SOME PROPERTIES OF KIRKHAM ’ S METHOD FOR DRAIN SPACING DETERMINATION IN MARSHY-GLEY SOIL

The aim of this work is to present some pecularity of Kirkham’s method applied in drain spacing determination in steady state water flow in eugley soil type. The analysis was based on data obtained by measuring water discharge from drains and water table depth. Measurements was carried out on drainage field with drain spacing of 10 m, 20 m and 30 m, representing drainage treatments I, II and III, respectively. The estimation of drain spacing is moved to lower value in all treatments. The results of analysis show meaningful limitation of method, especially in the treatments with wider drain spacing as well as in the cases of deeper ground water.


Materials and Method
Experimental field was set up in Radmilovac near Belgrade.It occupies the area of 1.5 ha.The type of soil is eugley.The field is divided into three plots.Each plot is drained by subsurface pipe drains.Treatment I, II and III represent plots with drain spacing 10 m, 20 m and 30 m, respectively.The depth of drains is 0.9 m on average, and mean hydraulic conductivity of the experimental field is 0.6 m⋅day -1 .The equivalent drain depths by treatments I, II and III are d 1 =0.45 m d 2 = 1.06 m d 3 =1.51m, respectively.The experiment was carried out during three successive seasons out of vegetation from 1995 -1997.(D j u r o v i ć , 1999).
The measurements of water table depth and drain discharge shown in Table 1 were selected being considered as representative measurements in steady state water flow conditions.The absence of groups of data around different values during the whole study period was found by the previous analysis (D j u r o v i ć , 2000).There is no group of data more frequent than others, therefore the assumptions on moving of statistics (mean value, median or mode) during successive seasons out of vegetation is not sustainable.Then, the data are considered as a part of one unit of time series.

Index of measurements
Note: K -hydraulic conductivity (m⋅day -1 ); hwater table depth (m), qdrain discharge (m/day), Ldrain spacing (m) where: The values of F k are shown in Table 2 (cit.W e s s e l i n g , 1977).
In his later papers (1960) Kirkham took into account head loss due to vertical flow above the drains.When drains are set up in the interfaces of two soil layers, general equation which includes vertical recharge of water is as follows: where: K a -hydraulic conductivity above the drain (m⋅day -1 ); K b -hydraulic conductivity below the drain (m⋅day

Results and Discussion
The obtained results of drain spacing determination by applying Kirkham's method for treatment I (L = 10 m) are shown in Figs. 1 -4.In the cluster of measurements, shown in fig. 1 two segments single out.The first segment is for k< 28 and the second one for k>28.The first interval is characterised by deeper ground water, whereas the second one for shallow water table.Kirkham's method generates more precisely the estimation in the second interval, admittedly with a bit increased spreading out that will be reflected in higher standard deviation.Histogram is one type of estimation of density probability function which, in fact, contains maximal information of random variable (Fig. 2).Statistics such as mean value, variance or moments of higher order are very often insufficient to identify the nature of some variables.Histogram in itself shows the magnitude of measured data, and it is very simple to show mean value, dispersion around mean value, but much more.Histogram clearly shows the most frequently measured data (so-called mode which exactly means the point of maximum in the function of density probability) but a realm of possible value with the highest concentration or most frequent occurrence (V u k a d i n o v i ć , 1986; FAO, 1976).Mean value of error for the treatment I is -1.518 m, median -2.42 m, mode of Histogram of error (Fig. 2) is utterly irregular with the highest concentration of data at point -5 m.The same trend of error normally distributed is shown in  Histogram of error shows a high negative error, considering that the most concentration of data are in the realm of around -10 m.Error of estimation plotted on normal probability paper (Fig. 8) is not linear due to concavity of data trend.
Very similar trend of negative value of error has been observed in treatment III.The results of this analysis are shown in figs.8 -12.Mean value of error is much higher according to the absolute value of mode (-20.35m).However, it should be pointed out that on histogram of error estimation (Fig. 10) two peaks exist, therefore the mentioned value of mode is under question.

C o n c l u s i o n
Kirkham's method, as one of the representative methods for drain spacing determination in steady state water flow in marshy-clay type of soil on the experimental field with three drainage treatments, estimated better drain spacing in treatment I with narrower drains (L= 10 m).In treatment III, Kirkham's method more precisely generates estimation in the case of shallower ground water.In all drainage treatments estimations of drain spacing are moved toward lower values.Histogram of error explicates high negative value of error as well.Moving toward lower values is illustrated through the negative values of mean, median and mode of error of estimation.

T a b. 1 .
-Measurements in steady state water flow

- 1 )
Drain spacing calculation by Kirkham's method is based on equation 1.The values of F k are shown in Table 2 (cit.Wessiling) and the values of function for the facts corresponding to drainage treatments I, II and III obtained by interpolation are: F k(10) =2.8714, F k(20) =2.4414, F k(30) =2.5364.