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AXIOMATIC FORMULATIONS
Graciela Herrera Zamarrón
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SCIENTIFIC PARADIGMS
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•Generality •Clarity •Simplicity
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
4
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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6
BALANCES ARE THE BASIS OF
THE AXIOMATIC FORMULATION
OF MODELS
EXTENSIVE AND INTENSIVE PROPERTIES
B t
,B t
E t x t dx
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B
“Estensive property”: Any that can be expressed as a volume integral
“Intensive proporty”: Any extensive per unit volumen; this is, ψ
FUNDAMENTAL AXIOMA BALANCE CONDITION
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An extensive property can change in
time, exclusively, because it enters into
the body through its boundary or it is
produced in its interior.
BALANCE CONDITIONS IN TERMS OF THE EXTENSIVE PROPERTY
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)()(
),(),(tBtB
xdntxxdtxgdt
dE
property extensive theof flux"" theis ),(
property extensive theof "generation" theis ),(
tx
txg
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BALANCE CONDITIONSIN TERMS OF THE INTENSIVE PROPERTY
gvt
)(
Balance differential equation
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
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Ngvt
,...,1;)(
Balance differential equations
Intensive properties
N,...,1,
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
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Are the relationships that incorporate
the scientific and technological
knowledge available about the system
in question
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 16
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) 17
MATHEMATICAL EXPRESSION OF THE FAMILY OF EXTENSIVE PROPERTIES
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w
o
o
g
ww wB t
oo oB t
dgo dgB t
gg gB t
M t S dx
M t S dx
M t S dx
M t S dx
- porosidad- saturación fase (fracción de volumen ocupado por la fase)
- densidad de la fase, , densidad neta del aceite
o
oo
Sm
V
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BASIC MATHEMATICAL MODEL
ggggg
dgdgwdgdg
ooooo
wwwww
gt
gt
gt
gt
v
v
v
v
FAMILY OF INTENSIVE PROPERTIES
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ww w
oo o
dgo dg
gg g
S
S
S
S
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BASIC MATHEMATICAL MODEL
ggggg
gg
dgdgwdgo
dgo
ooooo
oo
wwwww
ww
gSt
S
gSt
S
gSt
S
gSt
S
v
v
v
v
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
1 1
1
1 1 1
0,
0,
0,
0,
aS aSu ag and ju DVS BDDC
S jS j S jS jg and aS Primal DVS
jS jS jS jg and a DVS FETI DP
SaS a SaS aS jg and jS Dual DVS
v v