A New Method For Estimating Relative Permeabilitiies From Unstabilized Displacement Data
Authors: Sarma Hemanta K. , Bentsen Ramon G. 
Published: March 2010

Abstract
The water-oil relative permeability characteristics oj a reservoir rock are among the more important flow properties needed to undertake a water flood prediction. The explicit external-drive (unsteady-state) methods provide a quick means for obtaining such in formation. However, because of the rapid changes in saturation which take place in the immediate vicinity of a displacement front, it is not possible to estimate relative permeabilities over the entire saturation range of interest when using these methods. This limitation can be removed, if data from an unstabilized displacement may be used. Use of such data is permissible provided (i) saturation profiles and pressure gradients in both phases can be measured directly, (ii) the explicit external-drive theory is modified to account for the existence of capillary pressure gradients, and (iii) and the analysis is carried out from a Lagrangian rather than an Eulerian point of view.
This paper shows how unstabilized displacement data may be used to generate relative permeability curves over the entire saturation range of interest. Also, it is demonstrated that it is acceptable to use equilibrium capillary pressure data to predict the pressure difference between water and oil in a dynamic displacement provided the displacement is stable. Moreover, it is verified that, if a displacement is unstabilized, the fraction of water flowing at a particular location along the core is a function of both time and saturation. In addition, the techniques which enabled implementation of the new method are used to investigate the conditions under which the external-drive techniques are valid. Finally, relative permeability curves obtained using the new method are presented and compared to those obtained using the explicit external-drive method.

Introduction
One of the more important properties needed to undertake a waterflood prediction are the water-oil relative permeability characteristics of the reservoir rock. Either steady-state or external-drive techniques may be used to obtain such information. However, because of the long time needed to obtain stabilization when using steady-state methods, the external-drive (unsteady-state) methods are usually preferred.
Unsteady-state methods are based on one-dimensional theory(1, 2, 3). As a consequence, displacements used to measure relative permeability must be conducted in systems which are linear and homogeneous. Moreover, the pressure and saturation must be uniform at each cross section along the length of the core(4). Finally, if a Lagrangian formulation is to be used, it is important that saturation decreases monotonically along the length of the core(5).
External-drive techniques require numerical or graphical differentiation of experimental data. Because inaccuracies in data measurement become amplified by the process of differentiation, there is a need to deal effectively with this problem. In this regard, two approaches may be taken, the implicit and the explicit. If the displacement is unstabilized, or if the endpoint mobility ratio is near one, resort must be had to one of the implicit approaches(6–9), or to the new method developed in this paper. If the displacement is stabilized, it becomes possible to use the simpler, explicit approaches(10–12). Regardless of the approach employed, it is important that the procedures used to implement the method be selected carefully, if the problems associated with differentiating experimental data are to be avoided(12).