Consider an unconfined aquifer system with localized recharge over a rectangular area on the ground surface of the system. The origin of the Cartesian coordinate system is located at the center of the recharge area as illustrated in Fig. 1a. The area has a size of 2l by 2w on the x-y plane. The shortest distance between an observation point (x, y) and a point (xe, ye) on the edge of the area is defined as d=min⁡((x-xe)2+(y-ye)2). The initial water table separates the unsaturated and saturated zones as shown in Fig. 1b and is chosen as the reference datum of the coordinate system. The initial thicknesses of the unsaturated and saturated zones prior to the recharge are denoted as b and B, respectively.

The mathematical model for the aquifer system comprises two simultaneous equations for unsaturated and saturated flows. The equation for saturated flow in homogeneous and anisotropic aquifers is expressed as Kx∂2h∂x2+Ky∂2h∂y2+Kz∂2h∂z2=Ss∂h∂tfor-B≤z≤0, where h(x, y, z, t) is the hydraulic head in the saturated zone; t is elapsed time since recharge began; Kx, Ky, and Kz are, respectively, the saturated hydraulic conductivities in the x, y, and z directions; Ss is the specific storage. Richards' equation for unsaturated flow is expressed as (Richards, 1931) Kx∂∂xkrϕ∂ϕ∂x+Ky∂∂ykrϕ∂ϕ∂y+Kz∂∂zkrϕ∂ϕ∂z=Cϕ∂ϕ∂t, where ϕ(x, y, z, t) is the hydraulic head in the unsaturated zone. The relative hydraulic conductivity kr(ϕ) and specific moisture capacity C(ϕ) are defined by the Gardner constitutive model (Gardner, 1958) as krϕ=ea(ϕ-z) and Cϕ=aSyea(ϕ-z), where Sy is the specific yield and a is the unsaturated exponent related to the pore-size distribution of a medium ranging from 0.2 to 5 m-1 (Philip, 1969). Substituting Eqs. (3) and (4) into Eq. (2) leads to Kx∂2ϕ∂x2+Ky∂2ϕ∂y2+Kz∂2ϕ∂z2-aKz∂ϕ∂z+aKx∂ϕ∂x2+Ky∂ϕ∂y2+Kz∂ϕ∂z2=aSy∂ϕ∂t. It is essentially nonlinear and solved with difficulty by analytical methods. Kroszynski and Dagan (1975) employed the approach of perturbation expansion to simplify Richards' equation as a first-order linearized equation and developed an approximate solution for unsaturated–saturated flow induced by well pumping. The approach is extensively used in many studies on unsaturated–saturated flow (e.g., Mathias and Butler, 2006; Tartakovsky and Neuman, 2007; Mishra et al., 2012; Liang et al., 2017a). The linearized version of Richards' equation is written as Kx∂2ϕ∂x2+Ky∂2ϕ∂y2+Kz∂2ϕ∂z2-aKz∂ϕ∂z=aSy∂ϕ∂tfor0≤z≤b. The initial conditions for those two zones are ϕ=h=0att=0. Because of symmetry of the recharge area along the x and y axes, the first quadrant (i.e., x≥0 and y≥0) of the flow domain is considered. Thus, all the horizontal outer boundaries are specified as the no-flow condition expressed as ∂ϕ∂u=∂h∂u=0atu=0,lim⁡u→∞∂ϕ∂u=lim⁡u→∞∂h∂u=0, where u∈(x,y). The top boundary condition for the recharge area is denoted as Kze-az∂ϕ∂z=I[Hx-Hx-l][Hy-Hy-w]atz=b, where I is a constant recharge rate and H() is the Heaviside step function. Note that Eq. (10) can be written as Kzexp⁡(-az)∂ϕ/∂z=I inside the recharge area 0≤x≤l and 0≤y≤w and denoted as ∂ϕ/∂z=0 outside that area. The impermeable boundary condition at the aquifer bottom is written as ∂h∂z=0atz=-B. The two continuity requirements of the hydraulic head and flux at the water table are expressed, respectively, as ϕ=hatz=0 and ∂ϕ∂z=∂h∂zatz=0. The continuity conditions are valid when the water table change is less than 50 % of the initial saturated aquifer thickness, which is certified by a Hele-Shaw experiment (Marino, 1967).

Define the dimensionless variables and parameters as follows: h‾=hB,ϕ‾=ϕB,x‾=xd,y‾=yd,z‾=zB,t‾=KxtSsd2,l‾=ld,w‾=wd,b‾=bB,κz=Kzd2KxB2,κy=KyKx,α=aB,σ=SySsB,ξ=IKz, where the overbar represents a dimensionless variable or parameter. According to Eq. (14), the unsaturated–saturated flow model is rewritten as ∂2ϕ‾∂x‾2+κy∂2ϕ‾∂y‾2+κz∂2ϕ‾∂z‾2-ακz∂ϕ‾∂z‾=ασ∂ϕ‾∂t‾for0≤z‾≤b‾∂2h‾∂x‾2+κy∂2h‾∂y‾2+κz∂2h‾∂z‾2=∂h‾∂t‾for-1≤z‾≤0ϕ‾=h‾=0att‾=0,∂ϕ‾∂u‾=∂h‾∂u‾=0atu‾=0,lim⁡u‾→∞∂ϕ‾∂u‾=lim⁡u‾→∞∂h‾∂u‾=0,exp⁡(-αz‾)∂ϕ‾∂z‾=ξ[Hx‾-Hx‾-l‾][Hy‾-Hy‾-w‾]atz‾=b‾,∂h‾∂z‾=0atz‾=-1,ϕ‾=h‾atz‾=0,∂ϕ‾∂z‾=∂h‾∂z‾atz‾=0, where u‾∈(x‾,y‾).

The unsaturated–saturated flow model composed of Eqs. (15)–(23) is solved by the methods of Laplace and Fourier cosine transforms. The Laplace transform is defined as f̃=∫0∞f‾exp(-pt‾)dt‾ with the property that ∫0∞∂f‾∂t‾exp(-pt‾)dt‾=pf̃-f‾|t‾=0, where f̃∈(ϕ̃,h̃) represents the dimensionless hydraulic head in the Laplace domain, p is the Laplace transform parameter, f‾∈(ϕ‾,h‾), and f‾|t‾=0=0 is from Eq. (17). Using Eqs. (24) and (25) converts ϕ‾(x‾,y‾,z‾,t‾) into ϕ̃(x‾,y‾,z‾,p), ∂ϕ‾/∂t‾ into pϕ̃, h‾(x‾,y‾,z‾,t‾) into h̃(x‾,y‾,z‾,p), ∂h‾/∂t‾ into ph̃, and ξ into ξ/p. The model then becomes ∂2ϕ̃∂x‾2+κy∂2ϕ̃∂y‾2+κz∂2ϕ̃∂z‾2-ακz∂ϕ̃∂z‾=ασpϕ̃ and ∂2h̃∂x‾2+κy∂2h̃∂y‾2+κz∂2h̃∂z‾2=ph̃ subject to the transformed boundary conditions of lim⁡u‾→0,∞∂ϕ̃/∂u‾=lim⁡u‾→0,∞∂h̃/∂u‾=0 with u‾∈(x‾,y‾), exp⁡-αz‾∂ϕ̃/∂z‾=(ξ/p)[Hx‾-Hx‾-l‾][Hy‾-Hy‾-w‾] at z‾=b‾, and ∂h̃/∂z‾=0 at z‾=-1. Moreover, the transformed continuity conditions are ϕ̃=h̃ and ∂ϕ̃/∂z‾=∂h̃/∂z‾ at z‾=0.

Afterward, one may take the double Fourier cosine transform that provides f^=∫0∞∫0∞f̃cos⁡ω1x‾cos⁡ω2y‾dx‾dy‾ and ∫0∞∫0∞∂2f̃∂x‾2+κy∂2f̃∂y‾2cos⁡ω1x‾cos⁡ω2y‾dx‾dy‾=-(ω12+κyω22)f^, where f^∈(ϕ^,h^) represents the dimensionless hydraulic head in the Fourier domain; ω1 and ω2 are the Fourier cosine transform parameters. The transform converts ϕ̃(x‾,y‾,z‾,p) into ϕ^(ω1,ω2,z‾,p), h̃(x‾,y‾,z‾,p) into h^(ω1,ω2,z‾,p),

∂2ϕ̃/∂x‾2+κy(∂2ϕ̃/∂y‾2) into -(ω12+κyω22)ϕ^, ∂2h̃/∂x‾2+κy(∂2h̃/∂y‾2) into -(ω12+κyω22)h^, and (ξ/p)[Hx‾-Hx‾-l‾][Hy‾-Hy‾-w‾] into ξsin⁡ω1l‾sin⁡ω2w‾/(pω1ω2). Equations (26) and (27) hence become ordinary differential equations in terms of z‾ denoted, respectively, as κz∂2ϕ^∂z‾2-ακz∂ϕ^∂z‾-ασp+ω12+κyω22ϕ^=0 and κz∂2h^∂z‾2-p+ω12+κyω22h^=0. Similarly, the transformed boundary conditions are expressed as ∂h^∂z‾=0atz‾=-1 and exp⁡(-αz‾)∂ϕ^∂z‾=ξsin⁡ω1l‾sin⁡ω2w‾/(pω1ω2)atz‾=b‾. The transformed continuity conditions are written as ϕ^=h^atz‾=0 and ∂ϕ^∂z‾=∂h^∂z‾atz‾=0. Solving Eqs. (30) and (31) subject to Eqs. (32)–(35) and then taking the inverse Fourier cosine transform leads to the solutions in the Laplace domain written as ϕ̃x‾,y‾,z‾,p=4π2∫0∞∫0∞Re(x‾,y‾,ω1,ω2)Ωϕ(ω1,ω2,z‾,p)dω1dω2for0≤z‾≤b‾ and h̃x‾,y‾,z‾,p=4π2∫0∞∫0∞Re(x‾,y,‾ω1,ω2)Ωh(ω1,ω2,z‾,p)dω1dω2for-1≤z‾≤0 with Ωϕ=4p(μ1+μ2)exp[αz‾+α+λ2b‾2][λ2cosh⁡λ1cosh⁡λ2z‾2+2λ1sinh⁡λ1-αcosh⁡λ1sinh⁡λ2z‾2],Ωh=4p(μ1+μ2)λ2exp⁡[(α+λ2)b‾2]cosh1+z‾λ1,Re=ξsin⁡ω1l‾sin⁡ω2w‾cos⁡ω1x‾cos⁡ω2y‾/(ω1ω2)forω1≠0andω2≠0,ξw‾sin⁡ω1l‾cos⁡ω1x‾/ω1forω1≠0andω2=0,ξl‾sin⁡ω2w‾cos⁡ω2y‾/ω2forω1=0andω2≠0,ξw‾l‾forω1=0andω2=0,μ1=[exp⁡b‾λ2-1]λ22-α2cosh⁡λ1,μ2=2λ1[λ2+αexp⁡b‾λ2+λ2-α]sinh⁡λ1,λ1=(p+ω12+κyω22)/κz,λ2=α2+4(ασp+ω12+κyω22)/κz. Note that Eq. (36a) is the solution for unsaturated flow, while Eq. (36b) is that for saturated flow. The inverse Laplace transform to both solutions may not be tractable. The numerical inversion of Laplace transform proposed by Stehfest (1970) is therefore used to obtain time-domain results of the solutions. The double integrals in the solutions can be evaluated numerically by the Gaussian quadrature (e.g., Gerald and Wheatley, 2004) using the dblquad Matlab built-in function (Gilat and Subramaniam, 2007) or the NIntegrate Mathematica built-in function (Wolfram, 1996).

The present solution can be applied to the problem of time-varying recharge rates based on Duhamel's integral (Bear, 1979, p. 158). The dimensionless transient head solution g‾t subject to the dimensionless time-varying recharge rate ξtt‾ can be expressed as g‾t=g‾0+∫0t‾∂ξtτ∂τg‾t‾-τdτ, where τ is a dummy variable, g‾0 denotes ϕ‾ or h‾ for the initial dimensionless recharge rate ξtt‾=0, and g‾(t‾-τ) represents ϕ‾ or h‾ with t‾ replaced by t‾-τ. If Eq. (37) is not an integrable function, we can evaluate numerically through the discretization method that (Singh, 2005) g‾N=g‾0+∑i=1NΔξiΔt‾G(N-i+1),Δξi=ξi-ξi-1,GM=∫0Δt‾g‾(MΔt‾-τ)dτ, where g‾N signifies the dimensionless head solution at t‾=Δt‾×N; Δt‾ is a dimensionless time step; GM is called the ramp kernel; ξi and ξi-1 are, respectively, dimensionless recharge rates at t‾=Δt‾×i and t‾=Δt‾×(i-1).

The percentage of the water from the localized recharge reaching the water table is defined as recharge efficiency (RE) (Munevar and Marino, 1999) written as REt=Kz∫0∞∫0∞∂h∂zdxdy/(Ilw)atz=0, where the denominator I×l×w is the volumetric rate of the water entering the aquifer system from the recharge, and the double integral is the sum of the infiltration flux at the water table. According to the dimensionless quantities defined in Eq. (14), Eq. (39) becomes RẼp=∫0∞∫0∞∂h̃∂z‾dx‾dy‾/(ξl‾w‾)atz‾=0, where RẼ represents RE in the Laplace domain and h̃ is defined in Eq. (36b). The RE increases from zero to a value equal to or below unity. The infiltration process does not affect the water table when RE =0. On the other hand, the water from the surface recharge totally arrives at the aquifer when RE =1.

The sensitivity analysis is commonly used to assess the change in the hydraulic head in response to a small change in a hydraulic parameter. The normalized sensitivity coefficient based on the present solution is defined as Si,t=∂O∂Pi/Pi, where O represents the present solution for the unsaturated or saturated flow and Pi is the ith parameter. Equation (41) can be approximated as Si,t=OPi+ΔPi-O(Pi)ΔPi/Pi, where ΔPi is an increment set to 10-3Pi (Yeh et al., 2008). Note that a large value of |Si,t| indicates that the head is sensitive to the change in the target parameter.

An iterative algorithm based on an implicit finite-difference approximation to Eq. (5) is developed to solve the nonlinear unsaturated–saturated flow model. Figure 2 shows the finite-difference grids in the simulation domains of 0≤x≤500 m, 0≤y≤500 m, and -20 m ≤z≤10 m discretized by a non-uniform grid with small grid sizes near the recharge area of 0≤x≤50 m and 0≤y≤50 m and large grid sizes away from that area. The domain falls in the first quadrant due to symmetrical flow to the x-axis and y-axis. The saturated thickness is 20 m and the unsaturated thickness is 10 m. All the boundaries except the recharge region are therefore under the no-flow condition. Equation (5) is approximated as KxΔxwΔXϕi+1,j,km+1-2ΔxwΔxeϕi,j,km+1+ΔxeΔXϕi-1,j,km+1+KyΔysΔYϕi,j+1,km+1-2ΔysΔynϕi,j,km+1+ΔynΔYϕi,j-1,km+1+KzΔz2ϕi,j,k+1m+1-2ϕi,j,km+1+ϕi,j,k-1m+1-aKzΔzϕi,j,k+1m+1-ϕi,j,km+1+aKxΔxwϕi+1,j,km+1-(Δxw+Δxe)ϕi,j,km+1+Δxeϕi-1,j,km+12+KyΔysϕi,j+1,km+1-(Δys+Δyn)ϕi,j,km+1+Δynϕi,j-1,km+12+KzΔz2ϕi,j,k+1m+1-ϕi,j,km+12=aSyΔtϕi,j,km+1-ϕi,j,km,ΔX=ΔxwΔxe(Δxw+Δxe)/2,ΔY=ΔysΔyn(Δys+Δyn)/2, where ϕi,j,km is the hydraulic head in the unsaturated zone at a nodal point (i, j, k); superscript m represents one time step earlier than the present time denoted as superscript m+1; Δxw, Δxe, Δyn and Δys are grid sizes beside a nodal point (i, j, k) in the west, east, north and south, respectively; Δz is the grid size on the z-axis; Δt is the time step. Note that Eq. (43) reduces to the discretized expression of Eq. (6) when the quadratic terms are neglected. Similarly, Eq. (1) is approximated as KxΔxwΔXhi+1,j,km+1-2ΔxwΔxehi,j,km+1+ΔxeΔXhi-1,j,km+1+KyΔysΔYhi,j+1,km+1-2ΔysΔynhi,j,km+1+ΔynΔYhi,j-1,km+1+KzΔz2hi,j,k+1m+1-2hi,j,km+1+hi,j,k-1m+1=SsΔthi,j,km+1-hi,j,km, where hi,j,km is the hydraulic head in the saturated zone at a nodal point (i, j, k). The initial condition for each nodal point is expressed as ϕi,j,k1=hi,j,k1=0 at each(i,j,k). The no-flow condition specified at the outer boundaries shown in Fig. 2a and the bottom can be written as ϕi-1,j,km+1=ϕi+1,j,km+1andhi-1,j,km+1=hi+1,j,km+1ati=1,nx,ϕi,j-1,km+1=ϕi,j+1,km+1andhi,j-1,km+1=hi,j+1,km+1atj=1,ny,hi,j,k-1m+1=hi,j,k+1m+1atk=1, where nx and ny are the total number of grids on the x- and y-axes, respectively. The top boundary condition is approximated as ϕi,j,k-1m+1=ϕi,j,k+1m+1outside the recharge area,Kze-abΔzϕi,j,k+1m+1-ϕi,j,km+1=Iinside the recharge areaatk=nz, where nz is the total number of grids on the z-axis. The grid sizes Δxw, Δxe, Δys, and Δyn are all 5 m inside the recharge area, while outside the area they gradually increase according to the formulas Δxe=1.2Δxw and Δyn=1.2Δys starting from Δxw=Δys=5 m and Δxe=Δyn=1.2×5 m =6 m. Note that the largest grid size is set equal to 25 m for good accuracy in solution prediction. The grid size Δz is set to 0.1 m and the time step Δt is chosen as 0.1 days for the period of 0–2.5 days and 0.25 days for 2.5–100 days. The total number of nodal points is 327 789. The values of the hydraulic parameters are shown in Table 1.

The head solution to the nonlinear model of Eqs. (43)–(49) is obtained by an iteration method. Initially, the quadratic terms in Eq. (43) are assumed as Kxgx(n-1)Gx(n)+Kygy(n-1)Gy(n)+Kzgz(n-1)Gz(n) with Gx(n)=Δxwϕi+1,j,km+1,(n)-Δxw+Δxeϕi,j,km+1,(n)+Δxeϕi-1,j,km+1,(n),Gy(n)=Δysϕi,j+1,km+1,(n)-(Δys+Δyn)ϕi,j,km+1,(n)+Δynϕi,j-1,km+1,(n),Gz(n)=(ϕi,j,k+1m+1,(n)-ϕi,j,km+1,(n))/Δz,gx(n-1)=Δxwϕi+1,j,km+1,(n-1)-Δxw+Δxeϕi,j,km+1,(n-1)+Δxeϕi-1,j,km+1,(n-1),gy(n-1)=Δysϕi,j+1,km+1,(n-1)-(Δys+Δyn)ϕi,j,km+1,(n-1)+Δynϕi,j-1,km+1,(n-1),gz(n-1)=(ϕi,j,k+1m+1,(n-1)-ϕi,j,km+1,(n-1))/Δz, where superscript (n) represents the nth iteration and gradients gx(n-1), gy(n-1), and gz(n-1) cause a linearized Eq. (43) because they are known head values from the previous iteration. At the first time step (i.e., t=Δt, m=2), the first iteration solves a system of Eqs. (44)–(49) and the linearized Eq. (43) with gx(0)=gy(0)=gz(0)=1 and obtains the numerical solution of ϕi,j,k2,(1) at each nodal point. The second iteration obtains ϕi,j,k2,(2) with updated values of gx(1), gy(1), and gz(1) from the previous result of ϕi,j,k2,(1). Repeat this iteration process for n≥3 until the convergence condition of ϕi,j,k2,(n)-ϕi,j,k2,(n-1)<10-4 at each nodal point in the unsaturated zone is satisfied. The last result of ϕi,j,k2,(n) is therefore the head solution to the nonlinear model. Similarly, the iteration process is applied to obtain ϕi,j,km,(n) for m≥3 at the other time steps (i.e., t=2Δt, 3Δt,…) with the convergence condition ϕi,j,km,(n)-ϕi,j,km,(n-1)<10-4. Note that the first iteration at each time step calculates gx(0), gy(0), and gz(0) using ϕi,j,km,(n) obtained at the previous time step.

Here we investigate the difference between the present solution and Chang et al.'s (2016) analytical solution to explore the effect of unsaturated flow on the head distributions in the aquifer. Chang et al.'s (2016) solution considers 3-D saturated flow in an unconfined aquifer with localized recharge but neglects the effect of unsaturated flow. One might expect that the difference will mainly be dominated by the magnitudes of parameters α (dimensionless unsaturated exponent) and b‾ (dimensionless unsaturated thickness). Figure 3 displays the predicted temporal head distributions at (x‾,y‾,z‾)=(2,0,-0.5) by their solution and the present solution, Eq. (36b), for different pairs of (α, b‾) with α=102 or 103 and b‾ from 10-3 to one. Significant difference in h‾ predicted by both solutions can be seen except for the cases (α, b‾)=(102,10-1), (103, 10-1), and (103, 10-2) shown in the figure. The result indicates that the thickness of the unsaturated zone is less than 10 % of the saturated aquifer thickness (i.e., b‾≤0.1) for obtaining close predictions from both solutions. When b‾=10-2, both solutions disagree if α=102 and agree well if α=103, indicating that the magnitude of the product αb‾ (=ab) should at least be 10 (i.e., αb‾≥10) for good agreement of both solutions. It is worth noting that both solutions disagree even for a much thinner unsaturated zone as compared with the aquifer (i.e., b‾=10-3) because of αb‾<10. Additionally, the curves for α=10 all disagree with Chang et al.'s (2016) solution, whereas the curves for α=104 match with this solution except for two cases where (α, b‾)=(104,0.5) and (104, 1) (not shown in the figure). Judging from the above, one can recognize that the effect of unsaturated flow on the predicted head in saturated aquifers is negligible when αb‾≥10 and b‾≤0.1. A great number of existing analytical solutions ignoring unsaturated flow give accurate predictions only when those two inequalities are satisfied (e.g., Chang and Yeh, 2007; Illas et al., 2008; Bansal and Teloglou, 2013). Otherwise, significant deviations will happen in their predictions.

The effect of the unsaturated zone on the RE is explored based on the curves of the RE versus t‾ shown in Fig. 4 plotted using Eq. (40) for different pairs of (α,b‾)=(100,0.5), (10, 0.5), (1, 0.01), (1, 0.5), and (1, 1). For a given t‾, the RE increases with α for a fixed b‾ and decreasing b‾ for a fixed α. After t‾=106, the RE approaches an ultimate value equalling unity when (α,b‾)=(100,0.5) and (1, 0.01), 0.9 when (α,b‾)=(10,0.5), 0.7 when (α,b‾)=(1,0.5), and 0.6 when (α,b‾)=(1,1). Those results imply that the ultimate recharge efficiency (URE) depends on the magnitudes of both α and b‾. Figure 5 illustrates contours of the URE at t‾=107 in the ranges of 0.01≤b‾≤1 and 1≤α≤100. The URE approaches unity when b‾<0.05 or α>20. In contrast, it is below 0.9 and related to a given pair (α, b‾) when b‾>0.1 and α<10. It is clearly seen that the RE is great for a large α and/or a small b‾. Those results provide useful information in the estimation of the amount of water from the recharge entering the aquifer. Notice that the case of URE <1 may be due to the problem that unsaturated flow is influenced by the water retention capacity and diffusivity in the horizontal direction.

Chang et al. (2016) performed the sensitivity analysis to investigate the sensitivity of the hydraulic head in saturated aquifers to the change in each of the aquifer parameters. This section focuses on the sensitivity analysis of the head in the unsaturated zone. Consider the recharge area of 0≤x≤50 m and 0≤y≤50 m and the observation points A at (0, 0, 5 m) under the area and B at (100 m, 0, 5 m) beside the area. Other values of the parameters are given in Table 1. The temporal distribution curves of the normalized sensitivity coefficient Si,t predicted by Eq. (42) to each of the parameters a, l, w, Ss, Sy, Kx, Ky, and Kz are exhibited in Fig. 6a for point A and Fig. 6b for point B. At a given time, a positive Si,t means that the change in the specific parameter causes the increase in the head. In contrast, a negative Si,t signifies that the change leads to the head decrease. The magnitude of the head remains unchanged when Si,t=0. Obviously, the parameters l, w, Sy, Kx, and Ky are important factors affecting the predicted head observed at points A and B, revealing that those parameters should be included in the flow model. The head at point A is sensitive to the changes in a and Kz, but that at point B is insensitive. The result implies that unsaturated flow prevails under the recharge area but does not away from the area. In addition, the coefficient Si,t to Ss almost equals zero over the entire recharge period, indicating that the change in Ss does not affect the predicted head in the unsaturated zone.

The finite-difference solution to the unsaturated–saturated flow model based on the nonlinear and linearized versions of Richards' equation, Eqs. (5) and (6), has been developed and described in Sect. 2.6. It is used to validate the present solution. Figure 7 demonstrates temporal head distributions observed at (25 m, 0, 5 m) and (25 m, 0, -10 m) under the recharge area and at (94.65 m, 0, 5 m) and (94.65 m, 0, -10 m) beside the area. The figure displays good agreement on the predicted head distributions from both the solutions. It is noteworthy that numerous attempts had been made by scholars to examine the accuracy of the linearized version of Richards' equation (e.g., Kroszynski and Dagan, 1975; Mishra and Neuman, 2010; Liang et al., 2017b). They also revealed that the linearized equation causes insignificant deviation in model predictions. We therefore conclude that the present solution is correctly developed and fairly predicts the hydraulic head for the unsaturated–saturated flow induced by localized recharge.