Archie's empirical resistivity-saturation relation of 1942 is widely applied in the petroleum industry. Despite its shortcomings, Archie's equation is the basis for inferring water saturation even in carbonates with complex pore structure, albeit with empirical tweaking of cementation and saturation exponents. Industry literature is replete with examples of why this approach leads to erroneous estimates of the water saturation, and methods have been proposed where the range of saturation present in the reservoir has been subdivided into segments each having a different set of exponents. Here, based on a homogenization methodology, we propose a simplified model for effective resistivity of an inter- and intra-granular vuggy carbonate, when the pore sizes in the subsystems are well separated. Methodologies for apportioning pore fractions and their sizes depend on the open-hole logs and/or core data. Computed results show significant deviations from Archie correlations in microporous or vuggy intervals.
Archie (1942) proposed a relationship between the formation water saturation, Sw, and its true resistivity, Rt, porosity, ϕ, and the resistivity of the aqueous phase, Rw, as shown in Eq. 1:
where m and n are the cementation and saturation exponents, respectively. Archie suggested that both m and n be set to a value of 2. Commonly, the ratio of R0, the resistivity for aqueous saturation of unity, to Rw is termed the formation factor (Equation).
Archie further shows that the ratio of the resistivity, R, of a rock partially saturated with brine to R0 is equal to the inverse of the water saturation, Sw, raised to the exponent, n. Subsequently this ratio, Rt/R0, came to be known as resistivity index.
In many carbonates, m and n take on values much different than 2, and are thought to be saturation dependent (Sweeney and Jennings, 1960; Lucia, 1983; Focke and Munn, 1987; Dixon and Marek, 1990). Keller (1953), Sweeney and Jennings (1960), Morgan and Pirson (1964), Mungan and Moore (1968), Donaldson and Siddiqui (1989) and Tsakiroglou and Fleury (1999) advocate relating saturation exponent to wettability state. Dixon and Marek (1990) and Ma et al. (2005) suggested that the saturation cycle in the reservoir be first identified and different set of Archie parameters be applied to each identified cycle. Although the exponent may differ with wettability and, more importantly, with directionality of saturation change, particularly in oil-wet media, the apparent variability of the exponent due to microscopic structure remains to be quantified.