Compositional simulators have most often used the IMPES (implicit pressure, explicit saturation/concentration) formulation, due to the computational burden of generating and solving fully implicit multicomponent equations, though implicit codes also date back many years [ Coats, 1980; Chien et al., 1985]. With LEA, to our knowledge universally assumed in petroleum codes, the number of partial differential equations is the number of components, often 8 to 10 in applications. The IMPES pressure equation, with Newton-Raphson iteration, has averaged 1 to 1.5 iterations to converge to a solution [ Young, 1992], and satisfactory results may often be obtained with one iteration per time step [ Watts, 1986]. The principal limitation is numerical stability, with the time step constrained by some form of Courant condition.
Groundwater modeling has generally proceeded from an implicit perspective.
Reeves and Abriola [1994] proposed a ``set-iterative'' two-phase
compositional scheme that separated bulk-phase balances (two without LEA, one
overall balance with LEA) from component balances in phases (
without
LEA, n-1 with LEA for n components), with the two sets weakly coupled
through composition-dependent fluid properties and mass-transfer terms. The
equations were solved in a sequentially implicit fashion, passing from the
most-volatile component to the least. A common difficulty with such
formulations is the inconsistency that arises from the treatment of
composition dependencies in transport terms: explicit in the phase balances,
implicit in the component balances. The authors avoided a simultaneous fully
implicit approach because of its computational requirements and considered an
``adaptive IMPES'' formulation impractical. This method, called ``adaptive
implicit'' in the petroleum literature, adaptively treats
some grid cells as IMPES and others implicitly as determined
by an automatic switching
criterion, ideally incurring the expense of implicitness only where stability
demands it. The conclusion of impracticality was based on the difficulty
of substituting primary variables with many components as phases appear and
disappear, which the use of total concentrations or total mole numbers should
avoid, and on
the expensive need to make most cells implicit after mass transfer
[ Forsyth, 1988]. The latter need arose from switching criteria based
solely on observed changes in primary variables; criteria that also incorporate
numerical stability analysis in a Young-Stephenson code have required at most
15 to 20 percent of cells to be implicit, even in high-velocity near-well
applications such as coning [ Young and Russell, 1993]. These criteria
allow a cell to switch from implicit to IMPES, which often happens after a
front passes, and have proved useful in selecting time steps for efficient
IMPES simulations.