Surrogate Models by Means of Functional Kriging to Estimate Uncertainty in Reservoir and Geomechanical Simulations

Accurate prediction tools for land subsidence induced by oil/gas production are fundamental for energy companies to identify possible affected areas and to propose measures to counteract its potentially adverse effects. The hydraulic and mechanical problems are physically coupled, therefore a set of partial differential equations (PDEs) for multiphase flow and stress-strain evolution in porous media should be solved. The flow-deformation models, no matter the degree of coupling between fluid flow and stress-strain used in the simulation, are computationally demanding so their use in a data assimilation workflow would be unfeasible for large models. Indeed, the calibration process requires running the simulation for several values of the parameters of the PDE to find the sets that best reproduce the observed data. Parameters are petrophysical but also mechanical properties of the rocks. Observed data are production data (well flow rates and pressures) and geodetic measurements, like for instance data from global positioning system (GPS). GPS stations, installed on production platforms, receive signals from satellites continuously monitoring their three-dimensional positions and provide very accurate measurements of vertical displacements. The challenge concerns the simultaneous calibration of the flow and geomechanical models integrating the aforementioned different data sources. In this paper, we propose an approach for fluid and subsidence estimation based on functional kriging surrogate models and Ensemble Smoother with Multiple Data Assimilation (ES-MDA). Surrogate models are approximations of numerical models built upon running the simulation for a limited number of values of the PDE parameters, selected based on appropriate experimental design techniques. The functional formulation of universal kriging allows to reproduce curves as a function of time and, in our case, allows to jointly replicate the temporal behavior of fluid production and subsidence with a very low computational cost. Using the ES-MDA is then possible to automatically assimilate multiple data sources rapidly and to predict the forecast uncertainty. The workflow has been applied on two test cases: a synthetic reservoir which is a benchmark in the reservoir modeling community and a real field. In both cases, surrogate models proved to be computationally efficient and data assimilation led to an ensemble of models able to quantify prediction uncertainties accurately.