CHAPTER 1 INTRODUCTION
3.7 List of Notations
The following notations are used in this chapter
) =
3 ( ) 3 ( ) 2 ( ) 3 ( ) 2 ( ) 1 ( ) 3 ( ) 2 ( ) 1
( , mn , mn , p , p , p , q , q , r
mn A A C C C B B D
A constants with m =1,2,
..., ,
3 n=1,2,3,..., p=1,2,3 ,..., q=1,2,3,..., r=1,2,3,...
h= depth of the ditch drains as measured from surface of the soil for the flow problems as shown in Figs. 3.1, 3.2 and 3.3, respectively [L];
1 =
H depth of water in the left trench as measured from the surface of the soil for the flow problems as shown in Figs. 3.1, 3.2 and 3.3, respectively [L];
3 =
H depth of water in the right trench as measured from the surface of the soil for the flow problems as shown in Figs. 3.1, 3.2 and 3.3, respectively [L];
=
= KxKy
K equivalent hydraulic of soil [LT-1];
=
= x y
a K K
K ) /
( 2 anisotropy ratio of soil (dimensionless);
x =
K horizontal hydraulic conductivity of soil [LT-1];
y =
K vertical hydraulic conductivity of soil [LT-1];
2 / 1
1) ( / )
(K = Ss Ky [T1/2 L-1];
M, N, P, Q, R = integers which can take values values 1, 2, 3, …;
0 =
N number of divisions of the ponding surface at the top of the soil;
shalf =
Q discharge per unit length through the face OA of Fig. 3.1 [L3T-1L-1];
side =
Q total discharge per unit length from both the sides of a ditch for the flow problem of Fig. 3.1 [L3T-1L-1];
) =
3 ( ) 2 ( , l
l Q
Q discharge per unit length of the ditches through the face OA of Figs. 3.2 and 3.3, respectively [L3T-1L-1];
) =
3 ( ) 2 ( ) 1
( , top , top
top Q Q
Q discharge per unit length of the ditches through the surface GE of Figs. 3.1, 3.2 and 3.3, respectively [L3T-1L-1];
) =
3 (
QtopX top discharge function defined on the surface GE of Fig. 3.3, [L3T-1L-1];
f =
QtopX(3) top discharge function expressed as a percentage of Qtop(3), dimensionless;
) =
(st
Qside steady state total discharge per unit length from both the sides of a ditch for the flow problem of Fig. 3.1, [L3T-1L-1];
) =
( ) 3 ( ) (
) 2 (st , lst
l Q
Q steady state discharge per unit length of the ditches through face OA of Figs.
3.1, 3.2 and 3.3, respectively [L3T-1L-1];
) =
( ) 3 ( ) (
) 2 ( , rst
st
r Q
Q steady state discharge per unit length of the ditches through face CD of Figs.
3.1, 3.2 and 3.3, respectively [L3T-1L-1];
stot =
V volume of water seeping through the surface GE of Fig. 3.1 per unit length of the ditches in time t, [L3L-1];
) =
3 ( ) 2 ( ) 1
( , top , top
top V V
V volume of water seeping per unit length of the ditches through the surface GE of Figs. 3.1, 3.2 and 3.3, respectively [L3L-1];
) =
( ) 3 ( )
( ) 2 ( ) (
) 1
( , topst , topst
st
top Q Q
Q steady state discharge per unit length of the ditches through the surface GE of Figs. 3.1, 3.2 and 3.3, respectively [L3T-1L-1];
) =
1 (
Sh semi-spacing between the adjacent ditches in the real plane of Fig. 3.1 [L];
h =
S semi-spacing between the adjacent ditches in the computational plane of Fig. 3.1 [L];
) =
3 ( ) 2 ( ), 1
( a , a
a S S
S distance between the adjacent drains in the real plane for the flow problems as shown in Figs. 3.1, 3.2 and 3.3, respectively [L];
vai =
S distance of the ith (1≤i≤N0 −1) inner bund from the origin O in the real plane for the flow problem of Fig. 3.3 [ L ];
=
= vai a
vi S K
S / distance of the ith (1≤i≤N0 −1) inner bund from the origin O in the computational plane for the flow problem of Fig. 3.3 [ L ];
s =
S specific storage of soil [L-1];
=
t time variable for the flow problems of Figs. 3.1, 3.2 and 3.3, respectively [T];
=
x horizontal coordinate as measured from the origin O for the flow problems of Figs. 3.1, 3.2 and 3.3, respectively, in the real plane [L];
=
X horizontal coordinate as measured from the origin O for the flow problems of Figs. 3.1, 3.2 and 3.3, respectively, in the computational plane [L];
=
y vertical coordinate as measured from the origin O for the flow problems of Figs. 3.1, 3.2 and 3.3, respectively, in the real plane [L];
0 =
δ depth of ponding at the surface of the soil for the flow problems of Figs. 3.1 and 3.2, respectively [L];
i =
δ ponding depth at the ith segment on the surface of the soil for the flow problem of Fig.
3.3 [L];
a =
ε width of the ditch banks in the real planefor the flow problems of Figs. 3.1, 3.2 and 3.3, respectively [L];
=
=εa /Ka
ε width of the ditch banks in the computational planefor the flow problems of Figs. 3.1, 3.2 and 3.3, respectively [L];
) =
3 ( ) 2 ( ) 1
( ,φ , φ
φ hydraulic head distribution corresponding the flow domains of Figs. 3.1, 3.2 and 3.3, respectively [L];
) =
( ) 3 ( ) (
) 2 ( ) (
) 1
(st ,φ st , φ st
φ steady state hydraulic head distribution corresponding the flow domains of Figs. 3.1, 3.2 and 3.3, respectively [L];
) =
3 ( ) 2 ( ), 1
( ψ , ψ
ψ steady state stream function corresponding the flow domains of Figs. 3.1, 3.2 and 3.3, respectively [L2T-1];
n =
n n
) 3 ( ) 2 , ( ) 1
( ψ , ψ
ψ steady state stream normalized function corresponding the flow domains of Figs. 3.1, 3.2 and 3.3, respectively [dimensionless]
CHAPTER 4
SUMMARY AND CONCLUSIONS
Subsurface drainage is provided in many humid areas of the world for reclaiming waterlogged soils and in restoring flooded areas in a desired frame of time. In the irrigated fields in semi-arid and arid areas, drainage is also extensively provided to maintain a proper salt balance in the root zones of plants. Drainage also helps in establishing a suitable air- water balance in soils. One of the methods of cleaning a salt affected soil is to subject the soil to a ponding field at the surface of the soil with the help of embankments so that water is forced though it and in the process wash away a part of the salt present in the soil profile, the salt rich water is then being drained by a series of ditch drains installed for the purpose.
Thus, in order to have a better design of a subsurface ditch drainage network for cleaning a salt affected soil, it is essential that the underlying subsurface hydraulics of flow to the drains be thoroughly studied. Further, since subsurface drainage depends greatly on the direction dependent water transmitting capacities of a soil, it is crucial that these properties of soil be accurately estimated in the field. In this study, analytical models for predicting flow into fully and partially penetrating auger holes underlain by an impervious layer have been obtained for an unconfined aquifer of finite horizontal and vertical extents. These solutions have been developed with the intention that the horizontal and vertical saturated conductivities of a phreatic aquifer as well as the horizontal influence of an auger hole test be accurately estimated utilizing experimental data obtained from the test. Further, generalized analytical solutions of the transient ditch drainage problem have also been obtained both for the cases when the flow field over the surface of a ponded field is being subjected to a uniform as well as a variable depth of ponding. The basic objective of obtaining a general solution to the transient ditch drainage problem is to have a mathematical tool which can be utilized to provide better designs of subsurface ditch networks for cleaning salt affected soils as well as for reclaiming waterlogged areas. We now give below, in brief, the rundowns of each of our investigations.
4.1 Hydraulics of an Auger Hole in an Unconfined Aquifer of Finite Horizontal