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Soil Physics

Beerkan Estimation of Soil Transfer Parameters through Infiltration Experiments—BEST

^a Laboratoire des Sciences de l'Environnement, Ecole Nationale des Travaux Publics de l'Etat, Rue Maurice Audin, 69518 Vaulx-en-Velin (France); L. Lassabatère currently at: Division for Water and Environment, LCPC Nantes, Route de Bouaye, BP 4129, 44341 Bouguenais cedex, France
^b Laboratoire d'Etude des Transferts en Hydrologie et Environnement, LTHE (UMR 5564, CNRS, INPG, IRD, UJF), BP 53, 38041 Grenoble Cedex 9 (France)
^c Bioengineering Dep., Oregon State Univ., 116 Gilmore Hall, Corvallis, OR 97331-3906, USA, I. Braud currently at: CEMAGREF, UR Hydrologie-Hydraulique, 3bis Quai Chauveau, 69336 Lyon Cédex 9 (France)

^* Corresponding author (rafael.angulo{at}hmg.inpg.fr)

	ABSTRACT

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX A APPENDIX B REFERENCES

 
Studying soil hydrological processes requires the determination of soil hydraulic parameters whose assessment using traditional methods is expensive and time-consuming. A specific method, Beerkan estimation of soil transfer parameters referred to as BEST was developed to facilitate the determination of both the water retention curve, (h), and the hydraulic conductivity curve, K(), defined by their shape and scale parameters. BEST estimates shape parameters from particle-size distribution analysis and scale parameters from infiltration experiments at null pressure head. Saturated water content is measured directly at the end of infiltration. Hydraulic conductivity and water pressure scale parameters are calculated from the steady-state infiltration rate and prior estimation of sorptivity (S) This is provided by fitting transient infiltration data on the classical two-term equations with values from zero to a maximum corresponding to null hydraulic conductivity and using a data subset for which the two-term infiltration equations are verified as valid. BEST was compared with other fitting methods to estimate sorptivity and hydraulic conductivity from infiltration modeling data on the basis of the same infiltration equations for three contrasting soils: agricultural soil, sandy soil, and a coarser fluvioglacial deposit. The other methods failed sometimes to model accurately experimental data and to provide values in agreement with physical principles of water infiltration (negative values for hydraulic conductivity, too high steady-state infiltration rate). None of these anomalies was encountered when modeling cumulative infiltration with BEST. BEST appears to be a promising, easy, robust, and inexpensive way of characterizing the hydraulic behavior of soil.

Abbreviations: BEST; Beerkan estimation of soil transfer parameters • BEST/I, BEST applied to cumulative infiltration I • BEST/q, BEST applied to infiltration flux q • CI, cumulative infiltration • CL, cumulative linearization • DL, derivative linearization • IF, infiltration flux

	INTRODUCTION

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX A APPENDIX B REFERENCES

 
THE WORK presented here contributes toward understanding the hydrology of the vadose zone that forms a link between the surface water component and the groundwater component. This is important for understanding and characterizing the hydrological cycle and transfer of contaminants transported by water. Such understanding relies on determining the hydraulic characteristics curves of the soils, that is, water retention (h), also referred to as h(), and hydraulic conductivity K() curves.

Several methods have been developed to determine these hydraulic characteristic curves, from the simplest that require only readily available information, such as particle-size distribution and physicochemical characteristics or simple field measurements (Jarvis et al., 2002), to the most sophisticated that require the full experimental determination of the water retention curve h() and the hydraulic conductivity K() using laboratory apparatus (Raimbault, 1986; Mallants et al., 1997). Some methods have been based on field experiments such as infiltration experiments (Simunek et al., 1998; Angulo-Jaramillo et al., 2000; Jacques et al., 2002). These are usually performed by imposing a given pressure head (h_f), corresponding to a specific water content (_f), through either simple rings or disc infiltrometers, depending on the sign of the pressure heads h_f (Angulo-Jaramillo et al., 2000). Field based methods appear to be more advantageous insofar as (i) they are less expensive and time-consuming, (ii) the gain in precision in the laboratory is limited because soil cores usually provide a poor picture of the real soil, and (iii) laboratory devices are badly adapted for revealing the hydraulic behavior of watersheds at field scale (Minasny and McBratney, 2002).

Haverkamp et al. (1996) pioneered a specific methodology known as the "Beerkan Method." The Beerkan Method considers certain analytic formulae for the hydraulic characteristic curves and estimates their shape parameters, which are texture dependent, from simple particle-size analysis, and their scale parameters, which are structure dependent, from field infiltration experiments at null pressure head (Haverkamp et al., 1999). This method was applied to real watersheds (Galle et al., 2001; Braud et al., 2003) and has recently been improved and assessed with simulated data (Braud et al., 2005).

This paper presents a specific method (BEST) based on the Beerkan Method. The proposed algorithm was developed to improve robustness of deriving scale parameters from a single infiltration experiment performed at null pressure head. BEST fitting algorithm, which estimates sorptivity and hydraulic conductivity through fitting infiltration data, was also compared with four fitting methods described in Vandervaere et al. (2000a). We considered three contrasting soils: an agricultural soil, a sandy soil, and a coarse fluvioglacial deposit.

	THEORY

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX A APPENDIX B REFERENCES

 
This paper focuses specifically on the van Genuchten relation for the water retention curve (Eq. [1a]), with the Burdine condition (Eq. [1b]) and the Brooks and Corey relation (Eq. [2]) for hydraulic conductivity (Burdine, 1953; Brooks and Corey, 1964; van Genuchten, 1980):

[1a]

[1b]

[2]

where n, m and are shape parameters and h_g, _s, _r, and K_s are scale parameters. Usually, _r is very low and thus considered to be zero. These relations were proved accurate and relevant for describing the hydraulic behavior of most soils (Fuentes et al., 1992; Haverkamp et al., 1999).

Fitting infiltration experiment data on either numerical (Simunek et al., 1998) or analytical expressions (Haverkamp et al., 1994) can provide estimations of scale parameters and combinations of them, such as sorptivity, as described below. Considering an infiltration experiment with zero water pressure on an r_d-in-radius circular surface above a uniform soil with a uniform initial water content (₀), the three dimensional cumulative infiltration I(t) and the infiltration rate q(t) can be approached by the following explicit transient two-term (Eq. [3a] and [3b]) and steady-state expansions (Eq. [3c] and [3d]) (Haverkamp et al., 1994):

[3a]

[3b]

[3c]

[3d]

where constants A, B, and C can be defined for the specific case of a Brooks and Corey relation (Eq. [2]) and taking into account initial conditions as (Haverkamp et al., 1994):

[4a]

[4b]

[4c]

where ß 0.6 and 0.75, which apply for most soils when ₀ < 0.25 _s (Smetten et al., 1994; Haverkamp et al., 1994). Considering that the shape parameter and the initial and saturated water content were determined beforehand, the fit of experimental data on Eq. [3] provides estimations of saturated hydraulic conductivity (K_s) and sorptivity (S).

Once estimated, sorptivity [S = S(₀,_s)] can be expressed as a function of the scale parameters. Sorptivity can first be estimated from initial and final water contents, respectively ₀ and _f, and hydraulic characteristic curves by the following relation (Parlange, 1975, Elrick and Robin, 1981):

[5]

For the specific case of Eq. [1] and Eq. [2], a zero initial water content (₀ = 0) and a saturated final water content (_f = _s), this relation can be simplified as follows (Haverkamp et al., 1999):

[6a]

[6b]

where stands for the usual Gamma function. Usually, the initial water content differs from zero (₀ 0). In this case, S(₀,_s) can be quite accurately approximated from S(0, _s) through the following relations (Haverkamp et al., 1999):

[7a]

[7b]

where K₀ stands for hydraulic conductivity at ₀. At least, considering Eq. [6] and [7], sorptivity can be expressed as a function of the scale parameters by the following relation:

[8]

BEST estimates shape parameters on the basis of the particle-size analysis. This kind of identification has led to numerous articles (Haverkamp et al., 1999) and is described in more detail in the material and methods section. Thus it estimates scale parameters from infiltration experiments by using a specific algorithm whose main characteristics are also briefly described in the material and methods section. The saturated water content is measured at the end of the infiltration. BEST first estimates sorptivity by fitting the transient cumulative infiltrations or infiltration rates on the two-term equations Eq. [3]. The fit is based on the replacement of hydraulic conductivity K_s by its sorptivity function S and the experimental steady-state infiltration rate (q₊ q₊^exp) through Eq. [3d] and the following specificities: an accurate reproduction of experimental data, a fit for S between zero and a maximum value that corresponds to a null hydraulic conductivity (capillary driven flow), and the use of restricted data subsets to ensure the validity of Eq. [3]. As suggested by Jacques et al. (2002), once sorptivity is estimated, the saturated hydraulic conductivity is driven through Eq. [3d], assuming that steady state has been reached. The pressure head scale parameter (h_g) is then estimated from the other hydraulic parameters through Eq. [8].

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX A APPENDIX B REFERENCES

 
The Beerkan Infiltration Method and Materials
The Beerkan infiltration method uses a simple annular ring. The surface vegetation is removed while the roots remain in situ. A soil sample is collected for particle-size analysis and to determine its initial gravimetric water content. Another sample of known volume is extracted to determine its bulk density (_d). Then, the cylinder is positioned at the soil surface and inserted to a depth of about 1 cm to avoid lateral loss of the ponded water at the soil surface (Fig. 1 ). A fixed volume of water is poured into the cylinder at time zero, and the time elapsed during the infiltration of the known volume of water is measured. When the first volume has completely infiltrated, a second known volume of water is added to the cylinder, and the time needed for it to infiltrate is measured (cumulative time). The procedure is repeated for a series of about 8 to 15 known volumes and cumulative infiltration is recorded (Fig. 1). Finally, the data set is made up of a number of N_tot discrete points (t_i, I_i) and a smooth form of I^exp(t) is provided by these experimental points. At the end of the experiment, the saturated soil is sampled to determine the saturated gravimetric water content (w_s) and thus the saturated volumetric water content (_s) from the bulk density (_d) and the gravimetric water content, considering that water density is 1 g cm ^–3:

[9]

We studied three soils: an agricultural soil from Roujan, near Montpellier (Hébrard, 2004), a sandy soil from Chernobyl (Dewière, 2004) and a fluvioglacial deposit at the experimental Django Reinhardt site, east of Lyon (Barraud et al., 2002). These soils are referred to respectively as Roujan, Chernobyl, and Django Reinhardt. They represent three different types: agricultural soil, sandy soil, and gravely deposit and have very different textures (Fig. 2 ) and structures. Roujan can be considered as a well graded silt and is far finer and better graded than the other soils. Django Reinhardt is the most structured and the coarsest with particles larger than 2 mm. These soils were chosen because they represent three contrasting types of soils and because Chernobyl has the same texture as Django Reinhardt and the same structure as Roujan. The soil main characteristics and infiltration parameters are synthesized in Table 1. We used normal rings for the Beerkan Method (Angulo-Jaramillo et al., 2000) with a larger ring for the coarser soil to improve the representation of the soil's behavior.

View larger version (154K): [in this window] [in a new window]   Fig. 1. Beerkan infiltration. Known volumes of water prepared in the cups are successively poured through the ring and time is measured.

View larger version (18K): [in this window] [in a new window]   Fig. 2. Particle-size distributions of the soils studied (<2 mm). Experimental data (points) and modeling (solid lines).

View this table: [in this window] [in a new window]   Table 1. Beerkan infiltration parameters: ring radius (rd), water volume (Vw), dry bulk density (d), initial volumetric water content (0), and saturated volumetric water content (s).

[10a]

[10b]

where F(D) is the cumulative frequency associated with diameter D, M and N are two shape parameters, and D_g is a scale parameter. This fit provides the estimations of parameters M, N, and D_g. This optimization is performed by the least square technique, while convergence stability is achieved by using a change of variable x = 1/D. The shape index (p_m) of the media can then be estimated from M and N by (Zatarain et al., 2003):

[11a]

[11b]

[11c]

where coefficient is defined as (Fuentes et al., 1998):

[12]

where s is the fractal dimension of the media defined as the root of the following equation (Fuentes et al., 1998):

[13]

where is the soil porosity.

The shape parameters of the retention curve (m, n) are derived from the values of the shape index considering the positive root of Eq. [11b] for m and by deriving n from m by Burdine's condition:

[14]

[15]

Shape parameter of the hydraulic conductivity can be estimated from the capillary models (Haverkamp et al., 1999):

[16]

where is the product of m x n, and p is a tortuosity parameter that depends on the capillary model, that is, zero (Childs and Collis-George, 1950), 0.5 (Mualem, 1976), 1 (Burdine, 1953), or 1.33 (Millington and Quirk, 1961). A priori, tortuosity p should depend on the kind of soil (Haverkamp et al., 1999). Failing further information, we decided to use Burdine's condition in the same way as for the water retention curve.

Scale Parameter Estimation by Infiltration Modeling with BEST
The first scale parameter, that is, the saturated volumetric water content, was estimated from the saturated gravimetric water content and the dry bulk density through Eq. [9]. The other scale parameters are derived from modeling experimental infiltration. BEST uses equations equivalent to Eq. [3a] and [3b]. These equations were obtained by the replacement of hydraulic conductivity K_s by its function S and steady-state infiltration rate q₊ in Eq. [3]:

[17a]

[17b]

The four main steps of the BEST method are described below.

Step 1. Data
BEST estimates the experimental transient cumulative infiltration I^exp(t) and infiltration rate q^exp(t) and asymptotic infiltration rate q₊^exp from the whole data set (t_i,I_i), as follows:

[18a]

[18b]

[18c]

[18d]

where N_tot is the total number of the whole data set and N_end is the number of points considered for the linear regression. N_end has to be chosen large enough to provide an accurate estimate of the asymptotic infiltration rate. The value of the experimental infiltration rate estimated with the right term of Eq. [18b] is associated with time t_I* defined as the square mean root time (Eq. [18c]) instead of the usual arithmetic mean time. We show in Appendix A that time t_I* is the exact antecedent of the right term of Eq. [18b] when transient infiltration is ruled by Eq. [3a] and [3b].

Step 2. Constants
BEST calculates constants A, B, and C from the shape parameters (m, n, and ), constants and ß, and the initial and saturated water contents (₀, _s) with Eq. [4].

Step 3. Maximum Sorptivity
BEST estimates a maximum sorptivity referred to as S_MAX by fitting experimental data on Eq. [17], assuming that B = 0, that is, that water flow is capillary driven only. The fit is performed by minimizing the classical objective functions for cumulative infiltration I(t) and infiltration rate q(t):

[19a]

[19b]

where k is the number of data points considered for the transient state. As Eq. [3a], [3b], and thus Eq. [17] are valid only at transient state, the fit may not be valid for large values of k. Then, sorptivity is estimated for all values of k from a minimum of five points to a maximum N_tot (whole data set). S_MAX is assumed to be the maximum value of the whole sequence. The requirement to obtain positive values for K_s leads to an additional condition: S_MAX² must be less than the steady-state infiltration rate (q₊) divided by Coefficient A. Thus maximum sorptivity S_MAX is defined by:

[20]

Step 4. Hydraulic Conductivity and Water Pressure Scale Parameters Estimation
Whereas in Step 3 BEST fits experimental data considering a null value for B for Eq. [17], in Step 4 BEST considers the real value for B, that is, the value determined with Eq. [4b]. The fit is performed by minimizing the objective functions defined by Eq. [19]. As Eq. [17] may not be valid for all the points, BEST fits data for a minimum of five points to a maximum of N_tot (whole data set). For each data subset containing the k first points, BEST estimates sorptivity S(k), hydraulic conductivity K_s(k) from S(k) by Eq. [3d] and a maximum time t_max(k) defined as follows:

[21]

where tgrav is the gravity time defined by Philip (1969). Time t_max(k) is considered as the maximum time for which transient expressions can be considered valid (See Appendix B). Then, the longest time of data subset t_k is compared with t_max(k). The values of S(k) and K_s(k) are not considered valid unless t_k is lower than t_max(k). Amongst all the values of S(k) and K_s(k) that fulfill this condition, we retain for S and K_s the S(k_step) and K_s(k_step) values that correspond to the largest k, referred to as k_step (gain in precision). The scale parameter for water pressure (h_g) is then estimated from sorptivity (S) and hydraulic conductivity (K_s) by the following relation obtained from Eq. [8]:

[22]

This specific method is referred to as either "BEST/I" or "BEST/q," according to the choice of time series I or q for the fitting, respectively. BEST is coded with Mathcad 11 (Mathsoft Engineering and Education, 2002).

Estimating Scale Parameters from Infiltration Modeling by using the Four Different Fitting Methods
The different fitting methods referred to as cumulative linearization (CL, Smiles and Knight, 1976), derivative linearization (DL, Vandervaere et al., 2000a), cumulative infiltration (CI) and infiltration flux (IF) were analyzed by Vandervaere et al. (2000a). All are based on the assumption that transient infiltration is ruled by the two-term relations Eq. [3a] and [3b]. To apply these methods, we derived the cumulative infiltration I(t), infiltration rate q(t), and the derivative of infiltration with respect to the square root time from (t_i,I_i), as proposed in (Smith et al., 2002):

[23a]

[23b]

[23c]

[23d]

We fitted the experimental data to their corresponding functional relations (see Table 2) by a classical least square optimization to obtain the coefficients C1 and C2. We used the whole dataset for methods CI and IF, and only the appropriate dataset for CL and DL as described in Smith et al. (2002). The sorptivity and saturated hydraulic conductivity were then derived from the values C1 and C2 through the relations defined in Table 2 (column "Parameters") as proposed by Vandervaere et al. (2000b) for single infiltration experiments. We took into account the hydraulic relationships (Eq [1] and [2]) and the initial water content in the expressions of A and B in the relations that define S and K_s from coefficients C1 and C2. Then, these estimations for sorptivity and saturated hydraulic conductivity were used along with shape parameters estimated from fitting particle size distribution to estimate water pressure scale head parameter through relation [8].

	RESULTS AND DISCUSSION

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX A APPENDIX B REFERENCES

View this table: [in this window] [in a new window]   Table 3. Shape (M and N) and scale (Dg) parameters of particle-size distribution, fit relative errors (Er), and estimated hydraulic shape parameters (m, n and ).

View larger version (27K): [in this window] [in a new window]   Fig. 3. Modeling with BEST/I cumulative infiltration rate for the three soils ([a] Roujan, [b] Chernobyl, [c] Django Reinhardt).

View larger version (14K): [in this window] [in a new window]   Fig. 4. Example of BEST estimation of the (a) sorptivity (S) and hydraulic conductivity (Ks) and (b) longest time of the data subset (tk) and maximum time (tmax) versus the number of points used for the fit (k) for the case of Chernobyl.

Initially, the estimated values of sorptivity and hydraulic conductivity must be positive:

[24a]

[24b]

Some methods led to significantly negative values, mostly for hydraulic conductivity (Table 4). In this case, the water pressure scale parameters could not be determined and were referred to as "n.d." (Table 4).

Second, the sorptivity and hydraulic conductivity values must define modeled infiltration rates higher at transient than at steady state:

[25]

For all the methods, the transient modeled data was considered only over the time interval for which they are valid, where the upper limit was defined as the maximum time of the dataset that served for the estimation of S and K_s. Equation [25] was not verified for certain methods, such as CI applied to infiltration through Roujan (Fig. 5a ). This discrepancy also has consequences on the modeled cumulative infiltration and causes an over-steep slope at steady state. The S and K_s values that led to these problems could not be considered as valid.

View larger version (25K): [in this window] [in a new window]   Fig. 5. Examples of inadequate modeling: (a) overestimation of steady state infiltration rate (CI method applied to Roujan) and (b) overestimation of transient cumulative infiltration (IF and BEST/q methods applied to Chernobyl).

[26]

where y_i^exp (i = 1..k) are the experimental data that served for the estimation of S and K_s, y_i are the corresponding modeled data. Some methods such as IF and BEST/q led to unacceptable errors for transient cumulative infiltration (Table 4, Er_I, underlined) due to either under or overestimation (Fig. 5b). Concerning infiltration rates, the worst methods were IF and DL (Table 4, Er_q, underlined). This may be due, among other things, to wrong estimations of times t_i^o and y_i*, which were associated with the experimental infiltration rate and the derivative of infiltration with respect to the square root time (Eq. [23b] and [23d]).

Lastly, the modeled cumulative infiltrations I₊(t) and infiltration rates q₊ at steady state must be in agreement with the experimental data. In particular, the experimental infiltration rate q^exp(t) must converge and stay higher than or equal to q₊:

[27]

This was not verified for several methods, such as CI applied to infiltration through Roujan (Fig. 5a) with a modeled steady-state rate q₊ about 50% higher than the experimental steady-state infiltration q₊^exp. This resulted in an over-steep slope for the modeled steady-state cumulative infiltration I₊(t) (Fig. 5a).

We tested all the methods, including BEST, for the three cases studied taking into account the validity criteria described above (Table 5). BEST/I satisfied all the validity criteria for the three soils studied (Table 5). BEST/q showed robustness but over or underestimated cumulative infiltrations for Roujan and Chernobyl. The robustness of BEST/I may result from its construction. Initially, it fits for cumulative infiltration. Estimating experimental cumulative infiltration may appear easier and more accurate than any derivative of infiltration. Second, BEST subtracts the sorptivity term (A S²) to the steady state infiltration rate instead of coefficient C2. Considering that steady state is higher than coefficient C2, that is, A S² is a lower percentage of q₊, the robustness of the subsequent estimation of K_s is then improved. Thirdly, sorptivity is kept lower than a maximum value to ensure hydraulic conductivity remains positive. Lastly, BEST estimates sorptivity by using only the data for which Eq. [3] are valid.

View this table: [in this window] [in a new window]   Table 5. Validity criteria for the determination of physically meaningful values of sorptivity (S) and hydraulic conductivity at saturation (Ks).

View this table: [in this window] [in a new window]   Table 6. Hydraulic shape parameters (m, n, and ) and scale parameters (hg, s, Ks) estimated for the three soils.

Analysis of water retention curves facilitates understanding of the hydraulic behavior of the three soils. The water retention curves of both Chernobyl and Django Reinhardt show a sudden change of water content close to water pressure step (Fig. 6a ). The increase of hydraulic conductivity also seems to form a step shape, that is, a sharp increase below a given point and a plateau (Fig. 6b). This may result from the particle-size distributions of Django Reinhardt and Chernobyl that are both close to being unimodal with little deviation (Fig. 2). When water pressure increases and borders a pressure step corresponding to the mean pore size, most of the pores may saturate. Consequently, both water content and hydraulic conductivity increase considerably. Chernobyl and Django Reinhardt differ only with respect to their mean pore sizes, Django Reinhardt being much coarser. As a result, it saturates at a higher water pressure (Fig. 6a) and has a higher saturated hydraulic conductivity (Fig. 6b). The case of Roujan is quite different. Regarding its well-graded particle-size distribution (Fig. 2), its pore-size distribution is much more dispersed around the mean. As a result, when the water pressure increases, the water content may also increase gradually, filling the different pore-size modes, and leading to a possible gradual increase of hydraulic conductivity.

View larger version (19K): [in this window] [in a new window]   Fig. 6. Soil hydraulic characteristics curves: (a) water retention curve h() and (b) hydraulic conductivity K().

	CONCLUSIONS

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX A APPENDIX B REFERENCES

	APPENDIX A

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX A APPENDIX B REFERENCES

 
The right term of Eq. [18b] can be written as follows:

[28]

Considering that the cumulative infiltration follows Eq. [3a] exactly, it can be replaced by its function to times at t_i and t₁₊₁ and simplified as follows:

[29]

[30]

[31]

[32]

[33]

The right term of Eq. [32] corresponds to the expression of infiltration rate q(t) (Eq. [3b]) at the time t_i*:

[34]

	APPENDIX B

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX A APPENDIX B REFERENCES

 
To be considered as valid values with a physical meaning sorptivity (S) and hydraulic conductivity (K_s) must define an infiltration rate that verifies:

[35]

if Eq. [3a], [3b], [3c], and [3d] are valid for all the times belonging to the interval [0,t_max], Eq. [3b] and [3d] imply that:

[36]

The simplification of Eq. [36] leads to the following equation:

[37]

This condition leads to the definition of the maximum time (t_max) for which Eq. [3] and thus [17] can be considered as valid. This maximum time is defined by the second term of the inequality that is:

[38a]

where t_grav corresponds to the gravity time defined by Philip (1969):

[38b]

Received for publication January 20, 2005.

	REFERENCES

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX A APPENDIX B REFERENCES

 

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