A review of mathematical models for geothermal wellbore simulation

Abstract

A wellbore simulator is an important tool for understanding flow in geothermal wells, however, the literature on the topic is confusing because previous wellbore simulators have been based on a variety of versions of the equations for conservation of mass, momentum and energy. This lack of consistency in describing the behaviour of flow in a geothermal well makes it unclear as to whether some models correctly conserve key quantities and presents difficulties when comparing models. This study aims to give clarity by providing a comprehensive review of conservation equations used in modelling both transient and steady multiphase flow in a geothermal wellbore.We present as a standard for comparison a complete, physically accurate three-equation model for transient flow in a geothermal wellbore. Two of the four previous transient geothermal wellbore simulators use this three-equation model, while the other two simulators are based on approximate equations that do not fully conserve all quantities.We also review previous steady-state geothermal wellbore simulators and find that some are based on mathematical models that do not correctly conserve key quantities. Some models approximate the momentum flux term using average variables, while others incorrectly account for the effect of changes in cross-sectional area on momentum flux. Our analysis shows that both assumptions can have a significant impact on simulation results under certain conditions, however, under normal operating conditions, these differences are relatively small.