Page 206 - gas transport in porous media

CHAPTER 11

SCALING ISSUES IN POROUS AND
FRACTURED MEDIA

VINCENT C. TIDWELL
Sandia National Laboratories, P. O. Box 5800, MS 0735, Albuquerque, NM 87185, USA

The continuum hypothesis of rational mechanics forms the basis of most subsurface
flow and transport models. In this approach, the time-and-space dependence of state
variables is expressed in the form of differential balance equations formulated on
the principles of mass, momentum, and energy conservation. To achieve tractable
solutions for the resulting balance equations, simplifying assumptions are used that
typically introduce constitutive properties into the flow/transport equations. These
properties (e.g., permeability, dispersivity) account for the integrated effects of het-
erogeneities and physical processes that occur at scales much smaller than the desired
scale of analysis. Constitutive properties are related not to a discrete point within the
porous media but to a control volume or sample support and are assumed to vary
smoothly enough in time and space so that the resulting balance equations can be
solved by standard analytical/numerical methods of differential equations.
Because of technological and computational constraints, it is rarely possible to
measure constitutive properties at the desired scale of analysis. For this reason, some
averaging or scaling model is required to transfer information from the scale of mea-
surement to the desired scale of analysis. If the averaging process of the particular
property under study were known, the problem would be alleviated. For example,
the average porosity of a volume is simply the arithmetic average of the porosities of
all the samples that constitute it. The simple arithmetic averaging process holds true
for additive variables such as porosity and ore grade. Unfortunately, many constitu-
tive properties (e.g., permeability) are not additive; that is, the scaling process not
only depends on the volume fraction present but other factors as well. These factors
include, but are not limited to, the heterogeneous characteristics (i.e., length scales,
variance, spatial patterns) of the medium (e.g., Gelhar and Axness, 1983; Dagan,
1984; Fogg, 1986), the nature (e.g., linear vs. convergent flow) of the flow field (e.g.,
Desbarats, 1992a; Indelman and Abramovich, 1994; Tidwell et al., 1999), and scale

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C. Ho and S. Webb (eds.), Gas Transport in Porous Media, 201–212.
© 2006 Springer.