Ok, so I will talk about these multi-phase flows in porous media.

UFHA means the Universidade Federal do Rio de Janeiro, which is the oldest university

in Brazil.

Well, it's not to say too much because just 104 years, so 4 years is not so much.

These are joined to work with André de Oliveira Gomes.

He is a stochastic mathematician and came to Brazil to work with me as a postdoc in

stochastic transport equations, but we decided to work on these, what I have been working

for sometimes.

So, I will give you an introduction of this problem, I will give you the model, maybe

you have already seen this model with Professor Joaquim Escher, more related to the Sharpe

interface, when you have two fluids for instance.

Then I am going to present you some difficulties to solve, to show the existence of solutions

to this problem, to this system.

And previous work, somehow people have modified the system to get solutions, so I am going

to talk in particular, not to talk, but to show you, to recall a paper from Khrushkov

and another paper from myself.

Then I am going to present you a new approach that you have used here, which is the stiffness

limits.

We are going to see what this means, which have been used, some nowadays by the group

of Beno Apertam, but it is an old idea from Benoit, Philippe Benoit.

And then I will try to give you more or less the sketch of the proof of the existence of

this system, the existence of weak solutions of this system.

The Boccalier system has been studied by many authors because it plays an important role

in wide range of planning, operation of oil wheels to restore oil, injecting water in

the porous media, so it means that you have a bounded domain, injecting water and try

to get more oil.

But it is also related to filtration problems, dam study, capture of CO2, for instance,

in the porous media, so it is a very interesting problem related to these multiphase flows

in porous media.

You could see this in the old book from Chavet and Chavrier, it is a classical book for multiphase

flows and also this book from Bustos, Khrushkov and Burger.

But I am going to present you how it is modeled.

So the most important example is the encroachment of water in oil sands, you have the flow with

a mixture of oil and water and how this Boccalier system consists.

It consists in particular of a conservation law that can express the mass balance of some density

which is somehow related with the density of the oil in the gas, oil in the water, sorry.

The flow of the mixture involving a nonlinear continuum equation, so you have a scalar conservation

law, so G of u, v is your flux function, so this flux function is not homogeneous, depends

explicitly in T and X given by the velocity vector field v.

G of u represents an additional bone filling in the term of the density and v as I said before

is the velocity field used to see, called the seepage velocity because it is the mixture of the flow.

And it is given somehow by the Darcy law, so it means that you have somehow here mobility,

inverse of the mobility of velocity which is somehow proportional to the gradient of the pressure.

So this is not given exactly by the Darcy law, but it is from the empirical Darcy law,

the Darcy law applies for each component of the mixture.

So you combine then you get this velocities for the flow, which describes the dynamics of the flow

relation of scalar function pressure, which is called pressure.

I put here use called pressure because since this model is incompressible, you are considered

that the oil and the water is incompressible, so this looks like the pressure here is a multiplier

related to the constraints of incompressibility.