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Thank you also for inviting me. It's certainly a pleasure to be here and to be able to participate

in this celebration of the 60th birthday of Peter, especially since it's such a nicely

organised and very nice conference.

Okay, well I wanted to talk about space-time domain decomposition methods for mixed formulations,

for flow and transport in porous media, and in particular with fractures.

And this is joint work with Jerome, who I think most of you know who's here,

and Michelle Cairn, who is also in our group and would also have liked to be here,

but had a conflict and could not be.

And Caroline Jaffe, who was in our group for two years.

Well, she's become an unofficial member of our group, but she's her position's at Paris 13.

And then most of all with Feng, Titao Feng-Huang, who did her thesis on the topic

and has done almost all of the work.

Okay, so in our group at ENRIA, what we've long been interested in is numerical methods

for flow and transport in heterogeneous porous media.

And one main of the examples that we've worked on is porous media around nuclear waste deposits sites,

modeling flow and transport there, and that's why UNDRA,

which is the national agency for radioactive waste, the management of radioactive waste,

that's why they financed quite a few theses for us, including this one.

And of course, their problem, they had the deep underground repository site in here,

where there's high-level waste, and here you see a far-field example.

Here it is blown up about 50 times so that you can see the heterogeneity of the medium.

Also, they're interested in subsurface waste storage,

and this is a near-field simulation, and they have different materials,

strongly heterogeneous media, and this calls for different time scales in different parts of the domain.

There are large differences in spatial scales and their long-term computations,

so all of this makes these problems quite complicated.

Now, another application that we've looked at for some time now is porous media with fractures.

Now, what do we mean by that? By a fracture, we mean part of the domain.

We mean a particular type of heterogeneity in the domain where the permeability is either much, much higher,

several orders of magnitude higher or several orders of magnitude lower, forming in a barrier.

But also, the fractures are much smaller width than any of your other parameters, spatial measures in the domain.

So, this also complicates modeling them.

Now, some different types of models that are commonly used for fractures are double continuum models derived from homogenization.

A lot of times, people use discrete fracture networks with no exchange with the surrounding rock,

that is just looking at flow in the fractures, but not looking at what any exchange with the rock,

and of course, both of these types of models are very good for certain applications.

But there are other applications where what we call a reduced fracture model with exchange between, with a matrix rock is important.

Okay, in these reduced fracture models, the fracture is modeled as an object of co-dimension one.

If this is the fracture blown up, meaning orders of magnitude so that you can see it,

then by formal asymptotic calculations or simply by averaging across cross-sections of the fracture,

we come up with what we call the reduced models.

Now, these were described in, well, first of all, in this paper for our highly permeable fracture,

because that was our notion of what a fracture was, was a privileged channel.

But then talking about it once, we were told, well, they can be barriers.

So we extended the model to take care of high or low permeability fractures.

And so in these three papers, we introduced these models and analyzed them.

The models we gave were in mixed formulation because also in our group at INRIA, we're always looking at mixed finite element models.

So these analyses were all in for the mixed formulation.

And in fact, having nothing to, not talking at all about fractures of porous media back in 74