Defining macro-scale pressure from the micro-scale

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XVI International Conference on Computational Methods in Water Resources (CMWR-XVI) Ingeniørhuset

Defining macro-scale pressure from the micro-scale
Paper
Author:Helge K. Dahle <helge.dahle@mi.uib.no> (University of Bergen)
S. Majid Hassanizadeh <hassanizadeh@geo.uu.nl> (Utrecht University)
Jan M. Nordbotten <jan.nordbotten@mi.uib.no> (University of Bergen)
Michael A. Celia <celia@princeton.edu> (Princeton University)
Presenter:Jan M. Nordbotten <jan.nordbotten@mi.uib.no> (University of Bergen)
Date: 2006-06-18     Track: Special Sessions     Session: Pore-Scale Modelling: New Developments And Applications
DOI:10.4122/1.1000000713
DOI:10.4122/1.1000000714

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.