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Analysis of three-dimensional ponded drainage of a multi-layered soil underlain by an impervious barrier

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Abstract

A general analytical model is proposed for predicting three-dimensional seepage into ditch drains through a soil column comprising of three distinct vertical anisotropic soil layers and underlain by an impervious barrier, the drains being fed by a distributed ponding head introduced at the surface of the soil column. The problem is solved for three different situations resulting from three different locations of the water table in the ditches, namely, when the water level lies in the first layer, when it lies in the second layer and finally when it falls in the third layer. The derived solutions are validated by comparing with analytical solutions of others for a few drainage scenarios; in addition, a few numerical checks on them have also been carried out by making use of the Processing MODFLOW environment. From the study, it is seen that ponded drainage of a multi-layered soil is mostly three-dimensional in nature, particularly in locations close to the drains and that the directional conductivities of the layers play a pivotal role in deciding the hydraulics of flow associated with such a system. Further, it has also come out of the study that by suitably altering the ponding distribution at the surface of the soil, the uniformity of water movement in a multi-layered drainage system can be considerably improved mainly if the directional conductivities of the bottom layers are relatively lower than those of the top layer. As soils in nature are mostly stratified and as no analytical solution to the three-dimensional ponded ditch drainage problem currently exists for a layered soil, the proposed solutions are expected to be important additions to the already existing repertoire of drainage solutions on the subject, particularly when looked in the context of reclamation of water-logged and saline soils in layered field situations.

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Correspondence to Gautam Barua.

Appendices

Appendix A: Determination of Fourier coefficients

1.1 A.1 Case 1

We first seek to determine the coefficients \( C_{{p_{2} q_{2} (1)}} \) of Eq. (2). Towards realizing this, we apply boundary conditions (Va) and (Vb) to this equation; this yields the following relations

$$ \sum\limits_{{p_{2} = 1}}^{{P_{2} }} {\sum\limits_{{q_{2} = 1}}^{{Q_{2} }} {C_{{p_{2} q_{2} (1)}} } } \sin \left[ {\left( {\frac{{p_{2} \pi }}{{S_{2} }}} \right)y} \right]\sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{2} }}{2}} \right)\frac{\pi }{{H_{2} }}} \right]z} \right\} = - z,\quad x = 0,\quad 0 < y < S_{2} ,\quad 0 < z \le H_{1} , $$
(A1)
$$ \sum\limits_{{p_{2} = 1}}^{{P_{2} }} {\sum\limits_{{q_{2} = 1}}^{{Q_{2} }} {C_{{p_{2} q_{2} (1)}} } } \sin \left[ {\left( {\frac{{p_{2} \pi }}{{S_{2} }}} \right)y} \right]\sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{2} }}{2}} \right)\frac{\pi }{{H_{2} }}} \right]z} \right\} = - H_{1} ,\quad x = 0,\quad 0 < y < S_{2} ,\quad H_{1} \le z \le H_{2} . $$
(A2)

Running a Fourier series in the space defined by \( 0 < y < S_{2} \) and \( 0 < z \le H_{2} , \) an expression for evaluation of \( C_{{p_{2} q_{2} (1)}} \) can thus be written as

$$ C_{{p_{2} q_{2} (1)}} = \left( {\frac{4}{{S_{2} H_{2} }}} \right) \times \int\limits_{0}^{{S_{2} }} {\sin \left[ {\left( {\frac{{p_{2} \pi }}{{S_{2} }}} \right)y} \right]} dy \times \left\{ {\int\limits_{0}^{{H_{1} }} { - z\sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{2} }}{2}} \right)\frac{\pi }{{H_{2} }}} \right]z} \right\}dz} + \int\limits_{{H_{1} }}^{{H_{2} }} { - H_{1} \sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{2} }}{2}} \right)\frac{\pi }{{H_{2} }}} \right]z} \right\}dz} } \right\}. $$

Simplifying the above integrals, we finally get \( C_{{p_{2} q_{2} (1)}} \) as

$$ C_{{p_{2} q_{2} (1)}} = \left( {\frac{4}{{S_{2} H_{2} }}} \right)\left\{ {\frac{{1 - \cos \left[ {\left( {\frac{{p_{2} \pi }}{{S_{2} }}} \right)S_{2} } \right]}}{{\left( {\frac{{p_{2} \pi }}{{S_{2} }}} \right)}}} \right\}\left\{ {\frac{{ - \sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{2} }}{2}} \right)\frac{\pi }{{H_{2} }}} \right]H_{1} } \right\}}}{{\left[ {\left( {\frac{{1 - 2q_{2} }}{2}} \right)\frac{\pi }{{H_{2} }}} \right]}}} \right\}. $$
(A3)

Likewise, implementation of boundary conditions (VIIIa) and (VIIIb) on Eq. (2) followed by Fourier expansion in the space defined by \( 0 < y < S_{2} \) and \( 0 < z \le H_{2} \) give \( B_{{p_{1} q_{1} (1)}} \) as

$$ B_{{p_{1} q_{1} (1)}} = \left( {\frac{4}{{S_{2} H_{2} }}} \right)\left\{ {\frac{{1 - \cos \left[ {\left( {\frac{{p_{1} \pi }}{{S_{2} }}} \right)S_{2} } \right]}}{{\left( {\frac{{p_{1} \pi }}{{S_{2} }}} \right)}}} \right\}\left\{ {\frac{{ - \sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{1} }}{2}} \right)\frac{\pi }{{H_{2} }}} \right]H_{1} } \right\}}}{{\left[ {\left( {\frac{{1 - 2q_{1} }}{2}} \right)\frac{\pi }{{H_{2} }}} \right]}}} \right\}. $$
(A4)

Similarly, the expressions for the coefficients \( D_{{p_{3} q_{3} (1)}} \) and \( F_{{p_{4} q_{4} (1)}} \) can be obtained by making use of boundary conditions (XIVa), (XIVb), (XIa) and (XIb) and by performing appropriate Fourier expansions in the relevant spaces; all these give \( D_{{p_{3} q_{3} (1)}} \) and \( F_{{p_{4} q_{4} (1)}} \) as

$$ D_{{p_{3} q_{3} (1)}} = \left( {\frac{4}{{S_{1} H_{2} }}} \right)\left\{ {\frac{{1 - \cos \left[ {\left( {\frac{{p_{3} \pi }}{{S_{1} }}} \right)S_{1} } \right]}}{{\left( {\frac{{p_{3} \pi }}{{S_{1} }}} \right)}}} \right\}\left\{ {\frac{{ - \sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{3} }}{2}} \right)\frac{\pi }{{H_{2} }}} \right]H_{1} } \right\}}}{{\left[ {\left( {\frac{{1 - 2q_{3} }}{2}} \right)\frac{\pi }{{H_{2} }}} \right]}}} \right\} $$
(A5)

and

$$ F_{{p_{4} q_{4} (1)}} = \left( {\frac{4}{{S_{1} H_{2} }}} \right)\left\{ {\frac{{1 - \cos \left[ {\left( {\frac{{p_{4} \pi }}{{S_{1} }}} \right)S_{1} } \right]}}{{\left( {\frac{{p_{4} \pi }}{{S_{1} }}} \right)}}} \right\}\left\{ {\frac{{ - \sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{4} }}{2}} \right)\frac{\pi }{{H_{2} }}} \right]H_{1} } \right\}}}{{\left[ {\left( {\frac{{1 - 2q_{4} }}{2}} \right)\frac{\pi }{{H_{2} }}} \right]}}} \right\}. $$
(A6)

To evaluate \( Q_{uv(1)} , \) we now apply boundary conditions (IIIa), (IIIb), (IIIc), (IIId), (IIIe), (IIIf), (IIIg), (IIIh) and (IIIi) to Eq. (2); this gives us the following equations

$$ \sum\limits_{u = 1}^{U} {\sum\limits_{v = 1}^{V} {Q_{uv(1)} } } \sin \left( {N_{u} x} \right)\sin \left( {N_{v} y} \right) = \delta_{1} ,\quad 0 < x < S_{1} ,\quad 0 < y < d_{y1} ,\quad z = 0, $$
(A7)
$$ \sum\limits_{u = 1}^{U} {\sum\limits_{v = 1}^{V} {Q_{uv(1)} } } \sin \left( {N_{u} x} \right)\sin \left( {N_{v} y} \right) = \delta_{1} ,\quad 0 < x < S_{1} ,\quad d_{{y(2N_{0} - 2)}} < y < S_{2} ,\quad z = 0, $$
(A8)
$$ \sum\limits_{u = 1}^{U} {\sum\limits_{v = 1}^{V} {Q_{uv(1)} } } \sin \left( {N_{u} x} \right)\sin \left( {N_{v} y} \right) = \delta_{1} ,\quad 0 < x < d_{x1} ,\quad d_{y1} < y < d_{{y(2N_{0} - 2)}} ,\quad z = 0, $$
(A9)
$$ \sum\limits_{u = 1}^{U} {\sum\limits_{v = 1}^{V} {Q_{uv(1)} } } \sin \left( {N_{u} x} \right)\sin \left( {N_{v} y} \right) = \delta_{1} ,\quad d_{{x(2N_{0} - 2)}} < x < S_{1} ,\quad d_{y1} < y < d_{{y(2N_{0} - 2)}} ,\quad z = 0, $$
(A10)
$$ \sum\limits_{u = 1}^{U} {\sum\limits_{v = 1}^{V} {Q_{uv(1)} } } \sin \left( {N_{u} x} \right)\sin \left( {N_{v} y} \right) = \delta_{i} ,\quad d_{x(i - 1)} < x < d_{{x(2N_{0} - i)}} ,\quad d_{y(i - 1)} < y < d_{yi} ,\quad z = 0, $$
(A11)
$$ \sum\limits_{u = 1}^{U} {\sum\limits_{v = 1}^{V} {Q_{uv(1)} } } \sin \left( {N_{u} x} \right)\sin \left( {N_{v} y} \right) = \delta_{i} ,\quad d_{x(i - 1)} < x < d_{{x(2N_{0} - i)}} ,\quad d_{{y(2N_{0} - i - 1)}} < y < d_{{y(2N_{0} - i)}} ,\quad z = 0, $$
(A12)
$$ \sum\limits_{u = 1}^{U} {\sum\limits_{v = 1}^{V} {Q_{uv(1)} } } \sin \left( {N_{u} x} \right)\sin \left( {N_{v} y} \right) = \delta_{i} ,\quad d_{x(i - 1)} < x < d_{xi} ,\quad d_{yi} < y < d_{{y(2N_{0} - i - 1)}} ,\quad z = 0, $$
(A13)
$$ \sum\limits_{u = 1}^{U} {\sum\limits_{v = 1}^{V} {Q_{uv(1)} } } \sin \left( {N_{u} x} \right)\sin \left( {N_{v} y} \right) = \delta_{i} ,\;d_{{x(2N_{0} - i - 1)}} < x < d_{{x(2N_{0} - i)}} ,\quad d_{yi} < y < d_{{y(2N_{0} - i - 1)}} ,\quad z = 0,\quad 2 \le i \le \left( {N_{0} - 1} \right),\quad N_{0} > 2, $$
(A14)
$$ \sum\limits_{u = 1}^{U} {\sum\limits_{v = 1}^{V} {Q_{uv(1)} } } \sin \left( {N_{u} x} \right)\sin \left( {N_{v} y} \right) = \delta_{{N_{0} }} ,\quad d_{{x(N_{0} - 1)}} < x < d_{{xN_{0} }} ,\quad d_{{y(N_{0} - 1)}} < y < d_{{yN_{0} }} ,\quad z = 0, $$
(A15)

where \( N_{u} = \left( {{{u\pi } \mathord{\left/ {\vphantom {{u\pi } {S_{1} }}} \right. \kern-0pt} {S_{1} }}} \right) \) and \( N_{v} = \left( {{{v\pi } \mathord{\left/ {\vphantom {{v\pi } {S_{2} }}} \right. \kern-0pt} {S_{2} }}} \right) \). Thus, \( Q_{uv(1)} \) can be evaluated by carrying out a double Fourier expansion in \( 0 < x < S_{1} , \) \( 0 < y < S_{2} ; \) carrying out such an expansion, we get an expression for \( Q_{uv(1)} \) as

$$ \begin{aligned} & Q_{uv(1)} = \left( {\frac{4}{{S_{1} S_{2} }}} \right) \times \left\{ {\delta_{1} \left[ {\int\limits_{0}^{{S_{1} }} {\int\limits_{0}^{{d_{x1} }} {\sin (N_{u} } } x)\sin (N_{v} y)dxdy + \int\limits_{0}^{{S_{1} }} {\int\limits_{{d_{{y(2N_{0} - 2)}} }}^{{S_{2} }} {\sin (N_{u} x)} } \sin (N_{v} y)dxdy} \right.} \right. \\ & + \left. {\int\limits_{0}^{{d_{x1} }} {\int\limits_{{d_{y1} }}^{{d_{{y(2N_{0} - 2)}} }} {\sin (N_{u} } } x)\sin (N_{v} y)dxdy + \int\limits_{{d_{{x(2N_{0} - 2)}} }}^{{S_{1} }} {\int\limits_{{d_{y1} }}^{{d_{{y(2N_{0} - 2)}} }} {\sin (N_{u} } } x)\sin (N_{v} y)dxdy} \right] \\ & + \sum\limits_{i = 2}^{{I = (N_{0} - 1)}} {\delta_{i} } \left[ {\int\limits_{{d_{x(i - 1)} }}^{{d_{{x(2N_{0} - i)}} }} {\int\limits_{{d_{y(i - 1)} }}^{{d_{yi} }} {\sin (N_{u} x)} } } \right.\sin (N_{v} y)dxdy + \int\limits_{{d_{x(i - 1)} }}^{{d_{{x(2N_{0} - i)}} }} {\int\limits_{{d_{{y(2N_{0} - i - 1)}} }}^{{d_{{y(2N_{0} - i)}} }} {\sin (N_{u} x)} } \sin (N_{v} y)dxdy \\ & \left. { + \int\limits_{{d_{x(i - 1)} }}^{{d_{xi} }} {\int\limits_{{d_{yi} }}^{{d_{{y(2N_{0} - i - 1)}} }} {\sin (N_{u} x)\sin (N_{v} y)dxdy + } } \int\limits_{{d_{{x(2N_{0} - i - 1)}} }}^{{d_{{x(2N_{0} - i)}} }} {\int\limits_{{d_{yi} }}^{{d_{{y(2N_{0} - i - 1)}} }} {\sin (N_{u} x)} } \sin (N_{v} y)dxdy} \right] \\ & \left. { + \delta_{{N_{0} }} \int\limits_{{d_{{x(N_{0} - 1)}} }}^{{d_{{xN_{0} }} }} {\int\limits_{{d_{{y(N_{0} - 1)}} }}^{{d_{{yN_{0} }} }} {\sin (N_{u} x)\sin (N_{v} y)dxdy} } } \right\}. \\ \end{aligned} $$

These integrals, upon simplification, give

$$ \begin{aligned} & Q_{uv(1)} = \left( {\frac{4}{{S_{1} S_{2} }}} \right) \\ & \times \left\{ {\delta_{1} \left\{ {\left[ {\frac{{1 - \cos \left( {N_{u} S_{1} } \right)}}{{N_{u} }}} \right]} \right.} \right.\left[ {\frac{{1 - \cos \left( {N_{v} d_{y1} } \right)}}{{N_{v} }} + \frac{{\cos \left( {N_{v} d_{{y(2N_{0} - 2)}} } \right) - \cos \left( {N_{v} S_{2} } \right)}}{{N_{v} }}} \right] \\ & + \left. {\left[ {\frac{{\cos \left( {N_{v} d_{y1} } \right) - \cos \left( {N_{v} d_{{y(2N_{0} - 2)}} } \right)}}{{N_{v} }}} \right] \times \left[ {\frac{{1 - \cos \left( {N_{u} d_{x1} } \right)}}{{N_{u} }} + \frac{{\cos \left( {N_{u} d} \right)_{{x(2N_{0} - 2)}} - \cos \left( {N_{u} S_{1} } \right)}}{{N_{u} }}} \right]} \right\} \\ & + \sum\limits_{i = 2}^{{I = N_{0} - 1}} {\delta_{i} } \left\{ {\left[ {\frac{{\cos \left( {N_{u} d_{x(i - 1)} } \right) - \cos \left( {N_{u} d_{{x(2N_{0} - i)}} } \right)}}{{N_{u} }}} \right]} \right. \\ & \times \left[ {\frac{{\cos \left( {N_{v} d_{y(i - 1)} } \right) - \cos \left( {N_{v} d_{yi} } \right)}}{{N_{v} }} + \frac{{\cos \left( {N_{v} d_{{y(2N_{0} - i - 1)}} } \right) - \cos \left( {N_{v} d_{{y(2N_{0} - i)}} } \right)}}{{N_{v} }}} \right] \\ & + \left[ {\frac{{\cos \left( {N_{v} d_{yi} } \right) - \cos \left( {N_{v} d_{{y(2N_{0} - i - 1)}} } \right)}}{{N_{v} }}} \right] \\ & \times \left. {\left[ {\frac{{\cos \left( {N_{u} d_{x(i - 1)} } \right) - \cos \left( {N_{u} d_{xi} } \right)}}{{N_{u} }} + \frac{{\cos \left( {N_{u} d_{{x(2N_{0} - i - 1)}} } \right) - \cos \left( {N_{u} d_{{x(2N_{0} - i)}} } \right)}}{{N_{u} }}} \right]} \right\} \\ & + \delta_{{N_{0} }} \left. {\left\{ {\left[ {\frac{{\cos \left( {N_{u} d_{{x(N_{0} - 1)}} } \right) - \cos \left( {N_{u} d_{{xN_{0} }} } \right)}}{{N_{u} }}} \right] \times \left[ {\frac{{\cos \left( {N_{v} d_{{y(N_{0} - 1)}} } \right) - \cos \left( {N_{v} d_{{yN_{0} }} } \right)}}{{N_{v} }}} \right]} \right\}} \right\} \\ \end{aligned} $$
(A16)

To satisfy the intermediate condition (Ia), the head expressions for the top and middle soil layers are next equated at \( z = H_{2} ; \) this results in the equation

$$ \begin{aligned} & \sum\limits_{{p_{1} = 1}}^{{P_{1} }} {\sum\limits_{{q_{1} = 1}}^{{Q_{1} }} {B_{{p_{1} q_{1} (1)}} } } \sin \left[ {\left( {\frac{{p_{1} \pi }}{{S_{2} }}} \right)y} \right]\sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{1} }}{2}} \right)\frac{\pi }{{H_{2} }}} \right]H_{2} } \right\}\left[ {\frac{{\sinh (\lambda_{{p_{1} q_{1} }} x)}}{{\sinh (\lambda_{{p_{1} q_{1} }} S_{1} )}}} \right] \\ & + \sum\limits_{{p_{2} = 1}}^{{P_{2} }} {\sum\limits_{{q_{2} = 1}}^{{Q_{2} }} {C_{{p_{2} q_{2} (1)}} } } \sin \left[ {\left( {\frac{{p_{2} \pi }}{{S_{2} }}} \right)y} \right]\sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{2} }}{2}} \right)\frac{\pi }{{H_{2} }}} \right]H_{2} } \right\}\left\{ {\frac{{\sinh [\lambda_{{p_{2} q_{2} }} (S_{1} - x)]}}{{\sinh (\lambda_{{p_{2} q_{2} }} S_{1} )}}} \right\} \\ & + \sum\limits_{{p_{3} = 1}}^{{P_{3} }} {\sum\limits_{{q_{3} = 1}}^{{Q_{3} }} {D_{{p_{3} q_{3} (1)}} } } \sin \left[ {\left( {\frac{{p_{3} \pi }}{{S_{1} }}} \right)x} \right]\sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{3} }}{2}} \right)\frac{\pi }{{H_{2} }}} \right]H_{2} } \right\}\left[ {\frac{{\sinh (\lambda_{{p_{3} q_{3} }} y)}}{{\sinh (\lambda_{{p_{3} q_{3} }} S_{2} )}}} \right] \\ & + \sum\limits_{{p_{4} = 1}}^{{P_{4} }} {\sum\limits_{{q_{4} = 1}}^{{Q_{4} }} {F_{{p_{4} q_{4} (1)}} } } \sin \left[ {\left( {\frac{{p_{4} \pi }}{{S_{1} }}} \right)x} \right]\sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{4} }}{2}} \right)\frac{\pi }{{H_{2} }}} \right]H_{2} } \right\}\left\{ {\frac{{\sinh [\lambda_{{p_{4} q_{4} }} (S_{2} - y)]}}{{\sinh (\lambda_{{p_{4} q_{4} }} S_{2} )}}} \right\} \\ & + \sum\limits_{k = 1}^{K} {\sum\limits_{l = 1}^{L} {E_{kl(1)} } } \sin \left[ {\left( {\frac{k\pi }{{S_{1} }}} \right)x} \right]\sin \left[ {\left( {\frac{l\pi }{{S_{2} }}} \right)y} \right]\tanh \left( {\lambda_{kl} H_{2} } \right) + \sum\limits_{u = 1}^{U} {\sum\limits_{v = 1}^{V} {Q_{uv(1)} } } \sin \left[ {\left( {\frac{u\pi }{{S_{1} }}} \right)x} \right]\sin \left[ {\left( {\frac{v\pi }{{S_{2} }}} \right)y} \right]\frac{1}{{\cosh \left( {\lambda_{uv} H_{2} } \right)}} \\ & = \sum\limits_{{i_{1} = 1}}^{{I_{1} }} {\sum\limits_{{j_{1} = 1}}^{{J_{1} }} {G_{{i_{1} j_{1} (1)}} } } \sin \left[ {\left( {\frac{{i_{1} \pi }}{{S_{1} }}} \right)x} \right]\sin \left[ {\left( {\frac{{j_{1} \pi }}{{S_{2} }}} \right)y} \right]\frac{1}{{\cosh [\lambda_{{i_{1} j_{1} }} (H_{3} - H_{2} )]}} \\ & + \sum\limits_{{i_{2} = 1}}^{{I_{2} }} {\sum\limits_{{j_{2} = 1}}^{{J_{2} }} {H_{{i_{2} j_{2} (1)}} } } \sin \left[ {\left( {\frac{{i_{2} \pi }}{{S_{1} }}} \right)x} \right]\sin \left[ {\left( {\frac{{j_{2} \pi }}{{S_{2} }}} \right)y} \right] - H_{1} . \\ \end{aligned} $$
(A17)

Multiplying both sides of the above equation by \( \sin \left[ {\left( {\frac{{u_{1} \pi }}{{S_{1} }}} \right)x} \right] \times \sin \left[ {\left( {\frac{{v_{1} \pi }}{{S_{2} }}} \right)y} \right] \) and then making a Fourier run in the space \( 0 < x < S_{1} \) and \( 0 < y < S_{2} \) and evaluating the resulting integrals, we get an equation linking the coefficients of Eq. (2) and Eq. (3) as

$$ \begin{aligned} & E_{kl(1)} \tanh \left( {\lambda_{kl} H_{2} } \right) - G_{{i_{1} j_{1} (1)}} \frac{1}{{\cosh [\lambda_{{i_{1} j_{1} }} (H_{3} - H_{2} )]}} - H_{{i_{2} j_{2} (1)}} + Q_{uv(1)} \frac{1}{{\cosh \left( {\lambda_{uv} H_{2} } \right)}} \\ & = - \left( {\frac{{4H_{1} }}{{S_{1} S_{2} }}} \right)\left[ {\frac{{1 - \cos (N_{{u_{1} }} S_{1} )}}{{N_{{u_{1} }} }}} \right]\left[ {\frac{{1 - \cos \left( {N_{{v_{1} }} S_{2} } \right)}}{{N_{{v_{1} }} }}} \right] + \left( {\frac{2}{{S_{1} }}} \right)\sum\limits_{{q_{1} = 1}}^{{Q_{1} }} {B_{{p_{1} q_{1} (1)}} } \sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{1} }}{2}} \right)\frac{\pi }{{H_{2} }}} \right]H_{2} } \right\}\left[ {\frac{{N_{{u_{1} }} \cos (N_{{u_{1} }} S_{1} )}}{{N_{{u_{1} }}^{2} + \lambda_{{p_{1} q_{1} }}^{2} }}} \right] \\ & - \left( {\frac{2}{{S_{1} }}} \right)\sum\limits_{{q_{2} = 1}}^{{Q_{2} }} {C_{{p_{2} q_{2} (1)}} } \sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{2} }}{2}} \right)\frac{\pi }{{H_{2} }}} \right]H_{2} } \right\}\left[ {\frac{{N_{{u_{1} }} }}{{N_{{u_{1} }}^{2} + \lambda_{{p_{2} q_{2} }}^{2} }}} \right] \\ & + \left( {\frac{2}{{S_{2} }}} \right)\sum\limits_{{q_{3} = 1}}^{{Q_{3} }} {D_{{p_{3} q_{3} (1)}} } \sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{3} }}{2}} \right)\frac{\pi }{{H_{2} }}} \right]H_{2} } \right\}\left[ {\frac{{N_{{v_{1} }} \cos (N_{{v_{1} }} S_{2} )}}{{N_{{v_{1} }}^{2} + \lambda_{{p_{3} q_{3} }}^{2} }}} \right] \\ & - \left( {\frac{2}{{S_{2} }}} \right)\sum\limits_{{q_{4} = 1}}^{{Q_{4} }} {F_{{p_{4} q_{4} (1)}} } \sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{4} }}{2}} \right)\frac{\pi }{{H_{2} }}} \right]H_{2} } \right\}\left[ {\frac{{N_{{v_{1} }} }}{{N_{{v_{1} }}^{2} + \lambda_{{p_{4} q_{4} }}^{2} }}} \right], \\ \end{aligned} $$
(A18)

where \( N_{{u_{1} }} = \left( {\frac{{u_{1} \pi }}{{S_{1} }}} \right), \) \( N_{{v_{1} }} = \left( {\frac{{v_{1} \pi }}{{S_{2} }}} \right), \) \( u_{1} = k = u = i_{1} = i_{2} = p_{3} = p_{4} , \) \( v_{1} = l = v = j_{1} = j_{2} = p_{1} = p_{2} \) and \( Q_{1} = Q_{2} = Q_{3} = Q_{4} \to \infty . \)

We next equate the flux expressions for the top and the middle layers at \( z = H_{2} ; \) this takes care of intermediate condition (Ib) and yields the equation

$$ \begin{aligned} & - K_{{z_{1} }} \sum\limits_{k = 1}^{K} {\sum\limits_{l = 1}^{L} {E_{kl(1)} \lambda_{kl} } } \sin \left[ {\left( {\frac{k\pi }{{S_{1} }}} \right)x} \right]\sin \left[ {\left( {\frac{l\pi }{{S_{2} }}} \right)y} \right] \\ & = K_{{z_{2} }} \sum\limits_{{i_{2} = 1}}^{{I_{2} }} {\sum\limits_{{j_{2} = 1}}^{{J_{2} }} {H_{{i_{2} j_{2} (1)}} } } \sin \left[ {\left( {\frac{{i_{2} \pi }}{{S_{1} }}} \right)x} \right]\sin \left[ {\left( {\frac{{j_{2} \pi }}{{S_{2} }}} \right)y} \right]\lambda_{{i_{2} j_{2} }} \tanh [\lambda_{{i_{2} j_{2} }} (H_{3} - H_{2} )]. \\ \end{aligned} $$
(A19)

Now, performing a Fourier expansion in \( 0 < x < S_{1} \) and \( 0 < y < S_{2} \), we get from the above equation the relation

$$ K_{{z_{1} }} E_{kl(1)} \lambda_{kl} + K_{{z_{2} }} H_{{i_{2} j_{2} (1)}} \lambda_{{i_{2} j_{2} }} \tanh [\lambda_{{i_{2} j_{2} }} (H_{3} - H_{2} )] = 0, $$
(A20)

where \( k = i_{2} \) and \( l = j_{2} . \) Also, to satisfy intermediate condition (IIa), we next equate the hydraulic heads of the middle and the bottom layers at \( z = H_{3} ; \) this results in the equation

$$ \begin{aligned} & \sum\limits_{{i_{1} = 1}}^{{I_{1} }} {\sum\limits_{{j_{1} = 1}}^{{J_{1} }} {G_{{i_{1} j_{1} (1)}} } } \sin \left[ {\left( {\frac{{i_{1} \pi }}{{S_{1} }}} \right)x} \right]\sin \left[ {\left( {\frac{{j_{1} \pi }}{{S_{2} }}} \right)y} \right] + \sum\limits_{{i_{2} = 1}}^{{I_{2} }} {\sum\limits_{{j_{2} = 1}}^{{J_{2} }} {H_{{i_{2} j_{2} (1)}} } } \sin \left[ {\left( {\frac{{i_{2} \pi }}{{S_{1} }}} \right)x} \right]\sin \left[ {\left( {\frac{{j_{2} \pi }}{{S_{2} }}} \right)y} \right]\left\{ {\frac{1}{{\cosh [\lambda_{{i_{2} j_{2} }} (H_{3} - H_{2} )]}}} \right\} \\ & = \sum\limits_{{i_{3} = 1}}^{{I_{3} }} {\sum\limits_{{j_{3} = 1}}^{{J_{3} }} {P_{{i_{3} j_{3} (1)}} } } \sin \left[ {\left( {\frac{{i_{3} \pi }}{{S_{1} }}} \right)x} \right]\sin \left[ {\left( {\frac{{j_{3} \pi }}{{S_{2} }}} \right)y} \right]\left\{ {\frac{{\cosh [\lambda_{{i_{3} j_{3} }} (h - H_{3} )]}}{{\sinh [\lambda_{{i_{3} j_{3} }} (h - H_{3} )]}}} \right\}. \\ \end{aligned} $$
(A21)

A double Fourier series expansion on the above equation in the space \( 0 < x < S_{1} \) and \( 0 < y < S_{2} \) followed by simplification of the ensuing integrals lead to the relation

$$ G_{{i_{1} j_{1} (1)}} + H_{{i_{2} j_{2} (1)}} \left\{ {\frac{1}{{\cosh [\lambda_{{i_{2} j_{2} }} (H_{3} - H_{2} )]}}} \right\} - P_{{i_{3} j_{3} (1)}} \left\{ {\frac{{\cosh [\lambda_{{i_{3} j_{3} }} (h - H_{3} )]}}{{\sinh [\lambda_{{i_{3} j_{3} }} (h - H_{3} )]}}} \right\} = 0, $$
(A22)

where \( i_{1} = i_{2} = i_{3} \) and \( j_{1} = j_{2} = j_{3} . \) Finally, by equating the fluxes of the middle and bottom layers at \( z = H_{3} \) [so as to satisfy intermediate condition (IIb)] gives us the equation

$$ \begin{aligned} & - K_{{z_{2} }} \sum\limits_{{i_{1} = 1}}^{{I_{1} }} {\sum\limits_{{j_{1} = 1}}^{{J_{1} }} {G_{{i_{1} j_{1} (1)}} } } \sin \left[ {\left( {\frac{{i_{1} \pi }}{{S_{1} }}} \right)x} \right]\sin \left[ {\left( {\frac{{j_{1} \pi }}{{S_{2} }}} \right)y} \right]\lambda_{{i_{1} j_{1} }} \tanh [\lambda_{{i_{1} j_{1} }} (H_{3} - H_{2} )] \\ & = K_{{z_{3} }} \sum\limits_{{i_{3} = 1}}^{{I_{3} }} {\sum\limits_{{j_{3} = 1}}^{{J_{3} }} {P_{{i_{3} j_{3} (1)}} } } \sin \left[ {\left( {\frac{{i_{3} \pi }}{{S_{1} }}} \right)x} \right]\sin \left[ {\left( {\frac{{j_{3} \pi }}{{S_{2} }}} \right)y} \right]\lambda_{{i_{3} j_{3} }} . \\ \end{aligned} $$
(A23)

On simplification, this yields

$$ K_{{z_{2} }} G_{{i_{1} j_{1} (1)}} \lambda_{{i_{1} j_{1} }} \tanh [\lambda_{{i_{1} j_{1} }} (H_{3} - H_{2} )] + K_{{z_{3} }} P_{{i_{3} j_{3} (1)}} \lambda_{{i_{3} j_{3} }} = 0, $$
(A24)

where \( i_{1} = i_{3} \) and \( j_{1} = j_{3} . \)Linear equations arising out of Eqs. (A18), (A20), (A22) and (A24) corresponding to a drainage scenario of figure 1 (with the level of water in the ditches lying in the top layer) can now be solved using Gauss elimination or a similar procedure [78] to evaluate the coefficients \( E_{kl(1)} , \) \( G_{{i_{1} j_{1} (1)}} , \) \( H_{{i_{2} j_{2} (1)}} \) and \( P_{{i_{3} j_{3} (1)}} . \) In this context, it should be noted that all other coefficients of Eqs. (2), (3) and (4) corresponding to the studied drainage situation can be directly evaluated using Eqs. (A3), (A4), (A5), (A6) and (A16). In fact, these values need to be determined first before proceeding to solve Eqs. (A18), (A20), (A22) and (A24) as these coefficients are required for evaluating these constants.

1.2 A.2 Case 2

To evaluate the constants \( C_{{p_{2} q_{2} (2)}} \) for this drainage situation, we introduce boundary condition (XVII) in Eq. (20); this leads to a relation involving the coefficients \( C_{{p_{2} q_{2} (2)}} \) as

$$ \sum\limits_{{p_{2} = 1}}^{{P_{2} }} {\sum\limits_{{q_{2} = 1}}^{{Q_{2} }} {C_{{p_{2} q_{2} (2)}} } } \sin \left[ {\left( {\frac{{p_{2} \pi }}{{S_{2} }}} \right)y} \right]\sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{2} }}{2}} \right)\frac{\pi }{{H_{2} }}} \right]z} \right\} = - z,\quad x = 0,\quad 0 < y < S_{2} ,\quad 0 < z \le H_{2} , $$
(A25)

Thus, \( C_{{p_{2} q_{2} (2)}} \) can be determined by performing a double Fourier run in \( 0 < y < S_{2} \) and \( 0 < z \le H_{2} ; \) after carrying out the concerned integral, we find \( C_{{p_{2} q_{2} (2)}} \) as

$$ C_{{p_{2} q_{2} (2)}} = \left( {\frac{4}{{S_{2} H_{2} }}} \right)\left\{ {\frac{{1 - \cos \left[ {\left( {\frac{{p_{2} \pi }}{{S_{2} }}} \right)S_{2} } \right]}}{{\left( {\frac{{p_{2} \pi }}{{S_{2} }}} \right)}}} \right\}\left\{ {\frac{{ - \sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{2} }}{2}} \right)\frac{\pi }{{H_{2} }}} \right]H_{2} } \right\}}}{{\left[ {\left( {\frac{{1 - 2q_{2} }}{2}} \right)\frac{\pi }{{H_{2} }}} \right]}}} \right\}. $$
(A26)

Also, incorporation of boundary conditions (XX), (XXVI) and (XXIII) in the head functions gives, after carrying out the relevant Fourier expansions, expressions for the coefficients \( B_{{p_{1} q_{1} (2)}} , \) \( D_{{p_{3} q_{3} (2)}} \) and \( F_{{p_{4} q_{4} (2)}} \) as

$$ B_{{p_{1} q_{1} (2)}} = \left( {\frac{4}{{S_{2} H_{2} }}} \right)\left\{ {\frac{{1 - \cos \left[ {\left( {\frac{{p_{1} \pi }}{{S_{2} }}} \right)S_{2} } \right]}}{{\left( {\frac{{p_{1} \pi }}{{S_{2} }}} \right)}}} \right\}\left\{ {\frac{{ - \sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{1} }}{2}} \right)\frac{\pi }{{H_{2} }}} \right]H_{2} } \right\}}}{{\left[ {\left( {\frac{{1 - 2q_{1} }}{2}} \right)\frac{\pi }{{H_{2} }}} \right]}}} \right\}, $$
(A27)
$$ D_{{p_{3} q_{3} (2)}} = \left( {\frac{4}{{S_{1} H_{2} }}} \right)\left\{ {\frac{{1 - \cos \left[ {\left( {\frac{{p_{3} \pi }}{{S_{1} }}} \right)S_{1} } \right]}}{{\left( {\frac{{p_{3} \pi }}{{S_{1} }}} \right)}}} \right\}\left\{ {\frac{{ - \sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{3} }}{2}} \right)\frac{\pi }{{H_{2} }}} \right]H_{2} } \right\}}}{{\left[ {\left( {\frac{{1 - 2q_{3} }}{2}} \right)\frac{\pi }{{H_{2} }}} \right]}}} \right\} $$
(A28)

and

$$ F_{{p_{4} q_{4} (2)}} = \left( {\frac{4}{{S_{1} H_{2} }}} \right)\left\{ {\frac{{1 - \cos \left[ {\left( {\frac{{p_{4} \pi }}{{S_{1} }}} \right)S_{1} } \right]}}{{\left( {\frac{{p_{4} \pi }}{{S_{1} }}} \right)}}} \right\}\left\{ {\frac{{ - \sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{4} }}{2}} \right)\frac{\pi }{{H_{2} }}} \right]H_{2} } \right\}}}{{\left[ {\left( {\frac{{1 - 2q_{4} }}{2}} \right)\frac{\pi }{{H_{2} }}} \right]}}} \right\}, $$
(A29)

respectively. Also, by applying boundary conditions (XVIIIa) and (XVIIIb) to Eq. (21) and then carrying out a double Fourier run in \( 0 < x < S_{1} \) and \( 0 < y < S_{2} \) on the resultant equations, we get a relation for evaluation of coefficients \( J_{{p_{6} q_{6} (2)}} \) as

$$ J_{{p_{6} q_{6} (2)}} = \left[ {\frac{4}{{S_{2} (H_{3} - H_{2} )}}} \right]\left\{ {\frac{{1 - \cos \left[ {\left( {\frac{{p_{6} \pi }}{{S_{2} }}} \right)S_{2} } \right]}}{{\left( {\frac{{p_{6} \pi }}{{S_{2} }}} \right)}}} \right\}\left\{ {\frac{{ - \sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{6} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right](H_{1} - H_{2} )} \right\}}}{{\left[ {\left( {\frac{{1 - 2q_{6} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right]}}} \right\}. $$
(A30)

Similarly, from boundary conditions (XXIa), (XXIb) and Eq. (21), we arrive at the following expression for the coefficients \( I_{{p_{5} q_{5} (2)}} \)

$$ I_{{p_{5} q_{5} (2)}} = \left[ {\frac{4}{{S_{2} (H_{3} - H_{2} )}}} \right]\left\{ {\frac{{1 - \cos \left[ {\left( {\frac{{p_{5} \pi }}{{S_{2} }}} \right)S_{2} } \right]}}{{\left( {\frac{{p_{5} \pi }}{{S_{2} }}} \right)}}} \right\}\left\{ {\frac{{ - \sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{5} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right](H_{1} - H_{2} )} \right\}}}{{\left[ {\left( {\frac{{1 - 2q_{5} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right]}}} \right\}. $$
(A31)

Also, implementation of boundary conditions (XXIVa) and (XXIVb) for the middle soil layer at the ditch face \( y = 0 \) gives us a relation describing the coefficients \( L_{{p_{8} q_{8} (2)}} \) as

$$ L_{{p_{8} q_{8} (2)}} = \left[ {\frac{4}{{S_{1} (H_{3} - H_{2} )}}} \right]\left\{ {\frac{{1 - \cos \left[ {\left( {\frac{{p_{8} \pi }}{{S_{2} }}} \right)S_{1} } \right]}}{{\left( {\frac{{p_{8} \pi }}{{S_{1} }}} \right)}}} \right\}\left\{ {\frac{{ - \sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{8} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right](H_{1} - H_{2} )} \right\}}}{{\left[ {\left( {\frac{{1 - 2q_{8} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right]}}} \right\}. $$
(A32)

Further, by incorporating boundary conditions (XXVIIa) and (XXVIIb) in Eq. (21) and carrying out the necessary Fourier expansion on the resultant equations, we get an expression for evaluating \( K_{{p_{7} q_{7} (2)}} \) as

$$ K_{{p_{7} q_{7} (2)}} = \left[ {\frac{4}{{S_{1} (H_{3} - H_{2} )}}} \right]\left\{ {\frac{{1 - \cos \left[ {\left( {\frac{{p_{7} \pi }}{{S_{2} }}} \right)S_{1} } \right]}}{{\left( {\frac{{p_{7} \pi }}{{S_{1} }}} \right)}}} \right\}\left\{ {\frac{{ - \sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{7} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right](H_{1} - H_{2} )} \right\}}}{{\left[ {\left( {\frac{{1 - 2q_{7} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right]}}} \right\}. $$
(A33)

The expression for \( Q_{uv(2)} \) remains the same as in \( Q_{uv(1)} \) [Eq. (A16)] since they are both derived from the same boundary conditions.

Also, by equating the hydraulic heads for the top and the middle layers at \( z = H_{2} \) [intermediate condition (Ia)] and carrying out the necessary Fourier expansion on the resulting equation, we get the relation

$$ \begin{aligned} & E_{kl(2)} \tanh \left( {\lambda_{kl} H_{2} } \right) - G_{{i_{1} j_{1} (2)}} \frac{1}{{\cosh [\lambda_{{i_{1} j_{1} }} (H_{3} - H_{2} )]}} - H_{{i_{2} j_{2} (2)}} + Q_{uv(2)} \frac{1}{{\cosh \left( {\lambda_{uv} H_{2} } \right)}} \\ & = - \left( {\frac{{4H_{2} }}{{S_{1} S_{2} }}} \right)\left[ {\frac{{1 - \cos (N_{{u_{1} }} S_{1} )}}{{N_{{u_{1} }} }}} \right]\left[ {\frac{{1 - \cos (N_{{v_{1} }} S_{2} )}}{{N_{{v_{1} }} }}} \right] \\ & + \left( {\frac{2}{{S_{1} }}} \right)\sum\limits_{{q_{1} = 1}}^{{Q_{1} }} {B_{{p_{1} q_{1} (2)}} } \sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{1} }}{2}} \right)\frac{\pi }{{H_{2} }}} \right]H_{2} } \right\}\left[ {\frac{{N_{{u_{1} }} \cos (N_{{u_{1} }} S_{1} )}}{{N_{{u_{1} }}^{2} + \lambda_{{p_{1} q_{1} }}^{2} }}} \right] \\ & - \left( {\frac{2}{{S_{1} }}} \right)\sum\limits_{{q_{2} = 1}}^{{Q_{2} }} {C_{{p_{2} q_{2} (2)}} } \sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{2} }}{2}} \right)\frac{\pi }{{H_{2} }}} \right]H_{2} } \right\}\left[ {\frac{{N_{{u_{1} }} }}{{N_{{u_{1} }}^{2} + \lambda_{{p_{2} q_{2} }}^{2} }}} \right] \\ & + \left( {\frac{2}{{S_{2} }}} \right)\sum\limits_{{q_{3} = 1}}^{{Q_{3} }} {D_{{p_{3} q_{3} (2)}} } \sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{3} }}{2}} \right)\frac{\pi }{{H_{2} }}} \right]H_{2} } \right\}\left[ {\frac{{N_{{v_{1} }} \cos (N_{{v_{1} }} S_{2} )}}{{N_{{v_{1} }}^{2} + \lambda_{{p_{3} q_{3} }}^{2} }}} \right] \\ & - \left( {\frac{2}{{S_{2} }}} \right)\sum\limits_{{q_{4} = 1}}^{{Q_{4} }} {F_{{p_{4} q_{4} (2)}} } \sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{4} }}{2}} \right)\frac{\pi }{{H_{2} }}} \right]H_{2} } \right\}\left[ {\frac{{N_{{v_{1} }} }}{{N_{{v_{1} }}^{2} + \lambda_{{p_{4} q_{4} }}^{2} }}} \right], \\ \end{aligned} $$
(A34)

where \( N_{{u_{1} }} \) and \( N_{{v_{1} }} \) are as defined before, \( u_{1} = k = u = i_{1} = i_{2} = p_{3} = p_{4} , \) \( v_{1} = l = v = j_{1} = j_{2} = p_{1} = p_{2} \)and \( Q_{1} = Q_{2} = Q_{3} = Q_{4} \to \infty . \) Further, the flux equality of the top and middle soil layers at \( z = H_{2} \)[intermediate condition (Ib)] followed by appropriate Fourier expansion of the resultant equation give us the relation

$$ \begin{aligned} & \left( {\frac{{K_{{z_{1} }} }}{{K_{{z_{2} }} }}} \right)E_{kl(2)} \lambda_{kl} + H_{{i_{2} j_{2} (2)}} \lambda_{{i_{2} j_{2} }} \tanh [\lambda_{{i_{2} j_{2} }} (H_{3} - H_{2} )] = \\ & - \left( {\frac{2}{{S_{1} }}} \right)\sum\limits_{{q_{5} = 1}}^{{Q_{5} }} {I_{{p_{5} q_{5} (2)}} } \left[ {\left( {\frac{{1 - 2q_{5} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right]\left[ {\frac{{N_{{u_{1} }} \cos (N_{{u_{1} }} S_{1} )}}{{N_{{u_{1} }}^{2} + \lambda_{{p_{5} q_{5} }}^{2} }}} \right] \\ & + \left( {\frac{2}{{S_{1} }}} \right)\sum\limits_{{q_{6} = 1}}^{{Q_{6} }} {J_{{p_{6} q_{6} (2)}} } \left[ {\left( {\frac{{1 - 2q_{6} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right]\left[ {\frac{{N_{{u_{1} }} }}{{N_{{u_{1} }}^{2} + \lambda_{{p_{6} q_{6} }}^{2} }}} \right] \\ & - \left( {\frac{2}{{S_{2} }}} \right)\sum\limits_{{q_{7} = 1}}^{{Q_{7} }} {K_{{p_{7} q_{7} (2)}} } \left[ {\left( {\frac{{1 - 2q_{7} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right]\left[ {\frac{{N_{{v_{1} }} \cos (N_{{v_{1} }} S_{2} )}}{{N_{{v_{1} }}^{2} + \lambda_{{p_{7} q_{7} }}^{2} }}} \right] \\ & + \left( {\frac{2}{{S_{2} }}} \right)\sum\limits_{{q_{8} = 1}}^{{Q_{8} }} {L_{{p_{8} q_{8} (2)}} } \left[ {\left( {\frac{{1 - 2q_{8} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right]\left[ {\frac{{N_{{v_{1} }} }}{{N_{{v_{1} }}^{2} + \lambda_{{p_{8} q_{8} }}^{2} }}} \right], \\ \end{aligned} $$
(A35)

where \( u_{1} = k = i_{2} = p_{7} = p_{8} , \) \( v_{1} = l = j_{2} = p_{5} = p_{6} \) and \( Q_{5} = Q_{6} = Q_{7} = Q_{8} \to \infty . \) Also, from the conditions of head and flux equality of the middle and bottom soil layers at \( z = H_{3} \) [intermediate conditions (IIa) and (IIb)], we get by carrying out appropriate Fourier expansions, the relations

$$ \begin{aligned} & - G_{{i_{1} j_{1} (2)}} - H_{{i_{2} j_{2} (2)}} \frac{1}{{\cosh [\lambda_{{i_{2} j_{2} }} (H_{3} - H_{2} )]}} + P_{{i_{3} j_{3} (2)}} \frac{{\cosh [\lambda_{{i_{3} j_{3} }} (h - H_{3} )]}}{{\sinh [\lambda_{{i_{3} j_{3} }} (h - H_{3} )]}} = \\ & - \left( {\frac{2}{{S_{1} }}} \right)\sum\limits_{{q_{5} = 1}}^{{Q_{5} }} {I_{{p_{5} q_{5} (2)}} } \sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{5} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right](H_{3} - H_{2} )} \right\}\left[ {\frac{{N_{{u_{1} }} \cos (N_{{u_{1} }} S_{1} )}}{{N_{{u_{1} }}^{2} + \lambda_{{p_{5} q_{5} }}^{2} }}} \right] \\ & + \left( {\frac{2}{{S_{1} }}} \right)\sum\limits_{{q_{6} = 1}}^{{Q_{6} }} {J_{{p_{6} q_{6} (2)}} } \sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{6} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right](H_{3} - H_{2} )} \right\}\left[ {\frac{{N_{{u_{1} }} }}{{N_{{u_{1} }}^{2} + \lambda_{{p_{6} q_{6} }}^{2} }}} \right] \\ & - \left( {\frac{2}{{S_{2} }}} \right)\sum\limits_{{q_{7} = 1}}^{{Q_{7} }} {K_{{p_{7} q_{7} (2)}} } \sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{7} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right](H_{3} - H_{2} )} \right\}\left[ {\frac{{N_{{v_{1} }} \cos (N_{{v_{1} }} S_{2} )}}{{N_{{v_{1} }}^{2} + \lambda_{{p_{7} q_{7} }}^{2} }}} \right] \\ & + \left( {\frac{2}{{S_{2} }}} \right)\sum\limits_{{q_{8} = 1}}^{{Q_{8} }} {L_{{p_{8} q_{8} (2)}} } \sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{8} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right](H_{3} - H_{2} )} \right\}\left[ {\frac{{N_{{v_{1} }} }}{{N_{{v_{1} }}^{2} + \lambda_{{p_{8} q_{8} }}^{2} }}} \right] \\ & + \left( {\frac{4}{{S_{1} S_{2} }}} \right)(H_{1} - H_{2} )\left[ {\frac{{1 - \cos (N_{{u_{1} }} S_{1} )}}{{N_{{u_{1} }} }}} \right]\left[ {\frac{{1 - \cos (N_{{v_{1} }} S_{2} )}}{{N_{{v_{1} }} }}} \right] \\ \end{aligned} $$
(A36)

where \( u_{1} = i_{1} = i_{2} = i_{3} = p_{7} = p_{8} , \) \( v_{1} = j_{1} = j_{2} = j_{3} = p_{5} = p_{6} , \) \( Q_{5} = Q_{6} = Q_{7} = Q_{8} \to \infty \) and \( K_{{z_{2} }} G_{{i_{1} j_{1} (2)}} \lambda_{{i_{1} j_{1} }} \tanh [\lambda_{{i_{1} j_{1} }} (H_{3} - H_{2} )] + K_{{z_{3} }} P_{{i_{3} j_{3} (2)}} \lambda_{{i_{3} j_{3} }} = 0, \) (A37)

where \( i_{1} = i_{3} \) and \( j_{1} = j_{3} . \)

Equations (A26), (A27), (A28), (A29), (A16), (A30), (A31), (A32) and (A33) can now be used to determine the coefficients \( B_{{p_{1} q_{1} (2)}} , \) \( C_{{p_{2} q_{2} (2)}} , \) \( D_{{p_{3} q_{3} (2)}} , \) \( F_{{p_{4} q_{4} (2)}} , \) \( Q_{uv(2)} , \) \( I_{{p_{5} q_{5} (2)}} , \) \( J_{{p_{6} q_{6} (2)}} , \) \( K_{{p_{7} q_{7} (2)}} \) and \( L_{{p_{8} q_{8} (2)}} \) corresponding to any ditch drainage scenario of figure 1 when the level of water in the ditches lie in the middle layer; these values can then be used in Eqs. (A34), (A35), (A36) and (A37) to evaluate the remaining Fourier coefficients \( E_{kl(2)} , \) \( G_{{i_{1} j_{1} (2)}} , \) \( H_{{i_{2} j_{2} (2)}} \) and \( P_{{i_{3} j_{3} (2)}} \) corresponding to the studied drainage situation.

1.3 A.3 Case 3

The expressions for determining the coefficients \( B_{{p_{1} q_{1} (3)}} , \) \( C_{{p_{2} q_{2} (3)}} , \) \( D_{{p_{3} q_{3} (3)}} \) and \( F_{{p_{4} q_{4} (3)}} \) work out to be the same as those of \( B_{{p_{1} q_{1} (2)}} , \) \( C_{{p_{2} q_{2} (2)}} , \) \( D_{{p_{3} q_{3} (2)}} \) and \( F_{{p_{4} q_{4} (2)}} , \) respectively of the previous problem. Also, the expression for \( Q_{uv(3)} \) remains the same as that of \( Q_{uv(1)} \) and \( Q_{uv(2)} \) of the previous two problems. However, the relations for evaluation of \( I_{{p_{5} q_{5} (3)}} , \) \( J_{{p_{6} q_{6} (3)}} , \) \( K_{{p_{7} q_{7} (3)}} \) and \( L_{{p_{8} q_{8} (3)}} \) will now be different from those of the second problem; here they can be obtained by making use of the conditions (XXXIII), (XXX), (XXXIX) and (XXXVI). Using these conditions, the relations describing these coefficients for the present drainage situation can be expressed as

$$ I_{{p_{5} q_{5} (3)}} = \left[ {\frac{4}{{S_{2} (H_{3} - H_{2} )}}} \right]\left\{ {\frac{{1 - \cos \left[ {\left( {\frac{{p_{5} \pi }}{{S_{2} }}} \right)S_{2} } \right]}}{{\left( {\frac{{p_{5} \pi }}{{S_{2} }}} \right)}}} \right\}\left\{ {\frac{{ - \sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{5} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right](H_{3} - H_{2} )} \right\}}}{{\left[ {\left( {\frac{{1 - 2q_{5} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right]}}} \right\}, $$
(A38)
$$ J_{{p_{6} q_{6} (3)}} = \left[ {\frac{4}{{S_{2} (H_{3} - H_{2} )}}} \right]\left\{ {\frac{{1 - \cos \left[ {\left( {\frac{{p_{6} \pi }}{{S_{2} }}} \right)S_{2} } \right]}}{{\left( {\frac{{p_{6} \pi }}{{S_{2} }}} \right)}}} \right\}\left\{ {\frac{{ - \sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{6} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right](H_{3} - H_{2} )} \right\}}}{{\left[ {\left( {\frac{{1 - 2q_{6} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right]}}} \right\}, $$
(A39)
$$ K_{{p_{7} q_{7} (3)}} = \left[ {\frac{4}{{S_{1} (H_{3} - H_{2} )}}} \right]\left\{ {\frac{{1 - \cos \left[ {\left( {\frac{{p_{7} \pi }}{{S_{2} }}} \right)S_{1} } \right]}}{{\left( {\frac{{p_{7} \pi }}{{S_{1} }}} \right)}}} \right\}\left\{ {\frac{{ - \sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{7} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right](H_{3} - H_{2} )} \right\}}}{{\left[ {\left( {\frac{{1 - 2q_{7} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right]}}} \right\} $$
(A40)

and

$$ L_{{p_{8} q_{8} (3)}} = \left[ {\frac{4}{{S_{1} (H_{3} - H_{2} )}}} \right]\left\{ {\frac{{1 - \cos \left[ {\left( {\frac{{p_{8} \pi }}{{S_{2} }}} \right)S_{1} } \right]}}{{\left( {\frac{{p_{8} \pi }}{{S_{1} }}} \right)}}} \right\}\left\{ {\frac{{ - \sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{8} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right](H_{3} - H_{2} )} \right\}}}{{\left[ {\left( {\frac{{1 - 2q_{8} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right]}}} \right\}. $$
(A41)

Also, implementation of boundary conditions (XXXIa) and (XXXIb) on ϕ3(3) and then carrying out the necessary Fourier runs on the resultant equations gives us the relation for determination of \( N_{{p_{10} q_{10} (3)}} \) as

$$ N_{{p_{10} q_{10} (3)}} = \left[ {\frac{4}{{S_{2} (h - H_{3} )}}} \right]\left\{ {\frac{{1 - \cos \left[ {\left( {\frac{{p_{10} \pi }}{{S_{2} }}} \right)S_{2} } \right]}}{{\left( {\frac{{p_{10} \pi }}{{S_{2} }}} \right)}}} \right\}\left\{ {\frac{{ - \sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{10} }}{2}} \right)\left( {\frac{\pi }{{h - H_{3} }}} \right)} \right](H_{1} - H_{3} )} \right\}}}{{\left[ {\left( {\frac{{1 - 2q_{10} }}{2}} \right)\left( {\frac{\pi }{{h - H_{3} }}} \right)} \right]}}} \right\}. $$
(A42)

In the same way, by imposing boundary conditions (XXXIVa) and (XXXIVb) on Eq. (29), we get, after making the necessary Fourier runs, an equation for evaluating the coefficients \( M_{{p_{9} q_{9} (3)}} \) as

$$ M_{{p_{9} q_{9} (3)}} = \left[ {\frac{4}{{S_{2} (h - H_{3} )}}} \right]\left\{ {\frac{{1 - \cos \left[ {\left( {\frac{{p_{9} \pi }}{{S_{2} }}} \right)S_{2} } \right]}}{{\left( {\frac{{p_{9} \pi }}{{S_{2} }}} \right)}}} \right\}\left\{ {\frac{{ - \sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{9} }}{2}} \right)\left( {\frac{\pi }{{h - H_{3} }}} \right)} \right](H_{1} - H_{3} )} \right\}}}{{\left[ {\left( {\frac{{1 - 2q_{9} }}{2}} \right)\left( {\frac{\pi }{{h - H_{3} }}} \right)} \right]}}} \right\}. $$
(A43)

Further, implementation of boundary conditions (XXXVIIa), (XXXVIIb), (XLa) and (XLb) on Eq. (29) give us the relations for evaluation of \( V_{{p_{12} q_{12} (3)}} \) and \( U_{{p_{11} q_{11} (3)}} \) as

$$ V_{{p_{12} q_{12} (3)}} = \left[ {\frac{4}{{S_{1} (h - H_{3} )}}} \right]\left\{ {\frac{{1 - \cos \left[ {\left( {\frac{{p_{12} \pi }}{{S_{1} }}} \right)S_{1} } \right]}}{{\left( {\frac{{p_{12} \pi }}{{S_{1} }}} \right)}}} \right\}\left\{ {\frac{{ - \sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{12} }}{2}} \right)\left( {\frac{\pi }{{h - H_{3} }}} \right)} \right](H_{1} - H_{3} )} \right\}}}{{\left[ {\left( {\frac{{1 - 2q_{12} }}{2}} \right)\left( {\frac{\pi }{{h - H_{3} }}} \right)} \right]}}} \right\} $$
(A44)

and

$$ U_{{p_{11} q_{11} (3)}} = \left[ {\frac{4}{{S_{1} (h - H_{3} )}}} \right]\left\{ {\frac{{1 - \cos \left[ {\left( {\frac{{p_{11} \pi }}{{S_{1} }}} \right)S_{1} } \right]}}{{\left( {\frac{{p_{11} \pi }}{{S_{1} }}} \right)}}} \right\}\left\{ {\frac{{ - \sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{11} }}{2}} \right)\left( {\frac{\pi }{{h - H_{3} }}} \right)} \right](H_{1} - H_{3} )} \right\}}}{{\left[ {\left( {\frac{{1 - 2q_{11} }}{2}} \right)\left( {\frac{\pi }{{h - H_{3} }}} \right)} \right]}}} \right\}, $$
(A45)

respectively. Also, by carrying out head and flux equality of the top and middle layers at \( z = H_{2} \) [intermediate conditions (Ia) and (Ib)], we have for this flow situation the relations

$$ \begin{aligned} & E_{kl(3)} \tanh \left( {\lambda_{kl} H_{2} } \right) - G_{{i_{1} j_{1} (3)}} \frac{1}{{\cosh [\lambda_{{i_{1} j_{1} }} (H_{3} - H_{2} )]}} - H_{{i_{2} j_{2} (3)}} + Q_{uv(3)} \frac{1}{{\cosh (\lambda_{uv} H_{2} )}} \\ & = - \left( {\frac{{4H_{2} }}{{S_{1} S_{2} }}} \right)\left[ {\frac{{1 - \cos (N_{{u_{1} }} S_{1} )}}{{N_{{u_{1} }} }}} \right]\left[ {\frac{{1 - \cos \left( {N_{{v_{1} }} S_{2} } \right)}}{{N_{{v_{1} }} }}} \right] \\ & + \left( {\frac{2}{{S_{1} }}} \right)\sum\limits_{{q_{1} = 1}}^{{Q_{1} }} {B_{{p_{1} q_{1} (3)}} } \sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{1} }}{2}} \right)\frac{\pi }{{H_{2} }}} \right]H_{2} } \right\}\left[ {\frac{{N_{{u_{1} }} \cos (N_{{u_{1} }} S_{1} )}}{{N_{{u_{1} }}^{2} + \lambda_{{p_{1} q_{1} }}^{2} }}} \right] \\ & - \left( {\frac{2}{{S_{1} }}} \right)\sum\limits_{{q_{2} = 1}}^{{Q_{2} }} {C_{{p_{2} q_{2} (3)}} } \sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{2} }}{2}} \right)\frac{\pi }{{H_{2} }}} \right]H_{2} } \right\}\left[ {\frac{{N_{{u_{1} }} }}{{N_{{u_{1} }}^{2} + \lambda_{{p_{2} q_{2} }}^{2} }}} \right] \\ & + \left( {\frac{2}{{S_{2} }}} \right)\sum\limits_{{q_{3} = 1}}^{{Q_{3} }} {D_{{p_{3} q_{3} (3)}} } \sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{3} }}{2}} \right)\frac{\pi }{{H_{2} }}} \right]H_{2} } \right\}\left[ {\frac{{N_{{v_{1} }} \cos (N_{{v_{1} }} S_{2} )}}{{N_{{v_{1} }}^{2} + \lambda_{{p_{3} q_{3} }}^{2} }}} \right] \\ & - \left( {\frac{2}{{S_{2} }}} \right)\sum\limits_{{q_{4} = 1}}^{{Q_{4} }} {F_{{p_{4} q_{4} (3)}} } \sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{4} }}{2}} \right)\frac{\pi }{{H_{2} }}} \right]H_{2} } \right\}\left[ {\frac{{N_{{v_{1} }} }}{{N_{{v_{1} }}^{2} + \lambda_{{p_{4} q_{4} }}^{2} }}} \right] \\ \end{aligned} $$
(A46)

and

$$ \begin{aligned} & \left( {\frac{{K_{{z_{1} }} }}{{K_{{z_{2} }} }}} \right)E_{kl(3)} \lambda_{kl} + H_{{i_{2} j_{2} (3)}} \lambda_{{i_{2} j_{2} }} \tanh [\lambda_{{i_{2} j_{2} }} (H_{3} - H_{2} )] = \\ & - \left( {\frac{2}{{S_{1} }}} \right)\sum\limits_{{q_{5} = 1}}^{{Q_{5} }} {I_{{p_{5} q_{5} (3)}} } \left[ {\left( {\frac{{1 - 2q_{5} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right]\left[ {\frac{{N_{{u_{1} }} \cos (N_{{u_{1} }} S_{1} )}}{{N_{{u_{1} }}^{2} + \lambda_{{p_{5} q_{5} }}^{2} }}} \right] \\ & + \left( {\frac{2}{{S_{1} }}} \right)\sum\limits_{{q_{6} = 1}}^{{Q_{6} }} {J_{{p_{6} q_{6} (3)}} } \left[ {\left( {\frac{{1 - 2q_{6} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right]\left[ {\frac{{N_{{u_{1} }} }}{{N_{{u_{1} }}^{2} + \lambda_{{p_{6} q_{6} }}^{2} }}} \right] \\ & - \left( {\frac{2}{{S_{2} }}} \right)\sum\limits_{{q_{7} = 1}}^{{Q_{7} }} {K_{{p_{7} q_{7} (3)}} } \left[ {\left( {\frac{{1 - 2q_{7} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right]\left[ {\frac{{N_{{v_{1} }} \cos (N_{{v_{1} }} S_{2} )}}{{N_{{v_{1} }}^{2} + \lambda_{{p_{7} q_{7} }}^{2} }}} \right] \\ & + \left( {\frac{2}{{S_{2} }}} \right)\sum\limits_{{q_{8} = 1}}^{{Q_{8} }} {L_{{p_{8} q_{8} (3)}} } \left[ {\left( {\frac{{1 - 2q_{8} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right]\left[ {\frac{{N_{{v_{1} }} }}{{N_{{v_{1} }}^{2} + \lambda_{{p_{8} q_{8} }}^{2} }}} \right], \\ \end{aligned} $$
(A47)

respectively, where \( N_{{u_{1} }} \) and \( N_{{v_{1} }} \) are as defined before, \( u_{1} = k = u = i_{1} = i_{2} = p_{3} = p_{4} = p_{7} = p_{8} , \) \( v_{1} = l = v = j_{1} = j_{2} = p_{1} = p_{2} = p_{5} = p_{6} \) and \( Q_{1} = Q_{2} = Q_{3} = Q_{4} = Q_{5} = Q_{6} = Q_{7} = Q_{8} \to \infty . \) Further, equating the hydraulic heads and fluxes for the middle and bottom soil layers at the boundary \( z = H_{3} \) give us two additional equations linking the coefficients of these layers; they are as

$$ \begin{aligned} & - G_{{i_{1} j_{1} (3)}} - H_{{i_{2} j_{2} (3)}} \frac{1}{{\cosh [\lambda_{{i_{2} j_{2} }} (H_{3} - H_{2} )]}} + P_{{i_{3} j_{3} (3)}} \frac{{\cosh [\lambda_{{i_{3} j_{3} }} (h - H_{3} )]}}{{\sinh [\lambda_{{i_{3} j_{3} }} (h - H_{3} )]}} = \\ & - \left( {\frac{2}{{S_{1} }}} \right)\sum\limits_{{q_{5} = 1}}^{{Q_{5} }} {I_{{p_{5} q_{5} (3)}} } \sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{5} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right](H_{3} - H_{2} )} \right\}\left[ {\frac{{N_{{u_{1} }} \cos (N_{{u_{1} }} S_{1} )}}{{N_{{u_{1} }}^{2} + \lambda_{{p_{5} q_{5} }}^{2} }}} \right] \\ & + \left( {\frac{2}{{S_{1} }}} \right)\sum\limits_{{q_{6} = 1}}^{{Q_{6} }} {J_{{p_{6} q_{6} (3)}} } \sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{6} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right](H_{3} - H_{2} )} \right\}\left[ {\frac{{N_{{u_{1} }} }}{{N_{{u_{1} }}^{2} + \lambda_{{p_{6} q_{6} }}^{2} }}} \right] \\ & - \left( {\frac{2}{{S_{2} }}} \right)\sum\limits_{{q_{7} = 1}}^{{Q_{7} }} {K_{{p_{7} q_{7} (3)}} } \sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{7} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right](H_{3} - H_{2} )} \right\}\left[ {\frac{{N_{{v_{1} }} \cos (N_{{v_{1} }} S_{2} )}}{{N_{{v_{1} }}^{2} + \lambda_{{p_{7} q_{7} }}^{2} }}} \right] \\ & + \left( {\frac{2}{{S_{2} }}} \right)\sum\limits_{{q_{8} = 1}}^{{Q_{8} }} {L_{{p_{8} q_{8} (3)}} } \sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{8} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right](H_{3} - H_{2} )} \right\}\left[ {\frac{{N_{{v_{1} }} }}{{N_{{v_{1} }}^{2} + \lambda_{{p_{8} q_{8} }}^{2} }}} \right] \\ & + \left( {\frac{4}{{S_{1} S_{2} }}} \right)(H_{3} - H_{2} )\left[ {\frac{{1 - \cos (N_{{u_{1} }} S_{1} )}}{{N_{{u_{1} }} }}} \right]\left[ {\frac{{1 - \cos (N_{{v_{1} }} S_{2} )}}{{N_{{v_{1} }} }}} \right] \\ \end{aligned} $$
(A48)

and

$$ \begin{aligned} & \left( {\frac{{K_{{z_{2} }} }}{{K_{{z_{3} }} }}} \right)G_{{i_{1} j_{1} (3)}} \lambda_{{i_{1} j_{1} }} \tanh [\lambda_{{i_{1} j_{1} }} (H_{3} - H_{2} )] + P_{{i_{3} j_{3} (3)}} \lambda_{{i_{3} j_{3} }} = \\ & - \left( {\frac{2}{{S_{1} }}} \right)\sum\limits_{{q_{9} = 1}}^{{Q_{9} }} {M_{{p_{9} q_{9} (3)}} \left[ {\left( {\frac{{1 - 2q_{9} }}{2}} \right)\left( {\frac{\pi }{{h - H_{3} }}} \right)} \right]} \left[ {\frac{{N_{{u_{1} }} \cos (N_{{u_{1} }} S_{1} )}}{{N_{{u_{1} }}^{2} + \lambda_{{p_{9} q_{9} }}^{2} }}} \right] \\ & + \left( {\frac{2}{{S_{1} }}} \right)\sum\limits_{{q_{10} = 1}}^{{Q_{10} }} {N_{{p_{10} q_{10} (3)}} } \left[ {\left( {\frac{{1 - 2q_{10} }}{2}} \right)\left( {\frac{\pi }{{h - H_{3} }}} \right)} \right]\left[ {\frac{{N_{{u_{1} }} }}{{N_{{u_{1} }}^{2} + \lambda_{{p_{10} q_{10} }}^{2} }}} \right] \\ & - \left( {\frac{2}{{S_{2} }}} \right)\sum\limits_{{q_{11} = 1}}^{{Q_{11} }} {U_{{p_{11} q_{11} (3)}} } \left[ {\left( {\frac{{1 - 2q_{11} }}{2}} \right)\left( {\frac{\pi }{{h - H_{3} }}} \right)} \right]\left[ {\frac{{N_{{v_{1} }} \cos (N_{{v_{1} }} S_{2} )}}{{N_{{v_{1} }}^{2} + \lambda_{{p_{11} q_{11} }}^{2} }}} \right] \\ & + \left( {\frac{2}{{S_{2} }}} \right)\sum\limits_{{q_{12} = 1}}^{{Q_{12} }} {V_{{p_{12} q_{12} (3)}} \left[ {\left( {\frac{{1 - 2q_{12} }}{2}} \right)\left( {\frac{\pi }{{h - H_{3} }}} \right)} \right]\left[ {\frac{{N_{{v_{1} }} }}{{N_{{v_{1} }}^{2} + \lambda_{{p_{12} q_{12} }}^{2} }}} \right]} , \\ \end{aligned} $$
(A49)

where \( u_{1} = i_{1} = i_{3} = p_{11} = p_{12} = p_{7} = p_{8} , \) \( v_{1} = j_{1} = j_{2} = j_{3} = p_{9} = p_{10} = p_{5} = p_{6} \) and \( Q_{5} = Q_{6} = Q_{7} = Q_{8} \) \( = Q_{9} = Q_{10} = Q_{11} = Q_{12} \to \infty . \) Now, Eqs. (A46), (A47), (A48) and (A49) can be solved simultaneously to evaluate the coefficients \( E_{kl(3)} , \) \( G_{{i_{1} j_{1} (3)}} , \) \( H_{{i_{2} j_{2} (3)}} \) and \( P_{{i_{3} j_{3} (3)}} \) pertaining to any drainage situation of figure 1 when the level of water in the drains lie solely on the bottom layer. It again remains without saying that before Eqs. (A46), (A47), (A48) and (A49) could be utilized to determine these constants for a drainage scenario, here also it is incumbent that the coefficients \( B_{{p_{1} q_{1} (3)}} , \) \( C_{{p_{2} q_{2} (3)}} , \) \( D_{{p_{3} q_{3} (3)}} , \) \( F_{{p_{4} q_{4} (3)}} , \) \( Q_{uv(3)} , \) \( I_{{p_{5} q_{5} (3)}} , \) \( J_{{p_{6} q_{6} (3)}} , \) \( K_{{p_{7} q_{7} (3)}} , \) \( L_{{p_{8} q_{8} (3)}} , \) \( M_{{p_{9} q_{9} (3)}} , \) \( N_{{p_{10} q_{10} (3)}} , \) \( U_{{p_{11} q_{11} (3)}} \) and \( V_{{p_{12} q_{12} (3)}} \) be first worked out [by utilizing Eqs. (A26), (A27), (A28), (A29), (A16), (A38), (A39), (A40), (A41), (A42), (A43), (A44) and (A45), respectively] for the studied situation as without these values \( E_{kl(3)} , \) \( G_{{i_{1} j_{1} (3)}} , \) \( H_{{i_{2} j_{2} (3)}} \) and \( P_{{i_{3} j_{3} (3)}} \) cannot not be determined using these equations.

Appendix B: Divergence of the top discharge function [i.e., Eq. (17)] when calculated at a point separating two unequal ponding depths at the surface of the soil

To demonstrate the above observation, we first consider a situation where the upper limit of the integral of Eq. (17) in the \( x - \) direction is taken at \( x = d_{x1} . \) This condition is chosen since the inner bund at this location, as may be observed (figure 1), separates two unequal ponding depths \( \delta_{1} \) and \( \delta_{2} \) at the surface of the soil. Incorporating this and the expression for \( Q_{uv(1)} \) of Eq. (A16) in Eq. (18), we get, after some mathematical simplifications, a term like

$$ \begin{aligned} & \left( {\frac{4}{{S_{1} S_{2} }}} \right)\sqrt {\frac{{K_{{x_{1} }} }}{{K_{{z_{1} }} }}} \sum\limits_{u = 1}^{U} {\sum\limits_{v = 1}^{V} {\delta_{1} } } \left( {\frac{1}{{N_{u} N_{v}^{2} }}} \right)\sqrt {\left[ {1 + \frac{{N_{v}^{2} }}{{N_{u}^{2} }}\left( {\frac{{K_{{y_{1} }} }}{{K_{{x_{1} }} }}} \right)} \right]} \tanh \left\{ {\left[ {\sqrt {N_{u}^{2} \left( {\frac{{K_{{x_{1} }} }}{{K_{{z_{1} }} }}} \right) + N_{v}^{2} \left( {\frac{{K_{{y_{1} }} }}{{K_{{z_{1} }} }}} \right)} } \right]H_{2} } \right\} \\ & \times \cos^{2} (N_{u} d_{x1} )\cos (N_{v} y)\cos (N_{v} d_{{y(2N_{0} - 2)}} ) \\ \end{aligned} $$

It is easily discernable that for any specified value of v, both \( \sqrt {\left[ {1 + \frac{{N_{v}^{2} }}{{N_{u}^{2} }}\left( {\frac{{K_{{y_{1} }} }}{{K_{{x_{1} }} }}} \right)} \right]} \) and \( \tanh \left\{ {\left[ {\sqrt {N_{u}^{2} \left( {\frac{{K_{{x_{1} }} }}{{K_{{z_{1} }} }}} \right) + N_{v}^{2} \left( {\frac{{K_{{y_{1} }} }}{{K_{{z_{1} }} }}} \right)} } \right]H_{2} } \right\} \) tend to 1 when we allow u, and consequently \( N_{u} , \) to increase sans any upper limit. This can be mathematically represented as

$$ \mathop {Lim}\limits_{u \to \infty } \, \sqrt {\left[ {1 + \left( {\frac{{S_{1} N_{v} }}{u\pi }} \right)^{2} \left( {\frac{{K_{{y_{1} }} }}{{K_{{x_{1} }} }}} \right)} \right]} = 1 $$

and

$$ \mathop {Lim}\limits_{u \to \infty } \, \tanh \left\{ {\left[ {\sqrt {\left( {\frac{u\pi }{{S_{1} }}} \right)^{2} \left( {\frac{{K_{{x_{1} }} }}{{K_{{z_{1} }} }}} \right) + N_{v}^{2} \left( {\frac{{K_{{y_{1} }} }}{{K_{{z_{1} }} }}} \right)} } \right]H_{2} } \right\} = 1. $$

If we assume that \( \tanh \left\{ {\left[ {\sqrt {N_{u}^{2} \left( {\frac{{K_{{x_{1} }} }}{{K_{{z_{1} }} }}} \right) + N_{v}^{2} \left( {\frac{{K_{{y_{1} }} }}{{K_{{z_{1} }} }}} \right)} } \right]H_{2} } \right\} \) and \( \sqrt {\left[ {1 + \frac{{N_{v}^{2} }}{{N_{u}^{2} }}\left( {\frac{{K_{{y_{1} }} }}{{K_{{x_{1} }} }}} \right)} \right]} \)reach approximately 1 after the first four terms of u for a specified value of v, then the rest of the aforementioned series, after some mathematical manipulations, can be expressed as

$$ \left( {\frac{{4\delta_{1} }}{{\pi S_{2} }}} \right)\sqrt {\frac{{K_{{x_{1} }} }}{{K_{{z_{1} }} }}} \sum\limits_{u = 5}^{U \to \infty } {\left( {\frac{1}{{uN_{v}^{2} }}} \right)} \cos^{2} (u\pi \alpha )\cos (N_{v} y)\cos (N_{v} d_{{y(2N_{0} - 2)}} ), $$

where \( \alpha = d_{x1} /S_{1} \, (0 < \alpha < 1). \) It is to be noted that \( \cos^{2} (u\pi \alpha ) \) will lie between 0 and 1 for all possible values of \( u \) \( \left[ {{\text{i}} . {\text{e}} . ,0 \le \cos^{2} (u\pi \alpha ) \le 1, u \in \left\{ {5,6,7 \ldots } \right\}} \right] \). Thus, there can be two possibilities for a chosen \( \alpha , \) namely \( \cos^{2} (u\pi \alpha ) \ne 0 \) for any value of \( u \in \left\{ {5,6,7 \ldots } \right\} \) or \( \cos^{2} (u\pi \alpha ) = 0 \) for a subset of positive integral values represented as \( \left\{ {u_{n1} ,u_{n2} ,u_{n3} \ldots } \right\}, \) which belongs to the set \( u \in \left\{ {5,6,7 \ldots } \right\} \). For the first case, if we conveniently assume that \( M_{\hbox{min} } = {\text{ minimum of cos}}^{ 2} (u\pi \alpha ) \) for any \( u \in \left\{ {5,6,7 \ldots } \right\} \), then we can write \( M_{\hbox{min} } \left( {\frac{{4\delta_{1} }}{{\pi S_{2} }}} \right)\frac{1}{{N_{v}^{2} }}\cos (N_{v} y)\cos (N_{v} d_{{y(2N_{0} - 2)}} )\sum\limits_{u = 5}^{U \to \infty } {\frac{1}{u}} < \left( {\frac{{4\delta_{1} }}{{\pi S_{2} }}} \right)\frac{1}{{N_{v}^{2} }}\cos (N_{v} y)\cos (N_{v} d_{{y(2N_{0} - 2)}} )\sum\limits_{u = 5}^{U \to \infty } {\frac{{\cos^{2} (u\pi \alpha )}}{u}} \)

However, since the series \( \sum\limits_{u = 5}^{U \to \infty } {\left( {\frac{1}{u}} \right)} \) diverges, we can safely infer that so does the series on the right hand side of the inequality.

Now, we need to prove the divergence of the series for the second case, where \( \cos^{2} (u\pi \alpha ) = 0 \) for \( u \in \left\{ {u_{n1} ,u_{n2} ,u_{n3} \ldots } \right\} \). For our convenience, we divide the series into two sub-series as follows

$$ \begin{aligned} & \left( {\frac{{4\delta_{1} }}{{\pi S_{2} }}} \right)\frac{1}{{N_{v}^{2} }}\cos (N_{v} y)\cos (N_{v} d_{{y(2N_{0} - 2)}} )\sum\limits_{{u_{ni} }}^{U \to \infty } {\frac{{\cos^{2} (u\pi \alpha )}}{u}} \\ & + \left( {\frac{{4\delta_{1} }}{{\pi S_{2} }}} \right)\frac{1}{{N_{v}^{2} }}\cos (N_{v} y)\cos (N_{v} d_{{y(2N_{0} - 2)}} )\sum\limits_{u = 5}^{U \to \infty } {\frac{{\cos^{2} (u\pi \alpha )}}{u}} , \\ \end{aligned} $$

where the summation index \( u \) of the second term of the series belongs to the set \( \left\{ {5,6,7 \ldots } \right\}\backslash \left\{ {u_{n1} ,u_{n2} ,u_{n3} \ldots } \right\} \) \( = u \in \left\{ {5,6,7 \ldots } \right\}\backslash \left\{ {u_{n1} ,u_{n2} ,u_{n3} \ldots } \right\} = \left\{ {u_{p1} ,u_{p2} ,u_{p3} \ldots } \right\} \) (say). If we now take \( M_{\hbox{min} }^{'} = {\text{ minimum of }}\cos^{2} (u\pi \alpha ) \) for all \( u \in \left\{ {u_{p1} ,u_{p2} ,u_{p3} \ldots } \right\} \), we will then have the inequality

$$ M_{\hbox{min} }^{'} \left( {\frac{{4\delta_{1} }}{{\pi S_{2} }}} \right)\frac{1}{{N_{v}^{2} }}\cos (N_{v} y)\cos (N_{v} d_{{y(2N_{0} - 2)}} )\sum\limits_{u = 5}^{U \to \infty } {\frac{1}{u}} < \left( {\frac{{4\delta_{1} }}{{\pi S_{2} }}} \right)\frac{1}{{N_{v}^{2} }}\cos (N_{v} y)\cos (N_{v} d_{{y(2N_{0} - 2)}} )\sum\limits_{u = 5}^{U \to \infty } {\frac{{\cos^{2} (u\pi \alpha )}}{u}} $$

But even in this case, the series in the right hand side of the above inequality diverges. Hence, we can state with conviction that \( Q_{top(1)}^{f} \) diverges at this bund location. In an analogous fashion, it can be proved that \( Q_{top(1)}^{f} \) happens to diverge at other inner bund locations too when they divide two unequal ponding depths at the surface of the soil.

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Chakrabarti, S., Barua, G. Analysis of three-dimensional ponded drainage of a multi-layered soil underlain by an impervious barrier. Sādhanā 45, 234 (2020). https://doi.org/10.1007/s12046-020-01445-8

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