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

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.

1 Introduction

A considerable proportion of the world’s population still depends on agriculture for their sustenance and about 70% of all abstracted water is being currently utilized for agriculture to enhance crop production [1, 2]. The global area under agriculture, however, is expected to go down in future as a result of mostly wide spread urbanization of existing agricultural lands [3]. Also, bringing in more freshwater into agriculture in future will be a challenge as the water demands by other competing sectors like municipality, industry and environment are also likely to increase in future as well. Thus, efforts must be directed to augment agricultural productivity per unit area of cultivable lands to meet the increasing food demand of an expanding global population [4] and to meet this objective, improved irrigation and drainage practices on existing agricultural lands are expected to play a leading part [5, 6]. Irrigation, as a practice for augmenting agricultural productivity, has been there for a long time now and its role in future for enhancing agricultural outputs is only expected to increase [7]. However, irrigation often leads to water-logging and salinity in irrigated lands [6, 8, 9] – problems which must be negotiated for competitive agriculture to prevail. One of the most potent ways of arresting irrigation-induced salinity and water-logging in agricultural fields is through introduction of subsurface drains in these fields – several studies on the subject have proven the veracity of this argument ([6, 10,11,12,13,14,15,16,17] – to name a few). The process generally involves forcing good quality irrigation water through a salt affected soil so as to wash and remove the salts present in the soil profile to a desirable level and then draining and collecting this salt-laden water with the help of a network of subsurface drains installed for the purpose [18,19,20,21,22,23,24,25]. The head necessary to force water through the salt-laden soil is often provided by introducing a uniform ponding head at the surface of the soil. Subsurface drains may be buried pipelines or open ditches but ditch drains are mainly preferred in locations where the soil conductivity is relatively low and the topography comparatively flat [26, 27]. Subsurface drainage is now also becoming increasingly important in paddy fields as controlled drainage of these fields has been found to greatly reduce the emissions of methane and nitrous oxide from these fields – two atmospheric trace gases which contribute greatly to the cause of global warming and ozone depletion [28,29,30,31,32]. Thus, the role of subsurface drainage now-a-days is not only restricted for mitigating water-logging and salinity of irrigated soils but in few other areas of environmental remediation as well.

Most of the analytical studies related to open drains are done by assuming the flow to be either one or two-dimensional in nature and by assuming the soil to be homogeneous and isotropic.Over the years, several investigators ([33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49] – to cite a few) have provided analytical models describing steady two-dimensional groundwater flow to subsurface drains from ponded fields under different hydraulic settings of the problem. However, apart from Sarmah and Barua’s 2015 [49] solution, none of the other solutions can account for stratification of soils. Sarmah and Barua [25] provided an analytical model for the ponded drainage problem which can accommodate three-dimensional flow situations but this solution is also not for multi-layered soils. Soils in nature, however, are mostly stratified rather than uniform and soil stratification may greatly affect the distribution and movement of water and contaminant through them [50,51,52,53,54]. The conductivity contrast of the soil layers may be noticed in its extreme form in a paddy field where, very often, an extremely low conductivity top plow-sole layer can be seen to exist over relatively more conductive layer(s). Also, numerous modeling studies have categorically demonstrated that the conductivity and thickness of this plow-sole layer play a pivotal role in deciding overall subsurface water dynamics of a paddy field [55,56,57,58]. Further, within a layer also, the conductivity of a soil may vary substantially with the direction of flow [59,60,61] and layered [62] or compacted soils [63, 64] may exhibit a higher conductivity in the horizontal direction than that in the vertical direction. However, the situation may be a reversed one for a well-structured soil, where the soil may instead exhibit a higher conductivity in the vertical direction in comparison to that in the horizontal direction [65, 66]. It is worth noting that information on saturated hydraulic conductivity and anisotropy of the comprising layers of a stratified soil column are fundamental for successful modelling of two- and three-dimensional transport of water and contaminant in a heterogeneous porous formation [67]. While a drainage model with the two-dimensional flow assumption may loosely hold for large-sized fields, it may fall shy in many real field situations where the size of a field may be quite small. This is because subsurface flow to a trench/stream is mostly three-dimensional in nature and serious error in the simulation results may incur if this aspect of flow is not being considered during the development of a subterranean flow model [68,69,70]. This has also been one of the findings of Sarmah and Barua’s [25] analytical works where it is seen that seepage to subsurface drains from a small-sized ponded field is mostly three-dimensional in nature, particularly in areas close to the drains. Thus, noting the fact that soils in general are mostly stratified and that three-dimensional flows are a common norm in a ponded drainage system, it is felt that there is a need to develop a comprehensive solution to the ponded drainage problem by including all these factors in the problem’s framework. This study attempts to address this need.

2 Problem statement and solution

Figure 1 shows the geometry of the flow problem under consideration. As can be seen, the flow domain considered for analysis is assumed as a rectangular ponded field $S_{1} \times S_{2}$ in horizontal extent and underlain by an impervious barrier at a depth of h from the surface of the soil. The field is being drained on all of its sides by four vertical drains running all the way up to the impervious barrier. The depths of the bottom of the first and the second layers are taken as $H_{2}$ and $H_{3} ,$ respectively – all these distances being measured with respect to the surface of the soil. The water level is assumed to be the same for all the drains and is denoted as $H_{1} ,$ where again the distance is considered from the surface of the soil. For mathematical convenience, a specific coordinate system with the origin located at O is been fixed to the flow domain as shown. A variable ponding pyramid is introduced at the top of the field by making use of a network of thin bunds placed at the surface of the soil as illustrated in the $x^{{\prime }} - x^{{{\prime \prime }}}$ and $y^{{\prime }} - y^{{{\prime \prime }}}$ cross-sectional views of the problem figure. For simplicity, it is also assumed that a continuous refilling in the field is keeping this ponding field unchanging with time. $N_{0}$ denotes the number of ponding strips while $\delta_{ps}$ $(1 \le ps \le N_{0} )$ represents the depth of ponding of the $ps^{th}$ strip as measured from the surface of the soil. Ditch bunds of width $\varepsilon_{x}$ and $\varepsilon_{y}$ are being placed at the edges of the field so as to prevent the ponded water from flowing directly into the ditches. The distances of the $i^{th}$ inner bund from the origin in the $x -$ and $y -$ directions are taken as $d_{xi}$ and $d_{yi} ,$ respectively. The steady state governing equations for the three layers of the flow domain, based on the principle of continuity and on the assumptions of incompressibility of soil and the traversing fluid, can be expressed as [71]

Figure 1

General geometry of a three-dimensional ponded ditch drainage system subject to a variable ponding distribution at the surface of the soil.

$$ K_{{x_{hc} }} \frac{{\partial^{2} \phi_{hc(j)} }}{{\partial x^{2} }} + K_{{y_{hc} }} \frac{{\partial^{2} \phi_{hc(j)} }}{{\partial y^{2} }} + K_{{z_{hc} }} \frac{{\partial^{2} \phi_{hc(j)} }}{{\partial z^{2} }} = 0,\quad (hc = 1,{ 2, 3)} $$

(1)

where $K_{{x_{hc} }} ,$ $K_{{y_{hc} }}$ and $K_{{z_{hc} }}$ are the directional hydraulic conductivities of the soil layers in the $x - ,$ $y -$ and $z -$ directions, respectively and $\phi_{hc(j)}$ are the hydraulic head functions in the three aforementioned soil layers. The index j takes on the values 1, 2 and 3 depending on the location of water level in the ditches, namely, it is 1 if the level of water in the ditches is anywhere in the top layer, 2 if it is anywhere in the intermediate layer and 3 if it is lying anywhere in the bottom layer. Thus, the flow problem of figure 1 will have three independent parts depending on the level of water in the drains. However, whatever may be the location of the water level in the drains, there will be a few conditions which will be common for all these flow situations; they can be expressed as

$$ \phi_{1(j)} \left( {x,y,z} \right) = \phi_{2(j)} \left( {x,y,z} \right),\quad 0 < x < S_{1} ,\quad 0 < y < S_{2} ,\quad z = H_{2} , $$

(Ia)

$$ - K_{{z_{1} }} \frac{{\partial \phi_{1(j)} \left( {x,y,z} \right)}}{\partial z} = - K_{{z_{2} }} \frac{{\partial \phi_{2(j)} \left( {x,y,z} \right)}}{\partial z},\quad 0 < x < S_{1} ,\quad 0 < y < S_{2} ,\quad z = H_{2} , $$

(Ib)

$$ \phi_{2(j)} \left( {x,y,z} \right) = \phi_{3(j)} \left( {x,y,z} \right),\quad 0 < x < S_{1} ,\quad 0 < y < S_{2} ,\quad z = H_{3} , $$

(IIa)

$$ - K_{{z_{2} }} \frac{{\partial \phi_{2(j)} \left( {x,y,z} \right)}}{\partial z} = - K_{{z_{3} }} \frac{{\partial \phi_{3(j)} \left( {x,y,z} \right)}}{\partial z},\quad 0 < x < S_{1} ,\quad 0 < y < S_{2} ,\quad z = H_{3} , $$

(IIb)

$$ \phi_{1(j)} \left( {x,y,z} \right) = \delta_{1} ,\quad 0 < x < S_{1} ,\quad 0 < y < d_{y1} ,\quad z = 0, $$

(IIIa)

$$ \phi_{1(j)} \left( {x,y,z} \right) = \delta_{1} ,\quad 0 < x < S_{1} ,\quad d_{{y(2N_{0} - 2)}} < y < S_{2} ,\quad z = 0, $$

(IIIb)

$$ \phi_{1(j)} \left( {x,y,z} \right) = \delta_{1} ,\quad 0 < x < d_{x1} ,\quad d_{y1} < y < d_{{y(2N_{0} - 2)}} ,\quad z = 0, $$

(IIIc)

$$ \phi_{1(j)} \left( {x,y,z} \right) = \delta_{1} ,\quad d_{{x(2N_{0} - 2)}} < x < S_{1} ,\quad d_{y1} < y < d_{{y(2N_{0} - 2)}} ,\quad z = 0, $$

(IIId)

$$ \phi_{1(j)} \left( {x,y,z} \right) = \delta_{i} ,\quad d_{x(i - 1)} < x < d_{{x(2N_{0} - i)}} ,\quad d_{y(i - 1)} < y < d_{yi} ,\quad z = 0, $$

(IIIe)

$$ \phi_{1(j)} \left( {x,y,z} \right) = \delta_{i} ,\quad d_{x(i - 1)} < x < d_{{x(2N_{0} - i)}} ,\quad d_{{y(2N_{0} - i - 1)}} < y < d_{{y(2N_{0} - i)}} ,\quad z = 0, $$

(IIIf)

$$ \phi_{1(j)} \left( {x,y,z} \right) = \delta_{i} ,\quad d_{x(i - 1)} < x < d_{xi} ,\quad d_{yi} < y < d_{{y(2N_{0} - i - 1)}} ,\quad z = 0, $$

(IIIg)

$$ \phi_{1(j)} \left( {x,y,z} \right) = \delta_{i} ,\quad d_{{x(2N_{0} - i - 1)}} < x < d_{{x(2N_{0} - i)}} ,\quad d_{yi} < y < d_{{y(2N_{0} - i - 1)}} ,\quad z = 0, $$

(IIIh)

$$ \phi_{1(j)} \left( {x,y,z} \right) = \delta_{{N_{0} }} ,\quad d_{{x(N_{0} - 1)}} < x < d_{{xN_{0} }} ,\quad d_{{y(N_{0} - 1)}} < y < d_{{yN_{0} }} ,\quad z = 0, $$

(IIIi)

$$ - K_{{z_{3} }} \frac{{\partial \phi_{3(j)} \left( {x,y,z} \right)}}{\partial z} = 0,\quad 0 < x < S_{1} ,\quad 0 < y < S_{2} ,\quad z = h. $$

(IV)

where $2 \le i \le N_{0} - 1$ and $N_{0} > 2.$ It should be noted that $N_{0} = 1$ will mean no inner bunds and the water level will then be a constant over the entire surface of the soil. Also, for $N_{0} = 2$ there will be two inner bunds both in the $x -$ and $y -$ directions and naturally for these situations boundary conditions (IIIe) to (IIIh) will not come into picture. We will now attempt to obtain solutions to the different flow situations of figure 1 arising from different locations of water level in the ditches starting with the case when the level of water in the drains lies in the top layer.

2.1 Case 1: Level of water in the ditches is on or above the boundary between the top and the middle soil layers

The specific boundary conditions pertaining to this flow situation can be represented as

$$ \phi_{1(1)} \left( {x,y,z} \right) = - z,\quad x = 0,\quad 0 < y < S_{2} ,\quad 0 < z \le H_{1} , $$

(Va)

$$ \phi_{1(1)} \left( {x,y,z} \right) = - H_{1} ,\quad x = 0,\quad 0 < y < S_{2} ,\quad H_{1} \le z \le H_{2} , $$

(Vb)

$$ \phi_{2(1)} \left( {x,y,z} \right) = - H_{1} ,\quad x = 0,\quad 0 < y < S_{2} ,\quad H_{2} \le z \le H_{3} , $$

(VI)

$$ \phi_{3(1)} \left( {x,y,z} \right) = - H_{1} ,\quad x = 0,\quad 0 < y < S_{2} \quad H_{3} \le z < h, $$

(VII)

$$ \phi_{1(1)} \left( {x,y,z} \right) = - z,\quad x = S_{1} ,\quad 0 < y < S_{2} ,\quad 0 < z \le H_{1} , $$

(VIIIa)

$$ \phi_{1(1)} \left( {x,y,z} \right) = - H_{1} ,\quad x = S_{1} ,\quad 0 < y < S_{2} ,\quad H_{1} \le z \le H_{2} , $$

(VIIIb)

$$ \phi_{2(1)} \left( {x,y,z} \right) = - H_{1} ,\quad x = S_{1} ,\quad 0 < y < S_{2} ,\quad H_{2} \le z \le H_{3} , $$

(IX)

$$ \phi_{3(1)} \left( {x,y,z} \right) = - H_{1} ,\quad x = S_{1} ,\quad 0 < y < S_{2} ,\quad H_{3} \le z < h, $$

(X)

$$ \phi_{1(1)} \left( {x,y,z} \right) = - z,\quad 0 < x < S_{1} ,\quad y = 0,\quad 0 < z \le H_{1} , $$

(XIa)

$$ \phi_{1(1)} \left( {x,y,z} \right) = - H_{1} ,\quad 0 < x < S_{1} ,\quad y = 0,\quad H_{1} \le z \le H_{2} , $$

(XIb)

$$ \phi_{2(1)} \left( {x,y,z} \right) = - H_{1} ,\quad 0 < x < S_{1} ,\quad y = 0,\quad H_{2} \le z \le H_{3} , $$

(XII)

$$ \phi_{3(1)} \left( {x,y,z} \right) = - H_{1} ,\quad 0 < x < S_{1} ,\quad y = 0,\quad H_{3} \le z < h, $$

(XIII)

$$ \phi_{1(1)} \left( {x,y,z} \right) = - z,\quad 0 < x < S_{1} ,\quad y = S_{2} ,\quad 0 < z \le H_{1} , $$

(XIVa)

$$ \phi_{1(1)} \left( {x,y,z} \right) = - H_{1} ,\quad 0 < x < S_{1} ,\quad y = S_{2} ,\quad H_{1} \le z \le H_{2} , $$

(XIVb)

$$ \phi_{2(1)} \left( {x,y,z} \right) = - H_{1} ,\quad 0 < x < S_{1} ,\quad y = S_{2} ,\quad H_{2} \le z \le H_{3} , $$

(XV)

$$ \phi_{3(1)} \left( {x,y,z} \right) = - H_{1} ,\quad 0 < x < S_{1} ,\quad y = S_{2} ,\quad H_{3} \le z < h. $$

(XVI)

Assuming the solution of Eq. (1) as a product of three distinct functions of the space variables $x,$ $y$ and $z,$ respectively and then applying the same on it and separating the variables out [72], the hydraulic head expressions for the three soil layers for this flow problem, in view of the above boundary conditions, can be written as

$$ \begin{aligned} & \phi_{1(1)} \left( {x,y,z} \right) = \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]z} \right\} \times \frac{{\sinh (\lambda_{{p_{1} q_{1} }} x)}}{{\sinh (\lambda_{{p_{1} q_{1} }} S_{1} )}} \\ & + \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\} \times \frac{{\sinh [\lambda_{{p_{2} q_{2} }} (S_{1} - x)]}}{{\sinh (\lambda_{{p_{2} q_{2} }} S_{1} )}} \\ & + \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]z} \right\} \times \frac{{\sinh (\lambda_{{p_{3} q_{3} }} y)}}{{\sinh (\lambda_{{p_{3} q_{3} }} S_{2} )}} \\ & + \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]z} \right\} \times \frac{{\sinh [\lambda_{{p_{4} q_{4} }} (S_{2} - y)]}}{{\sinh (\lambda_{{p_{4} q_{4} }} S_{2} )}} \\ & + \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] \times \frac{{\sinh (\lambda_{kl} z)}}{{\cosh (\lambda_{kl} H_{2} )}} \\ & + \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] \times \frac{{\cosh \left[ {\lambda_{uv} (H_{2} - z)} \right]}}{{\cosh (\lambda_{uv} H_{2} )}}, \\ \end{aligned} $$

(2)

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

(3)

and

$$ \phi_{3(1)} \left( {x,y,z} \right) = \sum\limits_{{i_{3} = 1}}^{{I_{3} }} {\sum\limits_{{j_{3} = 1}}^{{J_{3} }} {P_{{i_{3} j_{3} (1)}} } } \sin \left[ {\left( {\frac{{i_{3} \pi }}{{S_{1} }}} \right)x} \right]\sin \left[ {\left( {\frac{{j_{3} \pi }}{{S_{2} }}} \right)y} \right] \times \frac{{\cosh [\lambda_{{i_{3} j_{3} }} (h - z)]}}{{\sinh [\lambda_{{i_{3} j_{3} }} (h - H_{3} )]}} - H_{1} , $$

(4)

where

$$ (\lambda_{{p_{1} q_{1} }} )^{2} = \left\{ {\left( {\frac{{p_{1} \pi }}{{S_{2} }}} \right)^{2} \left( {\frac{{K_{{y_{1} }} }}{{K_{{x_{1} }} }}} \right) + \left[ {\left( {\frac{{1 - 2q_{1} }}{2}} \right)\frac{\pi }{{H_{2} }}} \right]^{2} \left( {\frac{{K_{{z_{1} }} }}{{K_{{x_{1} }} }}} \right)} \right\}, $$

(5)

$$ (\lambda_{{p_{2} q_{2} }} )^{2} = \left\{ {\left( {\frac{{p_{2} \pi }}{{S_{2} }}} \right)^{2} \left( {\frac{{K_{{y_{1} }} }}{{K_{{x_{1} }} }}} \right) + \left[ {\left( {\frac{{1 - 2q_{2} }}{2}} \right)\frac{\pi }{{H_{2} }}} \right]^{2} \left( {\frac{{K_{{z_{1} }} }}{{K_{{x_{1} }} }}} \right)} \right\}, $$

(6)

$$ (\lambda_{{p_{3} q_{3} }} )^{2} = \left\{ {\left( {\frac{{p_{3} \pi }}{{S_{1} }}} \right)^{2} \left( {\frac{{K_{{x_{1} }} }}{{K_{{y_{1} }} }}} \right) + \left[ {\left( {\frac{{1 - 2q_{3} }}{2}} \right)\frac{\pi }{{H_{2} }}} \right]^{2} \left( {\frac{{K_{{z_{1} }} }}{{K_{{y_{1} }} }}} \right)} \right\}, $$

(7)

$$ (\lambda_{{p_{4} q_{4} }} )^{ \, 2} = \left\{ {\left( {\frac{{p_{4} \pi }}{{S_{1} }}} \right)^{2} \left( {\frac{{K_{{x_{1} }} }}{{K_{{y_{1} }} }}} \right) + \left[ {\left( {\frac{{1 - 2q_{4} }}{2}} \right)\frac{\pi }{{H_{2} }}} \right]^{2} \left( {\frac{{K_{{z_{1} }} }}{{K_{{y_{1} }} }}} \right)} \right\}, $$

(8)

$$ (\lambda_{kl} )^{2} = \left[ {\left( {\frac{k\pi }{{S_{1} }}} \right)^{2} \left( {\frac{{K_{{x_{1} }} }}{{K_{{z_{1} }} }}} \right) + \left( {\frac{l\pi }{{S_{2} }}} \right)^{2} \left( {\frac{{K_{{y_{1} }} }}{{K_{{z_{1} }} }}} \right)} \right], $$

(9)

$$ (\lambda_{uv} )^{2} = \left[ {\left( {\frac{u\pi }{{S_{1} }}} \right)^{2} \left( {\frac{{K_{{x_{1} }} }}{{K_{{z_{1} }} }}} \right) + \left( {\frac{v\pi }{{S_{2} }}} \right)^{2} \left( {\frac{{K_{{y_{1} }} }}{{K_{{z_{1} }} }}} \right)} \right], $$

(10)

$$ (\lambda_{{i_{1} j_{1} }} )^{2} = \left[ {\left( {\frac{{i_{1} \pi }}{{S_{1} }}} \right)^{2} \left( {\frac{{K_{{x_{2} }} }}{{K_{{z_{2} }} }}} \right) + \left( {\frac{{j_{1} \pi }}{{S_{2} }}} \right)^{2} \left( {\frac{{K_{{y_{2} }} }}{{K_{{z_{2} }} }}} \right)} \right], $$

(11)

$$ (\lambda_{{i_{2} j_{2} }} )^{2} = \left[ {\left( {\frac{{i_{2} \pi }}{{S_{1} }}} \right)^{2} \left( {\frac{{K_{{x_{2} }} }}{{K_{{z_{2} }} }}} \right) + \left( {\frac{{j_{2} \pi }}{{S_{2} }}} \right)^{2} \left( {\frac{{K_{{y_{2} }} }}{{K_{{z_{2} }} }}} \right)} \right], $$

(12)

$$ (\lambda_{{i_{3} j_{3} }} )^{2} = \left[ {\left( {\frac{{i_{3} \pi }}{{S_{1} }}} \right)^{2} \left( {\frac{{K_{{x_{3} }} }}{{K_{{z_{3} }} }}} \right) + \left( {\frac{{j_{3} \pi }}{{S_{2} }}} \right)^{2} \left( {\frac{{K_{{y_{3} }} }}{{K_{{z_{3} }} }}} \right)} \right] $$

(13)

and $B_{{p_{1} q_{1} (1)}} ,$ $C_{{p_{2} q_{2} (1)}} ,$ $D_{{p_{3} q_{3} (1)}} ,$ $F_{{p_{4} q_{4} (1)}} ,$ $E_{kl(1)} ,$ $Q_{uv(1)} ,$ $G_{{i_{1} j_{1} (1)}} ,$ $H_{{i_{2} j_{2} (1)}}$ and $P_{{i_{3} j_{3} (1)}}$ are all constants. These coefficients can now be estimated by utilizing the boundary and intermediate conditions of the problem as demonstrated in “Appendix A”.

For evaluating the velocity distributions, Darcy’s law can next be applied. Thus, the expressions for determining the velocity functions in the three soil layers can be expressed as

$$ V_{xhc(1)} = - K_{{x_{hc} }} \frac{{\partial \phi_{hc(1)} (x,y,z)}}{\partial x}, $$

(14)

$$ V_{yhc(1)} = - K_{{y_{hc} }} \frac{{\partial \phi_{hc(1)} (x,y,z)}}{\partial y}, $$

(15)

and

$$ V_{zhc(1)} = - K_{{z_{hc} }} \frac{{\partial \phi_{hc(1)} (x,y,z)}}{\partial z},\quad (hc = 1,{ 2, 3)} $$

(16)

where $V_{xhc(1)} ,$ $V_{yhc(1)}$ and $V_{zhc(1)}$ are the velocity distributions in the $x - ,$ $y -$ and $z -$directions, respectively. The top discharge function, $Q_{top(1)}^{f} (x,y),$ for any point $(x,y)$ at the top of the flow domain is evaluated as

$$ Q_{top(1)}^{f} (x,y) = - K_{{z_{1} }} \int\limits_{{\varepsilon_{x} }}^{x} {\int\limits_{{\varepsilon_{y} }}^{y} {\left( {\frac{{\partial \phi_{1(1)} }}{\partial z}} \right)_{z = 0} } } dxdy. $$

(17)

Solving the above equation utilizing the head expression for the top layer [(Eq. (2)], we get the final expression for $Q_{top(1)}^{f}$ as

$$ \begin{aligned} & Q_{top(1)}^{f} (x,y) = - K_{{z_{1} }} \left\{ {\sum\limits_{{p_{1} = 1}}^{{P_{1} }} {\sum\limits_{{q_{1} = 1}}^{{Q_{1} }} {B_{{p_{1} q_{1} (1)}} } } \left[ {\left( {\frac{{1 - 2q_{1} }}{2}} \right)\left( {\frac{\pi }{{H_{2} \lambda_{{p_{1} q_{1} }} }}} \right)} \right]\left\{ {\frac{{\cos \left[ {\left( {\frac{{p_{1} \pi }}{{S_{2} }}} \right)\varepsilon_{y} } \right] - \cos \left[ {\left( {\frac{{p_{1} \pi }}{{S_{2} }}} \right)y} \right]}}{{\left( {\frac{{p_{1} \pi }}{{S_{2} }}} \right)}}} \right\}} \right. \times \left\{ {\frac{{\cosh (\lambda_{{p_{1} q_{1} }} x) - \cosh (\lambda_{{p_{1} q_{1} }} \varepsilon_{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)}} } } \left[ {\left( {\frac{{1 - 2q_{2} }}{2}} \right)\left( {\frac{\pi }{{H_{2} \lambda_{{p_{2} q_{2} }} }}} \right)} \right]\left\{ {\frac{{\cos \left[ {\left( {\frac{{p_{2} \pi }}{{S_{2} }}} \right)\varepsilon_{y} } \right] - \cos \left[ {\left( {\frac{{p_{2} \pi }}{{S_{2} }}} \right)y} \right]}}{{\left( {\frac{{p_{2} \pi }}{{S_{2} }}} \right)}}} \right\} \times \left\{ {\frac{{\cosh [\lambda_{{p_{2} q_{2} }} (S_{1} - \varepsilon_{x} )] - \cosh [\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)}} } } \left[ {\left( {\frac{{1 - 2q_{3} }}{2}} \right)\left( {\frac{\pi }{{H_{2} \lambda_{{p_{3} q_{3} }} }}} \right)} \right]\left\{ {\frac{{\cos \left[ {\left( {\frac{{p_{3} \pi }}{{S_{1} }}} \right)\varepsilon_{x} } \right] - \cos \left[ {\left( {\frac{{p_{3} \pi }}{{S_{1} }}} \right)x} \right]}}{{\left( {\frac{{p_{3} \pi }}{{S_{1} }}} \right)}}} \right\} \times \left\{ {\frac{{\cosh (\lambda_{{p_{3} q_{3} }} y) - \cosh (\lambda_{{p_{3} q_{3} }} \varepsilon_{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)}} } } \left[ {\left( {\frac{{1 - 2q_{4} }}{2}} \right)\left( {\frac{\pi }{{H_{2} \lambda_{{p_{4} q_{4} }} }}} \right)} \right]\left\{ {\frac{{\cos \left[ {\left( {\frac{{p_{4} \pi }}{{S_{1} }}} \right)\varepsilon_{x} } \right] - \cos \left[ {\left( {\frac{{p_{4} \pi }}{{S_{1} }}} \right)x} \right]}}{{\left( {\frac{{p_{4} \pi }}{{S_{1} }}} \right)}}} \right\} \times \left\{ {\frac{{\cosh [\lambda_{{p_{4} q_{4} }} (S_{2} - \varepsilon_{y} )] - \cosh [\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)} } } \left[ {\frac{{\lambda_{kl} }}{{\cosh (\lambda_{kl} H_{2} )}}} \right]\left\{ {\frac{{\cos \left[ {\left( {\frac{k\pi }{{S_{1} }}} \right)\varepsilon_{x} } \right] - \cos \left[ {\left( {\frac{k\pi }{{S_{1} }}} \right)x} \right]}}{{\left( {\frac{k\pi }{{S_{1} }}} \right)}}} \right\} \times \left\{ {\frac{{\cos \left[ {\left( {\frac{l\pi }{{S_{2} }}} \right)\varepsilon_{y} } \right] - \cos \left[ {\left( {\frac{l\pi }{{S_{2} }}} \right)y} \right]}}{{\left( {\frac{l\pi }{{S_{2} }}} \right)}}} \right\} \\ & - \sum\limits_{u = 1}^{U} {\sum\limits_{v = 1}^{V} {Q_{uv(1)} } } \lambda_{uv} \tanh (\lambda_{uv} H_{2} )\left\{ {\frac{{\cos \left[ {\left( {\frac{u\pi }{{S_{1} }}} \right)\varepsilon_{x} } \right] - \cos \left[ {\left( {\frac{u\pi }{{S_{1} }}} \right)x} \right]}}{{\left( {\frac{u\pi }{{S_{1} }}} \right)}}} \right\} \times \left. {\left\{ {\frac{{\cos \left[ {\left( {\frac{v\pi }{{S_{2} }}} \right)\varepsilon_{y} } \right] - \cos \left[ {\left( {\frac{v\pi }{{S_{2} }}} \right)y} \right]}}{{\left( {\frac{v\pi }{{S_{2} }}} \right)}}} \right\}} \right\}. \\ \end{aligned} $$

(18)

Here, we need to note that $Q_{top(1)}^{f}$ diverges (“Appendix B”) when it is being calculated exactly at a location separating two unequal ponding depths at the top of the field. Also, in order to get the total discharge for a flow scenario, we need to simply alter the upper limits of integration from $x$ and $y$ to $S_{1} - \varepsilon_{x}$ and $S_{2} - \varepsilon_{y} ,$ respectively in the integrals of Eq. (17). Thus, we get the expression for the total discharge, $Q_{top(1)}^{{}} ,$ from the top of the soil as

$$ Q_{top(1)} = - K_{{z_{1} }} \int\limits_{{\varepsilon_{x} }}^{{S_{1} - \varepsilon_{x} }} {\int\limits_{{\varepsilon_{y} }}^{{S_{2} - \varepsilon_{y} }} {\left( {\frac{{\partial \phi_{1(1)} }}{\partial z}} \right)_{z = 0} } } dxdy. $$

(19)

Like the top discharge, the discharge through the sides of the drains can also be easily worked out by making use of Darcy’s law and the hydraulic head expressions of the layers. As these expressions can be straightway obtained by applying Darcy’s law on the head functions, we are not giving them here. We next proceed to obtain solution of the flow problem of figure 1 for the situation when the water level of the drains lies in the middle layer.

2.2 Case 2: Level of water in the ditches is below the boundary between the top and the middle soil layers but on or above the boundary between the middle and bottom soil layers

The boundary conditions applicable to this drainage situation can be expressed as

$$ \phi_{1(2)} \left( {x,y,z} \right) = - z,\quad x = 0,\quad 0 < y < S_{2} ,\quad 0 < z \le H_{2} , $$

(XVII)

$$ \phi_{2(2)} \left( {x,y,z} \right) = - z,\quad x = 0,\quad 0 < y < S_{2} ,\quad H_{2} \le z \le H_{1} , $$

(XVIIIa)

$$ \phi_{2(2)} \left( {x,y,z} \right) = - H_{1} ,\quad x = 0,\quad 0 < y < S_{2} ,\quad H_{1} \le z \le H_{3} , $$

(XVIIIb)

$$ \phi_{3(2)} \left( {x,y,z} \right) = - H_{1} ,\quad x = 0,\quad 0 < y < S_{2} ,\quad H_{3} \le z < h, $$

(XIX)

$$ \phi_{1(2)} \left( {x,y,z} \right) = - z,\quad x = S_{1} ,\quad 0 < y < S_{2} ,\quad 0 < z \le H_{2} , $$

(XX)

$$ \phi_{2(2)} \left( {x,y,z} \right) = - z,\quad x = S_{1} ,\quad 0 < y < S_{2} ,\quad H_{2} \le z \le H_{1} , $$

(XXIa)

$$ \phi_{2(2)} \left( {x,y,z} \right) = - H_{1} ,\quad x = S_{1} ,\quad 0 < y < S_{2} ,\quad H_{1} \le z \le H_{3} , $$

(XXIb)

$$ \phi_{3(2)} \left( {x,y,z} \right) = - H_{1} ,\quad x = S_{1} ,\quad 0 < y < S_{2} ,\quad H_{3} \le z < h, $$

(XXII)

$$ \phi_{1(2)} \left( {x,y,z} \right) = - z,\quad 0 < x < S_{1} ,\quad y = 0,\quad 0 < z \le H_{2} , $$

(XXIII)

$$ \phi_{2(2)} \left( {x,y,z} \right) = - z,\quad 0 < x < S_{1} ,\quad y = 0,\quad H_{2} \le z \le H_{1} , $$

(XXIVa)

$$ \phi_{2(2)} \left( {x,y,z} \right) = - H_{1} ,\quad 0 < x < S_{1} ,\quad y = 0,\quad H_{1} \le z \le H_{3} , $$

(XXIVb)

$$ \phi_{3(2)} \left( {x,y,z} \right) = - H_{1} ,\quad 0 < x < S_{1} ,\quad y = 0,\quad H_{3} \le z < h, $$

(XXV)

$$ \phi_{1(2)} \left( {x,y,z} \right) = - z,\quad 0 < x < S_{1} ,\quad y = S_{2} ,\quad 0 < z \le H_{2} , $$

(XXVI)

$$ \phi_{2(2)} \left( {x,y,z} \right) = - z,\quad 0 < x < S_{1} ,\quad y = S_{2} ,\quad H_{2} \le z \le H_{1} , $$

(XXVIIa)

$$ \phi_{2(2)} \left( {x,y,z} \right) = - H_{1} ,\quad 0 < x < S_{1} ,\quad y = S_{2} ,\quad H_{1} \le z \le H_{3} , $$

(XXVIIb)

$$ \phi_{3(2)} \left( {x,y,z} \right) = - H_{1} ,\quad 0 < x < S_{1} ,\quad y = S_{2} ,\quad H_{3} \le z < h. $$

(XXVIII)

Falling again on the separation of variables method, the hydraulic head expressions for the layers for this case, in view of the above boundary conditions, can be written as

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

(20)

$$ \begin{aligned} & \phi_{2(2)} \left( {x,y,z} \right) = \sum\limits_{{i_{1} = 1}}^{{I_{1} }} {\sum\limits_{{j_{1} = 1}}^{{J_{1} }} {G_{{i_{1} j_{1} (2)}} } } \sin \left[ {\left( {\frac{{i_{1} \pi }}{{S_{1} }}} \right)x} \right]\sin \left[ {\left( {\frac{{j_{1} \pi }}{{S_{2} }}} \right)y} \right] \times \frac{{\cosh [\lambda_{{i_{1} j_{1} }} (z - H_{2} )]}}{{\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} (2)}} } } \sin \left[ {\left( {\frac{{i_{2} \pi }}{{S_{1} }}} \right)x} \right]\sin \left[ {\left( {\frac{{j_{2} \pi }}{{S_{2} }}} \right)y} \right] \times \frac{{\cosh [\lambda_{{i_{2} j_{2} }} (H_{3} - z)]}}{{\cosh [\lambda_{{i_{2} j_{2} }} (H_{3} - H_{2} )]}} \\ & + \sum\limits_{{p_{5} = 1}}^{{P_{5} }} {\sum\limits_{{q_{5} = 1}}^{{Q_{5} }} {I_{{p_{5} q_{5} (2)}} } } \sin \left[ {\left( {\frac{{p_{5} \pi }}{{S_{2} }}} \right)y} \right]\sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{5} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right](z - H_{2} )} \right\} \times \frac{{\sinh (\lambda_{{p_{5} q_{5} }} x)}}{{\sinh (\lambda_{{p_{5} q_{5} }} S_{1} )}} \\ & + \sum\limits_{{p_{6} = 1}}^{{P_{6} }} {\sum\limits_{{q_{6} = 1}}^{{Q_{6} }} {J_{{p_{6} q_{6} (2)}} } } \sin \left[ {\left( {\frac{{p_{6} \pi }}{{S_{2} }}} \right)y} \right]\sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{6} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right](z - H_{2} )} \right\} \times \frac{{\sinh [\lambda_{{p_{6} q_{6} }} (S_{1} - x)]}}{{\sinh (\lambda_{{p_{5} q_{5} }} S_{1} )}} \\ & + \sum\limits_{{p_{7} = 1}}^{{P_{7} }} {\sum\limits_{{q_{7} = 1}}^{{Q_{7} }} {K_{{p_{7} q_{7} (2)}} } } \sin \left[ {\left( {\frac{{p_{7} \pi }}{{S_{1} }}} \right)x} \right]\sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{7} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right](z - H_{2} )} \right\} \times \frac{{\sinh (\lambda_{{p_{7} q_{7} }} y)}}{{\sinh (\lambda_{{p_{7} q_{7} }} S_{2} )}} \\ & + \sum\limits_{{p_{8} = 1}}^{{P_{8} }} {\sum\limits_{{q_{8} = 1}}^{{Q_{8} }} {L_{{p_{8} q_{8} (2)}} } } \sin \left[ {\left( {\frac{{p_{8} \pi }}{{S_{1} }}} \right)x} \right]\sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{8} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right](z - H_{2} )} \right\} \times \frac{{\sinh [\lambda_{{p_{8} q_{8} }} (S_{2} - y)]}}{{\sinh (\lambda_{{p_{8} q_{8} }} S_{2} )}} \\ & - H_{2} \\ \end{aligned} $$

(21)

and

$$ \phi_{3(2)} \left( {x,y,z} \right) = \sum\limits_{{i_{3} = 1}}^{{I_{3} }} {\sum\limits_{{j_{3} = 1}}^{{J_{3} }} {P_{{i_{3} j_{3} (2)}} } } \sin \left[ {\left( {\frac{{i_{3} \pi }}{{S_{1} }}} \right)x} \right]\sin \left[ {\left( {\frac{{j_{3} \pi }}{{S_{2} }}} \right)y} \right] \times \frac{{\cosh [\lambda_{{i_{3} j_{3} }} (h - z)]}}{{\sinh [\lambda_{{i_{3} j_{3} }} (h - H_{3} )]}} - H_{1} , $$

(22)

where $\lambda_{{p_{1} q_{1} }} ,$ $\lambda_{{p_{2} q_{2} }} ,$ $\lambda_{{p_{3} q_{3} }} ,$ $\lambda_{{p_{4} q_{4} }} ,$ $\lambda_{kl} ,$ $\lambda_{uv} ,$ $\lambda_{{i_{1} j_{1} }} ,$ $\lambda_{{i_{2} j_{2} }}$ and $\lambda_{{i_{3} j_{3} }}$are the same as mentioned in Eqs. (5), (6), (7), (8), (9), (10), (11), (12) and (13), respectively and

$$ (\lambda_{{p_{5} q_{5} }} )^{2} = \left\{ {\left( {\frac{{p_{5} \pi }}{{S_{2} }}} \right)^{2} \left( {\frac{{K_{{y_{2} }} }}{{K_{{x_{2} }} }}} \right) + \left[ {\left( {\frac{{1 - 2q_{5} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right]^{2} \left( {\frac{{K_{{z_{2} }} }}{{K_{{x_{2} }} }}} \right)} \right\}, $$

(23)

$$ (\lambda_{{p_{6} q_{6} }} )^{2} = \left\{ {\left( {\frac{{p_{6} \pi }}{{S_{2} }}} \right)^{2} \left( {\frac{{K_{{y_{2} }} }}{{K_{{x_{2} }} }}} \right) + \left[ {\left( {\frac{{1 - 2q_{6} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right]^{2} \left( {\frac{{K_{{z_{2} }} }}{{K_{{x_{2} }} }}} \right)} \right\}, $$

(24)

$$ (\lambda_{{p_{7} q_{7} }} )^{2} = \left\{ {\left( {\frac{{p_{7} \pi }}{{S_{1} }}} \right)^{2} \left( {\frac{{K_{{x_{2} }} }}{{K_{{y_{2} }} }}} \right) + \left[ {\left( {\frac{{1 - 2q_{7} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right]^{2} \left( {\frac{{K_{{z_{2} }} }}{{K_{{y_{2} }} }}} \right)} \right\}, $$

(25)

$$ (\lambda_{{p_{8} q_{8} }} )^{2} = \left\{ {\left( {\frac{{p_{8} \pi }}{{S_{1} }}} \right)^{2} \left( {\frac{{K_{{x_{2} }} }}{{K_{{y_{2} }} }}} \right) + \left[ {\left( {\frac{{1 - 2q_{8} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right]^{2} \left( {\frac{{K_{{z_{2} }} }}{{K_{{y_{2} }} }}} \right)} \right\}. $$

(26)

Further, as shown in “Appendix A”, here also the constants $B_{{p_{1} q_{1} (2)}} ,$ $C_{{p_{2} q_{2} (2)}} ,$ $D_{{p_{3} q_{3} (2)}} ,$ $F_{{p_{4} q_{4} (2)}} ,$ $E_{kl(2)} ,$ $Q_{uv(2)} ,$ $G_{{i_{1} j_{1} (2)}} ,$ $H_{{i_{2} j_{2} (2)}} ,$ $I_{{p_{5} q_{5} (2)}} ,$ $J_{{p_{6} q_{6} (2)}} ,$ $K_{{p_{7} q_{7} (2)}} ,$ $L_{{p_{8} q_{8} (2)}}$ and $P_{{i_{3} j_{3} (2)}}$ of the head expressions can be determined by making use of the boundary and intermediate conditions pertaining to the problem. Also, since the hydraulic head expressions for the top and the bottom layers for this drainage situation are similar to those of the previous case, the Darcian velocity expressions for these layers for the present case (i.e., $V_{x1\left( 2 \right)} ,$ $V_{y1\left( 2 \right)} ,$ $V_{z1\left( 2 \right)} ,$ $V_{x3\left( 2 \right)} ,$ $V_{y3\left( 2 \right)}$ and $V_{z3\left( 2 \right)}$) would then naturally be also similar to the corresponding expressions of the previous case (i.e., to $V_{x1\left( 1 \right)} ,$ $V_{y1\left( 1 \right)} ,$ $V_{z1\left( 1 \right)} ,$ $V_{x3\left( 1 \right)} ,$ $V_{y3\left( 1 \right)}$ and $V_{z3\left( 1 \right)}$). However, it must be noted that the relevant Fourier coefficients to be used in these expressions are now $B_{{p_{1} q_{1} (2)}} ,$ $C_{{p_{2} q_{2} (2)}} ,$ $D_{{p_{3} q_{3} (2)}} ,$ $F_{{p_{4} q_{4} (2)}} ,$ $E_{kl(2)} ,$ $Q_{uv(2)} ,$ $P_{{i_{3} j_{3} (2)}}$ and not $B_{{p_{1} q_{1} (1)}} ,$ $C_{{p_{2} q_{2} (1)}} ,$ $D_{{p_{3} q_{3} (1)}} ,$ $F_{{p_{4} q_{4} (1)}} ,$ $E_{kl(1)} ,$ $Q_{uv(1)}$ and $P_{{i_{3} j_{3} (1)}}$ of the previous case. Also, as the hydraulic head function of the middle layer for this drainage situation is different from that of the previous case, the directional velocity functions for this layer, naturally, would then also be different from those of the previous case. But, like before, these functions can also be easily worked out by a direct application of Darcy’s law on the hydraulic head expression of the middle layer [i.e., on Eq. (21)]. Finally, since the hydraulic head function for the top layer for this drainage situation is similar to the previous case, the expression for the top discharge function, $Q_{top(2)}^{f} ,$ for this case, will hence be also identical to that of the previous case; however, while using this expression, care need to be exercised to see that the Fourier coefficients corresponding to this case are only being used (i.e., $B_{{p_{1} q_{1} (2)}} ,$ $C_{{p_{2} q_{2} (2)}} ,$ $D_{{p_{3} q_{3} (2)}} ,$ $F_{{p_{4} q_{4} (2)}} ,$ $E_{kl(2)}$ and $Q_{uv(2)}$) and not the coefficients pertaining to the previous case.

We will now address the last of our problem for the case when the level of water in the drains lies solely in the bottom layer.

2.3 Case 3: Level of water in the ditches is below the boundary between the middle and bottom soil layers

The boundary condition pertaining to this flow situation can be represented as

$$ \phi_{1(3)} \left( {x,y,z} \right) = - z,\quad x = 0,\quad 0 < y < S_{2} ,\quad 0 < z \le H_{2} , $$

(XXIX)

$$ \phi_{2(3)} \left( {x,y,z} \right) = - z,\quad x = 0,\quad 0 < y < S_{2} ,\quad H_{2} \le z \le H_{3} , $$

(XXX)

$$ \phi_{3(3)} \left( {x,y,z} \right) = - z,\quad x = 0,\quad 0 < y < S_{2} ,\quad H_{3} \le z \le H_{1} , $$

(XXXIa)

$$ \phi_{3(3)} \left( {x,y,z} \right) = - H_{1} ,\quad x = 0,\quad 0 < y < S_{2} ,\quad H_{1} \le z < h, $$

(XXXIb)

$$ \phi_{1(3)} \left( {x,y,z} \right) = - z,\quad x = S_{1} ,\quad 0 < y < S_{2} ,\quad 0 < z \le H_{2} , $$

(XXXII)

$$ \phi_{2(3)} \left( {x,y,z} \right) = - z,\;x = S_{1} ,\quad 0 < y < S_{2} ,\quad H_{2} \le z \le H_{3} , $$

(XXXIII)

$$ \phi_{3(3)} \left( {x,y,z} \right) = - z,\quad x = S_{1} ,\quad 0 < y < S_{2} ,\quad H_{3} \le z \le H_{1} , $$

(XXXIVa)

$$ \phi_{3(3)} \left( {x,y,z} \right) = - H_{1} ,\quad x = S_{1} ,\quad 0 < y < S_{2} ,\quad H_{1} \le z < h, $$

(XXXIVb)

$$ \phi_{1(3)} \left( {x,y,z} \right) = - z,\quad 0 < x < S_{1} ,\quad y = 0,\quad 0 < z \le H_{2} , $$

(XXXV)

$$ \phi_{2(3)} \left( {x,y,z} \right) = - z,\quad 0 < x < S_{1} ,\quad y = 0,\quad H_{2} \le z \le H_{3} , $$

(XXXVI)

$$ \phi_{3(3)} \left( {x,y,z} \right) = - z,\quad 0 < x < S_{1} ,\quad y = 0,\quad H_{3} \le z \le H_{1} , $$

(XXXVIIa)

$$ \phi_{3(3)} \left( {x,y,z} \right) = - H_{1} ,\quad 0 < x < S_{1} ,\quad y = 0,\quad H_{1} \le z < h, $$

(XXXVIIb)

$$ \phi_{1(3)} \left( {x,y,z} \right) = - z,\quad 0 < x < S_{1} ,\quad y = S_{2} ,\quad 0 < z \le H_{2} , $$

(XXXVIII)

$$ \phi_{2(3)} \left( {x,y,z} \right) = - z,\quad 0 < x < S_{1} ,\quad y = S_{2} ,\quad H_{2} \le z \le H_{3} , $$

(XXXIX)

$$ \phi_{3(3)} \left( {x,y,z} \right) = - z,\quad 0 < x < S_{1} ,\quad y = S_{2} ,\quad H_{3} \le z \le H_{1} , $$

(XLa)

$$ \phi_{3(3)} \left( {x,y,z} \right) = - H_{1} ,\quad 0 < x < S_{1} ,\quad y = S_{2} ,\quad H_{1} \le z < h. $$

(XLb)

Taking recourse again to the separation of variables method, the hydraulic head expressions for the layers for this drainage situation, considering the conditions as listed above, can be expressed as

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

(27)

$$ \begin{aligned} & \phi_{2(3)} \left( {x,y,z} \right) = \sum\limits_{{i_{1} = 1}}^{{I_{1} }} {\sum\limits_{{j_{1} = 1}}^{{J_{1} }} {G_{{i_{1} j_{1} (3)}} } } \sin \left[ {\left( {\frac{{i_{1} \pi }}{{S_{1} }}} \right)x} \right]\sin \left[ {\left( {\frac{{j_{1} \pi }}{{S_{2} }}} \right)y} \right] \times \frac{{\cosh [\lambda_{{i_{1} j_{1} }} (z - H_{2} )]}}{{\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} (3)}} } } \sin \left[ {\left( {\frac{{i_{2} \pi }}{{S_{1} }}} \right)x} \right]\sin \left[ {\left( {\frac{{j_{2} \pi }}{{S_{2} }}} \right)y} \right] \times \frac{{\cosh [\lambda_{{i_{2} j_{2} }} (H_{3} - z)]}}{{\cosh [\lambda_{{i_{2} j_{2} }} (H_{3} - H_{2} )]}} \\ & + \sum\limits_{{p_{5} = 1}}^{{P_{5} }} {\sum\limits_{{q_{5} = 1}}^{{Q_{5} }} {I_{{p_{5} q_{5} (3)}} } } \sin \left[ {\left( {\frac{{p_{5} \pi }}{{S_{2} }}} \right)y} \right]\sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{5} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right](z - H_{2} )} \right\} \times \frac{{\sinh (\lambda_{{p_{5} q_{5} }} x)}}{{\sinh (\lambda_{{p_{5} q_{5} }} S_{1} )}} \\ & + \sum\limits_{{p_{6} = 1}}^{{P_{6} }} {\sum\limits_{{q_{6} = 1}}^{{Q_{6} }} {J_{{p_{6} q_{6} (3)}} } } \sin \left[ {\left( {\frac{{p_{6} \pi }}{{S_{2} }}} \right)y} \right]\sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{6} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right](z - H_{2} )} \right\} \times \frac{{\sinh [\lambda_{{p_{6} q_{6} }} (S_{1} - x)]}}{{\sinh (\lambda_{{p_{6} q_{6} }} S_{1} )}} \\ & + \sum\limits_{{p_{7} = 1}}^{{P_{7} }} {\sum\limits_{{q_{7} = 1}}^{{Q_{7} }} {K_{{p_{7} q_{7} (3)}} } } \sin \left[ {\left( {\frac{{p_{7} \pi }}{{S_{1} }}} \right)x} \right]\sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{7} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right](z - H_{2} )} \right\} \times \frac{{\sinh (\lambda_{{p_{7} q_{7} }} y)}}{{\sinh (\lambda_{{p_{7} q_{7} }} S_{2} )}} \\ & + \sum\limits_{{p_{8} = 1}}^{{P_{8} }} {\sum\limits_{{q_{8} = 1}}^{{Q_{8} }} {L_{{p_{8} q_{8} (3)}} } } \sin \left[ {\left( {\frac{{p_{8} \pi }}{{S_{1} }}} \right)x} \right]\sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{8} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right](z - H_{2} )} \right\} \times \frac{{\sinh [\lambda_{{p_{8} q_{8} }} (S_{2} - y)]}}{{\sinh (\lambda_{{p_{8} q_{8} }} S_{2} )}} \\ & - H_{2} \\ \end{aligned} $$

(28)

and

$$ \begin{aligned} & \phi_{3(3)} \left( {x,y,z} \right) = \sum\limits_{{i_{3} = 1}}^{{I_{3} }} {\sum\limits_{{j_{3} = 1}}^{{J_{3} }} {P_{{i_{3} j_{3} (3)}} } } \sin \left[ {\left( {\frac{{i_{3} \pi }}{{S_{1} }}} \right)x} \right]\sin \left[ {\left( {\frac{{j_{3} \pi }}{{S_{2} }}} \right)y} \right] \times \frac{{\cosh [\lambda_{{i_{3} j_{3} }} (h - z)]}}{{\sinh [\lambda_{{i_{3} j_{3} }} (h - H_{3} )]}} \\ & + \sum\limits_{{p_{9} = 1}}^{{P_{9} }} {\sum\limits_{{q_{9} = 1}}^{{Q_{9} }} {M_{{p_{9} q_{9} (3)}} } } \sin \left[ {\left( {\frac{{p_{9} \pi }}{{S_{2} }}} \right)y} \right]\sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{9} }}{2}} \right)\left( {\frac{\pi }{{h - H_{3} }}} \right)} \right](z - H_{3} )} \right\} \times \frac{{\sinh (\lambda_{{p_{9} q_{9} }} x)}}{{\sinh (\lambda_{{p_{9} q_{9} }} S_{1} )}} \\ & + \sum\limits_{{p_{10} = 1}}^{{P_{10} }} {\sum\limits_{{q_{10} = 1}}^{{Q_{10} }} {N_{{p_{10} q_{10} (3)}} } } \sin \left[ {\left( {\frac{{p_{10} \pi }}{{S_{2} }}} \right)y} \right]\sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{10} }}{2}} \right)\left( {\frac{\pi }{{h - H_{3} }}} \right)} \right](z - H_{3} )} \right\} \times \frac{{\sinh [\lambda_{{p_{10} q_{10} }} (S_{1} - x)]}}{{\sinh (\lambda_{{p_{10} q_{10} }} S_{1} )}} \\ & + \sum\limits_{{p_{11} = 1}}^{{P_{11} }} {\sum\limits_{{q_{11} = 1}}^{{Q_{11} }} {U_{{p_{11} q_{11} (3)}} } } \sin \left[ {\left( {\frac{{p_{11} \pi }}{{S_{1} }}} \right)x} \right]\sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{11} }}{2}} \right)\left( {\frac{\pi }{{h - H_{3} }}} \right)} \right](z - H_{3} )} \right\} \times \frac{{\sinh (\lambda_{{p_{11} q_{11} }} y)}}{{\sinh (\lambda_{{p_{11} q_{11} }} S_{2} )}} \\ & + \sum\limits_{{p_{12} = 1}}^{{P_{12} }} {\sum\limits_{{q_{12} = 1}}^{{Q_{12} }} {V_{{p_{12} q_{12} (3)}} } } \sin \left[ {\left( {\frac{{p_{12} \pi }}{{S_{1} }}} \right)x} \right]\sin \left\{ {\left[ {\left( {\frac{{1 - 2q_{12} }}{2}} \right)\left( {\frac{\pi }{{h - H_{3} }}} \right)} \right](z - H_{3} )} \right\} \times \frac{{\sinh [\lambda_{{p_{12} q_{12} }} (S_{2} - y)]}}{{\sinh (\lambda_{{p_{12} q_{12} }} S_{2} )}} \\ & - H_{3} , \\ \end{aligned} $$

(29)

where $\lambda_{{p_{1} q_{1} }} ,$ $\lambda_{{p_{2} q_{2} }} ,$ $\lambda_{{p_{3} q_{3} }} ,$ $\lambda_{{p_{4} q_{4} }}$ $\lambda_{kl} ,$ $\lambda_{uv} ,$ $\lambda_{{i_{1} j_{1} }} ,$ $\lambda_{{i_{2} j_{2} }} ,$ $\lambda_{{p_{5} q_{5} }} ,$ $\lambda_{{p_{6} q_{6} }} ,$ $\lambda_{{p_{7} q_{7} }} ,$ $\lambda_{{p_{8} q_{8} }}$ and $\lambda_{{i_{3} j_{3} }}$ are as given in Eqs. (5), (6), (7), (8), (9), (10), (11), (12), (23), (24), (25), (26) and (13), respectively and

$$ (\lambda_{{p_{9} q_{9} }} )^{2} = \left\{ {\left( {\frac{{p_{9} \pi }}{{S_{2} }}} \right)^{2} \left( {\frac{{K_{{y_{3} }} }}{{K_{{x_{3} }} }}} \right) + \left[ {\left( {\frac{{1 - 2q_{9} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right]^{2} \left( {\frac{{K_{{z_{3} }} }}{{K_{{x_{3} }} }}} \right)} \right\}, $$

(30)

$$ (\lambda_{{p_{10} q_{10} }} )^{2} = \left\{ {\left( {\frac{{p_{10} \pi }}{{S_{2} }}} \right)^{2} \left( {\frac{{K_{{y_{3} }} }}{{K_{{x_{3} }} }}} \right) + \left[ {\left( {\frac{{1 - 2q_{10} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right]^{2} \left( {\frac{{K_{{z_{3} }} }}{{K_{{x_{3} }} }}} \right)} \right\}, $$

(31)

$$ (\lambda_{{p_{11} q_{11} }} )^{2} = \left\{ {\left( {\frac{{p_{11} \pi }}{{S_{1} }}} \right)^{2} \left( {\frac{{K_{{x_{3} }} }}{{K_{{y_{3} }} }}} \right) + \left[ {\left( {\frac{{1 - 2q_{11} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right]^{2} \left( {\frac{{K_{{z_{3} }} }}{{K_{{y_{3} }} }}} \right)} \right\}, $$

(32)

$$ (\lambda_{{p_{12} q_{12} }} )^{2} = \left\{ {\left( {\frac{{p_{12} \pi }}{{S_{1} }}} \right)^{2} \left( {\frac{{K_{{x_{3} }} }}{{K_{{y_{3} }} }}} \right) + \left[ {\left( {\frac{{1 - 2q_{12} }}{2}} \right)\left( {\frac{\pi }{{H_{3} - H_{2} }}} \right)} \right]^{2} \left( {\frac{{K_{{z_{3} }} }}{{K_{{y_{3} }} }}} \right)} \right\}. $$

(33)

Like in the previous cases, the Fourier constants $B_{{p_{1} q_{1} (3)}} ,$ $C_{{p_{2} q_{2} (3)}} ,$ $D_{{p_{3} q_{3} (3)}} ,$ $F_{{p_{4} q_{4} (3)}} ,$ $E_{kl(3)} ,$ $Q_{uv(3)} ,$ $G_{{i_{1} j_{1} (3)}} ,$ $H_{{i_{2} j_{2} (3)}} ,$ $I_{{p_{5} q_{5} (3)}} ,$ $J_{{p_{6} q_{6} (3)}} ,$ $K_{{p_{7} q_{7} (3)}} ,$ $L_{{p_{8} q_{8} (3)}} ,$ $P_{{i_{3} j_{3} (3)}} ,$ $M_{{p_{9} q_{9} (3)}} ,$ $N_{{p_{10} q_{10} (3)}} ,$ $U_{{p_{11} q_{11} (3)}}$ and $V_{{p_{12} q_{12} (3)}}$ for this situation can also be estimated by making use of the relevant boundary and intermediate conditions of the problem (“Appendix A”). Further, here also, Darcy’s Law can be applied to work out the directional velocity functions of the layers. It is also evident that the top discharge function, $Q_{top(3)}^{f} ,$ retains the same expression as mentioned in Eq. (19) since the hydraulic head expression of the first layer for this case is also similar to the corresponding expressions of the previous two cases (i.e., Case 1 and Case 2).

3 Verification of the proposed solutions

3.1 Case 1

We now proceed to check the veracity of the proposed solution by comparing with relevant analytical, experimental and numerical outputs for a few specified drainage situations of figure 1. We start with first of our solutions which, as explained before, is for a situation where the level of water in the drains lies on the first layer. It may be observed that if one of the horizontal dimensions of figure 1 is taken much larger (theoretically infinite) than the vertical and the other horizontal dimensions, then flow in a vertical section located further away from both the boundaries of this longer dimension will be closely two-dimensional in nature; thus, the three-dimensional solution provided here for the drainage problem of figure 1 can also be applied, after appropriate modifications, for predicting two-dimensional flow to a ponded ditch drainage system in a stratified soil as well. Adopting such a reduced procedure, we determine the $Q_{top(1)}^{{}} /2Kh$ ratio at a vertical section located 500 m from the Northern and Southern boundaries of the flow domain for a drainage situation with the flow parameters of figure 1 taken as $S_{1} = 1000{\text{ m}},$ $S_{2} = 100{\text{ m}},$ $h = 3{\text{ m,}}$ $H_{1} = 2.55{\text{ m,}}$ $H_{2} = 2.7{\text{ m,}}$ $H_{3} = 2.85{\text{ m,}}$ $\delta_{i} = 0{\text{ m,}}$ $\varepsilon_{x} = \varepsilon_{y} = 0{\text{ m}}$ and $K = K_{{x_{1} }} = K_{{y_{1} }}$ $= K_{{z_{1} }} = K_{{x_{2} }}$ $= K_{{y_{2} }} = K_{{z_{2} }}$ $= K_{{x_{3} }} = K_{{y_{3} }} = K_{{z_{3} }} = 0.05\,{\text{ m/day}} .$ From our solution, we find this ratio as 0.7197 for the considered flow situation. For a parallel situation but with the drains running totally empty, this value from Fukuda’s [38] and Youngs’ [41] solutions, works out as 0.743 and 0.742, respectively. In this context, it should be noted that the water level in our computation for the concerned flow situation is taken as 2.55 m from the top and not 3 m since the proposed drainage solution of figure 1 is only valid for situations where the level of water in the ditches is on or above the bottom of the first layer and not when it is below it. Even then, as may be observed, our $Q_{top(1)}^{{}} /2Kh$ ratio for the concerned situation is matching very closely with the ones obtained from Fukuda’s and Youngs’ [38, 41] solutions thereby showing that the proposed solution for the first case of the problem has been correctly developed. Further, from his experimental results, Fukuda [38] found this ratio as 0.72, which again can be seen to agree very closely with the value as predicted by our analytical solution. Thus, this matching of our result with the experimentally observed results of Fukuda can also be treated as an experimental verification of our solution.

As a further check of our analytical solution, we again compare our steady state hydraulic heads for a two-dimensional flow situation of figure 1 with the corresponding results as obtained from the analytical works of Kirkham [36] when the flow parameters are as shown in figure 2. It should be noted that here also, before comparison with Kirkham’s [36] solution, our model is first approximately reduced to a two-dimensional one by considering the head distribution only at a section located further away from both the Northern and the Southern boundaries of the flow domain (i.e., at a section $S_{1} /2 = { 7} . 5 {\text{ m)}} .$ It is, thus, because of this, the third dimension (i.e., along the $x -$ axis) is not appearing in figure 2. As can be seen from figure 2, the steady state hydraulic heads as predicted by our reduced two-dimensional model for this drainage configuration are in good agreement with the identical heads as obtained from Kirkham’s two-dimensional solution [36] of the problem thereby showing once again that our solution for the first case of the problem has been rightly developed.

Figure 2

Comparison of steady state hydraulic heads as obtained from the proposed solution at a vertical cross-section located half-way (i.e., at $S_{1} /2 = { 7} . 5 {\text{ m)}}$ between the Northern and the Southern boundaries of figure 1 with the corresponding values as obtained from Kirkham’s 1965 [36] steady state solution when the flow parameters of figure 1 are are taken as $S_{1} = 15{\text{ m,}}$ $S_{2} = 5{\text{ m,}}$ $h = 1{\text{ m,}}$ $H_{1} = 0.4$ ${\text{m,}}$ $H_{2} =$ $0.6{\text{ m,}}$ $H_{3} =$ $0.8{\text{ m,}}$ $\delta_{i} = 0{\text{ m,}}$ $\varepsilon_{x} = \varepsilon_{y} = 0{\text{ m}}$ and $K_{{x_{1} }} =$ $K_{{y_{1} }} =$ $K_{{z_{1} }} =$ $K_{{x_{2} }} =$ $K_{{y_{2} }} =$ $K_{{z_{2} }} =$ $K_{{x_{3} }} =$ $K_{{y_{3} }} =$ $K_{{z_{3} }} =$ $1{\text{ m/day}} .$

Sarmah and Barua [25] provided a solution to the three-dimensional ponded drainage problem of figure 1 for a single-layered soil and since our multi-layered solution can be easily reduced to a single-layered one by treating the corresponding directional conductivities of the layers as same, we can, thus, also use their solution for comparison with that of ours for such single-layered ponded drainage situations. Figure 3 shows comparison results for such a situation when the flow parameters of figure 1 are taken as shown. As can be seen, the hydraulic heads as obtained from our solution for the concerned drainage scenario are in near perfect agreement with the corresponding values as obtained from Sarmah and Barua’s solution [25] thereby providing us with a verification of our three-dimensional ponded solution with yet another existing analytical solution to the problem.

Figure 3

Comparison of steady state hydraulic heads as obtained from the proposed solution with the corresponding values as obtained from Sarmah and Barua’s 2017 [25] three-dimensional solution to the problem for a single-layered soil when the flow parameters of figure 1 are taken as $S_{1} = 6{\text{ m,}}$ $S_{2} =$ $5{\text{ m,}}$ $h = 1{\text{ m,}}$  $H_{1} =$ $0.35{\text{ m,}}$ $\delta_{i} = 0{\text{ m,}}$ $\varepsilon_{x} = \varepsilon_{y} = 0.05{\text{ m,}}$ $K_{{x_{1} }} = K_{{x_{2} }} =$ $K_{{x_{3} }} = 0.6{\text{ m/day,}}$ $K_{{y_{1} }} =$ $K_{{y_{2} }} =$ $K_{{y_{3} }} =$ $0.5{\text{ m/day}}$ and $K_{{z_{1} }} =$ $K_{{z_{2} }} = K_{{z_{3} }} =$ $0.2{\text{ m/day}} .$

As a further check of our analytical model, a numerical model was also drawn for a typical drainage setting of figure 1 utilizing the Processing MODFLOW environment [73]. To perform a MODFLOW simulation of the flow situation of figure 1, a flow domain of 10 m × 5 m × 1 m was first considered and then it was subdivided into a network of grids containing 201 rows, 101 columns and 42 layers. A row spacing of 0.05 m was considered while the column spacing for the grid was kept at 0.05 m. The thickness of each vertical layer of the grid was assigned as 0.025 m. For simulating the impervious layer underlying the flow domain, all the cells of the 42^nd layer were kept inactive. The cells of the 1^st row, 201^st row, 1^st column and 101^st column were kept as fixed head cells to model the ditch drains. The cells of the topmost layer were made as fixed head cells and assigned a value of 0.03 m to replicate the constant ponding atop the field. All the other cells in the model were assigned as active and the head value for them was kept as 0 m. The water level in the ditches was simulated by assigning a 0 m head to cells in the 1^st column, 101^st column, 1^st row and 201^st row and then proceeding to decrease it by 0.025 for every next layer encountered in the vertical direction till it reached the 15^th layer where a head value of -0.35 m was assigned. This head value at the drains had then been continued up to the 41^th layer. The horizontal and vertical hydraulic conductivities of the cells of the first layer and extending up to the 19^th layer were assigned values of 2 m/day and 0.5 m/day, respectively and the anisotropy ratio of these cells was assigned a value of 0.5; all these provided the conductivity information for the first soil layer of the model. The middle soil layer was made up of MODFLOW layers starting from the 20^th layer and extending up to the 31^th layer. For these layers, the horizontal and vertical hydraulic conductivities were inputted as 1.8 m/day and 0.8 m/day, respectively and the anisotropy ratio as 0.6666. To simulate the conductivities of the bottom soil layer, a value of 1.4 m/day was assigned for the horizontal hydraulic conductivity and a value of 1 m/day for the vertical hydraulic conductivity to all the MODFLOW layers starting from the 32^th layer and extending up to the 42^nd layer while maintaining an anisotropy ratio of unity in these layers. With these settings in place, a steady state MODFLOW run was carried and the hydraulic head contours obtained for a few heads compared with the corresponding contours obtained from our proposed solution. Figure 4 shows such a comparison. For clearness, comparisons of heads at a few locations in the studied drainage space are also shown in table 1. As may be observed, the predictions from our solution are matching very closely – differing mostly in the third places of decimals only – with the corresponding MODFLOW results thereby providing us with a numerical verification of our solution.

Figure 4

Comparison of steady state hydraulic heads as obtained from the proposed solution with the corresponding values as obtained from MODFLOW when the flow parameters of figure 1 are taken as in table 1.

Table 1 Comparison of hydraulic heads as obtained from the proposed solution with those obtained from MODFLOW at a few coordinate locations when the flow parameters of figure 1 are taken as $S_{1} = 10{\text{ m,}}$ $S_{2} = 5{\text{ m,}}$ $h = 1{\text{ m,}}$ $H_{1} = 0.35{\text{ m,}}$ $H_{2} = 0.45{\text{ m,}}$ $H_{3} = 0.75{\text{ m,}}$ $\delta_{i} =$ $0.03{\text{ m,}}$ $\varepsilon_{x} = \varepsilon_{y} =$ $0.05{\text{ m,}}$ $K_{{x_{1} }} =$ $1 {\text{ m/day,}}$ $K_{{y_{1} }} = 2$ ${\text{m/day,}}$ $K_{{z_{1} }} = 0.5 \,$ ${\text{m/day,}}$ $K_{{x_{2} }} = 1.2$ ${\text{m/day,}}$ $K_{{y_{2} }} = 1.8 \,$ ${\text{m/day,}}$ $K_{{z_{2} }} =$ $0.8{\text{ m/day,}}$ $K_{{x_{3} }} = 1.4$ ${\text{m/day,}}$ $K_{{y_{3} }}$ $= 1.4{\text{ m/day}}$ and $K_{{z_{3} }} = 1$ ${\text{m/day}} .$

3.2 Case 2

We will now make efforts to check the validity of our solution for the case when the level of water in the ditches of the flow problem of figure 1 lies on the middle layer. Like in the previous solution, here also a battery of checks has been carried out to determine the validity of this solution. Further, this three-dimensional solution can also be made applicable for a two-dimensional situation by following exactly a similar procedure as has been mentioned in the previous problem. The value of $Q_{top(2)} /2Kh$ ratio for a two-dimensional flow situation located at a section 500 m mid-way between the Northern and Southern boundaries of figure 1 is now working out as 0.732 when the other flow parameters of figure 1 are taken as $S_{1} = 1000\,{\text{ m,}}$ $S_{2} = 100{\text{ m,}}$ $h = 3{\text{ m,}}$ $H_{1} = 2.7{\text{ m,}}$ $H_{2} = 2.55\,{\text{ m,}}$ $H_{3} = 2.85{\text{ m,}}$ $\delta_{i} = 0{\text{ m,}}$ $\varepsilon_{x} = \varepsilon_{y} = 0{\text{ m}}$ and$K = K_{{x_{1} }} = K_{{y_{1} }} = K_{{z_{1} }} = K_{{x_{2} }} = K_{{y_{2} }} = K_{{z_{2} }} = K_{{x_{3} }} = K_{{y_{3} }} = K_{{z_{3} }} = 0.05{\text{ m/day}} .$As this ratio from Fukuda’s and Youngs’ [38, 41] solutions for the same flow setting are turning out to be 0.743 and 0.742, respectively – values quite close to the one predicted by our model – the developed solution, thus, for this case can also be considered as correctly developed. It is worth noting that Fukuda found this value experimentally to be 0.72 and the close matching of this with our predicted value can, thus, also be taken as an experimental verification of the developed solution for this case as well. Figures 5 and 6 also show comparison of our solution with the analytical works of Kirkham and Sarmah and Barua [25, 36] for a few drainage settings of figure 1 for a single-layered soil. As can be seen, in all of these cases, the predictions from our solution could successfully reproduce the corresponding predictions as obtained from the analytical works of others thereby providing us with a further proof about the accuracy of our solution for the second case. To have a further confirmation of our solution for this case, a MODFLOW check on it has also been carried for a particular setting of figure 1; table 2 and figure 7 show the comparison results of our solution with that of MODFLOW for the studied drainage scenario. As can be seen, our solution could match the hydraulic heads as obtained from MODFLOW for all the tested points thereby providing us with a further proof about the veracity of our model for this particular case.

Figure 5

Comparison of steady state hydraulic heads as obtained from the proposed solution at a vertical cross-section located half-way (i.e., at $S_{1} /2 = { 7} . 5 {\text{ m)}}$ between the Northern and the Southern boundaries of figure 1 with the corresponding values as obtained from Kirkham’s 1965 [36] steady state solution when the flow parameters of figure 1 are taken as $S_{1} = 15{\text{ m,}}$ $S_{2} = 5{\text{ m,}}$ $h = 1{\text{ m,}}$ $H_{1} =$ $0.7{\text{ m,}}$ $H_{2} = 0.4{\text{ m,}}$ $H_{3} = 0.8{\text{ m,}}$ $\delta_{i} =$ $0{\text{ m,}}$ $\varepsilon_{x} = \varepsilon_{y} = 0{\text{ m}}$ and $K_{{x_{1} }}$ $= K_{{y_{1} }} =$ $K_{{z_{1} }} =$ $K_{{x_{2} }} =$ $K_{{y_{2} }} =$ $K_{{z_{2} }} =$ $K_{{x_{3} }}$ $= K_{{y_{3} }} = K_{{z_{3} }} = K_{{y_{3} }} = K_{{z_{3} }}$ $= 1{\text{ m/day}} .$

Figure 6

Comparison of steady state hydraulic heads as obtained from the proposed solution with the corresponding values as obtained from Sarmah and Barua’s 2017 [25] three-dimensional solution to the problem for a single-layered soil when the flow parameters of figure 1 are taken as $S_{1} = 6{\text{ m,}}$ $S_{2} = 5{\text{ m,}}$ $h = 1{\text{ m,}}$ $H_{1} =$ $0.65$ ${\text{m,}}$ $\delta_{i} =$ $0.03{\text{ m,}}$ $\varepsilon_{x} =$ $\varepsilon_{y} = 0.05$ ${\text{m,}}$ $K_{{x_{1} }} =$ $K_{{x_{2} }} =$ $K_{{x_{3} }}$ $= 0.8{\text{ m/day,}}$ $K_{{y_{1} }} =$ $K_{{y_{2} }} =$ $K_{{y_{3} }} =$ $1.6{\text{ m/day}}$ and $K_{{z_{1} }} = K_{{z_{2} }} =$ $K_{{z_{3} }} = 0.4$ ${\text{m/day}} .$

Table 2 Comparison of hydraulic heads as obtained from the proposed solution with those obtained from MODFLOW at a few coordinate locations when the flow parameters of figure 1 are taken as $S_{1} = 10{\text{ m,}}$ $S_{2} = 5{\text{ m,}}$ $h = 1{\text{ m,}}$ $H_{1} = 0.6$ ${\text{m,}}$ $H_{2} = 0.35{\text{ m,}}$ $H_{3} = 0.75{\text{ m,}}$ $\delta_{i} = 0.05$ ${\text{m,}}$ $\varepsilon_{x} = \varepsilon_{y}$ $= 0.05{\text{ m}}$, $K_{{x_{1} }} = 1 \,$ ${\text{m/day,}}$ $K_{{y_{1} }} = 2$ ${\text{m/day,}}$ $K_{{z_{1} }} =$ $1. 5 {\text{ m/day,}}$ $K_{{x_{2} }} =$ $1. 2 {\text{ m/day,}}$ $K_{{y_{2} }} = 0.8$ ${\text{m/day,}}$ $K_{{z_{2} }} =$ $0.9 \,$ ${\text{m/day,}}$ $K_{{x_{3} }} =$ $0. 5 {\text{ m/day,}}$ $K_{{y_{3} }}$ $= 0.5 \,$ ${\text{m/day}}$ and $K_{{z_{3} }} =$ $0. 6 {\text{ m/day}} .$

Figure 7

Comparison of steady state hydraulic heads as obtained from the proposed solution with the corresponding values as obtained from MODFLOW when the flow parameters of figure 1 are taken as mentioned in table 2.

3.3 Case 3

We will now provide a few checks on the validity of last of our solutions for the case when the level of water in the ditches lies on the bottom layer. To do that, we first reduce our three-dimensional flow situation, like in the previous two solutions, to approximately a two-dimensional one in the $y - z$ plane by assigning a large separation distance (theoretically infinite) between the Northern and Southern boundaries of the model and then considering a $y - z$ section located further away from both these boundaries. Considering such a reduced two-dimensional model, the $Q_{top(3)} /2Kh$ ratio at a vertical section located mid-way between the Northern and Southern boundaries of figure 1 is next been worked with the flow parameters of the problem taken as $S_{1} = 1000{\text{ m}},$ $S_{2} = 100{\text{ m}},$ $h = 3{\text{ m,}}$ $H_{1} = 3{\text{ m,}}$ $H_{2} = 2{\text{ m,}}$ $H_{3} = 2.5{\text{ m,}}$ $\delta_{i} = 0{\text{ m,}}$ $\varepsilon_{x} = \varepsilon_{y} = 0{\text{ m}}$ and $K = K_{{x_{1} }} = K_{{y_{1} }} = K_{{z_{1} }} = K_{{x_{2} }} = K_{{y_{2} }} = K_{{z_{2} }} = K_{{x_{3} }} =$ $K_{{y_{3} }} = K_{{z_{3} }} = 0.05{\text{ m/day;}}$we find this value for this situation as 0.744. For the same drainage configuration, as mentioned before, this ratio has been predicted as 0.743 and 0.742, respectively by Fukuda’s (1957) and Youngs’ [38, 41] solutions–values very close to our predicted value of 0.744. Also, Fukuda [38] found this ratio as 0.72 from his experimental results; thus, the close matching of this result with our value of 0.744 can also be taken as an experimental verification of our proposed solution for this case as well. In this context, we would like to point out that for this case of the drainage configuration of figure 1, simulating a drainage system with the drains running totally empty is a possibility since we are now assuming the water level of the drains to be lying below the upper boundary of the bottom layer. However, simulating a drainage system with the ditches running totally empty cannot be done using our solutions for the previous two configurations of the system as in both of these drainage settings, the water level in the ditches is assumed to lie either in the top or middle layer of the stratified soil and not on the bottom layer. Nevertheless, simulating a system with the ditches running nearly empty (but not totally empty) is still nearly possible with our previous solutions since we can always take the thickness of the bottom and/or the middle layer to be appreciably small (but not totally zero and that is what we have done in the previous two cases) while imitating such a system. The validity of the current solution is also been extensively tested, like in the previous two solutions, by making comparisons with the analytical works of Kirkham and Sarmah and Barua [25, 36] for a few drainage situations of figure 1. Also, to a have further proof about the accuracy of our model for this case, a numerical check on it has also been carried out for a drainage setting of figure 1 utilizing again the Processing MODFLOW platform. As can be seen from figures 8, 9 and 10, in all the tested situations, our predictions are found to match closely with their analytical and numerical counterparts thereby establishing that our solution for this drainage setting has also been correctly developed. The numerical matching of our solution is also been amply demonstrated in table 3, where, as may be observed, the heads predicted by our model are in close proximities with the ones obtained from MODFLOW for the concerned drainage situation.

Figure 8

Comparison of steady state hydraulic heads as obtained from the proposed solution at a vertical cross-section located half-way (i.e. at $S_{1} /2 = { 7} . 5 {\text{ m)}}$ between the Northern and the Southern boundaries of figure 1 with the corresponding values as obtained from Kirkham’s 1965 [36] steady state solution when the flow parameters of figure are taken as $S_{1} = 15{\text{ m,}}$ $S_{2} = 5{\text{ m,}}$ $h = 1{\text{ m,}}$ $H_{1} =$ $0.9{\text{ m,}}$ $H_{2} = 0.4{\text{ m,}}$ $H_{3} = 0.8{\text{ m,}}$ $\delta_{i} = 0{\text{ m,}}$ $\varepsilon_{x} = \varepsilon_{y} = 0{\text{ m}}$ and $K_{{x_{1} }} = K_{{y_{1} }} = K_{{z_{1} }} = K_{{x_{2} }} = K_{{y_{2} }} = K_{{z_{2} }} = K_{{x_{3} }} = K_{{y_{3} }} = K_{{z_{3} }} = 1 {\text{ m/day}} .$

Figure 9

Comparison of steady state hydraulic heads as obtained from the proposed solution with the corresponding values as obtained from Sarmah and Barua’s 2017 [25] three-dimensional solution to the problem for a single-layered soil when the flow parameters of figure 1 are taken as $S_{1} = 6{\text{ m, }}\,S_{2} = 5{\text{ m, }}\,h = 1{\text{ m, }}\,H_{1} = 1{\text{ m, }}\,\delta_{i} = 0.05{\text{m, }}$ $\varepsilon_{x} = \varepsilon_{y} = 0.05{\text{ m,}}$ $K_{{x_{1} }} = K_{{x_{2} }} = K_{{x_{3} }} = 2{\text{ m/day,}}$ $K_{{y_{1} }} = K_{{y_{2} }} = K_{{y_{3} }} = 0.5{\text{ m/day}}$ and $K_{{z_{1} }} = K_{{z_{2} }} = K_{{z_{3} }} = 1 {\text{ m/day}} .$

Figure 10

Comparison of steady state hydraulic heads as obtained from the proposed solution with the corresponding values as obtained from MODFLOW when the flow parameters of figure 1 are taken as in table 3.

Table 3 Comparison of hydraulic heads as obtained from the proposed solution with those obtained from MODFLOW at a few coordinate locations when the flow parameters of figure 1 are taken as $S_{1} = 10{\text{ m,}}$ $S_{2} = 5{\text{ m,}}$ $h = 1{\text{ m,}}$ $H_{1} = 0.9{\text{m,}}$ $H_{2} = 0.35{\text{ m,}}$ $H_{3} = 0.7{\text{ m,}}$ $\delta_{1} = 0.03{\text{m,}}$ $\delta_{2} = 0.06{\text{ m,}}$ $d_{x1} = 2{\text{ m,}}$ $d_{x2} = 8{\text{ m,}}$ $d_{y1} = 1{\text{ m,}}$ $d_{y2} = 4{\text{ m,}}$ $\varepsilon_{x} = \varepsilon_{y} = 0.05{\text{ m,}}$ $K_{{x_{1} }} = 0.8{\text{ m/day,}}$ $K_{{y_{1} }} = 1{\text{m/day,}}$ $K_{{z_{1} }} = 0.5{\text{m/day,}}$ $K_{{x_{2} }} = 1{\text{m/day,}}$ $K_{{y_{2} }} = 1.5{\text{ m/day,}}$ $K_{{z_{2} }} = 0.75{\text{ m/day,}}$ $K_{{x_{3} }} = 1{\text{ m/day,}}$ $K_{{y_{3} }} = 2{\text{m/day}}$ and $K_{{z_{3} }} = 1{\text{m/day}} .$

4 Discussions

We will now make use of our developed solutions to study a few three-dimensional ponded drainage situations. Firstly, we consider the variation of top discharge with change in hydraulic conductivities of the constituent layers by treating each of the layers as isotropic. Figures 11(a) and (b) show variations of top discharge with the conductivity ratio $(K_{1} /K_{2} = K_{2} /K_{3} ,$ where $K_{{x_{1} }} = K_{{y_{1} }} = K_{{z_{1} }} = K_{1} ,$ $K_{{x_{2} }} = K_{{y_{2} }}$ $= K_{{z_{2} }} = K_{2}$ and $K_{{x_{3} }} = K_{{y_{3} }} = K_{{z_{3} }} = K_{3} )$ of the layers when the flow parameters of figure 1 are taken as shown. As may be observed from these figures, the top discharge for the studied situations is varying mostly non-linearly with the increase of conductivity ratio of the layers when this ratio is less than unity; however, for values of this ratio higher than unity, the top discharge can be observed to follow approximately a linear trend with the increase of this ratio. Thus, top discharge is pretty sensitive to the conductivity of the top layer, particularly for higher values. This is understandable since with the increase in conductivity of the top layer, resistance offered to the movement of water particles through it progressively decreases thereby causing more water to seep through it for the same head difference between the top of the soil and the recipient ditches. It can also be observed from figure 11(c) that even though $Q_{top(3)}$ is increasing with the increase in $K_{1} ,$ the rate of increase, however, is not keeping pace with the increase of $K_{1}$ with the result that $Q_{top(3)} /(K_{{z_{1} }} = K_{1} )$ ratio is actually found decreasing with the increase in conductivity ratio of the soil layers for this drainage situation. In this context, it should also be noted that since we have fixed the conductivity of the middle layer $K_{2}$ as 1 m/day for this flow situation, an increase in the $K_{2} /K_{3}$ ratio actually signifies a decrease in the conductivity of the bottom layer; thus, the resistance offered by this layer to water movement, in reality, is getting progressively increased with the increase of this ratio and not the other way round.

Figure 11

Variation of top discharge with conductivity ratio of the soil layers when flow parameters of figure 1 are taken as $S_{1} = 8{\text{ m,}}$ $S_{2} = 8{\text{ m,}}$ $h = 1.5{\text{ m,}}$ $H_{2} = 0.5{\text{ m,}}$ $H_{3} = 1{\text{ m,}}$ $\delta_{i} = 0{\text{ m,}}$ $\varepsilon_{x} = \varepsilon_{y} = 0{\text{ m}}$ and $K_{{x_{2} }} = K_{{y_{2} }} = K_{{z_{2} }} = 1{\text{ m/day}} .$

We next study the relationship between the top discharge and the anisotropy of the soil layers. Figure 12(a) shows variation of top discharge with the anisotropy ratio$(K_{{x_{1} }} /K_{{z_{1} }} = K_{{y_{1} }} /K_{{z_{1} }} =$ $K_{{x_{2} }} /K_{{z_{2} }} = K_{{y_{2} }} /K_{{z_{2} }} = K_{{x_{3} }} /K_{{z_{3} }} = K_{{y_{3} }} /K_{{z_{3} }}$) of the layers for a drainage situation of figure 1 where the layers are all taken of equal thickness (0.5 m) and where their vertical hydraulic conductivities are allowed to increase progressively with depth with $K_{{z_{1} }} ,$ $K_{{z_{2} }}$ and $K_{{z_{3} }}$ taken as 0.5 m/day, 1 m/day and 1.5 m/day, respectively. In figure 12(b), a similar variation is studied but the order of the vertical conductivities of the layers is now reversed with $K_{{z_{1} }} ,$ $K_{{z_{2} }}$ and $K_{{z_{3} }}$ taken as 1.5 m/day, 1 m/day and 0.5 m/day, respectively. As may be observed, for the considered flow scenarios, the top discharge is more sensitive to change in the anisotropy ratio of the layers at low values of this ratio and the slope of the discharge variation curve tends to decrease with the increase in the anisotropy ratio of the layers. This can be seen to be true either when the vertical conductivities of the layers increase or decrease with depth. Thus, a mere change in the vertical conductivity of the layers alone may result in a considerable change in the top discharge of a multi-layered ponded ditch drainage system, particularly if the vertical conductivity of the top layer is much higher than that of the vertical conductivities of the lower layers. An increase in the anisotropy ratio of the layers actually means that the overall conductivities of the layers are getting increased (since the vertical conductivities of the layers are kept fixed here) and hence it is not surprising that the top discharge is getting increased with the increase in anisotropy ratio of the layers. Also, for situations where the vertical conductivity of the top layer is relatively higher than the vertical conductivities of the lower layers [figure 12(b)], the increase in anisotropy ratio of the layers is causing the already relatively higher directional conductivities of the top layer to increase still further thereby triggering a greater short-circuiting of flow to the drains through the top layer of the soil profile under the same hydraulic gradient than for a situation where the anisotropy ratio of the layers is low.

Figure 12

Variation of top discharge with anisotropy ratio of the soil layers when flow parameters of figure 1 are taken as $S_{1} = 8\,{\text{m,}}$ $S_{2} = 8\,{\text{m,}}$ $h = 1.5\,{\text{m,}}$ $H_{1} = 1.5\,{\text{m,}}$ $H_{2} = 0.5\,{\text{m,}}$ $H_{3} = 1\,{\text{m,}}$ $\delta_{i} = 0{\text{ m,}}$ $\varepsilon_{x} = \varepsilon_{y} = 0{\text{ m}}$ and (a) $K_{{z_{1} }} = 0.5\,{\text{m/day,}}$ $K_{{z_{2} }} = 1\,{\text{m/day,}}$ $K_{{z_{3} }} = 1.5\,{\text{m/day,}}$ (b) $K_{{z_{1} }} = 1.5\,{\text{m/day,}}$ $K_{{z_{2} }} = 1\,{\text{m/day}}$ and $K_{{z_{3} }} = 0.5{\text{ m/day}} .$

The travel times of water particles from top of the field to recipient ditch drains are next investigated for a few drainage scenarios using the velocity expressions derived from the proposed solutions and the technique suggested by Grove et al [74]. From the cases studied [figures 13(a) and 14(a)], it is seen that when hydraulic conductivities of the soil layers decrease with depth, travel times of water particles from far away locations to the drains are relatively much higher than those for situations when the hydraulic conductivities of soil layers increase with depth. However, no such drastic disparity in travel time values are observed for these cases when water particles located closer to the ditches are being traced. This is because water particles originating from central locations between the ditches have to traverse in longer arcs, mostly crisscrossing all the layers of the soil profile, on their journey to the ditches and hence if the conductivities of the bottom layers are low, more time is required to move through them under a given hydraulic gradient. On the other hand, for water particles starting from locations close to the ditches, the pathlines to the ditches are generally much shorter and can even be confined to the top layer alone depending on a drainage situation and hence for particles traversing in these paths, the travel times are not getting unduly affected by the small changes in the conductivities of the layers. It can also be observed from figure 14(b) that by adopting a staggered ponding distribution at the surface of the soil using inner bunds with progressively increasing ponding depths towards the centre of the field, considerable decrease in travel times of water particles from locations away from the ditch faces can be brought about for drainage situations where directional hydraulic conductivities of the soil layers decrease with depth. This is because for such a ponding distribution, heads available to push the water particles travelling in longer arcs to the drains are relatively higher than when the field is subjected to a uniform ponding depth at the surface of the soil. In all these figures, it is to be noted that $\eta_{1} ,$ $\eta_{2}$ and $\eta_{3}$ represent the porosity of the first, second and third layers, respectively.

Figure 13

Travel time (in days) of water particles starting from the surface of a three-dimensional ponded ditch drainage system to the recipient drains when the flow parameters of figure 1 are taken as $S_{1} = 8{\text{ m,}}$ $S_{2} = 8{\text{ m,}}$ $h = 1.5{\text{ m,}}$ $H_{1} = 1.5{\text{ m,}}$ $H_{2} = 0.5{\text{ m,}}$ $H_{3} = 1.1{\text{ m,}}$ $\eta_{1} = \eta_{2} = \eta_{3} = 0.3,$ $\varepsilon_{x} = \varepsilon_{y} = 0.05{\text{ m,}}$ $K_{{x_{1} }} = K_{{y_{1} }} = K_{{z_{1} }} =$ $1{\text{ m/day,}}$ $K_{{x_{2} }} = K_{{y_{2} }} = K_{{z_{2} }} = 2{\text{ m/day,}}$ $K_{{x_{3} }} = K_{{y_{3} }} = K_{{z_{3} }} = 3{\text{ m/day}}$ and (a) $\delta_{i} = 0{\text{ m,}}$ (b) $\delta_{1} = 0{\text{ m,}}$ $\delta_{2} = 0.1{\text{ m,}}$ $\delta_{3} = 0.25{\text{ m,}}$ $\delta_{4} = 0.4{\text{ m,}}$ $d_{x1} = 0.5{\text{ m,}}$ $d_{x2} = 1.25{\text{ m,}}$ $d_{x3} = 2{\text{ m,}}$ $d_{x4} = 6{\text{ m,}}$ $d_{x5} = 6.75{\text{ m,}}$ $d_{x6} = 7.5{\text{ m,}}$ $d_{y1} = 0.5{\text{ m,}}$ $d_{y2} = 1.25{\text{ m,}}$ $d_{y3} = 2{\text{ m,}}$ $d_{y4} = 6{\text{ m,}}$ $d_{y5} = 6.75{\text{ m}}$ and $d_{y6} = 7.5{\text{ m}} .$

Figure 14

Travel time (in days) of water particles starting from the surface of a three-dimensional ponded ditch drainage system to the recipient drains when the flow parameters of Fig. 1 are taken as $S_{1} = 8{\text{ m,}}$ $S_{2} = 8{\text{ m,}}$ $h = 1.5{\text{ m,}}$ $H_{1} = 1.5{\text{ m,}}$ $H_{2} = 0.5{\text{ m,}}$ $H_{3} = 1.1{\text{ m,}}$ $\eta_{1} = \eta_{2} = \eta_{3} = 0.3,$ $\varepsilon_{x} = \varepsilon_{y} = 0.05{\text{ m,}}$ $K_{{x_{1} }} = K_{{y_{1} }} = K_{{z_{1} }} =$ $3{\text{ m/day,}}$ $K_{{x_{2} }} = K_{{y_{2} }} = K_{{z_{2} }} = 2{\text{ m/day,}}$ $K_{{x_{3} }} = K_{{y_{3} }} = K_{{z_{3} }} = 1{\text{ m/day}}$ and (a) $\delta_{i} = 0{\text{ m,}}$ (b) $\delta_{1} = 0{\text{ m,}}$ $\delta_{2} = 0.1{\text{ m,}}$ $\delta_{3} = 0.25{\text{ m,}}$ $\delta_{4} = 0.4{\text{ m,}}$ $d_{x1} = 0.5{\text{ m,}}$ $d_{x2} = 1.25{\text{ m,}}$ $d_{x3} = 2{\text{ m,}}$ $d_{x4} = 6{\text{ m,}}$ $d_{x5} = 6.75{\text{ m,}}$ $d_{x6} = 7.5{\text{ m,}}$ $d_{y1} = 0.5{\text{ m,}}$ $d_{y2} = 1.25{\text{ m,}}$ $d_{y3} = 2{\text{ m,}}$ $d_{y4} = 6{\text{ m,}}$ $d_{y5} = 6.75{\text{ m}}$ and $d_{y6} = 7.5{\text{ m}} .$

Further, as can be inferred from figure 15, the level of water in the drains plays a crucial role in determining travel times of water particles in a three-dimensional ponded ditch drainage system – travel times of the particles for the same ponded drainage scenario tend to decrease with the decrease of water level depths in the ditches. This is because, in comparison to a situation where the water level in the drains is high, a low water level in the drains results in the availability of a relatively higher potential difference between the ponding field and the drains. Thus, for the same ponding head, travel times of water particles to the low flowing drains will be relatively lesser than the corresponding travel times to high water level drains. From all these figures, it has also become clear that ponded drainage in a stratified soil is mostly three-dimensional in nature and the pathlines may exhibit considerable curvature, particularly in locations close to the ditches. Further, it can also be seen that an increase in the horizontal and lateral conductivities has a tendency to flatten and an increase in the vertical conductivity has a tendency to straighten the pathlines within a layer of a ponded drainage system. This is because an increase in the horizontal and lateral conductivities of a layer, allowing all other factors to remain the same, results in a proportionate increase in the components of the velocity vector in these directions in relation to its vertical component thereby causing the pathlines to flatten more towards the horizontal when measured with respect to their initial state. The situation, however, gets reversed with the increase in vertical conductivity of a layer as in such a situation, the share of the vertical component of the velocity vector gets proportionally increased with respect to the other two components of the velocity vector thereby causing the pathlines to straighten up within the layer.

Figure 15

Travel time (in days) of water particles starting from the surface of a three-dimensional ponded ditch drainage system to the recipient drains when the flow parameters of figure 1 are taken as $S_{1} = 8{\text{ m,}}$ $S_{2} = 8{\text{ m,}}$ $h = 1.5{\text{ m,}}$ $H_{2} = 0.5{\text{ m,}}$ $H_{3} = 1.1{\text{ m,}}$ $\delta_{i} = 0{\text{ m,}}$ $\varepsilon_{x} = \varepsilon_{y} = 0{\text{ m,}}$ $\eta_{1} = 0.35,$ $\eta_{2} = \eta_{3} = 0.3,$ $K_{{x_{1} }} = K_{{y_{1} }} = 0.3{\text{ m/day,}}$ $K_{{z_{1} }} = 0.2 \,$ ${\text{m/day,}}$ $K_{{x_{2} }} = K_{{y_{2} }} = 1.5{\text{ m/day,}}$ $K_{{z_{2} }} = 0.5{\text{ m/day,}}$ $K_{{x_{3} }} = K_{{y_{3} }} = 2{\text{ m/day,}}$ $K_{{z_{3} }} = 1{\text{ m/day}}$ and (a) $H_{1} = 1.5{\text{ m,}}$ (b) $H_{1} = 0.75{\text{ m}} .$

The presence of muddy and plow sole soil layers in paddy fields may greatly inhibit the movement of infiltration water in such fields [57, 75, 76]. In the flow situations of figure 16, an attempt is being made to study how the presence or absence of a plow sole layer in a ponded paddy field affects the movement of drainage water in a three-dimensional ditch drainage system. As may be observed from these figures, the occurrence of muddy and plow sole layers close to the surface of a drained field, owing to their very low conductivities, may greatly extend the travel times of water particles to the drains as compared to transport times of particles under normal soil conditions. This is particularly true for particles originating from surficial locations close to the drains. These layers provide considerable resistance to movement of water through them and since they are common in paddy fields, subsurface drainage of these fields may require considerably more time as compared to drainage of normal soils.

Figure 16

Travel time (in days) of water particles starting from the surface of a three-dimensional ponded ditch drainage system to the recipient drains when the flow parameters of figure 1 are taken as (a) $S_{1} = 5{\text{ m,}}$ $S_{2} = 5{\text{ m,}}$ $h = 1{\text{ m,}}$ $H_{1} = 0.8{\text{ m,}}$ $H_{2} = 0.25{\text{ m,}}$ $H_{3} = 0.35{\text{ m,}}$ $\delta_{i} = 0{\text{ m,}}$ $\varepsilon_{x} = \varepsilon_{y} = 0{\text{ m,}}$ $\eta_{1} = \eta_{2} = 0.55,$ $\eta_{3} = 0.45,$ $K_{{x_{1} }} = K_{{y_{1} }} = K_{{z_{1} }} =$ $0.005{\text{ m/day,}}$ $K_{{x_{2} }} = K_{{y_{2} }} = K_{{z_{2} }} = 0.003{\text{ m/day,}}$ $K_{{x_{3} }} = K_{{y_{3} }} = K_{{z_{3} }} = 0.03{\text{ m/day,}}$ (b) $S_{1} = 5{\text{ m,}}$ $S_{2} = 5{\text{ m,}}$ $h = 1{\text{ m,}}$ $H_{1} = 0.8{\text{ m,}}$ $\delta_{i} = 0{\text{ m,}}$ $\varepsilon_{x} = \varepsilon_{y} = 0{\text{ m,}}$ $\eta_{i} = 0.45,$ $K_{{x_{1} }} = K_{{y_{1} }} = K_{{z_{1} }} = 0.03{\text{ m/day,}}$ $K_{{x_{2} }} = K_{{y_{2} }} = K_{{z_{2} }} = 0.03{\text{ m/day}}$ and $K_{{x_{3} }} = K_{{y_{3} }} = K_{{z_{3} }} = 0.03{\text{ m/day}} .$.

The non-symmetrical ponding distribution drainage scenario of figure 17 further corroborates that the travel time of water particles in a pathline, depends, among other factors, on the ponding head that it has been subjected to at the surface of the soil and that a higher head causes this time to decrease and a lower head to increase.

Figure 17

Travel time (in days) of water particles starting from the surface of a three-dimensional ponded ditch drainage system to the recipient drains when the flow parameters of figure 1 are taken as $S_{1} = 5{\text{ m,}}$ $S_{2} = 5{\text{ m,}}$ $h = 1{\text{ m,}}$ $H_{1} = 0.5{\text{ m,}}$ $H_{2} = 0.4{\text{ m,}}$ $H_{3} = 0.75{\text{ m,}}$ $\eta_{1} = \eta_{2} = \eta_{3} = 0.3,$ $\varepsilon_{x} = \varepsilon_{y} = 0.05{\text{ m,}}$ $K_{{x_{1} }} = K_{{y_{1} }} = 2{\text{ m/day,}}$ $K_{{z_{1} }} = 1.5{\text{ m/day,}}$ $K_{{x_{2} }} = K_{{y_{2} }} = 1.5{\text{ m/day,}}$ $K_{{z_{2} }} = 1.2{\text{ m/day,}}$ $K_{{x_{3} }} = K_{{y_{3} }} = 1.2{\text{ m/day,}}$ $K_{{z_{3} }} = 1{\text{ m/day,}}$ $\delta_{1} = 0{\text{ m,}}$ $\delta_{2} = 0.04{\text{ m,}}$ $\delta_{3} = 0.08{\text{ m,}}$ $d_{x1} = 0.5{\text{ m,}}$ $d_{x2} = 1.5{\text{ m,}}$ $d_{x3} = 4.5{\text{ m,}}$ $d_{x4} = 4.75{\text{ m,}}$ $d_{y1} = 0.5{\text{ m,}}$ $d_{y2} = 1.5{\text{ m,}}$ $d_{y3} = 3.5{\text{ m}}$ and $d_{y4} = 4.5{\text{ m}} .$

An effort is also been made to study the distribution of top discharge at the surface of a three-dimensional ponded drainage system for a few drainage settings of figure 1 – figures 18, 19 and 20 show such distributions for the studied situations. As can be seen, we are now expressing the top discharges within predefined surface locations as a percentage of the total discharge through the top of the soil; also, the stream surfaces corresponding to these situations have also been plotted. From these figures, it is clear that the top discharge distribution in a three-dimensional ponded ditch drainage system is profoundly impacted by the directional conductivities of the constituent layers of a multi-layered soil and that neglecting the stratification of a soil profile may lead to an erroneous reading of movement of drainage water through such a soil column. Also, from figures 18 and 19, we see that introduction of a gradually increasing ponding pyramid towards the centre of a three-dimensional ponded ditch drainage system may result in a considerable improvement in the uniformity of water movement in such a system as compared to a situation where the ponding head at the surface of the field is kept a constant one; this is particularly true for a stratified soil where the conductivities of the layers decrease with depth. Thus, for draining such a soil, a progressively increasing ponding distribution away from the drains may be adopted on the surface of the soil for achieving a better uniformity of water movement in the drained space. In this context, it is worth noting that drainage with a uniform ponding system often leads to non-uniform movement of water in a drained space with most of the water to the drains coming from areas close to the drains only ([23, 25, 36, 41, 49, 77] – to cite a few).

Figure 18

Top discharge distribution for ponded drainage of a single-layered soil when the flow parameters of figure 1 are taken as $S_{1} = 16{\text{ m,}}$ $S_{2} = 10{\text{ m,}}$ $h = 1.5{\text{ m,}}$ $H_{1} = 1.2{\text{ m,}}$ $H_{2} = 0.5{\text{ m,}}$ $H_{3} = 1{\text{ m,}}$ $\delta_{i} = 0{\text{ m,}}$ $\varepsilon_{x} = \varepsilon_{y} = 0{\text{ m,}}$ $K_{{x_{1} }} = K_{{y_{1} }} = 1{\text{ m/day,}}$ $K_{{z_{1} }} = 0.5{\text{ m/day,}}$ $K_{{x_{2} }} = K_{{y_{2} }} = 2{\text{ m/day,}}$ $K_{{z_{2} }} = 1{\text{ m/day,}}$ $K_{{x_{3} }} = K_{{y_{3} }} =$ $3{\text{ m/day}}$ and $K_{{z_{3} }} = 2{\text{ m/day}} .$

Figure 19

Top discharge distribution for ponded drainage of a stratified soil when the flow parameters of figure 1 are taken as $S_{1} = 8{\text{ m,}}$ $S_{2} = 8{\text{ m,}}$ $h = 1.5{\text{ m,}}$ $H_{1} = 1.2{\text{ m,}}$ $H_{2} = 0.5{\text{ m,}}$ $H_{3} = 1{\text{ m,}}$ $\varepsilon_{x} = \varepsilon_{y} = 0.05{\text{ m,}}$ $K_{{x_{1} }} = K_{{y_{1} }} = 1.5{\text{ m/day,}}$ $K_{{z_{1} }} = 0.5{\text{ m/day,}}$ $K_{{x_{2} }} = K_{{y_{2} }} = 2{\text{ m/day,}}$ $K_{{z_{2} }} = 1{\text{ m/day,}}$ $K_{{x_{3} }} = K_{{y_{3} }} = 3{\text{ m/day,}}$ $K_{{z_{3} }} = 2{\text{ m/day}}$ and (a) $\delta_{i} = 0{\text{ m,}}$ (b) $\delta_{1} = 0{\text{ m,}}$ $\delta_{2} = 0.1{\text{ m,}}$ $\delta_{3} = 0.2{\text{ m,}}$ $\delta_{4} = 0.4{\text{ m,}}$ $d_{x1} = 0.5{\text{ m,}}$ $d_{x2} = 1.5{\text{ m,}}$ $d_{x3} = 2.5{\text{ m,}}$ $d_{x4} = 5.5{\text{ m,}}$ $d_{x5} = 6.5{\text{ m,}}$ $d_{x6} = 7.5{\text{ m,}}$ $d_{y1} = 0.5{\text{ m,}}$ $d_{y2} = 1.5{\text{ m,}}$ $d_{y3} = 2.5{\text{ m,}}$ $d_{y4} = 5.5{\text{ m,}}$ $d_{y5} = 6.5{\text{ m}}$ and $d_{y6} = 7.5{\text{ m}} .$

Figure 20

Top discharge distribution for ponded drainage of a stratified soil when the flow parameters of figure 1 are taken as $S_{1} = 8{\text{ m,}}$ $S_{2} = 8{\text{ m,}}$ $h = 1.5{\text{ m,}}$ $H_{1} = 1.2{\text{ m,}}$ $H_{2} = 0.5{\text{ m,}}$ $H_{3} = 1{\text{ m,}}$ $\varepsilon_{x} = \varepsilon_{y} = 0.05{\text{ m,}}$ $K_{{x_{1} }} = K_{{y_{1} }} = 3{\text{ m/day,}}$ $K_{{z_{1} }} = 2{\text{ m/day,}}$ $K_{{x_{2} }} = K_{{y_{2} }} = 2{\text{ m/day,}}$ $K_{{z_{2} }} = 1{\text{ m/day,}}$ $K_{{x_{3} }} = K_{{y_{3} }} = 1.5{\text{ m/day,}}$ $K_{{z_{3} }} = 0.5{\text{ m/day}}$ and (a) $\delta_{i} = 0{\text{ m,}}$ (b) $\delta_{1} = 0{\text{ m,}}$ $\delta_{2} = 0.1{\text{ m,}}$ $\delta_{3} = 0.2{\text{ m,}}$ $\delta_{4} = 0.4{\text{ m,}}$ $d_{x1} = 0.5{\text{ m,}}$ $d_{x2} = 1.5{\text{ m,}}$ $d_{x3} = 2.5{\text{ m,}}$ $d_{x4} = 5.5{\text{ m,}}$ $d_{x5} = 6.5{\text{ m,}}$ $d_{x6} = 7.5{\text{ m,}}$ $d_{y1} = 0.5{\text{ m,}}$ $d_{y2} = 1.5{\text{ m,}}$ $d_{y3} = 2.5{\text{ m,}}$ $d_{y4} = 5.5{\text{ m,}}$ $d_{y5} = 6.5{\text{ m}}$ and $d_{y6} = 7.5{\text{ m}} .$

It may also be noted that the difference in volume of water seeping into the drains between a uniform and a variable distribution at the surface of the soil – assuming all other factors to remain the same – may not be appreciable for many drainage situations and a variable distribution – depending on the magnitude of the uniform and distributed ponding depths and the other parameters of the problem – may actually lead to a lesser volume of water seeping into the drains than a constant depth of ponding at the surface of the soil. For example, for the drainage situation of figure 18, the volume of water seeping into the drains from the surface of the soil in 1 hour is 1.4335 m³, the corresponding figures, had the ponding depth at the surface of the soil been a uniform one with 0.1 m and 0.2 m depths, respectively would have been 1.3348 m³ and 1.5211 m³. Thus, if we have taken a constant depth of ponding of 0.2 m (i.e., by considering half of the maximum ponding depth of 0.4 m at the central strip) in the drainage situation of figure 18, more water with lesser uniformity would then have actually seeped into the drains as compared to the variable ponding distribution scenario as shown in the figure. For the flow situation of figure 19 where the directional conductivities are decreasing with depth, these figures are (i.e., volume of water seeping in 1 hour) 2.7773 m³, 2.6569 m³ and 3.1642 m³, respectively, where again, as can be seen, a constant ponding depth of 0.2 m is actually causing much more water to seep into the drains as compared to the imposed variable ponding distribution as shown. The increase in uniformity with a staggered ponding distribution of pyramidal nature away from the drains can be explained by observing that a distribution like this provides a greater water head for forcing water particles travelling in longer arcs to the drains as compared to situations where the ponding distribution is a flat one. We would also like to add here that the pyramidal shape of surface ponding distribution considered in these examples is simply to see how such distributions are affecting the uniformity of water movement in the studied drainage situations vis-à-vis a constant depth of ponding over the soil. These are just a few examples only; our solution can very well be used to analyze ponded drainage situations for a multitude of other ponding distributions as well (i.e., apart from the ones as shown) since by suitably altering the locations of the inner bunds and the ponding depths within these bunds, a wide range of ponding distributions at the surface of the soil can be suitably imposed. From figures 18 and 20(a), we also see that when the hydraulic conductivities of the upper soil layer is lower than that of the other soil layers for constant ponding situations, uniformity of top discharge distribution of a three-dimensional drainage system is relatively better in comparison to the case where the top soil layer has the same or larger hydraulic conductivity values than the lower layers. This is due to the fact that when the conductivity of the top layer is low, the water particles are not getting readily infiltrated into the soil even from closer locations to the drains thereby resulting in relatively less volume of water to seep through a surface area close to the ditches as compared to seepage through the same area when the conductivity of the top layer is high. Thus, the conductivity contrasts of a multi-layered soil alone may have a significant effect on the overall dynamics of ground water movement of a ponded ditch drainage system in a stratified soil.

5 Conclusions

A comprehensive analytical model has been proposed for predicting groundwater seepage into a network of ditch drains in a three-layered soil column underlain by an impervious barrier, the drains being fed by a variable ponding distribution introduced at the surface of the soil. The problem has been solved by first assuming the level of water in the drains to lie on the top layer, then on the middle layer and finally on the bottom layer. In all these cases, it has been assumed that the level of water in all the drains is the same. Further, it has also been assumed in all these derivations that the flow is steady and the stratified drainage space as fully saturated. These, thus, can be taken as limitations of the proposed solutions as they are not valid for transient as well as variably saturated soil conditions. It should, however, be noted that as there currently no analytical solution to the 3D ponded ditch drainage problem for a stratified soil even with these limitations, all the solutions proposed here, thus, can be considered as new. These solutions can also be regarded as an extension of the analytical works of Sarmah and Barua [25] from a single-layered soil to that of a multi-layered soil system under steady state situation. In addition, it is also worth noting that the mathematical procedure outlined here is quite general and has the inherent flexibility to solve the multi-layered drainage problem considered here with more than three layers as well.

From the study, it has become clear that ponded drainage of a multi-layered soil is mostly three-dimensional in nature, particularly in locations close to the drains where the curvature of the streamlines may be quite noticeable. However, if the length of the drainage lines on one side is taken much longer than on a side orthogonal to it, then the flow in a vertical section located further away from the longer boundaries of the flow domain can then be modeled using the two-dimensional flow assumption without introducing appreciable error. Further, similar to a two-dimensional ponded drainage situation, three-dimensional ponded drainage in a stratified soil with a constant ponding depth at the surface of the soil is also mostly restricted to areas close to the drains with minimal contribution of flow coming to the drains from locations further away from the drains. However, by providing a progressively increasing ponding distribution away from the ditches and towards the centre of the ponding field, considerable improvement on the uniformity of water movement in a stratified ponded space can be brought about in comparison to a situation where the imposed ponding head over the field is a uniform one. The solution proposed here can be suitably applied for the design of subsurface of drains in both single as well as multi-layered soils; this is an advantage since soils in actual field situations are seldom homogeneous. Further, because of the ability of these new solutions to account for complexities like soil stratifications and three-dimensional flows of the ponded drainage problem, they may also be used for verifying complex numerical codes related to subsurface drainage and related areas after reducing these codes to relatively simpler hydrogeological settings for which solutions have been obtained in the current study. Thus, these solutions are expected to be important additions to the already available collection of analytical solutions related to subsurface drainage under ponded conditions.

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.