hydro-mechanical element coupled with xfem

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Titre : Élément hydro-mécanique couplé avec XFEM Date : 05/12/2017 Page : 1/32Responsable : COLOMBO Daniele Clé : R7.02.18 Révision :

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Élément Hydraulic coupled with XFEM

Summary:

In this document we present the formulation of a new type of finite elements. It is about a hydraulic elementcrossed by a discontinuity (an interface or a crack) whose design rests on the use of the finite element methodextended in the formulation of the discretized equations of the model poro-mechanics for a porous environmentsaturated with fluid.

Warning : The translation process used on this website is a "Machine Translation". It may be imprecise and inaccurate in whole or in part and isprovided as a convenience.Copyright 2021 EDF R&D - Licensed under the terms of the GNU FDL (http://www.gnu.org/copyleft/fdl.html)



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Contents1 Introduction........................................................................................................................................... 4

2 Presentation of the problem.................................................................................................................5

2.1 Definition of the field of study........................................................................................................5

2.2 Assumptions and notations............................................................................................................5

3 Équations constitutive of model HM-XFEM..........................................................................................6

3.1 Équations for mechanics................................................................................................................6

3.1.1 Équation of balance (case of the solid mass).......................................................................6

3.1.2 Model of cohesive zones (case of the interface)..................................................................7

3.1.3 Boundary conditions for mechanics......................................................................................8

3.2 É quations for the hydrodynamics..................................................................................................8

3.2.1 Équation of conservation of the mass (case of the solid mass)...........................................8

3.2.2 Équation of conservation of the mass (case of the interface)..............................................8

3.2.3 Boundary conditions for the hydrodynamics.........................................................................9

3.3 Équations of the model poro-mechanics........................................................................................9

3.3.1 Expression of the mass contributions...................................................................................9

3.3.2 Expression of mass flows.....................................................................................................9

3.3.3 Evolution of the variable of porosity...................................................................................10

3.3.4 Evolution of the density of the fluid....................................................................................10

3.3.5 Derived from the mass contributions..................................................................................10

3.3.6 Derived from mass flows....................................................................................................11

4 Variational formulation........................................................................................................................ 11

4.1 Weak formulation of the mechanical problem.............................................................................12

4.2 Weak formulations of the hydrodynamic problem.......................................................................13

4.2.1 Weak formulation for the solid mass..................................................................................13

4.2.2 Weak formulation for the interface.....................................................................................13

5 Discretization of the problem..............................................................................................................13

5.1 Temporal discretization................................................................................................................13

5.1.1 Discretization of the mechanical equation..........................................................................14

5.1.2 Discretization of the equations of the hydrodynamics........................................................14

5.1.2.1 Case of the solid mass...........................................................................................14

5.1.2.2 Case of the interface..............................................................................................14

5.1.3 Discretization of the equations of the model poro-mechanics............................................14

5.1.3.1 Case of the mass contributions..............................................................................14

5.2 Discretization with XFEM.............................................................................................................15

5.2.1 Representation of associated element HM-XFEM and ddls...............................................15

5.2.2 Space reduced for the discretization of the fields associated with discontinuity................16

5.2.3 Approximations of the fields with XFEM.............................................................................17

5.3 Extension to the multi-fissured case............................................................................................18




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5.3.1 Representation of the element multi-fissured associated HM-XFEM and ddls...................18

5.3.2 Approximations of the fields with XFEM in the multi-fissured case....................................19

5.3.3 Junction of hydraulic fractures............................................................................................19

5.4 Imposition of a flow in an interface hydraulics.............................................................................20

6 Resolution of the coupled problem.....................................................................................................22

6.1 Linearization of the coupled problem...........................................................................................22

6.1.1 Linearization by the method of Newton-Raphson...............................................................22

6.1.2 É integral criture of the linearized problem.........................................................................23

6.2 Writing of the elementary terms with XFEM................................................................................25

6.2.1 Writing in matric form of the coupled problem...................................................................25

27

6.2.2 Form of the elementary matrices for mechanics................................................................27

6.2.3 Expression of the second members for mechanics............................................................28

6.2.4 Form of the elementary matrices for the hydrodynamics...................................................29



6.2.4.3 Continuity of the pressure.......................................................................................30

6.2.5 Expression of the second elementary members for the hydrodynamics............................30



6.2.5.3 Continuity of the pressure.......................................................................................31

7 Bibliography........................................................................................................................................ 31

8 Description of the versions of the document......................................................................................32




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1 Introduction

The formulation of the model coupled HM-XFEM is based on the equations of the model of joint[R7.02.15] like on those of the model of behavior THHM [R7.01.11] in the saturated case. Classically,the model of joint is used to model in 2D the behavior of a water seal or a discontinuity in thepresence of a flow of fluid in her centre, thus generating a pressure of fluid. The model of joint makesit possible to take into account:

• the preferential flow of the fluid in discontinuity conditioned by the opening of this one,• the exchange of fluid enters the porous environment and discontinuity the beam,• propagation of discontinuity,• deformation of the porous environment induced by the pressure of fluid.

Within the framework of the classical finite element method, the use of such a model presents a majordrawback. Indeed it is necessary to explicitly represent discontinuity in the grid and to make agree thelips of this one with the edges of the elements constituting the grid. That implies that for its evolution, itis necessary to resort to algorithms of projection worked out for the actualization of its geometry. Thisstage can prove very expensive in computing times for complex geometries. In order to free itself from this constraint related to the grid, we plan the introduction of a new hydraulicelement (HM) coupled with the wide finite element method (XFEM) [1,2,3]. This method, based on theprinciple of partition of the unit [4], ensures a greater flexibility for modelings utilizing more or lesscomplex geometries. Indeed with this method, discontinuity is not represented any more physically inthe grid but in manner symbolic system while enriching with additional degrees of freedom, theapproximated fields. For more details on the finite element method extended in Code_Aster, the usercan refer to documentation [R7.02.12] (mechanical case only).

In the literature certain authors already considered the coupling of method XFEM with the model poro-mechanics coupled HM. It is the case of [5] in the case of the dynamic analysis of the porousenvironments in unsaturated conditions. The extension to case THM for a saturated medium crossedby an impermeable interface is considered in [6]. Other authors as [7] proposed a model which cantake into account the singularity of the field of pressure at a peak of discontinuity, by adapting theexpression of the singular functions for the bottom of discontinuity. However, the models developed bythese authors take into account neither the phenomena of exchange which can exist between thesurrounding medium (named massive in the continuation of this document) and discontinuity, nor thepropagation of the latter within the porous environment. This point will be taken into account in themodel developed in present documentation by the introduction of the cohesive laws regularized intothe weak formulation of the mechanical equilibrium equation (see § 3.1.2).

From a point of view practises, the main difficulty in the formulation of element HM-XFEM is theconstruction of various spaces of approximation of the mechanical magnitudes and hydraulic. Therespect of the stability condition LBB is essential in order to obtain a single and convergent solution[8.9]. The violation of this condition (i.e. of not choosing the good degree of interpolation of the variousfields approximated to see § 5.2.1) involve oscillations of the solution. The cohabitation of elementsHM-XFEM and the classical elements HM (those whose documentations refer [R7.01.11] and[R7.01.10]) is also a delicate point, in particular with regard to the distribution of the degrees offreedom to the various nodes (tops or mediums) of each type of element (see § 5.2.1).

In the continuation of this documentation we will recall the framework of study of the problem, then inthe second time the fundamental equations of the model poro-mechanics implied in the formulation ofmodel HM-XFEM. Finally we will proceed to the discretization of the variational forms of theequilibrium equations, at the same time in time (thanks to one θ - diagram) and spaces some (thanksto method XFEM).

The presentation of model HM-XFEM and its validation were the object of a scientific publication [15].




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2 Presentation of the problem

2.1 Definition of the field of study

That is to say Ω⊂ℝd with d∈{2, 3} a field entirely crossed by a crack or a permeable interface.

That is to say n the normal external with the border ∂Ω field, and nc that of the interface Γc .

It is possible to break up:• the border of the field Ω in ∂Ω=Γu∪Γ t∪Γp∪ΓF where the boundary conditions are

imposed (of type Dirichlet and Neumann) for the hydrodynamics (on Γp and ΓF ) and formechanics (on Γu and Γt ),

• the interface in Γc=Γ f∪Γ1∪Γ2 where Γ1 and Γ2 the lips of discontinuity represent. Oneimpose conditions of flow on Γ1 , Γ2 and Γ f and of the cohesive surface efforts on Γ1 andΓ2 .

Figure 2.1-1 give a notation symbolic of the conditions imposed on the border of the field.

Figure 2.1-1: Imposition of the boundary conditions

2.2 Assumptions and notations

One considers a porous environment saturated with liquid (in general of water). The rate mixingassociated in Code_Aster is thus LIQU_SATU (for more details to refer to the note of model THM[U2.04.05]). In addition there exists a longitudinal flow of fluid on the level of the interface. Bysupposing that this one is permeable, the exchanges of fluid take place between the solid mass (leftΩ who is not the interface) and interfaces it. They are represented on Figure 2.2-1.

Figure 2.2-1: Orientation of exchanges

between the solid mass and discontinuity q1 and q2 are flows due to the exchanges between the solid mass and the interface, and are

expressed in kg.m-2 . s-1 . These flows come from the interface and are directed respectively towards

the parts higher and lower of the solid mass than the level of the lips Γ1 and Γ2 interface. They aredirected discontinuity towards the solid mass.




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The field of pressure on the level of the solid mass is noted p and that on the level of the interface is

noted pf (field induced by the fluid circulating on the level of the interface). The field of

displacements is noted u and the jump of displacement on the level of the interface is noted ⟦u⟧ .

That is to say P1 a point of Γ1 and P2 a point of Γ2 and nc = nc1 the normal external with Γ1

and nc2 the normal external with Γ2 . The jump of normal displacement (taken negative or null in

opening and positive in interpenetration of the lips) is thus defined in the following way:

⟦u⟧ .nc=(u(P1)−u (P2)). nc⩽0

On Figure 2.2-2, one indicates the conventions adopted for the taking into account of the jump ofdisplacement to the level of the interface.

Figure 2.2-2: Definition of the jump ofdisplacement on the level of discontinuity

For the solid mass, as for the interface, the assumption of the effective constraints is taken intoaccount. As follows:

• the total constraint in the solid mass is noted σ , • the total constraint (related to the cohesive efforts) on the level of the interface is noted t c .

The assumption of the small disturbances is allowed. In addition it is considered that the sizesmechanics and hydraulics are isotropic.

3 Équations constitutive of model HM-XFEM

3.1 Équations for mechanics

3.1.1 Équation of balance (case of the solid mass)

In the case of the solid mass the conservation equation of the momentum (by taking account of thevoluminal efforts) can be put in the form:

Div (σ )+r Fm=0

with:• σ=σ

'−bp1 (under the assumption of the effective constraints) where b is the coefficient of

Biot,• r density homogenized such as in the saturated case r=r 0+mw where r0 is the density

homogenized in the configuration of reference and mw mass water contributions,

• r Fm represent the voluminal efforts acting on Ω (in practice it is the forces of gravity).




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3.1.2 Model of cohesive zones (case of the interface)

In order to take into account the propagation of discontinuity, the irreversibility of the fracturing and thenot-interpenetration of the lips of discontinuity, one chooses to model the behavior of the interface orthe crack using a cohesive law. For more information concerning the establishment of these laws, the user can refer to documentation[R7.02.11]. In what concerns us, on the level of discontinuity, it is possible to distinguish 3 zones:

• a completely open zone, on the level of which the total constraint on the lips of discontinuity isequal to pf nc on Γ2 and equalizes with −p f nc on Γ1 . In this zone the value of theconstraint is due mainly to the there circulating fluid,

• a cohesive zone (or Processing Fractures Zones (FPZ)) whose opening depends on the value of

the total constraint which is then equal to t c=t c'− p f n , n being the normal external to

surface concerned Γ1 or Γ2 . Beyond a certain opening w c , the constraint corresponds tothat of the preceding zone,

• and an adherent healthy zone on the level of which the lips of discontinuity are in contact anddo not interpenetrate.

On Figure 3.1.2-1 one locates the various zones of constraints associated with the model of cohesivezones. The face of crack is then naturally localised at the border between the cohesive zone and theadherent zone.

The law of behavior used for the cohesive laws takes the shape of a lenitive relation between thecohesive constraint t c ' and the jump of displacement ⟦u⟧ on the level as of lips of discontinuity. Thus one poses:

t c ' =∂ ψ

∂⟦u⟧

with ψ the energy of surface, whose expression depends on the cohesive law used.

The cohesive law adopted for hydraulic elements XFEM is the law CZM_LIN_MIX detailed in[R7.02.19]. It is about a not regularized cohesive law (thus with an infinite initial slope). σ c is the

critical stress from which the damage starts in the cohesive zone. Gc=σc w c

2 corresponds to the

quantity of energy required to break a unit of area completely.••


Figure 3.1.2-1: Distribution of the constraints on the level of discontinuity



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Figure 3.1.2-2 : Mixed cohesive law.

3.1.3 Boundary conditions for mechanics

The writing of the boundary conditions for mechanics on the border of the field Ω and on the level ofthe interface are written:

Boundary conditions for mechanics

• u = 0 on Γu (imposed displacements) • σ⋅n = t on Γt (imposed surface efforts)

• σ⋅nc1= t c

1 on Γ1 (imposed cohesive surface efforts)

• σ⋅nc2= t c

2 on Γ2 (efforts surface cohesive imposed)

3.2 É quations for the hydrodynamics

3.2.1 Équation of conservation of the mass (case of the solid mass)

In the case of the solid mass the conservation equation of the mass is put in the form:

∂mw

∂ t + Div (M )=0

with:• mw mass contributions (by unit of volume) expressed S in kg.m-3 ,

• M expressed mass flows in kg.m-2 . s-1 .

3.2.2 Équation of conservation of the mass (case of the interface)

In the case of the interface the conservation equation of the mass is put in the form:

∂w∂ t

+ Div (W )=0

with:• w mass contributions (by unit of area) in the crack expressed in kg.m-2 ,


σc

w c

⟦u⟧n

Mode I

σc

wc

−wc

tc ,τ

⟦u⟧τ

Mode II or mode III

−σc

αwc

w c:=2Gcσc

tc ,n



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• W mass flows S in the crack expressed in kg.m-1 . s-1 ,

3.2.3 Boundary conditions for the hydrodynamics

The writing of the boundary conditions for the hydrodynamics on the border of the field Ω and thelevel of the interface allows to obtain:

Boundary conditions for the hydrodynamics

• p = 0 on Γp (pressure imposed)

• M⋅n = M ext on ΓF (for the solid mass)

• W⋅nc = W ext on Γ f (for discontinuity)

• M⋅nc1= q1

on Γ1 (equality of massive flow/discontinuity)

• M⋅nc2= q2

on Γ2 (equality of massive flow/discontinuity)

An additional boundary condition to take into account is the continuity of the pressure pf on the levelof each lip of the interface. This condition is necessary because of very low thickness of the interface.It results in the following linear relation:

• pinf= p f on Γ1

• psup= p f on Γ2

with:• psup pressure of fluid above the interface (field enriched by XFEM),

• pinf pressure of fluid below the interface (field enriched by XFEM).

3.3 Équations of the model poro-mechanics

In this part one does nothing but point out the useful equations of the model poro-mechanicsdeveloped in documentation [R7.01.11].

3.3.1 Expression of the mass contributions

• Case of the solid mass The mass contributions can be put in the form (with S lq=1 ):

mw=ρϕ(1+εv) with:

• ρ ,ϕ respectively density of water (in kg.m-3 ) , the variable of porosity eulérienne,

• εv=Tr (ε )=Tr (∇ su) voluminal deformation ( where Tr is the linear application traces).

• Case of the interfaceThe mass contributions can be put in the form:

w=ρ⟦u⟧. nc with:

• ρ respectively density of water, • ⟦u⟧ .nc the normal opening of discontinuity (in m ) .

3.3.2 Expression of mass flows

• Case of the solid mass Mass flow in the solid mass follows the law of Darcy. Thus one poses:




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M=λ ρ(−∇ p+ρFm)

with λ =K int

μ L mobility has liquid where K int is the intrinsic permeability of the solid mass (in m2 )

and μ the dynamic viscosity of the fluid (in Pa.s ).

Note:

expression of mobility utilizes actually the relative permeability of the fluid kwrel

(Sw) (which is

a function of water saturation and is given by the law of Mualem/Van Genuchten [1 0 ] ) i.e.

λ =K int k w

rel

μ . É as well given as the medium is saturated ( Sw=1 ) , the permeability

relating to water is thus equal to 1 .

• Case of the interfaceMass flow in discontinuity can be written according to the cubic law [11] (one neglects the effects ofgravity). Thus one poses:

W=−ρ(⟦u⟧. nc )

3

12μ∇ p f

with μ the dynamic viscosity of the fluid (in Pa.s ) .

3.3.3 Evolution of the variable of porosity

The evolution of the variable of porosity (eulérienne) characterizing the solid mass is given in theisothermal case saturated by:

d ϕ=(b−ϕ)(d εv +dpK s

)

with K s the module of compressibility of the solid matrix (in Pa ) and b the coefficient of Biot.

3.3.4 Evolution of the density of the fluid

• Case of the solid massThe evolution of the density of the fluid in the solid mass is given in the isothermal casesaturated by:

dρ

ρ=

dpKw

with Kw the coefficient of compressibility of the liquid (in Pa ).

• Case of the interfaceThe evolution of the density of the fluid on the level of the interface is given in the isothermalcase saturated by:

dρ

ρ=

dp f

Kw

3.3.5 Derived from the mass contributions




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Éas well given as differential mass contributions intervenes in the expression of the tangent operatorassociated with the system linearized (see § 6 ), its expression is pointed out here in the isothermalsaturated case.

• Case of the solid massFirst of all differential of the contributions mass can be written:

dmw=ρϕ d εv + ρ(1+εv )d ϕ + ϕ(1+εv )dρ Taking into account the assumption of the small disturbances, by preoccupation with asimplification, one supposes that (1+εv)≈1 . With final (while replacing d ϕ and dρ bytheir expressions):

d mw = ρb dεv + (( ρ(b−ϕ)

K s

+ρ ϕ

Kw))dp

• Case of the interfaceDifferential of the contributions mass is written:

dw = ρ d (⟦u⟧⋅nc ) + (⟦u⟧⋅nc)ρd p f

K w

3.3.6 Derived from mass flows

Since differential flows mass intervenes in the expression of the tangent operator associated with thesystem linearized (see § 6 ) its expression is pointed out here in the isothermal saturated case.

• Case of the solid massThe differential of mass flows in the case of the solid mass is written:

d M=( Mρ + ρλ Fm)ρ d p

Kw

+Mλ d λ − ρλ d ( ∇ p )

• Case of the interfaceThe differential of mass flows in the case of the interface is written:

d W = (Wρ )ρ d pKw

−ρ

12μ∇ p f (3 (⟦u⟧⋅nc )

2 )d (⟦u⟧⋅nc ) −ρ(⟦u⟧⋅nc)

3

12μd ( ∇ pf )

4 Variational formulation

Before giving the expression of the variational formulations of the equilibrium equations presentedabove, we will give the definition of spaces of approximation of the fields of displacements, ofpressures (of the solid mass and the interface), of flows q1 and q2 and of the multipliers of Lagrange

λ ,μ and of the jump of displacement w useful for the model of cohesive zone:

• the space of the fields of displacements kinematically acceptable on the border of the field Ωis such as:

U0={u*∈H1(Ω)/u* discontinu à traversΓc , u*

=0 sur Γu}




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• the space of the acceptable fields of pressure on the border of the field Ω is such as:

P0={p*∈H 1(Ω)/ p* discontinue à travers Γc , p*

=0 sur Γp}

• the space of the unknown factors q1 and q2 is such as:

Q1={q1*∈H−1 /2

(Γc)/q1*∈Γ1} and Q2={q2

*∈H−1 /2

(Γc)/q2*∈Γ2}

• the space of the unknown factors of pressure pf is such as:

F0={p f*∈H−1 /2

(Γc)/ pf* continue sur Γc}

•

• the space of the unknown factors of the multipliers of Lagrange λ ,μ and of the jump ofdisplacement w is such as:

L0={λ*∈H−1 /2

(Γc)/ λ* continue sur Γc}

4.1 Weak formulation of the mechanical problem

As explained within [R7.02.19], the framework of the formulation of type “mortar” for the model ofcohesive zone, the jump of displacement w is introduced like a new unknown factor of the problem,which will not be discretized like ⟦u⟧ but will be a projection on a reduced space M h (see § 5.2.2).The total energy of the problem is written then:

E(u , λ ,w )=12∫Ω

ϵ(u ):C :ϵ(u)dΩ−∫Γt

t⋅udΓt+∫Γ c

Π (w , λ )dΓc

Π(w ,λ ) is the density of energy of surface and t surface efforts imposed on Γt . The multiplier

of Lagrange λ will be discretized on same space as w (confer [R7.02.19]).

The solution of the continuous problem consists of a minimization under constraints of equality

(u ,w ,λ)=argminw*=⟦u*⟧

E (u* ,λ* ,w*) . We can write the Lagrangian associated one like:

L(u ,w ,λ ,μ )=12∫Ω

ϵ(u) :C :ϵ(u )dΩ−∫Γ t

t⋅ud Γt+∫Γc

Π(w ,λ )dΓc+∫Γc

μ⋅(⟦u⟧−w )d Γc

The multiplier of Lagrange μ will be also discretized on reduced space M h .The writing of theconditions of optimality of this Lagrangian led to the following variational formulation:

Equilibrium equation ∀u*∈U 0,∫Ωσ(u ) :ϵ(u*

)dΩ−∫Γt

t⋅u*d Γ t+∫Γc

μ⋅⟦u*⟧d Γc=0

Projection of the jump of displacement

∀μ*∈L0 ,∫Γc

(⟦u⟧−w )⋅μ*d Γc=0

Expression of the cohesive force

∀ w*∈L0,−∫Γc

[μ−t c (λ+rw )]⋅w*dΓc=0

Law of interface∀ λ

*∈L0,−∫

Γc

[λ −t c (λ+rw )]r

⋅λ*d Γc=0

r ESt the parameter of increase (confer [R7.02.19]). Let us recall that:




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• σ=σ'−bp1

• t c=t c'− p f n

4.2 Weak formulations of the hydrodynamic problem

4.2.1 Weak formulation for the solid mass

The weak formulation of the conservation equation of the mass in the case of the solid mass is written:

−∫Ω

∂mw

∂ tp* dΩ +∫

ΩM⋅∇ p*d Ω

=∫ΓF

M ext p* d ΓF −∫Γ 1

q1 p*d Γ1 −∫Γ2

q2 p* d Γ2

∀ p*∈P0

with M ext the normal flows imposed on the part ΓF of ∂Ω .

4.2.2 Weak formulation for the interface

The weak formulation of the conservation equation of the mass in the case of the interface is written:

−∫Γc

∂ w∂ t

p f* d Γc +∫Γc

W⋅∇ p f*d Γc

=∫Γf

W ext p f* d Γf +∫

Γ1

q1 p f* d Γ1 +∫

Γ2

q2 p f*d Γ2

∀ p f*∈F0

with W ext the normal flows imposed on the part Γ f of Γc .

The weak formulation of the condition of continuity of the pressure pf on the level of the interface iswritten:

∫Γ1

( psup− pf ) q1* d Γ1=0 ∀ q1

*∈Q1

∫Γ2

( pinf− p f ) q2

* d Γ2=0 ∀ q2*∈Q 2

Note:

The condition of continuity of the pressure pf to the level of each lip of the interface brings

concerned two linear relations of the type psup−p f = 0 on Γ1 and pinf

− p f = 0 on Γ2 .

In Code_Aster, in order to manage this kind of relation (which is in fact a boundary conditionand not an equilibrium equation, the such conservation equation of the mass), we resort tothe introduction of fields of multipliers of Lagrange. In fact, the multipliers (which one namesin the continuation hydraulic multipliers of Lagrange) concerned in these two variational

formulations are in fact virtual flows q1* and q2

* .

5 Discretization of the problem

5.1 Temporal discretization

The conservation equations of the mass in the case of the solid mass and the interface utilizeexplicitly in their formulations the temporal variable t . In order to discretize these equations, we useone θ - diagram.




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In term of notation a subscripted variable by one + is a variable taken at the end of the step of timeand a subscripted variable by one - is a variable taken at the beginning of step of time which is a

priori known. One poses Δ t =t+−t - .

5.1.1 Discretization of the mechanical equation

The discretization in time of the equations for mechanics does not utilize of θ - diagram. They areexpressed at the moment + (i.e. after the phase of prediction) and are written:

Equilibrium equation ∀u*∈U 0,∫Ωσ+(u) :ϵ(u*

)dΩ−∫Γt

t+⋅u*d Γt+∫Γ c

μ+⋅⟦u*

⟧d Γc=0


∀μ*∈L0 ,∫Γc

(⟦u+⟧−w+)⋅μ*dΓc=0

Expression of the cohesive force

∀ w*∈L0,−∫Γc

[μ +−t c

+(λ

++rw+)]⋅w*d Γc=0

Law of interface∀ λ

*∈L0,−∫Γc

[ λ+−t c+(λ++r w+)]r

⋅λ*dΓc=0

5.1.2 Discretization of the equations of the hydrodynamics

5.1.2.1 Case of the solid mass

Discretization in time of the conservation equation of the mass using one θ - diagram is written:

−∫Ω

mw+−mw

-

Δ tp* dΩ + θ∫

ΩM+

⋅∇ p* dΩ + (1−θ )∫ΩM -

⋅∇ p* dΩ

= θ∫ΓF

M ext+ p*d ΓF + (1−θ )∫

ΓF

M ext- p* d ΓF − θ∫

Γ1

q1+ p*d Γ1

− (1−θ )∫Γ1

q1- p* d Γ1 − θ∫

Γ2

q2+ p* dΓ2 − (1−θ)∫

Γ2

q2- p* d Γ2

∀ p*∈P0

5.1.2.2 Case of the interface

Discretization in time of the conservation equation of the mass using one θ - diagram is written:

−∫Γ c

w+−w -

Δ tp f

* d Γc + θ∫Γ c

W +⋅∇ p f

* d Γc + (1−θ)∫Γc

W -⋅∇ p f

* dΓc

= θ∫Γf

W ext+ pf

* d Γf + (1−θ )∫Γ f

W ext- p f

* d Γ f + θ∫Γ1

q1+ p f

* d Γ1

+ (1−θ )∫Γ 1

q1- p f

*d Γ1 + θ∫Γ2

q2+ p f

*d Γ2

+ (1−θ)∫Γ2

q2- pf

* d Γ2

∀ p f*∈F0

5.1.3 Discretization of the equations of the model poro-mechanics

The equations presented in this part correspond to the equations of the § 3.3 expressed in anincremental way. These equations are developed because they are affected by the discretization withXFEM.

5.1.3.1 Case of the mass contributions




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• Case of the solid mass The mass contributions are written then in an incremental way:

mw+−mw

-= ρ+

ϕ+ (1+εv

+ ) − ρ-ϕ

- (1+εv- )

• Case of the interfaceThe mass contributions are written then in an incremental way:

w+−w-

= ρ+(⟦u⟧⋅nc)

+− ρ-

(⟦u⟧⋅nc) -

5.2 Discretization with XFEM

5.2.1 Representation of associated element HM-XFEM and ddls

To represent element HM-XFEM, we chose to use quadratic elements which can be either quadrangleswith 8 nodes (QUAD8) or triangles with 6 nodes (TRIA6), or hexahedrons with 20 nodes (HEXA20), orpentahedrons with 15 nodes (PENTA15), or pyramids with 13 nodes (PYRA13), or tetrahedrons with10 nodes (TETRA10). We consider that any element of the grid crossed by the interface is of type HM-XFEM. This element bathes around elements HM not nouveau riches. They are those used classicallyfor modelings HM.

The Figure 5.2.1-1 represent element HM-XFEM with a mesh QUAD8.

Figure 5.2.1-1 : Representation of element HM-XFEM with QUAD8

On each figure presented previously, the lists in each node of the element contain the degrees offreedom (ddls) associated with each category of nodes (nodes tops or nodes mediums).

Degrees of freedom: • a and b are respectively associated with the part classical and enriched by the approximation

of the field of displacements uh ,

• c and d are respectively associated with the part classical and enriched by the approximation

of the field of pressure in the solid mass ph ,

• pf are associated with the approximation of the field of pressure pfh on the level of the

interface,• q1 and q2 are associated with the approximation of the fields of hydraulic multipliers of

Lagrange q1h and q2

h .

• λ , μ and w are respectively associated with the approximation of the multipliers of

Lagrange λh and μ

h and of the jump of displacement wh .

Hydraulic elements present in Code_Aster use a “mixed interpolation”, in order to reduce theoscillations of the digital solution [14]. The field of displacements is thus interpolated in a quadraticway while the field of pressure of pore is interpolated in a linear way. Degrees of freedom c and dare thus carried only by the nodes tops while the degrees of freedom a and b are carried at the




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same time by the nodes mediums and the nodes tops. With regard to the fields associated

with discontinuities: pfh , q1

h , q2h , λ

h , μh and wh , degrees of freedoms are carried only by the

nodes tops. In addition, their space of approximation is reduced (see § 5.2.2) in order to observe thestability condition LBB [8.9]. This condition indeed imposes a hierarchy between spaces ofapproximation, without what one observes the appearance of oscillations and it not unicity of thesolution of the coupled problem.

In addition it is important to note that the elements HM-XFEM which share nodes tops with classicalelements HM, must undergo an additional treatment. It is indeed necessary to put at zero degrees offreedom nouveau riches (for the fields of pressure in the solid mass and displacements) but also to putat zero degrees of freedom pf , q1 , q2 , λ , μ and w on the level of the nodes joint topsbetween these two types of element. The procedure of elimination used is that described in the § 4.4of documentation [R5.03.54].

On Figure 5.2.1-1 one summarizes the problematic points associated with the cohabitation betweenthe elements HM-XFEM and elements HM classical.

Figure 5.2.1-1: Cohabitation of element HM-XFEM withclassical element HM

5.2.2 Space reduced for the discretization of the fields associated with discontinuity

For a detailed description of the discretization of the unknown factors of contact (and a fortiori of allthe unknown factors associated with discontinuity), the reader can refer to documentation [R5.03.54],§5. In short, them initial components multiplier are defined on Nœuds tops K elements parentsintersected (see fig. 5.2.2-1 ). Implementation such elements of contact is detailed in [R5.03.54], §4.One imposes then relations of equality between some of these initial components in order to lead to alower number N

λ of degrees of freedom indeed independent. These relations are carried by certain

intersected edges V , known as vital edges: a degree of freedom I really independent is divided by

a group of Nœuds of K (see fig. 5.2.2-1 ), which produces a function of wide form of contact

ψI :=∑i∈ I

N i (see fig. 5.2.2-1 ). The algorithm of selection of such vital edges, and thus of

construction of reduced space, is detailed in documentation [R5.03.54], §6. The field of multipliers isthen obtained by interpolation on the elements parents and the discrete multiplier is the trace of thisfield on the interface:

Mh :={∑I μ Iψ I |Γ ,μ I∈ℝd

}




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Figure 5.2.2-1 : Grid nonin conformity with the crack and space reduced for the interface

In our case, the fields associated with discontinuity are not only the multipliers of Lagrange λ ,μ andthe jump of displacement w useful for the cohesive law CZM_LIN_MIX but also hydraulic fields

associated with discontinuity: pressure of fluid in discontinuity pf , and flows q1 and q2 . These

fields all are discretized on reduced space Mh .

5.2.3 Approximations of the fields with XFEM

The approximation of the involved fields can thus be put in the form (the fields tests used in the weakforms of the equilibrium equations are approximated same manner):

• for the field of displacements:

uh(x )=∑i=1

nn

ai φi(x ) + ∑j=1

nne

b j φ j(x )~H (lsn(x ))

• for the field of pressure in the solid mass:

ph(x)=∑

i=1

nns

c i ψi(x ) + ∑j=1

nnse

d j ψ j(x)~H (lsn(x ))

• for the field of pressure in the interface:

pfh(x)=∑

i=1

nnc

( p f )i~ψi(x )

• for the fields of hydraulic multipliers of Lagrange:

q1h(x)=∑

i=1

nnc

(q1)i~ψi(x )

q2h(x)=∑

i=1

nnc

(q2)i~ψi( x)

• for the fields of multipliers of Lagrange and the useful jump of displacement for the model ofcohesive zone:

λh(x )=∑

i=1

nnc

λ i~ψi(x )

μ h(x )=∑i=1

nnc

μi~ψi(x )


Group of nodes dividing a common DDL of interface

Support of the function of not-local form ψ 2

Surface of cracking

Vital edge: edge carrying a relation of equality

Another intersected edge 1

2

3

4



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wh (x)=∑i=1

nnc

wi~ψi(x )

with:• lsn level set normal whose Iso-zero represents the interface. For more precise details on the

use of level set with XFEM in Code_Aster, the reader will be able to refer to documentation[R7.02.13],

• ~H the discontinuous function through the discontinuity located by lsn(x)=0 (confer

[R7.02.12])• φi(x) functions of form of the quadratic element used for the interpolation of the field of

displacements, • ψi(x ) the function of form of the linear element relative used for the interpolation of the field

of pressure in the solid mass.• ~ψi( x) linear functions of form used for the interpolation of the fields associated with

discontinuity. They differ from the functions of form of the linear element relative ψi(x ) ifnodes tops of the linear element relative do not belong to any edge intersected by discontinuity(see § 5.4 and [R5.03.54] for more details). This situation appears only for the nonsimpliciauxelements.

• nn the whole of the nodes of the grid and nne the whole of the nodes nouveau riches of thegrid.

• nn the whole of the nodes tops of the grid and nne the whole of the nodes tops nouveauriches of the grid.

• nnc the nodes tops belonging to an edge intersected by discontinuity Γc .

Note:

As we can notice it in the definition of the approximations of the fields of displacements andpressure of the solid mass, enrichment due to the presence of a point of crack is not takeninto account. In the formulation of the mechanical problem, we base ourselves on a model ofcohesive zones regularized. The advantage of such a model (historically introduced byBarenblatt [12] in order to improve the theory of Griffith concerning the fractured mediums[13]) is of stage the fact that the constraints are infinite at a peak of crack. Consequently,enrichment by singular functions of the approximation of the field of displacements isinappropriate here, but completely possible with regard to the enrichment of theapproximation of the field of pressure in the solid mass as suggests it [7].

5.3 Extension to the multi-fissured case

5.3.1 Representation of the element multi-fissured associated HM-XFEM and ddls

In Code_Aster, elements XFEM can support to 4 cracks per element (see [R7.02.12]). It is alsopossible to have connected cracks. These features are in particular available for elements HM-XFEM.With each additional crack, one adds a set of degrees of freedom nouveau riches for the field ofdisplacement and of pressure of pore as well as a set of degrees of freedom associated with additionaldiscontinuity. On the Figure 5.3.1-1, one represents an element HM-XFEM QUAD8 intersected by twointerfaces (off-line on the left and connected on the right) as well as the degrees of freedom carried byeach node.




3d95ce8d4773

Figure 5.3.1-1 : Representation of an element HM-XFEM QUAD8 intersected by twodiscontinuities off-line (on the left) and connected (on the right)

5.3.2 Approximations of the fields with XFEM in the multi-fissured case

If several cracks intersect the same mesh, for each node, a correspondence is established betweeneach interface and the degrees of freedom nouveau riches and interface (see [R7.02.12]). Thus theapproximation of the involved fields can be put in the following form:

• for the field of displacements:

uh(x )=∑i=1

nn

ai φi(x ) + ∑ifiss=1

nfiss

∑j=1

nne

b jα(ifiss , j ) φ j(x )~H (lsnifiss(x ))

• for the field of pressure in the solid mass:

ph(x)=∑i=1

nns

c i ψi(x ) + ∑ifiss

nfiss

∑j=1

nnse

d jα( ifiss , j)ψ j(x) ~H (lsnifiss(x))

• for the degrees of freedom of interface, one has for example for the pressure of fluid pf in

the interface ifiss :

pfh ,ifiss(x )=∑

i=1

nnc

( p fα(ifiss ,i ))i

~ψi(x )

with:• α the function which with each interface associates the number of associated degree of

freedom enriched or interface for each Nœud.

5.3.3 Junction of hydraulic fractures

The fields associated with each interface are thus discretized with sets of distinct degrees of freedom.The interfaces thus function independently, even if they exert their influence one on the other via theporous matrix. In the case of a hydraulic junction of interface, it is however appropriate to impose ahydraulic connection, in order to allow the exchanges of fluid the level of the junction (see FigureFigure 5.3.3-1) . For this purpose, one can is to impose the conservation of the mass on the level ofthe junction (law of the nodes on flows W in each branch of the junction), that is to say to impose

the continuity of the pressure pf in the cracks on the level of the junction.




3d95ce8d4773

Figure 5.3.3-1: Hydraulic junction of crack

Being given the reduced space of approximation which we have for the field pf , it is preferable to

impose the continuity of the pressure pf on the level of the junction, rather than to impose an

equality on flows W who utilize the gradient of pf . For this purpose, one identifies a node carrying

the degree of freedom pf at the same time for the principal crack and the connected crack and oneforces the equality of these two degrees of freedom (see Figure Figure 5.3.3-2).

Figure 5.3.3-2: Imposition of the continuity of the pressurepf in the hydraulic cracks on the level of a junction.

5.4 Imposition of a flow in an interface hydraulics

With regard to the surface flows of fluid injected into the porous matrix, the integration of thesecond member is done naturally on the edge ΓF on which flow is imposed M ext (see § 4.2.1 ), of

dimension ndim−1 if the dimension of the field Ω is ndim . On the other hand, when it is aquestion of imposing a flow of fluid directly in the fracture, the dimension of the mouth Γ f is

ndim−2 if the dimension of the field Ω is ndim . The integration of the loading flow W ext




3d95ce8d4773

requires thus more attention. Let us interest initially if the dimension of Ω is 2 . Flow W ext express

yourself then in kg . s−1 and Γ f is tiny room to a point. Let us recall that the space of approximation

of the fields associated with the cohesive interface like pf be based only on the nodes tops of theedges strictly intersected by the interface and on the nodes on which the interface passes. On theFigure Figure 5.4-1, nodes indeed carrying the degree of freedom pf are marked by white squares

and reds. The integration of the term ∫Γf

W ext p f* d Γ f will thus be done on the restriction of this

space of approximation at the edge of the field. If the cohesive interface is in conformity at the edge ofthe field, one will impose directly W ext on the node N1 (see Figure 5.4-1 right-hand side). In the

case nonin conformity, one will have to determine the relative distance α between the mouth I and

the nodes tops of the edge of edge intersected N1 and N2 (see Figure 5.4-1 left). One will impose

then (1−α)W ext on the node N1 and αW ext on the node N2 .

Figure 5.4-1: Imposition of a flow in a fracture for the models 2D plan in the case noninconformity (on the left) and in the case conforms (on the right).

If the dimension of Ω is 3 , flow W ext express yourself in kg .m−1. s−1 and Γ f is a curve. In

order to integrate the term ∫Γf

W ext p f* d Γ f , it is necessary to have a support which approximates the

curve Γ f . For this purpose, one rebuilds an approximation of the interface on the edge of the fieldlike a chain of segments to 3 nodes. In each face of edge, the interface Γc is discretized by asegment with 3 nodes. With this intention, one uses the same procedure as for the creation of underelements of integration XFEM (see [R7.02.12]). Faces of edge intersected by Γc are rocked within

the space of reference (see Figure 5.4-2). The intersections then are determined I1 and I 2 with theedges of the face which will constitute the nodes ends of the segment with 3 nodes. Then onedetermines the position of the node medium I3 on the mediator of the segment [I 1 I 2] . One then

obtains a quadratic approximation of the interface at the edge of the field Ω . One is then capable to

evaluate ∫Γf

W ext p f* d Γ f by carrying out an integration on the chain of the segments with 3 Nœuds

which approximates Γ f . It is necessary however well to take care of the space of approximation ofpf on the intersected faces. As recalled previously, the space of approximation of the field pf be

based only on the nodes tops of the edges strictly intersected by discontinuity and the nodes on whichdiscontinuity passes. On the FigureFigure 5.4-2, only nodes N1 , N 3 and N 4 face quadrangle of

edge carry the degree of freedom pf . In order to satisfy all the same the partition with the unit in the

face quadrangle, one uses functions of form modified for the field pf :




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~ψN 1=ψN 1

+ψN2

3~ψN2

=0

~ψN 3=ψN 3

+ψN2

3~ψN 4

=ψN 4+

ψN2

3

with ψ functions of form of the linear element relative and ~ψ functions of form modified to adapt to

the space of approximation of the field pf .

Figure 5.4-2: Imposition of a flow in a fracture for the models 3D

6 Resolution of the coupled problem

6.1 Linearization of the coupled problem

6.1.1 Linearization by the method of Newton-Raphson

The coupled problem being nonlinear (it not linearity of the problem is due under the terms of mass ofthe variational formulations of the conservation equations of the mass for the solid mass and theinterface and under the cohesive terms for mechanics) we carry out its linearization using the methodof Newton-Raphson.

That is to say F the nonlinear system associated with the variational formulations of theconservation equations of the mass (for the solid mass and the interface), of the mechanicalequilibrium equation and the condition of continuity of the pressure pf on the level of the interface.

That is to say xk the vector of the nodal unknown factors to the iteration of Newton k such as:

xk= {uk pk pf

k q1k q2

kλ μ w }

T




3d95ce8d4773

With the iteration k+1 (the vector of the nodal unknown factors xk+ 1 ) one is not known poses:

F(xk+1)=0

In order to be able to determine xk+ 1 , we resort to a development of Taylor of F (which is a

presumedly continuous and derivable vector function) in the vicinity of xk (then known with the

iteration k+1 ). Thus the linear system with the iteration k+1 is written:

−F (xk) =∂F( xk

)

∂ xk ⋅δ xk

with δ xk=xk+ 1

− xk the increment of the values of the nodal unknown factors between two

successive iterations (which is an unknown factor with the iteration k+1 ), ∂F(xk

)

∂ xk the tangent

matrix and F(xk) the second member. These the last two terms are known with the iteration k+1

and are functions of xk .

6.1.2 É integral criture of the linearized problem

In the continuation one considers that the unknown factors are noted with one δ and the fields testswith one * while exposing.

The linear system with the iteration of Newton k+1 is written (for a step of time): • mechanical equilibrium equation

∀u*∈U 0 :

∫Ωσ

' +(δu) :ε (u*

)dΩ −∫Ωbδ p (1:ε (u*

))dΩ −∫Ωρ+b Fm+ Tr (∇(δu))u* dΩ

−∫Ω (ρ+(b−ϕ

+)

K s

+ρ+

ϕ+

Kw)δ p Fm+u*d Ω+∫Γc

δμ⋅⟦u*⟧dΓc

= −∫Ω

σ' +

(u) :ε (u*)dΩ +∫

Ωb p+

(1:ε (u*))dΩ +∫

Ωr+ Fm+u* dΩ +∫

Γt

t+u*d Γt

−∫Γc

μ⋅⟦u*⟧dΓc

• projection of the jump of displacement

∀μ*∈L0 :

∫Γc

(⟦δu⟧−δw )⋅μ*d Γc=−∫Γc

(⟦u⟧−w )⋅μ*d Γc

• constraint cohesive

∀ w*∈L0 :

−∫Γ c [δμ−

∂ t ' c∂(λ+r w)

⋅(δ λ +r δw ) ]⋅w*d Γc−∫Γ c

δ p f nc⋅w*d Γc

=∫Γc

[μ−t ' c (λ+rw )+ p f nc]⋅w*d Γc




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• law of interface

∀ λ*∈L0 :

−∫Γ c [δλr

−∂ t ' c∂(λ +r w)

⋅(δλr

+δw )]⋅λ*d Γc

=∫Γc

[ λ−t ' c (λ+rw) ]r

⋅λ*dΓc

• conservation equation of the mass (case of the solid mass)

Thus ∀ p*∈P0 :

−∫Ω

ρ+ bTr (∇ (δu)) p* dΩ −∫Ω ((ρ

+(b−ϕ

+)

K s

+ρ+

ϕ+

K w)δ p) p*d Ω +

Δ t θ [∫Ω(λ+ (−∇ p+

+ ρ+ Fm+ ) + λ+ρ+ Fm +) ρ+

Kw

δ p ∇ p* dΩ] +

Δ t θ [∫Ωρ+ (−∇ p+

+ ρ+Fm+ )∂λ

+

∂ p+ δ p ∇ p* dΩ −∫Ω

ρ+λ

+∇ (δ p) ∇ p* dΩ] +

Δ t θ [∫Ωρ+ (−∇ p+ + ρ+Fm+ )

∂ λ+

∂εv+ Tr (∇ (δu))∇ p* dΩ ]

+ Δ t θ∫Γ1

δq1 p* d Γ1 + Δ t θ∫Γ2

δq2 p* d Γ2 =∫Ω

(mw+

− mw- ) p* dΩ

− Δ t θ∫ΩM+

∇ p* dΩ − Δ t(1 − θ)∫ΩM -

∇ p*d Ω

+ Δ t θ∫ΓF

M ext+ p* d ΓF + Δ t(1 − θ)∫

ΓF

M ext- p* d ΓF

− Δ t θ∫Γ 1

q1+ p* d Γ1 − Δ t (1 − θ)∫

Γ1

q1- p* dΓ1

− Δ t θ∫Γ 2

q2+ p* d Γ2 − Δ t (1 − θ)∫Γ 2

q2- p* d Γ2

• conservation equation of the mass (case of the interface)

Thus ∀ p f*∈F0 :




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−∫Γc

ρ+δ⟦u⟧⋅nc p f* d Γc −∫Γc (ρ

+⟦u+

⟧⋅nc

Kw

δ pf ) pf* d Γc

+ Δ t θ [∫Γ c

−ρ+

∇ pf+

12μ(3 (⟦u+

⟧⋅nc)2)δ ⟦u⟧⋅nc ∇ p f

* d Γc ]+ Δ t θ [∫Γ c

−(⟦u+

⟧⋅nc)3∇ pf

+

12μ

ρ+

Kw

δ p f ∇ p f* d Γc]

+ Δ t θ [∫Γ c

−ρ+ (⟦u+

⟧⋅nc )3

12 μ∇ (δ pf ) ∇ p f

* d Γc ]− Δ t θ[∫Γ1

δ q1 p f* d Γ1 ] − Δ t θ [∫Γ 2

δ q2 p f* dΓ2 ]

=∫Γc

(w+− w -

) p f* d Γc − Δ t θ∫

Γc

W +∇ p f

* d Γc − Δ t (1−θ)∫Γc

W -∇ p f

*d Γc

+ Δ t θ [∫Γ1

q1+ p f

*d Γ1] + Δ t(1−θ)[∫Γ1

q1- p f

* d Γ1 ]+ Δ t θ [∫Γ2

q2+ p f

*d Γ2 ] + Δ t(1−θ)[∫Γ2

q2- p f

* d Γ2 ]+ Δ t θ∫

Γ f

W ext+ p f

* d Γ f + Δ t (1−θ)∫Γ f

W ext - p f

*d Γ f

• equation of continuity of the pressure pf on the level of the interface

Thus ∀ q1*∈Q 1 and ∀ q2

*∈Q2

∫Γ1

δ p inf q1* d Γ1 −∫

Γ1

δ p f q1* d Γ1 = −∫

Γ 1

( pinf − p f )q1* d Γ1 on Γ1

∫Γ2

δ psup q2* d Γ2 −∫

Γ2

δ p f q2* d Γ2 = −∫

Γ 2( psup

− p f )q2* dΓ2 on Γ2

6.2 Writing of the elementary terms with XFEM

6.2.1 Writing in matric form of the coupled problem The system of equations previously discretized to the iteration of Newton k+1 can put itself in the

form (where δ u , δ p , δ pf , δq1 , δq2 , δλ , δμ and δ w are the unknown factors of theproblem to be solved):

Équilibre mechanical {u*}[ Kmeca1 ](δu) + {u*}[ Kmeca

2 ](δ p) + {u*}[ A ] (δu) +

{u*}[ B ](δ p) + {⟦u⟧

*} [C1 ] {δμ}=

{u*}(Lmeca) + {⟦u⟧*}(L1)


{μ*} [ Kμ u ](δu) + {μ

*} [−Kw μ ]

T(δ w) =

{μ*}(Lu) − {μ

*}(Lw)

Cohesive constraint{w*

}[−Kwμ ](δμ) + {w*} [Dw w ](δ w) + {w*

} [ Kw p ](δ pf ) +

{w*}[ Dλw ]

T(δλ ) = −{w*

}(Lμ2) + {w*

}(Lcohe1

) + {w*}( Lp)

Law of interface{λ

*}[ Dλ w ](δ w) + {λ

*}[ Dλ λ ] (δλ ) =

−{λ*}(Lμ

1) + {λ

*}(Lcohe

2)




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Conservation of mass (case of the solidmass)

{p*}[M hydro

1](δ u) + {p*

}[M hydro2

](δ p) + Δ t θ{p*}[ Khydro

1](δ p) +

Δ t θ{p*}[ K hydro

2](δ p) + Δ t θ{p*

}[K hydro3

](δ p) +

Δ t θ{p*}[ K hydro

4](δ u) + Δ t θ{p*

}[ E1] (δ q1) + Δ t θ{ p*

}[ E2](δ q2)

= {p*}( Lhydro

1) + Δ t {p*

}(Lhydro2

)θ + Δ t {p*}(Lhydro

3)θ +

Δ t {p*}( Lhydro

4)θ + Δ t {p*

}( Lhydro5

)θ

Conservation of mass (case of the interface)

{p f*}[W hydro

1](δ ⟦u⟧) + {p f

*}[W hydro

2](δ p f ) + Δ t θ{p f

*}[H hydro

1](δ ⟦u⟧) +

Δ t θ{p f*}[ H hydro

2](δ p f ) + Δ t θ{p f

*}[ H hydro

3](δ p f ) +

Δ t θ{p f*}[ D1

](δ q1) + Δ t θ{p f*}[ D2

](δ ⟦u⟧)

Δ t θ{p f*}[ D3

](δ q2) + Δ t θ{p f*}[ D 4

](δ ⟦u⟧) =

{p f*}(Lhydro

6) + Δ t {p f

*}(Lhydro

7)θ + Δ t {p f

*}( Lhydro

8)θ +

Δ t {p f*}(Lhydro

9)θ + Δ t {p f

*}(Lhydro

10)θ

Continuity of the pressure {q1

*}[D1

] (δ p) + {q1*}[D2

] (δ pf ) = {q1*}(J cont

1)

{q2*}[D3

] (δ p) + {q2*}[D4

](δ pf ) = {q2*}(J cont

2)

Kmeca1 is the elementary matrix of mechanical rigidity classically met in mechanics,

Kmeca2 is due to the decomposition of the tensor of the constraints of the solid mass (assumption of

the effective constraints), A and B are due to the taking into account of the mass contributions in the expression of the

homogenized density intervening in the expression of the voluminal efforts on Ω ,

C1 is the elementary matrix of rigidity for the interface,

Kμu and K

wμ are matrices discretizing the operators “mortar”, the last one also managing the basic

change.

Kw p is an elementary matrix of rigidity for the interface,

Matrices D are all diagonal per blocks: for I and J two distinct DDL of Lagrange, they checkD IJ=0 ,

M hydro1 and M hydro

2 are the elementary matrices of mass in the case of the solid mass for the

hydrodynamics,

W hydro1 and W hydro

2 are the elementary matrices of mass in the case of the interface for the

hydrodynamics ,

Khydro1 , K hydro

2 , Khydro3 and Khydro

4 are the elementary matrices of rigidity for the hydrodynamics in

the case of the solid mass,

H hydro1 , H hydro

2 and Hhydro3 are the elementary matrices of rigidity for the hydrodynamics in the case

of the interface,

E1 and E2 are the matrices of exchanges in the case of the solid mass,

D1 and D2 are the matrices of exchanges in the case of the interface,

F1 , F2 , F3 and F4 are the matrices of continuity of the pressure on the level of the interface,




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Lmeca is the second member of the forces of volumes and surface applied to the field and to itsborder,

L1 is the second member for the interface,

Lu , Lw and Lp are second members for the projection of the jump of displacements,

Lμ2 , and Lcohe

1 are second members for the cohesive constraint,

Lμ1 and Lcohe

2 are second members for the law of interface,

Lhydroi with i∈⟦1,5⟧ second members due to the mass contributions and flows in the case of solid

mass for the hydrodynamics,

Lhydroi with i∈⟦6,10⟧ second members due to the mass contributions and flows in the case of the

interface for the hydrodynamics,

J hydro1 and J hydro

2 second members due to the exchanges on Γ1 and Γ2 .

Note:

As we can notice it in L be forms of elementary matrices definite Ci afterwards , quantities

ρ+ , ϕ+ (functions displacements and of the pressure), p+ and u+(ou ⟦u+

⟧) are left inthe state (not discretized), because they are sizes obtained during the preceding iteration ofNewton (for the step of current time + ) . They are thus a priori known. In the form of theelementary matrices (for the mechanical and hydrodynamic case) we will not voluntarilyindicate the number of the iteration of Newton on these quantities to avoid overloading theexpressions.

6.2.2 Form of the elementary matrices for mechanics

Matrices elementary of mechanical rigidity to the iteration of Newton k+1 is written:

{u*}[ Kmeca1 ](δu) =∫Ω

(a i* + ~H b i

*)∇ φi Ci j ∇ φ j(δa j + ~H δ b j)dΩ

The elementary matrices due to the decomposition of the tensor of the constraints of the solid mass(assumption of the effective constraints) is written with the iteration of Newton k+1 :

{u*}[ Kmeca

2 ](δ p) = −∫Ωb(ai

*+

~H bi*)∇ φi[Id ]ψ j(δc j +

~H δd j)d Ω

The matrix A with the iteration of Newton k+1 is written:

{u*}[ A ] (δu) = −∫Ω

ρ + b(a i*

+~H b i

*)φi[ Id ] ∇ φ j(δ a j +

~H δb j)Fm+d Ω

The matrix B with the iteration of Newton k+1 is written:

{u*}[ B ](δ p) = −∫

Ω ((ρ+(b−ϕ+

)

K s

+ρ+ϕ+

Kw))(a i

*+

~H b i*)φi ψ j(δ c j +

~H δd j)Fm+ dΩ

The matrix associated with projection with the jump with displacements to the iteration with Newtonk+1 is written:

{⟦u⟧*} [C1 ](δμ) =∫Γ c

(2 ~H b i* )φi

~ψ j (δμ j ) d Γc




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Components of the unknown factors u and μ are defined in a fixed base (e X ,eY ,eZ ) , while

components of w and λ are defined in the local base (n , τ 1 , τ 2) on fissured surface Γc in each

point x∈Γc , so that :

w (x)=∑i=1

N λ

ψI (x) (w I ,n n(x )+w I , τ1 τ1(x )+w I ,τ 2 τ 2(x ) )

A similar definition is worth for λ . For a degree of freedom I reduced space (see §5.2.2), it is

possible to determine the components t ' c ,nI ,t 't , τ1

I , t 'c , τ 2I

cohesive force from (w I ,n , w I , τ1 , w I , τ2 ) ,

(λI ,n ,λI ,τ1 ,λI , τ2) and of the cohesive law. These components are not intended to be associatedwith a particular direction I degree of freedom, but intended to be connected to the weak direction

for the total constraint μ

written in fixed base (confer [R7.02.19]).

{u}* [Kμ u ](δμ)=∫Γ c

~ψ jφi2~H bi

*⋅(δμ j)dΓc

{w }* [Kw μ ](δμ)=∫Γ c

~ψi~ψ j w j

*⋅Q⋅(δμJ )d Γc with Q the matrix of basic change orthonormal

definite like previously.

{w }* [Kw p ](δ p f)=∫Γc

~ψ j~ψi wi

*⋅(δ pf ) jnc dΓc

{w }* [Dww ](δw)=∫Γ c

w i*(δw i)r

∂ t ' c∂(λ +r w)

(λ i+r wi)~ψi2dΓc

{λ }* [Dλw ](δw)=∫

Γc

λ i*(δwi)

∂ t ' c∂(λ+r w)

(λ i+r wi)~ψi2d Γc

{λ }* [Dλλ ](δ λ )=∫

Γc

λ i*(δ λ i)

1r ( ∂ t ' c

∂(λ+rw)(λ i+r w i)−1) ~

ψi2dΓc

6.2.3 Expression of the second members for mechanics

In the expressions of the second members introduced here one indicates the number of the precedingiteration of Newton k .

{u*}(Lmeca) = −∫Ω

(a i*

+~H b i

*) ∇ φi (σ

'+(u))

k d Ω +∫Ωb(a i

*+

~H b i*)∇ φi [ Id ]( p+

)k d Ω

+∫Ω

(a i*

+~H b i

*)φi(r0 + (mw

+)

k)Fm+ dΩ +∫

Γt

(a i*+

~H b i*)φi ( t+

)k d Γt

{⟦u⟧*}(L1

) =−∫Γc

(2 ~H bi* )φi

~ψ j μ jk d Γc

{u}i* (Lμ

1 )i=−bi*⋅∫

Γc

2 φiμ d Γc

{μ}*

( Lu )=−μ i*⋅∫Γc

~ψi⟦u⟧d Γc

{w }*

(Lp )=−wi*⋅∫Γ c

~ψi pf nc dΓc




3d95ce8d4773

{μ}*

( Lw )=−μ i*⋅∫Γc

~ψiQT⋅wdΓc

{w }* (Lμ2 )=w i

*⋅∫Γc

~ψiQ⋅μ dΓc

{w }* (Lcoh1 )=w i

*⋅t ' c (λ i+r wi)∫Γc

~ψid Γc

{w }* (Lcoh

2 )I=w i*

rt ' c (λ i+r w i)∫Γc

~ψI dΓc

6.2.4 Form of the elementary matrices for the hydrodynamics


Elementary matrices of mass with the iteration of Newton k+1 are written:

{p*} [ M hydro1 ](δu) = −∫Ω

bρ+(c i* + ~H d i

*)ψi [ Id ] ∇ φ j(δa j + ~H δb j)dΩ

{p*} [ M hydro

2 ](δ p) = −∫Ω ((ρ+

(b−ϕ+)

K s

+ρ+ϕ+

Kw))(c i

*+

~H d i*)ψi ψ j(δ c j +

~H δ d j)dΩ

Elementary matrices of rigidity to the iteration of Newton k+1 are written:

{p*}[ Khydro1 ](δ p) =∫

Ω(λ+ (−∇ p++ρ+ Fm+ ) + ρ+λ+ Fm +) ρ+

Kw

(c i* + ~H d i

*) ∇ ψi ψ j(δc j + ~H δd j)d Ω

{p*}[ Khydro2 ](δ p) =∫

Ωρ+ (−∇ p++ρ+Fm+ )

∂ λ+

∂ p+ (c i* + ~H d i

*)∇ ψi ψ j(δc j + ~H δ d j)d Ω

{p*}[ Khydro

3](δ p) = −∫Ω

ρ+ λ+(c i

*+

~H d i*)∇ ψi ∇ ψ j(δ c j +

~H δ d j)d Ω

{p*}[ K hydro

4](δu) =∫

Ωρ+ (−∇ p+

+ρ+ Fm +)∂ λ+

∂ εv+ (ci

*+

~H d i*) ∇ ψi [ Id ] ∇ φ j(δa j +

~H δb j)dΩ

Elementary matrices of exchange for the solid mass with the iteration of Newton k+1 are written:

{p*}[ E1

](δq1) =∫Γ1

(c i*

+~H d i

*) ψi

~ψ j(δ q1)j d Γ1

{p*}[ E2

](δq2) =∫Γ2

(c i*

+~H d i

*) ψi

~ψ j(δ q2) j d Γ2


Elementary matrices of mass with the iteration of Newton k+1 are written:

{p f*}[W hydro

1] (δ⟦u⟧) = −∫Γc

ρ+( p f

*)i

~ψi φ j ( 2~H δb j ) nc dΓc




3d95ce8d4773

{p f*}[W hydro

2] (δ p f ) = −∫

Γ c

ρ+⟦u⟧⋅nc

K w

( p f*)i

~ψi~ψ j ( δ p f ) j d Γc

Elementary matrices of rigidity to the iteration of Newton k+1 are written:

{p f*}[ Hhydro

1](δ⟦u⟧) = −∫

Γc

ρ+ (3 (⟦u⟧⋅nc)2 ) ∇ p f

+

12μ( p f

*)i ∇

~ψi φ j ( 2~H δb j ) nc dΓc

{p f*}[ Hhydro

2](δ p f ) = −∫Γc

(⟦u⟧⋅nc )3∇ p f

+

12μ

ρ+

Kw

(p f*)i ∇

~ψi~ψ j(δ p f )j d Γc

{p f*}[ Hhydro

3](δ p f ) = −∫Γc

ρ+(⟦u⟧⋅nc )

3

12μ( p f

*)i ∇

~ψi ∇~ψ j(δ p f )j d Γc

Elementary matrices of exchange for the interface with the iteration of Newton k+1 are written:

{p f*}[ D1

](δq1) = −∫Γ1

( p f*)i

~ψi~ψ j(δq1) jd Γ1

{p f*}[ D2

](δ q2) = −∫Γ2

( p f*)i

~ψi~ψ j(δ q2)j d Γ2

6.2.4.3 Continuity of the pressure

Elementary matrices of exchange (for the equation of continuity of the pressure on the level of theinterface) to the iteration of Newton k+1 are written:

{q1*}[F1

](δ pinf) =∫Γ1

(q1*)i

~ψi ψ j(δc j +~H δd j)d Γ1

{q1*}[F2

](δ p f ) =−∫Γ1

(q1*)i

~ψi~ψ j(δ pf ) j dΓ1

{q2*}[F3

](δ psup) =∫Γ2

(q2*)i

~ψi ψ j(δc j +~H δ d j)d Γ2

{q2*}[F4

] (δ p f ) = −∫Γ 2

(q2*)i

~ψi~ψ j(δ p f )j d Γ2

Note:

In the form of the elementary matrices defined in this part, one can note the presence of twoadditional “unknown factors”, δ pinf and δ psup . They correspond in fact to the unknown

factor relating to the fields of pressure δ p defined respectively on Γ1 (i.e. δ pinf ) and onΓ2 (i.e. δ psup ).

6.2.5 Expression of the second elementary members for the hydrodynamics





3d95ce8d4773

In the expressions of the second members introduced here one indicates the number of the precedingiteration of Newton:

{p*}(Lhydro1 ) =∫

Ω(ci

* + ~H d i*)ψi ((mw

+ )k − (mw- )k) dΩ

{p*}(Lhydro

2)θ = −θ∫Ω

(c i*

+~H d i

*)∇ ψi(M

+ )

k dΩ − (1−θ)∫Ω(c i

*+

~H d i*)∇ ψi(M

- )

k dΩ

{p*}(Lhydro

3)θ = θ∫Γ F

(ci*

+~H d i

*)ψi(M ext

+ )

k d ΓF + (1−θ)∫Γ F

(c i*

+~H d i

*) ψi(M ext

- )

k d ΓF

{p*}(Lhydro

4)θ =− θ∫Γ1

(c i*

+~H d i

*) ψi(q1

+ )

k dΓ1 − (1−θ)∫Γ1

(ci*

+~H di

*)ψi(q1

- )

k d Γ1

{p*}(Lhydro

5)θ =− θ∫Γ2

(c i*

+~H d i

*) ψi(q2

+ )

k dΓ2 − (1−θ)∫Γ2

(c i*+

~H di*)ψi(q2

- )

k d Γ2


In the expressions of the second members introduced here one indicates the number of the precedingiteration of Newton.

{p f*}(Lhydro

6) =∫Γc

( p f*)i

~ψi ((w+ )

k− (w -

)k ) d Γc

{p f*}(Lhydro

7)θ = −θ∫Γ c

( p f*)i ∇

~ψi(W +

)k dΓc − (1−θ)∫Γc

(p f*)i ∇

~ψi(W -

)k d Γc

{p f*}(Lhydro

8)θ = θ∫Γf

( p f*)i

~ψi(W ext +

)k d Γ f + (1−θ)∫Γf

( p f*)i

~ψi(W ext -

)k d Γ f

{p f*}(Lhydro

9)θ = θ∫Γ1

( p f*)i

~ψi(q1+ )

k d Γ1 + (1−θ)∫Γ1

(p f*)i

~ψi(q1- )

k d Γ1

{p f*}(Lhydro

10)θ = θ∫Γ2

( p f*)i

~ψi(q2+ )

k d Γ2 + (1−θ)∫Γ2

(p f*)i

~ψi(q2- )

k d Γ2

6.2.5.3 Continuity of the pressure

In the expressions of the second members introduced here (for the equation of continuity of thepressure on the level of the interface), one indicates the number of the preceding iteration of Newton.

{q1*}(J cont

1) = −∫

Γ1

(q1*)i

~ψi ((p inf)

k− ( p f

+)

k )d Γ1

{q2*}(J cont

2 ) = −∫Γ1

(q2*)i

~ψi ((psup)k − ( p f+)k) d Γ2

7 Bibliography

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[2] Belytschko. T., Black. T. Elastic minimal ace growth in finite elements with remeshing. InternationalNewspaper for Numerical Methods in Engineering, 45 (1999) 601-620.

[3] Moës. NR., Doldbow. J., Belytschko. T.A finite element method for ace growth without remeshing.International Newspaper for Numerical Methods in Engineering, 46 (1999) 131-150.

[4] Melenk.J.M, Babuška.I. The partition of unity finite element method: BASIC theory and applications.Methods computer in Applied Mechanics and Engineering, 139 (1996) 289-314.

[5] Khoei. A.R., Haghighat. E. Extended finite deformable element modeling of porous media with arbitraryinterfaces. Applied Mathematical Modeling, 35 (2011) 5426-5441.




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[6] Khoei. A.R., Moallemi. S., Haghighat. E. Thermo-hydro-mechanical modeling of impermable discontinuityin saturated porous media technical with XFEM. Engineering Fractures Mechanics, 96 (2012) 701-723.

[7] Lecampion. B. Year extended finite element method for hydraulic fracture problems. Communications inNumerical Methods in Engineering, 25 (2) 2009.

[8] Bathe.K.J, Brezzi.F. Stability of finite element mixed interpolations for contact problems. Returns.Chechmate. Acc. Lincei., 12 (9): 167-183, 2001

[9] Babuška.I. The finite element method with lagrangian multipliers. Numerische Mathematik. 20 (3 (179-192,1973.

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[11] Witherspoon. P., Wang. J., Iwai. K., Scale. J. Validity of cubic law for fluid flow in has deformable rock'n'rollfractures. Resour toilets. Abstr., 16 (6) 1980.

[12] Barenblatt. G. The mathematical theory of equilibrium aces in brittle fracture. Advances in AppliedMechanics. 7 (1962) 55-129.

[13] Homand. F., Duffaut. P., Handbook of rock mechanics. Volume 1: Bases, Presses of the École des Mines,Paris, 2000

[14] A. Ern and S. Meunier. A posteriori error analysis of eurler-galerkin approximations to coupled elliptic-parabolic problems. SWARM: M2AN, 43:353-375, 2009

[15] Mr. Faivre, B. Paul, F. Golfier, R.Giot, P. Massin and D? Colombo: 2D coupled HM-XFEM modeling withcohesive zone model and applications to fluid driven fracture network.

8 Description of the versions of the document

Version Aster

Author (S) Organization (S)

Description of the modifications

12 Maxime Faivre (ENSG) Initial version13 Bertrand PAUL (IFPEN) Version 2


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