The bulk volume fraction of conducting brine b in a reservoir rock is given by the product of porosity and water saturation, b = φSw, symmetric in φ and Sw. Bulk conductivity σt of a rock varies as this product varies. When b = 1 bulk conductivity necessarily equals brine conductivity, σt = σw; conversely, bulk conductivity typically vanishes at some b 1 0. A mathematical description of the physical relationship underlying bulk rock conductivity and brine conductivity and the bulk volume fraction of brine must be con strained by these properties.
The classical three-parameter relationship connecting bulk formation conductivity to porosity and water saturation is an ad hoc formula comprising two separate power laws proposed by Archie in 1941 and modified by Winsauer et al. in 1952. The three adjust able parameters a, m, and n, in the usual formulation of these laws are chosen, for particular data sets, by the method of least squares to minimize sums of squared residuals between observed and predicted data. The resulting theory depends upon the term φmswn, breaking the φ–Sw symmetry inherent in the bulk volume brine fraction. While the laws have proved useful for predicting water saturations by interpolation within the bounds of observed data sets, they do not usually extrapolate to correct values at the limits of their domain of applicability, suggesting that the Archie double-power law formulation cannot correctly represent the underlying physics.
The classical power laws devolve from an arbitrary selection of one particular class of fitting function from among several possible choices. It is possible to use the same porosity-resistivity data sets used by Archie and Winsauer to define an alternative three-parameter model. The proposed model's adjust able parameters are critical values of porosity, water saturation and conductivity and can be interpreted directly in terms of the percolation thresh old of the medium. The proposed model simultaneously (1) treats the fractional brine volume b as the fundamental variable, (2) accommodates the boundary condition σt = 0 at b ¹ 0 and satisfies σt = σw at b = 1, and (3) gives a smaller sum of squared residuals using the classical data sets than the classical models. Many additional insights into the conductivity-porosity-water saturation relationship follow from this new model.
The efficacy of the model is illustrated using Archie's and Winsauer's original data by better predicting conductivity from porosity, demonstrated by a smaller sum of squared residuals using those data, and using conductivity-porosity-water saturation data published by Hamada and others in 2002.
In situ estimates of bulk formation resistivity made on petroleum reservoirs by remote sensing instruments located in wellbores (i.e., resistivity logs) have been among the primary observations used in evaluating hydro carbon volumes for more than a half-century, and resistivity estimates continue to remain an important element of formation evaluation. However, whereas the engineering of the data acquisition technology has kept pace with developments in new materials and new electronic methods, interpretation technology remains—perhaps arguably—unimproved for more than 60 years.