Efficient Multivariate Property Modeling with Seismic Data

The joint modeling of the petrophysical properties (i.e., porosity, permeability) from wells in the presence of one or more seismic attributes (i.e., impedance) may be cumbersome, as the linear model of coregionalization needs to be simultaneously fitted to all experimental direct and cross-variograms, and the strong assumptions are required in the collocated cokriging system. Transforming each petrophysical property to an uncorrelated factor through the projection-pursuit multivariate transform allows each uncorrelated factor to be simulated independently. However, considering the case where there is an exhaustive secondary variable, the uncorrelated factors can no longer be simulated independently, as they are still conditionally dependent through the secondary variable. The aim of this paper is to provide a solution to this problem through the simulation of each uncorrelated factor in a subsequent fashion; that is, the first uncorrelated factor is cosimulated using the available secondary variable as a covariate; the second uncorrelated factor is cosimulated using a super-secondary variable generated by merging the previously simulated first uncorrelated factor and the secondary variable, and the kth uncorrelated factor is cosimulated using a super-secondary variable generated by merging all previously simulated uncorrelated factors (1,..., k - 1) as well as the secondary variable. This hierarchical simulation framework preserves the correlation structure between the uncorrelated factors themselves and between the uncorrelated factors and the secondary variable. The methodology is demonstrated in case studies using synthetic and real reservoir datasets. It is shown that the use of PPMT approach and the hierarchical simulation workflow in combination achieves: (1) multivariate complexity in the data is accounted for through the PPMT approach, and (2) the reproduction of the observed bivariate relationships in the simulated realizations of the petrophysical properties themselves and the secondary information is ensured by the hierarchical simulation workflow.

Automated summary

The paper addresses the issue of joint modeling of petrophysical and seismic properties through a hierarchical simulation framework, preserving correlation structures and enabling independent simulation. By utilizing the projection-pursuit multivariate transform, each uncorrelated factor can be simulated independently, with super-secondary variables generated from previously simulated factors and secondary variables for co-simulation.