Parameter Estimation of a Population-balance Foam Model Using Two-step Multi-variable Search

Summary Evaluation of modeling techniques for foam flow through porous media requires reliable laboratory measurements. Previously, a set of experimental data points have been collected in steady-state foam flood. Significant efforts have been made to mitigate foam hysteresis and to ensure experimental repeatability in each run by properly restoring the system. In this work, we have investigated the steady-state behavior of the Chen et al population-balance foam model in porous media. The classic Nelder-Mead search algorithm is used to estimate the foam-model parameters from the abovementioned experimental data with a variety of total fluid velocities and foam qualities. Our results show that this foam model does not correctly model the high-quality foam regime as the limiting capillary pressure is not reached. Further analysis reveals that, depending on the initial guesses, two different steady-state saturations at the same foam quality can be obtained. We have identified that the quadratic formula in the foam coalescence equation is the source of the issue, with which the same foam coalescence rate results in two saturation values. Therefore, we have resolved the problem with significantly reduced bubble density when the capillary pressure exceeds the limiting value. The improvement in this model results in physically meaningful fit to the steady-state data with a unique solution. Additionally, sensitivity studies of the parameters indicate that the trapped gas function could be combined with other parameters in the model based on our steady-state data fit. During this investigation we have discovered that the lack of proper initial guesses frequently causes convergence issues of the Nelder-Mead search algorithm. A new two-step approach is therefore developed with a combination of direct calculation and Nelder-Mead search to estimate the foam-model parameters. The new approach greatly reduces the parameter space explored in the algorithm, thus it significantly improves the computational efficiency and the convenience of probing a suitable set of initial guesses to mitigate convergence issues. For the first time, we have provided methodology with improved multi-variable parameter search and evaluation of hysteresis-free steady-state foam data with a population-balance foam model. The improvement in the model makes it not only correctly simulate the effect of the limiting capillary pressure but also potentially more stable in reservoir simulation practices due to the elimination of non-physical solutions.