``Limited'' refers to models that do not allow for at least three phases and three components. A mainly qualitative screening model of Weaver et al. [1994] simulated transport of a partitionable NAPL contaminant in the vadose zone, neglecting capillary-pressure gradients and reducing the problem to a first-order hyperbolic equation that could be solved by the method of characteristics. Mayer and Miller [1992] reported a two-dimensional two-phase two-component model allowing for nonequilibrium dissolution of NAPL into the aqueous phase, resulting in three equations solved for two pressures (p-p formulation) and one concentration. A Galerkin finite-element method with lumped storage terms was used for the two flow equations, which were solved simultaneously by Picard iteration. A Petrov-Galerkin scheme, again with Picard iteration, was sequentially applied to transport. Geostatistical heterogeneity was included. Guarnaccia and Pinder [1992] considered the same situation except that water was also allowed to dissolve in the NAPL phase, leading to four equations. Finite elements with orthogonal collocation and mass lumping yielded two flow equations, solved simultaneously for one pressure and one saturation (p-s formulation) by modified Picard iteration [ Celia et al., 1990], and two decoupled transport equations solved for two concentrations. The model simulated emplacement and removal of dense NAPL in a saturated system. Kueper and Frind [1991a] developed a finite-difference model for the same problem without mass transfer, solving implicit p-s equations simultaneously by Newton-Raphson iteration with an Orthomin iterative linear-equation solver preconditioned by the Dupont-Kendall-Rachford (DKR) incomplete factorization. Heterogeneity was emphasized, with a grid aligned with geological bedding.
Essaid et al. [1993] developed a two-phase two-component upstream finite-difference p-p formulation. There was no mass transfer, and three-phase saturation-dependent properties were used in the unsaturated zone, where air pressure was constant. The implicit nonlinear equations were solved by Picard iteration. Faust et al. [1989] presented the only three-dimensional model in this group for the same type of problem, using a p-s formulation. The implicit nonlinear equations were solved simultaneously with Newton-Raphson iteration, using a slice successive overrelaxation (SOR) scheme for linear equations.
Parker et al. [1994] and Wu et al. [1994] described two-dimensional areal models of three-dimensional systems with local gravity-capillary vertical equilibrium. Parker et al. [1994], using the model of Kaluarachchi et al. [1990], simulated water and light-hydrocarbon components with no mass transfer, sequentially solving triangular or quadrilateral finite-element equations with preconditioned conjugate-gradient iteration. The model was derived from vertical integration of air, oil, and water phases. Wu et al. [1994] pointed out difficulties with such a vertical-equilibrium assumption in the vadose zone, such as inapplicability to descending NAPLs, and integrated only in the saturated zone. The two-phase two-component model allowed a nonzero residual NAPL saturation through history-dependent pseudo-functions, using the vertical distribution of NAPL as well as its vertically averaged saturation. The mass-conservative finite-element equations included fully implicit well terms based on the equivalent-radius formulation of Peaceman [1983] and were solved by Newton-Raphson iteration. A limitation of the model was that it required an immobile zone above the water table, even if the water table changed with time.