Evaluation of Normalized Stone's Methods for Estimating Three-Phase Relative Permeabilities

Abstract

This paper examines normalized forms of Stone's two methods for predicting three-phase relative permeabilities. Recommendations are made on selection of the residual oil parameter, S om, in Method I. The methods are tested against selected published three-phase experimental data, using the plotting program called CPS-1 to infer improved data fitting. It is concluded that the normalized Method I with the recommended form for S om, is superior to Method II. Introduction Stone has produced two methods for estimating three-phase relative permeability from two-phase data. Both models assume a dominant wetting phase (usually water), a dominant nonwetting phase (gas), and an intermediate wetting phase (usually oil). The relative permeabilities for the water and gas are assumed to permeabilities for the water and gas are assumed to depend entirely on their individual saturations because they occupy the smallest and largest pores, respectively. The oil occupies the intermediate-size pores so that the oil relative permeability is an unknown function of water and gas saturation. For his first method, Stone proposed a formula for oil relative, permeability that was a product of oil relative permeability in the absence of gas, oil relative permeability in the absence of gas, oil relative permeability in the absence of mobile water, and some permeability in the absence of mobile water, and some variable scaling factors. He compared this formula with the experimental results of Corey et al., Dalton et al., and Saraf and Fatt. The formula is likely to be most in error at low oil relative permeability where more data are needed that show the behavior of residual oil saturation as a function of mixed gas and water saturations. In particular, the best value for the parameter S om that occurs in the model is not well resolved. In his second method, Stone developed a new formula and compared it against the data of Corey et al., Dalton et al., Saraf And Fatt, and some residual oil data from Holmgren and Morse. Stone suggested that his second method gave reasonable agreement with experiments without the need to include the parameter S om. If in the absence of residual oil data, S om = 0 is used in the first method, the second method is then better than the first method, although it tends to under predict relative permeability. Dietrich and Bondor later showed that Stone's second method did not adequately approximate the two-phase data unless the oil relative permeability at connate water saturation, k rocw, was close to unity. Dietrich and Bondor suggested a normalization that achieved consistency with the two-phase data when k rocw, was not unity. This normalization can be unsatisfactory because k roc an exceed unity in some saturation ranges if k rocw is small. More recently this objection has been overcome by the normalization of Method II introduced by Aziz and Settari. Aziz and Settari also pointed out a similar normalization problem with Stone's first method and suggested an alternative to overcome the deficiency. However, no attempt was made to investigate the accuracy of these normalized formulas with respect to experimental data. In the next section of the paper we review the principal forms of Stone's formulas, and introduce some new ideas on the use and choice of the parameter S om. Later we examine the first of Stone's assumptions that water and gas relative permeabilities are functions only of their respective saturations for a water-wet system. This involves a critical review of all the published experimental measurements. Earlier reviews did not take into account some of the available data. Last, we examine the predictions of normalized Stone's methods for oil relative permeability against the more reliable experimental results. It is concluded that the normalized Stone's Method I with the improved definition of S om is more accurate than the normalized Method II. Mathematical Definition of Three-Phase Relative Permeabilities We briefly review the principal forms of the Stone's formulas that use the two-phase relative permeabilities defined by water/oil displacement in the absence of gas, k rw = k rw (S w) and k row = k row (S w) and gas/oil displacement in the presence of connate water, k rg = k rg (S g) and k rog = k rog (S g). SPEJ p. 224

Compositional Three-Phase Relative Permeability and Capillary Pressure Models Using Gibbs Free Energy

“…This simplification requires the two-phase parameters such as residual saturations to be corrected before being applied in a three-phase model (e.g. Fayers and Matthews 1984, Aleman 1986, Fayers 2000and Yuan and Pope 2012. The other unavoidable modification is related to hysteresis.…”

“…This modification was extended later by Beygi et al (2015) to reflect cycle dependency of saturation hysteresis in a three-phase flow. Fayers et al (1984) added the compositional effect to a modified Baker's model and used phase densities to ensure a smooth transition in oil-water and gas-water relative permeabilities.…”

“…Three phase residual saturations are typically less than two phase residual saturations (Fayers andMatthews 1984, Aleman 1986). Yuan and Pope (2012) modeled three phase residuals based on two phase residual saturations and the saturations of the other two phases as follows:…”

Multiple Viscous Solutions for Systems of Conservation Laws

“…Systems of conservation laws of mixed elliptic-hyperbolic type arise in the theory of three-phase flow in porous media (for example in Stone's permeability model) [4], widely employed in petroleum reservoir engineering [10]. They arise in many other applications as well, including transonic flow in gas dynamics [9].…”

“…The first is that the viscous profile entropy criterion which we employ is the most selective in common usage and takes into account effects usually neglected when physical systems are modeled by conservation laws. The second reason is that this nonuniqueness occurs in a relatively simple mixed-type model with quadratic polynomial flux functions, which captures essential features of Stone's permeability model ( [10], [4]). This model is a description of immiscible three-phase flow commonly used in petroleum reservoir engineering.…”