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Abstract
Reservoir and well simulations are often coupled to allow more realistic production forecasts. Proxy functions are typically generated by a well simulator as a preprocessing step to characterize the response from the production system when the reservoir condition changes. With proxy functions, a reservoir simulator can take into account the response from the well without simultaneously performing the well simulation, which can be time prohibitive. This indirect coupling is effective only when the interpolated values from the proxy function closely match the results from the surface simulation. Typical implementations of proxy functions require characterizations at each grid point, which can be expensive to compute for a fine grid. Due to the intrinsic curvilinearity of well performance curves, the accuracy of proxy functions highly depends on the number of sampling points.

In this paper, we discuss a new method for generating proxy functions that improves accuracy without sacrificing performance. We propose to use kriging to enhance the efficiency and accuracy of characterization of the surface simulation. Kriging interpolation provides a grid-free representation of the proxy function, which allows the flexibility of choosing the sampling points for well simulations. We propose to put more sampling points where the rate of change of the proxy function is relatively large.

Our numerical results show that the grid-free method reduces the interpolation error by over 40%, without increasing the number of sampling points. Furthermore, with a comparable level of accuracy, simulation time for generating the proxy function is reduced by 50%.

Introduction
Coupled reservoir and well/network simulations provide valuable insight for realistic production forecast. Proxy functions are typically generated by a well/network simulator as a pre-processing step to characterize the response from the production system when the reservoir condition changes. With proxy functions, a reservoir simulator can take into account the response from the well without simultaneously performing the well simulation, which could be time prohibitive. This indirect coupling is effective only when the interpolated values from the proxy function closely match the results from the surface simulation. Due to the intrinsic curvilinearity of well performance curves, the accuracy of proxy functions highly depends on the number of sampling points. Typical implementations of proxy functions require characterizations at each grid point, which consists of a set of regularly spaced sampling variables, including flow rate, water cut, GOR, outlet pressure, gas injection rate, etc., as the coordinates. A naive way of improving the accuracy of the proxy function is to sample the surface simulation result on a denser grid in the parameter space. This can be expensive to compute for a fine grid, especially with high dimensionalities. For example, 20 evenly spaced samples for each dimension for a five-dimensional parameter space would require over 32 million simulations.