Letter to the Editor
Comment on the paper ``An analytical solution for
design of bi-level drainage systems'' by A.K. Verma,
S.K. Gupta, K.K. Singh, H.S. Chauhan
$
A.R. Kacimov
Department of Soil and Water Sciences, Sultan Quboos University, P.O. Box 34, Al-Khad 123, Sultanate of Oman
Accepted 11 January 2000
Abstract
The hierarchy of models applicable to transient free surface problems of ground water ¯ow to drains is discussed. The model choice is related to drain type and dewatering stage. Flow pattern is illustrated to vary with decrease of the water table. Simulation of drains and corresponding boundary conditions are discussed.#2000 Elsevier Science B.V. All rights reserved.
Keywords:Drainage; Water table; Ground water; Transient ¯ow
1. Introduction
Recently, Verma et al., 1998 (VEA in what follows) analyzed a transient flow to a two-depth periodic system of drains dewatering an initially saturated porous layer. VEA used the simplest model based on a linear diffusion equation, from which the water table dynamics was calculated. However, the results of VEA should be discussed in connection to the following questions: (1) What does the model describe? (2) Can more general models be used, and, if so, how to choose the appropriate model depending on drain construction and stage of dewatering?
2. Hierarchy of models
Bi-level drains were a subject of recent theoretical analyses (Hathoot, 1998; Kirkham and Horton, 1992, 1993). Several models were developed for non-steady flows to drains.
$Agric. Water Manage. 1998; 37, 75±92.
E-mail address: anvar.kasimov@ksu.ru (A.R. Kacimov).
The most general saturated±unsaturated flow model is based on the Richards equation (Ahmad et al., 1993; Fipps and Skaggs, 1991). The model requires a knowledge of two basic functions in soil (pressure±moisture content and hydraulic conductivity±moisture content). The governing equation forH(x,y,t) (His the hydraulic head,xthe horizontal coordinate,ythe vertical coordinate andtthe time) is nonlinear and numerical methods were implemented to solve the problem.
In a simpler nonlinear potential model conductivity and moisture content are constant, the saturated zone is restricted from above by a phreatic surfaceFAG(our Fig. 1a), the shape of which, F(x,t), is a priori unknown. The head H(x,y,t) satisfies the Laplace equation and is searched in a domain with the moving free boundaryF. The influence of the unsaturated zone is often taken into account by `effective' infiltration±evaporation distributed along the phreatic surface as a boundary condition alongF(keeping the head as a harmonic function). The Laplace equation was solved analytically for a single drain, a groundwater mound decaying with time (Polubarinova-Kochina, 1977), and mathematically equivalent Hele±Shaw problems (Hohlov and Howison, 1993). The transient nature of the problem appears through a nonlinear boundary condition alongF
(the Laplace equation is time independent). The diffusion equation with a free boundary for periodic drains can be solved only numerically (Ahmad et al., 1991; Trivellato, 1993). Note, that for shallow agricultural drains with small spacing (unlike extensive deep aquifers or oil deposits subject to high pressure variations) the specific storage (compressibility) can be neglected.
A simpler linear potential model linearizes the free boundary condition. The model is valid for approximately horizontal phreatic surfaces. The governing equation remains Laplacian. A harmonic functionH(x,y,t) needs to be found in a domain with a fixed upper boundary. The shapeF(x,t) being reconstructed a posteriori as a linear disturbation of the upper flow boundary. Tube and ditch drains were investigated in terms of this linear model (El-Nimr, 1973; Polubarinova-Kochina, 1977; Savci, 1990).
Another simplification of the nonlinear potential model is based on the Dupuit± Forchheimer (hydraulic) approximation which assumes that the slope of the water table is small and the horizontal components of the Darcian velocity do not depend on the vertical coordinate (Polubarinova-Kochina, 1977). In this way the vertical coordinateydisappears from the governing equation. Factually, the nonlinear condition along F becomes the governing equation, which is called the Boussinesq equation. Therefore, the search forH
distribution is reduced to determining the saturated zone thicknessF(x,t). However, the Boussinesq equation is still nonlinear and calls for sophisticated mathematical techniques if an analytical solution is necessary (Barenblatt et al., 1990; Basak and Murty, 1981; Collins, 1976; Gupta et al., 1994). Even for a single ditch drain in a semi-infinite aquifer the equation was solved only for specific transient regimes such as a sudden increase or decrease of the water level in the ditch (Polubarinova-Kochina, 1977). Approximations of
F by polynomials with time-dependent coefficients were developed to solve the Boussinesq equation for a system of drains (Singh and Rai, 1989).
problem (2) and (3) of VEA was solved by Crank (1975) in a more general form, for an arbitrary initial elevation of the water tableF(x,0). All functions in Eq. (4) of VEA, their derivatives, and extrema can be easily treated by standard symbolic software like Wolfram (1991) and, hence, the program description of VEA is redundant. Note that linearization is sensitive to the boundary and initial conditions (e.g. Barenblatt et al., 1990; Polubarinova-Kochina, 1977 showed that linearization may lead to poor predictions ofF).
3. Model choice
What model to select? Selection depends on many factors including type, size, depth, and spacing of drains, the stage of the process, the availability of soil data, among others. VEA sidestepped the issue, concerning which h0, h1, L, d1, and t in their model is
applicable and how their model relates others in the existing hierarchy. The results obtained from a simple model should be compared with a more general model and vice versa. For example, the nonlinear potential model should be compared against the Richards equation, solutions of the Boussinesq equation against the nonlinear potential model, any linearization against the corresponding nonlinear model. Fipps and Skaggs (1991) investigated a problem similar to VEA in terms of the Richards equation and matched the numerical modeling with a simple kinematic approach and empiric formulae; Kirkham et al. (1974) scrutinized the restrictions and advantages of the Dupuit±Forchheimer approximation in comparison with the potential theory; McEnroe (1993) used the hydraulic approximation to calculate the maximum of F and then compared the results againstf(x) derived from the nonlinear potential model; Koussis et al. (1998) utilized a linearised Boussinesq equation and discussed the range of its validity and boundary conditions in comparison with the Boussinesq equation and the potential model; Upadhyaya and Chauhan (1998) compared an analytical solution of the Boussinesq equation both with numerical results and linearised solutions. VEA related their calculations to the results of Kumar et al. (1994), which were obtained from the same linear diffusion equation, that can hardly be accepted as a verification. Eventually, any model should be verified under laboratory and field conditions. VEA presented only one example of this type (their Fig. 2). It would have been better to restrict themselves to purely mathematical derivations since the standard protocol of modeling (Anderson and Woessner, 1992, pp. 6±9) includes the analysis of sensitivity to all model parameters under several flow conditions and, hence only one example is insufficient.
4. Flow pattern and boundary conditions
flow into two neighboring drains, bifurcating along the bed into BI and BJ, as in Kirkham et al. (1974), Figs. 10±23. Hence, it is important to note that both drains are gaining. Therefore, att0 the potential model (nonlinear or linear) should be implemented for analytic analysis, as was shown in numerous books and papers (e.g. Hathoot, 1998; Kirkham et al., 1974; Kacimov, 1993; Polubarinova-Kochina, 1977; Strack, 1989). Schilfgaarde (1974) adduced how the potential model can be matched through approximate techniques with the falling water table conditions. Rehbinder (1997) illustrated (just for VEA flow pattern) how a sequence of solutions of potential problems can incorporate the falling water table, which was assumed to be horizontal.
After a while, the separatrice changes its shape to AMB (Fig. 1b) and the upper drain becomes partially gaining, partially losing behaving hydrodynamically as a dipole. In this case, even in potential models the drain can not be simulated as a hydrodynamic sink.
Finally, att!1the separatrice disappears (Fig. 1c) the whole `slug' is `digested' by drains, and seepage is controlled by the head difference in the drains. The upper drain (source) loses water and feeds the lower drain (sink). The water table FG then stabilizes and flow becomes quasi-horizontal. VEA solution is again not applicable. Indeed, Eq. (4) of VEA att!1, as a solution of the equation d2h/dx20, predicts a straight tilted line FG. This is obviously incorrect because a steady state flow between two constant head boundaries gives the Dupuit parabola forF(x,t). Moreover, even in the case of a mono-level drain system (h10) the large time limit of Eq. (4), or an equivalent equation of
Kumar et al., 1994, is questionable. According to Boussinesq (1904) (see also Lorre et al., 1994) the water table drops asF(x,t)A/(1Bt) whereAhmG(x/L),B4.46Khm/(4fL2),
hm is the maximal water table elevation between two drains, and G the Boussinesq
function (Polubarinova-Kochina, 1977). The water table in VEA solution does not follow this time dependence, while from other models it does (Lorre et al., 1994).
The scenario above is, of course, imaginable and happens at sufficiently small values of drain spacing, high initial elevations of the water table, and constant heads in the drains.
Strictly speaking, Fig. 1 of VEA does not correspond to their boundary-value problems (2) and (3). Conditions (3a) and (3c) may model ideal ditches (so called chimney drains, Cedergren, 1989), which are gravel packed and penetrate the whole depth of the soil layer as is shown in our Fig. 2 or in Singh et al. (1991). A tube diverting seeped water is bedded in the ditch trough. The tube can operate under either free drainage conditions, or a backwater effect (McEnroe, 1993). For ditches with narrow spacing and low super-elevation above the tube apex (Kirkham and Horton, 1992) the boundary EC may not be equipotential but an isobaric line (seepage face). In this case, the known solutions for flow in a dam (Polubarinova-Kochina, 1977), obtained in terms of the nonlinear potential model, should be applied, in the manner of Koussis et al. (1998) who justified the drain boundary condition. Moreover, if the conductivity of the drain packing is not high enough (as compared with the soil conductivity) i.e. the drain is hydraulically `imperfect', seepage both in the soil and in the packing should be studied (Cedergren, 1989). In this case the fourth type boundary conditions (continuity of the normal flux and pressure) should be imposed along the soil-gravel interface (Ligget and Liu, 1983).
While chimney drains are common in dams (Cedergren, 1989), in agricultural applications drain ditches not always tap the whole soil layer. In this case, ditch geometrical `imperfectness' should be taken into account. VEA solution includes, neither the drain diameter, nor the ditch sizes. Recall, that usually, if the whole flow problem can not be solved in terms of a potential model, the flow domain is decomposed into a quasi-horizontal fragment (where the Dupuit-Forchheimer approximation holds) and a near-drain radial flow zone (such decomposition is possible if L@d1h1). A drain is then
modeled by a sink and one of the near-sink equipotentials is considered to be the drain radius. Numerical modeling (Fipps and Skaggs, 1991), experiments (Lennoz-Gratin, 1989), and analytical solutions (Fujii and Kacimov, 1998) proved real flows not to be purely radial near the drain. Fipps and Skaggs (1991) concluded that for transient regimes (similar to VEA) the deviation from radial conditions, commonly used in pre-computer drainage models, is more pronounced.
In the vicinity of drain tubes, the boundary conditions should be chosen according to the type of the drain envelope, gravel pack, or perforation. For example, the size and shape of an equipotential line encompassing the drain was related to the drain wrapping (Lennoz-Gratin, 1989). For mole drains the drain circumference (EDGE in our Fig. 1) may consist of a constant head line (DC) and an isobaric line (ED). The latter was modeled by Ilyinsky and Kacimov (1992), and Fujii and Kacimov (1998). For perforated tubes the Richards equation (Fipps and Skaggs, 1991) with small-size finite elements near the drain circumference and a potential model for flow into periodic pipe slots (Panu and Filice, 1992) were used. Obviously, none of these `fine' pictures can be studied in terms of the model used by VEA.
The linear diffusion equation for flows, as in our Fig. 2, was solved in hydrological applications. For example, Onder (1994) considered flow towards a reservoir from a confined aquifer under the same boundary and initial conditions as VEA. Note, Onder (1994) studied precisely 1-D system and, hence, his model is sound (other flow regimes of this type were investigated by Singh et al. (1991) and Workman et al., (1997).
hydrological data obfuscates the difference between the results obtained from mathematical models of varying complexity. Nonetheless, modelers should strive for veracity according to their own standards avoiding any sloppiness. The model derived (including the governing equations and boundary conditions), the predictions it renders, and the implied recommendations should then be thoroughly tested. Ideally, modeling should encompass collating solutions developed in terms of several analytical approaches, running numerical programs like MODFLOW, verifying in the field the results so that the mooted conclusions can be instructive for drainage engineers.
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