A multiscale method for modeling transport in porous media with multiscale structures
Creators: name: Aarnes, Jørg
name: Efendiev, Yalchin
name: Lie, Knut-Andreas
name: Aarnes, Jørg
Publisher: name: XVI International Conference on Computational Methods in Water Resources
Year: 2006

description: This talk will focus on simulation of flow in strongly heterogeneous oil reservoirs.
The main purpose of reservoir simulation studies is to visualize flow scenarios and
estimate production characteristics associated with a particular well configuration.
This information is then used to determine well locations that maximize oil recovery.

Oil reservoirs may stretch across a few kilometers horizontally,
but are usually no more than one hundred meters in the vertical direction.
To produce oil from the reservoir, one drills wells into the rock formations.
At specified locations, water or gas is pumped into the reservoir creating
a pressure build up that drives the reservoir liquids towards production wells.
To conduct reservoir flow simulations, one starts with a geological model that
specifies the distribution of permeability and porosity in the reservoir. A
geological model may consist of 10--100 million grid blocks that can be
10--50 m in the horizontal direction, but only 0.1--1 m in the vertical direction.
The distribution of permeability in geological models is often characterized by a
very large span of scales. In fact, it is not uncommon that permeability values span
across multiple orders of magnitude across short distances. This poses a
continuing challenge to flow simulation since both small-scale and large-scale
heterogeneous structures can have a strong impact on global flow patterns.

Reservoir simulation studies are mostly conducted on
coarsened models with less parameters. These models are obtained
by using so-called upscaling procedures that are designed specifically for coarse
scale simulations of flow in porous media. Along with the current enthusiasm for
multiscale methodologies, attempts have been made to avoid upscaling by using
multiscale numerical methods. Several promising multiscale methods have been proposed
for the pressure equation that produce accurate velocity fields on a fine scale.
Combined with the use of an efficient numerical technique for solving the saturation
equation at the fine scale, this
approach provides an appealing alternative to upscaling based simulation.
However, the computational cost of solving the saturation equation at the fine scale
may be prohibitively expensive.

Here we propose a multiscale method for the saturation equation. The basic idea is
to model the transport on a coarse grid, but unlike upscaling methods, we use
velocities and saturation values at the fine scale in a rigorous manner to account
for fine scale features in the flow scenario. Hence, once a new saturation
solution is obtained on the coarse grid, it is mapped onto a plausible fine scale
saturation profile. Another key difference between this approach and upscaling
techniques for the transport problem, is that here both gravity and capillary forces
are modeled. Numerical results are presented for a difficult
Benchmark test-case to demonstrate the capabilities of the method.