On the applicability of a particle method in models of increasing complexity: a case study for modelling subsidence.

Subsidence may occur as a result of hydrocarbon production, when the reservoir compacts because of the pressure reduction in the gas-producing layer. Geertsma et al. 1973 modelled the subsidence above compacting reservoirs, in particular the Groningen gas field in the Netherlands. Today, geodetic techniques such as InSAR (Interferometric synthetic-aperture radar) provide high-quality measurements of the rate of land subsidence. To analyse the process of subsidence, assimilation of geodetic data can help estimate geomechanical parameters. Here, we investigate the applicability of a particle method for parameter estimation by studying cases with varying degrees of freedom. With a conceptual model of a heterogeneous reservoir, we show that in this more complex case the exponential relationship for independent and identically distributed (i.i.d.) prior parameter components (Snyder et al. 2008) does not hold because the observations depend on multiple parameters. Instead, we derive an empirical relationship between entropy and state dimension. The concept of Shannon entropy gives the level of information of a given system. By adopting this concept, we evaluate the ensemble size required to conserve the necessary information in the posterior probability distribution of the estimation problem. We illustrate how a measure of entropy helps choose the required ensemble size taking into account both 1) the dimension and 2) the degree of freedom of the system as well as the quality of the observations. The results show that the posterior entropy can be used as an indicator for the quality of the estimate and its posterior distribution. In the i.i.d. case, there is a straightforward exponential relationship between entropy and state dimension rendering the ensemble size necessary for the particle method to be effective. In more complex problems, the required ensemble size is shown to depend on the quality of the observations and the degree of freedom of the system. Our results show that for the non i.i.d. problem of the subsidence modelling of a heterogeneous reservoir, the equivalent entropy provides information on the efficacy of the particle method and helps determine the required ensemble size.