Conditioning of Marked Point Processes within a Bayesian Framework

Shale units with low permeability create barriers to fluid flow in a sandstone reservoir. A spatial stochastic model for the location of shale units in a reservoir is defined. The model is based on a marked point process formulation, where the marks are parameterized by random functions for the shape of a shale unit. This extends the traditional formulation in the sense that conditioning on the actual observations of the shale units is allowed in an arbitrary number of wells penetrating the reservoir. The marked point process for the shale units includes spatial interaction of units and allows a random number of units to be present. The model is defined in a Bayesian setting with prior pdfs assigned to size–shape parameters of shale units. The observations of shales in wells are associated with a likelihood function. The posterior pdf of the marked point process can only partially be developed analytically; the final solution must be determined by sampling using the Metropolis–Hastings algorithm. An example is presented, demonstrating the consequences of increasing the number of wells in which observations are made.