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### Abstract

This work presents new Kriging-based algorithms that allow incorporating nonlinear constraints such as a given average and variance into the kriged field. The first of these constraints allows one to obtain, for example, permeability fields whose average matches well-test-derived permeability; the second one enables one to generate fields that have a desired variability. Therefore, a well-known drawback of Kriging, namely excessive smoothness, is overcome. As these constraints are applied, the Kriging estimates of all blocks become mutually dependent and must be solved simultaneously. This fact led to the development of a new concept, that of Simultaneous Kriging. Furthermore, because permeability is not an additive variable, the problem becomes nonlinear; an optimization procedure based on Newton's method is used to solve it. Both average and variance constraints are applied through Lagrange multiplier technique. Power averaging with a ω exponent that varies from -1 to 1 is used to estimate field-generated permeability within the well-test drainage area. The optimization algorithm searches for an optimum ω value as well as a set of gridblock values such that the desired field average and variance are met. Starting from an initial guess and tolerance, the algorithm will reach the closest minimum within the solution space. Unlike other Kriging methods, the solution is not unique; in fact, infinite, equiprobable solutions can be found. From this point of view, the proposed algorithms become more like conditional simulation. A number of permeability fields were generated and used as input of a numerical simulator to calculate their actual well-test permeability. The simulation results proved that the fields generated by the algorithm actually matched well-test permeability and field data variance within a reasonable tolerance. The same procedure can be used to incorporate other nonlinear constraints, such as facies proportion.

/content/papers/10.3997/2214-4609.20146398
2008-09-08
2021-09-25