The reconstruction of subsurface property maps using projection onto convex sets

Obtaining a map of the spatial distribution of some physical property of the Earth subsurface from sparse data is a fundamental problem in aquifer and petroleum reservoir characterization. This problem can be stated in terms of an inverse problem as the search for a map that satisfies certain constraints, i.e., measurements and information on the spatial statistics of the property. We present the results of numerical experiments applying the method of projection onto convex sets, or POCS, that functions both as an estimation procedure, where the objective is to obtain a unique guess of the unknown spatial distribution given some constraints, and as a simulation procedure, where the objective is to obtain a set of nonunique reconstructions that are consistent with some measurements and have given spatial statistics. Numerical experiments that use point measurements at wellbores and volume integral measurements show that incorporating data from the region of the reservoir between wellbores allows better localization of large-scale heterogeneities in the reservoir volume