[16] The governing equation for vertical one-dimensional airflow is given by generalized Darcy's law for multiphase case

where *q*_{a} is the volumetric flux of air, *k* is the intrinsic permeability of the porous medium, *k*_{ra} is the relative permeability of the porous medium to air (assumed to be a constant), *μ*_{a} is the dynamic viscosity of air, and *ρ*_{a} is the density of air. Compared to the force caused by pressure gradients, the force due to gravity is much smaller so it can be neglected under most conditions [*Charbeneau*, 2000]. Hence, equation (7) can be simplified to

From (8), the mass flux of air for a column of cross-sectional area *A* is [*Charbeneau*, 2000]

In order to integrate (9), the equation of state *ρ*_{a}(*p*_{a}) is needed. Here air is assumed to be an ideal gas. Under isothermal condition, Boyle's law holds and

where *ρ*_{a0} is the air density at reference state (the local atmospheric condition and temperature). With the equation of state given by (10), equation (9) can be integrated from the fine sand surface to its bottom to find

where *L*_{1} is the thickness of the fine sand layer, which is assumed to be relatively thin so that *k*_{ra} can be approximated as a constant. The surface of the fine sand layer is in contact with atmosphere, so the air pressure at the surface is atmospheric pressure *p*_{a0}. From (3),

with *h*_{a} being a small correction term (i.e., *ρ*_{w0}*gh*_{a} ≪ *p*_{a0}). Substitute (12) into (11), and expanding and keeping only the leading order term for *h*_{a} leads to

Equation (13) describes the air mass flux through the fine sand layer from the external atmosphere into the coarse sand layer. The minus sign indicates that the flow is downward. It should be noted that under this condition, the air mass flux *m*_{f} is a linear function of the vacuum *h*_{a}.

[17] The mass of air in the vadose zone at any given time is

Substituting the equation of state (10) into (14), and then differentiating the resulting equation with respect to *t* leads to

Substituting (3) into (15) leads to

where the atmospheric pressure *p*_{a0} is expressed as a height of water column of the reference water density, which is written as

Equation (16) is the rate of change of the air mass in the vadose zone in the column. It must be equal to the air mass flux into the column, i.e., the absolute value of (13). Hence, equating (16) and the absolute value of (13) and rearranging leads to

The derivative *dh*/*dt* in the first term of the right-hand side of (18) can be replaced by (5). The resulting equation is

Equation (19) is the ODE describing the time variation of vacuum in the vadose zone. The initial condition is

Equations (5) and (19) constitute the set of governing equations describing the flow of air and water.

[18] Substitute (12) into (11) and expand, and if both the terms are kept, then the air mass flux changes into

Equating the absolute value of (21) and (16) and then replacing *dh*/*dt* by (5) leads to

Then the set of governing equations becomes (5) and (22).