Numerical Modelling of Coupled
Poromechanics Processes
Robert Charlier*  Lyesse Laloui*** – Frédéric Collin *,**
* Université de Liège
Institut de Génie Civil et de Mécanique
Chemin des Chevreuils 1, 4000 Liège
Sart Tilman, Belgique
Robert.Charlier@ulg.ac.be
**Chargé de Recherches, Fonds National Belge de la Recherche Scientifique
*** Ecole polytechnique fédérale de Lausanne, EPFL
Soil mechanics laboratory, Station 18
CH-1015 Lausanne, Switzerland
This paper is the basis for a course dedicated to geomechanics for energy
production. A number of different coupling are discussed, with respectively the fluid flow
(saturated and unsaturated) and the thermal transfers in deformable porous media.
Eventually some aspects on the numerical modelling with the finite element method are
discussed.
ABSTRACT.
RÉSUMÉ.
Cet article est la base d’un cours relatif à la géomécanique appliquée à la
production d’énergie. Divers couplages avec des écoulements de fluides en régime saturé ou
non et avec les transferts thermiques sont pris en compte dans des milieux poreux
déformables. Enfin, certains aspects spécifiques relatifs à la modélisation aux éléments finis
sont discutés.
KEY WORDS:
geomechanics, Biot, partial saturation, thermo-plasticity, finite elements, multiphysics coupling.
MOTS-CLÉS:
géomécanique, Biot, saturation partielle, thermoplasticité, méthode des éléments
finis, couplages multiphysiques.
Title of the journal. Volume X – no X/2002, pages 1 to n
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Title of the journal. Volume X – no X/2002
1. Introduction
This paper is intended as the basis for a course on coupled poromechanics processes
and their numerical modelling with the finite element method. The paper is
concerned with some application in geomechanics for energy production: mainly oil
and gas reservoir engineering, but also concrete for energy transformation structures
(power plants), nuclear waste disposal (lot of developments have been published),
geothermal energy production …
The following aspects will be successively treated:


Poroelasticity, following the Biot pioneering work;
Non-linear multiphysics coupling (hydromechanics, thermomechanics,
thermo-hydro-mechanics, partial saturation …)
 Coupled constitutive models
 Strain localisation and failure, joints and fractures
 Finite element modelling of coupled processes
Few constitutive models will be presented hereafter. Numerous other valuable
models have been published and are of interest. This paper is only an introduction to
constitutive modelling.
The ALERT Autumn Schools 2001 and 2005 have been devoted to Multiphysics
coupling in geomaterials. The comprehensive courses notes have been published
(Laloui, Charlier & Pijaudier-Cabot, 2005) and (Gens & Charlier, 2001). The reader
may find much more details then in this paper which party a synthesis from these
books.
2. Poroelasticity
Soils and rocks are generally considered as porous materials filled by one or several
fluids. Interactions between the porous skeleton and the fluid phases are very
important in many engineering problems. These coupling effects are the key issue in
petroleum engineering, if we want to understand phenomena like reservoir
compaction, oil recovery operations, casing collapse … It is therefore necessary to
develop numerical tools able to model these multiphasic materials. Different
approaches have been proposed in the past to tackle such kind of problems. The first
one is the general theory of porous media (based on a mixture theory) which could
be referred to the works of Bowen (Bowen, 1980) ), Hutter et al. (1999) and de Boer
(De Boer, 2000). Using an averaging theory (Hassanizadeh et al., 1979a, 1979b),
Lewis and Schrefler (2000) propose the general equations for multiphasic media.
Finally, a third approach follows the pioneer ideas of Biot (1941) on poroelasticity
and has been developed by Coussy (1995, 2004) in the framework of
thermodynamics of irreversible processes.
Geomaterials like soils, rocks and concrete are porous media generally
considered as the superposition of several continua (Coussy, 1995): the solid
Numerical poromechanics 3
skeleton (grains assembly) and the fluid phases (water, air, oil, gas). In this section,
our analysis is restricted to isothermal saturated conditions. It is assumed that there
are two immiscible phases: the porous skeleton and one fluid (water in the usual
case), their respective behaviour remains elastic. Phase changes like evaporation and
dissolution are not considered.
Based on averaging theories, Lewis and Schrefler (2000) proposed the governing
equations for the full dynamic behaviour of a partially saturated porous medium.
Hereafter, these equations are restricted for quasi-static problem in saturated
conditions. In the following developments, the balance equations of the coupled
problem are written in the current solid configuration denoted Ω t. Therefore the
reference configuration is changing all along the modelling
2.1. Balance equations and state equations
Two kinds of balance equations are necessary to describe the poromechanical
behaviour of a porous medium (in isothermal conditions): the momentum and the
mass balance equations. They can be written separately for the two phases or
alternatively for the whole mixture on one hand and another phase on the other
hand. Moreover it is necessary to define the state equation of the fluid and the solid
skeleton..
Mass balance equation
Since the current configuration used is defined following the skeleton movement,
the mass balance equation of the solid skeleton is necessarily met. For a given
mixture volume Ωt, solid mass balance equation reads:
   s ,t (1  nt )t 
t
0
[1]
where s,t is the solid grain density, nt is the porosity defined by the ratio of the
void volume Ωvt and the mixture volume Ωt, and t is the time.
From Equation 1, the evolution of the porosity can be computed:
t
 s ,t
t
t 
n  s ,t (1  n )  (1  n ) t


t
[2]
Fluid mass balance equation includes fluid flows contribution and a sink/source
term. For a unit mixture volume, fluid mass balance equation reads:
4
Title of the journal. Volume X – no X/2002
   w ,t n t 
t
 
 div j w  f w
[3]
where w,t is the fluid density, jw is the fluid mass flow (jw = w,t qw) and f w is a
mass rate of production/injection of fluid. In the following the fluid mass will be
denoted mw = w,tnt.
qw is the relative fluid flow with respect to the solid phase and is given by the
well known Darcy’s law:
qw  
kw
 grad pw   w,t g 
w
where
[4]
k w is the intrinsic permeability tensor, w is the fluid viscosity, pw is the
fluid pressure and g is the gravity vector.
Momentum balance equation
In the mixture balance of momentum equation, the interaction forces between fluid
phase and grain skeleton cancelled. In a strong form, this equation reads:
[5]
div   G  0
 
where ij is the total stress tensor and G is the body force vector.
State equations
To ensure the closure of the balance equations, it is necessary to define the
constitutive behaviour of each constituent. As far as a saturated porous medium is
considered in the following, the different constituents are the fluid phase, the solid
grains and the solid grain skeleton.
The fluid is a compressible material and the following relationship holds:
 w,t pw

 w,t k w
where kw is the fluid compressibility modulus.
[6]
Numerical poromechanics 5
When grains are assumed as a compressible material, their density depends on
the fluid pressure and the stresses acting on the solid grains. The following state
equation is proposed (Lewis et al., 2000):
 s ,t pw
 'TZ
 s 
s ,t

k
(1  nt )k s
[7]
where ks is the solid grain compressibility modulus and ’TZ is the Terzaghi’s
mean effective stress.
Even if the grains are incompressible, the solid skeleton can deform due to grains
arrangement. In poroelasticity, an elastic constitutive law is used to link the strain
tensor to the effective stress tensor. Note that we do not specify for the moment what
kind of effective stress we use:

2

 'ij   2G ik  jl  ( K  G ) ij kl   kl
3


[8]
where K is the bulk modulus and G is the shear modulus.
It should be pointed out that the bulk modulus defined above is a drained
modulus. This means that this property is measured during an experiment in drained
conditions without any overpressure of the pore fluid.
Equations [3] and [5] may not solve independently because strong interactions
exist between fluid flow and the mechanical behaviour. The numerical treatment of
this system of equations will be described in section 6. Coupling effects are present
on one hand in the storage term of the mass balance equation and in the computation
of the total stress tensor on the other hand. Another coupling effect can also be taken
into account: the influence the material damage on the permeability. In order to
exhibit these effects, two cases are considered in the following with an increasing
complexity.
2.2. Case 1: Incompressibility of the solid grains
In soil mechanics, it is commonly admitted that the solid grains are incompressible
for the applied stress state. Under this assumption, the porosity evolution reduces to
the following relationship:
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6
t
n  (1  n ) t

t
t
[9]
where the contribution of the solid grain compressibility vanishes. The porosity
variation is then only due to the solid skeleton rearrangement.
Moreover, most problems in geotechnics are solved under the hypothesis of the
Terzaghi postulate (Terzaghi, 1943):
 kl   kl  pw kl
[10]
where  denotes the total stress, ’ the effective stress and pw the pore pressure.
This postulate may be justified by different ways, among which the simple
equilibrium scheme:
F
1
1
F
Figure 1. Schematic view of Terzaghi postulate concepts for a granular (after
Schlosser, 1988)
From the previous equations, it comes that the fluid mass evolution and the total
stress tensor have the following expressions:
Numerical poromechanics 7
mw
 w ,t
pw t
 w t
k

e
 ij  Cijkl
 kl  pwij
[11]
[12]
This allows solving the poromechanical problem under the assumption of grain
incompressibility.
2.2. Case 2: Compressibility of the solid grains
At higher stress level, the solid phase (grains) is no more incompressible. (Biot,
1941) formulated first the equations of the coupled problem in this latter case.
(Detournay & Cheng, 1993) and (Coussy, 2004) founded their works on this theory.
The Biot’s effective stress is used in poromechanics instead of the Terzaghi’s one.
The following definition holds:
 kl   kl  bpw kl
[13]
where b is the so-called Biot’s coefficient, which is defined as follow:
b  1
K
ks
[14]
where K is the drained bulk modulus and ks is the solid grain bulk modulus.
The porosity evolution includes thus two contributions: the skeleton and the solid
grain compressibility. Using equations [2] and [7], the porosity evolution has the
following expression under the assumption of an elastic behaviour for the skeleton:
p
t 
nt  (b  nt )  ws  t 
 
k
[15]
From the previous equations, it comes that the fluid mass evolution and the total
stress tensor have the following expressions:
mw
 w ,t

pw
t
b t
M

e
 ij  Cijkl
 kl  bpwij
where M is the Biot’s modulus defined as follow:
[16]
[17]
8
Title of the journal. Volume X – no X/2002
1
nt b  nt

 s
M kw
k
[18]
Note that the Hook tensor Ce relies here the strain increment to the Biot’s
effective stress increment, instead of the Terzaghi’s effective stress in the previous
section.
2.3. Extension to poroplasticity
When the solid grains are incompressible, the extension to poroplasticity is
straightforward because it is only necessary to replace the Hook matrix by an
elastoplastic tensor in the equation [12], which gives the relation between total stress
increment and the strain increment.
However, Biot’s theory has been developed only for elastic materials and the
final formulation of the fluid mass evolution is only valid under this assumption.
Extension to elastoplasticity is thus not straightforward. Coussy (1995) proposes a
thermodynamic framework for poroplasticity. However such developments remain
matter of questions.
3. Plasticity for coupled processes: Partial saturation
3.1. Introduction
Most oil reservoir rocks have experimented different fluids: they may have
deposited under sea, and so saturated by (salted) water. Later, they have been
covered by a quite impervious (clayey, argillaceous) rock-sealing layer. Oil and gas
were migrating from the mother rock to the reservoir rock, pushing partly out of the
pores the water. The reservoir rock pores may then be partly saturated by residual
(sea) water, by oil and by gas. The wetting fluid (indices w, often water) is in full
contact with the rock skeleton, while the non-wetting fluid (indices nw, often oil)
remains captive in bubbles, or in a continuous phase, depending of the saturation
degree, whose boundary are stable thanks to superficial tensions and capillary
menisci.
Numerical poromechanics 9
Figure 2. Schematic view of granular geomaterial with two fluids
These superficial tensions are related to the so-called suction s:
s  pnw  pw
[19]
Suction is not only due to capillary effects (called matrix suction) but also to
osmotic effects and more generally to chemical disequilibria.
The superficial tensions are attracting the soil grains and creating an additional
inter-granular stress. During reservoir production (oil production, water injection),
the saturation variation and the depletion induce a variation of suction, of internal
stress state and of strain.
Figure 3 (Laloui & Nuth, 2005) shows a representation of the possible stress
paths that the material may experience (respecting the traditional soil mechanics
sign convention: compression being positive). In addition to the usual triaxial plane
expressed in terms of mean effective stress, p’, and deviatoric stress, q, a third axis
is added to capture the variations of suction s.
10
Title of the journal. Volume X – no X/2002
s=c
onst
q
3
V
4
2
s
B
1
p'
st A
con
=
'
p
Figure 3. Possible stress paths in (p’,q, s)
space
Non-wetting
fluid, nw
Vnw
Wetting
fluid, w
Vw
Solid
grains
Vs
Figure 4. Volume fractions :
distribution of solid, wetting and nonwetting fluids in unsaturated soil
Assuming the volume distribution in the porous medium as illustrated in Figure
4, these concepts have been formalised for a long time by e.g. by Bishops (1959),
who proposed to write the effective stress as:
 ij  ( ij  pnw ij )   ( pnw  pw ) ij
[20]
The effective stress parameter  takes into account the volume ratio between the
phases (degree of saturation Sr) as well as their compressiblities and surface
tensions. A simplified version of this Bishop’s effective stress was proposed by
Schrefler (1984):
 ij  ( ij  pnw ij )  Sr ( pnw  pw ) ij  ( ij  pnw ij )  Sr s ij
[21]
A wetting process (e.g. a water injection) corresponds to a suction decrease.
Following equations [20] or [21], at a constant total stress level, the Bishop’s
effective stress decreases and the strain is a dilatation or a swelling one in the case of
the poroelasticity, what is already observed for soils and rock at a given stress
conditions (e.g. reduced stress level). For other stress conditions (e.g. higher stress
level), the wetting strains are compressive ones, what is generally called collapse
strains. To explain the collapse in the framework of Bishop effective stress, this
latter should be coupled to a poroplastic constitutive behaviour. Such collapse strain
could probably explain some reservoir subsidence, among which the Ekofisk one
(Charlier et al, 2001).
Numerical poromechanics 11
To properly describe the role of the effective stress parameter
Morgenstern (1977) suggested adopting, instead the Bishop effective stress, two
independent stress variables among the total stress, the pore wetting and pore nonwetting stresses. Alonso et al. (1990) have first proposed an integrated elastoplastic
model for taking into account such wetting – collapse behaviour using the net stress
(total stress minus the pore non-wetting stress) and the suction. Their model, also
called Barcelona Basic Model, is shortly described in the following.
Elastoplasticity is considered, i.e. strain supports an additive decomposition into
reversible – elastic and irreversible – plastic parts:
 ij   ije   ijp
[22]
Strain variations are induced by either stress variation (denoted by m exponent)
or by a variation of the suction (denoted by s exponent). The strain rate writes:
 ij   ijm   ijs
 ijm   ije,m   ijp,m
 
s
ij
e,s
ij

[23]
p ,s
ij
3.2. Elasticity
The elastic strain induced by the coupling variable s rate is a purely volumetric one:
 ije,s  K s sij
[
24]
It may be a non-linear relation, i.e. Ks may depend on the stress and coupling
variable states. Often, for clayey materials, an oedometer like law is considered, but
this has to be adapted to the actual material answer:
v 
e,s
s
s
1  e0 s
[25]
3.3. Plasticity
As shown in figure 5, on an isotropic stress path, elastic as well as plastic strains
appear. Generally, for clays, the preconsolidation stress, i.e. the yield stress, and the
Title of the journal. Volume X – no X/2002
12
Specific volume
plastic slope depends on the suction level. This gives the shape of the yield surface
(elastic domain) in the p (net stress)-s (suction) space.
Suction
sy
s1
p01
s2
s3
p02 p03
ln p
SI
s3
Elastic
domain
s2
LC
s1
p*0
p01 p02 p03
Figure 5. Isotropic mechanical loading at constant suctions: determination of the
Loading Collapse (LC) yield curve in the (p-s) plane.
The yield surface has to take into account three types of mechanical behaviours:

Nearly isotropic compression, characterised by a pre-consolidation pressure
as plasticity threshold p0 ;
Deviatoric shear failure based on internal friction model and friction angle
or p (mean stress) – q (second invariant of the stress tensor) slope M;
Traction strength t , isotropic traction ps or cohesion c.


A schematic view of such a yield surface is given in figure 6-a. In the p-q
invariant plane, it is composed of an elliptic cap, a friction line and a traction cut-off.
q
(a)

(b)

M

p
p
p
-ps t
c0
p0
(p0-ps)/2
Figure 6. Yield surface for purely mechanical problems (a) and for Mechanical –
environmental coupling (b).
p
Numerical poromechanics 13
This yield surface writes here:
f1  q2  M 2  p  ps  p  p0   0 ; p   p0 - ps  /2
f 2  q  M  p  ps   0 ;  t  p   p0 - ps  / 2
[26]
f3  p   t  0
The slope M may depend on the Lode angle, as proposed e.g. by Van Eekelen
(1980). Additionally, a specific plasticity threshold may be written for the suction
variable:
s  s0
[27]
The mechanical – suction coupling is schematised for the yield surface in the
figure 5 and Figure 6-b, where the coupling between the suction and respectively the
pre-consolidation pressure p0, and the traction strength t are expressed (Collin et
al., 2005).
Generally speaking, it appears the following suction dependences of the plastic
material parameters, more precisely the yield surface parameters:



The pre-consolidation pressure p0 is highly dependent on the suction
(Laloui et al, 2001; Laloui & Nuth, 2005)
The friction angle at the critical state  seems quite independent of the
suction (Laloui et al., 2001)
The cohesion c and the traction strength t depend on the suction.
However the know-how on these aspects is not highly developed.
Following these observations, most papers have been devoted to the cap of the
yield surface, i.e. the Cam-Clay equation f1 (see Equation [26]), and to the preconsolidation pressure suction evolution. The hardening law will govern this
evolution. It depends on two internal variables:
p0   ( p0* , s )
[28]
where p0* is the p0 value when the suction vanishes. Classically, p0* evolves
with the volumetric plastic strain, following the Cam-Clay concept:
p0* 
1 e * p
p
  0 v
[29]
Title of the journal. Volume X – no X/2002
14
But any other hardening law may be used. The plastic strain rate derives from a
flow potential, which often differs from the yield surface (non-associative plasticity)
in the stress plane (figure 6-a) but generally coincides with the yield surface in the
coupling plane (figure 6-b). Once again the coupling term is a purely volumetric
one. This writes:
 * p   p ,m   vp , s
I
g
p
*
3

g  g 1 g g 1 g g 1 g g g g g 


,

,

,
,
,
, 
*

 1 3 s  2 3 s  3 3 s  4  5  6 s 
[30]
[31]
Deriving equation [28] with respect to the time:
p0 
 * 
 1  e * p 
p0 
s *
p0 v 
s
*
p0
s
p0   
s
[32]
The preconsolidation pressure rate depends on the plastic volumetric strain vp
and on the suction. If one considers a plastic loading where the stresses do not vary,
following the consistency condition:
f 
f
f
 ij 
p0  0
 ij
p0
p0  0
[33]
[34]
then the plastic strain rate induced by a purely suction loading writes:

s

p
v 
 1  e *
p0
p0*   

[35]
3.4. Fluid transfers
Two or more fluids flow in the pore space. It is generally considered that each one
follows a Darcy like equation, generalised by Richards for the water – air mixture.
The relative fluid velocity qw writes:
Numerical poromechanics 15
q (i )  
k (i )
 grad p( i )   (i ) g 
( i ) 
[36]
where the subscript (i) indicates the fluid (wetting versus non-wetting, oil versus
gas, oil versus water…), q is the fluid Darcy velocity, p the pressure,
viscosity, ρ the fluid density and g the gravity vector. The intrinsic permeability
tensor is noted k(i). The main change in the Darcy law for partial saturation is a
dependency of the permeability k(i) on the saturation degree Sr(i) :
k (i )  k (i ) ( S r(i ) )
[37]
Fluid mass balance equation includes classically fluid flows contribution and a
sink/source term. For a unit mixture volume, fluid mass balance equation reads:
  Sr(i )  (i ) n 
t
 
 div j (i )  f (i )
[38]
where j is the fluid mass flow (j =  q) and f is a mass rate of
production/injection of fluid. The fluid storage Srn depends on the saturation
degree, on the fluid density and on the change of the pore volume, i.e. on the strain.
The porosity evolution is a term coupling the mechanics and the flow. If the fluid
density changes, it has to be added in the storage evolution. The saturation degree
depends itself on the suction:
S r(i )  S r(i ) ( s)
[39]
This function is highly non-linear. A typical curve is given in figure 7.
Moreover, wetting and drying paths give different curves. In soil mechanics, a
number of expressions have been given for eq. [39] by various authors.
16
Title of the journal. Volume X – no X/2002
SUCTION, s (MPa)
1.5
Retention curves of Lixhe
chalk (oil-water)
OSMOTIC TECHNIQUE
(imbibition, various samples)
1.0
MERCURY INTRUSION
POROSIMETRY
(drainage, one sample)
OVERPRESSURE
(drainage, one sample)
0.5
0.0
0
20
40
60
80
100
WATER SATURATION, Srw (%)
Figure 7. Oil-water retention of Lixhe Chalk (Priol et al. 2004)
4. Plasticity for coupled processes: Thermoplasticity
4.1. Introduction
The development of the thermoplasticity follows intentionally a framework as
similar as possible to the unsaturated elastoplasticity of §3. Effectively, the two
formalisms are very similar, for the transfer problem and for the constitutive law
itself (Collin et al, 2005).
The geological layers are at higher temperature at great depth then at surface. For
example, the Ekofisk reservoir temperature is about 130°C. The Ekofisk clayey
capping rock layers are also at high temperature. When drilling a well, a mud is
circulated to transport the cutting to the surface and to cool the drilling head. This
mud is often at surface temperature and induces a temperature gradient. Also when
stimulating a reservoir by seawater injection, cold water at surface temperature is
injected in hot deep geological layers. For these two cases a thermal transfer arrives.
What is its effect on the rock mechanical behaviour? Most published research on
thermo-mechanics of geomaterials concerns water saturated clay material. What
follows is partly issued from a synthesis done by Laloui et al (2005).
Numerical poromechanics 17
When saturated geomaterials are heated, all of the solid and fluid constituents
dilate. In the case of cohesive soils (silt and clay), this dilation produces a decrease
in the strength of the adsorbed layers and a modification of the distance between the
clay particles (Fleureau, 1979). In normally consolidated conditions (NC), the
cohesive soils contract when they are heated and a significant part of this
deformation is irreversible upon cooling. This thermal contraction is an unusual
behaviour for any material. It is related to the microstructure and to the equilibrium
of absorbed water layers. Figure 8 illustrates such results: the response to a thermal
heating-cooling cycle at constant isotropic stress of a sample of saturated, drained
clay (Cekerevac, 2003).
100
Normally consolidated state
Overconsolidated state (OCR=12)
80
60
40
Dilatation
Compaction
20
-1
-0.5
0
0.5
1
Volumetric strain,  [%]
v
Figure 8. Typical thermal behaviour of fine soils during a thermal heating-cooling
cycle – Kaolin clay (Cekerevac, 2003)
As introduced in the section 3.3, preconsolidation pressure, p0, is considered here
as the stress yield limit which separates “elastic” pre-yield from “plastic” post-yield
behaviour in isotropic or oedometer conditions. Several results from the literature
show a decreasing of preconsolidation pressure with increasing temperature, figure 9
(Laloui & Cekerevac, 2003).
100
Tidfors & Sälfords (1989)
80
Eriksson (1989)
60
Moritz (1995)
Moritz (1995)
40
Boudali et al. (1994)
20
0
20
40
60
80
Temperature, T [°C]
Figure 9. Influence of temperature on the preconsolidation pressure.
18
Title of the journal. Volume X – no X/2002
The thermal strain rate tensor,  T , due to thermo-mechanical loading can be
split into thermo-elastic,  e,T , and thermo-plastic,  p ,T , components :
 ijT   ije,T   ijp ,T
[40]
4.2. Thermo-elasticity
The thermo-elastic strain,  ijeT , is composed of the superposition of a mechanical
elastic strain under adiabatic conditions,  ije, m , and a reversible thermal strain,  ije,T :
 ve,T   s' T ;
[41]
where  ve ,T is the thermo-elastic volumetric strain rate (  ve,T  tr ( ije,T ) ).The
thermal expansion coefficient of the solid skeleton,  s' , has to be explicated : here it
is the skeleton coefficient, which differs from the saturating fluid one.
Experimentally, one can measure the drained skeleton coefficient, the water
coefficient, the grain coefficient, the undrained bi- or tri-phases material, or any
other mixture, depending on the stress and pore pressure control.  s' , varies strongly
with temperature and slightly with pressure according to:
 s'  (  s' 0   T ) 
[42]
in which  s' 0 is the isotropic thermal expansion coefficient at a reference
temperature, T0 (usually ambient temperature), and  the ratio between the critical
 , and the mean effective pressure, p ' , at
state pressure for the initial state, pco
ambient temperature:

p 'c 0
p
[43]
 corresponds to the slope of the variation of  s' with respect to the current
temperature, T , at  1 .
4.3. Thermo-plasticity
Generally speaking, the yield surface for an isotropic loading may be written as:
p0  F1 (T ) F2    vp 
[44]
Numerical poromechanics 19
this means that two isotropic hardening mechanisms exist, a thermal one and a
mechanical / strain one. These processes are not coupled but only combined through
a multiplicative function. The mechanical hardening takes the classical form for
cohesive soils, as indicated in oedometer tests:
p0  F1 (T ) exp    vp
p
c0
(T ) exp    vp

[45]
pc 0 (T ) is the value of the preconsolidation pressure at temperature T ,

 
1 e
is the plastic compressibility modulus. A dependency law for the
thermal evolution of the preconsolidation pressure, pc 0 (T ) , is introduced to take into
account the thermal effect on the yield limit (Laloui & Cekerevac, 2003):
pc 0 (T )  pc 0 (T0 ) 1   log T / T0 
[46]
where pc 0 (T0 ) is the value of the preconsolidation pressure at the reference
temperature and  a material parameter. The expression of the isotropic thermoplastic yield limit is thus given by:
fTi  p ' pc 0 (T0 ) exp    vP
1   log T T 
[47]
0
Equation [47] contains two material parameters:  expressing the evolution of
mechanical hardening and  controlling the evolution of thermal hardening. This
latter one defines the shape of the yield limit as shown in Figure 10.
T1
Yield limit fTi
Thermo-elastic
domain
T0
Temperature, T [°C]
Temperature
100
80
60
=0.4
=0.3
=0.15
=0.075
40
20
0
Mean effective stress
'c (T1)
'c (T0)
70
80
90
100
110
Preconsolidation pressure, ' [kPa]
c
Figure 10. Isotropic thermo-plastic yield limit and its dependency on the
parameter  .
The flow rule is associated:
Title of the journal. Volume X – no X/2002
20
fTi ( p, T , vp )  gTi ( p, T , vp )
g

 iip  i Ti  i
 'ii
3
[48]
The plastic multiplier, i , is determined using the consistency equation.
Few results exist on a friction angle dependency on temperature (Laloui, 2001).
Some results indicate that the cohesion depends probably on the temperature.
However, no clear results have been presently published on these parameters
temperature dependency (Cekerevac & Laloui, 2004).
4.4. Heat transfers
Heat transfer in solids follows a Fourier’s like model for conduction. The
conduction heat flux q writes:
q
cond
  (T )
[49]
where T denotes the temperature. The conductibility is noted λ. It may be a
tensor if the anisotropy has to be taken into account. It depends on the saturation
degree and can be evaluated based on a kind of mixture formula between the phases
(solid and fluids). It depends on the temperature:
   (T )
[50]
In rocks saturated by one or two moving fluids, heat is also transported by fluid
(flux q(i)), this is the advection process.
q
ad ( i )
q
tot
 Hq
(i )
[51]
H is the fluid enthalpy, which depends on the temperature and on the saturation
degree. The total heat flux is then:
q
cond
q
ad (1)
q
ad ( 2 )
 ...
[52]
The heat balance equation writes classically:
   Ht  0
div q
tot
[53]
Heat transfer in soils and rocks is generally not as nonlinear as the fluid flow or
the solid mechanics. Then numerical convergence is easier. Moreover the coupling
Numerical poromechanics 21
of mechanics and fluid flow on heat transfer is lower then the coupling in the
opposite direction. Heat transfer has then to be analysed first.
5. Strain localisation, rupture, faults and fractures
5.1. Introduction – strain localisation
When considering the hydromechanical coupling, the permeability dependency on
the mechanical strains may be described by the Kozeny Carman equation or any
equivalent. However, the permeability variations obtained under large strain are not
very large, generally under one order of magnitude. Higher variations of strains or of
relative displacement are obtained in narrow zones (where strain localised) or in
fracture (where strains have fully localised). Such significant evolution of
permeability becomes an important subject of researches, for, e.g. characterisation of
the Excavated Damaged Zone – EDZ around tunnels (especially for underground
storage of nuclear wastes), or for evaluation of an oil reservoir evolution.
Strain localisation is a bifurcation of the stress and strain path. The bifurcation
has been analysed e.g. by Rice and co-authors. The result is the so-called Rice
criterion, which indicates if a shear band localisation is possible:
det(Cijkl ni nk )  0
[54]
In this equation, Cijkl is the constitutive tensor, which depends on the stress state,
and ni is the normal to the shear band.
On the one hand, only some constitutive models allow the Rice criterion to be
met. Among elastoplasticity, non-associated models and strain softening models
allow strain localisation. Hypoplastic models and damage models allow it also.
On the other hand, if the criterion [54] is verified for some stress states, the
direction of the potential shear bands is given. In such (2D) case, the criterion
reduces to a fourth order equation in the tangent of the band direction.
The criterion gives the band potentiality but not effectiveness. Only a map of
strains, strain rate, relative strain rate, or velocity may give information about an
actual banding.
Strain localisation depends on fluid pressure. Desrues (2005) has shown
experimental results on undrained sand samples and numerical models. Charlier and
Radu (1997) and Sanavia et al (2005) have shown finite element modelling of shear
banding with hydromechanical coupling and partial saturation. Large strain finite
22
Title of the journal. Volume X – no X/2002
element codes (Charlier & Radu 1997) allow to models very large strains in shear
bands (over 100%).
However, with classical finite element, the strain intensity in shear bands
depends on the mesh size. Similarly the load – displacements curve depends also on
the mesh size. To avoid such pathological dependency, it is needed to improve the
constitutive model: non-local models (as for concrete damage models, Bazant et al
1984, Pijaudier-Cabot et al 1987) or models with microstructure (micro-stresses and
micro-strains, Chambon et al, 2001, Collin et al, 2006) or Cosserat models (1909).
5.2. Fracture model for hydromechanical coupling
The ultimate state of a shear band in rock material, after very high strain level, is a
fracture, a joint, or any other discontinuity. Most oil/gas reservoir contains a number
of such discontinuities. This section is devoted to the fracture / discontinuities
modelling. The basic idea is to consider the fracture as an interface where two rock
parts contact. Then one can evaluate the contact stresses and their evolution, and the
fluid flow through the interface. In some sense, this is a derivation of a unilateral
contact with friction model. The hereunder formalism is not the only one for fracture
modelling. Other kinds of joint element have also been proposed in the literature
(Goodman et al, 1968 & 1974).
Interface laws – Mechanical law
A particular constitutive law describes the links between the contact stress rate and
the contact strain rate of an interface element subjected to normal loading. This
relation, firstly deduced by Goodman’s experiments, shows that the fracture closure
Vj changes, under increasing normal stress (n), in a non-linear way, closing
resembling a hyperbola. A characteristic example is illustrated in Figure 11.
From a physical point of view this behaviour can be explained with the
progressive mobilization of the fracture asperities. At the beginning of the
compression, few points are in contact and the deformations related to small
imposed stress are important. With the progressive fracture closure, with the
increasing augmentation of contact between asperities, the relative displacements
become smaller with progressing stress applied, until an asymptotic fracture closure
value is reached for very high values of stress.
The present behaviour can be attributed to the influence of different factors:
a)
initial actual contact area and vertical distribution of the aperture between
fracture walls;
b) strength and deformability of asperities;
Numerical poromechanics 23
c) fracture wall roughness;
d) thickness and physical properties of the unfilling material, if present.
Figure 11. Normal stress-deformation relations of intact and fractured rock
The fracture non-linearity in the n-Vj relation doesn’t take into account the
effect of fracture wall aperture, strength, roughness and unfilled materials. Then, the
strength-deformation relation is expressed through the empirical hyperbolic function
proposed by Bart (2000) (Fig. 11.):
 n ' 
K ni D0
V 

1  j 
D0 


V j
[55]
where:
- Kni is the normal initial stiffness associated to the fracture;
- D0 is the asymptotical fracture opening, related to the fracture when stresses
equal to zero are applied;
-  is an empirical coefficient ranging between 2 and 6, its value is increasing
with the fracture roughness. For that parameter Bandis et al. (1983) proposed
the value 2 that seems giving a correct description of the mechanical behaviour
of the fracture.
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Title of the journal. Volume X – no X/2002
Interface laws – Flow law
3
interface
element
S
{u, v,p f }3
I
pF}5
3’
I
{p f
2
S
fl
S
f
{u, v, p }2
ft2
1
S
{u, v, p f }1
F
p }5
I
2’
I
{p f }2’
F
}6
p }5
5
F
{u, v,p f
ft1
6
{u, pv,f
F
F
p }5
}3’
F
}5
1’
{ p f }1’
4
{u, v,
p Ff
foundation
}4
pF}5
Figure 12. Flow and displacements in a 2-D parabolic interface element.
To model the hydraulic behavior of a loaded fracture, an anisotropic description of
water flows through the interface element is proposed. The fluid flow is described in
its transversal and longitudinal path referred to the fracture. According to the
definition of a transverse transmissivity Tt; two transverse fluid flows ft1 and ft2 can
be described by the following relations (Fig. 12):



ft1  Tt p Ff  p If and ft 2  Tt p If  p Sf

[56]
describing respectively with ft1 the fluid path from one fracture wall to the
internal element I describing the interface and with ft2 the fluid path from the internal
element to the other fracture wall (flow moving from one boundary to the other
transversally to the fracture).
At the same time, if the interface longitudinal permeability kl is not nil, the
longitudinal fluid mass flow fl must be considered. In particular, longitudinal flow
through fractures can be assumed analogous to laminar flow between two perfectly
smooth parallel plates separated by a uniform distance. This assumption is based on
the observation that most natural fractures are approximately planar on the scale of
the fracture length. So for laminar flow between two parallel plates, the longitudinal
fluid flow referred to the fracture is given by the so called “cubic equation” (Tsang
and Witherspoon, 1981):
Numerical poromechanics 25
f l  d 3
f g
h f 
12  f
[57]
where fl varies as the cube of the distance between the plates d; f is the fluid
viscosity, f the constant fluid density and h is the change in hydraulic head across
the boundaries of the flow domain. As it can be seen in this case the hydraulic
conductivity of a fracture with aperture d is given by:
kl  d 2  f g / 12 f
[58]
In this case, to describe the fluid flow distribution into the fracture, equations
[56] and [57] are introduced into the Reynolds equation, this leads to the following
expression:
div  f l  0
[59]
Interface finite element – General concept of a contact problem
Consider two deformable solids (or domains) U and D with boundaries U and
D (Fig. 13). They are contacting through boundaries CU and CD.
U
U
e U2
e U3
SU
SD
S
x2

x3
D
 UC
e U1
 CD
e 3D
e1D
e 2D
D
x1
Figure 13. Contact between 2 deformable solids
In the local referential plane (e1,e2,e3) (Fig. 13), for a plane or axi-symmetrical
problem, the stress tensor in each solid reduces to a contact stress vector C defined
by two components :
   p 
C   1   
 2    
[60]
26
Title of the journal. Volume X – no X/2002
where p and  are the pressure and the shear objective stress vector (Charlier &
Cescotto, 1988). The perfectly sticking contact condition is enforced numerically
using the classical penalty method, which allows a small relative velocity between
points SU and SD.

U
Nodes
U
e1
e U3
Integration points
D
eU2

U
C
xd
C
x2
x3
xc

D
x1
Figure 14. Parabolic interface finite elements (U<0, i.e. no contact).
 C
The contact stress vector C is associated with the relative displacement velocity
(through the interface mechanical constitutive law) defined as the time
derivative of the distance vector u between CU and CD (Fig. 14). The contact
side of each body U and D can be discretised with interface isoparametric
elements, which are compatible (same degree and common nodes) with the solid
finite elements used to discretise the corresponding body (Fig. 14). The frictional
interface elements used here are based on mixed variational formulation (Cescotto &
Charlier 1993): contact stresses are computed at contact element integration points
whereas displacements of the solid boundary are computed at nodal points
The contact normal strain is simply obtained locally from geometrical
computation of the distance C between the two contact interfaces CU and CD
with C  u  e1 . For more details see (Habraken & Cescotto 1996). 2-D finite
element are isoparametric elements (Fig. 14), with 2 (linear) or 3 (parabolic) nodes
describing the interface element, the rock side, with 3 degrees of freedom (d.o.f) per
node (2 mechanical displacements u and v, and the fluid pressure on the structure
side pfS). To describe the seepage flow inside the interface pfI, 2 or 3 further nodes
are added with only 1 d.o.f. per node, the fluid pressure inside the interface; these
nodes are thus the same co-ordinates that the corresponding nodes on the interface
element.
Numerical poromechanics 27
6. Finite element for coupled processes
This paragraph is part of (Charlier et al, 2001).
6.1. Finite element modelling: monolithical approach
Modelling the coupling between different phenomena should imply to model each of
them and, simultaneously, all the interactions between them. A first approach
consists in developing new finite elements and constitutive laws especially dedicated
to the physical coupled problem to be modelled. This approach allows taking
accurately all the coupling terms into account. However there are some drawbacks
that will be discussed in a later section.
Constitutive equations for coupled phenomena will be shortly discussed in the
following sections. The number of basic unknowns and following the number of
degrees of freedom – dof per node are increased. This has a direct effect on the
computer time used for solving the equation system (up to the third power of the
total dof number). Coupled problems are highly time consuming. Isoparametric
finite element will often be considered. However some specific difficulties may be
encountered for specific problems. Nodal forces or fluxes are computed in the same
way as for decoupled problems. However stiffness matrix evaluation is much more
complex, as interactions between the different phenomena are to be taken into
account. Remember that the Newton – Raphson stiffness or iteration matrix is the
derivative of internal nodal forces / fluxes with respect to the nodal unknowns
(displacements / pressures / temperature…). The complexity is illustrated by the
following scheme of the stiffness matrix, restricted to the coupling between two
problems:
Derivative of problem 1 nodal forces
with respect to problem 1 nodal
unknowns
Derivative of problem 2 nodal forces
with respect to problem 1 nodal
unknowns
Derivative of problem 1 nodal forces
with respect to problem 2 nodal
unknowns
Derivative of problem 2 nodal forces
with respect to problem 2 nodal
unknowns
The part of the stiffness matrix in cells 1-1 and 2-2 are similar or simpler to the
ones involved in uncoupled problems. The two other cells 1-2 and 2-1 are specific to
coupling and may be of certain complexity. Remember also that the derivative
consider internal nodal forces / fluxes as obtained numerically, i.e. taking into
account all numerical integration / derivation procedures. On the other hand, large
difference of orders of magnitude between different terms may cause troubles in
solving the problem and so need to be checked.
Numerical convergence of the Newton – Raphson process has to be evaluated
carefully. It is generally based on some norms of the out-of-balance forces / fluxes.
28
Title of the journal. Volume X – no X/2002
However, coupling implies often mixing of different kinds of dof, which may not be
compared without precaution. Convergence has to be obtained for each basic
problem modelled, not only for one, which would then predominate in the computed
indicator.
6.2. Physical aspects: various terms of coupling
A large number of different phenomena may be coupled. It is impossible to discuss
here all potential terms of coupling, and we will restrict ourselves to some basic
cases. In the following paragraphs, some fundamental aspects of potential coupling
are briefly described.
6.2.1 Hydromechanical coupling
Number of dof per node : 3 (2 displacements + 1 pore pressure) for 2D analysis and
4 (3 displacements + 1 pore pressure) for 3D analysis.
Coupling mechanical deformation of soils or rock mass and one fluid flow in
pores is a frequent problem in geomechanics. The first coupling terms are related to
the influence of pore pressure on mechanical equilibrium through the Terzaghi's
postulate
     pI
[61]
with the effective stress tensor ’ related to the strain rate tensor thanks to the
constitutive equation, and the unity tensor I.
The second type of coupling concerns the influence of the solid mechanics
behaviour on the flow process, which comes first through the storage term. Storage
of water in saturated media is mainly due to pores strains, i.e. to volumetric changes
in soil / rock matrix :
S  v
[62]
Another effect, which may be considered, is the permeability change related to
the pore volume change, which may for example be modelled by the Kozeni –
Carman law as a function of the porosity k = k(n). This effect is much more
significant for fractures (see next paragraph).
The time dimension may cause some problems. First implicit scheme are used
for the solid mechanics equilibrium and various solutions are possible for the pore
pressure diffusion process. Consistency would imply to use fully implicit schemes
for the two problems. Moreover, it has been shown that time oscillations of the pore
pressure may occur for other time schemes. Associated to the Terzaghi's postulate,
Numerical poromechanics 29
oscillations could appear also on the stress tensor, what can quickly degrade the
numerical convergence rate for elastoplastic or elastoviscoplastic constitutive laws.
When using isoparametric finite elements, the shape function for geometry and
for pore pressure are identical. Let us consider for example a second order finite
element. As the displacement field is of second order, the strain rate field is linear.
For an elastic material, the effective stress tensor rate is then also linear. However
the pore pressure field is quadratic. Then the Terzaghi's postulate mixes linear and
quadratic field, which is not highly consistent. Some authors have then proposed to
mix in one element quadratic shape functions for the geometry and linear shape
functions for pore pressure. But then problems arrive with the choice of spatial
integration points (1 or 4 Gauss points?).
Numerical locking problems may also appear for isoparametric finite element
when the two phases material (fluid + rock) is quite incompressible, i.e. for very
short time steps with respect with the fluid diffusion time scale. Specific elements
have to be developed for such problems.
6.2.2. Two fluids flow in rigid porous media coupling
Flow in partly saturated rigid media is here considered. Flow in oil or gas reservoirs
two or three fluids among oil, gas, condensates and water. Partial solving or mixture
between different fluids is sometimes possible. Then two or more dof per node are
to be considered. The permeability and storage equation of each phase are
depending on the suction or saturation level, and so the problem may be highly nonlinear. However, coupling is not difficult to numerically be developed, as the
formulation is similar for each phase.
6.2.3. Diffusion and transport coupling
Heat and one fluid flow in a rigid porous media are concerned here. The fluid
specific weight and viscosity is depending on the temperature, and the heat transport
by advection – diffusion process is depending on the fluid flow. Then a diffusion
process and an advection – diffusion process have to be solved simultaneously.
Number of dof per node: 2 (fluid pore pressure and salt concentration of
temperature).
6.2.4. Thermo-hydro-mechanical coupling
The phenomena considered here are much more complex as they associate
multiphase fluid flow, hydromechanical coupling and temperature effects. All the
features described in the preceding sections are to be considered here, associated to
some new points.
30
Title of the journal. Volume X – no X/2002
Heat diffusion has to be modelled. Temperature variation affects fluid flow, by a
modification of the fluid specific weight or viscosity. Moreover, if the two fluids
concerned are a liquid and a gas (e.g. water and air), then equilibrium between the
phases has to be modelled: dry air – vapour equilibrium.
Heat transfer is governed not only by conduction but also by advection by the
liquid and gas movements. Similarly transfers of vapour and gas species in the gas
phase are governed by diffusion and gradient of species density, but also by
advection with the global gas movements. If the concerned geomaterials has a very
low permeability (like clay for capping barriers), then the diffusion effects will
predominate and advection doesn’t necessitate specific formulation (Collin et al,
1999).
Finally the total number of dof per node is 5 for a 2D problem: 2 displacements,
2 fluid pore pressures and the temperature.
6.3. Finite element modelling: staggered approach
Monolithical approach of coupled phenomena implies identical space and time
meshes for each phenomenon. This is not always possible, for various reasons. The
coupled problems may have different numerical convergence properties, generally
associated to different physical scales or non-linearity. For example, a coupled
hydromechanical problem may need large time steps for the fluid diffusion problem,
in order to allow, in each step, fluid diffusion along distance of the order of
magnitude of the finite elements. In the same time, strong non-linearity may occur in
solid mechanics behaviour (strong elastoplasticity changes, interface behaviour,
strain localisation…) and then the numerical convergence needs short time – loading
steps, which should be adapted automatically to the rate of convergence. Then it is
quite impossible to obtain numerical convergence for identical time and space
meshes.
Research teams of different physical and numerical culture have progressively
developed different problems modelling. As an example, fluid flow has been largely
developed using the finite difference method for oil reservoir engineering taking
multiphase fluid flow (oil, gas, condensate, water …) into account. Coupling such
fluid flow with geomechanics in a monolithical approach would imply to implement
all the physical features already developed respectively in finite elements and finite
differences codes. The global human effort would be very important.
Coupled problems are generally presenting a higher non-linearity level then
uncoupled ones. Then inaccuracy in parameters or in the problem idealisation may
cause degradations of the convergence performance.
Numerical poromechanics 31
How can we solve such problems and obtain a convincing solution? First of all, a
good strategy would be to start with the uncoupled modelling of the leading process,
and to try to obtain a first not too bad solution. Then one can add a first level of
coupling and complexity, followed by a second one… until the full solution is
obtained.
However such trick is not always sufficient. Staggered approaches may then give
an interesting solution. In a staggered scheme, the different problems to be coupled
are solved separately, with (depending on the cases) different space or time mesh, or
different numerical codes. However, the coupling is ensured thank to transfer of
information between the separated models at regular meeting points. This concept is
summarised on the Figure 15. It allows theoretically coupling any models together.
Mechanical
problem
N1 steps
Mechanical
problem
N2 steps
Fluid flow
problem
M1 steps
Fluid flow
problem
M2 steps
Time
Figure 15. Scheme of a staggered coupling
When using different spatial meshes, or when coupling finite elements and finite
differences codes, the transfer of information needs often an interpolation procedure,
as the information to be exchanged are not defined at the points in the different
meshes.
The accuracy of the coupling scheme will mainly depend on the information
exchanges frequency (which is limited by the lower time step that can be used) and
by the type of information exchanged. The stability and accuracy of the process has
been checked by different authors (Turska et al 1993, Zienckiewicz et al, 1988). It
has been shown that a good choice of the information exchange may improve highly
the procedure efficiency.
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Title of the journal. Volume X – no X/2002
7. Acknowledgements
Authors are grateful to FNRS, La Communauté Française de Belgique and the
European Commission for the financial help to their research projects.
8. Bibliography/References
Alonso E.E., Gens A. and Josa A., A constitutive model for partially saturated soils,
Géotechnique, vol. 40 no. 3, 1990, p. 405-430.
Bandis, S. Lumsden, A.C. & Barton, N. 1983. Foundamentals of rock joints deformation. Int.
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