Summary
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.
Introduction
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.
© 2009. Society of Petroleum Engineers
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History
- Original manuscript received:
6 June 2007
- Revised manuscript received:
19 May 2008
- Manuscript approved:
20 May 2008
- Published online:
1 June 2009
- Version of record:
1 June 2009
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