Abstract:Computer powers are often not sufficient to model highly complex fine-scale processes on large scales.
Our aim is therefore to develop "upscaling" techniques to allow a simulation of simpler physical processes including subgrid terms which account for the more complex fine-scale processes.
We want to consider two different directions, as you can see in the figure: first, we will concentrate on the upscaling of heterogeneous formations to study the processes and implement
the problem structure and second, we will deal with the upscaling of complex processes such as two-phase - two-component processes for a two-phase simulation in homogeneous media.
The final goal is to unify the two directions to consider the upscaling of complex processes in heterogeneous media.
Further extensions of this model concept are possible.
Especially, we could consider complex fine-scale processes other than the treated two-phase - two-component processes, such as isothermal - non-isothermal systems.
We want to use a higher order discontinuous Galerkin (DG) method for an IMPES (implicit pressure - explicit saturation) formulation of two-phase flow,
respectively an IMPESC (implicit pressure - explicit saturation and concentration) formulation of two-phase -- two-component flow.
For the solution of the pressure equation, fast multigrid methods exist.
The solution of the saturation equation will be performed using the DG space discretization with higher order explicit Runge-Kutta schemes for the time discretization.
The advantages of the DG method are not only a fast solution with low numerical dispersion which is an especially crucial point for the simulation on coarse meshes, but also local mass conservativity,
flexible mesh handling and applicability to elliptic, parabolic, and hyperbolic problems.
Oscillations due to the higher order schemes will be overcome using slope limiters.
To achieve our first goal concerning the upscaling of two-phase flow in heterogeneous media, we will take the following steps.
First, we will consider a linear two-phase problem, without gravity, capillary effects, sources, and with constant phase densities. In the second step, we will extend it to a nonlinear problem.
Next, we will include capillarity and in the last step, we will extend our model to the compressible case.
Part I: Upscaling of Heterogeneous Formations
Concerning the upscaling of these equations, approaches for the first two steps exist for the linear case and for the nonlinear case.
For the other two steps, approaches still have to be developped.
To achieve our second goal, we want to develop an upscaling technique based on the IMPESC formulation of the two-phase - two-component model problem.
We will first develop a model concept allowing the description of two-component processes in a two-phase framework and then implement it into the upscaling framework prepared in part I.
Part II: Upscaling of complex fine-scale processes