Macroscopic equations for flow in unsaturated porous media

M.R. Hoffmann

Research output: Thesisinternal PhD, WU


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.

Original languageEnglish
QualificationDoctor of Philosophy
Awarding Institution
  • Wageningen University
  • Leijnse, Toon, Promotor
  • Troch, P.A.A., Promotor, External person
Award date24 Sept 2003
Place of Publication[S.l.]
Print ISBNs9789058088932
Publication statusPublished - 24 Sept 2003


  • fluid mechanics
  • porous media
  • unsaturated flow
  • flow
  • pore volume
  • mathematical models
  • two-phase systems


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