Analysis of Simplified

ANALYSIS OF SIMPLIFIED

INFILTRATION TESTS AND

RETRIEVAL OF SOIL

HYDRODYNAMIC PROPERTIES

IN THE CONTEXT OF THE

ALPILLES 1997 EXPERIMENT

Isabelle BRAUD

LTHE, BP 53, 38041 Grenoble Cedex 9, France.

October 1998

1. INTRODUCTION.

In the framework of the Alpilles 1997 experiment (Prévot et al., 1998), it was attempted to test a simplified method for the estimation of soil hydrodynamic properties in the field. The methodology was proposed by Haverkamp et al. (1998) and is based on a non-dimensional form of the infiltration equation. In the field, a minimum of activity is required: simple infiltration test, measure of the porosity, particle size distribution and of the initial soil moisture content. The methodology had been tested on a catalogue of soils (GRIZZLY, Haverkamp et al, 1997) but had not been used extensively in the field.

The purpose of the experimental work conducted during the Alpilles experiment was i) to apply the methodology under real conditions and develop the algorithms necessary to retrieve the soil hydrodynamic properties ii) compare the results of the method to other techniques such as the use of infiltrometers or the Wind method for the estimation of soil hydrodynamic properties (retention and hydraulic conductivity curves) iii) use the portability and the economy of human resources of the method to explore the spatial variability of the soil hydrodynamic properties.

In the following, the theory underlying the method, the field activity, the development of mathematical algorithms for the estimation of the various parameters and the results will be presented.

1. THEORY.

2.1.Non-dimensional form of the infiltration equation.

The following presentation will be restricted to the Green and Ampt (1911) model of infiltration for clarity. Haverkamp et al. (1998) gave the generalisation of the theory to other forms of the infiltration equation, but the basic conclusions remain the same as with the Green and Ampt case. Furthermore, when trying to apply the method to the field, the Green and Ampt model, which has the smallest number of parameters is the only one usable in practice. The Green and Ampt model assumes that the soil profile and the distribution of the soil moisture are homogeneous. The infiltration process is modelled as a saturation wetting front crossing the soil profile vertically. The soil moisture above the wetting front is at saturation and the soil moisture below the wetting front is equal to the initial soil moisture. The retention curve is modelled as a step function from the value at the initial soil moisture to the value of the wetting front, while the hydraulic conductivity is modelled as a point value.

In dimensional form, the cumulative 1-D infiltration I is given by Green and Ampt (1911):

(1)

where I (m) is the cumulative infiltration, Ksis the saturated hydraulic conductivity (m s-1), t (min) is the time since the infiltration began, s is the saturated water content (m3 m-3), o is the initial soil moisture (m3 m-3), hsurf(m) is the pressure applied at the soil surface (assumed to be  0 in the following) and hfis the soil water pressure at the wetting front.

By dividing (1) by , and by denoting:

(2)

(3)

where S2 is the square of the sorptivity, Eq. (1) can be rewritten:

(4)

or(5)

if we denote:

(6)

Eq. (5) is the non-dimensional form of the Green and Ampt equation.

The Green and Ampt model assumes that the initial hydraulic conductivity Ko is zero. Haverkamp et al. (1998) modified the following equation to take a non-zero value of Ko into account. The cumulative infiltration I reads:

(7)

However, the form of Eq. (4) and (5) remains unchanged. Only the scaling factors are modified and read:

(8)

(9)

This last form will be used in the developments below.

2.2.Possible use of the non-dimensional infiltration equation for the derivation of field hydrodynamic properties.

When interpreting the non-dimensional 1-D infiltration equation (5) graphically, it appears that couples of values (time from the beginning of infiltration, cumulative infiltration) (t, I) lie on a unique and invariant curve defined by (5) when they are scaled respectively by tand I. As (5) defines a monotonic increasing curve, the values of the two scaling factors verifying (5) are unique (Haverkamp et al., 1998).

This feature can be used in an inverse procedure to analyse the results of infiltration tests and try to derive the soil hydrodynamic properties. In theory, with two points on the curve defined by (5), it is possible to derive the two scaling factors and the procedure converges rapidly (Haverkamp et al., 1998).

The hydrodynamic properties of a soil can be represented by the retention curve h() modelled with the Van Genuchten (1980) model Eq. (10) and the hydraulic conductivity K() modelled by the Brooks and Corey (1964) model Eq. (11).

(10)

(11)

These equations contain two shape parameters m and  and three scale parameters: the saturated water content s, the scale parameter for the water pressure hg and the saturated hydraulic conductivity Ks.

Haverkamp et al. (1997) show that the shape parameters m and  can be determined from the particle size distribution. The cumulative particle size distribution can be modelled with a function similar to the Van Genuchten (1980) model by:

(12)

where d is the particle diameter, dg is the scale for the diameters.

Then the shape parameter of the Van Genuchten model is obtained by:

(13)

The model defined by (12) can be adjusted to the data or a simplified formulation function of the clay content C (%) and of the silt content Si (%) can be used to derive M (Bouraoui et al., 1998)

(14)

Haverkamp et al. (1997) have shown that the shape parameter of the Brooks and Corey model was linked to the shape parameter of the Van Genuchten model with the following empirical relationship, fitted on the available GRIZZLY soil data base:

(15)

To get the saturated water content, some knowledge of the porosity is needed. The porosity  can be deduced from dry bulk density d by:

(16)

and the saturated water content from porosity and particle size distribution by the following empirical relationship, shown by Haverkamp et al. (1997) to be a good approximation of the theoretical relationships proposed by Fuentes-Ruiz (1992):

(17)

If the initial and saturated moisture contents o and s as well as the shape parameter  of the K() relationship (11) are known, the use of the non-dimensional infiltration equation can lead to the value of the saturated hydraulic conductivity. Indeed, by identifying the value of the scale parameters I and t or more precisely I and Ks-Ko which are independent, such that at least two couples (t, I) of a simple infiltration test lie on the non-dimensional infiltration curve (5), the value of Ks can be deduced by:

(18)

If the scale parameters are known, as well as the surface water pressure applied during the infiltration test hsurf, the value of the pressure at the wetting front hfcan be calculated from:

(19)

By writing that the sorptivity calculated with the Green and Ampt model or the Van Genuchten model were identical, Haverkamp et al. (1996) managed to relate the pressure of the wetting front to the pressure of the wetting front for an initial water content equal to zero and finally to the scale parameter of the Van Genuchten model hg:

(20)

(21)

where  is the Gamma function.

In summary, the measurements needed to determine the parameters of the retention and hydraulic conductivity curves in the field are the following:

-particle size distribution analysis (practically the clay, silt and sand contents are sufficient)

-dry bulk density measurements

-simplified infiltration test with at least two measures of the time of infiltration and the cumulative infiltrated height.

The procedure described below allows then to derive the various parameters of the hydrodynamic properties. The methodology for the shape parameters had been tested by Haverkamp et al. (1997) on a catalogue of soils. In the framework of the Alpilles experiment, the applicability of the method in the field was tested on the various instrumented fields. The field activity, the optimisation technique developed to estimate the scale parameters and the results are presented in the next section.

1. EXPERIMENTAL WORK.

3.1.Description of the simplified infiltration tests.

The simplified infiltration tests were designed to minimise the time needed per test, in order to be able to cover larger surfaces. Indeed, traditional methodologies: double ring infiltration tests or use of infiltrometers require a long time and it is not possible to repeat them several times in order to investigate the spatial variability of the soil properties, which is known to be large. With the simplified tests which require a minimum of material and are in general of lower duration, such a study becomes possible.

The methodology used during the Alpilles experiment was the following.

The material used was a ring, which diameter was 13 cm. The ring was put at the surface of the soil and buried a minimum in order to avoid lateral losses of water. It was not necessary to cut the vegetation or to have a very plane surface. Of course, significant slope was avoided, because the infiltration would not have been uniform anymore into the ring. Then a first volume of water was poured into the ring and the chronometer was started at the time the water was poured. The time needed to infiltrate the first volume was read on the chronometer. When the first volume was completely infiltrated, a second volume was poured into the ring and the time needed to infiltrate this second volume was also determined (cumulative time). The procedure was reiterated for a third volume, and if necessary a fourth volume. The volumes used were generally 500, 300, 200 cm3 for the first, second and third volumes. This lead to a maximum value of the surface pressure of 4 cm. With the procedure used, the surface pressure is no more constant, as it is assumed into the theory but Haverkamp et al. (1998) showed that small variations of hsurf were not influencing the results very much. Therefore, the methodology described above was acceptable.

A gravimetric sample was collected in order to determine the initial moisture content. At the end of the infiltration test, an undisturbed sample, with a known volume was extracted in order to determine the porosity. Finally, a sample of the soil was collected for particle size distribution analysis.

3.2.Location of the infiltration tests and presentation of the rough data.

Infiltration tests were performed on all the instrumented fields. For field 101 (non irrigated wheat), it was intended to sample the spatial variability of the properties. Two axes were determined corresponding to the axes of the radar flights. Axis A was parallel to the rows and Axis B was perpendicular to the rows. The field was about 300x500 m. The sample points were equally located on those axis, with an average distance of about 25 m. On axis A, the points were numbered A0 to A12 from South to North. On axis B, they were numbered B1 to B18 from East to West. On this field a pit was also opened down to 1.2 m. Some tests were also performed into this pit at various depths.

For the other fields, the tests were in general performed randomly near the mast measuring the climatic data and the surface fluxes.

Table 1 summarises the various sampled fields, the way the tests were labelled and the type of vegetation covering the field at the time the tests were performed.

Field number / Numbering of the tests / Vegetation / Date of tests
101 / A0 to A12 (axis parallel to the rows)
B1 to B18 (axis perpendicular to the row)
C1 to C6 (pit)
C7-C8 (Wind sample)* / Non irrigated winter wheat / April 15 1997
April 16-17 1997
May 14 1997
March 27 1997
214 / D1 to D10 / Non irrigated spring wheat / April 22 1997
203 / F1 to F3
F4 to F8 (Wind sample)* / Alfafa / April 18 1997
April 24 1997
120 / G1 to G9 / Irrigated winter wheat / April 22 1997
102 / H1 to H6 (Wind sample)*
H7 to H12 / Bare soil
Sunflower / April 23 1997
June 9 1997
501 / I1 to I3 / Sunflower / April 24 1997
121 / J1 to J8 / Sunflower / June 10 1997

* (Wind sample) means that sample for application of the Wind method were extracted in the vicinity of the corresponding test sites.

Rough data collected at the various sites are presented in Appendix 1.

3.3.Discussion on the applicability of the method to the Alpilles experiment.

Several features need to be underlined concerning the application of the methodology to the Alpilles experiment.

Most of the infiltration tests were performed in April 1997 between April 14 and 25 1997. It had not rained for the beginning of January. Therefore, the soil, especially at the surface was very dry. As the soils in the experimental area contained a large fraction of clay (about 40%), cracks had developed in most of the fields. Some of them could reach 2 cm in width and a meter in depth. The theory presented in section 1 assumed that the soil was homogeneous, which was obviously not the case for the Alpilles fields. It was intended to install the ring in places without faults, but during the infiltration it was obvious that both processes could occur: a new crack could open or a crack could be closed due to swelling of the soil. In both cases the velocity of the infiltration were different and it will be seen below that those cases could be detected easily with the non-dimensional form of the infiltration equation. Half of the infiltration tests had to be discarded because of the cracks, which considerably reduced the final number of exploitable tests. Furthermore, the presence of cracks prevented the use of a ring with a larger diameter, which would have covered a more representative surface of the field. But it was not possible to find places without cracks to install them.

Another feature which was obvious on some of the tests, is that the clay soils were swelling. But no precise measure of the phenomenon could be done in the field.

In conclusion, it appeared that the types of soils encountered during the Alpilles experiment were not the most adapted ones to test the methodology in optimal conditions. However, the test of the method was performed up to the end, in order to define the limitations and the conditions of applicability of the method. Other methods such as the use of infiltrometers (Vauclin and Chopart, 1984) and the Wind method (Ref) were used in parallel on some of the sites in order to compare the methodologies.

1. OPTIMISATION PROCEDURE FOR THE DETERMINATION OF THE SCALE PARAMETERS.

4.1.Mathematical formulation of the problem and convergence criterion.

The infiltration tests of the Alpilles experiment were interpreted using the Green and Ampt model and by assuming that the flow was vertical. Haverkamp et al. (1994) proposed a correction accounting for three directional effects (lateral flow), but it was not found necessary to perform such a correction, given the texture of the soil and that no lateral development of the wetted front was observed in the field. The basic equation from which the independent parameters- the scale parameter I and (Ks – Ko) - were retrieved was given by (see section 1.) :

(22)

This equation was rewritten:

(23)

If we denoted by ti, the measured infiltration times, their average, and gIi(I, (Ks – Ko)) the calculated values for infiltration Ii, the function which was optimised to estimate I and (Ks – Ko) was:

(24)

This criterion was chosen after various trials. The advantage of using the efficiency is that for all the tests it has the same order of magnitude [0, 1] contrarily to the sum of squares of time differences between observation and model, where the observed range can vary from less than a minute to several hours. Furthermore, it was observed that this criterion could become negative in some cases. It was then clear that the corresponding infiltration test should be rejected.

The maximum of (24) was determined using a Gauss-Marquart algorithm. The details of the derivations are given in Appendix 2. The robustness of the method was tested using simulated cases with some noise on the values of the time and infiltration heights. It showed that in order to converge towards the true value of the parameters, it was necessary to start the optimisation process close to this solution.

A procedure was therefore developed in order to get this first guess of the parameters. This procedure used only two measured couples of time and infiltration. The first couple was used to calculate a first value of (Ks – Ko), assuming an arbitrary value for I (mathematically, it is possible to determine a lower bound for I and an upper bound for (Ks – Ko)). Then the second couple was used to estimate a new value for I with the value of (Ks – Ko) previously estimated and so on until convergence. The procedure was applied to the three possible combinations of couples corresponding respectively to first and second volumes, first and third volumes and second and third volumes. The solution leading to the minimum value of  was retained as the initial value for the optimisation of (24) using the three available couples. For some of the tests, the first procedure based on two couples failed for the three combinations. The corresponding tests were later on discarded in the analysis.

For some tests, the procedure only converged with the first two couples, the third point appearing then under the non-dimensional curve. These tests were associated to the following physical process: the first two points corresponded to an infiltration within cracks. These cracks were closed due to swelling when the third volume was applied, lowering the velocity of the infiltration. For such tests, only the first two points were considered, and it was assumed that the results in terms of hydraulic conductivity could be associated to the flow within the cracks, or at least to an equivalent porous medium.