WATER RESOURCES BULLETIN VOL. l l ,NO.6 AMERICAN WATER RESOURCES ASSOCIATION DECEMBER 1975

MEASUREMENT OF HYDRAULIC CONDUCTIVITY AND DIFFUSIVITY FOR PREDICTING THE PROCESS OF SOIL

WATER INFILTRATION FROM A TRICKLE SOURCE'

J. Ben-Asher, N. Diner, A . Brandt and D. Goldberg'

ABSTRACT: Two soil water functions, hydraulic conductivity K(6) and diffusivity D(d), were estimated by two methods In one method D(@ was estimated according to Bruce and Klute (1956), and K(d) was calculated from D(6) and the retention curve. In the second, K(@ was obtained by field estimation, with D(6) being calculated from K(6) and the retention curve.

The criterion of reliability for both methods was agreement between experimental and predicted distribution of soil water content. The prediction was made using the functions K(8) and D(6) as soil water parameters in both methods.

Theoretical and experimental agreement was generally good. The f i s t method, however, was found to be best for high soil water content and the second for low soil water content. In addition, the water content at the end of the monotonic increase of function D(6) (estimated according to Bruce and Klute 1956) was found to be about the upper limit of field soil water content. It can be used as a boundary condition in the numerical solution of a cylindrical model of infiltration from a trickle source. It was concluded that the best agreement between theory and experiment can be found when the combined values of D(@ and K(@ from both methods of estimation are used.

INTRODUCTION Since it is important to describe and predict the transient infiltration of water through

soil, particular attention should be given to methods of measuring the functions of the most important parameters of infiltration, namely, hydraulic conductivity and diffusivity. Many methods for measurement of these parameters as a function of water content of a porosic medium such as soil have been reported (Bruce and Klute 1956, Childs and Collis George 1950, Klute, et a1 1956, and Rose, et a1 1965), but the results are not always reliable.3 The problem sometimes lies in the fine measurements required in the wetting front zone. In this area the water content and its functions change markedly over short distances. The phenomenon is particularly noticeable in sandy soils where hydraulic conductivity decreases sharply with a slight decrease in water content.

Paper No. 75051 of the Water Resources Bulletin. Discussions are open until August 1, 1976. *Respectively, Research and Development Authority of the University of the Negev, Beer Sheva;

Israel Institute for Biological Research, Ness Ziona; Weizmann Institute of Science, Rehovoth; Hebrew University, Rehovoth, Israel.

3 N . Diner, 1970. M.Sc. Thesis, Weizmann Institute of Science, Rehovoth, Israel (in Hebrew).

1187

1188 Ben-Asher, Diner, Brandt and Goldberg

To determine the described soil water parameters as functions of water content, two different approaches were tested on the sandy soil of Nahal Sinai. These functions were used for numerical prediction of water movement in the soil in a cylindrical model of infiltration from a trickle source (Brandt, ef a1 1971).

Theoretical considemtions One-dimensional vertical flow of water may be described by the equation

where 8 = soil water content (cm3 ~ m - ~ ) t = time (min) z = soil depth (cm) h = hydraulic head (cm) K = hydraulic conductivity (cm min-' )

By integration of equation ( I ) , with respect t o z, assuming that there is no water movement through the soil surface, that is

av(z,t) ah we have - = K ( - - I )

at a Z

where V, having the units of length, indicates the total volume of water per unit area in the range (0, z) and at time (t), that is:

The diffusivity function may be estimated by horizontal one-dimensional flow of water through a column, having constant air-dried initial water content. It is possible to calculate D(0) (Bruce and Klute 1956) from measurement of water content (0 ) as a function of distance from source (x) at a given time (t) in the interval O n < 8 < Bs where en and 8, are the initial and saturated water contents, respectively. Equation (3) was used for this calculation.

1 dX 2 dB D(e) = - - - I dB

e n (3)

where X = XT

Predicting Process of Soil Water Infiltration 1189

and x is the distance from the source of the water.

D(0) might indeed be calculated numerically if the function A(0) could be measured accurately. However, since extensive errors in measuring 0 affect the values of (dX/d0), D(0) would then be incorrectly estimated. To overcome this difficulty the function D(0) was calculated according to Bressler, et al 1972, with equation 5 (see Materials and Methods) being the numerical solution to equation 3. The numerical solution of the function A(O ), which is based on D(0), was compared repeatedly with the experimental flow profile 0(A) until agreement was reached. (See Bresler, et a1 1971 and figure 1.)

I I I I I I I 1 1 I I I I 1

Figure 1 . Agreement Between the Experimental Data and the Theoretical Prediction Using D(0) according to Bruce and Klute (1 956) and Philip’s (1 955)

Numerical Solution of the Water Flow Equation.

MATERIALS AND METHODS Estimation of Hydraulic Conductivity

Direct field estimation of unsaturated hydraulic conductivity was made at Nahal Sinai. The sand consisted of 97% fine and coarse sand and 3% clay and silt, and had a bulk density of 1.63 f 0.04 g/cm3. A square pond with an area of about 2 mz was filled with about 1 m3 water: two tensiometers placed at depths of 5, 10, 30, 60, and 90 cm respectively, measured the hydraulic head. The soil water content was sampled with a Veihmeyer tube from the area around the tensiometers. Each moisture sample was taken

1190 Ben-Asher, Diner, Brandt and Goldberg

twice at the end of infiltration and again after 2, 5, 10, 24, and 48 hr., respectively. Tensiometers were read ever 1 or 2 hr. during the first 24 hr. after the end of infiltration.

According to equation (2), K(0) may be evaluted as

where the values on the right were determined in field experiment by measuring the soil water content and hydraulic head simultaneously.

The changes in water content (equation 1) between samplings were determined graphically from the area between the lines representing the water content at given times.

The gradient in the hydraulic head was calculated from differences between tensiometer readings taken at the same time in adjacent layers. The calculated hydraulic conductivity was related to the simultaneous average soil water content at depth of sampling.

Estimation of the Diffsivity The experimental soil was packed in columns to the desired density. The diffusivity

function D(0) within the interval On < 0 < O s was determined from equation (3), and solved numerically according to equation (5).

Linear extrapolation gives 0 *o as

The final estimation of the diffusivity function was done in four correction steps according to Bresler, et al(1971).

The hydraulic conductivity was calculated from direct measurement of the diffusivity and the diffusivity was calculated also from field measurements of the hydraulic conductivity. These calculations were made from the definition of the diffusivity function expressed by :

dh D(0) = K(0) -

de

dh - was read from the retention curve described by Bresler, et al. de

Predicting Process of Soil Water Infiltration 1191

Checking the Functions D(0) and K(8) by Field Measurement The functions were checked by comparing the distribution of soil moisture content

during infiltration with the predicted results derived from the cylindrical flow model of infiltration from a trickle source (Brandt, et a1 1971).

The soil was wetted by trickling without evaporation. The commercial tricklers discharged 2000 cm3/hr. During the 300-min irrigation the soil moisture was sampled by Veihmeyer tube every 10 cm to a depth of 60 cm below the trickle source, and horizontally to a distance of 25 cm at a depth of 0-3 cm.

DISCUSSION Determination of Soil Water Parameters

The function D(e) is shown in figure 2, where the curve indicates that the diffusivity function has a characteristic form. The measurements taken during horizontal infiltration showed that emax (the value of 0 for which the function D(e) reaches its maximum value) is about the maximum value of 8 that can be measured in the field. Values of 8 higher than emax were found in the laboratory experiments, and seem to be the result of changes in the soil structure at the interface of soil and water and its affected zone. The characteristic form of the function D(e) which reaches a maximum for emax and then decreases sharply with increasing 8 , apparently describes soil resistance to a moisture content higher than Omax. This may be described by the classic flow equation, where

aK(e ) ae dD - - - div [D(e) grad 01 -- = D(0) div grad t9 t - (grad 8)2 - - az (7) at az ae

Whenever f3 reaches values higher than Omax, and grad 0 f 0, the sharp decrease of D dD dtJ as a function of 0 makes the term - (grad 0)’ strongly negative and concomitantly -

also becomes negative. dB dt Thus the right side of the equation gives a sharp decrease of moisture content t o a

stable value 0 < Omax. For this reason, and for practical purposes a moisture content higher than emax is

unimportant. Diffusivity is used only within the range of monotonic increase and Omax may be used at the upper limit of the soil moisture content. Therefore Omax could be one of the boundary conditions for the numerical solution of the flow equation.

Figure 1 indicates that a high density of points describes the experimental function O(h) in the range of relatively high water content (.23 < e <ed. On the other hand, there is a small number of measured points at low water content (0, < 6 < .23) near the wetting front. The water movement in a horizontal column enables the function D(0) to be calculated more accurately at high water content where the points 8*i are dense, rather than at low water content where the points 8*i are dispersed. When using the direct measurement of the function K(0) (figure 3) to calculate the function D(e), the differences between diffusivities found with the two methods increase as the soil water content decreases. The values of the two diffusivity functions are similar only in the range of.21 < e <.27.

1192 Ben-Asher, Diner, Brandt and Goldberg

I c .- E

E N

0

Figure 2. Diffusivity as a Function of Soil Water Content. (Solid Line Estimated according to Bruce and Klute

and Dashed Line Calculated from Field Estimation of K[B]).

The unsaturated hydraulic conductivity K(0) calculated from the functions D(0) and K(0) and measured directly in the field had the same characteristics as the diffusivity. There were differences of some orders of magnitude in the range of low water content between the values of the same function.

Direct estimation of K(0) is preferable for two reasons: As opposed to D(Q, it has physical significance, and it is difficult accurately to' derive the term dh/d0 required for the calculation of K(0). As stated previously, it is also difficult accurately to derive the term dh/d0 required for the calculation of D(0). Hence the value of K(0) estimated from equation ( 6 ) is influenced by two possible errors; that of D(0) and that of h(0) estimated after smoothing of the experimental function.

Predicting Rocess of Soil Water Infiltration

~

I I

I I a I I I

1193

Figure 3. Hydraulic Conductivity as a Function of Soil Water Content. (Solid Line Calculated from D[6] and

Dashed Line Estimated from Direct Field Estimation of K [ 6 ] ) .

Application of the Parameters for Predicting Water Infiltration into Soil Since there is no standard method of measuring the functions D(0) and K(0), it is

impossible to find an objective criterion for measuring the accuracy of the two methods of estimating these functions.

A good criterion of the reliability of the functions is to use them for numerical solutions of problems of soil water flow and to compare the theoretical with the experimental results.

Figure 1 shows the final step of evaluation of D(6). It is a numerical solution of the horizontal column problem carried out according to Philip (1955). Using this procedure,

1194 Ben-Asher, Diner, Brandt and Coldberg

the function D(8) was improved until the numerical solution reached the experimental results (Bresler, et a1 1971). This procedure gives no indication of the reliability of K(8) nor of the reliability of the two functions when using them to predict results of independent horizontal flow, vertical infdtration or two-dimensional infiltration as predicted by the cylindrical model of infdtration (Brandt et a1 1971).

The relationship between the experimental and the predicted results is given in figures 4 and 5. The experimental results were found during infiltration from a trickle source, and predicted results were obtained from the cylindrical model when using the functions K(8) and D(6) according to equations (4) and (5). Figure 4 indicates that vertical water content distribution near the trickle source is similar to one-dimensional vertical infiltration. The predicted dashed curve shows the three typical zones: saturated zone, transition zone, and wetting zone. This last zone is clearly shown to be limited by the wetting front. Quantitatively, there was satisfactory agreement between the two predicted curves and the measured water distribution in the soil. The solid curve decreases gradually with soil depth, within the cubic components of the histogram. Hence the calculated solid curve of moisture content frequently agreed with the experimental points on the histogram.

Figure 5 indicates that the calculated horizontal water distribution on the approximate soil surface was not significantly influenced by the changes of the soil-water parameters D(8) and K(8).

8 (cm3. ~ r n - ~ )

0

20 O F f - O F W a 20

--- - d

I t=30 t = 6 0 - 40 E

= 60 V v

0

n

-----' t:120 ,! t=300 40 /

I I

60 I

Figure 4. Vertical Distribution of Soil Water Content Near the Trickle Source (t-time from the Beginning of Infiltration [ min]). Dashed and Solid Curves are Soil Water Content Calculated from the 'Cylindrical Flow Model (Brandt, et a2 1971) using K(8) and D(8) according to Method of Bruce and Klute (1956). and from the Field Measurement, Respectively. Histogram is Experimental Soil Water Content.

Predicting Process of Soil Water Infiltration

0.3 - - \ t = 6 0

3 -. - 0.2 -

- 0.1 -

0

1195

h

m

E u m ’ E

0.3 r 1

0.2

0.1

0

CD 0.3 r 1

0 10 20 30 The distance from a trickle source ( c m )

Figure 5. Horizontal Distribution of Soil Water Content as a Function of Distance from Trickle Source on Soil Surface (t-time from Beginning of Infiltration [min]). Solid and Dashed Curves are Soil Water Content Distribution Calculated from the Cylindrical Flow Model (Brandt, et a1 1971) using K(8) and D(8) according to the Method of Bruce and Klute (1956), and from Field Measurement, Respectively. Histogram Describes the Experimental Soil Water Content.

The model described by Brandt, et a1 1971, uses the function K(8) for the vertical calculation as a term describing the effect of gravitation on the flow of water, hence it does not significantly affect the results of horizontal water distribution. This is probably due to the fact that the changes of D(8) at low water content (found by extrapolation of the curve shown in figure 2) are slighter than the changes of K(8) at the same range.

From figures 4 and 5 it also appears that the upper limit of the experimental water content was approximately equal to Omax.

CONCLUSIONS 1. Measurement of the function D(8) according to Bruce and Klute (1956), is

well-adapted to the cylindrical model of trickle source infiltration for describing the

1196 Ben-Asher, Diner, Brandt and Goldberg

distribution of soil water content at relatively high level. It fails, however, to describe the distribution at low soil water content.

2. An alternative method is suggested for obtaining the values of the functions at low range. This method is based on direct measurement of hydraulic conductivity in the field.

3. Omax may be regarded as the upper limit of soil moisture content in the field and may be used as a boundary condition for the numerical solution of the water flow equation.

LITERATURE CITED

Brandt, A., E. Bresler, N. Diner, J. Ben-Asher, J. Heller, and D. Coldberg, 1971. Infiltration from a Trickle Source: I Mathematical Models. Soil Sci. SOC. Amer. Proc. 35, pp. 675482.

Bresler, E., J. Heller, N. Diner, J. Ben-Asher, A. Brandt, and D. Coldberg, 1971. Infiltration from a Trickle Source: 11. Experimental Data and Theoretical Prediction. Soil Sci. SOC. Amer. Proc 35,

Bruce, R. R. and A. Klute, 1956. The Measurement of Soil Moisture Diffusivity. Soil Sci. SOC. Amer.

Chiids, E. C. and N. Collis George, 1950. The Permeability of Porous Materials. Proc. Roy. Soc. A201,

Philip, J. R.. 1955. Numerical Solution of Equations of the Diffusion Type with Diffusivity

Rose, C. W., W. R. Stern, and J. E. Drummond, 1965. Determination of Hydraulic Conductivity as a

pp. 683-689.

Roc. 20, pp. 458462.

pp. 392-405.

Concentrated Dependent. Trans Faraday SOC. 51, pp. 885-892.

Function of Depth and Water Content for Soil in situ. Aust. I. Soil Res. 3, pp. 1-9.