We apply different particle swarm optimizers (PSO) to a history matching problem for the synthetic Stanford VI sand-and-shale reservoir. The ill-posed character of this inverse problem is attenuated by reducing the model complexity using a Spatial Principal Component base and by combining as observables flow production measurements and cross-well seismic data. Additionally the inverse problem is solved in a stochastic framework searching for the family of reservoir models that equally fit the data. We show that PSO algorithms have a very good convergence rate and in addition provide approximate measures of uncertainty around the optimum facies model. The uncertainty estimation, although it is a proxy for the true posterior distribution of model parameters, allow us to perform risk analysis..
Characterizing the spatial distribution of heterogeneous reservoir properties is one of the major challenges in reservoir modeling for optimizing production. Well data together with seismic data are typically used to infer the spatial distribution of properties such as facies, porosity and permeability. The seismic history matching problem consists then in obtaining reservoir models that match production data as well as seismic time lapse data. The main challenge of this inverse problem is that the production data alone does not uniquely constrain the porosity and permeability of the reservoir. Combining flow production measurements with time lapse seismic data has been useful for better constraining the history matching (Huang et al., 1997; EcheverrÍa and Mukerji, 2009; Xia and Huang, 2009; and Dadashpour et al., 2009, among others). Furthermore, to run the flow simulator and produce accurate results a detailed description of the reservoir is needed. This causes the inversion problem to be highly illposed due to its underdetermined character. One solution commonly found in the literature is to use non-linear leastsquares methods with Tikhonov regularization around a reservoir reference model that is constructed using prior geological and geophysical knowledge. The result is a unique reservoir model that shows the best trade-off between the data prediction and the model complexity. No uncertainty estimation is usually performed around this model due to the high computational cost.. Stochastic approaches to inverse problems consist in shifting attention to the probability of existence of certain interesting subsurface structures instead of looking for a unique model. Global optimization methods are well suited to perform this task. Although global algorithms can be used in exploitative form, their main advantage is that they can potentially address the inverse problem as a sampling problem. Their use as samplers requires a reasonably fast forward modeling and a small number of independent parameters. The use of Particle Swarm Optimization in geosciences still remains restrained (Shaw and Srivastava, 2007; Fernández-MartÍnez et al., 2010 a,b). In reservoir engineering PSO has been used to determine the optimum well location and type in very heterogeneous reservoirs (Onwunalu and Durlofsky, 2009). The use of global optimization techniques is hampered in real history matching problems by the large number of parameters needed to accurately describe the reservoir and by the very costly forward solutions.