This paper presents an extension of Yang's Y-function method (a function of oil fractional flow) aiming to improve performance analysis of a mature waterflood or waterdrive reservoir for any oil-water viscosity ratio condition. The new approach is based on an appropriate mathematical expression that substitutes the traditional semilog linear relationship between the oil-water relative permeability ratio and water saturation. The production decline analytical method to diagnose waterflood performance as proposed by Yang (2009a and 2009b) and derived from the solution of the one-dimensional (1D) frontal advance Buckley-Leverett (BL) equation, applies to reservoirs with water breakthrough starting from semilog linear section of the kro/krw vs water saturation plot and with moderate oil-water viscosity ratios. To solve this flow equation, the fractional flow curve, which is function of oil and water viscosity and also oil and water relative permeability ratio, must be known. The traditional semilog linear relationship of oil-water relative permeability ratio for intermediate water saturation range has been consistently used to solve fractional flow and then the BL equation. Because the water saturation at breakthrough (Swbt) moves to the boundaries of the water saturation range at both very unfavorable and favorable oil-water viscosity ratio, the use of the traditional semilog linear trend to both extreme conditions of oil-water viscosity ratio could lead to unrealistic oil prediction during the productive period of waterflooding or waterdrive. Therefore, a representative oil-water relative permeability ratio and water saturation approach of the reservoir rock is necessary to obtain reliable predictions at and after water breakthrough.
This paper presents the use of the resulting expression of solving the derivative (with respect to water saturation) of the quotient between individual power-law expressions of relative permeability for both oil and water phases, specifically from Corey's correlations of relative permeability curves. The resulting term named B' that substitutes the constant parameter B in the Y-function is presented in this work. Theoretical and field data examples are also presented to illustrate this approach.
Results show that the new approach (the comprehensive Y-function, cY) derives a solution with a slope m that can be different than -1 for a normal displacement efficiency from the log-log plot of Y vs. tD, therefore m value from real field data will reflect the displacement efficiency imposed by reservoir characteristics and/or operational conditions. This new approach also improves reliability when calculating the volumetric sweep efficiency (areal sweep efficiency times vertical sweep efficiency) and predicting both water and oil cut of mature waterflood or waterdrive reservoirs. It also expands the Yang's model application by improving its performance during both very unfavorable and favorable oil-water viscosity ratio, particularly at and after water breakthrough.