WATER FLOW IN SOIL MACROPORES 11. A COMBINED FLOW MODEL

K. BEVEN’ and P. GERMANN’

(Institute of Hydrology, Wallingford, Oxon OX1 0 8BB)

Summary This paper presents a one-dimensional model of bulk flow in a combined

micropore/macropore system, which has been developed as a result of the experimental work described in Part I. The problems posed by the presence of macropores to model development and validation are discussed and one exploratory model formulation is described. The results of several simulations are presented and used to demonstrate the effect of macropores on infiltration rates in soils of different hydraulic conductivity.

Introduction PART I of this paper (Germann and Beven, 1981) reviewed the need for more information on the importance of macropores in hydrology and presented the results of laboratory experiments on large undisturbed cores of two soils. The results demonstrated the effects of different macropore structures on the processes of water flow through the soil. This paper aims to provide a model for bulk flow through a combined micropore/macropore system that utilizes a minimum of information about the complexities of water flow through the soil, much of which is obtainable from the experiments described in Part I. The complexity of the model is in keeping with the current state of knowledge of the hydrological role of macropores. The model is used to demonstrate how macropores may affect infiltration, redistribution and runoff of water under different circumstances.

Any approach to modelling water flow in field soils cannot reject approaches based on Darcy’s Law and the Richards (1 93 1) equation because, although Darcy’s Law is itself an empirical approximation to the complexities of flow, such models have been well proven in the past (e.g., Haverkamp et al., 1977; Bresler et al., 1979). Thus any model of a combined micropore/macropore system should reduce to a Darcy-type model when there are no macropores. This suggests the introduction of a domain concept in modelling a combined system, with the micropores as one domain that conforms to hydraulic principles based on Darcy’s law. The next level of model complexity is to introduce a second domain representing the bulk flow in macropores, and to allow interaction between the two domains in some physically realistic manner. There is some justification in

‘Now at Department of Environmental Sciences, University of Virginia, Charlottesville, Virginia 22903 U.S.A. 2 0 n study leave from Laboratories of Hydraulics, Hydrology and Glaciology (VAW) annexed to the Federal Institute of Technology (ETH), CH-8092 Zurich

Journal of Soil Science, 1981, 32, 15-29

16 K. BEVEN AND P. GERMANN

the use of such a domain concept from experimental evidence (as discussed in Beven and Germann, 1980; Germann and Beven, 1981).

The use of a domain concept to model water flow in a combined micropore/macropore system was suggested at least as long ago as Muskat (1946) in a study of saturated flow through fissured limestone. Philip (1968) used a domain concept to analyze flow in aggregated media, in which both domains were subject to capillarity effects. More recently Scotter (1978) has produced a model for solute flows in a saturated macropore system and Kutilek and Novak (1976) have examined the effect of saturated macropores on infiltration into unsaturated soil. The model of Edwards et al. (1979) comes closest to the approach adopted in the present work. They restricted their analysis to flow in a cylindrical column around a single pore. Flow in the micropores is treated in two dimensions (vertically and radially away from the macropore) using a solution of the Richards equation. Flow in the macropores is modelled using a simple accounting procedure for given depth increments with input at the surface and infiltration losses to the micropores at the side of the macropore. Velocities of flow in the macropore are assumed to be non-limiting and surface runoff is assumed to occur only after the macropores fill to the surface. Numerical experiments with this model clearly demonstrate the importance of macropores on infiltration and runoff, at least under conditions of heavy rainfall.

However a more general model is required to simulate the effect of macropores at scales larger than a single pore. The results from such a model should integrate the effect of a realistic distribution of pores at given depths in the profile of a field soil. A first attempt at such a model is described below.

Spatial averaging in a combined microporelmacropore system We shall treat both micropore and macropore domains in terms of a

macroscopic continuum approach, in which the volumetric water contents in both are characterized as spatial averages over some finite volume of soil around a point of interest, P. The size of this volume, the representative elementary volume (REV; see for example Bear, 1972), should be sufficiently large to integrate the effects of the microscopic changes in individual pores, but much smaller than the flow domain of interest, so that the representation of conditions at P is meaningful (Fig. 1). For a micropore system without macropores the REV may be relatively small for broadly homogeneous soil. This volume would usually, for instance, be within the scale of the spatial sampling range of a neutron probe.

For a combined micropore/macropore system, because of the greater spacing between the larger pores, a suitable REV will be much large] (perhaps 1 to 10 m2 in area, with a depth corresponding to changes in the distribution of the macropores in the soil profile) to obtain a spatial average that is statistically characteristic of the soil around a point. It will therefore be much more difficult to measure the average characteristics and nature of the response of the macropore system than of a pure micropore system, particularly as the macropores will generally constitute only a small volume of the soil (perhaps in the rage of 1 to 5 per cent). Because of the interaction between the macropore and micropore domains, the REV for

=. In

0 (1

0 Q

0

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0

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Q

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r"

0

WATER FLOW IN SOIL MACROPORES 11.

2

e U >

0 Q

Domain of 5 Domain of homogenous microscopic I 'porous medium' variability I

I

-- 16-1 i b o 1b1 162 6 3

Approximate space scale (cm)

17

FIG. 1. The variation of porosity with spatial scale and the definition of representative elementary volume.

the micropores in a combined model should then be of the same order as that of the macropores.

If predictions are therefore made at the scale of the REV for the macropores, it will not be easy to validate those predictions by measurements in the field. One response to this is to restrict the scale of interest to that corresponding to the REV for the micropore system, with single macropores, then imposing special boundary conditions for a micropore solution. This is the approach adopted by Edwards et al. (1979) and may be suitable for the simulation of single soil profiles.

If, on the other hand, the scale of interest is of a hillslope or catchment, then the REV characteristic of the macropores, with the area of a small plot, remains relatively small. The construction of a model on the basis of the macropore REV may then be worthwhile and may be expected to generate the development of measurement techniques suitable for model calibration and validation. This is the approach developed, on an exploratory basis, in this paper.

18 K. BEVEN AND P. GERMANN

Flow equations for the micropore system The development of the flow equation for the micropore system follows

the usual formulation for a one-dimensional vertical system, combining a continuity equation:

where Omi(cm3 cm-') is the volumetric water content in the micropores, Qmi(cm s-') is the volume flux density in the micropores, S(cm3 s-' ~ m - ~ ) is an exchange flow between micropores and macro-

pores treated as a source/sink term, z(cm) is elevation above an arbitrary datum, and t(s) is time.

where K(cm s-') is the hydraulic conductivity function and 4J(cm) is the total hydraulic potential (4 = I,$ + z , where I,$ is the capillary potential in the micropores). Substituting (2) into (1)

or

where D(cm2s-') is the diffusivity function. In the present model (3) is solved by an implicit finite difference scheme for a number of points in the profile spaced at depth increments Az. The model assumes single valued (non-hysteretic) functions for D (6,;) and K(Omi).

Flow equations for the macropore system In the macropore system, it is assumed that unless all the pores are

saturated, the ambient pressures are atmospheric throughout. For the one-dimensional representation of the present model it is further assumed that, below the level of saturation in the macropores, the pressures are hydrostatic above the lower limit of the macropores. This allows a simple separation of the macropore flow model into unsaturated and saturated zones but implies that all the macropores are sufficiently interconnected to minimize the effects of restrictions on flow. No attempt is made in this exploratory model to take account specifically of the effects of 'necking' (Bouma et al., 1977) or of air pressure effects (Linden and Dixon, 1976) on flow rates. In addition, the macropore structure is assumed to be constant over time.

WATER Flow IN SOIL MACROPORES 11. 19

The macropore system must also conform to the general continuity equation

where Om,(cm3 cm-’) is the volumetric water content in the macropores, Qma(cm s-’) is the volume flux density in the macropores, and

S(cm3 s-’ cm-’) is an exchange term corresponding to that of Equation (1).

There is little information available on the nature of flow in macropores but it seems reasonable to assume that in general

The complexities of the macropore network are such that it is impossible to specify the relationship on theoretical grounds alone. The approach here is to make some assumptions about the nature of flow in the macropores in order to specify the form of Equation 5 . Given that Q,, = 0 when Om, = 0, a measurement of fully saturated flow through the macropore system will provide, to a first approximation, an upper limit on flow rates and can be used as a scaling factor to complete the specification of ( 5 ) . It is expected, however, that the precise form of ( 5 ) will not be essential to building a useful model of the joint macropore/micropore system since, assuming that periods of flow in the macropores will be relatively short, it will he more important to predict the volume of water supplied to and lost from the macropores correctly.

It is assumed that an approximate pore size distribution can be specified for the macropores. This distribution may be made up of a collection of N class sizes of circular pores containing n, pores, each of average radius r, , i = 1 . . . N , and M class sizes of cracks containing m, cracks of average width D, and length L,, j = 1 . . . M. More complex shapes can be simplified or decomposed into these two types (Dolezal, 1976a, b) . We shall further assume that flow in the macropores remains laminar and that as the pores become wet and then dry there is always an equal depth of water flowing on the walls of all size classes unless a given class is saturated. Thus, during wetting, the largest pores will saturate last.

Fully saturated laminar flow in vertical cylindrical pores and straight-sided cracks has been treated by Childs (1969). We can follow a similar development for partially saturated flow. For volume flux, qp(cm’ s-’) in a vertical annular cylinder of inner radius r , in a pore of radius R (Fig. 2)

r4 - r2R2 + r4 In R - r4 In

where g(cm s-’) is the acceleration due to gravity, p(g cm-’) is the density of the water and p(g cm s-’) is the dynamic viscosity of the water. Equation (6) reduces to Childs’ formula when r = 0.

20 K. BEVEN AND P. GERMANN R

FIG. 2. Definition diagram for film flow in a cylindrical pore; R is radius of the pore, r is radius of the air filled cylinder.

For the case of volume flux, qc, in a partially saturated crack of width, D, and length L with a symmetrical depth of flow, d , on each side (Fig. 3)

4 s 3 2 P

qc = - L - d 3 (71

which reduces to Childs' formula for d = D / 2 . A final relationship between flow and water content in the macropores may be defined as follows:

FIG. 3. Definition diagram of film flow in a planar crack; D is width of the crack, d is thickness of the water film, L is length of the crack.

WATER FLOW IN SOIL MACROPORES 11. 21

where Q(cm s-I) is a volume flux density and remembering that once pores of a given size are saturated (i.e. rj = 0, or dj - Ill,*) they make no further contribution to Q or Omo. The specification of ( 5 ) IS then completed by the scaling

where K,, (cm s-I) is a measured saturated hydraulic conductivity and Q,,, is the fully saturated flow predicted from (8). This scaling will be necessary to take account empirically of tortuosity, form roughness, ‘necking’, dead ended pores and other irregularities in the flow pattern.

As an example, consider a hypothetical porous medium with microporosity of 0.49 and a macroporosity of 0.01. Assume that .the micropores are just saturated, so that the addition of further water will cause the water in the macropores to flow. In this example, interaction between the two domains is ignored and a unit hydraulic gradient is assumed in the

I I I I I I l l 1 0 - ~ 2 3 4 5 6 7 8 9

e,,(cm3 ~ m - ~ )

FIG. 4. Volume flux density for different types of macropores versus their water content. (a) Cylindrical pores, diameter 1.0 cm; (b) cracks 0.5 x 5.0 cm; (c) cylindrical pores, diameter 0.2 an; (d) cracks 0.1 x 1.0 cm; (e) mixture of pores (diameter from 0.2 to 1.0 cm) and

cracks (length from 1.0 to 5.0 cm, width from 0.1 to 0.5 cm).

22 K. BEVEN AND P. GERMANN

micropores so that a steady flow rate is maintained equal to the saturated hydraulic conductivity, say 1 x cm s- ’ . Fig. 4 compares the micropore discharge with predicted volume flux densities, Q, for various arrangements of macropores at different moisture contents, emu, on the basis of (8) and (9) alone with no scaling as suggested above.

These results suggest that a further generalization can be introduced into the model by approximating the curves of Fig. 4 by a function of the form

Q = a(emu)’

so that from (10)

Qmu = K m u a ’ ( ~ m u ) ’

where

a’ = a/Q,,.

Introducing (1 1) into (4) leads to

or

where

This has the form of a kinematic wave equation (Lighthill and Whitham, 1955) with

representing the non-linear kinematic wave velocity. Equation (12) is used in the present model to predict flow in partially saturated macropores, and is solved by an implicit finite difference scheme. Because of the shock front nature of the infiltration of water into the macropores it is necessary to maintain careful control of the time step use in the solution to ensure reasonable accuracy.

When saturated conditions build up from the lower limit of the macropores, a simple accounting procedure is used at each time step to calculate the rise of fall of the water table in the macropores depending on the rate of inflow from the unsaturated part above, the rate of loss to the micropore system and the current storage capacity of the macropores immediately above the water table. When there is no inflow from above,

WATER FLOW IN SOIL MACROPORES 11. 23

and the macropore water table is falling, it is assumed that complete drainage of the macropores takes place.

Interaction between the domains To complete the combined model it is necessary to specify the boundary

conditions and in particular to consider the S term of Equations (1) and (4). This must also be treated as a macroscopic average and it is convenient to assume that the S term is controlled within the micropore system with the form of a Darcy-type flow in a horizontal direction, such that

where A$/Ax is a representative hydraulic gradient and AX is a characteristic length which will reflect the average spacing of the macropores. For different internal boundary conditions, the hydraulic gradient may be taken as

0 7 Oma = 0, e m i < emi..mr (14a) ( * ( e m ; ) - OVAX 7 0 < Oma < ~m,..s,r (1 4b) ( * ( e m ; ) - dw,) /A~, Oma = emusut (1 4c)

where the subscript sat represents fully saturated conditions and d,(cm) is depth below a water table in the macropores. The range of conditions that can be handled by the models is illustrated in Fig. 5 .

The external boundary conditions for the model have been treated as follows for the initial simulations. With reference to Fig. 5 ,

6 . = e . ( 2 m a . L = I - Qmi.L 9 mr.L mizaf

Qmo.L < Qmasm (16b) where I(cm s-') is the rainfall input at the soil surface,

L is the surface node in the finite difference solution, and Az(cm) is the spacing between nodes.

Rainfall in excess of infiltration into both macropores and micropores is assumed to be removed instantaneously as surface runoff. At the lower boundary a condition of constant water content is assumed.

Results of model simulations The first model simulations have been aimed at demonstrating the relative

importance of the micropore and macropore domains in soils of different permeabilities. The soils were assumed to have a constant microporosity of 0.495 and a macroporosity of 0.01. The relative soil characteristics used are

24 1 1””c 1

I > O

I=O

...

(e )

1 1 ” O l 1

1-0

w

FIG. 5 . Different stages of water flow in a macropore. (a) Infiltration into a dry macropore; (b) rising water table in the macropore; (c) completely unsaturated macropore, start of surface runoff; (d) falling water table in the macropore - after precipitation has stopped; (e) water flow from micropores into macropores in the case of a rising overall water table. I is the rainfall rate input at the surface and Of is surface runoff. Solid shading represents water in the macropores, heavy dots saturated micropores and light dots unsaturated

micropores.

t 1

0-0

/ - -/A- 2

I 1 I I I

25

-0.4 Y“

Y ‘ o.2 . 0.0

FIG. 6. Relative soil water properties used by the model; A relative hydraulic conductivity of the micropores (Kmi/Kmi,s,,,) versus soil water tension. 0 relative water content of the

micropores pores (Omi/Omi,so,) versus soil water tension.

shown in Fig. 6 and were kept constant for all simulations. The macropores were assumed to extend to a depth of 100 cm with constant parameters over that depth of K,, = 2.78 x cm s-’, a’ = 1.58 x lo4, b = 3.60, Ax c 10.4 cm. The results of eight runs are reported here, using 4 different values of K(O,,,,Ja,), with and without macropores. The values of K(O ) changed in steps of one order of magnitude from 1.23 x cm s-”l’”t“o 1.23 x cm s-’. A hypothetical rainstorm lasting 266 s with a peak density of 2.5 x cm s-‘ between 25 s and 260 s was used to define the surface boundary conditions.

The results for the soil of the lowest hydraulic conductivity are shown in Fig. 7. Fig. 7b shows the run without macropores demonstrating the limited depth of infiltration over a period of infiltration. Fig. 7a shows the run with macropores, showing how the macropores rapidly fill to the surface and allow additional infiltration into the micropores at depth in the profile. In contrast, Fig. 8 shows the results for the soil of highest conductivity: although the micropores become saturated at the surface and there is a small amount of infiltration into the macropores, this is rapidly lost into the sur- rounding micropores and the effect on the evolution of the total soil mois- ture profile is minimal.

The results of all the runs are summarized in Fig. 9 in terms of cumulative infiltration. This also shows how the presence of macropores will generally increase the overall losses to infiltration. However, it is interesting to note that the greatest effect is for those soils with intermediate hydraulic conductivities. This is because the total available storage in the macropores is relatively small and once they are saturated, further infiltration into the macropores is governed by the rate of loss into the surrounding micropores. This type of behaviour may be expected to hold in general but the relative magnitude of the macropore effect will vary with the specific nature of the

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WATER FLOW IN SOIL MACROPORES I1

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Time (s) FIG. 9. Cumulative precipitation and infiltration versus time; + precipitation. Hydraulic conductivity at saturation Kmi,so, (cm s-'): V I 1.23 x AA 1.23 x 1.23 x 0. 1.23 x lo-*. Open symbols: without macropores, closed symbols: with

macropores.

macropore system (depth, macroporosity, variability with depth) and the boundary conditions imposed at the surface.

The model is necessarily simplistic and exploratory and with present measurement techniques cannot easily be validated in the field. However, it is hoped that this type of approach will stimulate developments in considering the effects of spatially variable discontinuities, such as macropores, on the overall soil hydrology at a site.

Acknowledgments We have had stimulating discussions about soil macropores with a number

of people, but it was Steve Trudgill who planted the idea of modelling bulk flows through the soil. P. G. thanks the Director of the Institute of Hydrology for making the facilities of the Institute available during his year of study leave as well as the Swiss National Foundation for Scientific Research.

REFERENCES BEAR, J. 1972. Dynamics offluids in porous media. New York: Elsevier. BEVEN, K., and GERMANN, P. 1980. The role of macropores in the hydrology of field soils.

Institute of Hydrology, Report Number 69, Wallingford, United Kingdom. BOUMA, J., JONGERIUS, A., BOERSMA, 0.. JAGER, A., and SCHOONDERBEEK, D.

1977. The function of different types of macropores during saturated flow through four swelling soil horizons. Soil Science Society America Journal 41, 945-950.

WATER FLOW IN SOIL MACROPORES 11. 29

BRESLER, E., BIELORIA, H., and LAUJER, A. 1979. Field test of solution flow models in a heterogeneous irrigated cropped soil. Water Resources Research 15, 465-652.

CHILDS, E. C. 1969. An introduction 10 the physical basis of soil water phenomena. New York: Wiley.

DOLEZAL, F. 19760. The hydraulic efficiency of soil macropores. Proceedings of the Bratislava Symposium on Water in heavy soils (eds Kutilek, M. and Sutor, J.), Vol. I.,

DOLEZAL, F. 19766. The measurement of hydrostatic and hydrodynamic parameters of a swelling soil. Proceedings of the Bratislava Symposium on Water in heavy soils (eds Kutilek, M. and Sutor, J.), Vol. I, 80-90.

EDWARDS, W. M., VAN DER PLOEG, R. R. and EHLERS, W. 1979. A numerical study of the effects of non-capillary-sized pores upon infiltration. Soil Science Society America Journal 43, 851-856.

GERMANN, P. and BEVEN, K. 1981. Water4low in soil macrowres. I. An exwrimental

185- 195.

approach. Journal of Soil Science 32, 1-13. HAVERKAMP. R.. VAUCLIN. M.. TOURNA. J.. WIERENGA. P. J.. and VACHAUD. G..

1977. A compa'rison of numerical simulation models for one-dimensional infiltration. 'Soil Science Society America Journal 41, 285-294.

KUTILEK, M. and NOVAK, V. 1976. The influence of soil cracks upon infiltration and ponding time. Proceedings of the Bratislava Symposium on Water in heavy soils (eds Kutilek, M. and Sutor, J.), Voi. I, 126-134.

LIGHTHILL, M. J. and WHITHAM, G. B. 1955. On kinematic waves, I. Flood movement in long rivers. Proceedings of the Royal Society, London, 229 Series A, 229-281.

LINDEN, D. R. and DIXON, R. M. 1976. Soil air pressure effects on route and rate of infiltration. Soil Science Society America Journal 40, 963-965.

MUSKAT, M. 1946. The flow of homogeneous fluid through porous media. Ann Arbor, Michigan: J. W. Edwards, Inc.

PHILIP, J. R. 1968. The theory of absorption in aggregated media, Australian Journal of Soil Research 6, 1-20.

RICHARDS, L. A. 1931. Capillary conduction of liquids through porous medium. Physics 1,

SCOTTER, D. R. 1978. Preferential solute movement through larger soil voids. I. Some 318-333.

computations using simple theory. Australian Journal of Soil Research 16, 257-267.

(Received 30 November 1979)