Infiltration into a class of vertically nonuniform

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1 Hydrological Sciences - Journal - des Sciences Hydrologiques, 29, 4,12/1984 Infiltration into a class of vertically nonuniform soils KEITH BEVEN Institute of Hydrology, Wallingford, Oxfordshire OXIO 8BB, UK ABSTRACT An infiltration model based on the Green-Ampt assumptions is developed for a class of non-uniform soils in which saturated hydraulic conductivity decreases as an exponential function of depth, and the storage-suction factor C = A6Ati> is a constant. An analysis of measured hydraulic conductivity and porosity data suggests that the model may be useful in some soils where the changes of hydraulic characteristics with depth are of this form. A method for using the model with time variable rainfall rates is given. The model is applied to simulate the experimental results of Childs & Bybordi (1969) who measured infiltration into layered sand profiles. Agreement was satisfactory even for this case. Infiltration dans une catégorie de sols non uniforme suivant un profil vertical RESUME Un modèle d'infiltration basé sur les hypothèses de Green-Ampt a été mis au point pour une classe de sols non uniformes dans lesquels la conductivité hydraulique saturée décroît comme une fonction exponentielle de la profondeur. Une analyse des mesures de conductivité hydraulique et des données de porosité suggère que le modèle peut être utile pour certains sols où les variations de caractéristiques hydrauliques avec la profondeur sont de cette forme. On donne une méthode pour employer le modèle pour le cas d'intensités de précipitations variables dans le temps. Le modèle est appliqué pour simuler les résultats expérimentaux de Childs & Bybordi (1969) qui ont mesuré l'infiltration dans des profils avec des couches de sable. L'accord a été satisfaisant même pour ce cas. I PRODUCTION Field soils are non-uniform in space. This paper will consider only non-uniformities in the hydraulic characteristics with depth in the soil profile. Traditionally, non-uniformities with depth have been classified with respect to layers of the soil. In some cases these layers correspond to well defined horizons recognized by the soil taxonomist and may, such as in a well developed podsol, correspond to abrupt changes in soil hydraulic characteristics associated with changes in texture, structure and organic matter. In other cases, horizons may be less well defined and the "layers" may then be more related to the depths at which samples were taken 425

2 426 Keith Beven for analysis in the laboratory than to any distinct pedological units. In the analysis of flow through soils that are non-uniform with depth,it is often assumed that soil hydraulic properties are more homogeneous within horizons or layers than between layers. It will be argued here that a more continuous approach to changes in hydraulic characteristics with depth may be more appropriate in some soils, particularly in the case of near surface properties affecting infiltration. This case is made on the basis that where rainfall or irrigation rates are such that the soil surface is at or close to saturation, then rates of flow will be governed primarily by the largest continuous pores, which may be in the macropore size range (see discussion in Beven & Germann, 1982). These large pores or channels for the infiltrating water may be due to natural or artificial mechanical action (e.g. drying cycles or ploughing) or have biotic origins (e.g. earthworm or root channels). In most cases, except perhaps ploughing and some other agricultural practices, it would not be expected that these processes would lead to relatively homogeneous hydraulic characteristics in the surface layer. It would be more likely that the influence of such processes would decrease with depth in some regular way. A simple analysis of flows in such soils based on kinematic theory has been presented by Beven (1982a,b) using functional relationships for the change of saturated hydraulic conductivity, K s, and porosity, 9 g, with soil depth such as: K s = K Q e fz (la) where z is depth below the soil surface and f and g are coefficients. There are now many numerical schemes for the solution of the equations of flow through unsaturated soils that are non-uniform with depth. Such solutions can take account of any arbitrary variation with depth. Justification for the use of simple functional relationships such as equations (1) is based on a continuing need for computationally inexpensive models that remain good approximations to detailed physically-based models. The infiltration model that is reported here is part of a wider study of water flows at the hillslope and catchment scale (e.g. Beven, 1982a,b). One of the aims of this work is an examination of the effect of spatial heterogeneity on water flows at different scales, a problem necessitating the use of computationally efficient models. This paper provides further evidence for the use of equations (1) and develops analytical theory to describe infiltration into such soils. SOILS DATA Beven (1982b) has demonstrated that relationships of the form of equations (1) could be used to describe the characteristics of 24 soil types of hydraulic conductivity decreasing with depth. The range of parameter values obtained by regression of observed depth

3 Infiltration into vertically non-uniform soils 427 and porosity values is summarized in Table 1. Additional evidence for the utility of equations (1) has been obtained from an examination of the data presented by Holtan et al. (1968) for 177 soil types in the United States. These data have been used in a number of recent studies aimed at providing parameter values for infiltration and soil water flow calculations (e.g. Clapp & Hornberger, 1978; Brakensiek et al., 1981). Soils for which both porosity and vertical saturated hydraulic conductivity data Table 1 Values of parameters of equations (1) obtained from regressions of hydraulic conductivity and porosity with depth K Q f 0 O g Source (m h" 1 ) (n-t 1 ) (rrt 1 ) 0.02 to to to to-0.2 Beven (1982b) Table 3 24 soils 0.01 to to to to soils taken from Holtan etal. (1968) were reported at three or more depths were considered for analysis. Thirty-eight soil types showing a general decrease in saturated hydraulic conductivity with depth were chosen subjectively and the data were fitted to equations (1) using least squares regression. Where more than one value for porosity or saturated conductivity were reported at any depth, these were averaged (arithmetic average) beforehand. Of the 38 soil types chosen, 27 showed at least one regression with a regression coefficient significant at the 0.1 level. Twenty-four showed significant relationships between K g and depth while only 13 showed significant relationships between 6 S and depth. This may in part result from the subjective selection procedure based on the pattern of K g values rather than that of 9 g values. It will be seen from Table 1 that the range of K Q values representing a saturated hydraulic conductivity at the soil surface is low, and the highest value is two orders of magnitude lower than the highest value of 91.2 m h reported in Table 3 of Beven (1982b) and derived from the data given in Whipkey (1965). This is almost certainly due to the experimental procedure used by Holtan et al. (1968). The hydraulic conductivities were measured in the laboratory on samples cut from intact fist-sized fragments taken from the field, in a 75 mm diameter, 25 mm deep permeameter ring. Such a procedure would allow only a limited influence of secondary or structural porosity on hydraulic conductivity measurements. The influence of the secondary porosity would be expected to increase the vertical non-uniformity of the soil hydraulic characteristics.

4 428 Keith Beven A plot of derived K Q and 9 values for the combined data sets (Fig.l) reveals a scattered set of points with the data of Whipkey (1965) as an outlier of high K. There is a statistically significant relationship between the two parameters but given the nature of the data involved little physical significance can be attached to this relationship. K <mhr-') 100-, Fig. 1 Plot of K 0 and B Q values for the soils of Table 1. 0c A plot (Fig.2) of the coefficients f and g in equations (1) for the combined data sets shows that the decrease of K g with depth is, as would be expected, very much more rapid than that of 6. There is no significant relationship between the values of f and g and the mean value of g is not significantly different from zero. THEORY In previous papers, Beven (1982a,b) has developed a theory of flow through the soil based on simplifying assumptions that neglect gradients of capillary potential in the unsaturated zone. This theory will be most applicable where surface hydraulic conductivities are high relative to rainfall rates such that there is no control imposed by the soil on infiltration rates. The simplified theory leads to a kinematic wave equation for the progress of a wetting front into the soil into which non-uniform soil hydraulic characteristics described for example by equations (1) are easily incorporated. In this paper we shall consider the case where the rainfall rates might exceed the infiltration capacity of the soil. Under such conditions, the gradient of capillary potential has an important effect during the early stages of infiltration and cannot be neglected. We shall here pursue an analysis that incoporates the effects of capillarity based on the assumption of a piston-like wetting front. For uniform soils, this type of model is known as

5 Infiltration into vertically non-uniform soils i g (m-0 Fig. 2 Plot of f and g values for the soils of Table 1 and Table 3 of Beven (1982b). the Green & Ampt (1911) model or the "similarity" or "delta function" model (Philip, 1969). Philip (1954) showed that the Green-Ampt model provides an exact solution for the infiltration rate, i(t), if the moisture content profiles at successive times preserve similarity, whatever their shape. The Green & Ampt model has been applied to soils that are non-uniform with depth for layered profiles (Bouwer, 1969; Childs & Bybordi, 1969) and crusted profiles (Hillel & Gardner, 1970; Ahuja, 1974). Childs & Bybordi (1969) showed how the approximate Green & Ampt theory accurately predicts infiltration into layered sand columns in the laboratory. Onstad et al. (1973) have applied the layered model to results from five in situ soil monoliths with partial success. It is worth noting that equations (1) would not fit their published K g values for the monoliths. Philip (1969) suggests that the Green & Ampt model will generally overestimate infiltration rates. The model is, however, easy to apply and has been widely used. We shall consider first the case where the rainfall, which will be assumed of constant intensity, does not immediately saturate the soil surface. The time taken to reach saturation at the surface is the time to ponding, t p. Naturally ponding will occur only where K Q S r, the incident rainfall rate. We shall further simplify the problem by utilizing the results of the analysis of the soils data above and assume that the coefficient g is zero, so that porosity is a constant 9 g = 9 with depth. Under the piston displacement assumption of the Green-Ampt model, it can be assumed that the vertical flow velocity, while varying with time, is always constant with depth. Then following the procedure of Neuman (1976), Darcy's Law can be integrated over the depth to the wetting front, yielding an expression for the effective wetting front suction or capillary drive, Alp as: Atp = J k(if)) dijj *i where ip. is the soil potential at the initial moisture content 6^ and k(tjj) is the relative hydraulic conductivity function. To obtain an analytical solution to the infiltration problem it will be

6 430 Keith Beven necessary to assume that the "storage-suction factor" of Morel- Seytoux & Khanji (1974), given by C = A9A^ (where AG = Q s - Q ± ) is constant with depth. For A6 constant with depth, this implies that the shape of the relative hydraulic conductivity function is constant with depth. Clearly, if Aip decreases with depth along with the saturated hydraulic conductivity, the implication is the rather unrealistic assumption that A9 increases with depth. It is physically plausible that Atjj may increase with decreasing K g (see for example Table 2), so that A9 must decrease with depth to keep C constant. Morel-Seytoux & Khanji (1974) suggest that C should vary over the rather narrow range of 0 to 0.1 m and, except at short times, predictions should be relatively insensitive to the value of C. Table 2 Data on sand layers used in Childs & Bybordi (1969) infiltration experiments Layer (m) K s Imh" Ai// A0 C (m) (m) In general, the effective conductivity behind the wetting front, K, will be less than the saturated conductivity, K. We shall, however, assume a relationship between K and depth of the same form as equations (1) so that: K(z) = K Q e fz (2) If the wetting front has reached a depth Z, then, under the assumptions given above, the infiltration rate, i, is given by a generalization of the relationship for a layered soil (Childs & Bybordi, 1969) as: _dl Aij; + Z dt Z Z ~ Z r "^~ ~-, / 2 =Jdz/K(z)] where I is cumulative infiltration. Following Mein & Larson (1973) we shall assume that at the point of ponding at time t, the cumulative infiltration, I, has penetrated as a wetting front to a depth Z where: and: Z p = I p /A6 (4) I = r t p (5) (3)

7 Infiltration into vertically non-uniform soils 431 Substitution in equation (4) given i = r at the onset of ponding and using (2) gives: K n f(aifj + I p /A6) -fi /A6 (6) Equation (6) is most easily solved for I_ and therefore t p by a numerical technique such as Newton-Raphson iteration. After ponding, equation (3) continues to apply. Rewriting equation (3) in terms of I: ctl K n f(àji + I/A6) dt e-fi/a (7) Equation (7) may be integrated given I = I_ at t = t for AIJJA constant to give: t - t, ln(i + C) "FE {f*(i + C)P + T, m=l - ^ in (I C) (8) where C = AI(JA6, the storage suction factor of Morel-Seytoux & Khanji (1974) f* = -f/a6, and X = ln(i p + C) = a constant,f*c (f*(ip + C) F ln(ip + C) + I m=l m!m This solution is more complex than Green-Ampt infiltration for a uniform soil. However, the series term converges rapidly and the solution is easily applied by substituting values of I into equation (8) and calculating corresponding values of t. Alternatively, if infiltration rates are required at fixed times, equations (7) and (8) can be used in a Newton-Raphson iterative procedure to calculate I given t. EXTENSION TO TIME VARIABLE RAINFALL RATES For many applications it is not possible to assume a constant rainfall rate and the time variability of rainfall must be taken into account. Following the approach of Morel-Seytoux (1981) this may be done in the present model by application of equation (6) in the form: K Q f (All; (Ij + AIj)/A9),f(I + AIj)/A6 (9) where r.; is the rainfall rate at time step j, ZJ is the cumulative infiltration in time steps 1 to j - 1, and AI^ is the infiltration volume prior to ponding in time step j. If the solution yields AI J > r-jat, where At is the time step, then there is no ponding

8 432 Keith Beven in the time step and all the rainfall infiltrates (i.e. IJ + J = IJ n i-jat). If ponding does occur in the time step then equation (8) can be applied for the remainder of the time step to determine the total infiltration volume at the end of the time step. S 0.05 Fig. 3 Predicted cumulative infiltration I for two soils with K Q = 0.02 m h -1. Dashed line f = 12.0 m~ s ; dot-dash line f = 2.8 m" 1 ; solid line is cumulative rainfall. Arrows indicate predicted time to ponding, (a) r = 0.05 m rf ', (b) r = 0.02 m If 1. RESULTS Figure 3 shows the predicted infiltration rates for different assumed rainfall rates for two soils taken from the analysis of the Holtan et al. (1968) data. These soils have similar K Q values but very different fitted f coefficients. It has been assumed in these predictions that K Q = K Q and C is a constant for both soils. Figure 3 shows how infiltration is limited by the decrease in conductivity with depth in the soil of lower (more negative) f value. The infiltration model developed above has also been applied to the experimental data of Childs & Bybordi (1969) who infiltrated water into layered columns of different sands (Table 2). For the Hours Fig. 4 Predicted infiltration into the layered sand profile at Table 2 (using parameters K 0 = mh"',f = m -1 ), compared with the experimental data of Childs & Bybordi (1969, circles). Solid line, C = 0.2 m; dashed line, C = 0.6 m (1969).

9 Infiltration into vertically non-uniform soils 433 purposes of the present model equation (2a) was fitted to the K g values by assigning a given K value to the geometric average i depth of each layer giving the coefficients K Q = m h and f = nf 1 (r = 0.98, N =6, significant at a = 0.01). Cumulative infiltration calculated using different constant values of C = AI(JA9 for all layers is shown in Fig.4. The experimental data were collected under ponded conditions so a time to ponding of zero was assumed. While this profile of layers with different K s and C values is not an ideal test of the current model, it is probably representative of the nearest that could be achieved in the laboratory to the ideally non-uniform profile assumed by the model. Despite this limitation, agreement between observed and simulated infiltration rates is satisfactory, and sensitivity to the range of variation in C is small. CONCLUSIONS This paper has developed an infiltration model based on the Green- Ampt assumptions for a class of non-uniform soils in which saturated hydraulic conductivity decreases as an exponential function of depth. Arguments for the utility of the model have been based on an analysis of measured hydraulic conductivity and porosity data. This type of variation in hydraulic conductivity with depth is not uncommon, but does not appear to be strongly related to variations in porosity, providing further evidence that the largest pores, which may make up only a small proportion of the pore space, may dominate in the transmission of water under wet conditions. The particular assumptions about soil hydraulic characteristics implicit in the model are that saturated hydraulic conductivity should be adequately described by a functional relationship of the form of equation (la), and the product of potential difference at the wetting front, kip, and the difference in the moisture content across the wetting front, A9, should be constant with depth. The evidence presented here suggests that the functional relationships of equations (1) may be useful for describing the change of hydraulic properties with depth in some soil profiles. This conclusion needs further substantiation using experimental techniques that allow the influence of structural porosity on hydraulic conductivity to be determined. The utility of the Green- Ampt based infiltration equation under such conditions also needs verification by field experiment. It is hoped that the relatively simple approach suggested here that takes account of the vertical non-uniformity of soil characteristics may be more appropriate in certain situations than traditional infiltration equations that assume homogeneous soil properties. REFERENCES Ahuja, L.R. (1974) Applicability of Green-Ampt approach to water infiltration through surface crust. Soil Sci. 118, Beven, K.J. (1982a) On subsurface stormflow: predictions with

10 434 Keith Beven simple kinematic theory for saturated and unsaturated flows. Wat. Resour. Res. 18 (6), Beven, K.J. (1982b) On subsurface stormflow: an analysis of response times. Hydrol. Sci. J. 27 (4), Beven, K.J. & Germann, P.F. (1982) Macropores and water flow in soils. Wat. Resour. Res. 18 (5), Bouwer, H. (1969) Infiltration of water into non-uniform soil. J. Irrig. Drain. Div. ASCE 95 (IR4), Brakensiek, O.L., Engleman, R.L. & Rawls, W.J. (1981) Variation within texture classes of soil water parameters. Trans. Am. Soc. Agric. Engrs 24 (2), Childs, E.C. & Bybordi, M. (1969) The vertical movement of water in stratified porous material 1. Infiltration. Wat. Resour. Res. 5 (2), Clapp, R.B. & Hornberger, G.M. (1978) Empirical equations for some soil hydraulic properties. Wat. Resour. Res. 14 (4), Green, W.H. & Ampt, G.A. (1911) Studies in soil physics. I. The flow of air and water through soils. J. Agric. Sci. 4 (1), Hillel, D. & Gardner, W.R. (1970) Transient infiltration into crust topped profiles. Soil Sci. 109, Holtan, H.N., England, C.B., Lawless, G.P. & Schumaker, G.A. (1968) Moisture-tension data for selected soils on experimental watersheds. USDA, ARS, Publ. RS Mein, R.G. & Larson, C.L. (1973) Modeling infiltration during a steady rain. Wat. Resour. Res. 9, Morel-Seytoux, H.J. (1981) Application of infiltration theory for determination of excess rainfall hyetographs. Wat. Resour. Bull. 17 (6), Morel-Seytoux, H.J. & Khanji, J. (1974) Derivation of an equation of infiltration. Wat. Resour. Res. 10, Neuman, S.P. (1976) Wetting front pressure head in the infiltration model of Green and Ampt. Wat. Resour. Res. 12, Onstad, C.A., Olson, T.C. & Stone, L.R. (1973) An infiltration model tested with monolith moisture measurements. Soil Sci. 116, Philip, J.R. (1954) An infiltration equation with physical significance. Soil Sci. 77, Philip, J.R. (1969) Theory of infiltration. Advances in Hydroscience 5, Whipkey, R.Z. (1965) Subsurface stormflow on forested slopes. Bull. Int. Ass. Sci. Hydrol. 10 (2), Received 29 November 1983; accepted 24 September 1984.