Derivative Computation using an Adjoint-Based Goal-Oriented Iterative Multiscale Lagrangian Framework

Abstract

Even though negative effects on the use of crude oil have surfaced over the past years, our energy matrix still largely relies on this energy source. The production of oil, therefore, plays an important role in our society. Unfortunately, the process of oil production is highly uncertain. There are uncertainties associated on the production strategy, e.g. where and how many wells should be drilled and how these wells should operate because of the uncertainties associated with the limited knowledge about the subsurface. In this thesis we are dealing with the uncertainty of the rock permeability distribution. Typically, rock permeabilities in the rock vary, but from the outside this can’t be perceived. If these rock permeabilities are estimated inaccurately, they will result in inaccurate pressure solutions. Then, this can lead to faulty decisions regarding the oil exploitation. To resolve this issue, a data assimilation technique may be applied to correct these model parameters based on mismatch of simulated data and observations. For this optimization technique, often gradient information is required. Since
in reservoir simulation the number of parameters generally is extremely high, computation of this information is computationally expensive. Therefore, a multiscale framework is employed to improve the
computational efficiency of the forward simulation. Multiscale methods are able to solve the model equations at a computationally efficient coarse scale and can easily interpolate this solution to the fine
scale resolution. Next, we use a Lagrangian set-up together with a multiscale framework to re-derive an efficient formulation for the derivative computation. However, as the multiscale method is prone to errors, this derivative computation formulation is recast in an iterative fashion, using a residual based
iterative multiscale method to provide control of these errors. In this thesis we show that this method generates accurate gradients. In contract to the high accuracy of the method, this method comprises a computationally heavy smoothing step. This issue can be resolved by making smart use of the Lagrange
multipliers, to re-derive an efficient iterative multiscale solution strategy. The multipliers are used to identify important domains of the region for which smoothing is required and for which regions we may neglect the smoothing. We show that the newly proposed iterative multiscale goal oriented method is computationally more efficient and we show that method is promising for efficient derivative computation, but that more work is required to fully demonstrate the benefit of this method.