Quantification of uncertainty in coarse-scale relative permeabilities due to sub-grid heterogeneity

Reservoir production forecasts are essentially uncertain due to the lack of data. Specifically, it is impossible to estimate detailed heterogeneity in a reservoir. In order to mitigate the ambiguity of a model, production data is incorporated into a history-matching process. However, there is insufficient data to constrain the subsurface properties all over the field.

We investigated the parameterisation of coarse-scale relative permeabilities for history-matching and uncertainty quantification. Coarse-scale models are often employed in history matching, because of computational cost. The results of the investigation provided us with guidelines for history-matching a coarse-scale model to the observed data by adjusting relative permeabilities.

This paper addresses two issues. Firstly, because the coarse-scale model inevitably misses out sub-grid heterogeneity, physical dispersion is ignored in the simulation. Secondly, the small-scale heterogeneity is not explicitly known and can only be inferred by history-matching. To solve these problems, local features in the coarse-scale relative permeability curves were adjusted in history-matching to capture the effect of physical dispersion and to compensate for the effect of numerical dispersion.

The success of history-matching relative permeabilities depends on the flexibility of the saturation function. We applied the flexible B-spline function as well as a conventional power or exponential function, namely the Corey or Chierici functions, respectively. We compared these parameterisations in terms of the resulting relative permeabilities during history-matching and uncertainty appraisal.

The history-matched relative permeabilities and their uncertainty envelopes were examined in comparison with the two-phase upscaling results. We used a synthetic data set for which the true solution is known. The two-phase upscaling was conducted using the truth model to give a reference set of coarse-scale relative permeability curves. We also compared the truth production profiles with the uncertainty envelopes which were quantified in Bayesian framework. Our results highlight the fact that the parameterisation affects the width of uncertainty envelope.