Comparisons of Calculated and Measured Capillary Potentials from Line Sources1

ADRIAN W. THOMAS, HAROLD R. DUKE, DAVID W. ZACHMANN, AND E. GORDON KRUSEZ

ABSTRACTThis paper is a sequel to D. W. Zachmann and A. W. Thomas'

(1973) development of equations describing steady infiltration fromline sources. Calculated distributions of capillary potentials are com-pared with those measured in a soil bin designed to model the waterdistribution from a subsurface irrigation system. Limitations of thesteady-state equations for predicting capillary potentials are given. Aprocedure for selecting lateral depth and spacing to attain a given ma-tric flux potential is also given.

Additional index words: subsurface irrigation, hydraulic conduc-tivity, infiltration.

MUCH LITERATURE has been written during the past dec-ade describing theoretical investigation of point and

line sources. Generally, these solutions have been analyticaland apply only to steady-state systems. Typical examples ofsuch studies are by Philip (7), Raats (8), Gilley and Allred(3), Warrick (10), Zachmann and Thomas (11), and Thom-as, Kruse, and Duke (9). Recently, Warrick (10) andLomen and Warrick (5) developed analytical solutionswhere the strengths (unit rate of discharge) of point and linesources are cyclic (transient infiltration). These investiga-tors have made and acknowledged assumptions limiting theusefulness of their solutions. Even so, they have impliedthat the solutions would provide a basis for designing trickleor subsurface irrigation systems.

Testing and verifying of mathematical models is often themost difficult aspect in developing them as useful predictivetools. However, unless experimentation coexists with de-velopment, the models' reliability will be uncertain andtheir usefulness limited. Very little experimental evidencehas been reported to indicate the merits and limitation of themathematical solutions associated with point and line sour-ces. Exceptions, where experimentation complemented themathematical development, are Brandt et al. (1) and Bresleret al. (2) who developed mathematical solutions for describ-ing infiltration from trickle irrigation sources (point source)and conducted laboratory and field experiments for verifica-tion.

This paper compares measured and calculated capillarypotentials using the mathematical solution developed byZachmann and Thomas (11). Their solutions, based on anexponential relationship between hydraulic conductivityand capillary potential, is applicable to steady infiltrationfrom a distribution of line sources (buried or on the surface)which lie in a horizontal plane and are parallel and equallyspaced. The measured potentials were obtained from an ex-

'Contribution from Southern Piedmont Conservation Research Center,Watkinsville, GA 30677, Athens, Georgia Area, Southern Region, Agri-cultural Research Service, USDA, and Colorado-Wyoming Area Offices,in cooperation with the Colorado State University Experiment Station.Received 28 April 1975. Approved 2 Oct. 1975.

Agricultural Engineer, USDA, Watkinsville, GA 30677; AgriculturalEngineer, USDA, Fort Collins, CO 80521; Assistant Professor, Depart-ment of Mathematics, Colorado State Univ., Fort Collins, Colo.: and Ag-ricultural Engineer, USDA, Fort Collins, Colo., respectively.

perimental system designed to model the water distributionfrom subsurface irrigation. In addition, a procedure forselecting lateral depth and spacing to attain a given matricflux potential is given, if hydraulic characteristics of the soilare known. A future paper will show the implications of thisselection in designing an irrigation system.

GOVERNING EQUATIONS AND SOLUTIONInfiltration, as used in this paper, is the physical process of

water being emitted from line sources into the soil and its con-sequent movement due to pressure and gravity gradients. Thelinearized flux equation used to describe the water movement, asgiven by Philip (6), is

d20a*2

_~ a

30 [i]where x and z form a two-dimensional flow plane (z positive in thedirection of gravity); 0 is the matric flux potential, and a is a con-stant. Two important assumptions required to obtain the linearizedequation are

= IJ

and

K « exp (aip),

[2]

[3]

where ̂ is the capillary potential; ^o is an arbitrarily small value ofcapillary potential, and K is the hydraulic conductivity. Althoughthe exponential relationship between K and ̂ is not exact, it doesmodel the nonlinear decrease of K with ¥ for soils over relativelysmall ranges of ^P. The significance of the assumption implied inEq. [3] is discussed in more detail by Philip (6) and Thomas et al.(9).

Zachmann and Thomas (11) presented the boundary value prob-lem which 0 must satisfy as

920 30

SOIL SURFACE

DEEP SOIL MEDIUM WITH DEEP WATER TABLEFig. 1.—Schematic of semi-infinite strip showing lateral depth and

half spacing

10

THOMAS ET AL.: COMPARISONS OF CAPILLARY POTENTIALS FROM LINE SOURCES 11

subject to boundary conditions

~~ [6]

where za and x0 are the depth and half spacing of the line sources,respectively; 8 is the Dirac delta function; Q is the strength of theline source, and v0 is the constant velocity flux at z0- The origin ischosen at the line source (Fig. 1). A negative value of va representsa constant evaporation rate at the surface. The solution for Eq. [4-7] is

0 fr. z) = -0a ^ ffl + a-n/3

[<rn cosh vn(z - Z0) + ft sinh an(z - Z0)] — cos -— [8]X0 X0

(Z0 0)

[a~n cosh crnZo - ft sinh crnZo] — cos —— , (0 =s z)XQ XO

[9]

where /3 = a/2 and o-n = [j82 + (mr/x0)2]1/2. Equations [8] and[9] are applicable to a single semi-infinite strip (crosshatched area,Fig. 1). The solution can be expressed in terms of capillary poten-tial by substituting Eq. [8] or [9] into Eq. [10]

ln(Ks/a) [10]

where Ks is the saturated conductivity. Equation [10] was devel-oped by Raats (8), and recently Thomas et al. (9) utilized it in ex-ample calculations.

LABORATORY MODELThe experimental study was conducted to model subsurface irri-

gation by injecting water uniformly from horizontal tubes buried insoils. A soil bin was divided into two compartments, each 122 cmdeep, 122 cm wide, and 15 cm thick. Two field soils of differenttextures were selected for the experiments—clay loam, sieved topass a no. 20 (841 micron) screen, was placed in one compart-ment; sandy loam, sieved to pass a 0.32-cm screen, in the other.The soils were placed in their respective compartments with afunnel-top tremie to minimize particle-size separation. Soil wasadded in 5-cm layers to provide a homogeneous medium. Averagebulk densities of the clay loam and sandy loam soils were 1.00 and1.32 g/cm3, respectively. The particle-size distribution of each soilwas determined by the hydrometer method, (Table 1).

A Mariotte siphon provided a constant head of water for the in-jection tube in each compartment. Small plastic tubing of appro-priate length was used to control the rate of application of demin-eralized water. The injectors, buried 15 cm below and parallel tothe surface, extended through the soil mass. The injectors were

Table 1.—Classification and particle-size distribution of soils

Clay loam Sandy loam

Clay (< 2/to)SiltSand (>50/Lttn)

26.837.635.6

15.222.862.0

perforated and wrapped with glass fiber mat to allow uniformwater distribution.

Four hollow ceramic candles (25 cm long and 5 cm diameter),with bubbling pressures of 25 psi, were placed in the bottom ofeach compartment. A constant suction was maintained on thecandles to simulate a deep water table. A vacuum system con-trolled the suction on the candles at 488 cm H2O, simulating awater table 6.6m (20 ft) below the soil surface. The water ex-tracted from the candles (deep seepage) was collected in calibratedplastic cylinders.

Capillary potentials were measured with 83 tensiometers in eachcompartment. Each tensiometer was fabricated from a 1-cmdiameter ceramic disk with a bubbling pressure of 25 psi. The ten-siometers, mounted through capscrews, were under compressionto maintain the ceramic in contact with the soil. Symmetricalarrangement of tensiometers about the source permitted readingsfrom corresponding pairs to be averaged to reduce data scatter.

To establish steady-state flow from the simulated subsurface ir-rigation laterals, the soil masses were first flooded with water onthe surface. After the top of the soil bin was covered with plastic toprevent evaporation, a steady injection of water through the later-als was initiated. The soil drained monotonically until steady flowwas attained (approximately 5 days), i.e., the rate of water in-jected through the laterals equalled the rate extracted from thecandles. The capillary potential distributions were measured dur-ing steady flow conditions.

The functional relationships between hydraulic conductivity (K)and capillary potential (SP) were determined in situ for the soils ineach compartment. Hillel (4) has discussed such steady-state tech-niques. The soil mass was flooded with water on the surface untilpositive pressures were attained, as evidenced by excessive leak-age from the compartments. Subsequently, the flow rate was peri-odically reduced to allow the soil to drain monotonically untilsteady flow was established. Hydraulic gradients were measured atseveral positions throughout the soil mass. The flow rate wasdecreased, and the process repeated until flow rates were so smallthat applying water uniformly from the drip tubes became dif-ficult. The experimental conductivity-capillary potential rela-tionships calculated from these data are shown in Fig. 2.

RESULTS AND DISCUSSIONCapillary potentials were calculated using [10] where the

functions of @(.t,z) are given in Eq. [8] and [9]. The values

SANDY LOAM—•Ks=I.OOXIO"3cm/sec

-40 -30 -20 -10 0CAPILLARY POTENTIAL (cm H20)

Fig. 2.—Hydraulic conductivity and capillary potential relationshipsfor soils

12 SOIL SCI. SOC. AMER. J., VOL. 40, 1976

CALCULATED CAPILLARY POTENTIALS

I \ \I > \/ . , «--"' , I \-35/ V v -25-y i y

-30-35-

SCALE: i20cmi

MEASURED CAPILLARY POTENTIALSFig. 3.—Calculated and measured capillary potential distributions (-*,

cm H2O) for clay loam

of a and Ks were obtained from Fig. 2, where a is the slopeof the curve (d K(^)ld^), and Ks is the value of K ex-trapolated to ^P = 0. Since the soil bin was covered toprevent evaporation, the flux normal to the soil surface waszero, i.e., v0 = 0. The source strengths (Q) were 1.05 x10r3 crtr cm sec"1 and 1.08 X 10~3 cm3 cm-sec"1 for the clay loam and sandy loam, respectively.These values represent water application (irrigation rate) ofapproximately 0.75 cm/day. The calculated capillary poten-tials for the clay loam and sandy loam are presented in Fig,3 and 4, respectively. Distributions of measured potentialsfor the soils are also plotted for comparison.

Errors can be expected in using Eq. [10] with an ex-trapolated value of Ks when soils remain saturated over alarge range of M*. A graphical technique can be used to over-come this limitation of Eq. [10]. An arbitrarily small valueof capillary potential, which corresponds to a zero value ofmatric flux potential (i.e., the area under a rectilinear plot ofK(ty) vs. W is zero at the arbitrarily small value of ^P) is selec-ted. The area under the curve is then summed from the arbi-trarily small value of ̂ until this area equals the calculatedmatric flux potential. The corresponding value of ^ is thecapillary potential for the point in question. Therefore, thecapillary potential corresponding to the matric flux potentialis selected from the actual K(ty) curve rather than computedwith Eq. [10]. The graphical technique should not be usedfor large values of capillary potential when the exponentialrelationship of K and ̂ is no longer valid.

The advantage of the graphical technique is that the satu-rated hydraulic conductivity, Ks, is not required to computevalues of capillary potential. However, the simplicity of

-60

-40'

SCALE: i20cml

CALCULATED CAPILLARY POTENTIALS

LATERAL\\\\ \-40'

SCALE

*-35

20cm.

MEASURED CAPILLARY POTENTIALSFig. 4.—Calculated and measured capillary potential distributions (-^,

cm H2O) for sandy loam

using the equation to calculate capillary potentials is a dis-tinct advantage. Comparison of the two techniques showedvery little difference in computed potentials for these twosoils.

When comparing the measured and calculated distribu-tions of capillary potential, we observed that in mostregions of the soil mass, the calculated capillary potentialswere within 25% of the measured values. In the regionabove the laterals (z0 ̂ z< 0), the calculated capillary po-tentials were within 10% of the measured values. Directlybelow the laterals (z> 0), however, the calculated capillarypotentials were approximately two times larger than themeasured values.

There are several reasonable explanations for the dif-ferences between measured and calculated potentials belowthe laterals. First, there was a tendency for the soil to settle,thereby increasing in density at depth but creating voids di-rectly below the laterals (some soil was added to fill voidsbefore steady-state measurements). Consequently, inregions of increased density the saturated hydraulic conduc-tivity decreased and a higher capillary potential wasrequired to maintain the flow. For example, the measuredand calculated capillary potential decreases (from pointbelow and near lateral to near bottom of soil bin) were ap-proximately 20 to 30 and 7 to 8 cm H2O, respectively. Alateral is not likely to uniformly release water about its cir-cumference if part is in contact with unsaturated soil andpart is exposed to atmospheric conditions. This problemillustrates some of the difficulty of representing a lateralwith a mathematical line source. A second reason for the

\

THOMAS ET AL.: COMPARISONS OF CAPILLARY POTENTIALS FROM LINE SOURCES

Table 2.—Dimensionless matric flux potential for the upper corners as a function of lateral depth and half spacing

0.02 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90

13

1.00 10.6290.900.800.700.600.50

Z0 0.400.300.200.100.02

12.98315.85719.36823.65628.89335.29043.10452.64764.30375.295

2.1262.5973.1712.8744.7315.7797.0588.621

10.52612.78714.311

1.0631.2981.5861.9372.3652.8893.5274.2985.2006.1126.529

0.7090.8651.0571.2911.5751.920

2.3352.8173.3393.7984.004

0.5310.6490.7910.9651.1741.4241.7162.0402.3652.6282.720

0.4240.5170.6300.7650.9261.1141.3261.5511.7651.9302.001

0.3510.4270.5180.6260.7520.8961.0531.2141.3621.4711.504

0.2980.3610.4350.5220.6220.7330.8520.9701.0751.1511.173

0.2560.3080.3700.4410.5200.6070.6980.7860.8630.9180.943

0.2220.2660.3170.3740.4380.5070.5770.6440.7020.7430.753

0.1930.2300.2720.3200.3710.4260.4810.5330.5770.6080.623

1.00

differences between measured and calculated capillary po-tentials below the laterals is the exponential approximationof the hydraulic conductivity and capillary potential rela-tionship. Although a saturated soil remains essentially satu-rated after some finite decrease in capillary potential, this isnot represented in the exponential relationship. Con-sequently, the actual values of Ks are somewhat less thanthose given in Fig. 2. For a relatively large value of matricflux potential, there corresponds a larger value of capillarypotential for an actual K(^) relationship than for the ex-ponential approximation. Therefore, for a large value of 0,the calculated capillary potential will be smaller than thecapillary potential measured with a tensiometer.

A third reason may be that the lower boundary conditionin the experiment is not well simulated by the mathematicalsolution. However, the authors believe that the constantsuction by the candles has more influence on the shape ofthe contours than on the magnitude of the capillary poten-tials at any particular point. The general shape and distribu-tion of the contours for the measured and calculated valuesof capillary potential are quite similar in the upper regionwhich is of primary interest for maintaining adequate soilwater for plants. The contours of the measured capillary po-tentials below the laterals are almost flat on the bottom. Incontrast, the contours of the calculated capillary potentialsare elongated. One reason for the difference in shape of thecontours below the laterals is due to the differences in the1

gradients of the capillary potential for the mathematical andexperimental cases, as described earlier. A second reasonfor the almost flat bottom of the contours of the measuredvalues of capillary potential is the influence of the constantsuction produced by the candles along the bottom of the soilbin, i.e., candles were not deep enough to simulate a watertable at infinity.

As indicated earlier, the measured and calculated capil-lary potentials compared best in the region above the later-als. For example, the measured and calculated capillary po-tentials in the upper corners of the soil mass for the clayloam were —66 and —67 cm H2O, respectively. The mea-sured and calculated capillary potentials in the uppercorners of the soil mass for the sandy loam were —70 and— 71 cm H2O, respectively. This was the region, above andaway from the laterals, where the capillary potentials werethe smallest, i.e., the soil was the driest. The authorsbelieve that the upper corners are logical points to base thedesign of subsurface irrigation (however, any point can bechosen). Table 2 gives the dimensionless matric flux poten-

tial (A/) for the upper corners as a function of dimensionlesslateral depth (Zo) and half spacing (A). The definitions ofthe dimensionless terms as given by Zachmann and Thomas(11) are N = (2ir@lQ); Ztt = PZO, and Z = ($x0: The capillarypotential at the surface and midway between laterals can becomputed by using the information in Table 2 and Eq. [10],if the lateral strength, Q, and the soil characteristics, a andKs, are known.

Table 2 can also be used to select lateral depth and spac-ing for a given matric flux potential, if an application ratecan be chosen and the hydraulic characteristics K(ty) of thesoil are known. The following example demonstrates theprocedure for selecting the lateral depth and spacing. First,an expression for A/ is developed by using Eq. [10] and thedefinition of the dimensionless matric flux potential,

[H]

a and Ks, are 0.05respectively. Also,

Assume that the soil characteristics,cm * and 0.5 Xassuming that the

10- cm seclateral strength is 0.5 X 10 3 cm3

cm~' sec"1 and that the capillary potential at the sur-face and midway between the laterals is -100 cm, N iscomputed as 0.85. There are infinite combinations of lateraldepth and spacing which can be selected from Table 2 whenN = 0.85. Two choices with a wide range are

1) Z0 = 0.20A = 0.81

2) Z0 = 0.90A = 0.31

Lateral depth = 8 cmLateral spacing = 65 cmLateral depth = 36 cmLateral spacing = 25 cm

Although the selection is based on a given set of data, thechoice of lateral depth and spacing will greatly influence theapplication rate. For example, the application rates (Q/2x0)for choice 1 and 2 are 0.67 and 1.73 cm/day, respectively.

SUMMARYThis study compared calculated and measured distribu-

tions of capillary potential in a soil bin designed to modelthe water distribution from a subsurface irrigation system.Capillary potentials were computed from mathematicalequations developed to model steady infiltration from a dis-tribution of line sources which lie in a horizontal plane andare parallel and equally spaced. Capillary potentials weremeasured from a steady flow regime created in a soil bin fortwo soils.

14 SOIL SCI. SOC. AMER. J., VOL. 40, 1976

This study has shown promise that the equations can soil. A later paper will deal with the influence of water ex-predict the capillary potentials in most regions of a soil mass traction by plants and the implications of subsurface irriga-which is irrigated with a subsurface system. tion design.

The region of soil where the calculated and measuredcapillary potentials compared best was above and awayfrom the laterals. The measured capillary potentials in theupper corners of the soil mass were — 66 and —70 cm H2Ofor the clay loam and sandy loam, respectively, while thecalculated potentials were within 1 cm H2O of the measuredvalues in each case. The measured and calculated capillarypotentials compared poorly in the region directly beneaththe laterals. Difficulties in representing a finite irrigation lat-eral with a line source, limitations of the assumed exponen-tial relationship of hydraulic conductivity and capillary po-tential, and inadequate simulation of lower boundarycondition are possible reasons for the poor comparison.

A procedure for selecting lateral depth and spacing to at-tain a given capillary potential between laterals was given.The designer must know the hydraulic conductivity-capillary potential relationship of the soil and must selectthe lateral strength and capillary potential at the surface andmidway between the laterals. Although there are infinitecombinations of lateral depth and spacing which will satisfythe procedure, each combination will require careful con-sideration. The designer must be aware of the water require-ments of the crop, practical limitations of lateral depth andspacing, and deep seepage.

The study does not consider evaporation from the soilsurface nor the influence of plants extracting water from the