Estimation Of Petrophysical Parameters Using A Robust Levenberg-marquardt Procedure

A newly developed procedure for minimizing nonlinear sum-of-squares functions provides more accurate estimates of petrophysical parameters than linear techniques. It uses robust weighting factors with a Levenberg-Marquardt algorithm that can be applied to both nonlinear and linear least-squares problems. The advantage of using robust weighting is the inherent resistance of the weighted procedures to the effects of outliers or any particular data points with large systematic error. In the presence of questionable data, this modified Levenberg-Marquardt algorithm for solving nonlinear least-squares problems uses only 30 to 50% of the computer time required for a conventional Levenberg-Marquardt type algorithm to converge to an answer. The new algorithm will solve some problems where a conventional Levenberg-Marquardt-type algorithm fails to converge or converges to an erroneous value. For a Mesaverde sandstone containing hydrocarbon- and water-bearing zones the robust nonlinear least-squares algorithm produced reasonable estimates for the cementation exponent (2.13) and water resistivity (0.14 ohm-meters). A robust linear least-squares fit of the logarithm of porosity on the logarithm of resistivity gave 2.8 for the cementation exponent, and a formation-water resistivity of 0.0352 ohmmeters. These unacceptable results for the linear analysis occur because the data exhibit too limited a range to establish a linear trend on a Pickett plot reliably. With more water-bearing zones than hydrocarbon-bearing zones present in the data, the new algorithm also identifies hydrocarbon-bearing zones by applying low weights to them. They are treated as the "outliers" in the data set. In the Mesaverde example, several points with low apparent water-resistivity values were identified as possible shaly zones by their low robust weights. Linear methods must be used in a time-consuming, trial and error fashion for this type of analysis. The new algorithm can also estimate matrix density, formation-water resistivity, and the cementation exponent simultaneously. This is similar to a Hingle-plot analysis except that the cementation exponent is allowed to vary to optimize the result. The Mesaverde example illustrates how this procedure can identify a possible log-calibration problem. Linear least-squares algorithms cannot be used for this type of multiple-parameter nonlinear analysis. The power of this method over conventional methods results from the techniques of including robust weighting factors for the gradient, as well as the least-squares error residuals, and using the Choleski matrix decomposition, which avoids inversion of a singular matrix during any given iteration. By simultaneously attacking the problems of singularity and outliers, the new algorithm will solve most problems with equal or better accuracy and precision than conventional nonlinear algorithms, as well as solve some problems where conventional algorithms fail to converge or converge to the wrong answer.