Description
Speaker: Jerome Droniou, Monash university, Melbourne.
Title: ELLAM schemes for a model of miscible flow in porous medium: design
and analysis.
Abstract: Tertiary oil recovery is the process which consists in injecting
a solvent through a well in an underground oil reservoir, that will mix
with the oil and reduce its viscosity, thus enabling it to flow towards a
second reservoir. Mathematically, this process is represented by a coupled
system of an elliptic equation (for the pressure) and a parabolic equation
(for the concentration).
The parabolic equation is strongly convection-dominated, and discretising
the convection term properly is therefore essential to obtain accurate
numerical representations of the solution. One of the possible
discretisation techniques for this term involves using characteristic
methods, applied on the test functions. This is called the
Eulerian-Lagrangian Localised Adjoint Method (ELLAM).
In practice, due to the ground heterogeneities, the available grids can be
non-conforming and have cells of various geometries, including generic
polytopal cells. Along with the non-linear and heterogeneous/anisotropic
diffusion tensors present in the model, this creates issues in the
discretisation of the diffusion terms.
In this talk, we will present a generic framework, agnostic to the
specific discretisation of the diffusion terms, to design and analyse
ELLAM schemes. Our convergence result applies to a range of possible
schemes for the diffusion terms, such as finite elements, finite volumes,
discontinuous Galerkin, etc. Numerical results will be presented on
various grid geometries.