Vadose Zone Journal

Correction of the Vimoke–Taylor Concept Representing Drains in a Numerical Simulation ModelMarius HeinenDrains are special internal or boundary conditions in numerical simulation models. Instead of approximating them by a hole surrounded by a very dense grid of finite elements, the single node or single element approach based on the theory of Vimoke and Taylor proposed in 1962 offers a good alternative. Several authors have suggested that the Vimoke and Taylor constant to adapt the hydraulic conductivity of the element representing the drain should be changed by a certain factor. However, different correction factors have been given. Here this correction factor is derived for a control volume finite element numerical simulation model for different ratios of the size of the control volume representing the drain and the effective drain radius. It is shown that this relationship is dependent on the way the hydraulic conductivity is averaged at the interfaces between the neighboring control volumes. The relationships were obtained by optimization against an analytical solution for a steady-state, saturated situation. By applying the additional correction to a hypothetical transient situation for four soil types it was shown that the application of the additional correction factor resulted in 3 to 13% lower drain discharges compared to uncorrected simulations. Consequently, higher groundwater levels of on average 2 to 4 cm were obtained when applying the additional correction. For situations where exact predictions of drain discharge are needed, typically when solute transport is considered, it is advised to make use of the additional correction. Model specific correction factors may be required.

Abbreviations: CV, control volume; VT, Vimoke and Taylor.

Boundary conditions have a large impact on water flow in soils. Drain tubes repre-sent a special boundary. In literature several approaches are described to incorporate a drain in simulation models for water flow. For example, Fipps et al. (1986) compared the follow-ing four drain representations: a multinode approach using a model drain, a single node with specified flux, a single node with a specified pressure head, and the resistance adjust-ment method proposed by Vimoke and Taylor (1962) and Vimoke et al. (1962). Fipps et al. (1986) concluded that the Vimoke and Taylor (VT) concept results in accurate predictions of both hydraulic head and drain flow rates. Moreover, it is simply implemented and does not need a fine numerical grid around the drain location. It has been used in a diversity of studies, including but not limited to, Skaggs and Tang, (1979, referring to Fausey, 1975), Fipps et al. (1986), Rogers and Fouss (1989), Rogers and Selim (1989), Rogers et al. (1989; 1995), Nieber and Warner (1991), Gallichand (1993, 1994), Mohanty et al. (1997; 1998a,b), Tarboton and Wallender (2000), Jaynes et al. (2001), Buyuktas and Wallender (2002), Gärdenäs et al. (2006), Carlier et al. (2007), and Castanheira and Santos (2009). Multiple drains in a soil domain can be considered (e.g., Fipps and Skaggs, 1986, 1989).

In their analysis, Vimoke and Taylor (1962) and Vimoke et al. (1962) represented a drain in an electric analogue by introducing a drain resistance. The value of the drain resis-tance was calculated as the product of the default resistance times a constant (Cd) that was derived from an analogy with the impedance theory of a transmission line inside a square insulator. When implementing this theory into numerical simulation models, sev-eral authors concluded that this constant Cd was not sufficient to obtain an exact match

Proper prediction of drain dis-charge using the single node Vimoke and Taylor approach in a numerical simulation model requires an additional correction factor. This correction factor was calibrated and dependent on node size, drain radius and the way of averaging the hydraulic con-ductivity. For a transient situation the additional correction factor resulted in higher groundwater levels and lower drain discharges.

Alterra, Wageningen UR, Team Soil Physics and Land Management, P.O. Box 47, 6700 AA Wageningen, The Netherlands (marius.heinen@wur.nl).

Vadose Zone J. doi:10.2136/vzj2014.06.0066Received 6 June 2014.

Original Research

© Soil Science Society of America 5585 Guilford Rd., Madison, WI 53711 USA.

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Published October 14, 2014

Vadose Zone Journal p. 2 of 9

between simulated drain flow rates or pressure head fields around the drain and the analytical solution of Kirkham (1949, 1957). Rogers and Fouss (1989) used one-half of the correction factor combined with some denser grid. According to these authors, VT used two resistors in parallel, resulting in a net resistance equal to one-half of the resistance of the resistors used. The use of half of the correction factor is, therefore, consistent with the VT electrical analog model. Half the correction factor was used also by Rogers et al. (1989, 1995), Rogers (1994), Rogers and Selim (1989). In the user guides of the SWMS_2D model (Šimůnek et al., 1994) and the CHAIN_2D model (Šimůnek and van Genuchten, 1994) it is mentioned that the VT correction factor should be reduced by a factor 2 to 4. Kohler et al. (2001) stated that, “the effective drain radius…and the geometry of the finite element grid deter-mine the value of the drain factor.” Gallichand (1993, 1994) stated that the correction factor for block-centered situations is differ-ent from point-centered situations (as used by VT and Rogers and Fouss, 1989). He developed a third-order polynomial relationship between the corrected VT constant and the natural logarithm of s/r using the MODFLOW model (Harbaugh, 2005), where s is a typical element size and r is the drain radius.

Apparently there is no consensus about the exactness of the VT factor and a further correction thereof. Furthermore, so far analy-ses on such a correction have been performed only for steady-state, saturated conditions. According to Šimůnek et al. (1994, p. 30):

“… further studies of the single grid point representation of a subsurface drain may be needed, especially for transient variably-saturated flow conditions.”

The aim of this study was (i) to determine the correction factor for a rectangular control volume finite element model, (ii) to determine how this additional correction factor depends on the way hydraulic conductivity averaging is done in the numerical discretization, and (iii) to determine the effect of the additional correction factor on drain discharge under transient conditions for different soil types.

6TheorySeveral decades ago, when computer simulation models were not yet used for studies of water movement in soils, electrical analog experiments were performed instead. Vimoke and Taylor (1962) and Vimoke et al. (1962) reasoned that drains could be represented by nodal points in a regular finite element mesh.

According to the VT concept, the resistance to flow towards the node that represents the drain must be adapted from that of the surrounding conducting material, according to

Rdrain = RCd [1]

where Rdrain is the drain resistance (W), R is the resistance of the surrounding conducting material (W), and Cd is a dimensionless correction factor based on the characteristic impedance of a single conductor in a square conductive enclosure. They defined Cd as

0d

0

ZC

Z=

¢ [2]

where Z0 is the characteristic impedance of a transmission line analog of the drain (W), and Z¢0 is the characteristic impedance of vacuum (W) given by

0 0 0Z ¢ = m e [3]

where m0 is the permeability of vacuum (1.2566 10−6 H m−1), e0 is the permittivity of vacuum (8.8542 ´ 10−12 F m−1), so that Z¢0 approximately equals 376.7 W. Vimoke and Taylor (1962) and Vimoke et al. (1962) approximated Z0 as

0 10138log 6.48 2.34 0.48 0.12Z a b c» r+ - - - [4]

where r equals the ratio s/r (dimensionless), s is the size of the square that represents the drain (cm), r is the (effective) radius of the drain (cm), and a, b, and c (all dimensionless) are given by

4

41 0.4051 0.405

a-

-

+ r=

- r [5a]

8

81 0.1631 0.163

b-

-

+ r=

- r [5b]

12

121 0.0671 0.067

c-

-

+ r=

- r [5c]

The approximate expression for Z0 is valid for s >> r.

Real drain tubes have a limited amount of openings through which water can enter the drain. Therefore, the effective r will be smaller than the actual drain radius. The effective r can be calculated from the number and size of the small openings in the drain tube (Mohammad and Skaggs, 1983) or from the entrance resistance of drains (see, e.g., Dierckx, 1999). Its value may range between a few centimeters and a few millimeters.

The above concept can be used to model a drain in a numerical simulation model. Here the two-dimensional FUSSIM2 model (Heinen, 1997, 2001; Heinen and de Willigen, 1998) was used, which is a control volume finite element model, in which the soil domain is divided in rectangular control volumes (CV; see e.g., Patankar, 1980).

Vadose Zone Journal p. 3 of 9

At the location of a drain, a square element or CV is chosen for which the hydraulic conductivity is adapted with respect to that of the surrounding soil (Fig. 1). Since hydraulic conductivity can be regarded as the inverse of resistance, we then have

draind

KKC

= [6]

This formulation was used by, for example, Fipps et al. (1986) and Gallichand (1993, 1994). It is noted here that in SWMS_2D (Šimůnek et al., 1994; see also Gärdenäs et al., 2006) and CHAIN_2D (Šimůnek and van Genuchten, 1994; see also Mohanty et al., 1997) Kdrain is calculated as the product of K and a coefficient named Cd; however, there the definition of that Cd parameter was the reciprocal of the one given here in Eq. [2]. This is confusing, but the final result is the same. For r < 506.1 the value of Cd (Eq. [2]) is less than one, so that Kdrain > K.

Different researchers have pointed out that the correction for K according to Eq. [2] is not sufficient. This is based on a compari-son between simulated outflow or simulated total heads fields and the analytical solution by Kirkham (1949, 1957). Here Eq. [6] is reformulated, introducing a correction factor

drain dd d d

K KK fC f C

= = [7]

where fd is the dimensionless correction factor. Following Gallichand (1993), the additional correction is assumed to be dependent on the ratio r = s/r.

Implementation in Numerical CodeIn FUSSIM2 the net flow into the CV representing the drain is calculated each time step as

Qdrain = (qtop − qbottom + qleft − qright)s [8]

where Qdrain is the drain discharge rate (cm3 cm−1 drain length d−1) and q is the Darcy water flux density (cm3 cm−2 d−1) split in contributions from the four sides (top, bottom, left, and right) and is given as

( )nb drainnb avg

nb

h hq K

s-

=D [9]

where subscript “nb” refers to the neighboring (top, bottom, left, right) control volume, hdrain is the pressure head of the CV rep-resenting the drain, and Ds is the distance between the centers of the neighboring CVs (for equal size CVs around the drain Ds = s). In case Qdrain is computed as positive (i.e., flow into the drain) and the CV representing the drain is saturated, hdrain is set equal to an internal condition equal to zero (i.e., free draining drain); otherwise, no drain outflow occurs, and the program continues with hdrain (<0) as following from the general numerical solution. The model has also an option to handle submerged drains (i.e., with a positive pressure inside the drain), but this is not considered in this study.

In principle this approach can be implemented in three-dimen-sional codes as well. In that case the line drain can be represented by a (line) series of (cubical) voxels, each treated as described above. The total discharge of this simulated (part of the) drain is then obtained as the sum of the discharge in each of these voxels.

Calibration of fd Based on Steady-State Saturated Kirkham SolutionKirkham (1949, repeated in Kirkham, 1957) gave an analytical solution for steady-state water flow during ponding into equally spaced drain tubes in homogeneous soil overlying an imper-vious layer. Here this situation was considered to calibrate fd. Calibrations were performed on (i) the pressure head (h, cm) field in the simulated soil domain (Kirkham, 1949, his equations [6,8,9] gives the total head H from which h = H − z, where z is the gravi-tational head, cm), and (ii) on the drain discharge (Qdrain, cm3 cm−1 drain length) (Kirkham, 1949, his Eq. [11]). The pressure head distribution is given by

h(x, z) = t + d + fG(x, z) − z [10]

where f is a flux coefficient (cm) given by

( )t d rF

+ -j=- [11]

Fig. 1. A square element of soil surrounding a drain, indicating the square size s and drain radius r (left, middle), and the square representing the drain surrounded by control volumes in the numerical model with an adapted value for hydraulic conductivity Kdrain (right).

Vadose Zone Journal p. 4 of 9

the function G(x,z) is given by

( )

( )

( )

( ) ( )

( ) ( )

cosh cos2 2

cosh cos2 2

, ln2

cosh cos2 2

2cosh cos

2 2

x ma zH H

x ma zH H

G x zx ma z d

H Hx ma z d

H H

é ùæ öp - æ öp÷ç ÷ê úç÷-ç ÷ç÷ ÷çê úç ÷ç è øè øê ú´ê úæ öp - æ öp÷ê úç ÷ç÷+ç ÷çê ú÷ ÷çç ÷ç è øè øê úë û=æ ö æ öp - p -÷ ÷ç ç÷ ÷+ç ç÷ ÷ç ç÷ ÷ç çè ø è øæ ö æ öp - p -÷ ÷ç ç÷ ÷-ç ç÷ ÷ç ç÷ ÷ç çè ø è ø

m

¥

=-¥

ì üï ïï ïï ïï ïï ïï ïï ïï ïï ïï ïï ïï ïï ïí ýï ïé ùï ïê úï ïï ïê úï ïï ïê úï ïê úï ïï ïê úï ïï ïê úï ïê úï ïë ûî þ

å

[12]

and F is given by

( )

( )

2cosh coscosh cos 2 22 2ln

2cosh cos cosh cos2 2 2 2

d rmama rH HH HF

ma r d rmaH H H H

ì é ùæ öï p -æ öpé ùæ ö æ öp pï ÷ç÷ê úç÷ ÷ ÷ç ç +ê ú ç÷- ç÷ ÷ ÷ç ç ÷çê úç÷ ÷ ÷ç ç çè øê úè ø è ø è øê ú= ê úí ê úæ ö æ ö æ öê úp p p -æ öp÷ ÷ ÷ê úç ç ç÷ç+ê ú÷ ÷ ÷-ç ç ç÷ç÷ ÷ ê úç ç ÷÷ç çè ø è ø ÷ê ú çè ø è øë û ê úë û( )2

tan4

2 lntan

4

coshcosh cos 22 22 lncosh cos

2 2

m

d rHr

H

mama rHH H

ma rH H

¥

=-¥

üïïï ïï ïï ïï ïýï ïï ïï ïï ïï ïï ïî þé ùæ öp - ÷çê ú÷ç ÷ê úç ÷çè øê ú=- +ê úæ öp ÷ê úç ÷ç ÷ê úçè øê úë û

æpé ùæ ö æ öp p÷ ÷ç çê ú+÷ ÷ç ç÷ ÷ç ç èê úè ø è ø- ê ú

æ ö æ öê úp p÷ ÷ç ç-ê ú÷ ÷ç ç÷ ÷ç çè ø è øê úë û

å

( )

( )1

2cos

22

cosh cos2 2

m

d rHd rma

H H

¥

=

ì üé ùæ öï ïp -öï ï÷ç÷ê úç ÷-ï ïç÷ç ÷ï ï÷çê úç ÷çø è øï ïï ïê úí ýê úæ öï ïp -æ öpï ï÷ê úç÷ç ÷+ï ïç÷çê ú÷ï ï÷ç ç ÷çè ø è øï ïê úë ûï ïî þ

å

[13]

where a is the drain spacing (cm), d is the drain depth (cm), H is the soil depth (cm), r is the drain radius (cm), t is the height of the ponded water layer (cm), x is the horizontal distance (cm), z is the vertical distance (cm), and m is the summation counter. The origin of the (x, z) coordinate system is located at the center of the drain, with x positive to the right, and z positive upward. In case it is assumed that r is small compared the other length measures, i.e., r << {H, H − d, d, a}, the drainage outflow rate Q (cm3 water cm−1 length of the drain d−1) is estimated from

Q = 4pfKs [14]

where Ks is the hydraulic conductivity at saturation (cm d−1).

In previous studies where the analytical solution of Kirkham (1949, 1957) was used, comparison between simulated and analytic drain discharge was performed (e.g., Fipps et al., 1986; Nieber and Warner, 1991; Gallichand, 1993; Tarboton and Wallender, 2000). However, the analytic expression for drain discharge is an approxi-mation and holds for a small drain radius compared to the distance to adjacent boundaries (i.e., soil surface, impermeable layer, drain spacing). When the numerical simulation model is calibrated, typi-cally the correction value fd, this potentially introduces an error in fd. Therefore, in this study calibration was performed both on the h(x,z) and on Q.

In the numerical simulation model the water flux density across the interfaces of neighboring control volumes is calculated as the hydraulic conductivity (K) times the gradient in pressure head across this interface. The K at the interface is computed as the average of the K values of the two neighboring CVs. Since the way of averaging determines the magnitude of the average K (Kavg; Fig. 1), this will also be of influence on the calibration of fd. Therefore, the calibration was performed for three averaging schemes (i.e., arithmetic averaging):

( )arithmetic drain1K w K wK= - + [15]

geometric averaging

1geometric drain

w wK K K-= [16]

and harmonic averaging

( ) 1

harmonicdrain

1 w wKK K

-é ù-ê ú= +ê úë û [17]

where w is a weighting factor accounting for possible differences in size of the control volumes representing the drain and its neigh-bors; for equal grid sizes w = 0.5.

The soil considered was a loam soil (Table 1). The drain spacing was 1200 cm, soil profile depth was 200 cm, drain depth was 97.5 cm, and s = 5 cm. The (effective) drain radius r was varied in the range 1.0 to 0.01 cm; thus, r ranged from 5 to 500. Calibration was performed by minimizing the sum of squared relative differ-ences (SSQ) given by

2

1SSQ

ni i

ii

M OO=

æ ö- ÷ç ÷= ç ÷ç ÷çè øå [18]

where O and M represent the analytic and simulated data (h with n = 2520 or Qdrain with n = 1), respectively. Brent’s method (Press et al., 1992) was used for minimization of SSQ.

To test if the calibration is dependent on s, drain spacing, soil depth, and drain depth, additional calibrations were performed with s =

Table 1. Hydraulic properties in terms of the Mualem (1976)–van Genuchten (1980) parameters for four soils (source: Wösten et al., 1994).

Soil qr qs a n (m = 1–1/n) l Ks

—— cm 3 cm−3 —— cm−1 cm d−1

Sand 0.01 0.45 0.0152 1.412 −0.213 17.81

Loam 0.0 0.40 0.0194 1.250 −0.802 14.07

Clay 0.0 0.60 0.0243 1.111 −5.395 5.26

Peat 0.0 0.73 0.0134 1.320 0.534 13.44

Vadose Zone Journal p. 5 of 9

10, drain spacing of 2400 cm, soil depth equal to 400 cm, and soil depth equal to 400 cm combined with drain depth equal to 197.5 cm. These extra calibrations were only performed for h and for the case with harmonic averaging of K.

Effect of fd in a Transient SituationThe ultimate test would be a situation where simulated results are compared to error-free field data. However, this requires exact knowledge of the build-up of the soil layering of the test site and their exact hydraulic properties. De Vos et al. (2000) (also, De Vos, 1997) made an attempt to model drain discharge in a layered soil profile and experienced great difficulties due to uncertainties in the soil hydraulic properties. If such a situation would be used to verify the correctness of the fd factor obtained above, the outcome is likely to be influenced by uncertainties in soil hydraulic properties rather than in the drainage concept studied here. Therefore, the influence of fd on a transient situation was tested for a uniform soil with imposed hydraulic properties by comparing simulation with and without the inclusion of fd. A 59-d period (February–March 2009, bare soil) was used with daily rainfall data (www.knmi.nl [accessed 29 Sept. 2014], location Lelystad) and an effective drain radius equal to 0.05 cm. The total rainfall was 111.1 mm, and potential evaporation from the soil surface was 44.3 mm. Other parameters were: drain spacing 12 m, drain center at depth 97.5 cm, size of control volume representing drain s = 5 cm, impermeable layer at the 200-cm depth, soil water initially in equilibrium with h = 100 cm at the bottom. Four soil types were considered (Table 1).

6Results and DiscussionCalibration fdCurvilinear relationships were obtained for fd as a function of ln(r) (Fig. 2); for convenience the relationship between Cd/fd and r was considered too (Fig. 2). In most cases these relationships could be described by power functions, and in a few cases by a linear rela-tionship (Table 2). The resulting relationships were nearly identical for the situations where optimization was done on pressure head and on drain discharge (Fig. 2, Table 2).

For the case with arithmetic averaging of K no feasible solutions could be obtained for r > 20. Therefore, it is advised against arith-metic averaging, at least for the region surrounding the drain. For the case with geometric averaging, both fd or Cd/fd could be repre-sented by a power function. For the case with harmonic averaging fd was poorly related to ln(r), but Cd/fd could be represented very well by a linear relationship. For comparison the analytical rela-tionship obtained by Gallichand (1993) is also given in Fig. 2 and Table 2. Remarkably, his third-order polynomial relationship for Cd as a function of ln(r) could here be approximated well by a linear relationship. Apparently his polynomial relationship was influenced by his data for r < 5, while here only situations with r ³ 5 were considered. Gallichand (1993) used the finite difference

model MODFLOW, in which the averaging of the hydraulic con-ductivity is done harmonically (Harbaugh, 2005).

Fig. 2. Relationships between calibrated fd (top) or Cd/fd (bottom) as a function of ln(r) for situations with calibration based on pressure head fields (h) or based on drain discharge (Qdrain) for three ways of averaging of K. For comparison the relationship obtained by Gallichand (1993) is presented as well (dashed lines).

Table 2. Linear or power relationships between fd or Cd/fd as a function of x = ln(r) (r = 5–500) and the corresponding coefficient of determination R2 for the calibration of fd based on either pressure head fields or drain discharge and for three ways of averaging the hydraulic conductivity.

fd Cd/fdPressure head

Arithmetic† 0.581 − 0.184x R2 = 0.9999 0.109x3.972 R2 = 0.9070

Geometric 0.501x−1.162 R2 = 0.9997 0.331x2.142 R2 = 0.9999

Harmonic 0.325x−0.478 R2 = 0.9552 −1.104 + 1.257x R2 = 1.0000

Drain discharge

Arithmetic† 0.595 − 0.188x R2 = 0.9999 0.113x3.871 R2 = 0.9158

Geometric 0.521x−1.181 R2 = 0.9998 0.319x2.161 R2 = 0.9999

Harmonic 0.338x−0.498 R2 = 0.9524 −1.135 + 1.258x R2 = 1.0000

Gallichand (1993)

0.420x−0.434 R2 = 0.9682 −0.750 + 0.903x R2 = 0.9999

† Only valid for r in range 5 to 20.

Vadose Zone Journal p. 6 of 9

As stated before, in the VT approach the approximate expression for Z0 is valid for s >> r. In other words, the drain should be fully surrounded by soil inside the element, which means that at least r should be larger than 2. Here data were presented for r ³ 5. For the case of harmonic averaging calibration based on the h field, additional optimizations were performed for r = 2.5 and 3.33. These results were in line with the linear relationship presented in Fig. 2 and Table 2.

Figure 3 presents the pressure head fields around the drain for r = 5 and geometric averaging for situations with fd = 1 (i.e., no additional correction on Cd), fd = 0.1, and fd = 0.2833 (optimized value), as well as the one obtained from the analytical solution of Kirkham (1949). Differences in pressure head distributions are small but noticeable in the direct zone around the center of the drain. The effect of these small h differences results, however, in different drain discharges: +10.0%, −19.9%, and −0.5% for fd = 1.0, 0.1, and 0.2833, respectively, compared to the analytical discharge of 1553 cm3 cm−1 drain length (or 12.94 mm).

Results for calibration with s = 10 cm, a greater drain spacing, a deeper soil profile with drains at depth 97.5 or 197.5 cm, all with harmonic averaging, did result in approximately the same results for Cd/fd as a linear function of ln(r) (Table 3). Thus, the calibra-tion of the fd parameter is independent of the system dimensions at least for this limited test.

Like others it was shown that for the control volume finite element model FUSSIM2 the single CV representation of the VT drain concept can be used to describe the analytical solution of Kirkham (1949), provided that the correction factor Cd introduced by VT is adapted (divided) by an additional correction factor fd.

Šimůnek et al. (1994) indicated that their VT correction factor should be divided by a factor 2 to 4. Due to their different defini-tion of Cd (as mentioned above) this corresponds to a value for fd equal to 0.25 to 0.5. For small values of r this is in the same range as obtained in this study and that by Gallichand (1993). Rogers and Fouss (1989) referred to Fipps et al. (1986), who used an expression equal to that given by Eq. [6], and they reasoned that the correc-tion factor, here given as fd, should be equal to 2. Surprisingly, this is much larger than the values obtained in this study. However, it is unknown whether they used the original expression for Cd (here given by Eq. [2]), or if they used the VT Cd values in their Table 1,

which were eight times larger than the values obtained from Eq. [2]. In the latter case, this would correspond to fd = 1/(8/2) = 0.25, which then would be more in line with the findings of this study and those of Gallichand (1993).

The treatment of the correction factor fd thus far has been empirical. An approximate expression for the impedance (Z0) for a transmis-sion line in a square isolator formed the basis for the VT analysis (they referred to Westman, 1959). They used this analogy to set up an electrical analogue model for a resistance network to repre-sent a drain. Next, modelers used the analogy between resistance flow and flow through porous media to transform this into a drain representation in numerical simulation models. Since these are not exact steps taken, it is not surprising that the match is inexact. Furthermore, the expression for Z0 given by VT did not mention the dependency on the dielectric permittivity of the insulating material. According to Middleton (2002) a better approximation for Z0 is given by (again for r >> 1)

100

138log 6.48 2.34 0.48 0.12a b cZ

r+ - - -»

e [19]

Fig. 3. Pressure head distribution in the neighborhood of the drain (indicated by the + sign in the middle) for three values of fd and the analytical solution according to Kirkham (1949).

Table 3. Linear or power relationships between fd or Cd/fd as a function of x = ln(r) (r = 5–500) and the corresponding coefficient of determination R2 for the calibration of fd based on pressure head fields for harmonic averaging of the hydraulic conductivity for different system dimensioning.

fd Cd/fds = 10 cm 0.261x−0.322, R2 = 0.9789 −1.081 + 1.258x, R2 = 1.0000

s = 10 cm + large drain spacing 0.261x−0.322, R2 = 0.9789 −1.081 + 1.258x, R2 = 1.0000

s = 10 cm + deep soil profile 0.261x−0.322, R2 = 0.9789 −1.080 + 1.258x, R2 = 1.0000

s = 10 cm + deep soil profile + deep drain 0.272x−0.346, R2 = 0.9736 −1.144 + 1.267x, R2 = 1.0000

Vadose Zone Journal p. 7 of 9

where e is the dielectric permittivity of the insulating medium surrounding the conductor. In case of air e = 1, so that Eq. [14] reduces to [4], which may have been assumed by VT. When drains are functioning, the soil is saturated. The e of saturated mineral soils would be in the range 20 to 40 (e.g., Topp et al., 1980); this would be larger for soils with higher porosities, such as peat soils. When applying Eq. [14], this would result in a Cd value being about 4.5 to 6.5 lower than the one according to VT (Eq. [2]), corre-sponding to fd values between 0.15 and 0.22. The approximation of Z0 holds for s >> r. This would mean that for large values of r the approximation would be good. In other words, at large r one would expect fd to become constant, which can be seen here for the harmonic averaging scheme and to a lesser extent also for the case with geometric averaging.

Due to the approximations underlying the theory and the numeri-cal implementation in a simulation model it seems that there doesn’t exist an analytical way to obtain a correction factor for K of the square element representing the drain. Therefore, one must rely on pre-calibrated correction factors. When using harmonic averaging of K the overall correction factor (Cd/fd) can be well obtained from a linear relationship with ln(r). As this relationship differs from a node-centered finite difference model (Gallichand, 1993) and a control volume finite element model (this study), it is advised that such a relationship is determined for each numerical simulation model.

The calibration was performed for steady-state, saturated con-ditions, so that the shapes of the water retention and hydraulic conductivity functions do not matter. The only parameter that could matter is the hydraulic conductivity at saturation (Ks). Since both the analytic and the numerical model are functions of Ks one would not expect any effect of soil type (i.e., Ks) on the outcome of the calibration procedure. This was verified by repeating the calibration for a sand soil with harmonic averaging and clay soil for geometric averaging. The resulting calibration curves were identi-cal for those obtained with the loam soil.

Effect fd in a Transient SituationFor the transient case the drain discharge rates obtained with fd = 0.085 (r = 100, geometric averaging; see Table 2) were generally lower than when no additional correction was employed ( fd = 1), for all soil types (Fig. 4). Cumulatively, these differences showed the following range: −3.2% (clay), −3.6% (loam), −5.2% (sand),

−12.9% (peat) (Table 4). The clay soil showed lower drain discharges than the other three soils, but resulted in more storage of water at the end of the simulation period. The relative differences in daily discharge was largest at peak discharge events (Fig. 4). For situations where exact predictions of drain discharge are needed, typically when solute transport is considered, it is wise to make use of fd. The inclusion of fd in the numerical computations has no influence on the computational time of the numerical model.

This exercise was repeated for a larger drain radius (resulting in r = 5). In that case the difference in drain discharge between uncor-rected and corrected situations was much smaller, ranging between

−0.6% (clay) and −2.3% (peat) for the 59-d period considered.

The difference in drain discharge must also be seen in the pres-sure head distribution in the soil, or typically in the position of the groundwater level (Fig. 5). The lower drain discharge due to the presence of fd logically resulted in shallower groundwater levels. Based on the hourly data the average difference was −2.1,

−2.4, −4.1, and −3.9 cm for the sand, loam, clay, and peat soils, respectively. The maximum difference seen at the hourly data was approximately −3.3, −4.1, −28.8, and −6.4 cm for the sand, loam, clay, and peat soils, respectively.

An additional simulation for the loam soil was performed with har-monic averaging for K at the sides of the CV representing the drain and geometric averaging elsewhere. For this case fd = 0.1566 (r = 100; see Table 2). The discharges rates, cumulative discharge and position of the groundwater level were not different from those pre-sented in Fig. 4 and 5. For example, cumulative discharge equaled

Fig. 4. Drain discharge rates as a function of time for fd = 1 and fd = 0.085 (r = 100) under transient rainfall rates (top graphs) for four soil types.

Vadose Zone Journal p. 8 of 9

51.2 mm, compared to 51.5 mm in case geometric averaging was done everywhere in the soil domain.

To my best knowledge no other studies have been reported indi-cating the difference between drain discharge with and without a correction factor fd. Therefore, it is not possible to compare these findings with other findings. The VT theory including the new correction factor fd resulted in lower discharges (rates, cumula-tive) and consequently in higher groundwater levels predicted by the numerical simulation model. These findings were more pronounced for small values of r and were seen both for the steady-state and for the transient cases considered in this study.

6ConclusionsIn this study it was shown that for a control volume finite element numerical simulation model the Vimoke and Taylor (1962) and Vimoke et al. (1962) drain concept can be employed, provided that an additional correction factor is included. The VT coefficient Cd should be divided by a correction factor fd. This correction factor fd is dependent on the ratio size of the square control volume rep-resenting the drain s and the effective drain radius r.

The relationship between fd and r (= s/r) is dependent on the type of averaging used for computing the average hydraulic conductiv-ity at the boundaries of the control volume representing the drain. It is advised against using arithmetic averaging because for larger ratios of r no feasible correction factors could be obtained. For the case of harmonic averaging a linear relationship was found between Cd/fd and the natural logarithm of r in the range 5 £ r £ 500. Calibration was performed against the analytical solution of Kirkham (1949), both for the pressure head field and for the drain discharge, resulting in approximately equal relationships.

For a transient situation it was seen that disregarding the additional correction factor (i.e., taking fd = 1) resulted in an overestimation of drain discharge and consequently an underestimation of the position

of the groundwater level. These differences differed between four major soil types and were more obvious for large values of r.

For situations where exact predictions of drain discharge are needed, typically when solute transport is considered, it is wise to make use of fd. The inclusion of fd in the numerical computations is simple and has no influence on the computational time of the numerical model.

As the relationship between fd and r differs from that obtained for a block-centered finite difference model (Gallichand, 1993), it is advised that after implementation of the VT concept in a numerical model, a calibration must be performed to determine the relationship between fd and r. Such a calibration will not cost much computational time because steady state under saturated conditions is almost instantaneously achieved using a numerical model. The calibration may result in different outcomes for dif-ferent numerical methods, and it may be influenced by the exact implementation of the head-flux calculation of the element repre-senting the drain.

AcknowledgmentsThanks to Falentijn Assinck for critically reading the manuscript.

Table 4. Change in storage (DS), rainfall (R), actual evaporation (E), drain discharge (Q), and relative drain discharge after the 59 d for the transient case for four soil types for two values of fd

DS R E Q Qrel

————————— mm ———————— %

Sand fd = 1 14.0 111.1 44.3 52.8

fd = 0.085 16.6 111.1 44.3 50.2 −5.2

Loam fd = 1 15.3 111.1 44.3 51.6

fd = 0.085 17.1 111.1 44.3 49.8 −3.6

Clay fd = 1 29.7 111.1 44.2 37.2

fd = 0.085 30.8 111.1 44.2 36.1 −3.2

Peat fd = 1 10.4 111.1 44.3 56.4

fd = 0.085 16.9 111.1 44.3 49.9 −12.9

Fig. 5. Groundwater levels as a function of time for fd = 1 and fd = 0.085 (r = 100) under transient rainfall rates (top graphs) for four soil types.

Vadose Zone Journal p. 9 of 9

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