Assessment of uncertainty in saturation estimated from archie's equation

Volumetric methods are used to estimate the hydrocarbons in place of a reservoir. They require petrophysical data including porosity 0 and water saturation S_w which can not 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, φ, cementation factor, m, tortuosity factor, a, saturation exponent, n, true reservoir resistivity, R_t, and resistivity of formation water, R _w. 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 S_w estimated by Archie's equation. Error propagation equations (based on a Taylor series expansion of S_w around the mean values of the input paramaters) 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 faciés, can increase or decrease uncertainty in S _w and hence, ignoring existing correlation among the input parameters may lead to incorrectly characterizing the uncertainty in S_w. 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 S _w 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 S_w distributions are normal. However, it is well known that Monte Carlo methods are more general as they provide rigorous sampling of the response function (S_w 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.