Abstract

Hoffmann, M.R., 2003. Macroscopic Equations for Flow in
  Unsaturated Porous Media.  Dissertation, Wageningen University, The
Netherlands.


This dissertation describes averaging of microscale flow equations to
obtain a consistent description of liquid flow in unsaturated porous
media on the macroscale.  It introduces a new method of averaging the
pressure term and a unit cell model capable of describing
unsaturated flow.

Starting from the description of liquid flow through individual pores,
a macroscopic equation for flow of a liquid in a porous medium in the
presence of a gas is derived.  The flow is directly influenced by
phase interfaces, i.e.\ solid-liquid or gas-liquid.  By including
these pore scale phenomena in a continuum description of fluid
transport in porous media, equations for liquid flow on the macroscale
are obtained.

The unit cell model is based on a simplified geometric representation
of a porous medium.  It allows for the modeling of the important
characteristics of a porous medium for unsaturated flow.
Through the use of volume averaging and direct integration macroscale
momentum and mass balance equations are derived from the microscale
momentum and mass balance equations, resulting in a novel form of the
macroscale pressure term.  The macroscale flow of liquids in
unsaturated porous media can be written proportional to a driving
force, which is proportional to the difference of the inverse area
averaged liquid pressures across an averaging volume.  In principle
the flow is driven by gradients in liquid pressure, but due to the
nonlinear coupling between capillary forces and liquid pressure the
driving force becomes nonlinear.

Two dynamic terms were derived by simplifying the flow dynamics in a
porous medium.  They remain to be tested quantitatively and still have
considerable uncertainty concerning their exact form and/or magnitude.

Comparison of the newly proposed macroscale equations with the
Buckingham-Darcy equation shows that, using reasonable assumptions,
the newly proposed macroscale equations can be written in a form
similar to the Buckingham-Darcy equation.  The newly proposed
macroscale equations are compared to an experiment and satisfactory
agreement between experiment and calculations was observed.

keywords: momentum balance, nonlocal equations, porous media,
upscaling, unsaturated flow, volume averaging.