#### 2.1. Model Description for Numerical Simulation

[6] Consider an air-water system in a coastal area composed of two horizontal layers. The upper layer lies in the unsaturated zone. The lower layer forms an unconfined aquifer which abuts a tidal water body. The water table of the unconfined aquifer, which fluctuates within the lower layer in response to the tidal level fluctuations, forms the lower boundary of the system and causes airflow therein. When the water table increases, it expels the pore air so that the ground surface exhales. When the water table decreases, it leaves extra pore space for the pore air so that the ground surface inhales. Assume that (1) the subsurface system is in an isothermal condition at 25°C, (2) the water and air are immiscible, the vaporized water in the air and the dissolution of air by the water are negligible, (3) there are no sinks or sources, and (4) the airflow in the unsaturated zone induced by the water table fluctuation is one-dimensional (vertical). Horizontal airflow can be neglected.

[7] At room temperature, assumption 2 is reasonable because the air dissolves into water at most by about 1.5% by volume [*Weir and Kissling*, 1992] and the water vaporization is very slow. In reality, the water table fluctuates with different amplitudes and phase shifts at different locations. Therefore the airflow driven by the fluctuating water table is not strictly vertical. However, if the depth of the water table is small enough and the attenuation of the water table fluctuation is not significant, the airflow driven by the fluctuating water table can be approximately regarded as vertical flow.

[8] Let the *z* axis be vertical, positive upward with the origin at the ground surface. The respective thickness of the lower and upper layer is *b*_{L} and *b*_{U}. Let *W*(*t*) be the water table of the unconfined aquifer, i.e., the real elevation of the air-water interface of the unconfined aquifer, and *H*_{P}(*t*) be the hydraulic head of the unconfined aquifer measured in a piezometer screened only at the bottom of the lower layer. The tide-induced local fluctuation of the head of the unconfined aquifer at a fixed inland location can be usually expressed as [e.g., *Nielsen* 1990]

where *D* is the distance from the surface to the mean head (see Figure 1), *N* is the number of sinusoidal components of the tide-induced local head fluctuation, *A*_{i}, ω_{i} and *c*_{i} are the amplitude, frequency and phase shift of the *i*th component, respectively.

[9] When the water table rises, the air pressure in the pores immediately above the water table will be higher than the atmosphere pressure, in this case *W*(*t*) will be lower than *H*_{P}(*t*) (see Figure 1). When the water table falls, the process is reversed and *W*(*t*) will be higher than *H*_{P}(*t*). The water table *W*(*t*) is unknown because it is related to the unknown air pressure .

[10] Under the assumptions 1–4, the above one-dimensional air-water two-phase flow system can be simulated using TOUGH2 numerical simulator [*Pruess et al.*, 1999]. In the TOUGH2, the air pressure *P*^{a} and the degree of gas saturation *S*^{a} are chosen to be the primary variables. The nonlinear governing equations of the air-water two-phase flow are given by *Pruess et al.* [1999] and will not given here. As far as the boundary conditions are concerned, the air pressure *P*^{a} equals the atmospheric pressure *P*_{atm} on the surface and *P*_{atm} + ρ^{w}*gH*_{P}(*t*) at the bottom of the lower layer. Here ρ^{w} is the water density [ML^{−3}], *g* is the gravitational acceleration [LT^{−2}]. The gas saturation *S*^{a} equals 0 at the bottom of the lower layer and the ratio of the effective to the total air-filled porosity of the upper layer on the soil surface.

[11] The air pressure in the unsaturated zone is usually affected significantly by barometric pressure fluctuations, but not always so, e.g., in the coastal area considered by *Jiao and Li* [2004] the water table fluctuation dominates the air flow. During the 72-hour period from 7 to 9 February 2001 the amplitude of the daily fluctuation of the barometric pressure is only about 0.5 kPa, but the amplitude of the daily fluctuation of the air pressure observed in the subsoil, which is mainly induced by the water table fluctuation, is as much as 2.5 kPa [see *Jiao and Li*, 2004, Figure 2a]. Our numerical simulation based on TOUGH2 simulator also indicated that the barometric pressure fluctuation can be considered in a very simple way just by subtracting the barometric fluctuation data from the subsoil air pressure without loss of much accuracy. On the basis of this reason, in this paper the atmospheric pressure *P*_{atm} is assumed to be a constant.

[12] The initial conditions of *P*^{a} and *S*^{a} are replaced by the periodicity of *P*^{a} and *S*^{a}: *P*^{a}(*z*, *t*) = *P*^{a}(*z*, *t* + *t*_{P}), *S*^{a} = S^{a}(*z*, *t* + *t*_{P}), where *t*_{P} is the period of the periodic function (1). The definition of *t*_{P} is given by *Li and Jiao* [2003, equation (9c)]. However, for numerical solutions a hypothetical initial condition is necessary. To obtain periodic numerical solutions, the simulations were run for sufficiently large number (5 ∼ 20) of tidal cycles until the effects of the initial conditions become negligible and the numerical solution becomes periodical. The results were then recorded and analyzed.

#### 2.2. Simplified Model for Analytical Solution

[13] In order to obtain the analytical solutions, the nonlinear air-water two-phase flow model has to be simplified using some extra assumptions. These assumptions are the gravitational effect of air phase is negligible (assumption 5); the lower layer is so permeable that the spatial variation of the air pressure *P*^{a}(*z*, *t*) within the range *W*(*t*) ≤ *z* ≤ −*b*_{U} is negligible, the air pressure therein only depends on time and can be defined by *P*_{0}(*t*) (assumption 6); the lower layer is so permeable that the vertical hydraulic gradient of the unconfined aquifer is negligible, the head at different depth is therefore equal to *H*_{P}(*t*) (assumption 7); the capillary effects in the lower layer is negligible (assumption 8). With these assumptions, the study domain is restricted within the upper layer (−*b*_{U} < *z* < 0) with a new unknown *P*_{0}(*t*). The reasonability of these assumptions will be examined by numerical solutions in section 3.2. If the variation range of *P*^{a} is less than one tenth of the atmospheric pressure *P*_{atm}, based on the ideal gas law, the governing equation for air in the upper layer can be linearized as [e.g., *Shan*, 1995]

where *n*_{aU} is air-filled effective porosity of the upper layer (see Figure 1), μ^{a} is the viscosity [ML^{−1} T^{−1}] of air, *k*_{U} is the air permeability [L^{2}] of the upper layer. The boundary condition at the ground surface is

In order to derive the boundary conditions at the bottom of the upper layer, consider *q*(−*b*_{U}, *t*), the vertical isothermal volumetric flux [LT^{−1}] of air through the soil cross section at *z* = −*b*_{U}. According to the extended Darcy's law for gases [*Stonestrom and Rubin*, 1989], one has

[14] Because of pressure balance and assumptions 6–8, the head *H*_{P}(*t*) and the water table *W*(*t*) of the unconfined aquifer satisfy (see Figure 1)

Consider a vertical column within the range *W*(*t*) ≤ *z* ≤ −*b*_{U} and with unit horizontal cross-section area. According to the mass conservation law, the initial air mass *M*_{0} in the vertical column at initial time *t* = 0 equals the sum of the air mass contained in the layer at time *t* and the air mass through the bottom *z* = −*b*_{U} during the period [0, *t*], i.e.,

where *n*_{aL} is the air-filled effective porosity of the lower layer (see Figure 1), ρ^{a} is the air density [ML^{−3}]. The second term represents the air mass through the upper layer's bottom during the period [0, *t*] (positive for mass loss and negative for mass gain). Differentiating equation (3c) with respect to time, substituting the expression of *W*(*t*) obtained from (3b) into the resultant equation, then using the ideal gas law, and substituting (3a) and (1) into the resultant equation, yield

where

Equation (3d) is the boundary condition at *z* = −*b*_{U} for the governing equation (2a). Because the coefficient ξ(*t*) depends on *P*_{0}(*t*), (3d) is nonlinear and it is difficult to find the accurate analytical solution of (2a), (2b) and (3d). Replacing *P*_{0}(*t*) by *P*_{atm}, and *A*_{i} cos (ω_{i}*t* + *c*_{i}) by its temporal mean, i.e., zero, then the right-hand side of (3e) can be simplified into a constant defined as

When ρ^{w}*g**A*_{i} ≪ *P*_{atm} and the variation magnitude of *P*_{0}(*t*) is less than 10% of the atmosphere pressure *P*_{atm}, *d* is an adequate approximation of ξ(*t*). Replacing ξ(*t*) in the nonlinear boundary condition (3d) by *d*, yields

Equation (3g) is an approximate linearization of equation (3d).