Important decisions in the oil industry rely on reservoir simulation predictions. Unfortunately, most of the information available to build the necessary reservoir simulation models are uncertain, and one must quantify how this uncertainty propagates to the reservoir predictions. Recently, ensemble methods based on the Kalman filter have become very popular due to its relatively easy implementation and computational efficiency. However, ensemble methods based on the Kalman filter are developed based on an assumption of a linear relationship between reservoir parameters and reservoir simulation predictions as well as the assumption that the reservoir parameters follows a Gaussian distribution, and these assumptions do not hold for most practical applications. When these assumptions do not hold, ensemble methods only provide a rough approximation of the posterior probability density functions (pdf's) for model parameters and predictions of future reservoir performance. However, in cases where the posterior pdf for the reservoir model parameters conditioned to dynamic observed data can be constructed from Bayes' theorem, uncertainty quantification can be accomplished by sampling the posterior pdf. The Markov chain Monte Carlos (MCMC) method provides the means to sample the posterior pdf, although with an extremely high computational cost because, for each new state proposed in the Markov chain, the evaluation of the acceptance probability requires one reservoir simulation run. The primary objective of this work is to obtain a reliable least-squares support vector regression (LS-SVR) proxy to replace the reservoir simulator as the forward model when MCMC is used for sampling the posterior pdf of reservoir model parameters in order to characterize the uncertainty in reservoir parameters and future reservoir performance predictions using a practically feasible number of reservoir simulation runs. Application of LS-SVR to history-matching is also investigated.
Even when written in terms of a pseudo-pressure function, the diffusivity equation for flow of gases through porous media is, rigorously speaking, non-linear because the viscosity-compressibility product is pseudo-pressure dependent. However, several techniques and analysis procedures neglect such non-linearity. A new methodology for constructing solutions for gas reservoirs through the Green’s Function technique has been recently proposed in the literature. Such methodology handles the viscosity-compressibility product variation rigorously and it has been applied to solve several gas well tests problems successfully. However, wellbore storage and skin effects have not been considered yet in this new approach.
In this work, the Green’s Function technique is applied to obtain a new solution for an infinite homogeneous isotropic gas reservoir being produced through a single vertical well with wellbore storage and skin. The solution, however, does not consider non-Darcy flow effects. Even though the wellbore storage introduces a new non-linearity to an already non-linear problem, this work presents two accurate approximate solutions compared to the results from a commercial numerical simulator.
This work also shows that wellbore pseudo-pressure dimensionless solution is a function of the correlating groups CD exp (2S) and tD/CD, exactly like the wellbore dimensionless liquid solutions are. Liquid and gas dimensionless solutions under these correlating groups are not equal, though.