# Effect of Soil Air Movement and Compressibility on Infiltration Rates1

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Effect of Soil Air Movement and Compressibility on Infiltration Rates1

LE VAN PHUC AND H. J. MoREL-SEYioux2

ABSTRACTThe purpose of the study was to determine the hydrologic

significance of the soil air flow and of its compressibility in theprocess of infiltration and moisture redistribution. Results of anumerical solution of the equations describing the simultaneousflow of air and water in a soil are presented. The results showthat the air flow and compressibility may affect significantly theinfiltration rates and the saturation profiles.

Additional Index Words: water infiltration, two-phase flow,saturation profiles, redistribution.

E LECTRONIC COMPUTERS have stimulated a great deal ofinterest in numerical solutions to unsteady unsaturatedflow. The application of unsaturated flow theory, devel-oped by soil scientists, to infiltration (11) has providedvaluable information to the hydrologist.

However, the work with unsaturated flow which has beenapplied to hydrology to date has been based on the assump-tion that air in the soil offered no dynamic resistance toflow. The assumption that there is only a negligible resist-ance to infiltration due to air flow has resulted in a relativelack of attention paid to the presence of air in the soil incases when it may have a significant effect on infiltration.

Laboratory work on infiltration has shown that air indeedhas a significant effect especially when a permeable soil isunderlain by a relatively impermeable layer and there is nolateral passage to permit the air to escape (2, 15). Results(1) for a two-phase flow treatment of infiltration, using asimplified flow model, show that the infiltration rate aftera certain time is well below the saturated hydraulic conduc-tivity which was considered a lower limit by all the previ-ous authors. A moisture profile cannot be obtained by thismethod (1), but it is clear that the profile must differ fromthe one-phase flow profile especially in the upper partwhere air is flowing out and eventually bubbling out whenponding occurs.

Two-phase flow problems are commonly solved in thepetroleum literature by finite difference techniques (6, 7,9). The particular finite difference technique used in thisstudy differs only in minor ways from the methods cur-rently used by the petroleum industry (3,4).

This study is in the one-dimensional space. Air is in-cluded as a second phase, with the assumption that it can-not escape laterally. Hysteresis effects are neglectedthroughout. A separate paper (10) will report the effect ofhysteresis on infiltration.

1 Contribution from Dep. of Civil Engineering, Colorado

State Univ., Fort Collins 80521. This work was supported bythe Dep. of Interior, Office of Water Resources Research,under OWRR B-033-Colorado. Received Aug. 20, 1971. Ap-proved Aug. 31, 1971.2

M.S. graduate of Colorado State Univ., Dep. of Civil En-gineering, and Associate Professor of Civil Engineering, respec-tively.

TWO-PHASE FLOW EQUATIONSContinuity equations for the two phases are

d(qw)/dz = OSw/dt)

and

[1]

[2]

where z is the vertical coordinate oriented positive down-ward, q is velocity, 0 is porosity, S is saturation, t is time andp is specific mass. The subscripts a and w refer to air andwater, respectively. Water has been assumed incompressibleand therefore its specific mass does not appear in equation[1]. The velocities (in the Darcy sense) are related to pressureby Darcy's equations:

_ , r w f p w .= k (- pwg)

w OZ

ra ,pa . ( - Pa8)pa OZ

[3]

[4]

where k is intrinsic permeability, kri is relative permeability ofphase i, pi is pressure of phase i, and g is acceleration of grav-ity. Equation [3] can be rewritten equivalently:

^ dz Sz [5]

where pc = capillary pressure = pa pw a known functionof Sw. Equation [2] can be expanded and modified, using theperfect gas law of an isothermal process (i.e., taking pa = paRT)and also utilizing the relation of Sw + Sa = 1. As a result,equation [2] takes the form:

(1-SJ dpaRT dt dz ' Pa dz

where R is the perfect gas constant. Altogether there are fiveequations (equations [1], [4], [5], [6], and the perfect gaslaw) in the two fundamental unknowns Sw and pa; and threeintermediate unknowns, pa, qa and qw. Of course, the threeunknowns, pa, qa, and qw, can be eliminated to obtain twoequations in the two unknowns, Sw and pa. However, thisis not a good procedure for obtaining the finite differenceapproximations.

NUMERICAL FINITE-DIFFERENCE PROCEDUREThe fundamental equations are equation [6] and

dt dz L [7]

The velocities qa and qw are obtained from equations [4] and[5]. Equation [7] was obtained by substituting equation [5]into equation [1].

Equations [6] and [7] are nonlinear parabolic differential237

238 SOIL SCI. SOC. AMER. PROC., VOL. 36, 1972Infiltration rote

IO"4 cm/s

.2 .3

Fig. 1Effective permeabilities of water and air.

.9 10 SWaterSaturation

equations and no exact analytical solutions are known. How-ever, it is possible to approximate their solutions by finitedifference techniques.

The numerical method used to solve the problem above isan implicit (12) scheme for equation [6] and an explicit (12)scheme for equation [7]. The purpose of this paper is not todiscuss the pros and cons of a particular scheme of finite dif-ferencing, but to show that the neglect of the viscous resist-ance of air movement and of the compressibility of air in thegoverning equations, which are presumed to represent thephysical phenomena, will cause inaccurate prediction of infil-tration rates in some instances. For this reason, the reader isreferred to Van Phuc, 1969s, for further details on thenumerical procedures.

PROBLEMS CONSIDEREDInitial Conditions

Initially, the air pressure is assumed atmospheric through-out the medium. In all cases a uniform condition of saturation,with value 0.30, is assumed to exist initially in the soil.

Boundary ConditionsBoundary conditions must be defined at the soil surface and

at the lower boundary of the medium.At the lower boundary, two cases must be distinguished

depending on whether the medium is semi-infinite or of finiteextent. For the semi-infinite case at the lower boundary, watersaturation and air pressure retain their initial values at all times.For the case of a medium of finite extent, the lower boundaryis assumed impervious and the boundary conditions are con-ditions of zero velocity for both water and air.

At the soil surface the boundary conditions are time-depen-dent. They depend on the time and intensity characteristicsof the rainfall hyetograph. If water is available at the soil sur-face due to rainfall or ponding, the soil is assumed temporarilysaturated and the air pressure equals atmospheric pressureplus ponding pressure. Ponding pressure is the pressure of thedepth of water over the soil surface. No runoff of excess wateris permitted. Darcy's equation is then used to calculate thewater velocity at the surface (the infiltration rate). If the cal-culated value exceeds the "feasible" infiltration rate (i.e., cur-rent water availability at the surface/time increment), theboundary condition is changed to the condition that the waterinflux equals the feasible rate and air pressure equals atmo-spheric pressure. If, on the other hand, the calculated value is

3 Van Phuc, Le, 1969. General one-dimensional model for

infiltration. M.S. Thesis, Dep. of Civil Engineering, ColoradoState Univ., Ft. Collins, Colo. 83 p.

R,,= Gravity flow

Impervious lower boundary - Depth 495 cm

Gravity flow

0 5OOO IOOOO

Rainfall rate - I0~4 cm/s3OOOO 35000 t sec

Hyetograph- lOh-roin

356OO t secFig. 2Hyetograph and corresponding infiltration curve.

less than the feasible infiltration, the assumed boundary condi-tions are proper provided that no air escapes upward. If theair pressure immediately below the surface does not exceed thesum: atmospheric pressure plus ponding pressure plus entrypressure (the "threshold" pressure); then the assumed bound-ary conditions were proper and the calculations proceed. If, onthe contrary, that air pressure exceeds the threshold value, avery small desaturation at the surface is assumed and calcula-tions proceed.

In the case of no water availability at the surface (redistri-bution stage), the boundary conditions are the usual condi-tions of zero velocity for water and of atmospheric pressurefor air.

Rainfall Hyetograph CharacteristicThe hyetograph selected for this study is one of relatively

low rainfall intensity (at most twice the soil saturated hydrau-lic conductivity) and long duration (10 hours). All resultsreported in this paper were obtained as responses to this singlehyetograph.

Soil CharacteristicsIdealized soil characteristics were used. The curve of effec-

tive permeability for water (Fig. 1) has a realistic shape.Clearly that for air does not. However, actual curves wouldalways lie below and to the left of the straight line. As a con-sequence, the results of the study provide a conservative esti-mate of the effect of including air presence in the predictionof infiltration rates. With more realistic curves of air permea-bility, the corresponding infiltration rates would be lower thanthe ones presented in this study.

Similarly, an idealized curve of capillary pressure was used.For wetting it was defined by the equation:

.02 3.0 [8]

There is no fundamental justification for this equation ex-cept that it fitted (least square) the data very well. The unitsof pc are c.g.s. units (dynes/cm2).

Sw* is the normalized (5) saturation (expressed in percent)namely:

5* = 100 = 100 0.892

VAC PHUC & MOREL-SEYTOUX: EFFECT OF SOIL AIR MOVEMENT AND COMPRE

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