Journal of Hydrology (2007) 332, 374–380

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Non-linear radiation in the Boussinesq equationof the agricultural drainage

Manuel Zavala *, Carlos Fuentes, Heber Saucedo

Instituto Mexicano de Tecnologia del Agua (Mexican Institute of water Technology), 62550 Jiutepec, Morelos, Mexico

Received 23 June 2005; received in revised form 20 May 2006; accepted 24 July 2006

Summary Water flow to subterranean drains is described by using the Boussinesq equationsubjected to radiation type boundary conditions; these conditions establish a relationshipbetween drainage flux and water head at the drain. Analyzed radiation conditions are a qua-dratic polynomial equivalent to the Hooghoudt equation and a power function; both conditionscontain linear and quadratic radiations. Evidence is given that these last radiations correspondto the extreme probabilistic models of water flow proposed, respectively, by Purcell [Purcell,W.R., 1949. Capillary pressures their measurement using mercury and the calculation of perme-ability thereform. Petr. Trans., Amer. Inst. Mining Metall. Eng. 186, 39–48] and Childs and Col-lis-George [Childs, E.C., Collis-George, N., 1950. The permeability of porous materials. Proc. R.Soc., Ser. A 201, 392–405].

Hooghoudt type radiation results from a convex combination of extreme radiations using areference flux and an interpolation factor as parameters. Power radiation is established fromboth fractal geometry concepts and partially correlated water flow by soil structure. Thisfractal radiation contains a reference flux and an exponent equal to double of the surfacesoil-particles fractal dimension with regards to Euclidean soil dimension. Considering thatconvex radiation is an approximation to fractal radiation, the least square method allowsus to establish a relationship between the interpolation factor and the exponent of the fractalradiation.

We use a numerical solution to the Boussinesq equation subjected to radiation conditionsto describe a drainage experiment performed in the laboratory. The results shows that thecumulative drained depth is better represented by fractal and convex radiations rather thanby extreme ones.ª 2006 Elsevier B.V. All rights reserved.

KEYWORDSConvex radiation;Fractal radiation;Water flow probabilisticmodels;Soil–drain interface

0d

022-1694/$ - see front matter ª 2006 Elsevier B.V. All rights reservedoi:10.1016/j.jhydrol.2006.07.009

* Corresponding author. Tel./fax: +77 73 29 36 58.E-mail address: mzavala73@yahoo.com.mx (M. Zavala).

Introduction

Mathematical modeling of water flow to drains can be per-formed with the Boussinesq equation (1904):

.

z=0

Soil surface

q

Free surface

Column base

z=h

z=P

z=0

Soil surface

Free surface

Column base

z=h

z=P

Figure 1 Soil column on the drain.

Non-linear radiation in the Boussinesq equation of the agricultural drainage 375

loh

ot¼ r � ½KsðDo þ hÞrh� þ R ð1Þ

where t is time, h is the pressure head or the water-tableelevation above the drains, Ks is the saturated hydraulicconductivity, Do is the distance from the impermeable bar-rier to the drains, R is the recharge and l is the storagecapacity.

To study the agricultural drainage with Eq. (1), the initialand boundary conditions should be defined at the domain.The initial condition is established from the water table po-sition at the initial time. Dirichlet and Neumann boundarytype conditions can be used on drains to solve Eq. (1); thepressure head on the drains is required in the first conditionwhereas the drainage flux is required in the second one. Athird type of boundary condition is a linear combination ofthe precedent conditions; this condition includes a resis-tance parameter to the flow at the soil–drain interface.Null resistance corresponds to the Dirichlet condition andinfinite resistance corresponds to Neumann condition. Thethird condition is a radiation type condition (Carslaw andJaeger, 1959).

The approximated Glover–Dumm analytic solution is ob-tained from the one-dimensional Boussinesq equation takinginto account the fact that the pressure head on the drain in-stantly reaches zero value (Dumm, 1954). However, thisconsideration is satisfied only if the drainpipe walls offerno resistance to water flow.

In some studies, experimental head on drain variationhas been used as a boundary condition to estimate Bous-sinesq equation parameters by means of reproducing obser-vations regarding head between drains or inverse problem(Pandey et al., 1991; Gupta et al., 1994). In steady-statethe pressure head on the drain has been calculated fromthe Hebert equation and then it has been integrated tothe Hooghoudt solution (1940) of Eq. (1) (e.g., Miles and Kit-mitto, 1989). Flux condition at the boundary can be used ininverse problems to identify parameters. However, in directproblems, it is the most interesting variable.

In the case of drainage, the radiation condition estab-lishes that drainage flux is directly proportional to the pres-sure head on the drain and inversely proportional to theresistance in the interface between soil and the drainpipewall in concordance to the Ohm law. With this condition,Fuentes et al. (1997) have obtained to an approximatedanalytic solution of the one-dimensional Boussinesq equa-tion, which contains the Glover–Dumm solution a particu-lar case (see Fragoza et al., 2003). Radiation condition isconsidered in the numerical solution proposed by MacDon-ald and Harbaugh (1988). Due to the fact that in the Hebertequation the drainage flux and the pressure head are pro-portional, this equation can be adapted as a radiationcondition.

In the work by Kohler et al. (2001) about water flow todrains simulation using the Richards equation (1931), a gen-eralization of the Hooghoudt solution, presented by Ooster-baan et al. (1989) as a boundary condition in the drains, isused. The Hooghoudt solution establishes a second gradepolynomial between drainage flux and head at the inter-drains center. Following these ideas, a second grade polyno-mial can be proposed for dependency between flux andhead at the drains:

q ¼ bhþ ch2 ð2Þ

which contains the typical linear radiation with c = 0. Withb = 0 a quadratic radiation would be obtained.

Linear and quadratic radiation behaviors can also be ob-tained from a power radiation:

q ¼ ahn ð3Þ

where n = 1 corresponds to linear radiation and n = 2 to qua-dratic radiation.

The aim of this work is to justify the non-linear radiationcondition in subterranean agricultural drainage taking intoaccount capillary probabilistic extreme models of soil waterflow according to Purcell (1949) and Childs and Collis-George (1950) and considering fractal geometry concepts.

Extreme probabilistic models for drainageflux

Consider a soil column of a P height placed over the drain(see Fig. 1). If h is the pressure head on the drain and AT

the transversal area of the column, then the total volumeof the column is given by VT = ATP while water volume inthe saturated zone is Vw = Awh where Aw is the total porearea. Water volume in the saturated area relative to the to-tal soil volume (V* = Vw/VT) is given by V* = A* (h/P), whereA* = Aw/AT is the total pore area relative to the total soilarea.

If the whole pores set is represented as a parallel capil-lary tube system then A* = /, where / is the volumetric soilporosity, that is V* = /(h/P). Purcell (1949) shows that thehydraulic conductivity of such system is proportional to /(e.g., Fuentes et al., 2001). This leads to establish the pro-portion q / Ks(h/P), which corresponds to the linear radia-tion condition.

A different expression can be obtained for water flowfrom the probabilistic approach of Childs and Collis-George(1950). Performing a cut perpendicular to the macroscopicflow direction in a neighborhood near the column bottom,two porosity sections / are obtained. Each one is placedz1 = 0 and z2 = dP, where dP << P is a small positive amountof the pore size order. A water particle located at a poreof section z2 can continue its course through the same cap-illary pore or to change into another pore of a different sizelocated at the z1 position. Modeling of these possibilities of

376 M. Zavala et al.

change can be done accordingly to the introduction of ameeting probability of the two sections at a intermediateposition z12 ¼ 1

2dP. The probability that these sections meet

totally random way at this position is the productV*V* = /2(h/P)2, which also represents the effective flowarea. Because in this model the hydraulic conductivity isproportional to /2 (e.g., Fuentes et al., 2001), the propor-tion q / Ks(h/P)

2 corresponds to the quadratic radiationcondition.

Linear and quadratic radiations represent, from a proba-bilistic approach, the possible extreme behaviors for drain-age flux. The first one corresponds to a completelydeterministic movement of soil water (that is a completecorrelation of the capillary pores). The second one corre-sponds to a completely random movement (that is a com-pletely random meeting of the capillary pores).

The maximum drainage flux (qs) corresponds to h = P.This leads to formulate, respectively, linear and quadraticradiation conditions, in the following way:

q ¼ qs

h

P

� �ð4Þ

q ¼ qs

h

P

� �2

ð5Þ

where qs = cKs, in which c is a non-dimensional conductancecoefficient.

Convex radiation condition

The Hooghoudt radiation type (Eq. (2)) can be formulated asa convex combination of the both linear and quadratic radi-ations, represented by Eqs. (4) and (5) namely

q ¼ qs ð1� xÞ h

P

� �þ x

h

P

� �2" #

ð6Þ

which will be called convex radiation condition. In this one,the interpolation factor (x) is such that 0 < x < 1. Clearlyx = 0 corresponds to linear radiation and x = 1 to the qua-dratic radiation. Eqs. (2) and (6) are equivalent by makingb = qs(1 � x)/P and c = qsx/P2. It must be observed thatthe parameters number in Eqs. (2) and (6) is the same, band c in the first and qs and x in the second.

Fractal radiation condition

From a probabilistic approach it is reasonable that the soilbehavior is between the extremes behaviors of Purcell(1949) and Childs and Collis-George (1950). This is due tothe fact that soil water flow does not occur as in parallelcapillary system nor does it behave in a random way (Mil-lington and Quirk, 1961). This occurs because the soil geom-etry determines the water movement. An intermediatebehavior can be obtained if the soil is considered as a fractalobject (Mandelbrot, 1982; Rieu and Sposito, 1991a,b; Oles-chko et al., 1997; Fuentes et al., 1998, 2001). In order toachieve this, the Mandelbrot area–volume relationship willbe used.

We have that V / L3 and A / L2 in Euclidean geometry.Here V, A and L represent, respectively, volume, area andlength of an object, for a sphere V ¼ 4

3pr3, A = 4pr2, L = r

where r its radius; since L / V1/3 and L / A1/2 then A / V2/3

is obtained. In fractal geometry A / VD/E where D is the sur-face fractal dimension of the object and E = 3 is the spaceEuclidean dimension where the object is embedded. If l rep-resents the total areal porosity, then ls = 1 � l is the solidtotal area relative to the soil total area or total areal solidityand /s = 1 � / is the solid total volume relative to the soil to-tal volume or total volumetric solidity, therefore ls ¼ /s

s,that is 1 � l = (1 � /)s where s = D/E is the fractal dimensionrelated to Euclidean dimension. According to the probabilis-tic idea, the relationship between areal and volumetricporosities is l = / s/s = /2s; s = 1/2 corresponds to Purcellmodel and s = 1 to Childs and Collis-George model.

Because of ls + l = 1, the relationship between relativefractal dimension and total volumetric porosity is implicitlydefined by the equation:

ð1� /Þs þ /2s ¼ 1; s ¼ D=E ð7Þ

It can be demonstrated that l 6 /, s! 1/2 when /! 0 ands! 1 when /! 1, in other words we have 1/2 < s < 1 when0 < / < 1. The complete correlation is presented in soils witha porosity that tends to zero and the complete decorrela-tion in soils whose porosity tends to unity. Considering thatl = /2s, the equation that defines the relationship between sand l is obtained:

ð1� lÞ1=s þ l1=2s ¼ 1 ð8Þ

At the soil column base, in accordance to the probailisticidea, the relative flow area is Vs

�Vs� ¼ /2sðh=PÞ2s and since

the hydraulic conductivity is proportional to l = /2s (Fuen-tes et al., 1998, 2001), q / Ks(h/P)

2s is obtained. Therefore

q ¼ qs

h

P

� �2s

ð9Þ

where qs = cKs corresponds to h = P.When the drainage analysis are performed with the Bous-

sinesq equation, it is not possible to introduce drainpipegeometrical characteristics such as its diameter and wallorifice density and size. The drains are represented as dotsor lines. However, the precedent analysis can at least allowthe introduction of the wall geometrical characteristicswhen the wall is considered a porous medium.

If the porous media located at the z1 and z2 positions aredifferent, that is the case at the column base (soil–drainwall), the procedure to obtain Eq. (9) must be modified. Ifthe {/m, sm, lm, Ks} notation is used for soil propertiesand {/d, sd, ld, Kd} for the properties of the drainpipe wall,the effective flow area Vsm

� Vsd� ¼ /sm

m ðh=PÞsm/sd

d ðh=PÞsd is ob-

tained. Considering /smm /

ffiffiffiffiffiKs

pand /sd

d /ffiffiffiffiffiffiKd

p, we deduce

the proportion q /ffiffiffiffiffiffiffiffiffiffiKsKd

pðh=PÞsmþsd . Then qs ¼ c

ffiffiffiffiffiffiffiffiffiffiKsKd

pand

s ¼ 12ðsm þ sdÞ.

The fractal radiation defined by Eq. (9), unlike Eq. (3), orEqs. (2) and (6), has one unknown parameter (qs) instead oftwo, since the power s is a known function of porosity; thisradiation justifies Eq. (3) by making a = qs/P

2s and n = 2s.The fractal radiation can be considered as an interpolationformula of the drainage flux extreme models; making2s = 1 + d in such a way that 0 < d = 2s � 1 < 1, the linearradiation is represented by d = 0 while the quadratic radia-tion is represented by d = 1.

Non-linear radiation in the Boussinesq equation of the agricultural drainage 377

To estimate the drainage flux with Eq. (9) when soil sat-urated hydraulic conductivity (Ks) and the porosity of bothporous media are known, it is necessary to supply bothparameters c and Kd. The first one can be estimated bymeans of the Boussinesq equation, subject to fractal radia-tion, calibrated in a drainage test. The second one can beestimated considering the flux on the drain wall as closelydescribed by a Poiseuille regime. If this is the case, conduc-tivity Kd in circular pores is defined by (Fuentes et al., 2001)

Kd ¼1

8

g

m

ZXT

R2s dxd ð10Þ

where g is the gravitational acceleration, m is the waterkinematic viscosity coefficient, XT is the whole pore domain(saturated flow), Rs is the radius defining the pores area anddxd is the elemental porosity area to which the pores of ra-dius Rs contribute.

ConsideringR

XTdxd ¼ ld, and since the drainpipe walls

usually present openings of equal size (Rs = Rsd), Eq. (10) istransformed into Kd ¼ 1

8ðg=mÞR2

sdld. The hydraulic radius(RHd), defined as the ratio of the hydraulic area to the wet-ted perimeter, may be used to apply the formula to non-cir-cular openings; since Rsd = 2RHd in a water filled circularopening, the following is true: Kd ¼ 1

2ðg=mÞR2

Hdld. The12coef-

ficient can be replaced by the Kozeny coefficient (Cf) ofwhich values resulting from the different shapes of theopenings (Bear, 1972), nevertheless in this case the possibledisagreements with respect to circular shape can beabsorbed by the coefficient c, since it appears to be a mul-tiplicative factor. It can therefore be accepted that thedrain-wall conductivity is by definition:

Kd ¼1

2

g

mR2Hdld ð11Þ

In the fractal radiation condition, the only parameter inneed of calibration is the interface soil–drain conductancenon-dimensional coefficient (c).

An equivalence between convex and fractalradiation

The Hooghoudt radiation type (Eq. (2)) has been used indrainage transitory regime studies, and has been adaptedin the current study as a convex radiation condition, Eq.(6), using a reference drain flux (qs) and an interpolationfactor (x) as parameters. Considering that the fractal radi-ation in Eq. (9) contains the parameters (qs,s), where s is re-lated to the porosity through Eq. (7), it is convenient toestablish a relationship between x and s, with the purposeof characterizing the interpolation coefficient in functionof the soil porosity.

The non-dimensional variables h* = h/P and q* = q/qs areintroduced in order to study the possible relationships be-tween x and s. These variables allow Eqs. (6) and (9) tobe rewritten in the following way:

q�c ¼ ð1� xÞh� þ xh2�; 0 < x < 1 ð12Þ

q�f ¼ h1þd� ; 0 < d ¼ 2s� 1 < 1 ð13Þ

where 0 < h* < 1, 0 < q�c;f < 1, q�cð0Þ ¼ q�f ð0Þ ¼ 0 and q�cð1Þ ¼q�f ð1Þ ¼ 1; c and f subscripts indicate, respectively, convexand fractal.

Considering that parameters x and d vary in the samesense, that is x = 0 corresponds to d = 0, and x = 1 tod = 1, their general relationship can be written as follows:

x ¼ kdd ð14Þ

where the coefficient kd P 0 can depend on d, and take thevalue k1 = 1.

The kd coefficient shape depends on the method used tobring functions (12) and (13) near to each other. With thesecond order expansion of hd

� ¼ 1� dþ dh� around h* = 1,it can be deduced that q�c ¼ ð1� dÞh� þ dh2

� and thuskd = 1; the result provides the exact fluxes at the domainends of h*, nevertheless convex radiation overestimatesfractal radiation in accordance to h1þd

� 6 ð1� dÞh� þ dh2�,

this overestimation is more clearly emphasized whenh*! 0 than when h*! 1. On the assumption that functions(12) and (13) agree at point h*p, kd ¼ ð1� hd

�pÞ=½dð1� h�pÞ� isobtained, which includes the previous case when h*p! 1.The equality of the areas under both curves leads tokd = 3/(2 + d), from which kd P 1 and k0 ¼ 3

2.

The classic least squares method is used considering theconvex radiation as an estimator, q�cðh�; xÞ, of the fractalradiation q�f ðh�Þ. The value of x which minimizes the squareerror integral on the domain of h*, is searched for a given d.This is defined by

IðxÞ ¼Z 1

0

½q�f ðh�Þ � q�cðh�; xÞ�2dh� ð15Þ

The condition dI/dx = 0 allows to obtain:

kd ¼5ð7þ dÞ

2ð3þ dÞð4þ dÞ ð16Þ

where it can be observed that k0 ¼ 3524, is relatively close to

the value obtained by the equality of the areas criterionðk0 ¼ 3

2Þ.

The approximation degree can be estimated by means ofthe correlation coefficient, defined by r ¼ r2

fc=ðrfrcÞ, or ofthe determination coefficient (r2), where r2

fc stands for thecovariance between fractal and convex fluxes; and rf andrc, respectively, represent the fractal and convex fluxesstandard deviations. The covariance is defined by

r2fc ¼

Z 1

0

q�f ðh�Þq�cðh�Þdh� �Z 1

0

q�f ðh�Þdh�Z 1

0

q�cðh�Þdh� ð17Þ

and the definitions of the standard deviations are obtainedwith Eq. (17) by the rules rf ¼

ffiffiffiffiffiffir2ff

pand rc ¼

ffiffiffiffiffiffiffir2cc

p.

The introduction of Eqs. (12) and (13) in Eq. (17) allowsobtaining the determination coefficient (r2):

r2 ¼ 5ð3þ 2dÞð12þ 3dþ xdÞ2

ð3þ dÞ2ð4þ dÞ2ð15þ x2Þð18Þ

Taking into account Eqs. (14) and (16), the r2 curve as afunction of d presents a minimal point in (0.4335, 0.9996).This indicates that the convex radiation is close to fractalradiation with r2 P 0.9996, if the least squares method isused.

Application

To evaluate the descriptive capacity of the convex radiation(Eq. 6) as well as of the fractal radiation (Eq. 9), and also to

378 M. Zavala et al.

estimate conductance parameter (c), a drainage experi-ment was modeled with a numerical solution of the one-dimensional Boussinesq equation:

loh

ot¼ o

oxKsðDo þ hÞ oh

ox

� �þ R ð19Þ

subject to the initial and boundary conditions:

hðx; 0Þ ¼ hsðxÞ ð20Þ

� Ksoh

ox� q ¼ 0 ð21Þ

The positive sign in Eq. (21) is taken for the drain located atx = 0, while the negative one is for the drain located atx = L, where L is the drains separation. In Eq. (21), the con-vex radiation (6) or the fractal radiation (9) is introducedaccording to each case.

The drained water flow by length unit at each drain is ob-tained with

Q ¼ 2qðhþ DoÞ ð22Þ

and the cumulative drained depth by

‘ðtÞ ¼ 1

L

Z t

0

QðsÞds ð23Þ

The system (19)–(23) is numerically solved by executing thespatial discretisation with the Galerkin finite element meth-od, the temporal discretisation with an implicit finite differ-ences method, linearising the resulting system with Picarditerative method and solving the whole algebraic equationssystem with a preconditioned conjugated gradient method(Noor and Peters, 1987). These methods are well docu-

2L

EDo

PSoil

drains

L/2 L L/2

x

yz

x=-L/2 x=3L/2

x=0 x=L

Soil surface

h = 0

h = PBl

Figure 2 Drainage module scheme.

Table 1 Root mean square error (RMSE) between the calculatcorresponding to the different radiation conditions. Conductanceshown

Radiation type Conductance coefficient (c) Maximum draina

Linear 0.0227 5.07Convex 0.0260 5.80Fractal 0.0256 5.71Quadratic 0.0396 8.84

mented; for instance, by Pinder and Gray (1977), Mori(1986) and Zienkiewicz et al. (2005).

The experiment was carried out in a drainage modulemade with acrylic sheets in which two PVC drains were in-stalled (Fig. 2). The drain diameter is d = 5 cm, the totalnumber of circular openings in the wall is No = 233 and theopening diameter is do = 1.58 mm. The module dimensionsare: L = 100 cm, P = 120 cm, Do = 25 cm, E = 30 cm andBl = 35 cm.

The module was filled with an altered sample of sandysoil of the Mexican region of Tezoyuca, Morelos, passedthrough a 2 mm sieve; the soil was disposed on 20 cm thicklayers, seeking to maintain a constant apparent density. Thesoil was saturated by applying a constant water head on itssurface until the entrapped air was virtually removed. Oncethe drains were closed, the water head was removed fromthe soil surface; the surface of the module was then coveredwith a plastic in order to avoid evaporation. Finally, thedrains were opened to measure the drained water volume;it is worth noticing that the initial condition was equivalentto h(x,0) = P and the recharge was null (R = 0) during thedrainage phase.

Soil porosity (/m) is calculated with the formula/m = 1 � qt/qs, where qt is the bulk density and qs is theparticles density; with the bulk density determined fromthe weight and volume of the soil of drainage moduleqt = 1.22 g/cm3 and the particles density qs = 2.65 g/cm3,/m = 0.5396 cm3/cm3 is obtained. With this value and Eq.(7) sm = 0.7026 is then obtained. The value of Ks = 18.3 cm/h saturated hydraulic conductivity was estimated from aconstant head test (Zavala, 2003). The storage capacity iscalculated with l = ‘1/P where ‘1 is the experimental max-imum drained depth; in the carried out test it was‘1 = 23.92 cm and P = 120 cm, from which l = 0.1993 cm3/cm3.

The drain wall area is Ad = pdE = 471.24 cm2, the totalarea of the circular openings is Ao ¼ No

14pd2

o ¼ 4:57 cm2,the areal porosity ld = Ao/Ad = 0.0098 and according to rela-tionship (8) sd = 0.5689. The hydraulic radius of the openingsis RHd = 0.0397 cm, and considering g = 981 cm/s2 and m =0.01 cm2/s, the drain wall conductivity Kd = 2,721.5 cm/his obtained.

The mean values of saturated conductivity and therelative dimension are Ks ¼

ffiffiffiffiffiffiffiffiffiffiKsKd

p¼ 223:2 cm=h and s ¼

12ðsm þ sdÞ ¼ 0:6358, respectively. Using Eqs. (14) and (16)and the parameter value d = 2s � 1 = 0.2716, we obtainthe convex radiation interpolation factor x = 0.3533.

To minimize the root mean square error (RMSE) betweenthe calculated drained depth with Boussinesq equation andthe experimental drained depth, the conductance parame-

ed drained depth and the experimental drained depth (cm)coefficients and maximum fluxes obtained in calibration are

ge flux (qs) (cm/h) Root mean square error (RMSE) (cm)

0.9400.7830.7251.060

0

5

10

15

20

25

0 6 12 18 24Time (h)

)mc(

htpe

d de

niard evital

um

uC

.

o Experimental

Quadratic radiation

Linear radiation

0

5

10

15

20

25

0 60 120 180 240Time (h)

)mc(

htpe

d de

niard evital

um

uC

.

o Experimental

Quadratic radiationLinear radiation

Figure 3 Comparison between the experimental drained depth and the calculated drained depths with linear radiation c = 0.0227(RMSE = 0.94 cm) and with quadratic radiation c = 0.0396 (RMSE = 1.06 cm).

0

5

10

15

20

25

0 6 12 18 24Time (h)

)mc(

htpe

d de

niard evital

um

uC

.

o

Experimental

Fractal radiation

Convex radiation

0

5

10

15

20

25

0 60 120 180 240Time (h)

)mc(

htpe

d de

niard evital

um

uC

.

o Experimental

Fractal radiationConvex radiation

Figure 4 Comparison between the experimental drained depth and the calculated drained depths with fractal radiationc = 0.0256, s = 0.6358 (RMSE = 0.725 cm), and convex radiation c = 0.0260, x = 0.3533 (RMSE = 0.783 cm).

Non-linear radiation in the Boussinesq equation of the agricultural drainage 379

ter and the drainage maximum flux corresponding to eachradiation condition, were calibrated; Table 1 shows the ob-tained results. Fig. 3 presents the experimental draineddepth compared to the calculated drained depth in functionof time corresponding to linear and quadratic radiations,respectively. Comparison shows that the drained depthexperimental evolution presents a noticeably different cur-vature from both curvatures obtained with the linear andquadratic radiation; the soil water movement is not com-pletely correlated nor completely random.

Fig. 4 compares the experimental drained depth with thecalculated drained depth according to convex and fractalradiations. It is worth noticing that the calculated curvesare closer to the experimental curve rather than the calcu-lated curves with radiations extreme. These results indicatethat water dynamics in a subterranean drainage system canbe studied with the Boussinesq equation subject to fractalor convex radiations at drains.

Conclusions

The water flow towards subterranean drains has been de-scribed with the Boussinesq equation subject to a radiation

type boundary condition, which relates the drainage fluxwith the pressure head on the drain.

The relationship between the drainage flux and the pres-sure head on the drain can be described with a quadraticpolynomial, analog to Hooghoudt equation which relatesthe drainage flux with to the head in the middle of the sep-aration between the drains; this radiation condition in-cludes the classic linear radiation and a quadraticradiation. The relationship can also be described with apower function; this power radiation includes also the linearand quadratic radiations.

The drainage analysis of the soil column above a drain, inthe context of the water flow capillary models proposed byPurcell (1949) and Childs and Collis-George (1950), it has al-lowed to establish from a probabilistic point of view, the ex-treme behaviors of the radiation condition. The linearradiation corresponds to the Purcell model, where poresare conceived as a parallel capillary tube system throughwhich the water particles flow without possibility of capil-lary changing. The quadratic radiation corresponds to Childsand Collis-George model in which, by means of a transversecutting of the capillary system, the water particles changefrom one capillary to another in a random way.

380 M. Zavala et al.

The Hooghoudt type radiation is obtained as a convexcombination of the extreme probabilistic behaviors. Thisconvex radiation contains two unknown parameters, a ref-erence flux and an interpolation factor. Power radiation isestablished considering on the one hand, that water parti-cles flow is not completely correlated nor completely ran-dom, since the trajectories are determined by the soilstructure, and on the other hand fractal geometry concepts.This fractal radiation contains a reference flux and an expo-nent, equal to double of the surface soil-particles fractaldimension relative to soil Euclidean dimension; the fractalradiation interpolates the linear and quadratic radiationsthrough its exponent. Considering that convex radiation isan approximation to fractal radiation, a relationship be-tween the interpolation factor and the fractal exponentwas established using the least square method.

A numerical solution of the Boussinesq equation, subjectto the different radiation conditions was used to describe alaboratory drainage experiment. The accumulated draineddepth was better represented by the fractal and convexradiations than by the extreme radiations. This allows usto conclude that both the fractal and the convex radiationsmay be applied in the study of the water flow in subterra-nean drainage systems.

Acknowledgements

This work was partially supported by CONACYT (ConsejoNacional de Ciencia y Tecnologıa) through its project‘‘SEP-2004-C01-47083/A-1’’. We greatly thank the anony-mous reviewers for their insightful comments andsuggestions.

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