Abstract
An analytical solution in the form of infinite series is developed for predicting time-dependent three-dimensional seepage into ditch drains from a flat, homogeneous and anisotropic ponded field of finite size, the field being assumed to be surrounded on all its vertical faces by ditch drains with unequal water level heights in them. It is also assumed that the field is being underlain by a horizontal impervious barrier at a finite distance from the surface of the soil and that all the ditches are being dug all the way up to this barrier. The solution can account for a variable ponding distribution at the surface of the field. The correctness of the proposed solution for a few simplified situations is tested by comparing predictions obtained from it with the corresponding values attained from the analytical and experimental works of others. Further, a numerical check on it is also performed using the Processing MODFLOW environment. It is noticed that considerable improvement on the uniformity of the distribution of the flow lines in a three-dimensional ponded drainage space can be achieved by suitably altering the ponding distribution at the surface of the soil. As the developed three-dimensional ditch drainage model is pretty general in nature and includes most of the common variables of a ditch drainage system, it is hoped that the drainage designs based on it for reclaiming salt-affected and water-logged soils would prove to be more efficient and cost-effective as compared with designs based on solutions developed by making use of more restrictive assumptions. Also, as the developed model can handle three-dimensional flow situations, it is expected to provide reliable and realistic drainage solutions to real field situations than models being developed utilizing the two-dimensional flow assumption. This is because the existing two-dimensional solutions to the problem are actually valid not for a field of finite size but for an infinite one only.
Similar content being viewed by others
References
Hoorn Van J W and Alphen Van J G 1994 Salinity control. In: Ritzema H P (Ed.) Drainage principles and applications. ILRI 533-600. Wageningen: ILRI
Ritzema H P, Satyanarayana T V, Raman S and Boonstra J 2008 Subsurface drainage to combat water logging and salinity in irrigated lands in India: lessons learned in farmer’s field. Agric. Water Manage. 95: 179–189
Youngs E G 1994 Seepage to ditches from a ponded surface. J. Hydrol. 161: 145–154
Smedema L K, Abdel-Dayem S and Ochs W J 2000 Drainage and agricultural development. Irrig. Drain. Syst. 14: 223–235
Wichelns D, Cone D and Stuhr G 2002 Evaluating the impact of irrigation and drainage policies on agricultural sustainability. Irrig. Drain. Syst. 16: 1–14
Ghassemi F, Jakeman A J and Nix H A 1995 Salinisation of land and water resource: human causes, extent, management and case studies. Sydney: University of New South Wales Press
Khan S, Tariq R, Yuanlai C and Blackwell J 2004 Can irrigation be sustainable? In: New directions for a diverse planet: Proceedings of the 4th International Crop Science Congress 2004, Brisbane, Australia, 26 September–1 October 2004.
Martinez-Beltran J 2002 World food summit. Food and Agriculture Organization of the United Nations (http: //www. fao.org /worldfoodsummit /english/newsroom/focus/focus1.htm)
Rhoades J D 1997 Sustainability of irrigation: an overview of salinity problems and control strategies. In: Proceedings of the 1997 Annual Conference of the Canadian Water Resources Association, Footprints of Humanity: Reflections on Fifty Years of Water Resources Developments, Lethbridge, Alberta, Canada, June 3–6, 1997
Barua G and Alam W 2013 An analytical solution for predicting transient seepage into ditch drains from a ponded field. Adv. Water Resour. 52: 78–92
Rao K V G K and Leeds-Harrison P B 1991 Desalination with subsurface drainage. Agric. Water Manage. 19: 303–311
Sarmah R and Barua G 2015 Hydraulics of a partially penetrating ditch drainage system in a layered soil receiving water from a ponded field. J. Irrig. Drain. Eng. 141: 04015001
Xin P, Yu X, Lu C and Li L 2016 Effects of macro-pores on water flow in coastal subsurface drainage systems. Adv. Water Resour. 87: 56–67
Youngs E G and Leeds-Harrison R B 2000 Improving efficiency of desalinization with subsurface drainage. J. Irrig. Drain. Eng. 126(6): 375–380
Groenigen K J V, Kessel C V and Hungate B A 2013 Increased greenhouse-gas intensity of rice production under future atmospheric conditions. Nat. Clim. Chang. 3: 288–291
Qiu J 2009 China cuts methane emissions from rice fields. Nature News, doi:10.1038/news.2009.833
Brainard E C and Gelhar L W 1991 Influence of vertical flow on ground-water transport. Ground Water 29(5): 693–701
Kirkham D 1950 Seepage into ditches in the case of a plane water table and an impervious substratum. Trans. Am. Geophys. Union 31(3): 425–430
Kirkham D 1960 Seepage into ditches in the case of a plane water table overlying a gravel substratum. Trans. Am. Geophys. Union 65(4): 1267–1272
Kirkham D 1965 Seepage of leaching water into drainage ditches of unequal water level height. J. Hydrol. 3: 207–224
Kirkham D, Toksoz S and Van der Ploeg R R 1974 Steady flow to drains and wells. Drainage for Agriculture. USA: American Society of Agronomy
Fukuda H 1975 Underdrainage into ditches in soil overlying an impervious substratum. Trans. Am. Geophys. Union 38(5): 730–739
Barua G and Tiwari K N 1995 Analytical solutions of seepage into ditches from ponded fields. J. Irrig. Drain. Eng. 121(6): 396–404
Bereslavskii E N 2006 Groundwater flow to a system of drainage canals. Water Resour. 33(4): 417–420
Ilyinsky N B and Kacimov A R 1992 Problems of seepage to empty ditch and drain. Water Resour. Res. 28(3): 871–877
Römkens M J M 2009 Estimating seepage and hydraulic potentials near incised ditches in a homogeneous, isotropic aquifer. Earth Surf. Process. Landf. 34: 1903–1914
Warrick A W and Kirkham D 1969 Two-dimensional seepage of ponded water to full ditch drains. Water Resour. Res. 5(3): 685–693
Afruzi A, Nazemi A H and Sadraddini A A 2014 Steady-state subsurface drainage of ponded fields by rectangular ditch drains. Irrig. Drain. 63(5): 668–681
Barua G and Sarmah R 2016 An analytical solution for predicting transient seepage into partially penetrating ditch drains receiving water from a ponded field. Acta Geophys. 64(1): 149–205
Chahar B R and Vadodaria G P 2008 Steady subsurface drainage of homogeneous soil by ditches. Water Manage. 161(WM6): 303–311 (Proceedings of ICE)
Chahar B R and Vadodaria G P 2008 Drainage of ponded surface by an array of ditches. J. Irrig. Drain. Eng. 134(6): 815–823
Chahar B R and Vadodaria G P 2012 Steady subsurface drainage of ponded surface by an array of parallel ditches. J. Hydrol. Eng. 17(8): 895–908
Voss C I and Provost A M 2008 SUTRA: a model for saturated–unsaturated variable-density ground-water flow with solute or energy transport. U.S. Geological Survey Water-Resources Investigations Report 02-4231
Dielman P J 1973 Reclamation of salt-affected soils in Iraq. Publication No. 11, Wageningen, The Netherlands: International Institute for Land Reclamation and Improvement
Martinez-Beltran J 1978 Drainage and reclamation of salt affected soils in the Bardenas area, Spain. ILRI Publication 24. Wageningen, The Netherlands: ILRI
Mirjat M S and Rose D A 2009 Streamline pattern and salt leaching through progressive flooding between subsurface drains. Irrig. Drain. 58: 199–208
Murdoch L C 1994 Transient analysis of an interceptor trench. Water Resour. Res. 30(11): 3023–3031
Meigs L C and Bahr J M 1995 Three-dimensional groundwater flow near narrow surface water bodies. Water Resour. Res. 31(12): 3299–3307
Hossain M and Narciso J 2004 Long-term prospects for the global rice economy. In: Proceedings of FAO Rice Conference, Rome, Italy
Hantush M S 1964 Hydraulics of well. In: Chow V T (Ed.) Advances in hydroscience. New York: Academic Press pp. 281–442
Kirkham D and Powers W L 1972 Advanced soil physics. Wiley-Interscience, New York
Sarmah R 2015 Two-and three-dimensional analysis of flow into ditch drains from a ponded field. Ph.D. Thesis, Indian Institute of Technology Guwahati, India (thesis accepted)
Grove D B, Beatam W A and Sower F B 1970 Fluid travel time between a recharging well pair in an aquifer having a uniform regional flow field. Water Resour. Res. 6(5): 1404–1410
Hultquist J P M 1992 Constructing stream surfaces in steady 3D vector fields. In: Proceedings of Visualization’92, pp. 171–198
Steward D R 1998 Stream surfaces in two-dimensional and three-dimensional divergence-free flows. Water Resour. Res. 34(5): 1345–1350
Chiang W and Kinzelbach W 2001 3D-groundwater modeling with PMWIN: a simulation system for modeling groundwater flow and pollution. Berlin: Springer-Verlag
Maasland M 1957 Soil anisotropy and land drainage. In: Luthin J N (Ed.) Drainage of agricultural lands. Madison, Wisconsin: American Society of Agronomy, pp. 216–285
Schafer D C 1996 Determining capture zones in homogeneous anisotropic aquifers. Ground Water 34(4): 628–639
Chen S K and Liu C W 2002 Analysis of water movement in paddy rice fields (I) experimental studies. J. Hydrol. 260: 206–215
MacDonald A M, Bonsor H C, Dochartaigh B E O and Taylor R G 2012 Quantitative maps of groundwater resources in Africa. Environ. Res. Lett. 7(2): 024009
Stibinger J 2009 Terrain experimental measurement of saturated hydraulic conductivity on paddy fields in Taoyuan (Taiwan) during the cycle of flooded period. Agric. Trop. Subtrop. 42(2): 82–89
Tabuchi T 2004 Improvement of paddy field drainage for mechanization. Paddy Water Environ. 2: 5–10
Chen C S and Chang C C 2003 Well hydraulics theory and data analysis of the constant head test in an unconfined aquifer with the skin effect. Water Resour. Res. 39(5): 1121
Grisak G E and Cherry J A 1975 Hydrologic characteristics and response of fractured till and clay confining a shallow aquifer. Can. Geotech. J. 12(1): 23–43
Neuman S P 1975 Analysis of pumping test data from anisotropic unconfined aquifers considering delayed gravity response. Water Resour. Res. 11(2): 329–342
Sharp J M 1984 Hydrogeologic characteristics of shallow glacial drift aquifers in dissected till plains (North-Central Missouri). Ground Water 22: 683–689
Shaver R B 1998 The determination of glacial till specific storage in north Dakota. Ground Water 36: 552–557
Lundstrom D R and Stegman E C 1988 Irrigation scheduling by the checkbook method. Bulletin AE-792 (Rev.), North Dakota State University Extension Service, Fargo
Ayars J E, Corwin D L and Hoffman G J 2012 Leaching and root zone salinity control. In: Wallender W W and Tanji K K (Eds.) Agricultural salinity assessment and management, 2nd ed., ASCE Manual and Reports on Engineering Practice No. 71. Reston, VA: ASCE. Chapter 12, pp. 371–403
FAO 1985 Water quality for agriculture. Irrigation and Drainage, Paper 29, Rev. 1. Rome: FAO
Author information
Authors and Affiliations
Corresponding author
Appendices
List of notations
- \( A_{{m_{1} ,n_{1} }} ,B_{{m_{2} ,n_{2} }} ,C_{{m_{1} ,n_{1} }} ,D_{{m_{4} ,n_{4} }} ,E_{pqr} ,F_{{m_{1} ,n_{1} }} : \) :
-
\({\text{constants}}\;{\text{with}}\;m_{1} = 1,2,3, \ldots , n_{1} = 1,2,3, \ldots ,\quad m_{2} = 1,2,3, \ldots ,\quad n_{2} = 1,2,3, \ldots , m_{3} = 1,2,3, \ldots ,\quad n_{3} = 1,2,3, \ldots ,\quad m_{4} = 1,2,3, \ldots , n_{4} = 1,2,3, \ldots ,\;m_{5} = 1,2,3, \ldots ,\;n_{5} = 1,2,3, \ldots , p = 1,2,3, \ldots ,\;q = 1,2,3, \ldots ,\;r = 1,2,3, \ldots \)
- h :
-
depth of the soil column, L
- \( H_{1} \) :
-
height of water in the Northern ditch as measured from the surface of the soil, L
- \( H_{2} \) :
-
height of water in the Southern ditch as measured from the surface of the soil, L
- \( H_{3} \) :
-
height of water in the Eastern ditch as measured from the surface of the soil, L
- \( H_{4} \) :
-
height of water in the Western ditch as measured from the surface of the soil, L
- K :
-
=(K x K y K z )1/3 equivalent hydraulic conductivity of soil, LT−1
- \( \left( {K_{x}^{a} } \right)^{2} \) :
-
=K x /K z , anisotropy ratio of soil in the x-direction, dimensionless
- \( \left( {K_{y}^{a} } \right)^{2} \) :
-
=K y /K z , anisotropy ratio of soil in the y-direction, dimensionless
- K x :
-
hydraulic conductivity of soil in the x-direction, LT−1
- K y :
-
hydraulic conductivity of soil in the y-direction, LT−1
- K z :
-
hydraulic conductivity of soil in the z-direction, LT−1
- (K1):
-
=\( \sqrt {S_{s} /K_{z} ,} \) L−1 T1/2
- M:
-
distance in metres, L
- M 1, N 1, M 2, N 2, M 3, N 3, M 4, N 4, M 5, N 5, P, Q and R: :
-
number of terms to be summed in the series solution, 1,2,3,…
- \( N_{0} \) :
-
number of divisions of the ponding surface at the top of the soil
- \( Q_{N} ,Q_{S} ,Q_{E} ,Q_{W}: \) :
-
discharge through the Northern, Southern, Eastern, Western faces of the soil column of figure 1, L3T−1
- \( Q_{\text {top}}^{f} \) :
-
top discharge function defined for the surface of the soil of figure 1, L3T−1
- \( Q_{\text {top}} \) :
-
discharge through the top surface of the soil of figure 1, L3T−1
- \( Q_{\text {top}}^{nf} \) :
-
top discharge function expressed as a percentage of \( Q_{\text {top}} \), dimensionless
- L :
-
distance between the adjacent drains in the x-direction in the real space of figure 1, L
- L X :
-
distance between the adjacent drains in the X-direction in the computational space of figure 1, L
- B :
-
distance between the adjacent drains in the y-direction in the real space of figure 1, L
- B Y :
-
distance between the adjacent drains in the Y-direction in the computational space of figure 1, L
- \( d_{xi} \) :
-
distance of the ith \( (1 \le i \le N_{0} {-}1) \) inner bund from the origin O in the x-direction of figure 1 in the real space, L
- \( S_{Xi} \) :
-
=\( d_{xi} /K_{x}^{a} , \) L
- \( d_{yi} \) :
-
distance of the ith \( (1 \le i \le N_{0} {-}1) \) inner bund from the origin O in the y-direction of figure 1 in the real space, L
- \( S_{Yi} \) :
-
\( d_{yi} /K_{y}^{a} , \) L
- \( S_{s} \) :
-
specific storage of soil, L−1
- \( V_{x} \) :
-
velocity distribution for the flow domain of figure 1 in the x-direction, LT−1
- \( V_{y} \) :
-
velocity distribution for the flow domain of figure 1 in the y-direction, LT−1
- \( V_{z} \) :
-
velocity distribution for the flow domain of figure 1 in the z-direction, LT−1
- t :
-
time variable for the flow problem of figure 1, T
- x :
-
coordinate as measured from the origin O of figure 1 in the East-West dirction in the real space
- X :
-
\( = x/K_{x}^{a} , \) L
- y :
-
coordinate as measured from the origin O of figure 1 in the North-South dirction in the real space
- Y :
-
\( = y/K_{y}^{a} , \) L
- z :
-
coordinate as measured from the origin O of figure 1 in the downward direction in the real space, L
- \( \delta_{i} \) :
-
ponding depth at the ith segment on the surface of the soil, L
- \( \varepsilon_{x} \) :
-
width of the ditch banks in the x-direction in the real space of figure 1, L
- \( \varepsilon_{y} \) :
-
width of the ditch banks in the y-direction in the real space of figure 1, L
- \( \phi \) :
-
hydraulic head distribution for the flow domain of figure 1 (with th Northern boundary as a ditch drainage boundary), L
Appendix 1. Determination of coefficients of the hydraulic head function of Eq. (2)
In this section, the coefficients appearing in Eq. (2) will be determined utilizing the appropriate initial and boundary value conditions mentioned in the definition of the problem. To evaluate \( A_{{m_{{_{1} }} n_{{_{1}}}}}, \) boundary conditions (IIIa) and (IIIb) can be made use of; application of the same to Eq. (2) at \( Y = 0 \) gives
Thus, \( A_{{m_{{_{1} }} n_{{_{1} }} }} \)can be evaluated by running a double Fourier series in the domain covered by \( 0 < X < L_{X} \) and \( 0 < z < h; \) this yields an expression for \( A_{{m_{{_{1} }} n_{{_{1} }} }} \)as
Simplification of the above integrals yields
Similarly, an application of boundary conditions (IIa) and (IIb) to Eq. (2) gives \( B_{{m_{{_{2} }} n_{{_{2} }} }} \) as
Likewise, boundary conditions (Va) and (Vb) and (IVa) and (IVb) can be utilized to evaluate the constants \( C_{{m_{3} n_{3} }} \) and \( D_{{m_{4} n_{4} }} \) of Eq. (2); the relevant expressions for the same can be expressed as
and
Next, to work out the constants \( F_{{m_{5} n_{5} }} \)of Eq. (2), boundary conditions (VIIa) to (VIIj) can be made use of; applying the same to Eq. (2), the following set of equations can be realized:
where
and
Thus, \( F_{{m_{5} n_{5} }} \) can be evaluated by running a double Fourier series in the space defined by the intervals \( 0 < X < L_{X} \) and \( 0 < Y < B_{Y} ; \) this yields an equation for evaluating \( F_{{m_{5} n_{5} }} \) as
Simplification of the above integrals gives an expression for \( F_{{m_{5} n_{5} }} \) as
There still remain the constants \( E_{pqr} \) to be determined. Towards this end, the initial condItion (I) can be applied to Eq. (2); the pertinent expression for evaluating these constants can then be expressed as
Now, performing a triple Fourier run in the space defined by the intervals \( 0 < X < L_{X} , \) \( 0 < Y < B_{Y} \) and \( 0 < z < h, \) an expression for the constants \( E_{pqr} \) can then be worked out as
Identifying the first, second, third, fourth and fifth triple-integrals of Eq. (41) as \( I^{(1)} , \) \( I^{(2)} , \) \( I^{(3)} , \) \( I^{(4)} \) and \( I^{(5)} , \) respectively, and then simplifying them yields an expression for \( E_{pqr} \) as
where
For \( N_{{m_{{_{1} }} }} = N_{p} \)
and for \( N_{{m_{{_{1} }} }} \ne N_{p} \)
For \( N_{{n_{{_{1} }} }} = N_{r} \)
and for \( N_{{n_{{_{1} }} }} \ne N_{r} \)
For \( N_{{m_{{_{2} }} }} = N_{p} \)
and for \( N_{{m_{{_{2} }} }} \ne N_{p} \)
For \( N_{{n_{{_{2} }} }} = N_{r} \)
and for \( N_{{n_{{_{2} }} }} \ne N_{r} \)
where
For \( N_{{m_{{_{3} }} }} = N_{q} \)
and for \( N_{{m_{{_{3} }} }} \ne N_{q} \)
For \( N_{{n_{{_{3} }} }} = N_{r} \)
and for \( N_{{n_{{_{3} }} }} \ne N_{r} \)
where
For \( N_{{m_{{_{4} }} }} = N_{q} \)
and for \( N_{{m_{{_{4} }} }} \ne N_{q} \)
For \( N_{{n_{4} }} = N_{r} \)
and for \( N_{{n_{4} }} \ne N_{r} \)
where for \( N_{{m_{{_{5} }} }} = N_{p} \)
and for \( N_{{m_{{_{5} }} }} \ne N_{p} \)
For \( N_{{n_{5} }} = N_{q} \)
and for \( N_{{n_{5} }} \ne N_{q} \)
and
All the coefficients of Eq. (2) are thus determined and the boundary value problem of figure 1 hence stands solved.
Further, like in the determination of the top discharge function, Darcy’s law can also be applied to evaluate the time-dependent discharges being received through the Northern, Southern, Eastern and Western faces of the ditches; naming these discharges as \( Q_{N} , \) \( Q_{S} , \) \( Q_{E} \) and \( Q_{W} , \) respectively, their expressions, thus, can be represented as
and
Further, by performing time integrals on the concerned discharges functions, the volume of water seeping through the top and vertical faces of the studied ponded system within a desired time interval can also be worked out.
Rights and permissions
About this article
Cite this article
Sarmah, R., Barua, G. Analysis of three-dimensional transient seepage into ditch drains from a ponded field. Sādhanā 42, 769–793 (2017). https://doi.org/10.1007/s12046-017-0628-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12046-017-0628-6