In multi-phase flow simulation, the prevailing approach to discretizing flux
terms treats the elliptic and the hyperbolic terms in the equations separately.
With this concept, the flux is calculated analogously as for single phase flow,
and later multiplied by the upstream mobility. This approach is valid when the
mobility is a scalar quantity, which is the case for most traditional models.
However, tensorial relative permeability (and thus tensorial mobilities) in
general arise on all scales, as is seen in both laboratory and field
experiments. Furthermore, upscaling methods almost invariably lead to
anisotropic relative permeabilities. The saturation dependency of the fluid
permeability tensor means that the upstream direction is no longer uniquely
defined, which challenges common numerical schemes. In this work, we give
examples of how the relative permeability has preferential flow directions on
different scales. We then study how to incorporate tensorial relative
permeability fields into control volume methods. In particular, we address two
immediate challenges: Firstly, the non-determinacy of the upstream direction
invalidates the use of upstream weighting for the saturation equation. We
discuss the applicability of Godunov methods to handle flow cases where
interfaces have either no or two upstream directions. Secondly, there is a
marked increase in computational complexity associated with the pressure
equation. We discuss the possibility of mitigating this computational
complexity through mass lumping methods. The validity and computational
efficiency of our approaches is discussed on theoretical terms and with the
support of numerical implementations.
Several studies have shown that the relative permeability function may be
anisotropic at scales ranging from the Representative Elementary Volume (REV)
scale (Corey and Rathjens, 1956), through the laboratory scale (Eichel et al
2005), to the field scale (Yeh et al., 1985a,b; Rustad et al 2008).
Furthermore, in these studies it is frequently observed that a full description
of relative permeability give significant qualitative differences from scalar
models. Nevertheless relative permeability is almost invariably modeled as a
scalar. This is due to two main reasons: Firstly, anisotropic relative
permeability is poorly characterized due to difficulties in conducting both
relative permeability measurements and multi-phase upscaling. Secondly,
saturation dependent anisotropy leads to significant added complexity in terms
of numerical methods and simulation.
In this paper, we begin to address both these points. In Section 2, we discuss
the understanding that exists regarding relative permeability (in the
continuation, we will always imply that relative permeability is anisotropic,
unless otherwise stated). In particular, we show using a network flow model how
anisotropy arises even at the finest continuum scale, and subsequently use a
vertically segregated system to highlight how not only the aspect ratio of
anisotropy can change with saturation, but also the principle directions.
In Section 3, we highlight some of the main challenges that arise with a full
description of relative permeability, using IMPES time-stepping with
control-volume spatial discretization as a backdrop for our discussion.
Subsequently, we introduce new ideas for accelerating the pressure solver, as
well as an approach that allows for the saturation equation to be solved
Section 4 combines the work of the preceding two sections within numerical
implementations. These illustrate A) Flow patterns that are unique to full
relative permeability systems and B) The performance of our proposed solution
approaches. The paper is concluded in Section 5.