Data Assimilation in Reservoir Management Using the Representer Method and the Ensemble Kalman Filter

The aim of data assimilation is to improve numerical models by adding measurement information. In case of petroleum engineering, the model might be the combination of a reservoir simulator, a rock-physics model and a wave propagation package. Measurements can originate from geology, seismics, petrophysics, down-hole sensors and surface facilities. In theory, one tries to maximize the likelihood of the parameters given the measurements, where the numerical model is used as a weak constraint. In practice the problem is often reduced to a least squares problem by assuming Gaussian error statistics, resulting in a variety of related data assimilation algorithms. In this paper the Ensemble Kalman Filter (EnKF) and the Representer Method (RM) are compared. For linear systems they solve the same least squares problem; for non-linear systems, like multiphase flow in porous media, they have their own peculiarities and utilization. A variational method like the RM might get stuck in a local minimum of the squared data misfit objective function, whereas this objective does not even have a physical or probabilistic interpretation for non-linear models or non-Gaussian probability distributions. The measurement update of a filter overestimates the jump of the forecasted reservoir states towards the observed reservoir states. These errors accumulate and cannot be corrected in an iterative process, unlike what can be done in a variational method. The RM is computationally more demanding than an ENKF, especially when the number of measurements increases. Unlike the ENKF, the RM not only calculates estimates of state variables and model parameters, but it also quantifies what the isolated effect of every measurement in space and time is on the final estimate. It is therefore a promising method to quantify the “value of information” of a specific measurement.