Validity of Archie Equation in Carbonate Rocks

Abstract

Archie (1941) found an empirical equation for consolidate sandstone relating some formation parameters, porosity and water saturation. Archie equation is not easy to apply to rocks because formation parameters (a, m and n) are functions of electrical tortuosity. Electrical tortuosity is determined by pore geometry, tortuousity of the pore system and wettability which, affects oil - water distribution in the pores. Carbonate rocks with their complex pore systems and intermediate to oil wettability are difficult to describe petrophysically with a single a, m and n over a reservoir interval. Formation Resistivity Factor (FR) is highly variable. There is often no linear and direct relation between the resistivity index (IR) and formation water saturation (Sw) as the variably sized pores desaturate at different rates. These cases are labeled as non-Archie rocks. Archie is strictly valid only when the rock is strongly water wet, clay free and has a more or less uniform pore size distribution. An important issue that Archie equation misses is the fluid critical point for multi - component state in which different phases co-exist. It is not clear how at reservoir conditions oil and water coexist in the pore system. Introduction Archie (1941)(1) introduced an equation, which combines resistivity index (IR) and Formation Resistivity Factor (FR) in order to calculate water saturation. Using this equation, water saturation is commonly computed. Archie equation requires the values of cementation exponent ‘m’, saturation exponent ‘n’, and the rock consolidation factor ‘a’. He derived two empirical relationships; the formation resistivity factor (FR) which is related to porosity, and the resistivity index (IR) which is related to the water saturation. The equation was not a precise one as he mentioned and was only an approximate relationship. The conventional procedure to determine ‘m’ and ‘n’ parameters is by crossplot techniques. Plotting Formation Resistivity Factor (FR) versus core or log porosity on the log-log paper is used to find ‘a’ and ‘m’ values. The value of ‘m’ is the slope and ‘a’ is the intercept. However, in carbonate rocks most of the points are scattered and one slope cannot be driven. The same situation occurs when water saturation is plotted against resistivity index (IR) to find the value of ‘n’. In his paper, Archie mentioned that formation resistivity (Rt), as a quantitative value is not accurate as a result of the effect of borehole size, mud and its filtrate, bed thickness, wellbore deviation and connate water salinity. The logging industry has made great strides in obtaining accurate information on porosity and resistivity but has failed to narrow the uncertainty for ‘a’, ‘m’, and ‘n’ parameters. In clean, clastic quartz reservoirs, the industry has been able to apply a constant ‘a’, ‘m’ and ‘n’ with some success. Carbonates present a much more difficult situation. Their pore geometry is often complex and infinitely variable foot by foot in the reservoir. The application of constant Archie parameters in carbonates results in inaccurate and occasionally "just plain wrong" water saturation. The industry has for several decades sought a down-hole cementation exponent predictor through various mathematical and empirical models. One method determined ‘m’ in the flushed zone and then applied the results to the virgin zone(2). This method assumed the saturation exponent was equal to the cementation exponent. The effect of the mud cake, the borehole conditions and incomplete invasion complicated the results. The presumption of ‘n’ equal to ‘m’ has proved false(3). Table 1 presents a failed attempt to relate visual examinations of the core pore systems to the cementation exponent in lieu of special core analysis (SCAL) data. There is currently no reliable way to determine in-situ what the continuing varying values of this cementation exponent are. Some researcher(4) found that the values of ‘m’ and ‘n’ only represent two numbers that minimize the error function of the equation. In fact Archie(5) emphasized that rocks are heterogeneous and their characteristics cannot be expressed as a real mathematical equation. The effect of pore structure(6), rock wettability(7), hysteresis, type of oil(8), dispersion(9), pressure, temperature, and effective stress(10) on ‘m’ and ‘n’ parameters were variably addressed by other researchers.

A cohesive approach at estimating water saturation in a low-resistivity pay carbonate reservoir and its validation

“…Archie is only valid when the rock is water wet and free of clay with a uniform pore size distribution (Talabani et al 2000). Archie equation parameters (a, m, n) are functions of electrical tortuosity, which is related to the pore geometry and wettability (Talabani et al 2000).…”

“…Archie equation parameters (a, m, n) are functions of electrical tortuosity, which is related to the pore geometry and wettability (Talabani et al 2000). Carbonates are well known with their complex pore geometry and varying wetting characteristics, resulting in variations of the Archie exponents across the reservoir.…”

On the Concept and Challenges of Water Saturation Determination and Modeling in Carbonate Reservoirs

“…Extensive laboratory studies has shown that Non-Archie behaviour could be attributed to multiple rock types, heterogeneous formations and anisotropic properties, multiniodal pore size, intragranular or micro-porosity, wettability, saturation history and hysteresis effect (drainage vs. imbibition) in the reservoir [3,4]. The main difficulty is to relate the electrical response from logging suites to the appropriate rock parameters and the in-situ dynamic mechanisms.…”

Enhanced characterization of reservoir hydrocarbon components using electromagnetic data attributes

“…Using three different techniques, conventional Archie's parameter techniques, core Archie's parameter estimation technique and a three-dimensional technique, the resulting profiles exhibited significantly differing water saturation values that were attributed to the uncertainty levels in the determination of Archie's parameters. Talabani et al [36] investigated the validity of Archie's equation for carbonate rock formations and concluded that the so called cementation factor m in Archie's equation is influenced by multiple factors, and may significantly differ for complex pore systems. Maute [37] outlined a data-analysis method for obtaining optimal Archie's parameter with reduced uncertainty for the general formation, outlining the challenges and variation in the parameters for a general rock reservoir formation.…”