Analytical modeling of flow into open drains in a layered soil receiving water from a ponded field

Comparison of transient decay values as obtained from the proposed solution at times t100 s and 500 s with the corresponding values as obtained from the single-layer transient solution of Sarmah and Barua (2017) of the problem for anisotropic soils when the flow parameters of the problem ( Fig. 3.4) are considered as S16 m, S25 m, h1 m.

Fig. 2.15 Comparison of steady state hydraulic heads as obtained from the proposed solution with corresponding values as obtained from MODFLOW when the flow parameters of the problem are considered as

Literature Review

Bereslavskii (2006) used the Riemann-Schwartz symmetry principle to propose an analytical solution to the fully penetrating trench drainage problem defined for homogeneous and isotropic soils. Solutions based on conformal transformations have also been proposed by Chahar and Vadodaria (2008a, 2008b, 2012) for different versions of the trench drainage problem.

Objectives

For the transient case as well, analytical verifications of the proposed solution for some drainage scenarios with respect to the single-layer soil were performed by first transforming the model to a single-layer anisotropic soil and then comparing the heads obtained from it with the corresponding values obtained from the analytical works of Barua and Alam (2013). Furthermore, numerical verifications of the developed solutions were also done for some two-layer trench drainage configurations by drawing appropriate models using the Processing MODFLOW (Chiang and Kinzelbach 2001) environment and then comparing the analytical outputs with their numerical equivalents.

Solutions for the Two-Dimensional Continuity Equation of Groundwater flow for a Homogeneous and Anisotropic Soil

The solution proposed in equation 2.11) is a general form of the fundamental solution of the leading partial differential equation. By changing the values of arbitrary constants  and , we get countless such solutions to the governing equation.

Mathematical Formulation and Solution

Level of Water in the Ditches is Different and Above the Boundary between the Soil Layers

Determination of Coefficients of the Steady State Solution (steady state solution is exact and valid for all possible arrangement of parameters of Fig. 2.2)
Determination of the Transient Solution Coefficients [transient state solution is approximate and is applicable only when conductivity and specific storage of the layers
Verifications of the Proposed Solution
MODFLOW Verification of the Proposed Solution

We are now concerned with determining the transient discharge function at the top of the flow domain. We apply Darcy's law at the surface of the earth for the hydraulic head term 1(1).

Fig. 2.2 illustrates the height of water, and consequently, the boundary conditions at the drainage ditches for this particular case.The boundary conditions specific to the ditch drain boundaries for this particular case are

Level of Water in the Ditches is Different and Below the Boundary between the Soil Layers

Determination of the Steady State Solution Coefficients (steady state solution is exact and valid for all possible arrangement of parameters of Fig. 2.10)

Evaluation of the above integral gives Gj(2) as. 2.131) Similarly, by boundary conditions (XIIIa) and (XIIIb) on Eq. The expression for the coefficients Amn( 2) (with the Fourier coefficients Bp(2), Cq(2), Ek(2) and Dr(2)) is identical to that of Amn(1); however, coefficients Zuv(2) will now have a different expression than that of Zuv(1), since the steady state head distribution of the lower layer is now different from that of the previous case. Variation of Qtop(2)/2Kh ratios with time as obtained from the proposed solution for different h and H1H2h values when the other parameters of the flow problem are considered.

Comparison of steady state hydraulic heads as obtained from the proposed solution with the corresponding values as obtained from Kirkham's (1965) single layer steady state solution of the problem for isotropic soil when the flow parameters of the problem are considered as S5m, h 1 m, H10 .75 m, H2 0.85 m,  0 m,. Comparison of transient hydraulic heads as obtained from the proposed solution with corresponding values as obtained by MODFLOW at two different times of simulation of the system when the flow parameters of the problem are considered as S5.0 m, h1.0 m, H30.55 m,.

Fig. 2.10 Ditch drainage system for a two-layered soil when height of water in the ditches is below the bottom surface of the top soil layer

Level of Water in the Left Ditch is Below the Boundary between the Soil Layers while Level of Water in the Right Ditch is Above the Boundary between the Soil Layers

Variation of Qtop(3)/2Kh ratio with time as obtained from the proposed solution for different h and H1h values when the other parameters of the flow problem are considered as. Comparison of steady state hydraulic heads as obtained from the proposed solution with the corresponding values as obtained from Kirkham's (1965) single layer steady state solution of the problem for isotropic soils when the flow parameters of the problem are considered as S 5 m, . Plots of Qtop(3)/2Kh versus hH1/h as obtained from the proposed solution for various times of simulation of the system with the corresponding steady state plot as obtained from Young's (1994) single-layer solution of the problem for isotropic soil, when the pond depth is considered zero and the other flow parameters for the problem are taken as S 100 m, h2.0 m,.

Comparison of transient hydraulic heads as obtained from the proposed solution with the corresponding values obtained by MODFLOW at two different times of system simulation when the flow parameters of the problem are considered as S5.0 m, h1.0 m, H10.65 m, . Comparison of transient hydraulic heads as obtained from the proposed solution with the corresponding values obtained by MODFLOW at two different times of system simulation when the flow parameters of the problem are considered as S5.0 m, h1.0 m, H20.45 m, .

Fig. 2.19. Variation of Q top (3) / 2 Kh ratios with time as obtained from the proposed solution for different h and H 1  h values when the other parameters of the flow problem are considered as

Level of Water in the Left Ditch is Above the Boundary Between the Soil Layers while Level of Water in the Right Ditch is Below the Boundary Between the Soil Layers

As in the last two cases, the expressions for the directional velocity functions Vx1( 4) and Vy1 ( 4) for the upper layer in this drainage situation will also be similar to that in the first case, but here again the Fourier coefficients in these expressions must be used Bp (4), Cq(4), Ek(4), Dr(4), Amn(4) and Zuv(4) and not the corresponding Fourier coefficient of the previous cases. Variation of Qtop(4)/ 2Kh ratios with time as obtained from the proposed solution for different values of h and H2 h when other parameters of the flow problem are considered. Plots of Qtop(4) / 2Kh vs hH1/h obtained from the proposed solution for different simulation times of the system with the corresponding steady-state plot as obtained from Youngs (1994) one-layer solution to the isotropic soil problem when the depth of the lake is considered zero and the other flow parameters of the problem are taken as S100 m, h2.0 m,.

Comparison of the steady-state head values obtained from the proposed solution with the corresponding values obtained from Kirkham's (1965) single-layer steady-state solution of the problem for isotropic soils when the flow parameters of the problem are considered. Comparison of transient decays as obtained from the proposed solution with corresponding values as obtained from Barua and Alam's (2012) single-layer transient solution of the problem for anisotropic soils at two different times of simulation of the system when the flow parameters for the problem are considered as S 5 m, h1 m, H10.75 m, H20.25 m.

Discussions

Plot of steady state upper discharge values versus the thickness of the upper soil layer when the other parameters of the flow problem of Fig. Plot of steady state upper discharge versus the thickness of the upper soil layer when the other parameters of the flow problem of Fig. Similar behavior of the peak discharge function can also be observed if the dam system of Fig.

Travel times (in days) of water particles starting from the surface of a two-layer pond ditch drainage system to the receiving drain and the distribution function of the upper discharge when the system flow parameters (Fig. 2.2) are taken as S 6 m, h2 m, H31.2 m, H1H2 1 m,. Travel times (in days) of water particles starting from the surface of a two-layer pond ditch drainage system to the receiving drain and the distribution function of the upper discharge when the system flow parameters (Fig. 2.10) are taken as S 6 m, h2 m, H31.2 m, H1H2.

Fig 2.34. Variation of steady state horizontal gradient with depth of water at the face of a ditch when the parameters of the flow problem of Fig

Conclusions

The study shows that the hydraulics of flow around a drainage system of ditches with lakes in a two-layer soil is largely influenced by the directional conductivities of the constituent soil layers and the lake field located on the soil surface, among other factors. . It also became clear from the study that lacustrine drainage of stratified soils with a uniform depth of lacustrine on the soil surface would lead to uneven movement of water into the soil - areas close to the drains will have a much higher proportion of flow through them compared to regions further away from the ditches. Thus, cleaning the soil profile through a drainage system with ponds with a single ponding field will result in uneven soil profile cleaning with areas near the drains being overwashed and areas further away from the drains underwashed.

Therefore, these solutions are expected to be more versatile in their applications in drainage design than existing solutions to the pond drainage problem. Another limitation of the developed solutions is that the drains are assumed to be excavated up to an impermeable barrier, which, in actual field situations, is hardly the case.

List of Notations

It should also be noted that we have already assumed that the inner edges at the surface of the soil have infinitesimally small thickness in order to reduce the surface boundary for mathematical convenience to strictly a Dirichlet boundary. Si  horizontal distance of the ith inner beam from the origin 'O' for the flow problem of Fig. Full  volume of water seeping through the side faces of the left ditch for the flow problem shown by Fig.

Vol  volume of water seeping through the sides of the right ditch for the flow problem shown in figure. The drains are fed by a distributed water column introduced at the surface of the soil column.

Solutions for the Three-Dimensional Continuity Equation of Groundwater flow for a Homogeneous and Anisotropic Soil

The solution proposed in Eq. 3.12) is the general form of a fundamental solution of the governing partial differential equation. By varying the values of the arbitrary constants   and  , we can obtain countless such solutions to the governing equation. Since the sum of all these solutions will also satisfy Eq. 3.2), a solution of the governing equation can thus be expressed as

Assigning summation indices to the constants  and  and then summing several fundamental solutions of Eq. 3.15), we can have another solution of the differential equation as Similarly, after dividing Eq. In view of Eq. and (3.29) will now be used to solve the three different variants of the three-dimensional multi-layer dam drainage problem in the present study.

Mathematical Formulation and Solution

Level of Water in the Ditches is above the Boundary between the Top and the Middle Soil Layers

Determination of Coefficients of the Transient State Part of the Solution [transient state solution is approximate and is applicable only when directional
MODFLOW Verifications of the Proposed Solution

In order to establish that the SR1 triple Fourier series exists only in the top soil layer of the aquifer, we ensure that the sinusoidal bases associated with the series vanish for the middle layer and the bottom layer. Execution of a triple Fourier series in the specified domain, an expression for evaluating the coefficients. Now, we need to prove the divergence of the series for the second case, where cos2u.

But even in this case, the series on the right side of the above. The expression for discharge through the north ditch face of the flow domain can be expressed as. By solving the above integrals, we get QNorth(1) t as. 3.135) The discharge expression, QSouth(1) t , for the southern trench face of the flow domain can also be expressed as.

S S S then the transient solution is correct for all possible combinations of the parameters of Fig.

Fig. 3.4. Three dimensional ditch drainage system for a three-layered soil when height of water in the ditches is above the boundary between the top and the middle soil layers

Level of Water in the Ditches is below the Boundary between the Top and the Middle Soil Layers and above the Boundary between the Middle and the Bottom Soil

Determination of Coefficients of the Steady State Part of the Solution (steady state solution is exact and valid for all possible arrangement of parameters of Fig. 3.10)

Considering the general solutions of the three-dimensional continuity equation as expressed in Eqs. As can be seen, the hydraulic head expressions for the upper and lower layers for this drainage situation are similar to those of the previous case. As a function of the hydraulic head of the middle layer for the drainage situation of Fig.

It should be noted that Quv(2) will have an identical expression to Quv(1) in the previous problem, since the boundary condition at the top of the field is the same as in the previous problem. By performing a double Fourier run in the relevant domains, we obtain the relevant integrals for evaluation.