A Multi-dimensional Approach to Scaled Immiscible Fluid Displacement
Authors: Demetre George P. , Bentsen Ramon G. , Flock Donald L. 
Published: March 2010

Abstract
Laboratory studies must be scaled, if they are to be useful in predicting fluid flow behaviour in petroleum reservoirs. To this end, an extended set of scaling groups has been derived which includes all of the variables in the immiscible fluid displacement problem. Analysis of the conditions under which various scaling groups may be neglected suggests that the geometric factor need not be considered when comparing a model to its prototype, provided the displacement is stable. To verify this result, and to investigate the effect of stability on the immiscible displacement problem, a series of immiscible displacements was carried out, using cores having various lengths and diameters, in both water-wet and oil-wet systems. The experimental results show that the geometric factor is not important, provided that the displacement is either stable or pseudo-stable. Moreover, it is demonstrated that neither the linear nor the radial scaling groups were, by themselves, adequate as correlating parameters in the transition zone between stable and pseudo-stable displacement (13.56&lt;N,&lt;900). Finally, the experimental results corroborate earlier experimental work, delimiting stable and unstable displacements, undertaken in this laboratory.

Introduction
If model studies are to provide useful information concerning the displacement of one fluid by another in a petroleum reservoir, flow in the model and the prototype must be similar. Thus, in order that kinematic similarity may be achieved, it is usual to require that the model be geometrically similar to the prototype. Moreover, if transient phenomena occur in the model, it is important to bear in mind that phenomena in the model and the prototype must be compared at homologous times.
Also of importance is the requirement that the model and the prototype be dynamically similar. In the fluid displacement studies considered here, this can be achieved by requiring that the ratio of the viscous forces (M r), the ratio of the gravitational forces to the viscous forces (Ng), and the ratio of the capillary forces to the viscous forces (N) have the same values in the model and the prototype.
Model studies of the kind described above always contain implicit assumptions as to the nature of the physical system under study. Thus, it is implicitly assumed that the flow regime is the same in the model and the prototype. Moreover, the implicit assumption is made that Darcy's law is obeyed in both systems. It is important that such assumptions be recognized.
Ideally, the model and the prototype should be completely similar. That is, every dimensionless product should have the same value for the model as for the prototype. Usually it is not feasible to impose complete similarity in a model test. Consequently, some of the independent dimensionless variables, which are believed to have secondary influences, are allowed to deviate from their correct values. It is important to know when such departures from complete similarity are justified.
Dimensionless products may be derived by either dimensional analysis or inspectional analysis techniques.