Trade-off between stability and bias in a history match problem using smoothness constraint

The history matching (HM) hydrocarbon reservoir inverse problem is ill-posed because its solution may be non-unique or unstable. Variability of acceptable solutions around local minima may be very large. Nonetheless, it is a common practice in HM to find a unique reservoir model, although there is no guarantee that this model represents the geology. Qualitative geological information is generally not considered because of difficulties to express them mathematically. Here, we incorporate 'smoothness' in the spatial variation of physical properties as an example of a geological qualitative constraint that can be mathematically incorporated in an objective function also honouring the data. The constraint is valid if lateral continuity exists (e.g. fluvial-deltaic siliciclastic reservoirs). We mean smoothness by conditioning the permeability and/or porosity difference between adjacent grid blocks to be small. We use a synthetic 2D water-oil model and a solution search technique allowing characterising both the optimum solution and its variability. The smoothness constraint reduces the variance of the estimates by introducing bias in the solutions still preserving good match. The key point to achieve an optimum trade-off between stability and data match is the tuning of the parameter, controlling the relative importance of the constraint in the objective function. The smoothness constraint cannot be applied to all reservoirs. However, we corroborate the idea that a 'tool box' of HM can be designed; each 'tool' can incorporate a different constraint. Therefore, the interpreter can judiciously choose a specific tool from this tool box according to the adherence of its constraint to the particular reservoir being studied.