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DIVISION S-1-SOIL PHYSICS

Transient Flow from Tension Infiltrometers

II. Four Methods to Determine Sorptivity and Conductivity

^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, Guelph, ON, Canada N1G 2W1

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

	ABSTRACT

TOP ABSTRACT INTRODUCTION Theory Discussion Conclusions REFERENCES

 
In Vandervaere et al. (2000) it was shown that the transient regime of axisymmetric infiltration can be described by a two-term equation with one term proportional to the square root of time and the other term proportional to time. The two corresponding coefficients, C₁ and C₂, are functions of the hydraulic conductivity, K, and the sorptivity, S. In this paper we propose four different methods to achieve the determination of S and K. The four methods differ by the number of disk radii and the number of supply pressure head values which are utilized. We show that the accuracy of a given method is highly dependent on the combination of S and K values obtained. Three situations can be distinguished, depending on the disk radius: (i) the flow is dominated by the lateral capillary term; (ii) the flow is dominated by the gravity term; (iii) lateral capillary and gravity terms have equivalent weights. The seven model soils tested here all correspond to the first situation with usual disk radius values. This tends to show that a precise estimation of K is unlikely from disk infiltrometer data. We introduce a new time scale, t_stab, which generalizes the concepts corresponding to the two well known time scales t_grav and t_geom. We propose a guideline for the investigator to choose between all existing methods of analysis that use steady or transient flow. Finally, the four new methods are tested against numerically simulated tests with Grenoble sand and Yolo light clay.

Abbreviations: GS, Grenoble sand • MP, multi-potential • MR, multi-radii, MS, multi-sorptivity • MS1, MS method with a one-disk experiment • MS2, MS method with a multi-radii experiment • Sc, Scotter • ST, single test • WS, White and Sully • YLC, Yolo light clay

	INTRODUCTION

TOP ABSTRACT INTRODUCTION Theory Discussion Conclusions REFERENCES

 
IN VANDERVAERE ET AL. (2000), the first paper of this series, it was shown that the transient regime of axisymmetric infiltration from tension disk infiltrometers is adequately described by a two-term equation similar in form to the Philip's (1957b) one-dimensional vertical infiltration equation:

(1)

where I is cumulative infiltration depth (L), t is elapsed time (T).

The following expressions for coefficients C₁ and C₂ were proposed by Haverkamp et al. (1994):

(2)

(3)

where S is the capillary sorptivity (L T^-1/2), K is the hydraulic conductivity (L T^-1), ß is a constant in the interval (0, 1) (see Eq. [7] in Haverkamp et al., 1994), is a constant equal to 0.75, R is the disk radius (L) and is volumetric water content. Subscripts n and 0 denote, respectively, initial and surface boundary conditions.

Equations [1] through [3] were obtained under the following assumptions: (i) the soil is homogeneous and isotropic; (ii) the initial water content _n or the initial pressure head h_n is uniform; (iii) the initial pressure head h_n is sufficiently small for the condition K(h_n) << K(h₀) to be fulfilled.

These are the same assumptions as those made by Wooding (1968) to establish the following equation of steady state infiltration flux density q:

(4)

where is the Kirchhoff transform (L² T^-1) defined by

(5)

where K(h) is the hydraulic conductivity (K)–pressure head (h) relationship.

A remarkable feature of Eq. [1] is that the coefficient C₁ does not depend on the disk radius R, and it is well verified both numerically and experimentally (Smettem et al., 1994, 1995; Vandervaere et al, 2000). Once the coefficients C₁ and C₂ are known for a given infiltration test, the task of interest for the investigator is to determine S and K. It is obvious that this can be achieved by solving a two-equation ([2] and [3]) and two-unknown (S–K) system. Equations can also be obtained by using two or more disk radii or by applying two or more pressure heads at the soil surface. It is the aim of this paper to propose, evaluate and compare several new methods for the determination of S and K. All these methods have in common the fact that steady state of infiltration from the disk is not required.

We will pay particular attention to the standard deviation error associated with parameter estimations. The estimate uncertainties are often disregarded in the literature dealing with disk infiltrometer studies; however, it is sometimes essential to evaluate the measurement error variance of S and/or K. This is the case, for example, when comparisons are made on values obtained at different locations (spatial variability studies) or at different instants (temporal variability studies). Such comparisons are meaningless if differences found between values are equal to or smaller than their standard deviation errors.

	Theory

TOP ABSTRACT INTRODUCTION Theory Discussion Conclusions REFERENCES

 
Single Test (ST) Method
Determination of Sorptivity
The capillary sorptivity, S, is commonly estimated by assuming that both gravity and lateral capillarity effects can be neglected at the beginning of the axisymmetric infiltration process. Cumulative infiltration is then approximated by Philip's (1957a) equation established for one-dimensional horizontal infiltration:

(6)

and S can be determined as the slope of I plotted as a function of during a time interval T for which Eq. [6] is considered valid (Fig. 1)

(7)

View larger version (12K): [in this window] [in a new window]   Fig. 1 Principle of the classic determination of S at short times: a case without a sand contact layer

(i) It is difficult, if not impossible, to know how long Eq. [6] can be considered as valid for estimating S with a desired degree of accuracy. The calculated S value will be overestimated because of neglected gravity and lateral diffusion effects (Smettem et al., 1995), but determining how large this error is requires knowledge of soil physical properties. As a matter of fact, the chosen time interval T is likely to influence strongly the calculated S value (Bonnell and Williams, 1986).

(ii) Tension disk infiltrometers are usually placed on a layer of sand to ensure hydraulic contact between the infiltrometer and the soil. The sand is normally chosen for its high conductivity so that no impeding effect modifies the steady state infiltration flux; however, the effects of this layer on the first stages of infiltration can be influential (Vandervaere et al., 1997; 2000) and should be eliminated for the determination of the underlying soil sorptivity. Three stages are generally observed in the field (Fig. 2) . The slope of the I– graph is high during the first stage, which corresponds to infiltration into the sand, then it is minimal when the sand influence is terminated (Stage 2), and finally, it increases again when the effects of gravity and/or geometry become important (Stage 3).

View larger version (17K): [in this window] [in a new window]   Fig. 2 Principle of the classic determination of S at short times: a case with a sand contact layer

In fact, Eq. [7] is an unnecessary approximation. Equation [6] does not need to be valid for any time interval T if the following limit is used:

(8)

The solution of Eq. [8] is unique because the flow equation [1] is easily linearized into

(9)

In Vandervaere et al. (2000) we showed that linear regression technique using Eq. [9] was the most robust method for the determination of the coefficients C₁ and C₂ as the intercept and the half-slope of the regression line, respectively. This method of determination of the coefficient C₁ (which is equal to S) gives an unbiased estimation of sorptivity because the validity of Eq. [6] is not needed. Indeed, C₁ is extrapolated at with Eq. [8], and this extrapolated value is unique because Eq. [9] is linear in .

Determination of Hydraulic Conductivity
After coefficients C₁ and C₂ have been determined, hydraulic conductivity can be easily calculated using Eq. [2] and [3] if ₀ and _n were measured:

(10)

Because ß lies in the interval (0, 1) and is a priori unknown, only an interval of K values can be determined, with the minimum and the maximum corresponding to the cases , respectively. For a lognormally distributed error (for example), ß can be set to 0.6, which yields a K value with an error of a factor 1.4. Such an error remains acceptable for a field estimation. A potentially more important source of error comes from the uncertainty in the predetermined C₁ value. Without considering the uncertainty in ß, Eq. [10] yields

(11)

which clearly shows that the error in K will increase when C₁ (i.e., S) gets larger as compared to K, i.e., for soils with important capillary effects as compared to gravity forces. The fact that it is the squared value of S that is needed to calculate K amplifies the error. Consequently, it must be kept in mind that a precise estimation of the hydraulic conductivity of a soil is unlikely when the observed infiltration process is essentially driven by capillarity.

Re-Examination of the White and Sully (WS) Method
Another "single test" method was proposed by White and Sully (1987) that uses a steady state regime instead of a transient regime of axisymmetric infiltration. White and Sully (1987) established the following very useful relationship between S and :

(12)

where b is a constant lying in the interval (1/2, /4) depending on the shape of the capillary diffusivity function D(). The coefficient b can be set to an average value of 0.55 for most situations (White and Sully, 1987; Smettem and Clothier, 1989; Warrick and Broadbridge, 1992). Combining Eq. [4] and [12] yields

(13)

allowing K to be calculated if S is determined by analysis of the early stages of infiltration. As in the case of our ST method, in order to determine K with reasonable accuracy, the contribution of S must not be too large relative to the contribution of K, i.e., gravity forces must play an important role in the steady state infiltration as compared with the capillary forces. However, an unbiased application of the WS method is possible when calculating sorptivity with Eq. [8] rather than with the traditionally used Eq. [7] which overestimates S.

Multi-Radii (MR) Method
When the early-time estimation of sorptivity (i.e., coefficient C₁) proves difficult or doubtful, as when the contact layer masks the initial infiltration regime for example, an interesting alternative is offered by using more than one disk radius. When two infiltration tests are conducted with disks of radius R_A and R_B (R_B < R_A), determination of the coefficient C₂ for each of them, C_2A and C_2B respectively, gives the two-equation system:

(14)

(15)

which can be solved easily for calculating sorptivity and hydraulic conductivity:

(16)

(17)

In Eq. [16], _n can be chosen as the average of final water contents measured for the two (or more) disk experiments. In Eq. [17], ß is unknown, lying in the interval (0, 1). Thus, only an interval can be established for K with the limiting cases .

One can note the similarity between the equations [14] and [17] and those established by Scotter et al. (1982) for steady state fluxes emanating from ponded rings of different radii. The main difference is that, with our method, the steady state regime is not needed, thus avoiding the usual questions about the steady regime being "sufficiently steady" or not, questions which are even more important when two experimental values have to be utilized jointly to yield a sorptivity or conductivity estimation.

The MR method has two advantages over the ST method: (i) it is the squared value of sorptivity which is calculated by Eq. [16], which minimizes the error in S; and (ii) the K estimation (Eq. [17]) does not require volumetric water content measurements. On the other hand, the drawback of the MR method lies in the fact that the two (or more) disk experiments must be performed at different locations, which introduces complications from the short-distance spatial variability of soil properties. As for the steady state multi-radii method of Scotter et al. (1982), several replications for each radius are necessary to measure average C₂ coefficients with reasonable confidence.

Recent findings by Wang et al. (1998), who performed steady state experiments with disks of different diameters, show that despite large coefficients of variation between experiments, the disk radius does not introduce any systematic bias in the single-radius analysis results. In other words, the average results obtained by using a large disk alone are the same as those obtained by using a small disk alone. Thus, it is theoretically correct to develop methods using combinations of different disk radii.

If three different disk radii are available, the MR method can be applied graphically by plotting experimental values of the coefficient C₂ as a function of /[R (₀ - _n)] (Fig. 3) . The graph obtained should be linear (see Eq. [3]) with the slope equal to S² and the intercept equal to (2 - ß)/3 K. As data points are usually scattered by the effects of measurement errors and spatial variability of soil properties, the slope and intercept can be obtained by a simple linear regression technique. The degree of data scattering indicates the necessity for extra replications in order to reduce parameter uncertainty. Moreover, classical regression analysis formulas provide confidence intervals on K and S estimates. Note that in the case of widely scattered data, negative values of K may be obtained.

View larger version (11K): [in this window] [in a new window]   Fig. 3 Principles of the multi-radii (MR) method with three disk radii

Assuming that h_n does not vary much between different experiments performed at different h₀ values, Eq. [5] yields

(18)

Combining Eq. [12] and [18] gives

(19)

If two experiments are performed at supply pressure heads h₀₁ and h₀₂ (h₀₁ < h₀₂), K can be estimated with Eq. [19] in a finite differences form:

(20)

neglecting the variations of b with h₀.

It appears that, with the MS method, only sorptivity measurements performed at two (or more) supply potentials are necessary to estimate the soil hydraulic conductivity. Obviously, the MS method seems to be an interesting alternative to ST and MR methods when K (gravity forces) plays a minor role in the axisymmetric infiltration process relative to S (capillary forces); i.e., in the situations where ST and MR methods are likely to be inaccurate.

The sorptivity estimation at any given pressure head value can be obtained with a one-disk experiment (Eq. [8], Method MS1) or with a multi-radii experiment (Eq. [16], Method MS2).

	Discussion

TOP ABSTRACT INTRODUCTION Theory Discussion Conclusions REFERENCES

 
Which Method Should One Opt For?
Apart from inverse modeling (Quadri et al., 1994; Simunek and van Genuchten, 1996; 1997), there exists a large number of available analytical methods for analyzing disk infiltrometer tests. All these methods are based on solving a two-equation system to determine soil sorptivity and hydraulic conductivity. The methods differ in the technique used to obtain these two equations, whether use is made of the steady or the transient infiltration regime, one or several disk radii, and/or one or several supply pressure heads. The methods using Wooding's (1968) steady state flux equation (Eq. [4]) have been widely used and compared in the last few years (Hussen and Warrick, 1993; Logsdon and Jaynes, 1993; Cook and Broeren, 1994), but the results from these studies are difficult to interpret. The conclusions of a given study also seem difficult to extend to other field situations and no general experimental strategy can be established due to the lack of a theoretical background with which to evaluate the suitability and accuracy of the different methods.

Three well known methods make use of Wooding's steady state flux expression. These are (i) the WS method using one disk radius and one pressure head; (ii) the Scotter et al. (1982) method (Sc method) using two (or more) disk radii and one pressure head; (iii) the multi-potential (MP) method proposed by Reynolds and Elrick (1991) and Ankeny et al. (1991), which uses one disk radius and two (or more) pressure heads.

The four methods presented in this study make use of the transient infiltration equation of Haverkamp et al. (1994): (iv) the ST method, which uses one disk radius and one pressure head; (v) the MR method, which uses two (or more) disk radii and one pressure head; (vi) the MS1 method, which uses one disk radius and two (or more) pressure heads; (vii) the MS2 method, which uses two (or more) disk radii and two (or more) pressure heads. The operating conditions of these seven methods are summarized in Table 1 .

• (i) the vertical capillary term: S
  
• (ii) the (vertical) gravity term: (2 - ß)/3 Kt
  
• (iii) the lateral capillary term: S²/[R (₀ - _n)] t
  

For the sake of simplicity, we put

(21)

(22)

The importance of the vertical capillary term, S (coefficient C₁), is maximal at the beginning of an experiment and is progressively dominated by the two other terms (coefficient C₂) as time increases. The time, t_stab, after which the vertical capillary term S becomes dominated by the combination of the two other terms, linear with t, is defined by

(23)

which yields

(24)

We propose the term stability time for t_stab, because it is the magnitude of the combination of the two terms linear with t as compared with the linear term in that stabilizes the infiltration flux. This stability time generalizes the well known t_grav and t_geom notions (Philip, 1969; White and Sully, 1987). Indeed, it is easily seen that

(24a)

(24b)

The order of magnitude of t_stab varies widely with the soil type. For example, with a radius of 125 mm and h₀ = 0, t_stab is only 450 s for Grenoble sand, while t_stab reaches 43 h for Yolo light clay.

For any experimental time, it is obvious that the accuracy of the C₁ estimation is maximized with the ratio C₁/C₂:

(25)

This ratio is maximal for a particular S value, S_opt, given by

(26)

which yields

(27)

Equation [27] gives, for a given set of values of R, , A, and , the sorptivity value S_opt for which it is easiest to determine S as the coefficient C₁. In the linear graphic representation using Eq. [9], the intercept (C₁) is then maximal compared with the slope (C₂).

The variations of C₁/C₂ with S are shown in Fig. 4 for the Grenoble sand . The first observation which can be made is that, the larger the disk radius, the higher the ratio C₁/C₂. This means that the estimation of the coefficient C₁ is optimized by using a disk infiltrometer as large as possible to minimize the lateral capillary term. For a given disk radius, the ratio C₁/C₂ has its maximum for . Three different situations can be distinguished:

View larger version (16K): [in this window] [in a new window]   Fig. 4 Variations of the C1/C2 ratio with S. The dashed line represents the location of Sopt. The represented example corresponds to the Grenoble sand (A = 2.12 x 10-2 mm s-1; = 0.312 m3 m-3)

2. Lateral capillarity domain. If S > S_opt (on the right side of Fig. 4), then the estimation of the coefficient C₁ is made difficult by the importance of the lateral capillary flow. This corresponds to soils having a relatively low hydraulic conductivity as compared with their sorptivity. A precise estimation of K is unlikely.

3. S_opt border. If (dashed line in Fig. 4), then the estimation of the coefficient C₁ is optimized. Early-time vertical capillary flow is maximal. Gravity and lateral capillary terms have equivalent weights in the flow process. A precise estimation of K is possible. The equation of this border is given by the combination of Eq. [25] and [27]:

(28)

For a given soil, it is interesting to know in which of these three regions the data is situated for a given disk radius. Taking (our largest disk radius), we examined this issue for seven model soils of very different hydraulic properties given in Fuentes et al. (1992) and for two boundary conditions . For each of these 14 situations, we calculated the optimal S_opt value (Eq. [27]) and the true value of sorptivity according to the Parlange (1975b) expression:

(29)

Results given in Table 2 show that for all soils and for both pressure head values, S is larger than its optimal value, S_opt. The ratio S/S_opt varies between 1.4 and 9.5 with an average of 4.2. This means that all these soils, including sands, loams, and clays, are situated in the lateral capillarity domain.

View larger version (28K): [in this window] [in a new window]   Fig. 5 S–K diagram of seven model soils and two supply potential values. The frontier between the two domains, shown for three values of the disk radius R, corresponds to ß = 0.6 and = 0.4

(30)

Values of R_opt, given in Table 3 , range from 24 cm (for Grenoble sand) to 10 m (for Beit Nefota clay), which is much larger than the radius of most available devices for field use.

Still, despite covering a large range of texture, the seven soils considered here are only a small sample of existing soils. Moreover, most of these soils are in fact repacked materials that were studied in the laboratory. Thus, they may not be representative of soils encountered in field situations where the effects of structure (i.e., macropores, cultural practices, presence of roots and stones, etc.) on the hydraulic properties can be considerable, especially close to saturation. In the next part of this series, we will reexamine the question of which domain the soils are situated in with the results obtained from different field studies.

Of course, the values of S and K are not chosen by the investigator. They are physical properties that we must cope with and the problem is to obtain their estimations as precisely as possible. The analysis of preliminary results obtained at the beginning of a field investigation can be a great help in establishing a strategy for the forthcoming tests. The main idea of such a strategy is to opt for methods that are based on using the dominating terms in the flow process. We propose the following guideline:

Step 1. Initial observation of the soil profile should give information pertinent to a decision between steady state and transient flow methods. The former can be used when the soil appears to be reasonably homogeneous for at least 10 cm depth and when initial water content gradients close to the surface are not too steep. In the opposite case and/or when the time necessary to reach steady state is a limiting factor, transient flow methods are preferred. Of course, transient flow analysis is also compatible with situations where steady state methods are used. Among steady state methods, the MP method, which consists in applying several pressure head values successively at the same location, is the most demanding in terms of vertical homogeneity; indeed, the differences in the observed flux values that correspond to the different h₀ values must be the result of the pressure head variation and not the result of soil layering.

Step 2. Run a few experiments with the largest disk available. Use a sand contact layer only if necessary. Plot the data in the form of Eq. [9]. Does the data appear linear (apart from the first readings perturbed by the contact layer) for a certain time interval (say, at least 3–5 min)?

Step 2.1. If not, then the two-parameter transient flow equation (Eq. [1]) should not be used because it is not valid and thus, would yield meaningless values for C₁ and C₂. Use steady state methods.

Step 2.2. If yes, then determine the coefficients C₁ and C₂ by linear regression, without considering the first few points affected by the contact layer. Then, calculate S and K with Eq. [2] and [3] (ST method) and calculate S_opt (Eq. [27]) taking .

Step 2.2a. If S >> S_opt, then the estimation of K is very sensitive to measurement errors. The WS, ST, Sc, and MR methods should be avoided. We recommend the MP method (if steady state is used) and/or the MS methods (if transient flow is used). The drawback of the MP method is that sorptivity cannot be estimated. Between MS1 and MS2, choose the MS1 method if the estimation of the coefficient C₁ appears correct (i.e., if the contact layer influence is not too important); choose the MS2 method in the opposite case. An interesting strategy for improving the K estimation consists of working in relatively wet initial conditions that reduce sorptivity [keeping in mind that the condition K(h_n) << K(h₀) must be fulfilled]. The experimenter can also consider using a twin-disk system (Smettem et al., 1995) that reduces the effects of water moving laterally.

Step 2.2b. If S S_opt, then the estimation of sorptivity as equal to the coefficient C₁ is reliable. K can be reasonably well estimated with WS and ST methods. It is the most desirable situation if both S and K are to be estimated. Methods Sc and MR will provide acceptable estimates for both K and S. To estimate K accurately, the disk radius must be as large as possible.

Step 2.2c. If S << S_opt, then the conditions are very good for estimating K with WS and ST methods, but a precise estimation of sorptivity is unlikely. Consequently, the S estimation that has been used to establish that S < S_opt must be considered with care. Some supplementary tests are recommended to confirm or deny the statement that S < S_opt. The Sc and MR methods will provide good K estimates but poor S estimates. The MP, MS1 and MS2 methods are not recommended. In order to improve the S² estimation, the disk radius chosen must be small enough to enhance the lateral capillary term. The estimation of S will also be improved when initial conditions are the driest.

Evaluation of the Methods with Simulated Infiltration Tests
Axisymmetric infiltration tests were performed with the numerical code 3DFLOW (Ross, 1990) described in Vandervaere et al. (2000) of this series. Two soils were used for the study: Grenoble sand (GS) and Yolo light clay (YLC). Their hydrodynamic properties are given in Fuentes et al. (1992) using the van Genuchten (1980) equation for water retention:

(31)

where _r, h_G, and m are fitting parameters, and the Brooks and Corey (1964) equation is used for hydraulic conductivity

(32)

where is a fitting parameter.

The 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. Following Smettem et al. (1995), the duration of the simulations was set to a value T₁₀, which corresponds to an approximate depth of 10 cm being wetted by infiltration:

(33)

This yields T₁₀ values of 5 min for GS and 20 h for YLC, which is less than the time necessary to approach steady state.

Quasi-exact values of sorptivity for the different boundary conditions were calculated with Eq. [29] and used as a reference to evaluate the accuracy of the ST and MR methods. Reference values of S and K are given in Tables 4 and 5 .

View larger version (29K): [in this window] [in a new window]   Fig. 6 Coefficients C1 and C2 estimated by the single test method. Estimates for R = 125 mm (squares), for R = 40 mm (circles), and for R = 24.25 mm (triangles) are compared with exact values (lines) for Grenoble sand (left) and Yolo light clay (right)

Single Test (ST) Method
Determination of Sorptivity
With the ST and WS methods, which use one single disk radius and one single pressure head value, S is simply equal to the coefficient C₁. As explained above, this estimation must be made with the largest disk radius that enhances the term in C₁ by minimizing the coefficient C₂ (see Eq. [1–3]). Thus, the error in S is the error in the C₁ value obtained with the disk radius of 125 mm; i.e., an average underestimation of 8.6%, which is acceptable in the field where spatial variability generally induces much higher coefficients of variation. Note that this error would be reduced with a shorter duration of experiment.

This imprecision can be compared with the one arising when using the classical early-time determination of S; i.e., when using Eq. [7] instead of Eq. [8]. This error depends, of course, on the time interval T throughout which the data are analyzed. We tested Eq. [8] for GS with T values ranging from 30 to 150 s and for YLC with T values ranging from 50 to 200 min. Times are much larger with YLC, as readings in the field are expected to be much less frequent with low permeable soils.

Results are shown in Fig. 7 defining the error as

(34)

View larger version (27K): [in this window] [in a new window]   Fig. 7 Classic determination of S at short times: estimation error as a function of the time T taken into account (Eq. [7])

Determination of Hydraulic Conductivity
The estimation of the hydraulic conductivity requires the determination of the coefficients C₁ and C₂. Thus, the precision of K is affected by the errors in both C₁ and C₂. Moreover, K is calculated by subtracting the lateral capillary term from C₂ (Eq. [10]). Consequently, the accuracy of this K estimate is highly dependent on the domain in which the soil is situated:

(34a)

The flow is gravity-driven and K is the dominant term in C₂. The subtraction in Eq. [10] is a well conditioned operation and the K estimate is reliable.

(34b)

The flow is driven by lateral capillary forces and K is a minor term in C₂. The subtraction in Eq. [10] is a badly conditioned operation and the K estimate may be unreliable.

Considering the fact that GS and YLC are both in the lateral capillarity domain, we foresee that the ST method would provide poor K estimates. Values of K obtained with the ST method are compared with exact values in Fig. 8 , showing that the hydraulic conductivity is systematically overestimated. The error ranges from 0% to +115% for GS and from +30% to +90% for YLC. This error tends to be larger when closer to saturation. It can be noted that the error in K is larger than the errors in the C₁ and C₂ coefficients used for its calculation; again, this is due to the minor role played by gravity flow in the infiltration process, which is observed in soils such as GS and YLC. It has been verified that the WS method, which uses the same C₁ estimate for sorptivity but uses Wooding's steady flux equation (Eq. [4]) instead of the coefficient C₂, leads to similar errors in K (note the similarity between Eq. [10] and [13]).

View larger version (31K): [in this window] [in a new window]   Fig. 8 K estimates by the single test (ST) method (squares) compared with exact value (plain line) for Grenoble sand (top) and Yolo light clay (bottom)

Results are presented for the three disk radii and the six pressure heads in Fig. 9 (GS) and 10 (YLC) that demonstrate the excellent linearity of the coefficient C₂ vs. 1/R (correlation coefficients ranges from 0.9991–0.999999). On the other hand, Fig. 9 and 10 clearly show the minor role played by the hydraulic conductivity (corresponding to the intercept) as compared with lateral capillarity (corresponding to the slope). As for the ST method, this makes the problem of determining K very badly conditioned. Indeed, it is clear from Fig. 9 and 10 that a small perturbation of the C₂ values will not greatly affect the slopes (S²), but it may change the intercept (K) drastically. Thus, it is predictable that the small errors in the C₂ data will be amplified when calculating K, but not when calculating S².

View larger version (23K): [in this window] [in a new window]   Fig. 9 The multi-radii method for Grenoble sand. Symbols correspond to estimates and plain lines correspond to regression lines

View larger version (25K): [in this window] [in a new window]   Fig. 10 The multi-radii method for Yolo light clay. Symbols correspond to estimates and plain lines correspond to regression lines

View larger version (29K): [in this window] [in a new window]   Fig. 11 K and S estimates by the multi-radii method (squares) compared with exact values (plain lines) for Grenoble sand (top) and Yolo light clay (bottom)

Using Matric Flux Potential–Pressure Head Relationship: Multi-Sorptivity (MS) Methods
All of the soils represented in Fig. 5 are in the lateral capillarity domain. Thus, it is more appropriate for these soils to determine sorptivity, which is the dominant parameter, rather than hydraulic conductivity. Using Eq. [20], hydraulic conductivity estimates are obtained from sequential, paired sorptivity values at two different pressure heads. The sorptivity values can be obtained by using only coefficients C₁ (Eq. [8], Method MS1) or by using only coefficients C₂ (Eq. [16], Method MS2). With both methods, the error in K is of the order of 10%, Method MS2 being slightly more accurate (Fig. 12) . The error is always lower than 20%, which is surprisingly good for an approach based on differentiation.

View larger version (36K): [in this window] [in a new window]   Fig. 12 K estimates by the multi-sorptivity method with a one-disk experiment (MS1) (triangles) and the multi-sorptivity method with a multi-radii experiment (MS2) (squares) compared with exact value (plain line) for Grenoble sand (top) and Yolo light clay (bottom)

	Conclusions

TOP ABSTRACT INTRODUCTION Theory Discussion Conclusions REFERENCES

The suitability of those methods according to the experimental context was carefully examined. In particular, method appropriateness was shown to be highly dependent on the ratio of K and S. A good experimental strategy consists of chosing the method of analysis based on the dominant term of the flow in use: vertical capillary flow, vertical gravity flow, or lateral capillary flow. Placing the results of a test in an S–K diagram (Fig. 5) is an uncomplicated way of appreciating the relative importance of these three components. It is also an important thing to do in order to estimate the confidence in the S and K values. Taking into account the three flow components, we proposed a new time scale, t_stab, called stability time, which combines and replaces the two classical time scales t_grav and t_geom.

The study of seven model soils that represent a wide range of textural characteristics showed that all these soils are in the lateral capillarity domain; i.e., the lateral capillary flow term (S²) dominates the gravity flow term (K). The disk radius R_opt for which these two terms would have equivalent weights is, for all soils, larger than reasonably feasible devices. This has several consequences:

• (i) It is the lateral component of the flow and not the gravity which stabilizes the infiltration flux;
  
• (ii) The effects of the source geometry cannot be neglected at short times for the determination of sorptivity;
  
• (iii) During a disk infiltrometer experiment, the observed phenomena are mainly due to capillarity-driven flow; hydraulic conductivity must then be considered as a variable playing a minor role, and thus, it is difficult to determine accurately.
  

We proposed an unbiased method of determining S at early times (Eq. [8]) with which it is not necessary to assume that gravity and/or geometry have a negligible impact at the beginning of experiments. The classical method that uses Eq. [7] may lead to a dramatic overestimation of sorptivity, because the lateral flow effects cannot in fact be neglected.

The four proposed methods were evaluated with simulated axisymmetric infiltration tests for two well known soils, GS and YLC. Sorptivity was always better estimated than hydraulic conductivity, which is in concord with the fact that both soils are in the lateral capillarity domain. The mean absolute value of the error in S is 8% with Method ST and 2% with Method MR. The mean absolute value of the error in K is 60% with Method ST, 16% with Method MR, and 10% with Methods MS1 and MS2.

Nevertheless, the seven soils presented in this study, despite covering a large range of textural characteristics, cannot pretend to be a complete representation of the considerable variety of existing soils. Moreover, these model soils are in fact repacked materials with their hydraulic characteristics determined in the laboratory. Many authors claimed that real soils encountered in situ may have structural features that can modify their hydraulic properties, especially close to saturation. Consequently, it remains to be studied now how real soils, with hydraulic parameters measured in situ, would be placed on the S–K diagram (Fig. 5). The main question is: are structural effects and enhanced gravity flow sufficient to place these soils in the gravity domain instead of in the lateral capillarity domain? In a forthcoming paper, we will try to answer this question by analyzing results obtained from various experimental trials.

	ACKNOWLEDGMENTS

 
The authors wish to thank P.J. Ross (CSIRO, Townsville, Australia) for making the numerical code 3DFLOW available to us.

	REFERENCES

TOP ABSTRACT INTRODUCTION Theory Discussion Conclusions REFERENCES

 

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