Transcript macroscopic physics

AXIOMATIC FORMULATIONS
Graciela Herrera Zamarrón
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SCIENTIFIC PARADIGMS
•Generality
•Clarity
•Simplicity
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AXIOMATIC FORMULATION OF
MODELS
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MACROSCOPIC PHYSICS
There are two major branches of Physics:
•Microscopic
•Macroscopic
The approach presented belongs to the field
of Macroscopic Physics
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GENERALITY
• The axiomatic method is the key to developing
effective procedures to model many different
systems
• In the second half of the twentieth century the
axiomatic method was developed for macroscopic
physics
• The axiomatic formulation is presented in the books:
– Allen, Herrera and Pinder "Numerical modeling in science
and engineering", John Wiley, 1988
– Herrera and Pinder "Fundamentals of Mathematical and
computational modeling", John Wiley, in press
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BALANCES ARE THE BASIS OF
THE AXIOMATIC FORMULATION
OF MODELS
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EXTENSIVE AND INTENSIVE PROPERTIES
B t 
B
E  t      x, t dx
B t 
“Estensive property”: Any that can be expressed as a
volume integral
“Intensive proporty”: Any extensive per unit volumen; this
is, ψ
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FUNDAMENTAL AXIOMA
BALANCE CONDITION
An extensive property can change in
time, exclusively, because it enters into
the body through its boundary or it is
produced in its interior.
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BALANCE CONDITIONS
IN TERMS OF THE EXTENSIVE PROPERTY
dE
  g ( x, t )d x    ( x, t )  nd x
dt B (t )
B ( t )
g ( x, t ) is the" generation" of theextensiveproperty
 ( x, t ) is the" flux"of theextensiveproperty
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BALANCE CONDITIONS
IN TERMS OF THE INTENSIVE PROPERTY
Balance differential equation

   (v )     g
t
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THE GENERAL MODEL OF
MACROSCOPIC MULTIPHASE
SYSTEMS
• Any continuous system is characterized by a
family of extensive properties and a family of
phases
• Each extensive property is associated with
one and only one phase
• The basic mathematical model is obtained by
applying to each of the intensive properties
the corresponding balance conditions
• Each phase moves with its own velocity
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THE GENERAL MODEL OF
MACROSCOPIC SYSTEMS
Intensive properties

 ,  1,...,N
Balance differential equations




   (v  )     g  ;   1,..., N
t

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SIMPLICITY
PROTOCOL OF THE AXIOMATIC METHOD FOR
MAKING MODELS OF MACROSCOPIC PHYSICS:
• Identificate the family of extensive properties
• Get a basic model for the system
– Express the balance condition of each extensive property in
terms of the intensive properties
– It consists of the system of partial differential equations
obtained
– The properties associated with the same phase move with the
same velocity
• Incorporate the physical knowledge of the system
through the “Constitutive Relations”
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CONSTITUTIVE EQUATIONS
Are the relationships that incorporate
the scientific and technological
knowledge available about the system
in question
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THE BLACK OIL MODEL
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GENERAL CHARACTERISTICS OF
THE BLACK-OIL MODEL
• It has three phases: water, oil and gas
• In the oil phase there are two components:
non-volatile oil and dissolved gas
• In each of the other two phases there is only
one component
• There is exchange between the oil and gas
phases: the dissolved gas may become oil and
vice versa
• Diffusion is neglected
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FAMILY OF EXTENSIVE PROPERTIES OF THE
BLACK-OIL MODEL
• Water mass (in the water phase)
• Non-volatile oil mass (in the oil
phase)
• Dissolved gas mass (in the oil
phase)
• Gas mass (in the gas phase)
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MATHEMATICAL EXPRESSION OF THE
FAMILY OF EXTENSIVE PROPERTIES
M w  t  
 S w  w dx

Bw  t 
 o
 M  t     So o dx
Bo  t 
 dg
 M  t   Bo t   So  dg dx
M g  t  
 S g  g dx

Bg  t 

 - porosidad
S - saturación fase  (fracción de volumen ocupado por la fase)
mo
 - densidad de la fase, o 
, densidad neta del aceite
Vo
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BASIC MATHEMATICAL MODEL





w
w w
w
    v     g
t
o
o
o

o o
    v     g
t
dg

dg
dg w
dg
    v     g
t
g

g
g g
g
    v     g
t
w




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FAMILY OF INTENSIVE PROPERTIES
 w   S 
w w

 o   S 
o o


dg
   So  dg

 g   S 
g g

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BASIC MATHEMATICAL MODEL






S w  w
w
w
w
   S w  w v     g
t
o
o
S o  o
o
   S o  o v     g
t
S o  dg
w
dg
dg
   S o  dg v     g
t
S g  g
g
g
g
   S g  g v     g
t


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AXIOMATIC FORMULATION OF
DOMAIN DECOMPOSITION METHOD
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PARALELIZATION METHODS
• Domain decomposition methods are the
most effective way to parallelize
boundary value problems
– Split the problem into smaller boundary
value problems on subdomains
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DOMAIN DECOMPOSITION METHODS
1
aS aSu  ag and ju  0, DVS  BDDC
1
1
S jS jv  S jS j g and aS v  0, Primal  DVS
1
jS jS   jS j g and a   0, DVS  FETI  DP
1
1
1
S aS a   S aS aS j g and jS   0, Dual  DVS