Petro-seismic inversion for sandstone properties

We use empirical relations derived from laboratory and log data for sandstones to estimate unknown parameters from given parameters. The known and unknown parameters may be any of the following: compressional and shear wave velocities (V_p and V_s), density (ρ), compressional and shear wave quality factors (Q_p and Q_s), effective pressure (P), frequency (f), water saturation (S), porosity (ϕ), clay content (C), fluid permeability (k), and V_p/V_s and Q_s/Q_p ratios. The goal is to obtain estimates of all thirteen of these petro-seismic variables from subsets of these as input; whether this is achievable depends on the particular combination of input variables.

The inversion typically proceeds through a hierarchy of three levels of parameter estimation, in order of the expected reliability of the estimates. First, we solve for the parameters that can be obtained directly from the fitted empirical equations. Then, a grid search is performed to simultanously fit more than one unknown parameter by finding values that best predict the known parameters. Finally, various approximations are invoked to predict values when the inputs are insufficient to use the direct equations or a grid search. These three procedures can be initiated in any desired order and any number of times.

Even if the complete set of seismic parameters (V_p, V_s, ρ, Q_p, Q_s, and f) are given, it is not possible to uniquely obtain any of the remaining petrophysical variables (S, ϕ, C, P, or k) directly from the fitted equations without constraints on (or assumptions about) some of the other variables. There is a trade-off between porosity and clay content that can be resolved by simultaneous fitting of multiple seismic parameters (i.e., V_p, V_s, and V_p/V_s). The predictability of the parameters from various combinations of the available data is expressed using correlation coefficients (R²) and standard deviations. With optimal input combinations, parameters with R² ≥ 0.8 are (in order from best to worst) V_p, 100/Q_p, ρ, V_s, and S. Other parameters with 0.63 ≥ R² ≤ 0.79 are (in order from best to worst) ϕ, Q_s/Q_p, V_p/V_s, ln k, 100/Q_s, and C. P and f are the least reliably estimated. All of the seismic parameters can be accurately estimated if all of the petrophysical parameters are known.