Uncertainty Updating in Oil Reservoir Simulation

It is generally accepted that in order to make reliable predictions a reservoir model must be able to reproduce the available measurement and production history of the real reservoir. The standard procedure to achieve this is to take an initial model, created from seismic data and the geological knowledge of an engineer, and condition it to the measurements. The conditioning is typically performed by minimising a quadratic mismatch function of the measurements over the most sensitive uncertain parameters of the model. This inverse problem, in the oil reservoir sector called history matching, is an ill-posed problem in the sense of Hadamard [1, p. 31]. Optimisation techniques are usual methods to solve it, for example gradient based ones or Evolutionary Algorithms like Evolution Strategies (ES) and Genetic Algorithms (GA). Also manual optimisation by a specialist is still a common approach. These methods are limited to handling a small number of parameters, typically on the order of 30-50, but industry scale reservoir models contain much higher numbers. Thus often a suitable parametrisation of the model has to be found, as a manual task by reservoir engineers. The history matching problem can also be formulated in a probabilistic setting [2], which is not an ill-posed problem in general: the knowledge about the problem is contained in an a priori probability density rM(m) over the full model parameter space. New knowledge enters through measurements, which are given by a probability density on the data space rD(d). These two sources of information can be combined into Bayesian-like updated or a posteriori knowledge sM(m) on the problem itself according to