Surface prediction using rejection sampling to handle non-linear constraints

We demonstrate accurate prediction of geological surfaces by imposing consistent physical and stochastic relationships between surfaces. The accuracy is improved by using all relevant information collected in wells: well points, zonation in horizontal sections, and gas/fluid content along wells. The conditioned surfaces are used to provide estimates of gross rock volumes of oil and gas reservoirs. In particular, it is shown how knowledge of spill point and zonation along well paths affect trapped volumes. A plain rejection sampling technique is used to deal with the highly non-linear relationships between a surface and its spill point. For well path conditioning, an extension of kriging to treat inequality constraints is proposed. It is based on efficient rejection sampling from a high dimensional truncated multivariate Gaussian distribution. The impact on gross rock volume distributions from different assumptions and data types is demonstrated by examples and the uncertainties in all the involved data types are consistently handled and quantified.