A finite analytic algorithm for modeling variably saturated flows. Duan a,b, Li Chen a,b. Chang an University, Ministry of Education, P. R.

Transcription

1 1 2 3 A finite analytic algorithm for modeling variably saturated flows Zaiyong Zhang a,b,c, Wenke Wang a,b,*, Chengcheng Gong a,b, Zhoufeng Wang a,b, Lei Duan a,b, Li Chen a,b a Key Laboratory of Subsurface Hydrology and Ecological Effects in Arid Region, Chang an University, Ministry of Education, P. R. China b School of Environmental Science and Engineering, Chang an University, P. R. China c Department of Hydrology and Water Resources, University of Arizona, Tucson, Arizona 85721, USA *Corresponding author: Present/permanent address: School of Environment Science and Engineering, Chang an University, Yanta Road 126, Xi an, Shaanxi, the People s Republic of China Tel.: , address: wenkew@chd.edu.cn; wenkew@gmail.com (W. Wang)

2 Abstract A general numerical algorithm in the context of finite analytic method has been developed to accurately and efficiently simulate variably saturated flow. By separating the nonlinear hyperbolic characteristic from the linear parabolic part, the governing equation in the developed method has been transformed by Kirchhoff transformation. The transformed governing equation can be applied to variably saturated systems with arbitrary constitutive relationships between pressure head and the secondary variables. Subsequently, the finite analytic method (FAM) has been used to solve the transformed governing equation. The stability of FAM has been proven by a rigorous mathematical analysis. Further, numerical experiments have been used to show that FAM is accurate by comparing its results with the analytical solution and those obtained from other numerical approaches in 1D and 2D computational domains. These numerical experiments show that the developed method is not only more accurate but also more efficient than other numerical methods. Finally, it has also been shown that FAM can reproduce the results of a field experiment. Therefore, there are advantages to simulate water flow in variably saturated zone with the developed method

3 Keywords: Richards equation, variably saturated flow, Finite analytic method, Kirchhoff transformation 1. Introduction The process of water flowing through variably saturated zone is a crucial problem in agricultural and environmental engineering, soil science, theoretical and applied hydrology as the results can affect the water management (Serrano, 2004; Kim et al., 2005; Twarakavi et al., 2008). Experimentation and mathematical modeling are two effective approaches to study the process of water flowing through variably saturated zone. While experimentation (Wang et al., 2011) can better reflect the micro- and macro-environment, it is however expensive, tedious and time-consuming. Hence, numerical modeling is generally a more popular approach for evaluating variably saturated flow and assessing the sensitivity of model output to various model parameters (Celia et al., 1990; Srivastava and Yeh, 1992; Yeh et al., 1993; Harter and Yeh, 1996; Yeh and Zhang, 1996; Yeh and Simunek, 2002; Zhang et al., 2015, 2016a and 2016b). The numerical modeling is undoubtedly a cost-effective approach to study and analyze flow under various scenarios. Due to the nonlinear physical processes, it is difficult to obtain analytical solutions for variably saturated flow problems, without making some simplifications. Thus, to the best of our knowledge, the numerical approach can only be applied to variably saturated zone where the water flow or solute transport is fully described. It is well

4 known that Richards equation can be solved using finite difference method (FDM) or finite element method (FEM). Further, Richards equation can be expressed in three forms: i.e. moisture content form, pressure head form and the mixed form in which both variables are employed. The pressure head form has been widely used to simulate variably saturated flow in many studies (e.g. Srivastava and Yeh, 1992; Therrien and Sudicky, 2001; Mayer et al., 2002). However, there are some situations where modeling variably saturated flow is still a problem. For example, when water infiltrates into very dry soils, one may encounter poor iterative efficiency, mass-balance error and convergence problems. Specifically, Celia et al. (1990) pointed out that the time derivative term can cause poor mass balance. In particular, h although the time term of the pressure head form ( C( h) ) and the moisture content t θ form are mathematically equivalent in the continuous partial differential t h equation, the discrete form of the pressure head form ( C( h) ) may cause large t computational error due to high nonlinearity of C (h). Moreover, the mass balance error grows with increasing time-step size. To overcome this problem, Milly (1985) introduced a modified definition of the capacity term. By coupling with mass-lumping, it ensures effective global mass balance in the pressure head form equation. Further, Rathfelder and Abriola (1994) presented a scheme called the standard chord slope approximation, which can evaluate a specific soil moisture capacity ( C ( h) ). This scheme is able to provide accurate mass balance. As for the moisture content form, there is less nonlinearity in hydraulic diffusivity D ( θ ) than hydraulic conductivity

5 k ( θ ). Hence, the mass balance error is smaller and better suited for modeling infiltration into an initially dry soil. However, the moisture content form also has some serious limitations. For example, it cannot simulate water flowing in a saturated zone, since the pressure-saturation relationship does not exist. It cannot apply to layered soils since the moisture content is not continuous at the interfaces between different soil types. In order to overcome those shortcomings, based on the moisture content form, Kirkland et al. (1992) developed methods which not only conform to the mass balance but also describe the saturated flow accurately. However, this method is numerically unstable, because the values of the parameters are discontinuous at the saturated-unsaturated interface. In addition, many studies (Hills et al., 1989; Matthews et al., 2004, 2005; Zha et al., 2012) have used the moisture content form equation to numerically simulate water flowing into layered soils. Schaudt and Morrill (2002) combined the method developed by Romano et al. (1998) with the inherent numerical efficiency of the θ -based RE. However, it cannot guarantee conservation of the global mass when a wetting front reaches a heterogeneous interface under high infiltration. Further, using the Method of Lines (MOL), Matthews et al. (2004, 2005) investigated the θ -based RE to layered soils. However, by assuming flux continuity at the interface, it may introduce significance error in the storage term. In order to minimize the mass balance error, numerical methods using the mixed form equation have been developed. Celia et al. (1990) developed a modified Picard iteration scheme which can solve the mixed form

6 equation with excellent mass balance. However, the mass-conservative methods cannot ensure the accuracy of the solution. No matter which form of the equation is used, the main difficulty in solving the highly nonlinear Richards equation comes from its hyperbolic characteristic, despite its parabolic form, counting on the degree of saturation in the solution domain (Ji et al., 2007). Some researchers (Williams et al., 2000; Wu and Forsyth, 2001; Farthing et al., 2003) further pointed out that the degree of nonlinearity is determined from the constitutive relationships between the pressure head, the saturation permeability and the relative permeability, which may introduce major difficulties in the numerical solution procedures. This unique mathematical characteristic of Richards equation needs a special method, such as Picard or Newton-Raphson iterative procedure, which can solve mixed elliptic, parabolic, and hyperbolic partial differential equations (Baca et al., 1997). However, Smith et al. (2002) indicated that the transform approaches, such as Kirchhoff transform, have attempted to linearize Richards equation using specific constitutive relationships. As such, the nonlinear hyperbolic characteristic of Richards equation is separated from the linear parabolic part in each iterative solution stage. Ross and Bristow (1990) pointed out that the Kirchhoff transform is a tool simulating water flow in nonhysteretic. The Kirchhoff transformation has been used to eliminate the square of the first derivatives which is in the expanded Richards equation. In the vicinity of the sharp wetting front, a satisfactory solution cannot be

7 achieved unless very small space and time increments are used. Alternatively, it can be solved using a round-about method, such as the one used by Hanks and Bowers (1962). In this method, the diffusivity is computed first and then based on the relative hydraulic conductivity from the computed diffusivity, the coefficients are determined. Using Kirchhoff transformation to transform the equation makes it easier to solve numerically than the mixed and pressure-head forms of Richards equation. It is also different from the saturated-based form as it can be applied to the fully saturated conditions. Therefore, several authors have developed efficient numerical algorithms to solve this form of Richards equation (Vauclin et al., 1979; Ji et al., 2007; Zhang et al., 2015). Moreover, even if the mixed or pressure-based form of Richards equation is solved, the Kirchhoff transform can be a useful tool to approximate the effective conductivity between the adjacent nodes (Szymkiewicz, 2013). Vauclin et al. (1979) also showed that Kirchhoff transformation is advantageous in solving the infiltration equation in unsaturated media. They showed that the discretized equation after transformation does not require the estimation of the coefficient K at the intermodal calculation points where the solution is unknown. Additionally, because of its integral nature, variations in the U function are much smaller than those in h. This reduces the numerical errors that are with the discretization. However, to the best of our knowledge, transformation approaches have rarely been applied to FAM for simulating variably saturated flow.

8 A finite analytic method (FAM) was presented by Chen and Chen (1981, 1984) to solve the heat conduction and Navier-Strokes equations. Subsequently, FAM has been used to simulate water flow in the unsaturated zones (Tsai et al., 1993, 2000; Zhang et al., 2015, 2016). The basic idea of the FAM is the incorporation of the analytic solution in a small local element to formulate the algebraic representation of the partial differential equation of the unsaturated flow. Therefore, FAM can effectively control the numerical dispersion and oscillation (Hwang et al., 1985). The objective of this study is to generalize the Kirchhoff transformation approach, based on the finite analytic method (FAM), for solving the two-dimensional variably saturated flow equation. First, the Kirchhoff transformation has been used to separate the nonlinear hyperbolic characteristic from the linear parabolic part in each iterative solution stage. Then, the equation has been solved using FAM. The stability of FAM has also been proven. Finally, in order to evaluate the numerical performance of FAM, several typical problems of variably saturated flow from the literature have been simulated using FAM and VSAFT2 software (Yeh et al., 1993) Generalized Transformation Approach Governing Equation The governing equation describing variably saturated zone can be expressed as:

9 h h h k h c h ( ( ) = k( h) + k( h) + ) (1) t x x z z h in which t is time (T), h is the pressure head (L), k (h) is the hydraulic conductivity (L/T), C ( h) is the specific moisture capacity function (1/cm), x and z are the Cartesian coordinates (L), in which upward and to the right are the positive direction. Prior to obtaining the numerical solution of Richards equation, the hydraulic conductivity and soil water retention curve have to be available first. In this study, three different types of soil hydraulic function have been used. Type 1 is Gardner model (1958), 179 k k( h) = k s s exp( βh) h < 0 h 0 and S e θ -θ exp( βh) r = = θ s θr 1 h < 0 h 0 (2) Type 2 is the combined Gardner model (Gardner, 1958) and van Genuchten model (van Genuchten, 1980). It has been used to describe the hydraulic conductivity and moisture content on the pressure head, as follows: 183 k k( h) = k s s exp( βh) h < 0 h 0 θ -θ θ θ 1 r and Se = = 1+ ( α h ) s r 1 n 1 m h < 0 h 0 (3) 184 Type 3 is Brooks and Corey model (1964), as follows: 185 (4) 186 where is the degree of effective saturation; θ is the volumetric water content (L 3 /L 3 ); k (h) is the hydraulic conductivity (L/T); θ s and k s are the water content (L 3 /L 3 ) and conductivity at natural saturation (L/T), respectively; θ r is the residual

10 189 water content (L 3 /L 3 ); (-), α (1/L), n (-), and m (-) are the soil parameters with m = /n; h e is the pressure head at the air entry (negative) (L). λ and η are the parameters that define the shapes of the water retention and conductivity curves and they are generally related by η = 2 λ + 3 (Brooks and Corey, 1964) Formulae of FAM Using the relationship of the hydraulic conductivity and the Kirchhoff transformation, Eq. (1) can be written as: 2 2 u u u u F ( u) = + + G( u) (5) 2 2 t x z z C ( ) ( h) F u = and G( u) k( h) ( h) ( h) 1 dk = (6) k dh where u and v are the Kirchhoff transform variables (Zhang et al., 2015) u = v v h h max h = min h max k k( α ) dα ( α ) dα (8) (7) 203 where h max and h min are the maximum pressure head (cm) and the minimum 204 pressure head (cm), respectively. In order to simplify the computations, h max and h min have been assigned as infinity and zero, respectively (Zhang et al., 2015). The formulae of FAM, based on Brooks and Corey model, are shown in Appendix A

11 209 Fig. 1 The finite analytic grids of a two-dimensional domain and the finite analytic 210 nodes in the element E i, j The computational formulae of FAM can be obtained using Eq. (5). The details are as follows. The computational domain of Eq. (5) can be discretized into small 215 elements using the Δ x Δz rectangles, as shown in Fig. 1. A local element E i, j of 216 FAM with 2 Δx 2Δz and a linear-exponential function has been assumed as the 217 boundary condition of the local element (Wang et al., 2012). To obtain the analytical 218 solution of Eq. (5), the unsteady term u t has been approximated by n n ( u u ) Δt i 1. Then, Eq. (5) in each local element and at a given time can be i expressed as follows: u u u + 2B x z z 2 2 n f (10) 222 n 1 where B = 0.5G, ( u), i j f = F ( u n-1 i, j ) u n i, j u Δt n 1 i, j, n is the number of elements at the n th time step. With the boundary condition in the local element E,, u( Δx, y) = ae u( Δx, y) = a u( x, Δy) = an u( x, Δy) = as 2By ( e 1) + 2By W ( e 1) 2 Ax ( e 1) + 2 Ax ( e 1) b y + c E + b N y + c + b x + c S W E b x + c N S W (11) i j 225 where the constants a, b, c, a,... andc can be specified by the eight nodal values E E E W s 226 of the pressure head on the boundary, as follows:

12 227 a b c N N N = 2 ( u + u 2u )/ 4sh ( AΔx) NE 1 = 2Δx = u NC [ u u cth( AΔx)( u + u 2u )] NE NW NW NC NE NW NC (12) Using the method of variables separation with the boundary conditions and the eight boundary nodal values, the calculated formulae of FAM can be derived by evaluating at the interior node R(x = 0, z = 0) of the local element (Fig. 1). 231 u n p = c u n NE NE + c NW u n NW + c u n SE SE + c SW u n SW + c u n NC NC + c u n SC SC + c u n EC EC + c u n WC WC c f f (13) 232 where c NE BΔz BΔz BΔz BΔz = c = e E, c = c = e E, c = c = e E, c = e EA, NW SE SW SE SW NC 233 c SC BΔz = e EA, c EC = EB, c WC = EB. By integrating f n n 1 ui j u n-1, i, j = F i, j ( u) into Eq. Δt 234 (13), we obtain u n n n n n n n n = ( c u + c u + c u + c u + c u + c u + c u + c u + X ) /(1 X ) n p NE NE NW NW SE SE SW SW NC NC SC SC EC EC WC WC + (14) n 1 n 1 n 1 where X c ΔzF ( u) u /( ΔtG ( )) = p i, j p i, j u, c p = cne 2cSE + cnc csc 2, EA = 2E2, 238 ( BΔz) ( BΔz) 2 ch 1 EB = 2BΔz E2, ch 1 1 E = BΔzcth( BΔz) E2 E2, 4ch ( BΔz) E 2 = i= 1 μ i B + λ i i+ 1 ( 1) 3 ( γ ) ch( μ Δz) i 2 2 =, λi π i, 2i 1 = 2 Δx E = 2 Δx Δz Δzth E2 + 4Δ ( BΔz) BΔx z Bch( BΔz) 2 2 1, γ = i π 2 i,, i = 1,2,.... Note that Eq. (14) reduces to Laplace equation when dealing with fully saturated problem, while the coefficients become

13 242 c 1 = cnw = cse = csw E, cnc = csc = EA, cec = cwc = EB, E = 0.25 E2 E2, NE = 243 E Δx Δz Δx = E2 +, 2 Δz 4Δz E 2 = i= 1 i+ 1 ( 1) 3 ( γ ) ch( λ Δz) i i, EA = 2E2, 1 EB = 2E 2, γ = i 2i 1 i π, λi = π, i = 1,2, Δx Using the preceding approach, a set of algebraic equations associated with the boundary conditions can be derived for each interior node in the domain. These equations can then be solved using SOR iterative method. 2.3 The stability of FAM The coefficients of FAM have three characteristics in unsaturated region. First, all 250 of the coefficients cne, cnw, cse, csw, cnc, csc, cec, cwc are greater than zero and smaller 251 than one and c = 2c 2c + c c 0 p NE SE NC SC due to B < 0. Second, 252 cne + cnw + cse + csw + cnc + csc + cec + cwc = 1. Finally, FG F ( u) ch ( ) = = is always greater or equal to zero due to Ch ( ) 0 ( u) dk( h) dh n 1 n 1 i, j i, j( u) n 1 Gi, j i, j dk( h) and 0 dh >. For the error between the exact solution and the finite analytic numerical solution in vadose zone, the error equation of the finite analytic numerical model is n n n n n n n n n Δz n 1 n 1 Δz n 1 ξp = cneξne + cnwξnw + cseξse + cswξsw + cncξnc + cscξsc + cecξec + cwcξwc + cp FG ( u) ξp /(1 + cp FG ( u)) Δt Δt (15) Since all the coefficients in Eq. (14) are positive,

14 ξ n p = Δz c ξ + c ξ + c ξ + c ξ + c ξ + c ξ + c ξ + c ξ + c FG ( u) ξ Δt Δz n cp FG ( u) Δt n n n n n n n n n 1 n 1 NE NE NW NW SE SE SW SW NC NC SC SC EC EC WC WC p p (16) 263 Based on the character of inequality, Eq. (16) becomes ξ n p (17) n n n n n n n n n 1 n 1 ( cneξne + cnwξnw + cseξse + cswξsw + cncξnc + cscξsc + cecξec + cwcξwc ) + cp FG ( u) ξp Δz Δt n cp FG ( u) Δz Δt n sup ξ (18) p Δz c + c + c + c + c + c + c + c + c FG u Δt Δz n cp FG ( u) Δt n n n 1 sup ξi NE NW SE SW NC SC EC WC sup ξp p ( ) 268 By applying characteristics of FAM 1 and 2, Eq. (18) reduces to 269 n sup ξ p + c Δz FG Δt u + c Δz FG Δt u n n n 1 sup ξi sup ξp p ( ) n 1 1 p ( ) (19) Since Point P is an arbitrary point in the computational domain, Eq. (19) can be expressed as n n n ξpmax = max sup ξp,sup ξ i (20) 273 n 1 n ξpmax = max sup ξ p (21) 274 ξ n 1 p max Δz Δz 1 + c FG ( u) 1 c FG ( u) ξ Δt Δt Δz Δz + c FG u + c FG u Δt Δt n 1 n 1 n 1 p p pmax n 1 n 1 1 p ( ) 1 p ( ) (22)

15 275 By applying the characteristics of FAM 1 and 3, we obtain c p 0, 276 FG F ( u) Ch ( ) ( ) = = 0 ( u) dk( h) dh n 1 n 1 i, j i, j u n 1 Gi, j i, j 277 ξ n pmax (23) n 1 ξ pmax Therefore, using Kirchhoff transform and the exponential and linear functions as boundary functions, the finite analytic-numerical computational format is unconditionally stable. In saturated zone, the coefficients of FAM also have three characteristics. The first and second are the same as those in vadose zone. Based on the first and second characteristics, the third characteristic in the algebraic equation in saturated zone computational domain is the diagonally dominant equations. Noting that A = ( a i, j ) is a real number matrix, which is aii, > ai, j, and A 0. Therefore, based on the j i Cramer s rule, the results of the matrix are unique Model verifications In this study, the performance of FAM has been tested by solving four cases. The first three are from earlier studies and the fourth is a field experiment designed to test the developed method Case 1: comparison with one-dimensional analytical solution and FAM

16 In this case, the exact analytical solution derived by Srivastava and Yeh (1991) has been used to assess the accuracy of the developed method (Romano et al., 1998; Brunone et al., 2003; Matthews et al., 2004). Type 1 Gardner constitutive model has been utilized to describe the flow through a 100 cm long 1D profile. The parameters for the Type 1 model are 1 cm/h, 0.1 1/cm, 0.4 cm 3 /cm 3 and 0.06 cm 3 /cm 3 for,,, and, respectively (Table 1). The upper and lower boundary conditions are constant infiltration rate ( ( z = L, t) = 0. 9 q cm/h) and constant head ( h ( z = 0, t) = 0 cm). The initial pressure head profile is the steady state corresponding to the infiltration rate of 0.1 cm/h at the top boundary and the constant head ( h ( z = 0, t) = 0 cm) at the bottom boundary. The spatial increment Δz = 1 cm, and the total simulation time = 100 h. The local error and mass balance error are calculated using: ( z t) h( z, t) ( z, t) FAM δ, = h Analyticla solution t 0 M numerical M numerical ψ () t = t 0 M M analytical analytical (25) (24) Fig. 2 Comparison of pressure heads from the analytical solutions and FAM, and local error and mass balance error distributions Table 1 Parameters describing the soil hydraulic models in the numerical case studies Fig. 2a shows the pressure head profile of FAM and the analytical solutions at 1, 5, 10, 20, 30 and 100 h. As shown in Fig. 2a, the pressure head profiles of FAM are in

17 agreement with those derived from the analytical solutions. In order to investigate the numerical behavior of FAM, the local error and the mass balance error are shown in Figs. 2b and 2c, respectively. At the early infiltration stage, FAM induces a relatively large error at the top of the column (i.e. 3.5%). On the other hand, the local error of the entire profile decreases with increasing time. However, there is a mass balance error in the performance with increasing time, as shown in Fig. 2c. The maximum balance error (i.e. 2.7%) occurs during the early infiltration stage, and decreases thereafter Case 2: comparison between two-dimensional analytical solution, FAM and VSAFT Tracy (2007) used Type 1 model to derive an analytical solution for transient 2D flow with specific initial and boundary conditions. The parameters for Type 1 model are 0.42 cm/h, /cm, 0.45 cm 3 /cm 3 and 0.15 cm 3 /cm 3 for,,, and, 330 respectively (Table 1). A very dry a L (L 2 ) rectangular block with initial condition 331 h ( x, z,0) = (26) h r By applying water at the top boundary, the pressure head becomes zero in the center a L of the top boundary (, ) 2 2, while the pressure head gradually decreases to h = hr at the outer edges of the top boundary. The detail formula is (27)

18 The boundary conditions are h = hr for the other boundaries. In this study, we set a = L = 10 m and h r = -10m. The grid size, applied for FAM and VSAFT2 in the x and z directions, are both 0.2 m. The simulation duration is two days with a constant time step hour. In this case, the local error of the numerical result is also calculated by using Eq. (24). To have a better understanding of the computational numerical behavior (Lehmann and Ackerer, 1988; Rathfelder and Abriola, 1994; Hyunuk, 2011), the average relative error (ARE) and the maximum absolute error (MAE) are estimated as follows: 344 Average relative error = i, j ( hi, j h ) i, j h i, j i, j 2 2 (28) 345 Maximum absolute error = max (29) i, j h i, j hi, j 346 where h, is the pressure head obtained using numerical methods, while h i, j is the i j 347 analytical solution Fig. 3 (a) Pressure heads obtained from the analytical solution and FAM, (b) local error of VSAFT 2 and (c) local error of FAM

19 Fig. 3a shows the contours of the pressure head at the end of Day 2 obtained from the analytical solution and FAM. Compared to the analytical solution, FAM can accuracy predict the depth of the wetting front. As shown in Figs. 3b and 3c, the maximum local errors for the two numerical methods, VSAFT2 and FAM, are less than 0.1%. Although both numerical methods perform well in terms of accuracy, the spatial distributions of local error are significantly different. The maximum local error of FAM is mainly located near the top boundary and only confined to a small area. On the contrary, the maximum local error of VSAFT2 occupies half of the computational domain. Noting that Table 2 also shows the CPU time, the total number of iteration, average relative error and maximum absolute error for both numerical methods. For FAM, it only takes nearly one-third of CPU time as compared to use VSAFT2. Moreover, the average relative error of FAM is seven times smaller than that of VSAFT2. Hence, it can be seen that FAM is faster, requires less iterations, both the average relative error and maximum absolute error are smaller than those of VSAFT Table 2 Comparison of CPU, iteration number, between FAM and VSAFT Case 3: Stream-aquifer system 374

20 The performance of the numerical model has been evaluated by applying it to a hypothetical stream-aquifer system (Wang et al., 2016). We generated a m rectangular domain, as shown in Fig. 4. The stream is 0.6 m width and is located at the center of the top boundary. The right and left boundaries are both 1.0 m deep ditches. The porous media of the entire system is silt-fine sand. The parameters for the Type 1 model are 7.13 cm/h, 6.6 1/cm, 0.43 cm 3 /cm 3 and cm 3 /cm 3 for,,, and, respectively (Table 1). There are two parts in the top boundary. First, the depth of the stream is 0.4 m. Second, a negative pressure head of -0.5 m has been applied to the remainder of the boundary. The bottom boundary is impermeable, and prescribed pressure heads have been applied to the two side boundaries. The boundary of the unsaturated zones on both sides from the ground surface to the water table has been determined using linear interpolation between the negative pressure head at the ground surface and zero pressure. The boundary of saturated zones on both ditches is equal to the actual water head in the ditches. In both FAM and VSAFT2, the lengths of the grid in the x and z directions are both 0.1 m. The simulation duration is 5 hours with a constant time step hour. Figure 5 shows the pressure heads at 0.2, 0.4, 0.8, 1.4, 2.0 and 5.0 hours for both FAM and VSAFT2. In general, the numerical solutions of FAM are in good agreement with those obtained from VSAFT2 at different simulation time. Further, Fig. 5 shows that the contours of the pressure head (m) from FAM are smooth throughout the domain including the area near the water table. This is an indication

21 that there is no numerical oscillation. At t = 2.0 h, the water table from FAM is smoother than that from VSAFT2. This may be due to the coarse grids used in VSAFT2. However, Zhang et al. (2016) pointed out that FAM can still give relatively accurate results even when coarse grids are used Fig. 4 Model of a hypothetical laboratory-scale stream-aquifer system Fig. 5 Comparison of pressure heads from FAM and VASFT This test shows that the process of surface water recharging the groundwater. At the early infiltration stage (i.e. t = 0.2, 0.4, 0.8, 1.2 h), there is no significant change in the water table, since the moisture content is mainly transported to the unsaturated zone. At t = 2.0 h, the surface water recharges the middle of the water table. Based on the results of this test, there is a lag time before the groundwater starts to be recharged. This lag time may depend on the depth of vadose zone, porous media etc. In practice, there are approaches which simplify the function of vadose zone. They are based on the following assumptions (Vauclin et al., 1979) (1) the transfer is instantaneous, (2) at any time, the distribution of flux reaching the aquifer is equal to the distribution of flux applied at the soil surface, and (3) above the water table, the soil has a uniform water content, which equals to the so-call residual water content. However, Vauclin et al. (1979) pointed out that these assumptions have no physical basis. Therefore, if the transfer of water through vadose zone is neglected, it has a major effect on the

22 prediction of the recharge of a water table aquifer. A better approach is to apply a unified numerical treatment to variably saturated flow For this case, both the average relative error and the maximum absolute error have not been considered. On the other hand, the CPU time has been considered. The CPU time for FAM is 3.75 s which is less than s for VSAFT Fig. 6 Sketch of the field experiment Case 4: field experiment For this case, FAM has been used to simulate a field test. The field test has been carried out in the Ordos Basin, China. The objective is to check if there is an inverted water table beneath the stream (Wang et al., 2016). A site, where the depth of groundwater is about 8 m according to observation wells near the testing site, has been chosen in an area with uniform fine sand in the unsaturated zone. A quadrate test tank (1 1 1 m 3 ) has been designed to hold a bed of uniform fine sand. There are three 40 mm diameter boreholes along the center line of the tank. One borehole is at the centre of the tank and two are at 0.3 m and 0.8 m from one side of the tank, as

23 shown in Fig. 6a. The boreholes have been drilled to a depth of 2.7 m from the ground surface, as shown in Fig. 6b. These boreholes have been used to measure the moisture content by TDR during the test period. At the soil surface, a constant water level (0.4 m in this test) has been applied for 2.0 h. While there are changes in the measured moisture content in the centre borehole, there is no change in the other two boreholes throughout the test. Therefore, the flow condition is considered as a one-dimensional homogeneous variably saturated flow. The grid has been discretized into uniform cells of 0.02 m, and the time step is 0.01 h. As shown by the field test results, the surface water level does not affect the moisture content at the depth of 2.6 m. Therefore, the pressure head at the bottom of the field test is considered a constant. The parameters for the Type 2 model are 1.67 cm/h, /cm, 0.38 cm 3 /cm 3 and 0.05 cm 3 /cm /cm, 4.89 for,,,, and, respectively (Table 1). The physical properties of the fine sand are shown in Table 2. The related parameters of the Type 2 model are shown in Table Fig. 7 shows the measured and the simulated moisture contents at 0.5, 1, 1.5, and 2 hours, and they are in good agreements. This proves that FAM can be applied to solving practical problem. 453 Table 3 Medium properties 454 Fig. 7 Comparison of moisture contents from FAM and measurements 455

24 Conclusions It is well known that the highly nonlinear Richards equation has hyperbolic and parabolic characteristics. Due to its hyperbolic characteristic, it is difficult to use numerical methods of reasonable time increment to obtain an accurate spatial-temporal solution for the flow. In order to overcome this problem, we have developed a 2D FAM to solve Richards equation. FAM has been developed based on separating the nonlinear hyperbolic characteristic of Richard s equation from the linear parabolic part in each iterative solution stage using Kirchhoff transformation. FAM has been shown to be superior to other numerical methods. The case studies have shown that FAM can effectively simulate water flow in variably saturated zone. Under relatively dry initial condition, FAM can determine the sharp wetting front better than the analytical solution. Furthermore, FAM uses less CPU time and can obtain relatively accurate solution. The outcomes may be attributed to three factors. First, the Kirchhoff transformation can be used to eliminate the square of the first derivatives which are in the expanded Richards equation. Therefore, the Kirchhoff transformation can solve the infiltration equation in unsaturated media (Vauclin et al., 1979). Second, the discretization of the equation, which has been transformed by Kirchhoff, does not require the estimation of the hydraulic conductivity at the intermodal calculation points. In addition, the nonlinearity of variations in the U function, because of its integral nature, is much slower than that in pressure head. Fig.

25 shows the variations of U and k (h) against pressure head. It can be seen that k (h) is highly nonlinear, with the value changes from 10-6 to 10 cm/hr. Hence, even for a small change in the pressure head, there is a large change in the hydraulic conductivity. However, the nonlinearity of the variation of U is much weaker than that of k (h) Fig. 8 Variations of U and k (h) against pressure head Finally, the finite analytic method has been incorporated into the analytical solution in a small local element to formulate the algebraic representation of the partial different equation of variably saturated flow. The coefficients of the spatial term, which are in the governing equation, are regardless of the soil parameters, and the governing equation also is relative simple. In addition, the value of the definition 490 variable ranges from -1 to 1. Hence, it can be easily solved. Furthermore, FAM can effectively control the numerical oscillation and dispersion (Hwang et al., 1985; Tsai et al., 1993; Zhang et al., 2015) Acknowledgments 497

26 This study was supported by the Key Program of National Natural Science Foundation of China (No ) and National Natural Science Foundation of China (No , ). The analysis was also partially supported by the program for Changjiang Scholars and Innovative Research Team of the Chinese Ministry of Education (IRT0811). The first author is grateful to the Chinese Scholarship Council (Project number: ) for providing an opportunity to be a Visiting Research Student at the University of Arizona, USA Appendix A This appendix contains a derivation of the formulation of FAM based on Brooks and Corey model. The Brooks and Corey model:

27 519 (A1) Using the relationship of the hydraulic conductivity and the Kirchhoff transformation, Eq. (1) can be written as: 2 2 u u u u F ( u) = + + G( u) (A2) 2 2 t x z z where 524 and (A3) 525 (A4) 526 and (A5) The other derived processes are the same as the formulation of FAM based on Gardner model References 535

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35

36 Table 1 Parameters describing the soil hydraulic models in the numerical case studies Soil Sources (cm 3 /cm 3 ) (cm 3 /cm 3 ) (cm -1 ) (cm -1 ) n Ks (cm/h) Type 1 [44,45] Type 1 [32,45] Type 1 [38] Type 2 [38]

37 Table 2 Comparison of CPU, iteration number, between FAM and VSAFT2 Scheme CPU(s) Nb.Iter ARE(%) MAE(m) FAM VSAFT

38 Table 3 Medium properties Particle composition(%) Bulk density Porosity (cm 3 /cm 3 ) (g/cm 3 ) 0.5~ ~ ~

39 Figure 1.

40 Fig. 1 The finite analytic grids of a two-dimensional domain and the finite analytic nodes in the element E i, j

41 Figure 2.

42 Fig. 2 Comparison of pressure heads from the analytical solutions and FAM, and local error and mass balance error distributions

43 Figure 3.

44 Fig. 3 Comparison of pressure heads from the analytical solution and FAM, and comparison of FAM and VSAFT2 local errors

45 Figure 4.

46 Fig. 4 Model of a hypothetical laboratory-scale stream-aquifer system

47 Figure 5.

48

49

50

51

52

53 Fig. 5 Comparison of pressure heads from FAM and VASFT2

54 Figure 6.

55 Fig. 6 Sketch of the field experiment

56 Figure 7.

57 Fig. 7 Comparison of moisture contents from FAM and measurements (Lines represent the results of FAM; dots represent the measured values)

58 Figure 8.

59 Fig. 8 Variations of U and k(h) against pressure head