10.4122/1.1000000714
Dahle, Helge K.
Helge K.
Dahle
helge.dahle@mi.uib.no
Hassanizadeh, S. Majid
S. Majid
Hassanizadeh
hassanizadeh@geo.uu.nl
Nordbotten, Jan M.
Jan M.
Nordbotten
jan.nordbotten@mi.uib.no
Celia, Michael A.
Michael A.
Celia
celia@princeton.edu
Nordbotten, Jan M.
Jan M.
Nordbotten
jan.nordbotten@mi.uib.no
Defining macro-scale pressure from the micro-scale
XVI International Conference on Computational Methods in Water Resources
2006
2006
Micro-scale models have proven to be powerful theoretical tools in groundwater flow
and transport modelling. In addition to being useful in estimating traditional
parameters, such as (relative) permeability and capillary pressure functions, micro-
scale models have recently provided insight into complex multi-phase flow
phenomena, such as the so-called dynamic capillary pressure, and are central in
investigating theoretical developments in multi-phase flow modelling.
To transfer the results of a micro-scale model to larger scales, a proper
definition of macro-scale variables in terms of micro-scale quantities is crucial.
One such variable is the pressure. Traditionally, macro-scale pressure of a given
phase is defined in terms of the intrinsic phase average; i.e. the average of micro-
scale pressure weighted by the volume of the phase. We show, by averaging of micro-
scale momentum equations, that the macro-scale pressure in the Darcy equation is
not necessarily the intrinsic phase average of its micro-scale equivalent. This
will be the case if there are gradients of porosity or saturation in the system,
and these gradients lead to non-negligible changes on the scale of the averaging
volume. We have formulated a modified interpretation of macro-scale pressure. The
implications of this modification for parameters on the macro-scale are
significant, in particular for dynamic relative permeability and capillary
pressure. We show that recent interpretations of dynamic capillary pressure can
change significantly when this modified definition of macro-scale pressure is used.
We also show, through simple example calculations, that inadmissible relative
permeability values (e.g. values larger than 1) can result when using the standard
average to define macro-scale phase pressures, but that no such problems arise with
the new pressure definition. These simple calculations also imply that dynamic
capillary pressure effects may arise with the standard average, which do not appear
with the new pressure definition. The new pressure definition is further
investigated in the more complex setting of bundle-of-tubes and dynamic network
models.