The governing equation for vertical one-dimensional airflow is given by generalized Darcy's law for multiphase case
where qa is the volumetric flux of air, k is the intrinsic permeability of the porous medium, kra is the relative permeability of the porous medium to air (assumed to be a constant), μa is the dynamic viscosity of air, and ρa 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 ρa(pa) is needed. Here air is assumed to be an ideal gas. Under isothermal condition, Boyle's law holds and
where ρa0 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 L1 is the thickness of the fine sand layer, which is assumed to be relatively thin so that kra 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 pa0. From (3),
with ha being a small correction term (i.e., ρw0gha ≪ pa0). Substitute (12) into (11), and expanding and keeping only the leading order term for ha 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 mf is a linear function of the vacuum ha.
 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 pa0 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.
 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).