HDTHM associate team
Mathematical and numerical methods for thermohydromechanical models in porous media with discontinuities
 Teams: the project is a collaboration between the joint InriaLJAD team Coffee, Nice, France and the School of Mathematics at Monash University, Australia
 Members of InriaLJAD team Coffee involved in the project:
Roland Masson (PI):
 Professor at the department of Mathematics J.A. Dieudonné, University Nice Sophia Antipolis.
 Member of the joint InriaLJAD team Coffee.
 Domain of expertise: finite volume discretization of PDEs, discretization of multiphase Darcy flows in heterogeneous and fractured porous media, formulation and discretization of multiphase compositional Darcy flows, multiphysics coupling algorithms and domain decomposition methods, applications to reservoir, basin, geothermal systems, and geological storage modelling.
Konstantin Brenner:
 MCF at the department of Mathematics J.A. Dieudonné, University Nice Sophia Antipolis.
 Member of the joint InriaLJAD team Coffee.
 Domain of expertise: finite volume discretization of PDEs, convergence of numerical schemes, modeling and discretization of multiphase Darcy flow with strong capillary pressure heterogeneities, accelerated Newton's method for nonlinear PDEs.
Laurent Monasse :
 CR at InriaSophia Antipolis Méditerranée.
 Member of the joint InriaLJAD team Coffee.
 Domain of expertise: finite volume discretization of PDEs,
fluidstructure interaction, solid mechanics, fracture dynamics,
Discrete Element methods.
El Houssaine Quenjel :
 Postdoctoral researcher at the department of Mathematics J.A. Dieudonné, University Nice Sophia Antipolis. Funded by the ANR project CHARMS.
 Member of the joint InriaLJAD team Coffee.
 Domain of expertise: discretization and numerical analysis for compressible gas liquid Darcy flows in porous media, positive finite volume schemes for nonlinear degenerate parabolic equations and two phase Darcy flows.
 Members of the School of Mathematics, Monash University involved in the project:
Jérome Droniou (coPI) :
 Associate Professor in the School of Mathematics, Monash University, Australia
 Domain of expertise: finite volume and hybrid highorder methods for linear and nonlinear models, convergence analysis of schemes for nonlinear and coupled models  including multiphasic flows in fractured networks, discretisation and analysis of advectiondominated models, theoretical analysis of partial differential equations and its discrete translations to numerical schemes.
KimNgan Le :
 PostDoc researcher in the School of Mathematics, Monash University, Australia. Funded by the Australian Research Council (Discovery Project number DP170100605; lead CI: A/Prof. J. Droniou), 01/12/1731/12/19 with options for extension.
 Domain of expertise: theoretical and numerical analysis of stochastic partial differential equations, numerical methods for miscible porous media flows, characteristic methods for advection models.

Context of the HDTHM project:
Many real life applications in the geosciences involve processes like multiphase, nonisothermal flow or hydromechanical coupling in heterogeneous porous media. Such mathematical models are commonly coupled systems of partial differential equations, including nonlinear and possibly degenerate parabolic ones. Next to the inherent difficulties posed by such equations, further challenges are due to the heterogeneity of the medium and the presence of discontinuities like fractures. This has a strong impact on the complexity of the models, challenging their mathematical and numerical analysis and the development of efficient simulation tools.
This collaboration focuses on the so called hybriddimensional matrix fracture models obtained by averaging both the unknowns and the equations in the fracture width and by imposing appropriate transmission conditions at both sides of the matrix fracture interfaces. Given the high geometrical complexity of real life fracture networks, the main advantages of these hybriddimensional compared with fulldimensional models are to both facilitate the mesh generation and the discretisation of the model, and to reduce the computational cost of the resulting schemes. This type of hybriddimensional models is the object of intensive researches since the last 15 years due to the ubiquity of fractures in geology and their considerable impact on the flow and transport of mass and energy in porous media, and on the mechanical behavior of the rocks.

Objectives of the HDTHM project: Hybriddimensional matrix fracture models combine geometrical complexity with highly contrasted properties and constitutive laws at the matrix fracture interfaces leading to strong nonlinear couplings and a large range of space and time scales. It leads to new challenges in terms of mathematical analysis, discretization, nonlinear solvers and preconditioners.
Building up on our successful collaboration on hybriddimensional twophase flow models, the objective of this project is to design novel numerical methods for two classes of hybriddimensional matrix fracture models, motivated by applications to geothermal systems.
On the one hand, we will focus on thermohydro models with nonlinear coupling of the porous media variable density flow with the energy conservation equation.
On the other hand, we will consider hydromechanical models that
couple the hybriddimensional porous media flow with the mechanical
deformation of the matrix. For such models, the flow in the
fractures has a strong nonlinear dependence upon the fracture width,
resulting from the matrix mechanical deformation which itself depends
on the fluid pressure in the fractures.

Visits:
 Jérome Droniou, El Houssaine Quenjel, Konstantin Brenner and Roland Masson will jointly participate to the workshop POEMS (POlytopal Element Methods in Mathematics and Engineering), 29th april  3rd may 2019, CIRM, Marseille, France.
 Konstantin Brenner and Roland Masson will visit Jérome Droniou at Monash University, Australia, for one month in july 2019

Joint publications:

Related projects:
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