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Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
Document: R7.01.11
Models of behavior THHM
Summary:
This note introduces a family of laws of behavior
THM
for the saturated and unsaturated mediums. One y
described the relations allowing to calculate the hydraulic and thermal quantities, by taking account of
strong couplings between these phenomena and also with the mechanical deformations. Relations presented here
can be coupled with any law of mechanical behavior, subject making the assumption
said effective stresses of Bishop and that the mechanical law of behavior defines constants
rubber bands (useful for the coupled terms). The purely mechanical part of the laws is not presented.
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Count
matters
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Appendix 4
Derived seconds from air and steam pressures dissolved according to
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1 Introduction
We introduce here a family of laws of behavior
THM
for the saturated and unsaturated mediums.
We describe the relations allowing to calculate the hydraulic and thermal quantities, in
taking account of strong couplings between these phenomena and also with the mechanical deformations.
The relations presented here can be coupled with any law of behavior
mechanics, subject making the assumption known as of the effective stresses of Bishop and that the law of
mechanical behavior defines constant rubber bands (useful for the coupled terms). For
this reason, the purely mechanical part of the laws is not presented here.
Modelings selected are based on the presentation of the porous environments elaborate in particular by
O. Coussy [bib1]. The relations of behavior are obtained starting from considerations
thermodynamic and with arguments of homogenization that we do not present here, and which
are entirely described in the document of P. Charles [bib2]. In the same way the general writing of
equilibrium equations and conservation is not detailed, and one returns the reader to the documents
[R5.03.01] [bib3] and [R7.01.10] [bib4], which contain definitions useful for the comprehension of
present document.
The mechanics of the porous environments gathers a very exhaustive collection of physical phenomena
concerning with the solids and the fluids. It makes the assumption of a coupling between the mechanical evolutions
solids and fluids, seen like continuous mediums, with the hydraulic evolutions, which
regulate the problems of diffusion of fluids within walls or volumes, and the evolutions
thermics.
Each component of the porous environment thus has a mechanical, hydraulic behavior and
thermics. The theory tries to gather all these physical phenomena. Phenomena
chemical (transformations of the components, reactions producing of components etc…), in the same way
that the radiological phenomena are not taken into account at this stage of the development of
Code_Aster. The mechanical, hydraulic and thermal phenomena are taken into account or not
according to the behavior called upon by the user in the control
STAT_NON_LINE
, according to
following nomenclature:
Modeling
Phenomena taken into account
KIT_HM
Mechanics, hydraulics with an unknown pressure
KIT_HHM
Mechanics, hydraulics with two unknown pressures
KIT_THH
Thermics, hydraulics with two unknown pressures
KIT_THM
Thermics, mechanics, hydraulics with an unknown pressure
KIT_THHM
Thermics, mechanics, hydraulics with two unknown pressures
The document present describes the laws for the most general case said
THHM
. Simpler cases
are obtained starting from the general case by simply cancelling the quantity absent.
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2
Presentation of the problem: Assumptions, Notations
In this chapter, one mainly endeavors to present the porous environment and its
characteristics.
2.1
Description of the porous environment
The porous environment considered is a volume made up of a more or less homogeneous solid matrix,
more or less coherent (very coherent in the case of concrete, little in the case of sand). Between
solid elements, pores are found. One distinguishes the closed pores which do not exchange anything with theirs
neighbors and the connected pores in which the exchanges are numerous. When one speaks about
porosity, it is well of these connected pores about which one speaks.
Inside these pores are a certain number of fluids (one excludes solidification from these
fluids), present possibly under several phases (fluid or gas exclusively), and
presenting an interface with the other components. To simplify the problem and to take in
count the relative importance of the physical phenomena, the only interface considered is that enters
fluid and the gas, the interfaces solid fluid/being neglected.
2.2 Notations
We suppose that the pores of the solid are occupied by with more the two components, each one
coexisting in two phases to the maximum, one liquidates and the other gas one. Sizes X associated
with the phase J (j=1,2) of fluid I will be noted:
ij
X
. When there are two components in addition to the solid, it
are a fluid (typically water) and a gas (typically dry air), knowing that the fluid can be
present in gas form (vapor) in the gas mixture and that the air can be present under
form dissolved in water. When there is one component in addition to the solid, that can be one
fluid or a gas.
The porous environment at the current moment is noted
, its border
. It is noted
0
0
,
at the initial moment.
The medium is defined by:
·
parameters (vector position
X
, time T),
·
variables (displacements, pressures, temperature),
·
intrinsic sizes (forced and mass deformations, contributions, heat,
hydraulic enthali, flows, thermics…).
The general assumptions carried out are as follows:
·
assumption of small displacements,
·
reversible thermodynamic evolutions (not necessarily for mechanics),
·
isotropic behavior,
·
the gases are perfect gases,
·
mix ideal perfect gases (total pressure = nap of the partial pressures),
·
balance thermodynamic between the phases of the same component.
The various notations are clarified hereafter.
2.2.1 Descriptive variables of the medium
These are the variables whose knowledge according to time and of the place make it possible to know
completely the state of the medium. These variables break up into two categories:
·
geometrical variables,
·
variables of thermodynamic state.
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2.2.1.1 Variables
geometrical
In all that follows, one adopts a Lagrangian representation compared to the skeleton (within the meaning of
[bib1]) and co-ordinates
()
T
S
X
X
=
are those of a material point attached to the skeleton. All them
space operators of derivation are defined compared to these co-ordinates.
Displacements of the skeleton are noted
()
=
Z
y
X
U
U
U
T
,
X
U
.
2.2.1.2 Variables of thermodynamic state
In a general way, the following indices are used:
W
for liquid water
AD
for the dissolved air
have
for the dry air
vp
for the water vapor
The thermodynamic variables are:
·
pressures of the components:
()
T
p
W
,
X
,
()
T
p
AD
,
X
,
()
T
p
vp
,
X
,
()
T
p
have
,
X
,
·
the temperature of the medium
()
T
T
X,
.
These various variables are not completely independent. Indeed, if only one is considered
component, thermodynamic balance between its phases imposes a relation between the pressure of
vapor and pressure of the fluid of this component. Finally, there is only one pressure
independent by component, just as there is only one conservation equation of the mass.
The number of independent pressures is thus equal to the number of independent components.
choice of these pressures is free (combinations of the pressures of the components) provided that them
pressures chosen, associated the temperature, form a system of independent variables.
For the case known as saturated (only one component air or water) we chose the pressure of this single
component.
For the case says unsaturated (presence of air and water), we chose like variables
independent:
·
total pressure of gas
()
have
vp
gz
p
p
T
p
+
=
,
X
,
·
capillary pressure
()
AD
W
gz
lq
gz
C
p
p
p
p
p
T
p
-
-
=
-
=
,
X
.
These pressures have a very strong physical interpretation, the total gas pressure for reasons
obvious, and pressure capillary, also called suction, because capillary phenomena
are very important in modeling presented here. It would have been possible also to choose
steam pressure or the percentage of relative moisture (relationship between the steam pressure and the pressure
of vapor saturated at the same temperature) physically accessible. Modeling becomes then
more complex and in any event, capillary pressure, gas pressure and percentage of relative moisture
(relationship between the steam pressure and the saturating steam pressure) are connected by the law of Kelvin.
For the particular case of the behavior known as `
LIQU_GAZ_ATM
'one makes the assumption known as of Richards:
pores are not saturated by the fluid, but the pressure of gas is supposed to be constant and only
variable of pressure is the pressure of fluid.
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2.2.1.3 Descriptive fields of the medium
The main unknown factors, which are also the nodal unknown factors (noted
()
T
,
X
U
in this document)
are:
·
2 or 3 (according to the dimension of space) displacements
)
,
(
,
)
,
(
,
)
,
(
T
U
T
U
T
U
Z
y
X
X
X
X
for
modelings
KIT_HM
,
KIT_HHM
,
KIT_THM
,
KIT_THHM
,
·
the temperature
()
T
T,
X
for modelings
KIT_THH
,
KIT_THM
,
KIT_THHM
,
·
two pressures
)
,
(
),
,
(
2
1
T
p
T
p
X
X
(which is
()
T
p
C
,
X
,
()
T
p
gz
,
X
in the case studied) for
modelings
KIT_HHM
,
KIT_THH
,
KIT_THHM
,
·
a pressure
)
,
(
1
T
p X
(which is
()
T
p
W
,
X
or
()
T
p
gz
,
X
according to whether the medium is saturated by one
fluid or a gas) for modelings
KIT_HM
,
KIT_THM
.
2.2.2 Derived
particulate
This paragraph partly shows the paragraph “derived particulate, densities voluminal and
mass “of the document [R7.01.10]. Description that we make of the medium is Lagrangian by
report/ratio with the skeleton.
That is to say
has
an unspecified field on
, that is to say
()
T
S
X
the punctual coordinate attached to the skeleton that
we follow in his movement and is
()
T
fl
X
the punctual coordinate attached to the fluid. One notes
dt
has
D
has
S
=
&
the temporal derivative in the movement of the skeleton:
(
)
(
)
()
(
)
T
T
T
has
T
T
T
T
has
dt
has
D
has
S
S
T
S
-
+
+
=
=
,
,
lim
0
X
X
&
has
&
is called particulate and often noted derivative
dt
da
. We prefer to use a notation which
recall that the configuration used to identify a particle is that of the skeleton by report/ratio
to which a particle of fluid has a relative speed. For a particle of fluid the identification
()
T
S
X
is
unspecified, i.e. that the particle of fluid which occupies the position
()
T
S
X
at the moment
T
is not
even as that which occupies the position
()
'
T
S
X
at another moment
'
T
.
2.2.3 Sizes
The equilibrium equations are:
·
conservation of the momentum for mechanics,
·
conservation of the masses of fluid for hydraulics,
·
conservation of energy for thermics.
The writing of these equations is given in the document [R7.01.10] [bib4], which defines also what
we call in a general way a law of behavior
THM
and gives the definitions of
generalized stresses and deformations. This document uses these definitions. Equations
of balance utilize directly the generalized stresses.
The generalized stresses are connected to the deformations generalized by the laws of behavior.
The generalized deformations are calculated directly starting from the variables of state and theirs
temporal space gradients.
The laws of behaviors can use additional quantities, often arranged in the variables
interns. We gather here under the term of size at the same time the stresses, the deformations and
additional sizes.
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2.2.3.1 Sizes characteristic of the heterogeneous medium
·
Porosity eulérienne:
.
If one notes
the part of volume
occupied by the vacuums in the current configuration, one
a:
=
The definition of porosity is thus that of porosity eulérienne.
·
Saturation in fluid:
lq
S
If one notes
lq
the total volume occupied by the fluid, in the current configuration, one has by
definition:
=
lq
lq
S
This saturation is thus finally a proportion varying between 0 and 1.
·
Densities eulériennes of water
W
, of the dissolved air
AD
, of the dry air
have
, of
vapor
vp
, of gas
gz
.
If one notes
W
(resp
AD
,
have
,
vp
) water masses (resp of dissolved air, dry air and of
vapor) contents in a volume
skeleton in the current configuration, one has by
definition:
(
)
(
)
-
=
-
=
=
=
D
S
D
S
D
S
D
S
lq
vp
lq
have
lq
AD
lq
W
1
1
vp
have
AD
W
The density of the gas mixture is simply the sum of the densities
dry air and vapor:
vp
have
gz
+
=
In the same way for the liquid mixture:
AD
W
lq
+
=
One notes
0
0
0
0
,
,
,
have
vp
AD
W
initial values of the densities.
·
Lagrangian homogenized density:
R
.
At the moment running the mass of volume
,
M
, is given by:
0
0
=
rd
M
.
2.2.3.2 Sizes
mechanics
·
The tensor of the deformations
()
(
)
U
U
X
U
+
=
T
T
2
1
,
)
(
.
One will note
()
V
tr
=
.
·
The tensor of the stresses which are exerted on the porous environment:
.
This tensor breaks up into a tensor of the effective stresses plus a tensor of
stresses of pressure
1
p
+
=
'
.
p
and
'
are components of the stresses
generalized. This cutting is finally rather arbitrary, but corresponds all the same to
an assumption rather commonly allowed, at least for the mediums saturated with fluid.
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2.2.3.3 Sizes
hydraulics
·
Mass contributions in components
have
vp
AD
W
m
m
m
m
,
,
,
(unit: kilogram per meter
cubic).
They represent the mass of fluid brought between the initial and current moments. They form part
generalized stresses.
·
Hydraulic flows
have
vp
AD
W
M
M
M
M
,
,
,
(unit: kilogram/second/square meter).
One could not give very well no more precise definition of the contributions of mass and of
flow, considering that their definition is summarized to check the equilibrium equations
hydraulics:
(
)
(
)
=
+
+
+
=
+
+
+
0
0
AD
have
AD
have
vp
W
vp
W
Div
m
m
Div
m
m
M
M
M
M
&
& &
&
éq
2.2.3.3-1
We nevertheless will specify the physical direction as of the these sizes, knowing that what
we write now is already a law of behavior.
Speeds of the components are measured in a fixed reference frame in space and time.
One notes
W
v
the speed of water,
AD
v
that of the dissolved air,
vp
v
that of the vapor,
have
v
that
dry air, and
dt
D
S
U
v
=
that of the skeleton.
The mass contributions are defined by:
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
lq
vp
lq
V
vp
vp
lq
have
lq
V
have
have
lq
AD
lq
V
AD
AD
lq
W
lq
V
W
W
S
S
m
S
S
m
S
S
m
S
S
m
-
-
-
+
=
-
-
-
+
=
-
+
=
-
+
=
éq
2.2.3.3-2
Mass flows are defined by:
(
)
(
)
(
) (
)
(
)
(
)
S
vp
L
vp
vp
S
have
L
have
have
S
AD
L
AD
AD
S
W
L
W
W
S
S
S
S
v
v
M
v
v
M
v
v
M
v
v
M
-
-
=
-
-
=
-
=
-
=
1
1
éq
2.2.3.3-3
The mass contributions make it possible to define the total density seen compared to
configuration of reference:
have
vp
AD
W
m
m
m
m
R
R
+
+
+
+
=
0
, where
0
R
indicate the density
homogenized at the initial moment.
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Other intermediate hydraulic sizes are introduced:
·
concentration of the vapor in gas
gz
vp
vp
p
p
C
=
,
·
gas flow:
(
)
vp
vp
vp
have
have
vp
gz
gz
C
C
+
-
=
M
M
M
1
. This equation specifies that the speed of
gas is obtained by making an average (balanced sum) speeds of different
gas according to their concentration,
·
the steam pressure
vp
p
.
2.2.3.4 Sizes
thermics
·
not convectée heat
Q
(see further) (unit: Joule),
·
mass enthali of the components
m
ij
H
(
m
have
m
vp
m
AD
m
W
H
H
H
H
,
,
,
) (unit
:
Joule/Kelvin/kilogram),
·
heat transfer rate:
Q
(unit: J/S/square meter).
All these sizes belong to the stresses generalized within the meaning of the document [R7.01.10]
[bib4].
2.2.4 Data
external
·
the mass force
m
F
(in practice gravity),
·
heat sources
,
·
boundary conditions relating either to variables imposed, or on imposed flows.
3 Equations
constitutive
3.1
Conservation equations
It is here only about one recall, the way of establishing them is presented in [R7.01.10] [bib4].
3.1.1 Balance
mechanics
While noting
the tensor of the total mechanical stresses and
R
density homogenized of
medium, mechanical balance is written:
()
0
=
+
m
R
F
Div
éq
3.1.1-1
We remind the meeting that
R
is connected to the variations of fluid mass by the relation:
have
vp
AD
W
m
m
m
m
R
R
+
+
+
+
=
0
éq
3.1.1-2
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3.1.2 Conservation of the fluid masses
For the fluid the derivative
dt
has
D
has
S
=
&
in fact a derivative eulérienne is and the equations which we write
for the fluid comprise terms of transport, even if they can be hidden by the choice of
unknown factors. The conservation equations of the fluid masses are written then:
(
)
(
)
=
+
+
+
=
+
+
+
0
0
AD
have
AD
have
vp
W
vp
W
Div
m
m
Div
m
m
M
M
M
M
&
& &
&
éq
3.1.2-1
3.1.3 Conservation of energy: thermal equation
(
)
()
(
)
+
+
+
+
=
+
+
+
+
+
+
+
+
+
m
have
vp
AD
W
have
m
have
vp
m
vp
AD
m
AD
W
m
W
have
m
have
vp
m
vp
AD
m
AD
W
m
W
Div
H
H
H
H
Div
Q
m
H
m
H
m
H
m
H
F
M
M
M
M
Q
M
M
M
M
'&
&
&
&
&
éq 3.1.3-1
3.2
Equations of behavior
3.2.1 Evolution of porosity
(
)
-
+
-
-
=
S
C
lq
gz
V
K
dp
S
dp
dT
D
B
D
0
3
éq
3.2.1-1
In this equation, one sees appearing the coefficients
B
and
S
K
.
B
is the coefficient of Biot and
S
K
is
the module of compressibility of the solid matter constituents. If
0
K
indicate the module of compressibility
“drained” of the porous environment, one with the relation:
S
K
K
B
0
1
-
=
éq
3.2.1-2
3.2.2 Evolution of the contributions of fluid mass
By using the definition of the contributions of fluid mass and while putting forward arguments purely
geometrical, one finds:
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
lq
vp
lq
V
vp
vp
lq
have
lq
V
have
have
lq
AD
lq
V
AD
AD
lq
W
lq
V
W
W
S
S
m
S
S
m
S
S
m
S
S
m
-
-
-
+
=
-
-
-
+
=
-
+
=
-
+
=
éq
3.2.2-1
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HT-66/05/002/A
As example, we show the first relation in the saturated case
1
=
lq
S
(with
W
lq
=
).
That is to say an elementary field of porous environment of volume
. One notes
S
the volume occupied by
solid matter constituents and
L
the volume occupied by the fluid and gas. One notes
0
,
0
S
,
0
L
the same ones
volumes in an initial state. We remind the meeting that
V
note the variation of volume of the porous environment and us
let us note
S
V
voluminal variation of the solid matter constituents.
One has by definition:
=
L
(
)
(
)
Vs
L
S
S
+
=
-
=
-
=
1
1
0
But
(
)
(
) (
)
-
+
=
-
1
1
1
0
V
One deduces some:
(
) (
)
(
)
Vs
V
S
+
=
-
+
1
1
1
0
0
It is enough to write then
(
)
0
0
0
1
-
=
S
to obtain:
(
) (
)
(
)
(
)
Vs
V
+
-
=
-
+
1
1
1
1
0
0
0
From where one deduces:
(
)
(
)
(
)
0
0
1
1
-
-
-
=
-
V
Vs
One uses the eulérienne definition homogenized density
R
(not to be confused with
the Lagrangian definition of the equation [éq 3.1.1-2]):
(
)
+
-
=
lq
S
R
1
and the definition of the mass contribution in fluid:
(
)
0
0
+
=
lq
m
R
R
One obtains:
(
)
(
)
0
0
0
0
0
0
0
1
1
+
+
-
=
+
-
lq
S
lq
S
m
lq
that is to say still:
(
)
0
0
0
0
0
0
0
1
+
+
=
+
+
lq
S
V
lq
S
S
m
lq
S
Using the conservation of the mass of the solid matter constituents:
0
0
S
S
S
S
=
one obtains finally:
(
)
lq
V
lq
m
lq
+
=
+
0
0
1
Code_Aster
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HT-66/05/002/A
3.2.3 Laws of behavior of the fluids
3.2.3.1 Fluid
dT
K
dp
D
W
W
W
W
W
3
-
=
éq
3.2.3.1-1
One sees appearing the module of compressibility of water
W
K
and its module of dilation
W
.
3.2.3.2 Gas
For the equations of reaction of gases, one takes the law of perfect gases:
T
M
R
p
ol
vp
vp
vp
=
éq
3.2.3.2-1
T
M
R
p
ol
have
have
have
=
éq
3.2.3.2-2
One sees appearing the molar mass of the vapor,
ol
vp
M
, and that of the dry air,
ol
have
M
.
3.2.4 Evolution of the enthali
3.2.4.1 Enthalpy
fluid
(
)
W
W
W
p
W
m
W
dp
T
dT
C
dh
3
1
-
+
=
éq
3.2.4.1-1
One sees appearing the specific heat with constant pressure of water:
p
W
C
.
By replacing in this expression the pressure of the fluid by its value according to the pressure
thin cable and of the pressure of gas, one a:
(
)
dT
C
dp
dp
dp
T
dh
p
W
W
AD
C
gz
W
m
W
+
-
-
-
=
3
1
éq
3.2.4.1-2
While noting
p
AD
C
specific heat with constant pressure of the dissolved air, one a:
dT
C
dh
p
AD
m
AD
=
éq
3.2.4.1-3
3.2.4.2 Enthalpy of gases
dT
C
dh
p
vp
m
vp
=
éq
3.2.4.2-1
dT
C
dh
p
have
m
have
=
éq
3.2.4.2-2
One sees appearing the specific heat with constant pressure of the dry air
p
have
C
and that of the vapor
p
vp
C
.
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HT-66/05/002/A
3.2.4.3 Contribution of heat except fluids
It is the quantity
Q
who represents the heat received by the system except contribution enthalpic of
fluids.
(
)
dT
C
Tdp
Tdp
Td
K
Q
gz
m
lq
m
gz
C
m
lq
V
0
0
0
3
3
3
3
+
+
-
+
=
éq 3.2.4.3-1
One sees appearing several expansion factors:
m
gz
m
lq
,
,
0
. The coefficient
0
is a data:
it corresponds at the same time to the expansion factor of the porous environment and to that of the solid matter constituents (which
find being inevitably equal in the theory which we present here).
m
gz
m
lq
,
are given by the relations:
(
)
(
)
(
)
T
S
B
S
lq
lq
m
gz
3
1
1
0
-
+
-
-
=
éq
3.2.4.3-2
(
)
lq
lq
lq
m
lq
S
B
S
+
-
=
0
éq
3.2.4.3-3
One also sees appearing in [éq 3.2.4.3-1] the specific heat to constant deformation of
porous environment
0
C
, which depends on the specific heat to constant stress of the porous environment
0
C
by
the relation:
2
0
0
0
0
9
-
=
TK
C
C
éq
3.2.4.3-4
0
C
is given by a law of mixture:
(
)
(
)
(
)
p
have
have
p
vp
vp
lq
p
AD
AD
p
W
W
lq
S
S
C
C
S
C
C
S
C
C
+
-
+
+
+
-
=
1
)
(
1
0
éq
3.2.4.3-5
where
S
C
represent the specific heat to constant stress of the solid matter constituents and
S
mass
voluminal of the solid matter constituents. For the calculation of
S
, one neglects the deformation of the solid matter constituents, one
thus confuses
S
with its initial value
0
S
, which is calculated in fact starting from the specific mass
initial of the porous environment
0
R
by the following formula of the mixtures:
(
)
(
) (
)
0
0
0
0
0
0
0
0
0
0
0
1
)
(
1
have
vp
lq
lq
AD
W
S
S
R
S
+
-
-
+
-
=
-
éq
3.2.4.3-6
3.2.5 Laws of dissemination
3.2.5.1 Dissemination of heat
One takes the conventional law of Furrier:
T
T
-
=
.
Q
éq
3.2.5.1-1
where one sees appearing the thermal coefficient of conductivity
T
.
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T
is a function of porosity, saturation and temperature and is given in the form of
product of three functions plus a constant:
T
cte
T
T
lq
T
S
T
T
T
S
+
=
)
(
).
(
).
(
éq
3.2.5.1-2
3.2.5.2 Dissemination of the fluids
They are the laws of Darcy, to which one adds the law of Fick in the presence of vapor.
The laws of Darcy are written for gas and the fluid:
(
)
m
lq
lq
H
lq
lq
p
lq
F
M
+
-
=
éq
3.2.5.2-1
(
)
m
gz
gz
H
gz
gz
p
gz
F
M
+
-
=
éq
3.2.5.2-2
where we see appearing hydraulic conductivities
H
lq
and
H
gz
for the fluid and gas
respectively.
One makes the approximation that
(
)
m
lq
lq
H
W
W
p
W
F
M
+
-
=
The dissemination in the gas mixture is given by the law of Fick thanks to the relation:
-
-
=
-
gz
vp
vp
vp
vp
have
have
vp
vp
p
p
C
C
D
)
1
(
M
M
éq
3.2.5.2-3
where
vp
D
is the coefficient of dissemination of Fick of the gas mixture (L
2
.T
- 1
), one notes thereafter
vp
F
such
that:
)
1
(
vp
vp
vp
vp
C
C
D
F
-
=
éq
3.2.5.2-4
and with
gz
vp
vp
p
p
C
=
éq
3.2.5.2-5
One thus has:
vp
vp
have
have
vp
vp
C
F
-
=
-
M
M
éq
3.2.5.2-6
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Moreover, one a:
(
)
vp
vp
vp
have
have
vp
gz
gz
C
C
M
M
M
+
-
=
1
éq
3.2.5.2-7
and:
have
vp
gz
+
=
éq
3.2.5.2-8
For the dissemination of the liquid mixture, the usual writing is as follows:
AD
AD
W
AD
D
-
=
-
M
M
éq
3.2.5.2-9
where
AD
D
is the coefficient of dissemination of Fick of the liquid mixture. In order to keep a writing
homogeneous with that of the gas mixture one notes thereafter
AD
F
such as:
AD
AD
D
F
=
éq
3.2.5.2-10
And concentration
AD
C
corresponds here to the density of the dissolved air:
AD
AD
C
=
éq
3.2.5.2-11
AD
AD
W
AD
C
F
-
=
-
M
M
éq
3.2.5.2-12
Concerning the fluid, one admitted that liquid Darcy applies at the speed of liquid water. There is not
thus not to define mean velocity of the fluid.
(
)
m
lq
lq
H
lq
W
W
p
F
M
+
-
=
éq
3.2.5.2-13
and:
AD
W
lq
+
=
éq
3.2.5.2-14
By combining these relations, one finds then:
(
)
vp
vp
vp
m
gz
gz
H
have
have
C
F
C
p
gz
+
+
-
=
F
M
éq
3.2.5.2-15
(
)
vp
vp
vp
m
gz
gz
H
vp
vp
C
F
C
p
gz
-
-
+
-
=
)
1
(
F
M
éq
3.2.5.2-16
(
)
m
lq
lq
H
lq
W
W
p
F
M
+
-
=
éq
3.2.5.2-17
(
)
AD
AD
m
lq
lq
H
lq
AD
AD
C
F
p
-
+
-
=
F
M
éq
3.2.5.2-18
Code_Aster
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Hydraulic conductivities
H
lq
and
H
gz
are not directly data and their value is
known starting from the formulas:
()
()
T
S
K
K
W
lq
rel
lq
H
lq
µ
).
(
int
=
éq
3.2.5.2-19
(
)
()
T
p
S
K
K
gz
gz
lq
rel
gz
H
gz
µ
,
).
(
int
=
éq
3.2.5.2-20
int
K
is the intrinsic permeability, characteristic of the porous environment and user datum, function
unspecified of porosity;
W
µ
is the dynamic viscosity of water, characteristic of water and user datum, function
unspecified of the temperature;
gz
µ
is the dynamic viscosity of gas, characteristic of gas and user datum, function
unspecified of the temperature;
rel
lq
K
is the permeability relating to the fluid, characteristic of the porous environment and user datum,
unspecified function of saturation in fluid;
rel
gz
K
is the permeability relating to gas, characteristic of the porous environment and user datum, function
unspecified of saturation in fluid and the gas pressure.
Note:
Here definite hydraulic conductivities are not inevitably very familiar for
mechanics of grounds, which usually use for the saturated mediums the permeability
K
,
which is homogeneous at a speed.
The relation enters
K
and
H
lq
is as follows:
G
K
W
H
lq
=
where
G
is the acceleration of gravity.
The coefficient of dissemination of Fick of the gas mixture
vp
F
is a characteristic of the porous environment,
unspecified user datum function of the steam pressure, the gas pressure, of
saturation and of the temperature which one will write like a product of function of each one of these
variables:
)
(
).
(
).
(
).
(
)
,
,
,
(
S
F
T
F
P
F
P
F
S
T
P
P
F
S
vp
T
vp
gz
gz
vp
vp
vp
vp
gz
vp
vp
=
one will neglect the derivative by
report/ratio with the steam pressure and saturation. Same manner for the coefficient of dissemination
of Fick of the liquid medium:
)
(
).
(
).
(
).
(
)
,
,
,
(
S
F
T
F
P
F
P
F
S
T
P
P
F
S
AD
T
AD
lq
lq
AD
AD
AD
AD
lq
AD
AD
=
, one does not take
in account that the derivative according to the temperature.
3.2.6 Balance
water-steam
This relation is essential and it results in to reduce the number of unknown factors of
pressure.
One notes
m
W
H
mass enthalpy of water,
m
W
S
its entropy and
m
W
m
W
m
W
Ts
H
G
-
=
its free enthalpy.
One notes
m
vp
H
mass enthalpy of the vapor,
m
vp
S
its entropy and
m
vp
m
vp
m
vp
Ts
H
G
-
=
its enthalpy
free.
Balance water vapor is written:
m
W
m
vp
G
G
=
éq
3.2.6-1
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HT-66/05/002/A
Who gives:
(
)
m
W
m
vp
m
W
m
vp
S
S
T
H
H
-
=
-
éq
3.2.6-2
In addition, the definition of the free enthalpy teaches us that:
sdT
dp
dg
-
=
, which, applied to
vapor and with water, compound with the relation
m
m
W
vp
dg
dg
=
and while using [éq 3.2.6-2] gives:
(
)
T
dT
H
H
dp
dp
m
W
m
vp
W
W
vp
vp
-
+
=
éq
3.2.6-3
This relation can be expressed according to the capillary pressure and of the gas pressure:
(
)
(
)
T
dT
H
H
dp
dp
dp
dp
m
W
m
vp
vp
AD
C
gz
W
vp
vp
-
+
-
-
=
éq
3.2.6-4
3.2.7 Balance air dissolved dryness-air
The dissolved air is defined via the constant of Henry
H
K
, which connects the molar concentration of dissolved air
ol
AD
C
(moles/m3) with the pressure of dry air:
H
have
ol
AD
K
p
C
=
éq
3.2.7-1
with
ol
AD
AD
ol
AD
M
C
=
éq
3.2.7-2
Molar mass of the dissolved air,
ol
AD
M
is logically the same one as that of the dry air
ol
have
M
. For
the dissolved air, one takes the law of perfect gas:
T
M
R
p
ol
have
AD
AD
=
éq
3.2.7-3
The pressure of dissolved air is thus connected to that of dry air by:
RT
K
p
p
H
have
AD
=
éq
3.2.7-4
3.2.8 The mechanical behavior
One will write it in differential form:
1
p
D
D
D
+
=
'
éq
3.2.8-1
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By using a formulation of Bishop [bib10] extended to the unsaturated mediums one writes:
(
)
C
lq
gz
p
dp
S
dp
B
D
-
-
=
éq
3.2.8-2
In the relation [éq 3.2.8-1] evolution of the tensor of effective stress
is supposed
to depend that displacement of the skeleton and variables internal
. Usual terms related to
thermal deformation are integrated into the calculation of the effective stress:
(
)
D
dT
D
F
D
,
0
'
I
-
=
éq
3.2.8-3
The reason of this choice is to be able to use any law of conventional thermomechanics for
calculation of the effective stresses, laws which, in more the share of the writings are in conformity with [éq 3.2.8-3].
3.2.9 The isotherm of sorption
To close the system, there remains still a relation to be written, connecting saturation and the pressures. Us
chose to consider that saturation in fluid was an unspecified function of the pressure
thin cable, that this function was a characteristic of the porous environment and provided in data by
the user.
Since the user can provide a function very well
()
C
lq
p
S
refine per pieces, and being
given that the derivative of this function,
C
lq
p
S
, plays an essential physical role, we chose
to ask the user to also provide this curve, remainder with its load to ensure itself of
coherence of the data thus specified.
It is noticed that in the approach present, one speaks about a bi-univocal relation between saturation and
capillary pressure. It is known that for the majority of the porous environments, it is not the same relation which
must be used for the paths of drying and the paths of hydration. It is one of the limits
approach present.
3.2.10 Summary of the characteristics of material and the user data
·
The Young modulus
0
E
and the drained Poisson's ratio
0
allow to calculate it
modulate compressibility drained of the porous environment by
(
)
0
0
0
2
1
3
-
=
E
K
,
·
the coefficient of Biot
S
K
K
B
0
1
-
=
allows to calculate the module of compressibility of the grains
solids
S
K
,
·
the module of compressibility of water
W
K
,
·
the expansion factor of water
W
,
·
the constant of perfect gases
R
,
·
molar mass of the vapor
ol
vp
M
,
·
molar mass of the dry air
ol
have
M
, (=
ol
AD
M
)
·
specific heat with constant pressure of water
p
W
C
,
·
specific heat with constant pressure of the dissolved air
p
AD
C
·
specific heat with constant pressure of the dry air
p
have
C
,
·
specific heat with constant pressure of the vapor
p
vp
C
,
Code_Aster
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Titrate:
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HT-66/05/002/A
·
the expansion factor of the porous environment
0
who is also that of the solid matter constituents,
·
specific heat with constant stress of the solid matter constituents
S
C
,
·
the thermal coefficient of conductivity of the solid matter constituents only,
T
S
, unspecified function of
the temperature,
·
the thermal coefficient of conductivity of the fluid,
lq
T
, unspecified function of
temperature,
·
the thermal coefficient of conductivity of the dry air,
have
T
, unspecified function of
temperature,
·
the coefficient of dissemination of fick for the gas mixture,
vp
F
, unspecified function of
temperature, of the gas pressure, the steam pressure and saturation
·
the coefficient of dissemination of fick for the liquid mixture,
AD
F
, unspecified function of
temperature and of the pressure of fluid, the pressure of the dissolved air and saturation.
·
The constant of Henry
H
K
unspecified function of the temperature,
·
the intrinsic permeability,
int
K
, unspecified function of porosity,
·
the dynamic viscosity of water,
W
µ
, unspecified function of the temperature,
·
the dynamic viscosity of gas,
gz
µ
, unspecified function of the temperature,
·
the permeability relating to the fluid,
rel
lq
K
, unspecified function of saturation in fluid,
·
the permeability relating to gas,
rel
gz
K
, unspecified function of saturation in fluid and of
gas pressure,
·
the relation capillary saturation/pressure,
()
C
lq
p
S
, unspecified function of the pressure
thin cable,
·
in a general way the initial state is characterized by:
-
the initial temperature,
-
initial pressures from where initial saturation is deduced
()
0
0
C
lq
p
S
,
-
initial specific mass of water
0
W
,
-
initial porosity
0
,
-
initial pressure of the vapor
0
vp
p
from where one deduces the initial density from
vapor
0
vp
,
-
initial pressure of the dry air
0
have
p
from where one deduces the initial density from L `dry air
0
have
.
-
initial density of the porous environment
0
R
,
- density of the solid matter constituents
0
S
. This data is useful only for
modelings including of thermics, and one will have to take care that it is coherent
with the other data, through the relation
:
(
)
(
) (
)
0
0
0
0
0
0
0
0
0
0
0
1
)
(
1
have
vp
lq
lq
AD
W
S
S
R
S
+
-
-
+
-
=
-
,
-
initial enthali of water, the dissolved air, the vapor and the dry air.
Code_Aster
®
Version
7.4
Titrate:
Models of behavior THHM
Date:
01/09/05
Author (S):
C. CHAVANT, S. GRANET
Key
:
R7.01.11-B
Page
:
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Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
3.3
The state of reference and the initial state
The introduction of the initial conditions is very important, in particular for the enthali.
In practice, one can reason by considering that one has three states for the fluids:
·
the state running,
·
the state of reference: it is that of the fluids in a free state. Very often one will take for
pressures of water and air atmospheric pressure. In this state of reference, one can
to consider that the enthali are null,
·
the initial state: it is important to note that, in an initial state of the porous environment, water is in a state
hygroscopic different from that of interstitial water. For the enthali of water and vapor one
will have to take:
0
0
ion
vaporisat
of
latent
heat
)
(
=
=
=
=
-
=
-
=
m
AD
init
m
have
init
init
m
vp
init
W
atm
init
W
W
ref.
L
init
W
m
W
init
H
H
T
L
H
p
p
p
p
H
Note:
The initial vapor pressure will have to be taken in coherence with these choices. Very often, one
leaves the knowledge of an initial state of hygroscopy. The relative humidity is the report/ratio
between the steam pressure and the steam pressure saturating at the temperature considered.
One then uses the law of Kelvin which gives the pressure of the fluid according to the pressure of
vapor, of the temperature and the steam pressure
saturating:
=
-
)
(
ln
0
T
p
p
T
M
R
p
p
sat
vp
vp
ol
vp
W
W
W
. This relation is valid only for
isothermal evolutions. For evolutions with variation in temperature, the formula
[éq 4.1.4-1] further established gives a more complete expression but in fact incremental.
Knowing a law giving the steam pressure saturating to the temperature
0
T
, by
example:
(
)
-
+
-
+
=
5
.
273
1354
.
559
.
31
5
.
273
7858
.
2
0
0
0
10
)
(
T
T
sat
vp
T
p
, and a degree of hygroscopy
HR
, one deduces some
the steam pressure thanks to
)
(
)
(
0
0
T
p
HR
T
p
sat
vp
vp
=
.
3.4
Nodal unknown factors, initial values and values of reference
We approach here a point which is due more to choices of programming than to true aspects of
formulation. Nevertheless, we expose it because it has important practical consequences.
main unknown factors which are also the values of the degrees of freedom, are noted:
{}
=
ddl
ddl
ddl
Z
y
X
ddl
T
U
U
U
U
2
PRE
1
PRE
Code_Aster
®
Version
7.4
Titrate:
Models of behavior THHM
Date:
01/09/05
Author (S):
C. CHAVANT, S. GRANET
Key
:
R7.01.11-B
Page
:
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Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
According to modeling, they can have different significances:
LIQU_SATU LIQU_VAPE LIQU_GAZ_ATM
GAS
LIQU_VAPE_GAZ
PRE1
W
p
W
p
W
C
p
p
-
=
gz
p
W
gz
C
p
p
p
-
=
PRE2
gz
p
LIQU_GAZ LIQU_AD_GAZ_VAPE
PRE1
W
gz
C
p
p
p
-
=
AD
W
gz
C
p
p
p
p
-
-
=
PRE2
gz
p
gz
p
One will then define the real pressures and the real temperature by:
init
ddl
p
p
p
+
=
for the pressures
PRE1
and
PRE2
and
init
ddl
T
T
T
+
=
for the temperatures, where
init
p
and
init
T
are defined under the key word
THM_INIT
control
DEFI_MATERIAU
.
Values written by
IMPR_RESU
are the values of
{}
ddl
U
. The boundary conditions are defined
for
{}
ddl
U
. The key word
DEPL
key word factor
ETAT_INIT
control
STAT_NON_LINE
defines
initial values of
{}
ddl
U
. Initial values of the enthali, which belong to the stresses
generalized are definite starting from the key word
SIGM
key word factor
ETAT_INIT
control
STAT_NON_LINE
. The real pressures and the real temperature are used in the laws of
behavior, in particular laws of the type
()
C
lq
p
F
S
=
or
T
dT
D
p
dp
+
=
. Initial values of
densities of the vapor and the dry air are defined starting from the initial values of the pressures
gas and of vapor (values read under the key word
THM_INIT
control
DEFI_MATERIAU
). One
notice that, for displacements, the decomposition
init
ddl
U
U
U
+
=
is not made: the key word
THM_INIT
control
DEFI_MATERIAU
thus does not allow to define displacements
initial. The only way of initializing displacements is thus to give them an initial value by
key word factor
ETAT_INIT
control
STAT_NON_LINE
.
3.5 Effective stresses and total stresses. Boundary conditions
of stress
The partition of the stresses in stresses total and effective is written:
1
p
+
=
'
is the total stress, a.c. D that which checks:
()
0
=
+
m
R
F
Div
is the effective stress. For the laws of effective stresses, it checks
:
(
)
0
,
dT
D
F
D
-
=
, where
(
)
U
U
+
=
T
2
1
and
represent the internal variables.
p
is calculated according to the water pressures. The adopted writing is incremental and, if one
wants that the value of
p
that is to say coherent with the value
init
p
defined under the key word
THM_INIT
, it is necessary
to initialize
p
by the key word
SIGM
key word factor
ETAT_INIT
control
STAT_NON_LINE
.
In the files results, one finds the stresses effective
under the names of components
SIXX
… and
p
under the name
SIP
. The boundary conditions in stresses are written in stresses
total.
Code_Aster
®
Version
7.4
Titrate:
Models of behavior THHM
Date:
01/09/05
Author (S):
C. CHAVANT, S. GRANET
Key
:
R7.01.11-B
Page
:
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Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
3.6
Some numerical values
We give here some reasonable values for certain coefficients. These values are not
programmed in Code_Aster, they are provided here as an indication:
For perfect gases, one retains the following values:
1
.
3144
.
8
-
=
K
J
R
1
3
.
10
.
18
-
-
=
mole
kg
M
ol
vp
1
3
.
10
96
.
28
-
-
=
mole
kg
M
ol
have
For CO2, the value of the constant of Henry with 20°C is of:
1
3
.
3162
-
=
mole
m
AP
K
H
For liquid water, one a:
3
/
1000
m
kg
W
=
MPa
K
W
2000
=
The thermal expansion factor of water is correctly approached by the formula:
(
)
4
5
10
19
.
2
273
Ln
10
52
.
9
-
-
-
-
=
T
W
(
1
-
K
)
The heat-storage capacities have as values:
1
1
1
1
1
1
1
1
1000
1870
4180
800
-
-
-
-
-
-
-
-
=
=
=
=
K
JKg
C
K
JKg
C
K
JKg
C
K
JKg
C
p
have
p
vp
p
lq
S
One also gives a law of evolution of the latent heat of liquid phase shift
vapor:
()
(
)
Kg
J
T
T
L
/
15
.
273
2443
2500800
-
-
=
.
4
Calculation of the generalized stresses
In this chapter, we specify how are integrated the relations described into chapter 3. More
precisely still, we give the expressions of the stresses generalized within the meaning of
document [R7.01.10] [bib4] when laws of behaviors
THM
are called for the option
RAPH_MECA
within the meaning of the document [R5.03.01] [bib3]. So that this document follows of readier the command of
programming, we will consider two cases: the case without dissolved air and that with.
The generalized stresses are:
Q
M
M
M
M
,
;
,
,
;
,
,
;
,
,
;
,
,
;
,
'
Q
H
m
H
m
H
m
H
m
m
AD
AD
AD
m
have
have
have
m
vp
vp
vp
m
W
W
W
p
The generalized deformations, from which the generalized stresses are calculated are:
()
T
T
p
p
p
p
gz
gz
C
C
,
;
,
;
,
;
,
U
U
Code_Aster
®
Version
7.4
Titrate:
Models of behavior THHM
Date:
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Author (S):
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Key
:
R7.01.11-B
Page
:
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R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
The variables intern that we retained are:
In the case without vapor:
lq
W
S
,
,
In the case with vapor and without dissolved air:
lq
vp
W
S
p,
,
,
In the case with dissolved vapor and air:
lq
AD
vp
W
S
p
p
,
,
,
,
In this chapter, we adopt the usual notations Aster, namely the indices + for the values
quantities at the end of the pitch of time and indices - for the quantities at the beginning of the pitch of time.
Thus, the known quantities are:
·
generalized stresses, deformations and internal variables at the beginning of the pitch of time:
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
Q
M
M
M
M
,
;
,
,
;
,
,
;
,
,
;
,
,
;
,
'
Q
H
m
H
m
H
m
H
m
m
AD
AD
AD
m
have
have
have
m
vp
vp
vp
m
W
W
W
p
-
()
-
-
-
-
-
-
-
-
T
T
p
p
p
p
gz
gz
C
C
,
;
,
;
,
;
,
U
U
-
-
-
-
-
AD
vp
p
p
W
,
,
,
·
deformations generalized at the end of the pitch of time:
-
()
+
+
+
+
+
+
+
+
T
T
p
p
p
p
gz
gz
C
C
,
;
,
;
,
;
,
U
U
·
The unknown quantities are the stresses, and variables intern at the end of the pitch of time:
-
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
Q
M
M
M
M
,
;
,
,
;
,
,
;
,
,
;
,
,
;
,
'
Q
H
m
H
m
H
m
H
m
m
AD
AD
AD
m
have
have
have
m
vp
vp
vp
m
W
W
W
p
-
+
+
+
+
AD
vp
p
p
W
,
,
,
4.1
Case without dissolved air
4.1.1 Calculation of porosity and it it density of the fluid
The first thing to be made is of course to calculate saturation at the end of the pitch of time
()
+
+
=
C
lq
lq
p
S
S
. Porosity is by integrating on the pitch of time the equation [éq 3.2.1-1].
One obtains then:
(
)
(
)
(
)
(
)
-
-
-
-
-
+
-
-
=
-
-
-
+
+
-
+
-
+
-
+
-
+
S
C
C
gz
gz
V
V
K
p
p
S
p
p
T
T
B
B
lq
0
3
ln
éq
4.1.1-1
The density of the fluid is by integrating on the pitch of time the equation [éq 3.2.3.1-1].
What gives:
(
)
-
+
-
+
-
+
-
+
-
-
+
-
-
=
T
T
K
p
p
p
p
W
W
C
C
gz
gz
W
W
3
ln
éq
4.1.1-2
Code_Aster
®
Version
7.4
Titrate:
Models of behavior THHM
Date:
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Author (S):
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:
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Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
4.1.2 Calculation of the expansion factors
It is about a simple application of the formulas [éq 3.2.4.3-2] and [éq 3.2.4.3-3], which are evaluated
at the end of the pitch of time:
()
(
)
(
)
+
+
+
+
+
+
+
+
-
+
-
-
=
=
=
T
S
B
S
lq
m
gz
m
have
m
vp
lq
3
1
1
0
éq
4.1.2-1
(
)
+
+
+
+
+
+
-
=
lq
lq
lq
m
W
S
B
S
0
éq
4.1.2-2
4.1.3 Calculation of the fluid enthali
The fluid enthali are calculated by integration of the equations [éq 3.2.4.1-1], [éq 3.2.4.2-1],
[éq 3.2.4.2-2].
(
) (
)
(
)
-
+
-
+
+
+
-
+
-
+
+
-
-
-
+
-
+
=
C
C
gz
gz
W
p
W
m
W
m
W
p
p
p
p
T
T
T
C
H
H
W
3
1
éq
4.1.3-1
(
)
-
+
-
+
-
+
=
T
T
C
H
H
p
vp
m
vp
m
vp
éq
4.1.3-2
(
)
-
+
-
+
-
+
=
T
T
C
H
H
p
have
m
have
m
have
éq
4.1.3-3
4.1.4 Air and steam pressures
On the basis of the relation [éq 3.2.6-4] in which one carries the law of reaction of perfect gases
[éq 3.2.3.2-1], one finds
(
)
-
+
-
=
T
dT
H
H
dp
dp
RT
M
p
dp
m
W
m
vp
C
W
gz
W
ol
vp
vp
vp
1
1
that one integrates by one
path initially at constant temperature (one then considers the density of constant water),
then of
-
T
with
+
T
with constant pressures.
(
) (
)
[
]
(
)
-
+
-
-
-
=
+
-
-
+
-
+
+
+
-
+
T
T
m
W
m
vp
ol
vp
C
C
gz
gz
ol
vp
vp
vp
T
dT
H
H
R
M
p
p
p
p
RT
M
p
p
W
2
1
ln
The first term corresponds to the path at constant temperature, the second with the path with pressures
constants. By using the definitions [éq 3.2.4.1-1] and [éq 3.2.4.2-1] of the enthali, one sees that for
an evolution with constant pressures:
(
) (
)
2
2
2
T
T
T
C
C
T
H
H
T
H
H
p
W
p
vp
m
W
m
vp
m
W
m
vp
-
-
-
-
-
+
-
=
-
One thus has for such a path:
(
)
(
)
(
)
-
+
-
+
-
-
=
-
-
+
-
-
+
+
-
-
-
+
-
T
T
T
T
T
C
C
T
T
H
H
T
dT
H
H
p
W
p
vp
m
W
m
vp
T
T
m
W
m
vp
1
1
ln
1
1
2
Code_Aster
®
Version
7.4
Titrate:
Models of behavior THHM
Date:
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Author (S):
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Key
:
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:
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R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
That is to say finally:
(
) (
)
[
]
(
)
(
)
-
+
-
+
-
-
+
-
-
-
=
+
-
-
+
+
-
-
-
-
+
-
+
+
+
-
+
1
ln
1
1
1
ln
T
T
T
T
C
C
R
M
T
T
H
H
R
M
p
p
p
p
RT
M
p
p
p
W
p
vp
ol
vp
m
W
m
vp
ol
vp
C
C
gz
gz
ol
vp
vp
vp
W
éq
4.1.4-1
One can then calculate the densities of the vapor and the air by the relations [éq 3.2.3.2-1] and
[éq 3.2.3.2-2]:
+
+
+
=
T
p
R
M
vp
ol
vp
vp
éq
4.1.4-2
(
)
+
+
+
+
-
=
T
p
p
R
M
vp
gz
ol
have
have
éq
4.1.4-3
4.1.5 Calculation of the mass contributions
The equations [éq 3.2.2-1] give null mass contributions to moment 0. Way is written
incremental the equations [éq 3.2.2-1]:
(
)
(
)
(
)
(
)
(
)
(
)
)
1
(
1
)
1
(
1
)
1
(
1
)
1
(
1
1
1
-
-
-
-
+
+
+
+
-
+
-
-
-
-
+
+
+
+
-
+
-
-
-
-
+
+
+
+
-
+
-
+
-
-
+
+
=
-
+
-
-
+
+
=
+
-
+
+
=
lq
V
vp
lq
V
vp
vp
vp
lq
V
have
lq
V
have
have
have
lq
V
W
lq
V
W
W
W
S
S
m
m
S
S
m
m
S
S
m
m
éq 4.1.5-1
4.1.6 Calculation of the heat-storage capacity and Q' heat
There are now all the elements to apply at the end of the pitch of time the formula [éq 3.2.4.3-5]:
(
)
p
have
have
p
vp
vp
lq
p
W
W
lq
S
S
C
C
S
C
S
C
C
+
+
+
+
+
+
+
+
+
+
-
+
+
-
=
)
1
(
)
1
(
0
éq
4.1.6-1
One uses of course [éq 3.2.4.3-4] who gives:
2
0
0
0
0
9
-
=
+
+
+
K
T
C
C
éq
4.1.6-2
Although variation of heat
Q
is not a total differential, it is nevertheless licit of
to integrate on the pitch of time and one obtains while integrating [éq 3.2.4.3-1].
(
)
(
)
(
)
(
)
(
)
-
+
+
-
+
+
+
-
+
+
-
+
-
+
-
+
-
-
+
-
+
-
=
+
T
T
C
p
p
T
p
p
T
T
K
Q
Q
gz
gz
m
lq
m
gz
C
C
m
lq
V
V
0
2
1
2
1
2
1
0
0
3
3
3
3
'
'
éq 4.1.6-3
where we noted:
2
2
1
-
+
+
=
T
T
T
. We chose here a formula of “point medium” for
variable temperature.
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Titrate:
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Key
:
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HT-66/05/002/A
4.1.7 Calculation of the mechanical stresses
The calculation of the effective stresses is done by calling upon the incremental laws of mechanics selected
by the user. One integrates on the pitch of time [éq 3.2.8-2] and one a:
(
)
(
)
-
+
+
-
+
-
+
-
+
-
-
=
C
C
lq
gz
gz
p
p
p
p
bS
p
p
B
éq
4.1.7-1
4.1.8 Calculation of hydrous and thermal flows
It is necessary well could calculate all the coefficients of dissemination:
The coefficient of Fick
(
)
+
+
+
+
=
gz
C
p
p
T
F
F
,
,
Thermal diffusivity
T
cte
T
T
lq
T
S
T
T
T
S
+
=
+
+
+
+
)
(
).
(
).
(
Hydraulic permeabilities and conductivities:
()
()
(
)
()
+
+
+
+
+
+
+
+
+
=
=
T
p
S
K
K
T
S
K
K
gz
lq
rel
gz
H
gz
W
lq
rel
W
H
lq
gz
µ
µ
,
).
(
).
(
int
int
Vapor concentrations:
+
+
+
=
vp
vp
vp
p
p
C
It does not remain any more whereas to apply the formulas [éq 3.2.5.1-1], [éq 3.2.5.2-15], [éq 3.2.5.2-16] and
[éq 3.2.5.2-17] to find:
+
+
+
-
=
T
T
Q
éq 4.1.8-1
(
)
[
]
+
+
+
+
+
+
+
+
+
+
+
+
-
=
vp
vp
vp
m
H
gz
have
have
C
F
C
p
vp
have
gz
F
M
éq
4.1.8-2
(
)
[
]
+
+
+
+
+
+
+
+
+
-
-
+
+
-
=
vp
vp
vp
m
H
gz
vp
vp
C
F
C
p
vp
have
gz
)
1
(
F
M
éq
4.1.8-3
[
]
m
W
lq
H
lq
W
W
p
F
M
+
+
+
+
+
+
-
=
éq
4.1.8-4
4.2
Case with dissolved air
4.2.1 Calculation of porosity
Same manner, the first thing to be made is to calculate saturation at the end of the pitch of time
()
+
+
=
C
lq
lq
p
S
S
. Porosity is by integrating on the pitch of time the equation [éq 3.2.1-1]. One
thus recall that:
(
)
(
)
(
)
(
)
-
-
-
-
-
+
-
-
=
-
-
-
+
+
-
+
-
+
-
+
-
+
S
C
C
gz
gz
V
V
K
p
p
S
p
p
T
T
B
B
lq
0
3
ln
Code_Aster
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Titrate:
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Key
:
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:
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HT-66/05/002/A
4.2.2 Calculation of the expansion factors
It is about a simple application of the formulas [éq 3.2.4.3-2] and [éq 3.2.4.3-3], which are evaluated
at the end of the pitch of time:
(
) (
)
(
)
+
+
+
+
+
+
+
+
-
+
-
-
=
=
=
T
S
B
S
has
has
lq
lq
m
gz
m
have
m
vp
3
1
1
0
éq
4.2.2-1
(
)
+
+
+
+
+
+
-
=
lq
lq
lq
m
W
S
B
S
0
éq
4.2.2-2
(
)
+
+
+
+
+
+
+
-
=
T
S
B
S
lq
lq
m
AD
3
0
éq
4.2.2-3
4.2.3 Calculation of density and dissolved and dry air, steam pressures
On the basis of the relation [éq 3.2.6-4] in which one carries the law of reaction of perfect gases
[éq 3.2.3.2-1], one finds:
(
)
-
+
=
T
dT
H
H
dp
RT
M
p
dp
m
W
m
vp
W
W
ol
vp
vp
vp
1
éq
4.2.3-1
Contrary to the case without dissolved air
W
p
is not now known any more.
)
(
vp
gz
H
C
gz
have
H
C
gz
AD
lq
W
p
p
K
RT
p
p
p
K
RT
p
p
p
p
p
-
-
-
=
-
-
=
-
=
thus
dT
p
p
K
R
dp
dp
K
RT
dp
dp
dp
vp
gz
H
vp
gz
H
C
gz
W
)
(
)
(
-
-
-
-
-
=
éq
4.2.3-2
One integrates [éq 4.2.3.1] while including there [éq 4.2.3.2] by a path initially into constant temperature (one
then consider the density of constant water), then of
-
T
with
+
T
with constant pressures. With
final one obtains:
(
)
(
)
(
)
(
)
(
)
-
+
-
+
-
-
-
+
-
-
=
+
-
-
+
+
+
-
-
+
+
-
-
+
-
-
+
+
-
-
+
T
T
m
W
m
vp
ol
vp
gz
vp
H
W
ol
vp
C
C
W
ol
vp
vp
vp
H
W
ol
vp
gz
gz
H
W
ol
vp
vp
vp
T
dT
H
H
R
M
T
T
p
p
K
R
M
p
p
RT
M
p
p
K
M
p
p
K
RT
M
p
p
2
ln
)
1
1
(
ln
éq 4.2.3-3
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HT-66/05/002/A
Contrary to the preceding case, there is here a nonlinear equation to solve. One will make for that
a method of the corrector-predictor type. One poses
vp
p~
such as:
(
)
(
)
(
)
-
+
-
-
-
-
=
+
-
-
+
+
-
-
+
+
-
-
T
T
m
W
m
vp
ol
vp
C
C
ol
vp
gz
gz
H
ol
vp
vp
vp
T
dT
H
H
R
M
p
p
RT
M
p
p
K
RT
M
p
p
W
W
2
)
1
1
(
~
ln
éq
4.2.3-4
and thus
(
)
(
)
(
)
-
+
-
-
-
-
=
+
-
-
+
+
-
-
+
+
-
-
T
T
m
W
m
vp
C
C
ol
vp
gz
gz
H
ol
vp
vp
vp
T
dT
H
H
p
p
RT
M
p
p
K
RT
M
p
p
W
W
2
)
1
1
(
exp
.
~
éq 4.2.3-5
Moreover, as in the section [§4.1.4], one recalls that
(
)
(
)
(
)
-
+
-
+
-
-
=
-
-
+
-
-
+
+
-
-
-
+
-
T
T
T
T
T
C
C
T
T
H
H
T
dT
H
H
p
W
p
vp
m
W
m
vp
T
T
m
W
m
vp
1
1
ln
1
1
2
Like
+
=
+
-
-
+
vp
vp
vp
vp
vp
vp
p
p
p
p
p
p
~
ln
~
ln
ln
and that by D.L
1
~
~
~
1
ln
~
ln
-
-
+
=
+
+
+
vp
vp
vp
vp
vp
vp
vp
p
p
p
p
p
p
p
+
vp
p
will thus be given by the following linear expression:
(
)
(
)
-
+
-
+
=
-
+
-
+
-
-
+
-
+
T
T
p
p
K
R
M
p
p
K
M
p
p
gz
vp
H
ol
vp
vp
vp
H
ol
vp
vp
vp
W
W
ln
1
~
éq 4.2.3-6
from where
+
-
+
-
=
-
+
-
-
+
-
-
-
+
T
T
R
M
p
K
T
T
R
p
p
M
K
p
ol
vp
vp
H
gz
vp
ol
vp
H
vp
W
W
ln
1
~
ln
éq
4.2.3-7
From there the other pressures are calculated easily:
+
+
+
-
=
vp
gz
have
p
p
p
+
+
+
=
RT
K
p
p
H
have
AD
+
+
+
+
-
-
=
AD
C
gz
W
p
p
p
p
Code_Aster
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Titrate:
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:
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:
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R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
One can then calculate the densities of the vapor and the air by the relations [éq 3.2.3.2-1],
[éq 3.2.3.2-2] and [éq 3.2.7-3]:
+
+
+
=
T
p
R
M
vp
ol
vp
vp
éq
4.2.3-8
(
)
+
+
+
+
-
=
T
p
p
R
M
vp
gz
ol
have
have
éq
4.2.3-9
+
+
+
=
RT
M
p
ol
have
AD
AD
éq
4.2.3-10
The density of water is by integrating on the pitch of time the equation [éq 3.2.3.1-1].
What gives:
(
)
-
+
-
+
-
+
-
+
-
+
-
-
+
-
+
-
-
=
T
T
K
p
p
p
p
p
p
W
W
AD
AD
C
C
gz
gz
W
W
3
ln
éq 4.2.3-11
4.2.4 Calculation of the fluid enthali
The fluid enthali are calculated by integration of the equations [éq 3.2.4.1-1], [éq 3.2.4.1-3],
[éq 3.2.4.2-1], [éq 3.2.4.2-2].
(
) (
)
(
)
-
+
-
+
-
+
+
+
-
+
-
+
+
-
+
-
-
-
+
-
+
=
AD
AD
C
C
gz
gz
W
p
W
m
W
m
W
p
p
p
p
p
p
T
T
T
C
H
H
W
3
1
éq 4.2.4-1
(
)
-
+
-
+
-
+
=
T
T
C
H
H
p
AD
m
AD
m
AD
éq
4.2.4-2
(
)
-
+
-
+
-
+
=
T
T
C
H
H
p
vp
m
vp
m
vp
éq
4.2.4-3
(
)
-
+
-
+
-
+
=
T
T
C
H
H
p
have
m
have
m
have
éq
4.2.4-4
4.2.5 Calculation of the mass contributions
The equations [éq 3.2.2-1] give null mass contributions to moment 0. Way is written
incremental the equations [éq 3.2.2-1]:
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
)
1
(
1
)
1
(
1
)
1
(
1
)
1
(
1
1
1
1
1
-
-
-
-
+
+
+
+
-
+
-
-
-
-
+
+
+
+
-
+
-
-
-
-
+
+
+
+
-
+
-
-
-
-
+
+
+
+
-
+
-
+
-
-
+
+
=
-
+
-
-
+
+
=
+
-
+
+
=
+
-
+
+
=
lq
V
vp
lq
V
vp
vp
vp
lq
V
have
lq
V
have
have
have
lq
V
AD
lq
V
AD
AD
AD
lq
V
W
lq
V
W
W
W
S
S
m
m
S
S
m
m
S
S
m
m
S
S
m
m
éq
4.2.5-1
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Titrate:
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:
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HT-66/05/002/A
4.2.6 Calculation of the heat-storage capacity and Q' heat
There are now all the elements to apply at the end of the pitch of time the formula [éq 3.2.4.3-5]:
(
)
p
have
have
p
vp
vp
lq
p
AD
AD
p
W
W
lq
S
S
C
C
S
C
C
S
C
C
+
+
+
+
+
+
+
+
+
+
+
-
+
+
+
-
=
)
1
(
)
(
)
1
(
0
éq 4.2.6-1
One uses of course [éq 3.2.4.3-4] who gives:
2
0
0
0
0
9
-
=
+
+
+
K
T
C
C
éq
4.2.6-2
Although variation of heat
Q
is not a total differential, it is nevertheless licit of
to integrate on the pitch of time and one obtains while integrating [éq 3.2.4.3-1].
(
)
(
)
(
)
(
)
(
)
-
+
+
-
+
+
+
-
+
+
-
+
-
+
-
+
-
-
+
-
+
-
=
+
T
T
C
p
p
T
p
p
T
T
K
Q
Q
gz
gz
m
lq
m
gz
C
C
m
lq
V
V
0
2
1
2
1
2
1
0
0
3
3
3
3
'
'
éq 4.2.6-3
where we noted:
2
2
1
-
+
+
=
T
T
T
. We chose here a formula of “point medium” for
variable temperature.
4.2.7 Calculation of the mechanical stresses
The calculation of the effective stresses is done by calling upon the incremental laws of mechanics selected
by the user. One integrates on the pitch of time [éq 3.2.8-2] and one a:
(
)
(
)
-
+
+
-
+
-
+
-
+
-
-
=
C
C
lq
gz
gz
p
p
p
p
bS
p
p
B
éq
4.2.7-1
4.2.8 Calculation of hydrous and thermal flows
It is of course necessary to calculate all the coefficients of dissemination:
Coefficients of Fick
)
,
,
,
(
+
+
+
+
+
S
T
P
P
F
gz
vp
vp
and
)
,
,
,
(
+
+
+
+
+
S
T
P
P
F
lq
AD
AD
Thermal diffusivity
T
cte
T
T
lq
T
S
T
T
T
S
+
=
+
+
+
+
)
(
).
(
).
(
Hydraulic permeabilities and conductivities:
()
()
(
)
()
+
+
+
+
+
+
+
+
+
=
=
T
p
S
K
K
T
S
K
K
gz
lq
rel
gz
H
gz
W
lq
rel
W
H
lq
gz
µ
µ
,
).
(
).
(
int
int
Concentrations out of vapor and air dissolved:
+
+
+
=
gz
vp
vp
p
p
C
and
+
+
=
AD
AD
C
Code_Aster
®
Version
7.4
Titrate:
Models of behavior THHM
Date:
01/09/05
Author (S):
C. CHAVANT, S. GRANET
Key
:
R7.01.11-B
Page
:
32/52
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
It does not remain any more whereas to apply the formulas [éq 3.2.5.1-1], [éq 3.2.5.2-15], [éq 3.2.5.2-16],
[éq 3.2.5.2-17] and [éq 3.2.5.2-18] to find:
+
+
+
-
=
T
T
Q
éq 4.2.8-1
(
)
[
]
+
+
+
+
+
+
+
+
+
+
+
+
-
=
vp
vp
vp
m
H
gz
have
have
C
F
C
p
vp
have
gz
F
M
éq
4.2.8-2
(
)
[
]
+
+
+
+
+
+
+
+
+
-
-
+
+
-
=
vp
vp
vp
m
H
gz
vp
vp
C
F
C
p
vp
have
gz
)
1
(
F
M
éq
4.2.8-3
[
]
m
AD
W
lq
H
lq
W
W
p
F
M
)
(
+
+
+
+
+
+
+
+
-
=
éq
4.2.8-4
[
]
+
+
+
+
+
+
-
+
+
-
=
AD
AD
m
AD
W
lq
H
lq
AD
AD
C
F
p
F
M
)
(
éq 4.2.8-5
5
Calculation of derived from the generalized stresses
In this chapter, we give the expressions of derived from the stresses generalized by
report/ratio with the deformations generalized within the meaning of the document [R7.01.10] [bib4], C `be-with-statement terms
who are calculated when the laws of behaviors
THM
are called for the option
RIGI_MECA_TANG
within the meaning of the document [R5.03.01] [bib3].
In order not to weigh down the talk, we give the expression of the differentials of the stresses
generalized, knowing that the derivative partial result some directly.
5.1
Derived from the stresses
The calculation of the differential of the effective stresses is left with the load of the law of behavior
purely mechanical, that we do not describe in this document. Differential of the stress
p
is given directly by the expression [éq 3.2.8-2].
5.2
Derived from the mass contributions
While differentiating [éq 3.2.2-1], one a:
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
lq
V
lq
V
lq
V
lq
V
vp
vp
vp
vp
lq
V
lq
V
lq
V
lq
V
have
have
have
have
lq
V
lq
V
lq
V
lq
V
AD
AD
AD
AD
lq
V
lq
V
lq
V
lq
V
W
W
W
W
dS
S
D
S
D
S
D
DM
dS
S
D
S
D
S
D
DM
dS
S
D
S
D
S
D
DM
dS
S
D
S
D
S
D
DM
+
-
-
+
+
-
+
-
+
=
+
-
-
+
+
-
+
-
+
=
+
+
+
+
+
+
=
+
+
+
+
+
+
=
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
éq
5.2-1
Code_Aster
®
Version
7.4
Titrate:
Models of behavior THHM
Date:
01/09/05
Author (S):
C. CHAVANT, S. GRANET
Key
:
R7.01.11-B
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:
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Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
5.2.1 Case without dissolved air
by taking account of [éq 3.2.1-1] and of [éq 3.2.3.1-1], [éq 3.2.3.2-1], [éq 3.2.3.2-2] and while supposing
1
1
+
V
one finds:
-
-
+
-
-
+
-
-
-
-
+
-
=
-
-
+
-
-
+
-
-
-
-
+
-
=
-
-
+
+
-
-
-
+
=
dT
p
dp
S
dp
K
S
B
dp
K
B
S
S
P
S
D
S
B
DM
dT
p
dp
S
dp
K
S
B
dp
K
B
S
S
P
S
D
S
B
DM
dT
dp
K
B
K
S
dp
K
B
S
K
S
P
S
D
bS
DM
m
have
have
have
lq
gz
S
lq
C
S
lq
lq
C
lq
V
lq
have
have
m
vp
vp
vp
lq
gz
S
lq
C
S
lq
lq
C
lq
V
lq
vp
vp
m
W
gz
S
W
lq
C
S
lq
W
lq
C
lq
V
lq
W
W
3
)
1
(
)
1
) (
(
)
(
)
1
(
)
1
(
3
)
1
(
)
1
) (
(
)
(
)
1
(
)
1
(
3
)
(
)
(
2
éq 5.2.1-1
One sees appearing the derivative of saturation in fluid compared to the capillary pressure, quantity which
play an essential part.
The expression [éq 3.2.6-4] of the differential of the steam pressure also makes it possible to calculate
pressure of dry air:
(
)
T
dT
H
H
RT
M
dp
RT
M
dp
RT
p
M
dp
m
W
m
vp
ol
vp
C
W
ol
vp
gz
W
vp
ol
vp
have
-
-
-
-
=
)
1
(
éq
5.2.1-2
One defers [éq 5.2.1-2] and [éq 3.2.6-4] in [éq 5.2.1-1] and one finds:
dT
dp
K
B
K
S
dp
K
B
S
K
S
P
S
D
bS
DM
m
W
gz
S
W
lq
C
S
lq
W
lq
C
lq
V
lq
W
W
3
)
(
)
(
2
-
-
+
+
-
-
-
+
=
éq 5.2.1-3
(
)
(
)
(
)
(
)
(
)
(
) (
)
(
)
(
)
dT
T
p
H
H
S
dp
p
S
K
S
B
dp
p
S
K
B
S
S
p
S
D
S
B
DM
vp
m
lq
m
vp
lq
vp
m
vp
gz
lq
vp
vp
lq
S
lq
C
lq
vp
vp
lq
S
lq
lq
C
lq
V
lq
vp
vp
-
-
+
-
+
-
+
-
-
+
-
-
-
-
-
-
+
-
=
1
3
1
1
1
1
1
éq 5.2.1-4
(
)
(
)
(
)
(
)
(
)
(
) (
)
(
)
(
)
dT
T
p
H
H
S
dp
p
S
K
S
B
dp
p
S
K
B
S
S
p
S
D
S
B
DM
have
m
lq
m
vp
lq
vp
m
have
gz
lq
vp
lq
have
lq
S
lq
C
lq
vp
have
lq
S
lq
lq
C
lq
V
lq
have
have
-
-
-
-
+
-
-
+
-
-
+
-
+
-
-
-
-
+
-
=
1
3
1
1
1
1
1
éq 5.2.1-5
Code_Aster
®
Version
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Titrate:
Models of behavior THHM
Date:
01/09/05
Author (S):
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Key
:
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:
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Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
5.2.2 Case with dissolved air
As previously, by taking account of [éq 3.2.1-1] and of [éq 3.2.3.1-1], [éq 3.2.3.2-1],
[éq 3.2.3.2-2], [éq 3.2.7.3] and while supposing
1
1
+
V
one finds:
dT
T
p
K
S
dp
K
B
P
p
K
S
dp
P
S
K
B
S
P
p
K
S
D
bS
DM
m
W
W
W
lq
gz
S
gz
W
W
lq
C
C
lq
S
lq
C
W
W
lq
V
lq
W
W
-
+
-
+
+
+
-
-
+
=
3
)
(
)
(
2
éq 5.2.2-1
dT
B
T
p
K
M
S
dp
K
B
P
p
K
M
S
dp
P
S
K
B
S
P
p
K
M
S
D
bS
DM
have
H
AD
ol
have
lq
gz
S
gz
have
H
AD
ol
have
lq
C
C
lq
S
lq
C
have
H
AD
ol
have
lq
V
lq
AD
AD
-
-
+
-
+
+
+
-
-
+
=
)
(
3
)
(
)
(
0
2
éq 5.2.2-2
dT
T
p
H
H
S
dp
p
S
K
S
B
dp
p
S
K
B
S
S
P
S
D
S
B
DM
vp
m
lq
m
vp
lq
vp
m
vp
gz
lq
vp
vp
lq
S
lq
C
lq
vp
vp
lq
S
lq
lq
C
lq
V
lq
vp
vp
-
-
+
-
+
+
-
+
-
-
+
-
-
-
-
-
-
+
-
=
)
) (
1
(
3
)
1
(
)
1
) (
(
)
1
(
)
(
)
1
(
)
1
(
éq 5.2.2-3
dT
T
p
H
H
S
dp
p
S
K
S
B
dp
p
S
K
B
S
S
P
S
D
S
B
DM
have
m
lq
m
vp
lq
vp
m
vp
gz
lq
vp
lq
have
lq
S
lq
C
lq
vp
have
lq
S
lq
lq
C
lq
V
lq
have
have
-
-
-
-
+
-
-
+
-
-
+
-
+
-
-
-
-
+
-
=
)
) (
1
(
3
)
1
(
)
1
) (
(
)
1
(
)
(
)
1
(
)
1
(
éq
5.2.2-4
The derivative partial are given in [§Annexe 3].
5.3
Derived from the enthali and Q' heat
There still, we do nothing but point out expressions already provided to chapter 2:
5.3.1 Case without dissolved air
(
)
dT
C
dp
dp
T
dh
p
W
W
C
gz
W
m
W
+
-
-
=
3
1
dT
C
dh
p
vp
m
vp
=
dT
C
dh
p
have
m
have
=
(
)
dT
C
Tdp
Tdp
Td
K
Q
gz
m
lq
m
gz
C
m
lq
V
0
0
0
3
3
3
3
'
+
+
-
+
=
Code_Aster
®
Version
7.4
Titrate:
Models of behavior THHM
Date:
01/09/05
Author (S):
C. CHAVANT, S. GRANET
Key
:
R7.01.11-B
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:
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Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
5.3.2 Case with dissolved air
(
)
dT
T
p
T
C
dp
p
p
dp
p
p
T
dT
C
dp
dp
dp
T
dh
AD
W
W
p
W
C
C
AD
gz
gz
AD
W
W
p
W
W
AD
C
gz
W
m
W
-
-
+
+
-
-
-
=
+
-
-
-
=
3
1
1
1
3
1
3
1
dT
C
dh
p
AD
m
AD
=
dT
C
dh
p
vp
m
vp
=
dT
C
dh
p
have
m
have
=
(
)
dT
C
Tdp
Tdp
Td
K
Q
gz
m
lq
m
gz
C
m
lq
V
0
0
0
3
3
3
3
'
+
+
-
+
=
5.4
Derived from the heat transfer rate
One leaves [éq 3.2.5.1-1] and [éq 3.2.5.1-2].
While differentiating [éq 3.2.5.1-2] and while using [éq 3.2.1-1], one finds:
(
)
dT
T
S
B
T
S
dp
T
S
K
B
S
p
S
T
S
dp
T
S
K
B
D
T
S
B
D
T
T
lq
T
S
T
T
T
lq
T
S
T
C
T
T
lq
T
S
T
S
lq
C
lq
T
T
lq
T
S
T
gz
T
T
lq
T
S
T
S
v
T
T
lq
T
S
T
T
)
(
).
(
).
(
'
3
).
(
)
(
'
).
(
).
(
)
(
).
(
).
(
'
)
(
).
(
).
(
'
).
(
)
(
).
(
).
(
'
)
(
)
(
).
(
).
(
'
)
(
0
-
-
+
-
-
+
-
+
-
=
That is to say finally:
(
)
TdT
T
S
B
T
S
Tdp
T
S
K
B
S
p
S
T
S
Tdp
T
S
K
B
Td
T
S
B
D
T
T
lq
T
S
T
T
T
lq
T
S
T
C
T
T
lq
T
S
T
S
lq
C
lq
T
T
lq
T
S
T
gz
T
T
lq
T
S
T
S
v
T
T
lq
T
S
T
-
-
-
-
-
-
-
-
-
-
=
)
(
).
(
).
(
'
3
).
(
)
(
'
).
(
).
(
)
(
).
(
).
(
'
)
(
).
(
).
(
'
).
(
)
(
).
(
).
(
'
)
(
)
(
).
(
).
(
'
)
(
0
Q
éq
5.4-1
5.5
Derived from hydrous flows
It is of course necessary to set out again of the equations [éq 3.2.5.2-15], [éq 3.2.5.2-16], [éq 3.2.5.2-17] and [éq 3.2.5.2-18]
that one differentiates.
Code_Aster
®
Version
7.4
Titrate:
Models of behavior THHM
Date:
01/09/05
Author (S):
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Key
:
R7.01.11-B
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:
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Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
5.5.1 Case without dissolved air
(
)
vp
vp
vp
have
vp
vp
vp
have
vp
gz
gz
vp
vp
vp
have
m
vp
gz
H
gz
have
H
gz
H
lq
vp
vp
vp
have
have
have
m
H
gz
have
have
have
have
C
D
F
C
C
F
cd.
C
p
p
F
dT
T
F
C
D
p
D
D
C
F
C
D
D
+
+
+
+
+
-
+
-
+
+
=
F
M
F
M
M
éq
5.5.1-1
(
)
(
)
vp
vp
vp
vp
vp
vp
vp
vp
vp
gz
gz
vp
vp
vp
vp
m
have
gz
H
gz
vp
H
gz
H
gz
vp
vp
vp
vp
vp
vp
m
H
gz
vp
vp
vp
vp
C
D
F
C
C
F
cd.
C
p
p
F
dT
T
F
C
D
p
D
D
C
F
C
D
D
-
-
+
+
-
-
+
-
+
-
+
+
+
=
)
1
(
)
1
(
1
F
M
F
M
M
éq 5.5.1-2
(
)
C
gz
H
lq
W
H
lq
H
lq
W
W
m
H
lq
W
W
W
W
p
D
p
D
D
D
D
-
-
+
+
=
M
F
M
M
éq
5.5.1-3
In order to clarify these differentials completely, it is necessary to know the differentials of the masses
voluminal of the fluids, as well as the differentials of
gz
vp
vp
p
p
C
=
and of its gradient. Knowing
[éq 3.2.6-4], one can then calculate the differential of the pressure of dry air:
(
)
T
dT
H
H
dp
dp
dp
dp
dp
m
W
m
vp
vp
C
W
vp
gz
W
vp
W
vp
gz
have
-
-
+
-
=
-
=
éq
5.5.1-4
By deriving the relation from perfect gases one a:
T
dT
p
dp
D
have
have
have
have
-
=
and
T
dT
p
dp
D
vp
vp
vp
vp
-
=
, which, in
using [éq 3.2.6-4] and [éq 5.5.1-4] gives:
(
)
dT
T
Tp
H
H
dp
p
dp
p
D
vp
vp
m
W
m
vp
vp
C
vp
W
vp
gz
vp
W
vp
vp
-
-
+
-
=
2
2
2
éq
5.5.1-5
(
)
dT
T
Tp
H
H
dp
p
dp
p
D
have
have
m
W
m
vp
vp
have
C
W
vp
have
have
gz
W
vp
W
have
have
have
-
-
-
+
+
-
=
éq 5.5.1-6
Code_Aster
®
Version
7.4
Titrate:
Models of behavior THHM
Date:
01/09/05
Author (S):
C. CHAVANT, S. GRANET
Key
:
R7.01.11-B
Page
:
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Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
[éq 3.2.6-4] allows to express the gradient of the steam pressure:
(
)
(
)
T
T
H
H
p
p
p
m
lq
m
vp
vp
C
gz
lq
vp
vp
-
+
-
=
éq
5.5.1-7
While differentiating [éq 5.5.1-7] one finds:
(
)
(
)
T
T
H
H
D
T
D
T
H
H
p
p
D
D
p
D
p
D
p
D
m
W
m
vp
vp
m
W
m
vp
vp
C
gz
W
W
vp
W
vp
C
gz
W
vp
vp
-
+
-
+
-
-
+
-
=
2
éq
5.5.1-8
The last term of [éq 5.5.1-8] is written:
(
)
(
)
2
T
dT
T
H
H
dh
dh
T
T
D
T
T
H
H
T
T
H
H
D
m
W
m
vp
vp
m
W
m
vp
vp
vp
m
W
m
vp
m
W
m
vp
vp
-
-
-
+
-
=
-
éq 5.5.1-9
Knowing the differentials of the its gradient and steam pressure, the expressions of
differentials of
vp
C
and of its gradient are easy to calculate:
gz
gz
vp
gz
vp
vp
dp
p
p
p
dp
cd.
2
-
=
who gives:
gz
gz
vp
gz
vp
vp
p
p
p
p
p
C
-
=
2
and that one differentiates in:
gz
gz
vp
gz
vp
gz
gz
vp
vp
gz
gz
gz
vp
gz
gz
vp
gz
vp
vp
p
D
p
p
dp
p
p
p
p
p
dp
p
p
p
p
D
p
p
p
p
p
D
C
D
gz
-
-
+
-
=
-
=
2
2
3
2
2
2
vp
dp
is given by [éq 3.2.6-4] and
vp
p
D
by [éq 5.5.1-8].
5.5.2 Case with dissolved air
(
)
vp
vp
vp
have
vp
vp
vp
have
vp
gz
gz
vp
vp
vp
have
m
vp
gz
H
gz
have
H
gz
H
gz
vp
vp
vp
have
have
have
m
H
gz
have
have
have
have
C
D
F
C
C
F
cd.
C
dp
p
F
dT
T
F
C
D
p
D
D
C
F
C
D
D
+
+
+
+
+
-
+
-
+
+
=
F
M
F
M
M
éq 5.5.2-1
Code_Aster
®
Version
7.4
Titrate:
Models of behavior THHM
Date:
01/09/05
Author (S):
C. CHAVANT, S. GRANET
Key
:
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:
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HT-66/05/002/A
(
)
(
)
vp
vp
vp
vp
vp
vp
vp
vp
vp
gz
gz
vp
vp
vp
vp
m
have
gz
H
gz
vp
H
gz
H
gz
vp
vp
vp
vp
vp
vp
m
H
gz
vp
vp
vp
vp
C
D
F
C
C
F
cd.
C
dp
p
F
dT
T
F
C
D
p
D
D
C
F
C
D
D
-
-
+
+
-
-
+
-
+
-
+
+
+
=
)
1
(
)
1
(
1
F
M
F
M
M
éq 5.5.2-2
(
)
m
AD
lq
H
lq
W
H
lq
H
lq
W
W
m
H
lq
W
W
W
W
D
p
D
D
D
D
F
M
F
M
M
+
-
+
+
+
=
éq 5.5.2-3
(
)
(
)
(
)
AD
AD
AD
C
C
AD
AD
m
W
lq
H
lq
AD
H
lq
m
lq
lq
H
lq
AD
AD
m
H
lq
lq
lq
H
lq
AD
C
D
F
C
dp
p
F
dT
T
F
D
p
D
D
p
D
p
D
-
+
-
+
-
+
+
-
+
+
-
=
F
F
F
M
)
(
éq 5.5.2-4
It is necessary to know the differentials of the densities of the fluids, as well as the differentials of
gz
vp
vp
p
p
C
=
,
AD
AD
C
=
and of their gradient. One first of all will calculate the differentials of the masses
voluminal by using the derivative partial of pressures given in [§Annexe 3].
By deriving the relation from perfect gases one a:
T
dT
p
dp
D
have
have
have
have
-
=
and
T
dT
p
dp
D
vp
vp
vp
vp
-
=
, that one
can express in the form:
dT
T
T
p
p
dp
p
p
dp
p
p
p
D
have
have
have
have
gz
gz
have
C
C
have
have
have
have
-
+
+
=
éq 5.5.2-5
dT
T
T
p
p
dp
p
p
dp
p
p
p
D
vp
vp
vp
vp
gz
gz
vp
C
C
vp
vp
vp
vp
-
+
+
=
éq 5.5.2-6
By using the relation [éq 3.2.3.1-1], one obtains:
dT
T
p
K
dp
p
p
dp
p
p
K
D
W
W
W
W
W
gz
gz
W
C
C
W
W
W
W
-
+
+
=
3
éq
5.5.2-7
Code_Aster
®
Version
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Titrate:
Models of behavior THHM
Date:
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Key
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:
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HT-66/05/002/A
And like
have
H
ol
have
AD
dp
K
M
D
=
,
+
+
=
dT
T
p
dp
p
p
dp
p
p
K
M
D
have
gz
gz
have
C
C
have
H
ol
have
AD
éq
5.5.2-8
As previously, the expressions are used
gz
gz
vp
gz
vp
vp
dp
p
p
p
dp
cd.
2
-
=
who gives
gz
gz
vp
gz
vp
gz
vp
C
C
vp
gz
vp
dp
p
p
p
p
p
dT
T
p
dp
p
p
p
cd.
-
+
+
=
2
1
1
éq
5.5.2-9
and
gz
gz
vp
gz
vp
vp
p
p
p
p
p
C
-
=
2
and that one differentiates in:
gz
gz
vp
gz
vp
gz
gz
vp
vp
gz
gz
gz
vp
gz
gz
vp
gz
vp
vp
p
D
p
p
dp
p
p
p
p
p
dp
p
p
p
p
D
p
p
p
p
p
D
C
D
gz
-
-
+
-
=
-
=
2
2
3
2
2
2
éq 5.5.2-10
with
T
T
p
p
p
p
p
p
p
p
vp
C
C
vp
gz
gz
vp
vp
+
+
=
éq
5.5.2-11
and
vp
p
D
that one differentiates in the following way:
T
D
T
p
p
D
p
p
p
D
p
p
dT
T
T
p
T
p
p
p
T
p
p
p
T
dp
T
T
p
p
p
p
p
p
p
p
p
p
dp
T
T
p
p
p
p
p
p
p
p
p
p
T
D
T
p
p
D
p
p
p
D
p
p
T
T
p
D
p
p
p
D
p
p
p
D
p
D
vp
C
C
vp
gz
gz
vp
vp
C
C
vp
gz
gz
vp
gz
vp
gz
C
C
vp
gz
gz
gz
vp
gz
C
vp
C
C
C
vp
C
gz
gz
vp
C
vp
C
C
vp
gz
gz
vp
vp
C
C
vp
gz
gz
vp
vp
+
+
+
+
+
+
+
+
+
+
+
=
+
+
+
+
+
=
éq 5.5.2-12
Code_Aster
®
Version
7.4
Titrate:
Models of behavior THHM
Date:
01/09/05
Author (S):
C. CHAVANT, S. GRANET
Key
:
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:
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HT-66/05/002/A
The derivative partial of the second command are developed in [§Annexe 4].
For the dissolved air, one proceeds with the same stages:
AD
AD
C
=
RT
p
M
AD
ol
AD
.
=
thus
-
=
dT
RT
p
RT
dp
M
cd.
AD
AD
ol
AD
AD
2
.
who gives
-
+
+
=
dT
p
RT
P
T
p
dp
p
p
RT
dp
p
p
RT
M
cd.
lq
AD
AD
gz
gz
AD
C
C
AD
ol
AD
AD
1
1
1
.
2
éq
5.5.2-13
and
-
=
T
RT
p
RT
p
M
C
AD
AD
ol
AD
AD
2
.
and that one differentiates in:
-
-
+
-
=
T
D
RT
p
dT
RT
p
T
RT
p
dp
RT
T
p
D
RT
M
C
D
AD
AD
AD
AD
AD
ol
AD
AD
2
2
3
2
2
1
1
éq 5.5.2-14
with
T
T
p
p
p
p
p
p
p
p
AD
C
C
AD
gz
gz
AD
AD
+
+
=
éq
5.5.2-15
and
AD
p
D
that one differentiates in the following way:
T
D
T
p
p
D
p
p
p
D
p
p
dT
T
T
p
T
p
p
p
T
p
p
p
T
dp
T
T
p
p
p
p
p
p
p
p
p
p
dp
T
T
p
p
p
p
p
p
p
p
p
p
AD
C
C
AD
gz
gz
AD
AD
C
C
AD
gz
gz
AD
gz
AD
gz
C
C
AD
gz
gz
gz
AD
gz
C
AD
C
C
C
AD
C
gz
gz
AD
C
+
+
+
+
+
+
+
+
+
+
+
=
éq
5.5.2-16
The derivative partial of the second command are developed in [§Annexe 4].
Code_Aster
®
Version
7.4
Titrate:
Models of behavior THHM
Date:
01/09/05
Author (S):
C. CHAVANT, S. GRANET
Key
:
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:
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Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
6 Bibliography
[1]
O. COUSSY: “Mechanical of the porous environments”. Editions TECHNIP.
[2]
C. CHAVANT, P. CHARLES, Th. DUFORESTEL, F. VOLDOIRE
: “
Thermo hydro
mechanics of the porous environments unsaturated in Code_Aster “. Note HI-74/99/011/A.
[3]
I. VAUTIER, P. MIALON, E. LORENTZ: “Quasi static nonlinear Algorithm (operator
STAT_NON_LINE
) “. Document Aster [R5.03.01].
[4]
C. CHAVANT: “Modelings THHM general information and algorithms”. Document Aster
[R7.01.10].
[5]
A. GIRAUD: “Adaptation to the nonlinear model poroelastic of Lassabatère-Coussy to
modeling porous environment unsaturated “. (ENSG).
[6]
J. WABINSKI, F. VOLDOIRE: Thermohydromécanique in saturated medium. Note EDF/DER
HI/74/96/010, of September 1996.
[7]
T. LASSABATERE: “Hydraulic Couplings in porous environment unsaturated with
phase shift: application to the withdrawal of desiccation of the concrete “. Thesis ENPC.
[8]
PH. MESTAT, Mr. PRAT: “Works in interaction”. Hermes.
[9]
J.F. THILUS et al.: “Poro-mechanics”, Biot Conference 1998.
[10]
A.W. BISHOP & G.E. BLIGHT: “Nap Aspects off Effective Stress in Saturated and Partly
Saturated Soils “, Geotechnics n° 3 flight. XIII, pp. 177-197. 1963.
Code_Aster
®
Version
7.4
Titrate:
Models of behavior THHM
Date:
01/09/05
Author (S):
C. CHAVANT, S. GRANET
Key
:
R7.01.11-B
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:
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Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
Internal appendix 1 generalized Stresses and variables
Stresses:
Number
Name of component Aster
Contents
1
SIXX
xx
2
SIYY
yy
3
SIZZ
zz
4
SIXY
xy
5
SIXZ
xz
6
SIYZ
yz
7
SIP
p
8
M11
W
m
9
FH11X
X
W
M
10
FH11Y
y
W
M
11
FH11Z
Z
W
M
12
ENT11
m
W
H
13
M12
vp
m
14
FH12X
X
vp
M
15
FH12Y
y
vp
M
16
FH12Z
Z
vp
M
17
ENT12
m
vp
H
18
M21
have
m
19
FH21X
X
have
M
20
FH21Y
y
have
M
21
FH21Z
Z
have
M
22
ENT21
m
have
H
18
M22
AD
m
19
FH22X
X
AD
M
20
FH22Y
y
AD
M
21
FH22Z
Z
AD
M
22
ENT22
m
AD
H
23
QPRIM
'
Q
24
FHTX
X
Q
25
FHTY
y
Q
26
FHTZ
Z
Q
Code_Aster
®
Version
7.4
Titrate:
Models of behavior THHM
Date:
01/09/05
Author (S):
C. CHAVANT, S. GRANET
Key
:
R7.01.11-B
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:
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Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
The variables intern if one takes account of the saturation (of type “..HH.” or THV):
Number
Name component Aster
Contents
1 V1
0
-
2 V2
0
lq
lq
-
3 V3
0
vp
vp
p
p
-
4 V4
lq
S
And in the other cases:
Number
Name component Aster
Contents
1 V1
0
-
2 V2
0
lq
lq
-
3 V3
0
vp
vp
p
p
-
Appendix 2 Data material
One gives here the correspondence between the vocabulary of the Aster controls and the notations used in
present note for the various sizes characteristic of materials.
A2.1 key Word factor
THM_LIQU
RHO
0
lq
UN_SUR_K
lq
K
1
ALPHA
lq
CP
p
lq
C
VISC
()
T
lq
µ
D_VISC_TEMP
()
T
T
lq
µ
A2.2 key Word factor
THM_GAZ
MASS_MOL
ol
have
M
CP
p
have
C
VISC
()
T
have
µ
D_VISC_TEMP
()
T
T
have
µ
Code_Aster
®
Version
7.4
Titrate:
Models of behavior THHM
Date:
01/09/05
Author (S):
C. CHAVANT, S. GRANET
Key
:
R7.01.11-B
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:
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Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
A2.3 key Word factor
THM_VAPE_GAZ
MASS_MOL
ol
VP
M
CP
p
vp
C
VISC
()
T
vp
µ
D_VISC_TEMP
()
T
T
vp
µ
A2.4 key Word factor
THM_AIR_DISS
CP
p
AD
C
COEF_HENRY
H
K
A2.5 key Word factor
THM_INIT
TEMP
T
init
PRE1
1
P
init
PRE2
2
P
init
PORO
0
NEAR
_
VAPE
0
vp
p
One recalls that, according to modeling, the two pressures
1
P
and
2
P
represent:
LIQU_SATU LIQU_VAPE LIQU_GAZ_ATM GAS
LIQU_VAPE_GAZ
1
P
W
p
W
p
W
C
p
p
-
=
gz
p
W
gz
C
p
p
p
-
=
2
P
gz
p
LIQU_GAZ LIQU_AD_GAZ_VAPE
1
P
W
gz
C
p
p
p
-
=
AD
W
gz
C
p
p
p
p
-
-
=
2
P
gz
p
gz
p
A2.6 key Word factor
THM_DIFFU
R_GAZ
R
RHO
0
R
CP
S
C
BIOT_COEF
B
SATU_PRES
()
C
lq
p
S
D_SATU_PRES
()
C
C
lq
p
p
S
PESA_X
m
X
F
PESA_Y
m
y
F
Code_Aster
®
Version
7.4
Titrate:
Models of behavior THHM
Date:
01/09/05
Author (S):
C. CHAVANT, S. GRANET
Key
:
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:
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Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
PESA_Z
m
Z
F
PERM_IN
)
(
int
K
PERM_LIQU
()
lq
rel
lq
S
K
D_PERM_LIQU_SATU
()
lq
lq
rel
lq
S
S
K
PERM_GAZ
(
)
gz
lq
rel
gz
p
S
K
,
D_PERM_SATU_GAZ
(
)
gz
lq
lq
rel
gz
p
S
S
K
,
D_PERM_PRES_GAZ
(
)
gz
lq
gz
rel
gz
p
S
p
K
,
FICKV_T
)
(T
F
T
vp
FICKV_S
)
(S
F
S
vp
FICKV_PG
)
(
G
gz
vp
P
F
FICKV_PV
)
(
vp
vp
vp
P
F
D_FV_T
)
(T
T
F
T
vp
D_FV_PG
)
(
gz
gz
gz
vp
P
P
F
FICKA_T
)
(T
F
T
AD
FICKA_S
)
(S
F
S
AD
FICKA_PA
)
(
AD
AD
AD
P
F
FICKA_PL
)
(
lq
lq
AD
P
F
D_FA_T
)
(T
T
F
T
vp
LAMB_T
)
(T
T
T
DLAMB_T
T
T
T
T
)
(
LAMB_PHI
)
(
T
DLAMBPHI
)
(
T
LAMB_S
)
(S
T
S
DLAMBS
S
S
T
S
)
(
LAMB_CT
T
CT
Code_Aster
®
Version
7.4
Titrate:
Models of behavior THHM
Date:
01/09/05
Author (S):
C. CHAVANT, S. GRANET
Key
:
R7.01.11-B
Page
:
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Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
Note:
For modelings utilizing it thermal, and for the calculation of the specific heat
homogenized, the relation is used:
(
)
(
)
(
)
p
have
have
p
vp
vp
lq
p
lq
lq
lq
S
S
C
C
S
C
S
C
C
+
-
+
+
-
=
1
1
0
.
In this formula, one confuses
S
with its initial value
0
S
whose value is read under the key word
RHO of the key word factor ELAS.
Appendix 3 Derived from the pressures according to the deformations
generalized
One details here the calculation of derived from pressure according to the generalized deformations. It is reminded the meeting that
the equation [éq 3.2.6.3] is
T
dT
L
dp
dp
W
W
vp
vp
+
=
with
m
W
m
vp
H
H
L
-
=
. Moreover
have
H
have
H
W
lq
AD
dp
K
RT
dT
p
K
R
dp
dp
dp
+
=
-
=
and
vp
gz
have
dp
dp
dp
-
=
. By combining these equations one obtains:
(
)
+
-
+
+
-
=
-
+
-
+
+
-
=
-
C
gz
H
AD
H
vp
H
W
vp
W
C
gz
H
AD
W
vp
W
H
vp
dp
dp
K
RT
dT
T
p
K
LR
K
RT
dp
dp
dp
K
RT
T
dT
p
L
K
RT
dp
1
1
1
One can thus write the derivative partial of water and the vapor according to the generalized deformations:
1
1
;
1
1
;
1
-
=
-
-
=
-
+
-
=
H
W
vp
C
W
H
W
vp
H
gz
W
H
W
vp
AD
H
vp
W
K
RT
p
p
K
RT
K
RT
p
p
K
RT
T
p
K
LR
T
p
(
)
vp
W
H
C
vp
vp
W
H
H
gz
vp
vp
W
H
AD
W
vp
K
RT
p
p
K
RT
K
RT
p
p
T
K
RT
p
L
T
p
-
=
-
-
=
-
+
-
=
1
;
1
;
1
.
Relations
vp
gz
have
dp
dp
dp
-
=
and
W
C
gz
AD
dp
dp
dp
dp
-
-
=
allow to derive all the pressures,
since one will have
C
vp
C
have
gz
vp
gz
have
vp
have
p
p
p
p
p
p
p
p
T
p
T
p
-
=
-
=
-
=
;
1
;
and
C
W
C
AD
gz
W
gz
AD
W
AD
p
p
p
p
p
p
p
p
T
p
T
p
-
-
=
-
=
-
=
1
;
1
;
Code_Aster
®
Version
7.4
Titrate:
Models of behavior THHM
Date:
01/09/05
Author (S):
C. CHAVANT, S. GRANET
Key
:
R7.01.11-B
Page
:
47/52
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
Appendix 4 Derived seconds of air and steam pressures
dissolved according to the generalized deformations
One calculates here the derivative partial of the second command of the steam pressure necessary to the section [§5.5.2].
One will note thereafter:
vp
W
H
K
RT
With
-
=
1
and
-
=
1
2
H
vp
W
K
RT
With
W
W
W
vp
vp
T
p
K
T
T
p
p
With
3
1
1
1
3
-
-
-
=
W
W
W
vp
vp
T
p
K
T
T
p
p
With
3
1
1
1
4
-
+
+
-
=
Derived seconds from the steam pressure:
-
=
C
vp
vp
C
W
W
gz
vp
C
p
p
p
p
p
K
With
With
p
p
p
1
1
1
2
2
-
=
gz
vp
vp
gz
W
W
gz
vp
gz
p
p
p
p
p
K
With
With
p
p
p
1
1
1
2
2
-
-
-
=
4
1
1
1
1
2
With
K
R
K
RT
With
With
K
R
p
p
T
vp
W
H
H
H
gz
vp
-
-
=
C
W
W
C
vp
vp
vp
W
C
vp
C
p
p
K
p
p
p
With
p
p
p
1
1
1
1
2
-
-
=
gz
W
W
gz
vp
vp
vp
W
C
vp
gz
p
p
K
p
p
p
With
p
p
p
1
1
1
1
2
-
-
=
4
1
1
2
With
K
R
With
p
p
T
vp
W
H
C
vp
-
-
+
+
-
-
=
C
vp
vp
C
W
W
vp
W
W
AD
W
W
C
W
vp
C
p
p
p
p
p
K
L
p
K
L
p
p
With
With
T
T
p
p
1
1
)
(
)
1
(
1
1
1
1
1
2
-
-
+
+
-
-
=
gz
vp
vp
gz
W
W
vp
W
W
AD
W
W
gz
W
vp
gz
p
p
p
p
p
K
L
p
K
L
p
p
With
With
T
T
p
p
1
1
)
(
)
1
(
1
1
1
1
1
2
)
(
)
4
(
1
.
1
3
1
.
1
2
2
W
AD
vp
W
H
vp
W
H
W
W
W
W
W
AD
vp
L
p
With
K
R
T
K
RT
With
T
T
p
K
L
T
p
With
T
T
p
T
-
-
+
-
-
-
-
=
Code_Aster
®
Version
7.4
Titrate:
Models of behavior THHM
Date:
01/09/05
Author (S):
C. CHAVANT, S. GRANET
Key
:
R7.01.11-B
Page
:
48/52
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
Derived seconds from the pressure of dissolved air:
-
=
C
W
W
C
vp
vp
H
gz
AD
C
p
p
K
p
p
p
With
With
K
RT
p
p
p
1
1
1
2
2
-
=
gz
W
W
gz
vp
vp
H
gz
AD
gz
p
p
K
p
p
p
With
With
K
RT
p
p
p
1
1
1
2
2
(
)
T
With
With
With
K
R
With
K
R
p
p
T
H
vp
W
vp
W
H
gz
AD
.
3
1
1
2
1
1
2
2
+
+
-
=
-
=
C
W
W
C
vp
vp
vp
W
H
C
AD
C
p
p
K
p
p
p
With
K
RT
p
p
p
1
1
1
1
2
-
=
gz
W
W
gz
vp
vp
vp
W
H
C
AD
gz
p
p
K
p
p
p
With
K
RT
p
p
p
1
1
1
2
(
)
T
With
With
K
R
p
p
T
vp
W
H
C
AD
.
3
1
1
1
2
+
=
3
.
1
1
1
1
1
2
With
T
p
K
LR
With
K
RT
p
p
T
p
p
p
K
LR
With
T
p
p
AD
vp
H
vp
W
H
C
AD
C
vp
vp
vp
H
vp
W
AD
C
-
-
-
=
3
.
1
1
1
1
1
2
With
T
p
K
LR
With
K
RT
p
p
T
p
p
p
K
LR
With
T
p
p
AD
vp
H
vp
W
H
gz
AD
gz
vp
vp
vp
H
vp
W
AD
gz
-
-
-
=
+
-
-
-
+
-
=
T
With
T
p
K
LR
With
K
RT
T
p
T
T
p
T
T
p
p
K
LR
With
T
p
T
AD
vp
H
vp
W
H
AD
AD
vp
vp
H
vp
vp
W
AD
1
3
.
1
1
1
1
1
1
1
2
2
Code_Aster
®
Version
7.4
Titrate:
Models of behavior THHM
Date:
01/09/05
Author (S):
C. CHAVANT, S. GRANET
Key
:
R7.01.11-B
Page
:
49/52
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
Appendix 5 Equivalence with formulations ANDRA
In order to be able to fit in the platform ALLIANCE, it is necessary to be coherent with the formulations
posed by the ANDRA in the document [bib11]. We propose here an equivalence between the notations which
would be dissimilar. These differences relate to only the writing of:
·
The equation of energy
·
The law of Henry
·
Dissemination in the fluid
·
Dissemination in gas
Notice concerning the enthali:
It is required to have coherence between the two models that
the user of Aster returns:
0
0
=
m
lq
H
and
0
0
L
H
m
vp
=
,
A5.1 Equation of energy
The table above points out the two formulations:
Notations Aster
Notations ANDRA
m
lq
H
N
S
W
W
W
=
m
vp
H
(
)
N
S
W
v
v
-
=
1
m
have
H
(
)
N
S
W
have
have
-
=
1
lq
M
W
W
F
=
have
M
have
have
F
=
vp
M
v
v
F
=
By rewriting the equation of energy of Aster with these notations, one finds:
(
)
(
)
()
(
)
(
)
[
]
(
)
(
)
+
+
+
=
-
+
-
+
+
-
-
-
+
-
+
-
+
+
G
Q
have
have
v
v
W
W
gz
m
gz
m
lq
C
m
lq
v
S
S
W
v
v
W
have
have
W
W
W
F
F
F
dt
dT
TK
dt
dp
T
dt
dp
T
dt
D
T
K
C
N
dt
D
T
T
Div
N
S
F
N
S
F
N
S
F
Div
dt
D
2
0
0
0
0
0
9
3
3
3
1
1
1
The first line being that of the ANDRA and others being a priori negligible.
Code_Aster
®
Version
7.4
Titrate:
Models of behavior THHM
Date:
01/09/05
Author (S):
C. CHAVANT, S. GRANET
Key
:
R7.01.11-B
Page
:
50/52
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
A5.2 Law of Henry
In the formulation of the ANDRA, the formulation of Henry is given by
W
W
ol
have
have
has
L
M
M
H
P
=
with
concentration of air in water that have it can bring back to a density such as
AD
has
L
=
. H
express yourself out of AP.
In the ASTER formulation, one recalls that the law of Henry is expressed in the form:
ol
AD
AD
ol
AD
M
C
=
with
H
have
ol
AD
K
p
C
=
.
H
K
express yourself in Pa.m
3
.mol
- 1
.
There is thus equivalence:
W
W
H
M
H
K
=
A5.3 Dissemination of the vapor in the air
In formulation ANDRA the water vapor flow in the air according to the concentration of
water vapor d' in the air or the relative humidity is noted:
E
G
v
v
Diff
D
F
-
=
.
_
with the concentration defined as the molar report/ratio in gas:
G
E
G
N
N
=
.
In Aster, this same flow is written
:
vp
vp
v
Diff
C
F
F
=
_
with the coefficient of Fick vapor
)
1
(
vp
vp
vp
vp
C
C
D
F
-
=
and D
vp
the coefficient of dissemination of Fick of the gas mixture. C
vp
is defined like
the report/ratio of the pressures such as:
gz
vp
vp
p
p
C
=
.
The law of perfect gases makes it possible to write that
E
G
vp
C
=
thus
vp
E
G
C
=
and
vp
v
v
Diff
C
D
F
=
.
_
.
Thus equivalence ASTER/ANDRA is written simply:
v
vp
D
F
=
.
A5.4 Dissemination of the air dissolved in water
In formulation ANDRA the flow of air dissolved in water is expressed
has
L
has
E
ds
has
D
F
=
.
_
_
with
ol
AD
AD
has
L
M
=
.
Code_Aster
®
Version
7.4
Titrate:
Models of behavior THHM
Date:
01/09/05
Author (S):
C. CHAVANT, S. GRANET
Key
:
R7.01.11-B
Page
:
51/52
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
In Aster, this same flow is written:
AD
AD
v
ds
has
C
F
F
=
_
_
with the coefficient of air-dissolved Fick
)
1
(
AD
AD
AD
AD
C
C
D
F
-
=
and D
AD
the coefficient of dissemination of Fick of the liquid mixture. C
AD
is definite such
that:
has
L
AD
W
C
=
. Thus:
has
AD
D
F
=
.
Code_Aster
®
Version
7.4
Titrate:
Models of behavior THHM
Date:
01/09/05
Author (S):
C. CHAVANT, S. GRANET
Key
:
R7.01.11-B
Page
:
52/52
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
Intentionally white left page.