Application of an Ensemble Based Method for Pressure-Rate Deconvolution

Abstract This paper employs an ensemble-based method to address and assess the uncertainty in solutions to the single- and multi-well pressure-rate deconvolution problem. The method is implemented in two field cases from the Campos Basin. The first field case comprises a single vertical well producing from a virgin reservoir with continuous recording of bottom-hole pressure but uncertain flow-rate history. The second field case comprises a pair of horizontal wells, one acting as a producer and the other as an injector, with similar uncertainties in the flow-rate history and the additional complexity of a multi-well problem. We extend the investigation into the solution of the pressure-rate deconvolution proposed in a previous work (Kubota and Piccinini, 2022), which relies on the fundamental solution of the diffusion equation for bounded domains. In the present study, the deconvolved pressure response is obtained using the Ensemble Smoother with Multiple Data Assimilation (ES-MDA), a popular method for history matching numerical reservoir simulation models. ES-MDA provides an uncertainty quantification of the deconvolved pressure response and the investigated porous volume, along with other reservoir properties. Because ES-MDA uses a Bayesian formulation, data are weighted according to the inverse of the error variance. We explore this feature to include data from flowing periods, a desirable feature for the analysis of long-term pressure data with few well shut-ins. The results obtained for the two field cases consist of the posterior ensemble of model parameters, estimated flow rates, investigated pore volume, productivity index, single-well deconvolved response, as well as interference responses for the multi-well case.