Scaled Fluid-Flow Models With Permeabilities Differing From That Of The Prototype
Authors: Bentsen R.G. 
Published: March 2010

Abstract
Laboratory studies, if they are to be useful in predicting fluid flow behaviour in petroleum reservoirs, must be scaled. 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 prototype. This paper shows that this restriction may be eliminated, provided certain conditions are met. In particular, the breakthrough displacement efficiency in a linear system may be considered to be a function of mobility ratio only, providedthe transition zone is short, and provided the assumption underlying Bukley-Levrett theory are not seriously violated. The paper also demonstrates that evaluating the displacing-phase mobility at the average saturation in the swept region is equivalent (albeit approximately) to accounting for two-phase flow in that region.

Introduction
If information gathered through laboratory model studies is to be useful in predicting fluid flow behaviour 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 Rapoport(1) 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, because the more permeable model sands may have irreducible water and residual oil saturation very different from those in the prototype. Moreover, if model fluids areselected 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, very different flow regimesmay exist in the model and the prototype. It is just this problem which led Craig, et al.(4) 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 this regard, it is suggested that normalized rather than relative permeabilities be used, as this will result in the automatic choice of the mobility ratio as a scaling parameter. It is the purpose of this paper to show that the mobility ratio has a preponderant effect on the displacement process in a linear system, and that, as a consequence, it is possible to eliminate, under certain circumstances, the requirement that the relative per meabilities (or their ratio) be the same in the model ami the prototype.

Derivation of Scaling Criteria
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.