GRADUAL DEFORMATION OF BOOLEAN SIMULATIONS

Gaussian random simulations are used in petroleum engineering and hydrology to describe permeability and porosity distributions in subsurface reservoirs. Except by luck, the generated simulations do not yield numerical flow answers consistent with the measured production data. Thus, they have to be modified, which can be done by running an optimization process. The gradual deformation method was introduced to modify Gaussian simulations. As the resulting variations are continuous, this technique is of interest for gradient-based optimizations. Based upon the gradual deformation method, a preliminary approach was suggested to modify also Boolean simulations. In this paper, we aim at going one step further. First, the gradual deformation scheme, initially developed for Gaussian probabilities, is reformulated for Poisson probabilities. It provides a new tool for varying the number of objects populating a Boolean simulation. Up to now, changing this number induced sudden object appearance or disappearance, which produced strong objective function discontinuities. Such a behavior is especially undesired when running gradient-based optimizations. Thus, we extend the proposed approach to continuously add or remove objects from Boolean simulations. The resulting algorithm integrates easily into optimization procedures and reduces, at least partially, the objective function discontinuities due to the appearance or disappearance of objects.