Challenges in adjoint-based well location optimization when using well models

Abstract

There is a general consensus that the most efficient method for large-scale well location optimization is gradient-based with gradients computed with an adjoint formulation. Handels et al. (2007) (later published in journal form as Zandvliet et al., 2008), were the first to use the adjoint method for well placement optimization for which they introduced the concept of ‘pseudo wells’ surrounding the well to be optimized. Sarma et al. (2008) presented a method to determine the sensitivity of the objective function with respect to the actual well locations directly from the adjoint gradients. The direct dependency of the objective function on the well location comes from weighing the well indices of the pseudo wells by a continuous well-location-dependent function. However, this method is not consistent with the use of the Peaceman well-inflow model. In this work we utilize the Ding well-inflow model (1994), which adjusts the transmissibilities of the adjacent grid blocks of off-centered wells. The basic underlying idea is that the explicit dependency of the flow equations on the well location, as formulated in the Ding model, would enable a direct calculation of the adjoint gradients of the objective function with respect to the well location. Unfortunately, attempts to implement this idea resulted in significant challenges. Using a simple homogenous 2-D reservoir example, we demonstrate how the non-smoothness of the objective function with the change in the well location, (resulting from assumptions in the Ding model) especially around the grid block borders can lead to incorrect adjoint gradients. We then show that this problem persists for a smoother objective function in which the Ding method is applied to a larger neighborhood around the well block. We conclude that irregularities in the objective function resulting from the original Ding well-inflow model adversely affect gradient-based well location optimization and that modifications to the well model will be required to develop a robust Ding model-based well location optimization method.