Assessment of Uncertainty in Saturation Estimated from Archie's Equation

Abstract

Volumetric methods are used to estimate the hydrocarbons in place of a reservoir. They require petrophysical data including porosity ø and water saturation Sw which cannot be directly measured but must be inferred from other measurements. For example, water saturation is conventionally obtained from the Archie's equation, which requires inputs of porosity, ø, cementation factor, m, tortuosity factor, a, saturation exponent, n, true reservoir resistivity, Rt, and resistivity of formation water, Rw. Uncertainties in these input values directly affect the accuracy of water saturation estimation. In this paper, we investigate the propagation of uncertainties of these input parameters into Sw estimated by Archie's equation. Error propagation equations (based on a Taylor series expansion of Sw around the mean values of the input parameters) are derived for uncertainty characterization. Two cases are considered for the relationship between the input parameters; correlated or completely independent. It is shown that correlation among the input parameters, which may be due to different rock facies, can increase or decrease uncertainty in Sw and hence, ignoring existing correlation among the input parameters may lead to incorrectly characterizing the uncertainty in Sw. In addition, the dimensionless sensitivities and relative uncertainties of the input parameters, derived from the error propagation equations, clearly identify which of the input parameters will dominate the total uncertainty in Sw computations. Monte Carlo methods were used to verify the developed error propagation equations. A comparative study shows that the error propagation method is a good first approximation for uncertainty analysis, especially if the resulting Sw distributions are normal. However, it is well known that Monte Carlo methods are more general as they provide rigorous sampling of the response function (Sw in our problem) by accounting for the nonlinearity existing between the response function and its input parameters, and thus should be used for accurately quantifying the uncertainty. Introduction Uncertainties in petrophysical properties are widely accepted, but rarely applied in formation evaluation and reservoir characterization. Without proper consideration of these uncertainties, estimation of original oil in place (OOIP) can be in error (Cronquist 2001). This is particularly important for reservoirs with large reserves, such as those super giant fields in the Middle East. In order to calculate water saturation from Archie's equation (Archie 1942), the porosity, f, cementation factor, m, tortuosity factor, a, saturation exponent, n, true reservoir resistivity, Rt, and resistivity of formation water, Rw, are needed. These input values have uncertainties associated with them, which will result in an uncertainty in the estimated Sw. Main sources of uncertainty in the determination of porosity from well logs are measurement errors, selected models, and uncertainties in input parameters related to the models (Verga et al. 2002; Adams 2005). The parameters m and n are normally obtained from electrical measurements made on core plugs. For microscopically heterogeneous media, particularly carbonates, expressions for m have been derived in the literature in terms of the intragranular and vuggy components (Ramakrishnan et al. 2001). Changes in the cementation factor, m, and saturation exponent, n, are often difficult to quantify (Eyvazzadeh et al. 2003). The uncertainties in m and n will propagate into the uncertainty in Sw. Rt must be determined from log data, and quantification of uncertainties in Rt may be difficult. Rw is one of the most difficult variables to obtain and may have a high degree of uncertainty. Eyvazzadeh et al. (2005, 2007) demonstrate how the errors in saturation values can be calculated based on the resistivity measurements. The parameters in the Archie equation such as f, m, n, Rt, and Rw may be associated with high levels of uncertainty. Generally, petrophysical interpretations do not account for uncertainty thus decisions made based on deterministic values may result in improper engineering decisions such as design of facilities.