Analytical Study of Airflow Induced by Barometric Pressure and Groundwater Head Fluctuations in a Two-Layered Unsaturated Zone

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1 Analytical Study of Airflo Induced by Barometric Pressure and Groundater Head Fluctuations in a To-Layered nsaturated Zone by Jin-Ying Song, Hailong Li, and Li Wan Abstract A study on subsurface airflo plays a vital role in quantifying the effectiveness of natural attenuation of volatile organic compounds (VOCs) or in determining the need of engineering systems (e.g., soil vapor extraction of VOCs). Here, e present a ne analytical solution for describing the subsurface airflo induced by barometric pressure and groundater head fluctuations. The solution improves a previously published semi-analytical solution into a fully explicit expression and can save much computation efforts hen it as used to estimate the soil permeability and porosity, hich as demonstrated by a hypothetical example. If the groundater head and barometric pressure fluctuations have the same frequency and the same order of magnitude for the amplitudes, each or the combination of both fluctuations ill generate the air exchange volumes of the same order of magnitude through the ground surface. Particularly, the air exchange volume caused by the combined fluctuations increases ith the upper layer s permeability and loer layer s porosity and decreases ith the phase difference beteen these to fluctuations, fluctuation frequency, and the upper layer s thickness. The air exchange volume may decrease quickly to zero essentially hen the upper layer s permeability decreases -fold and decrease fourfold to fivefold hen the phase difference decreases from π to zero. Introduction The study of airflo in unsaturated soils plays a crucial role in solving various engineering problems in agriculture, the nuclear industry, and environmentology (Shan 995). Quantifying subsurface airflo is important in designing a natural (passive) vapor extraction system to remove volatile organic compounds (VOCs) from contaminated sites. Buckingham (94) first described and analyzed airflo in the unsaturated zone in response to barometric pressure variations. Stallman (ritten communication, 96) proposed a method to estimate air permeability of the unsaturated soils via air pressure observations at any depth. Subsequently, this method as applied in single-layer (Stallman 967; Stallman and Weeks 969) and multilayer (Weeks 978) unsaturated systems. By the 98s, subsurface airflo became a hot topic attributed to the significance of environmental protection and control. Analytical solutions describing subsurface airflo ere derived and used to determine the air permeability of the unsaturated zone (Shan 995; Lu 999; Szilagyi 4)., The Author(s) Groundater Monitoring & Remediation, National Ground Water Association. doi:./ x In most previous studies on subsurface airflo, the ater table as regarded as a fixed impervious boundary. In reality, hoever, the ater table may fluctuate in response to various periodic natural forces such as earth tides (Hsieh et al. 987), sea tides (Li et al. ; Jeng et al. 5), aves (Li and Barry ), and barometric pressure fluctuations (Rostaczer 988). The fluctuations of the ater table induce airflo in the unsaturated zone (Jiao and Li 4; Li and Jiao 5; Xia et al. ). Li and Jiao (5) considered the effects of tide-induced ater table fluctuations on subsurface airflo. They developed an explicit analytical solution to the vertical airflo driven by tide-induced groundater head fluctuations in a coastal to-layered system consisting of a less permeable upper layer and a highly permeable loer layer. Xia et al. () derived an analytical solution to describe the air pressure variation in a coastal three-layered unsaturated zone caused by tideinduced ater table fluctuations. Their solutions ignored the effects of barometric pressure fluctuations. In reality, hoever, airflo in the unsaturated zone is affected not only by groundater head fluctuations but also by barometric pressure fluctuations. Li et al. () expanded the solution of Li and Jiao (5) by considering both the barometric pressure fluctuations and the tide-induced head fluctuations. Based on the Laplace transform, Li et al. () obtained a 4 Groundater Monitoring & Remediation 33, no. / Winter 3/pages 4 47 NGWA.org

2 semi-analytical solution starting from a hypothetical zero initial condition. In order to obtain a strictly periodic solution, one has to run the inverse numerical Laplace transform until the effects of the zero initial condition die out, hich makes calculation more time-consuming. The obective of this paper is to develop an explicit analytical solution for describing the subsurface airflo induced by barometric pressure and general groundater head fluctuations. Our solution expands on the previous solution of Li and Jiao (5) by including the barometric pressure fluctuations and improves the semi-analytical solution of Li et al. () into an explicit, periodic analytical solution ithout any initial condition. In addition, e generalize the concept of ater table fluctuations as the driving force. Then, a hypothetical inverse problem is presented to demonstrate the fact that our solution can be conveniently used to estimate the porosity and air permeability of an unsaturated zone. Finally, e calculate and discuss the rate of air exchange beteen the atmosphere and soil pores using the analytical solution. Mathematical Model We consider the vertical subsurface airflo in a tolayered system affected by both the barometric pressure fluctuations on the ground surface and groundater head fluctuations beneath the ater table. The latter may be induced by various periodic natural forces such as earth tides, sea tides, and barometric pressure fluctuations. The model consists of to subsurface horizontal layers. The upper layer lies in the unsaturated zone ith lo permeability and the loer layer forms an unconfined aquifer ith high permeability. Let the z axis be vertical, positive upard, and the intersection ith the surface be the origin (elevation datum) of the axis (Figure ). The thickness of the loer and upper layer is b L and b, respectively. Let W(t) be the elevation of the ater table in the unconfined aquifer, and G(t) be the total groundater head measured by a pressure transducer installed at the bottom of the loer layer (Figure ). It is assumed that even at the loest ater table the pressure transducer is in the saturated zone. Folloing Li and Jiao (5), the system is assumed to meet the folloing assumptions: () the system is in an isothermal condition at a temperature of 5 C and has no sinks or sources; () the ater and air are immiscible, and the air is an ideal gas; (3) the airflo in the unsaturated zone is one dimensional (vertical) and the horizontal component can be neglected; (4) the gravitational effect of air phase is negligible in comparison ith the applied pressure gradient; and (5) the permeability of the loer layer is so high that the spatial variation of the air pressure P(z, t) in the unsaturated zone of the loer layer (i.e., hen the elevation z is in the range of W(t) z b ) is negligible. The air pressure therein only depends on time and can be denoted as P (t) and equals P(z, t) z b. Similarly, the groundater head in the unconfined aquifer at any depth only depends on time and is defined as G(t). The fluctuations of the hydraulic head G(t) can be caused by various periodic natural forces such as earth tides, sea tides, and barometric pressure fluctuations. Folloing the arguments and assumptions in previous studies (Weeks 978; Nielsen 99; Shan et al. 99; Li and Barry ), it is assumed that G(t) can be expressed as P Gt D A t c () N () atm + + ρ g i icos( ωi + i) here D is the mean depth of the ater table from the surface [L], P atm is the mean atmospheric pressure [M/L/ T ], r W is the ater density [M/L 3 ], g is the gravitational acceleration [L/T ], t is time [T], and N is the number of sinusoidal components of the groundater head fluctuations observed by the pressure transducer. The parameters A i, i, and c i are the amplitude, frequency, and phase shift of the ith component, respectively. The barometric pressure fluctuations on the ground surface can be defined as N P f () t Patm + Bicos( mt i + ni) () i here N P is the number of sinusoidal components of the barometric pressure fluctuations, and the parameters B i, m i, and n i are the amplitude, frequency, and phase shift of the relevant ith component, respectively. Based on the ideal gas la, hen the range of variations in the air pressure is less than one-tenth of the atmospheric pressure P atm, the governing equation for the air pressure in the upper layer can be ritten as (Shan 995) nμ P P, b < z < Patmk t z (3) here n is air-filled effective porosity of the upper layer, m is the dynamic viscosity [M/L/T] of air, and k is air permeability of the upper layer [L ]. The boundary condition on the ground surface (z ) is given by P( z, t) f ( t) z (4) Figure. Schematic of periodic subsurface airflo in the unsaturated zone induced by barometric pressure and groundater head fluctuations. The boundary condition at the loer boundary z b is similar to that in Li and Jiao (5) and can be ritten as P ρ gk P ρ g Aiωisin( ωit+ ci) (5a) t dn z d N Lμ i b NGWA.org J.-Y. Song et al./ Groundater Monitoring & Remediationn 33, no. :

3 here n L is air-filled effective porosity of the loer layer, and the constant d is defined as d + ρg( D b)/ Patm (5b) Because of pressure balance and assumption (5), the head G(t) and the ater table W(t) of the unconfined aquifer satisfy the folloing (Figure ): P (,) () t Pzt b Gt () Wt () + Wt () + (5c) ρ g ρ g Analytical Solution sing the superposition principle, the solution to the boundary value problem (3) to (5) can be explicitly expressed as the sum of to terms P P + Pb (6) here P W is the air pressure induced by the groundater head fluctuations at the loer boundary hen one ignores the barometric pressure fluctuations on the ground surface (i.e., hen the term f(t) on the right-hand side of Equation 4 vanishes). The governing equation and boundary conditions for the air pressure P W have the same forms as those for the air pressure P a in Li and Jiao (5). Therefore, P W has exactly the same mathematical expression as P a given by Equation 5 in Li and Jiao (5), namely N P ( z, t; d, r, θ ) ρ g Aσ ( z/ b ; d, r, θ ) cos[ ω t+ τ( z/ b; d, r, θ ) + c ], b < z < (7a) here σ ( z/ b ; d, r, θ ) C ( z/ b ; d, r, θ ),,..., N (7b) π τ ( z/ b; d, r, θ ) + Arg[ C( z/ b; d, r, θ )],,..., N (7c) and C (z/b ; d, r, q W ) is a complex function defined as θ sinh[ ( + i) θ z/ b] C( z / b; d, r, θ ), ( + i) r cosh[( + i) θ ] + idθ sinh[( + i) θ ],..., N (7d) here q W is the air-leaking resistance ith respect to the th sinusoidal component of the ater head fluctuations (Li and Jiao 5) defined as b ω n μ θ (7e) Patmk and r is the dimensionless air-filled porosity (Li and Jiao 5) defined as r ρ gb Patm n (7f). L The difference beteen P W here and P a in Li and Jiao (5) lies only in the driving forces. The groundater n head fluctuations in Li and Jiao (5) are induced by sea tides only. Here, it is assumed that the driving forces of the groundater head fluctuations include not only sea tides but also earth tides and barometric pressure fluctuations. As a given boundary condition, the groundater head is measured by a pressure transducer installed in the saturated zone. The term P b in Equation 6 is the air pressure induced by the barometric pressure fluctuations on the ground surface hen the groundater head fluctuations at the loer boundary z b are neglected (i.e., hen the right-hand side of Equation 5a vanishes). Namely, P b satisfies the folloing governing equation and boundary conditions n μ P P, b < z < P k t z b b atm N P i (8a) Pb (,) z t B cos( ) z i mt i n + i (8b) Pb ρgk P b t dnlμ z b (8c) The solution to Equations 8a to 8c is (see Appendix for the derivation) here N P Pb (,; ztdr,, θb ) Bσ( z/ b ; dr,, θb ) cos[ mt + τ( z/ b ; d, r, θ b ) + n ], b < z < (9a) and C(z/b ; d, r, q b ) is a complex function defined as Cz ( / b; drθ,, ) d ϕ sinh( ϕ + ϕ z/ b ) + rcosh( ϕ + ϕ z/ b ), d ϕ sinh( ϕ ) + rcosh( ϕ ) (9b) (9c),..., NP (9d) In Equation 9d, is an intermediate variable defined as ϕ ( + i) θb (9e) here q b is the air-leaking resistance ith respect to the th sinusoidal component of the barometric pressure fluctuations and is given by θ b b mn μ P k atm (9f) The solution (9a) to (9d) indicates that, for each sinusoidal component, the periodic air pressure P b in the unsaturated zone is determined by to factors: the dimensionless amplitude s and the phase shift t. Both of these to factors are determined by the three dimensionless parameters q b 4 J.-Y. Song et al./ Groundater Monitoring & Remediation 33, no. : 4 47 NGWA.org

4 (if the component subscript is ignored), r, and d. Similarly, the periodic air pressure P W in the unsaturated zone is also determined by the three dimensionless parameters q W, r, and d. Therefore, Equation 6 is determined by the four dimensionless parameters q b, q W, r, and d. The solution given here improves the semi-analytical solution of Li et al. () into an explicit analytical solution ithout the requirement for initial conditions. The curves in Figures 3 and 4 of Li et al. (), hich ere calculated from the asymptotic, periodic semi-analytical solution of Li et al. (), ere calculated using our ne solution. The results of the ne solution given by our solution are coincident ith Li et al. () (not shon here for the sake of succinctness). Discussion Hypothetical Example of Soil Parameter Estimation As Li et al. () have discussed the asymptotic, periodic solution in detail, particularly in their Figures 3 and 4, here e ill focus on the applicability of our ne solution to estimate unsaturated zone parameters. To this end, a hypothetical example as designed to illuminate ho the porosity n and air permeability k of the upper layer can be determined. The approach used is described as follos. Recall that Equation 6 is determined by the four dimensionless parameters q W, q b, r, and d, and e no assume that the barometric pressure and groundater head have the same period, q W q b q. We then use a given set of physical parameters to determine the true values of the parameters q, r and hich d are used to generate exact air pressure fluctuation data immediately beneath the interface beteen the upper and loer layers forced by a given sinusoidal barometric pressure fluctuations on the ground surface and a sinusoidal groundater head fluctuations in the unconfined aquifer. These generated data are treated as observed pressure data and random noise are added to represent observation errors. Then, e assume that the parameters q and r are unknon, the boundary conditions, the system setup, and the observed air pressure fluctuation data are knon. The inverse problem is solved to estimate the to unknon parameters q and r based on Equation 6. By comparing the estimated and the true values of parameters q and r, the applicability of Equation 6 in estimating the parameters of the unsaturated zone is examined. The unsaturated zone is assumed to have the folloing parameter values: b 3.3 m, k. m, n.5, D 6.5 m, and n L.4. The angular velocity of the groundater head and barometric pressure fluctuations is π/4 h, barometric pressure fluctuation amplitude is B Pa, and groundater head fluctuation amplitude is A.5 m. Other physical parameter values used are as follos: average atmospheric pressure P atm.35 kpa, ater density r W kg/m 3, gravitational acceleration g 9.8 m/s, and air viscosity m.76 5 kg/(m s). sing Equations 5b, 9b, and 9c and the above model parameter values, the true values of the three basic parameters ere calculated as: d.66, q.3, and r.998, yielding an air pressure time series at the interface beteen the upper and loer layers for 73 h shon in Figure. These Figure 3. Changes of the ratio Q in ith air-leaking- resistance q for four different cases hen r. and d.3. Case : Both the barometric pressure and ground ater head fluctuate ith a phase difference of o (A. m, B Pa, and n o); Case : Only ground ater head fluctuates (A. m and B Pa); Case 3: Only barometric pressure fluctuates (A m and B Pa); and Case 4: Both the barometric pressure and ground ater head fluctuate ith zero phase difference (A. m, B Pa, and n ). Figure. Air pressure variation ith time at the interface beteen the upper and loer layers. The solid line is the true value generated from the true model parameter values using Equation 6. The triangles represent the observed data, hich ere perturbed by numerically generated noise ranging from 3 to 3 Pa. The dots represent the air pressures predicted by solution 6 using the estimated values of p and r in Table. Figure 4. Changes of the ratio Q in ith p for different values of r hen d.3, A. m, B Pa, and n o. NGWA.org J.-Y. Song et al./ Groundater Monitoring & Remediationn 33, no. :

5 exact air pressure data ere then perturbed into observed air pressure data by numerically generated random numbers that are uniformly distributed from 3 to 3 Pa representing the observation errors (Figure ). In order to estimate the parameters r and q, the folloing least-squares problem 7 min [ Pzt (, ; r, θ, d) P ] () r,lgθ b, d.66 is solved, here P * (,, 7) are the 73 hourly observed air pressure at the interface beteen the to layers, P(z, t ; r, q, d) is the air pressure given by the analytical solution (6) at the location z, and the observation time t (,, 7), corresponding to the three dimensionless parameters q, r, and d. Here, the logarithmic transformation of q is used to replace the parameter itself in order to speed up the convergence and avoid the computation of negative parameter values (Carrera and Neuman 986a, 986b; Li and Yang ; Li et al. 7). The results are summarized in Table. The least-squares residual (i.e., the minimized value of expression ()) ranges from to Pa for different initial guess values of the parameters, almost independent of the variation of the initial values. From Table, it can be seen that the estimated parameter values of r and q, and the fitting beteen the observed data and the analytical solution are adequately close to their respective true values and independent of the initial parameter values. From Figure it can be seen that the air pressure values predicted by solution (6) based on the estimated parameters are almost coincident ith the true air pressure values. From Equations 7e, 7f, and 9f, parameter values of q and r can be equivalently transformed into those of the air permeability k and porosity n of the upper layer, respectively, if the other model parameters are given. Table lists the corresponding values of k and n. One can see that the inverse problem defined by the minimizing problem () converges for a reasonably sufficiently ide initial value ranges of k and n. The aforementioned example shos that the explicit analytical solution can be used conveniently, reliably, and efficiently to estimate the air permeability and porosity of the upper layer. Breathing of the Ground Surface The fluctuation of the groundater head and the barometric pressure ill cause the breathing of the ground surface, that is, the air exchange beteen the atmosphere and soil pore air. Quantifying such subsurface airflo is important in remediating soils contaminated by VOCs. In this section, e use an example to demonstrate ho the model parameters and phase difference beteen the fluctuations of the groundater head and the barometric pressure affect the air exchange occurring on the ground surface. For convenience of discussion, e only consider one single sinusoidal component of the groundater head and the barometric pressure fluctuations, respectively. Therefore, e have dropped the subscript that is used to denote the number of sinusoidal component for the model parameters such as q b, q W, A, B, n, and c. As the daily (diurnal) periodic fluctuations are common and significant both for the barometric pressure and groundater head fluctuations, a period of t day is applied both for the barometric pressure and groundater head fluctuations. Thus, e have θ θ θ b ω n μ d b () Patmk here d π/t [T ] is the frequency of daily fluctuations of the barometric pressure and groundater head. On the basis of the generalized Darcy s La, e can obtain the air flo rate q [L/T ] at the ground surface z k q P μ z () When the air flos into the unsaturated zone from the ground surface, the ground surface inhales and e have q <. When the air flos out of the ground surface from the soil, the ground surface exhales and e have q >. So, the volume of air flos into the ground surface over a fluctuation period per unit area is given by: Q in k P t t min{, q}dt min, d μ z k t P max, dt μ z (3a) Here, the volume of air Q in is assumed to be positive value. se Equation 6, solution expression (7) for P W and expression (9) for P b, and select the initial time so that the phase shift c, then t (3b) Table Values of Model Parameters Estimated by Least-Squares Fitting k (L ) n log (p) r Initial value ranges [ 5, 9 ] [.,.4] [.93,.79] [.3,.66] True values Estimated values Relative error (%) a The initial value ranges of log (q) and r are calculated using Equations 9b and 9c by the value ranges of k and n. a The relative error is defined as the ratio of the difference beteen the true and estimated values to the true value. 44 J.-Y. Song et al./ Groundater Monitoring & Remediation 33, no. : 4 47 NGWA.org

6 So, using the definition Equations 9d and 7d of C and C, e have P iωdt in Re{e [ Be C + ρ gac ]} z (3c) here r(+ i) θsinh[(+ i) θ] + iθ d cosh[(+ i) θ] C b (+ i) θdsinh[(+ i) θ] + rcosh[(+ i) θ] (3d) (+ i) θ C b (+ i) rcosh[( + i) θ] + idθsinh[(+ i) θ] (3e) Substituting Equations 3c to 3e into Equation 3a yields Q rn ω b { t L d iωdt in in B C + ρ z gac θ ρ g max, Re{e [ e ] dt } (3f) Equation 3 indicates that the air exchange rate through the ground surface Q in during a day depends on seven variable parameters: q, r, d, A, B, n, and n L, and the ratio Q in / n L depends on six variable parameters: q, r, d, A, B, and n. Hence, e ill discuss Q in instead of Q in. The ratio Q in / n L has the same units as Q in, for example, m 3 per m of the ground surface and per day. Folloing Li and Jiao (5) and Li et al. (), the folloing value ranges ill be used in discussion for the three main dimensionless parameters,.8 q.4,.4 r., and.5 d.97. The integral in Equation 3f as calculated by using numerical integral based on trapezoidal formula. The convergence criterion for the numerical integral as set to be 6 m. Figure 3 shos ho the ratio Q in changes ith the airleaking-resistance q for four typical cases hen the parameters r and d are fixed. The folloing observations can be made from Figure 3:. The ratio Q in increases as q decreases for each case. Note that the parameter q is dominated by the soil air permeability k and is proportional to / k, so Q in increases ith the air permeability k.. When both the groundater head and barometric pressure fluctuate (Case and Case 4), the air exchange through the ground surface is significantly affected by the phase difference beteen the groundater head and barometric pressure fluctuations. When the groundater head and barometric pressure fluctuate out of phase ith a phase difference of π (Case ), the inhaling of the soil is significantly enhanced because the maximum barometric pressure and the minimum groundater head happen simultaneously. Similarly, the exhaling of the soil is also significantly enhanced because the minimum barometric pressure and the maximum groundater head happen simultaneously. This increases the vertical pressure gradient in the soil significantly, and the ratio Q in becomes the largest among the four cases. When the groundater head G(t) and barometric pressure fluctuate synchronously ith zero phase difference (Case 4), the inhaling of soil is significantly dampened because the Figure 5. Changes of the ratio Q in ith p for different values of d hen r., A. m, B Pa, and n o. maxima happen simultaneously both for the barometric pressure and the groundater head G(t). Similarly, the exhaling of soil is also significantly dampened because the minima occur simultaneously both for the barometric pressure and the groundater head G(t). This significantly decreases the vertical pressure gradient in the soil, and the ratio Q in becomes the smallest among the four cases. 3. The air exchange rate for Case (groundater head fluctuations ith an amplitude of. m) is significantly greater than Case 3 (barometric pressure fluctuations ith an amplitude of. kpa). This indicates that the groundater head fluctuations may induce more efficient airflo for soil ventilation than the barometric pressure fluctuations. This situation may occur in coastal areas because groundater head there often fluctuates ith an amplitude not less than. m. Figure 4 shos ho the ratio Q in changes ith the dimensionless parameter q for different r hen d.3, A. m, B. kpa, and n π. For any fixed value of r, the ratio Q in decreases ith q and approaches zero as q becomes sufficiently large. When q is sufficiently small, the relative air inflo tends to a constant, hich means that hen the air permeability of the upper layer is sufficiently large or the thickness of the upper layer is sufficiently small, the volume of air exchange through the ground surface reaches maximum. The ratio Q in increases ith r for any fixed value of q. Figure 5 shos ho the ratio Q in changes ith the dimensionless parameter q for different values of d hen r., A. m, B. kpa, and n π. For any fixed value of d, the ratio Q in decreases ith q from a positive constant for sufficiently small q to zero for sufficiently large q. When lgq is less than about.6, the ratio Q in increases ith d, but very sloly. When lgq is larger than about.6, the ratio Q in becomes insensitive to the variation of d. In general, the impact of the variation of d on the air exchange rate is very limited. This may be explained by the definition of d (Equation 5b). One can see that the larger the value of d, the larger the thickness of the unsaturated buffer zone beneath the upper layer. Therefore, the parameter d only affects the air volume of the buffer zone rather than the air exchange through the ground surface. NGWA.org J.-Y. Song et al./ Groundater Monitoring & Remediationn 33, no. :

7 It ould be interesting to find the general conditions here the airflo in the shallo soil is more influenced by barometric pressure fluctuations than groundater head fluctuations. In the folloing, this ill be briefly discussed. Based on expression (3f), it is easy to see that the contribution of the barometric pressure fluctuations to the air exchange through the ground surface is proportional to the fluctuation amplitude B (in Pa), and the contribution of the groundater head fluctuations, the fluctuation amplitude r W ga (in Pa). Therefore, the airflo in the shallo soil ill be more influenced by barometric pressure fluctuations than groundater head fluctuations if the amplitude of groundater head fluctuations is much smaller than that of the barometric pressure fluctuations. In expression (3f), let b, D, and d + r g(d b )/P atm be a constant, then the contribution of the groundater head fluctuations to the air exchange through the ground surface (the term containing C ' z ) tends to zero because the term C ' z tends to zero as b and d remains a constant. On the other hand, the contribution of the barometric pressure fluctuations to the air exchange through the ground surface Q (the term containing C ' z ) tends to the folloing positive constant ρ gn k + n d P ω μn k i ρ gn k L atm d Qin B P atmωμ d n L d + ρ g P atmωμ d n k + i P atmωμ d n L d because the term C ' z tends to the folloing constant lim C b (4a) ρgn ωdnμk + i( ρgn ωdnμk + ωdnμnld Patm) ( nldpatm ωdnμk + ρgnk Patm) + inldpatm ωdnμk (4b) Therefore, the airflo in the shallo soil ill be more influenced by barometric pressure fluctuations than groundater head fluctuations if the thickness of the upper layer is large enough so that the influence of the groundater head fluctuations fades out in the shallo part of the upper layer. Summary We have developed an explicit analytical solution that describes the subsurface airflo in a to-layered unsaturated system subected to barometric pressure and general groundater head fluctuations. Our solution generalizes the analytical solution of Li and Jiao (5) in the sense that the barometric pressure fluctuations are included. Our solution also improves the semi-analytical solution of Li et al. () into an explicit analytical expression. An example of inverse problem demonstrates that our solution can be conveniently and reliably used to estimate the porosity and air permeability of the unsaturated zone on the scale of hole thickness of the upper layer. Evaluations of the basic soil parameters are fundamental for the assessment of various concerns related to engineering, ecology, and environmentology. The air exchange rate through the ground surface, hich is driven by the barometric pressure and groundater head fluctuations, is quantitatively discussed through an example. nder reasonable assumptions that the groundater head and barometric pressure fluctuations have the same frequency and the same order of magnitude for the amplitudes, the folloing conclusions can be made:. The groundater head fluctuation alone, or the barometric pressure fluctuation alone, or the combination of both fluctuations ill generate the air exchange volumes of the same order of magnitude through the ground surface.. Particularly, the air exchange volume caused by the combined fluctuations increases ith the porosity of the loer layer and decreases ith the phase difference beteen both fluctuations and the air-leaking-resistance parameter q. The air exchange volume may decrease quickly to essentially zero hen q increases from about. to The air exchange volume is affected by the phase difference beteen the groundater head and barometric pressure fluctuations by a factor of 4 to 5. It decreases from its maximum to minimum hen the phase difference ranges from π to zero. In addition, the airflo in the shallo soil ill be more influenced by barometric pressure fluctuations than groundater head fluctuations if at least one of the folloing to conditions is satisfied: () the amplitude of the groundater head fluctuations is much smaller than that of the barometric pressure fluctuations; () the thickness of the upper layer is so large that the influence of the groundater head fluctuations fades out in the shallo portion of the upper layer. The analytical solution presented here can be used to understand conditions here the air ventilation ithin the layer is affected by atmospheric and groundater head fluctuations and the model parameters such as air permeability and thickness of upper layer. This information may be useful to optimize data collection strategies, understand subsurface airflo patterns, and remediating of VOCscontaminated sites. Acknoledgments This ork as supported by the NSFC (National Natural Science Foundation of China) Outstanding Young Scientist Grant (No. 459). We thank the Editor, the AE, Dr. Fred D. Tillman, and other to anonymous revieers for their thoughtful comments. Appendix: Derivation of the Solution Because of the linearity of boundary value problem (7a) to (7c), using the superposition principle, the solution P b (z, t) can be expressed as N P Pb (,) z t P (,) z t (A) here P (z, t) is the th sinusoidal component of P b (z, t). Let C (z, t) be a complex function so that P ( z, t) Re{ C ( z, t)} (A) here Re denotes the real part of the complex expression mentioned. 46 J.-Y. Song et al./ Groundater Monitoring & Remediation 33, no. : 4 47 NGWA.org

8 Then, the boundary value problem can be simplified into nμ C C, b < z< P k t z atm (A3a) C ( z, t) B exp[i( )] z m t + n (A3b) C ρgk C t dnlμ z (A3c) b No suppose C ( z, t) Z( z) exp[ i( mt + n )] (A4) here Z(z) is an unknon function of z, i. Substituting Equation A4 into Equations A3a to A3c, then dividing the results by exp[i(m t + n )], yield nμ Z ( z) i m Z( z) P k (A5a) Z atm ( z) B (A5b) ρ gk (i m Z( z) Z ) dn Lμ b The solution of (A5) is given by Equation 9d. (A5c) References Buckingham, E. 94. Contributions to our knoledge of the aeration of soils. nited States Department of Agriculture Bureau of Soils Bulletin 5. Carrera, J., and S.P. Neuman. 986a. Estimation of aquifer parameters under transient and steady state conditions:. Maximum likelihood method incorporating prior information. Water Resources Research, no. : 99. Carrera, J., and S.P. Neuman. 986b. Estimation of aquifer parameters under transient and steady state conditions: 3. niqueness, stability, and solution algorithms. Water Resources Research, no. : 7. Hsieh, P.A., J.D. Bredehoeft, and J.M. Farr Determination of aquifer transmissivity from earth tide analysis. Water Resources Research 3, no. : Jeng, D.S., X. Mao, P. Enot, D.A. Barry, and L. Li. 5. Springneap tide-induced beach ater table fluctuations in a sloping coastal aquifer. Water Resources Research 4, no. 7: W76. DOI:.9/5WR3945. Jiao, J. J., and H.L. Li. 4. Breathing of coastal vadose zone induced by sea level fluctuations. Geophysical Research Letters 3, no. : L5. DOI:.9/4GL957. Li, J., H.B. Zhan, G.H. Huang, and K.H. You.. Tide-induced airflo in a to-layered coastal land ith atmospheric pressure fluctuations. Advances in Water Resources 34: DOI:.6/. advaters..4. Li, H.L., L. Li, L. David, C.B. Michel, and L. Guanyi. 7. Modeling tidal signals enhanced by a submarine spring in a coastal confined aquifer extending under the sea. Advances in Water Resources 3: Li, H.L., and J.J. Jiao. 5. One-dimensional airflo in unsaturated zone induced by periodic ater table fluctuation. Water Resources Research 4, no. 4: W47. DOI:.9/4WR396. Li, L., and D.A. Barry.. Wave-induced beach groundater flo. Advances in Water Resources 3, no. 4: Li, H.L., and Q.C. Yang.. A least-squares penalty method algorithm for inverse problems of steady-state aquifer models. Advances in Water Resources 3, no. 8: Li, L., D.A. Barry, F. Stagnitti, J.Y. Parlange, and D.S. Jeng.. Beach ater table fluctuations due to spring-neap tides: moving boundary effects. Advances in Water Resources 3, no. 8: Lu, N Time-series analysis for determining vertical air permeability in unsaturated zones. Journal of Geotechnical and Geoenvironmental Engineering 5, no. : Nielsen, P. 99. Tidal dynamics of the ater-table in beaches. Water Resources Research 6, no. 9: Rostaczer, S Determination of fluid-flo properties from the response of ater levels in ells to atmospheric loading. Water Resources Research 4, no. : Shan, C Analytical solutions for determining vertical air permeability in unsaturated soils. Water Resources Research 3, no. 9: 93. Shan, C., R.W. Falta, and I. Javandel. 99. Analytical solutions for steady-state gas-flo to a soil vapor extraction ell. Water Resources Research 8, no. 4: 5. Stallman, R.W Flo in the zone of aeration. Advances in Hydroscience 4: Stallman, R.W., and E.P. Weeks The use of atmospherically induced gas-pressure fluctuations for computing hydraulic conductivity of the unsaturated zone (abstract). Geological Society of America Abstract Programs Part 7: 3. Szilagyi, J. 4. Vadose zone influences on aquifer parameter estimates of saturated-zone hydraulic theory. Journal of Hydrology 86, no. 4: DOI:.6/.hydrol Weeks, E.P Field determination of vertical permeability to air in the unsaturated zone. nited States Geological Survey Professional Paper, 5. Xia, Y.Q., H.L. Li, and L. Wang.. Tide-induced air pressure fluctuations in a coastal unsaturated zone: effects of thin lo-permeability pavements. Ground Water Monitoring Remediation 3, no. : DOI:./ Biographical Sketches J.-Y. Song is at State Key Laboratory of Biogeology and Environmental Geology, China niversity of Geosciences-Beiing, Beiing 83, China and School of Water Resources and Environmental Science, China niversity of Geosciences-Beiing, Beiing 83, China; ysong9@gmail.com. H. Li, corresponding author, is at School of Water Resources and Environmental Science, China niversity of Geosciences- Beiing, Beiing 83, China; (86) ; fax: (86) -83-8; hailongli@cugb.edu.cn. L. Wan is at State Key Laboratory of Biogeology and Environmental Geology, China niversity of Geosciences-Beiing, Beiing 83, China and School of Water Resources and Environmental Science, China niversity of Geosciences-Beiing, Beiing 83, China; anli@cugb.edu.cn. NGWA.org J.-Y. Song et al./ Groundater Monitoring & Remediationn 33, no. :