Analysis of three-dimensional transient seepage into ditch drains from a ponded field

Abstract

An analytical solution in the form of infinite series is developed for predicting time-dependent three-dimensional seepage into ditch drains from a flat, homogeneous and anisotropic ponded field of finite size, the field being assumed to be surrounded on all its vertical faces by ditch drains with unequal water level heights in them. It is also assumed that the field is being underlain by a horizontal impervious barrier at a finite distance from the surface of the soil and that all the ditches are being dug all the way up to this barrier. The solution can account for a variable ponding distribution at the surface of the field. The correctness of the proposed solution for a few simplified situations is tested by comparing predictions obtained from it with the corresponding values attained from the analytical and experimental works of others. Further, a numerical check on it is also performed using the Processing MODFLOW environment. It is noticed that considerable improvement on the uniformity of the distribution of the flow lines in a three-dimensional ponded drainage space can be achieved by suitably altering the ponding distribution at the surface of the soil. As the developed three-dimensional ditch drainage model is pretty general in nature and includes most of the common variables of a ditch drainage system, it is hoped that the drainage designs based on it for reclaiming salt-affected and water-logged soils would prove to be more efficient and cost-effective as compared with designs based on solutions developed by making use of more restrictive assumptions. Also, as the developed model can handle three-dimensional flow situations, it is expected to provide reliable and realistic drainage solutions to real field situations than models being developed utilizing the two-dimensional flow assumption. This is because the existing two-dimensional solutions to the problem are actually valid not for a field of finite size but for an infinite one only.

Appendices

List of notations

$A_{{m_{1} ,n_{1} }} ,B_{{m_{2} ,n_{2} }} ,C_{{m_{1} ,n_{1} }} ,D_{{m_{4} ,n_{4} }} ,E_{pqr} ,F_{{m_{1} ,n_{1} }} :$ :

${\text{constants}}\;{\text{with}}\;m_{1} = 1,2,3, \ldots , n_{1} = 1,2,3, \ldots ,\quad m_{2} = 1,2,3, \ldots ,\quad n_{2} = 1,2,3, \ldots , m_{3} = 1,2,3, \ldots ,\quad n_{3} = 1,2,3, \ldots ,\quad m_{4} = 1,2,3, \ldots , n_{4} = 1,2,3, \ldots ,\;m_{5} = 1,2,3, \ldots ,\;n_{5} = 1,2,3, \ldots , p = 1,2,3, \ldots ,\;q = 1,2,3, \ldots ,\;r = 1,2,3, \ldots$

h :

depth of the soil column, L

$H_{1}$ :

height of water in the Northern ditch as measured from the surface of the soil, L

$H_{2}$ :

height of water in the Southern ditch as measured from the surface of the soil, L

$H_{3}$ :

height of water in the Eastern ditch as measured from the surface of the soil, L

$H_{4}$ :

height of water in the Western ditch as measured from the surface of the soil, L

K :

=(K _x K _y K _z )^1/3 equivalent hydraulic conductivity of soil, LT⁻¹

$\left( {K_{x}^{a} } \right)^{2}$ :

=K _x /K _z , anisotropy ratio of soil in the x-direction, dimensionless

$\left( {K_{y}^{a} } \right)^{2}$ :

=K _y /K _z , anisotropy ratio of soil in the y-direction, dimensionless

K _x :

hydraulic conductivity of soil in the x-direction, LT⁻¹

K _y :

hydraulic conductivity of soil in the y-direction, LT⁻¹

K _z :

hydraulic conductivity of soil in the z-direction, LT⁻¹

(K₁):

=$\sqrt {S_{s} /K_{z} ,}$ L⁻¹ T^1/2

M:

distance in metres, L

M _1, N _1, M _2, N _2, M _3, N _3, M _4, N _4, M _5, N _5, P, Q and R: :

number of terms to be summed in the series solution, 1,2,3,…

$N_{0}$ :

number of divisions of the ponding surface at the top of the soil

$Q_{N} ,Q_{S} ,Q_{E} ,Q_{W}:$ :

discharge through the Northern, Southern, Eastern, Western faces of the soil column of figure 1, L³T⁻¹

$Q_{\text {top}}^{f}$ :

top discharge function defined for the surface of the soil of figure 1, L³T⁻¹

$Q_{\text {top}}$ :

discharge through the top surface of the soil of figure 1, L³T⁻¹

$Q_{\text {top}}^{nf}$ :

top discharge function expressed as a percentage of $Q_{\text {top}}$, dimensionless

L :

distance between the adjacent drains in the x-direction in the real space of figure 1, L

L _X :

distance between the adjacent drains in the X-direction in the computational space of figure 1, L

B :

distance between the adjacent drains in the y-direction in the real space of figure 1, L

B _Y :

distance between the adjacent drains in the Y-direction in the computational space of figure 1, L

$d_{xi}$ :

distance of the ith $(1 \le i \le N_{0} {-}1)$ inner bund from the origin O in the x-direction of figure 1 in the real space, L

$S_{Xi}$ :

=$d_{xi} /K_{x}^{a} ,$ L

$d_{yi}$ :

distance of the ith $(1 \le i \le N_{0} {-}1)$ inner bund from the origin O in the y-direction of figure 1 in the real space, L

$S_{Yi}$ :

$d_{yi} /K_{y}^{a} ,$ L

$S_{s}$ :

specific storage of soil, L⁻¹

$V_{x}$ :

velocity distribution for the flow domain of figure 1 in the x-direction, LT⁻¹

$V_{y}$ :

velocity distribution for the flow domain of figure 1 in the y-direction, LT⁻¹

$V_{z}$ :

velocity distribution for the flow domain of figure 1 in the z-direction, LT⁻¹

t :

time variable for the flow problem of figure 1, T

x :

coordinate as measured from the origin O of figure 1 in the East-West dirction in the real space

X :

$= x/K_{x}^{a} ,$ L

y :

coordinate as measured from the origin O of figure 1 in the North-South dirction in the real space

Y :

$= y/K_{y}^{a} ,$ L

z :

coordinate as measured from the origin O of figure 1 in the downward direction in the real space, L

$\delta_{i}$ :

ponding depth at the ith segment on the surface of the soil, L

$\varepsilon_{x}$ :

width of the ditch banks in the x-direction in the real space of figure 1, L

$\varepsilon_{y}$ :

width of the ditch banks in the y-direction in the real space of figure 1, L

$\phi$ :

hydraulic head distribution for the flow domain of figure 1 (with th Northern boundary as a ditch drainage boundary), L

Appendix 1. Determination of coefficients of the hydraulic head function of Eq. (2)

In this section, the coefficients appearing in Eq. (2) will be determined utilizing the appropriate initial and boundary value conditions mentioned in the definition of the problem. To evaluate $A_{{m_{{_{1} }} n_{{_{1}}}}},$ boundary conditions (IIIa) and (IIIb) can be made use of; application of the same to Eq. (2) at $Y = 0$ gives

$$ \sum\limits_{{m_{1} = 1}}^{{M_{1} }} { \, \sum\limits_{{n_{1} = 1}}^{{N_{1} }} {A_{{m_{{_{1} }} n_{{_{1} }} }} \sin (N_{{m_{{_{1} }} }} X)\sin (N_{{n_{{_{1} }} }} z)} } = - z,\quad 0 < X < L_{X} ,\quad 0 < z < H_{2} , $$

$$ \sum\limits_{{m_{1} = 1}}^{{M_{1} }} { \, \sum\limits_{{n_{1} = 1}}^{{N_{1} }} {A_{{m_{{_{1} }} n_{{_{1} }} }} \sin (N_{{m_{{_{1} }} }} X)\sin (N_{{n_{{_{1} }} }} z)} } = - H_{2} ,\quad 0 < X < L_{X} ,\quad H_{2} \le z < h. $$

Thus, $A_{{m_{{_{1} }} n_{{_{1} }} }}$can be evaluated by running a double Fourier series in the domain covered by $0 < X < L_{X}$ and $0 < z < h;$ this yields an expression for $A_{{m_{{_{1} }} n_{{_{1} }} }}$as

$$ A_{{m_{1} n_{1} }} = - \left( {\frac{2}{{L_{x} }}} \right)\left( {\frac{2}{h}} \right)\left[ {\int\limits_{0}^{{L_{X} }} {\int\limits_{0}^{{H_{2} }} {z\sin (N_{{m_{1} }} X)\sin (N_{{n_{1} }} z)dXdz\left. { + \int\limits_{0}^{{L_{X} }} {\int\limits_{{H_{2} }}^{h} {H_{2} \sin (N_{{m_{1} }} X)\sin (N_{{n_{1} }} z)dXdz} } } \right].} } } \right. $$

(31)

Simplification of the above integrals yields

$$ A_{{m_{{_{1} }} n_{{_{1} }} }} = - \left( {\frac{2}{{L_{x} }}} \right)\left( {\frac{2}{h}} \right)\left[ {\frac{{1 - \cos (N_{{m_{1} }} L_{X} )}}{{N_{{m_{1} }} }}} \right]\left[ {\frac{{\sin (N_{{n_{1} }} H_{2} )}}{{(N_{{n_{1} }} )^{2} }}} \right]. $$

(32)

Similarly, an application of boundary conditions (IIa) and (IIb) to Eq. (2) gives $B_{{m_{{_{2} }} n_{{_{2} }} }}$ as

$$ B_{{m_{{_{2} }} n_{{_{2} }} }} = - \left( {\frac{2}{{L_{x} }}} \right)\left( {\frac{2}{h}} \right)\left[ {\frac{{1 - \cos (N_{{m_{2} }} L_{X} )}}{{N_{{m_{2} }} }}} \right]\left[ {\frac{{\sin (N_{{n_{2} }} H_{1} )}}{{(N_{{n_{2} }} )^{2} }}} \right]. $$

(33)

Likewise, boundary conditions (Va) and (Vb) and (IVa) and (IVb) can be utilized to evaluate the constants $C_{{m_{3} n_{3} }}$ and $D_{{m_{4} n_{4} }}$ of Eq. (2); the relevant expressions for the same can be expressed as

$$ C_{{m_{{_{3} }} n_{{_{3} }} }} = - \left( {\frac{2}{{B_{Y} }}} \right)\left( {\frac{2}{h}} \right)\left[ {\frac{{1 - \cos (N_{{m_{3} }} B_{Y} )}}{{N_{{m_{3} }} }}} \right]\left[ {\frac{{\sin (N_{{n_{3} }} H_{4} )}}{{(N_{{n_{3} }} )^{2} }}} \right] $$

(34)

and

$$ D_{{m_{{_{4} }} n_{{_{4} }} }} = - \left( {\frac{2}{{B_{Y} }}} \right)\left( {\frac{2}{h}} \right)\left[ {\frac{{1 - \cos (N_{{m_{4} }} B_{Y} )}}{{N_{{m_{4} }} }}} \right]\left[ {\frac{{\sin (N_{{n_{4} }} H_{3} )}}{{(N_{{n_{4} }} )^{2} }}} \right]. $$

(35)

Next, to work out the constants $F_{{m_{5} n_{5} }}$of Eq. (2), boundary conditions (VIIa) to (VIIj) can be made use of; applying the same to Eq. (2), the following set of equations can be realized:

$$ \sum\limits_{{m_{5} = 1}}^{{M_{5} }} { \, \sum\limits_{{n_{5} = 1}}^{{N_{5} }} {F_{{m_{5} n_{5} }} \sin (N_{{m_{5} }} X)\sin (N_{{n_{5} }} Y)} } = \delta_{1} ,\quad 0 < X < L_{X} ,\quad 0 < Y < S_{Y1} ,\quad z = 0, $$

$$ \sum\limits_{{m_{5} = 1}}^{{M_{5} }} { \, \sum\limits_{{n_{5} = 1}}^{{N_{5} }} {F_{{m_{5} n_{5} }} \sin (N_{{m_{5} }} X)\sin (N_{{n_{5} }} Y)} } = \delta_{1} ,\quad 0 < X < L_{X} ,\quad S_{{Y(2N_{0} - 2)}} < Y < B_{Y} ,\quad z = 0, $$

$$ \sum\limits_{{m_{5} = 1}}^{{M_{5} }} { \, \sum\limits_{{n_{5} = 1}}^{{N_{5} }} {F_{{m_{5} n_{5} }} \sin (N_{{m_{5} }} X)\sin (N_{{n_{5} }} Y)} } = \delta_{1} ,\quad 0 < X < S_{X1} ,\quad S_{Y1} < Y < S_{{Y(2N_{0} - 2)}} ,\quad z = 0, $$

$$ \sum\limits_{{m_{5} = 1}}^{{M_{5} }} { \, \sum\limits_{{n_{5} = 1}}^{{N_{5} }} {F_{{m_{5} n_{5} }} \sin (N_{{m_{5} }} X)\sin (N_{{n_{5} }} Y)} } = \delta_{1} ,\quad S_{{X(2N_{0} - 2)}} < X < L_{X} ,\quad S_{Y1} < Y < S_{{Y(2N_{0} - 2)}} ,\quad z = 0, $$

$$ \sum\limits_{{m_{5} = 1}}^{{M_{5} }} { \, \sum\limits_{{n_{5} = 1}}^{{N_{5} }} {F_{{m_{5} n_{5} }} \sin (N_{{m_{5} }} X)\sin (N_{{n_{5} }} Y)} } = \delta_{j} ,\quad S_{X(j - 1)} < X < S_{{X(2N_{0} - j)}} ,\quad S_{Y(j - 1)} < Y < S_{Yj} ,\quad z = 0, $$

$$ \sum\limits_{{m_{5} = 1}}^{{M_{5} }} { \, \sum\limits_{{n_{5} = 1}}^{{N_{5} }} {F_{{m_{5} n_{5} }} \sin (N_{{m_{5} }} X)\sin (N_{{n_{5} }} Y)} } = \delta_{j} ,\quad S_{X(j - 1)} < X < S_{{X(2N_{0} - j)}} ,\quad S_{{Y(2N_{0} - j - 1)}} < Y < S_{{Y(2N_{0} - j)}} ,\quad z = 0, $$

$$ \sum\limits_{{m_{5} = 1}}^{{M_{5} }} { \, \sum\limits_{{n_{5} = 1}}^{{N_{5} }} {F_{{m_{5} n_{5} }} \sin (N_{{m_{5} }} X)\sin (N_{{n_{5} }} Y)} } = \delta_{j} ,\quad S_{X(j - 1)} < X < S_{Xj} ,\quad S_{Yj} < Y < S_{{Y(2N_{0} - j - 1)}} ,\quad z = 0, $$

$$ \sum\limits_{{m_{5} = 1}}^{{M_{5} }} { \, \sum\limits_{{n_{5} = 1}}^{{N_{5} }} {F_{{m_{5} n_{5} }} \sin (N_{{m_{5} }} X)\sin (N_{{n_{5} }} Y)} } = \delta_{j} ,\quad S_{{X(2N_{0} - j - 1)}} < X < S_{{X(2N_{0} - j)}} ,\quad S_{Yj} < Y < S_{{Y(2N_{0} - j - 1)}} ,\quad z = 0, $$

$$ (2 \le j \le N_{0} - 1) $$

$$ \sum\limits_{{m_{5} = 1}}^{{M_{5} }} { \, \sum\limits_{{n_{5} = 1}}^{{N_{5} }} {F_{{m_{5} n_{5} }} \sin (N_{{m_{5} }} X)\sin (N_{{n_{5} }} Y)} } = \delta_{{N_{0} }} ,\quad S_{{X(N_{0} - 1)}} < X < S_{{XN_{0} }} ,\quad S_{{Y(N_{0} - 1)}} < Y < S_{{YN_{0} }} ,\quad z = 0, $$

where

$$ S_{Xi} = \left( {\sqrt {\frac{{K_{z} }}{{K_{x} }}} } \right)d_{xi} $$

(36)

and

$$ S_{Yi} = \left( {\sqrt {\frac{{K_{z} }}{{K_{y} }}} } \right)d_{yi} ,\quad [i = 1,2,3, \ldots ,(2N_{0} - 2)] $$

(37)

Thus, $F_{{m_{5} n_{5} }}$ can be evaluated by running a double Fourier series in the space defined by the intervals $0 < X < L_{X}$ and $0 < Y < B_{Y} ;$ this yields an equation for evaluating $F_{{m_{5} n_{5} }}$ as

$$ \begin{aligned} F_{{m_{5} n_{5} }} = \left( {\frac{2}{{B_{Y} }}} \right)\left( {\frac{2}{{L_{X} }}} \right)\left\{ {\delta_{1} \left[ {\int\limits_{0}^{{L_{X} }} {\int\limits_{0}^{{S_{Y1} }} {\sin (N_{{m_{5} }} X)\sin (N_{{n_{5} }} Y)dXdY} } } \right.} \right. \hfill \\ + \int\limits_{0}^{{L_{X} }} {\int\limits_{{S_{{Y(2N_{0} - 2)}} }}^{{B_{Y} }} {\sin (N_{{m_{5} }} X)\sin (N_{{n_{5} }} Y)dXdY} } + \int\limits_{0}^{{S_{X1} }} {\int\limits_{{S_{Y1} }}^{{S_{{Y(2N_{0} - 2)}} }} {\sin (N_{{m_{5} }} X)\sin (N_{{n_{5} }} Y)dXdY} } \hfill \\ \left. { + \int\limits_{{S_{{X(2N_{0} - 2)}} }}^{{L_{X} }} {\int\limits_{{S_{Y1} }}^{{S_{{Y(2N_{0} - 2)}} }} {\sin (N_{{m_{5} }} X)\sin (N_{{n_{5} }} Y)dXdY} } } \right] + \sum\limits_{j = 2}^{{j = N_{0} - 1}} {\delta_{j} } \left[ {\int\limits_{{S_{X(j - 1)} }}^{{S_{{X(2N_{0} - j)}} }} {\int\limits_{{S_{Y(j - 1)} }}^{{S_{Yj} }} {\sin (N_{{m_{5} }} X)\sin (N_{{n_{5} }} Y)dXdY} } } \right. \hfill \\ + \int\limits_{{S_{X(j - 1)} }}^{{S_{{X(2N_{0} - j)}} }} {\int\limits_{{S_{{Y(2N_{0} - j - 1)}} }}^{{S_{{Y(2N_{0} - j)}} }} {\sin (N_{{m_{5} }} X)\sin (N_{{n_{5} }} Y)dXdY} } + \int\limits_{{S_{X(j - 1)} }}^{{S_{Xj} }} {\int\limits_{{S_{Yj} }}^{{S_{{Y(2N_{0} - j - 1)}} }} {\sin (N_{{m_{5} }} X)\sin (N_{{n_{5} }} Y)dXdY} } \hfill \\ \left. { + \int\limits_{{S_{{X(2N_{0} - j - 1)}} }}^{{S_{{X(2N_{0} - j)}} }} {\int\limits_{{S_{Yj} }}^{{S_{{Y(2N_{0} - j - 1)}} }} {\sin (N_{{m_{5} }} X)\sin (N_{{n_{5} }} Y)dXdY} } } \right]\left. { + \delta_{{N_{0} }} \int\limits_{{S_{{X(N_{0} - 1)}} }}^{{S_{{XN_{0} }} }} {\int\limits_{{S_{{Y(N_{0} - 1)}} }}^{{S_{{YN_{0} }} }} {\sin (N_{{m_{5} }} X)\sin (N_{{n_{5} }} Y)dXdY} } } \right\}. \hfill \\ \end{aligned} $$

(38)

Simplification of the above integrals gives an expression for $F_{{m_{5} n_{5} }}$ as

$$ F_{{m_{5} n_{5} }} = \left( {\frac{2}{{B_{Y} }}} \right)\left( {\frac{2}{{L_{X} }}} \right)\left\{ {\delta_{1} \left[ {\left( {\frac{{1 - \cos (N_{{m_{5} }} L_{X} )}}{{N_{{m_{5} }} }}} \right)\left[ {\frac{{1 - \cos (N_{{n_{5} }} S_{Y1} )}}{{N_{{n_{5} }} }} + \frac{{\cos (N_{{n_{5} }} S_{{Y(2N_{0} - 2)}} ) - \cos (N_{{n_{5} }} B_{Y} )}}{{N_{{n_{5} }} }}} \right]} \right.} \right. + \left( {\frac{{\cos (N_{{n_{5} }} S_{Y1} ) - \cos (N_{{n_{5} }} S_{{Y(2N_{0} - 2)}} )}}{{N_{{n_{5} }} }}} \right)\left. {\left[ {\frac{{1 - \cos (N_{{m_{5} }} S_{X1} )}}{{N_{{m_{5} }} }} + \frac{{\cos (N_{{m_{5} }} S_{{X(2N_{0} - 2)}} ) - \cos (N_{{m_{5} }} L_{X} )}}{{N_{{m_{5} }} }}} \right]} \right] + \sum\limits_{j = 2}^{{j = N_{0} - 1}} {\delta_{j} \left[ {\left( {\frac{{\cos (N_{{m_{5} }} S_{X(j - 1)} ) - \cos (N_{{m_{5} }} S_{{X(2N_{0} - j)}} )}}{{N_{{m_{5} }} }}} \right)} \right]} \times \left[ {\frac{{\cos (N_{{n_{5} }} S_{Y(j - 1)} ) - \cos (N_{{n_{5} }} S_{Yj} )}}{{N_{{n_{5} }} }} + \frac{{\cos (N_{{n_{5} }} S_{{Y(2N_{0} - j - 1)}} ) - \cos (N_{{n_{5} }} S_{{Y(2N_{0} - j)}} )}}{{N_{{n_{5} }} }}} \right] + \left( {\frac{{\cos (N_{{n_{5} }} S_{Yj} ) - \cos (N_{{n_{5} }} S_{{Y(2N_{0} - j - 1)}} )}}{{N_{{n_{5} }} }}} \right)\left. { \times \left[ {\frac{{\cos (N_{{m_{5} }} S_{X(j - 1)} ) - \cos (N_{{m_{5} }} S_{Xj} )}}{{N_{{m_{5} }} }} + \frac{{\cos (N_{{m_{5} }} S_{{X(2N_{0} - j - 1)}} ) - \cos (N_{{m_{5} }} S_{{X{{(2N_{0} - j)}} }} )}}{{N_{{m_{5} }} }}} \right]} \right] + \delta_{{N_{0} }} \left. {\left[ {\frac{{\cos (N_{{m_{5} }} S_{{X(N_{0} - 1)}} ) - \cos (N_{{m_{5} }} S_{{XN_{0} }} )}}{{N_{{m_{5} }} }} \times \frac{{\cos (N_{{n_{5} }} S_{{Y(N_{0} - 1)}} ) - \cos (N_{{n_{5} }} S_{{YN_{0} }} )}}{{N_{{n_{5} }} }}} \right]} \right\}. $$

(39)

There still remain the constants $E_{pqr}$ to be determined. Towards this end, the initial condItion (I) can be applied to Eq. (2); the pertinent expression for evaluating these constants can then be expressed as

$$ \sum\limits_{p = 1}^{P} {\sum\limits_{q = 1}^{Q} {\sum\limits_{r = 1}^{R} {E_{pqr} \sin (N_{p} X)\sin (N_{q} Y)\sin (N_{r} z)} } } = - \sum\limits_{{m_{1} = 1}}^{{M_{1} }} { \, \sum\limits_{{n_{1} = 1}}^{{N_{1} }} {A_{{m_{1} n_{1} }} \frac{{\sinh \left[ {\sqrt {(N_{{m_{1} }} )^{2} + (N_{{n_{1} }} )^{2} } \, (B_{Y} - Y)} \right]}}{{\sinh \left[ {\sqrt {(N_{{m_{1} }} )^{2} + (N_{{n_{1} }} )^{2} } \, B_{Y} } \right]}}\sin (N_{{m_{1} }} X)\sin (N_{{n_{1} }} z)} } - \sum\limits_{{m_{2} = 1}}^{{M_{2} }} { \, \sum\limits_{{n_{2} = 1}}^{{N_{2} }} {B_{{m_{2} n_{2} }} \frac{{\sinh \left[ {\sqrt {(N_{{m_{2} }} )^{2} + (N_{{n_{2} }} )^{2} } \, Y} \right]}}{{\sinh \left[ {\sqrt {(N_{{m_{2} }} )^{2} + (N_{{n_{2} }} )^{2} } \, B_{Y} } \right]}}\sin (N_{{m_{2} }} X)\sin (N_{{n_{2} }} z)} } - \sum\limits_{{m_{3} = 1}}^{{M_{3} }} { \, \sum\limits_{{n_{3} = 1}}^{{N_{3} }} {C_{{m_{3} n_{3} }} \frac{{\sinh \left[ {\sqrt {(N_{{m_{3} }} )^{2} + (N_{{n_{3} }} )^{2} } \, (L_{X} - X)} \right]}}{{\sinh \left[ {\sqrt {(N_{{m_{3} }} )^{2} + (N_{{n_{3} }} )^{2} } \, L_{X} } \right]}}\sin (N_{{m_{3} }} Y)\sin (N_{{n_{3} }} z)} } - \sum\limits_{{m_{4} = 1}}^{{M_{4} }} { \, \sum\limits_{{n_{4} = 1}}^{{N_{4} }} {D_{{m_{4} n_{4} }} \frac{{\sinh \left[ {\sqrt {(N_{{m_{4} }} )^{2} + (N_{{n_{4} }} )^{2} } \, X} \right]}}{{\sinh \left[ {\sqrt {(N_{{m_{4} }} )^{2} + (N_{{n_{4} }} )^{2} } \, L_{X} } \right]}}\sin (N_{{m_{4} }} Y)\sin (N_{{n_{4} }} z)} } - \sum\limits_{{m_{5} = 1}}^{{M_{5} }} { \, \sum\limits_{{n_{5} = 1}}^{{N_{5} }} {F_{{m_{5} n_{5} }} \frac{{\cosh \left[ {\sqrt {(N_{{m_{5} }} )^{2} + (N_{{n_{5} }} )^{2} } \, (h - z)} \right]}}{{\cosh \left[ {\sqrt {(N_{{m_{5} }} )^{2} + (N_{{n_{5} }} )^{2} } \, h} \right]}}\sin (N_{{m_{5} }} X)\sin (N_{{n_{5} }} Y)} } . $$

(40)

Now, performing a triple Fourier run in the space defined by the intervals $0 < X < L_{X} ,$ $0 < Y < B_{Y}$ and $0 < z < h,$ an expression for the constants $E_{pqr}$ can then be worked out as

$$ E_{pqr} = - \left( {\frac{2}{{L_{X} }}} \right)\left( {\frac{2}{{B_{Y} }}} \right)\left( {\frac{2}{h}} \right)\left\{ {\int\limits_{0}^{{L_{X} }} {\int\limits_{0}^{{B_{Y} }} {\int\limits_{0}^{h} {\sum\limits_{{m_{1} = 1}}^{{M_{1} }} { \, \sum\limits_{{n_{1} = 1}}^{{N_{1} }} {A_{{m_{1} n_{1} }} } } } } } \frac{{\sinh \left[ {\sqrt {(N_{{m_{1} }} )^{2} + (N_{{n_{1} }} )^{2} } \, (B_{Y} - Y)} \right]}}{{\sinh \left[ {\sqrt {(N_{{m_{1} }} )^{2} + (N_{{n_{1} }} )^{2} } \, B_{Y} } \right]}}} \right. \times \sin (N_{{m_{1} }} X)\sin (N_{{n_{1} }} z)\sin (N_{p} X)\sin (N_{q} Y)\sin (N_{r} z)dXdYdz + \int\limits_{0}^{{L_{X} }} {\int\limits_{0}^{{B_{Y} }} {\int\limits_{0}^{h} {\sum\limits_{{m_{2} = 1}}^{{M_{2} }} { \, \sum\limits_{{n_{2} = 1}}^{{N_{2} }} {B_{{m_{2} n_{{_{2} }} }} } } \frac{{\sinh \left[ {\sqrt {(N_{{m_{2} }} )^{2} + (N_{{n_{2} }} )^{2} } \, Y} \right]}}{{\sinh \left[ {\sqrt {(N_{{m_{2} }} )^{2} + (N_{{n_{2} }} )^{2} } \, B_{Y} } \right]}}} } } \times \sin (N_{{m_{2} }} X)\sin (N_{{n_{2} }} z)\sin (N_{p} X)\sin (N_{q} Y)\sin (N_{r} z)dXdYdz + \int\limits_{0}^{{L_{X} }} {\int\limits_{0}^{{B_{Y} }} {\int\limits_{0}^{h} {\sum\limits_{{m_{3} = 1}}^{{M_{3} }} { \, \sum\limits_{{n_{3} = 1}}^{{N_{3} }} {C_{{m_{3} n_{3} }} } } \frac{{\sinh \left[ {\sqrt {(N_{{m_{3} }} )^{2} + (N_{{n_{3} }} )^{2} } \, (L_{X} - X)} \right]}}{{\sinh \left[ {\sqrt {(N_{{m_{3} }} )^{2} + (N_{{n_{3} }} )^{2} } \, L_{X} } \right]}}} } } \times \sin (N_{{m_{3} }} Y)\sin (N_{{n_{3} }} z)\sin (N_{p} X)\sin (N_{q} Y)\sin (N_{r} z)dXdYdz + \int\limits_{0}^{{L_{X} }} {\int\limits_{0}^{{B_{Y} }} {\int\limits_{0}^{h} {\sum\limits_{{m_{4} = 1}}^{{M_{4} }} { \, \sum\limits_{{n_{4} = 1}}^{{N_{4} }} {D_{{m_{4} n_{4} }} } } \frac{{\sinh \left[ {\sqrt {(N_{{m_{4} }} )^{2} + (N_{{n_{4} }} )^{2} } \, X} \right]}}{{\sinh \left[ {\sqrt {(N_{{m_{4} }} )^{2} + (N_{{n_{4} }} )^{2} } \, L_{X} } \right]}}} } } \times \sin (N_{{m_{4} }} Y)\sin (N_{{n_{4} }} z)\sin (N_{p} X)\sin (N_{q} Y)\sin (N_{r} z)dXdYdz + \int\limits_{0}^{{L_{X} }} {\int\limits_{0}^{{B_{Y} }} {\int\limits_{0}^{h} {\sum\limits_{{m_{5} = 1}}^{{M_{5} }} { \, \sum\limits_{{n_{5} = 1}}^{{N_{5} }} {F_{{m_{5} n_{5} }} } } \frac{{\cosh \left[ {\sqrt {(N_{{m_{5} }} )^{2} + (N_{{n_{5} }} )^{2} } \, (h - z)} \right]}}{{\cosh \left[ {\sqrt {(N_{{m_{5} }} )^{2} + (N_{{n_{5} }} )^{2} } \, h} \right]}}} } } \left. { \times \sin (N_{{m_{5} }} X)\sin (N_{{n_{5} }} Y)\sin (N_{p} X)\sin (N_{q} Y)\sin (N_{r} z)dXdYdz} \right\}. $$

(41)

Identifying the first, second, third, fourth and fifth triple-integrals of Eq. (41) as $I^{(1)} ,$ $I^{(2)} ,$ $I^{(3)} ,$ $I^{(4)}$ and $I^{(5)} ,$ respectively, and then simplifying them yields an expression for $E_{pqr}$ as

$$ E_{pqr} = - \left( {\frac{2}{{L_{X} }}} \right)\left( {\frac{2}{{B_{Y} }}} \right)\left( {\frac{2}{h}} \right)\left[ {I^{(1)} + } \right.I^{(2)} + I^{(3)} + I^{(4)} \left. { + I^{(5)} } \right], $$

(42)

where

$$ I^{(1)} = \sum\limits_{{m_{1} = 1}}^{{M_{1} }} { \, \sum\limits_{{n_{1} = 1}}^{{N_{1} }} {A_{{m_{{_{1} }} n_{{_{1} }} }} } } I_{a}^{(1)} I_{b}^{(1)} I_{c}^{(1)} . $$

(43)

For $N_{{m_{{_{1} }} }} = N_{p}$

$$ I_{a}^{(1)} = \frac{{L_{X} }}{2} $$

(44)

and for $N_{{m_{{_{1} }} }} \ne N_{p}$

$$ I_{a}^{(1)} = 0, $$

(45)

$$ I_{b}^{(1)} = \frac{{N_{q} }}{{(N_{{m_{{_{1} }} }} )^{2} + (N_{{n_{{_{1} }} }} )^{2} + (N_{q} )^{2} }}. $$

(46)

For $N_{{n_{{_{1} }} }} = N_{r}$

$$ I_{c}^{(1)} = \frac{h}{2} $$

(47)

and for $N_{{n_{{_{1} }} }} \ne N_{r}$

$$ I_{c}^{(1)} = 0. $$

(48)

$$ I^{(2)} = \sum\limits_{{m_{2} = 1}}^{{M_{2} }} { \, \sum\limits_{{n_{2} = 1}}^{{N_{2} }} {B_{{m_{2} n_{{_{2} }} }} } } I_{a}^{(2)} I_{b}^{(2)} I_{c}^{(2)} . $$

(49)

For $N_{{m_{{_{2} }} }} = N_{p}$

$$ I_{a}^{(2)} = \frac{{L_{X} }}{2} $$

(50)

and for $N_{{m_{{_{2} }} }} \ne N_{p}$

$$ I_{a}^{(2)} = 0, $$

(51)

$$ I_{b}^{(2)} = - \frac{{N_{q} \cos (N_{q} B_{Y} )}}{{(N_{{m_{{_{2} }} }} )^{2} + (N_{{n_{{_{2} }} }} )^{2} + (N_{q} )^{2} }}. $$

(52)

For $N_{{n_{{_{2} }} }} = N_{r}$

$$ I_{c}^{(2)} = \frac{h}{2} $$

(53)

and for $N_{{n_{{_{2} }} }} \ne N_{r}$

$$ I_{c}^{(2)} = 0. $$

(54)

$$ I^{(3)} = \sum\limits_{{m_{3} = 1}}^{{M_{3} }} { \, \sum\limits_{{n_{3} = 1}}^{{N_{3} }} {C_{{m_{3} n_{3} }} } } I_{a}^{(3)} I_{b}^{(3)} I_{c}^{(3)} , $$

(55)

where

$$ I_{a}^{(3)} = \frac{{N_{p} }}{{(N_{{m_{3} }} )^{2} + (N_{{n_{3} }} )^{2} + (N_{p} )^{2} }}. $$

(56)

For $N_{{m_{{_{3} }} }} = N_{q}$

$$ I_{b}^{(3)} = \frac{{B_{Y} }}{2} $$

(57)

and for $N_{{m_{{_{3} }} }} \ne N_{q}$

$$ I_{3}^{(b)} = 0. $$

(58)

For $N_{{n_{{_{3} }} }} = N_{r}$

$$ I_{c}^{(3)} = \frac{h}{2} $$

(59)

and for $N_{{n_{{_{3} }} }} \ne N_{r}$

$$ I_{c}^{(3)} = 0. $$

(60)

$$ I^{(4)} = \sum\limits_{{m_{4} = 1}}^{{M_{4} }} { \, \sum\limits_{{n_{4} = 1}}^{{N_{4} }} {D_{{m_{4} n_{{_{4} }} }} } } I_{a}^{(4)} I_{b}^{(4)} I_{c}^{(4)} , $$

(61)

where

$$ I_{a}^{(4)} = - \frac{{N_{p} \cos (N_{p} L_{X} )}}{{(N_{{m_{4} }} )^{2} + (N_{{n_{4} }} )^{2} + (N_{p} )^{2} }}. $$

(62)

For $N_{{m_{{_{4} }} }} = N_{q}$

$$ I_{b}^{(4)} = \frac{{B_{Y} }}{2} $$

(63)

and for $N_{{m_{{_{4} }} }} \ne N_{q}$

$$ I_{b}^{(4)} = 0. $$

(64)

For $N_{{n_{4} }} = N_{r}$

$$ I_{c}^{(4)} = \frac{h}{2} $$

(65)

and for $N_{{n_{4} }} \ne N_{r}$

$$ I_{c}^{(4)} = 0. $$

(66)

$$ I^{(5)} = \sum\limits_{{m_{5} = 1}}^{{M_{5} }} { \, \sum\limits_{{n_{5} = 1}}^{{N_{5} }} {F_{{m_{5} n_{5} }} } } I_{a}^{(5)} I_{b}^{(5)} I_{c}^{(5)} , $$

(67)

where for $N_{{m_{{_{5} }} }} = N_{p}$

$$ I_{a}^{(5)} = \frac{{L_{X} }}{2} $$

(68)

and for $N_{{m_{{_{5} }} }} \ne N_{p}$

$$ I_{a}^{(5)} = 0. $$

(69)

For $N_{{n_{5} }} = N_{q}$

$$ I_{b}^{(5)} = \frac{{B_{Y} }}{2} $$

(70)

and for $N_{{n_{5} }} \ne N_{q}$

$$ I_{b}^{(5)} = 0 $$

(71)

and

$$ I_{c}^{(5)} = \frac{{N_{r} }}{{(N_{{m_{5} }} )^{2} + (N_{{n_{5} }} )^{2} + (N_{r} )^{2} }}. $$

(72)

All the coefficients of Eq. (2) are thus determined and the boundary value problem of figure 1 hence stands solved.

Further, like in the determination of the top discharge function, Darcy’s law can also be applied to evaluate the time-dependent discharges being received through the Northern, Southern, Eastern and Western faces of the ditches; naming these discharges as $Q_{N} ,$ $Q_{S} ,$ $Q_{E}$ and $Q_{W} ,$ respectively, their expressions, thus, can be represented as

$$ Q_{N} (t) = - K_{y} \int\limits_{0}^{h} {\int\limits_{0}^{L} {\left( {\frac{\partial \phi }{\partial y}} \right)} }_{y = B} dxdz, $$

(73)

$$ Q_{S} (t) = K_{y} \int\limits_{0}^{h} {\int\limits_{0}^{L} {\left( {\frac{\partial \phi }{\partial y}} \right)} }_{y = 0} dxdz, $$

(74)

$$ Q_{E} (t) = - K_{x} \int\limits_{0}^{h} {\int\limits_{0}^{B} {\left( {\frac{\partial \phi }{\partial y}} \right)} }_{x = L} dydz $$

(75)

and

$$ Q_{W} (t) = K_{x} \int\limits_{0}^{h} {\int\limits_{0}^{B} {\left( {\frac{\partial \phi }{\partial y}} \right)} }_{x = 0} dydz. $$

(76)

Further, by performing time integrals on the concerned discharges functions, the volume of water seeping through the top and vertical faces of the studied ponded system within a desired time interval can also be worked out.