Splines as an Optimization Tool in Petroleum Engineering

摘要

Planning well placements and assisted history matching are examples of some of the difficult optimization problems that occur in petroleum engineering. In both problems objective function evaluations are computationally expensive because they involve forward simulations using reservoir simulators. When gradient based methods are applied to these problems derivatives of the objective function with respect to the parameters (i.e. well location or reservoir properties) arerequired. The simplest way to calculate these derivatives is by perturbation, which requires further expensive simulation runs. More mathematically complex methods can also be used but these can be difficult to implement. Response surfaces are used to extract the most value from simulation runs that are made. Prior work has used kriging and least-squares methods to fit such surfaces. This work utilizes splines as a surface fitting tool. Splines are a useful tool because the derivatives required for a gradient based optimizer can be readily computed using various packages. Splines are shape preserving (i.e. convexity properties are unchanged). Constraints can be implemented when fitting splines. This work demonstrates the application of the new splinebased optimization procedure on a well placement problem. This example is based on a real 17 MMSTB oil field. The operators of the field planned to begin waterflooding. The optimization tool was used to plan the location (areally) of the water injection well. The well location determined using the tool is predicted to produce over 800 MSTB of incremental oil when compared to the original location proposed by the operators. The CPU requirements were modest, under 3% of the possible well locations were simulated. A history matching process is also demonstrated which is used to determine the ratio of vertical to horizontal permeability, a global porosity multiplier and the gas-oil contact depth.