6 Infiltration Brent E. Clothier HortResearch, Palmerston North, New Zealand 1. INTRODUCTION A water droplet incident at the soil surface has just two options: it can infiltrate the soil or it can run off. This partitioning process is critical. Infiltration, and its complement runoff, are of interest to hydrologists who study runoff generation, river flow, and groundwater recharge. The entry of water through the surface con- cerns soil scientists, for infiltration replenishes the soil’s store of water. The par- titioning process is critically dependent of the physical state of the surface. Fur- thermore, infiltrating water acts as the vehicle for both nutrients and chemical contaminants. Infiltration, because it is both a key soil process and an important hydro- logical mechanism, has been twice studied: once by soil physicists and again by hydrologists. Historically, their approaches have been quite different. In the for- mer case, infiltration was the prime focus of detailed study of small-scale soil processes, and in the latter, infiltration was just one mechanism in a complicated cascade of processes operating across the scale of a catchment. Latterly, access to powerful computers has meant that hydrologists have been able to incorporate the soil physicists’ detailed mechanistic descriptions of infiltration into their hydro- logical models of watershed functioning. This has increased the need to measure the parameters that control infiltration. 239 To the memory of John Philip (1927–1999), for without his endeavors this would have been a very short chapter. Copyright © 2000 Marcel Dekker, Inc. In this chapter, I first describe the development of one-dimensional ponded infiltration theory, discussing both analytical and quasi-analytical solutions. In passing, I mention empirical descriptions of infiltration before discussing the key development of a simple algebraic expression for infiltration that employs physi- cally based parameters. Emphasis is placed on theoretical approaches, for they can predict infiltration through having parameters capable of field measurement. The preeminent roles of the physical state of the soil surface and the nature of the upper boundary condition are stressed. Infiltration of water into soil can occur as a result of there being a pond of free water on the soil surface, so that the soil controls the amount infiltrated, or water can be supplied to the surface at a given rate, say by rainfall, so the soil only controls the profile of wetting, not the amount infiltrated. Next, I show how measurement of infiltration can be used, in an inverse sense, to determine the soil’s hydraulic properties. In this way, it is possible to predict infiltration into the soil, and general prediction of water movement through soil can also be made using these measured properties. Hydraulic interpretation of the theoretical parameters in the governing equations is outlined, as is the impact of infiltration on solute transport through soil. A list is presented of the various devices that have been developed to measure, in the field, the soil’s capillary and conductive properties that control infiltration. An outline of their respective merits is presented, as is a comparative ranking of utility. Finally, I conclude with a pre- sentation of some illustrative results and identify some issues that still remain problematic. Elsewhere in this book, there are complementary chapters on measurement of the soil’s saturated conductivity (Chap. 4) and the unsaturated hydraulic con- ductivity function (Chap. 5). Here the emphasis is on devices capable of in situ observation of infiltration, and the measurement in the field of those saturated and unsaturated properties that control the time course, and quantity, of infiltration. II. THEORY A. One-Dimensional Ponded Infiltration Significant theoretical description of water flow through a porous medium began in 1856 with Henry Darcy’s observations of saturated flow through a filter bed of sand in Dijon, France (Philip, 1995). Darcy found that the rate of flow of water, J (m s Ϫ1 ), through his saturated column of sand of length L (m), was proportional to the difference in the hydrostatic head, H (m), between the upper water surface and the outlet: DH J ϭ K (1) ͩͪ L 240 Clothier Copyright © 2000 Marcel Dekker, Inc. in which Darcy called K ‘‘un coefficient de´pendent du degre´ de perme´abilite´du sable.’’ We now call this the saturated hydraulic conductivity K s (m s Ϫ1 ) (Chap. 4). In 1907, Edgar Buckingham of the USDA Bureau of Soils established the theo- retical basis of unsaturated soil water flow. He noted that the capillary conduc- tance of water through soil, now known as the unsaturated hydraulic conductivity, was a function of the soil’s water content, u (m 3 m Ϫ3 ), or the capillary pressure head of water in the soil, h (m). The characteristic relationship between h and u (Chap. 3) was also noted by Buckingham (1907), so that he could write K ϭ K(h), or if so desired, K ϭ K(u). The total head of water at any point in the soil, H,is the sum of the gravitational head due to its depth z below some datum, conve- niently here taken as the soil surface, and the capillary pressure head of water in the soil, h: H ϭ h Ϫ z. Here, h is a negative quantity in unsaturated soil, due to the capillary attractiveness of water for the many nooks and crannies of the soil pore system. Thus locally in the soil, Buckingham found that the flow of water could be described by dH dH J ϭϪK(h) ϭϪK(h) ϩ K(h) (2) ͩͪ dz dz which identifies the roles played by capillarity, the first term on the right hand side, and gravity, the second term. These two forces combine to move water through unsaturated soil (Chap. 5). In deference to the discoverers of the satu- rated form, Eq. 1, and the unsaturated form, Eq. 2, the equation describing water flow at any point in the soil is generally referred to as the Darcy–Buckingham equation. L. A. Richards (1931) combined the mass-balance expression that the tem- poral change in the water content of the soil at any point is due to the local flux divergence, ץu ץJ ϭϪ (3) ͩͪ ͩͪ ץt ץz zt with the Darcy–Buckingham description of the water flux J, to arrive at the gen- eral equation of soil water flow, ץu ץץhdK(h) ץh ϭ K(h) Ϫ · (4) ͩͪ ץt ץz ץzdhץz where t (s) is time. Unfortunately, this formula, known as Richards’ equation, does not have a common dependent variable, for u appears on the left and h on the right. The British physicist E. C. Childs ‘‘decided to try some other hypothe- sis thatwatermovement is decided by the moisture concentration gradient . . . [and] that water moves according to diffusion equations’’ (Childs, 1936). Childs and Collis-George (1948) noted that the diffusion coefficient for water in soil Infiltration 241 Copyright © 2000 Marcel Dekker, Inc. could be written as K(u) dh/du. From this, in 1952 the American soil physicist Arnold Klute wrote Richards’ equation in the diffusion form of ץu ץץu dK ץu ϭ D(u) Ϫ · (5) ͩͪ ץt ץz ץzdu ץz This description shows soil water flow to be dependent on both the soil water diffusivity function D(u), and the hydraulic conductivity function K(u), but this nonlinear partial differential equation is of the Fokker–Planck form, which is no- toriously difficult to solve. Klute (1952) developed a similarity solution to the gravity-free form of Eq. 5, subject to ponding of free water at one end of a soil column. Five years later, the Australian John Philip developed a power-series solu- tion to the full form of Eq. (5), subject to the ponding of water at the surface of a vertical column of soil, initially at some low water content u n (Philip, 1957a). This general solution predicts the rate of water entry through the soil surface, i(t) (m s Ϫ1 ), following ponding on the surface. The surface water content is main- tained at its saturated value, u s . The cumulative amount of water infiltrated is I (m), being the integral of the rate of infiltration since ponding was established. As well, I can be found from the changing water content profile in the soil, t ϱ u s I ϭ ͵ i(tЈ) dtЈ ϭ ͵ u(zЈ) Ϫ u dzЈ ϭ ͵ z(u) du (6) ͫͬ n 00 u n Philip’s (1957a) series solution for I(t) can be written 1/2 3/2 4/2 ··· I(t) ϭ St ϩ At ϩ At ϩ At ϩ (7) 34 where the sorptivity S (m s Ϫ1/2 ) and the coefficients A, A 3 , A 4 , canbeitera- tively calculated from the diffusivity and conductivity functions, D(u) and K(u). The form of Eq. 7 indicates that I increases with time, but at an ever-decreasing rate. In other words, the rate of infiltration i ϭ dI/dt is high initially, due to the capillary pull of the dry soil. But with time the rate declines to an asymptote. Special analytical solutions can be found for cases where certain assump- tions are made about the soil’s hydraulic properties. When the soil water diffusiv- ity can be considered to be a constant, and K varies linearly with u, an analytical solution is possible. This is because Eq. 5 becomes linearized (Philip, 1969) and so there is an analytical solution for infiltration into a soil whose hydraulic prop- erties can be considered only weakly dependent on u. At the other end of the scale of possible behavior, Philip (1969) presented an analytical solution for a soil whose diffusivity could be considered a Dirac d-function, in which D is zero, except at u s where it goes to infinity. For the analytical solution, this so-called delta-soil, or Green and Ampt soil, also needs to have K ϭ K s at u s , and K ϭ 0 for all other u. 242 Clothier Copyright © 2000 Marcel Dekker, Inc. Philip and Knight (1974) showed that the Dirac d-function solution pro- duces a rectangular profile of wetting (shown later in Fig. 4). It was this geometric form of wetting that was used as the physical basis for Green and Ampt’s (1911) functional model of infiltration. If a rectangular profile of wetting is assumed, then behind the wetting front located at depth z f , u(z) ϭ u s , for 0 Ͻ z Ͻ z f ; and beyond the wetting front, u(z) ϭ u n , z Ͼ z f . If the soil has a shallow free-water pond at the surface, and if it is considered that there is a wetting front potential head, h f , at z f , then the Darcy–Buckingham law (Eq. 2) predicts the rate of water infiltrat- ing through the surface as K (z Ϫ h ) sf f J ϭ (8) z f The rectangular profile of wetting allows easy evaluation of the mass balance integral of Eq. 6, and its differentiation to provide the rate of infiltration into the soil, dI d(z (u Ϫ u )) dz fs n f i ϭϭ ϭ ·(u Ϫ u ) (9) sn dt dt dt Equating Eqs. 8 and 9 provides a variables-separable ordinary differential equa- tion in z f , (z Ϫ h ) dz ff f K ϭ (u Ϫ u ) (10) ssn zdt f which can be solved to provide the penetration of the wetting front into the soil with time, (u Ϫ u ) z sn f t ϭ z ϩ h ln 1 Ϫ (11) ͫͩͪͬ ff Kh sf Althrough this expression is not explicit, it does allow implicit prediction of I(t) from basic soil properties. By considering flow in the absence of gravity, z f is eliminated from the numerator of Eq. 8, and an explicit expression for gravity-free infiltration is arrived at that only contains a square-root-of-time term, as would be expected for a diffusion-like process. By comparing coefficients with Eq. 7, it is found that a Green and Ampt soil must have the sorptivity 2 S ϭϪ2Kh(u Ϫ u ) (12) sf s n So, if the soil is considered to have the characteristics that lead to a rectangular profile of wetting (shown in Fig. 4), and there is a constant pressure-potential head, h f , always associated with the wetting front, then simple expressions can be derived to predict infiltration into such a soil (Eqs. 11 and 12). More recently, Parlange (1971) developed a new and general quasi-analyti- cal solution for infiltration into any soil (Eq. 5). This was extended by Philip and Infiltration 243 Copyright © 2000 Marcel Dekker, Inc. Knight (1974) using a flux– concentration relationship, F(Ѳ), that hides much of the nonlinearity in the soil’s hydraulic properties of D and K. Here Ѳ is the nor- malized water content. Considering these mathematical solutions to the flow equation for infiltra- tion I, subject to ponding, Childs (1967) commented that ‘‘further investigations to throw yet more light on the basic principles of the flow of water tendtobe matters of crossing t’s and dotting i’s . . . serious difficulties remain in the path of practical application of theory . . . [being] held back by the inadequate develop- ment of methods of assessment of the relevant parameters.’’ These analytical or quasi-analytical solutions are seldom used to predict infiltration directly from the soil’s hydraulic properties. The theory and its de- velopment are presented here, for they identify the underpinning physics of infil- tration. Nowadays, however, the current power of computers, coupled with the burgeoning growth of numerical recipes for solving nonlinear partial differen- tial equations, has meant that brute-force numerical solutions to Eq. 5 for infil- tration are easily obtained, provided that the functional properties of D and K are known. Thus given a knowledge of the soil’s hydraulic properties, it is a reasonably straightforward exercise to predict infiltration, either analytically or numerically. Infiltration measurements hold the key to obtaining in situ characterization of these soil properties. It is possible to use Eq. 7 or the like in an inverse sense, to use infiltration observations to infer the soil’s hydraulic character. The time course of water entry into soil, I(t), depends, as Eq. 7 shows, on coefficients that relate to the hydraulic properties of D(u) and K(u). Infiltration can quite easily be measured in the field. Hence, I will proceed to show how this measurement of I can be used to extract in situ information about the soil’s capillary and conductive properties. 1. Empirical Descriptions Before passing to the discussion of the developments that have led to the use of measurements to predict one-dimensional infiltration behavior, I sidetrack a little to review some of the empirical descriptions of the shape of i(t). This digression is simply to complete our historical record of the study of infiltration, for such empirical equations have little merit nowadays. The Kostiakov–Lewis equation, I ϭ at b (Swartzendruber, 1993), has descriptive merit through its simplicity, yet comparison with Eq. 7 indicates the inadequacy of this power-law form, for in reality b needs to be a function of time. The hydrologist Horton (1940) pro- posed that the decline in the infiltration rate could be described using i ϭ i ϱ ϩ (i o Ϫ i ϱ ) exp(Ϫbt), where the subscripts o and ϱ refer to the initial and final rates. If description is all that is sought, then the three-term expression will perform better due to its greater fitting ‘‘flexibility.’’ In neither case do the fitted parameters 244 Clothier Copyright © 2000 Marcel Dekker, Inc. have physical meaning, so care needs to be exercised in their extrapolation beyond the fitted range. 2. Physically Based Descriptions However, the two-term algebraic equation of Philip (1957b) is different from other empirical descriptions. It rationally incorporates physical notions. Simply by trun- cating the power series of Eq. 7, Philip (1957a) arrived at the expression for the infiltration rate of 1 Ϫ1/2 i ϭ St ϩ A (13) 2 which will be applicable at short and intermediate times. However at longer times, we know for ponded infiltration that lim i(t) ϭ K (14) s → t ϱ The means by which these two expressions can best be joined has worried some soil physicists, with A/K s having been found to be bracketed between 1 Ϫ 2/p and 2/3, but probably lying nearer the lower limit (Philip, 1988). However, as Philip (1987) noted, relative to practical incertitudes, a two-term algebraic expres- sion often suffices, with both terms having physical meaning, plus correct short- and long-time behavior, viz. 1 Ϫ1/2 i ϭ St ϩ K (15) s 2 The coefficient of the square-root-of-time term, the sorptivity S, integrates the capillary attractiveness of the soil. Mathematically, as we will see later, this can be linked to the soil water diffusivity function D(u). The role of capillarity de- clines with the square root of time. The second term, which is time independent, is the saturated conductivity K s , which is the maximum value of the conductivity function K(h) that occurs when the soil is saturated, h ϭ 0. If the soil is initially saturated (S ϭ 0), or if infiltration has been going on for a long time, then gravity will alone be drawing water into the soil at the steady rate of K s . Eq. 15 is aptly named Philip’s equation. To understand Eq. 15 is to understand the basics of infiltration. B. Multidimensional Ponded Infiltration However, a one-dimensional geometric description is not always appropriate. For example, infiltration into soil might be from a buried and leaking pipe, or it might be from a finite surface puddle of water. In these cases, the physics previously Infiltration 245 Copyright © 2000 Marcel Dekker, Inc. described above must now be referenced to the geometry of the source. The re- spective roles of capillarity and gravity in establishing the rate of multidimen- sional infiltration, v o (t)(ms Ϫ1 ), through a surface of radius of curvature r o (m), are now more complex. Following Philip (1966), let m be the number of spatial dimensions required for geometric description of the flow. The geometry depicted in Fig. 1 might be a transverse section through a cylindrical channel. This would be a 2-D source with m ϭ 2. Or it could be a diametric cut through a spherical pond that would be represent a 3-D geometry. So here m ϭ 3. The more curved the wetted perimeter of the source, the smaller is r o , and the greater is the role of the soil’s capillarity relative to gravity. In the limit as r o → ϱ, the geometry be- comes one-dimensional (m ϭ 1), and the source spreads right across the soil’s surface. As already noted, if the soil is considered to have a constant diffusivity D, and a linear K(u), then ananalytical solution can be found for one-dimensional infiltration because the governing equation is linearized. This also applies to multidimensional infiltration, if the flow description of Eq. 5, which has m ϭ 1, is written in a form appropriate to a flow geometry with either m ϭ 2orm ϭ 3 (Philip, 1966). Philip’s (1966) linearized multidimensional infiltration results are illustrative and are presented in Fig. 2. There, the infiltration rate through the wetted perimeter, v o , is normalized with respect to the saturated conductivity K s , and the time is normalized by a nondimensional time, t grav ϭ (S/K s ) 2 . To allow 246 Clothier Fig. 1. An idealized multidimensional infiltration source, in which water infiltrates into the soil through a wetter perimeter of radius of curvature r o . Capillarity and gravity com- bine to draw water into the dry soil. Copyright © 2000 Marcel Dekker, Inc. easy comparison, the radius of curvature is also normalized, and given as R o ϭ r o [K s (u s Ϫ u n ) 2 /pS 2 ]. For the one-dimensional case in Fig. 2 (m ϭ 1), the infiltration rate can be seen to fall, as the effects of capillarity fade with the square, and higher, roots of time (Eq. 7). At around t/pt grav , the infiltration rate is virtually the asymptote of v o ϭ K s . Such behavior is predicted by Eq. 15. Two cases are given for two- dimensional flow from cylindrical channels, m ϭ 2. For the tightly curved channel (R o ϭ 0.05), the effect of the source geometry on capillarity is clearly seen, and the infiltration rate is nearly two times K s at dimensionless time 10. For the less- curved channel (R o ϭ 0.25), the geometry-induced enhancement of capillary ef- fects is correspondingly less. In the three-dimensional case (m ϭ 3), for the curved spherical pond with R o ϭ 0.05, the effect of capillarity is so enhanced by the 3-D source geometry that the infiltration rate through the pond walls achieves a steady flux of over 5K s by unit time. Whereas infiltration in one dimension (m ϭ 1) gradually approaches K s ,the source geometry in 2-D and 3-D (m ϭ 2 and 3) ensures that the infiltration rate finally arrives at a true steady-state value, v ϱ . In Fig. 2, the time taken to realise v ϱ Infiltration 247 Fig. 2. The normalized temporal decline in the rate of infiltration through the ponded surface into a one-dimensional soil profile (m ϭ 1), and from two cylindrical channels (m ϭ 2) of contrasting radii of curvature (r o ), as well as from two spherical ponds (m ϭ 3) of different radii. To allow comparison of one-, two-, and three-dimensional flows, the infiltration rate, time, and radii of source curvature have all been normalized. (Redrawn from Philip, 1966.) Copyright © 2000 Marcel Dekker, Inc. is more rapid in 3-D than it is for m ϭ 2. This achievement of a steady flow rate in 3-D is, as we will see later, a major advantage for certain devices in the field measurement of infiltration. In this device-context, it is useful to consider in more detail the three- dimensional flow from a shallow, circular pond of water of radius r o . The history of the study of this problem is given in Clothier et al. (1995), so here we need only concern ourselves with the seminal result of Wooding (1968). The New Zealander Robin Wooding was concerned about the radius requirements for double-ring in- filtrometers (shown later in Fig. 5), and he found a complex-series solution for the steady flow from a shallow, circular surface pond of free water. However, he did note that the steady flow could be approximated by a simple equation in which capillary effects were added to the gravitational flow in inverse proportion to the length of the wetted perimeter of the pond, 4f s v ϭ K ϩ (16) ϱ s pr o Here the sum effect of the soil’s capillarity is expressed in terms of the integrals of the hydraulic properties of D and K, the so-called matric flux potential u 0 s f ϭ ͵ D(u)du ϭ ͵ K(h)dh (17) s u h n It was necessary for Wooding (1968) to consider that the soil’s unsaturated hy- draulic conductivity function could be given by the exponential form K(h) ϭ K exp(ah) (18) s with the unsaturated slope a (m Ϫ1 ), so that K s f ϭ (19) s a This formulation allows Wooding’s equation (Eq. 16) for the steady volumetric infiltration from the circular pond, Q ϱ ϭ pr o 2 v ϱ (m 3 s Ϫ1 ), to be written as 4r o 2 Q ϭ K pr ϩ (20) ͩͪ ϱ so a In this way we can see the role of the pond’s area in generating the gravitational component of infiltration, and that of the perimeter in creating a capillary contri- bution. We will return later to this special form of multidimensional ponded infiltration. C. Boundary Conditions Thus far, we have only considered the case where water is supplied by a surface pond of free water, namely 248 Clothier Copyright © 2000 Marcel Dekker, Inc. [...]... Bouwer (1 961 ) and Youngs (1972) used an electrical-analog approach, whereas Wooding (1 968 ) provided a simple expression based on the properties of the soil (Eq 16) The ASTM standard double-ring infiltrometer has radii of 150 and 300 mm (Lukens, 1981), although the correct ratio will be soil dependent, and related to the relative sizes of the conductivity K and the sorptivity S (Eq 16) The flows vo and vo*... combined with Eq 26 to obtain the slope, aϭ Ks K (u Ϫ u ) ϭ s s 2 n fs bS Copyright © 2000 Marcel Dekker, Inc (28) 252 Clothier So by monitoring infiltration to infer both K s and S (Eq 15), and by measuring us and un , Eqs 26 and 28 give us functional descriptions of the soil s D(u) and K(h) These capillary and gravity properties allow us to infer some pore-geometric characteristics of the soil s hydraulic... so that the mean values of v1 and v2 are used in Eqs 38 and 39 Scotter et al (1982) showed { { how the variance in S and K s can be calculated This twin-ring technique allows both the soil s capillarity and its conductivity to be measured, and here the disturbance to the soil s structure is minimal It is only necessary to press the rings gently into the soil surface, and a mud caulking can be used... in soil Here, for simplicity, we consider a soil lying horizontally with water being absorbed in the x direction During infiltration, water-borne chemicals are transported into the soil The entry of water into soil is a hydrodynamic phenomenon: the wetting front rides into the soil on ‘‘top’’ of the antecedent water content, un (Fig 4) For the case of a d-function soil, that is, one possessing Green and. .. knowledge of the soil s sorptivity S, and conductivity K s , given vo (Eq 23) So it is imperative that S and K s be measured for field soils D Hydraulic Characteristics of Soil There are three functional properties necessary to describe completely the hydraulic character of the soil: the soil water diffusivity function D(u), the unsaturated hydraulic conductivity function K(h), and the soil water characteristic... tension s and density r of water, and for the acceleration due to gravity White and Sully (1987) called l m a ‘‘physically plausible estimate of flow-weighted mean pore dimensions.’’ By combining Eqs 30 and 31 it is possible to use properties measured during infiltration (S and K s ; us and un ) to deduce something dynamic about the magnitude of the pore size class involved in drawing water into the soil. .. Inc Infiltration 261 D Closed-Top Permeameters 1 Air-Entry Permeameter There is a seductive utility in Green and Ampt’s Eq 8, for if we could find both the saturated conductivity K s and the wetting front potential h f , we would be able to describe infiltration using Eq 11 Bouwer (1 966 ) described a device that allowed this, his so-called air entry permeameter A ring is driven into the soil to a depth... the measurement of un , and final observation of the water content uo just under the disk, then Eq 26 gives the unsaturated matric flux potential fo ϭ f(uo ) Thus the short-time observations of 3-D infiltration from the disk can be used in a manner similar to that employed in 1-D by Talsma (1 969 ) (Sec III.A) However, it can be difficult to ensure that only the true square-root-of-time signal is observed... (1 969 ), the approach of Perroux and White (1988) permits measurement of both the soil s capillarity and its conductivity from observations of 3-D infiltration from a disk permeameter set at h o 2 Twin and Multiple Disks To get around the problem of finding the sorptivity from the short-time infiltration curve, the twin ponded ring technique of Scotter et al (1982) (Eqs 38 and 39) was used by Smettem and. .. Furthermore, it is the surface-ventedness and connectedness of the mesopores and the macropores that operate at near-saturated heads that play dominant roles in establishing the shape of the soil s near-saturated K(h) (Clothier, 1990) Disk permeameters, which operate in the range of Ϫ150 Ͻ h (mm) Ͻ 0, are useful tools with which to observe the soil s near-saturated conductivity Messing and Jarvis (1993) used . (1 961 ) and Youngs (1972) used an elec- trical-analog approach, whereas Wooding (1 968 ) provided a simple expression based on the properties of the soil (Eq. 16) . The ASTM standard double-ring in- filtrometer. u s and u n , Eqs. 26 and 28 give us functional descriptions of the soil s D(u) and K(h). These capillary and gravity properties allow us to infer some pore-geo- metric characteristics of the soil s. allow comparison of one-, two-, and three-dimensional flows, the infiltration rate, time, and radii of source curvature have all been normalized. (Redrawn from Philip, 1 966 .) Copyright © 2000 Marcel