Water Resour Manage (2013) 27:2429–2445 DOI 10.1007/s11269-013-0295-2
Generalized Explicit Models for Estimation of Wetting Front Length and Potential Recharge Shakir Ali & Narayan C. Ghosh & Ranvir Singh & B. K. Sethy
Received: 7 April 2011 / Accepted: 22 January 2013 / Published online: 7 February 2013 # Springer Science+Business Media Dordrecht 2013
Abstract Determination of length of advancement of wetting front is prerequisite for estimation of potential recharge. The advancement of wetting front is a time varying function governs by depth of ponding and suction head. Use of the Green-Ampt (GA) model for determining time varying length of wetting front involves a trial and error iterative method and hence, a tedious procedure. Replacing the logarithmic term of the GA model by sequential segmental second order polynomial, generalized algebraic equation based models for estimating time varying length of advancement of wetting front and potential recharge rates have been developed. Unlike following a trial and error method as involve in the GA model, the proposed model provides an explicit equation with no restriction to infiltration time period and depth .of ponding. The universal values of the models coefficients for different ranges of Lf
H þ yf
[Lf = length of advance of wetting front, H = depth of
ponding, and yf = suction head at the wetting front] have been determined with the help of the GA model by numerical experiments. Validity of the model has also been tested with the published laboratory experimental data. Analyzed results showed, the proposed models have similar responses as that of the GA model within a maximum relative error of 0.5 % for length of wetting front and 1.2 % for potential recharge estimate, and the corresponding percent bias has been found 0.20 % and 0.12 %, respectively. The proposed models can S. Ali (*) : B. K. Sethy Central Soil and Water Conservation Research and Training Institute, Research Centre, Kota 324 002, Rajasthan, India e-mail:
[email protected] B. K. Sethy e-mail:
[email protected] N. C. Ghosh National Institute of Hydrology, Roorkee 247 667, India e-mail:
[email protected] R. Singh Department of Hydrology, Indian Institute of Technology Roorkee, Roorkee 247667, India e-mail:
[email protected]
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successfully be used as alternate to the GA model to design artificial groundwater recharge structures, irrigation systems and resolving solute transport problems. Keywords Green-Ampt . Explicit equation . Potential recharge . Wetting front
1 Introduction Over the years mainly since 1990s, phenomenal growth of groundwater uses in many countries, in which India ranks number one, (Shah 2009) has posed problem of continuous depletion of groundwater table and deteriorating groundwater quality (CGWB, 2011a), which has also raised concerns on sustainability of the groundwater system and livelihood it supports (Shah 2009). To obviate such situation, Managed Aquifer Recharge (MAR) (Dillon et al. 2009) [also known as Artificial Groundwater Recharge (AGR), in many places] by rainwater harvesting through engineered recharge techniques is promoted as an effective tool to enhance groundwater resources, to restore back depleted groundwater table and to improve groundwater quality(Dillon et al. 2009; CGWB, 2011b). Various MAR techniques are practiced worldwide, which include; recharge pond, recharge basin, check dam, furrow, ditch, and sub-surface barrier, etc. Selection of specific MAR schemes depends on hydrologic and hydrogeologic conditions of the area in which these are to be implemented. In areas where depth to groundwater level is more and rainfall is less, such as, arid region in Rajasthan (India), one is more concern about effectiveness of these recharge schemes towards enhancement of groundwater resources. This is possible to ascertain when a reliable and an effective method for estimation of potential recharge are available. Need of accurate estimation of potential recharge and it’s time to reach water table is further aggravated due to increasing environmental quality awareness and desire to control the groundwater contamination (Govindaraju et al. 1996). Recharge processes in MAR has two distinguish components; one that considers advancement of wetting front by vertical movement of water through the unsaturated zone and continues till wetting front touches the water table, known as potential recharge; other one is subsequent recharge after the wetting front touches the groundwater table, known as actual recharge (Ali 2009). Groundwater table evolves after start of the actual recharge process. According to the Rushton (1997), arrival of potential recharge component to water table depends on unsaturated zone processes and ability of saturated zone to accept it. The potential recharge rate depends on difference of heads of water above ground surface, depth of wetting front below the ground surface and underneath soil properties. Based on flow dimensions and dynamics, depth of ponding, hydraulic conductivity and initial and boundary conditions, a numerous physically and empirical infiltration models of varying complexity had been developed in the past and are in use for estimation of potential recharge. These models are: Green and Ampt (1911), Richards (1931), Philip (1957), Mein and Larson (1973), Smith (1972), Morel-Seytoux and Khanji (1974), Smith and Parlange (1978) and empirical infiltration equations (Kostiakov 1932; Horton 1933). Most of these infiltration models do not explicitly consider depth of ponding over the surface except the Green-Ampt (GA) and Richards’ model (Bouwer 1978; Abdulrazzak and Morel-Seytoux 1983; Dagan and Bresler 1983; Govindaraju et al. 1996). The Richards’ equation that contains non-linear functions related to soil water potential, unsaturated hydraulic conductivity, and soil water content has the difficulty to express by an explicit analytic solution unless a simplified linear case is considered. The GA model, which is considered to be simpler than the Richards’, is widely used for simulating one-dimensional
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vertical potential recharge from homogeneous soil under the assumption of constant suction head at wetting front, constant saturated hydraulic conductivity behind the wetting front, uniform soil moisture content, ponded depth of water and sharp wetting front. The GA model has been the subject of investigations on theoretical (Philip 1969; Morel-Seytoux and Khanji 1974; Kale and Sahoo 2011) and empirical (Aggelides and Youngs 1978) fronts owing to its simplicity and satisfactory performance for a variety of hydrological problems. Very recently, Kale and Sahoo (2011) reviewed the variants of the Green-Ampt infiltration models in terms of their degree of accuracy, complexity, applicability limits and usefulness in rainfall-runoff and irrigation modeling studies. A numerous studies (Richards 1931; Govindaraju et al. 1996) demonstrated similar performances by the GA and Richards’ model in estimation of potential recharge. Bouwer (1978) and Freyberg et al. (1980) indicated that the GA model is adequate for estimate of the potential recharge when the depth of impoundment over an area is significant. Philip (1992) reported that the accuracy of the GA model increases with the increase of ponding depth. The GA model basically represents an equation with time varying length of advancement of wetting front appearing implicitly under an algebraic and logarithmic term. Estimate of length of wetting front thus by the GA model is a trial and error method, and is generally made using numerical iteration technique, e.g. Newton-Rapson method (Rao et al. 2009), Runge-Kutta method (Enciso-Medina et al. 1998) etc. To avoid trial and error method, many investigators (Salvucci and Entekhabi 1994; Barry et al. 1995; Serrano 2003; Chen and Young 2006; Mailapalli et al. 2009; Ali 2009) gave explicit solutions of the GA model in different forms. For example, Fok (1967) expressed by power function, Salvucci and Entekhabi (1994) by rapidly varying power series, Barry et al. (1995, and 2005) by Lambert W function, Serrano (2001 and 2003) by decomposition series, Ramos (2007) and Mailapalli et al. (2009) by nonstandard explicit integration. Each approximation has its own merits and demerits in terms of relative complexity and errors. For example, approximation by Salvucci and Entekhabi (1994) showed a maximum relative error of 2.3 %, Li et al.(1976) approximation has maximum relative errors of 8 %, Swartzendruber (1974) approximation gave a maximum relative error of 11 %. Very recently, Barry et al.(2005) derived an approximation of the GA model by neglecting depth of ponding from Lambert W function that showed a maximum relative errors less than 0.5 %. Barry et al.(2005) approximation can thus be considered accurate than the other approximate models if the depth of ponding is negligible. Barry et al. (2005) model fails to produce reliable estimate of potential recharge when depth of ponding is high (3–4 m) as normally occurs in the recharge scheme. Several investigators (Reeder et al. 1980; Philip 1992; Warrick et al. 2005) reported that potential recharge varied with depth of ponding over the soil surface. It thus calls for an explicit derivation for estimating potential recharge, which is simple to use, relatively accurate and also preserves all the characteristics as depicts by the GA model. The present paper is focused to address the above concerns by a developing an alternate explicit model based on the in-depth study of the GA model. The paper also focuses on validity testing of the derived model in estimation of length of advancement of wetting front and potential recharge for different ranges of depth of ponding and depth to groundwater table.
2 Theoretical Considerations The GA model for estimation of time varying potential recharge rate is expressed as (Bouwer 1978):
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H þ yf Rp ¼ K s 1 þ Lf
ð1Þ
where Rp is the time dependent potential infiltration/recharge rate per unit area [LT−1]; Ks is the saturated hydraulic conductivity of the transmission zone, ½LT 1 ; H is the depth of ponding [L]; y f is the suction head at the wetting front (negative pressure head) [L]; and Lf is the length of advancement of wetting front [L]. The wetting front is the interface between the wetted and non-wetted zone. For constant H, replacing Rp by η (dLf /dt), Eq. (1) after integration leads to the well known GA infiltration equation (Bouwer 1978): " # Lf Ks ð2Þ t ¼ Lf H þ y f ln 1 þ η H þ yf in which η is the fillable porosity = θs − θi, where θi is the initial volumetric moisture content [L3L−3]; θs is the volumetric moisture content at near saturation [L3L−3]. 2.1 Approximation of Logarithmic Term of Eq. (2) n . o H þ y f 1 can be expressed by the algebraic The logarithm term in Eq. (2) for Lf infinite series as: h i 2 3 4 Lf Lf Lf Lf Lf þ 13 Hþy 14 Hþy þ ::::: ¼ Hþy 12 Hþy ln 1 þ Hþy f
f
f
f
f
ð3Þ
Many investigators (Philip 1969; Bouwer 1978; Dagan and Bresler 1983) approximated the logarithmic term of the GA model by retaining the first two terms of Eq. (3). These approximations are valid for short time period of infiltration (Talsma 1969; Dagan and Bresler 1983). n . o H þ yf For Lf > 1 , the series described by Eq. (3) is not valid. The logarithmic term in Eq. (2) for all practical purposes specially for groundwater recharging can be approximated by a second order polynomial of the following form (Ali 2009): h i Lf ln 1 þ Hþy f
Ai þ Bi
Lf Hþy f
þ Ci
Lf Hþy f
2 ð4Þ
value where Ai, Bi and Ci are the coefficients of the polynomial n . whose o depend on the logarithmic distribution of the operating variable Lf H þ y f , and are to be determined; suffix ‘i’ refers number of polynomials, an integer number. For evaluating the performance of replacement of logarithm term by an algebraic equation as in Eq. (4), the parameters, Ai, Bi and Ci are to be known apriori for .all practical purposes. Values of the parameters, Ai, Bi and Ci for different ranges of Lf
H þ yf
are obtained by
numerical experiments. For 1 þ xÞ for different .numerical experiment, synthetic values of lnð. x, ∣x∣>0, where, x ¼ Lf H þ y f are generated. The range of x ¼ Lf H þ y f 150 was considered . for estimating values of the parameters of Eq. (4). The reason for considering Lf H þ y f up to a value of 150 is that, depth to groundwater table (which is equal to Lf) in
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arid and semi-arid region normally remains very high, usually ranges from 50 m to 300 m and depth of ponding varies .from 1 to3 m. The synthetic data generated using L.H.S of Eq. (4) for H þ yf
different ranges of Lf
are thereafter utilized to fit to a second-order polynomial.
The values of the coefficients corresponding to the best-fit curve are considered to be the values of Ai, Bi, and Ci. The coefficient of determination, R2, and the standard error of estimate, SE are chosen as the decision variables, i.e., higher the value of R2 (close to 1) and lower the value of SE (close to zero), better is the approximation. It is found that the logarithmic distribution is fitted closely to second order polynomial (Eq. 4) in five sequential segments; for operating variable, (i) ∣x∣≤1; (ii) 1 H, are chosen. Applying the values of Ks, =f, η of the four selected soil textural classes namely; loamy sand, loam, clay loam and sandy clay loam and using H= 2.0 m in the GA model (Eq. 2), time varying Lf is computed by trial and error method for time period of 180 days. Using the same value of Ks, =f, η and H to the respective segmental 750 675
160
Loamy sand (K s = 1. 466 m/day)
600
Proposed model GA model
450 375
Proposed model GA model
100
300
80 60
225
40
150
20
75 0
Loam (K s = 0.0317 m/day)
120
Lf (m)
525
Lf (m)
140
0
20
40
60
80
100
120
140
160
180
0
200
0
20
40
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120
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200
25
35
Sandy clay (K s= 0.029 m/day)
Clay loam (K s = 0.055 m/day)
30
20
Proposed model GA model
Lf(m)
25
Lf(m)
80
Time (days)
Time (days)
20 15
Proposed model GA model
15
10
10 5 5 0
0
20
40
60
80
100
Time (days)
120
140
160
180
200
0
0
20
40
60
80
100
Time (days)
Fig. 3 Comparison of the length of advancement of wetting front estimated by the proposed model (Eq. 8) and the GA model (Eq. 2) for 4 USDA soil textural classes
Generalized Explicit Models for Estimation of Wetting Front 1.0
Loamy sand
0.8 0.6 0.4 0.2 0.0 -0.2
0
20
40
60
80
100
120
140
160
180
200
Time(days)
-0.4 -0.6 -0.8
Percent relative error(%)
Percent relative error(%)
1.0
-1.0
Clay loam
0.6 0.4 0.2 0
20
40
60
80
100
120
140
160
180
200
Time(days)
-0.4 -0.6 -0.8 -1.0
Percent relative error(%)
Percent relative error(%)
0.6 0.4 0.2 0.0 0
20
40
60
80
100
120
140
160
180
200
120
140
160
180
200
-0.2
Time(days) -0.4 -0.6 -0.8
1.0
0.8
-0.2
Loam
0.8
-1.0
1.0
0.0
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Sandy clay
0.8 0.6 0.4 0.2 0.0 0
20
40
60
80
100
-0.2
Time(days) -0.4 -0.6 -0.8 -1.0
Fig. 4 Percent relative errors of the proposed model with respect to the GA model for length of advancement of wetting front for 4 USDA soil textural classes
model [Eq. (8)] and satisfying the condition of time period, values of Lf are computed. A comparison of the computed values of Lf obtained using Eq. 8 and Eq. 2 is shown in Fig. 3. Comparison shows a close match (Fig. 3) between the distributions of the proposed and GA model for all the four soil textural classes. The distributions of the PRE values of Lf for four soil classes are shown in Fig. 4. The profiles of PRE (Fig. 4) showed a maximum relative error bound of ±0.5 %. The quantitative statistics of the percent bias (PB), coefficient of determination (R2) and index of agreement (d) (Legates and McCabe 1999) are given in Table 5a. Results revealed (Table 5a) that the PB is very low (≤ 0.20 %), index of agreement, d and R2 are perfectly 1, for all four classes of soil, which indicate perfect agreement between the distribution of the proposed and GA model. The performance of the proposed model (Eq. 8) is also tested for short period using the laboratory experiment data of Fok and Chiang (1984). The data presented by Fok and Chiang (1984) were for Makiki clay soil (Andic Ustic Humitropepts) under different initial soil moisture levels (i.e. dry and wet). For dry soil samples, the value of Ks, H + y f and η were taken as 3.9×10−5 m/min, 0.683 m and 0.187 m3/m3, respectively, and for wet soil samples, these values were 3.9×10−5 m/min, 0.683 m and 0.187 m3/m3 (Fok and Chiang 1984). Figure 5 showed the comparison of distributions of Lf estimated using the data of Fok and Chiang (1984) for both dry and wet soils in the proposed (Eq. 8) and GA model (Eq. 2). Table 5 Goodness-of-fit measures for values of Lf computed by the proposed model (Eq. 8) and the GA model (Eq. 2) for four selected soil textural classes
Textural classes of soils
R2
d
PB (%)
Loamy sand
1.0000
1.0000
−0.20
Loam Clay loam
1.0000 1.0000
1.0000 1.0000
−0.03 0.04
Sandy clay
1.0000
1.0000
0.01
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250
Wet soil Measured 200 Estimated by proposed model Estimated by the GA model 175 225
150
150
L f (mm)
L f (mm)
Dry soil Measured 200 Estimated by proposed model Estimated by the GA model 175 225
125 100
125 100
75
75
50
50
25
25
0
0 0
10 20 30 40 50 60 70 80
90
0
10 20 30 40 50 60 70 80
Time (min)
90
Time (min)
Fig. 5 Comparison of responses of the Lf, estimated by the proposed model (Eq. 8) and the GA model (Eq. 2) with the measured values for dry and wet soil samples
In the case of dry soil, the deviation between the distributions of Lf by the proposed model and the measured values is more than the distribution depicted by the GA model (Fig. 5), while for the wet soil (Fig. 5) it is reversed. These discrepancies may be because of estimates obtained under their unsaturated and saturated conditions of soils. In fact, a transmission wetting zones exist in the field conditions. The statistical analysis of the distributions, however, showed a near close performance between the proposed and GA models, d and R2 being≈1 for both the samples (Table 6). It is further noted that proposed model slightly overestimated (PB0) the values than the measured values (Table 6). The analyses of both long as well as short infiltration periods showed that the proposed Lf model (Eq. 8) possesses all characteristics and potentials represent by the GA model. 3.2 Estimation of Potential Recharge The values of the cumulative potential recharge, Rc for all the four selected soil textural classes are estimated using Eq. (11). The statistical parameters R2, D, PB and PRE calculated from the outputs of the two models are found similar as obtained in the case of Lf. Time varying potential recharge rates are also estimated using the GA model (Eq. 1) and the proposed models (Eq. 12a, b) satisfying the conditions of segmental time periods using the same data as used for Lf estimation, i.e., four soil textural classes and constant head, H=2 m. Table 6 Goodness-of-fit measures for values of Lf computed by the proposed model (Eq. 8) and the GA model(Eq. 2) with respect to the measured values for dry and wet soil samples
R2
d
Measured versus GA model
0.9983
0.9976
2.70
Measured versus proposed model
0.9932
0.9926
−3.25
Wet soil Measured versus GA model
0.9989
0.9943
5.42
Measured versus proposed model
0.9991
0.9991
−0.54
Soil sample
PB (%)
Dry soil
Generalized Explicit Models for Estimation of Wetting Front
Table 7 Statistics of different goodness-of-fit measures for values of Rp computed by the proposed and GA models for four USDA soil textural classes
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R2
Textural classes of soils
d
PB (%)
Proposed (Eq. 12a) versus GA(Eq. 1) model Loamy sand Loam
0.9984 0.9804
0.9964 0.9807
−0.22 −1.13
Clay loam
0.9938
0.9956
0.05
Sandy clay
0.9966
0.9965
0.49 −0.01
Proposed (Eq. 12b) versus GA(Eq. 1) model Loamy sand
1.0000
1.0000
Loam
1.0000
1.0000
0.01
Clay loam
0.9999
1.0000
−0.12
Sandy clay
0.9998
0.9999
0.09
The quantitative agreements and errors between the GA and proposed models are given in Table 7. The statistical parameters (Table 7) showed a very good agreement between the two models, R2 ≈1 and d≈1. The PB of distribution depicted by Eq. (12b) is found lower than by Eq. (12a) in comparison to the GA model (Eq. 1) for all the four soil textural classes except clay loam soil (Table 6). This means that the model represented by Eq. (12b) is more accurate than the equation obtained by differentiation of cumulative potential recharge. The reason is that Eq. (12b) has similar mathematical structure as of the GA model. Moreover, Eq. (12b) can more conveniently be used if the Lf is known priori. Figure 6 shows the comparison of distribution of Rp estimated using the GA and the proposed model (Eq. 12b) that showed a close match. Figure 6 also showed that for a particular Ks, η and =f, the 1.0
Loamy sand (K s = 1.466 m/day)
1.9
Infiltration rates (m/day)
Infiltration rates (m/day)
2.0
Proposed model(Eq.13b) GA model(Eq.1)
1.8 1.7 1.6 1.5 1.4
Loam (K s= 0.317 m/day)
0.9
Proposed model(Eq.13b) GA model(Eq.1)
0.8 0.7 0.6 0.5 0.4 0.3
1.3 0
20
40
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0
20
40
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Time (days)
100
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0.26
0.50
Clay loam (K s = 0.055 m/day)
0.45 0.40
Infiltration rates (m/day)
Infiltration rates (m/day)
80
Time (days)
Proposed model(Eq.13b) GA model(Eq.1)
0.35 0.30 0.25 0.20 0.15 0.10 0.05
Sandy clay (K s = 0.029 m/day)
0.24 0.22 0.20
Proposed model(Eq.13b) GA model(Eq.1)
0.18 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02
0
20
40
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80
100
Time (days)
120
140
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200
0
20
40
60
80
100
Time (days)
Fig. 6 Comparison of responses of infiltration rates by the proposed model (Eq. 12b) and the GA model (Eq. 1)
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0.08
Loamy sand
0.06 0.04 0.02 0.00 0
20
40
60
80
100
120
140
160
180
200
-0.02
Time(days) -0.04 -0.06
Percent relative error(%)
Percent relative error(%)
0.10
-0.08
1.6 1.4 Clay loam 1.2 1.0 0.8 0.6 0.4 0.2 0.0 20 40 -0.2 0 -0.4 -0.6 -0.8 -1.0 -1.2 -1.4 -1.6
60
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Time(days)
180
200
Percent relative error(%)
Percent relative error(%)
-0.10
1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 -0.20
Loam
20
40
60
-0.4 -0.6 -0.8 -1.0 -1.2 -1.4 -1.6
1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 -0.20 -0.4 -0.6 -0.8 -1.0 -1.2 -1.4 -1.6
80
100
120
140
160
180
120
140
160 180
200
Time(days)
Sandy clay
20
40
60
80
100
200
Time(days)
Fig. 7 Percent relative errors of the proposed model (Eq. 12b) with respect to the GA model (Eq. 1) for the potential recharge rates into selected soils
potential recharge rates is high on the onset of the recharge process and reduces gradually with time and reaches nearly to a steady state at a longer time. Further, as the value of Ks increases, the potential recharge rates, Rp also increases. These characteristics are on the expected lines. The profiles of the PRE for all the selected soil textures (Fig. 7) showed a maximum relative error of ±1.2 % between the proposed and the GA model. The trends of the PREs are similar as found in the case of Lf (Fig. 4). Table 8 Time delay for wetting front to reach the depth to water table for a constant depth of water (H=2 m) used in 11 USDA different soil texture classes Soil textural classes
Time delay for reaching wetting front to a given depth to water table (days) 1m
5m
10 m
25 m
50 m
Sand
0.02
0.17
0.52
1.61
3.53
Loamy sand
0.05
0.56
1.72
5.31
11.68
Sandy loam
0.12
1.32
4.12
12.75
28.09
Loam
0.24
2.79
8.60
26.57
58.48
Silt loam Sandy clay loam
0.51 0.54
5.77 5.96
18.34 19.45
57.02 60.79
125.88 134.54
Clay loam
1.19
13.32
43.10
134.52
297.48
Silt clay loam
1.97
21.74
72.10
226.06
501.08
Sandy clay
1.84
20.47
67.06
209.77
464.44
Silty clay
3.15
34.60
115.46
362.41
803.78
Clay
4.47
48.62
164.23
516.60
1146.97
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3.3 Estimation of Time for Wetting Front to Reach Water Table . Based on the described criteria of Lf
H þ yf
(Eqs. 10(a)–(e)), time delays for wetting
front to reach various depth to water table (Dw =1, 5, 10, 25 and 50) for a constant water depth (H=2 m) calculated for 11 USDA soil textural classes are presented in Table 8. For shallow depth to water table (1–5 m), the time delay, Td is found from 1 h to 13 days in most of the soil classes except medium and fine texture soils (Table 8). For medium depth to water table (10–25 m), Td ranged from 1 to 135 days in all soils except fine texture soils. For larger depth to water table (≥ 50 m), Td is found between 1 month to several months in most of the soil texture except very course textures as sand and loamy sand. Results revealed that Td ≥ 1 day usually corresponds to shallow to deep water table in real life problems indicating that larger infiltration opportunity time would require for recharging groundwater.
4 Conclusions There have been continuous efforts in the past to derive alternate simplified models as replacement to the Greem-Ampt model for estimation of length of advancement of wetting front and potential recharge rate. The derived alternate models had been found to have some restrictions to use. By replacing the logarithmic term of the Green-Ampt (GA) model by sequential segmental second order polynomials, generalized algebraic equation based models for estimating the length of advancement of wetting front, cumulative potential recharge and potential recharge rate have been developed. The universal values of the models parameters values have been obtained by comparing performances of the proposed models with the GA model for varied ranges of values and soil texture classes. Numerical experiments and published laboratory experimental data have been used for testing the validity of the model. The quantitative statistical measures, namely; percent relative error, percent bias, coefficient of determination and index of agreement have been utilized to assess the performances of the proposed models. It is observed that the proposed models possess similar potential as that of the GA model in estimation of length of advancement of wetting front, cumulative recharge and recharge rates for short as well long time periods. The proposed model for length of advancement of wetting front and potential recharge rates showed a maximum relative error of 0.5 % and 1.2 %, respectively, and the percent bias are found less than 0.20 and 0.12 % in comparison to the GA model. The proposed models are simple, do not require iteration, easily programmable and give reasonable accurate estimate of time varying length of advancement of wetting front, cumulative recharge and potential recharge rate with no restriction to time periods and soil texture classes. The proposed models can successfully be extended to design groundwater recharging schemes, irrigation systems and solute transport and to compute length of advancement of the wetting front, cumulative recharge and potential recharge rate.
References Abdulrazzak MJ, Morel-Seytoux HJ (1983) Recharge from an ephemeral stream following wetting front arrival to water table. Water Resour Res 19:194–200 Aggelides S, Youngs EG (1978) The dependence of the parameters in the Green and Ampt infiltration equation on the initial water content in draining and wetting states. Water Resour Res 14(5):857–862
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