Thermal adaptive implicit method: Time step selection

Abstract

Abstract We present new linear-stability criteria for the Thermal Adaptive Implicit Method (TAIM). The analysis is applied to the mass and energy conservation equations that describe the flow and transport of an arbitrary number of components, which can partition across multiple fluid phases, in the presence of thermal effects. The existing TAIM criteria ( Moncorge and Tchelepi, 2009 ) do not guarantee oscillation-free numerical solutions for thermal compositional displacement processes that involve very steep temperature fronts. We derive a new stability limit on temperature that overcomes these numerical problems. The methodology is based on linear-stability analysis of the standard low-order space and time discretization schemes of the conservation laws used in general-purpose thermal-compositional reservoir simulators. Specifically, for spatial discretization, phase-based, upstream weighting is used for first derivatives and central differencing is used for second derivatives. In terms of the robustness and accuracy of the TAIM stability limits, our analysis and computational results indicate that honoring the divergence of the total-velocity in the linearized system of coupled mass and energy conservation equations is more important than accounting for the rock and fluids compressibility effects. Moreover, we demonstrate through scaling analysis and numerical examples that for most problems of practical interest, a simple stability criterion obtained by assuming incompressible multiphase flow is quite robust. The relationship between the full and simplified stability criteria is analyzed in detail. The methodology is demonstrated using several thermal–compositional examples, including Steam Assisted Gravity Drainage (SAGD). Finally, the criterion for the numerical stability of the temperature is divided into convection and conduction parts. Detailed testing using several simulation models shows clearly that the conduction part of the criterion is quite important across the parameter space of practical interest. Thus, in order to simulate the flow dynamics of large, thermal–compositional reservoir models, the conduction term should always be discretized implicitly, and the TAIM stability criteria should be applied to the mass conservation equations and the convection terms in the energy balance. This means that temperature, like pressure, is an unknown variable in every gridlock.