Gaussian Process Models for Well Placement Optimisation

Gaussian Process models have been proposed as statistical models that allow interpolation between existing data points. One advantage of this approach is that the Gaussian process model includes an estimate of the accuracy of the predicted expected value at any point within the parameter space, unlike direct interpolators often used as proxy models. When used as part of an optimisation process we can use the Gaussian process model to eliminate those areas where we have high confidence that the optimal solution will not be found. This allows the efficient targeting of resources on those areas of parameter space that could yield the optimal solution, and also facilitates a more global analysis of the parameter space. Gaussian Processes naturally provide clear visualisation of the objective surface at various stages of the optimisation, which generates insight into the optimisation process sometimes lacking in alternative approaches, and thus facilitate human validation/intervention if desired.

To understand the theoretical basis of the approach requires a level of statistical knowledge that is not commonly found outside of the statistics community, which may have inhibited uptake. However, the approach can be easily implemented from first principles in python, using a recipe by Rasmussen and Williams, without needing a deep understand of the theoretical underpinning. The recipe has a small number of controls that need to be set by the user. In this paper we construct empirical models of the effect of these controls on the interpolation and explain their limitations from a theoretical perspective. We explore how dynamic adjustment of the controls might be used as part of an optimisation scheme.

We apply our approach to the well placement optimisation problem. The reservoir model used for the exercise is the PUNQ Complex Model, which is a 2.4 million cell representation of a BRENT sequence reservoir. A combination of producers and injectors are sequentially placed in the model using a greedy algorithm with the optimal position at each iteration being selected using the Gaussian Process model as a proxy for the true objective surface. The result is compared to a manually derived solution by an experienced reservoir engineer which required 22 wells. The result obtained by this approach reaches the same level of performance using only 18 wells.