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 unsaturated
flow. 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, ColoradoState 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 = — <t>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 dpa
RT 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
<bM»=_d_r_>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, 1972
Infiltration roteIO"4 cm/s
.2 .3
Fig. 1—Effective permeabilities of water and air.
.9 10 S«WaterSaturation
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 CONSIDERED
Initial ConditionsInitially, 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 forinfiltration. 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/s
3OOOO 35000 t sec
Hyetograph- lOh-roin
356OO t secFig. 2—Hyetograph 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 Characteristic
The hyetograph selected for this study is one of relativelylow 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 Characteristics
Idealized 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 = 1000.892
VAC PHUC & MOREL-SEYTOUX: EFFECT OF SOIL AIR MOVEMENT AND COMPRESSIBILITY 239
.5 .6 .7 .8 .9 I. S«.
Sar .'Rttiouol air saturationDepth! 495 cm10 h - rainfatl
10s PATM
Fig. 3—Infiltration process in the case of an impervious lowerboundary.
where SWT is residual (5) water saturation and Sar is residual(5) air saturation. For drying the same equation was used,thus hysteresis is not considered except for a nominal thresholdvalue. The porosity of the medium was 10%. This ratheruntypical value was used to reduce computational costs.
RESULTS
In all cases to be discussed the hyetograph is the oneshown in Fig. 2. This figure shows the infiltration rateresulting from the rainfall episode for a medium of finiteextent, 495-cm deep. The infiltration rate equals the rain-fall rate for the first 1,550 sec. Beyond that time there isponding at the surface. In spite of some uncertainty due tothe visible scattering in the calculated points, the curveclearly dips below the saturated hydraulic conductivity("gravity flow"). All previous studies with a one-phase flowmodel, e.g., (8, 13), have shown the gravity flow as thelower limit for infiltration rate, even in the case of a soillayer underlain by an impermeable stratum or a watertable. Note that at the end of the infiltration period (time35,981 sec on Fig. 3) the wetting front has not reached thelower boundary.
Starting from the initial water saturation, the soil at thetop will wet. When water infiltrates the soil, the air in thesoil pores is pushed aside to make room for the water. Witha lower no-flow boundary, air can only escape at the top,but its flow is impeded by water entering the soil. Air pres-sure inside the medium builds up as the soil wets, as shownby the profiles of air pressure in the medium at varioustimes of Fig. 4.
49Zcm
Wetting -front
1.05 «10° 1.1 x 10s—Po
pressureCGS
Wetting -front
1/19 secondsL33£67 seconds
Fig. 4—Air pressure curves during infiltration with a no-flowlower boundary.
I06 PATM 105x10*———I————r
UxlO«
wetting -front
wetting''front
wetting-front
air pressureCGS
PATM = 1,012328. CGS
200
300
400
Zcm
Fig. 5—Air pressure curves during infiltration in a semi-infinitemedium.
It is interesting to see that the air pressure buildup ispresent even for a semi-infinite medium (Fig. 5). It is notas significant as the one shown on Fig. 4, but it will never-theless help to slow down infiltration. It must be remem-bered that the entry pressure seldom exceeds 10% of anatmosphere.
240 SOIL SCI. SOC. AMER. PROC., VOL. 36, 1972
Infiltration rote (units' I0~4cm/s)
-Ponding occurs 1800s ——— Semi-infinite medium
----- Impervajs lower boundary Depth =495 cm
5000 IQOOO 15,000 20POO 25pOO 3QOOO 35QOO t sec
Fig. 6—Comparison of infiltration rates for a semi-infinitemedium and a finite medium.
Figure 6 shows a comparison of infiltration curves forthe cases of a semi-infinite medium and a 495-cm-deepmedium. For the semi-infinite medium, ponding occursafter 1,800 seconds. It occurred at 1,550 sec for the 495-cm-deep one.
In the unsaturated (one-phase) flow theory two physicaleffects which both tend to reduce infiltration are neglected,namely, the viscous resistance to airflow and the air com-pression ahead of the wetting front. Thus the two-phase flowtheory, which includes these two effects, will necessarilypredict lower infiltration rates than the unsaturated flowtheory. So the infiltration curve for the semi-infinite me-dium calculated by the two-phase flow theory must, ofnecessity, lie below the curve that would be predicted bythe unsaturated (one-phase) flow theory. In the case of asemi-infinite medium, the difference between the twocurves is probably negligible in many instances becausethere is relatively little air compression (Fig. 5).
In the case of a bounded medium, on the other hand,the air compression is substantial. Because this effect isneglected in the unsaturated flow theory, the predicted in-fi l trat ion rates by this theory will be the same, whether themedium is semi-infinite or bounded, until the wetting frontreaches the lower boundary. In the numerical experimentsreported in this paper the wetting front never reached thelower boundary during the infiltration stage. On the otherhand, the infiltration curve predicted by the two-phase flowtheory is much affected by the air compression. Thus it canbe said that the difference between the two curves of Fig. 6estimates the error committed by predicting infiltrationrates for bounded media using the unsaturated flow theory.For the case studied, the error in the cumulative infiltrationat the end of the rainfall will be of the order of 12% (20.2cm vs. 16.9 cm). The error in the late infiltration rates is ofthe order of 20%.
Figure 7 shows an enlargement of Fig. 3 for the upperrange of saturations. It shows that between the times 1,556sec and 10,031 sec, air starts escaping through the surfaceand continues to do so until the end of infiltration, eventhough there is ponded water on the surface. From Fig. 7,it is seen that the grids (Az = 15 cm except for the firstgrid for which Az — 7.5 cm) wet behind the wetting front,nearly reach saturation (except for residual air), and thendry slightly. This is explained by the fact that air pressureincreases with the quantity of water infiltrated into theclosed-bottom soil. When the air gradient becomes suffi-cient, it will push aside the water in the soil pores behind
912 10
Sor
Residual air saturation
• 1556sA 10031sO 20092s6 35981sImpervious lower boundaryDepth 495 cm
Fig. 7—Enlargement of Fig. 3 showing effect of air escapingthrough the surface during infiltration process.
the wetting front to make a path to escape at the top. Thisis in qualitative agreement with results of experiments (14).
Figures 3 and 7 show examples of the water saturationprofiles during the infiltration stage. In all cases4 the soilnever reaches the fully saturated (except for residual air)value of 0.912. This is true even in the case of a semi-infinite medium.4
CONCLUSION
The use of the two-phase flow approach to infiltrationproblems yields interesting results. Due to the presence ofthe flow of air, the soil is no longer fully saturated, as itappears to be in the one-phase flow approach.
In the case of a semi-infinite medium, the unsaturatedflow approach still gives a quicker way to find an accept-able solution for infiltration, because in this case the influ-ence of the air movement and compressibility is not great.But, with an impervious lower boundary or a water table,a noticeable difference in the total amount of infiltratedwater occurs. The saturation profiles near the surface indi-cate a clear shift toward a drier state to allow air to escape,thereby reducing the capacity of the soil to imbibe water.
1 Ibid.
REICHARDT ET AL.: SCALING OF HORIZONTAL INFILTRATION INTO HOMOGENEOUS SOILS 241
