1. AXIOMATIC FORMULATIONS Graciela Herrera Zamarrón 1

2. SCIENTIFIC PARADIGMS 2 Generality Clarity Simplicity

3. AXIOMATIC FORMULATION OF MODELS 3

4. MACROSCOPIC PHYSICS There are two major branches of Physics: Microscopic Macroscopic The approach presented belongs to the field of Macroscopic Physics 4

5. 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 5

6. 6 BALANCES ARE THE BASIS OF THE AXIOMATIC FORMULATION OF MODELS

7. EXTENSIVE AND INTENSIVE PROPERTIES 7 “Estensive property”: Any that can be expressed as a volume integral “Intensive proporty”: Any extensive per unit volumen; this is, ψ

8. FUNDAMENTAL AXIOMA BALANCE CONDITION 8 An extensive property can change in time, exclusively, because it enters into the body through its boundary or it is produced in its interior.

9. BALANCE CONDITIONS IN TERMS OF THE EXTENSIVE PROPERTY 9

10. 10 BALANCE CONDITIONS IN TERMS OF THE INTENSIVE PROPERTY Balance differential equation

11. 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 11

12. THE GENERAL MODEL OF MACROSCOPIC SYSTEMS 12 Balance differential equations Intensive properties

13. 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” 13

14. CONSTITUTIVE EQUATIONS 14 Are the relationships that incorporate the scientific and technological knowledge available about the system in question

15. THE BLACK OIL MODEL 15

16. 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

17. 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

18. MATHEMATICAL EXPRESSION OF THE FAMILY OF EXTENSIVE PROPERTIES 18

19. 19 BASIC MATHEMATICAL MODEL

20. FAMILY OF INTENSIVE PROPERTIES 20

21. 21 BASIC MATHEMATICAL MODEL

22. AXIOMATIC FORMULATION OF DOMAIN DECOMPOSITION METHOD 22

23. 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 23

24. DOMAIN DECOMPOSITION METHODS