Relative Permeability Effects on the Migration of Steamflood Saturation Fronts
Authors: Erkal A. , Numbere D.T. 
Published: December 2007

Abstract
The effects of various relative permeability-saturation relationships on the movement of water saturation fronts during steamflooding is investigated. Empirical correlations, curve fitting and spline interpolation functions are used to represent the relative permeability data. The effects of these correlations on water saturation movement and ultimately on oil recovery performance are compared. It was observed that only the spline function is able to adequately represent a given set of data throughout its saturation range. All the other correlations fit the given data well only at high water saturations and fit poorly at the low saturations. Furthermore, the correlations converge at high water saturations. As a consequence, the water saturation distribution within the hot water zone and its migration through time are nearly identical for the different correlations except for California crude correlation. This is because for steamfloods, the operating water saturations are more likely to be in the high end where all correlations fit well.

Introduction
Oil recovery calculation during steamflooding requires a knowledge of the water saturation distribution in the water zone ahead of the steam zone. This in turn relies on relative permeability-saturation data. The Buckley-Leverett frontal advance equation is typically adapted to track the movements of certain pre-determined saturations through the variable-temperature hot water zone. This is tedious and an approximation that calculates the instantaneous saturations in each temperature zone at any time can be used. Two-phase relative permeability correlations are of three general types;when relative permeability data are available, curve fitting the data with an equation that contains unknown parameters allows the parameters to be determined. Any polynomial expression can be used but a commonly used method is based on the premise that a semi-log plot of relative permeability versus saturation can be approximated by a straight line. This leads to an exponential equation to fit the datathe second type of relative permeability correlations are based on the integration of the area under a given capillary pressure curve. Brooks-Core, Wyllie, Pursell, Burdine, Van Genuchten and many others have derived equations based on this concept. For example, the Brooks-Corey relationships are given in terms of a lithology factor which allows the correlations to be generalized to any lithologythe third type of relative permeability correlation are those that give generalized equations intended for universal application to a region or type of lithology. Honarpour et. al., Gomaa and Polikar present equations of this type. Honarpour et al. further subdivide their equations for different wettability conditions, Polikar gives relative permeability correlations for bitumen. Gomaa gives relative permeability relationships for California crude. Frizell gives expressions that apply to end points only.
For one dimensional analytical steamflood recovery models in homogeneous isotropic reservoirs, the equations by Marx and Langenheim can be used to calculate the position of the steamflood front at all times prior to the critical time tc. Beyond tc Prats and Vogiatzis modified the approximate equation given by Mandl and Volek for the steam from location. This equations was presented in graphical form by Myhill and Stegemeier. For simplicity, the temperature profile in the hot water zone is assumed to drop linearly from the steam temperature to the cold reservoir temperature. The saturation profile in the steam and hot water zones are calculated using the Buckley-Leverett theory where the non-isothermal hot water zone was first divided into an appropriate number of isothermal zones. Several procedures are available to calculate the advance of the saturation profile with time. One method determines a number of characteristic saturations and tracks their movements through time. Unfortunately, when small time steps are used together with a high number of isothermal zones, the tracking process is very time consuming and is best done by a computer especially for lengthy times. Erkal and Numbere have found that at any desired time, the saturations in each temperature zone can be well approximated as follows: For any temperature, equate the thermal front and the saturation front equations and solve for the saturation that satisfies the equality. This is the instantaneous saturation within that temperature at that instant of time. The authors found that this instantaneous saturation makes a very good approximation to the saturation that would have been calculated by tracking from beginning. P. 385^