Although it is possible to apply traditional optimization algorithms together with the Randomized Maximum Likelihood (RML) method to generate multiple conditional realizations, the computation cost is high. This paper presents a novel method that integrates the Distributed Gauss-Newton (DGN) method with the RML method to generate multiple realizations conditioned to production data synchronously.
RML generates samples from an approximate posterior by finding a large ensemble of maximum posteriori points, from a distribution function in which the data and prior mean values have been perturbed with Gaussian noise. Rather than performing these optimizations in isolation, using large sets of simulations to evaluate the finite difference approximations of the gradients used to optimize each perturbed realization, we use a concurrent implementation, in which simulation results are shared among optimizations whenever these results are helping to converge a specific optimization. In order to improve sharing of results, we relax the accuracy of the finite difference approximations for the gradients, by using more widely spaced simulation results. To avoid trapping in local optima, a novel global search algorithm integrated with DGN and RML is applied. In this way we can significantly increase the number of conditional realizations that sample the approximate posterior, while reducing the total number of simulations needed to converge the optimization processes needed to obtain these realizations.
The proposed workflow has been applied to field examples on liquid rich shale or tight oil reservoirs developed with hydraulically fractured horizontal wells. The uncertain parameters include stimulated rock volume (SRV) and matrix properties, such as permeability and porosity, and hydraulic-fracture properties, such as conductivity, height, and half length. The case studies involve a sensitivity analysis to identify key parameters, a history matching study to generate history-matched realizations with the proposed method, and an uncertainty quantification of production forecasting based on those conditioned models. The new approach is able to enhance the confidence level of the Estimated Ultimate Recovery (EUR) assessment by accounting for production forecasting results generated from all history-matched realizations. Numerical results indicate that the new method is very efficient compared with traditional methods. Hundreds of history-matched, or rather data-conditioned, realizations can be generated in parallel within 20-40 iterations. The computational cost (CPU usage) is reduced by a factor of 10 to 25 when compared to the traditional RML approach.