The simulation of flow and transport processes in a complex porous
media at pore-scale is prohibitively expensive for most real applications.
Two widely used alternatives to the simulation of Navier-Stokes at the
pore-scale are to (i) upscale to core-scale and simulate a Darcy or
non-Darcy model which averages geometric properties through a conductivity
tensor, and to (ii) construct a pore-network model which relies on
approximation of the pore-scale geometric features into pores and throats
along with a relationship between throat length, radius, and conductivity. A
better approach is to combine these three representations of a porous media
into a multi-scale simulation framework. An additional concern in simulating
pore-scale flow stems from geometric uncertainty, both from uncertainty in
imaging as well as when there is porescale modification due to, e.g., hydrate formation or biofilm growth.
This results in pore-scale problems posed on stochastic domains, an area of
numerical analysis that has received increasing attention in recent years.
In this talk we present a three-scale simulation framework for flow in
porous media, which, at the pore-scale, relies on accurate computations of
conductivity. At the pore-scale we model geometric uncertainty through an
immersed boundary representation of hydrate formation or biofilm growth.
Computing statistics of the quantity of interest, the conductivity, in the
setting of random domains is shown to be a considerable computational
undertaking, and we explore a reduced order model for conductivity in the
presence of growth within the fluid domain, which dramatically reduces the
computational complexity of the three-scale solver.
Part of this work is joint with M. Peszynska, A. Trykozko, K. Kennedy, M. Prodanovic, and D. Wildenschild.