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Soil Science Society of America Journal 64:1263-1272 (2000)
© 2000 Soil Science Society of America

DIVISION S-1-SOIL PHYSICS

Transient Flow from Tension Infiltrometers

I. The Two-Parameter Equation

Jean-Pierre Vandervaerea, Michel Vauclina and Dave E. Elrickb

a Lab. d'étude des Transferts en Hydrologie et Environ. (CNRS UMR 5564, INPG, IRD, UJF) BP 53, 38041 Grenoble Cedex 9, France
b Dep. of Land Resource Sci., Univ. of Guelph, Ontario, Canada N1G 2W1

jean-pierre.vandervaere{at}hmg.inpg.fr


   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 Theory
 Discussion
 Summary and conclusions
 REFERENCES
 
Tension disk infiltrometer experiments are generally conducted until apparent steady state is reached because most of the methods of analysis are based on Wooding's solution for steady state flux. However, the time necessary to reach steady state may be a penalizing aspect for soils with low permeability and the information contained in the transient stages is not utilized. Moreover, these methods assume homogeneous soil and a uniform initial water content, which may be unrealistic when a large volume of soil is sampled. In this series, we propose and compare several new methods of analysis that are based on the transient stage of axisymmetric infiltration. In the first part, we show that a two-parameter equation—one term linear in square root of time and one term linear in time—adequately describes the transient flow from the disk infiltrometer for both simulated and laboratory tests. The technique used for the determination of the two coefficients must meet two criteria; it must verify the validity of the two-term equation throughout the duration of the experiment, and it must account for the early-time perturbation that is induced by the sand-contact layer placed between the disk and the soil. We show that the best technique consists in linearizing the data by differentiating cumulative infiltration with respect to the square root of time. Direct nonlinear fitting on cumulative infiltration or infiltration flux is likely to lead to unacceptable errors, either because of the undetected invalidity of the two-parameter equation or arising from the influence of the contact layer.

Abbreviations: CI, cumulative influx • CL, cumulative linearization • DL, differentiated linearization • GS, Grenoble sand • IF, infiltration flux • YLC, Yolo light clay


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
Theory
 Discussion
 Summary and conclusions
 REFERENCES
 
KNOWLEDGE OF THE MECHANISMS OF WATER MOVEMENT in the upper layers of the soil is of central importance in many research areas, such as agronomy, civil engineering, hydrology, and environmental sciences. Hence, there is a large demand for simple, fast, and accurate methods of hydrodynamic characterization of soils. Over the last decade, tension disk infiltrometers (Perroux and White, 1988) have been increasingly utilized for the in situ estimation of hydraulic conductivity K (LT-1) and capillary sorptivity S (LT-1/2) as functions of water pressure head h (L) (e.g., Thony et al., 1991; Logsdon and Jaynes, 1993; Cook and Broeren, 1994; Jarvis and Messing, 1995). An important advantage of this technique over laboratory methods is that it is performed in situ, which allows exploration of the dependence of hydraulic properties on soil structure (Hussen and Warrick, 1993), on the presence of roots or macropores (Clothier and White, 1981; Logsdon and Jaynes, 1993; Lin and McInnes, 1995), or on agricultural practices (Vauclin and Chopart, 1992; Mohanty et al., 1996) without facing sampling difficulties. Disk infiltrometers are portable and use relatively small volumes of water, which makes them particularly suitable for spatial variability studies (Smettem, 1987; Mohanty et al., 1994; Jarvis and Messing, 1995; Shouse and Mohanty, 1998). Furthermore, the three-dimensional geometry of infiltration allows steady state to be reached much faster than in the case of one-dimensional experiments (Elrick et al., 1990). On the other hand, axisymmetric flow is more complex than one-dimensional flow, and solutions of the flow equations are more difficult from an analytical point of view.

A decisive step was made when Wooding (1968) proposed a remarkably simple equation for the steady state flux density q{infty} (LT-1) emanating from a circular source of radius R (L) at the soil surface:

(1)
where {phi} (L2 T-1) is the Kirchhoff transform defined by

(2)
where hydraulic conductivity K is expressed as a function of h <= 0 and where the subscripts n and 0 denote, respectively, initial and surface boundary conditions (it is assumed in Eq. [1] that the initial soil pressure head hn is sufficiently small for the condition K(hn) << K(h0) to be fulfilled). Equation [1] was obtained under the assumption of a quasilinear soil (Pullan, 1990), i.e., following Gardner's (1958) K(h) relation:

(3)
where Ks is the hydraulic conductivity at natural saturation and {alpha} is a fitting parameter [L-1]. From Eq. [3], Eq. [2] reduces to

(4)
and Eq. [1] can be transformed to

(5)

Part of the success of the disk infiltrometer is due to the relative simplicity of the associated methods of analysis; only two unknowns have to be determined: K and {phi} in Eq. [1] or K and {alpha} in Eq. [5]. This can be achieved by using two (or more) radii (Scotter et al., 1982; Smettem and Clothier, 1989), or by imposing two (or more) pressure heads at the soil surface (Reynolds and Elrick, 1991; Ankeny et al., 1991).

A third method of analysis (White et al., 1992) can be obtained based on Philip's (1957) one-dimensional horizontal infiltration equation:

(6)
where I is the cumulative infiltration depth (L), t is time (T). Equation [6] appears as a truncation of Philip's (1957) one-dimensional vertical infiltration equation

(7)
where A is a parameter (LT-1) lying in the interval (K/3, 2K/3) (Youngs, 1968; Talsma and Parlange, 1972; Parlange, 1977; Fuentes et al., 1992) corresponding to the situation where gravity effects can be neglected.

White and Sully (1987) established the following relation between {phi} and S:

(8)
where {theta} is volumetric water content and b is a shape parameter in the interval (1/2, {pi}/4). A reasonable intermediate value of 0.55 can be taken for most field soils (Smettem and Clothier, 1989; White et al., 1992), as well as theoretical situations (Warrick and Broadbridge, 1992). The third method of analysis for disk infiltrometer tests then consists in fitting Eq. [6] on early time data by assuming that gravity and lateral effects are small at the beginning of the process and by using the S value herewith obtained to calculate {phi} by Eq. [8]. Determination of K is then straightforward from Eq. [1] and the method uses only one radius and one pressure head.

These three methods of analysis (multi-radii, multi-potential, and single test) based on Wooding's (1968) equation have been widely used and compared during the last few years (e.g., Logsdon and Jaynes, 1993; Cook and Broeren, 1994); however, some limitations can restrict or even prohibit their use in field situations.

(i) Equation [1] was obtained by assuming the soil to be homogeneous and isotropic and the initial water content to be uniform. In practice, gradients in water content and soil bulk density, soil layering, and large changes in soil texture all occur near the soil surface (White et al., 1992). Hence, it is not uncommon to obtain negative values of K (e.g., Hussen and Warrick, 1993; Logsdon and Jaynes, 1993).

(ii) It is often questionable to assume that a real steady state was reached at the end of a test. The time taken to reach steady state must always be shorter than the attention span or limit of patience of the experimenter (White et al., 1992), which is a serious burden for soils with low permeability.

Determination of the soil hydraulic properties can also be made by inverse modelling of an axisymmetric infiltration experiment (Quadri et al., 1994; Simunek and van Genuchten, 1996 and 1997), but the method can be problematic because of difficulties in dealing with the nonuniqueness of the solution.

These reasons have led researchers to look for simple equations of transient flow from the disk infiltrometer (Warrick and Lomen, 1976; Warrick, 1992; Smettem et al., 1994; Haverkamp et al., 1994). Advantages of analyzing transient rather than steady flow are obvious: (i) assumptions of homogeneous soil and uniform water content become more realistic with a reduction of the volume of soil sampled with a shorter experiment; (ii) vertical variations of hydraulic properties can be assessed with a high resolution by carrying out short infiltration tests at closely spaced depths; (iii) shorter experiments allow one to increase the number of sampling points over a field, which is of particular interest for spatial variability studies; (iv) the transient regime of infiltration contains information that is not analyzed when using only the steady state regime.

The aim of this series is to propose several new methods of analysis for disk infiltrometers based on the transient regime of axisymmetric flow. After a careful examination of the appropriate technique to choose for the determination of the equation parameters (this paper), accuracy of the methods and experimental strategy (choice of initial moisture conditions, choice of a disk radius, pertinence of using several disk radii) will be tested and discussed depending on the variable of interest, S or K (Vandervaere et al., 2000; this issue). In a forthcoming paper, the results from two experimental investigations will be used to demonstrate to what extent the methods are applicable in situ and to illustrate differences appearing between numerical simulations with model soils and field situations with real structured soils.


   Theory
TOP
 ABSTRACT
 INTRODUCTION
 Theory
Discussion
 Summary and conclusions
 REFERENCES
 
Transient Flow Equation
Transient axisymmetric infiltration from a circular source at the soil surface has been described by several researchers. Turner and Parlange (1974) calculated an approximate analytical expression for the lateral flux at the periphery of the one-dimensional infiltration process. Warrick and Lomen (1976) proposed an expression for {phi} valid for soils described by Eq. [3]. Basing their study on the work of Turner and Parlange (1974), Smettem et al. (1994) showed that the additional term accounting for the side effects due the axisymmetric flow geometry is linear in time:

(9)
where the subscripts 3D and 1D refer to axisymmetric three-dimensional and one-dimensional processes respectively, and {gamma} is a constant theoretically equal to {surd}0.3 when gravity effects are neglected at the periphery of the disk. Through comparison with experimental results, Smettem et al. (1994) showed that an appropriate value for {gamma} is 0.75.

On the basis of Eq. [9], Haverkamp et al. (1994) established a physically based infiltration equation for disk infiltrometers valid for short to medium time:

(10)
where ß is a constant constrained to 0 < ß < 1 (see Eq. [7] in Haverkamp et al., 1994). For the sake of simplicity, we recast Eq. [10] in the form

(11)
where the subscript 3D is omitted and with

(12)

(13)

One of the most accurate (Elrick and Robin, 1981) and yet simple expressions for sorptivity was proposed by Parlange (1975):

(14)
in which D({theta}) is the hydraulic diffusivity (L2 T-1).

Two-term forms of equation similar to Eq. [11] were found by Warrick (1992) and more recently by Zhang (1997b), although in these two studies, expressions for the coefficients C1 and C2 were empirically obtained and are different from Eq. [12] and [13]. Zhang (1997b) relates C1 to capillary forces and C2 to gravity forces (see Eq. [7] and [8] in his paper). This seems questionable, as Smettem et al. (1994) showed that the term accounting for lateral capillary flow from the disk is linear in time (Eq. [9]) and hence should appear in the coefficient C2. This is consistent with Wooding's (1968) equation (Eq. [1]), which show that both gravity (K) and capillary ({phi}) forces play a role in the steady state flux.

Note that if R -> {infty}, then Eq. [10] reduces to the one-dimensional Eq. [7]. The expression of the infiltration flux density q is straightforward from Eq. [11]:

(15)

The time interval over which Eq. [11] and [15] are valid will be discussed in Vandervaere et al. (2000)(this issue), the second paper of this series.

Determination of the Coefficients C1 and C2
At first sight, one can imagine that fitting the two coefficients of a nonlinear equation on a data set is a trivial question and that any best fit technique based on a classical least squares optimization criterion would be adequate. Our study tends to show that this is far from being true for the case of Eq. [11].

Principles
The most natural technique to determine C1 and C2 for a given infiltration test is to use the least squares optimization technique (Marquart, 1963) of nonlinear fitting of Eq. [11] on the (Ii, ti) data set. Since a cumulative infiltration vs. time curve has two degrees of freedom (one for scale and one for shape), determination of C1 and C2 in Eq. [11] is a well posed problem. Nevertheless, it is an ill-conditioned problem because of the obvious intercompensation between {surd}t and t. For example, an increase in C1 can be compensated for by a decrease in C2. Thus, particular care must be given to the choice of a fitting technique among the four which we present.

Direct nonlinear fitting of Eq. [7] or [11] (which have the same form) on a (Ii, ti) data set has been undertaken by numerous researchers (e.g., Bonnell and Williams, 1986; Bristow and Savage, 1987; Zhang, 1997a and 1997b). It is referred to in the following as the "cumulative infiltration" (CI) method; however, this technique offers no check for the adequacy of the form of the two-term equation with the data. Indeed, fitting Eq. [11] on any monotonically increasing data set convex-upwards (as cumulative infiltration usually appears) will provide fitted values for the coefficients C1 and C2, even if these values have no physical meaning. In short, "best-fit" does not necessarily mean "good fit". A similar difficulty occurs when fitting Eq. [15] on a (qi, ti) data set ["infiltration flux" (IF) method] because any monotonically decreasing data set concave-upwards (as infiltration flux usually appears) will provide fitted values for the coefficients C1 and C2, even if physically meaningless. Visual observation of the quality of the fit cannot stand as a test for the adequacy of Eq. [11] and [15] because of the difficulty of distinguishing the possible inadequacy of the equation and the scatter of the data points. To account for this, Smiles and Knight (1976) proposed linearizing Eq. [11] by dividing both sides by {surd}t, giving

(16)
and then plotting I/{surd}t as a function of {surd}t. In this form, it is easy to determine C1 as the intercept and C2 as the slope and to test the adequacy of Eq. [11] by checking the linearity of the data set ["cumulative linearization" (CL) method]. Hence, the extent to which the best fit also corresponds to a physically realistic fit can be controlled.

Another linear fitting technique consists in differentiating the cumulative infiltration data with respect to the square root of time ["differentiated linearization" (DL) method; Vandervaere et al., 1997]. Performing this differentiation on Eq. [11] yields

(17)
where d I/d{surd}t is approximated by

(18)
where n is the number of data points, and the corresponding {surd}t is calculated as the geometric mean:

(19)

Equation [17] shows that plotting vs. should be linear, with C1 equal to the intercept and C2 equal to half of the slope. If the data set is not linear, Eq. [11] must be considered inappropriate, and it would be likely that fitted values of the coefficients C1 and C2 would have no physical meaning.

We will now discuss the suitability of these four methods (CI, IF, CL, and DL) to determine unbiased values for the coefficients C1 and C2.

Case Study
The dangers of an inadequate fitting technique are illustrated by Fig. 1 , which corresponds to an infiltration test (no. 103) carried out on a sandy soil on a windy day. According to the cumulative infiltration and the infiltration flux curves, the experiment appears normal (Fig. 1a). Yet, the representation using Eq. [17] (DL method, Fig. 1b) clearly shows that the regime of infiltration changed drastically at a time of {approx}3 min. The representation using Eq. [16] (CL method, Fig. 1c) also shows a defective linearity. The use of cumulative data gives a smoother appearance than with the DL method, but the instant corresponding to the slope discontinuity is more difficult to determinate. It is very likely that during the Infiltration Test 103 the wind-induced movement of the apparatus had broken the hydraulic contact between the disk and the soil. However, the exact cause of this "accident" is of little importance here and could well have been any other experimental problem (leak in the system, presence of a stone or an impervious soil layer, etc.). Our purpose is to point out the fact that classical cumulative infiltration and infiltration flux curves do not reveal such discontinuities, whatever their causes may be.





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  		Fig. 1 Determination of coefficients C1 and C2 for a disk infiltrometer test: (a) cumulative infiltration and infiltration flux, (b) differentiated linearization method, (c) cumulative linearization method, (d) cumulative influx method, and (e) infiltration flux method

 
Nonlinear fitting of the coefficients C1 and C2 in Eq. [11] and [15] (Fig. 1d and 1e) yields values given in Table 1 . Values of the coefficients C1 and C2 obtained by linear regression using Eq. [16] and [17] for only the increasing linear portion of the data set ({surd}t < 13 s1/2 in Fig. 1b and 1c), i.e., before the experimental discontinuity, are also given in Table 1. Comparison of the numerical results in Table 1 shows differences of 50 to 90% for the coefficient C1 and differences of a factor of 3 to 6 for the coefficient C2, depending on whether the discontinuity is ignored (CI and IF methods) or accounted for (CL and DL methods). Such large differences with CI and IF methods are due to the optimization procedure, which exaggerates the curvature of the data set when applied over the entire duration of the experiment. Visual observation does not allow this problem to be detected, simply because slope discontinuities are nearly impossible to see with curves (Fig. 1a), while they are evident with straight lines (Fig. 1b and 1c).


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  		Table 1 Determination of the coefficients C1 and C2 for Infiltration Test 103 by four techniques

 
Although the example above can be seen as a particular case, it is of general concern that the time of measurement considered for the fitting procedure must not exceed the time for which the two-coefficient equation is valid (Parlange, 1977). Unexplained variations in the observed infiltration flux are not rare in field studies (e.g., Logsdon, 1997). The linear regression technique using Eq. [16] or [17] (CL and DL methods) is an adequate tool for the choice of the correct time interval and avoids the determination of meaningless coefficients C1 and C2.

Cumulative linearization and DL methods have two other advantages over CI and IF methods: First, standard deviation errors are easily calculated for the two fitted coefficients C1 and C2 with classical linear regression formulae. Second, all the data points of the data set have equivalent weight in the least-squares optimization criteria.

Influence of the Contact Layer: Does it Really Matter?
In most field situations the hydraulic contact between the tension disk infiltrometer and the soil is ensured by a layer of fine sand of a few millimeters depth. If the hydraulic conductivity of the sand is high compared with that of the underlying soil, which is the usual case, effects of the contact layer on the steady state infiltration regime is neglected. On the other hand, the water initially stored in the sand layer during the early stages of infiltration is likely to influence markedly the shape of the infiltration curve; the risk of biased parameter estimations when ignoring the sand layer must then be evaluated. The simplest way to account for the sand layer influence is to modify Eq. [11] as follows:

(20)
where Is and ts are respectively the depth of water and the time necessary to wet the sand layer in equilibrium with h0. The infiltration into the soil is then described by Eq. [20] for t > ts. Equations [15], [16], and [17] are respectively modified to

(21)

(22)

(23)

Equation [22] shows that the CL method is compromised by the effects of the sand layer; the intercept is modified by both Is and ts, and the slope is modified by ts. This is due to the use of cumulative data. On the other hand, IF and DL methods using differentiated data, are not influenced by Is, and the effect of ts becomes smaller as time increases (see Eq. [21] and [23]). Consequently, the effects of the storage of water in the sand layer at short times prohibit the use of methods based on cumulative data.

Also note that the pressure head h0 imposed at the soil surface is different from that imposed by the apparatus, which must be taken into account in the case of a thick sand layer, especially if h0 is close to zero.

Test of the Four Methods and Conclusions
In order to test the ability of the four different methods of adjustment (CI, IF, CL and DL) to return the exact values of the coefficients C1 and C2, we constructed a simulated infiltration test precisely obeying Eq. [11] with known values of C1 and C2, but modified to account for the presence of a contact sand layer. For this test we considered the case of a silt loam referred to as GE3 overlain by a fine layer of Grenoble sand (GS). Hydraulic properties of these two soils are given in Fuentes et al. (1992). Their saturated hydraulic conductivities and sorptivities calculated by Eq. [14] are given in Table 2 together with values of the parameter ß calculated with Eq. [7] in Haverkamp et al. (1994). The ratio of approximately 100 between the hydraulic conductivities of the sand and the soil is not uncommon in field situations because it matches the experimenter's demand of a negligible impeding effect for the sand layer on infiltration at long times. On the other hand, it is obvious that the infiltration flux q is strongly influenced by the contact layer at the very beginning of the experiment. Hence, we suppose that q varies continuously from an initial value qs in the sand (one-dimensional and not yet influenced by the underlying soil), to a normal value q0 in the soil (axisymmetric and no longer influenced by the sand). Arbitrarily, we assume this transition to last 4 s, and we construct the evolution of the infiltration flux in a manner described in Table 3 . Flux values qs and q0 are calculated from Eq. [12], [13], and [15] with the parameters given in Table 2, and q is then calculated with the coefficients given in Table 3. It is clear that the durations and coefficients in Table 3 are chosen arbitrarily. Nevertheless, they reflect a realistic situation with 1.7 mm of supplementary infiltrated water (that is, for example, the volume of water stored in a sand layer of 4.25-mm thickness and a porosity equal to 0.4) compared with the case without sand .


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  		Table 2 Hydrodynamic parameters for the Grenoble sand (GS) and the silt loam GE3

 

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  		Table 3 Construction of the infiltration flux q as affected by a sand contact layer

 
The simulations, for the cases with and without a contact layer, are presented in Fig. 2 in the different forms (CI, IF, CL, and DL) and numerical results are given in Table 4 . The following comments can be made:



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  		Fig. 2 Simulated effect of the sand contact layer on infiltration. (a) cumulative infiltration, (b) infiltration flux, (c) cumulative linearization method, and (d) differentiated linearization method

 

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  		Table 4 Determination of the coefficients C1 and C2 for the sand-affected infiltration test by four techniques

 
(i) Cumulative infiltration (Eq. [11] and [20]) shows a simple shift of the data due to the initial volume of water stored in the sand layer (Fig. 2a). When performing nonlinear fitting of Eq. [11] on the data set, C1 is overestimated by 54%, and C2 is negative, which is physically meaningless.

(ii) Infiltration flux (Eq. [15] and [21]) exhibits a perturbation of the four first points only (Fig. 2b). However, nonlinear fitting of Eq. [15] on the entire data set provides a C1 value overestimated by a factor of 3 and a negative value for C2. This very high error in C1 is due to the predominance of the vertical part of the curve (t small) in the least-squares optimization criteria.

(iii) Cumulative linearization (Eq. [16] and [22]) shows an important perturbation that prohibits the application of this method (Fig. 2c). The whole data set is perturbated by the contact layer influence because of the use of cumulative data. No estimation of the parameters C1 and C2 is possible.

(iv) Differentiated linearization (Eq. [17] and [23]) is affected by the sand layer only for the four first points, which are easy to detect since they correspond to the initial, sharply decreasing part of the curve (Fig. 2d). Then, linear regression can be restricted to the undisturbed rest of the data set, providing values of C1 and C2 without bias.

This simple simulation shows that, even though the perturbation induced by the contact layer seems small and of short duration, direct nonlinear fitting on a CI curve and/or an IF curve may lead to errors on the estimated parameters C1 and C2 so large that their values become completely meaningless. This high instability of Eq. [11] is explained by the high degree of intercompensation between {surd}t and t. Only the DL method of adjustment is able to provide unbiased values of the parameters C1 and C2, which can then be analyzed for further determination of S or K. It should be emphasized that in real situations, the influence of the contact layer on the infiltration process can be longer (e.g., at low values of applied potential) and/or more accentuated (e.g., in the case of sand overlying clay soils) than in the theoretical example described above.

In conclusion, the DL method is the only one of the four methods of parameter adjustment that offers all of the following advantages: (i) Estimation, by visual observation of the linearity, of the duration over which the two-term Eq. [11] remains valid; (ii) provides standard deviation errors for the parameters C1 and C2; (iii) gives equivalent weight to all data points in the least-squares optimization procedure; (iv) is able to detect and eliminate the influence of the contact layer, which corresponds to the early portion of the data set falling out of the monotonically increasing linear behavior.

Thus, the DL method can be used as a tool, not only to determine best-fit values for C1 and C2, but also to assess the validity of the two-term Eq. [11] for axisymmetric infiltration experiments.


   Discussion
TOP
 ABSTRACT
 INTRODUCTION
 Theory
 Discussion
Summary and conclusions
 REFERENCES
 
Validity of the Two-Term Equation Form for Simulated Infiltration Tests
In order to assess the adequacy of Eq. [11] simulations were performed with 3DFLOW, the numerical code developed by P.J. Ross at CSIRO Townsville, Australia. Using an implicit scheme of finite differences (Ross, 1990), the program solves Richard's equation with Kirchhoff transforms. Internodal distance is vertically minimal at the soil surface and horizontally minimal at the edge of the disk and increases arithmetically away from these regions. The grid was set in such a manner that a minimum of six nodes were situated under the disk, which guaranteed numerical stability. The time step is also variable, set minimally at the beginning of the simulation (equal to the duration of the test divided by 104), and then increasing with time (up to a value equal to the duration of the test divided by 103). A detailed description of the principles of the code 3DFLOW can be found in Ross and Bristow (1990).

The soil description uses the van Genuchten (1980) equation for water retention:

(24)
where {theta}s is saturated volumetric water content and {theta}r, hG and m are fitting parameters, and the Brooks and Corey (1964) equation for hydraulic conductivity:

(25)
where {eta} is a fitting parameter.

Tests were conducted for two soils of very different properties: Grenoble sand (GS) and Yolo light clay (YLC), whose parameters of the van Genuchten (1980) and Brooks and Corey (1964) equations are given in Table 5 according to Fuentes et al. (1992). The initial value of the pressure head was set to of water, which corresponds to {theta}n {approx} {theta}r; that is, a very dry soil. This maximizes the lateral capillarity-driven term of flow, which corresponds to the most severe conditions for the test. Pressure head applied at the soil surface was set to . Disk radii were chosen equal to those of our experimental devices: 125, 40, and 24.25 mm. Choosing the duration of the simulations is the result of a compromise: experiments must not be so short that regression calculations of the DL method cannot be performed with reasonable accuracy. On the other hand, when approaching steady state of infiltration, the validity of Eq. [10] becomes questionable because it was developed for transient stages only (Haverkamp et al., 1994). Following Smettem et al. (1995), we estimated the approximate duration T15 necessary for the infiltration front to reach a depth of 15 cm by

(26)


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  		Table 5 Parameters of the van Genuchten (1980) and Brooks and Corey (1964) equations for Grenoble sand (GS) and Yolo light clay (YLC) (from Fuentes et al., 1992)

 
This yields T15 values of 0.18 h for GS and 43 h for YLC. These values are less than Philip's convergence time tgrav (0.53 h for GS and 645 h for YLC), classically considered as the time scale necessary to approach steady state.

Simulations are shown in Fig. 3 for GS and YLC with and for radii of 125 and 24.25 mm. Excellent linearity is obtained with the DL method, correlation coefficients ranging between 0.9987 and 0.9997. Similar results are obtained with other values of h0, which validates the two-term equation form [11] for simulated axisymmetric infiltration tests. Values of the coefficients C1 and C2 obtained by linear regression are given in Table 6 . According to Eq. [12] and [13], the effect of a change of disk radius for a given soil should only be a change in the coefficient C2, while the value for C1 should be the same for any radius. This is well verified, the difference in C2 values between radii 125 and 24.25 mm being 280% for GS and 370% for YLC, while the differences in C1 values are only 1.9% for GS and 6.8% for YLC.



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  		Fig. 3 Application of the differentiated linearization method for numerically simulated infiltration tests and for two disk radii: (top) Grenoble sand (GS), and (bottom) Yolo light clay (YLC)

 

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  		Table 6 Determination of the coefficients C1 and C2 by the differentiated linearization method for simulated infiltration tests

 
Validity of the Two-term Equation Form for Laboratory Infiltration Tests
Infiltration experiments were conducted in the laboratory with two aims: (i) to assess experimentally the adequacy of the two-term Eq. [11]; and (ii) to test the assumption of independence of C1 from disk radius by comparing axisymmetric infiltration with one-dimensional infiltration using the same soil.

Tests were performed with a sand of calibrated granulometry (170-mm average particle diameter) called S31. The advantage of using a sandy soil is that no contact layer is needed. Applied surface pressure head values were set to -40, -70, -100, -125, and -150 mm. One-dimensional infiltration experiments were conducted on soil columns of 25 mm radius and 210 mm length using a disk infiltrometer of 24.25 mm radius. Axisymmetric experiments were performed with a quarter-disk infiltrometer of 60 mm radius (Quadri et al., 1994) placed in the corner of a plexiglas cubic box (150 x 150 x 180 mm3) filled with S31 sand (Fig. 4) . Infiltration was carried out until the infiltration front reached either the bottom or the walls of the box.



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  		Fig. 4 Quarter-disk infiltrometer and the cubic plexiglas box

 
Results are presented in Fig. 5 for that show the good linearity of the data in the form of Eq. [17]. Similar results were obtained with the other values of h0. Estimates of C1 as a function of pressure head are presented in Fig. 6 for the column and for the box. In principle, C1 should increase with h0, but there appears to be an anomaly at , where C1 decreases by {approx}15% from the value at . This anomaly was attributed to experimental imprecision (possibly linked with packing differences between samples). Thus, the value of 15% can be interpreted as the minimal error in our estimations, accounting both for analysis imprecision and for replication variability between samples. The average difference between the 1D and 3D estimations of the coefficient C1 is found to be also equal to 15% and thus, cannot be considered as significant. The assumption of independence of C1 with R is consequently in reasonable agreement with laboratory experiments.



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  		Fig. 5 Application of the differentiated linearization method to laboratory infiltration tests: axisymmetric geometry (3D), and one-dimensional geometry (1D)

 


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  		Fig. 6 Comparison of C1 estimates as a function of pressure head: axisymmetric geometry (3D), and one-dimensional geometry (1D)

 

   Summary and conclusions
TOP
 ABSTRACT
 INTRODUCTION
 Theory
 Discussion
 Summary and conclusions
REFERENCES
 
The axisymmetric cumulative infiltration from the tension disk infiltrometer can be adequately described with the two-term equation:

where C1 and C2 are parameters that depend on soil type and initial and boundary conditions. The parameter C1 is found to be independent of the disk radius for both numerical and laboratory tests. As cumulative infiltration curves have two degrees of freedom (i.e., one for scale and one for shape), determination of C1 and C2 for any given infiltration test is a well posed problem. However, this problem is ill-conditioned because of the possible intercompensation between {surd}t and t. Thus, estimation of the values for C1 and C2 can be highly dependent on the chosen optimization technique. The DL technique is the only one of the four methods analyzed that allows a visual check of the validity and range of applicability of the two-term equation. It is especially good for revealing and eliminating, at the beginning of experiments, the influence of a sand contact layer that may have disastrous effects on parameter estimations when not taken into account.

Knowing that it is possible to determine two coefficients, C1 and C2, for accurately describing axisymmetric infiltration with a two-term equation, the next questions are: how are these parameters related to soil sorptivity and conductivity? and how can we use these relations to determine S and K in the field with reasonable precision? In the next paper of this series, we propose and test several methods for the determination of hydraulic conductivity and capillary sorptivity based on the use of the transient flow two-parameter equation.Vandervaere Peugeot Vauclin Angulo-Jaramillo Lebel 1997


   ACKNOWLEDGMENTS
 
The authors wish to thank B.E. Clothier (HortResearch, Palmerston North, New Zealand) for use of the quarter-disk infiltrometer during and after his stay at LTHE and P.J. Ross (CSIRO, Townsville, Australia) for the numerical code 3DFLOW.


   REFERENCES
TOP
 ABSTRACT
 INTRODUCTION
 Theory
 Discussion
 Summary and conclusions
 REFERENCES
 




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