Direct inversion by de-trending stochastic permeability fields

In this paper a new method for direct inversion of production data to a permeability field is proposed. With this direct inversion methodology, history matching is physics based, honouring both the measurements and retaining the full variability in geology. The method involves adjustment of the geological permeability realizations using the measurements. First, the realizations are split in a trend and in a residual. The trend is a smooth function satisfying the same boundary conditions as the original realization; the residual then contains the geological stochastic information and satisfies homogenous boundary conditions. The split is made both for the potential and for the stream function. The trends from a realization are in serious conflict with the measurements. Therefore we replace the wrong trends in the realization by trends calculated with the measurements. The result is a model that obeys all measurements as well as the geology, and the objective function in terms of indirect inversion is zero. Darcy's law is explicitly included, continuity in flow is guaranteed and the boundary conditions are fulfilled. The prize that has to be paid is anisotropy in the resulting permeability field. However, in general this anisotropy is not serious in most of the model cells. We are still working on solutions to suppress this anisotropy. A possible extension of our current method is to incorporate the water cut measurements by a multiplication factor applied directly to the stream function. Another extension is from 2D to 3D: the velocity can then replace the stream function and the permeability will be derived from the velocity vectors and the potential gradients.