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AXIOMATIC FORMULATIONS
Graciela Herrera Zamarrón
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SCIENTIFIC PARADIGMS
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•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 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
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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.
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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
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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
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Ngvt
,...,1;)(
Balance differential equations
Intensive properties
N,...,1,
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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
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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 16
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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) 17
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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
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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
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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
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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