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

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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

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

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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