Stochastic proxy modelling for coalbed methane production using orthogonal polynomials

Abstract

Uncertainty in data or in the parameters of models occurs in many real world applications. Quantifying this uncertainty and its effects is required for robust design, control and optimization. In this paper, we attempt to build a proxy model for the stochastic solutions of coupled governing equations describing coalbed methane (CBM) production at different well bottomhole pressures. To achieve this, monthly production from wells (output) is expanded as a linear combination of Legendre orthogonal polynomials in the input (well bottomhole pressure) and the Wiener-Askey polynomial chaos is used to propagate the uncertainty of the model parameters. A Gaussian quadrature technique is then employed to solve for the coefficients of the basis functions in the proxy model. Alternatively, nonlinear least squares curve fitting using the Levenberg-Marquardt algorithm (LMA) is also used with polynomial chaos expansion to generate the stochastic proxy model. The proxy model now enables robust optimization using statistical metrics of CBM production calculated over the entire parameter space. In the case of multiple decision variables, the appropriate proxy model built using these techniques will allow for robust optimization without the use of any search algorithms.