The following equations for the concentrations, ci, describe the transport of solutes in a variably saturated porous medium for the most general case, when the pore space is primarily filled with liquid but also contain pockets or immobile gas:

On the left-hand side of Equation 4-16, the first three terms correspond to the accumulation of species within the liquid, solid, and gas phases, while the last term describes the convection due to the velocity field u (SI unit: m/s).

In Equation 4-16 ci denotes the concentration of species i in the liquid (SI unit: mol/m3), cP, i the amount adsorbed to (or desorbed from) solid particles (moles per unit dry weight of the solid), and cG, i the concentration of species i in the gas phase.

The equation balances the mass transport throughout the porous medium using the porosity εp, the liquid volume fraction θ; the bulk (or drained) density, ρb = (1 − εp)ρ, and the solid phase density ρ (SI unit: kg/m3).

For saturated porous media, the liquid volume fraction θ is equal to the porosity εp, but for partially saturated porous media, they are related by the saturation s as θ = sεp. The resulting gas volume fraction is av = εp − θ = (1-s)εp.

On the right-hand side of Equation 4-16, the first term introduces the spreading of species due to mechanical mixing () as well as from diffusion and volatilization to the gas phase. The tensor is denoted DD (SI unit: m2/s) and the effective diffusion by De (SI unit: m2/s).

The last two terms on the right-hand side of Equation 4-16 describe production or consumption of the species; Ri is a reaction rate expression which can account for reactions in the liquid, solid, or gas phase, and Si is an arbitrary source term, for example due to a fluid flow source or sink.

In order to solve for the solute concentration of species i, ci, the solute mass sorbed to solids cP,i and dissolved in the gas-phase cG,i are assumed to be functions of ci. Expanding the time-dependent terms gives

where kP,i = ∂cP,i/∂ci is the adsorption isotherm and kG,i = ∂cG,i/∂ci is the linear volatilization. Equation 4-16 can then be written as

In the case of transport in a saturated porous medium, θ = εp and the governing equations are