The Flaw of Averages And the Pitfalls of Ignoring Variability In Rock Physics Interpretation

Summary Monte Carlo simulations, by taking in to account distributions of values instead of single average values, help to avoid the flaw of averages. Calculations based on a single “best guess” input may not give the “best guess” output. Ignoring the variability of rock properties in quantitative computations can drastically distort interpretation decisions. Monte Carlo simulations also give us confidence intervals and other measures of uncertainty. Computations using averages and average trends alone do not give any indication of the uncertainty due to the variability in the properties. Estimating the variability in attributes can be difficult from just visual examination of clusters in cross-plots. The density of points gets obscured in cross-plots. What appears to be the best choice of attributes in terms of visual separability of clusters may not be the best quantitative discriminants. The Flaw of Averages Ignoring the variability of rock properties in quantitative computations can lead to distortions in interpretations. Consider the following calculation as an example of the pitfall of ignoring distributions. We would like to compute the normal incidence reflectivity between an overlying shale layer and a packet of very thinly bedded, sub-seismic resolution sand/shale layers. The effective properties (velocity, density, and impedance) of the thin sand/shale layers are computed using Backus average. The contrast between the impedance of the shale and the effective impedance of the sand/shale packet then gives us the normal incidence reflectivity. The Backus average elastic modulus depends on the volumetric sand fraction. For normal incidence propagation perpendicular to the layers, the effective P-wave modulus is given by a volumetric harmonic average of the sand and shale moduli. The effective density is given by the usual volumetric arithmetic average of sand and shale densities. By doing the calculations for all sand/shale ratios (0 to 1) we get a relation between sand ratio and normal incidence reflectivity. We could do the computation in two different ways. We could take an average value for sand VP, density, and impedance, and another average value for the shale VP, density, and impedance, and use them in the equations for Backus average and reflectivity. The average values could be estimated from blocked well logs, for example. This average computation, ignoring the variability, gives us the single line shown in Figure 1, relating the computed reflectivity to the sand/shale fraction. One might use this line for interpreting observed reflectivities to estimate sand/shale ratio in the thinly layered packet. Instead of taking average sand and shale properties, another way to do the computation would be by Monte Carlo simulations, taking in to account the natural variability in the properties of the sand and shales (Avesth et al., 2005). Figure 1: Relation between normal-incidence reflectivity and sand/shale ratio in very thin-bedded sand-shale layers. The line is obtained using Backus average for thin layers, with blocked average values of sand and shale properties from well log. In this computation we ignored the variability of sand and shale velocity, density, and impedance.