Equations for Approximating Vertical Nonsteady-State Drainage of Soil Columns1

RAY D. JACKSON AND FRANK D. WHISLER 2

ABSTRACTEquations are derived, from a capillary tube model, that

predict vertical nonsteady-state drainage from soil columnsover the entire time of drainage for which experimental dataare available. The model assumes that a soil column is abundle of capillaries of various radii with the effective con-ductivity of the column decreasing as drainage progresses. Theconductivity is first considered to be a linear function of theaverage water level in the column, and then considered to bea quadratic function of the water level. Both resulting equa-tions predict drainage reasonably well when particular valuesare assigned to the ratio of the average height of capillary riseto the column length. A simplified expression is obtained byneglecting an insignificant term and assigning values to con-stants in the equation in which the conductivity was taken asa quadratic function of the average water level. The equationsare compared with experimental data.

Additional Key Words for Indexing: water flow, soil, variableconductivity.

THE NONSTEADY-STATE drainage of soil columns can bestbe predicted by numerical solution of a second-order,

nonlinear differential equation. Numerical results for spe-cific porous materials have been presented by Day andLuthin (1956), Liakopoulous (1964), Jensen and Hanks(1967), and Whisler and Watson (1968). In many prac-tical situations, the detailed information, such as the hy-draulic conductivity-soil water pressure function and thewater-characteristic relation, necessary for a numericalsolution, is not available. In these situations, less precisebut much simpler solutions are required. Approximatesolutions for nonsteady-state drainage have been reportedby Youngs (1960), Gardner (1962), and Ligon, Johnson,and Kirkham (1962). The various approximate solutionshave been examined and compared with experimental re-sults by Jensen and Hanks (1967) and Whisler and Bouwer(1970). These authors concluded that an equation due toYoungs best represented experimental data, but did notadequately predict drainage for long times, i.e. for timesafter 60% of the total drainage had taken place.

Youngs based his analysis on a capillary tube model,considering a soil column to be a bundle of capillaries ofequal radii. Vachaud (1968) extended Youngs analysisby considering tubes of equal radii grouped into bundles.He used Youngs equation for each bundle and summed thecontributions. For two or more bundles of different radiihis calculated results were less successful in predictingdrainage than Youngs equation during the early part ofdrainage but was more successful during the later stages.

1 Contribution from the Soil & Water Conservation ResearchDivision, ARS, USDA. Received Feb. 13, 1970. ApprovedMay 12, 1970.2 Research Physicist and Research Soil Scientist (Physics),respectively, US Water Conserv. Lab., Phoenix, Ariz.

In this paper, the analysis of Youngs (1960) is extendedby considering the hydraulic conductivity to be a functionof the average water level in a bundle of capillary tubes ofvarious radii. The resulting equations improve the predic-tion of drainage over the entire time period for whichexperimental results are available.

DEVELOPMENT OF EQUATIONS

Youngs EquationSince Youngs (1960) analysis forms the background of the

present development a brief review is warranted. The drainageof a fluid of density p from a capillary tube of radius r andlength L is described by Poisseuille's law

dz

where z is the vertical distance, / the time, i\ the viscosity ofthe fluid, and * is the potential difference between the heightz and the bottom of the tube, i.e., $ = pg(z — h). The termh represents the height of liquid left in the tube after infinitetime and g is the acceleration due to gravity. Substitutingpg(z — h) for * in equation [1] and rearranging yields

-dz r*pg (z - h)dt ~ 8r, z [2]

If there are n tubes per unit overall cross section, the initialflux q0 from the bottom of the bundle of tubes is nirpgr*(L — A) / (8 i jL) , the quantity of fluid Q drained at time / isnirr2(L — z), the quantity of fluid Qm drained at infinite timeis mrr2(L — h), and total quantity of fluid R initially held inthe tubes is nirr2L. Integrating equation [2], substituting theabove relations and rearranging gives

e [3]

Expanding the logarithm term in [3] into an infinite series,writing h/L = mrr2h/mrrzL = (R — Qas)/R, and rearrangingwill yield Youngs' equation [5]. The present notation is preferredto facilitate its comparison with equations to be derived here.

The curves in Fig. 1 were calculated using equation [3] forseveral values of h/L, The solid circles in Fig. 1 were obtainedfrom the line drawn through the experimental data points inYoungs' Fig. 2. The vertical bars approximate the range ofvariability of his data. These data are shown in this and somesubsequent figures to facilitate a comparison with theory andexperiment.

The line h/L = 0 shows accumulative drainage from a capil-lary tube sufficiently large that no liquid remains in the tubedue to capillary forces. The curve for h/L = 1 is the limit asthe total outflow becomes negligibly small in relation to thetotal water held in the tube. At this limit equation [3] reducesto

1 - QIQ. = [4]

where r = q0t/Qm. Equation [4] has become known as Youngs'equation. As Fig. 1 and the figures in Youngs' paper show,

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SOIL SCI. SOC. AMER. PROC., VOL. 34, 19701,0

Fig. 1—Outflow (Q/OOQ) versus T (T = q0 i/Qoo) for severalvalues of h/L calculated using equation [4], The symbolsindicate the range of experimental data reported by Youngs[I960].

equation [4] predicts experimental data quite well up to2/Goo = 0.6, or T= 1. For T> 1 the curve lies above thedata points, and therefore does not adequately predict drainagefor long times. Equation [4] was used by Vachaud [1968] inhis summation of contributions from bundles of tubes of twoor more radii.

The assumption that h/L == 1, necessary to obtain equation[4], is physically unrealistic. Youngs stressed the limitationsof a simple capillary tube model but pointed out the useful-ness of a simple expression such as [4]. By making additionalassumptions, some of which may also be physically unrealistic,equations are derived below that predict drainage from soilcolumns better than equation [4].

Conductivity Variable

Assume that a soil column can be represented by a largebundle of capillary tubes having a wide range of radii. Tubeshaving large radii will drain very rapidly while smaller tubesmay drain for long periods of time. We define / to be the sumof the areas of all tubes from which drainage occurs, dividedby the macroscopic area A. The average height of capillaryrise is h and the average water level is z in a column of lengthL. As the column drains, the effective conductivity of the entirecolumn is some average of the conductivities of all the capil-lary tubes. Initially the effective conductivity is high due totubes of large radii, but becomes quite low when the largetubes are drained. For this system, equation [2] can be written

(z - h) [5]

where Ke is the effective conductivity of the column. Theeffective conductivity Ke is composed of the variable conduc-tivity K ( z ) in the region L — h, and the constant conductivityK0 in the region h to O.

Linear Conductivity FunctionWe assume that the conductivity in the region L — h is initi-

ally K0 and falls linearly to zero as z falls to h, i.e. K(z) =K0(z — h)/(L — h). Adding the ratios of the length to the con-ductivity in the two regions (L/Ke — (L — h ) / K ( z ) + h/K0),and solving for the effective conductivity yields,

Fig. 2—Outflow versus T calculated using equation [7], Thesymbols indicate the range of experimental data reported byYoungs [I960].

Ke = (LLK0(z - h)

+ h(z - h) ' [6]

Substituting [6] in [5], integrating, and rearranging yields

[7]

Curves computed from equation [7] for several values ofh/L are shown in Fig. 2. For h/L = 0 or 1, equation [7]reduces to equation [4], Youngs' equation. The minimum curveis for h/L = 0.5. For 0.25 < h/L < 0.75 the curves are not verydifferent from the curve for h/L =: 0.5. The symbols represent-ing experimental data of Youngs are best predicted by thecurve for h/L = 0.5. Although equation [7] predicts the databetter than equation [4], it predicts values of 2/Goo greaterthan the data indicate for T > 2.

Quadratic Conductivity Function

We now assume that the conductivity in the region L — /;is initially K0 and decreases as a quadratic function of z,becoming 0 when z = h, i.e., K ( z ) = K0(z — /02/(L — /O2.The effective conductivity is then

Ke = LK0(z-[8](L - h)3 + h(z ~ h)2

Substituting [8] in [5], integrating and rearranging yields

A~L~ + 2Q/Q -

(A)' hi (1 - Q/QJ = r. [9]

JACKSON & WHISLER: VERTICAL NONSTEADY-STATE DRAINAGE OF SOIL COLUMNSi.o

717

Fig. 3—Outflow versus T calculated using equation [9], Thesymbols indicate the range of experimental data reported byYoungs [I960].

o GE 13-FINE SAND

• PULLMAN CLAY

o POUDRE UNIFORM SAND

R I V E R SAND

VOLCANIC SAND

17 SAND

ADELANTO LOAM

Fig. 4—Outflow versus r calculated using equations [4], [7],[9] and [10]. The symbols represent data from seven porousmaterials.

When h/L — 1 equation [9] reduces to [4], Youngs' equation.When h/L — 0, [9] reduces to

Q'Q. [10]

a result obtained statistically by the second author by findingthe best fit curve for drainage data from laboratory soilcolumns. It was this statistical result that led to the presenttheoretical development.

Curves calculated using equation [9] for various values ofh/L are shown in Fig. 3. For 0 < h/L < 0.7 the curves fallwithin a rather narrow region which can be approximated withequation [10]. The curves for h/L = 0.9 and 0.95 encompassthe experimental data quite well. Thus, for the coarse sandreported by Youngs [1960], equation [9], with h/L = 0.95,adequately predicts drainage for the entire time of drainagefor which experimental data were reported.

FURTHER COMPARISONS WITHEXPERIMENTAL DATA

Figure 4 shows experimental data for seven porousmaterials along with theoretical curves obtained using equa-tions [4], [7] (with h/L = 0.5), [9] (with h/L = 0.95), and[10]. Pertinent experimental parameters for the sevenporous materials are given in Table 1. A comparison of thetheoretical curves in Fig. 4 shows that equation [9] couldbest be used to predict drainage from the seven materials

Table 1—Pertinent experimental parameters for theseven porous materials

Porous material

Ge 13 fine sand*Pullman claytPoudre uniform sandtRiver sand*Volcanic sand*No. 17 sandtAdelanto loam§

KO(cm/day)

108698

2.677400700

1,1525

It(cm/day)

72.9512

1,9633246189344.6

Q.(cm')

20.643.255

11.332.0932.1733.65

5. 158

L(cm)126

28.543.5126136180170

h(cm)417.6

11.624163414

h/L

0.330.270.270.190.120.190.08

From M.E. Jensen personal communication.t From Corey et al. [1965].t From Whlsler and Watson [19681.5 From Van Bavel et al. [1968], Brust et al. [19681 and Bouwer [1969].

if a value of h/L = 0.95 were used. Interpreting h to bethe height of the equilibrium free water surface in a soilcolumn allowed to drain freely at its base (approximatelythe "air-entry value"), values of h/L for the seven porousmaterials were calculated and are given in Table 1. Thecalculated values of h/L range from 0.08 to 0.33 for theseven materials. Using these values of h/L, equation [9]would yield a group of curves all lying very close to theline for equation [10], which, for these materials is not animprovement over equation [4], Using these values of h/Lin equation [7] would produce a group of curves lyingbetween that for equation [4] and [7] (for h/L = 0.5).These curves would predict the data better than [4] or [10]but not as good as equation [9] if higher values of h/Lwere used.

In the theoretical development, h was assumed to be theaverage height of water held in the tubes due to capillaryforces. This height can be considerably greater than themeasured height of the "capillary fringe" in a soil column,especially if this height is estimated by the air-entry value.For this reason, the values of h/L given in Table 1 maynot be applicable to the capillary model upon which equa-tions [7] and [9] are based. Furthermore, there does notappear to be any systematic relationship between calcu-lated values of h/L and the position of the data pointsshown in Fig. 4. This becomes more evident when Youngs'[1960] Fig. 2 is examined. He presents data for waterdraining from a coarse sand for seven column lengths. Thecalculated values of h/L range from 0.15 to about 0.85,yet all data are grouped in a narrow region well withinexperimental error. Data presented by Vachaud [1968] inhis Table 1 for two column lengths when plotted as QIQ^versus T also do not appear to depend on the value of h/L.

SIMPLIFICATION OF EQUATION [9]

Since the parameter h/L does not appear to have a sig-nificant physical meaning, an attempt to simplify equation

718 SOIL SCI. SOC. AMER. PROC., VOL. 34, 1970

1.0

.8

.6

OO.4

.2

I T

Fig. 5—Outflow versus T calculated using equation [11]. Theupper and lover curves result from multiplying the constantin the first term in [11] by V2 and 2, respectively. The sym-bols indicate the range of experimental data reported byYoungs [I960].

[9] is warranted. As discussed above, equation [9] pre-dicts experimental data quite well if h/L = 0.95. At thisvalue of h/L the coefficient of the first term on the left,(1 — h/L)2, is about one-tenth of the coefficient of thesecond term on the left, 1/2 (h/L) (1 — h/L), and thevariable factor in the second term is always larger thanthe factor in the first term. Therefore, the first term willbe neglected. The factors within the bracket of the secondterm can be arranged to be

Q/Q(i - Q/X

Q^ W + a - e/ej2] - i} •The value of the factors within the braces range from 3when Q/QX = O to 1 when Q/Q^ = 1. The entire termis negligibly small for Q/Q^ < 0.5. For Q/QX > 0.5, thevalue of the factors within the parentheses ranges from1.5 to 1. Taking an average value of this factor to be 1.3and setting h/L = 0.95, we obtain

0.03 2/2M

(i - e/e32 " °'9 ln(1 " Q/QJ = T [11]

which is shown as the center line in Fig. 5. Equation [11]is considerably simpler than equation [7] and [9] and hasonly one more term than equation [4]. The line calculatedusing [11] best represents all of the data shown in Fig. 4,and as shown in Fig. 5 predicts the data given in Youngs'[1960] Fig. 2 very well. The upper and lower curves shown

in Fig. 5 were calculated assuming that the constant 0.03in the first term in [11] was in error by a factor of 1/2 and2, respectively. For the upper curve the constant is 0.015and for the lower, 0.06. Almost all of the data points inYoungs' Fig. 2 and large number of the points shown inour Fig. 4 lie within the region bounded by these curves.

It is concluded from this analysis that equation [11] isuseful for predicting the outflow from a freely drainingcolumn of porous material as a function of time. Therequired parameters are the total outflow Q^ and the ini-tial flux q0. The initial flux q0 can be estimated if the initialconductivity K0 and the length of the column is known. Itcan be obtained at the start of an experiment by measure-ment. The total outflow Qm can be estimated by graphicalintegration of a water retention curve. If the initial flux qais measured at the beginning of an experiment and theaccumulated outflow is measured at some time t, Q^ canbe computed using equation [11].