Stability Criteria for Thermal Adaptive Implicit Compositional Flows
- Arthur Moncorgé (Total E&P) | Hamdi A. Tchelepi (Stanford University)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- June 2009
- Document Type
- Journal Paper
- 311 - 322
- 2009. Society of Petroleum Engineers
- 6.5.2 Water use, produced water discharge and disposal, 5.4.2 Gas Injection Methods, 5.5 Reservoir Simulation, 7.4.4 Energy Policy and Regulation, 5.3.2 Multiphase Flow, 4.3.4 Scale, 5.4.6 Thermal Methods, 5.2.1 Phase Behavior and PVT Measurements, 5.3.1 Flow in Porous Media, 4.1.2 Separation and Treating, 5.2.2 Fluid Modeling, Equations of State, 5.3.9 Steam Assisted Gravity Drainage
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Linear stability analysis is performed for thermal/compositional displacement processes, and a concise statement of the stability limits is given. The discrete formulation is based on standard (low-order) space- and time-discretization schemes of the mass- and energy-conservation equations, which are widely used in general-purpose reservoir simulators. The analysis is applicable for thermal multicomponent multiphase flows and accounts for mass and heat convection, heat conduction, fluid compressibility, gravity, and capillarity. The derived stability limits reduce to those presented by Coats (2003a, 2003b) for isothermal compositional systems. The thermal-adaptive-implicit method (TAIM) stability criteria are tested thoroughly using a flexible thermal/compositional reservoir simulator based on the natural variable-set formulation. These numerical tests indicate that the obtained stability limits are quite sharp for a wide range of the parameter space. Specifically, small violations of these limits lead to unstable solutions for temperature and saturations. Moreover, small violations of the stability limits lead to significant deviations from reference solutions, and large persistent violations lead to completely unstable numerical results. Detailed analysis and extensive numerical testing indicate that a TAIM-based formulation, which uses the stability limits derived here as adaptive local criteria to decide whether to treat a variable as implicit or explicit, is a very promising approach for efficient simulation of thermal/compositional problems.
The flow and transport of energy and mass in porous media involve complex multicomponent, multiphase interactions and a wide range of length and time scales. Thermal-recovery processes of compositional fluids are usually described by conservation equations that are nonlinear and strongly coupled. The flow, transport, and phase behavior of these thermal/compositional systems, in which the components partition across multiple fluid phases as a function of composition pressure and temperature, can be quite difficult to model accurately. To resolve the length and time scales that govern the physics of thermal/compositional reservoir flows reasonably accurately, high-resolution discretizations, in both space and time, are usually required. As a result, it has proved quite difficult to simulate thermal/compositional processes in a scalable manner (i.e., efficient for highly detailed reservoir models with a large number of components).
In this paper, a TAIM for thermal/compositional displacements is presented. The fully implicit method (FIM), in which all the unknown variables and the coefficients that depend on them are treated implicitly, is the most common approach for the simulation of thermal-recovery processes. The main reason for using (first-order backward Euler) FIM is that the discretization scheme is unconditionally stable. Therefore, in theory, FIM allows for arbitrarily large timesteps. However, for detailed reservoir models with a large number of components, FIM can be very expensive computationally. Implicit pressure/explicit saturations (IMPES) (Stone and Garder 1961; Coats 2000) and implicit pressure and saturations, explicit compositions (IMPSAT) (Quandalle and Savary 1989; Cao and Aziz 2002; Haukås et al. 2007) methods treat some variables implicitly and others explicitly. Note that in all these mixed-implicit methods, pressure is always treated implicitly. Moreover, sources and sinks (wells) are treated in a fully implicit manner to deal effectively with the high throughput and large changes in composition, saturation, and temperature in the neighborhood of wells.
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