List of Notations

The following notations are used in this chapter

(1),

Bp C_q(1), E_k(1), D_r(1), H_w(1), A_mn(1), Z_uv(1), B_p(2), C_q(2), E_k(2), D_r(2), H_w(2), F_i(2), G_j(2),

(2),

Amn Z_uv(2), B_p(3), C_q(3), E_k(3), D_r(3), H_w(3), G_j(3), A_mn(3), Z_uv(3), B_p(4), C_q(4), E_k(4),

(4),

Dr H_w(4), F_i(4), A_mn(4), Z_uv(4) = constants with p1, 2, 3,...., , q1, 2, 3,...., , 1, 2, 3,...., ,

k  r1, 2, 3,...., , w1, 2, 3,...., , i1, 2, 3,...., , j1, 2, 3,...., , 1, 2, 3,...., ,

m  n1, 2, 3,...., , u1, 2, 3,...., and v1, 2, 3,...., ; =

h depth up to the impervious layer as measured from the surface of the soil [L];

H1 depth of water in the left ditch as measured from the surface of the soil [L];

H2  depth of water in the right ditch as measured from the surface of the soil [L];

H3 depth of the top soil layer as measured from the surface of the soil [L];

x1

K  horizontal hydraulic conductivity of the top soil layer as in Fig. 2.1 [LT^-1];

y1

K  vertical hydraulic conductivity of the top soil layer as in Fig. 2.1 [LT^-1];

x2

K  horizontal hydraulic conductivity of the bottom soil layer as in Fig. 2.1 [LT^-1];

y2

K  vertical hydraulic conductivity of the bottom soil layer as in Fig. 2.1 [LT^-1];

 

^{K1a 2 }



Kx1 Ky1



anisotropy ratio of the top soil layer (dimensionless);

 

^{K2a 2 }



Kx1 Ky1



anisotropy ratio of the bottom soil layer (dimensionless);

,

P Q, K, R, W, I, J, M, N, U, V = number of terms to be summed in the infinite series solutions, 1, 2, 3,….

Ni  



1 2 i



 2



hH3



 with i1, 2, 3,...., ;

i1

N 



i1 S



with i₁1, 2,3,...., ;

Nj  



1 2 j



 2



hH3



 with j1, 2, 3,...., ; Nk 



^k ^S



with k 1, 2, 3,...., ;

Nm 



^m ^S



with m1, 2, 3,...., ; Nn  ^_



^{1 2 n}



^{ 2h}_ ^with n1, 2, 3,...., ; Np  



1 2 p



 2H3 with p1, 2, 3,...., ; Nq  



1 2 q



 2H3 with q1, 2, 3,...., ; Nr 



^{r S}



^with r1, 2, 3,...., ;

Nu 



^u ^S



with u1, 2, 3,...., ; Nv ^_



^{1 2 v}



^{ 2h}_ ^with v1, 2, 3,...., ; Nw



^{w S}



^with w1, 2, 3,...., ;

N0  number of divisions of the ponding surface at the top of the soil (Fig. 2.1);

(1) leftside

Q  discharge through the side of the left ditch for Fig. 2.2 problem [L³T^-1];

(2) leftside

Q  discharge through the side of the left ditch for Fig. 2.10 problem [L³T^-1];

(3) leftside

Q  discharge through the side of the left ditch for Fig. 2.18 problem [L³T^-1];

(4) leftside

Q  discharge through the side of the left ditch for Fig. 2.26 problem [L³T^-1];

(1) rightside

Q  discharge through the side of the right ditch for Fig. 2.2 problem [L³T^-1];

(2) rightside

Q  discharge through the side of the right ditch for Fig. 2.10 problem [L³T^-1];

(3) rightside

Q  discharge through the side of the right ditch for Fig. 2.18 problem [L³T^-1];

(4) rightside

Q  discharge through the side of the right ditch for Fig. 2.26 problem [L³T^-1];

(1)

Qtop  discharge through the top surface for the flow problem of Fig. 2.2 [L³T^-1];

(2)

Qtop  discharge through the top surface for the flow problem of Fig. 2.10 [L³T^-1];

(3)

Qtop  discharge through the top surface for the flow problem of Fig. 2.18 [L³T^-1];

(4)

Qtop  discharge through the top surface for the flow problem of Fig. 2.26 [L³T^-1];

(1)

Qtopx  top discharge function for the flow problem of Fig. 2.2;

(2)

Qtopx  top discharge function for the flow problem of Fig. 2.10;

(3)

Qtopx  top discharge function for the flow problem of Fig. 2.18;

(4)

Qtopx  top discharge function for the flow problem of Fig. 2.26;

S spacing between any two adjacent ditches for the flow problem of Fig. 2.1 [L];

Si  horizontal distance of the i^th inner bund from the origin ‘O’ for the flow problem of Fig.

2.1 [L];

s1

S  specific storage of the top soil layer [L^-1];

s2

S  specific storage of the bottom soil layer [L^-1];

t  time variable [T];

1(1)

Vx  horizontal velocity distribution for the top layer of Fig. 2.2 [LT^-1];

1(2)

Vx  horizontal velocity distribution for the top layer of Fig. 2.10 [LT^-1];

1(3)

Vx  horizontal velocity distribution for the top layer of Fig. 2.18 [LT^-1];

1(4)

Vx  horizontal velocity distribution for the top layer of Fig. 2.26 [LT^-1];

2(1)

Vx  horizontal velocity distribution for the bottom layer of Fig. 2.2 [LT^-1];

2(2)

Vx  horizontal velocity distribution for the bottom layer of Fig. 2.10 [LT^-1];

2(3)

Vx  horizontal velocity distribution for the bottom layer of Fig. 2.18 [LT^-1];

2(4)

Vx  horizontal velocity distribution for the bottom layer of Fig. 2.26 [LT^-1];

1(1)

Vy  vertical velocity distribution for the top layer of Fig. 2.2 [LT^-1];

1(2)

Vy  vertical velocity distribution for the top layer of Fig. 2.10 [LT^-1];

1(3)

Vy  vertical velocity distribution for the top layer of Fig. 2.18 [LT^-1];

1(4)

Vy  vertical velocity distribution for the top layer of Fig. 2.26 [LT^-1];

2(1)

Vy  vertical velocity distribution for the bottom layer of Fig. 2.2 [LT^-1];

2(2)

Vy  vertical velocity distribution for the bottom layer of Fig. 2.10 [LT^-1];

2(3)

Vy  vertical velocity distribution for the bottom layer of Fig. 2.18 [LT^-1];

2(4)

Vy  vertical velocity distribution for the bottom layer of Fig. 2.26 [LT^-1];

(1) leftside

Vol  volume of water seeping through the side faces of the left ditch for the flow problem represented by Fig. 2.2 [L³];

(2) leftside

Vol  volume of water seeping through the side faces of the left ditch for the flow problem represented by Fig. 2.10 [L³];

(3) leftside

Vol  volume of water seeping through the side faces of the left ditch for the flow problem represented by Fig. 2.18 [L³];

(4) leftside

Vol  volume of water seeping through the side faces of the left ditch for the flow problem represented by Fig. 2.26 [L³];

(1) rightside

Vol  volume of water seeping through the side faces of the right ditch for the flow problem represented by Fig. 2.2 [L³];

(2) rightside

Vol  volume of water seeping through the side faces of the right ditch for the flow problem represented by Fig. 2.10 [L³];

(3) rightside

Vol  volume of water seeping through the side faces of the right ditch for the flow problem represented by Fig. 2.18 [L³];

(4) rightside

Vol  volume of water seeping through the side faces of the right ditch for the flow problem represented by Fig. 2.26 [L³];

(1)

Voltop  volume of water seeping through the top soil surface of Fig. 2.2 [L³];

(2)

Voltop  volume of water seeping through the top soil surface of Fig. 2.10 [L³];

(3)

Voltop  volume of water seeping through the top soil surface of Fig. 2.18 [L³];

(4)

Voltop  volume of water seeping through the top soil surface of Fig. 2.26 [L³];

x horizontal coordinate [L];

y vertical coordinate [L];

1(1)  hydraulic head distribution for the top soil layer for the flow problem of Fig. 2.2 [L];

1( 2)

  hydraulic head distribution for the top soil layer for the flow problem of Fig. 2.10 [L];

1(3)  hydraulic head distribution for the top soil layer for the flow problem of Fig. 2.18 [L];

1( 4)

  hydraulic head distribution for the top soil layer for the flow problem of Fig. 2.26 [L];

2(1)  hydraulic head distribution for the bottom soil layer for the flow problem of Fig. 2.2 [L];

2(2)  hydraulic head distribution for the bottom soil layer for the flow problem of Fig. 2.10 [L];

2(3)  hydraulic head distribution for the bottom soil layer for the flow problem of Fig. 2.18 [L];

2(4)  hydraulic head distribution for the bottom soil layer for the flow problem of Fig. 2.26 [L];

1 porosity of the top soil layer in Fig. 2.1 (dimensionless);

2  porosity of the bottom soil layer in Fig. 2.1 (dimensionless);

i  ponding depth of the i^th strip at the soil surface of Fig. 2.1 [L];

  width of the ditch banks of Fig. 2.1 [L];

  

1 1

 

1 1



2 2 2

;

mn Nm Kx Ss Nn Ky Ss

  ^  ^

  

2 2

 

2 2



2 2 2

;

uv Nu Kx Ss Nv Ky Ss

  ^  ^

CHAPTER 3

THREE-DIMENSIONAL SEEPAGE OF PONDED WATER INTO DITCH DRAINS IN A THREE-LAYERED SOIL COLUMN UNDERLAIN BY AN IMPERVIOUS SUBSTRATUM

In this chapter, an analytical solution is being suggested for predicting three-dimensional seepage into ditch drains through a soil column comprising of three distinct vertical soil layers and underlain by an impervious barrier. The drains are being fed by a distributed ponding water head introduced at the surface of the soil column. As in the case of the previous two-dimensional ditch drainage problem, the solutions proposed here for the three variants of the three-dimensional ponded ditch drainage problem are also exact and valid for any configuration of the flow parameters inherent in these problems for the steady state;

however, for the transient state, these solutions are strictly valid only when the directional conductivities and specific storage of the layers obey certain rules (as specified in the text) and not otherwise. All the derived solutions are being checked for their correctness by comparing with analytical solutions of others for specific situations; numerical checks on these solutions are also been carried out for a few drainage scenarios by making use of the Processing MODFLOW (Chiang and Kinzelbach 2001) environment.

3.1 Solutions for the Three-Dimensional Continuity Equation of Groundwater flow for