One of the more difficult scaling requirements to meet is that the relative permeabilities must be the same functions of saturation in the model and the prototype. Under certain circumstances, this restriction may be relaxed, provided the mobility ratio is defined in an appropriate manner. This paper shows that if normalized rather than relative permeabilities are used in the defining equation for mobility ratio, and if the normalized permeability to the displacing phase is evaluated at the average displacing fluid saturation, then the breakthrough displacement efficiency may be considered to be a function of mobility ratio only, provided the transition zone is short, and provided the assumptions underlying Buckley-Leverett Theory are not seriously violated.
If information gathered through laboratory model studies is to be useful in predicting fluid flow behavior in petroleum reservoirs, careful consideration must be given to scaling criteria.
Where the reservoir fluids may be considered as being immiscible and incompressible, the necessary criteria have been developed by Rapoport1 and others.2,3 Because of the large number of parameters to be scaled, it is not always possible to meet all of the scaling requirements. However, if some of the parameters have a preponderant effect on the process, it is usually possible to relax some of the scaling requirements.
Scaling requirements usually dictate a much higher permeability in the model than exists in the reservoir. Consequently, the scaling requirements pertaining to the dependence of relative permeabilities and dimensionless capillary pressures on saturation are usually not met, since the more permeable model sands may have irreducible water and residual oil saturations very different from those in the prototype. Moreover, if model fluids are selected to give equal viscosity ratios in the model and the prototype, one may find quite different mobility ratios in the two cases and, as a consequence, there may exist very different flow regimes in the model and the prototype. It is just this problem hich led Craig, et al. to correlate the results of their model studies with mobility ratio rather than viscosity ratio.
The difficulty of meeting these scaling requirements may be reduced, provided some care is taken in deriving the similarity groups used. In particular, care must be taken in deriving the particular form of the mobility ratio which is used as a correlating parameter. It is the purpose of this paper to show that, if normalized rather than relative permeabilities are used in the defining equation for mobility ratio, then the breakthrough displacement efficiency can be considered, under certain circumstances, to be a function of mobility ratio only.
For the purpose of this paper, it is sufficient to derive the scaling criteria for the linear displacement of one incompressible fluid by another in a homogeneous, isotropic, porous medium. More complex problems may be handled in an analogous manner.
The equations necessary for describing the displacement of one fluid by another are Darcy's law and the continuity equation. These equations for each phase are as follows: