In one dimension, Ryan and Cohen [1991] sequentially coupled two-phase (air and NAPL; immobile water) immiscible flow, with a NAPL phase in the unsaturated zone containing sparingly water-soluble organic compounds, to four-phase (air, water, NAPL, solid) nonequilibrium chemical transport. The model resulted in a flow equation for each phase and a transport equation for each solute in each phase. The numerical approach was notable for a front-tracking algorithm to locate the NAPL front as a function of penetration time without the front smearing typical of standard methods, yielding a discontinuity in NAPL saturation at the front. Abriola et al. [1993] studied surfactant-enhanced solubility of NAPLs with a one-dimensional finite-element model. Residual NAPL was immobile, trapped in spherical globules, with transport of dissolved organic and surfactant in the water phase and nonequilibrium solubility with linear driving force.
Brusseau [1992a] considered an immobile NAPL with rate-limited exchange with air, water, and solid phases in one dimension. The usual first-order mass-transfer model, which assumed that exchange was constrained by one factor (usually transfer at the interface), was extended to include a second resistance to allow for additional constraints such as liquid diffusion rates. This led to two mass-transfer equations, one for each concentration coupled to the interface.
Chen et al. [1992] modeled one-dimensional biodegradation and transport of benzene and toluene, with equilibrium mass exchange between some combinations of solid, liquid, gas, and biomass. Components included the two substrates, two electron acceptors, one trace nutrient, and two microbe populations, resulting in five transport equations, five algebraic mass-transfer equations, and two ordinary differential equations describing microbe growth. The finite-element transport equations were solved by Picard iteration, with Newton-Raphson for mass partitioning. Malone et al. [1993] considered a similar one-dimensional problem, with solid, fast oil (in terms of rate of mass transfer with water), slow oil, and water phases and n hydrocarbon components. The physical difference between the two oils was in the size of trapped globules. Equilibrium linear sorption and rate-limited linear oil-water mass transfer were assumed. The model included equations corresponding to intermediate reaction compounds, oxygen, and oil saturation. The equations were solved sequentially, with upstream block-centered finite differences for transport and ordinary differential equations for reactions and partitioning. Numerical dispersion was controlled by reducing the physical dispersion coefficient to compensate for upstream weighting, which is feasible in one dimension.