for which Eqs (10.52) and (10.91) are written

+ Po exp{ kOCkd

p

Po

p

Po +

By substituting led to

- Po exp{ koCl:B(T - To)}

~~(-Ckk) + Cl:jKj(T

(10.92)

- To)

the first Eq. (10.92) in the second and after differentiation,

one is (10.93)

208

Part II. Mechanism

of material strain

The second derivative consists in an undrained isothermal elastic transformation P O"kk

3 +P that is after substitution

=

=

Kj Po - -fkk

e

00

[O"Zk]

3 + Po

Chapter

10. Thermot

test. In diagram P (point C) will be I

(10.94) e

exp{kofkk}

r

of the first Eq. (10.94) in the second and derivation,

(10.95)

The last two partial derivatives [Eqs (8.58) and (8.63)] being identical, the entropy equation takes the form 8

=

80

+ m8~ + POCiBko

+ajKj

(f

kk-

:)

exp{a8ko(T

- TO)}fkk

+ ;:CEkk(T-To)

(10.96)

The thermal diffusivity equation is then easily obtained by substituting (10.96) in Eq. (8.107). Let us recall that this equation assumes that no thermal dissipation takes place in the plastic process.

10.17

e, ~

THE CONCEPT OF OVERCONSOLIDATION APPLICATION TO TRIAXIAL TESTS

1

Up to now, we have only considered normally consolidated samples: the beginning of the deviatoric phase always began for P = Po, (consolidation pressure of the sample). Other cases may however be encountered: let us imagine a consolidated clay under an overburden Po, then as a consequence of erosion, the lithostatic stress diminishes markedly. The material is therefore in the present state subjected to a loading P~ much lower than its actual consolidation pressure (maximum reached during its history): it is said to be overconsolidated and the ratio of overconsolidation is defined as N The value of the overconsolidation behaviour of a clay.

10.17.1

=

Po

(10.97)

p~

ratio plays a decisive part in the rheological

Undrained overconsolidated

test

Let us consider a normally consolidated clay sample under a mean effective stress Po (Fig. 10.12). Given eo the initial void ratio. From Po, let us carry out an undrained

R~

al'material

strain

Chapter 10. Thermoporoelastoplasticity

209

test. In diagram P, r, one then follows the curve PoBC [Eq. (10.89)]. The critical state (point C) will be reached, with a mean effective stress equal to (10.94)

P2

=

Po 2A

(10.98)

r

tlerivation,

(10.95) M

.meal, the entropy

(10.96)

o~-L

-J~

__ ~+-~

~-P

-L

e

i - the beginning '" pressure of the a consolidated lithostatic stress subjected to . - urn reached OIerconsolidation

critical

(10.97)

state

s

~

Fig. 10.12. Influence ratio

on the stress

of the overconsolidated path (after

Desptuz, 1987).

-p

r

210

Part II. Mechanism

of material strain

In diagram e, P, the undrained path, corresponds to line PC (initial consolidation follows path OP). Let us now imagine (still starting from Po) stopping the loading in B (still under undrained conditions). During path PoB, the material has hardened, and the initial elastic limit (passing through Po but not represented on the diagram) moves farther. In particular it passes through point B(r, Pd which is expressed by

(10.99) in which P~ is the new consolidation pressure. It can be calculated by taking account of the fact that B also belongs to the undrained path normally consolidated PoB that is [Eq. (10.89)]

Pl=

Po

[1+ ;2]

A

'h WIt

1]

=--r PI

(10.100)

Eliminating r between (10.99) and (10.100), one is led to

Chapter

10. Th~

If the test was: equal to 2, and III To study the ~ loading point IIJB! constant (and stiI not maintain the consolidate agaia PI (intersection p passing through , If the deviatOl travels along the w loading point rem the two critical sII point is not sit1Ul The represenfil the first plastic in one meets point (;

(10.101 ) An unloading from B being purely elastic the constitutive equation will be e + JCIn( -P)

= Const.

(10.102)

In this case till tion ratio compra

According to Eq. (10.102), as the void ratio does not vary, (it remains equal to the initial void ratio eo), the effective mean stress P remains therefore constant during this unloading (vertical path B Pion Fig. 10.12) while in diagram e, P, the figurative point remains fixed. To understand the effect of the overconsolidation ratio under undrained conditions (in other words at a constant void ratio equal to eo), let us reload the material starting from Pl' The present consolidation pressure being the overconsolidation ratio Nl is therefore such that

P;,

(10.103) Given the choice of Ps ; it can easily be verified that this ratio N; is comprised between 1 and 2. One can moreover, taking account of (10.101), relate pressure PI to the initial consolidation pressure Po and to the overconsolidation ratio Nl that is

(10.104) Starting from PI and reloading the material (still under undrained conditions with void ratio equal to eo), one follows a vertical elastic path PIB. At point B, the first plastic irreversibilities appear. One follows therefore, as far as the critical state in C, the undrained path normally consolidated BC. During this phase, the material hardens and the final yield locus corresponds to a consolidation pressure P; = 2P2.

A typical exau ical analysis (ovee

lllMerial strain

If the test was begun at point P2, the overconsolidation ratio would then had been equal to 2, and the critical state reached according to P2C, in a purely elastic path . To study the effect of an overconsolidation ratio greater than 2, the representative loading point must evolve towards P3 (Fig. 10.12) while maintaining the void ratio constant (and still equal to eo). Since from point P2 elastic unloading P2P3 would not maintain the void ratio constant (path CR in diagram e; P), it is necessary to consolidate again the sample (necessarily under drained conditions) until pressure PI (intersection point of the isotropic consolidation path and the elastic exponential passing through point eo, P3) then unload until P3 (elastic exponential SP3). If the deviator is gradually increased in P3, (under undrained conditions), one travels along the vertical line P3C3 in diagram P, r while in diagram e, P, the figurative loading point remains fixed. When the critical straight line is crossed in C3, one of the two critical state conditions is not respected. Indeed in diagram e, P, the loading point is not situated on the critical state path (point C for a void ratio equal to eo). The representative point crosses therefore the critical straight line r M P, and the first plastic irreversibilities appear only in B3. One can show experimentally that one meets point C by a straight line of slope m (which is a new material parameter)

. •• B (still under .C

211

Chapter 10. Thermoporoelastoplasticity

aad the initial moves farther.

(10.99)

(10.100)

=

(10.101)

(10.102)

r = m(-P)

I

+ (M

(-Po) - m)2A"

(10.105)

In this case the curve r, s, displays a peak (Fig. 10.13) while, for an overconsolidation ratio comprised between 1 and 2, it does not.

r

2
(10.103)

axi al strain

o~--------------------------~~ (10.104) Fig. 10.13. Influence of the overconsolidated ratio on the stress strain curve.

A typical example is presented in Fig. 10.14 and shows the validity of the theoretical analysis (overconsolidation ratios of 1, 2 and 12).

212

Part II. Mechanism of material strain

Chapter 10. Th~

M=O.90 2

~

•

•

Ii

•

OCR=l

• •

• -P(bar)

3 Fig. 10.14. Typical example (after

of an overconsolidated

clay

D'espax. 1987).

o

10.17.2

Drained overconsolidated test

In the case of drained triaxial tests (slope + 3 in diagram P - r) for samples previously consolidated under PI, the influence of the overconsolidation ratio can be summarized as follows (Fig. 10.15). For a normally consolidated sample one follows the elastoplastic path AlGI. For an initial mean stress comprised between P3 and PI (for example Pd, one follows the elastic exponential DIBI at the beginning then, as soon as one crosses the elastic limit in BI, the plastic path B, G1 as far as the critical state. The case of an initial mean stress equal to P3 poses no problem: the behaviour remains purely elastic as far as the critical state (point G3). Lastly, if the initial mean stress is smaller than P3 (for instance P2), it can be shown that one first follows an elastic exponential D2B2 with decrease of the void ratio until a peak is reached (point B2) followed by an increase in volume until the critical state is reached in point G2. Point B2 which corresponds to the elastic limit is situated on a curve OB2C3 of equation

r = m(-P)

+ (M

- m)

[(-;I)r

(_p)l-A

e~

(10.106) e

-

«material strain

Chapter

213

10. Thermoporoelastoplasticity

r

Cl

•..•..

)

O~~L-

~

~"_-P

-4__~~~~

critical

isotropic

consolidation

(10.106) e

Fig. 10.15. Influence under

drained

of consolidation

conditions

(after

ratio

on stress

Despax. 1987).

path

state

214

Chapter 10. Th~

Part II. Mechanism of material strain

C. THE CONCEPT OF INTERNAL FRICTION

10.19 THE]

THE MOHR-COULOMB CRITERION

In diagram tTr.p is known for ob. triaxial compressil cut the straight. iii

In numerous geomaterials, plastic strains take their origin from relative slippage of the grains. The most conventional criteria of rock mechanics derive from this concept. Therefore, although the subject has been developed by numerous authors it seemed essential to examine in detail what is perhaps one of the oldest concepts of rock mechanics: the Mohr-Coulomb criterion first formulated in 1773.

with the directioa criterion reveals a two extreme priDC

10.18 THE CONCEPT OF INTERNAL FRICTION

AND OF COHESION

The concept of slippage is intimately linked to that of friction. Let us imagine two grains 1 and 2 (Fig. 10.16) welded together by a glue of resistance c, and let us subject the whole to a normal stress a and to a shear stress T.

(J

• grain

glue (cohesion

1

t

c)

Equation (10.1 stresses (j[, UIII (. the triangles OO'(

gr ain 2

Fig 10.16. Physical

meaning

of Mohr-Coulomb

AB=~

criterion.

The contact between the two grains produces a frictional force proportional to the normal stress a and opposed to the motion of the two grains. The relative slippage will only be possible if (10.107) in which J1. is known as internal friction

coefficient and c cohesion of the material.

i

I

f

with

In Eq. (10.111),

Chapter

10.19

215

10. Thermoporoelastoplasticity

THE MOHR COULOMB STRAIGHT

LINE

=

In diagram U - T, Eq. (10.107) represents a straight line of slope Jl- tan