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A simple measure of geological continuity is provided in geostatistics through the variogram model. The variogram model is often estimated from well-log data. It provides a constraint on the spatial extent and direction of permeability anisotropy. The variogram, however, cannot capture strong reservoir connectivities or curvi-linear features.
In Caers et al. an iterative procedure is proposed for injecting the variogram continuity information into the streamline-based history matching method outlined above. An iterative Monte Carlo technique is used to constrain the current reservoir model kl(u) to the streamline effective permeabilities ?k(u) at the same time honoring the correct geological constraints. It is a Gauss-Markov method where each cell is visited randomly, and an attempts is made to change each reservoir grid-block such that the reservoir model honors better the streamline effective permeability constraint without destroying the geological continuity provided by the variogram.
Figures 1 and 2 display typical results from this approach. Figure 1 shows the reference field and 3 different matches constrained to the same data. It is a confined five spot with an injector in the center and producers at each corner. The mobility ratio is 5, and only 150 days of data are used to obtain a match, Figure 2. The predictive abilities of each of the 3 matched permeability fields are rather good even at producer 1 that has not broken through by 150 days. Further details are provided in Caers et al..
Reference: Caers, J., Krishnan, S., Wang, Y. & Kovscek, A. R. 2002. "A Geostatistical Approach to Streamline-based History Matching" SPE Journal. 7(3) September, 250-266. |