The petro-elastic model (PEM) represents an integral component in the closed-loop calibration of integrated four-dimensional (4D) solutions incorporating time-lapse seismic, elastic and petrophysical rock property modeling, and reservoir simulation. Calibration of the reservoir simulation model is needed so that it is not only consistent with production history but also with the contemporaneous subsurface description as characterized by time-lapse seismic. The PEM requires dry rock properties in its description, which are typically derived from mechanical rock tests. In the absence of those mechanical tests, a small data challenge is posed, whereby all necessary data is not available but the value of reconciling seismic attributes to simulated production remains. A seismic inversion-constrained n-dimensional metaheuristic optimization technique is employed directly on three-dimensional (3D) geocellular arrays to determine elastic and density properties for the PEM embedded in the commercial reservoir simulator.
Ill-posed dry elastic and density property models are considered in a field case where the seismic inversion and petrophysical property model constrained by seismic inversion exist. An n-dimensional design optimization technique is implemented to determine the optimal solution of a multidimensional pseudo-objective function comprised of multidimensional design variables. This study investigates the execution of a modified particle swarm optimization (PSO) method combined with an exterior penalty function (EPF) with varied constraints. The proposed technique involves using n-dimensional design optimization to solve the pseudo-objective function comprised of the PSO and EPF given limited availability of constraints. In this work, an examination of heavily and reduced-order penalized metaheuristic optimization processes, where the design variables and optimal solution are derived from 3D arrays, is conducted so that constraint applicability is quantified. While the process is examined specifically for PEM, it can be applied to other data-limited modeling techniques.